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the renormalization group ( rg ) has proved to be the most efficient tool for studying self - similar scaling behavior . first appeared within the context of quantum field theory @xcite , it was then successfully applied to a variety of problems as disparate as phase transitions , polymer dilutes , random walks , hydrodynamical turbulence , growth processes , and so on ; see , e.g. , the monorgaphs @xcite , the proceedings @xcite , and references therein . the most powerful and well - developed formulation of the rg is the field theoretic one ; see @xcite . it is this version of the rg that is simplest and most convenient in practical calculations , especially in higher orders . it is also important that it has a reliable basis in the form of quantum - field renormalization theory , including the renormalization of composite operators and operator product expansion . for this reason , the first step in the rg analysis of a given problem is to reformulate it as a field theoretic model . this means that the quantities under study should be represented as functional averages with the weight @xmath3 , where @xmath4 is a classical random field ( or set of fields ) and @xmath5 is certain action functional . for parabolic differential equations with an additive random source , such a formulation is provided by the well - known martin siggia rose formalism , see @xcite . in problems involving fluctuation effects in chemical reactions the somewhat more complicated approach of doi @xcite ( see also @xcite ) has also been widely used @xcite . no general recipe , however , seems to exist to cast a nonlinear problem to a field - theoretic form . such a reformulation , however , is by no means superfluous : once the field theoretic formulation has been found , it becomes possible to apply standard tools ( power counting of the 1-irreducible correlation functions etc ) to verify the renormalizability of the model , i.e. , the applicability of the rg technique , to derive corresponding rg equations , and to calculate its coefficients ( beta functions and anomalous dimensions ) within controlled approximations . an instructive example is provided by the model of the so - called true self - avoiding random walks @xcite . after its field theoretic formulation had been found @xcite , it became clear that the model in its original formulation was not renormalizable , and the direct application of the rg to it would lead to completely erroneous results . the renormalizable version of the model can be obtained by adding of infinitely many terms to the original action ; see @xcite . it has long been known , however , that symmetries of the rg type also appear in various physical problems described by ordinary or partial differential equations and integro - differential equations , whose solutions exhibit self - similar scaling behavior @xcite . since then , the list of such problems has been essentially increased ; see , e.g. , @xcite and references therein . as a rule , the field theoretic formulation for these models does not exist ( or , at least , is not known ) , and derivation of the corresponding rg equations is a nontrivial task . quoting the authors of @xcite , `` the procedure of revealing rg transformations @xmath6 in any partial case @xmath6 up to now is not a regular one . in practice , it needs some imagination and atypical manipulation ` invented ' for every particular case . '' in ref . @xcite , a general approach was proposed to constructing rg symmetries for certain classes of partial differential equations , but its relationship to the field theoretic rg techniques is not clear . the present paper is an attempt to ` bridge the gap ' between these two vast areas of the applicability of the rg : field theoretic models and partial differential equations . to be specific , we shall consider nonlinear diffusion equation of the form @xmath7 where @xmath8 is a scalar field , @xmath9 is the diffusion coefficient , @xmath10 is the laplace operator , and @xmath11 is some nonlinearity dependent on the field @xmath4 and its spatial derivatives . within the rg context , various special examples of eq . ( [ diff ] ) were studied earlier in @xcite . in practical calculations , we shall confine ourselves to the nonlinearity of the form @xmath12 , where @xmath2 is not necessarily integer . we shall show that the problem ( [ diff ] ) can be cast into a field theoretic model and apply the standard rg formalism to it to establish the scaling behavior and to calculate corresponding anomalous dimensions . then we shall discuss the range of applicability of the results obtained and their relationship to the previous rg treatments of the model . we begin the analysis of the cauchy problem ( [ diff ] ) with a localized initial condition which corresponds to the equation @xmath13 for the green function @xmath14 . it will be shown later that the large - scale asymptotic behavior of this problem survives for all integrable initial conditions ( i.e. such that @xmath15 converges ) . in eq . ( [ green ] ) we denote @xmath16 , where @xmath17 is the dimensionality of the @xmath18 space , and @xmath19 is the full set of parameters . the functional derivation of the msr formalism @xcite can be adopted to represent the solution of eq . ( [ green ] ) as a functional integral over the doubled set of fields , @xmath4 and @xmath20 : @xmath21 . \label{funi}\ ] ] here the normalization constant is included into the differential @xmath22 , the action functional has the form @xmath23 with @xmath24 . the last term in ( [ green ] ) can be treated as an addition to the ` interaction ' @xmath11 and gives rise to the last term in the exponential of eq . ( [ funi ] ) . the term quadratic in @xmath20 , typical to the msr actions , is absent in ( [ act ] ) owing to the absence of the random force in eq . ( [ diff ] ) . representation ( [ funi ] ) shows that the green function ( [ green ] ) can be viewed as the correlation function @xmath25 in the field - theoretic model with the action ( [ act ] ) . it is not convenient , however , to deal with the exponential composite operator @xmath26 . a more useful interpretation is the following : the integral ( [ funi ] ) describes the correlation function @xmath27 for the extended action @xmath28 with an ` ultralocal ' interaction term concentrated on a single spacetime point @xmath29 . the renormalization of field theoretic models with ultralocal terms , concentrated on surfaces , was studied in ref . @xcite in detail . the analysis of @xcite , which we also naturally take to apply to our case , has shown that the standard renormalization theory is applicable to such models , with some obvious modification ( see below ) . the analysis of ultraviolet ( uv ) divergences is based on the analysis of canonical dimensions ; see @xcite . dynamical models of the type ( [ act ] ) , in contrast to static models , have two scales , the length scale @xmath30 and the time scale @xmath31 . therefore , the canonical dimension of any quantity @xmath32 ( a field or a parameter in the action functional ) is described by two numbers , the momentum dimension @xmath33 and the frequency dimension @xmath34 , determined so that @xmath35 \sim [ l]^{-d_{f}^{k } } [ t]^{-d_{f}^{\omega}}$ ] . the dimensions are found from the obvious normalization conditions @xmath36 , @xmath37 , @xmath38 , @xmath39 , and from the requirement that each term of the action functional be dimensionless ( with respect to the momentum and frequency dimensions separately ) . then , based on @xmath33 and @xmath34 , one can introduce the total canonical dimension @xmath40 ( in the free theory , @xmath41 ) , which plays in the theory of renormalization of dynamical models the same role as the conventional ( momentum ) dimension does in static problems ; see @xcite . now let us turn to the special case of the model ( [ green ] ) with the extended action of the form @xmath42 where we have introduced the new parameter @xmath43 , which plays the part of the coupling constant ( a formal small parameter of the ordinary perturbation theory ) . canonical dimensions for the model ( [ extend ] ) are given in table [ table1 ] , including the dimensions of renormalized parameters , which will appear later on . from table [ table1 ] it follows that the model is logarithmic ( the coupling constant @xmath44 is dimensionless ) for @xmath45 . in what follows , we fix the exponent @xmath46 in eq . ( [ extend ] ) and consider the model in variable space dimension @xmath47 . then the uv divergences take on the form of the poles in @xmath48 in the correlation functions . the ` interaction ' is therefore irrelevant ( in the sense of wilson ) for @xmath49 , marginal ( logarithmic ) for @xmath50 , and relevant for @xmath51 ; cf . the analysis in ref . @xcite . this means that for @xmath52 , the ordinary perturbation expansion ( i.e. , series in @xmath44 ) fails to give correct infrared ( ir ) behavior and has to be summed up . the desired summation can be accomplished using the renormalization group . it is common wisdom of the renormalization theory that for the analysis of uv divergences of all correlation functions of the fields @xmath4 and @xmath20 it is sufficient to consider one - particle - irreducible ( 1pi ) correlation functions , whose graphical representation contains only graphs which remain connected after removal of one ( arbitrary ) line ( i.e. a free - field correlation or response function ) of the graph . the total canonical dimension of an arbitrary 1pi correlation function @xmath53 in the time coordinate representation is given by the relation @xmath54 where @xmath55 and @xmath56 are the numbers of corresponding fields . in ( [ gammag ] ) @xmath57 is the ( dimensionless ) generating functional of 1pi green functions . it should be noted , however , that due to the presence of the ultralocal term in the action , the functional @xmath57 is _ not _ the legendre transform of the functional @xmath58 , where @xmath59 $ ] is the generating functional of green functions of the model . moreover , contrary to the usual field theories , the functional @xmath60 does not include all connected graphs of @xmath61 . by definition of the generating functional , the 1pi green function with @xmath55 external @xmath4 legs and @xmath56 external @xmath20 legs may be obtained by @xmath55 functional differentiations of @xmath57 with respect to the field @xmath4 and @xmath56 differentiations with respect to @xmath20 . the canonical dimensions of the functional derivatives are related to the dimensions of the corresponding fields as @xmath62 = d - d^{k}_{\phi}$ ] , @xmath63 = 1 - d^{\omega}_{\phi}$ ] , and similarly for the auxiliary field @xmath20 . then the total canonical dimension of the function ( [ gammag ] ) in the frequency momentum representation ( obtained by the fourier transformation with respect to all @xmath64 independent differences of the time and coordinate arguments ) is obtained from ( [ deltac ] ) by subtracting the term @xmath65 and has the form @xmath66 where the data from table [ table1 ] are used in the last equality . the quantity ( [ deltacc ] ) is the formal index of the uv divergence for the function @xmath67 . like for usual ( local ) models , superficial uv divergences , whose removal requires counterterms , can be present only in those functions @xmath67 for which @xmath68 is a non - negative integer ; see @xcite . from eq . ( [ deltacc ] ) we conclude that for any positive @xmath17 , such divergences can exist only in the 1pi functions with @xmath69 and arbitrary value of @xmath56 . for all these functions @xmath70 , that is , the divergences are logarithmic and the corresponding counterterms in the frequency momentum representation are constants . at first glance , we have established that the model ( [ extend ] ) requires infinitely many counterterms , and hence it is not renormalizable . however , it turns out to be sufficient to renormalize the 1pi green function @xmath71 only to render the model finite , as we shall now show . the first few feynman diagrams of @xmath72 are shown in fig . i for @xmath73 ; the symmetry coefficients are shown for general @xmath46 ( it would be embarrassing to depict the diagrams for fractional @xmath46 , but the idea is the same ) . the lines with a slash denote the bare propagator @xmath74\over ( 4\pi\nu_{0}t)^{d/2}}\,,\ ] ] the end with a slash corresponds to the field @xmath20 , and the end without a slash corresponds to @xmath4 . the initial ( left ) point in each diagram corresponds to @xmath75 , and the final ( right ) point with variable number of attached lines corresponds to @xmath29 . the crucial point is that , as is easily seen from fig . i , all possible 1pi subdiagrams entering into the diagrams of @xmath72 belong to the only 1pi function @xmath71 ; no other 1pi functions are involved . the function @xmath72 appears to be ` closed with respect to the renormalization , ' i.e. , we can eliminate their uv divergences by the only counterterm corresponding to its 1pi part @xmath71 . + moreover , the renormalization of the only function @xmath71 is in fact sufficient to completely renormalize all functions with @xmath76 . a typical diagram for @xmath77 is shown in fig . it is clear that any such diagram reduces to a product of blocks that belong to the simplest function with @xmath78 ( we recall that there is no integration over @xmath29 , the only point that connects the blocks ) . therefore , the diagram contains no superficial divergences ; all its divergences are those of the subdiagrams and they are completely removed by the renormalization of the function with @xmath78 . this is equally true for any diagram of any function with @xmath76 . + in the generic case all the loops are created by the presence of a single local vertex with any number of @xmath20 legs , from which continous chains of retarded diffusion propagators emanate . due to the structure of the nonlinear term these chains do not branch , but they may merge ( the single @xmath20-field in the nonlinearity allows only one outgoing propagator from each ordinary vertex , whereas up to @xmath46 incoming chains are allowed ) . a little reflection along the lines sketched above shows then that all divergent 1pi green functions are factorized : @xmath79 thus , we are left with the only counterterm to the function @xmath71 . it is constant ( see above ) , which in the time coordinate representation corresponds to the function @xmath16 . in the action functional , after the integration over the field argument , this gives @xmath80 . such term is present in the extended action ( [ extend ] ) , so that our model is renormalized multiplicatively , with the only renormalization constant , which we denote @xmath81 . the renormalized action has the form @xmath82 here and below the @xmath83 and @xmath84 are the renormalized analogs of the bare parameters , @xmath85 is the reference mass in the minimal subtraction ( ms ) scheme , which we use in practical calculations , and the constant @xmath81 depends on the dimensionless parameters @xmath83 , @xmath46 and @xmath86 . the renormalized green function @xmath87 , which is finite for @xmath88 , is given by the representation ( [ funi ] ) with the substitution @xmath89 . if we now replace the local initial condition with an integrable one : @xmath90 , then after fourier transforming we obtain wave - vector integrals in which all the propagator lines starting from the initial condition contain a multiplicative factor @xmath91 . for the large - scale asymptotic analysis using rg it is sufficient to keep the leading small wave - number terms in all the lines which amounts to the replacement @xmath92 , and we thus return to loop integrals of the problem with localized initial condition in which @xmath93 is the amplitude of the initial @xmath94 function . to clarify the idea , consider the one - loop graph of fig . i , whose analytic expression with the initial condition @xmath90 is @xmath95 here , @xmath96 is the diffusion propagator ( [ bare ] ) . fourier transforming @xmath97 with respect to @xmath98 we arrive at the expression @xmath99 where @xmath100 is the spatial fourier transform of the diffusion kernel ( [ bare ] ) . from the point of view of rg , the ir relevant terms are given by the leading terms of the gradient expansion of the initial condition : @xmath101 . this allows to replace ( [ g01k ] ) by @xmath102 which corresponds to the localized initial condition with the amplitude @xmath103 . it follows from eqs . ( [ act ] ) , ( [ extend ] ) , and ( [ renac ] ) that the original and renormalized action functionals satisfy the relation @xmath104 , if the bare and renormalized parameters are related as follows : @xmath105 with the only renormalization constant @xmath81 from eq . ( [ renac ] ) . this implies the relation @xmath106 for the corresponding green functions in eq . ( [ funi ] ) ; i.e. , this quantity is multiplicatively renormalizable . we use @xmath107 to denote the differential operation @xmath108 for fixed @xmath109 and operate on both sides of this equation with it . this gives the basic rg equation : @xmath110 g_{r}(e,\mu ) = 0 , \label{rge}\ ] ] where @xmath111 is nothing else than the operation @xmath107 expressed in the renormalized variables . in eq . ( [ rge ] ) , we have written @xmath112 for any variable @xmath75 , and the rg functions ( the @xmath0 function and the anomalous dimensions @xmath113 ) are defined as @xmath114 . \label{rgf}\ ] ] the relation between @xmath0 and @xmath113 results from the definitions and the relations ( [ reno ] ) . we shall see below that , for small @xmath51 , an ir stable fixed point @xmath115 of the rg equation ( [ rge ] ) exists in the physical region @xmath116 i.e. , @xmath117 , @xmath118 . the functions @xmath72 and @xmath87 coincide up to a constant ( i.e. , independent of the time and space variables ) factor @xmath81 and the choice of the parameters ( bare @xmath109 or renormalized @xmath119 , @xmath85 ) and can equally be used in the analysis of the ir behavior . the general solution of the rg equations is discussed in detail , e.g. , in @xcite . it follows from this solution that , when an ir stable fixed point is present , the leading term of the ir behavior of the function @xmath120 satisfies eq . ( [ rge ] ) with the substitution @xmath121 : @xmath122 g_{r}(e,\mu ) = 0 . \label{rge2}\ ] ] in our case , the value of the anomalous dimension at the fixed point is found exactly owing to the relation between @xmath0 and @xmath113 in eq . ( [ rgf ] ) : @xmath123 dimensional considerations yield @xmath124 , where @xmath125 is some function of dimensionless variables . the dependence on @xmath83 is not displayed explicitly , because the derivatives with respect to this parameter do not enter into eq . ( [ rge2 ] ) . it follows from eq . ( [ rge2 ] ) that @xmath125 satisfies at the fixed point the equation @xmath126 \xi(s , y)=0 $ ] , its general solution is @xmath127 , where @xmath128 is an arbitrary function of the second variable @xmath129 . for the green function ( [ funi ] ) we then obtain @xmath130 where the form on the ` scaling function ' @xmath131 is not determined by the equation ( [ rge2 ] ) . the dependence on the parameters @xmath84 , @xmath85 can be easily restored from the dimensionality considerations ( see table i ) : @xmath132 although the value of @xmath133 in eq . ( [ exact ] ) and the solution ( [ solution ] ) have been obtained without practical calculation of the constant @xmath81 and functions ( [ rgf ] ) , such calculation is needed to check the existence , positivity and ir stability of the fixed point . within the @xmath86 expansion , these facts can be verified already in the simplest one - loop calculation . in order to check the validity and self - consistency of the approach , we calculated the constant @xmath81 up to the two - loop approximation . the calculation is performed in the frequency momentum ( @xmath134 , @xmath135 ) representation and calls for the formulas derived in ref . @xcite for a model of critical dynamics . two key points are as follows : the convolution of two functions of the form @xmath136 is a function of the same form , @xmath137 where @xmath138 and @xmath139 are both positive and the coefficient has the form : @xmath140 while the product of two such functions can be represented as a single integral of a function of the same form with the aid of the generalized feynman formula : @xmath141 for the sake of brevity , below we give only the final result : @xmath142 where we have introduced a new coupling constant , @xmath143 and have written @xmath144 with the convergent single integral @xmath145 in particular , @xmath146 and @xmath147 $ ] . then for the corresponding beta function we obtain @xmath148 $ ] , where we have used the last relation in eq . ( [ reno ] ) and the fact that @xmath149 for the functions dependent only on @xmath150 . this yields @xmath151 substituting eq . ( [ z ] ) into eq . ( [ beta1 ] ) gives @xmath152 + o(u^{4 } ) . \label{beta}\ ] ] note that the poles in @xmath86 in the constant @xmath81 cancel out in the function ( [ beta ] ) ; this is a manifestation of the general fact that the rg functions must be uv finite , i.e. , finite as @xmath88 . the cancellation is possible by virtue of the correlation that exists between the @xmath153 and @xmath154 terms in eq . ( [ z ] ) and can be used as an additional check of the consistency of the approach . the simple ( linear ) dependence on @xmath86 is a feature specific to the ms scheme . from eq . ( [ beta ] ) we find an explicit expression for the coordinate of the fixed point : @xmath155 as already said above , for small positive @xmath86 and @xmath2 the fixed point is positive and ir stable : @xmath156 . we have applied the field theoretic renormalization group to the non - stochastic differential equation ( [ green ] ) and established the scaling behavior in the ir asymptotic range , as a consequence of the existence of the ir stable fixed point in the physical range of parameters . the same asymptotic behavior is shown to be valid for integrable initial conditions which thus constitute the universality class of this fixed point . the key points are the formulation of the problem as a field theoretic model with an ultralocal term concentrated at a spacetime point and the fact that this model appears multiplicatively renormalizable , in spite of the naive power counting that indicates nonrenormalizability . simple explicit form of the scaling dimensions follows from the fact that there is only one independent renormalization constant in the problem . in particular , this explains a simple value @xmath157 of the exponent in the argument @xmath158 of the scaling function ( [ solution ] ) ( in models of dynamical critical phenomena @xcite and some models of nonlinear diffusion @xcite this exponent differs from two ) . recently , it has been conjectured @xcite that the dynamic exponent @xmath159 in the present problem . our asymptotic solution ( [ solution ] ) , however , does not predict any deviation from the canonical value @xmath157 , since there is no renormalization of the diffusion coefficient in the ms scheme we have used . in ref . @xcite with the use of a different renormalization procedure it was concluded that @xmath160 . we think , however , that it is not consistent to prescribe physical quantities values of the order @xmath161 on the basis of the _ one - loop _ calculation carried out in ref . @xcite , but a two - loop analysis is required for this accuracy . the rg analysis allows one to derive the rg equation rigorously and to prove that the behavior ( [ solution ] ) is indeed realized for @xmath51 , @xmath162 in the ir asymptotic range , specified by the relations @xmath163 and @xmath164 , where @xmath165 is the uv scale . the general solution of eq . ( [ rge ] ) interpolates between the ordinary perturbation theory for eq . ( [ green ] ) and the self - similar asymptotic expression ( [ solution ] ) . the scaling function @xmath166 can be calculated within the @xmath86 expansion ; in the lowest order one easily obtains @xmath167+o(\eps)$ ] . we hope that the ideas presented above might be useful in other models containing ultralocal contributions , which have several charges and hence richer ir behavior . another direction of generalization would be the analysis of green functions of vector quantities . we thank l. ts . adzhemyan , a. kupiainen , m. yu . nalimov and a. n. vasilev for discussions . the work was supported by the grant center for natural sciences ( grant no . e00 - 3 - 24 ) , the nordic grant for network cooperation with the baltic countries and northwest russia no . fin-18/2001 , and the academy of finland ( grant no . 79781 ) . a. n. vasilev , _ quantum - field renormalization group in the theory of critical phenomena and stochastic dynamics _ petersburg institute of nuclear physics , st . petersburg , 1998 ) [ in russian ; english translation : gordon & breach , in preparation ] . proceedings of the international conference `` renormalization group . '' d. v. shirkov , d. i. kazakov , and a. a. vladimirov ( eds . ) , world scientific , 1988 ; proceedings of the second international conference `` renormalization group 91 . '' d. v. shirkov and v. b. priezzhev ( eds . ) , world scientific , 1991 ; proceedings of the third international conference `` renormalization group 96 . '' d. v. shirkov , d. i. kazakov , and v. b. priezzhev ( eds . ) , jinr , dubna 1997 . m. doi , j. phys . * 9 * , 1465 ( 1976 ) ; * 9 * , 1479 ( 1976 ) . ya.b . zeldovich and a.a . ovchinnikov , zh . . fiz . * 74 * , 1588 ( 1978 ) . p. grassberger and m. scheunert , fortschr . phys . * 28 * , 547 ( 1980 ) . l. peliti , j. phys . a : math . gen . * 19 * , l365 ( 1986 ) . b. p. lee , j. phys . a : math . gen . * 27 * , 2633 ( 1994 ) . j. l. cardy and u. c. tuber , phys . * 77 * , 4780 ( 1996 ) ; j. stat . 90 * , 1 ( 1998 ) .

the paper is an attempt to relate two vast areas of the applicability of the renormalization group ( rg ) : field theoretic models and partial differential equations . it is shown that the green function of a nonlinear diffusion equation can be viewed as a correlation function in a field - theoretic model with an ultralocal term , concentrated at a spacetime point . this field theory is shown to be multiplicatively renormalizable , so that the rg equations can be derived in a standard fashion , and the rg functions ( the @xmath0 function and anomalous dimensions ) can be calculated within a controlled approximation . a direct calculation carried out in the two - loop approximation for the nonlinearity of the form @xmath1 , where @xmath2 is not necessarily integer , confirms the validity and self - consistency of the approach . the explicit self - similar solution is obtained for the infrared asymptotic region , with exactly known exponents ; its range of validity and relationship to previous treatments are briefly discussed .

nlin0207006

quantum coherence or quantum superposition is one of the most fundamental feature of quantum mechanics that distinguishes the quantum world from the classical world . it is one of the main manifestation of quantumness in a single quantum system . for a composite quantum system , due to its tensor structure , quantum superposition could directly lead to quantum entanglement , another intriguing feature of quantum mechanics and the very important physical resource in quantum information processing [ 1 ] . in fact , safely speaking , quantum coherence is one necessary condition for almost all the mysterious features of a quantum state . for example , both entanglement and quantum discord that has attracted much attention recently [ 1 - 10 ] , have been shown to be quantitatively related to some special quantum coherence [ 11,12 ] . however , when a quantum system undergoes a noisy quantum channel or equivalently interacts with its environment , the important quantum feature , i.e. , the quantum decoherence , could decrease . it is obvious that whether decoherence happens strongly depends on the quantum channel and the state itself , but definitely a quantum channel describes the fate of quantum information that is transmitted with some loss of fidelity from the sender to a receiver . in addition , if the subsystem of an entangled state passes through such a channel , disentangling could happen and go even more quickly . decoherence as well as disentangling for composite systems has never been lack of concern from the beginning . a lot of efforts have been paid to decoherence in a wide range such as the attempts of understanding of decoherence [ 13,14 ] , the dynamical behaviors of decoherence subject to different models [ 15 - 20 ] , the reduction or even prevention of decoherence [ 21 - 23 ] , the disentangling through various channels and so on [ 24 - 26 ] . actually , most of the jobs can be directly or indirectly summarized as the research on to what degree a noisy quantum channel or the environment influences ( mostly destroys ) the coherence , the fidelity or entanglement of the given quantum system of interests . so it is important and interesting to consider how to effectively evaluate the ability of a quantum channel that leads to decoherence , the loss of fidelity of a quantum state , or disentangling of a composite system , in particular , independent of the state . in this paper , we address the above issues by introducing three particular measure , the decoherence power , the infidelity power and the disentangling factor to describe the abilities , respectively . this is done by considering how much fidelity , coherence or entanglement ( for composite systems ) is decreased by the considered quantum channel on the average , acting on a given distribution of quantum state or the subsystem of an entangled state . this treatment has not been strange since the entangling power of a unitary operator as well as the similar applications in other cases was introduced [ 27,28 ] . however , because the calculation of the abilities of a quantum channel strongly depends on the structure of the quantum states which undergo the channel , the direct result is that only 2- dimensional quantum channel can be effectively considered . for the high dimensional quantum channels , one might have to consider these behaviors on a concrete state , which is analogous to that the entangling power can be only considered for the systems of two qubits [ 27 ] . these cases will not be covered in this paper . this paper is organized as follows . in sec . ii , we treat the quantum channel as the reduction mechanism of the fidelity and present the infidelity power accompanied by some concrete examples . in sec . iii , we consider how to influence the coherence of a state and give the decoherence power . some examples are also provided . in sec . iv , we analyze the potential confusion if we consider the decoherence of a mixed state and briefly discuss how to consider the influence of quantum channel on the subsystem of a composite quantum system . the conclusion is drawn in sec . when a quantum state undergoes a quantum channel , the state will generally be influenced . although some particular features of the state could not be changed , the concrete form of the state , i.e. , the fidelity , is usually changed . in order to give a description of the ability to which degree a quantum channel influences a quantum state , we would like to first consider the infidelity power of a quantum channel . with fidelity of two states @xmath0 and @xmath1 mentioned , one could immediately come up with the fidelity defined by @xmath2 or the trace distance defined by @xmath3 with @xmath4 [ 29 ] . however , consider some given distribution of state @xmath0 , one can find that the mentioned definitions are not convenient to derive a state - independent quantity . so we would like to consider another definition of the fidelity based on frobenius norm @xmath5 . _ definition 1_.-the fidelity of the state @xmath0 and @xmath6 is defined by @xmath7it is clear that if and only if @xmath0 and @xmath1 are the same , the fidelity @xmath8 . to proceed , we have to introduce a lemma . _ lemma 1_. for any @xmath9-dimensional matrix @xmath10 , and an @xmath11-dimensional maximally entangled state in the computational basis @xmath12 , the following relations hold:@xmath13and @xmath14__proof . _ _ the proof is direct , which is also implied in ref . [ 27].@xmath15 now , let @xmath16 denote a quantum channel and @xmath17 denote the final state of @xmath0 going through the channel , the fidelity given in eq . ( 1 ) can be rewritten as@xmath18based on lemma 1 , we can find that@xmath19 ^{t}\left\vert \phi _ { n}\right\rangle \notag \\ = ntr\left [ \rho \otimes \rho ^{\ast } \right ] \left ( \$\otimes \mathbf{1}% _ { n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \notag \\ = n^{3}\left\langle \phi _ { n^{2}}\right\vert \left [ \rho \otimes \rho ^{\ast } % \right ] \left ( \$\otimes \mathbf{1}_{n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \otimes \mathbf{1}% _ { n^{2}}\left\vert \phi _ { n^{2}}\right\rangle \notag \\ = n^{3}tr\left ( \$\otimes \mathbf{1}_{n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \otimes \rho ^{\ast } \otimes \rho \notag \\ \times s_{23}\left ( \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \otimes \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \right ) s_{23}\end{gathered}\]]and @xmath20 ^{2}\otimes \mathbf{1}_{n}\left\vert \phi _ { n}\right\rangle \notag \\ = ntr\left [ \$(\rho ) \otimes \rho ^{\ast } \right ] \left ( \$\otimes \mathbf{1}% _ { n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \notag \\ = n^{3}\left\langle \phi _ { n^{2}}\right\vert \left [ \$(\rho ) \otimes \rho ^{\ast } \right ] \left ( \$\otimes \mathbf{1}_{n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \otimes \mathbf{1}% _ { n^{2}}\left\vert \phi _ { n^{2}}\right\rangle \notag \\ = n^{3}\left\langle \phi _ { n^{2}}\right\vert \left ( \$\otimes \mathbf{1}% _ { n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \otimes \left [ \$(\rho ) \right ] ^{t}\otimes \rho \left\vert \phi _ { n^{2}}\right\rangle \notag \\ = n^{3}tr\left\ { \left ( \$\otimes \mathbf{1}_{n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \otimes \rho ^{\ast } \otimes \rho \right\ } \notag \\ \times s_{23}\left ( \$\otimes \mathbf{1}_{n}\right ) \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert \otimes \left\vert \phi _ { n}\right\rangle \left\langle \phi _ { n}\right\vert s_{23},\end{gathered}\]]where we consider @xmath21 with @xmath22 representing the swapping operations between the _ 2nd _ and _ 3rd _ subsystems . let@xmath23 s_{23 } \\ w_{2}=s_{23}\left [ \varrho _ { \phi } \otimes \varrho _ { \phi } \right ] s_{23 } \\ q_{1}=\tilde{\$}\left ( \varrho _ { \phi } \right ) \otimes \mathbf{1}_{n^{2 } } \\ q_{2}=\varrho _ { \phi } \otimes \mathbf{1}_{n^{2}}% \end{array}% \right.\]]with @xmath24 and @xmath25 , and substitute eq . ( 5 ) and eq . ( 6 ) into eq . ( 4 ) , @xmath26 can be rewritten as@xmath27with @xmath28 .\]]thus @xmath29 only depends on the influence of the maximally entangled state @xmath30 through the considered quantum channel instead of the information of the state of interests . the infidelity power can be directly defined as the difference of fidelity before and after going through the channel subject to some given distribution . however , different distributions would lead to quite different infidelity power . in this paper , we restrict ourselves to the uniform distribution . in order to describe the difference between the fidelities of the state @xmath31 with and without quantum channel @xmath16 , we can use @xmath32consider the uniform distribution @xmath33 of the state @xmath0 [ 27 ] , we can define the infidelity power as follows . _ definition 2_. the infidelity power of the @xmath9-dimensional quantum channel @xmath16 can be defined as@xmath34with @xmath35 where @xmath36 is some normalization factor , @xmath37 denotes the measure over the state @xmath0 induced by the uniform distribution @xmath33 . it is obvious that @xmath38 does not depend on the concrete form of a density matrix but only the dimension of the density matrices and the structure of the states . however , it is unfortunate that the structure of the high dimensional states is very complicated . it is difficult to describe a given distribution of such a state space . therefore , we only present the concrete expression of @xmath38 for a system of qubit . _ theorem 1_. the infidelity power of the quantum channel @xmath16 on a qubit can be given by@xmath39with @xmath40 _ proof . _ the density matrix of any qubit @xmath0 can be given in the bloch representation as@xmath41where @xmath42 $ ] is the pauli matrices , @xmath43 $ ] with @xmath44 the inclination and @xmath45 the azimuth in the bloch sphere , respectively , and @xmath46 is the radius . since we suppose the state is distributed uniformly , the state density can be given by @xmath47 . so one can easily find that@xmath48and @xmath49 d\varphi d\theta \notag \\ & = & \frac{1}{4}\left [ \mathbf{1}_{4}+\frac{1}{5}\left ( \sigma _ { x}\otimes \sigma _ { x}-\sigma _ { y}\otimes \sigma _ { y}+\sigma _ { z}\otimes \sigma _ { z}\right ) \right ] , \end{aligned}\]]which is consistent with eq . the proof is completed.@xmath50 as applications , we would like to calculate the infidelity power for a qubit of the depolarizing channel @xmath51 given in kraus representation as @xmath52the phase - damping channel @xmath53 given by @xmath54the amplitude - damping channel @xmath55 given by @xmath56and the generalized amplitude - damping channel @xmath57 given by@xmath58substitute these quantum channels [ 29 ] into @xmath59 , one can easily find that @xmath60 , \]]@xmath61 , \]]@xmath62 , \]]and@xmath63 , \]]consider the swap operator @xmath64 @xmath65 and substitute eqs ( 23 - 26 ) into eq . ( 12 ) , we have @xmath66 it is apparent that @xmath67 for @xmath68 , which means there is no quantum channel operating on them . @xmath69 , if @xmath70 , or @xmath71 . this is because @xmath57 will become @xmath72 or its dual quantum channel on this condition . it is a natural conclusion that @xmath73 reduce the fidelity due to @xmath74 , in particular , with the increase of the decay rate @xmath75 , the infidelity power is increasing . but it is not difficult to find that given @xmath75 , the ability of reducing the fidelity can be weakened if we adjust @xmath76 for the generalized amplitude - damping channel . the minimal infidelity power is obtained if @xmath77 . for a given density matrix , the definition of quantum coherence depends on not only the density matrix itself , but also the bases on which the density matrix is written . to evaluate whether there exists quantum coherence in a quantum state , physically one has to find some observables to reveal the interference by measuring them . it can be shown that all the quantum states ( density matrices ) but the maximally mixed state can demonstrate interference because one can always find such an observable that can reveal it so long as the observable does nt commute with the density matrix of interests . based on such a think , one could find various ways to defining quantum coherence . here we would like to define the maximal distance between the density matrix of interests and the corresponding maximally mixed state with all potential bases taken into account . the rigorous formulae can be given as follows . _ definition 3_.-quantum coherence for an @xmath9-dimensional density matrix @xmath0 is defined as the maximal distance between @xmath0 and the maximally mixed state @xmath78 with all potential bases considered , i.e. , @xmath79 where @xmath80 denotes the @xmath9-dimensional identity . @xmath81 is also the maximally potential coherence within all possible bases . _ the density matrix @xmath0 in a different framework can be given by @xmath82 where @xmath83 is the unitary transformation relating two different bases of @xmath0 . so the distance @xmath81 can be written as@xmath84where @xmath85 is the frobenius norm . in other words , given a basis in which the density matrix is written by @xmath31 , the coherence in this basis can be described completely by the off - diagonal entries of @xmath86 [ 11 ] . so if we extract the maximal contribution of coherence with different basis considered , we have @xmath87 _ { ij}^{2 } \notag \\ & = & tr\rho ^{2}-\min_{u}\sum_{k}\left [ u\rho u^{\dagger } \right ] _ { kk}^{2 } \notag \\ & = & tr\rho ^{2}-\frac{1}{n}.\end{aligned}\]]eq . ( 33 ) holds because one can always find a unitary matrix such that @xmath88 have the equal diagonal entries . it is obviously shown that our definition actually quantify the maximal coherence of a state by considering all the possible bases [ 11 ] . the proof is completed.@xmath89 from the definition , one will easily see that such a distance does not depend on the bases , i.e. , the unitary matrix @xmath83 . this does not contradict with the usual statement of the dependence of bases for quantum coherence . in fact , the dependence of bases is in that the different observables chosen to reveal the interference will lead to the different interference visibilities . this actually implies the ability of the observable that can reveal the quantum coherence of the given state , instead how much quantum coherence could be revealed for a state . in addition , it is obvious that the maximal value of the coherence measure is @xmath90 which can be attained by all the pure states . it is easy to understand because all the pure states are equivalent or interconverted under appropriate unitary transformations . it is obvious that eq . ( 31 ) is also closely related to the purity of a quantum state , the quantumness of a single state , so the coherence measure can also be understood as the purity measure or the quantumness measure etc . with a small potential deformation [ 30,31 ] . with this definition , we can proceed to consider a quantum system with the state @xmath0 undergoes a quantum channel @xmath16 with the final state can be given by @xmath91 . thus the coherence of the final state @xmath91 can be easily written as@xmath92 ^{2}-\frac{1}{n}.\]]for the latter use , next we would like to present a new form that separates the quantum state and the quantum channel in eq . ( 34 ) . using eqs . ( 6 ) and ( 7 ) , one can directly obtain@xmath93thus eq . ( 35 ) shows that the left quantum coherence of @xmath0 when it passes through the quantum channel @xmath16 . so the decoherence power that describes to what degree the decoherence has been reduced can be defined as the difference of the coherence before and after the channel subject to some distribution of quantum states . based on the above analysis , we can obtain the following definition for decoherence power . _ definition 4_. the decoherence power of the @xmath9-dimensional quantum channel @xmath16 can be defined as@xmath94with @xmath95 where @xmath96 @xmath97 denotes the measure over the pure state @xmath0 induced by the uniform distribution @xmath98 . from this definition , we should first note that we only consider the distribution of pure states instead of the whole state space . why we do nt cover mixed states will be analyzed in the latter part . in addition , one can find that @xmath99 analogous to infidelity power does not depend on the concrete form of a density matrix but only the dimension of the density matrices and its structure . however , due to the same reason as the infidelity power , we only present the concrete expression of @xmath100 for a system of qubit . _ theorem 2_. the decoherence power of the quantum channel @xmath16 on a qubit can be given by@xmath101with @xmath102 _ proof . _ the density matrix of any pure qubit can be given in the bloch representation as@xmath103where @xmath42 $ ] is the pauli matrices , @xmath43 $ ] with @xmath44 the inclination and @xmath45 the azimuth in the bloch sphere , respectively . since we suppose the state is distributed uniformly , the state density can be given by @xmath104 . so one can easily find that@xmath105thus , @xmath106it is equivalent to eq . the proof is completed.@xmath15 similarly , we also calculate the decoherence power for a qubit of the depolarizing channel @xmath51 , the phase - damping channel @xmath53 , the amplitude - damping channel @xmath55 and the generalized amplitude - damping channel @xmath57 . based on the analogous calculation to those in eqs . ( 27 - 30 ) , we have @xmath107as is expected again , we can find that @xmath108 will vanish if @xmath109 . however , unlike the infidelity power which is a monotone function of @xmath75 , the decoherence power of all the mentioned channels but the phase - damping channel will reach a maximum at some particular @xmath110 $ ] . this should be distinguished from the case where only the coherence in a fixed basis is considered . in fact , eq . ( 35 ) is suitable for all quantum states including mixed states when we consider the decoherence of a state through a channel . however , it is not hard to see that in the sense of the previous definition of coherence , the reduction of coherence depends on not only the quantum channel itself , but also the quantum state that undergoes the channel . so it is possible that a given quantum channel could reduce the coherence of some states and increase the coherence of the other states . thus the decoherence power will lead to confusion since it is defined as the average contribution of the reduction of coherence subject to a given state distribution . in order to demonstrate the inconsistent roles of a quantum channel on different states , we would like to take the above three quantum channels as examples to show the reduction and increase of the coherence . actually , one can find that the coherence of any states based on definition 1 will be reduced if they undergo the depolarizing channel @xmath51 and the phase - damping channel @xmath53 . but the amplitude - damping channel could lead to the increase of the coherence . in fig . 1 , we plot the two regions within and without which the coherence of all the states can be reduced and increased , respectively . . the outer ball represents all the states of a qubit . the inner spheriod denotes the states that loses coherence via the channel . the two balls inscribe at point @xmath111 ] since the direct consideration of decoherence of mixed state could lead to the confusion , we have to consider the mixed states in an indirect way , namely , we turn to an entangled bipartite composite system of a pure state with one of its subsystems undergoing a quantum channel . it is obvious that the reduced density matrix of any subsystem is mixed ; in addition , any local quantum channel on the subsystem will always reduce the entanglement of the composite system since a good entanglement measure should be an entanglement monotone . however , it should be noted that the decoherence based on disentanglement is different from our initial definition 1 . strictly speaking , we actually characterize the reduction of a special coherencequantum entanglement . in this sense , what we consider should be called as the disentangling power instead of the decoherence power of the original definition 1 . consider a bipartite pure state of two qubits @xmath112 , the entanglement can be reduced if one subsystem undergoes a quantum channel @xmath16 . if we employ concurrence [ 32 ] as entanglement measure , one can easily find that [ 33 ] @xmath113with @xmath114 . so no matter what distribution of quantum states is considered , the concurrence is directly reduced by the factor @xmath115 . thus the disentangling power of @xmath16 in this case can be directly characterized by @xmath116 . in addition , if @xmath117 is a mixed state , one can also consider the reduction of entanglement subject to some quantum channel . however , due to the complicated expression of concurrence for mixed states . so far there has not such a factorial form of the reduction of entanglement . in this paper , we study how a quantum channel influences the fidelity and the coherence of a state when the state goes through it and briefly discuss the reduction of entanglement when a subsystem undergoes a channel . we give the infidelity power and decoherence power of a quantum channel . they are independent of quantum states and describe an average contribution of the infidelity and the decoherence . as applications , we calculate the infidelity power and decoherence power of depolarizing channel , phase - damping channel , amplitude - damping channel and generalized amplitude - damping channel , respectively . we show that although quantum channels ( if it is not trivial ) definitely reduce the fidelity of the state through it and the entanglement with subsystems through it , for some channels , they could increase the coherence of some states . this work was supported by the national natural science foundation of china , under grant no . 11175033 and the fundamental research funds of the central universities , under grant no . dut12lk42 . t. werlang , s. souza , f. f. fanchini , and c. j. villasboas , phys . a , 80 , 024103 ( 2009 ) ; f. f. fanchini , t. werlang , c. a. brasil , l. g. e. arruda , and a. o. caldeira , _ ibid . _ , * 81 * , 052107 ( 2010 ) ; j. maziero , l.c . celeri , r. m. serra , and v. vedral , _ ibid . _ , * 80 * , 044102 ( 2009 ) ; a. ferraro , l. aolita , d. cavalcanti , f. m. cucchietti , and a. acin , _ ibid . _ , * 81 * , 052318 ( 2010 ) .

when a quantum state undergoes a quantum channel , the state will be inevitably influenced . in general , the fidelity of the state is reduced , so is the entanglement if the subsystems go through the channel . however , the influence on the coherence of the state is quite different . here we present some state - independent quantities to describe to what degree the fidelity , the entanglement and the coherence of the state are influenced . as applications , we consider some quantum channels on a qubit and find that the infidelity ability monotonically depends on the decay rate , but in usual the decoherence ability is not the case and strongly depends on the channel .

1411.0271

the magnetosphere of an accreting x - ray pulsar expands as the mass accretion rate decreases . as it grows beyond the co - rotation radius , centrifugal force prevents material from entering it . thus , accretion onto the magnetic poles ceases , and , consequently , x ray pulsations cease . this phenomenon has recently been observed , for the first time , in gx 1 + 4 and gro j1744 - 28 with rxte@xcite . here , we present further evidence to show that the phenomenon repeated itself for gro j1744 - 28 during the decaying phase of its latest x - ray outburst . the asm light curve ( as shown in the top panel of fig . 1 ) reveals that there have been two episodes of x - ray outburst in gro j1744 - 28 , separated by roughly one year . the source has been extensively monitored by the main instruments aboard rxte since its discovery@xcite . for detailed analyses , we have selected a number of pca observations , based on the asm light curve , to cover the decay phase of the outbursts . 1 ( bottom panel ) shows the pulsed fraction ( @xmath0 ) measured with each observation . for comparison , the published results@xcite for the first outburst are also presented here . a striking feature is the precipitous drop of the pulsed fraction as the source became `` quiescent '' both times . gro j1744 - 28 was generally not so quiet after the first outburst . in previous work@xcite , we happened to catch a brief period ( as indicated in fig . 1 ) when the pulsed emission became very weak or was not detected at all in some observations . following the latest ourburst , the source has shown little activity . its presence ( at about 20 - 30 mcrab ) has , however , been firmly established by the pca slew data . this provides a good opportunity to verify our previous interpretation of the phenomenon . we have searched for the known 2.14 hz pulse frequency , employing various techniques including ffts and epoch - folding , but have failed to detect it since the end of june 1997 ( as marked in fig . 1 ) . the results therefore argue strongly that the centrifugal barrier is active in this source during such faint period , as we have concluded previously@xcite . the source also shows interesting spectral evolution during the decay . the observed x - ray spectrum can be characterized by a simple power law with an exponential high - energy cutoff . as the quiescent state is approached , the spectrum softens significantly : the power - law becomes steeper , and more prominently , the cutoff energy decreases by roughly a factor of 2 ( see fig . 2 ) . at the end of the first `` quiescent '' period , the spectrum would recover to the bright - state shape . we have proposed before that the x - ray emission probably consists of two components : the emission from a large portion of the neutron star surface ( thus unpulsed ) , due to the `` leakage between field lines '' @xcite , and that from `` hot spots '' near the poles ( pulsed plus unpulsed ) . when the source was bright , the latter dominated , so the spectrum was hard ( corresponding to a much higher temperature of the hot spots ) . however , as soon as the centrifugal barrier took effect in the quiescent state , the observed x - rays were all due to the surface emission and their spectrum was therefore softer . it is interesting to note that the pileup of accreting matter on the neutron star surface might also cause unstable thermonuclear burning and produce type i bursts@xcite , like in x - ray bursters . the lack of such ( or does it ? ) in gro j1744 - 28 may be due to the suppression of this process by a significantly higher field@xcite . gro j1744 - 28 does produce x - ray bursts@xcite , unlike any other x - ray pulsars . the bursts are thought to be the product of accretion instability@xcite . they occurred at a rate of one to two dozen per hour near the peak of the outbursts@xcite , and the rate decreased as the x - ray flux decayed . at the start of the first quiescent period , the bursting activity ceased entirely@xcite for weeks before resuming again near the end@xcite . 3 ( the top panel ) shows an example of such activity ( with 7 major bursts ) on mjd 50260 ( @xmath1 26 june 1996 ) . we have separated the light curve of 26 june 1996 into burst and non - burst intervals . the x - ray pulsation is detected during the bursts but is _ not _ detected outside of them ( see fig . 3 ) . this is again consistent with the presence of the centrifugal barrier in gro j1744 - 28 . a sudden surge in the mass accretion rate that produces a burst would also momentarily push the magnetosphere inside the co - rotation radius and thus , the accretion to the poles would resume to produce the pulsed emission . as the system relaxes following a burst , the magnetosphere expands again ; the inhibition of accretion by the centrifugal barrier again suppresses the pulsation . we conclude by summarizing the main results as follows : * the results support our previous conclusion that the cessation of pulsed emission when the source becomes faint is a manifestation of the centrifugal barrier . * for gro j1744 - 28 , the x - ray emission in the quiescent state ( unpulsed ) likely comes from a large portion of the neutron star surface , due to the penetration of accretion flows through the magnetosphere . * accretion instability can still occur in the quiescent state ( less frequently ) , and produce type ii bursts . the pulsed emission was apparent during the bursts , presumably due to the resumption of accretion to the magnetic poles because of the momentary shrinkage of the magnetosphere . the pulsation stopped as the system recovered to the quiescent state .

we present further observational evidence of the effects of a centrifugal barrier in gro j1744 - 28 , based on continued monitoring of the source with rxte .

astro-ph9712175

in a 1959 paper m. e. fisher introduced the general bond - decorated ising model as one example of a set of exactly solvable transformations of spin-1/2 ising models.@xcite a bond - decorated ising model has an `` arbitrary statistical mechanical system '' inserted in every original ising bond . the partition function of this decorated model is related to the partition function of the original or bare ising model by the addition of a prefactor and a renormalization of the coupling constants and magnetic moments ( the ising model is supposed to be in a parallel magnetic field).@xcite knowledge of the partition function of a given ising model thus allows one to obtain the partition function of any bond - decorated version of that ising model . more recently , streka and jaur have used the method of bond - decoration to investigate the thermodynamics of mixed ising - heisenberg chains in parallel magnetic fields where the decorating unit is a spin dimer or trimer with anisotropic heisenberg coupling ( see fig . [ 4:fig : isingheisenberg]).@xcite the partition function of these chains is readily obtained from the known partition function of the ising chain and the energy levels of the decorating unit . therefore , they could calculate exact magnetic properties and theoretically show , for example , the existence of magnetization plateaus in certain bond - alternating chains . the ising - heisenberg chain discussed in ref . ( showing only two unit cells ) . the bonds of the ising chain ( 1-@xmath2-@xmath3- ) are decorated with heisenberg dimers @xmath4(2 - 3 ) , ( @xmath5-@xmath6 ) , the partition function of this chain is exactly solvable . ] the convenience of the decorated ising chain as a theoretical model for spin chains derives from the relative ease with which exact solutions are obtained , in contrast for example with the pure heisenberg chain , for which no exact partition function has been found . up to now , this property of solvability has been the prime motive for the study of these chains in the literature . indeed , in ref . the decorated ising chain was considered as a substitute for the intractable heisenberg model , and in ref . the principle reason for introducing ising bonds to replace the more reasonable heisenberg bonds in a chain of cu^2+^ ions was the desire to obtain a solvable model . this approach can be applied to any type of heisenberg chain with a repeating unit : replace enough preferably ferromagnetic heisenberg bonds by ising bonds to obtain a decorated ising chain that is solved easily and , in some cases , exhibits thermodynamic properties that are qualitatively comparable with those of the original chain.@xcite however , the role of the decorated ising chain in the field of one - dimensional magnetism is not confined to that of a simplified model of realistic quantum spin chains . in this paper we show that some new molecular rings and chains are real examples of decorated ising systems . concretely , we treat a [ dycumocu ] infinite chain@xcite and a ( dycr)@xmath8 tetrameric ring.@xcite these compounds were recently synthesized in the course of the ongoing synthetic efforts to make new and better single - chain magnets ( scms ) and single - molecule magnets ( smms ) , whose characteristic property is a blocking or slow relaxation of magnetization at low temperatures . we will not be concerned here with these dynamical aspects of their magnetism , but only with their static magnetic properties . a necessary property of smms and scms is a magnetic anisotropy . one line of approach is to introduce anisotropy by means of lanthanide ions , whether or not in combination with transition metal ions.@xcite the two compounds considered here are products of this approach , with dysprosium as lanthanide ion . the dy@xmath9 ion plays a crucial role in these systems ; the nature of the ground state of this ion in its ligand environment determines whether the system is a decorated ising system or not and consequently , whether its partition function is exactly solvable or not . the ground kramers doublet of dy@xmath9 must have complete uniaxial magnetic anisotropy - factor of the kramers doublet is not zero ; for example @xmath10 and @xmath11 . ] and must be separated from excited kramers doublets by an amount that is large compared with the exchange coupling ( typically , this separation must be 100 @xmath12 or more).@xcite the required information on the ground and excited doublets of the dy@xmath9 monomer can be derived from _ ab initio _ calculations on the monomer complex , isolated from the polynuclear compound.@xcite the [ dycumocu ] chain and the ( dycr)@xmath8 ring are shown to be decorated ising chains in an arbitrarily directed magnetic field . the magnetic properties , in particular powder magnetization and susceptibility , are calculated with the help of the transfer - matrix method , which is a bit more general and convenient for numerical computation than the renormalization of the ising parameters , which was used by fisher and by streka and jaur . the results compare well with experiment and allow to determine values for the exchange coupling constants . the excited crystal field kramers doublets ( or stark levels ) of dy@xmath9 are not included in the decorated ising model . because of their relatively low energetic position , these kramers doublets can have a non - negligible contribution to the magnetic properties of the chain . we account for this in a first approximation by adding this contribution , as calculated _ ab initio _ for the monomeric dy@xmath9 complex , to the results of the decorated ising model . part of the decorated ising chain . in each ising bond is inserted an arbitrary statistical mechanical system@xcite , called decorating unit , that interacts only with the two ising spins @xmath13 at the vertices of the bond . the ising spin variables commute with the hamiltonian and have definite values in the eigenstates of the chain.,width=325 ] the decorated ising chain may be divided into units delimited by the ising spins , as in fig . [ 4:fig : decisingchain ] . the hamiltonian of the decorated ising chain ( or ring ) of length @xmath14 may accordingly be written as follows : @xmath15 where the subhamiltonians @xmath16 correspond to units of the chain . if the chain is closed into a ring , periodic boundary conditions apply by identifying the last with the first spin : @xmath17 . the ising spins @xmath13 commute with the subhamiltonians and the subhamiltonians commute with each other : @xmath18=0 \quad \text{and } \quad [ \hat{h}_i,\hat{h}_j]=0.\ ] ] it is further assumed that there is no direct interaction between the decorating units themselves , i.e. , two different @xmath16 have no variables in common , except for an @xmath13 when they are neighbors . in writing eq . ( [ 4:eq : hamdecising ] ) we made use of the fact that the ising spins are conserved variables and may be considered as parameters rather than operators . in the case of spin-1/2 , they take on the values @xmath19 and @xmath20 , so that to each decorating unit there correspond four different subhamiltonians @xmath21 .. in fact , any parameter that takes on a finite number of values can serve as @xmath13 in eq . ( [ 4:eq : hamdecising ] ) . in practice however , @xmath13 represents most commonly a kramers doublet , which is most often described as an effective spin-1/2 . this is the case for the examples considered in this paper . ] since there is no direct interaction between the decorating units of the chain , the subhamiltonians @xmath16 in eq . ( [ 4:eq : hamdecising ] ) work for a given set of @xmath13 values on disjunct spaces . an eigenfunction of @xmath22 is then simply a direct product of @xmath14 independent eigenfunctions , one of each @xmath16 . the corresponding energy is : @xmath23 where @xmath24 is the @xmath25-th energy level of @xmath21 . following ref . we write the partition function of one decorating unit for fixed @xmath13 and @xmath26 on the bond vertices : @xmath27,\ ] ] where @xmath28 . the total partition function for the chain is then given by @xmath29 this is the well - known form of a transfer - matrix solution.@xcite each decorated bond ( or pair of neighboring ising spins ) is represented by a transfer matrix @xmath30 whose elements are the @xmath31 , the values of @xmath13 labeling the rows and the values of @xmath26 labeling the columns . explicitly for the spin-1/2 decorated ising chain : @xmath32 since most applications deal either with rings or very long chains , the boundaries can be identified ( @xmath17 ) and expression ( [ 4:eq : partitionfunction ] ) can be written as the trace of a matrix product : @xmath33 this is the most general expression for the partition function of a decorated ising chain . if the chain has translational symmetry the partition function is expressed in terms of eigenvalues.@xcite suppose that the chain is a repetition of identical decorated ising bonds , then the @xmath30 are all the same , so that @xmath34 where the @xmath35 are the eigenvalues of @xmath36 . according to perron s theorem on positive matrices , the eigenvalue @xmath37 with largest modulus is real , positive and nondegenerate.@xcite this yields the simple but exact result for the free energy per unit cell of the infinite chain : @xmath38 if a unit cell of the chain is spanned by @xmath39 decorated bonds instead of one then it is only necessary to combine @xmath39 transfer matrices into one new transfer matrix @xmath40 , with largest eigenvalue @xmath41 , to be used in eqs . ( [ 4:eq : partfunc_periodic ] ) and ( [ 4:eq : freenergy_infinite ] ) , where @xmath14 is to be replaced by @xmath42 . this situation arises , for example , when the local easy axes on the ising ions are not parallel but canted with respect to each other . the solution in terms of transfer matrices , eq . ( [ 4:eq : partfunc_transmatr ] ) , is not limited to decorated spin-1/2 ising chains . it is valid for chains having ising spins of any multiplicity or a combination of ising spins with different multiplicities , in which case the dimension of the transfer matrix is different from two by two . another advantage of the transfer matrix method is that it can be readily extended to include next - nearest - neighbor bonds between the ising spins . to this end , the transfer matrix has to be enlarged so that it does not jump from one ising spin to the next , but from one _ pair _ of ising spins to the next pair.@xcite we are interested in this paper in molecular chains or rings that can be described by a decorated ising model . this means that their low - energy spectrum can be modeled to satisfactory accuracy by an effective hamiltonian that has the properties described in section [ 4:sec : decising ] . it has been noted there that the composition of the decorating unit is basically arbitrary and we need therefore not consider the properties of that part . instead , our attention goes here to the molecular realization of the ising spin [ @xmath13 in eq . ] . we focus on because this ion is used in the two examples studied below , but the discussion applies equally well to several other trivalent lanthanides and also to some transition metal ions . it is known that lanthanide ions in a coordination environment are often well described by crystal field theory applied to the ground @xmath43 level . one assumes that @xmath44 , @xmath45 , and @xmath46 remain good quantum numbers . is a kramers ion that belongs to the second half of the lanthanide series , whose ground level is ( @xmath47 ) , with associated land factor @xmath48 . this multiplet splits into eight kramers doublets by the crystal field perturbation ( except for high - symmetric environments belonging to the @xmath49 , @xmath50 , or @xmath51 point groups , which split the multiplet in less than eight levels ) . it will be useful to view each kramers doublet as an effective spin-1/2 with its own @xmath1-factors ( 3 in number ) and corresponding magnetic axes . for example , take the kramers doublet @xmath52 ( of the level ) , quantized with respect to the @xmath53 axis . its @xmath1 factors are @xmath54 and @xmath10 . if the action of the crystal field on is such that the lowest kramers doublet is separated from the next one by an energy that is large compared to the energy of interaction with the magnetic field and the exchange interaction with neighboring ions , we can omit all excited kramers doublets from the hamiltonian and keep only the lowest doublet . in this way is described by a spin of 1/2 and every interaction in which it takes part enters the hamiltonian as a linear combination of the three spin operators @xmath55 , @xmath56 , and @xmath57 . if we now want to be an ising spin , defined by the first commutation relation in eq . , it is clear that only one of these spin components , say @xmath57 or simply @xmath58 , may actually appear in the hamiltonian . in other words , there must be no interaction that creates an off - diagonal matrix element between the two components of the kramers doublet . this can be shown to be true with high accuracy if the lowest kramers doublet is @xmath52 . the two interactions of importance here are the zeeman interaction with the magnetic field and the exchange interaction with other magnetic ions . the zeeman hamiltonian follows directly from the @xmath1 factors of the kramers doublet ( vide supra ) , and is @xmath59 where @xmath60 is the @xmath53 component of the applied magnetic field . note that the field may be applied in any direction , but it is only the @xmath53 component that interacts with the kramers doublet because @xmath10 . the vanishing of @xmath61 and @xmath62 in @xmath52 follows from the selection rule stating that a vector operator ( the magnetic moment in this case ) can not connect states for which @xmath63 differs by more than one unit . we will sometimes refer to the @xmath53 axis as the anisotropy axis , to stress that it is the only magnetic axis with nonvanishing @xmath1 factor . to evaluate the effect of exchange interaction , we must first take a closer look at the composition of the kramers doublet . in terms of the russel - saunders states @xmath64 we have @xmath65 in a basic ( super)exchange process between two magnetic centers , one electron of each center takes part . if we look at one center , the process removes an electron with certain spin projection ( up or down ) and puts it back on the center either with the same or with opposite spin projection.@xcite this gives rise to the selection rule @xmath66 . if the exchange interaction is to connect both components of the kramers doublet in eq . , at least five successive processes are needed , for @xmath67 . in other words , not one but five electron spins have to be flipped to connect @xmath68 with @xmath69 . if the basic exchange process ( i.e. the one for which @xmath70 ) occurs in , say , @xmath71th order of perturbation theory then an off - diagonal matrix element between the two components can only appear in @xmath72th order of perturbation theory . it is therefore reasonable to assume that the off - diagonal matrix element is negligibly small compared to the diagonal matrix elements ( @xmath73 ) so that the effect of exchange interaction on the kramers doublet is accurately described by the @xmath57 spin operator only . note that we derived the selection rule for exchange interaction on the basis of the spin quantum number @xmath74 only , without paying attention to the angular momentum quantum number @xmath75 , although @xmath75 changes even more than @xmath74 between the states of the kramers doublet . the existence and precise form of a selection rule for @xmath75 depends on the spatial symmetry of the exchange problem under consideration . @xmath76 is therefore not as useful as @xmath77 for predicting the vanishing of certain matrix elements of exchange interaction . note however that , even in the lowest symmetry , there is a maximum to the amount that @xmath75 can change in the basic exchange process described in the previous paragraph : within the @xmath78 orbitals , a one - electron process can bring about at most a change of @xmath79 . so we would have , in general , that a basic exchange process can induce the following changes in a lanthanide state : @xmath80 thus at least two steps of this kind are needed to bridge @xmath81 , but at least five are needed to bridge @xmath67 . so in this case , the selection rule of @xmath74 gives the stronger result and leads to the conclusions reached in the last paragraph on the matrix elements of exchange interaction in the kramers doublet . we can now derive the precise form of that part of the effective hamiltonian that refers to the exchange interaction between a ion [ with ground state ] and another magnetic center . we consider two cases , which we shall encounter in the examples in sections [ 4:sec : chain ] and [ 4:sec : ring ] : in the first case the other center is another ion ; in the second case the other center is an ion with an isotropic spin moment @xmath82 . consider first the exchange interaction with another ion . we assume that the second has the same property of uniaxiality as the first one and that it also shares all other relevant properties discussed in the previous paragraphs . then both ions are represented by a spin-1/2 doublet with a local anisotropy axis @xmath83 ( @xmath84 ) and we already know that the effect of exchange interaction in each doublet is proportional to @xmath85 . therefore the exchange hamiltonian is necessarily of the form @xmath86 where it is understood that the first spin belongs to ion 1 and the second to ion 2 . note that @xmath87 and @xmath88 need not be parallel with each other . as a second case , consider the interaction between ( anisotropy axis @xmath87 ) and an isotropic spin @xmath89 . the latter may typically be a transition metal ion with quenched orbital momentum . we found that the exchange processes that contribute do not change the spin projection on the ion ( @xmath90 , quantization axis @xmath87 ) , and that this result is independent of the exchange partner . it is also known that every exchange process commutes with the total spin ( the matrix elements involved are spin - independent matrix elements of kinetic , potential , and coulomb energy @xcite ) , so that @xmath91 . it follows then that @xmath92 , or , the exchange hamiltonian commutes with the @xmath93 component of @xmath89 . the simplest expression compatible with this requirement is @xmath94 where @xmath58 naturally represents and @xmath46 represents @xmath89 . the interaction is of ising form with the anisotropy axis of as ising axis . note that , when @xmath95 , higher powers of @xmath96 may enter the hamiltonian . considering exchange interaction as a perturbation however , one can usually assume that the lowest - order contribution , eq . , is the leading term . there are other cases conceivable , for example dipole - dipole interaction between the moment of and a neighboring moment . one will always find , as above , that the hamiltonian is a product of @xmath57 ( belonging to ) and a part that belongs to the other ion and whose form depends on the kind of the other ion and on the details of the interaction . we have now obtained that a ion , if its ground state is @xmath52 , fairly well separated from excited states , interacts with the magnetic field and with neighboring ions as an ising spin-1/2 , in the sense that the interaction is always proportional to @xmath57 , as expressed in the eqs . , , and . this means that a chain - like molecular structure having ions of this kind at regular positions in the chain would meet the requirements of a decorated ising chain , given in part by eqs . and . it remains of course to be shown that @xmath52 can indeed be the ground state of a coordinated ion in a polynuclear complex . at first sight , this seems rather unlikely . @xmath52 is an eigenstate of cylindrical symmetry . within lanthanide @xmath97 states , the crystal field is effectively cylindrical if there is , at least , an eightfold rotation axis ( @xmath98 ) or rotation - inversion axis ( @xmath99 ) . @xcite @xmath99 symmetry has been obtained , for example , in mononuclear bis(phtalocyaninato ) sandwich complexes of the lanthanides . @xcite even when such high symmetry is attained , the ground state is not necessarily the cylindrical doublet with highest @xmath100 value . @xcite apart from that , the symmetry of the coordination sphere of a lanthanide ion in a polynuclear , possibly heterometallic , complex or chain is usually much lower or even completely absent . such is the case for the two examples considered in this paper . on the basis of symmetry alone , there is thus no reason to expect @xmath52 to be an eigenstate , let alone the ground state . nevertheless , recent _ ab initio _ calculations have revealed the unexpected result that the ground state of several low - symmetry complexes of is very close to @xmath52 . @xcite they used the multiconfigurational , wavefunction - based casscf / rassi - so method , to obtain accurate wavefunctions for several of the lowest kramers doublets . calculation of the principal @xmath1 factors of these states gives an indication of their composition . it was found in several cases that the ground doublet has @xmath101 close to , but lower than 20 , and @xmath61 and @xmath62 close to 0 . this corresponds to a doublet mainly composed of @xmath52 . with evidence of _ ab initio _ calculations it is thus possible to identify in certain coordination environments as an ising spin-1/2 ( to a good approximation ) . we will use this information to identify the compounds in sections [ 4:sec : chain ] and [ 4:sec : ring ] as decorated ising chains ( or rings ) . to conclude this section we remark that one can not deduce from the vanishing of two @xmath1 factors alone that a kramers doublet will behave as an ising spin . it will , of course , in its interaction with the magnetic field [ eq . ] , but this will not , in general , be true for the exchange interaction . take again the example of , supposing the ground state is the kramers doublet @xmath102 . it is perfectly uniaxial because @xmath10 and @xmath103 . a closer look at the expansion of @xmath102 in terms of the russel - saunders states @xmath104 shows , however , that the selection rules in eq . permit a matrix element to exist between @xmath105 and @xmath106 , therefore introducing non - ising terms ( i.e. , @xmath55 and @xmath56 ) in the same order of perturbation theory as the ising term in the exchange hamiltonian . there exists a similarity between the spectra of the decorated and undecorated ising chains at which we want to take a closer look here . we suppose infinite , periodic chains , or periodic , even - membered rings ( in odd - membered rings spin - frustration complicates the picture ) . we also limit ourselves to chains with ising spins of 1/2 . the undecorated , or simple , ising chain in a magnetic field @xmath107 parallel with the @xmath53 axis is given by eq . and @xmath108 where @xmath13 is the @xmath53 component of @xmath109 ( @xmath110 ) . we assume , without loss of generality , @xmath111 . the eigenstates of the chain are spin _ configurations _ like ( @xmath112 ) , etc . of the @xmath113 eigenstates only two distinct ones can be the ground state : the ferromagnetic ( f ) and the antiferromagnetic ( af ) : @xmath114 when @xmath115 time - reversal symmetry makes every state degenerate with the state formed by flipping all the spins . this degeneracy is meant to be implied in , where only one of two states is shown in each case . only the two af states remain degenerate when @xmath116 . when @xmath117 a ground state level crossing occurs from af to f when @xmath107 is increased . at the point of crossing ( @xmath118 ) the two af states are degenerate together with all states derived from an af state by flipping one or more down - spins up . however , no other state than af or f can be the ground state at any other value of @xmath107 . thus the ground state of the ising chain is either f or af , and they are degenerate , together with an infinite number of other states , at the crossing point . we now decorate the ising bonds with identical but arbitrary units to obtain a periodic decorated ising chain . the spectrum is given by eq . . since the chain is periodic , the spectrum of the individual units @xmath16 is independent of @xmath119 and the energies may be written as @xmath120 , with @xmath71 ranging over the eigenstates of @xmath16 . there are four sets of @xmath120 : @xmath121 , @xmath122 , @xmath123 , and @xmath124 , in an obvious notation . notice that the eigenstates of this chain can still be classified according to the configuration of the _ ising _ spins : ( @xmath112 ) etc . , which follows from the fact that all the @xmath13 and @xmath22 form a commuting set of observables . an interesting question is whether the same rules hold for the ground state of the decorated ising chain as did for the simple ising chain . the answer is yes ; the ground state is either f or af ( referring to the ising spin configuration ) and a crossing between them is possible , with the same number and kind of degenerate states as in the simple ising chain . to show this , we have to consider only the lowest eigenstate belonging to each of the @xmath113 possible ising spin configurations . in these states every unit is in its lowest possible state for the given orientation of the neighboring ising spins : @xmath125 ( we assume this energy to be nondegenerate ) , so that the total energy of the chain state is @xmath126 where @xmath127 denotes the number of pairs of neighboring ising spins that are both spin up , etc . for example , in the f configuration in eq . , @xmath128 ( periodic boundary conditions are assumed ) , while in the af configuration , @xmath129 . the eigenstates we have just described , with energy , are in an obvious one - to - one correspondence with the eigenstates of the simple ising chain . the ground state is found by minimizing with respect to the @xmath130 , under the restrictions @xmath131 the first relation states that the total number of ising spins ( or , equivalently , unit cells ) is @xmath14 . the second relation follows from the fact that , in a cyclic spin configuration , every @xmath132 pair must eventually be followed by a @xmath133 pair , possibly after a number of @xmath134 pairs . another restriction is that whenever both @xmath127 and @xmath135 are not zero , @xmath136 [ and by eq . also @xmath137 must be at least one . using we can rewrite eq . as @xmath138\bigr)\\ & + n_{\downarrow\downarrow}\bigl(\varepsilon_1(\downarrow\downarrow)- \frac{1}{2}[\varepsilon_1(\uparrow\downarrow ) + \varepsilon_1(\downarrow\uparrow)]\bigr)\\ & + \frac{n}{2 } [ \varepsilon_1(\uparrow\downarrow)+\varepsilon_1(\downarrow\uparrow ) ] , \end{split}\ ] ] where we see that only the average @xmath139/2 $ ] of the `` antiparallel '' energies enters the equation . the last term is a constant and can be discarded for the purpose of relative energy considerations . we can now derive the values of @xmath127 and @xmath135 for the ground state of the chain , keeping in mind that @xmath125 is a function of the magnetic field @xmath140 . suppose then , first , that @xmath141 . time reversal symmetry asserts that @xmath142 and @xmath143 . it is simple to see that , depending on the relative ordering of @xmath144 and @xmath145 , @xmath146 is minimal in the f configuration ( @xmath128 or @xmath147 ; @xmath148 ) when @xmath149 or in the af configuration ( @xmath150 ; @xmath151 ) when @xmath152 ( we exclude the possibility of equality of both energies from the discussion ; @xmath153 would correspond , in the simple ising chain , with @xmath154 ) . when @xmath155 , time reversal symmetry is not operative , and we have , in general , four different energies @xmath125 . the equation shows that the configuration that minimizes @xmath146 is determined by the sign of the two terms in round brackets ; if both are positive , then @xmath150 ( af configuration ) ; if at least one of them is negative , then either @xmath128 or @xmath147 ( f configuration ) , depending on whether respectively @xmath144 or @xmath156 is lower . finally , the magnetic field can induce a transition from the af to an f ground state configuration , say with all ising spins up . this happens when @xmath157,\ ] ] and @xmath158 . at this point , the ground state configurations are all those for which @xmath159 , exactly the same as in the simple ising chain . we find thus a complete analogy between the simple and the decorated ising chain as far as the ground state ising spin configuration is concerned . the only possible configurations are the fully antiferromagnetically aligned and the fully ferromagnetically aligned configurations . no `` intermediate '' configuration can be the ground state . the only exception is the crossing point between af and f , where there is a high degeneracy of configurations . these conclusions are independent of the nature of the decorating unit . although the decorated ising model predicts that the af and f ground states are both doubly degenerate ( in zero field ) , this degeneracy is not a result of the spatial symmetry : in the cyclic group @xmath160 , the two af components combine into irreducible representations ( irreps ) @xmath161 and @xmath107 , while the two f components transform as two @xmath161 irreps . introduction of neglected terms in the hamiltonian , that destroy the ising property , could split these ground state components . the decorated ising chain affords two new kinds of ground state level crossings , not present in the simple ising chain . the first of these is the transition between one f configuration of the ising spins and the other : @xmath162 . this transition takes place when @xmath142 . this level crossing , induced by the magnetic field , can for example be encountered in frustrated ising - heisenberg chains.@xcite a second kind of new ground state transition arises from the crossing of levels _ within _ a decorating unit . a ground state crossing can result in which the ising spin configuration remains the same but the state corresponding to @xmath125 crosses with the state corresponding to @xmath163 . more precisely this happens in the f configuration when @xmath164 and in the af configuration when either @xmath165 or @xmath166 , or both . the previous paragraphs have shown that we do not need to consider configurations other than f and af for the ground state . level crossings are usually connected with the presence of good quantum numbers . for the ising - type crossings , the relevant conserved quantities are the @xmath14 ising spins @xmath167 . the crossing of energy levels within the decorating unit should be associated with a conserved variable that is _ internal _ to that unit , much the same as in isolated molecules . in section [ 4:sec : chain ] we will encounter an example where both transitions ising type and internal type occur in a magnetic field . in the following sections we will be comparing our theoretical results with measurements performed on powder samples of the crystalline compounds . in this section we consider the powder averaging of magnetization for the example of the simple ising chain . let @xmath168 and @xmath169 be the polar angles of the magnetic field vector with respect to a molecular reference frame , then the free energy is a function of @xmath168 , @xmath169 , and the strength of the field , @xmath107 : @xmath170 . the projection of the magnetization on the field direction @xmath171 is @xmath172 averaging over one hemisphere gives the powder magnetization @xmath173 let us see how the powder averaging affects the magnetization curve for a simple ising chain . take a spin-1/2 infinite antiferromagnetic ising chain with anisotropy axes parallel with each other and with the @xmath53 axis , and uniaxial @xmath1-factors ( @xmath10 and @xmath174 ) . this could for example be realized by a chain of identical units ( see section [ 4:sec : dy ] ) . the hamiltonian is given by eq . , substituting @xmath175 where @xmath176 is the @xmath53 component of the magnetic field , and @xmath117 . defining @xmath177 and @xmath178 , the magnetization , which has only a nonzero @xmath53 component , is@xcite @xmath179}{\sqrt{\sinh^2[b\cos\theta/2t]+e^{-j / t}}}.\ ] ] the projection on the field direction [ eq . ] is @xmath180 . plugging this in eq . and substituting @xmath181 yields the powder magnetization of the ising chain @xmath182}{\sqrt{\sinh^2[b u/2t]+e^{-j / t}}}u\,du.\ ] ] because the hamiltonian in eq . does not depend on @xmath169 , this variable has been integrated out in eq . . ] ] figs . [ 4:fig : magnising_zas ] and [ 4:fig : magnising_powder ] show plots of magnetization versus magnetic field , for coupling constant @xmath183 . the step - like appearance of @xmath184 is associated with the ground state crossing that occurs at @xmath185 . at that point , the antiferromagnetic ground state ( or rather ground state ising doublet ) is replaced by the ferromagnetic state ( all spins up ) . consequently , the magnetization jumps from zero to the saturation value of 0.5 , as seen in fig . [ 4:fig : magnising_zas ] . the magnetization of a powder sample of the same ising chain is shown in fig . [ 4:fig : magnising_powder ] . before the crossing point , @xmath186 behaves qualitatively the same as @xmath184 . after the crossing point however , @xmath186 is seen to reach only slowly its saturation value , which is half of the saturation value of @xmath184 , viz . the limiting curve of @xmath186 as @xmath187 can be calculated exactly from eq . : @xmath188 ( this is of course only valid for the antiferromagnetic case @xmath189 . ) in fig . [ 4:fig : magnising_powder ] , this limiting curve is very closely approximated by the curve at @xmath190 . clearly , the sharp step of @xmath184 transforms in the powder to the concave form displayed by @xmath186 . this is understood from the fact that , in a powder , for a given field @xmath191 , there is always a fraction of molecules that is not magnetized ( in the sense that they are in the antiferromagnetic ground state ) because they are oriented so with respect to the field , that @xmath192 ( see fig . [ 4:fig : magnising_zas ] ) . the powder saturates only when every molecule is fully magnetized , and this happens only for @xmath193 . therefore , @xmath186 ( at @xmath194 ) does not abruptly saturate at the crossover point , but increases slowly to saturation . in deriving the exchange hamiltonian in section [ 4:sec : dy ] we assumed that only the lowest kramers doublet on took part . this is certainly a good approximation when the gap between the lowest and the second - lowest kramers doublet is much larger then the strength of the exchange interaction . however , the excited kramers doublets often have to be taken into account to a certain degree of approximation if a comparison with experimental data on susceptibility and magnetization is desired . the crystal field splitting of the level is of the order of @xmath195 at room temperature . this gives rise to two effects : ( i ) a thermal population of excited kramers doublets , and ( ii ) a modification of the lowest kramers doublet as a function of the applied magnetic field by interaction with the excited doublets . effect ( i ) is mainly visible in the temperature dependence of @xmath196 , where @xmath197 is the powder magnetic susceptibility : for a single center , @xmath196 increases monotonically with increasing temperature , from the value of the ground doublet at 0 k to the saturation value of at higher temperatures . effect ( ii ) gives rise to temperature - independent paramagnetism ( tip ) . it is visible at temperatures sufficiently low so that only the ground doublet is occupied . it contributes a linear increase of @xmath196 with @xmath49 and a linear increase of the magnetization @xmath186 with the applied field @xmath107 . in the simplest approximation , the contribution of the excited kramers doublets to the magnetic properties of the chain is equal to the contribution they have to the properties of the single , isolated ion in the same ligand environment it has in the chain . let @xmath198 and @xmath199 denote susceptibility and magnetization derived from the decorated ising chain model , and let @xmath200 denote the susceptibility of the center and @xmath201 the magnetic moment induced by @xmath107 in the ground doublet of the center , then the corrected properties are ( supposing one ion per unit cell ) [ 4:eq : corrections ] @xmath202 the last equation assumes that only the ground doublet of is occupied . this is correct at the temperature at which magnetization curves are usually recorded ( e.g. , 2@xmath203 ) . if necessary , corrections due to other magnetic ions can be added in the same way . @xmath200 and @xmath204 can be obtained from multiconfigurational _ ab initio _ calculations @xcite , or , in oligonuclear complexes , experimentally by replacing certain magnetic ions with diamagnetic ions.@xcite the equations are evidently correct in the limit of vanishing exchange interactions . the assumption we make is that the exchange interactions are sufficiently small for these corrections to remain valid . a more accurate approach should take into consideration the fact that the excited kramers doublets also participate in the exchange interaction with neighbors . this is however out of the scope of the decorated ising model . we now turn our attention to two actual examples of decorated ising models based on : a chain , treated in this section , and a four - ring treated in the following section . the problem is approached as follows : the hamiltonian for the chain in a magnetic field is formulated , with the help of the considerations in section [ 4:sec : dy ] . values of the @xmath1 factors of the magnetic ions are taken directly from _ ab initio _ calculations , reported elsewhere.@xcite this leaves the exchange coupling constants as parameters of the model , to be fitted by comparison with experimental magnetization and susceptibility data . ( in section [ 4:sec : ring ] , the direction of the anisotropy axis contributes one extra parameter . ) @xmath0 chain , showing type of exchange interactions and labeling of exchange constants . see ref . for the complete molecular structure.[4:fig : chainscheme ] ] the [ dycumocu ] chain was recently synthesized and details of its chemical composition and structure are given in ref . . the crystal structure was found to consist of parallel linear chains each made of [ dycumocu ] unit cells . [ 4:fig : chainscheme ] shows how the metal ions are connected by ligand bridges . multiconfigurational casscf / rassi - so calculations have been performed on each of the four metal ions in their ligand environment , suitably disconnected from the rest of the chain ( details of the calculations can be found in ref . and the accompanying supplementary information ) . most important for us is that the center was found to have a ground kramers doublet , separated by 141 @xmath12 from the second doublet , and characterized by complete uniaxial anisotropy : @xmath205 ( actually @xmath61 and @xmath62 were calculated about 0.03 , which is small enough to be ignored . ) the value of @xmath206 shows that this doublet is only slightly perturbed from the @xmath52 doublet of the level , the latter having @xmath207 . together with the fact that the energy gap to the second kramers doublet is about ten times larger than the exchange interaction ( as we will find later ) , these results indicate that the ion will behave as an ising spin , as described in section [ 4:sec : dy ] . as a side - note we may add that the total splitting of the level was calculated to be 560 @xmath12 , which is indeed of the order of room - temperature @xmath195 ( see section [ 4:sec : corrections ] ) . both ( @xmath208 ) and ( @xmath209 ) have a spin = 1/2 , orbitally nondegenerate ground state , well separated ( @xmath210 ) from higher states . the two ions in the unit cell reside in almost identical environments@xcite and have therefore virtually the same properties . the calculated @xmath1 factors are tetragonal : @xmath211 for and @xmath212 for . to avoid unnecessary complications we will regard these ions as isotropic spins with root - mean - square @xmath1 factors @xmath213 this approximation will not have important consequences for the magnetic properties , which are largely dominated by the high moment anyway . we introduce exchange interaction between metal ions directly connected by ligand bridges . interacts with its three neighbors via the ising hamiltonian eq . . interaction between the isotropic spins is given by the heisenberg hamiltonian @xmath214 . [ 4:fig : chainscheme ] shows the exchange configuration , with single bonds representing ising interaction and double bonds representing heisenberg interaction . note that we have approximated the dy - cu@xmath215 and dy - cu@xmath216 coupling strengths to be equal ( @xmath217 ) , following the approximate local symmetry of the dy - cu pairs.@xcite it is now possible to see that the [ dycumocu]@xmath0 polymeric chain is indeed an experimental realization of an ising - heisenberg chain where the [ cumocu ] trimeric heisenberg units decorate the dy - dy bonds and the ising spins separate the [ cumocu ] heisenberg trimers from each other . the total hamiltonian in a magnetic field @xmath140 is then given by eq . and @xmath218 where @xmath219 is shorthand for @xmath220 and @xmath221 is the @xmath101 factor of [ eq . ] . the @xmath53 axis is the anisotropy axis of the center . we have not specified its direction with respect to the chain axis but this is not important here because there are no other axes in the problem ( the anisotropy axes are parallel by translational symmetry and we have assumed and isotropic ) . all the spins in eq . are spins of 1/2 . the hamiltonian exhibits some symmetry . it is rotationally invariant around @xmath53 , if @xmath140 is rotated simultaneously . we may therefore restrict @xmath140 to lie in a plane through @xmath53 , say the @xmath222 plane . this simplifies calculation of the powder magnetization eq . : one has to integrate only over @xmath168 . when @xmath140 is directed along the @xmath53 axis , the @xmath53 component of the total spin _ in _ each decorating unit is conserved : @xmath223=0.\ ] ] we also note that in this case the zeeman hamiltonian commutes _ almost _ with @xmath22 . it would commute exactly when @xmath224 , for then the last term in eq . reduces to @xmath225 . we let the length of the chain go to infinity : @xmath226 . to solve for the thermodynamic properties we are only required to find the eigenvalues @xmath227 of eq . ( see section [ 4:sec : decising ] ) , with @xmath228 , corresponding to the @xmath229 possible states of the [ cumocu ] spin unit . this is done by 4 numerical @xmath230 matrix diagonalizations , one for each @xmath231 pair . @xmath0 . ] we can now compare the theory with experiment . powder magnetization ( at 2@xmath203 ) and susceptibility data have been recorded.@xcite we recall that we have to correct the theoretical curves before comparing with experiment according to eq . . the corrections are provided by the _ ab initio _ calculations.@xcite @xmath204 turns out to be 0.03@xmath232 ; the accompanying correction in eq . is never more than 2.5% of @xmath186 . we ignore this correction . we do however correct @xmath197 as in eq . . the theoretical curve contains the correction for the contribution of excited kramers doublets . ] closest agreement with experiment was found for the following values of the exchange constants ( plots in figs . [ 4:fig : chainmp ] and [ 4:fig : chainchip ] ) : @xmath233 these were obtained by a least - squares fit of @xmath197 followed by a small manual adjustment to improve the fit of @xmath186 while not distorting that of @xmath197 appreciably . ( the least - squares fit of @xmath197 gave @xmath234 , @xmath235 , @xmath236 , @xmath237 . ) there are , as far as we know , no data in the literature with which to compare the values in eq . . however , an experimental study is available of a ( , ) dinuclear complex in which the bridging ligand is the same as in this chain.@xcite the authors found a ferromagnetic interaction . a superficial analysis of the susceptibility curve in that paper , using the ising hamiltonian we use in this paper , yields @xmath238 . the other values in are difficult to assess . for a discussion of these values and their relation with the molecular structure as well as some evidence from dft calculations , we refer to ref . . certainly , no confidence should be attached to the numbers in decimal places in . the correction refers to the contribution of excited kramers doublets , eq . . ] the effect of the excited kramers doublets of is most clearly seen in the @xmath196 curve ( fig . [ 4:fig : chainchitp ] ) . the curve shows a steady increase above 50@xmath203 which is not predicted by our decorated ising model , but is indeed due to the thermal population of the kramers doublets that originate from the level . we can obtain the expected high - temperature limit of @xmath196 by considering the metal ions as independent spins . the susceptibility components @xmath239 of an angular momentum multiplet @xmath44 with principal @xmath1-factors @xmath240 ( @xmath241 ) are given by @xcite @xmath242 summing over ( , @xmath243 ) , , , and ( all are isotropic ) gives @xmath244 a similar calculation , only including the lowest kramers doublet of , with @xmath1-factors as in eq . , gives 13.2@xmath245 . the correction supplied by the _ ab initio _ calculations to account for this difference , is seen to cover nicely the high - temperature part of the experimental curve . one notices that @xmath196 shows a slight depression around 40@xmath203 which is not entirely reproduced by the theory . this might indicate a failure of the simple approximation we used to include the excited kramers doublets . eq . is certainly correct at very high temperatures , when the exchange interactions are irrelevant , and at very low temperatures , when the excited kramers doublets are not occupied . if these two regions do not overlap , however , there is a temperature window between , in which excited doublets start to get occupied while exchange interaction is not quite negligible yet . in that case , the exchange interaction of the occupied excited doublet(s ) with other ions should be taken into account . such an interaction of antiferromagnetic type could possibly depress @xmath196 as observed . we shall now describe some features of the spectrum of the chain , paying attention to the properties described in section [ 4:sec : eigenstates ] . consider the chain without magnetic field . the exchange parameters in eq . predict a ground state that has an af ising spin configuration . this is in accordance with the susceptibility measurements , which show that @xmath246 as @xmath247 , requiring a nonmagnetic ground state ( fig . [ 4:fig : chainchitp ] ) . the ground state is indeed nonmagnetic because @xmath248 is the time - reversed state of @xmath249 . let @xmath74 denote an eigenvalue of @xmath250 [ eq . ] : @xmath251 . @xmath252 is conserved so @xmath74 may be used to label the eigenstates @xmath253 ( we may leave out the index @xmath119 because all units of the chain are identical ) . for the ground state , we find @xmath254 in @xmath248 and @xmath255 in @xmath249 . the powder magnetization ( see also fig . [ 4:fig : chainmp ] ) is compared with the projections of @xmath256 on the field direction [ eq . ] , for three different directions of the field ; @xmath168 is the angle between @xmath140 and the @xmath53 axis . ] same as fig . [ 4:fig : chainmcomponents ] but at lower temperature and to higher field . ] [ eq . ] in a magnetic field _ parallel _ with the @xmath53 axis ( @xmath257 ) . circles indicate ground state level crossings . the ground state of the chain is af in zero field ( left ) , switches to f at 0.64@xmath258 , and undergoes an internal level crossing at 6.3@xmath258 , marked by a change of the internal quantum number @xmath74 from 1/2 to 3/2 . both crossings can be seen in the @xmath257 magnetization curve in fig . [ 4:fig : chainmcomponentslowt ] . note that the energy curves appear as straight lines , although , with the exception of @xmath259 , they are not exactly straight , because the zeeman hamiltonian does not _ completely _ commute with the total hamiltonian . all @xmath121 decrease with increasing field strength because the large magnetic moment of dominates the smaller magnetic moments of the decorating unit . ] since the ground state is af , we might expect that in a magnetic field a crossover will occur to an f ground state . this is indeed what happens . the convex increase of @xmath186 in fig . [ 4:fig : chainmp ] points to a flip of the spins to a parallel configuration . this is inferred from the value of the magnetization , which approaches 6@xmath260 at 5@xmath258 . the [ cumocu ] unit alone can only contribute a maximum of @xmath261 . the strong increase must come from the contribution of the large moments . the behavior of magnetization along certain directions of applied field is shown in fig . [ 4:fig : chainmcomponents ] . the af @xmath262 f transition is most clearly seen when the field is applied along @xmath53 ( @xmath257 ) ; the transition occurs below 1@xmath258 . after 1@xmath258 , @xmath184 reaches an approximately constant plateau at @xmath263 . the saturation value of magnetization in direction @xmath264 is @xmath265 . this gives @xmath266 for @xmath257 , which shows that @xmath184 has not quite reached its maximum at 5@xmath258 yet . the positions of level crossings become more sharply defined on lowering the temperature ( fig . [ 4:fig : chainmcomponentslowt ] ) . here we also see that @xmath184 undergoes a second transition at 6.3@xmath258 , after which it reaches saturation . this transition is connected with a level crossing _ in _ the [ cumocu ] unit ( see section [ 4:sec : eigenstates ] ) from @xmath255 to @xmath267 , as opposed to the first transition , at 0.64@xmath258 , which is of the ising type , described by eq . . the latter is the analogue of the transition in the af simple ising chain ( fig . [ 4:fig : magnising_zas ] ) , while the `` internal '' transition has no such analogue but is unique to the decorated ising chain . the relevant energy level diagram is shown in fig . [ 4:fig : chainlevelcrossings ] . note that , for fields not parallel to @xmath53 ( for example , @xmath268 in fig . [ 4:fig : chainmcomponentslowt ] ) , @xmath74 is not a quantum number and the internal level crossing turns into an avoided crossing . this does not apply for the ising level crossing because the ising spins are always conserved . only when the field is applied perpendicular to @xmath53 ( @xmath269 in fig . [ 4:fig : chainmcomponentslowt ] ) does the af @xmath262 f transition not occur because the spins do not interact with perpendicular fields . @xmath0 , without correction for contribution of excited kramers doublets . the powder @xmath196 ( see also fig . [ 4:fig : chainchitp ] ) is compared with the cartesian components of @xmath270 . @xmath53 is the direction of the anisotropy axis of , @xmath271 is any direction perpendicular to @xmath53 . @xmath272 . ] the low - temperature limit of the powder magnetization in fig . [ 4:fig : chainmcomponentslowt ] may be compared with that of the simple ising chain in fig . [ 4:fig : magnising_powder ] . the resemblance is clear ; the decorated chain is different in the small linear increase of @xmath186 before the transition , and the more linear approach to saturation , which lies at ( @xmath273 . both are due to tip interaction in the [ cumocu ] unit , the effect of which is most clearly seen in the @xmath274 curve in fig . [ 4:fig : chainmcomponentslowt ] . to conclude this section we remark that the mentioned similarity with the magnetization of the simple ising chain is a consequence of the very high magnetic moment of the spins in comparison with the [ cumocu ] unit . the dominance of is most dramatically shown in the components of @xmath196 ( fig . [ 4:fig : chainchitcomponents ] ) . an application of eq . shows that the high - temperature limit of @xmath275 is @xmath276 , while that of @xmath277 is @xmath278 . cr@xmath8 molecule indicating numbering of atoms and exchange coupling constants . the boxes show the orientation of the local anisotropy axes on dy sites , when viewed from the poles of the @xmath279 and @xmath280 axes . the @xmath281 axis points out of the center of the scheme.,width=325 ] as a second example we describe in this section the application of the decorated ising model to a ring - shaped molecule.@xcite dy@xmath8cr@xmath8 consists of alternating and ions forming a closed ring . the four ions lie in a plane . the ions are positioned alternatingly above and below this plane , `` decorating '' the dy - dy bonds . the molecule has @xmath282 symmetry , the ions lying on @xmath283 axes and the ions lying on the mirror planes . we choose a molecular reference frame @xmath284 so that @xmath281 coincides with the @xmath285 axis and @xmath279 and @xmath280 coincide with the two @xmath283 axes of @xmath282 ( fig . [ 4:fig : ringscheme ] ) . _ ab initio _ calculations have been performed in the same way as for the [ dycumocu ] chain.@xcite from these , we take again the @xmath1-factors of the ground doublet of and of the isotropic ground state spin multiplet of ( @xmath286 , @xmath287 ) : @xmath288 the kramers doublet is again very close to the @xmath289 state , permitting the use of the ising model . however , the same calculation predicted the second kramers doublet at 30@xmath290 , not very high compared with exchange interaction , which we found in the previous section @xmath291 . this should be seen as a warning that our treatment of the excited kramers doublets as `` innocent '' may not be entirely correct here , and thus may lead to discrepancies with experiment . in this respect we must also note that , for such small excitation energies , the results of the _ ab initio _ calculations are not always conclusive on the nature of the ground state kramers doublet . in the present case , for instance , another set of calculations produced a ground state kramers doublet on that is not uniaxial as in eq . , having relatively large transversal @xmath1-factors.@xcite the decorated ising model would be unusable in this case . we find however that , assuming the axial g - factors in eq . , is an interesting example of a decorated ising ring , for which qualitative agreement with experimental magnetic properties can be obtained . direct ligand bridges connect each with two neighboring ions and two neighboring ions . exchange interaction between these pairs is introduced [ eqs and ] . the hamiltonian is then given by eq . : @xmath292 , and @xmath293 where @xmath294 denotes the ising spin-1/2 variable on dy@xmath295 and @xmath296 denotes the projection of the spin of cr@xmath295 on the magnetic anisotropy axis of dy@xmath295 ( for numbering , see fig [ 4:fig : ringscheme ] ) . similarly , @xmath297 is the projection of the magnetic field on the anisotropy axis of dy@xmath295 . @xmath221 is the @xmath101 factor of [ eq . ] . an interesting difference with the [ dycumocu ] chain is that here , in , the four anisotropy axes @xmath298 are not , in general , parallel , a result of point symmetry instead of translational symmetry . the orientation of the local anisotropy axis on , being one of the @xmath1-tensor main axes , is restricted by the local @xmath283 symmetry to be either parallel with , or orthogonal to the local @xmath283 axis . the first possibility can be excluded on the basis of the experiment ; with the @xmath298 pointing radially outwards at each dy@xmath295 , the ground state of the whole molecule is necessarily nonmagnetic , because the local moments add up to zero , independent of whether the ground state is f or af with respect to the ising spins . the experimental susceptibility measurement however indicates a magnetic ground state ( nonzero intercept on the vertical axis in fig . [ 4:fig : ringchitp ] ) . we must therefore choose the second case and let the anisotropy axis on each dy be orthogonal to the local @xmath283 axis and make an angle of @xmath299 with the molecular @xmath281-axis ( see fig . [ 4:fig : ringscheme ] ) . by applying the symmetry elements of @xmath282 to one of these anisotropy axes , one obtains the other three . when @xmath300 the four axes are parallel and point in the same direction as @xmath281 . we note that the _ ab initio _ calculations yielded @xmath301 . we will need some flexibility in our model however , so we leave @xmath299 as a parameter that will be determined from comparison with experiment . in terms of the molecular coordinate system , the projections on the local anisotropy axes are a function of @xmath299 : @xmath302 the same relations hold for the magnetic field , after replacing @xmath303 by @xmath107 . the fact that only exchange interactions of ising type appear in eq . makes it possible to find analytical solutions of the eigenvalues and the partition function . from eqs . and we see that the part of @xmath16 that involves @xmath303 is a projection of @xmath304 on the vector @xmath305 where @xmath83 is the unit vector along the anisotropy axis of dy@xmath295 ( the superscripts @xmath298 on @xmath13 are left out from now on ) . the vector defines the quantization axis of @xmath304 , which depends on the states on the neighboring sites ( @xmath13 , @xmath26 ) . the stronger the coupling ( @xmath217 ) with dy , the stronger will be the deviation of the quantization axis from the direction of @xmath140 . the eigenvalues of @xmath16 are then @xmath306 where @xmath307 , and @xmath308 is the length of the vector in eq . . some remarks should be made on the solutions . eqs . and ( replace @xmath303 by @xmath107 ) show that the spectrum in eq . is not the same for every unit @xmath119 , as it was in the [ dycumocu ] chain , unless @xmath140 is applied along the @xmath281 axis . this means that also the transfer matrices @xmath30 will be different and that we have to use eq . instead of eq . for the partition function . a second remark concerns the quantum number @xmath74 . the lowest energy in eq . is always given by @xmath309 , but note that the axis to which this quantization refers is not invariant ; in particular , it changes with strength and direction of applied field , so that @xmath74 does not represent a real conserved quantity that could be responsible for level crossings of the `` internal '' type . such crossings do not occur in . we conclude the solution by finding the partition function @xmath310 . substituting eq . in eq . we find @xmath311}{\sinh [ \beta b_i/2]}\\ & \quad \times \exp[\beta(j_2 s_is_{i+1 } + \mu_\mathrm{b } g_\mathrm{dy } s_i b^{z_i})]\end{aligned}\ ] ] with @xmath30 as defined in eq . , we obtain the partition function @xmath312 let us now compare the theoretical results with experiment . a great amount of information on the values of the parameters @xmath299 , @xmath217 and @xmath313 can be obtained by inspection of the powder @xmath196 curve ( fig . [ 4:fig : ringchitp ] ) . the nonzero intercept @xmath314 indicates a magnetic ground state.@xcite now from the general theory we know that the ground state is either f ( @xmath315 ) or af ( @xmath316 ) with respect to the spins ; for a periodic ring or chain , is valid here because we are considering the eigenstates of in the _ absence _ of magnetic field , in which case the ring is effectively cyclic symmetric . ] af is nonmagnetic so we decide that the ground state must be f. incidentally , we can precisely delineate the regions in parameter space where the ground state is f or af : [ 4:eq : ring_groundstates ] @xmath317 a second piece of information comes from the increase of @xmath196 with increasing temperature . this is partly but not completely due to the occupation of excited kramers doublets , as one can show by subtracting the contribution of the latter , obtained from the _ ab initio _ calculations ( not shown here ) . there must still be an antiferromagnetic interaction to explain the increase . since the are already known to be ferromagnetically aligned , the only possibility is that the spins couple antiferromagnetically with , or @xmath318 . with this information , we can determine the angle @xmath299 . at 0@xmath203 , @xmath196 is determined by the magnetic moment in the ground state only.@xcite in the f ( @xmath315 ) state , @xmath319 by symmetry and @xmath320 so @xmath321 . with the help of eqs . and the fact that , in the ground state , @xmath309 in eq . , we can evaluate eq . to find @xmath322 this is a strictly decreasing function of @xmath299 that can be used to derive @xmath299 from the experimental value @xmath323 , and the knowledge that @xmath324 . this gives @xmath325 . we derive values for @xmath217 , @xmath313 , and @xmath299 by a least - squares fitting of @xmath197 . as before , a correction for the contribution of excited kramers doublets is provided by the _ ab initio _ calculations and applied following eq . . the fitting yields @xmath326 the comparison with experiment is shown in figs . [ 4:fig : ringchitp ] and [ 4:fig : ringmp ] . note that the magnetic properties are reported per mole ( @xmath197 and @xmath196 ) or per molecule ( @xmath186 ) of and not per dycr unit . note also that @xmath324 , that @xmath313 satisfies eq . , and that @xmath299 agrees with the value derived above . has been added to the theoretical curve , according to eq . . ] the agreement of magnetization curves ( fig . [ 4:fig : ringmp ] ) is not as good as it was for the [ dycumocu]@xmath0 chain , although the qualitative properties seem to correspond . in particular , we mention the strong linear increase of @xmath186 at higher fields ( @xmath327 ) , which is due to the gradual orientation of the spins to the magnetic field [ see discussion connected with eq . ] , and , to a smaller extent , also to the correction of 0.5@xmath232 , a non - negligible linear contribution to magnetization , which is due to the low - lying excited kramers doublets . as was mentioned before , the discrepancies are not unexpected given a low - lying first excited kramers doublet of , which could undermine the assumptions underlying the decorated ising model . note also that we could not take the _ ab initio _ value of 37@xmath328 for @xmath299 . leaving @xmath299 as a parameter can be seen as a partial compensation for the inaccuracies of the model and the _ ab initio _ results . we have shown that the decorated ising model is a valid model for the magnetic properties of certain lanthanide - containing magnetic compounds , if the crystal field spectrum of the lanthanide ion satisfies certain properties . the most important of these is the requirement of a ground state kramers doublet with completely uniaxial magnetic anisotropy ( this statement is simplified , see section [ 4:sec : dy ] for the correct details ) . it is a remarkable fact that precisely this property has been established by multiconfigurational _ ab initio _ calculations on several centers that are part of polynuclear molecular magnets . perhaps the best known example is the triangle , where the ising properties of were used to explain the nature of the ground state.@xcite we have focused on as lanthanide ion because this is a much - used lanthanide in current synthetic research in molecular magnetism ( witness both compounds in this paper ) and because computational results showing that it meets the requirements for an ising spin are available . however , there is no reason to assume that the findings are unique to . we expect that other lanthanides with high momentum ( e.g. , ) will exhibit the same uniaxial anisotropy in certain ligand environments and that examples of decorated ising chains based on lanthanides other than will be found in the future . we thank liviu ungur for providing results of the _ ab initio _ calculations . we thank the referee for useful suggestions and comments . w. v.d.h . acknowledges financial support from the research foundation - flanders ( fwo ) .

it is shown that the bond - decorated ising model is a realistic model for certain real magnetic compounds containing lanthanide ions . the lanthanide ion plays the role of ising spin . the required conditions on the crystal - field spectrum of the lanthanide ion for the model to be valid are discussed and found to be in agreement with several recent _ ab initio _ calculations on centers . similarities and differences between the spectra of the simple ising chain and the decorated ising chain are discussed and illustrated , with attention to level crossings in a magnetic field . the magnetic properties of two actual examples ( a [ dycumocu]@xmath0 chain and a ring ) are obtained by a transfer - matrix solution of the decorated ising model . @xmath1-factors of the metal ions are directly imported from _ ab initio _ results , while exchange coupling constants are fitted to experiment . agreement with experiment is found to be satisfactory , provided one includes a correction ( from _ ab initio _ results ) for susceptibility and magnetization to account for the presence of excited kramers doublets on .

1011.4513

we present a detailed analysis of the regularity and decay properties of linear scalar waves near the cauchy horizon of cosmological black hole spacetimes . concretely , we study charged and non - rotating ( reissner nordstrm de sitter ) as well as uncharged and rotating ( kerr de sitter ) black hole spacetimes for which the cosmological constant @xmath0 is positive . see figure [ figintropenrose ] for their penrose diagrams . these spacetimes , in the region of interest for us , have the topology @xmath1 , where @xmath2 is an interval , and are equipped with a lorentzian metric @xmath3 of signature @xmath4 . the spacetimes have three horizons located at different values of the radial coordinate @xmath5 , namely the _ cauchy horizon _ at @xmath6 , the _ event horizon _ at @xmath7 and the _ cosmological horizon _ at @xmath8 , with @xmath9 . in order to measure decay , we use a time function @xmath10 , which is equivalent to the boyer lindquist coordinate @xmath11 away from the cosmological , event and cauchy horizons , i.e. @xmath10 differs from @xmath11 by a smooth function of the radial coordinate @xmath5 ; and @xmath10 is equivalent to the eddington finkelstein coordinate @xmath12 near the cauchy and cosmological horizons , and to the eddington finkelstein coordinate @xmath13 near the event horizon . we consider the cauchy problem for the linear wave equation with cauchy data posed on a surface @xmath14 as indicated in figure [ figintropenrose ] . slice of the kerr de sitter spacetime with angular momentum @xmath15 . indicated are the cauchy horizon @xmath16 , the event horizon @xmath17 and the cosmological horizon @xmath18 , as well as future timelike infinity @xmath19 . the coordinates @xmath20 are eddington finkelstein coordinates . _ right : _ the same penrose diagram . the region enclosed by the dashed lines is the domain of dependence of the cauchy surface @xmath14 . the dotted lines are two level sets of the function @xmath10 ; the smaller one of these corresponds to a larger value of @xmath10 . ] the study of asymptotics and decay for linear scalar ( and non - scalar ) wave equations in a neighborhood of the exterior region @xmath21 of such spacetimes has a long history . methods of scattering theory have proven very useful in this context , see @xcite and references therein ( we point out that near the black hole exterior , reissner nordstrm de sitter space can be studied using exactly the same methods as schwarzschild de sitter space ) ; see @xcite for a different approach using vector field commutators . there is also a substantial amount of literature on the case @xmath22 of the asymptotically flat reissner nordstrm and kerr spacetimes ; we refer the reader to @xcite and references therein . the purpose of the present work is to show how a uniform analysis of linear waves up to the cauchy horizon can be accomplished using methods from scattering theory and microlocal analysis . our main result is : [ thmintromain ] let @xmath3 be a non - degenerate reissner de sitter metric with non - zero charge @xmath23 , or a non - degenerate kerr de sitter metric with small non - zero angular momentum @xmath24 , with spacetime dimension @xmath25 . then there exists @xmath26 , only depending on the parameters of the spacetime , such that the following holds : if @xmath12 is the solution of the cauchy problem @xmath27 with smooth initial data , then there exists @xmath28 such that @xmath12 has a partial asymptotic expansion @xmath29 where @xmath30 , and @xmath31 uniformly in @xmath32 . the same bound , with a different constant @xmath33 , holds for derivatives of @xmath34 along any finite number of stationary vector fields which are tangent to the cauchy horizon . moreover , @xmath12 is continuous up to the cauchy horizon . more precisely , @xmath34 as well as all such derivatives of @xmath34 lie in the weighted spacetime sobolev space @xmath35 in @xmath36 , where @xmath37 is the surface gravity of the cauchy horizon . for the massive klein gordon equation @xmath38 , @xmath39 small , the same result holds true without the constant term @xmath40 . here , the spacetime sobolev space @xmath41 , for @xmath42 , consists of functions which remain in @xmath43 under the application of up to @xmath44 stationary vector fields ; for general @xmath45 , @xmath41 is defined using duality and interpolation . the final part of theorem [ thmintromain ] in particular implies that @xmath34 lies in @xmath46 near the cauchy horizon on any surface of fixed @xmath10 . after introducing the reissner de sitter and kerr de sitter metrics at the beginning of [ secrnds ] and [ seckds ] , we will prove theorem [ thmintromain ] in [ subsecrndsconormal ] and [ subseckdsres ] , see theorems [ thmrndspartialasympconormal ] and [ thmkdspartialasympconormal ] . our analysis carries over directly to non - scalar wave equations as well , as we discuss for differential forms in [ subsecrndsbundles ] ; however , we do not obtain uniform boundedness near the cauchy horizon in this case . furthermore , a substantial number of ideas in the present paper can be adapted to the study of asymptotically flat ( @xmath22 ) spacetimes ; corresponding boundedness , regularity and ( polynomial ) decay results on reissner nordstrm and kerr spacetimes will be discussed in the forthcoming paper @xcite . let us also mention that a minor extension of our arguments yield analogous boundedness , decay and regularity results for the cauchy problem with a ` two - ended ' cauchy surface @xmath14 up to the bifurcation sphere @xmath47 , see figure [ figintrobifurcation ] . . for solutions of the cauchy problem with initial data posed on @xmath14 , our methods imply boundedness and precise regularity results , as well as asymptotics and decay towards @xmath19 , in the causal past of @xmath47 . ] theorem [ thmintromain ] is the first result known to the authors establishing asymptotics and regularity near the cauchy horizon of rotating black holes . ( however , we point out that dafermos and luk have recently announced the @xmath48 stability of the cauchy horizon of the kerr spacetime for einstein s vacuum equations @xcite . ) in the case of @xmath22 and in spherical symmetry ( reissner nordstrm ) , franzen @xcite proved the uniform boundedness of waves in the black hole interior and @xmath49 regularity up to @xmath16 , while luk and oh @xcite showed that linear waves generically do not lie in @xmath50 at @xmath16 . there is also ongoing work by franzen on the analogue of her result for kerr spacetimes @xcite . gajic @xcite , based on previous work by aretakis @xcite , showed that for _ extremal _ reissner nordstrm spacetimes , waves _ do _ lie in @xmath50 . we do not present a microlocal study of the event horizon of extremal black holes here , however we remark that our analysis reveals certain high regularity phenomena at the cauchy horizon of _ near - extremal _ black holes , which we will discuss below . closely related to this , the study of costa , giro , natrio and silva @xcite of the nonlinear einstein scalar field system in spherical symmetry shows that , close to extremality , rather weak assumptions on initial data on a null hypersurface transversal to the event horizon guarantee @xmath50 regularity of the metric at @xmath16 ; however , they assume exact reissner nordstrm de sitter data on the event horizon , while in the present work , we link non - trivial decay rates of waves along the event horizon to the regularity of waves at @xmath16 . compare this also with the discussions in [ subsecrndshighreg ] and remark [ rmkrndshighreg ] . one could combine the treatment of reissner nordstrm de sitter and kerr de sitter spacetimes by studying the more general kerr newman de sitter family of charged and rotating black hole spacetimes , discovered by carter @xcite , which can be analyzed in a way that is entirely analogous to the kerr de sitter case . however , in order to prevent cumbersome algebraic manipulations from obstructing the flow of our analysis , we give all details for reissner nordstrm de sitter black holes , where the algebra is straightforward and where moreover mode stability can easily be shown to hold for subextremal spacetimes ; we then indicate rather briefly the ( mostly algebraic ) changes for kerr de sitter black holes , and leave the similar , general case of kerr newman de sitter black holes to the reader . in fact , our analysis is stable under suitable perturbations , and one can thus obtain results entirely analogous to theorem [ thmintromain ] for kerr newman de sitter metrics with small non - zero angular momentum @xmath24 and small charge @xmath23 ( depending on @xmath24 ) , or for small charge @xmath23 and small non - zero angular momentum @xmath24 ( depending on @xmath23 ) , by perturbative arguments : indeed , in these two cases , the kerr newman de sitter metric is a small stationary perturbation of the kerr de sitter , resp . reissner nordstrm de sitter metric , with the same structure at @xmath16 . in the statement of theorem [ thmintromain ] , we point out that the amount of regularity of the remainder term @xmath34 at the cauchy horizon is directly linked to the amount @xmath51 of exponential decay of @xmath34 : the more decay , the higher the regularity . this can intuitively be understood in terms of the _ blue - shift effect _ @xcite : the more a priori decay @xmath34 has along the cauchy horizon ( approaching @xmath19 ) , the less energy can accumulate at the horizon . the precise microlocal statement capturing this is a _ radial point estimate _ at the intersection of @xmath16 with the boundary at infinity of a compactification of the spacetime at @xmath52 , which we will discuss in [ subsecintrogeometry ] . now , @xmath51 can be any real number less than the _ spectral gap _ @xmath53 of the operator @xmath54 , which is the infimum of @xmath55 over all non - zero _ resonances _ ( or quasi - normal modes ) @xmath56 ; the resonance at @xmath57 gives rise to the constant @xmath40 term . ( we refer to @xcite and @xcite for the discussion of resonances for black hole spacetimes . ) due to the presence of a trapped region in the black hole spacetimes considered here , @xmath53 is bounded from above by a quantity @xmath58 associated with the null - geodesic dynamics near the trapped set , as proved by dyatlov @xcite in the present context following breakthrough work by wunsch and zworski @xcite , and by nonnenmacher and zworski @xcite : below ( resp . above ) any line @xmath59 , @xmath60 , there are infinitely ( resp . finitely ) many resonances . in principle however , one expects that there indeed exists a non - zero number of resonances above this line , and correspondingly the expansion can be refined to take these into account . ( in fact , one can obtain a full resonance expansion due to the complete integrability of the null - geodesic flow near the trapped set , see @xcite . ) since for the mode solution corresponding to a resonance at @xmath61 , @xmath62 , we obtain the regularity @xmath63 at @xmath16 , shallow resonances , i.e. those with small @xmath64 , give the dominant contribution to the solution @xmath12 both in terms of decay and regularity at @xmath16 . the authors are not aware of any rigorous results on shallow resonances , so we shall only discuss this briefly in remark [ rmkrndshighreg ] , taking into account insights from numerical results : these suggest the existence of resonant states with imaginary parts roughly equal to @xmath65 and @xmath66 , and hence the relative sizes of the surface gravities play a crucial role in determining the regularity at @xmath16 . whether resonant states are in fact no better than @xmath67 , and the existence of shallow resonances , which , if true , would yield a linear instability result for cosmological black hole spacetimes with cauchy horizons analogous to @xcite , will be studied in future work . once these questions have been addressed , one can conclude that the lack of , say , @xmath68 regularity at @xmath16 is caused precisely by shallow quasinormal modes . thus , somewhat surprisingly , the mechanism for the linear instability of the cauchy horizon of _ cosmological _ spacetimes is more subtle than for asymptotically flat spacetimes in that the presence of a cosmological horizon , which ultimately allows for a resonance expansion of linear waves @xmath12 , leads to a much more precise structure of @xmath12 at @xmath16 , with the regularity of @xmath12 directly tied to quasinormal modes of the black hole exterior . the interest in understanding the behavior of waves near the cauchy horizon has its roots in penrose s strong cosmic censorship conjecture , which asserts that maximally globally hyperbolic developments for the einstein maxwell or einstein vacuum equations ( depending on whether one considers charged or uncharged solutions ) with _ generic _ initial data ( and a complete initial surface , and/or under further conditions ) are inextendible as suitably regular lorentzian manifolds . in particular , the smooth , even analytic , extendability of the reissner nordstrm(de sitter ) and kerr(de sitter ) solutions past their cauchy horizons is conjectured to be an unstable phenomenon . it turns out that the question what should be meant by ` suitable regularity ' is very subtle ; we refer to works by christodoulou @xcite , dafermos @xcite , and costa , giro , natrio and silva @xcite in the spherically symmetric setting for positive and negative results for various notions of regularity . there is also work in progress by dafermos and luk on the @xmath48 stability of the kerr cauchy horizon , assuming a quantitative version of the non - linear stability of the exterior region . we refer to these works , as well as to the excellent introductions of @xcite , for a discussion of heuristic arguments and numerical experiments which sparked this line of investigation . here , however , we only consider linear equations , motivated by similar studies in the asymptotically flat case by dafermos @xcite ( see ( * ? ? ? * footnote 11 ) ) , franzen @xcite , sbierski @xcite , and luk and oh @xcite . the main insight of the present paper is that a uniform analysis up to @xmath16 can be achieved using by now standard methods of scattering theory and geometric microlocal analysis , in the spirit of recent works by vasy @xcite , baskin , vasy and wunsch @xcite and @xcite : the core of the precise estimates of theorem [ thmintromain ] are microlocal propagation results at ( generalized ) radial sets , as we will discuss in [ subsecintrostrategy ] . from this geometric microlocal perspective however , i.e. taking into account merely the phase space properties of the operator @xmath54 , it is both unnatural and technically inconvenient to view the cauchy horizon as a boundary ; after all , the metric @xmath3 is a non - degenerate lorentzian metric up to @xmath16 and beyond . thus , the most subtle step in our analysis is the formulation of a suitable extended problem ( in a neighborhood of @xmath69 ) which reduces to the equation of interest , namely the wave equation , in @xmath32 . the penrose diagram is rather singular at future timelike infinity @xmath19 , yet all relevant phenomena , in particular trapping and red-/blue - shift effects , should be thought of as taking place there , as we will see shortly ; therefore , we work instead with a compactification of the region of interest , the domain of dependence of @xmath14 in figure [ figintropenrose ] , in which the horizons as well as the trapped region remain separated , and the metric remains smooth , as @xmath70 . concretely , using the coordinate @xmath10 employed in theorem [ thmintromain ] , the radial variable @xmath5 and the spherical variable @xmath71 , we consider a region @xmath72 i.e. we add the ideal boundary at future infinity , @xmath73 , to the spacetime , and equip @xmath74 with the obvious smooth structure in which @xmath75 vanishes simply and non - degenerately at @xmath76 . ( it is tempting , and useful for purposes of intuition , to think of @xmath74 as being a submanifold of the blow - up of the compactification suggested by the penrose diagram adding an ` ideal sphere at infinity ' at @xmath19 at @xmath19 . however , the details are somewhat subtle ; see @xcite . ) due to the stationary nature of the metric @xmath3 , the ( null-)geodesic flow should be studied in a version of phase space which has a built - in uniformity as @xmath70 . a clean way of describing this uses the language of b - geometry ( and b - analysis ) ; we refer the reader to melrose @xcite for a detailed introduction , and @xcite and @xcite for brief overviews . we recall the most important features here : on @xmath74 , the metric @xmath3 is a non - degenerate lorentzian _ b - metric _ , i.e. a linear combination with smooth ( on @xmath74 ) coefficients of @xmath77 where @xmath78 are coordinates in @xmath79 ; in fact the coefficients are independent of @xmath75 . then , @xmath3 is a section of the symmetric second tensor power of a natural vector bundle on @xmath74 , the _ b - cotangent bundle _ @xmath80 , which is spanned by the sections @xmath81 . we stress that @xmath82 is a smooth , non - degenerate section of @xmath80 _ up to and including _ the boundary @xmath73 . likewise , the dual metric @xmath83 is a section of the second symmetric tensor power of the _ b - tangent bundle _ @xmath84 , which is the dual bundle of @xmath80 and thus spanned by @xmath85 . the dual metric function , which we also denote by @xmath86 by a slight abuse of notation , associates to @xmath87 the squared length @xmath88 . over @xmath89 , the b - cotangent bundle is naturally isomorphic to the standard cotangent bundle . the geodesic flow , lifted to the cotangent bundle , is generated by the hamilton vector field @xmath90 , which extends to a smooth vector field @xmath91 tangent to @xmath92 . now , @xmath93 is homogeneous of degree @xmath94 with respect to dilations in the fiber , and it is often convenient to rescale it by multiplication with a homogeneous degree @xmath95 function @xmath96 , obtaining the homogeneous degree @xmath97 vector field @xmath98 . as such , it extends smoothly to a vector field on the _ radial ( or projective ) compactification _ @xmath99 of @xmath80 , which is a ball bundle over @xmath74 , with fiber over @xmath100 given by the union of @xmath101 with the ` sphere at fiber infinity ' @xmath102 . the b - cosphere bundle @xmath103 is then conveniently viewed as the boundary @xmath104 of the compactified b - cotangent bundle at fiber infinity . the projection to the base @xmath74 of integral curves of @xmath93 or @xmath105 with null initial direction , i.e. starting at a point in @xmath106 , yields ( reparameterizations of ) null - geodesics on @xmath107 ; this is clear in the interior of @xmath74 , and the important observation is that this gives a well - defined notion of null - geodesics , or null - bicharacteristics , at the boundary at infinity , @xmath79 . we remark that the characteristic set @xmath108 has two components , the union of the future null cones @xmath109 and of the past null cones @xmath110 . the red - shift or blue - shift effect manifests itself in a special structure of the @xmath105 flow near the b - conormal bundles @xmath111 of the horizons @xmath112 , @xmath113 . ( here , @xmath114 for a boundary submanifold @xmath115 and @xmath116 is the annihilator of the space of all vectors in @xmath117 tangent to @xmath118 ; @xmath119 is naturally isomorphic to the conormal bundle of @xmath118 in @xmath79 . ) indeed , in the case of the reissner nordstrm de sitter metric , @xmath120 , more precisely its boundary at fiber infinity @xmath121 , is a _ saddle point _ for the @xmath105 flow , with stable ( or unstable , depending on which of the two components @xmath122 one is working on ) manifold contained in @xmath123 , and an unstable ( or stable ) manifold transversal to @xmath124 . in the kerr de sitter case , @xmath105 does not vanish everywhere on @xmath125 , but rather is non - zero and tangent to it , so there are non - trivial dynamics within @xmath125 , but the dynamics in the directions normal to @xmath125 still has the same saddle point structure . see figure [ figintroradial ] . is a radial null - geodesic , and @xmath47 is the projection of a non - radial geodesic . _ right : _ the compactification of the spacetime at future infinity , together with the same two null - geodesics . the null - geodesic flow , extended to the ( b - cotangent bundle over the ) boundary , has saddle points at the ( b - conormal bundles of the ) intersection of the horizons with the boundary at infinity @xmath79 . ] in order to take full advantage of the saddle point structure of the null - geodesic flow near the cauchy horizon , one would like to set up an initial value problem , or equivalently a forced forward problem @xmath126 , with vanishing initial data but non - trivial right hand side @xmath127 , on a domain which extends a bit past @xmath16 . because of the finite speed of propagation for the wave equation , one is free to modify the problem beyond @xmath16 in whichever way is technically most convenient ; waves in the region of interest @xmath128 are unaffected by the choice of extension . a natural idea then is to simply add a boundary @xmath129 , @xmath130 , which one could use to cap the problem off beyond @xmath16 ; now @xmath131 is _ timelike _ , hence , to obtain a well - posed problem , one needs to impose boundary conditions there . while perfectly feasible , the resulting analysis is technically rather involved as it necessitates studying the reflection of singularities at @xmath131 quantitatively in a uniform manner as @xmath132 . ( near @xmath131 , one does not need the precise , microlocal , control as in @xcite however . ) a technically much easier modification involves the use of a complex absorbing ` potential ' @xmath133 in the spirit of @xcite ; here @xmath133 is a second order b - pseudodifferential operator on @xmath74 which is elliptic in a large subset of @xmath134 near @xmath73 . ( without b - language , one can take @xmath133 for large @xmath10 to be a time translation - invariant , properly supported ps.d.o . on @xmath74 . ) one then considers the operator @xmath135 the point is that a suitable choice of the sign of @xmath133 on the two components @xmath136 of the characteristic set leads to an absorption of high frequencies along the future - directed null - geodesic flow over the support of @xmath133 , which allows one to control a solution @xmath12 of @xmath137 in terms of the right hand side @xmath127 there . however , since we are forced to work on a domain with boundary in order to study the forward problem , the _ pseudodifferential _ complex absorption does not make sense near the relevant boundary component , which is the extension of the left boundary in figure [ figintropenrose ] past @xmath6 . a doubling construction as in @xcite on the other hand , doubling the spacetime across the timelike surface @xmath131 , say , amounts to gluing an ` artificial exterior region ' to our spacetime , with one of the horizons identified with the original cauchy horizon ; this in particular creates another trapped region , which we can however easily hide using a complex absorbing potential ! we then cap off the thus extended spacetime beyond the cosmological horizon of the artificial exterior region , located at @xmath138 , by a _ spacelike _ hypersurface @xmath139 at @xmath140 , @xmath141 , at which the analysis is straightforward @xcite . see figure [ figintroextended ] . ( in the spherically symmetric setting , one could also replace the region @xmath134 beyond the cauchy horizon by a static de sitter type space , thus not generating any further trapping or horizons and obviating the need for complex absorption ; but for kerr de sitter , this gluing procedure is less straightforward to implement , hence we use the above doubling - type procedure for reissner nordstrm already . ) the construction of the extension is detailed in [ subsecrndsmfd ] . , creating an artificial horizon @xmath142 , and cap off beyond @xmath142 using a _ spacelike _ hypersurface @xmath139 . complicated dynamics in the extended region are hidden by a complex absorbing potential @xmath133 supported in the shaded region . ] we thus study the forcing problem @xmath143 with @xmath127 and @xmath12 supported in the future of the ` cauchy ' surface @xmath14 in @xmath144 , and in the future of @xmath139 in @xmath145 . the natural function spaces are _ weighted b - sobolev spaces _ @xmath146 where the spacetime sobolev space @xmath147 measures regularity relative to @xmath43 with respect to stationary vector fields , as defined after the statement of theorem [ thmintromain ] . more invariantly , @xmath148 , for integer @xmath44 , consists of @xmath43 functions which remain in @xmath43 upon applying up to @xmath44 b - vector fields ; the space @xmath149 of b - vector fields consists of all smooth vector fields on @xmath74 which are tangent to @xmath76 , and is equal to the space of smooth sections of the b - tangent bundle @xmath84 . now , is an equation on a compact space @xmath150 which degenerates at the boundary : the operator @xmath151 is a b - differential operator , i.e. a sum of products of b - vector fields , and @xmath152 is a b - ps.d.o .. ( note that this point of view is much more precise than merely stating that is an equation on a noncompact space @xmath153 ! ) thus , the analysis of the operator @xmath154 consists of two parts : _ firstly , _ the regularity analysis , in which one obtains precise regularity estimates for @xmath12 using microlocal elliptic regularity , propagation of singularities and radial point results , see [ subsecrndsregularity ] , which relies on the precise global structure of the null - geodesic flow discussed in [ subsecrndsflow ] ; and _ secondly , _ the asymptotic analysis of [ subsecrndsfredholm ] and [ subsecrndsasymp ] , which relies on the analysis of the mellin transformed in @xmath75 ( equivalently : fourier transformed in @xmath155 ) operator family @xmath156 , its high energy estimates as @xmath157 , and the structure of poles of @xmath158 , which are known as _ resonances _ or _ quasi - normal modes _ ; this last part , in which we use the shallow resonances to deduce asymptotic expansions of waves , is the only low frequency part of the analysis . the regularity one obtains for @xmath12 solving with , say , smooth compactly supported ( in @xmath153 ) forcing @xmath127 , is determined by the behavior of the null - geodesic flow near the trapping and near the horizons @xmath120 , @xmath113 . near the trapping , we use the aforementioned results @xcite , while near @xmath120 , we use radial point estimates , originating in work by melrose @xcite , and proved in the context relevant for us in @xcite ; we recall these in [ subsecrndsregularity ] . concretely , equation combines a forward problem for the wave equation near the black hole exterior region @xmath159 with a backward problem near the artificial exterior region @xmath160 , with hyperbolic propagation in the region between these two ( called ` no - shift region ' in @xcite ) . near @xmath7 and @xmath8 then , and by propagation estimates in any region @xmath161 , @xmath60 , the radial point estimate , encapsulating the red - shift effect , yields smoothness of @xmath12 relative to a b - sobolev space with weight @xmath162 , i.e. allowing for exponential growth ( in which case trapping is not an issue ) , while near @xmath6 , one is solving the equation _ away _ from the boundary @xmath79 at infinity , and hence the radial point estimate , encapsulating the blue - shift effect , there , yields an amount of regularity which is bounded from above by @xmath163 , where @xmath37 is the surface gravity of @xmath16 . in the extended region @xmath134 , the regularity analysis is very simple , since the complex absorption @xmath133 makes the problem elliptic at the trapping there and at @xmath142 , and one then only needs to use real principal type propagation together with standard energy estimates . combined with the analysis of @xmath156 , which relies on the same dynamical and geometric properties of the extended problem as the b - analysis , we deduce in [ subsecrndsfredholm ] that @xmath154 is fredholm on suitable weighted b - sobolev spaces ( and in fact solvable for any right hand side @xmath127 if one modifies @xmath127 in the unphysical region @xmath134 ) . in order to capture the high , resp . low , regularity near @xmath164 $ ] , resp . @xmath165 , these spaces have _ variable orders _ of differentiability depending on the location in @xmath74 . ( such spaces were used already by unterberger @xcite , and in a context closely related to the present paper in @xcite . we present results adapted to our needs in appendix [ secvariable ] . ) in [ subsecrndsasymp ] then , we show how the properties of the meromorphic family @xmath158 yield a partial asymptotic expansion of @xmath12 as in . using more refined regularity statements at @xmath166 , we show in [ subsecrndsconormal ] that the terms in this expansion are in fact _ conormal _ to @xmath6 , i.e. they do not become more singular upon applying vector fields tangent to the cauchy horizon . we stress that the analysis is conceptually very simple , and close to the analysis in @xcite , in that it relies on tools in microlocal analysis and scattering theory which have been frequently used in recent years . as a side note , we point out that one could have analyzed @xmath167 in @xmath32 only by proving very precise estimates for the operator @xmath167 , which is a hyperbolic ( wave - type ) operator in @xmath32 , near @xmath6 ; while this would have removed the necessity to construct and analyze an extended problem , the mechanism underlying our regularity and decay estimates , namely the radial point estimate at the cauchy horizon , would not have been apparent from this . moreover , the radial point estimate is very robust ; it works for kerr de sitter spaces just as it does for the spherically symmetric reissner nordstrm de sitter solutions . a more interesting modification of our argument relies on the observation that it is not necessary for us to incorporate the exterior region in our global analysis , since this has already been studied in detail before ; instead , one could start _ assuming _ asymptotics for a wave @xmath12 in the exterior region , and then relate @xmath12 to a solution of a global , extended problem , for which one has good regularity results , and deduce them for @xmath12 by restriction . such a strategy is in particular appealing in the study of spacetimes with vanishing cosmological constant using the analytic framework of the present paper , since the precise structure of the ` resolvent ' @xmath168 has not been analyzed so far , whereas boundedness and decay for scalar waves on the exterior regions of reissner nordstrm and kerr spacetimes are known by other methods ; see the references at the beginning of [ secintro ] . we discuss this in the forthcoming @xcite . in the remaining parts of [ secrnds ] , we analyze the essential spectral gap for near - extremal black holes in [ subsecrndshighreg ] ; we find that for _ any _ desired level of regularity , one can choose near - extremal parameters of the black hole such that solutions @xmath12 to with @xmath127 in a finite - codimensional space achieve this level of regularity at @xmath16 . however , as explained in the discussion of theorem [ thmintromain ] , it is very likely that shallow resonances cause the codimension to increase as the desired regularity increases . lastly , in [ subsecrndsbundles ] , we indicate the simple changes to our analysis needed to accommodate wave equations on natural tensor bundles . in [ seckds ] then , we show how kerr de sitter spacetimes fit directly into our framework : we analyze the flow on a suitable compactification and extension , constructed in [ subseckdsmfd ] , in [ subseckdsflow ] , and deduce results completely analogous to the reissner nordstrm de sitter case in [ subseckdsres ] . we are very grateful to jonathan luk and maciej zworski for many helpful discussions . we would also like to thank sung - jin oh for many helpful discussions and suggestions , for reading parts of the manuscript , and for pointing out a result in @xcite which led to the discussion in remark [ rmkrndshighreg ] ; thanks also to elmar schrohe for very useful discussions leading to appendix [ secsuppext ] . we are grateful for the hospitality of the erwin schrdinger institute in vienna , where part of this work was carried out . we gratefully acknowledge support by a.v.s national science foundation grants dms-1068742 and dms-1361432 . p.h . is a miller fellow and thanks the miller institute at the university of california , berkeley for support . we focus on the case of @xmath169 spacetime dimensions ; the analysis in more than @xmath169 dimensions is completely analogous . in the domain of outer communications of the 4-dimensional reissner nordstrm de sitter black hole , given by @xmath170 , with @xmath171 described below , the metric takes the form @xmath172 here @xmath173 and @xmath174 are the mass and the charge of the black hole , and @xmath175 , with @xmath176 the cosmological constant . setting @xmath177 , this reduces to the schwarzschild de sitter metric . we assume that the spacetime is non - degenerate : [ defrndsnondegenerate ] we say that the reissner de sitter spacetime with parameters @xmath178 is _ non - degenerate _ if @xmath179 has @xmath180 simple positive roots @xmath181 . since @xmath182 when @xmath183 , we see that @xmath184 the roots of @xmath179 are called _ cauchy horizon _ ( @xmath165 ) , _ event horizon _ ( @xmath185 ) and _ cosmological horizon _ ( @xmath186 ) , with the cauchy horizon being a feature of charged ( or rotating , see [ seckds ] ) solutions of einstein s field equations . to give a concrete example of a non - degenerate spacetime , let us check the non - degeneracy condition for black holes with small charge , and compute the location of the cauchy horizon : for fixed @xmath187 , let @xmath188 so @xmath189 . for @xmath190 , the function @xmath191 has a root at @xmath192 . since @xmath193 is negative for @xmath194 and for large @xmath195 but positive for large @xmath196 , the function @xmath197 has two simple positive roots if and only if @xmath198 , where @xmath199 is the unique positive critical point of @xmath200 ; but @xmath201 if and only if @xmath202 then : [ lemmarndsnondegenerate ] suppose @xmath203 satisfy the non - degeneracy condition , and denote the three non - negative roots of @xmath191 by @xmath204 . then for small @xmath174 , the function @xmath179 has three positive roots @xmath205 , @xmath113 , with @xmath206 , depending smoothly on @xmath23 , and @xmath207 . the existence of the functions @xmath205 follows from the implicit function theorem , taking into account the simplicity of the roots @xmath208 of @xmath191 . let us write @xmath209 ; these are smooth functions of @xmath210 . differentiating @xmath211 with respect to @xmath210 gives @xmath212 , hence @xmath213 , which yields the analogous expansion for @xmath214 . we now discuss the extension of the metric beyond the event and cosmological horizon , as well as beyond the cauchy horizon ; the purpose of the present section is to define the manifold on which our analysis of linear waves will take place . see proposition [ proprndsmfd ] for the final result . we begin by describing the extension of the metric beyond the event and the cosmological horizon , thereby repeating the arguments of @xcite ; see figure [ figrndsext23 ] . , the event horizon @xmath17 and the cauchy horizon @xmath16 . we first study a region @xmath215 bounded by an initial cauchy hypersurface @xmath14 and two final cauchy hypersurfaces @xmath216 and @xmath217 . _ right : _ the same region , compactified at infinity ( @xmath218 in the penrose diagram ) , with the artificial hypersurfaces put in . ] write @xmath219 , so @xmath220 we denote by @xmath221 a smooth function such that @xmath222 @xmath141 small , with @xmath223 , smooth near @xmath112 , to be specified momentarily . ( thus , @xmath224 as @xmath225 and @xmath226 . ) we then put @xmath227 and compute @xmath228 which is a non - degenerate lorentzian metric up to @xmath229 , with dual metric @xmath230 we can choose @xmath223 so as to make @xmath231 timelike , i.e. @xmath232 : indeed , choosing @xmath233 ( which undoes the coordinate change , up to an additive constant ) accomplishes this trivially in @xmath164 $ ] away from @xmath234 ; however , we need @xmath223 to be smooth at @xmath234 as well . now , @xmath231 is timelike in @xmath235 if and only if @xmath236 , which holds for any @xmath237 . therefore , we can choose @xmath238 smooth near @xmath185 , with @xmath239 for @xmath240 , and @xmath241 smooth near @xmath186 , with @xmath242 for @xmath243 , and thus a function @xmath244 , such that in the new coordinate system @xmath245 , the metric @xmath3 extends smoothly to @xmath229 , and @xmath231 is timelike for @xmath246 $ ] ; and furthermore we can arrange that @xmath247 in @xmath248 $ ] by possibly changing @xmath249 by an additive constant . extending @xmath223 smoothly beyond @xmath250 in an arbitrary manner , the expression makes sense for @xmath251 as well as for @xmath252 $ ] . we first notice that we can choose the extension @xmath223 such that @xmath231 is timelike also for @xmath253 : indeed , for such @xmath5 , we have @xmath254 , and the timelike condition becomes @xmath255 , which is satisfied as long as @xmath256 there . in particular , we can take @xmath257 for @xmath258 $ ] and @xmath259 for @xmath260 , in which case we get @xmath261 for @xmath258 $ ] with @xmath262 , and for @xmath263 with @xmath264 . we define @xmath265 beyond @xmath185 and @xmath186 by the same formula , using the extensions of @xmath238 and @xmath241 just described ; in particular @xmath266 in @xmath267 . we define a time orientation in @xmath268 by declaring @xmath231 to be future timelike . we introduce spacelike hypersurfaces in the thus extended spacetime as indicated in figure [ figrndsext23 ] , namely @xmath269 and @xmath270 [ rmkrndshypersurfacenotation ] here and below , the subscript ` i ' ( initial ) , resp . ` f ' ( final ) , indicates that outward pointing timelike vectors are past , resp . future , oriented . the number in the subscript denotes the horizon near which the surface is located . notice here that indeed @xmath271 and @xmath272 at @xmath217 , so @xmath231 and @xmath273 have opposite timelike character there , while likewise @xmath274 and @xmath275 at @xmath216 . the tilde indicates that @xmath216 will eventually be disposed of ; we only define it here to make the construction of the extended spacetime clearer . the region @xmath215 is now defined as @xmath276 bounded by three final cauchy hypersurfaces @xmath277 , @xmath278 and @xmath216 . a partial extension beyond the cauchy horizon is bounded by the final hypersurface @xmath279 and a timelike hypersurfaces @xmath131 . _ right : _ the same region , compactified at infinity , with the artificial hypersurfaces put in . ] next , we further extend the metric beyond the coordinate singularity of @xmath3 at @xmath6 when written in the coordinates , at @xmath6 ; see figure [ figrndsext12 ] : let @xmath280 where now @xmath281 , with @xmath282 for @xmath283 , and @xmath284 smooth down to @xmath6 . thus , by adjusting @xmath285 by an additive constant , we may arrange @xmath286 for @xmath287 $ ] . notice that ( formally ) @xmath288 , and @xmath289 in @xmath290 $ ] . thus , @xmath291,\ ] ] after extending @xmath284 smoothly into @xmath292 . this expression is of the form , with @xmath293 , @xmath294 and @xmath223 replaced by @xmath295 , @xmath296 and @xmath284 , respectively . in particular , by the same calculation as above , @xmath297 is timelike provided @xmath298 or @xmath299 in @xmath254 , while in @xmath235 , any @xmath300 works . however , since we need @xmath282 for @xmath5 near @xmath185 ( where @xmath254 ) , requiring @xmath297 to be timelike would force @xmath301 as @xmath302 , which is incompatible with @xmath284 being smooth down to @xmath6 . in view of the penrose diagram of the spacetime in figure [ figrndsext12 ] , it is clear that this must happen , since we can not make the level sets of @xmath295 ( which coincide with the level sets of @xmath293 , i.e. with parts of @xmath14 , near @xmath7 ) both remain spacelike and cross the cauchy horizon in the indicated manner . thus , we merely require @xmath298 for @xmath303 $ ] , making @xmath297 timelike there , but losing the timelike character of @xmath297 in a subset of the transition region @xmath304 . moreover , similarly to the choices of @xmath238 and @xmath241 above , we take @xmath305 in @xmath306 $ ] and @xmath307 in @xmath308 $ ] . using the coordinates @xmath309 , we thus have @xmath310 ; we further define @xmath311 thus , @xmath277 intersects @xmath14 at @xmath312 , @xmath313 . we choose @xmath314 as follows : we calculate the squared norm of the conormal of @xmath277 using as @xmath315 which is positive in @xmath316 $ ] provided @xmath60 , @xmath317 , since @xmath254 in this region . therefore , choosing @xmath314 so that it verifies these inequalities , @xmath277 is spacelike . put @xmath318 , so @xmath319 at @xmath320 , and define @xmath321 we note that @xmath278 is indeed spacelike , as @xmath271 there , and @xmath279 is spacelike by construction of @xmath295 . the surface @xmath131 is timelike ( hence the subscript ) . putting @xmath322 finishes the definition of all objects in figure [ figrndsext12 ] . in order to justify the subscripts ` f ' , we compute a smooth choice of time orientation : first of all , @xmath297 is future timelike ( by choice ) in @xmath268 ; furthermore , in @xmath323 , we have @xmath271 , so @xmath273 is timelike in @xmath323 . we then calculate @xmath324 in @xmath325 $ ] , so @xmath326 and @xmath297 are in the same causal cone there , in particular @xmath326 is future timelike in @xmath290 $ ] , which justifies the notation @xmath278 ; furthermore @xmath297 is timelike for @xmath327 , with @xmath328 in @xmath306 $ ] ( using the form of the metric with @xmath305 there ) , hence @xmath326 and @xmath329 are in the same causal cone here . thus , @xmath297 is _ past _ timelike in @xmath327 , justifying the notation @xmath279 . see also figure [ figrndsradial ] below . lastly , for @xmath277 , we compute @xmath330 by our choice of @xmath314 , hence the future timelike 1-form @xmath326 is indeed outward pointing at @xmath277 . we remark that from the perspective of @xmath331 , the surface @xmath216 is initial , but we keep the subscript ` f ' for consistency with the notation used in the discussion of @xmath332 . , bounded by the final cauchy hypersurface @xmath333 and two initial hypersurfaces @xmath139 and @xmath278 . the artificial extension in the region behind the cauchy horizon removes the curvature singularity and generates an artificial horizon @xmath142 . _ right : _ the same region , compactified at infinity , with the artificial hypersurfaces put in . ] one can now analyze linear waves on the spacetime @xmath334 if one uses the reflection of singularities at @xmath131 . ( we will describe the null - geodesic flow in [ subsecrndsflow ] . ) however , we proceed as explained in [ secintro ] and add an artificial exterior region to the region @xmath335 ; see figure [ figrndsext01 ] . we first note that the form of the metric in @xmath336 is @xmath337 thus of the same form as . define a function @xmath338 such that @xmath339 so @xmath340 on @xmath341 and @xmath342 on @xmath343 $ ] , see figure [ figrndsmustar ] . one can in fact drop the last assumption on @xmath344 , as we will do in the kerr de sitter discussion for simplicity , but in the present situation , this assumption allows for the nice interpretation of the appended region as a ` past ' or ` backwards ' version of the exterior region of a black hole . ( solid ) in the region @xmath134 beyond the cauchy horizon to a smooth function @xmath344 ( dashed where different from @xmath179 ) . notice that the @xmath344 has the same qualitative properties near @xmath345 $ ] as near @xmath164 $ ] . ] we extend the metric to @xmath346 by defining @xmath347 . we then extend @xmath3 beyond @xmath138 as in : put @xmath348 with @xmath349 , @xmath350 when @xmath351 , @xmath352 , where we set @xmath353 ; further let @xmath354 for @xmath355 , so @xmath356 in @xmath357 ( up to redefining @xmath358 by an additive constant ) . then , in @xmath359-coordinates , the metric @xmath3 takes the form near @xmath360 , with @xmath293 replaced by @xmath361 and @xmath362 ; hence @xmath3 extends across @xmath138 as a non - degenerate stationary lorentzian metric , and we can choose @xmath363 to be smooth across @xmath138 so that @xmath364 is timelike in @xmath365 , and such that moreover @xmath366 in @xmath367 , thus ensuring the form of the metric ( replacing @xmath293 and @xmath294 by @xmath361 and @xmath95 , respectively ) . we can glue the functions @xmath361 and @xmath295 together by defining the smooth function @xmath368 in @xmath369 to be equal to @xmath361 in @xmath365 and equal to @xmath295 in @xmath370 . define @xmath371 note here that @xmath364 is _ past _ timelike in @xmath365 . lastly , we put @xmath372 note that in the region @xmath332 , we have produced an artificial horizon @xmath142 at @xmath138 . again , the notation @xmath278 is incorrect from the perspective of @xmath332 , but is consistent with the notation used in the discussion of @xmath331 . let us summarize our construction : [ proprndsmfd ] fix parameters @xmath178 of a reissner de sitter spacetime which is non - degenerate in the sense of definition [ defrndsnondegenerate ] . let @xmath344 be a smooth function on @xmath373 satisfying , where @xmath179 is given by . for @xmath141 small , define the manifold @xmath374 and equip @xmath89 with a smooth , stationary , non - degenerate lorentzian metric @xmath3 , which has the form @xmath375\cup[r_2+\delta , r_3-\delta ] , \\ \begin{split } \label{eqrndsmetrictransition } g & = \mu_*\,dt_*^2 + 2s_j(1+\mu _ * c_j)\,dt_*\,dr+(2c_j+\mu _ * c_j^2)\,dr^2-r^2\,d\omega^2 , \\ & \qquad\qquad |r - r_j|\leq 2\delta , { \textnormal } { or } r\in[r_1 + 2\delta , r_2 - 2\delta],\ j=1 , \end{split } \\ \begin{split } \label{eqrndsmetricbeyond } g & = \mu_*\,dt_*^2 + 4s_j\,dt_*\,dr + 3\mu_*^{-1}\,dr^2-r^2\,d\omega^2 , \\ & \qquad\qquad r\in[r_j-2\delta , r_j-\delta],\ j=0,2,{\textnormal } { or } r\in[r_j+\delta , r_j+2\delta],\ j=1,3 , \end{split } \end{aligned}\ ] ] in @xmath376 $ ] , where @xmath377 . then the region @xmath378 is isometric to a region in the reissner nordstrm de sitter spacetime with parameters @xmath379 , with @xmath21 isometric to the exterior domain ( bounded by the event horizon @xmath17 at @xmath7 and the cosmological horizon @xmath18 at @xmath8 ) , @xmath380 isometric to the black hole region ( bounded by the future cauchy horizon @xmath16 at @xmath6 and the event horizon ) , and @xmath381 isometric to a region beyond the future cauchy horizon . ( see figure [ figrndsextfull ] . ) furthermore , @xmath89 is time - orientable . one can choose the smooth functions @xmath382 such that @xmath383 and @xmath384 the hypersurfaces @xmath385 are spacelike provided @xmath60 is sufficiently small ; here @xmath386 . they bound a domain @xmath153 , which is a submanifold of @xmath89 with corners . ( recall remark [ rmkrndshypersurfacenotation ] for our conventions in naming the hypersurfaces . ) @xmath89 and @xmath153 possess natural partial compactifications @xmath74 and @xmath150 , respectively , obtained by introducing @xmath387 and adding to them their ideal boundary at infinity , @xmath73 ; the metric @xmath3 is a non - degenerate lorentzian b - metric on @xmath74 and @xmath150 . adding @xmath73 to @xmath89 means defining @xmath388 where @xmath389 is identified with the point @xmath390 , and we define the smooth structure on @xmath74 by declaring @xmath75 to be a smooth boundary defining function . the extensions described above amount to a direct construction of a manifold @xmath391_r\times{\mathbb{s}}^2_\omega$ ] , where we obtained the function @xmath368 by gluing @xmath361 and @xmath295 in @xmath308 $ ] , and similarly @xmath295 and @xmath293 in @xmath325 $ ] ; we then extend the metric @xmath3 non - degenerately to a stationary metric in @xmath392 and @xmath393 , thus obtaining a metric @xmath3 on @xmath89 with the listed properties . , which is the diagram of reissner nordstrm de sitter in a neighborhood of the exterior domain and of the black hole region as well as near the cauchy horizon ; further beyond the cauchy horizon , we glue in an artificial exterior region , eliminating the singularity at @xmath194 . _ right : _ the compactification of @xmath153 to a manifold with corners @xmath150 ; the smooth structure of @xmath150 is the one induced by the embedding of @xmath150 into the plane ( cross @xmath394 ) as displayed here . ] we define the regions @xmath395 and @xmath215 as in , and , respectively , as submanifolds of @xmath153 with corners ; their boundary hypersurfaces are hypersurfaces within @xmath153 . we denote the closures of these domains and hypersurfaces in @xmath150 by the same names , but dropping the superscript ` @xmath396 ' . furthermore , we write @xmath397 for the ideal boundaries at infinity . one reason for constructing the compactification @xmath150 step by step is that the null - geodesic dynamics almost decouple in the subdomains @xmath398 , @xmath399 and @xmath400 , see figures [ figrndsext01 ] , [ figrndsext12 ] and [ figrndsext23 ] . we denote by @xmath83 the dual metric of @xmath3 . we recall that we can glue @xmath401 in @xmath398 , @xmath326 in @xmath402 $ ] and @xmath403 in @xmath400 together using a non - negative partition of unity and obtain a 1-form @xmath404 which is everywhere future timelike in @xmath150 . thus , the characteristic set of @xmath54 , @xmath405 with @xmath406 the dual metric function , globally splits into two connected components @xmath407 ( indeed , if @xmath408 , then @xmath409 , which is spacelike , so @xmath410 shows that @xmath411 . ) thus , @xmath110 , resp . @xmath109 , is the union of the past , resp . future , causal cones . we note that @xmath108 and @xmath136 are smooth codimension 1 submanifolds of @xmath412 in view of the lorentzian nature of the dual metric @xmath83 . moreover , @xmath136 is transversal to @xmath413 , in fact the differentials @xmath414 and @xmath415 ( @xmath75 lifted to a function on @xmath80 ) are linearly independent everywhere in @xmath412 . we begin by analyzing the null - geodesic flow ( in the b - cotangent bundle ) near the horizons : we will see that the hamilton vector field @xmath93 has critical points where the horizons intersect the ideal boundary @xmath416 of @xmath150 ; more precisely , @xmath93 is radial there . in order to simplify the calculations of the behavior of @xmath93 nearby , we observe that the smooth structure of the compactification @xmath150 , which is determined by the function @xmath387 , is unaffected by the choice of the functions @xmath223 in proposition [ proprndsmfd ] , since changing @xmath223 merely multiplies @xmath75 by a positive function that only depends on @xmath5 , hence is smooth on our initial compactification @xmath150 . now , the intersections @xmath417 are smooth boundary submanifolds of @xmath74 , and we define @xmath418 which is well - defined given merely the smooth structure on @xmath150 . the point of our observation then is that we can study the hamilton flow near @xmath120 using any choice of @xmath223 . thus , introducing @xmath419 , with @xmath420 near @xmath250 , we find from that @xmath421 let @xmath422 . then , with @xmath423 , and writing b - covectors as @xmath424 the dual metric function @xmath425 near @xmath120 is then given by @xmath426 correspondingly , the hamilton vector field is @xmath427 to study the @xmath93-flow in the radially compactified b - cotangent bundle near @xmath125 , we introduce rescaled coordinates @xmath428 we then compute the rescaled hamilton vector field in @xmath429 to be @xmath430 writing @xmath431 in a local coordinate chart on @xmath394 , we have @xmath432 . thus , @xmath433 at @xmath434 . in particular , @xmath435 have opposite signs ( by definition of @xmath294 ) , and the quantity which will control regularity and decay thresholds at the radial set @xmath120 is the quotient @xmath436 see definition [ defrndsorderfunctions ] and the proof of proposition [ proprndsglobalreg ] for their role . we remark that the reciprocal @xmath437 is equal to the _ surface gravity _ of the horizon at @xmath112 , see e.g. @xcite . we proceed to verify that @xmath438 is a source / sink for the @xmath105-flow within @xmath124 by constructing a quadratic defining function @xmath439 of @xmath125 within @xmath440 for which @xmath441 modulo terms which vanish cubically at @xmath120 ; note that @xmath442 has the same relative sign . now , @xmath125 is defined within @xmath443 by the vanishing of @xmath444 and @xmath445 , and we have @xmath446 , likewise for @xmath445 ; therefore @xmath447 satisfies . ( one can in fact easily diagonalize the linearization of @xmath105 at its critical set @xmath125 by observing that @xmath448 modulo quadratically vanishing terms . ) further studying the flow at @xmath112 , we note that @xmath273 is null there , and writing @xmath449 a covector @xmath450 is in the orthocomplement of @xmath273 if and only if @xmath451 ( using the form of the metric ) , which then implies @xmath452 in view of @xmath453 . since @xmath454 , we deduce that @xmath455 at @xmath456 , where we let @xmath457 we note that this set is invariant under the hamilton flow . more precisely , we have @xmath458 , so for @xmath264 , i.e. at @xmath8 , @xmath273 is in the same causal cone as @xmath459 , hence in the future null cone ; thus , letting @xmath460 and taking @xmath461 , we find that @xmath462 lies in the same causal cone as @xmath273 , but @xmath462 is not orthogonal to @xmath273 , hence we obtain @xmath463 ; more generally , @xmath464 it follows that forward null - bicharacteristics in @xmath110 can only cross @xmath8 in the inward direction ( @xmath5 decreasing ) , while those in @xmath109 can only cross in the outward direction ( @xmath5 increasing ) . at @xmath138 , there is a sign switch both in the definition of @xmath136 ( because there @xmath459 is _ past _ timelike ) and in @xmath465 , so the same statement holds there . at @xmath7 , there is a single sign switch in the calculation because of @xmath466 , and at @xmath6 there is a single sign switch because of the definition of @xmath136 there , so forward null - bicharacteristics in @xmath110 can only cross @xmath6 or @xmath7 in the inward direction ( @xmath5 decreasing ) , and forward bicharacteristics in @xmath109 only in the outward direction ( @xmath5 increasing ) . next , we locate the radial sets @xmath120 within the two components of the characteristic set , i.e. determining the components @xmath467 of the radial sets . the calculations verifying the initial / final character of the artificial hypersurfaces appearing in the arguments of the previous section show that @xmath468 at @xmath165 and @xmath186 , while @xmath469 at @xmath360 and @xmath185 , so since @xmath109 , resp . @xmath110 , is the union of the future , resp . past , null cones , we have @xmath470 in view of and taking into account that @xmath471 differs from @xmath75 by an @xmath5-dependent factor , while @xmath472 at @xmath120 , we thus have @xmath473 we connect this with figure [ figrndsextfull ] : namely , if we let @xmath460 , then @xmath474 is the unstable manifold at @xmath475 for @xmath352 and the stable manifold at @xmath475 for @xmath476 , and the other way around for @xmath477 . in view of , @xmath475 is a sink for the @xmath105 flow within @xmath478 for @xmath352 , while it is a source for @xmath476 , with sink / source switched for the ` @xmath479 ' sign . see figure [ figrndsradial ] . of the characteristic set , and the behavior of two null - geodesics . the arrows on the horizons are future timelike . in @xmath110 , all arrows are reversed . ] we next shift our attention to the two domains of outer communications , @xmath480 in @xmath398 and @xmath21 in @xmath400 , where we study the behavior of the radius function along the flow using the form of the metric : thus , at a point @xmath481 , we have @xmath482 , so @xmath472 necessitates @xmath483 , hence @xmath484 , and thus we get @xmath485 now for @xmath159 , @xmath486 vanishes at the radius @xmath487 of the _ photon sphere _ , and @xmath488 for @xmath489 ; likewise , for @xmath160 , by construction we have @xmath490 only at @xmath491 , and @xmath492 for @xmath493 . therefore , if @xmath472 , then @xmath494 unless @xmath495 , in which case @xmath462 lies in the _ trapped set _ @xmath496 restricting to bicharacteristics within @xmath497 ( which is invariant under the @xmath93-flow since @xmath498 there ) and defining @xmath499 we can conclude that all critical points of @xmath500 along null - geodesics in @xmath501 ( or @xmath346 ) are strict local minima : indeed , if @xmath502 at @xmath462 , then _ either _ @xmath495 , in which case @xmath503 unless @xmath472 , hence @xmath504 , _ or _ @xmath472 , in which case @xmath505 unless @xmath495 , hence again @xmath504 . as in @xcite , this implies that within @xmath79 , forward null - bicharacteristics in @xmath501 ( resp . @xmath346 ) either tend to @xmath506 ( resp . @xmath507 ) , or they reach @xmath7 or @xmath8 ( resp . @xmath138 or @xmath6 ) in finite time , while backward null - bicharacteristics either tend to @xmath508 ( resp . @xmath509 ) , or they reach @xmath7 or @xmath8 ( resp . @xmath138 or @xmath6 ) in finite time . ( for this argument , we make use of the source / sink dynamics at @xmath510 . ) further , they can not tend to @xmath511 , resp . @xmath512 , in both the forward and backward direction _ while remaining in @xmath346 , resp . @xmath501 , _ unless they are trapped , i.e. contained in @xmath511 , resp . @xmath512 , since otherwise @xmath513 would attain a local maximum along them . lastly , bicharacteristics reaching a horizon @xmath112 in finite time in fact cross the horizon by our earlier observation . the trapping at @xmath514 is in fact _ @xmath5-normally hyperbolic for every @xmath5 _ @xcite . next , in @xmath515 , we recall that @xmath273 is future , resp . past , timelike in @xmath516 and @xmath517 , resp . @xmath518 ; therefore , if @xmath453 lies in one of these three regions , @xmath454 implies @xmath519 ( this is consistent with and the paragraph following it . ) in order to describe the global structure of the null - bicharacteristic flow , we define the connected components of the trapped set in the exterior domain of the spacetime , @xmath520 then @xmath521 have stable / unstable manifolds @xmath522 , with the convention that @xmath523 , while @xmath524 is transversal to @xmath478 . concretely , @xmath525 is the union of forward trapped bicharacteristics , i.e. bicharacteristics which tend to @xmath526 in the forward direction , while @xmath527 is the union of backward trapped bicharacteristics , tending to @xmath526 in the backward direction ; further @xmath528 is the union of backward trapped bicharacteristics , and @xmath529 the union of forward trapped bicharacteristics , tending to @xmath530 . see figure [ figrndsflow ] . of the characteristic set and in the region @xmath531 of the reissner de sitter spacetime . the picture for @xmath110 is analogous , with the direction of the arrows reversed , and @xmath532 replaced by @xmath533 . ] the structure of the flow in the neighborhood @xmath398 of the artificial exterior region is the same as that in the neighborhood @xmath400 of the exterior domain , except the time orientation and thus the two components of the characteristic set are reversed . write @xmath534 a denote by @xmath535 the forward and backward trapped sets , with the same sign convention as for @xmath522 above . we note that backward , resp . forward , trapped null - bicharacteristics in @xmath536 , resp . @xmath537 , may be forward , resp . backward , trapped in the artificial exterior region , i.e. they may lie in @xmath538 , resp . @xmath539 , but this is the only additional trapping present in our setup . to state this succinctly , we write @xmath540 then : [ proprndsflow ] the null - bicharacteristic flow in @xmath541 has the following properties : 1 . [ itrndsflowbdy ] let @xmath542 be a null - bicharacteristic at infinity , @xmath543 , where @xmath544 . then in the backward direction , @xmath542 either crosses @xmath139 in finite time or tends to @xmath545 , while in the forward direction , @xmath542 either crosses @xmath217 in finite time or tends to @xmath546 . the curve @xmath542 can tend to @xmath526 in at most one direction , and likewise for @xmath547 . [ itrndsflowint ] let @xmath542 be a null - bicharacteristic in @xmath548 . then in the backward direction , @xmath542 either crosses @xmath549 in finite time or tends to @xmath550 , while in the forward direction , @xmath542 either crosses @xmath551 in finite time or tends to @xmath552 . [ itrndsflowhyp ] in both cases , in the region where @xmath518 , @xmath553 is strictly decreasing , resp . increasing , in the forward , resp . backward , direction in @xmath109 , while in the regions where @xmath516 or @xmath517 , @xmath553 is strictly increasing , resp . decreasing , in the forward , resp . backward , direction in @xmath109 . [ itrndsflowradialtrapped ] @xmath510 , @xmath554 as well as @xmath521 and @xmath555 are invariant under the flow . for null - bicharacteristics in @xmath110 , the analogous statements hold with ` backward ' and ` forward ' reversed and ` @xmath479 ' and ` @xmath556 ' switched . here @xmath139 etc . is a shorthand notation for @xmath557 . statement ( [ itrndsflowhyp ] ) follows from , and ( [ itrndsflowradialtrapped ] ) holds by the definition of the radial and trapped sets . to prove the ` backward ' part of ( [ itrndsflowbdy ] ) , note that if @xmath516 on @xmath542 , then @xmath542 crosses @xmath139 by ; if @xmath138 on @xmath542 , then @xmath542 crosses into @xmath516 since @xmath558 . if @xmath542 remains in @xmath559 in the backward direction , it either tends to @xmath547 , or it crosses @xmath6 since it can not tend to @xmath560 because of the sink nature of this set . once @xmath542 crosses into @xmath32 , it must tend to @xmath7 by ( [ itrndsflowhyp ] ) and hence either tend to the source @xmath561 or cross into @xmath562 . in @xmath562 , @xmath542 must tend to @xmath552 , as it can not cross @xmath7 or @xmath8 into @xmath563 or @xmath517 in the backward direction . the analogous statement for @xmath110 , now in the forward direction , is immediate , since reflecting @xmath542 pointwise across the origin in the b - cotangent bundle but keeping the affine parameter the same gives a bijection between backward bicharacteristics in @xmath109 and forward bicharacteristics in @xmath110 . the ` forward ' part of ( [ itrndsflowbdy ] ) is completely analogous . it remains to prove ( [ itrndsflowint ] ) . note that @xmath564 at @xmath565 ; thus in @xmath327 , where @xmath566 is future timelike , @xmath75 is strictly decreasing in the backward direction along bicharacteristics @xmath567 , hence the arguments for part ( [ itrndsflowbdy ] ) show that @xmath542 crosses @xmath139 , or tends to @xmath550 if it lies in @xmath568 ; otherwise it crosses into @xmath32 in the backward direction . in the latter case , recall that in @xmath380 , @xmath553 is monotonically increasing in the backward direction ; we claim that @xmath542 can not cross @xmath277 : with the defining function @xmath569 of @xmath277 , we arranged for @xmath570 to be past timelike , so @xmath571 for @xmath572 , i.e. @xmath127 is increasing in the backward direction along the @xmath93-integral curve @xmath542 near @xmath277 , which proves our claim . this now implies that @xmath542 enters @xmath268 in the backward direction , from which point on @xmath75 is strictly increasing , hence @xmath542 either crosses @xmath14 in @xmath573 , or it crosses into @xmath562 . in the latter case , it in fact crosses @xmath14 by the arguments proving ( [ itrndsflowbdy ] ) . the ` forward ' part is proved in a similar fashion . forward solutions to the wave equation @xmath126 in the domain of dependence of @xmath14 , i.e. in @xmath574 , are not affected by any modifications of the operator @xmath54 outside , i.e. in @xmath145 . as indicated in [ secintro ] , we are therefore free to place complex absorbing operators at @xmath512 and @xmath575 which obviate the need for delicate estimates at normally hyperbolic trapping ( see the proof of proposition [ proprndsglobalreg ] ) and for a treatment of regularity issues at the artificial horizon ( related to @xmath576 in , see also definition [ defrndsorderfunctions ] ) . concretely , let @xmath577 be a small neighborhood of @xmath578 , with @xmath579 the projection , so that @xmath580 in the notation of proposition [ proprndsmfd ] ; thus , @xmath577 stays away from @xmath581 . choose @xmath152 with schwartz kernel supported in @xmath582 and real principal principal symbol satisfying @xmath583 with the inequality strict at @xmath584 , thus @xmath133 is elliptic at @xmath585 . we then study the operator @xmath586 the convention for the sign of @xmath54 is such that @xmath587 . we will use weighted , variable order b - sobolev spaces , with weight @xmath588 and the order given by a function @xmath589 ; in fact , the regularity will vary only in the base , not in the fibers of the b - cotangent bundle . we refer the reader to ( * ? ? ? * appendix a ) and appendix [ secvariable ] for details on variable order spaces . we define the function space @xmath590 as the space of restrictions to @xmath150 of elements of @xmath591 which are supported in the causal future of @xmath592 ; thus , distributions in @xmath593 are supported distributions at @xmath592 and extendible distributions at @xmath551 ( and at @xmath76 ) , see ( * ? ? ? * appendix b ) ; in fact , on manifolds with corners , there are some subtleties concerning such mixed supported / extendible spaces and their duals , which we discuss in appendix [ secsuppext ] . the supported character at the initial surfaces , encoding vanishing cauchy data , is the reason for the subscript ` fw ' ( ` forward ' ) . the norm on @xmath593 is the quotient norm induced by the restriction map , which takes elements of @xmath594 with the stated support property to their restriction to @xmath150 . dually , we also consider the space @xmath595 consisting of restrictions to @xmath150 of distributions in @xmath594 which are supported in the causal past of @xmath551 . concretely , for the analysis of @xmath154 , we will work on slightly growing function spaces , i.e. allowing exponential growth of solutions in @xmath368 ; we will obtain precise asymptotics ( in particular , boundedness ) in the next section . thus , let us fix a weight @xmath596 the sobolev regularity is dictated by the radial sets @xmath597 and @xmath598 , as captured by the following definition : [ defrndsorderfunctions ] let @xmath588 . then a smooth function @xmath599 is called a _ forward order function for the weight @xmath51 _ if @xmath600 with @xmath576 defined in ; here @xmath601 is any small number . the function @xmath602 is called a _ backward order function for the weight @xmath51 _ if @xmath603 backward order functions will be used for the analysis of the dual problem . [ rmkrndsbeta1computation ] if @xmath604 ( and @xmath162 still ) , a forward order function @xmath602 can be taken constant , and thus one can work on fixed order sobolev spaces in proposition [ proprndsglobalreg ] below . this is the case for small charges @xmath174 : indeed , a straightforward computation in the variable @xmath605 using lemma [ lemmarndsnondegenerate ] shows that @xmath606 note that @xmath602 is a forward order function for the weight @xmath51 if and only if @xmath607 is a backward order function for the weight @xmath608 . the lower , resp . upper , bounds on the order functions at the radial sets are forced by the propagation estimate ( * ? ? ? * proposition 2.1 ) which will we use at the radial sets : one can propagate high regularity from @xmath609 into the radial set and into the boundary ( ` red - shift effect ' ) , while there is an upper limit on the regularity one can propagate out of the radial set and the boundary into the interior @xmath609 of the spacetime ( ` blue - shift effect ' ) ; the definition of order functions here reflects the precise relationship of the a priori decay or growth rate @xmath51 and the regularity @xmath602 ( i.e. the ` strength ' of the red- or blue - shift effect depending on a priori decay or growth along the horizon ) . we recall the radial point propagation result in a qualitative form ( the quantitative version of this , yielding estimates , follows from the proof of this result , or can be recovered from the qualitative statement using the closed graph theorem ) : [ proprndsradialrecall ] ( * ? ? ? * proposition 2.1 ) . suppose @xmath154 is as above , and let @xmath588 . let @xmath113 . if @xmath610 , @xmath611 , and if @xmath612 then @xmath510 ( and thus a neighborhood of @xmath510 ) is disjoint from @xmath613 provided @xmath614 , @xmath615 , and in a neighborhood of @xmath510 , @xmath616 is disjoint from @xmath613 . on the other hand , if @xmath617 , and if @xmath612 then @xmath510 ( and thus a neighborhood of @xmath510 ) is disjoint from @xmath613 provided @xmath614 and a punctured neighborhood of @xmath510 , with @xmath510 removed , in @xmath440 is disjoint from @xmath613 . we then have : [ proprndsglobalreg ] suppose @xmath162 and @xmath602 is a forward order function for the weight @xmath51 ; let @xmath618 be a forward order function for the weight @xmath51 with @xmath619 . then @xmath620 we also have the dual estimate @xmath621 for backward order functions @xmath622 and @xmath623 for the weight @xmath608 with @xmath624 . both estimates hold in the sense that if the quantities on the right hand side are finite , then so is the left hand side , and the inequality is valid . the arguments are very similar to the ones used in @xcite . the proof relies on standard energy estimates near the artificial hypersurfaces , various microlocal propagation estimates , and crucially relies on the description of the null - bicharacteristic flow given in proposition [ proprndsflow ] . let @xmath625 be such that @xmath626 . first of all , we can extend @xmath127 to @xmath627 , with @xmath628 supported in @xmath629 , @xmath630 still , and @xmath631 . near @xmath14 , we can then use the unique solvability of the forward problem for the wave equation @xmath632 to obtain an estimate for @xmath12 there : indeed , using an approximation argument , approximating @xmath628 by smooth functions @xmath633 , and using the propagation of singularities , propagating @xmath634-regularity from @xmath635 ( where the forward solution @xmath636 of @xmath637 vanishes ) , which can be done on this regularity scale uniformly in @xmath314 , we obtain an estimate @xmath638 since @xmath12 agrees with @xmath639 in the domain of dependence of @xmath14 . the same argument shows that we can control the @xmath640-norm of @xmath12 in a neighborhood of @xmath139 , say in @xmath641 , in terms of @xmath642 . then , in @xmath643 , we use the propagation of singularities ( forwards in @xmath109 , backwards in @xmath110 ) to obtain local @xmath634-regularity away from the boundary at infinity , @xmath73 . at the radial sets @xmath644 and @xmath598 , the radial point estimate , proposition [ proprndsradialrecall ] , allows us , using the a priori @xmath645-regularity of @xmath12 , to propagate @xmath646-regularity into @xmath647 ; propagation within @xmath648 then shows that we have @xmath646-control on @xmath12 on @xmath649 . since @xmath162 , we can then use ( * ? ? ? * theorem 3.2 ) to control @xmath12 in @xmath646 microlocally at @xmath511 and propagate this control along @xmath650 . near @xmath217 , the microlocal propagation of singularities only gives local control away from @xmath217 , but we can get uniform regularity up to @xmath217 by standard energy estimates , using a cutoff near @xmath217 and the propagation of singularities for an extended problem ( solving the forward wave equation with forcing @xmath628 , cut off near @xmath217 , plus an error term coming from the cutoff ) , see ( * ? ? ? * proposition 2.13 ) and the similar discussion around below in the present proof . we thus obtain an estimate for the @xmath646-norm of @xmath12 in @xmath268 . next , we propagate regularity in @xmath380 , using part ( [ itrndsflowhyp ] ) of proposition [ proprndsflow ] and our assumption @xmath651 ; the only technical issue is now at @xmath277 , where the microlocal propagation only gives local regularity away from @xmath277 ; this will be resolved shortly . focusing on the remaining region @xmath652 , we start with the control on @xmath12 near @xmath139 , which we propagate forwards in @xmath109 and backwards in @xmath110 , either up to @xmath333 or into the complex absorption hiding @xmath585 ; see @xcite for the propagation of singularities with complex absorption . moreover , at the elliptic set of the complex absorbing operator @xmath133 , we get @xmath653-control on @xmath12 , and we can propagate @xmath646-estimates from there . the result is that we get @xmath646-estimates of @xmath12 in a punctured neighborhood of @xmath166 within @xmath648 ; thus , the low regularity part of proposition [ proprndsradialrecall ] applies . we can then propagate regularity from a neighborhood @xmath166 along @xmath654 . this gives us local regularity away from @xmath655 , where the microlocal propagation results do not directly give uniform estimates . in order to obtain uniform regularity up to @xmath655 , we use the aforementioned cutoff argument for an extended problem near @xmath655 : choose @xmath656 such that @xmath657 for @xmath658 , @xmath659 , and such that @xmath660 if @xmath641 or @xmath240 or @xmath661 ; see figure [ figrndscutoff ] for an illustration . in particular , @xmath662=0 $ ] by the support properties of @xmath133 . therefore , we have @xmath663u,\quad u':=\chi u;\ ] ] note that we have ( uniform ) @xmath664-control on @xmath665u$ ] by the support properties of @xmath666 . extend @xmath667 beyond @xmath655 to @xmath668 with support in @xmath669 so that the global norm of @xmath670 is bounded by a fixed constant times the quotient norm of @xmath667 . the solution of the equation @xmath671 with support of @xmath672 in @xmath673 is unique ( it is simply the forward solution , taking into account the time orientation in the artificial exterior region ) ; but then the local regularity estimates for @xmath672 for the extended problem , which follow from the propagation of singularities ( using the approximation argument sketched above ) , give by restriction uniform regularity of @xmath12 up to @xmath655 . : the cutoff @xmath674 is supported in and below the shaded region ; the shaded region itself , containing @xmath675 , is where we have already established @xmath676-bounds for @xmath12 . ] putting all these estimates together , we obtain an estimate for @xmath677 in terms of @xmath678 . the proof of the dual estimate is completely analogous : we now obtain initial regularity ( that we can then propagate as above ) by solving the backward problem for @xmath54 near @xmath655 and @xmath217 . the estimates in proposition [ proprndsglobalreg ] do not yet yield the fredholm property of @xmath154 . as explained in @xcite , we therefore study the _ mellin - transformed normal operator family _ @xmath156 , see @xcite , which in the present ( dilation - invariant in @xmath75 , or translation - invariant in @xmath368 ) setting is simply obtained by conjugating @xmath154 by the mellin transform in @xmath75 , or equivalently the fourier transform in @xmath679 , i.e. @xmath680 , acting on functions on the boundary at infinity @xmath681 . concretely , we need to show that @xmath156 is invertible between suitable function spaces on @xmath682 for a weight @xmath162 , since this will allow us to improve the @xmath683 error term in by a space with an improved weight , so @xmath593 injects compactly into it ; an analogous procedure for the dual problem gives the full fredholm property for @xmath154 ; see @xcite and below for details . for any finite value of @xmath61 , we can analyze the operator @xmath684 , @xmath685 , using standard microlocal analysis ( and energy estimates near @xmath686 and @xmath687 ) . the natural function spaces are variable order sobolev spaces @xmath688 which we define to be the restrictions to @xmath544 of elements of @xmath689 with support in @xmath629 , and dually on @xmath690 , the restrictions to @xmath416 of elements of @xmath689 with support in @xmath691 , obtaining fredholm mapping properties between suitable function spaces . however , in order to obtain useful estimates for our global b - problem , we need uniform estimates for @xmath156 as @xmath157 in strips of bounded @xmath64 , on function spaces which are related to the variable order b - sobolev spaces on which we analyze @xmath154 . thus , let @xmath692 , @xmath693 , and consider the semiclassical rescaling @xcite @xmath694 we refer to @xcite for details on the relationship between the b - operator @xmath154 and its semiclassical rescaling ; in particular , we recall that the hamilton vector field of the semiclassical principal symbol of @xmath695 for @xmath696 is naturally identified with the hamilton vector field of the b - principal symbol of @xmath154 restricted to @xmath697 , where we use the coordinates in the b - cotangent bundle . for any sobolev order function @xmath698 and a weight @xmath588 , the mellin transform in @xmath75 gives an isomorphism @xmath699 where @xmath700 ( for @xmath701 ) is a semiclassical variable order sobolev space with a non - constant weighting in @xmath702 ; see appendix [ secvariable ] for definitions and properties of such spaces . the analysis of @xmath695 , @xmath703 , acting on @xmath704-type spaces is now straightforward , given the properties of the hamilton flow of @xmath154 . indeed , in view of the supported / extendible nature of the b - spaces @xmath705 and @xmath706 into account , we are led to define the corresponding semiclassical space @xmath707 to be the space of restrictions to @xmath416 of elements of @xmath704 with support in @xmath629 , resp . then , in the region where @xmath602 is not constant ( recall that this is a subset of @xmath709 ) , @xmath695 is a ( semiclassical ) real principal type operator , as follows from , and hence the only microlocal estimates we need there are elliptic regularity and the real principal type propagation for variable order semiclassical sobolev spaces ; these estimates are proved in propositions [ propvariablesclelliptic ] and [ propvariablesclpropagation ] . the more delicate estimates take place in standard semiclassical function spaces ; these are the radial point estimates near @xmath112 , in the present context proved in @xcite , and the semiclassical estimates of wunsch zworski @xcite and dyatlov @xcite ( microlocalized in @xcite ) at the normally hyperbolic trapping . near the artificial hypersurfaces @xmath710 , intersected with @xmath76 , the operator @xmath695 is a ( semiclassical ) wave operator , and we use standard energy estimates there similar to the proof of proposition [ proprndsglobalreg ] , but keeping track of powers of @xmath702 ; see @xcite for details . we thus obtain : [ proprndssemiclassical ] let @xmath711 . then for @xmath712 and @xmath713 $ ] , we have the estimate @xmath714 with a uniform constant @xmath33 ; here @xmath602 and @xmath619 are forward order functions for all weights in @xmath715 $ ] , see definition [ defrndsorderfunctions ] . for the dual problem , we similarly have @xmath716 where @xmath602 and @xmath619 are backward order functions for all weights in @xmath717 $ ] . notice here that if @xmath602 were constant , the estimate would read @xmath718 , which is the usual hyperbolic loss of one derivative and one power of @xmath702 . the estimate is conceptually the same , but in addition takes care of the variable orders . trapping causes no additional losses here , since @xmath719 . [ rmkrndssemiclassicaldual ] we have @xmath720 , and @xmath721 ; the change of sign in @xmath722 when going from to the dual estimate is analogous to the change of sign in the weight @xmath51 in proposition [ proprndsglobalreg ] . for future reference , we note that we still have high energy estimates for @xmath723 in strips including and extending _ below _ the real line : the only delicate part is the estimate at the normally hyperbolic trapping , more precisely at the semiclassical trapped set @xmath724 , which can be naturally identified with the intersection of the trapped set @xmath511 with @xmath725 for @xmath696 . thus , let @xmath726 be the minimal expansion rate at the semiclassical trapped set in the normal direction as in @xcite or @xcite ; let us then write @xmath727 for some real number @xmath58 ( in the kerr de sitter case discussed later , @xmath728 is a smooth function on @xmath724 ) ; see [ subsecrndshighreg ] , in particular and , for the ingredients for the calculation of @xmath728 in a limiting case . therefore , if @xmath729 , then @xmath730 . the reason for the ` @xmath731 ' appearing here is the following : for the ` @xmath556 ' case , note that for @xmath732 , corresponding to semiclassical analysis in @xmath733 , which near the trapped set @xmath511 intersects the _ forward _ light cone @xmath109 non - trivially , we propagate regularity _ forwards along the hamilton flow _ , while in the ` @xmath479 ' case , corresponding to propagation in the _ backward _ light cone @xmath110 , we propagate _ backwards along the flow_. using @xcite , see also the discussion in @xcite , we conclude : [ proprndssemiclassical2 ] using the above notation , the ( uniform ) estimates and hold with @xmath734 replaced by @xmath602 on the right hand sides , provided @xmath735 $ ] , where @xmath736 . the effect of replacing @xmath734 by @xmath602 is that this adds an additional @xmath737 to the right hand side , i.e. we get a weaker estimate ( which in the presence of trapping can not be avoided by @xcite ) ; the strengthening of the norm in the regularity sense is unnecessary , but does not affect our arguments later . we return to the case @xmath719 . if we define the space @xmath738 , then the estimates in proposition [ proprndssemiclassical ] imply that the map @xmath739 is fredholm for @xmath740 $ ] , with high energy estimates as @xmath157 . moreover , for small @xmath712 , the error terms on the right hand sides of and can be absorbed into the left hand sides , hence in this case we obtain the invertibility of the map . this implies that @xmath741 is invertible for @xmath740 $ ] , @xmath742 . since therefore there are only finitely many resonances ( poles of @xmath743 ) in @xmath744 for any @xmath711 , we may therefore pick a weight @xmath162 such that there are no resonances on the line @xmath682 , which in view of implies the estimate @xmath745 where @xmath746 is the manifold on which the dilation - invariant operator @xmath747 naturally lives ; here @xmath602 is a forward order function for the weight @xmath51 , and the subscript ` fw ' on the b - sobolev spaces denotes distributions with supported character at @xmath748 and extendible at @xmath749 . we point out that the choice @xmath75 of boundary defining function and the choice of @xmath75-dilation orbits fixes an isomorphism of a collar neighborhood of @xmath416 in @xmath150 with a neighborhood of @xmath750 in @xmath751 , and the two @xmath646-norms on functions supported in this neighborhood , given by the restriction of the @xmath752-norm and the restriction of the @xmath753-norm , respectively , are equivalent . equipped with , we can now improve proposition [ proprndsglobalreg ] to obtain the fredholm property of @xmath154 : first , we let @xmath602 be a forward order function for the weight @xmath51 , but with the more stringent requirement @xmath754 and we require that the forward order function @xmath755 satisfies @xmath756 . using with @xmath602 replaced by @xmath755 , and a cutoff @xmath656 , identically @xmath94 near @xmath416 and supported in a small collar neighborhood of @xmath416 , the estimate then implies ( as in @xcite ) @xmath757u\|_{{h_{{{\mathrm{b}}},{{\mathrm{fw}}}}}^{{\mathsf{s}}_0 - 1,\alpha } } \\ & \qquad\qquad + \|\chi({{\mathcal}p}-n({{\mathcal}p } ) ) u\|_{{h_{{{\mathrm{b}}},{{\mathrm{fw}}}}}^{{\mathsf{s}}_0 - 1,\alpha}}.\end{aligned}\ ] ] noting that @xmath758\in\tau{{\mathrm{diff}}_{{\mathrm{b}}}}^1 $ ] , the second to last term can be estimated by @xmath759 , while the last term can be estimated by @xmath760 ; thus , we obtain @xmath761 where the inclusion @xmath762 is now compact . this estimate implies that @xmath763 is finite - dimensional and @xmath764 is closed . the dual estimate is @xmath765 where now @xmath766 is a backward order function for the weight @xmath608 , and the backward order function @xmath622 satisfies the more stringent bound @xmath767 note that @xmath158 not having a pole on the line @xmath682 is equivalent to @xmath768 not having a pole on the line @xmath769 , since @xmath770 . we wish to take @xmath771 with @xmath602 as in the estimate ; so if we require in addition to that @xmath772 the estimates and for @xmath771 imply by a standard functional analytic argument , see e.g. ( * ? ? ? * proof of theorem 26.1.7 ) , that @xmath773 is fredholm , where @xmath774 and the range of @xmath154 is the annihilator of the kernel of @xmath775 acting on @xmath776 . we can strengthen the regularity at the cauchy horizon by dropping , cf . @xcite : [ thmrndsfredholm ] suppose @xmath162 is such that @xmath154 has no resonances on the line @xmath682 . let @xmath602 be a forward order function for the weight @xmath51 , and assume holds . then the map @xmath154 , defined in , is fredholm as a map , with range equal to the annihilator of @xmath777 . let @xmath778 be an order function satisfying both and , so by the above discussion , @xmath779 is fredholm . since @xmath780 , we a forteriori get the finite - dimensionality of @xmath781 . on the other hand , if @xmath782 annihilates @xmath777 , it also annihilates @xmath783 , hence we can find @xmath784 solving @xmath785 . the propagation of singularities , proposition [ proprndsglobalreg ] , implies @xmath677 , and the proof is complete . to obtain a better result , we need to study the structure of resonances . notice that for the purpose of dealing with a single resonance , one can simplify notation by working with the space @xmath786 , see , rather than @xmath787 , since the semiclassical ( high energy ) parameter is irrelevant then . [ lemmarndsresonancesupp ] 1 . [ itrndsresonancesupp ] every resonant state @xmath788 corresponding to a resonance @xmath61 with @xmath789 is supported in the artificial exterior region @xmath790 ; more precisely , every element in the range of the singular part of the laurent series expansion of @xmath158 at such a resonance @xmath61 is supported in @xmath790 . in fact , this holds more generally for any @xmath56 which is not a resonance of the forward problem for the wave equation in a neighborhood @xmath400 of the black hole exterior . [ itrndsresonancesupp0res ] if @xmath791 denotes the restriction of distributions on @xmath416 to @xmath32 , then the only pole of @xmath792 with @xmath793 is at @xmath57 , has rank @xmath94 , and the space of resonant states consists of constant functions . since @xmath12 has supported character at @xmath794 , we obtain @xmath795 in @xmath516 , since @xmath12 solves the _ wave _ equation @xmath796 there . on the other hand , the forward problem for the wave equation in the neighborhood @xmath400 of the black hole exterior does not have any resonances with positive imaginary part ; this is well - known for schwarzschild de sitter spacetimes @xcite and for slowly rotating kerr de sitter spacetimes , either by direct computation @xcite or by a perturbation argument @xcite . for the convenience of the reader , we recall the argument for the schwarzschild de sitter case , which applies without change in the present setting as well : a simple integration by parts argument , see e.g. @xcite or @xcite , shows that @xmath12 must vanish in @xmath21 . now the propagation of singularities at radial points implies that @xmath12 is smooth at @xmath7 and @xmath8 ( where the a priori regularity exceeds the threshold value ) , and hence in @xmath517 , @xmath12 is a solution to the homogeneous wave equation on an asymptotically de sitter space which decays rapidly at the conformal boundary ( which is @xmath8 ) , hence must vanish identically in @xmath517 ( see ( * ? ? ? * footnote 58 ) for details ) ; the same argument applies in @xmath380 , yielding @xmath795 there . therefore , @xmath797 , as claimed . an iterative argument , similar to ( * ? ? ? * proof of lemma 8.3 ) , yields the more precise result . the more general statement follows along the same lines ( and is in fact much easier to prove , since it does not entail a mode stability statement ) : suppose @xmath61 is not a resonance of the forward wave equation on @xmath400 , then a resonant state @xmath798 must vanish in @xmath400 , and we obtain @xmath797 as before ; likewise for the more precise result . this proves ( [ itrndsresonancesupp ] ) . for the proof of ( [ itrndsresonancesupp0res ] ) , it remains to study the resonance at @xmath97 , since the only @xmath400 resonance in the closed upper half plane is @xmath97 . note that an element in the range of the most singular laurent coefficient of @xmath792 at @xmath57 lies in @xmath799 ; but elements in @xmath799 which vanish near @xmath6 vanish identically in @xmath32 and hence are annihilated by @xmath791 , while elements which are not identically @xmath97 near @xmath6 are not identically @xmath97 in @xmath562 as well ; but the only non - trivial elements of @xmath799 ( which are smooth at @xmath185 and @xmath186 ) are constant in @xmath21 , and since @xmath800 in @xmath32 , we deduce ( by unique continuation ) that @xmath801 indeed consists of constant functions . but then the order of the pole of @xmath792 at @xmath57 equals the order of the @xmath97-resonance of the forward problem for @xmath54 in @xmath400 , which is known to be equal to @xmath94 , see the references above . the @xmath94-dimensionality of @xmath801 then implies that the rank of the pole of @xmath802 at @xmath97 indeed equals @xmath94 . since we are dealing with an extended global problem here , involving ( pseudodifferential ! ) complex absorption , solvability is not automatic , but it holds in the region of interest @xmath32 ; to show this , we first need : [ lemmarndssolvability ] recall the definition of the set @xmath803 , where the complex absorption is placed , from . under the assumptions of theorem [ thmrndsfredholm ] ( in particular , @xmath162 ) , there exists a linear map @xmath804 such that for all @xmath805 , the function @xmath806 lies in the range of the map @xmath154 in . by theorem [ thmrndsfredholm ] , the statement of the lemma is equivalent to @xmath807 let @xmath808 , @xmath809 . we claim that @xmath810 implies @xmath811 ; in other words , elements of @xmath812 are uniquely determined by their restriction to @xmath813 . to see this , note that @xmath814 on @xmath813 implies that in fact @xmath13 solves the homogeneous wave equation @xmath815 . thus , we conclude by the supported character of @xmath13 at @xmath655 and @xmath217 that @xmath13 in fact vanishes in @xmath563 and @xmath517 , so @xmath816 . using the high energy estimates , a contour shifting argument , see ( * ? ? ? * lemma 3.5 ) , and the fact that resonances of @xmath154 with @xmath789 have support disjoint from @xmath817 by lemma [ lemmarndsresonancesupp ] ( [ itrndsresonancesupp ] ) , we conclude that in fact @xmath818 , i.e. @xmath13 vanishes to infinite order at future infinity ; but then , radial point estimates and the simple version of propagation of singularities at the normally hyperbolic trapping ( since we are considering the _ backwards _ problem on _ decaying _ spaces ) see ( * ? ? ? * theorem 3.2 , estimate ( 3.10 ) ) imply that in fact @xmath819 . now the energy estimate in ( * ? ? ? * lemma 2.15 ) applies to @xmath13 and yields @xmath820 for @xmath821 , hence @xmath814 as claimed . therefore , if @xmath822 forms a basis of @xmath812 , then the restrictions @xmath823 are linearly independent elements of @xmath824 , and hence one can find @xmath825 with @xmath826 . the map @xmath827 then satisfies all requirements . we can then conclude : under the assumptions of theorem [ thmrndsfredholm ] , all elements in the kernel of @xmath154 in are supported in the artificial exterior domain @xmath790 . moreover , for all @xmath805 with support in @xmath32 , there exists @xmath828 such that @xmath137 in @xmath32 . if @xmath677 lies in @xmath763 , then the supported character of @xmath12 at @xmath592 together with uniqueness for the wave equation in @xmath516 and @xmath32 implies that @xmath12 vanishes identically there , giving the first statement . for the second statement , we use lemma [ lemmarndssolvability ] and solve the equation @xmath829 , which gives the desired @xmath12 . in particular , solutions of the equation @xmath137 exist and are unique in @xmath32 , which we of course already knew from standard hyperbolic theory in the region on ` our ' side @xmath32 of the cauchy horizon ; the point is that we now understand the regularity of @xmath12 _ up to _ the cauchy horizon . we can refine this result substantially for better - behaved forcing terms , e.g. for @xmath830 with support in @xmath32 ; we will discuss this in the next two sections . the only resonance of the forward problem in @xmath400 in @xmath793 is a simple resonance at @xmath57 , with resonant states equal to constants , see the references given in the proof of lemma [ lemmarndsresonancesupp ] , and there exists @xmath26 such that @xmath97 is the only resonance in @xmath831 . ( this does _ not _ mean that the global problem for @xmath154 does not have other resonances in this half space ! ) in the notation of proposition [ proprndssemiclassical2 ] , we may assume @xmath832 so that we have high energy estimates in @xmath831 . [ proprndspartialasymp ] let @xmath26 be as above . suppose @xmath12 is the forward solution of @xmath833 then @xmath12 has a partial asymptotic expansion @xmath834 with @xmath30 and @xmath657 near @xmath73 , @xmath660 away from @xmath73 , and @xmath34 is smooth in @xmath32 , while @xmath835 near @xmath6 . let @xmath836 , and let @xmath602 be a forward order function for the weight @xmath837 . using lemma [ lemmarndssolvability ] , we may assume that @xmath838 is solvable with @xmath839 by modifying @xmath127 in @xmath134 if necessary . in fact , by the propagation of singularities , theorem [ thmrndsfredholm ] , we may take @xmath602 to be arbitrarily large in compact subsets of @xmath32 . then , a standard contour shifting argument , using the high energy estimates for @xmath156 in @xmath831 , see ( * ? ? ? * lemma 3.5 ) or ( * ? ? ? * theorem 2.21 ) , implies that @xmath840 has an asymptotic expansion as @xmath132 @xmath841 where the @xmath842 are the resonances of @xmath154 in @xmath843 , the @xmath844 are their multiplicities , and the @xmath845 are resonant states corresponding to the resonance @xmath842 ; lastly , @xmath846 is the remainder term of the expansion . even though @xmath154 is dilation - invariant near @xmath73 , this argument requires a bit of care due to the _ extendible _ nature of @xmath12 at @xmath847 : one needs to consider the cutoff equation @xmath848u$ ] ; computing the inverse mellin transform of @xmath849 generates the expansion by a contour shifting argument , see ( * ? ? ? * lemma 3.1 ) . now @xmath154 annihilates the partial expansion , so @xmath850 on the set where @xmath657 ; by the propagation of singularities , proposition [ proprndsglobalreg ] , we can improve the regularity of @xmath34 on this set to @xmath835 . thus , we have shown regularity in the region where @xmath657 , i.e. where we did not cut off ; however , considering on an enlarged domain and running the argument there , with the cutoff @xmath674 supported in the enlarged domain and identically @xmath94 on @xmath150 , we obtain the full regularity result upon restricting to @xmath150 . now , by lemma [ lemmarndsresonancesupp ] ( [ itrndsresonancesupp ] ) , all resonant states of @xmath154 which are not resonant states of the forward problem in @xmath400 must in fact vanish in @xmath32 , and by part ( [ itrndsresonancesupp0res ] ) of lemma [ lemmarndsresonancesupp ] , the only term in that survives upon restriction to @xmath32 is the constant term . thus , we obtain a partial expansion with a remainder which decays exponentially in @xmath368 in an @xmath43 sense ; we will improve this in particular to @xmath851 decay in the next section . suppose @xmath12 solves , hence it has an expansion . for any killing vector field @xmath79 , we then have @xmath852 ; now if @xmath639 solves the global problem @xmath853 ( using the extension operator @xmath854 from lemma [ lemmarndssolvability ] ) , then @xmath855 in @xmath32 by the uniqueness for the cauchy problem in this region . but by proposition [ proprndspartialasymp ] , @xmath639 has an expansion like , with constant term vanishing because @xmath79 annihilates the constant term in the expansion of @xmath12 , and therefore @xmath639 lies in space @xmath856 near the cauchy horizon @xmath857 as well . this argument can be iterated , and we obtain @xmath858 any number @xmath859 of vector fields @xmath860 which are equal to @xmath861 or rotation vector fields on the @xmath394-factor of the spacetime which are independent of @xmath862 . these vector fields are all tangent to the cauchy horizon . we obtain for any small open interval @xmath2 containing @xmath165 that @xmath863 a posteriori , by sobolev embedding , this gives [ corrndsboundedness ] using the notation of proposition [ proprndspartialasymp ] , the solution @xmath12 of has an asymptotic expansion @xmath864 with @xmath30 , and there exists a constant @xmath28 such that @xmath865 . in particular , @xmath12 is uniformly bounded in @xmath32 and extends continuously to @xmath16 . translated back to @xmath866 , the estimate on the remainder states that for scalar waves , one has exponentially fast pointwise decay to a constant . this corollary recovers franzen s boundedness result @xcite for linear scalar waves on the reissner nordstrm spacetime near the cauchy horizon in the cosmological setting . the above argument is unsatisfactory in two ways : firstly , they are not robust and in particular do not quite apply in the kerr de sitter setting discussed in [ seckds ] ; however , see remark [ rmkkdsrescarter ] , which shows that using a ` hidden symmetry ' of kerr de sitter space related to the completely integrable nature of the geodesic equation , one can still conclude boundedness in this case . secondly , the regularity statement is somewhat unnatural from a pde perspective ; thus , we now give a more robust microlocal proof of the _ conormality _ of @xmath34 , i.e. iterative regularity under application of vector fields tangent to @xmath6 , which relies on the propagation of conormal regularity at the radial set @xmath166 , see proposition [ propbconormal ] . first however , we study conormal regularity properties of @xmath156 for fixed @xmath61 , in particular giving results for individual resonant states . _ from now on , we work locally near @xmath6 and microlocally near @xmath867 , and all pseudodifferential operators we consider implicitly have wavefront set localized near @xmath868 . _ as in [ subsecrndsflow ] , we use the function @xmath869 instead of @xmath75 , where @xmath419 , @xmath870 near @xmath6 , hence the dual metric function @xmath83 is given by . since @xmath471 is a smooth non - zero multiple of @xmath75 , this is inconsequential from the point of view of regularity , and it even is semiclassically harmless for @xmath703 . denote the conjugation of @xmath154 by the mellin transform in @xmath471 by @xmath871 with @xmath61 the mellin - dual variable to @xmath75 . we first study standard ( non - semiclassical ) conormality using techniques developed in @xcite and used in a context closely related to ours in @xcite . we note that the standard principal symbol of @xmath872 is given by @xmath873 then : [ lemmarndsmodule ] the @xmath874-module @xmath875 is closed under commutators . moreover , we can choose finitely many generators of @xmath876 over @xmath874 , denoted @xmath877 , @xmath878 and @xmath879 with @xmath880 elliptic , such that for all @xmath881 , we have @xmath882 = \sum_{\ell=0}^n c_{j\ell}a_\ell,\quad c_{j\ell}\in\psi^1(x),\ ] ] where @xmath883 for @xmath884 . since @xmath166 is lagrangian and thus in particular coisotropic , the first statement follows from the symbol calculus . further , is a symbolic statement as well ( since @xmath885\in\psi^2(x)$ ] , and the summand @xmath886 is a freely specifiable first order term ) , so we merely need to find symbols @xmath887 , homogeneous of degree @xmath94 , with @xmath888 , such that @xmath889 with @xmath890 for @xmath884 . note that this is clear for @xmath891 , since in this case @xmath892 . we then let @xmath893 , and we take @xmath894 to be linear in the fibers and such that they span the linear functions in @xmath895 over @xmath896 . we extend @xmath897 to linear functions on @xmath898 by taking them to be constant in @xmath5 and @xmath899 . ( thus , these @xmath900 are symbols of differential operators in the spherical variables . ) we then compute @xmath901 which is of the desired form since @xmath902 vanishes quadratically at @xmath166 ; moreover , for @xmath903 , one readily sees that @xmath904 vanishes quadratically at @xmath166 as well , finishing the proof . in the lagrangian setting , this is a general statement , as shown by haber and vasy , see ( * ? ? ? * lemma 2.1 , equation ( 6.1 ) ) . the positive commutator argument yielding the low regularity estimate at ( generalized ) radial sets , see ( * ? ? ? * proposition 2.4 ) , can now be improved to yield iterative regularity under the module @xmath876 : indeed , we can follow the proof of ( * ? ? ? * proposition 4.4 ) ( which is for a generalized radial source / sink in the b - setting , whereas we work on a manifold without boundary here , so the weights in the reference can be dropped ) or @xcite very closely ; we leave the details to the reader . in order to compress the notation for products of module derivatives , we denote @xmath905 in the notation of the lemma , and then use multiindex notation @xmath906 . the final result , reverting back to @xmath156 , is the following ; recall that @xmath907 is a source and @xmath908 is a sink for the hamilton flow within @xmath898 : [ lemmarndsmoduleestimate ] let @xmath909 be a vector of generators of the module @xmath876 as above . suppose @xmath910 . let @xmath911 be such that @xmath912 and @xmath83 are elliptic at @xmath907 , resp . @xmath908 , and all forward , resp . backward , null - bicharacteristics from @xmath913 , resp . @xmath914 , reach @xmath915 while remaining in @xmath916 . then @xmath917 in particular : [ corrndsresonantstateconormal ] if @xmath12 is a resonant state of @xmath154 , i.e. @xmath796 , then @xmath12 is conormal to @xmath6 relative to @xmath918 , i.e. for any number of vector fields @xmath919 on @xmath79 which are tangent to @xmath6 , we have @xmath920 . indeed , by the propagation of singularities , @xmath12 is smooth away from @xmath921 , and then lemma [ lemmarndsmoduleestimate ] implies the stated conormality property . we now turn to the conormal regularity estimate in the spacetime , b- , setting . let us define @xmath922 using the stationary ( @xmath75-invariant ) extensions of the vector field generators of the module @xmath876 defined in lemma [ lemmarndsmodule ] together with @xmath923 , one finds that the module @xmath924 is generated over @xmath925 by @xmath926 , @xmath927 and @xmath928 , with @xmath929 elliptic , satisfying @xmath930=\sum_{\ell=0}^{n+1 } c_{j\ell}a_\ell,\quad c_{j\ell}\in{\psi_{{\mathrm{b}}}}^1(m),\ ] ] with @xmath931 for @xmath932 . the proof of ( * ? ? ? * proposition 4.4 ) then carries over to the saddle point setting of proposition [ proprndsradialrecall ] and gives in the below - threshold case ( which is the relevant one at the cauchy horizon ) : [ propbconormal ] suppose @xmath154 is as above , and let @xmath588 , @xmath933 . if @xmath934 , and if @xmath612 then @xmath921 ( and thus a neighborhood of @xmath921 ) is disjoint from @xmath935 for all @xmath936 provided @xmath937 for @xmath936 , and provided a punctured neighborhood of @xmath921 , with @xmath921 removed , in @xmath440 is disjoint from @xmath938 . thus , if @xmath939 is conormal to @xmath654 , i.e. remains in @xmath940 microlocally under iterative applications of elements of @xmath924 this in particular holds if @xmath941 , then @xmath12 is conormal relative to @xmath942 , provided @xmath12 lies in @xmath943 in a punctured neighborhood of @xmath166 . using proposition [ propbconormal ] at the radial set @xmath166 in the part of the proof of proposition [ proprndspartialasymp ] where the regularity of @xmath34 is established , we obtain : [ thmrndspartialasympconormal ] let @xmath26 be as in proposition [ proprndspartialasymp ] , and suppose @xmath12 is the forward solution of @xmath944 then @xmath12 has a partial asymptotic expansion @xmath864 , where @xmath657 near @xmath73 , @xmath660 away from @xmath73 , and with @xmath30 , and @xmath945 for all @xmath859 and all vector fields @xmath946 which are tangent to the cauchy horizon @xmath6 ; here , @xmath947 is given by . the same result holds true , without the constant term @xmath40 , for the forward solution of the massive klein gordon equation @xmath948 , @xmath39 small . for the massive klein gordon equation , the only change in the analysis is that the simple resonance at @xmath97 moves into the lower half plane , see e.g. the perturbation computation in ( * ? ? ? * lemma 3.5 ) ; this leads to the constant term @xmath40 , which was caused by the resonance at @xmath97 , being absent . this implies the estimate and thus yields corollary [ corrndsboundedness ] as well . the amount of decay @xmath51 ( and thus the amount of regularity we obtain ) in theorem [ thmrndspartialasympconormal ] is directly linked to the size of the _ spectral gap _ , i.e. the size of the resonance - free strip below the real axis , as explained in [ subsecrndsasymp ] . due to the work of s barreto zworski @xcite in the spherically symmetric case and general results by dyatlov @xcite at ( @xmath5-)normally hyperbolic trapping ( for every @xmath5 ) , the size of the _ essential spectral gap _ is given in terms of dynamical quantities associated to the trapping , see proposition [ proprndssemiclassical2 ] ; we recall that the essential spectral gap is the supremum of all @xmath949 such that there are only finitely many resonances above the line @xmath950 . thus , the essential spectral gap only concerns the high energy regime , i.e. it does not give any information about low energy resonances . in this section , we compute the size of the essential spectral gap in some limiting cases ; the possibly remaining finitely many resonances between @xmath97 and the resonances caused by the trapping will be studied separately in future work . we give some indications of the expected results in remark [ rmkrndshighreg ] . in order to calculate the relevant dynamical quantities at the trapped set , we compute the linearization of the flow in the @xmath951 variables at the trapped set @xmath511 : we have @xmath952 modulo functions vanishing quadratically at @xmath511 , and in the same sense @xmath953 which in view of @xmath954 ( see also ) gives @xmath955 therefore , the expansion rate of the flow in the normal direction at @xmath511 is equal to @xmath956 to find the size of the essential spectral gap for the forward problem of @xmath54 , we need to compute the size of the imaginary part of the subprincipal symbol of the semiclassical rescaling of @xmath957 at the semiclassical trapped set . put @xmath958 , @xmath693 , then @xmath959 with @xmath960 , @xmath588 , we thus obtain @xmath961 the essential spectral gap thus has size at least @xmath51 provided @xmath962 , so @xmath963 we compute the quantity on the right for near - extremal reissner de sitter black holes with very small cosmological constant ; first , using the radius of the photon sphere for the reissner nordstrm black hole with @xmath22 , @xmath964 and the radius of the cauchy horizon @xmath965 we obtain @xmath966 for the size of the essential spectral gap for resonances caused by the trapping in the case @xmath22 . ( for @xmath177 , one finds @xmath967 , which agrees with ( * ? ? ? * equation ( 0.3 ) ) for @xmath22 . ) in the extremal case @xmath968 , we find @xmath969 . furthermore , we have @xmath970 thus , @xmath971 ; therefore , @xmath972 which blows up as @xmath973 ; this corresponds to the fact the surface gravity of extremal black holes vanishes . given @xmath45 , we can thus choose @xmath60 small enough so that @xmath974 , and then taking @xmath176 to be small , the same relation holds for the @xmath0-dependent quantities @xmath728 and @xmath947 . since there are only finitely many resonances in any strip @xmath975 , we conclude by theorem [ thmrndspartialasympconormal ] , taking @xmath976 close to @xmath728 , that for forcing terms @xmath127 which are orthogonal to a finite - dimensional space of dual resonant states ( corresponding to resonances in @xmath977 ) , the solution @xmath12 has regularity @xmath942 at the cauchy horizon . put differently , for near - extremal reissner nordstrm de sitter black holes with very small cosmological constant @xmath176 , waves with initial data in a finite codimensional space ( within the space of smooth functions ) achieve any fixed order of regularity at the cauchy horizon , in particular better than @xmath50 . [ rmkrndshighreg ] numerical investigations of linear scalar waves @xcite and arguments using approximations of the scattering matrix @xcite suggest that there are indeed resonances roughly at @xmath978 , @xmath476 , where @xmath979 and @xmath980 are the surface gravities of the cosmological horizon , see ; as @xmath981 , we have @xmath982 , and for extremal black holes with @xmath22 , we have @xmath983 . ( on the static de sitter spacetime , there is a resonance exactly at @xmath984 , as a rescaling shows : for @xmath985 , one has @xmath986 decay to constants away from the cosmological horizon , @xmath10 the static time coordinate , see e.g. @xcite ; now static de sitter space @xmath987 with cosmological constant @xmath176 can be mapped to @xmath988 via @xmath989 , @xmath990 , where @xmath991 is the surface gravity of the cosmological horizon , and @xmath992 , resp . @xmath993 , are static coordinates on @xmath988 , resp . @xmath987 . under this map , the metric on @xmath988 is pulled back to a constant multiple of the metric on @xmath987 . thus , waves on @xmath987 decay to constants with the speed @xmath994 , which corresponds to a resonance at @xmath984 . ) our analysis is consistent with the numerical results , _ assuming the existence of these resonances _ : we expect linear waves in this case to be generically no smoother than @xmath995 at the cauchy horizon , which highlights the importance of the relative sizes of the surface gravities for understanding the regularity at the cauchy horizon . for near - extremal black holes , where @xmath996 , this gives @xmath997 , thus the local energy measured by an observer crossing the cauchy horizon is of the order @xmath998 , which diverges in view of @xmath999 ; this agrees with ( * ? ? ? * equation 9 ) . we point out however that the waves are still in @xmath50 if @xmath1000 , which is satisfied for near - extremal black holes . this is analogous to sbierski s criterion @xcite for ensuring the finite energy of waves at the cauchy horizon of linear waves with fast decay along the event horizon . the rigorous study of resonances associated with the event and cosmological horizons will be subject of future work . the analysis presented in the previous sections goes through with only minor modifications if we consider the wave equation on natural vector bundles . for definiteness , we focus on the wave equation , more precisely the hodge dalembertian , on differential @xmath1001-forms , @xmath1002 . in this case , mode stability and asymptotic expansions up to decaying remainder terms in the region @xmath215 , a neighborhood of the black hole exterior region , were proved in @xcite . the previous arguments apply to @xmath1003 ; the only difference is that the threshold regularity at the radial points at the horizons shifts . at the event horizon and the cosmological horizon , this is inconsequential , as we may work in spaces of arbitrary high regularity there ; at the cauchy horizon however , one has , fixing a time - independent _ positive definite inner product _ on the fibers of the @xmath1001-form bundle with respect to which one computes adjoints : @xmath1004 at @xmath921 , with @xmath1005 , and @xmath1006 and endomorphism on the @xmath1001-form bundle ; and one can compute that the lowest eigenvalue of @xmath1006 ( which is self - adjoint with respect to the chosen inner product ) is equal to @xmath1007 . but then the regularity one can propagate into @xmath921 for @xmath1008 , @xmath588 , solving @xmath1009 , @xmath667 compactly supported and smooth , is @xmath1010 , as follows from ( * ? ? ? * proposition 2.1 and footnote 5 ) . thus , in the partial asymptotic expansion in theorem [ thmrndspartialasympconormal ] ( which has a different leading order term now , coming from stationary @xmath1001-form solutions of the wave equation ) , we can only establish conormal regularity of the remainder term @xmath34 at the cauchy horizon relative to the space @xmath1010 , which for small @xmath26 gives sobolev regularity @xmath1011 , for small @xmath60 . assuming that the leading order term is smooth at the cauchy horizon ( which is the case , for example , for 2-forms , see ( * ? ? ? * theorem 4.3 ) ) , we therefore conclude that , as soon as we consider @xmath1001-forms @xmath12 with @xmath1012 , our methods do not yield uniform boundedness of @xmath12 up to the cauchy horizon ; however , we remark that the conormality does imply uniform bounds as @xmath302 of the form @xmath1013 , @xmath60 small . a finer analysis would likely yield more precise results , in particular boundedness for certain components of @xmath12 ; and , as in the scalar setting , a converse result , namely showing that such a blow - up does happen , is much more subtle . we do not pursue these issues in the present work . we recall from @xcite the form of the kerr de sitter metric with parameters @xmath176 ( cosmological constant ) , @xmath173 ( black hole mass ) and @xmath24 ( angular momentum ) , are denoted @xmath1014 in @xcite , while our @xmath1015 are denoted @xmath1016 there . ] @xmath1017 where @xmath1018 in order to guarantee the existence of a cauchy horizon , we need to assume @xmath15 . analogous to definition [ defrndsnondegenerate ] , we make a non - degeneracy assumption : [ defkdsnondegenerate ] we say that the kerr de sitter spacetime with parameters @xmath1019 is _ non - degenerate _ if @xmath1020 has @xmath180 simple positive roots @xmath181 . one easily checks that @xmath1021 and again , @xmath6 ( in the analytic extension of the spacetime ) is called the _ cauchy horizon _ , @xmath7 the _ event horizon _ and @xmath8 the _ cosmological horizon_. we consider a simple case in which non - degeneracy can be checked immediately : [ lemmakdsnondegenerate ] suppose @xmath1022 , and denote the three non - negative roots of @xmath1023 by @xmath204 . then for small @xmath15 , @xmath1020 has three positive roots @xmath1024 , @xmath113 , with @xmath206 , depending smoothly on @xmath1025 , and @xmath1026 . we recall that the condition ensures the existence of the roots @xmath208 as stated . one then computes for @xmath1027 that @xmath1028 , giving the first statement . in order to state unconditional results later on , we in fact _ from now on assume to be in the setting of this lemma , i.e. we consider slowly rotating kerr de sitter black holes _ ; see remark [ rmkkdsresgeneral ] for further details . as in [ subsecrndsmfd ] , we discuss the smooth extension of the metric @xmath3 across the horizons and construct the manifold on which the linear analysis will take place ; all steps required for this construction are slightly more complicated algebraically but otherwise very similar to the ones in the reissner nordstrm de sitter setting , so we shall be brief . thus , with @xmath1029 we will take @xmath1030 for @xmath5 near @xmath250 , where @xmath1031 using @xmath1032 and @xmath1033 , one computes @xmath1034 using e.g. the frame @xmath1035 , @xmath1036 , @xmath1037 and @xmath1038 , one finds the volume density to be @xmath1039 moreover , the form of the dual metric is @xmath1040 this is a non - degenerate lorentzian metric apart from the usual singularity of the spherical coordinates @xmath1041 , which indeed is merely a coordinate singularity as shown by a change of coordinates @xcite , see also remark [ rmkkdsflowvalidcoord ] below . as in the reissner nordstrm de sitter case , one can start by choosing the functions @xmath238 and @xmath241 so that @xmath1042 for @xmath1043 and @xmath1044 for @xmath1045 , so that @xmath368 in is well - defined in a neighborhood of @xmath164 $ ] , and moreover one can choose @xmath238 and @xmath241 so that @xmath1046 is timelike in @xmath1047 $ ] : indeed , this is satisfied provided @xmath1048 we note that in @xmath1049 , we can take @xmath223 to be large and negative , and then at @xmath313 , we obtain @xmath1050 therefore , @xmath326 is future timelike for @xmath518 . near @xmath165 then , more precisely in @xmath1051 , we can arrange for @xmath1052 to be timelike again , and since @xmath1053 has the opposite sign , we find that @xmath1054 , i.e. @xmath1052 is future timelike there . in order to cap off the problem in @xmath134 , we again modify @xmath1020 to a smooth function @xmath1055 . since we can hide all the ( possibly complicated ) structure of the extension when @xmath1056 using complex absorption , we simply choose @xmath1055 such that @xmath1057 ( see also the discussion following . ) we can then extend the metric @xmath3 past @xmath360 by defining @xmath1058 near @xmath138 as in , with @xmath1020 replaced by @xmath1055 , and with @xmath1059 . we can then arrange @xmath1052 to be future timelike in @xmath652 , and @xmath273 is future timelike at @xmath1060 by a computation analogous to . we can now define spacelike hypersurfaces @xmath1061 exactly as in , bounding a domain with corners @xmath153 inside @xmath1062 and we will analyze the wave equation ( modified in @xmath134 ) on the compactified region @xmath1063 we further let @xmath685 , @xmath544 . since it simplifies a number of computations below , we will study the null - geodesic flow of @xmath1064 , i.e. the flow of @xmath1065 within the characteristic set @xmath1066 , where @xmath83 denotes the dual metric function . by pasting @xmath1052 in @xmath327 , @xmath326 in @xmath1067 and @xmath1046 in @xmath1068 together using a non - negative partition of unity , we can construct a smooth , globally future timelike covector field @xmath1069 on @xmath150 and use it to split the characteristic set into components @xmath136 as in . since the global dynamics of the null - geodesic flow in a neighborhood @xmath1068 of the exterior region are well - known , with saddle points of the flow ( generalized radial sets ) at @xmath1070 , where we define @xmath1071 , and a normally hyperbolically trapped set @xmath511 . as in parts of the discussion in [ subsecrndsflow ] , it is computationally convenient to work with @xmath1072 instead of @xmath368 near @xmath112 , where @xmath1073 ( i.e. effectively putting @xmath1074 ) . let @xmath422 and write b - covectors as @xmath1075 then the dual metric function reads @xmath1076 [ rmkkdsflowvalidcoord ] valid coordinates near the poles @xmath1077 are @xmath1078 and writing @xmath1079 , one finds @xmath1080 and @xmath1081 ; thus to see the smoothness of @xmath1082 near the poles , one merely needs to rewrite @xmath1083 as @xmath1084 and notice that @xmath1085 is smooth , as is @xmath1086 since this is simply the dual metric function on @xmath394 in spherical coordinates . we study the rescaled hamilton flow near @xmath120 using the coordinates and introducing @xmath1087 , @xmath1088 , @xmath1089 , @xmath1090 as the fiber variables similarly to : thus , @xmath1091 and we find that at @xmath434 , where @xmath1092 , @xmath1093 and thus the quantity controlling the threshold regularity at @xmath120 is @xmath1094 furthermore , if we put @xmath1095 then @xmath1096 , so the quadratic defining function @xmath1097 of @xmath120 within @xmath1098 satisfies @xmath1099 as in the reissner de sitter case , this implies that @xmath120 is a source or sink within @xmath1100 , with a stable or unstable manifold @xmath1101 transversal to the boundary . for @xmath1102 written as , one can check that @xmath1103 if and only if @xmath1104 , i.e. if and only if @xmath1105 ; in @xmath1106 , the quantity @xmath1107 therefore has a sign ( which is the same as in the discussion around ) , depending on the component of the characteristic set ; thus , null - geodesics in a fixed component @xmath136 of the characteristic set can only cross @xmath112 in one direction . furthermore , in the regions where @xmath1108 , and thus @xmath273 is timelike , we have @xmath1109 , see also . since we will place complex absorption immediately beyond @xmath6 , i.e. in @xmath1110 for @xmath1111 very small , it remains to check that at finite values of @xmath368 in this region , all null - geodesics escape either to @xmath73 or to @xmath333 ; but this follows from the timelike nature of @xmath1046 there , which gives that @xmath1112 is non - zero , in fact bounded away from zero . to summarize , the global behavior of the null - geodesic flow in @xmath531 is the same as that of the reissner nordstrm de sitter solution ; see figure [ figrndsflow ] . we point out that the existence of an ergoregion is irrelevant for our analysis : its manifestation is merely that null - geodesics tending to , say , the event horizon @xmath7 in the backward direction , may have a segment in @xmath562 before ( possibly ) crossing the event horizon into @xmath563 ; see also ( * ? ? ? * figure 8) . we use a complex absorbing operator @xmath152 as in [ subsecrndsregularity ] , with @xmath1113 on @xmath136 , and which is elliptic in @xmath1114 , @xmath1110 , where @xmath1111 is chosen sufficiently small to ensure that the dynamics near the generalized radial set @xmath166 control the dynamics in @xmath1115 : that is , null - geodesics near either tend to @xmath166 or enter the elliptic region of @xmath133 , i.e. @xmath1116 , in finite time , unless they cross @xmath1117 , i.e. @xmath333 , or @xmath6 . the analysis in [ subsecrndsregularity][subsecrndsconormal ] now goes through _ mutatis mutandis_. ( for completeness , we note that the threshold quantity @xmath947 , see , for small @xmath24 is given by @xmath1118 . ) in fact , to prove conormal regularity , we can use the same module generators as those constructed in the proof of lemma [ lemmarndsmodule ] , and the b - version , see the discussion around proposition [ propbconormal ] , goes through without changes as well . [ rmkkdsrescarter ] there exists a second order ` carter operator ' @xmath1119 , with principal symbol given by @xmath1120 in , that commutes with @xmath1064 ; concretely , in the coordinates used in ( which are valid near @xmath6 ) , @xmath1121 since @xmath1122 and @xmath1123 commute with @xmath1064 , and since moreover the sum of the first two terms of @xmath1124 is an elliptic operator on @xmath394 , we conclude , commuting @xmath1124 through the equation @xmath1125 , @xmath531 , that @xmath12 is smooth in @xmath10 and the angular variables . thus , we can deduce conormal regularity ( apart from iterative regularity under application of @xmath1126 ) for @xmath12 using such commutation arguments as well . note however that the existence of such the ` hidden symmetry ' @xmath1124 is closely linked to the complete integrability of the geodesic flow on kerr de sitter space , while the microlocal argument proving conormality applies in much more general situations and different contexts , see e.g. @xcite . we content ourselves with stating the analogues of theorem [ thmrndspartialasympconormal ] and corollary [ corrndsboundedness ] in the kerr de sitter setting : [ thmkdspartialasympconormal ] suppose the angular momentum @xmath15 is very small , such that there exists @xmath26 with the property that the forward problem for the wave equation in the neighborhood @xmath1068 of the domain of outer communications has no resonances in @xmath1127 other than the simple resonance at @xmath57 . let @xmath12 be the forward solution of @xmath944 then @xmath12 has a partial asymptotic expansion @xmath864 , with @xmath30 and @xmath657 near @xmath73 , @xmath660 away from @xmath73 , and @xmath945 for all @xmath859 and all vector fields @xmath946 which are tangent to the cauchy horizon @xmath6 ; here , @xmath947 is given by . in particular , there exists a constant @xmath28 such that @xmath865 , and @xmath12 is uniformly bounded in @xmath32 . again , the same result holds , without the constant term @xmath40 , for solutions of the massive klein gordon equation @xmath948 , @xmath39 small . [ rmkkdsresgeneral ] our arguments go through for general non - degenerate kerr de sitter spacetimes , _ assuming _ the ` resolvent ' family @xmath168 admits a meromorphic continuation to the complex plane with ( polynomially lossy ) high energy estimates in a strip below the real line , and the only resonance ( quasi - normal mode ) in @xmath793 is a simple resonance at @xmath97 ( ` mode stability ' ) . apart from the mode stability , these conditions hold for a large range of spacetime parameters @xcite , while the mode stability has only been proved for small @xmath24 . ( for the kerr family of black holes , mode stability is known , see @xcite . ) without the mode stability assumption , we still obtain a resonance expansion for linear waves up to the cauchy horizon , but boundedness does not follow due to the potential existence of resonances in @xmath789 or higher order resonances on the real line ; if such resonances should indeed exist , then boundedness would in fact be false for generic forcing terms or initial data . if on the other hand one _ assumes _ that the wave @xmath12 decays to a constant at some exponential rate @xmath26 in the black hole exterior region , the conclusion of theorem [ thmkdspartialasympconormal ] still holds . the analysis in [ secrnds ] and [ seckds ] relies on the propagation of singularities in b - sobolev spaces of variable order ; in fact , we only use microlocal elliptic regularity and real principal type propagation on such spaces . we recall some aspects of ( * ? ? ? * appendix a ) needed in the sequel , and refer the reader to @xcite for the proofs of elliptic regularity and real principal type propagation in this setting ; since all arguments presented there are purely symbolic , they go through in the b - setting with purely notational changes . moreover , we remark that adding ( constant ! ) weights to the variable order b - spaces does not affect any of the arguments . we use sobolev orders which vary only in the base , not in the fiber . in order to introduce the relevant notation , we consider the model case @xmath1128 of a manifold with boundary , and an order function @xmath1129 , constant outside a compact set ; recalling the symbol class @xmath1130 we then define @xmath1131 now @xmath1132 for @xmath1133 , provided @xmath1134 , due to derivatives falling on @xmath1135 , producing logarithmic terms . therefore , we can quantize symbols in @xmath1136 ; we denote the class of quantizations of such symbols by @xmath1137 . we will only work with @xmath1138 , @xmath1139 , in which case one can in particular transfer this space of operators to a manifold with boundary and obtain a b - pseudodifferential calculus ; see @xcite for the analogous case of manifolds without boundary . thus , if @xmath1140 and @xmath1141 for two order functions @xmath1142 , then @xmath1143 where @xmath61 denotes the principal symbol in the respective classes of operators ; the principal symbol of an element in @xmath1144 is well - defined in @xmath1145 . furthermore , we have @xmath1146\in\psi_{{{\mathrm{b}}},1-\delta,\delta}^{{\mathsf{s}}+{\mathsf{s}}'-(1 - 2\delta ) } , \quad \sigma(i[a , b])=h_{\sigma(a)}\sigma(b).\ ] ] for the purposes of the analysis in [ subsecrndsfredholm ] , we need to describe the relation of variable order b - sobolev spaces to semiclassical function spaces via the mellin transform . we work locally in @xmath1147 , and the variable order function is @xmath1148 , with @xmath602 constant outside a compact set . fixing a real number @xmath1149 and an elliptic , dilation - invariant operator @xmath1150 , @xmath1139 , the norm on @xmath1151 is given by @xmath1152 and all choices of @xmath1153 and @xmath909 give equivalent norms . ( this follows from elliptic regularity . ) since the @xmath1154-part of the norm is irrelevant in a certain sense ( it is only there to take care of a possible kernel of @xmath909 ) , we focus on the seminorm @xmath1155 we concretely take @xmath909 to be the left quantization of @xmath1156 , writing b-1-forms as @xmath1157 denote the mellin transform of @xmath12 in @xmath75 by @xmath1158 , and the fourier transform of @xmath1159 in @xmath1160 by @xmath1161 ; is the fourier transform of @xmath12 in @xmath1162 , where @xmath1163 . ] then by plancherel , @xmath1164 where @xmath1165 . using @xmath1166 , we can rewrite this integral as @xmath1167 this suggests : [ defvariablescl ] for @xmath1168 , constant outside a compact set , define the semiclassical sobolev space @xmath1169 , @xmath712 , by the norm @xmath1170 where @xmath1171 is a real number . the particular choice of the value of @xmath1153 is irrelevant , see remark [ rmkvariablesclnorm ] , where we also give a better , invariant , version of definition [ defvariablescl ] . thus , @xmath1172 as a space , but the semiclassical space captures the behavior of the norm as @xmath1173 . we remark that the space @xmath1174 becomes weaker as one increases @xmath1175 or decreases @xmath602 . [ rmkvariablesclconstorders ] if @xmath1176 and @xmath1177 are constants , we can use the equivalent norm @xmath1178 . using and taking the @xmath1179-term in into account , we thus have an equivalence of norms @xmath1180 the semiclassical analogues of the symbol spaces , which are adapted to working with the spaces @xmath1174 , are defined by @xmath1181 with @xmath1182 independent of @xmath702 . in our application , differentiation in @xmath1160 or @xmath899 will in fact at most produce a _ logarithmic _ loss , i.e. will produce a factor of @xmath1183 or @xmath1184 . for us , the main example of an element in @xmath1185 is the symbol @xmath1186 . quantizations of symbols in @xmath1185 are denoted @xmath1187 , and for @xmath1188 and @xmath1189 , we have @xmath1190 and @xmath1191 \in \psi_{h,1-\delta,\delta}^{{\mathsf{s}}+{\mathsf{s}}'-(1 - 2\delta),{\mathsf{w}}+{\mathsf{w}}'+2\delta},\ ] ] with principal symbols given by the product , resp . the poisson bracket , of the respective symbols . here , the principal symbol of an element of @xmath1187 is well - defined in @xmath1192 . [ rmkvariablesclnorm ] using elliptic regularity in the calculus @xmath1193 , we see that given @xmath1194 , @xmath1195 , we have @xmath1196 if and only if @xmath1197 , where @xmath1188 is a fixed elliptic operator ; i.e. we have an equivalence of norms @xmath1198 we next discuss microlocal regularity results for variable order operators ; general references for such results in the constant order ( semiclassical ) setting are @xcite and @xcite . working on a compact manifold @xmath79 without boundary now , @xmath1199 , suppose we are given a semiclassical ps.d.o . semiclassical elliptic regularity takes the following quantitative form on variable order spaces : [ propvariablesclelliptic ] if @xmath1201 are such that @xmath1202 ( the semiclassical elliptic set of @xmath1203 ) , and @xmath83 is elliptic on @xmath1204 , then @xmath1205 for any fixed @xmath1153 . this follows from the usual symbolic construction of a microlocal inverse of @xmath1203 near @xmath1204 . the semiclassical real principal type propagation of singularities requires a hamilton derivative condition on the orders @xmath1206 of the function space : let @xmath1200 with real - valued semiclassical principal symbol @xmath1207 , i.e. @xmath1208 is a classical symbol , which we assume for simplicity to be @xmath702-independent . let @xmath1209 be the rescaled hamilton vector field , with @xmath1210 is homogeneous of degree @xmath95 in the fibers of @xmath898 away from the zero section ; thus @xmath1211 is homogeneous of degree @xmath97 modulo vector fields vanishing at fiber infinity , and can thus be viewed as a smooth vector field on the radially compactified cotangent bundle @xmath1212 . at fiber infinity @xmath1213 , the @xmath1211 flow is simply the rescaled hamilton flow of the homogeneous principal part of @xmath1208 , while at finite points @xmath1214 , @xmath1211 is proportional to the semiclassical hamilton vector field . [ propvariablesclpropagation ] under these assumptions , let @xmath1215 be order functions , and let @xmath1216 be open ; suppose @xmath1217 and @xmath1218 in @xmath1219 . suppose @xmath1220 are such that @xmath83 is elliptic on @xmath1221 , and all backward null - bicharacteristics of @xmath1203 from @xmath1222 enter @xmath1223 while remaining in @xmath1224 . then @xmath1225 for any fixed @xmath1153 . for @xmath1226 , this gives the usual estimate of @xmath13 in @xmath1227 in terms of @xmath1228 in @xmath1229 , losing @xmath94 derivative and @xmath94 power of @xmath702 relative to the elliptic setting . the proof is almost the same as that of ( * ? ? ? * proposition a.1 ) , so we shall be brief . since the result states nothing about critical points of the hamilton flow , we may assume @xmath1230 on @xmath1219 ( at @xmath1231 , this means that @xmath1211 is _ non - radial _ ) . let @xmath1232 . let us first prove the propagation at fiber infinity : introduce coordinates @xmath1233 on @xmath1231 , @xmath1234 , centered at @xmath51 , such that @xmath1235 , and suppose @xmath1236 and the neighborhood @xmath1237 of @xmath1238 are such that @xmath1239_{q_1}\times{\overline}{u'}\subset u$ ] ; suppose we have a priori @xmath1174-regularity in @xmath1240_{q_1}\times{\overline}{u'}$ ] , i.e. @xmath1241 is elliptic there . we use a commutant ( omitting the necessary regularization in the weight @xmath96 for brevity ) @xmath1242 where @xmath1243 , @xmath1244 for @xmath1245 , @xmath1246 for @xmath1247 , with @xmath1248 large , and @xmath1249 near @xmath1250 , @xmath1251 near @xmath1252 $ ] ; moreover @xmath1253 , @xmath1254 . we then compute @xmath1255 now @xmath1256 , giving rise to the main ` good ' term , while the @xmath1257 term ( which has the opposite sign ) is supported where one has a priori regularity . the term on the second line can be absorbed into the first by making @xmath1258 large ( since @xmath1259 can then be dominated by a small multiple of @xmath1260 ) , while the last two terms have the same sign as the main term by our assumptions on @xmath602 and @xmath1175 . a positive commutator computation , a standard regularization argument , and absorbing the contribution of the imaginary part of @xmath1203 by making @xmath1258 larger if necessary , gives the desired result . for the propagation within @xmath898 , a similar argument applies ; we use local coordinates @xmath1233 in @xmath898 with @xmath1261 now , centered at @xmath51 , so that @xmath1235 ; and the differentiability order @xmath602 becomes irrelevant now , as we are away from fiber infinity . thus , we can use the commutant @xmath1262 , with @xmath674 exactly as above , and @xmath1263 localizing near @xmath97 ; the positive commutator argument then proceeds as usual . returning to , we observe that for @xmath1264 , we can apply this proposition to @xmath1265 under the single condition @xmath1217 , which is the same condition as for real principal type propagation for @xmath12 in b - sobolev spaces ( as it should ) . finally , we point out that completely analogous results hold for _ weighted _ b - sobolev spaces @xmath646 and their semiclassical analogues : the only necessary modification is that now we have to restrict the mellin - dual variable to @xmath75 , called @xmath61 here , to @xmath682 , since the mellin transform in @xmath75 induces an isometric isomorphism @xmath1266 we briefly recall supported and extendible distributions on manifolds with boundary , following ( * ? ? ? * appendix b ) . the model case is @xmath1267 , and we consider sobolev spaces with regularity @xmath45 . for notational brevity , we omit the factor @xmath1268 . thus , we let @xmath1269 called @xmath41 space with supported @xmath1270 , resp . extendible @xmath1271 , character at the boundary @xmath1272 . the hilbert norm on the supported space is defined by restriction from @xmath41 , while the hilbert norm on the extendible space comes from the isomorphism @xmath1273)^\bullet;\ ] ] since the supported space on the right hand side is a closed subspace of @xmath1274 , we immediately get an isometric extension map @xmath1275 which identifies @xmath1276 with the orthogonal complement of @xmath1277)^\bullet$ ] in @xmath1274 ; thus , @xmath1278 the dual spaces relative to @xmath43 are given by @xmath1279 we now discuss the case of codimension @xmath1280 corners , which is all we need for our application ; treating the case of higher codimension corners requires purely notational changes . we work locally on @xmath1281 , @xmath1282 , @xmath1283 . consider the domain @xmath1284 which is a submanifold of with corners of @xmath1285 . again , since the @xmath1286 variables will carry through our arguments below , we simplify notation by dropping them , i.e. by letting @xmath1287 . let @xmath45 . there are two natural ways to define a space @xmath1288 of distributions in @xmath41 with extendible character at @xmath1272 and supported character at @xmath1289 , which give rise to two a priori different norms and dual spaces : namely , @xmath1290\times[0,\infty))^{\bullet,\bullet } , \\ h^s((0,\infty)\times[0,\infty))^{-,\bullet}_2 & = \ { u\in h^s((0,\infty)\times{\mathbb{r}})^- \colon { \operatorname{supp}}u\subset(0,\infty)\times[0,\infty)\ } ; \end{split}\ ] ] we equip the first space with the quotient topology , and the second space with the subspace topology . the first space is the space of restrictions to @xmath1291 of distributions with support in @xmath1292 , while the second space is the space of extendible distributions in the half space @xmath1293 which have support in @xmath1291 ; see figure [ figsuppextdef ] . ( middle ) . left : choice @xmath94 . right : choice @xmath1280 . the supports of elements of the spaces that @xmath1294 , resp . @xmath1295 , are quotients , resp . subspaces of are shaded ; the ` @xmath97 ' indicates the vanishing condition in the definition of @xmath1295 . ] as in the case of manifolds with boundary discussed above , both spaces come equipped with isometric ( by the definition of their norms ) extension operators @xmath1296 with @xmath1297 and @xmath1298 contained in the space @xmath1299 of distributions in @xmath1300 with support in @xmath1301 , see figure [ figsuppextmaps ] . we can thus also describe the second variant of @xmath1302 equivalently as the quotient @xmath1303\times{\mathbb{r}})^\bullet.\ ] ] furthermore , the dual spaces are isometric to @xmath1304)^{\bullet,\bullet},\end{aligned}\ ] ] i.e. dualizing switches choices @xmath94 and @xmath1280 for the definition of the mixed supported and extendible spaces . since @xmath1307 is an isomorphism if and only if the dual map @xmath1310 is an isomorphism , it suffices to consider the case @xmath1311 . in view of the characterizations and of the two versions of @xmath1302 as quotients ( equipped with the quotient norm ! ) , it suffices to prove the existence of a bounded linear map @xmath1312 with @xmath1313 . the idea is to use the fact that for integer @xmath1314 , @xmath1315-spaces of extendible distributions are _ intrinsically _ defined : thus , for @xmath1316 , the restriction @xmath1317 to the lower half plane is an element of @xmath1318 with support in @xmath1319\times(-\infty,0)$ ] ; but then we can use an extension map @xmath1320 defined using reflections and rescalings ( see @xcite ) , which in addition preserves the property of being supported in @xmath1321 . we can then define the map @xmath1322 on @xmath41 by @xmath1323 for all integer @xmath1324 ; by interpolation , the same map in fact works for all @xmath1325 $ ] . since @xmath1001 can be chosen arbitrarily , this proves the existence of a map for any fixed real @xmath44 .

we show that linear scalar waves are bounded and continuous up to the cauchy horizon of reissner de sitter and kerr de sitter spacetimes , and in fact decay exponentially fast to a constant along the cauchy horizon . we obtain our results by modifying the spacetime beyond the cauchy horizon in a suitable manner , which puts the wave equation into a framework in which a number of standard as well as more recent microlocal regularity and scattering theory results apply . in particular , the conormal regularity of waves at the cauchy horizon which yields the boundedness statement is a consequence of radial point estimates , which are microlocal manifestations of the blue - shift and red - shift effects .

1512.08004

a massive star , perhaps 15 - 25 solar masses , evolves through hydrostatic burning to an `` onion - skin '' structure , with a inert iron core produced from the explosive burning of si . when that core reaches the chandresekar mass , the star begins to collapse . gravitational work is done on the infalling matter , the temperature increases , and the increased density and elevated electron chemical potential begin to favor weak - interaction conversion of protons to neutrons , with the emission of @xmath4s . neutrino emission is the mechanism by which the star radiates energy and lepton number . once the density exceeds @xmath5 10@xmath6 g/@xmath7 in the infall of a type ii supernova , however , neutrinos become trapped within the star by neutral - current scattering , @xmath8 that is , the time required for neutrinos to random walk out of the star exceeds @xmath9 . thus neither the remaining lepton number nor the gravitational energy released by further collapse can escape . after core bounce a hot , puffy protoneutron star remains . over times on the order of a few seconds , much longer than the 100s of milliseconds required for collapse , the star gradually cools by emission of neutrinos of all flavors . as the neutrinos diffuse outward , they tend to remain in flavor equilibrium through reactions such as @xmath10 producing a rough equipartition of energy / flavor . near the trapping density of 10@xmath6 g/@xmath7 the neutrinos decouple , and this decoupling depends on flavor due to the different neutrino - matter cross sections , @xmath11 one concludes that heavy - flavor neutrinos , because of their weaker cross sections for scattering off electrons ( and the absence of charged - current reactions off nucleons ) , will decouple at higher densities , deeper within the protoneutron star , where the temperature is higher . in the case of electron neutrinos , the @xmath4s are more tightly coupled to the matter than the @xmath12s , as the matter is neutron rich . the result is the expectation of spectral differences among the flavors . if spectral peaks are used to define an effective fermi - dirac temperature , then supernova models @xcite typically yield values such as @xmath13 some of the issues relevant to subsequent neutrino - induced nucleosynthesis include : + @xmath14 the @xmath4 and @xmath12 temperatures are important for the p / n chemistry of the `` hot bubble '' where the r - process is thought to occur . this is high - entropy material near the mass - cut that is blown off the protoneutron star by the neutrino wind . + @xmath14 matter - enhanced neutrino oscillations , in principle , could generate temperature inversions affecting p @xmath15 n charge - current balance , thus altering conditions in the `` hot bubble '' necessary for a successful @xmath3-process . + @xmath14 if the `` hot bubble '' is the @xmath3-process site , then synthesized nuclei are exposed to an intense neutrino fluence that could alter the r - process distribution . the relevant parameter is the neutrino fluence after r - process freezeout . following the chlorine , gallex / sage , and kamioka / super - kamiokande experiments , strong but circumstantial arguments led to the conclusion that the data indicated new physics . for example , it was observed that , even with arbitrary adjustments in the undistorted fluxes of pp , @xmath16be , and @xmath17b fluxes , the experimental results were poorly reproduced @xcite . when neutrino oscillations were included , however , several good fits to the data were found . these included the small - mixing - angle ( sma ) and large - mixing - angle ( lma ) msw solutions , the low solution , and even the possibility of `` just - so '' vacuum oscillations , where the oscillation length is comparable to the earth - sun separation . the ambiguities were convincingly removed by the charged- and neutral - current results of sno , which demonstrated that about 2/3rds of the solar neutrino flux was carried by heavy - flavor neutrinos @xcite . similarly , anomalies in atmospheric neutrino measurements a zenith - angle dependence in the ratio of electron - like to muon - like events indicated a distance - dependence in neutrino survival properties consistent with oscillations . the precise measurements of super - kamiokande provided convincing evidence for this conclusion , and thus for massive neutrinos @xcite . a summary of recent discoveries in neutrino physics include : + @xmath14 oscillations in matter can be strongly enhanced . + @xmath14 sno identified a unique two - flavor solar neutrino solution corresponding to @xmath18 and @xmath19 ev@xmath20 . + @xmath14 the kamland reactor @xmath12 disappearance experiment has confirmed the sno conclusions and narrowed the uncertainty on @xmath21 @xcite . + @xmath14 the super - kamiokande atmospheric neutrino results show that those data require a distinct @xmath22 ev@xmath20 and a mixing angle @xmath23 that is maximal , to within errors . + @xmath14 the kek - to - kamioka oscillation experiment k2k is consistent with the super - kamiokande atmospheric results , finding @xmath24 ev@xmath20 under the assumption of maximal mixing @xcite . + @xmath14 chooz and palo verde searches for reactor @xmath12 disappearance over the @xmath25 distance scale have provided null results , limiting @xmath26 @xcite . these results have determined two mass splittings , @xmath21 and the magnitude of @xmath27 . but as only mass differences are known , the overall scale is undetermined . likewise , because the sign of @xmath25 is so far unconstrained , two mass hierarchies are possible : the `` ordinary '' one where the nearly degenerate 1,2 mass eigenstates are light while eigenstate 3 is heavy , and the inverted case where the 1,2 mass eigenstates are heavy while eigenstate 3 is light . the relationship between the mass eigenstates @xmath28 and the flavor eigenstates @xmath29 is given by the mixing matrix , a product of the three rotations 1 - 2 ( solar ) , 1 - 3 , and 2 - 3 ( atmospheric ) : @xmath30 @xmath31 here @xmath32 , etc . we see , in addition to the unknown third mixing angle @xmath26 , this relationship depends on one dirac cp - violating phase parameterized by @xmath33 and two majorana cp - violating phases parameterized by @xmath34 and @xmath35 . the former could be measured in long - baseline neutrino oscillation experiments ( with the ease of this depending on the size of @xmath36 ) , while the latter could influence rates for double beta decay . these new neutrino physics discoveries could have a number of implications for supernova physics : + @xmath14 because of solar neutrinos , we have been able to probe matter effects up to densities @xmath37 g/@xmath7 characteristic of the solar core . as the density at the supernova neutrinosphere is @xmath38 g/@xmath7 , supernova neutrinos propagate in an msw potential that can be 10 orders of magnitude greater than any we have tested experimentally . in addition , the neutrinos propagate in a dense neutrino background , generating new msw potential contributions due to @xmath39 scattering . such effects , as well as the magnitude of the ordinary - matter msw effects , may be unique to the supernova environment . + @xmath14 we do not know @xmath26 , which is the crucial mixing angle for supernovae . this angle governs the @xmath4-heavy flavor level crossing encountered at depth in the star . this crossing occurs near the base of the carbon zone in the progenitor star , and remains adiabatic for @xmath40 . for the ordinary hierarchy , the resulting @xmath41 crossing would lead to a hotter @xmath4 spectrum . for an inverted hierarchy , the crossing would be @xmath42 . + @xmath14 presumably the position of this crossing will be influenced by the neutrino background contribution to the msw potential . this nonlinear problem is rather complicated because the flavor content of the background evolves with time ( being @xmath4-dominated at early times ) . inelastic neutrino - nucleus interactions are important to a range of supernova problems , including neutrino nucleosynthesis , the detection of supernova neutrinos in terrestrial detectors , and neutrino - matter heating that could boost the explosion . the heavy - flavor neutrinos have an average energy @xmath43 mev . however the most effective energy for generating nuclear transitions can be substantially higher because cross sections grow with energy and because nuclear thresholds are more easily overcome by neutrinos on the high - energy tail of the thermal distribution . for the neutrino energy range of interest the allowed approximation , which includes only the gamow - teller @xmath44 and fermi @xmath45 operators , is often not adequate . ( these are given for charge - current reactions ; the allowed operator for inelastic neutral current reactions is @xmath46 . ) important additional contributions come from operators that depend on the three - momentum transfer @xmath47 , which can approach twice the neutrino energy , for back - angle scattering . consequently @xmath48 , where @xmath49 is the nuclear size , may not be small . such first - forbidden contributions may be as important as the allowed contribution for supernova @xmath50s and @xmath51s . for all but the lightest nuclei , cross sections must be estimated from nuclear models , such as the shell model . shell model wave functions are generated by diagonalizing an effective interaction in some finite hilbert space of slater determinants formed from shells @xmath52 . the space may be adequate for describing the low - momentum components of the true wave function , but not the high - momentum components induced by the rather singular short - range nn potential . the effective interaction , usually determined empirically , is a low - momentum interaction that corrects for the effects of the excluded , high - momentum excitations . similarly , effective operators should be used in evaluating matrix elements , such as those of the weak interaction operators under discussion here , and the shell model wave function should have a nontrivial normalization . in effective interaction theory , that normalization is the overlap of the model - space wave function with the true wave function . because effective interaction theory is difficult to execute properly in some sense it is as difficult as solving the original problem in the full , infinite hilbert space nuclear modelers take short cuts . often all effective operator corrections are ignored : bare operators are used . in other cases , phenomenological operator corrections can been deduced from systematic comparisons of shell - model predictions and experimental data . the gamow - teller operator is an interesting case . rather thorough comparisons of @xmath53 and @xmath54 shell - model predictions with measured allowed @xmath55-decay rates have yielded a simple , phenomenological effective operator : the axial coupling @xmath56 should be used rather than the bare value @xcite . this observation is the basis for many shell - model estimates of the gamow - teller response that governs allowed neutrino cross sections . many of the shell - model techniques are quite powerful . moments techniques based on the lanczos algorithm @xcite have been used to treat spaces of dimension @xmath57 : important supernova neutrino cross sections for fe and ni isotopes have been derived in this way @xcite . another shell - model - based method uses monte carlo sampling @xcite . there are reasons to have less confidence in corresponding estimates of first - forbidden effective operators . the first - forbidden operators include the vector operator @xmath58 and the axial - vector operators @xmath59_{0,1,2}$ ] . electron scattering and photo - absorption provide tests of the vector operator , but direct probes of the axial responses are lacking . unitarity is also an issue . standard shell - model spaces satisfy the sum - rule contraints for the gamow - teller operator : the operator can not generate transitions outside a full shell , for example . in contrast , for harmonic - oscillator slater determinants , the first - forbidden operators generate transitions for which @xmath60 , where @xmath61 is the principal quantum number . thus , underlying sum rules are violated as the operators always connect either initial or final configurations to states outside the shell - model space . when the full momentum dependence of the weak interaction operators is included , the resulting spin - spatial structure includes forms such as @xmath62_{j m_j}\ ] ] where @xmath63 is a spherical bessel function , @xmath64 a spherical harmonic , and @xmath65 is a single - particle spin function . ( more complicated forms involve vector spherical harmonics combined with spatial operators such as @xmath66 . ) the fact that @xmath47 can not then be factored from the operator then makes lanczos moments techniques less useful : at every desired @xmath47 the lanczos procedure has to be repeated . ( there are techniques under development @xcite which exploit special properties of the harmonic oscillator to circumvent this problem . ) thus most calculations that treat the full momentum - dependence of the weak operators have used simple spaces , ones for which state - by - state summations of the weak transition strengths are practical . the approaches include truncated shell - model spaces , models based on the random phase approximation ( rpa ) , and even the highly schematic goldhaber - teller model . figure 2 compares continuum rpa results for charge - current reactions on @xmath67o with shell - model results of the sort described above @xcite . the quantities plotted are cross sections averaged over a thermal neutrino spectrum . this is an interesting test case because @xmath67o , naively a closed - shell nucleus , has a smaller gamow - teller response than most mid - shell nuclei . thus momentum - dependent contributions to the cross section should be more important than in many other cases . it is perhaps surprising , given the assumptions implicit in both the shell - model and crpa calculations , that the results agree so well over the full range of interesting supernova neutrino temperatures . the only significant discrepancy , at very low temperatures , is due to the inclusion of contributions from @xmath68o in the shell model calculation used in fig . 2 . ( the calculations were done for a natural oxygen target . ) due to the very low threshold for @xmath68o @xmath69 @xmath68f , this minor isotope ( 0.2% relative to @xmath67o ) dominates the o(@xmath4,e ) cross section at sufficiently low temperatures . the good agreement between the shell - model and crpa calculations , of course , could mask problems associated with common assumptions , such as the absence of a reliable procedure for assessing effective operators beyond the allowed approximation . several rare isotopes are thought to be created during a core - collapse supernova by neutrino reactions in the mantle of the star @xcite . the most common mechanism is inelastic neutral - current neutrino scattering off a target nucleus like @xmath70ne or @xmath6c , with significant energy transfer , e.g. , giant resonance excitation . the nucleus , excited above the continuum , then decays by nucleon or @xmath71 emission , leading to new nuclei . a nuclear network calculation is required to assess the survival of the neutrino - process products , such as @xmath72f in the ne shell and @xmath73b in the c shell . the co - produced nucleons can capture back on the daughter nucleus , destroying the product of interest . similarly , passage of the shock wave leads to heating that can destroy the product by @xmath74 and similar reactions . frequently the majority of the instantaneous production is lost due to such explosive processing . the enormous fluence of neutrinos can yield significant productions . typically 1% of the nuclei in the deep mantle of the star the c , ne , and o shells are transmuted by neutrinos . the most important products , like @xmath72f and @xmath73b , tend to be relatively rare odd - a isotopes neighboring very plentiful parent nuclei , such as @xmath70ne and @xmath6c . ( the parent isotopes are the hydrostatic burning products , typically . ) the natural abundances of such odd - a isotopes could , in principle , be due to neutrino processing . such nucleosynthesis calculations must be embedded in a model of the supernova event . important factors include : + @xmath14 a neutrino flux that tends to diminish exponentially , with a typical time scale @xmath75 @xmath5 3 sec ; + @xmath14 a pre - processing phase where nuclei in the mantle are exposed to the neutrino flux at some fixed radius @xmath3 , prior to shock arrival ; + @xmath14 a post - processing phase after shock wave arrival , where the material exposed to the neutrino flux is heated by the shock wave ( potentially destroying pre - processing productions ) , and then expands adiabatically off the star , with a temperature @xmath76 that consequently declines exponentially ; + @xmath14 integration of these neutrino contributions into an explosive nucleosynthesis network ; and + @xmath14 integration over a galactic model , with some assumptions on the range of stellar masses that will undergo core collapse and mantle ejection . calculations of this nature were done by woosley _ potentially significant neutrino - process productions include the nuclei @xmath72f , @xmath77b , @xmath16li , the gamma - ray sources @xmath78na and @xmath79al , @xmath80n , @xmath81p , @xmath82cl , @xmath83k , @xmath84v , and @xmath85sc . although the nuclear reaction network stopped at intermediate masses , the very rare isotopes @xmath1la and @xmath2ta were also identified as likely @xmath0-process candidates . recent work by heger _ _ @xcite extends these calculations in important ways . first , the evolution of the progenitor star includes the effects of mass loss . second , a reaction network is employed that includes all of the heavy elements through bi , using updated reaction rates . third , the nuclear evaporation process emission of a proton , neutron , or @xmath71 by the excited nucleus is treated in a more sophisticated statistical model that takes into account known nuclear levels and their spins and parities . while the calculations lack a full set of neutrino cross sections , those cross sections important to known ( _ e.g. _ , @xmath72f and @xmath73b ) and suspected ( _ e.g. _ , @xmath1la and @xmath2ta ) neutrino products were evaluated and incorporated into the network . the results are shown in fig . 3 , with production factors normalized to that of @xmath67o . thus a production factor of one would mean that the @xmath0-process would fully account for the observed abundance of that isotope . while @xmath73b might be slightly overproduced and @xmath72f slightly underproduced , given nuclear and astrophysics uncertainties , the @xmath0-process yields of these isotopes and @xmath1la are compatible with this being their primary origin . the case of @xmath1la is particularly interesting , as the primary channel for the production is charged - current reaction @xmath1ba(@xmath0,e)@xmath1la , where @xmath1ba is enhanced in the progenitor star by the @xmath86-process . this production is the only known case where a charged - current channel dominates the production . thus this yield is sensitive to the @xmath4 temperature a potential indicator for oscillations if the transformation occurs deep within the star . while @xmath2ta appears to be overproduced , the calculation does not distinguish production in the 9@xmath87 isomeric state from production in the 1@xmath88 ground state . only the isomeric fraction should be counted . an estimate of the @xmath0-process fraction that ends up in the isomeric state , following a @xmath89 cascade , has not been made . however , initiating reactions such as @xmath90hf(@xmath4,e@xmath87n ) involve low - spin parent isotopes , and the neutrino reaction transfers little angular momentum . thus one would anticipate that the majority of the yield would cascade to the ground state . the reduction factor of 3 - 4 required to bring the @xmath2ta production in line with the others of fig . 3 is compatible with this . there are other mechanisms for producing some of these @xmath0-process products . one interesting one , for example , is cosmic - ray spallation reactions on cno nuclei in the interstellar medium , which can produce @xmath77b and @xmath91li . some such process is required to explain the origin of @xmath92b , for example . cosmic - ray production , a secondary process , and the @xmath0-process , a primary mechanism , might be distinguished by measurements that would separately determine the evolution of @xmath92b and @xmath73b . if the @xmath0-process fraction of @xmath73b could be convincingly determined , this production would then become a more quantitative test of explosive conditions within the supernova carbon shell . we note two recent observational results relevant to the @xmath0-process . prochaska , howk , and wolfe @xcite recently observed over 25 elements in a galaxy at redshift @xmath93 = 2.626 , whose young age and high metallicity implies a nucleosynthetic pattern dominated by short - lived , massive stars . their finding of a solar b / o ratio in an approximately 1/3-solar - metallicity gas argues for a primary ( metal - independent ) production mechanism for b such as the @xmath0-process , rather than a secondary process . similarly , new f abundance data of cunha _ et al . _ show a low f / o ratio in two @xmath94 centauri stars , which argues against agb - star production of f ( one competing suggestion ) , but would be consistent with the @xmath0-process production @xcite . other speakers have discussed the @xmath3-process and the likelihood that the `` hot bubble '' the high - entropy nucleon gas that is blown off the protoneutron star surface by the neutrino wind is the primary site for the r - process . the nuclear physics of the @xmath3-process tells us that the synthesis occurs when the neutron - rich nucleon soup is in the temperature range of @xmath95k , which , in the hot bubble @xmath3-process , might correspond to a freeze - out radius of ( 600 - 1000 ) km and a time @xmath5 10 seconds after core collapse . the neutrino fluence after freeze - out ( when the temperature has dropped below 10@xmath96k and the @xmath3-process stops ) is then @xmath5 @xmath97 ergs/(100km)@xmath20 . thus , after completion of the @xmath3-process , the newly synthesized material experiences an intense flux of neutrinos . this suggests that @xmath0-process postprocessing could affect the @xmath3-process distribution . comparing to our earlier discussion of carbon- and neon - zone synthesis by the @xmath0-process , it is apparent that neutrino effects could be much larger in the hot bubble @xmath3-process : the synthesis occurs _ much _ closer to the star , at @xmath5 600 - 1000 km . ( the ne - shell radius is @xmath5 20,000 km . ) for this radius and a freezeout time of 10s , the post - processing " neutrino fluence the fluence that can alter the nuclear distribution after the @xmath3-process is completed is about 100 times larger than that responsible for fluorine production in the ne zone . as approximately 1/300 of the nuclei in the ne zone interact with neutrinos , and noting that the relevant neutrino - nucleus cross sections scale roughly as a ( a consequence of the sum rules that govern first - forbidden responses ) , one quickly sees that the probability of a heavy @xmath3-process nucleus interacting with the neutrino flux is approximately unity . because the hydrodynamic conditions of the @xmath3-process are highly uncertain , one way to attack this problem is to work backward @xcite . we know the final @xmath3-process distribution ( what nature gives us ) and we can calculate neutrino - nucleus interactions relatively well . thus by subtracting from the observed @xmath3-process distribution the neutrino post - processing effects , we can determine what the @xmath3-process distribution looked like at the point of freeze - out . in fig . 4 , the real " @xmath3-process distribution - that produced at freeze - out - is given by the dashed lines , while the solid lines show the effects of the neutrino post - processing for a particular choice of fluence . the nuclear physics input into these calculations is precisely that previously described : gt and first - forbidden cross sections , with the responses centered at excitation energies consistent with those found in ordinary , stable nuclei , taking into account the observed dependence on @xmath98 . one important aspect of fig . 4 is that the mass shift is significant . this has to do with the fact that a 20 mev excitation of a neutron - rich nucleus allows multiple neutrons ( @xmath5 5 ) to be emitted . ( the binding energy of the last neutron in an @xmath3-process neutron - rich nuclei is about 2 - 3 mev under typical @xmath3-process conditions . ) the second thing to notice is that the relative contribution of the neutrino process is particularly important in the valleys " beneath the mass peaks : the reason is that the parents on the mass peak are abundant , and the valley daughters rare . in fact , it follows from this that the neutrino process effects can be dominant for precisely seven isotopes ( te , re , etc . ) lying in the valleys below the a=130 ( not shown ) and a=195 ( fig . 4 ) mass peaks . furthermore if an appropriate neutrino fluence is picked , these isotope abundances are correctly produced ( within abundance errors ) . the fluences are @xmath99 values in fine agreement with those that would be found in a hot bubble @xmath3-process . so this is circumstantial but significant evidence that the material near the mass cut of a type ii supernova is the site of the @xmath3-process : there is a neutrino fingerprint . a more conservative interpretation of these results , however , places a bound on the @xmath3-process post - processing neutrino fluence by insisting that these isotopes not be overproduced . this bound will hold even if there are other mechanisms , such as neutron emission accompanying the @xmath55 decay of @xmath3-process parent nuclei as they move to the valley of stability , that contribute to the abundances of these rare isotopes . this bound is then an interesting constraint on supernova dynamics : the neutrino fluence after freezeout depends on the flux at the time of freezeout and on the dynamic time scale ( or the velocity of the material being expelled from the supernova ) . this constraint is plotted in fig . 5 along with one imposed by the observed @xmath55-flow equilibrium of nuclei near the mass peak . ( the @xmath55-flow equilibrium requires that the neutrino flux at freezeout not exceed the value where the neutrino reactions would compete with @xmath55 decay . this would destroy the observed correlation between abundance and @xmath55 decay lifetime . ) together these two constraints place upper bounds on the luminosity at freezeout ( equivalently , a lower bound on the freezeout radius ) and on the dynamic timescale . this goal of this talk is to make some connections between supernova neutrino physics , the nuclear structure governing neutrino - nucleus interactions , and new neutrino properties . the main example used here , the neutrino process , connects observable abundances with supernova properties , such as the @xmath3-process freezeout radius and dynamic timescale . thus by identifying @xmath0-process products and by reducing the associated nuclear structure uncertainties that govern their abundances , one may be able to place significant constraints on the explosion mechanism . the conditions for the @xmath3-process itself and for various @xmath0-process productions are set by neutrino physics . for example , the p / n chemistry of the `` hot bubble '' is largely governed by charged - current p @xmath15 n reactions , while the productions of @xmath1la and @xmath72f depend primarily on charged - current interactions on @xmath1ba and on neutral - current reactions on @xmath70ne , respectively . thus in principle such productions could be influenced by oscillations that invert @xmath4 and heavy - flavor neutrino spectra ( or , in the case of an inverted hierarchy , @xmath12 and heavy - flavor antineutrino spectra ) . this is another important reason for exploring the nucleosynthetic `` fingerprints '' of supernova neutrinos . the productions identified so far that would be influenced by neutrino oscillations , such as @xmath1la , are created at densities above those characterizing the naive 1 - 3 msw matter crossing , at least in the pre - processing phase . however , we have noted that the location of msw crossings could be perturbed by neutrino background effects . furthermore , in the post - processing phase , crossings will arise in the rarified matter that expands off the star . this is a fascinating question for the @xmath3-process , and perhaps also for certain @xmath0-process productions . these observations should motivated further studies of the potential effects of new neutrino physics on supernova nucleosynthesis . s. e. woosley and w. c. haxton , _ nature _ * 554 * , 45 ( 1988 ) ; s. e. woosley , d. h. hartmann , r. d. hoffman , and w. c. haxton , _ astroph . j. _ * 356 * , 272 ( 1990 ) ; g. v. domogatskii and d. k. nadyozhin , _ sov . * 22 * , 297 ( 1978 ) .

this talk reviews three inputs important to neutrino - induced nucleosynthesis in a supernova : 1 ) `` standard '' properties of the supernova neutrino flux , 2 ) effects of phenomena like neutrino oscillations on that flux , and 3 ) nuclear structure issues in estimating cross sections for neutrino - nucleus interactions . the resulting possibilities for neutrino - induced nucleosynthesis or the @xmath0-process in massive stars are discussed . this includes two relatively recent extensions of @xmath0-process calculations to heavier nuclei , one focused on understanding the origin of @xmath1la and @xmath2ta and the second on the effects following @xmath3-process freezeout . from calculations of the neutrino post - processing of the @xmath3-process distribution , limits can be placed on the neutrino fluence after freezeout and thus on the dynamic timescale for the expansion .

nucl-th0406012

quantum entanglement , predicted by einstein - podolsky - rosen ( epr ) in 1935 has fascinated many physicists because of its highly counterintuitive properties @xcite . for a long time , it was regarded as a problem of philosophy which could not be tested . in 1981 , however , aspect et al . succeeded in showing experimental evidence of quantum entanglement @xcite . since then , more interest has been paid to how to make use of the quantum entanglement of epr pairs in the fields of quantum cryptography , quantum information and quantum teleportation @xcite . quantum entanglement due to pair creation is well studied in non - relativistic regimes . one interesting feature of pair creation in relativistic quantum theory is that of observer dependence @xcite . for instance , the quantum entanglement between two free modes of a scalar field becomes less entangled if observers who detect each mode are relatively accelerated . in @xcite , they considered two free modes of a scalar field in flat space . one is detected by an observer in an inertial frame and the other by a uniformly accelerated observer . they evaluated the entanglement negativity , a measure of entanglement for mixed states , between the two free modes , which started in a maximally entangled state , to characterize the quantum entanglement and found that the entanglement disappeared for the observer in the limit of infinite acceleration . observer dependent entanglement can also be discussed in the context of an expanding universe . the expansion of the universe produces pair creation . quantum entanglement has been investigated in this situation in @xcite . to see if entanglement itself exists between causally disconnected regions of de sitter space , entanglement entropy , a measure of entanglement for pure states , has been studied in the bunch - davies vacuum @xcite and in @xmath3-vacua @xcite . entanglement negativity has been also studied in @xcite . since it was found that the entanglement between causally separated regions in de sitter space exists , in @xcite the question was asked whether there are observable effects of entanglement on the cosmic microwave background ( cmb ) in our universe . it was recently shown that quantum entanglement is not the only kind of quantum correlations possible , and are merely a particular characterization of quantumness . in fact , other quantum correlations have now been experimentally found , and quantum discord is known to be a measure of all quantum correlations , including entanglement @xcite . this measure can be non - zero even in the absence of entanglement . quantum discord is a useful measure to discuss the performance of quantum computers , which has lead to a lot of works on this topic @xcite . in order to see the observer dependence of all quantum correlations ( ie . the total quantumness ) , the quantum discord between two free modes of a scalar field in flat space , which are detected by two observers in inertial and non - inertial frames respectively , has also been discussed @xcite . they found that the quantum discord , in contrast to the entanglement , never disappears , even in the limit of infinite acceleration . to construct a theory of gravity compatible with quantum field theory is one of the biggest challenges in modern physics . hence , in trying to understand this problem fully it is important to study quantumness in the context of curved space . moreover , it is one of the cornerstones of inflationary cosmology that the large scale structure of our universe and temperature fluctuations of the cmb originate from quantum fluctuations during the initial inflationary era . therefore , this study of quantumness in curved spaces may itself lead to a better understanding of the initial stages of our universe and more precise predictions for cosmological observations . to this aim , quantum discord has recently been studied in a cosmological context @xcite . in this paper , we extend the study of quantum discord in rindler space to that in de sitter space . in the case of rindler space , the effect of a non - inertial frame on the quantum correlations has been studied . since the non - inertial observer in rindler space corresponds to the observer in an open chart of de sitter space , we will see the effect of the curvature of the open chart on the quantum discord between two free modes of a massive scalar field in de sitter space . the organization of the paper is as follows . in section 2 , we review the quantum information theoretic basis of quantum discord . in section 3 , we quantize a free massive scalar field in de sitter space and express the bunch - davies vacuum in a global chart in term of the fock space in open charts of de sitter space . in section 4 , we introduce two observers , alice and bob , who detect two free modes of the scalar field which start in an entangle state . we obtain the density operator for alice and bob . in section 5 , we compute the entanglement negativity . in section 6 , we evaluate the quantum discord in de sitter space . in the final section we summarize our results and provide an outlook for possible future calculations . quantum discord is a measure of all quantum correlations including entanglement for two subsystems @xcite . for a mixed state , this measure can be nonzero even if the state is unentangled . it is defined by quantum mutual information and computed by optimizing over all possible measurements that can be performed on one of the subsystems . in classical information theory , the mutual information between two random variables @xmath4 and @xmath5 is defined as @xmath6 where the shannon entropy @xmath7 is used to quantify the ignorance of information about the variable @xmath4 with probability @xmath8 , and the joint entropy @xmath9 with the joint probability @xmath10 of both events @xmath4 and @xmath5 occuring . the mutual information ( [ mi1 ] ) measures how much information @xmath4 and @xmath5 have in common . using bayes theorem , the joint probability can be written in terms of the conditional probability as @xmath11 where @xmath12 is the probability of @xmath4 given @xmath5 . the joint entropy can then be written as @xmath13 . plugging this into eq . ( [ mi1 ] ) , the mutual information can be expressed as @xmath14 where @xmath15 has been used , and the conditional entropy is defined as @xmath16 . the average over @xmath5 of shannon entropy of event @xmath4 , given @xmath5 . ( [ mi1 ] ) and ( [ mi2 ] ) are classically equivalent expressions for the mutual information . if we try to generalize the concept of mutual information to quantum system , the above two equivalent expressions do not yield identical results because measurements performed on subsystem @xmath5 disturb subsystem @xmath4 . in a quantum system , the shannon entropy is replaced by the von neumann entropy @xmath17 where @xmath18 is a density matrix . the probabilities @xmath10 , @xmath8 and @xmath19 are replaced respectively by the density matrix of the whole system @xmath20 , the reduced density matrix of subsystem @xmath4 , @xmath21 , and the reduced density matrix of subsystem @xmath5 , @xmath22 . in order to extend the idea of the conditional probability @xmath12 to the quantum system , we use projective measurements of @xmath5 described by a complete set of projectors @xmath23 , where @xmath24 distinguishes different outcomes of a measurement on @xmath5 . there are of course many different sets of measurements that we can make . then the state of @xmath4 after the measurement of @xmath5 is given by @xmath25 a quantum analog of the conditional entropy can then be defined as @xmath26 which corresponds to the measurement that least disturbs the overall quantum state , that is , to avoid dependence on the projectors . thus , the quantum mutual information corresponding to the two expressions eqs . ( [ mi1 ] ) and ( @xmath27 ) is defined respectively by @xmath28 the quantum discord is then defined as the difference between the above two expressions @xmath29 the quantum discord thus vanishes in classical mechanics , however it appears not to in some quantum systems . mode functions of a free massive scalar field in open charts of de sitter space were studied in detail in @xcite . by using the bogoliubov transformation between the global and open charts , the density matrix and entanglement entropy were calculated in @xcite . in this section , we review their results . and @xmath30 are the causally disconnected open charts of de sitter space.,height=415 ] we consider a free scalar field @xmath31 with mass @xmath32 in de sitter space represented by the metric @xmath33 . the action is given by @xmath34\,.\end{aligned}\ ] ] the coordinate systems of open charts in de sitter space with the hubble radius @xmath35 can be obtained by analytic continuation from the euclidean metric and divided into three parts which we call @xmath36 , @xmath37 and @xmath30 as shown in figure [ fig1 ] . their metrics are given respectively by @xmath38\,,\nonumber\\ ds^2_c&=&h^{-2}\left[dt_c^2+\cos^2t_c\left(-dr_c^2+\cosh^2r_c\,d\omega^2\right)\right]\,,\nonumber\\ ds^2_l&=&h^{-2}\left[-dt^2_l+\sinh^2t_l\left(dr^2_l+\sinh^2r_l\,d\omega^2\right ) \right]\,,\end{aligned}\ ] ] where @xmath39 is the metric on the two - sphere . note that the region @xmath30 and @xmath36 covered by the coordinates @xmath40 and @xmath41 respectively are the two causally disconnected open charts of de sitter space and @xmath30 regions is a part of the timelike infinity where infinite volume exists . ] . the region @xmath37 is covered by the coordinate @xmath42 . the solutions of the klein - gordon equation are expressed by @xmath43 where @xmath44 or @xmath45 and @xmath46 are harmonic functions on the three - dimensional hyperbolic space . the eigenvalues @xmath47 normalized by @xmath48 take positive real values . the positive frequency mode functions corresponding to the euclidean vacuum ( the bunch - davies vacuum ) that are supported both on the @xmath36 and @xmath30 regions are @xmath49 where @xmath50 are the associated legendre functions and the index @xmath51 takes the values @xmath52 which distinguishes two independent solutions for each region @xcite . note that the effect of the curvature of the three - dimensional hyperbolic space starts to appear around @xmath53 . and the effects gets stronger as @xmath47 becomes smaller than @xmath54 . thus we can probe the effect of the curvature on quantum entanglement by varying @xmath47 . we define a mass parameter @xmath55 the normalization factor for the solutions in eq . ( [ solutions ] ) is given by @xmath56 since they form a complete orthonormal set of modes , the field can be expanded in terms of the creation and annihilation operators @xmath57\,,\end{aligned}\ ] ] where @xmath58 satisfies @xmath59 and the commutation relations are @xmath60=\delta(p - p')\delta_{\sigma,\sigma'}\delta_{\ell,\ell'}\delta_{m , m'}$ ] . in the following the indices @xmath61 of the operators and mode functions are omitted for simplicity unless there may be any confusion . the bogoliubov transformation between the bunch - davies vacuum and @xmath36 , @xmath30 vacua , derived in @xcite , is expressed as @xmath62 where @xmath63 are annihilated by @xmath64 , @xmath65 respectively and @xmath66 note that in the case of conformal invariance ( @xmath67 ) and masslessness ( @xmath68 ) , we find @xmath69 . the relation of creation and annihilation operators between the bunch - davies vacuum and @xmath36 , @xmath30 vacua is given by @xmath70 \ , \label{ac}\end{aligned}\ ] ] where @xmath71 is the ratio of the normalizations of the mode functions in the bunch - davies vacuum eq . ( [ norm ] ) and in the @xmath36 , @xmath30 vacua @xmath72 the phase factor that appears in the bogoliubov transformation is @xmath73 and other variables which originally come from the coefficients of the legendre functions in eq . ( [ solutions ] ) are @xmath74 where @xmath75 . @xmath76 and @xmath77 are given by @xmath78 where @xmath79 the solutions in the bunch - davies vacuum ( [ solutions ] ) are related to those in the @xmath36 , @xmath30 vacua through bogoliubov transformations ( [ bogoliubov1 ] ) . using this transformation , we find that the ground state of a given mode seen by an observer in the global chart corresponds to a two - mode squeezed state in the open charts . these two modes correspond to the fields observed in the @xmath36 and @xmath30 charts . if we probe only one of the open charts , say @xmath30 , we have no access to the modes in the causally disconnected @xmath36 region and must therefore trace over the inaccessible region . this situation is analogous to the relationship between an observer in a minkowski chart and another in one of the two rindler charts in flat space , in the sense that the global chart and minkowski chart cover the whole spacetime geometry while open charts and rindler charts cover only a portion of the spacetime geometry and thus there exists horizons . we start with two maximally entangled modes with @xmath80 and @xmath81 of the free massive scalar field in de sitter space , @xmath82 we assume that alice has a detector which only detects mode @xmath81 and bob has a detector sensitive only to mode @xmath83 . when bob resides in the @xmath30 region , the bunch - davies vacuum with mode @xmath83 can be expressed as a two - mode squeezed state of the @xmath36 and @xmath30 vacua @xmath84 where we expanded the exponent in eq . ( [ bogoliubov1 ] ) . the @xmath85 and @xmath86 refer to the two modes of the @xmath30 and @xmath36 open charts . the single particle excitation state is then calculated by operating eq . ( [ ac ] ) on eq . ( [ bogoliubov2 ] ) , @xmath87\ , . \label{excitation}\end{aligned}\ ] ] note that the overall phase factor stemming from eq . ( [ phase ] ) can be absorbed in the definition of states . since the @xmath36 region is inaccessible to bob , we need to trace over the states in the @xmath36 region . if we plug eqs . ( [ bogoliubov2 ] ) and ( [ excitation ] ) into the initial maximally entangled state ( [ entangle ] ) , the reduced density matrix after tracing out the states in the @xmath36 region is represented as @xmath88 where @xmath89 with @xmath90 . this is a mixed state . as a function of @xmath91 and @xmath83 ( left ) . the logarithmic negativity vanishes only in the limit of @xmath92 for both @xmath67 and @xmath93 . ( right).,height=226 ] as a function of @xmath91 and @xmath83 ( left ) . the logarithmic negativity vanishes only in the limit of @xmath92 for both @xmath67 and @xmath93 . ( right).,height=226 ] let us calcuate the entanglement negativity , a measure of entanglement for mixed states , first . this measure is defined by the partial transpose , the non - vanishing of which provides a sufficient criterion for entanglement . if at least one eigenvalue of the partial transpose is negative , then the density matrix is entangled . if we take the partial transpose with respect to alice s subsystem , then we find @xmath94 we compute the eigenvalues @xmath95 numerically and the resultant negative eigenvalues are plotted in the left panel of figure [ fig2 ] . we see that the negative eigenvalues for @xmath67 and @xmath93 go to zero in the limit of @xmath96 in order to see if the entanglement vanishes clearly , we calculate the logarithmic negativity next . . the blue line is for @xmath97 , @xmath54 , the yellow is for @xmath98 , @xmath99 and the green is for @xmath67 , @xmath93 . ( left ) plot of the logarithmic negativity as a function of @xmath91 for @xmath100 . ( right),height=226 ] . the blue line is for @xmath97 , @xmath54 , the yellow is for @xmath98 , @xmath99 and the green is for @xmath67 , @xmath93 . ( left ) plot of the logarithmic negativity as a function of @xmath91 for @xmath100 . ( right),height=226 ] the negativity is defined by summing over all the negative eigenvalues @xmath101 and there exists no entanglement when @xmath102 . however , this measure is not additive and not suitable for multiple subsystems . the logarithmic negativity is thus a better measure than the negativity in this scenario and is defined as @xmath103 the state is entangled when @xmath104 . we sum over all negative eigenvalues and calculate the logarithmic negativity . we focus on alice and bob s detectors for modes of momentum @xmath81 and @xmath83 , and thus do nt integrate either over @xmath83 , nor a volume integral over the hyperboloid . we plot the logarithmic negativity in the right panel of figure [ fig2 ] . for @xmath67 and @xmath93 , the logarithmic negativity vanishes in the limit of @xmath92 , that is , in the limit of infinite curvature . this is consistent with the flat space result where the entanglement vanishes in the limit of infinite acceleration of the observer . here we stress that the entanglement weakens , but survives even in the limit of infinite curvature for a massive scalar field other than @xmath67 and @xmath68 as can be seen from the left panel of figure [ fig3 ] . the logarithmic negativity as a function of the mass parameter @xmath91 is plotted in figure [ fig3 ] where we take @xmath100 . the qualitative feature of it is similar to the left panel of figure [ fig2 ] and we can read off that the entanglement gets smaller for @xmath67 and @xmath93 . the oscillatory behavior comes from the factor @xmath105 in @xmath106 of eq . ( [ gammap ] ) @xmath107 . next we calculate the overall quantumness of the system given by the quantum discord . we will make our measurement on the alice s side . the quantum discord is @xmath108 to calculate the above , we rewrite the state ( [ rhoab1 ] ) as @xmath109 where we define @xmath110 the state is split into alice s subsystem ( two dimensions ) and bob s subsystem ( infinite dimensional ) . then alice s density matrix is easy to obtain as @xmath111 and we get @xmath112 . for the von neumann entropy of the whole system @xmath113 , we need to find the eigenvalues of @xmath20 numerically . in order to calculate the quantum conditional entropy @xmath114 , we restrict ourselves to projective measurements on alice s subsystem described by a complete set of projectors @xmath115 where @xmath116 , @xmath117 is the @xmath118 identity matrix and @xmath119 are the pauli matrices . here , a choice of the @xmath120 corresponds to a choice of measurement , and we will thus be interested in the particular measurement which minimises the disturbance on the system . then the density matrix after the measurement is @xmath121 the trace of it is calculated as @xmath122\,,\nonumber\end{aligned}\ ] ] where we used @xmath123 and @xmath124=\frac{1}{2}\,.\end{aligned}\ ] ] by using the parametrization @xmath125 @xmath126 is found to be independent of the phase factor @xmath31 : @xmath127\,.\end{aligned}\ ] ] thus , the quantum discord ( [ discord1 ] ) is now expressed as @xmath128 we will find eigenvalues of @xmath20 , @xmath129 and @xmath130 numerically and find @xmath131 that minimizes the above @xmath132 . the left panel of figure [ fig4 ] displays the result where the black line shows the minima of @xmath132 for different values of @xmath83 and we can read off that @xmath133 minimize @xmath132 . the right panel shows that @xmath132 has a finite value even in a limit , @xmath92 , in which the entanglement vanishes . note that the convergence of the sum for @xmath20 is not fast for @xmath67 and @xmath93 in the limit of @xmath92 because @xmath134 in the summation becomes @xmath54 , so we have truncated our plot for small @xmath83 . and @xmath131 for @xmath67 . the black line shows the minima for different values of @xmath83 . the right panel shows the quantum discord has a finite value even in a limit that entanglement vanishes @xmath135 for @xmath67.,height=226 ] and @xmath131 for @xmath67 . the black line shows the minima for different values of @xmath83 . the right panel shows the quantum discord has a finite value even in a limit that entanglement vanishes @xmath135 for @xmath67.,height=226 ] we can calculate the quantum discord ( [ discord2 ] ) analytically in the @xmath136 limit for any mass of the scalar field . since @xmath137 , the density matrix @xmath20 becomes @xmath138\nonumber\\ & \sim&\frac{1}{2}\,|00\rangle\langle 00|+\frac{1}{4}\,|10\rangle\langle 10| + \frac{1}{2\sqrt{2}}\,|00\rangle\langle 11|+\frac{1}{2\sqrt{2}}\,|11\rangle\langle 00| + \frac{1}{4}\,|11\rangle\langle 11|\,.\end{aligned}\ ] ] this can be factorized as @xmath139 which has eigenvalues of @xmath20 of @xmath140 and @xmath141 . on the other hand , for @xmath126 , we find @xmath142\nonumber\\ & \sim&\frac{1}{2}\left ( \begin{array}{cc } \frac{3}{2}\pm\frac{\cos\theta}{2 } & \pm\frac{\sin\theta}{\sqrt{2}}\\ \pm\frac{\sin\theta}{\sqrt{2 } } & \frac{1}{2}\mp\frac{\cos\theta}{2 } \end{array } \right)\,.\end{aligned}\ ] ] the eigenvalues of @xmath126 are then found to be @xmath143 $ ] . finally the quantum discord ( [ discord2 ] ) in this limit is found to be @xmath144\,.\nonumber\end{aligned}\ ] ] in this limit , the asymptotic value of the quantum discord @xmath145 becomes @xmath146 , which agrees with the numerical result in the right panel of figure [ fig4 ] . we also plot @xmath132 as a function of @xmath83 for different values of the mass parameter @xmath91 in the left panel of figure [ fig5 ] . we see that @xmath132 decreases faster in the cases of @xmath67 and @xmath93 compared with other masses of the scalar field as we go to large curvatures @xmath92 . in order to compare the dependence of @xmath91 for large scales with the entanglement negativity in figure [ fig3 ] , we plotted it in the right panel of figure [ fig5 ] . the features are similar to one another and the origin of the oscillatory behavior is again in the @xmath147 in @xmath106 of eq . ( [ gammap ] ) @xmath107 . . the blue line is for @xmath97 , @xmath54 , the yellow is for @xmath98 , @xmath99 and the green is for @xmath67 , @xmath93.(left ) the quantum discord as a function of @xmath91 when @xmath100 and @xmath148 . ( right),height=226 ] . the blue line is for @xmath97 , @xmath54 , the yellow is for @xmath98 , @xmath99 and the green is for @xmath67 , @xmath93.(left ) the quantum discord as a function of @xmath91 when @xmath100 and @xmath148 . ( right),height=226 ] in this work , we investigated quantum discord between two free modes of a massive scalar field in a maximally entangled state in de sitter space . we introduced two observers , one in a global chart and the other in an open chart of de sitter space , and then determined the quantum discord created by each detecting one of the modes . this situation is analogous to the relationship between an observer in minkowski space and another in one of the two rindler wedges in flat space . in the case of rindler space , it is known that the entanglement vanishes when the relative acceleration becomes infinite @xcite . in de sitter space , on the other hand , the observer s relative acceleration corresponds to the scale of the curvature of the open chart . we first evaluated entanglement negativity and then quantum discord in de sitter space . we found that the state becomes less entangled as the curvature of the open chart gets larger . in particular , for a massless scalar field @xmath68 and a conformally coupled scalar field @xmath67 , the entanglement negativity vanishes in the limit of infinite curvature . however , we showed that quantum discord does not disappear even in the limit that the entanglement negativity vanishes . in addition , we found that the entanglement negativity survives even in the limit of infinite curvature for a massive scalar field . in this paper , we considered a simple quantum state ( [ entangle ] ) . it would be interesting to discuss quantum discord in the context of the multiverse @xcite ( see also related works @xcite ) . it would also be desirable to find a way to prove if the large scale structure of our universe and temperature fluctuations of the cmb are originated from quantum fluctuations during the initial inflationary era . we will leave these issues for future work . sk was supported by ikerbasque , the basque foundation for science . jshock is grateful for the national research foundation ( nrf ) of south africa under grant number 87667 . jsoda was in part supported by mext kakenhi grant number 15h05895 . a. einstein , b. podolsky and n. rosen , `` can quantum mechanical description of physical reality be considered complete ? , '' phys . 47 * , 777 ( 1935 ) . a. aspect , p. grangier and g. roger , `` experimental tests of realistic local theories via bell s theorem , '' phys . * 47 * , 460 ( 1981 ) . a. aspect , j. dalibard and g. roger , `` experimental test of bell s inequalities using time varying analyzers , '' phys . lett . * 49 * , 1804 ( 1982 ) . r. horodecki , p. horodecki , m. horodecki and k. horodecki , `` quantum entanglement , '' rev . phys . * 81 * , 865 ( 2009 ) [ quant - ph/0702225 ] . j. garriga , s. kanno , m. sasaki , j. soda and a. vilenkin , `` observer dependence of bubble nucleation and schwinger pair production , '' jcap * 1212 * , 006 ( 2012 ) [ arxiv:1208.1335 [ hep - th ] ] . j. garriga , s. kanno and t. tanaka , `` rest frame of bubble nucleation , '' jcap * 1306 * , 034 ( 2013 ) [ arxiv:1304.6681 [ hep - th ] ] . m. b. frb , j. garriga , s. kanno , m. sasaki , j. soda , t. tanaka and a. vilenkin , `` schwinger effect in de sitter space , '' jcap * 1404 * , 009 ( 2014 ) [ arxiv:1401.4137 [ hep - th ] ] . i. fuentes - schuller and r. b. mann , `` alice falls into a black hole : entanglement in non - inertial frames , '' phys . lett . * 95 * , 120404 ( 2005 ) [ quant - ph/0410172 ] . m. alsing , i. fuentes - schuller , r. b. mann and t. e. tessier , `` entanglement of dirac fields in non - inertial frames , '' phys . a * 74 * , 032326 ( 2006 ) [ quant - ph/0603269 ] . h. ollivier and w. h. zurek , quantum discord : a measure of the quantumness of correlations , " phys . lett . * 88 * , ( 2001 ) 017901 . l. henderson and v. vedral , classical , quantum and total correlations , " j. phys . a : math . gen . * 34 * , ( 2001 ) 6899 . [ quant - ph/0105028 ] . a. datta , quantum discord between relatively accelerated observers , " phys . rev . a * 80 * , 052304 ( 2009 ) j. wang , j. deng and j. jing , `` classical correlation and quantum discord sharing of dirac fields in non - inertial frames , '' phys . a * 81 * , 052120 ( 2010 ) [ arxiv:0912.4129 [ quant - ph ] ] . j. l. ball , i. fuentes - schuller and f. p. schuller , `` entanglement in an expanding spacetime , '' phys . a * 359 * , 550 ( 2006 ) [ quant - ph/0506113 ] . i. fuentes , r. b. mann , e. martin - martinez and s. moradi , `` entanglement of dirac fields in an expanding spacetime , '' phys . rev . d * 82 * , 045030 ( 2010 ) [ arxiv:1007.1569 [ quant - ph ] ] . y. nambu and y. ohsumi , `` classical and quantum correlations of scalar field in the inflationary universe , '' phys . d * 84 * , 044028 ( 2011 ) [ arxiv:1105.5212 [ gr - qc ] ] . j. maldacena and g. l. pimentel , `` entanglement entropy in de sitter space , '' jhep * 1302 * , 038 ( 2013 ) [ arxiv:1210.7244 [ hep - th ] ] . s. kanno , j. murugan , j. p. shock and j. soda , `` entanglement entropy of @xmath3-vacua in de sitter space , '' jhep * 1407 * , 072 ( 2014 ) [ arxiv:1404.6815 [ hep - th ] ] . n. iizuka , t. noumi and n. ogawa , `` entanglement entropy of de sitter space @xmath3-vacua , '' nucl . b * 910 * , 23 ( 2016 ) [ arxiv:1404.7487 [ hep - th ] ] . s. kanno , j. p. shock and j. soda , `` entanglement negativity in the multiverse , '' jcap * 1503 * , no . 03 , 015 ( 2015 ) [ arxiv:1412.2838 [ hep - th ] ] . e. a. lim , `` quantum information of cosmological correlations , '' phys . d * 91 * , no . 8 , 083522 ( 2015 ) [ arxiv:1410.5508 [ hep - th ] ] . j. martin and v. vennin , `` quantum discord of cosmic inflation : can we show that cmb anisotropies are of quantum - mechanical origin ? , '' phys . d * 93 * , no . 2 , 023505 ( 2016 ) [ arxiv:1510.04038 [ astro-ph.co ] ] . m. sasaki , t. tanaka and k. yamamoto , `` euclidean vacuum mode functions for a scalar field on open de sitter space , '' phys . d * 51 * , 2979 ( 1995 ) [ gr - qc/9412025 ] . s. kanno , `` cosmological implications of quantum entanglement in the multiverse , '' phys . lett . b * 751 * , 316 ( 2015 ) [ arxiv:1506.07808 [ hep - th ] ] . a. albrecht , n. bolis and r. holman , `` cosmological consequences of initial state entanglement , '' jhep * 1411 * , 093 ( 2014 ) [ arxiv:1408.6859 [ hep - th ] ] . s. kanno , `` a note on initial state entanglement in inflationary cosmology , '' europhys . lett . * 111 * , no . 6 , 60007 ( 2015 ) [ arxiv:1507.04877 [ hep - th ] ] . h. collins and t. vardanyan , `` entangled scalar and tensor fluctuations during inflation , '' arxiv:1601.05415 [ hep - th ] . n. bolis , a. albrecht and r. holman , `` modifications to cosmological power spectra from scalar - tensor entanglement and their observational consequences , '' arxiv:1605.01008 [ hep - th ] .

we study quantum discord between two free modes of a massive scalar field in a maximally entangled state in de sitter space . we introduce two observers , one in a global chart and the other in an open chart of de sitter space , and the observers determine the quantum discord created by each detecting one of the modes this situation is analogous to the relationship between an observer in a minkowski chart and another in one of the two rindler charts in flat space . we find that the state becomes less entangled as the curvature of the open chart gets larger . in particular , for the cases of a massless , and a conformally coupled scalar field , the entanglement vanishes in the limit of infinite curvature . however , we find that the quantum discord never disappears even in the limit that entanglement disappears . = 1 mm kobe - cosmo-16 - 09 + sugumi kanno@xmath0 , jonathan p. shock@xmath1 and jiro soda@xmath2 1.5 cm

1608.02853

the higgs potential of the standard model ( sm ) , which is crucial in implementing the mechanism of spontaneous symmetry breaking , contains the unknown quartic coupling of the higgs field . as a consequence , the mass of the only higgs boson in the sm , which is determined by this quartic coupling , is not known @xcite . if a higgs boson is discovered and its mass measured , the higgs potential of the standard model can be uniquely determined . on the other hand , supersymmetry is at present the only known framework in which the higgs sector of the standard model ( sm ) , so crucial for its internal consistency , is natural @xcite . the minimal version of the supersymmetric standard model ( mssm ) contains two higgs doublets @xmath19 with opposite hypercharges : @xmath20 , @xmath21 , so as to generate masses for up- and down - type quarks ( and leptons ) , and to cancel gauge anomalies . after spontaneous symmetry breaking induced by the neutral components of @xmath22 and @xmath23 obtaining vacuum expectation values , @xmath24 , @xmath25 , @xmath26 , the mssm contains two neutral @xmath0-even - even higgs particles as @xmath1 and @xmath3 . ] ( @xmath1 , @xmath3 ) , one neutral @xmath0-odd ( @xmath5 ) , and two charged ( @xmath27 ) higgs bosons @xcite . although gauge invariance and supersymmetry fix the quartic couplings of the higgs bosons in the mssm in terms of @xmath28 and @xmath29 gauge couplings , @xmath30 and @xmath31 , respectively , there still remain two independent parameters that describe the higgs sector of the mssm . these are usually chosen to be @xmath32 and @xmath33 , the mass of the @xmath0-odd higgs boson . all the higgs masses and the higgs couplings in the mssm can be described ( at tree level ) in terms of these two parameters . in particular , all the trilinear self - couplings of the physical higgs particles can be predicted theoretically ( at the tree level ) in terms of @xmath33 and @xmath32 . once a light higgs boson is discovered , the measurement of these trilinear couplings can be used to reconstruct the higgs potential of the mssm . this will go a long way toward establishing the higgs mechanism as the basic mechanism of spontaneous symmetry breaking in gauge theories . although the measurement of all the higgs couplings in the mssm is a difficult task , preliminary theoretical investigations by plehn , spira and zerwas @xcite , and by djouadi , haber and zerwas @xcite ( referred to as ` dhz ' in the following ) , of the measurement of these couplings at the lhc and at a high - energy @xmath11 linear collider , respectively , are encouraging . in this paper we consider in detail the question of possible measurements of the trilinear higgs couplings of the mssm at a high - energy @xmath11 linear collider . we assume that such a facility will operate at an energy of 500 gev with an integrated luminosity per year of @xmath34 @xcite . ( this is a factor of 10 more than the earlier estimate . ) in a later phase one may envisage an upgrade to an energy of 1.5 tev . since the ` interesting ' cross sections fall off like @xmath35 , the luminosity should increase by a corresponding factor . an earlier estimated luminosity of @xmath36 at 1.5 tev may turn out to be too conservative . the trilinear higgs couplings that are of interest are @xmath9 , @xmath10 , and @xmath37 , involving both the @xmath0-even ( @xmath1 , @xmath3 ) and @xmath0-odd ( @xmath5 ) higgs bosons . the couplings @xmath9 and @xmath10 are rather small with respect to the corresponding trilinear coupling @xmath38 in the sm ( for a given mass of the lightest higgs boson @xmath39 ) , unless @xmath39 is close to the upper value ( decoupling limit ) . the coupling @xmath37 remains small for all parameters . throughout , we include one - loop radiative corrections @xcite to the higgs sector in the effective potential approximation . in particular , we take into account the parameters @xmath5 and @xmath40 , the soft supersymmetry breaking trilinear parameter and the bilinear higgs(ino ) parameter in the superpotential , respectively , and as a consequence the left right mixing in the squark sector , in our calculations . we thus include all the relevant parameters of the mssm in our study , which is more detailed than the preliminary one of dhz . for a given value of @xmath39 , the values of these couplings significantly depend on the soft supersymmetry - breaking trilinear parameter @xmath5 , as well as on @xmath40 , and thus on the resulting mixing in the squark sector . since the trilinear couplings tend to be small , and depend on several parameters , their effects are somewhat difficult to estimate . the plan of the paper is as follows . in section 2 we review the higgs sector of the mssm , including the radiative corrections to the masses . the trilinear couplings are presented in section 3 . in section 4 we review the possible production mechanisms for the multiple production of higgs bosons through which the trilinear higgs couplings can be measured at an @xmath11 linear collider . in section 5 we consider the dominant source of the multiple production of the higgs ( @xmath1 ) boson through higgs - strahlung of @xmath3 , and through production of @xmath3 in association with the @xmath0-odd higgs boson ( @xmath5 ) , and the background to these processes . this source of multiple production can be used to extract the trilinear higgs coupling @xmath9 . section 6 deals with a detailed calculation of the cross section for the double higgs - strahlung process @xmath41 . this process involves the trilinear couplings @xmath9 and @xmath10 of the @xmath0-even higgs bosons ( @xmath1 , @xmath3 ) . in section 7 we consider the different fusion mechanisms for multiple @xmath1 production , especially the non - resonant process @xmath42 , for which we present a detailed calculation of the cross section in the ` effective @xmath43 approximation ' . this process also involves the two trilinear higgs couplings , @xmath9 and @xmath10 , and is the most useful one for extracting the coupling @xmath10 . in section 8 we present , based on our calculations , the regions of the mssm parameter space in which the trilinear couplings @xmath9 and @xmath10 could be measured ; finally , in section 9 we present a summary of our results and conclusions . in this section we review the higgs sector of the minimal supersymmetric standard model in order to set the notation and to describe the approximations we use in our calculations . as mentioned in the introduction , we shall include the dependence on the parameters @xmath5 and @xmath40 through mixing in the squark sector . where there is an overlap , our notation and approach closely follow those of ref . @xcite . at the tree level , the higgs sector of the mssm is described by two parameters , which can be conveniently chosen as @xmath33 and @xmath32 @xcite . there are , however , substantial radiative corrections to the @xmath0-even neutral higgs masses and couplings @xcite . in the one - loop effective potential approximation , the radiatively corrected squared - mass matrix for the @xmath0-even higgs bosons can be written as @xcite @xmath44 \nonumber \\ & & + \frac{3 g^2}{16 \pi^{2 } m_w^2 } \left [ \begin{array}{cc } \delta_{11 } & \delta_{12}\\ \delta_{12 } & \delta_{22 } \end{array } \right ] , \end{aligned}\ ] ] where the second matrix represents the radiative corrections . the functions @xmath45 depend , besides the top- and bottom - quark masses , on the higgs bilinear parameter @xmath40 in the superpotential , the soft supersymmetry - breaking trilinear couplings ( @xmath46 , @xmath47 ) and soft scalar masses ( @xmath48 , @xmath49 , @xmath50 ) , as well as on @xmath32 . we shall ignore the @xmath51-quark mass effects in @xmath45 in our calculations , which is a reasonable approximation for moderate values of @xmath5230 . furthermore , we shall assume , as is often done , @xmath53 with these approximations we can write ( @xmath54 is the top quark mass ) @xcite : @xmath55 where @xmath56 and @xmath57 are squared stop masses given by @xmath58 ( we have ignored the small @xmath59-term contributions to the stop masses ) and @xmath60 the one - loop radiatively corrected masses ( @xmath61 , @xmath62 ; @xmath63 ) of the @xmath0-even higgs bosons ( @xmath1 , @xmath3 ) can be obtained by diagonalizing the @xmath64 mass matrix in eq . ( [ eq : m2 ] ) . the radiative corrections are , in general , positive , and they shift the mass of the lightest higgs boson upwards from its tree - level value . we show in fig . [ fig : masses ] the resulting mass of the lightest higgs boson , @xmath39 , as a function of @xmath40 and @xmath32 , for two values of @xmath5 and two values of @xmath33 , and for @xmath65 tev . with a wider range of parameter values , or when the squark mass scale is taken to be smaller , the dependence on @xmath40 and @xmath32 can be more dramatic @xcite . the higgs mass falls rapidly at small values of @xmath32 . since the lep experiments are obtaining lower bounds on the mass of the lightest higgs boson , they are beginning to rule out significant parts of the small-@xmath32 parameter space , depending on the model assumptions . for @xmath66 , aleph finds @xmath67 at 95% c.l . gev , irrespective of @xmath32 , and a limit of @xmath68 gev for @xmath69 @xcite . ] in our calculations , we shall therefore take @xmath70 to be a representative value . [ for a recent discussion on how the lower allowed value of @xmath32 depends on some of the model parameters , see ref . @xcite . ] the trilinear higgs couplings that are of interest can be written @xcite as a sum of the tree - level coupling and one - loop radiative corrections : @xmath71 in units of @xmath72 , the tree - level couplings are given by @xmath73 with @xmath74 the mixing angle in the @xmath0-even higgs sector , which can be calculated in terms of the parameters appearing in the @xmath0-even higgs mass matrix ( [ eq : m2 ] ) . the one - loop radiative corrections in ( [ eq : lambda - hhh])([eq : lambda - haa ] ) are ( in the above units ) : @xmath75 \left[1+(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^2 \right . \nonumber \\ & & + \left . \left.\frac{m_t^2}{m_{\tilde t_2}^2 } \left[1-(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t e_t\right ] \left[1-(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^2 - 2 \right ) \right ] , \label{17 } \\ \delta \lambda_{hhh } & = & \left ( \frac{3g^2 \cos^2\theta_w}{16 \pi^2 } \frac{m_t^4}{m_w^4 } \frac{\cos^3\alpha}{\sin^3\beta } \right ) \nonumber \\ & \times & \left [ 3\log\frac{m_{\tilde t_1}^2 m_{\tilde t_2}^2}{m_t^4 } + 3 ( m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t \log\frac{m_{\tilde t_1}^2}{m_{\tilde t_2}^2 } \right . \nonumber\\ & & + \left . 2\left(\frac{m_t^2}{m_{\tilde t_1}^2 } \left[1+(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^3 + \frac{m_t^2}{m_{\tilde t_2}^2 } \left[1-(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^3 -2 \right ) \right ] , \label{16 } \ ] ] @xmath76 , \label{21}\end{aligned}\ ] ] where @xmath77 and we have ignored the contributions from @xmath51-quarks and @xmath51-squarks , which are in general small with respect to those arising from @xmath78-quarks and @xmath78-squarks . we have also adopted the simplification described in eq . ( [ 2 ] ) in writing the above results . we shall make these approximations throughout this paper . we show in figs . [ fig : lamhhh2 ] , [ fig : lamhhh2 ] and [ fig : lamhaa2 ] the couplings @xmath9 , @xmath10 and @xmath37 as functions of @xmath40 and @xmath32 , for two values of @xmath5 and two values of @xmath33 , all for @xmath65 tev . the explicit dependence on @xmath5 and @xmath40 is not dramatic , but it should be kept in mind that unless @xmath33 is rather small , @xmath39 may change considerably with @xmath5 . the trilinear couplings change significantly with @xmath33 , and thus also with @xmath39 . this is shown more explicitly in fig . [ fig : lam - mh ] , where we compare @xmath9 , @xmath10 and @xmath37 for three different values of @xmath32 , and the sm quartic coupling @xmath79 . the sm quartic coupling includes one - loop radiative corrections @xcite , and its normalization is such that at the tree - level , it coincides with the trilinear coupling . at low values of @xmath39 , the mssm trilinear couplings are rather small . for some value of @xmath39 the couplings @xmath9 and @xmath10 start to increase in magnitude , whereas @xmath37 remains small . the values of @xmath39 at which they start becoming significant depend crucially on @xmath32 . for @xmath70 ( fig . [ fig : lam - mh]a ) this transition takes place around @xmath80100 gev , whereas for @xmath81 and 15 , the critical values of @xmath39 increase to 100110 and 120 gev , respectively ( see figs . [ fig : lam - mh]b and c ) . in this region , the actual values of @xmath9 and @xmath10 ( for a given value of @xmath39 ) change significantly if @xmath5 becomes large and positive . a non - vanishing squark - mixing parameter @xmath5 is thus seen to be quite important . also , we note that for special values of the parameters , the couplings may vanish @xcite . see also fig . 1 of ref . @xcite . to sum up the behaviour of the trilinear couplings , we note that @xmath9 and @xmath10 are small ( @xmath82 ) for @xmath83120 gev , depending on the value of @xmath32 . however , as @xmath39 approaches its maximum value , which is reached rapidly as @xmath33 becomes large , @xmath84 gev , these trilinear couplings become large ( @xmath85 ) . thus , as functions of @xmath33 , the trilinear couplings @xmath9 and @xmath10 are large for most of the parameter space . we also note that , for large values of @xmath32 , @xmath9 tends to be relatively small , whereas @xmath10 becomes large , if also @xmath33 ( or , equivalently , @xmath39 ) is large . we note that for a given higgs boson mass @xmath39 , the tree level sm trilinear higgs coupling is given by @xmath86 on the other hand , for large values of @xmath33 ( the decoupling limit ) the corresponding mssm trilinear coupling , eq . ( [ eq : lambda - hhh0 ] ) , becomes @xmath87 i.e. , it approaches the sm trilinear coupling . the different mechanisms for the multiple production of the mssm higgs bosons in @xmath11 collisions have been discussed by dhz . the dominant mechanism for the production of multiple @xmath0-even light higgs bosons ( @xmath1 ) is through the production of the heavy @xmath0-even higgs boson @xmath3 , which then decays by @xmath8 . the heavy higgs boson @xmath3 can be produced by @xmath3-strahlung , in association with @xmath5 , and by the resonant @xmath43 fusion mechanism . these mechanisms for multiple production of @xmath1 @xmath88 are shown in fig . [ fig : feynman - resonant ] . we note that all the diagrams of fig . [ fig : feynman - resonant ] involve the trilinear coupling @xmath9 . a background to ( [ eq : res - hhh ] ) comes from the production of the pseudoscalar @xmath5 in association with @xmath1 and its subsequent decay to @xmath89 @xmath90 leading to @xmath91 final states [ see fig . [ fig : feynman - nonres - zhh]d ] . a second mechanism for @xmath92 production is double higgs - strahlung in the continuum with a @xmath93 boson in the final state [ see fig . [ fig : feynman - nonres - zhh]a d ] , @xmath94 we note that the feynman diagram of fig . [ fig : feynman - nonres - zhh]c involves , apart from the coupling @xmath9 , the trilinear higgs coupling @xmath10 as well , whereas the other diagrams do not involve any of the trilinear higgs couplings . a third way of generating multiple higgs bosons in @xmath11 collisions is through associated production of ( @xmath92 ) with the pseudoscalar @xmath5 in the continuum [ see fig . [ fig : feynman - nonres - ahh ] ] : @xmath95 this process will be briefly discussed in section 6 . it involves , besides @xmath9 and @xmath10 , the trilinear coupling @xmath37 as well . it is , however , difficult @xcite to measure this coupling @xmath37 through the process ( [ eq : zstar - hha ] ) . finally , there is a mechanism of multiple production of the lightest higgs boson through non - resonant @xmath43 ( @xmath96 ) fusion in the continuum [ see fig . [ fig : feynman - nonres - ww ] ] : @xmath97 which will be discussed in section 7 . it is important to note that all the diagrams of fig . [ fig : feynman - resonant ] involve the trilinear coupling @xmath9 only . on the other hand , fig . [ fig : feynman - nonres - zhh]c , fig . [ fig : feynman - nonres - ahh]b and fig . [ fig : feynman - nonres - ww]c all involve both the trilinear higgs couplings @xmath9 and @xmath10 . as stated in section 4 , the dominant source for the production of multiple higgs bosons ( @xmath1 ) in @xmath11 collisions is through the production of the heavier @xmath0-even higgs boson @xmath3 either via higgs - strahlung or in association with @xmath5 @xcite , followed , if kinematically allowed , by the cascade decay @xmath8 . in terms of the @xmath93-electron couplings @xmath98 , @xmath99 , the cross sections for these processes can be written as @xcite @xmath100}{(1-m_z^2/s)^2 } , \label{eq : sigzh}\\ \sigma ( e^+e^- \rightarrow ah ) & = & \frac{g_f^2m_z^4}{96\pi s } ( v_e^2 + a_e^2)\sin^2(\beta - \alpha ) \frac{\lambda_a^{3/2}}{(1-m_z^2/s)^2 } , \label{eq : sigah}\end{aligned}\ ] ] where @xmath101 refers to @xmath102 , the two - body phase - space function , and is given as @xmath103 in fig . [ fig : sigma-500 - 1500 ] we plot the cross sections ( [ eq : sigzh ] ) and ( [ eq : sigah ] ) for the @xmath104 centre - of - mass energies @xmath105 and @xmath106 tev , as functions of the higgs mass @xmath107 and for @xmath108 . for large values of the mass @xmath33 of the pseudoscalar higgs boson , all the higgs bosons , except the lightest one ( @xmath1 ) , become heavy and decouple @xcite from the rest of the spectrum . in this case @xmath109 and the associated @xmath110 production ( [ eq : sigah ] ) becomes the dominant production mechanism for @xmath3 . at values of @xmath32 that are not too large , the trilinear @xmath111 coupling @xmath9 can be measured by the decay process @xmath8 , which has a width @xmath112 however , this is possible only if the decay is kinematically allowed , and the branching ratio is sizeable . in fig . [ fig : br - h - a-2 ] we show the branching ratios ( at @xmath70 ) for the main decay modes of the heavy @xmath0-even higgs boson as a function of the @xmath3 mass . apart from the @xmath92 decay mode , the other important decay modes are @xmath113 , @xmath114 . ( we have here disregarded decays to supersymmetric particles : charginos , stops , etc . if such particles are kinematically accessible , the @xmath115 and @xmath116 rates could be much smaller @xcite . ) we note that the couplings of @xmath3 to gauge bosons can be measured through the production cross sections for @xmath117 ; therefore the branching ratio @xmath118 can be used to measure the triple higgs coupling @xmath9 . the higgs - strahlung process [ fig . [ fig : feynman - resonant]a , eq . ( [ eq : sigzh ] ) ] gives rise to resonant two - higgs @xmath119 $ ] final states . this is to be contrasted with the associated production process [ fig . [ fig : feynman - resonant]b , eq . ( [ eq : sigah ] ) ] , which typically yields three higgs @xmath120 $ ] final states , since the channel @xmath121 is the dominant decay mode of @xmath5 in the mass range of interest . the decay width for @xmath121 can be written as @xcite @xmath122 where the @xmath123 are phase - space factors given by eq . ( [ 27 ] ) . in fig . [ fig : br - h - a-2 ] we show the branching ratios for the pseudoscalar @xmath5 for @xmath108 . a background to the multiple production of lighter higgs bosons @xmath1 comes from @xmath92 states generated in the sequential reaction @xmath124h$ ] [ see fig . [ fig : feynman - nonres - zhh]d ] . this is a genuine background in the sense that no higgs self - couplings are involved . but these background events are expected to be topologically very different from the signal events , since the two @xmath1 bosons do not form a resonance , whereas the @xmath125 $ ] does . the cross section for the process @xmath126 can be written as @xcite @xmath127 and is shown in fig . [ fig : sigma-500 - 1500 ] together with the signal cross sections ( [ eq : sigzh ] ) and ( [ eq : sigah ] ) . as a consequence of the decoupling theorem @xcite , the cross section becomes small for large values of @xmath107 . for increasing values of @xmath32 , the @xmath111 coupling gradually gets weaker ( see figs . [ fig : lamhhh2 ] and [ fig : lam - mh ] ) , and hence the prospects for measuring @xmath9 diminish . this is indicated by fig . [ fig : br - h - a-5 ] , where we show the @xmath3 and @xmath5 branching ratios for @xmath81 . there is in fact a sizeable region in the @xmath33@xmath32 plane where the decay @xmath115 is kinematically forbidden . this is indicated in fig . [ fig : hole ] . in this figure we also display the regions where the @xmath115 branching ratio is in the range 0.10.9 . clearly , in the forbidden region , the @xmath9 can not be determined from resonant production . for small and moderate values of @xmath32 , the study of decays of the heavy @xmath0-even higgs boson @xmath3 provides a means of determining the triple - higgs coupling @xmath9 . in order to extract the coupling @xmath10 , other processes involving two - higgs ( @xmath1 ) final states must be considered . the @xmath91 final states , which can be produced in the double higgs - strahlung @xmath128 of fig . [ fig : feynman - nonres - zhh ] , could provide one possible opportunity , since it involves the coupling @xmath10 through the mechanism of fig . [ fig : feynman - nonres - zhh]c . in this section we shall study these non - resonant processes in detail . the doubly differential cross section for the process @xmath128 shown in fig . [ fig : feynman - nonres - zhh ] can be written as @xcite @xmath129 where the couplings @xmath130 and @xmath131 have been defined at the beginning of section 5 . because of some misprints in the formulas given in @xcite for the coefficient @xmath132 , we have recalculated it . following @xcite , we introduce @xmath133 for the scaled energies of the higgs particles , @xmath134 for the scaled energy of the @xmath93 boson , and @xmath135 . also , we denote by @xmath136 the scaled squared masses of various particles : @xmath137 we can express our result in a compact form as follows : @xmath138\ , g_{ab } + { { \rm re}}[b(y_1)b^*(y_2)]\,g_{bb}\right\ } + \{x_1\leftrightarrow x_2\}. \label{eq : cala}\ ] ] here , @xmath139 + \biggl[\frac{\sin^2(\beta-\alpha)}{y_1+\mu_h-\tilde\mu_z } + \frac{\sin^2(\beta-\alpha)}{y_2+\mu_h-\tilde\mu_z}\biggr ] + \frac{1}{2\mu_z } \label{eq : lca}\ ] ] represents a contribution from diagram [ fig : feynman - nonres - zhh]a , where the lepton tensor couples directly to the final - state @xmath93 polarization tensor , as well as the contributions of diagrams [ fig : feynman - nonres - zhh]b and [ fig : feynman - nonres - zhh]c . similarly , @xmath140 represents the part of diagram [ fig : feynman - nonres - zhh]a where the lepton tensor couples to the final - state @xmath93 polarization tensor indirectly via the higgs momenta @xmath141 and @xmath142 , as well as diagram [ fig : feynman - nonres - zhh]d . the tildes on @xmath143 keep track of the widths , e.g.@xmath144 . the higgs self - couplings @xmath9 and @xmath10 occur only in the function @xmath145 , eq . ( [ eq : lca ] ) . the coefficients @xmath146 and @xmath30 , which do not involve any higgs couplings , can be expressed rather compactly as @xmath147 , \nonumber \\ g_{ab } & = & \mu_z[2(\mu_z-4\mu_h)+x_1 ^ 2+x_2(x_2+x_3 ) ] -y_1(2y_2-x_1x_3),\nonumber \\ g_{bb } & = & \mu_z^2(4\mu_h + 6 -x_1x_2 ) + 2\mu_z(\mu_z^2 + y_3 -4\mu_h ) \nonumber \\ & & + ( y_3-x_1x_2-x_3\mu_z-4\mu_h\mu_z)(2y_3-x_1x_2 - 4\mu_h+4\mu_z).\end{aligned}\ ] ] these coefficients ( we use a mixed notation , which involves both @xmath148 and @xmath149 ) correspond to those of @xcite as follows : @xmath150 . with this identification , we agree with the result given in the erratum to @xcite . in the limit of large @xmath33 , @xmath151 , the cross section reduces to the standard model cross section with @xmath152 + \frac{1}{2\mu_z } \\ b(y)&= & \frac{1}{2\mu_z}\ , \frac{1}{y+\mu_h-\tilde\mu_z},\end{aligned}\ ] ] where at the tree level , @xmath153 , as discussed in sec . 3 . we show in fig . [ fig : sig - zll-2]a the @xmath91 cross section , as given by eqs . ( [ eq : sigzh ] ) , ( [ eq : gamhhh ] ) and ( [ eq : sigzhh ] ) , in the limit of no squark mixing , and with @xmath154 . the structure around @xmath155 is due to the vanishing and near - vanishing of the trilinear coupling . in fig . [ fig : sig - zll-2]b d we have introduced squark mixing : @xmath156 , @xmath157 . ( for the decoupling - limit cross section , which is also shown , we use the mssm coupling , instead of the sm coupling , for the reason given in sec . 3 . ) in the case of no mixing , there is a broad minimum from @xmath158 to 90 gev , followed by an enhancement around @xmath159100 gev . this structure is due to the vanishing of the branching ratio for @xmath115 , which is kinematically forbidden in the region @xmath15890 gev , see fig . [ fig : hole ] ( this coincides with the opening up of the channel @xmath160 ) , followed by an increase of the trilinear couplings . this particular structure depends considerably on the exact mass values @xmath107 and @xmath39 . thus , it depends on details of the radiative corrections and the mixing parameters @xmath5 and @xmath40 . the @xmath161 channel contributes of the order of 20% in the region of the maximum at @xmath162100 gev . the resonant and non - resonant production of @xmath163 [ fig . [ fig : feynman - nonres - ahh ] ] can lead to three-@xmath1 final states in the region of @xmath33 , where @xmath5 has a significant branching ratio for decaying to @xmath164 , i.e. for @xmath33 below the @xmath165 threshold , and relatively low values of @xmath32 [ cf . [ fig : br - h - a-2 ] and [ fig : br - h - a-5 ] ] . in principle , this channel allows for a study of the coupling @xmath37 [ cf . [ fig : feynman - nonres - ahh]a ] . however , the prospects for measuring this coupling , which is rather small [ see fig . [ fig : lamhaa2 ] ] , was studied in ref . @xcite and found not to be very encouraging . as mentioned in section 4 , a double higgs ( @xmath92 ) final state in @xmath11 collisions can also result from the @xmath43 fusion mechanism , which can either be a resonant process as in ( [ eq : res - hhh ] ) , or a non - resonant one like ( [ eq : ww - fusion ] ) . since the neutral - current couplings are smaller than the charged - current ones , the cross section for the @xmath96 fusion mechanism in ( [ eq : res - hhh ] ) and ( [ eq : ww - fusion ] ) is an order of magnitude smaller than the @xmath43 fusion mechanism . we shall thus , in the following , ignore the @xmath96 fusion mechanism , and concentrate instead on the @xmath43 mechanism . the @xmath43 fusion mechanism provides another large cross section for the multiple production of @xmath1 bosons . the cross section for @xmath166 can be written as @xcite @xmath167 ^ 2}\ { \cal f}(x , y)\right ] \cos^2(\beta - \alpha ) , \label{eq : fusion - exact}\ ] ] where @xmath168 , \\ f(x , y ) & = & \left [ \frac{2x}{y^3 } - \frac{1 + 2x}{y^2 } + \frac{2 + x}{2y } - \frac{1}{2}\right ] \left[\frac{z}{1 + z } - \log(1 + z)\right ] + \frac{x}{y^3}\frac{z^2(1 - y)}{(1 + z ) } , \\ g(x , y ) & = & \left [ -\frac{x}{y^2 } + \frac{2 + x}{2y } - \frac{1}{2}\right ] \left[\frac{z}{1 + z } - \log(1 + z)\right],\end{aligned}\ ] ] with @xmath143 defined by eq . ( [ eq : mu ] ) and @xmath169 for @xmath170 , @xmath107 @xmath171 @xmath172 , and in the effective longitudinal @xmath173 approximation , the cross section ( [ eq : fusion - exact ] ) for @xmath166 can be written in the following simple form @xcite @xmath174\cos^2(\beta - \alpha ) . \label{32}\ ] ] however , in this approximation the cross section may be overestimated by a factor of @xmath175 for small values of masses and/or small centre - of - mass energies . for example , at @xmath176 gev the equivalent @xmath173 approximation gives a result that is twice as large as the exact cross section . therefore , we use the exact cross section ( [ eq : fusion - exact ] ) in our calculations . the cross section ( [ eq : fusion - exact ] ) is plotted in fig . [ fig : sigma-500 - 1500 ] for centre - of - mass energies , @xmath176 gev and @xmath106 tev , and for @xmath108 , as a function of @xmath107 . the resonant fusion mechanism , which leads to @xmath119 $ ] + [ missing energy ] final states is competitive with the process @xmath177 $ ] + [ missing energy ] , particularly at high energies . since the dominant decay of @xmath1 will be into @xmath178 pairs , the @xmath3-strahlung and the fusion mechanism will give rise to final states that will predominantly include four @xmath51-quarks . on the other hand , the process @xmath6 will give rise to six @xmath51-quarks in the final state , since the @xmath110 final state typically yields three - higgs @xmath120 $ ] final states . besides the resonant @xmath43 fusion mechanism for the multiple production of @xmath1 bosons , there is also a non - resonant @xmath43 fusion mechanism : @xmath180 through which the same final state of two @xmath1 bosons can be produced . the cross section for this process , which arises through @xmath43 exchange as indicated in fig . [ fig : feynman - nonres - ww ] , can be written in the ` effective @xmath43 approximation ' as approximation from the exact result . ] @xmath181 where @xmath182 . in the above , the cross section is written as a @xmath43 cross section , at invariant energy squared @xmath183 , folded with the @xmath43 ` luminosity ' @xcite : @xmath184 where @xmath185 . the @xmath43 cross section receives contributions from several amplitudes , according to the diagrams ( a)(d ) in fig . [ fig : feynman - nonres - ww ] . we have evaluated these contributions and express the result in a form analogous to that of ref . @xcite and @xmath186 reads @xmath187 ; it should be @xmath188 . with @xmath189 , the prefactors in their eq . ( 16 ) reduce to ours . ] : @xmath190 ^ 2\,g_0 \nonumber \\ & & \phantom{\frac{2}{\beta_h } } + \frac{2}{\beta_h}\biggl [ \frac{\hat\mu_z\sin(\beta-\alpha)}{1-\hat\mu_h}\,\lambda_{hhh } + \frac{\hat\mu_z\cos(\beta-\alpha)}{1-\hat\mu_h}\,\lambda_{hhh } + 1\biggr]\nonumber \\ & & \phantom{\frac{2}{\beta_h}aaaa } \times[\sin^2(\beta-\alpha)\,g_1 + \cos^2(\beta-\alpha)\,g_2 ] \nonumber \\ & & \phantom{\frac{2}{\beta_h } } + \frac{1}{\beta_h^2 } \{\sin^4(\beta-\alpha)\,g_3+\cos^4(\beta-\alpha)\,g_4 + \sin^2[2(\beta-\alpha)]\,g_5\}\biggr\},\end{aligned}\ ] ] where we have introduced ` reduced squared masses ' @xmath191 and the higgs velocity is @xmath192 . our approach differs from that of dhz in that we do not project out the longitudinal degrees of freedom of the intermediate @xmath173 bosons . instead , we follow the approach of ref . @xcite , where transverse momenta are ignored everywhere except in the @xmath173 propagators , the integrations over which are approximated as ( here @xmath193 and @xmath194 denote electron and neutrino momenta , respectively , in the process ( [ eq : ww - nonres ] ) ) : @xmath195 ^ 2 } \simeq\frac{\pi(1-x_1)}{m_w^2},\ ] ] where @xmath196 [ and @xmath197 represents the energy of the @xmath173 . the energy squared of the subprocess is given as @xmath198 . the contributions of diagrams ( b)+(c ) , ( a ) and ( d ) are given by the terms @xmath199 , @xmath200 and @xmath201 , respectively , with @xmath202 , and @xmath203 [ 2\hat\mu_w+1 - 3(\hat\mu_h-\hat\mu_w)^2]\frac{l_w}{a_w } \nonumber \\ & & + 16[2\hat\mu_w+(\hat\mu_h-\hat\mu_w)^2]^2y_w + 16\beta_h^2(1+a_w)^2 , \nonumber \\ g_4(x)&=&8\beta_h(\hat\mu_h-\hat\mu_c)^2[1 - 3(\hat\mu_h-\hat\mu_c)^2 ] \frac{l_c}{a_c } \nonumber \\ & & + 16(\hat\mu_h-\hat\mu_c)^4y_c + 16\beta_h^2(1+a_c)^2 , \ ] ] where @xmath204 and similarly @xmath205 , @xmath206 and @xmath207 , with @xmath208 replaced by @xmath209 , the latter being defined in terms of the charged higgs mass @xmath210 . the interference between diagrams ( b)+(c ) and ( a ) is given by the term @xmath211 , whereas the interferences between diagrams ( b)+(c ) and ( d ) , and between ( a ) and ( d ) are given by @xmath186 and @xmath212 , respectively . for these interference terms , we find @xmath213l_w -4\beta_h(1 + 2\hat\mu_h-2\hat\mu_w ) , \nonumber \\ g_2(x)&=&8(\hat\mu_h-\hat\mu_c)^2l_c -4\beta_h(1 + 2\hat\mu_h-2\hat\mu_c ) , \nonumber \\ g_5(x)&=&\frac{\beta_h}{4}(z_wl_w+z_cl_c)+8\beta^2_h(1+a_w)(1+a_c),\end{aligned}\ ] ] with @xmath214 + \frac{(1 - 2a_w)^2}{a_c+a_w}[8\hat\mu_w+(1 + 2a_w)^2 ] , \\ z_c&=&-\frac{(1 + 2a_c)^2}{a_c - a_w}[8\hat\mu_w+(1 + 2a_c)^2 ] + \frac{(1 + 2a_c)^2}{a_c+a_w}[8\hat\mu_w+(1 - 2a_c)^2].\end{aligned}\ ] ] our functions @xmath211@xmath212 correspond to those of dhz , cf . ( [ eq : sighat ] ) and their eq . ( 16 ) in @xcite . at small @xmath39 , the cross section is sensitive to small @xmath148 , where the ` effective @xmath43 approximation ' is not well defined , and our results differ from those of dhz . however , apart from the contributions from small @xmath148 , our results agree with those of dhz to a precision of 15% . we show in fig . [ fig : sig - ww-2 ] the @xmath43 fusion cross section , at @xmath215 , as given by eqs . ( [ eq : fusion - exact ] ) and ( [ eq : sigww - nonres ] ) , in the limit of no squark mixing , as well as with mixing ( as indicated ) , and with @xmath154 . the structure is very reminiscent of that of fig . [ fig : sig - zll-2 ] , and for the same reasons . however , the scale is different . following @xcite , we have indicated in the @xmath33@xmath32 plane the regions where @xmath9 and @xmath10 might be measurable , according to criteria analogous to those given there . in fig . [ fig : sensi-500 ] , we consider @xmath216 , and identify regions according to the following criteria : * regions where @xmath9 might become measurable are identified as those where @xmath217 ( solid ) , with the simultaneous requirement of @xmath218 [ see figs . [ fig : br - h - a-2][fig : hole ] ] . in view of the recent , more optimistic , view on the luminosity that might become available , we also give the corresponding contours for 0.05 fb ( dashed ) and 0.01 fb ( dotted ) . for @xmath219 we take the sum of ( [ eq : sigzh ] ) , ( [ eq : sigah ] ) and ( [ eq : fusion - exact ] ) . * regions where @xmath10 might become measurable are those where the _ continuum _ @xmath220 cross section [ eq . ( [ eq : sigww - nonres ] ) ] is larger than 0.1 fb ( solid ) . also included are contours at 0.05 ( dashed ) and 0.01 fb ( dotted ) . such regions are given for four cases of the mixing parameters @xmath5 and @xmath40 , as indicated . we have excluded from the plots the region where @xmath221 , according to the lep lower bound @xcite . this corresponds to low values of @xmath33 . we note that with an integrated luminosity of 500 fb@xmath222 , the contours at 0.1 fb correspond to 50 events per year . this will of course be reduced by efficiencies , but should indicate the order of magnitude that can be reached . at @xmath216 , with a luminosity of 500 fb@xmath222 per year , the trilinear coupling @xmath9 is accessible in a considerable part of the @xmath33@xmath32 parameter space : at @xmath33 of the order of 200300 gev and @xmath32 up to the order of 5 . with increasing luminosity , the region extends somewhat to higher values of @xmath33 . at values of @xmath33 below 100 gev , there is also a narrow band where @xmath9 is accessible . the ` steep ' edge around @xmath223 ( where increased luminosity does not help ) is determined by the vanishing of @xmath224 , see fig . [ fig : hole ] . the coupling @xmath10 is accessible in a much larger part of this parameter space , but with a moderate luminosity , ` large ' values of @xmath32 are accessible only if @xmath5 is small . in fig . [ fig : sensi-1500 ] , we consider @xmath225 tev , and present the analogous contours . here , for the case of @xmath9 we demand @xmath226 ( solid ) and 0.1 fb ( dashed ) , and for the case of @xmath10 we require the corresponding cross section [ eq . ( [ eq : sigww - nonres ] ) ] to be larger than 0.5 fb ( solid ) and 0.1 fb ( dashed ) . if a luminosity corresponding to these cross sections becomes available at @xmath225 tev , a somewhat larger region than at @xmath227 gev is accessible in the @xmath33@xmath32 plane . it should be stressed that the requirements discussed here are necessary , but not sufficient conditions for the trilinear couplings to be measurable . we also note that there might be sizable corrections to the @xmath43 approximation , and that it would be desirable to incorporate the dominant two - loop corrections to the trilinear couplings in the calculations . we have carried out a detailed investigation of the possibility of measuring the mssm trilinear couplings @xmath9 and @xmath10 at an @xmath11 collider . where there is an overlap , we have confirmed the results of ref . @xcite . our emphasis has been on taking into account all the parameters of the mssm higgs sector . we have studied the importance of mixing in the squark sector , as induced by the trilinear coupling @xmath5 and the bilinear coupling @xmath40 . at moderate energies ( @xmath216 ) the range in the @xmath33@xmath32 plane that is accessible for studying @xmath9 changes quantitatively for non - zero values of the parameters @xmath5 and @xmath40 . as far as the coupling @xmath10 is concerned , however , there is a qualitative change from the case of no mixing in the squark sector . if @xmath5 is large , then high luminosity is required to reach ` high ' values of @xmath32 . at higher energies ( @xmath215 ) , the mixing parameters @xmath5 and @xmath40 change the accessible region of the parameter space only in a quantitative manner . p. o. would like to thank the desy theory group and the cern theory division , whereas p. n. p. would like to thank the university of bergen , for kind hospitality while parts of this work were finished . it is also a pleasure to thank abdel djouadi , wolfgang hollik , bernd kniehl , conrad newton and peter zerwas for valuable discussions and advice . this research was supported by the research council of norway , and ( pnp ) by the university grants commission , india under project number 10 - 26/98(sr - i ) . j. ellis , g. ridolfi and f. zwirner , phys . * b257 * , 83 ( 1991 ) ; y. okada , m. yamaguchi and t. yanagida , prog . phys . * 85 * , 1 ( 1991 ) ; h. e. haber and r. hempfling , phys . lett . * 66 * , 1815 ( 1991 ) . r. hempfling and a. h. hoang , phys . lett . * b331 * , 99 ( 1994 ) ; + m. carena , j.r . espinosa , m. quirs and c.e.m . wagner , phys . * b355 * , 209 ( 1995 ) ; + m. carena , m. quirs and c.e.m . wagner , nucl . * b461 * , 407 ( 1996 ) ; + s. heinemeyer , w. hollik and g. weiglein , phys . * d58 * , 091701 ( 1998 ) . b. kileng , p. osland and p. n. pandita , z. phys . * c71 * , 87 ( 1996 ) ; b. kileng , p. osland and p. n. pandita , in _ xth international workshop : high energy physics and quantum field theory _ , proceedings of the conference , zvenigorod , russia , 1995 , ed . levtchenko and v.i . savrin ( moscow university press , moscow , 1996 ) , p. 167 ( hep - ph/9601284 ) ; b. kileng , p. osland and p. n. pandita , in proceedings of the international workshop on _ quantum systems : new trends and methods _ , minsk , belarus , 1996 , eds . y. s. kim , l. m. tomilchik , i. d. feranchuk and a. z. gazizov ( world scientific , singapore , 1997 ) , p. 231 ( hep - ph/9608315 ) . h. e. haber , in proceedings of the conference on _ perspectives for electroweak interactions in e@xmath228 e@xmath229 collisions _ , ringberg ( tegernsee ) , germany , 1995 ; ed . b. a. kniehl ( world scientific , singapore , 1995 ) p. 219 . for recent references , see : a. djouadi , p. janot , j. kalinowski , p.m. zerwas , phys . b376 * , 220 ( 1996 ) ; + a. bartl , h. eberl , k. hidaka , t. kon , w. majerotto and y. yamada , hep - ph/9709253 , in _ beyond the standard model , 5th international conference on physics beyond the standard model _ , balholm , norway , eds . g. eigen , p. osland and b. stugu ( aip , woodbury , n.y . , 1997 ) , p. 494 . r. n. cahn and s. dawson , phys . lett . * b136 * , 196 ( 1984 ) ; s. dawson , nucl . b249 * , 42 ( 1984 ) ; m. chanowitz and m. k. gaillard , phys . * b142 * , 85 ( 1984 ) ; i. kuss and h. spiesberger , phys . rev . * d53 * , 6078 ( 1996 ) .

we present a detailed analysis of multiple production of the lightest @xmath0-even higgs boson ( @xmath1 ) of the minimal supersymmetric standard model ( mssm ) at high - energy @xmath2 colliders . we consider the production of the heavier @xmath0-even higgs boson ( @xmath3 ) via higgs - strahlung @xmath4 , in association with the @xmath0-odd higgs boson ( @xmath5 ) in @xmath6 , or via the fusion mechanism @xmath7 , with @xmath3 subsequently decaying through @xmath8 , thereby resulting in a pair of lighter higgs bosons ( @xmath1 ) in the final state . these processes can enable one to measure the trilinear higgs couplings @xmath9 and @xmath10 , which can be used to theoretically reconstruct the higgs potential . we delineate the regions of the mssm parameter space in which these trilinear higgs couplings could be measured at a future @xmath11 collider . in our calculations , we include in detail the radiative corrections to the higgs sector of the mssm , especially the mixing in the squark sector . cern - th/98 - 189 + hep - ph/9806351 * measuring the trilinear couplings of mssm + neutral higgs bosons at high - energy @xmath2 colliders * 0.5 cm p. osland@xmath12 and p. n. pandita@xmath13 + @xmath14 department of physics , university of bergen , n-5007 bergen , norway@xmath15 + @xmath16 deutsches elektronen - synchrotron desy , d-22603 hamburg , germany + @xmath17 theoretical physics division , cern , ch 1211 geneva 23 , switzerland + @xmath18 department of physics , north eastern hill university , shillong 793 022 , india@xmath15 + pacs : 14.80.cp , 12.60.jv , 13.90.+i + * permanent addresses cern - th/98 - 189 + june 1998

hep-ph9806351

a liquid with gas bubbles has many applications in nature , industry and medicine @xcite . nonlinear wave processes in a gas liquid mixture were studied for the first time in works @xcite . the burgers , the korteweg de vries and the burgers korteweg de vries equations were obtained in @xcite for the description of long weakly nonlinear waves . the fourth order nonlinear evolution equation for nonlinear waves in a gas liquid mixture were obtained in @xcite taking into account an interphase heat transfer . nonlinear waves in a liquid with gas bubbles in the three dimensional case were considered in @xcite . linear waves in a gas liquid mixture under the van wijngaarden s theory were studied in @xcite . in @xcite propagation of linear waves in a liquid containing gas bubbles at finite volume fraction was considered . in previous studies of nonlinear waves in a liquid containing gas bubbles only the first order terms with respect to the small parameter were taken into account . on the other hand we know that using high order terms with respect to the small parameter at the derivation of nonlinear evolution equations allows us to obtain a more exact description of nonlinear waves @xcite . also taking into account high order correction in equations for nonlinear waves one can reveal important physical phenomena , such as interaction between dissipative and dispersive processes in a gas liquid mixture and its influence on waves propagation , new mechanisms of waves dispersion and dissipation . thus , it is important to study nonlinear waves in a liquid with gas bubbles taking into account second order terms in the asymptotic expansion . we investigate nonlinear waves in a liquid with gas bubbles taking into consideration not only high order terms with respect to the small parameter but the surface tension , liquid viscosity , intephase heat transfer and weak liquid compressibility as well . to the best of our knowledge the influence of these physical properties on nonlinear waves propagation simultaneously was not considered previously . the aim of our work is to study long weakly nonlinear waves in a liquid with gas bubbles taking into account both high order terms in the asymptotic expansion and the above mentioned physical properties in the model for nonlinear waves . we use the reductive perturbation method for the derivation of differential equations for nonlinear waves . we apply the concept of the asymptotic equivalence , asymptotic integrability and near identity transformations @xcite for studying nonlinear equations for long waves in a gas liquid mixture . asymptotically equivalent equations obtained in this work are connected to each other by a continuous group of non local transformations @xcite . these transformations are near identity transformations . thus , we introduce families of asymptotically equivalent equations for long weakly nonlinear waves in a liquid containing gas bubbles at quadratic order . as far as all these equations are equivalent we can use a more convenient and simple equation within this family . this equation is a normal form equation . such approach for the investigation of nonlinear evolution equation was proposed in works @xcite . near identity transformations are often named kodama s transformations . we derive two new nonlinear differential equations for long weakly nonlinear waves in a liquid with gas bubbles by the reductive perturbation method . in the case of dissipation main influence nonlinear waves are governed by the perturbation of the burgers equation . the perturbation of the burgers korteweg de vries equation corresponds to the main influence of dispersion on nonlinear waves propagation . we analyze dispersion relations for both equations . near identity transformations are used to obtain normal forms for the above mentioned equations . we show that a normal form for the equation in the dissipative case can be linearized under a certain condition on physical parameters . it is worth noting that this condition is realizable for physically meaningful values of parameters . analytical solution of the general dissipative equation in the form of a weak shock wave is obtained and analyzed . two cases of a normal form equation are analyzed provided that dispersion has the main influence . the first one is the case of negligible dissipation ( purely dispersive case ) where nonlinear waves are governed by the generalized korteweg de vries equation @xcite . we show that the generalized korteweg de vries equation for nonlinear waves in a liquid with gas bubbles is asymptotically equivalent to one of the integrable fifth order evolution equations that are the lax , the sawada kotera and the kaup kupershmidt equations . the general form of the dispersive nonlinear evolution equation seems to be nonintegrable . however , this equation admits analytical solitary wave solutions . the rest of this work is organized as follows . in section 2 we give the basic system of equations for nonlinear waves in a liquid with gas bubbles . we discuss the dispersion relation for linear waves as well . the main nonlinear differential equation for long weakly nonlinear waves is obtained by the reductive perturbation method . the nonlinear waves with the main influence of dissipation are studied in section 3 . section 4 is devoted to the investigation of nonlinear waves in the case of dispersion main influence . in section 5 we briefly discuss our results . for studying nonlinear waves in a liquid with gas bubbles we use the homogeneous model @xcite . we consider a bubble liquid mixture as a homogeneous media with an average pressure , an average density and an average velocity . we do not take into account interaction , formation , destruction and coalescence of bubbles . thus , the amount of gas bubbles in the mass unit is the constant @xmath0 . we assume that all gas bubbles are spherical . the nearly isothermal approximation @xcite is used for the modeling of heat transfer between a gas in bubbles and a liquid . in this approximation it is supposed that the temperature of the liquid is not changed and is equal to the temperature of the mixture in the unperturbed state ( @xmath1)@xcite . we consider influence of the liquid viscosity only at the interphase boundary . also we take into consideration the weak compressibility of the liquid using the keller miksis equation for the description of bubbles dynamics @xcite . also we consider the one dimensional case . in these assumptions we can use the following system of equations for the description of nonlinear waves in the liquid with gas bubbles @xcite : [ eq : main_dim_system ] @xmath2 @xmath3 @xmath4 @xmath5\bigg\},\hfill \label{eq : pressure_in_bubble}\end{aligned}\ ] ] @xmath6 @xmath7 we use the following notations in system : @xmath8 is cartesian coordinate , @xmath9 is time , @xmath10 is the density of the bubble liquid mixture , @xmath11 is the pressure of the mixture , @xmath12 is the velocity of the mixture , @xmath13 is the bubbles radius , @xmath14 are densities of the liquid and the gas respectively , @xmath15 is the pressure of the gas in bubbles , @xmath16 and @xmath17 are the pressure and the radius of bubbles in the unperturbed state , @xmath18 is the volume gas content , @xmath19 is the specific volume of the gas in the mixture , @xmath20 is the surface tension , @xmath21 is the kinematic liquid viscosity , @xmath22 is the ratio of the specific heats for the gas , @xmath23 is the thermal diffusivity of the gas , @xmath24 is the thermal conductivity of the gas , @xmath25 at @xmath26 , where @xmath1 is the temperature of the mixture in the unperturbed state . the first two equations from system are the continuity equation and the euler equation for the mixture . let us note that at the derivation of equation for bubbles dynamics the liquid viscosity at the inter phase boundary , the slight liquid compressibility and surface tension were taken into consideration @xcite . equation was obtained in @xcite under assumption that the gas temperature in the bubble deviates little from the temperature in the unperturbed state . equations , are definitions of the gas liquid mixture density , the volume gas content and the specific volume of the gas in the mixture correspondingly . let us note that the approach based on the theory of thermo microstretch fluid @xcite can be used for the description of a gas liquid continuum . for example , acceleration waves in a thermo microstretch fluid were studied in @xcite . we suppose that the pressure and density of the mixture in the unperturbed state are constants and all bubbles have the same radius and are uniformly distributed in the liquid . assuming that the volume gas content is small @xmath27 from we obtain @xmath28 this equation connects the density of the bubble liquid mixture with the bubbles radius . we use the following initial conditions : @xmath29 let us suppose that deviation of the mixture density is small : @xmath30 where @xmath31 is a small parameter corresponding to small deviations of the mixture density from its equilibrium value . using formula ( [ eq : rel1 ] ) from with accuracy up to @xmath32 we obtain @xmath33 substituting and into equations and using the dimensionless variables @xmath34 we have the following system of equations ( primes are omitted ) @xmath35 here @xmath36 and @xmath37 is the characteristic length scale and the characteristic time for our problem correspondingly and @xmath38 is the speed of linear waves in the liquid with gas bubbles : @xmath39 expressions for nondimensional parameters @xmath40 , @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath45 , @xmath46 , @xmath47 , @xmath48 , @xmath49 are presented in [ sec : a ] . in the case of incompressible liquid , system of equations is transformed to the system of equations considered in @xcite . if we neglect the liquid compressibility and the interphase heat transfer we obtain the model considered in works @xcite . system of equations consists of two conservation laws ( corresponding to conservation of mass and momentum ) and the dynamical state equation for the liquid containing gas bubbles . the complexity of is connected with high order derivatives and nonlinear terms in the dynamical state equation . systems similar to can be used for studying nonlinear waves in plasma physics , fluid dynamics , long waves in ferromagnetic media , nonlinear optics etc . @xcite . thus , one can consider system of equations as a prototypical system for the description of nonlinear waves in a nonlinear media with dispersion and dissipation . it is worth noting that there are two factors of waves damping in system : the first one mainly impacts on the damping of long waves and the second one mainly impacts on the damping of short waves . linear terms with the first and third order derivatives in the dynamical state equation correspond to these damping factors . the dispersion relation for the linearization of has the form @xmath50}. \label{eq : main_dispersion_relation}\ ] ] we see that system of equations in the linear case describes superposition of two waves propagating in opposite directions with the phase speed @xmath51 and the damping factor @xmath52 . using values of parameters one can obtain that the dispersion of waves is determined by the inertia of gas bubbles and the interphase heat transfer . analogously , we see that the dissipation of waves is determined by the liquid viscosity , liquid compressibility and interphase heat transfer . now we consider long weakly nonlinear waves governed by system of equations . we assume that the characteristic wavelength is much greater than the equilibrium radius of bubbles . thus , we have the small parameter @xmath53 equal to the ratio of bubbles radius in the unperturbed state @xmath17 to the characteristic wavelength @xmath36 ( @xmath54 ) . assuming that @xmath55 in is small we obtain @xmath56 from we see that system of equations contains at least two different time and length scales : that of the main influence of dissipation and that of the main influence of dispersion . below we consider nonlinear waves on these two scales and obtain governing nonlinear differential equations . we also assume that @xmath31 has the same order as @xmath53 ( @xmath57 ) . for the derivation of the equations for nonlinear waves we use the reductive perturbation method @xcite . we introduce the slow variables @xmath58 and search for the solutions of in the form of a series in the small parameter @xmath53 @xmath59 we introduce the parameter @xmath60 in variables in order to take into account two different time and length scales of waves in the bubble liquid mixture . varying the parameter @xmath60 we obtain nonlinear differential equations governing nonlinear waves on corresponding time and length scales . also the main influence of the different physical properties ( e.g. dissipation , dispersion ) corresponds to different values of @xmath60 . in this way we can reveal main physical properties affecting nonlinear waves at the corresponding length and time scales . substituting , into and collecting coefficients at @xmath61 we find @xmath62 collecting coefficients at @xmath53 and using relations we obtain the equation @xmath63 we see that right going nonlinear waves are governed by equation . below we consider two different cases of . the first one is the case of the main influence of dissipation ( @xmath64 ) . the other one is the case of the main influence of dispersion ( @xmath65 ) . thus , we see that dissipation mainly influences waves propagation at the longer length and time scales than the dispersion . we drop in three terms of order @xmath66 because they will not appear in our further consideration . later we will take into account in only terms to the first order of @xmath53 . thus , we consider the terms of the lowest order ( @xmath61 ) and the next one ( @xmath53 ) . at the lowest order in @xmath53 , one can obtain known equations for the waves in liquid with gas bubbles that is the burgers equation and the burgers korteweg de vries equation . at the next order we find new equations for more accurate description of nonlinear waves . also taking into consideration high order terms with respect to the small parameter we reveal new physical effects affecting nonlinear waves propagation . let us consider time and length scales where dissipation mainly influences nonlinear waves . substituting @xmath64 in and taking into account terms with @xmath53 in zero and first powers we obtain @xmath67 we see that is the generalization of the burgers equation @xcite for long weakly nonlinear waves in the liquid with gas bubbles . equation was derived for the first time in @xcite for the description of waves in the liquid containing gas bubbles . for waves in water containing carbon dioxide bubbles governed by for different values of the equilibrium bubbles radius @xmath68 mm ( curve 1 ) , @xmath69 mm ( curve 2 ) and @xmath70 mm ( curve 3 ) with @xmath71.,scaledwidth=55.0% ] let us use the following notations in : @xmath72 then we have @xmath73 linearizing under the trivial solution we find corresponding dispersion relation @xmath74\simeq \\ \simeq \varepsilon(\mu^{2}+\beta)k^{3}+i\mu k^{2}. \label{eq : ext_burgers_dispersion_relation } \end{gathered}\ ] ] from we see that waves attenuate with the linear damping coefficient proportional to the parameter @xmath75 . there is also weak dispersion of waves proportional to @xmath76 . using and we find that dissipation of waves governed by is determined by the liquid viscosity and thermal diffusivity of the gas . dispersion of waves governed by depends on the radius of bubbles in the unperturbed state , the volume gas content in the unperturbed state , the liquid viscosity and thermal diffusivity of the gas . we see that the phase and the group speed of nonlinear waves is the result of the interaction between the dispersion and the dissipation . one can take into account this phenomenon only if we consider high order corrections to the burgers equation . let us also note that according to dispersion relation the phase speed , the group speed and the linear damping coefficient are bounded for the all values of @xmath77 . let us consider the dependence of the linear damping coefficient ( @xmath78 ) on the wave number at different values of equilibrium bubble radius @xmath17 for water with bubbles of carbon dioxide . from fig.[fig:1.1 ] we see that the linear damping coefficient decreases when the equilibrium bubbles radius increases . this can be treated as the increase of the dispersion impact with the increase of bubbles radius . we also see from fig.[fig:1.1 ] that the attenuation of short waves for smaller values of @xmath17 is more intensive than the attenuation of long waves . it is worth noting that the main contribution to the damping parameter @xmath75 is given by the interphase heat transfer . the dependence of the phase speed ( @xmath79 ) on the wave number is very similar to the dependence of the linear damping coefficient on the wave number . equation is a prototypical equation at the order of @xmath53 for the nonlinear waves in media with dissipation and weak dispersion . however , for further investigation of it is better to convert it to the evolution form and to construct its normal form using the near identity transformations . applying the near identity transformations allow us to construct an equation which is asymptotically equivalent to and contains two arbitrary parameters . taking into account arbitrariness of these parameters we show below that one can use an integrable evolution equation for the description of nonlinear waves in the liquid with gas bubbles . even though the equation is integrable under a certain condition on physical parameters it can be used for description of waves because this condition is satisfied for real liquids with gas bubbles . in the general case ( without imposing any conditions on physical parameters ) one can use arbitrariness of the parameters introduced by the near identity transformations to obtain exact solutions in a more simple and convenient form . we differentiate once with respect to @xmath80 and substitute the result into . then , to the first order in @xmath53 , we obtain @xmath81 we see that is an evolution equation . now using near identity transformations we obtain a family of asymptotically equivalent equations for the description of nonlinear waves in the liquid with gas bubbles . substituting @xcite @xmath82 into , to the first order in @xmath53 , we have the equation @xmath83 here @xmath84 are arbitrary parameters of kodama transformations . choosing various values of parameters @xmath84 one can obtain different nonlinear evolution equations for long weakly nonlinear waves in the liquid containing gas bubbles . thus , equation is a family of asymptotically equivalent equations for nonlinear waves at the quadratic order in the case of dissipation main influence . let us note that equation was considered in works @xcite where applications of this equation to the acoustic waves and to the flow of viscous gas were considered . it is worth noting that the last equality from is exactly the dispersion relation for . thus , transformations with the procedure of the excluding mixed derivative do not change the dispersion relation to the first order in @xmath53 as it was pointed out in @xcite . and @xmath85 mm . , scaledwidth=55.0% ] one can find that is integrable under certain conditions on parameters @xmath86 , @xmath87 and @xmath88 . indeed , let us consider the following values of parameters @xmath84 : [ eq : sto_parameters3 ] @xmath89 @xmath90 then from after applying the following transformations @xmath91 we obtain the equation ( primes are omitted ) @xmath92 equation is the second member of the burgers hierarchy and is often called the sharma olver equation @xcite . this equation can be linearized by the cole hopf transformation . various exact solutions of were obtained in @xcite . equation was obtained under condition on the physical parameter @xmath88 . using we find that condition can be realized for the real liquid with gas bubbles . let us note that an analogous condition for in the case of waves in viscous gas leads to nonphysical values of parameters @xcite . .[tab : table1]values of bubbles radius @xmath17 in the unperturbed state obtained from condition . [ cols="<,^",options="header " , ] let us analyze condition in more detail . to simplify calculations we neglect the influence of the interphase heat transfer . using from we obtain an equation for the equilibrium bubbles radius @xmath17 . solving this equation for different carrier phases we obtain values of @xmath17 that are listed in tab.[tab : table1 ] . from tab.[tab : table1 ] one can see that these values of @xmath17 correspond to real liquids with gas bubbles . it is interesting to compare the one kink solution of equation under conditions with the one kink solution of the burgers equation . plots of these solutions at the time moment @xmath93 are presented in fig.[fig:1a ] . from fig.[fig:1a ] we see that the wave governed by the perturbed burgers equation has a larger width of the wave front than the wave governed by the burgers equation . this fact can be explained by the presence of additional nonlinear dissipative terms in . we can take into account these terms only if we consider high order corrections to the burgers equations . we also can see that dissipation connected with these terms is caused by both the interphase heat transfer and the liquid viscosity . let us construct solutions of without imposing condition . in this case equation is not integrable . thus , we have to choose parameters @xmath84 in such a way that the laurent expansion of the general solution has a simple form . let @xmath94 then equation has the following traveling wave solution @xmath95 where @xmath96 is an arbitrary traveling wave speed , @xmath97 is an arbitrary constant and @xmath98 is the parameter defining the wave amplitude and the width of the wave front . the parameter @xmath98 depends on physical parameters @xmath99 and the traveling wave speed @xmath96 . explicit expression for @xmath98 is presented in [ sec : b ] . solution can be obtained applying one of the ansatz methods for finding exact solutions . we used the simplest equation method @xcite . one can see that solution is a weak shock wave . using values of parameters @xmath99 presented in [ sec : a ] one can find that the width of the wave front of increases with the bubbles radius in the unperturbed state ( @xmath17 ) . the amplitude of solution is inversely related to @xmath17 . let us consider the length and time scales where dispersion mainly impacts on nonlinear waves propagation . in this case from we obtain the generalization of the burgers korteweg de vries equation . indeed , substituting @xmath65 into , to the first order in @xmath53 , we obtain @xmath100 where the following notations were used @xmath101 to the best of our knowledge equation is obtained for the first time . we see that equation at @xmath102 transforms the burgers korteweg de vries equation . particular cases of were considered in @xcite for waves in the gas liquid mixture . in the purely dispersive case ( i.e. @xmath103 ) equation is the generalization of the korteweg de vries equation @xcite . exact solutions for particular cases of this equation were considered in @xcite . for waves in water containing carbon dioxide bubbles governed by for different values of the equilibrium bubbles radius @xmath104 mm ( curve 1 ) , @xmath105 mm ( curve 2 ) and @xmath69 mm ( curve 3 ) with @xmath71 . , scaledwidth=55.0% ] the dispersion relation for has the form @xmath106+ik^{2}[\mu+\varepsilon(\mu\beta+\gamma)k^{2}+2\varepsilon^{2}\gamma\beta k^{4}]}{1+\varepsilon k^{2}[4\beta+\varepsilon(4k^{2}\beta^{2}+\mu^{2})]}\simeq\vspace{0.1cm}\\ \simeq \beta k^{3}+\varepsilon ( \mu^{2}-2\beta^{2}k^{2})k^{3}+i ( \mu k^{2}+\varepsilon ( \gamma-3\beta\mu ) k^{4 } ) . \label{eq : extended_bkdv_equation_dispersion_relation } \end{gathered}\ ] ] from we see that there are several mechanisms of waves dispersion and attenuation . dispersion of waves depends on the wavelength . one can see strong dispersion of long waves and weak dispersion of short waves . waves dispersion is governed by parameters @xmath107 and @xmath75 . we can see two mechanisms of waves attenuation as well : strong attenuation of long waves and weak attenuation of short waves . parameters @xmath75 , @xmath108 and @xmath107 define waves damping . using one can see that waves dispersion is caused by the presence of bubbles and by the interphase heat transfer . waves dissipation is connected with the liquid viscosity , interphase heat transfer and liquid compressibility . from fig.[fig:2b ] we see that when the radius of bubbles in the unperturbed state increases the phase speed decreases . this is due to the reducing of the heat transfer impact on nonlinear waves propagation . the dependence of the linear damping coefficient on the wave number is similar to the dependence of the phase speed on the wave number . let us construct a family of equations asymptotically equivalent to . in this way with the help of near identity transformations we will obtain a nonlinear evolution equation for waves in the gas liquid mixture that contains three arbitrary parameters . taking into consideration arbitrariness of these parameters one can find that in the case of negligible dissipation nonlinear waves are governed by one of the integrable fifth order evolution equations . in the general case one can use arbitrariness of these parameters for simplifying calculations during construction of exact solutions . excluding mixed derivative by the analogy with the previous case and applying near identity transformations @xcite : @xmath109 from we obtain @xmath110=0,\hfill \label{eq : extended_bkdv_equation_3 } \end{gathered}\ ] ] where @xmath111 are arbitrary parameters . equation is a family of the generalizations of the burgers korteweg de vries equation and is obtained for the first time . let us note that the dispersion relation for is equal to the right hand side of . thus , we again see that the kodama transformations do not change dispersion relations to the first order in @xmath53 @xcite . below we consider two different cases of . the first one is a purely dispersive case ( @xmath103 ) . the second one is the general case of . in the purely dispersive case ( @xmath103 ) from we obtain the generalized korteweg de vries equation @xcite : @xmath112=0 . \hfill \label{eq : extended_kdv_equation } \end{gathered}\ ] ] equation was well studied , for example exact solutions of were investigated in @xcite . the normal form equation for is not unique and depends on values of parameters @xmath111 . it can be one of the integrable fifth order evolution equations : the lax equation , the kaup kupershmidt equation or the sawada kotera equation . the cauchy problem for these equation can be solved with the help of the inverse scattering transform @xcite . indeed , let @xmath113 then from we obtain the lax equation @xmath114 + \beta v_{xxxxx}\bigg]=0 . \label{eq : kdv5 } \end{gathered}\ ] ] by the analogy at @xmath115 or at @xmath116 from we obtain the sawada kotera equation or the kaup kupershmidt equation correspondingly . the one soliton solution of the lax equation has the form @xmath117 , \label{eq : kdv5_soliton_solution}\ ] ] where @xmath96 and @xmath118 are arbitrary constants and the amplitude @xmath119 and the width @xmath120 are defined by @xmath121}{\alpha\varepsilon(5\sqrt{1 + 8\varepsilon c_{0}}-3 ) } , \quad \delta=\frac{4\beta\varepsilon}{[\beta\varepsilon(\sqrt{1 + 8\varepsilon c_{0}}-1)]^{1/2}}. \hfill \label{eq : kdv5_solitary_wave_speed } \end{gathered}\ ] ] ) for water containing carbon dioxide bubbles with @xmath122 and @xmath118=0.001 . , scaledwidth=55.0% ] from we see that the amplitude of the solitary wave depends on the wave speed and parameters @xmath123 and @xmath53 . the width of the solitary wave depends on the wave speed and parameters @xmath107 and @xmath53 . using values of @xmath123 , @xmath107 and @xmath53 we can find variations of @xmath119 and @xmath120 with radius of bubbles in the unperturbed state that are presented in fig.[fig:2.1 ] . from fig.[fig:2.1 ] we see that the amplitude @xmath119 slowly decreases when the radius of bubbles in the unperturbed state increases . the width of the solitary waves @xmath120 rapidly increases when the equilibrium bubbles radius increases . the generalized korteweg de vries equation and its particular cases have been studied by the present time . let us consider the general case of . equation is new and it is interesting to construct exact solutions for this equation . it is convenient to use in the values of @xmath111 the same as for the purely dispersive case ( equation ) , because we see that equations , have the same leading order terms . thus , using we guarantee a simple form for the laurent series of the solution . using in we obtain @xmath124=0 , \label{eq : extended_bkdv_equation_4 } \end{gathered}\ ] ] where @xmath125 for different values of radius of bubbles in the unperturbed state @xmath126 mm ( curve 1 ) , @xmath127 mm ( curve 2 ) and @xmath128 mm ( curve 3 ) for water containing carbon dioxide bubbles with @xmath71.,scaledwidth=55.0% ] equation admits elliptic , solitary wave and rational solutions . to construct these solutions we can use the simplest equation method @xcite . for example equation has a solution in the form @xmath129 where @xmath130 @xmath97 is an arbitrary constant . we do not present expressions for @xmath131 and @xmath132 due to their cumbersome form . there is a correlation on parameters @xmath133 for the existence of solution . we do not present this correlation either for the same reason , however we note that this correlation holds for physically relevant values of parameters @xmath133 . the plots of solution are presented in fig.[fig:2 ] for water with carbon dioxide bubbles of various radiuses and at @xmath134 . from fig.[fig:2 ] we see that solution is a weak shock wave . the amplitude of this wave decreases with the increasing of bubbles radius in the unperturbed state . the width of shock wave increases under the same condition . let us note that nonlinear waves similar to presented in fig.[fig:2 ] were observed experimentally ( e.g. see @xcite ) . the other solutions of have to be constructed elsewhere . we have studied long weakly nonlinear waves in the liquid containing gas bubbles . we have obtained equations for nonlinear waves taking into account high order terms in the asymptotic expansion ( equations , ) . these equations govern nonlinear waves at different length and time scales and are the generalizations of the burgers equation and the burgers korteweg de vries equation correspondingly . we have found normal form equation for using near identity transformations . we have shown that this normal form equation is integrable under a certain condition on physical parameters ( formula ) . it is worth noting that condition is realized for the liquid with gas bubbles . thus , we have found that equation is asymptotically integrable at @xmath64 with accuracy up to @xmath53 under condition for waves in the gas liquid mixture . in the general case of we have obtained and analyzed a solitary wave solution . the generalizations of the burgers korteweg de vries equation were analyzed by near identity transformations as well . we have found normal form equation . we have shown that in the purely dispersive case is the generalized korteweg de vries equation . using arbitrary parameters @xmath111 from near identity transformations we can transform the generalized korteweg de vries equation to one of the integrable fifth order evolution equations . thus , dynamics of nonlinear waves in the liquid with gas bubbles is asymptotically integrable in the purely dispersive case with accuracy up to @xmath53 . we have found exact special solutions of in the general case . let us note that system of equation used in the current work for the investigation of nonlinear waves in the liquid containing gas bubbles can be used for the investigation of nonlinear waves in other media . for example , one can use this system of equations for waves in visco elastic tubes @xcite , nonlinear waves in plasma @xcite and other applications @xcite . thus , the results of our work can be applied not only to the study of waves in the liquid with gas bubbles but also to the investigation of nonlinear waves in all media described by system of equations ( 5 ) . this research was partially supported by rfbr grant 14 - 01 - 00493-a , by grant for scientific schools 2296.2014.1 . and by grant for the state support of young russian scientists 3694.2014.1 . the nondimensional parameters of system of equations have the form @xmath135 @xmath136 @xmath137 @xmath138 @xmath139\rho_{0}^{2}}{525\gamma_{g}^{2}r_{0}^{2}d^{2 } } + \frac{\rho_{l}(4\chi_{1}r_{0}\chi^{2}+3\chi^{2 } ) \rho_{0}^{2 } c_{0}^{2}}{2p_{0}\,l^{2 } } , \label{eq : non - dim_parameters1}\end{aligned}\ ] ] @xmath140 @xmath141 @xmath142 @xmath143,\end{aligned}\ ] ] @xmath144 [ eq : non - dim_parameters ] we use in the following notations : @xmath145 nondimensional parameter @xmath146 correspond to the nonlinearity of waves , parameters @xmath147 correspond to the dissipation of the nonlinear waves , parameters @xmath148 correspond to the dispersion of the nonlinear waves . the parameter @xmath98 in solution has the form @xmath149 \times \hfill \\ \times \bigg[16\varepsilon^{2}\left(3\,\beta\,\alpha+3\,\mu^{2}\alpha+\mu\alpha^{2}+\nu\alpha+4\,\mu\,\beta+4\mu^{3}\right ) ^{2}\left ( 2\,\beta+2\,{\mu}^{2}+\mu\,\alpha+\nu \right ) \alpha\bigg]^{-1}. \end{gathered}\ ] ]

in this work we generalize the models for nonlinear waves in a gas liquid mixture taking into account an interphase heat transfer , a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for nonlinear waves . we also take into consideration high order terms with respect to the small parameter . two new nonlinear differential equations are derived for long weakly nonlinear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above mentioned physical properties . one of these equations is the perturbation of the burgers equation and corresponds to main influence of dissipation on nonlinear waves propagation . the other equation is the perturbation of the burgers korteweg de vries equation and corresponds to main influence of dispersion on nonlinear waves propagation . _ keywords : _ nonlinear waves ; liquid with gas bubbles ; reductive perturbation method ; perturbed burgers equation ; nonlinear evolution equations .

1610.04539

cosmic ray ( cr ) propagation is a complex process involving diffusion by magnetic field , energy losses and spallation by interactions with the interstellar medium ( ism ) . diffuse galactic @xmath0-rays are produced via the decay of neutral pion and kaon , which are generated by high energy cosmic nuclei interacting with interstellar gas , and via energetic electron inverse compton ( ic ) scattering and bremsstrahlung . the @xmath0 rays are not deflected by the magnetic field and the ism is transparent to @xmath0-rays below a few tev @xcite . therefore , the observation of the diffuse @xmath0-ray spectra and distribution is a valuable diagnosis of the self - consistency of propagation models , the distribution of cr sources and the ism . the galactic diffuse @xmath0 rays has been measured by egret @xcite and exhibits an excess above @xmath3 1 gev compared to prediction @xcite . the theoretical calculations are based on a conventional cr model , whose nuclei and electron spectra in the whole galaxy are taken to be the same as those observed locally . the discrepancy has attracted much attention @xcite since it was first raised . it may either indicate a non - ubiquitous proton or electron spectrum , or the existence of new exotic sources of diffuse @xmath0-ray emission . many efforts have been made to solve the `` gev excess '' problem within the frame of cr physics , such as adopting different cr spectra @xcite , or assuming more important contribution to diffuse @xmath0-rays from cr sources @xcite . a brief review of these efforts is given in @xcite . in that paper an `` optimized '' propagation model has been built by directly fitting the observed diffuse @xmath0-ray spectrum . this `` optimized '' model introduces interstellar electron and proton intensities that are different from the local ones and reproduces all the cr observational data at the same time . up to now , it seems to be the best model to explain the egret diffuse @xmath0-ray data based on cr physics . however , this `` optimized '' model is fine tuned by adjusting the electron and proton injection spectra , while keeping the injection spectra of heavier nuclei unchanged , as in the conventional model , so that the b / c ratio is not upset . furthermore a large scale proton spectrum different from the locally measured one might not be reasonable , since the proton diffusion time scale is much smaller than its energy loss time scale , which tends to result in a large scale universal proton spectrum within the galaxy apart from some specific sources . unlike protons , the electron spectrum may have large spatial fluctuation due to their fast energy losses from ic , bremsstrahlung , ionization and the stochastic sources @xcite . another interesting solution , given by de boer et al . @xcite , is that the `` gev excess '' is attributed to dark matter ( dm ) annihilation from the galactic halo , where the dm candidate is the neutralino from the supersymmetry ( susy ) . by fitting both the background spectrum shape from cosmic nuclei collisions and the signal spectrum shape from dark matter annihilation ( dma ) they found the egret data could be well explained @xcite . this suggestion is very interesting and impressive , due to the fact that in 180 independent sky regions and non - gaussian at low energy . ] , all the discrepancies between data and the standard theoretical prediction can be well explained by a single spectrum from dma with @xmath4 gev . furthermore , by fitting the spatial distribution of the diffuse @xmath0-ray emission they reconstructed the dm profile , with two rings supplemented to the smooth halo . the ring structure seems also necessary to explain the damping in the milky way rotation curve @xcite and the gas flaring @xcite . however , the dma solution to the `` gev excess '' also meets a great challenge because of its prediction of the antiproton flux . in de boer s model , this flux is more than one order of magnitude greater than data @xcite . the overproduction of antiprotons comes from two factors : a universal `` boost factor '' @xmath5 of the diffuse @xmath0-rays boosts the local antiproton flux by the same amount ; the two rings introduced to account for the diffuse @xmath0-ray flux enhance the antiproton flux greatly since they are near the solar system and are strong antiproton sources . in their work , de boer et al . did not try to develop a propagation model . instead they focused on _ reconstruction _ of the dm profile by fitting the egret data . they need a `` boost factor '' to enhance the contribution from dma . the background contribution from pion decay is arbitrarily normalized in order to fit data best . in the present work we try to build a propagation model to explain the egret diffuse @xmath0-ray data based on both strong s and de boer s models while overcoming their difficulties . in our model the diffuse @xmath0-ray comes from both crs and dma directly . on one hand we do not introduce a different interstellar proton spectrum from the local one ; on the other our model gives consistent @xmath6 flux even when including contribution from dma . furthermore we do not need the large `` boost factor '' to dma or renormalization factor to cr contribution . actually , the @xmath0-ray flux from dma is boosted by taking the subhalos into account . the diffuse @xmath0-ray spectra at different sky regions and its profiles as functions of galactic longitude and latitude are well consistent with egret data . in a previous paper @xcite , we have briefly introduced our model . full details are given in the present paper . the paper is organized as follows . we describe the calculation of the dma contribution in section ii . in section iii , we focus on the conventional cr model . as underlined , it explains the egret data , but produces too large @xmath6 flux . in section iv , we present our new propagation model and its predictions for the diffuse @xmath0-ray spectra and profiles , and the @xmath6 flux . finally we summarize and conclude in section v. here , we calculate the diffuse @xmath0-ray flux from dma . the frame of minimal supersymmetric extension of the standard model ( mssm ) is retained , where we assume that dm consists of the lightest neutralinos . a pair of neutralinos in the galactic halo can annihilate into leptons , quarks and gauge bosons . their decay products include a @xmath0-ray continuum and thus contribute to the @xmath0-rays diffuse emission produced by cr interaction with the ism . the flux of @xmath0 rays from the neutralino annihilation is given by @xmath7 where @xmath8 is the averaged neutralino annihilation cross section times relative velocity , @xmath9 is the differential flux in a single annihilation , @xmath10 is the mass of neutralino , @xmath11 is the distance between the detector and the @xmath0-ray source , and @xmath12 is the spherically - averaged dm distribution , determined by numerical simulation or by observations . the flux in eq . ( [ flux ] ) is determined by two independent factors , the first one only depending on the dm particle nature ( mass , strength of interaction and so on ) , the second one depending on the dm distribution only . the first factor is denoted as `` particle factor '' and the second as `` astrophysics factor '' . @xmath13 as a function of neutralino mass . thresholds for @xmath14 and @xmath0-ray kinetic energies are taken as @xmath15 gev . ] the `` particle factor '' is calculated by doing a random scan in the susy parameter space and choosing models which could satisfy all the collider and cosmological constraints . however , there are more than one hundred free susy breaking parameters , even for the r - parity conservative mssm . a general practice in phenomenological studies is to assume some simple relations between the parameters . following the assumptions in darksusy @xcite we take seven free parameters during the calculation of dm production and annihilation , i.e. , the higgsino mass parameter @xmath16 , the wino mass parameter @xmath17 , the mass of the cp - odd higgs boson @xmath18 , the ratio of the higgs vacuum expectation values @xmath19 , the scalar fermion mass parameter @xmath20 , the trilinear soft breaking parameter @xmath21 and @xmath22 . all the sfermions are taken with a common soft - breaking mass parameter @xmath20 ; all trilinear parameters are zero except those of the third family ; the bino and wino have the mass relation , @xmath23 , coming from the unification of the gaugino mass at the grand unification scale . a random scan in the 7-dimensional parameter space of mssm is performed using the package darksusy @xcite . we assume these parameters to range as follows : @xmath24 gev @xmath25 tev , @xmath26 , @xmath27 , @xmath28 . we find the @xmath0-ray spectrum with @xmath29 gev ( after being added to the background @xmath0-ray spectrum ) can fit the egret data very well , which is consistent with the result of de boer et al . @xcite . however , the branching ratio to @xmath0 rays varies by a few order of magnitude for different models , and only models with large branching ratio into @xmath0-rays can account for the `` gev excess '' . we also calculate the branching ratio into antiprotons since the @xmath6 flux is a sensitive test of the dm model . in fig . [ ratio ] we give the ratio of neutralino annihilation into @xmath14 and @xmath0-rays as a function of neutralino mass . we find @xmath30 above the threshold energy @xmath31 gev for @xmath32 gev . for light neutralinos , the variation of the ratio in the parameter space is at most as large as a factor of @xmath33 . for heavy neutralinos the variation is very small , and the reason is that @xmath14 and @xmath0-rays are coming from the same final states , thus they are closely related . below , we choose @xmath34 gev to explain the egret data which predicts @xmath35 and @xmath36 for the threshold energy @xmath31 gev . actually for @xmath37 between @xmath38 gev the annihilated @xmath0-ray spectra are almost identical . a few models also show similar branching ratio as in our chosen model ( e.g. , for @xmath39 gev ) . the astrophysics factor determining the annihilation fluxes is defined as @xmath40 with @xmath11 the distance to the source of @xmath0-ray emission , @xmath41 the density profile of dm , and @xmath42 the volume in which annihilation taking place . in the de boer model , the authors adopt a cored galactic halo model with two dm rings , boosting the @xmath0-ray flux by a factor of the order @xmath5 . in our work we take a cuspy nfw @xcite ( or moore @xcite ) profile . especially we take the contribution from subhalos into account to enhance the annihilation signals . high resolution simulations of cosmological structure evolution reveal that in the cold dm scenario the structures form hierarchically and a large number of substructures survive in the galactic halos . a fraction of about 10% of the total halo mass may have survived tidal disruption and appear as distinct and self - bound substructures inside the virialized host halos . the existence of substructures will enhance the annihilation rate greatly by enhancing the astrophysics factor in eq . ( [ flux ] ) . the mass function and spatial function of subhalos are given by n - body simulations . a simple analytical fit to simulations allows to write the probability of a subhalo with mass @xmath43 appearing at position @xmath44 as @xcite @xmath45 where @xmath46 is the normalization factor so that about @xmath47 of the halo mass is enclosed in subhalos , @xmath48 is the galactic halo mass , @xmath49 kpc is the core radius for the distribution of subhalos . the result given above agrees well with that of another recent simulation by gao et al . @xcite . the dm density profile within each subhalo is taken as the nfw @xcite , moore @xcite and a cuspier form @xcite as @xmath50 with @xmath51 . the last form is favored by the simulation conducted by reed et al . @xcite , which gives that @xmath52 , for the subhalo mass of @xmath53 with a large scattering , increasing as the subhalo mass decreases . small halos with large @xmath54 are also found by diemand et al . we take @xmath51 for the whole range of subhalo masses as a simple approximation . we calculate the concentration parameter @xmath55 by adopting the semi - analytic model of bullock et al . @xcite , which describes it as a function of the viral mass and redshift . we adopt the mean @xmath56 relation at redshift zero . the scale radius is then determined by @xmath57 , @xmath58 and @xmath59 for the three density profiles respectively . another factor determining the @xmath0-ray flux is the core radius , @xmath60 , within which the dm density should be kept constant due to the balance between the annihilation rate and the rate of dm particles falling into this region @xcite . the core radius @xmath60 is approximately @xmath61 kpc for @xmath51 and about @xmath62 kpc for the moore profile . along a direction @xmath63 , the subhalos contribute to the `` astrophysical factor '' @xmath64 , where @xmath65 is the astrophysical factor for a single subhalo , and @xmath66 is the number density of subhalos . when we calculate the integration along the line - of - sight starting from the sun , we get the jacobian determinant as @xmath67 , with @xmath68 the distance from the subhalo to the sun . the minimal subhalos can be as light as @xmath69 as shown by the recent simulation conducted by diemand et al . @xcite , while the maximal mass of substructures is taken to be @xmath70 @xcite . the tidal effects are taken into account based on the `` tidal approximation '' @xcite , where subhalos are disrupted near the galactic center ( gc ) . the total signal flux comes from the annihilation in the subhalos and in the smooth component . the astrophysical factor @xmath71 ( in unit of @xmath72 ) from different directions . the almost horizontal lines correspond to the contributions from subhalos only . ] in fig . [ density ] , we show the factor @xmath71 from the smooth component , the subhalos and the total contribution as a function of the direction to the gc . the @xmath71 from subhalos is almost isotropic in all directions as the sun is near the gc compared with the viral radius @xmath73 of the galactic halo . the largest enhancement for @xmath51 subhalos is observed at large angles and can reach 2 orders of magnitude . this enhancement depends on the value of @xmath60 , while for the moore profile the enhancement is about one order of magnitude , and for for the nfw profile only about @xmath74 times larger . the @xmath71 for moore and nfw profiles is not sensitive to @xmath60 @xcite . we also notice that near the gc there is no enhancement . this is actually a very important difference from the model given by de boer where the `` boost factor '' is universal @xcite . given the factor @xmath71 and the susy model we can predict the @xmath0-ray flux from neutralino annihilation . in the next section we give the background diffuse @xmath0-rays from cr interactions with the ism . the optimized model of strong et al . reproduces the diffuse @xmath0 rays assuming interstellar proton and electron spectra different from those locally measured . however , the required fluctuation of the proton spectrum may be not realistic . the works of de boer et al . have strongly indicated that dma may account for the diffuse @xmath0 ray excess @xcite . therefore our first attempt is to build a propagation model including contribution from dma based on the conventional cr model assuming universal cr spectra . cosmic ray results in our conventional model . lower and upper curves for b / c ratio correspond to solar modulated and local interstellar ( lis ) values respectively . for @xmath1be/@xmath75be ratio , the flat curve near 0.1 gev / nucleon corresponds to the modulated flux and the other one is lis . on the contrary , the lower curves for proton and electron spectra are the modulated ones and the upper are lis . b / c data are from ace@xcite , ulysses@xcite , voyager@xcite , heao-3@xcite and others @xcite ; @xmath1be/@xmath75be data from ulysses@xcite , ace@xcite and voyager@xcite ; proton data from bess98@xcite and ams@xcite ; electron data from caprice@xcite and heat@xcite . , title="fig : " ] cosmic ray results in our conventional model . lower and upper curves for b / c ratio correspond to solar modulated and local interstellar ( lis ) values respectively . for @xmath1be/@xmath75be ratio , the flat curve near 0.1 gev / nucleon corresponds to the modulated flux and the other one is lis . on the contrary , the lower curves for proton and electron spectra are the modulated ones and the upper are lis . b / c data are from ace@xcite , ulysses@xcite , voyager@xcite , heao-3@xcite and others @xcite ; @xmath1be/@xmath75be data from ulysses@xcite , ace@xcite and voyager@xcite ; proton data from bess98@xcite and ams@xcite ; electron data from caprice@xcite and heat@xcite . , title="fig : " ] cosmic ray results in our conventional model . lower and upper curves for b / c ratio correspond to solar modulated and local interstellar ( lis ) values respectively . for @xmath1be/@xmath75be ratio , the flat curve near 0.1 gev / nucleon corresponds to the modulated flux and the other one is lis . on the contrary , the lower curves for proton and electron spectra are the modulated ones and the upper are lis . b / c data are from ace@xcite , ulysses@xcite , voyager@xcite , heao-3@xcite and others @xcite ; @xmath1be/@xmath75be data from ulysses@xcite , ace@xcite and voyager@xcite ; proton data from bess98@xcite and ams@xcite ; electron data from caprice@xcite and heat@xcite . , title="fig : " ] cosmic ray results in our conventional model . lower and upper curves for b / c ratio correspond to solar modulated and local interstellar ( lis ) values respectively . for @xmath1be/@xmath75be ratio , the flat curve near 0.1 gev / nucleon corresponds to the modulated flux and the other one is lis . on the contrary , the lower curves for proton and electron spectra are the modulated ones and the upper are lis . b / c data are from ace@xcite , ulysses@xcite , voyager@xcite , heao-3@xcite and others @xcite ; @xmath1be/@xmath75be data from ulysses@xcite , ace@xcite and voyager@xcite ; proton data from bess98@xcite and ams@xcite ; electron data from caprice@xcite and heat@xcite . , title="fig : " ] we adopt the galprop package @xcite to calculate the propagation of crs and production of galactic background diffuse @xmath0 rays . it was shown that an explanation of the egret data in _ all _ sky directions in the conventional model is not an easy task , even including the contribution from dma|considering that dma provides only a single extra spectrum from neutralino annihilation . to give the best fit to the egret data , de boer et al . @xcite have to introduce arbitrary renormalization factors in different directions for the @xmath0 ray spectra given in the conventional model @xcite . since we are trying here to build a propagation model , we will not introduce any renormalization factors , but rather explain the @xmath0 ray data by adjusting the propagation parameters after adding the contribution from dma . after a lot of tests , we came to the following propagation parameters . the scale height of the propagation halo @xmath76 takes the same value 4kpc as that taken by strong et al . in their conventional and optimized models the nuclei injection spectra share the same power law form in rigidity , and nuclei up to @xmath77 and all relevant isotopes are included . the cr injection spectra are given in table [ inj_para ] . we adopt the diffusive reacceleration propagation model . the spatial diffusion coefficient is given as a function of rigidity in the form @xmath78 where @xmath79 , @xmath80@xmath81s@xmath82 , @xmath83gv , and @xmath84 . the alfvn speed to describe the reacceleration process is @xmath85 kms@xmath82 . these propagation parameters well describe the observed b / c ratio , the @xmath1be/@xmath75be ratio , and the local measured proton and electron spectra , as shown in fig . [ conv - crs ] . .[inj_para ] cosmic ray injection spectrum parameters . [ cols="<,^,^,^ " , ] a major uncertainty in calculating the diffuse galactic @xmath0-ray emission is the distribution of molecular hydrogen , since the derivation of h@xmath86 density from the co data is problematic @xcite . the co - to - h@xmath87 conversion factor @xmath88 from cobe / dirbe studies given by sodroski et al . @xcite is about @xmath89 times greater than the value given by boselli et al . @xcite in different galactocentric radius . the later is based on the measurement of galactic metallicity gradient and the inverse dependence of @xmath88 on metallicity @xcite . the value of @xmath88 is then normalized to the @xmath0-ray data @xcite . strong et al . have derived the @xmath88 by fitting the egret diffuse @xmath0-ray data directly @xcite . a constant @xmath90 @xmath91(k km s@xmath92 for @xmath93 gev was given @xcite . however , observations of particular local clouds yield lower values @xmath94 @xmath91(k km s@xmath92 @xcite . since the fit by strong et al . to the egret data in @xcite assumes only the background contributions , we expect that it gives a larger @xmath88 than is the case with the new dma component . we find that a smaller @xmath95 molecules @xmath91(k km s@xmath92 can give a much better description of the egret data below 1 gev . after taking dma into account , the global fitting is greatly improved . we assume for @xmath88 a constant value independent of the radius @xmath96 . as shown in ref . @xcite , the simple form of @xmath88 is compensated by an appropriate form of the cr sources . we have taken the radial distribution of cr sources in the form of @xmath97 , with @xmath98 , @xmath99 , @xmath100 kpc , and limiting the sources within @xmath101 kpc , which are adjusted to best describe the diffuse @xmath0-ray spectrum . the source distribution is shown in fig . [ s - d ] ( dashed line ) . spectra of diffuse @xmath0-rays for different sky regions ( top row , regions a , b , middle c , d , bottom e , f ) . the model components are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] spectra of diffuse @xmath0-rays for different sky regions ( top row , regions a , b , middle c , d , bottom e , f ) . the model components are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] spectra of diffuse @xmath0-rays for different sky regions ( top row , regions a , b , middle c , d , bottom e , f ) . the model components are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] spectra of diffuse @xmath0-rays for different sky regions ( top row , regions a , b , middle c , d , bottom e , f ) . the model components are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] spectra of diffuse @xmath0-rays for different sky regions ( top row , regions a , b , middle c , d , bottom e , f ) . the model components are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] spectra of diffuse @xmath0-rays for different sky regions ( top row , regions a , b , middle c , d , bottom e , f ) . the model components are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] following strong et al . @xcite , we divide the whole sky into six regions : region a corresponds to the `` inner radian '' , region b is the galactic plane excluding the inner radian , region c is the `` outer galaxy '' , regions d and e cover higher latitudes at all longitudes , and region f describes the `` galactic poles '' . the calculated diffuse @xmath0 spectra in the six different sky regions are given in fig . [ result ] . the diffuse @xmath0 ray background includes contributions from @xmath102 decays produced by nuclei collisions , ic scattering off the interstellar radiation field ( isrf ) , bremsstrahlung by electrons , and the isotropic extragalactic @xmath0 ray background ( egrb ) . for the regions a , b , c and d , i.e. the galactic plane and the intermediate latitude regions , the @xmath102 decay contribution is dominant . at high latitude regions e and f the egrb becomes more important . to account for the gev excess the peak of the dma @xmath0 ray spectrum has to have similar magnitude as the background . after adding the diffuse @xmath0 ray emission from dma to the galactic background , we obtain a perfect agreement with egret measurements for all sky directions . the result in fig . [ result ] is really a success , considering that we simply add the two kinds of diffuse @xmath0 ray contributions together , without introducing any arbitrary normalization for the background @xmath0 rays or `` boost factors '' for the dma contribution . it should be noted that including the enhancement by subhalos dose not exclude the ring - like structures proposed by de boer @xcite . that is natural since taking the subhalos into account only enhances the signals coming from the smooth component , but does not mimic the ring - like structure , which can fit the egret data at different directions @xcite . solar modulated flux of antiprotons in the conventional model . the upper / black curve represents the total @xmath6 flux . for the two lower curves , the one dominating at energies below several gev is the dma contribution and the other dominating at energies above several gev stands for the contribution from cr interactions . @xmath6 data are from wizard - mass91@xcite , caprice98@xcite , bess95 + 97@xcite , bess98@xcite , bess99@xcite , bess00@xcite , bess02@xcite , ams01@xcite . ] we have seen in fig . [ ratio ] that the branching ratios to @xmath0 rays and to @xmath6 are closely related . therefore the @xmath6 flux is a sensitive test of the dma scenario to solve the `` gev excess '' problem . in @xcite , bergstrm et al . have claimed that the de boer model has been ruled out due to the overproduction of @xmath6 flux . in their work they adopted a simple propagation model similar to that adopted in darksusy @xcite . we now check the @xmath6 flux in our propagation model . we first calculate the source term of @xmath6 produced by neutralino annihilation adopting the same susy model used for the diffuse @xmath0 ray data , @xmath103 where @xmath104 is the differential @xmath6 flux at energy @xmath105 due to a single annihilation , and @xmath106 are contributions from the smooth dm component and subhalos . the contribution from subhalos is given by @xmath107 where @xmath108 is the number density of subhalos with mass @xmath43 at radius @xmath44 , and @xmath109 is the astrophysical factor for a single subhalo with mass @xmath43 . all subhalo parameters are taken identical to those used to account for the diffuse @xmath0 rays . we then calculate , in the same propagation model , the @xmath14 spectrum at earth by incorporating the source term of eq . ( [ source ] ) in galprop . [ pbar ] shows the background , the signal and the total @xmath14 fluxes . the background antiproton flux s lower than the data ; this has been discussed in @xcite . this is another hint at the necessity of an exotic signals besides the ordinary cr secondaries . the @xmath14 flux in our model is about one order of magnitude greater than the measured data at energies lower than @xmath110 gev . however , we notice that our prediction is a few times lower than that by bergstrm et al . @xcite , when they adopted the median set of propagation parameters , which are similar to the propagation parameters in our conventional model . the difference may be due to two reasons : first we adopt a different propagation model ; second|and maybe more important| , we do not adopt a universal `` boost factor '' for the diffuse @xmath0 rays from dma as de boer and bergstrm et al . did . the enhancement by subhalos tends to boost the @xmath0 ray and @xmath6 fluxes at large radii , as shown in fig . [ density ] . therefore subhalos boost @xmath0 rays more than @xmath6 since only these @xmath6 produced within the galactic diffusion region contribute to the @xmath6 flux on earth . however , these effects are not enough to give a @xmath6 flux consistent with the data . this is related to the presence of two dm rings near the solar system , that enhance the antiproton flux greatly as they are strong @xmath6 sources . we have to resort to a new propagation model to suppress the @xmath6 further and give a consistent description of all data . inheriting the advantages of our conventional model in the last section , a new propagation model is intended to be built to account for the galactic diffuse @xmath0 rays , and at the same time the @xmath6 flux . degeneracies exist between the propagation parameters , mainly between the diffusion coefficient @xmath111 and the height of the diffusion region @xmath76 . different sets of parameters can all explain the cr data but lead to very different signals from dma @xcite . if we adopt smaller @xmath76 we can adjust @xmath111 at the same time to give similar prediction of the cr data . however adopting smaller @xmath76 leads to smaller @xmath6 flux from dma since only @xmath6 sources within the diffusion region contributes to the flux on the earth . the degeneracy leads to about one order of magnitude uncertainties of the @xmath6 flux when adopting different sets of propagation parameters in bergstrm et al . therefore it is straightforward to consider a smaller height of the diffusion region to suppress the dma signal . the diffusion halo height is determined by fitting the cr data . it is found that the average gas density crs crossed during their travel to the earth is about 0.2 atoms/@xmath112 , which is significantly lower than the average gas density in disk of about 1 atom/@xmath112 . this is explained by the possibility that crs are confined in a larger diffusion region than the gaseous disk , spending a longer time outside the gaseous disk . the height of the halo represents the volume of this diffusion region . in @xcite , strong and moskalenko derived @xmath113 kpc for the four types of radioactive nuclei isotopes of @xmath114mn / mn , @xmath115cl / cl , @xmath116al/@xmath117al and @xmath1be/@xmath75be , while the data of @xmath1be/@xmath75be favored a smaller halo of @xmath118 kpc . maurin et al . found that several settings of propagation parameters with halo height @xmath76 ranging from 1 kpc to 15 kpc could well fit the observational b / c ratio data @xcite . certainly the diffusion halo height should be at least as large as the scale height of the ionized component of the interstellar gas , @xmath119pc @xcite . the halo height is also constrained by the diffuse @xmath0 ray emission at the intermediate latitude , since too small halo height will lead to too low @xmath0 ray flux at at these latitudes . since the gas is not smoothly distributed in the disk , crs may travel in low density regions in the disk and may not diffuse to so large a region as usually considered . as shown in @xcite , the gas density and the strength of magnetic fields are higher than average within molecular clouds : the molecular clouds can act as magnetic mirrors to reflect and confine the charged cr particles . in this scenario crs may travel in a low gas density region rather than in a region with the average density . nevertheless , the effect of molecular cloud reflection is hard to quantify . we assume the gas density that crs crossed is lowered compared with the average gas density in the disk by a constant factor , which should be of the order of @xmath110 since the conventional diffusion model has been very successful in describing the cr transportation . we find that a reduction factor @xmath31.5 and @xmath120 kpc can reproduce all the data very well , which will be shown below . it should be noted that the reduction of the gas density crossed by crs is the key point of this propagation model . taking this smaller halo height , we then adjust the diffusion coefficient and the alfvn speed , so that we can reproduce the secondary to primary ratios , e.g. the b / c ratio . in the present model , parameters in eq . ( [ diff ] ) are taken as @xmath121 @xmath81 s@xmath82 , @xmath83gv and @xmath84 . the alfvn speed @xmath122 is taken as 19 km s@xmath82 . the injection spectra of nuclei and electrons are given in table [ inj_para ] . the cr source distribution adopted in this model is shown in fig . [ s - d ] , together with the source distribution adopted in @xcite , which is the same as the pulsar distribution @xcite , and that adopted in @xcite , which is obtained by fitting the egret data with a constant x@xmath123 value . the vertical bars are snr data points from @xcite . the flat source distribution derived by fitting the egret diffuse @xmath0 ray data is not in agreement with the distribution of snrs @xcite . as for the pulsar distribution , it is so peaked that it would aggravate the problem of reproducing the relatively smooth egret flux profile along the galactic plane @xcite . in addition , the pulsar distribution is probably not sufficiently reliable to trace the snrs accurately . in fact , only a proportion of about one in @xmath5 known pulsars appears to be convincingly associated directly with snrs @xcite . it is very interesting that in our new model , the source distribution consistent with snrs data ( solid line in fig . [ s - d ] ) can give a better description of the diffuse @xmath0 ray data with a constant co - to - h@xmath86 conversion factor x@xmath124 molecules @xmath91(k km s@xmath92 than the other two distribution functions . the radial source distribution has the same form as the previous one with @xmath125 , @xmath126 . the cr spectra in our new model . curves and data are the same as in fig . [ conv - crs ] . , title="fig : " ] the cr spectra in our new model . curves and data are the same as in fig . [ conv - crs ] . , title="fig : " ] the cr spectra in our new model . curves and data are the same as in fig . [ conv - crs ] . , title="fig : " ] the cr spectra in our new model . curves and data are the same as in fig . [ conv - crs ] . , title="fig : " ] in fig . [ cr - sp ] , we show the results of b / c , @xmath1be/@xmath2be , spectra of protons and electrons as predicted by this new propagation model . the stable secondary to primary ratio b / c and the unstable to stable secondaries @xmath1be/@xmath2be show that our model reproduces the experimental date very well . in this model the interstellar proton spectrum is taken to be the locally observed spectrum since the energy loss of proton is negligible . the interstellar electron spectrum has an intensity normalization at high energies different from the local one , similar to the electron spectrum adopted by strong et al . in @xcite but with smaller fluctuations . we will show below that this model can also reproduce the diffuse @xmath0 ray and @xmath6 data well . in order to give @xmath0 ray and @xmath6 fluxes consistent with experimental data , we also changed the dm ring parameters slightly : the inner ring is now located at r = 3.5 kpc and the outer ring is moved from r = 14kpc to 16kpc . on one hand , the position of rings are not crucial parameters to fit the egret data : indeed , a slight change will not vary the prediction of diffuse @xmath0 rays much . on the other hand , the rings are mainly helpful to explain the diffuse @xmath0 rays at intermediate latitudes ; there are not enough to account for the gev excess elsewhere , such as in regions d and e ( if only the smooth dm component is included ) . however , our consideration of subhalos helps to enhance @xmath0 ray emissions at large radii as shown in fig . [ density ] . therefore , we do not need so large @xmath0 ray emissions from the two rings to contribute to the intermediate latitudes as done in the de boer model , and we can move them slightly far away from the solar system . it is interesting to note that the analysis of the hi gas flaring by kalberla et al . also favored a dm ring located at a large radius of @xmath127 kpc @xcite . we have checked that these soft modulation of ring parameters did not change the rotation curve significantly . [ r - c ] shows the rotation curve for the present model . the contributions from the dm smooth halo , the two dm rings , and the bulge and disk are included . it can be seen that the rotation curve is consistent with data . diffuse gamma ray spectra in the six sky regions predicted by the new propagation model ( top row , regions a , b , middle c , d , bottom e , f ) . the various contributions are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] diffuse gamma ray spectra in the six sky regions predicted by the new propagation model ( top row , regions a , b , middle c , d , bottom e , f ) . the various contributions are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] diffuse gamma ray spectra in the six sky regions predicted by the new propagation model ( top row , regions a , b , middle c , d , bottom e , f ) . the various contributions are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] diffuse gamma ray spectra in the six sky regions predicted by the new propagation model ( top row , regions a , b , middle c , d , bottom e , f ) . the various contributions are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] diffuse gamma ray spectra in the six sky regions predicted by the new propagation model ( top row , regions a , b , middle c , d , bottom e , f ) . the various contributions are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] diffuse gamma ray spectra in the six sky regions predicted by the new propagation model ( top row , regions a , b , middle c , d , bottom e , f ) . the various contributions are @xmath102 decay , inverse compton , bremsstrahlung , egrb and dma ( dark red curve ) . , title="fig : " ] adding the background diffuse @xmath0 rays from crs and those from dma _ directly _ , we find the calculated diffuse @xmath0 rays are well consistent with observations . the diffuse @xmath0 ray spectra in the six sky regions are shown in fig . [ gamma ] . the prediction in the new propagation model is in good agreement with egret data . longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] longitude profiles at low latitudes ( @xmath128 ) in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the inner galaxy @xmath129 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] latitude profiles in the longitude ranges @xmath130 in the new propagation model , compared with egret data in 10 energy ranges from 30 mev to 10gev . , title="fig : " ] figs . [ l_pf ] , [ in_bpf ] and [ out_bpf ] display the diffuse @xmath0 longitudinal and latitudinal profiles in our present model . the line styles in these figures are the same as those given in figs . [ result ] and [ gamma ] , representing contributions from @xmath102 decay , ic , bremsstrahlung and egrb respectively . the solid red line is the total contribution from the background @xmath0 rays ( solid blue ) and dma signals . from these figures , it is obvious that the dma component is essential to explain the profiles above @xmath131 mev . we notice that the longitudinal profiles at low latitude ( @xmath132 ) and the latitudinal profiles in the outer galaxy with @xmath133 are in fairly good agreement with the egret data . however , for the latitudinal profiles in the inner galaxy ( @xmath134 ) , we also find that the intensities at intermediate latitudes @xmath135 are lower than measurements at energies from 500 mev to 4000 mev . a similar excess is present in the `` optimized model '' @xcite , in which it is pointed out that this may be related to an underestimate of the isrf in the galactic halo and that a factor of @xmath136 uncertainty on the isrf is quite possible due to the complexity of its calculation @xcite . furthermore , the smaller @xmath76 adopted here shallows the distribution of electrons , which may also contribute to this discrepancy . flux of antiprotons after solar modulation in the new propagation model . lines and data are the same as in fig . [ pbar ] . ] finally , the antiproton flux is given in fig . [ pbar2 ] . below @xmath137 gev , the @xmath6 flux from dma dominates the cr secondary @xmath6 . the total @xmath6 flux is a bit higher than the best fit value of the experimental data . mv . in fact , the modulation parameter is not a free parameter for different cr species , but depends on the solar activities . the measurements of the @xmath6 flux were taken from the solar maximum ( bess00 , bess02 ) to the solar minimum ( bess95 ) . ] however due to large errors of the present experimental data , the prediction is still consistent with data within @xmath138 . forthcoming high precision @xmath6 measurements by pamela @xcite and ams02 @xcite will be helpful to determine if the present model is viable . it should be noted that our model has the potential to further suppress the @xmath6 flux , by changing the rings position and the diffusion height @xmath76 . in @xcite , de boer et al . introduced an anisotropic propagation model to greatly suppress the @xmath6 flux . in their model , only the @xmath6 from dma within the gaseous disk is confined by magnetic field and contributes to the @xmath6 flux at earth . the @xmath6 from dma above the gaseous disk is blown away and has no contribution to the local @xmath6 flux . in order to reproduce the b / c data , they introduced a grammage parameter @xmath139 so that the secondaries and the resident time are increased by this factor , as a result of the molecular clouds confinement of charged particles . however , we think it may be hard to reproduce the diffuse @xmath0 ray data at intermediate and high latitudes in this model , since the crs above the gas disk are quickly blown away and produce very low @xmath0 ray emissivity . our model gives consistent galactic diffuse @xmath0 rays and @xmath6 flux with experimental data simultaneously without drastic modifications of the galprop model . in this work we propose to solve the `` gev excess '' problem of the galactic diffuse @xmath0 rays by developing a new propagation model which includes contributions from dma . we have shown that this propagation model can well reproduce the b / c , @xmath1be/@xmath75be data and spectra of protons and electrons . the galactic diffuse @xmath0 ray spectrum at different sky regions and its profile as a function of longitude and latitude are also shown to describe the egret data very well , if the dma contribution is included . the @xmath6 flux in this model is consistent with experimental data within @xmath138 . compared with previous works @xcite , our model does not introduce a normalization of the interstellar proton intensity different from the local one ( as it should be universal due to the negligible energy loss of protons ) . furthermore , neither the `` boost factors '' to the dma signals nor the arbitrary renormalization of the galactic @xmath0-ray background are needed in our model to explain the egret data . in our model , the @xmath6 flux from dma is suppressed by the following changes compared with the de boer model @xcite : 1 ) the smaller @xmath76 helps to suppress the @xmath6 flux from the smooth component of dm . the average gas density crs crossed may be smaller than the average gas density in the disk , as shown in @xcite . this fact tends to favor a smaller height of the cr diffusion region . 2 ) we do not adopt a universal `` boost factor '' for @xmath0 rays and @xmath6 . since the enhancement by subhalos is larger at large radii , the boost of @xmath6 at the solar neighborhood is smaller than the boost of @xmath0 rays . 3 ) the ring parameters are slightly adjusted , which does not change the @xmath0 ray profile while greatly suppressing the @xmath6 flux . this is because the distance dependence of the @xmath14 propagation is steeper ( exponential decrease ) than @xmath140 of @xmath0-rays @xcite . a potential problem of this model is related to the @xmath6 flux , which is consistent with data , but can not best fit the data . adjusting the propagation or dm parameters ( such as the rings ) can further lower the @xmath6 flux . however , more fine - tuning is required to fit the @xmath0-ray and rotation curve data . note a new explanation of the `` gev excess '' was given as an instrumental biases contaminating the egret data @xcite . however , whether this conjecture is correct or not can not be confirmed at the moment . forthcoming precise observations , such as glast @xcite , pamela @xcite and ams02 @xcite , will be decisive to validate or disprove this model . at present , because of the fundamental importance of the dm problem , we think that any possible implication of dm signals deserves a serious treatment . we thank the anonymous referee for helpful comments on the manuscript and d. maurin and j. lavalle for improvement on english . this work is supported by the nsf of china under the grant nos . 10575111 , 10773011 and supported in part by the chinese academy of sciences under the grant no . kjcx3-syw - n2 . p. gralewicz , j. wdowczyk , a. w. wolfendale , l. zhang , astron . & astrophysics * 318 * , 925 ( 1997 ) ; m. mori , astrophys . j. * 478 * , 225 ( 1997 ) ; t. a. porter , r. j. protheroe , j. phys . g * 23 * , 1765 ( 1997 ) ; m. pohl , j. a. esposito , astrophys . j. * 507 * , 327 ( 1998 ) ; f. a. aharonian , a. m. atoyan , astron . & astrophysics * 362 * , 937 ( 2000 ) . p. gondolo , j. edsjo , p. ullio , l. bergstrom , m. schelke , e.a . baltz , jcap 0407 , 008 ( 2004 ) . j. f. navarro , c. s. frenk s. d. m. white , mon . not . . soc . * 275 * , 56 ( 1995 ) ; j. f. navarro , c. s. frenk s. d. m. white , astrophys . j. * 462 * , 563 ( 1996 ) ; j. f. navarro , c. s. frenk s. d. m. white , astrophys . j. * 490 * , 493 ( 1997 ) . d. reed , f. governato , l. verde , j. gardner , t. quinn , j. stadel , d. merritt , g. lake , mon . not . r. astron . soc . * 357 * , 82 ( 2005 ) . j. diemand , b. moore , j. stade , nature * 433 * , 389 ( 2005 ) ; j. diemand , m. kuhlen , p. madau , astrophys . j. * 649 * , 1 ( 2006 ) . j. s. bullock , t. s. kolatt , y. sigad , r. s. somerville , a. v. kravtsov , a. a. klypin , j. r. primack , a. dekel , mon . not . . soc . * 321 * , 559 ( 2001 ) . j. h. chapell , & w. r. webber , proc . cosmic ray conf . ( paris ) , 2 , 59 ( 1981 ) ; r. dwyer , astrophys . j. * 224 * , 691 ( 1978 ) ; r. c. maehl , j. f. ormes , a. j. fisher , & f. a. hagen , ap&ss , 47 , 163 ( 1977 ) . j. j. connell , astrophys . j. * 501 * , l59 ( 1998 ) . aguilar _ et al . _ , physics reports 366 , 331 ( 2002 ) i. v. moskalenko , a. w. strong , j. f. ormes , m. s. potgieter , astrophys . j. * 565 * , 280 ( 2002 ) ; i. v. moskalenko , a. w. strong , s. g. mashnik , j. f. ormes , astrophys . j. * 586 * , 1050 ( 2003 ) . f. donato , n. fornengo , d. maurin , p. salati , r. taillet , phys . d**6**9 , 063501 ( 2004 ) . d. maurin , r. taillet , c. combet , astro - ph/0609522v3 ; d. maurin , r. taillet , c. combet , astro - ph/0612714v1 ; j. lavalle , q. yuan , d. maurin , x. j. bi , astron . astrophys . * 479 * , 427 ( 2008 ) . f. w. stecker , s. d. hunter , d. a. kniffen , arxiv:0705.4311

in this work we develop a new propagation model for the galactic cosmic rays based on the galprop code , including contributions from dark matter annihilation . its predictions of the galactic diffuse @xmath0 ray spectra are compatible with the egret data in all sky regions . it also gives consistent results about the diffuse @xmath0 ray longitude and latitude distributions . the b / c , @xmath1be/@xmath2be , proton , electron and antiproton spectra are in agreement with cosmic ray measurements as well . in this model we have taken a universal proton spectrum throughout the galaxy without introducing large fluctuation , considering the proton energy loss is negligible . the dark matter annihilation signals are `` boosted '' after taking the contributions from subhalos into account . another interesting feature of this model is that it gives better description of the diffuse @xmath0 rays when taking the source distribution compatible with supernova remnants data , which is different from previous studies .

0712.4038

this work is financially supported by the national science council of republic of china under the contract # : nsc-95 - 2112-m-007 - 059-my3 . c. k. chua , w. s. hou and s. y. tsai , phys . . * d66 * , 054004 ( 2002 ) ; h. y. cheng and k. c. yang , phys . rev . * d66 * , 014020 ( 2002 ) ; s. y. tsai , `` study of three - body baryonic b decays '' , ph . d thesis , national taiwan university ( 2005 ) . geng and y.k . hsiao , phys . d * 74 * , 094023 ( 2006 ) . c. q. geng , y. k. hsiao and j. n. ng , phys . lett . * 98 * , 011801 ( 2007 ) ; phys . d * 75 * , 094013 ( 2007 ) . geng and y.k . hsiao , phys . d * 75 * , 094005 ( 2007 ) . b. aubert _ et al . _ [ babar collaboration ] , phys . rev . * d72 * , 051101 ( 2005 ) ; phys . d * 76 * , 092004 ( 2007 ) ; see also t.b . hrynova , `` study of b meson decays to @xmath120 final states '' , ph.d . thesis , stanford university ( 2006 ) . c. q. geng and y. k. hsiao , phys . d * 72 * , 037901 ( 2005 ) ; int . j. mod . a * 21 * , 897 ( 2006 ) . m. z. wang _ et al . _ [ belle collaboration ] , phys . rev . d * 76 * , 052004 ( 2007 ) ; b. aubert [ babar collaboration ] , arxiv : hep - ex/0608020 . hou and a. soni , @xmath121 , 4247 ( 2001 ) . chua , w.s . hou and s.y . tsai , @xmath122 , 233 ( 2002 ) ; c. k. chua and w. s. hou , eur . j. c*29 * , 27 ( 2003 ) . f. piccinini and a. d. polosa , @xmath123 , 097508 ( 2002 ) ; h. y. cheng and k.c . yang , @xmath124 , 271 ( 2002 ) ; c. q. geng and y. k. hsiao , phys . b * 610 * , 67 ( 2005 ) . g. p. lepage and s. j. brodsky , phys . lett . * 43 * , 545(1979 ) [ erratum - ibid . * 43 * , 1625 ( 1979 ) ] ; g. p. lepage and s. j. brodsky , phys . rev . * d22 * , 2157 ( 1980 ) ; s. j. brodsky , g. p. lepage and s. a. a. zaidi , phys . * d23 * , 1152 ( 1981 ) .

we review the direct cp and t violation in the three - body baryonic @xmath0 decays in the standard model . in particular , we emphasize that the direct cp violating asymmetry in @xmath1 is around 22@xmath2 and the direct @xmath3 violating asymmetry in @xmath4 can be as large as @xmath5 , which are accessible to the current b factories at kek and slac as well as superb and lhcb . direct cp violation has been measured in both @xmath6 and @xmath7 systems @xcite , but it has not been observed and conclusive in @xmath8 and @xmath9 systems @xcite , respectively . on the other hand , t violation has been only seen in the @xmath6 process @xcite , related to the indirect cp violating parameter @xmath10 , whereas no t violating effect has been found in either @xmath8 or @xmath0 systems yet . in the standard model ( sm ) , it is clear that the unique phase of the cabbibo - kobayashi - maskawa ( ckm ) matrix @xcite is responsible for both observed cp and t violating effects . in this talk , we would like to explore the possibility to detect the direct cp and t violation in the @xmath0 systems in the current b - factories as well as the future ones such as superb and lhcb . in particular , we concentrate on the three - body charmless baryonic processes . our goal of the talk is to test the ckm paradigm of cp violation and unfold new physics . in the framework of local quantum field theories , t - violation implies cp - violation ( and vice versa ) , because of the cpt invariance of such theories . moreover , no violation of cpt symmetry has been found @xcite . still , it will be worthwhile to remember that outside this framework of local quantum field theories , there is no reason for the two symmetries to be linked @xcite . therefore , it would be interesting to directly investigate t violation in b decays , rather than inferring it as a consequence of cp - violation . the characteristic observables of the direct cp and t violation are rate asymmetries and momentum correlations , respectively . for example , in ( conjugate ) processes such as @xmath11 ( @xmath12 ) , the direct cp asymmetry arises if both the weak ( @xmath13 ) and strong ( @xmath14 ) phases are non - vanishing , given by a_cp&=&(b|*b*m)-(|b|m ) ( b|*b*m)+(|b|m ) , [ cpa ] whereas the direct t violation is related to the correlations known as triple product correlations ( tpc s ) , such as @xmath15 , given by _ t & = & 1 2(a_t-|a_t ) . where @xcite a_t = ( _ * b*(_*b*_m ) > 0 ) - ( _ * b*(_*b*_m ) < 0 ) ( _ * b*(_*b*_m ) > 0 ) + ( _ * b*(_*b*_m ) < 0 ) , [ atp ] and @xmath16 is the corresponding asymmetry of the conjugate process . it is interesting to note that to have a non - zero value of @xmath17 , both weak and strong phases are needed , whereas in the vanishing limit of the strong phase , @xmath18 is maximal . furthermore , there is no contribution @xcite to @xmath18 from final state interaction due to electromagnetic interaction . from the effective hamiltonian at the quark level for @xmath0 decays @xcite , the amplitudes of @xmath19 and @xmath20 are approximately given by @xcite @xmath21\ , , \nonumber\\ { \cal a}_{k^*}&\simeq&\frac{g_f}{\sqrt 2}m_{k^*}f_{k^*}\varepsilon^{\mu}\alpha_{k^*}\langle p\bar p|\bar u\gamma_\mu(1-\gamma_5 ) b|b^-\rangle\;,\end{aligned}\ ] ] respectively , where @xmath22 is the fermi constant , @xmath23 is the meson decay constant , given by @xmath24 ( @xmath25 ) with @xmath26 ( @xmath27 ) being the four momentum ( polarization ) of @xmath28 ( @xmath29 ) , and @xmath30 and @xmath31 are defined by @xmath32\ ; , \nonumber\\ % \beta_k&\equiv & v_{ub}v_{us}^*a_1-v_{tb}v_{ts}^*\bigg[a_4- a_6\frac{2 m_k^2}{m_b m_s}\bigg]\;,\nonumber\\ \alpha_{k^*}&\equiv & v_{ub}v_{us}^*a_1-v_{tb}v_{ts}^*a_4\;,\end{aligned}\ ] ] where @xmath33 are the ckm matrix elements and @xmath34 ( @xmath35 ) are given by @xmath36 with @xmath37 being effective wilson coefficients ( wc s ) shown in ref . @xcite and @xmath38 the color number for the color - octet terms . we note that for the decay amplitudes in eq . ( [ eq1 ] ) we have neglected the small contributions @xcite from @xmath39 involving the @xmath40 time - like baryonic form factors @xcite , where @xmath41 can be ( axial-)vector or ( pseudo)scalar currents . however , in our numerical analysis we will keep all amplitudes including the ones neglected in eq . ( [ eq1 ] ) . numerically , the ckm parameters are taken to be @xcite @xmath42 and @xmath43 with @xmath44 , @xmath45 , the values of @xmath46 are @xmath47 @xcite . we remark that @xmath34 contain both weak and strong phases , induced by @xmath48 and quark - loop rescatterings . explicitly , at the scale @xmath49 and @xmath38=3 , we obtain a set of @xmath50 , @xmath51 , and @xmath52 as follows : @xmath53\times 10^{-4 } \;,\nonumber\\ a_6&=&\big[(-595.5\mp 9.1\eta-3.9\rho)+i(-83.2\pm 3.9\eta-9.1\rho)\big]\times 10^{-4}\;,\end{aligned}\ ] ] for the @xmath54 ( @xmath55 ) transition . from eq . ( [ eq1 ] ) , we derive the simple results for the direct cp asymmetries of the @xmath56 modes as follows : @xmath57 where @xmath58 denote the values of the corresponding antiparticles . it is easy to see that @xmath59 are independent of the phase spaces as well as the hadronic matrix elements . as a result , the hadron parts along with their uncertainties in @xmath59 are divided out in eq . ( [ acp2 ] ) . we note that the cp asymmetries in eq . ( [ acp2 ] ) are related to the weak phase of @xmath60 @xcite . our results on the direct cp violation are summarized in table [ pre ] . in the table , we have included the current experimental data as well as the decay modes of @xmath61 . we note that the possible fluctuations induced from non - factorizable effects , time - like baryonic form factors and ckm matrix elements for @xmath59 are about @xmath62 ( 0.04 ) , 0.003 ( 0.01 ) and 0.01 ( 0.01 ) , respectively . the uncertainties from time - like baryonic form factors are constrained by the data of @xmath63 and @xmath64 @xcite and the errors on the ckm elements are from @xmath65 and @xmath48 given in ref . @xcite . it is interesting to point out that the large value of @xmath66=22% is in agreement with the babar data of @xmath67 . however , taken at face value ; the sign of our prediction @xmath68 is different from those by babar @xcite and belle @xcite collaborations . since the uncertainties of both experiments are still large it is too early to make a firm conclusion . for the direct t violation in the three - body charmless baryonic b decays @xcite . we concentrate on @xmath4 by looking for the tpc of the type @xmath69 . it is interesting to note that @xcite : [ exbr ] br ( b^0 | p^- ) = ( 3.290.47 ) 10 ^ -6 br(b^- |p ) & < & 4.6 10 ^ -7 . the enhancement of three - body decay over the two - body one is due to the reduced energy release in @xmath0 to @xmath70 transition by the fastly recoiling @xmath70 meson that favors the dibaryon production @xcite . theoretical estimations baryonic b decays are made @xcite , in consistent with the experimental observations . in the factorization method , the decay amplitude of @xmath71 contains the @xmath72 transition and @xmath73 baryon - pair inducing from the vacuum . the contributions to the decay at the quark level are mainly from @xmath74 , @xmath75 and @xmath76 operators . from these operators and the factorization approximation , the decay amplitude is given by @xcite @xmath77 where @xmath78 , @xmath79 and @xmath34 are defined in eq . ( [ a146 ] ) . from eq . ( [ m6 ] ) , the t - odd transverse polarization asymmetry @xmath80 is found to be @xmath81 where @xmath82 and @xmath83 are combinations of form factors , given by @xcite @xmath84\;,\ % \nonumber\\ a\;=\;f^{b \to \pi}_1(t)g_a(t)\;,\nonumber\\ s&=&\frac{m_b^2-m_\pi^2}{m_b - m_u}f^{b \to \pi}_0(t)f_s(t)\;,\ % \nonumber\\ p\;=\;\frac{m_b^2-m_\pi^2}{m_b - m_u}f^{b \to \pi}_0(t)g_p(t)\;.\end{aligned}\ ] ] it is noted that the @xmath85 ( @xmath86 ) term is from vector - scalar ( axialvector - pseudoscalar ) interference and there is no t - odd term from @xmath87 due to the same current structures . in eq . ( [ abfg ] ) , @xmath88 are the well known mesonic @xmath89 transition form factors @xcite , while @xmath90 , @xmath91 , @xmath92 , @xmath93 and @xmath94 are the @xmath95 time - like baryonic form factors , defined in ref . @xcite . based on the qcd counting rules @xcite and @xmath96 flavor symmetry , at @xmath97 one has that f_1(t)+ f_2(t)~g_a(t)~h_a(t)~f_s(t)~g_p(t)~ct^2 . in this limit @xmath98 and thus no t violation is expected . however , at the finite @xmath99 there are some high power terms of the @xmath99 expansion . a simple scenario of the power expansions for the baryonic form factors is as follows:@xcite [ ss ] f_1(t)+ f_2(t)=(ct^2+dt^3)^- , & & g_a(t)=(ct^2)^- , + f_s(t)=n_q(ct^2+dt^3)^- , & & g_p(t)=n_q(ct^2)^- , where @xmath100 , @xmath101 and @xmath102 and @xmath103 and @xmath104 are two new form factors . we now evaluate the numerical values for the tpcs in this simple power expansion scenario in eq . ( [ ss ] ) . our results of @xmath105 ( @xmath106 ) and @xmath107 are shown in table [ attable1 ] . [ attable1 ] as an illustration , in the table we have also turned off the strong phase ( @xmath108 ) by taking the imaginary parts of the quark - loop rescattering effects to be zero . from the table , we see explicitly that @xmath18 is indeed nonzero and maximal in the absence of the strong phase . we note that in our calculations we have neglected the final state interactions due to electromagnetic and strong interactions , which are believed to be small in three - body charmless baryonic decays @xcite . we also note that @xmath109 in @xmath110 can be induced but it is too small to be measured . it is interesting to point out that in order to observe @xmath105 ( @xmath16 ) in @xmath111 ( @xmath112 ) being at @xmath113% , we need to have about @xmath114 @xmath115 pairs at @xmath116 level . this is within the reach of the present day @xmath0 factories at kek and slac and others that would come up . it is clear that an experimental measurement of @xmath18 is a reliable test of the ckm mechanism of cp violation and , moreover , it could be the first evidence of the direct t violation in b decays . finally , we remark that we have also explored the direct cp and t violation in @xmath117 @xcite and we have found that the direct cp violating effect is small but the t violating one is as large as that in @xmath118 . in summary , we have shown that the direct cp violating asymmetry in @xmath1 is around 22@xmath2 and the direct @xmath3 violating asymmetry in @xmath4 can be as large as @xmath5 , which are accessible to the current b factories at kek and slac as well as the future ones such as superb and lhcb .

0801.0022

ngc 6388 and ngc 6441 are two of the most intriguing galactic globular clusters ( gc s ) . the integrated - light study by @xcite revealed a strong far - uv flux for these metal - rich ( @xmath0 } \simeq -0.60 $ ] and @xmath1 , respectively ; * ? ? ? * ) bulge gc s . the far - uv flux in resolved gc s is dominated by hot horizontal branch ( hb ) stars ( e.g. , * ? ? ? * ) , especially when rare uv - bright post - asymptotic giant branch stars are not present . accordingly , the most likely explanation for the far - uv flux in ngc 6388 and ngc 6441 was immediately recognized to be hot hb stars . however , while the uv - upturn phenomenon in elliptical galaxies is often attributed to blue hb stars ( e.g. , * ? ? ? * ) , no resolved metal - rich gc had been known with a blue hb morphology . the tendency for metal - rich gc s to have red hb s while metal - poor gc s have predominantly blue hb s reflects the `` first parameter '' of hb morphology . as a consequence , the presence of blue hb stars in ngc 6388 and ngc 6441 would represent an example of the so - called `` second - parameter ( 2@xmath2p ) phenomenon . '' the presence of blue hb stars extending almost as faint in @xmath3 as the turnoff ( to ) point in both ngc 6388 and ngc 6441 was confirmed by @xcite , who presented _ hubble space telescope _ ( _ hst _ ) photometry for both these gc s from the survey by @xcite . a remarkable feature of the published diagrams is the presence of a _ strongly sloped _ hb at colors where other gc s have a much more nearly `` horizontal '' hb . as emphasized by ( * , hereafter sc98 ) , such a sloped hb can not be simply the result of an older age or of enhanced mass loss along the red giant branch ( rgb ) : while these are able to move a star horizontally along the hb , neither is able to increase the luminosity of a blue hb star compared to the red hb or rr lyrae stars . likewise , while strong differential reddening might explain the sloping hb of a red hb cluster , it obviously can not produce rr lyrae and blue hb stars . sc98 conclude therefore that _ non - canonical _ 2@xmath2p candidates must be at play in ngc 6388 and ngc 6441 . however , these conclusions were challenged by ( * ? ? ? * , hereafter r02 ) , who computed models with non - standard values of the chemical abundance and mixing length parameter . some of their models did reveal sloped hb s , but only as a consequence of an anomalously _ faint red hb _ ( in @xmath3 ) , together with a blue hb having a @xmath3-band luminosity consistent with the canonical models ( see , e.g. , their fig . 2 ) . additional insights are provided by stellar variability and spectroscopic studies . @xcite , @xcite , @xcite , and @xcite have shown that the rr lyrae variable stars in these gc s , which occupy the normally `` horizontal '' part of the hb , have much longer periods than field rr lyrae of similar metallicity , thus strongly suggesting that they are intrinsically more luminous ( sc98 ) . moreover , theoretical calculations by @xcite have shown that , contrary to the suggestions by @xcite , the rr lyrae components in both clusters can not be explained in terms of evolution away from a position on the blue hb and neither can the sloping nature of the hb be reproduced in this way . on the other hand , the first spectroscopic measurements of the gravities of blue hb stars in both ngc 6388 and ngc 6441 @xcite revealed surface gravities that are higher than predicted by even the canonical models , thus arguing against an anomalously bright blue hb + rr lyrae component in these clusters . however , a recent reassessment of the spectroscopic gravities of blue hb stars in ngc 6388 by @xcite indicates that the actual gravities should , in fact , be lower than the canonical values . it appears that the 1999 values must have been in error by a substantial amount , probably due to unresolved blends in the crowded inner regions of these massive ( @xmath4 ; * ? ? ? * ) gc s . in an effort to shed light on this puzzling situation , we have made use of the data obtained for ngc 6388 in the course of our snapshot _ hst _ program to study stellar variability in the cluster , and also of archival data , to produce its deepest - ever color - magnitude diagram ( cmd ) . in 2 we describe this dataset and reduction procedures . in 3 we compare our cmd with 47 tucanae s ( ngc 104 ) . we close in 4 by discussing the implications of our results for our understanding of the origin of the peculiar hb morphology of ngc 6388 . symbols _ ) is overplotted on the 47 tuc cmd ( _ plus signs _ ) in the @xmath3 , @xmath5 plane . to produce this plot , the 47 tuc cmd was shifted by + 3.2 in @xmath3 and by + 0.85 in @xmath5 , so as to align their red hb components . , width=326 ] [ fig : cmd ] the ngc 6388 data used in this paper were obtained under _ hst _ program snap-9821 ( pi b. j. pritzl ) , which used the wide - field channel of the _ advanced camera for surveys _ ( acs ) to obtain six @xmath6 ( f435w ) , @xmath3 ( f555w ) , @xmath7 ( f814w ) exposure triptychs on separate dates ranging from 2003 october to 2004 june . in addition , we have retrieved data from the _ hst _ archives , as obtained under go-9835 ( pi g. drukier ) . these consist of 12 @xmath3 and 17 @xmath7 exposures obtained with the high - resolution channel of acs on 2003 october 30 . the results employed here for 47 tuc were obtained from the ground - based imagery used by stetson to define his secondary photometric standards . they consist of ( 125 , 136 , 94 ) images in ( @xmath6 , @xmath3 , @xmath7 ) from 12 distinct observing runs ; these photometric indices should be on the system of @xcite to well under 0.01@xmath8mag . stetson s ground - based data also include ( 84 , 137 , 99 ) images in ( @xmath6 , @xmath3 , @xmath7 ) for ngc 6388 . although these data are not included in the plots in this paper , they were used to establish a network of photometric standards in the cluster field which could be used to establish accurate photometric zero - points for the acs images . color transformations for the acs data were based on these local standards and similar acs - ground - based comparisons for 47 tuc , ngc 2419 , ngc 6341 ( m92 ) , and ngc 6752 . the data were reduced in the standard manner , using the daophot - allframe software packages , following commonly understood reduction procedures ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? a complete description of our dataset , reduction and calibration procedures will be described in a future paper ( stetson et al . 2006 , in preparation ) . before closing , we note that the acs filter bandpasses are not identical to the standard @xmath6 , @xmath3 , @xmath7 bandpasses of @xcite but then again , even among the various ground - based observing runs where the clusters were observed , the filters and detectors are not identical : bandpass mismatch is an unavoidable complication in filter photometry when one does not have a privately owned photometer . even _ with _ a private photometer , bandpasses can drift with atmospheric conditions , or as filters , detectors , and mirror coatings age . we have removed the effects of mismatch to second order by including linear and quadratic color terms , both in the transformation equations that relate our ground - based observations to landolt s photometric system _ and _ in the equations that relate the acs magnitudes to our ground - based system . extensive experience suggests that the residual effects of bandpass mismatch in these filters reach an irreducible minimum scatter of 0.01 0.02 mag on a star - by - star basis , and considerably less than this on an ensemble average of many stars . , but using the reddening - independent quantity @xmath9 instead of @xmath3 , enforcing a match of the to colors of the two clusters , and zooming in around the to / sgb level . , width=326 ] [ fig : qmd ] our deep @xmath3 , @xmath5 cmd of ngc 6388 is shown in figure [ fig : cmd ] ( _ red @xmath10 symbols _ ) , overplotted on the 47 tuc cmd ( _ plus signs _ ) . to obtain this plot , we have applied shifts of + 3.2 in @xmath3 and + 0.85 in to the 47 tuc data , in order to register its red hb to ngc 6388 s . the magnitude and color of the to in each cluster were determined by an iterative robust numerical fit of a parabola to the data in a magnitude range @xmath11mag of the to ; we obtained ( @xmath3 , ) @xmath12 for ngc 6388 , and ( 17.70 , 1.27 ) for 47 tuc . note that we obtained clean , sharp , well - populated sequences in the cmd by plotting only the stars with the best photometry . our selection criteria will be described in stetson et al . ( 2006 ) . the to points and sgb s of 47 tuc and ngc 6388 coincide in brightness rather well once their red hb s are registered . this suggests that any age difference between the two clusters is at the level of @xmath13 gyr , assuming that their red hb s have the same absolute luminosity . @xcite and @xcite computed isochrones for different values of [ fe / h ] , @xmath14 $ ] , helium abundance @xmath15 , the mixing length parameter @xmath16 , and rotation rates @xmath17 . as is already well known , the difference in color between the base of the rgb and the main - sequence to is a good relative age indicator , due to its small dependence on [ fe / h ] , @xmath14 $ ] , and @xmath17 . on the other hand , these calculations also show that this color difference , as well as the detailed shape of the sgb , do present some dependence on @xmath15 and @xmath16 . to perform a more meaningful differential comparison that minimizes the effects of differential reddening in ngc 6388 , we have replaced @xmath3 with the reddening - independent quantity @xmath18 , and then registered the to points of the two clusters in color . although numerical fitting of parabolas to @xmath9 as a function of @xmath5 at the to s of the two gc s suggests that their bluest colors differ by 0.77 mag , we still performed a sanity check by directly overplotting the two cmds with a range of horizontal shifts , and concluded that @xmath19 ( @xmath20 error bar ) appeared best to the eye ( see fig . [ fig : qmd ] ) . we should note that the detection limit in these data is near @xmath21 , and that past experience ( e.g. , * ? ? ? * ) leads us to expect that the observed colors may be biased too blue by @xmath220.1@xmath8mag as this limit is approached . accordingly , we have been careful to restrict our comparisons to the immediate vicinity of the to itself , @xmath23 , where any systematic bias is expected to be much smaller . figure [ fig : qmd ] shows that , when the to points of the clusters are registered , the difference in color @xmath24 between the base of the rgb and the to is very similar for both clusters . this quantity is only slightly larger for ngc 6388 than for 47 tuc , at a level @xmath25 mag ( @xmath20 error bar ) . if interpreted in terms of an age difference , this translates into @xmath26 gyr ( ngc 6388 being younger ) . assuming that the clusters have similar ages , [ fe / h ] , and @xmath14 $ ] , which appears most consistent with the to / red hb ( `` vertical method '' ) and the rgb data ( see also the next subsection ) , one also finds @xmath27 ( in the sense that ngc 6388 should have a _ lower _ @xmath15 ) , or @xmath28 ( again in the sense that ngc 6388 should have a lower @xmath16 ) . such a difference in @xmath15 between the two gc s would be the opposite of that needed to explain the ngc 6388 hb morphology . stronger constraints on @xmath15 and @xmath16 variations between the two clusters are derived from their rgb properties ( see 3.2 ) . the rgb s of 47 tuc and ngc 6388 have very similar morphologies , according to figure [ fig : cmd ] . the larger scatter in the ngc 6388 cmd than in 47 tuc s might be due in part to photometric errors and in part to differential reddening , thus making it difficult to determine if the stars scattered toward the red of the main rgb in ngc 6388 represent a minor metal - rich component . a sizeable metal - poor component is clearly not present ( see also r02 ) . a small deviation of the bulk of the brighter ngc 6388 rgb stars towards redder colors compared with the 47 tuc cmd , if real , would suggest that ngc 6388 might be slightly more metal - rich ( i.e. , by @xmath29 dex ) than 47 tuc ( @xmath0 } = -0.76 $ ] dex ; * ? ? ? when their red hb s are registered , one also finds that their to colors become somewhat offset ( with ngc 6388 s being bluer ) and that their rgb s actually cross just above the hb level ( fig . [ fig : cmd ] ) . theoretical isochrones show that these effects are to be expected if ngc 6388 is more metal - rich than 47 tuc at the level indicated above . the `` bump '' in the rgb luminosity function lies at a @xmath30 in ngc 6388 , and 14.50 in 47 tuc . thus , the to - to - bump @xmath3-magnitude differences are 3.26 and 3.20 , respectively . if real , this difference would suggest that ngc 6388 is more metal-_poor _ ( by @xmath31 dex ) than 47 tuc . on the other hand , the values of @xmath32 measured by @xcite differ by only 0.03 mag between 47 tuc and ngc 6388 ; this could be produced by a very small metallicity difference , by a @xmath33 ( ngc 6388 being more helium - rich ; see fig . 5 in r02 ) , and/or by a small difference in age of @xmath34 gyr ( ngc 6388 being younger ) . note , finally , that near - infrared photometry ( e.g. , * ? ? ? * ; * ? ? ? * ) shows that ngc 6388 and ngc 6441 have , if anything , slightly _ bluer _ rgb s than 47 tuc , contrary to what would be expected in the r02 scenario . indeed , @xcite find a normal value of @xmath35 for both ngc 6441 and 47 tuc ( ngc 6388 is not in their studied sample ) . the lack of a large difference in @xmath16 and [ fe / h ] between the two gc s , while not unexpected , already rules out the r02 scenario , in which a combination of high metallicity and low @xmath16 is invoked , leading to an underluminous red hb as mentioned in 1 . , but using @xmath9 instead of @xmath3 and focusing around the red hb region . , width=326 ] [ fig : cmd2 ] an overluminous red hb , as would be implied by a high primordial @xmath15 ( sc98 ) , is ruled out if , as appears likely , the two clusters have similar ages and metallicities . the hb luminosity function is also known to be affected by the stars evolutionary parameters ; in particular , a higher @xmath15 in ngc 6388 should produce more luminosity evolution away from the zero - age hb ( e.g. , * ? ? ? * ) , which is clearly not present in the observed cmd . likewise , the fact that there is no significant component fainter than the bulk of the ngc 6388 red hb stars suggests that any metal - rich component in this cluster ( i.e. , with @xmath0 } \gtrsim -0.5 $ ] dex ) should be minor . figure [ fig : cmd2 ] shows a more detailed comparison of the cmds of ngc 6388 and 47 tuc around the red hb region , again using the reddening - independent quantity @xmath9 . the 47 tuc @xmath9 values were shifted by @xmath36 mag . adding random scatter at a level of @xmath37 mag to the 47 tuc cmd allows one to reproduce the overall appearance of the rgb and the red hb quite well , thus implying a @xmath38 ( in the sense of ngc 6388 having a higher @xmath15 ) based on the luminosity width of the hb . the main difference between the two hb s is the tendency for the bluer of the ngc 6388 red hb stars to be slightly brighter than the average line that defines the 47 tuc red hb . this , along with the fact that the to points and sgb s of the two gc s match well in brightness when the two red hb s are registered , strongly suggests that both the rr lyrae and blue hb components ( and , to a lesser extent , also the bluer red hb stars ) of ngc 6388 are indeed intrinsically overluminous with respect to the 47 tuc red hb . well - populated agb `` clumps '' are seen in both the 47 tuc and ngc 6388 cmd s . @xcite have shown that , the bluer the hb morphology of a gc , the less pronounced the resulting agb clump . therefore , the majority of the stars in this phase originate from red hb stars . it is interesting to note that the difference in @xmath3 magnitude between the agb clump and the red hb is basically indistinguishable between ngc 6388 and 47 tuc . unfortunately , this does not provide us with strong constraints on the difference in @xmath15 or metallicity between the two gc s , since the difference in magnitude between the agb clump and the red hb is not very sensitive to these evolutionary parameters ( e.g. , * ? ? ? in the present _ letter _ , we have shown that , apart from the blue hb and rr lyrae components , the cmd s of ngc 6388 and 47 tuc are very similar , thus strongly suggesting that the bulk of the stars in these two clusters are very similar . differences in age , metallicity , @xmath14 $ ] , @xmath15 , and @xmath16 between the two gc s , if present , should be small . our results suggest that the red hb component of ngc 6388 is neither underluminous ( as in the `` canonical tilt '' scenario of r02 ) nor overluminous ( as in the high-@xmath15 scenario of sc98 ) , except for its bluest stars . on the other hand , both the rr lyrae and blue hb stars in ngc 6388 _ are _ significantly overluminous compared to field rr lyrae stars of similar [ fe / h ] , thus explaining ngc 6388 s anomalously long rr lyrae periods . the lack of a sizeable luminosity difference between the red hb s of ngc 6388 and 47 tuc indicates that _ the 2@xmath39p that leads to the production of an overluminous rr lyrae + blue hb component in ngc 6388 must be non - canonical in nature_that is , it must be neither age nor rgb mass loss , or else a sloped hb would not result ( sc98 ) . however , since only a relatively small fraction ( @xmath40 ; @xcite @xcite ) of the hb stars are in the rr lyrae strip or on the blue hb , only a minor fraction of the cluster stars should be affected by this non - canonical 2@xmath2p , which could be either @xmath15 or / and the helium - core mass at the he - flash . ( a fraction of the red hb stars , especially the bluer 25% or so , could also be affected , though to a much smaller extent . ) additional , detailed studies of the blue hb , rr lyrae , and main sequence components will be required before we are in a position to conclusively decide what 2@xmath2p(s ) is ( are ) responsible ( see * ? ? ? * for a recent review ) . we thank the referee , g. piotto , for a very helpful report . m.c . acknowledges support by proyecto fondecyt regular no . b.j.p . would like to thank nasa for the support for the snap-9821 project through a grant from the space telescope science institute . acknowledges support from the csce and nsf under ast 02 - 05813 . pritzl , b. j. , smith , h. a. , catelan , m. , & sweigart , a. v. 2001 , , 122 , 2600 pritzl , b. j. , smith , h. a. , catelan , m. , & sweigart , a. v. 2002 , , 124 , 949 pritzl , b. j. , smith , h. a. , stetson , p. b. , catelan , m. , sweigart , a. v. , layden , a. c. , & rich , r. m. 2003 , , 126 , 1381 ree , c. h. , yoon , s .- j . , rey , s .- c . , & lee , y .- w . 2002 , in omega centauri , a unique window into astrophysics , asp conf . ser . 265 , ed . f. van leeuwen , j. d. hughes , & g. piotto ( san francisco : asp ) , 101

using the _ hubble space telescope _ , we have obtained the first color - magnitude diagram ( cmd ) to reach the main - sequence turnoff of the galactic globular cluster ngc 6388 . from a comparison between the cluster cmd and 47 tucanae s , we find that the bulk of the stars in these two clusters have nearly the same age and chemical composition . on the other hand , our results indicate that the blue horizontal branch and rr lyrae components in ngc 6388 are intrinsically overluminous , which must be due to one or more , still undetermined , non - canonical second parameter(s ) affecting a relatively minor fraction of the stars in ngc 6388 .

astro-ph0609727

the neupert effect is the name given to the observational finding that the rising part of the soft ( sxr ) light curve often resembles the time integral of the hard x - ray ( hxr ) or microwave emission ( , 1968 ; dennis & zarro , 1993 ) . the physical relevance of the neupert effect basically arises from the fact that it is interpreted as a causal connection between the thermal and nonthermal flare emissions , which can be naturally explained within the nonthermal thick - target model ( brown , 1971 ) . in this model , the flare energy is released primarily in the form of nonthermal electrons , and hard x - rays are produced via electron - ion bremsstrahlung when the electron beams impinge on the lower corona , transition region and chromosphere . the model assumes that only a small fraction of the electron beam energy is lost through radiation ; most of the loss is due to coulomb collisions that serve to heat the ambient plasma . as a consequence of the rapid energy deposition a strong pressure imbalance develops between the dense , heated chromosphere and the tenuous corona . the high pressure gradients cause the heated plasma to convect into the corona in a process known as chromospheric evaporation ( antonucci et al . 1984 ; fisher et al . 1985 ) , where it gives rise to enhanced sxr emission via thermal bremsstrahlung . in this case , the hard x - ray flux is linked to the instantaneous rate of energy supplied by electron beams , whereas the soft x - ray flux is related to the accumulated energy deposited by the same electrons up to that time , and we can expect to see the neupert effect . any deviation from the neupert effect , in principle , suggests that the hot sxr emitting plasma is not heated exclusively by thermalization of the accelerated electrons that are responsible for the hxr emission . therefore , investigations of the neupert effect provide insight into the role of nonthermal electrons for the flare energetics . the neupert effect can be expressed as @xmath0 with @xmath1 the sxr peak flux and @xmath2 the hxr fluence , i.e. the hxr flux integrated over the event duration . the coefficient @xmath3 depends on several factors , as , e.g. , the magnetic field geometry and the viewing angle , and thus may vary from flare to flare ( lee et al . , however , if @xmath3 does not depend systematically on the flare intensity , then the sxr peak flux and the hxr fluence are linearly related . we utilize the sxr data from the _ geostationary operational environmental satellites _ ( goes ) and the hxr data from the _ burst and transient source experiment _ ( batse ) aboard the _ compton gamma ray observatory_. the x - ray sensor aboard goes consists of two ion chamber detectors , which provide whole - sun x - ray fluxes in the and 18 wavelength bands . batse is a whole - sky hxr flux monitor that , in part , consists of eight large - area detectors . from each detector there are hard x - ray measurements in four energy channels , 2550 , , 100300 and @xmath4300 kev ( schwartz et al . , 1992 ) . for the analysis , the 1-min averaged goes sxr data measured in the 18 channel and the hxr data collected in the batse solar flare catalog , archived in the solar data analysis center at nasa / goddard space flight center for the period 01/199706/2000 are used . the peak and total count rates are background subtracted for the flux below 100 kev . for the sxr events , we used the flux just before the flare start for background . to be identified as corresponding events we demand that the start time difference between a sxr and a hxr event does not exceed 10 min . overlapping events are excluded . applying these criteria , we obtained 1114 events that were observed in both hard and soft x - rays ( for details see veronig et al . , 2002a ) . for each event we determined the difference , @xmath5 , of the peak time of the sxr emission , @xmath6 , and the end time of the hxr emission , @xmath7 . furthermore , the time differences were normalized to the duration @xmath8 of the respective hxr event , i.e.@xmath9 figure [ fig_timing ] shows the histogram of the absolute and normalized time differences . both representations of the sxr hxr time difference have its mode at zero . 49% of the events lie within the range @xmath10 min , and 65% within @xmath11 min . for the normalized time differences , we obtain that 44% lie within @xmath12 hxr units , and 59% within @xmath13 hxr unit . this outcome suggests that certainly a considerable part of the events coincides well with the expectations from the effect regarding the relative timing . figure [ fig_fluencepeak ] shows the scatter plot of the sxr peak flux versus the hxr fluence for the complete sample , clearly revealing an increase of @xmath1 with increasing @xmath14 . it can also be inferred from the figure that the slope is not constant over the whole range but that it is larger for large hxr fluences than for small ones . for very large fluences , the slope approaches the value of 1 , indicative of a linear relation between the sxr peak flux and hxr fluence . we stress that the slope at small fluences might be affected by missing events with small sxr peak fluxes ( due to selection effects ) , and thus appear flatter than it is in fact . figure [ fig_fluencepeak ] reveals an interdependence between the importance of an event and the sign of the time difference . basically all large flares belong to the group of events with @xmath15 , i.e. the sxr peak occurs before the hxr end . moreover , the flares with @xmath15 reveal a strong tendency to be of long duration . we obtain a high cross - correlation coefficient for the sxr peak flux and hxr fluence relationship , @xmath16 . this coefficient is higher than those for the sxr peak flux and hxr peak flux , @xmath17 . this indicates that the correlation is primarily due to the hxr fluence sxr peak flux relationship , as predicted from the neupert effect , and not , e.g. , due to the fact that flares with large hxr peak fluxes also tend to have intense sxr counterparts . furthermore , it is important to note that the hxr fluence sxr peak flux correlation is higher for the events with negative time differences , @xmath18 , than for the events with positive time differences , @xmath19 . on the basis of the relative timing of the sxr peak and the hxr end , we extracted two subsets of events . the events of set 1 are roughly consistent with the timing expectations of the neupert effect , and the events of set 2 are inconsistent with it . the two sets are defined by the following conditions : @xmath20 the applied conditions represent a combination of absolute and normalized time differences in order to avoid as much as possible any a priori interdependence with the flare duration and/or flare intensity . out of the 1114 corresponding hxr and sxr flares , 485 ( 44% ) events fulfilled the timing criterion of set 1 ; 270 events ( 24% ) belong to set 2 ; 359 events ( 32% ) are neither attributed to set 1 nor to set 2 . in figure [ fig_fluencepeak2 ] , we plot the sxr peak flux versus the hxr fluence separately for set 1 and set 2 . the figure reveals that the two sets have very different characteristics besides the different timing behavior . set 1 contains many more large events and shows a steeper increase of @xmath1 with increasing @xmath2 than set 2 . moreover , set 1 contains more events with negative than positive time difference , whereas almost all events of set 2 are characterized by @xmath21 , i.e. increasing sxr emission beyond the end of the hard x - rays . on average , for small fluences the events belonging to set 2 have a larger sxr peak flux at a given hxr fluence than those of set 1 , indicating an excess " of sxr emission with respect to set 1 . the cross - correlation coefficients derived separately for the subsets reveal that the correlation among the sxr peak flux and hxr fluence is much more pronounced for the events of set 1 , @xmath22 , than those of set 2 , @xmath23 . 24% of the events have @xmath15 , i.e. the sxr maximum occurs before the end of the hxr emission . these events are preferentially of long duration . li et al . ( 1993 ) have calculated time profiles of soft and hard x - ray emission from a thick - target electron - heated model , finding that , in general , the time derivative of the sxr emission corresponds to the time profile of the hxr emission , as stated by the neupert effect . however , for gradual events they obtained that this relationship breaks down during the decay phase of the hxr event , in that the maximum of the sxr emission occurs before the end of the hxr emission . this phenomenon can be explained by the fact that the sxr emission starts to decrease if the evaporation - driven energy supply can not overcome the instantaneous cooling of the hot plasma , which is likely to happen in gradual flares . considering our observational findings together with the results from simulations by li et al . ( 1993 ) , presumably most of the events with @xmath15 are consistent with the electron - beam - driven evaporation model . in particular , the high correlation between @xmath1 and @xmath2 , @xmath24 , supports such interpretation . 56% of the events have @xmath21 ; these events are preferentially of short and weak hxr emission . in principle , the fact that the sxr emission is still increasing although the hxr emission , i.e. the electron input , has already stopped indicates that an additional agent besides the hxr emitting electrons is contributing to the energy input and prolonging the heating and/or evaporation . however , mctiernan et al . ( 1999 ) have shown that the sxr time profile depends on the temperature response of the used detector : an increase of the sxr emission of low - temperature flare plasma after the hxr end may arise due to cooling of high - temperature plasma . thus , we can not attribute all events with @xmath21 as inconsistent with the electron - beam - driven chromospheric evaporation model . instead , we consider as inconsistent only flares which show strong deviations from @xmath25 , i.e. the events belonging to set 2 . this means that for at least one fourth of the analyzed events an additional heating agent besides nonthermal electrons is suggested . a probable scenario is that energy is transported from the primary energy release site via thermal conduction fronts , which initiate chromospheric evaporation but do not produce hard . the finding that for a considerable fraction of flares , preferentially weak ones , an additional heating agent other than electron beams is suggested , is not only relevant for the flare energetics but also for parker s idea of coronal heating by nanoflares ( parker , 1988 ) . hudson ( 1991 ) pointed out that if the corona is heated by flare - like events of different sizes , then the flare energy distribution must have a power - law slope @xmath26 . if the sxr flux does not vary systematically with temperature and density , then the sxr peak flux is linearly related to the maximum thermal energy of the flare plasma ( see lee et al . , 1995 ; veronig et al . , 2002a ) . on the other hand , hxr fluence distributions can be considered as representative for the energy contained in nonthermal electrons . in figure [ fig_distr ] , we show the sxr peak flux and the hxr fluence distributions derived from 1114 corresponding sxr / hxr flares , finding @xmath27@xmath280.13 and @xmath29@xmath280.13 , respectively . the discrepancy between the slopes of the hxr fluence and the sxr peak flux distributions was already pointed out and discussed in lee et al . ( 1993 , 1995 ) and veronig et al . ( 2002b ) . the present analysis provides an explanation for this difference in power - law slopes : the relationship between the sxr peak flux and the hxr fluence is not linear , whereby the deviations from a linear correlation are strongest for weak flares ( cf . figure [ fig_fluencepeak ] ) . the soft x - ray flare emission increases due to energy supply by electron beams as well as any other heating agent , whereas the hard x - ray emission contains only information on the energy provided by electrons . together with our finding that particularly in weak events an additional heating agent besides electron beams is suggested , this strongly suggests that soft x - ray peak flux distributions are a more meaningful indicator of flare energy distributions than hard x - ray fluence distributions . furthermore , we have shown that weak flares have different characteristics than large flares , in the sense that electrons are less important for their energetics . therefore , it is possible that the power - law slope of nanoflare frequency distributions differs from that derived for observed flares , which is close to the critical value of @xmath30 . in this respect it should be worthwhile to investigate flare frequency distributions of sxr flares without hxr counterparts , i.e. without detectable particle acceleration , since these flares possibly provide the link to the smallest flare - like energy release events . a. v. , m. t. and a. h. gratefully the austrian _ fonds zur frderung der wissenschaftlichen forschung _ ( fwf grant p15344-phy ) for supporting this project . antonucci e. , gabriel a. h. , dennis b. r. , 1984 , apj 287 , 917 brown j. c. , 1971 , solar phys . 18 , 489 dennis b.r . , zarro d.m . , 1993 , solar phys.146 , 177 fisher g. h. , canfield r. c. , mcclymont a. n. , 1985 , apj 289 , 425 hudson h. s. , 1991 , solar phys . 133 , 357 lee t. t. , petrosian v. , mctiernan j. m. , 1993 , apj 412 , 401 lee t. t. , petrosian v. , mctiernan j. m. , 1995 , apj 448 , 915 li p. , emslie a.g . , mariska j.t . , 1993 , apj417 , 313 mctiernan j. m. , fisher g. h. , li p. , 1999 , apj 514 , 472 neupert w. m. , 1968 , apj 153 , l59 parker e. n. , 1988 , apj 330 , 474 schwartz r. a. , dennis b. r. , fishman g. j. , et al . , 1992 , nasa cp-3137 , 457 veronig a. , vrnak b. , dennis b. r. , temmer m. , hanslmeier a. , magdaleni j. , 2002a , a&a revised veronig a. , temmer m. , hanslmeier a. , otruba w. , messerotti m. , 2002b , a&a 382 , 1070

based on simultaneous observations of solar flares in hard and soft x - rays we studied several aspects of the neupert effect . about half of 1114 analyzed events show a timing behavior consistent with the effect . for these events , a high correlation between the soft peak flux and the hard fluence is obtained , being indicative of electron - beam - driven evaporation . however , for about one fourth of the events there is strong evidence for an additional heating agent other than electron beams . we discuss the relevance of these findings with respect to parker s idea of coronal heating by nanoflares .

astro-ph0208089

universal quantum cloning refers to the possibility of constructing unitary transformations which approximately copy an arbitrary quantum state and hence partially alleviate the limitations of the no - cloning theorem @xcite ( see also @xcite and @xcite ) . it was first achieved by buek and hillery in @xcite in which they proposed a cloning transformation which clones arbitrary states with equal fidelity @xmath1 . their pioneering work stimulated a lot of intense research in quantum cloning , a sample of which includes works on proofs of optimality @xcite , generalizations to @xmath2 cloning @xcite , cloning of @xmath3-level states @xcite , and finally experimental realization of cloning by various techniques @xcite . + since the optimal bound of @xmath4 for fidelity was set for universal cloning , attempts were also made to go beyond this limit by cloning special subsets of states for which we have some a priori partial information . this search was indeed successful and led to the so - called phase covariant quantum cloning @xcite . for two level states , or qubits , phase covariant quantum cloning means that a certain class of states , called ( @xmath5 ) equatorial states , defined as @xmath6 which is slightly higher than optimal bound achievable for universal quantum cloning . for @xmath3-level states it means that states of the form @xmath7 the crucial property of this class which allows for this higher fidelity is that all the coefficients in their expansion have equal norm . due to this property a state dependent term in the final density matrix of the clones in the cloning transformation , which is of the form @xmath8 , becomes automatically state independent ( universal ) , hence no need for making its coefficient vanish by tuning the parameters of the cloning transformation . with the automatic disappearance of this term and one more parameter at hand we find the chance to obtain higher fidelity than the optimal one this is all the technical point of the phase covariant quantum cloning . there is of course one motivation for studying these states which comes from quantum cryptography , since at least in the bb84 protocol , the states in transfer between the legitimate parties are of this form and an eavesdropper needs only to clone these kinds of states to threat the security of the communication . + however , when we think in terms of physical properties , the partial information that we have about these states is that the @xmath0 component of their spin is zero . therefore , it is natural to ask a more general question , that is , how well we can clone a spin states @xmath9 if we know the third component of its spin @xmath10 ? this question is specially interesting for those who try to achieve optimal cloning by nmr techniques @xcite . in fact this is precisely the state of a nuclear spin which is precessing in magnetic field with a definite energy . in this sense we not only generalize the concept of phase covariant quantum cloning , but describe it in a physically and experimentally interesting context . we show that there exist a one - parameter family of cloning transformations in which by tuning the parameter one can always clone such states with higher fidelity than the optimal one . furthermore we show that within this class , the case of equatorial states give a lower fidelity of cloning compared to other states . however they are unique in the sense that they are the only states in this class which give rise to separable density matrix for the outputs copies . we also show that our consideration can be readily generalized to @xmath3-level states . + the structure of this paper is as follows : in sec . [ sec2 ] we study the general properties of a one parameter family of cloning transformations of qubits . in sec . [ sec3 ] we make detailed comparison between different cloning transformations , namely the universal cloning machine proposed by buek and hillery , the phase covariant cloning proposed in @xcite and the one proposed in this paper . in sec . [ sec4 ] we briefly discuss the phase covariant cloning of @xmath11 level states @xcite in this new context . the paper ends with a conclusion . consider the following cloning transformation @xmath12 where on the left hand side we have not shown the blank state and initial state of the cloning machine and on the right hand side , the states from left to right correspond respectively to the input ( @xmath13 ) , the copy ( @xmath14 ) and the machine states ( @xmath15 ) . the states @xmath16 and @xmath17 , are also orthonormal regardless of their indices . the only requirement for this transformation to be unitary is that @xmath18 and @xmath19 be related as @xmath20 consider now a general two level state , i.e. a state with a definite spin in the direction @xmath21 , where @xmath22 and @xmath23 are the polar coordinates on the unit sphere . this state has the following form in the @xmath0-basis ( @xmath24 the output state of the composite system @xmath25 is obtained by tracing out the states of the machine @xmath15 , that is @xmath26 when acted on by the cloning machine ( [ basiccloner ] ) this state gives rise to the following density matrix for the output @xmath13 @xmath27 the new copy @xmath28 will also have the same density matrix . the fidelity of cloning defined by @xmath29 is found to be @xmath30 which after a little algebra using , the fact that @xmath31 , and the normalization condition @xmath32 , takes the form @xmath33 the last term clearly depends on the input state . all the states on the bloch sphere with the same value of @xmath34 are cloned with equal fidelity , a special subclass of these states are the so called equatorial states , those with @xmath35 . following buek and hillery @xcite it is useful to define and calculate two distances which characterize further the quality of cloning , namely @xmath36\end{aligned}\ ] ] which measures the degree of entanglement of the two output copies and @xmath37,\end{aligned}\ ] ] which measures the distance of the two mode output density matrix with the ideal situation of having two disentangled exact copies of the input states . the calculation of these distances are straightforward but rather lengthy . we give only the final results @xmath38 where @xmath39 and @xmath40 until now the value of @xmath18 has been kept arbitrary . we should now fix it and hence complete definition of our cloning transformation ( [ basiccloner ] ) . in the sequel , we consider three different cases . looking at eq . ( [ basicfidelity2 ] ) , we find that universality , in the sense of buek and hillery , is achieved only be setting @xmath41 which together with normalization yields @xmath42 here no optimization should be performed , since the demand of universality has fixed completely the parameter @xmath43 . it is interesting to note that in this case the two distances @xmath44 and @xmath45 are also state independent . in fact , by inserting the above value for @xmath18 in eqs . ( [ dab1final ] ) and ( [ dab2final ] ) one finds that @xmath46 in this part we are interested in cloning only the states with @xmath47 . thus the parameter @xmath48 is free and we can fix it by maximizing the value of @xmath49 . one thus finds @xmath50 which is slightly higher than the value for universal quantum cloning . + the distances are found to be @xmath51 although the distance @xmath52 for the equatorial states is appreciably higher than the universal value , as we will see below the equatorial states are separable when cloned phase covariantly @xcite while in universal cloning machine of buek and hillery the output states are not separable . in this case , we fix the value of @xmath22 and find from eq . ( [ basicfidelity2 ] ) that @xmath53 is extremized by two values of @xmath48 obtained from @xmath54 it turns out that the negative sign corresponds to the maximum fidelity . inserting this value of @xmath43 in eqs . ( [ basicfidelity2 ] ) , ( [ dab1final ] ) and ( [ dab2final ] ) will give us the optimal fidelity and the distances for this class of states . the results are shown in figs . [ fig1 ] , [ fig2 ] and [ fig3 ] . it is seen clearly that for each fixed @xmath22 , one can clone the spin state with a fidelity greater than the universal value and for most angles a higher fidelity can be obtained than for the equatorial states . if judged on the basis of the distances @xmath55 and @xmath56 , it also appears from figs . [ fig2 ] and [ fig3 ] , that there are other states which are closer to a product state than the equatorial ones . however the equatorial states are unique in one important respect which is discussed in the next subsection . .1 cm [ bc][][0.75][0]@xmath57 [ bc][][0.75][0]@xmath58 [ bc][][0.75][0]@xmath59 [ bc][][0.75][0]@xmath60 component as a function of @xmath61,title="fig:",width=226,height=188 ] -.9 cm [ bc][][0.75][0]@xmath62 [ bc][][0.75][0]@xmath63 [ bc][][0.75][0]@xmath59 [ bc][][0.75][0]@xmath64 as a function of @xmath65 . , title="fig:",width=264,height=200 ] -.9 cm [ bc][][0.75][0]@xmath66 [ bc][][0.75][0]@xmath67 [ bc][][0.75][0]@xmath59 [ bc][][0.75][0]@xmath68 as a function of @xmath65 . , title="fig:",width=264,height=200 ] for the universal cloning case , using peres - horodecki s positive partial transposition ( ppt ) criterion @xcite , it has been shown that two output modes are inseparable , while the phase covariant cloning of equatorial states lead to separable copies @xcite . to check separability for general angles , we have numerically computed the eigenvalues of the partial transpose of the output density matrix ( [ outputrho ] ) which is of the following form @xmath69^{t_a } = \left ( \begin{array}{cccc } \nu^2 \cos^{2}\frac{\theta}{2 } & \mu \nu e^{-i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } & \mu \nu e^{i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } & \mu^2\\ \mu \nu e^{i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } & \mu^2 & 0 & \mu \nu e^{i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } \\ \mu \nu e^{-i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } & 0 & \mu^2 & \mu \nu e^{-i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } \\ \mu^2 & \mu \nu e^{-i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } & \mu \nu e^{i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2 } & \nu^2 \sin^2 \frac{\theta}{2 } \end{array}\right),\end{aligned}\ ] ] and have found that three of the eigenvalues are always positive , while one of them is marginally negative and becomes zero only for the states on the north pole @xmath16 , the south pole @xmath70 and the equator of the bloch sphere . the values of this negative eigenvalue is shown in fig . [ fig4 ] and is compared to the negative eigenvalue of the universal machine which is @xmath71 , ( while the other eigenvalues of the universal cloning machine being @xmath72 and @xmath73 , all independent of @xmath22 ) . [ bc][bc][0.9][0]_nonpositive eigenvalue _ [ bc][][0.9][0]@xmath59 ^{t_a}$ ] vs. @xmath65 . the dashed line shows its value in universal cloning.,title="fig:",width=264,height=200 ] thus also in this general class of states , the equatorial states are special in that they are completely separable . however , if one considers the multiple criteria of high fidelity and approximate separability then it may be concluded from all the above figures that the states with angles less than @xmath74 radians around the north and south poles can be cloned with sufficiently high ( larger than 0.9 fidelity ) and rather good separability properties . in this section we address the question of optimality of the transformations ( [ basiccloner ] ) . the general form of these transformations are the same as the original cloning transformations found by buek and hillery @xcite and proved to be optimal for universal cloning @xcite . here we have shown that by adjusting the single parameter of these transformations , one can clone states with definite @xmath0 components of spin , with a higher than universal fidelity . however it may be possible to go beyond these one parameter family of transformations and obtain even higher fidelity . there is in fact a constructive procedure for deriving the trace preserving completely positive ( cp ) maps which perform a given task @xcite like cloning , to the best approximation . however we think that by following the procedure of @xcite our transformations may not retain their simple form that they have now . since phase covariant quantum cloning has been achieved for @xmath3-level states , again with a fidelity which is higher than the universal value , it is natural to ask how the above considerations extend to @xmath3-level states . consider the following cloning transformation @xcite , which is a simple and natural generalization of ( [ basiccloner ] ) @xmath75 it can be easily verified that this transformation is unitary provided that we have @xmath76 in particular for 3-level states or qutrits , the transformation is @xmath77 the cloning transformation ( [ basicclonerd ] ) transforms a pure state @xmath78 into a mixed state @xmath79 and in phase covariant cloning , one gets rid of the final term by considering only the equatorial states of the form given in eq . ( [ phasecovariantdlevel ] ) . clearly this is a heavy restriction on the states . to see what this implies for the states in terms of observables we note that the lie algebra of @xmath80 is spanned by traceless hermitian matrices . the cartan subalgebra of this lie algebra which generalizes the @xmath81 pauli matrix for spins , is spanned by diagonal traceless matrices , @xmath82 normalized to @xmath83 . one convenient choice is @xmath84 . for example , for qutrits we have @xmath85 thus phase covariant qutrit states are precisely those states for which @xmath86 in fact the most general state of a qutrit is given by @xmath87 and the fidelity of cloning of this state by the transformation ( [ basiccloner3 ] ) is found from eq . ( [ outputd ] ) to be equal to @xmath88 where @xmath89 for a phase covariant state where all the coefficients have equal amplitude we have @xmath90 and @xmath91 . for this very specific class of states with only two free parameters @xmath92 and @xmath93 , the fidelity is found from eq . ( [ fidelity3 ] ) to be @xmath94 which is optimized by taking @xmath95 giving a value of @xmath96 @xcite . ( there exists also another solution , namely @xmath97 , but for this particular situation it is the first solution which gives the higher fidelity , however see below and fig . [ fig5 ] . ) as noted above these states are those for which @xmath98 . + however instead of restricting oneself to this very specific class of states we can fix @xmath99 and then optimize the fidelity . in this case one finds that the optimum values of @xmath100 are obtained from [ bc][][0.75][0]@xmath60 [ bc][][0.75][0]@xmath101 [ bc][][0.75][0]@xmath102 [ bc][][0.75][0]@xmath103 . the two curves correspond to different choices of the signs for optimal @xmath104 ( eq . ( [ mu3])).,title="fig:",width=264,height=200 ] @xmath105 where @xmath106 . reinserting these optimal values of @xmath107 in eq . ( [ fidelity3 ] ) we obtain the optimal values of fidelity for each value of @xmath101 . the results are shown in fig . [ fig5 ] , where the two curves correspond to the two choices of sign in the expression for @xmath107 . it is seen that for obtaining the best possible cloning one should use either the plus or the minus sign depending on the value of @xmath101 . finally we observe that in general and for @xmath3 level states , the quantity @xmath108 , can actually be expressed in terms of the expectation values of the operators @xmath109 in the form @xmath110 this equation shows that the equatorial states are a very restricted class of states for which the expectation values of all the observables @xmath111 and @xmath112 have been fixed to zero . by fixing the value of the quantity @xmath113 which has the above simple expression in terms of these observables one can clone a much larger class of states with higher than universal fidelity . in particular one sees that while for two level states there is no difference in the number of parameters of the equatorial states and states with non - zero @xmath114 , the difference in the number of free parameters in the general @xmath3 level case can be quite large depending on the value of @xmath115 . we have described the true physical context for phase covariant quantum cloning , that is we have shown that once we have partial information about a state like the @xmath0 component of spin or the energy of a nuclear spin in a magnetic field , we can clone such a state with a fidelity higher than the optimal universal fidelity and higher than equatorial states . we have provided a one parameter family of cloning transformation so that for each value of the @xmath0 component , we can tune the parameter to obtain the maximum fidelity . we have also shown in this class the equatorial states are the only ones which give rise to separable density matrix for the outputs . however we have shown that it is possible to clone all the states in the vicinity of the north and south pole , for approximately ( @xmath116 radians or @xmath117 radians ) , with sufficiently high ( larger than 0.9 ) fidelity and rather good separability properties . the results of this paper may be useful for those who are interested in experimental realization of quantum cloning by using nuclear magnetic resonance ( nmr ) techniques . we have also discussed how phase covariant quantum cloning of @xmath3-level states can be generalized in the same way . ghirardi and t. weber , nuovo cimento b * 78 * , 9 ( 1983 ) . ( the impossiblity of exact cloning has been first mentioned in a referee report by g.c . ghirardi ( 1981 ) in response to a paper submitted to foundations of physics . ) d. bru , d. divincenzo , a.e . ekert , c.a . fuchs , c. macchaivello and j.a . smolin , phys . a * 57 * , 2368 ( 1998 ) . dariano and p. lo presti , phys . a * 64 * , 042308 ( 2001 ) . j.fiurasek , phys . a * 64 * , 062310 ( 2001 ) . n. gissin and s. massar , phys . . lett . * 79 * , 2153 ( 1997 ) .

it is known that in phase covariant quantum cloning the equatorial states on the bloch sphere can be cloned with a fidelity higher than the optimal bound established for universal quantum cloning . we generalize this concept to include other states on the bloch sphere with a definite @xmath0 component of spin . it is shown that once we know the @xmath0 component , we can always clone a state with a fidelity higher than the universal value and that of equatorial states . we also make a detailed study of the entanglement properties of the output copies and show that the equatorial states are the only states which give rise to separable density matrix for the outputs .

quant-ph0208080

if a black hole is a thing , does it have a boundary ? if so , where is it ? for stationary black holes the event horizon seems to be the obvious answer @xcite , but for evolving black holes the situation is less clear . obviously , all black holes must be formed ( a dynamical process ) and then undergo accretion and other evolutionary processes . furthermore , as a matter of principle isolated black holes always evolve through hawking radiation , and it has been argued that strictly speaking the event horizon may not even exist , due to quantum gravity effects close to the singularity @xcite . in any case it has been argued that the event horizon is ` unreasonably global ' @xcite . thus there are exact solutions such as the imploding vaidya spacetime @xcite where an observer can cross the event horizon even though her entire causal past is a piece of flat minkowski space @xcite . in numerical relativity this problem is posed sharply because one wants to identify the boundary of a black hole in an initial data set , not by inspection of the full solution . it is then natural to turn to closed trapped surfaces , the hallmark of gravitational collapse @xcite . efficient algorithms to identify the region where they occur within a given initial data set do exist @xcite . the boundary of such a spatial region is called the apparent horizon @xcite , and it is in itself a marginally outer trapped surface @xcite . it ` evolves ' to form hypersurfaces foliated by such surfaces @xcite , which we refer to as apparent 3-horizons . in general hypersurfaces foliated by marginally trapped surfaces are called marginally trapped tubes @xcite . there are still problems , because the location of the horizons obtained in this way depends strongly on an arbitrary slicing into space and time . in general there will be many marginally trapped tubes in a given black hole spacetime , and they weave through each other in a complicated way @xcite . another possibility is to locate the boundary of the region through which closed trapped surfaces pass , or the boundary of the region through which outer trapped surfaces pass . for closed trapped surfaces both null expansions are negative , while for outer trapped surfaces no condition is imposed on the inner expansion . it is important to make this distinction , because the two regions do not coincide in general . it was conjectured by eardley @xcite that the region defined by outer trapping coincides with the event horizon . ben - dov @xcite proved this for the vaidya spacetime , but he also proved that the region containing trapped surfaces with both expansions negative lies strictly inside it . nevertheless it can still happen that an observer can cross a closed trapped surface even though her entire causal past is a piece of flat minkowski spacetime , as we showed in an earlier paper @xcite . this proves that closed trapped surfaces have highly non - local properties too , they are _ clairvoyant _ : they are aware " of things that happen elsewhere , very far away with no causal connection . we focus on trapped surfaces with both expansions negative , partly because marginally trapped surfaces form marginally trapped tubes such as future outer trapping horizons @xcite and dynamical horizons @xcite having particularly interesting properties with regard to energy fluxes and the like , and partly simply for definiteness . we concentrate on spherically symmetric imploding spacetimes . we originally thought that this restriction would enable us to fully characterise the boundary @xmath1 of the trapped region the region through which closed trapped surfaces pass but in fact we did not succeed in this . still we are able to give what we think is a quite coherent picture of the trapped regions that occur in spherically symmetric spacetimes , and in particular we identify a past barrier for the location of closed trapped surfaces , and of marginally trapped tubes . this boundary turns out to have some quite non - local properties , and may penetrate flat regions of the spacetime . we will further prove that in spherically symmetric spacetimes the boundary @xmath1 can never in itself contain any marginally trapped surfaces , so that it can not be a marginally trapped tube . once we have learnt that the boundary @xmath1 of the trapped region suffers from non - local properties which are related to those of the event horizon , and that it is not a marginally trapped tube , we put forward a novel idea that may allow us to determine a preferred marginally trapped tube . we define the core @xmath2 of the trapped region as the region which is indispensable to maintain closed trapped surfaces . this core turns out to be generically smaller than the trapped region , and its boundary may thus be used as a definition of the black hole . we will actually identify a particular core in spherically symmetric spacetimes , and we will prove that its boundary is the unique spherically symmetric marginally trapped tube . it remains as an interesting question to know if this is the unique marginally trapped tube which is the boundary of a core . in outline , our paper is organised as follows : section [ sec : preli ] contains reminders about trapped surfaces , the fauna of different cases that can occur , and how they can be characterised in a way convenient for our purposes by the mean curvature vector . section [ sec : tandb ] defines the trapped region and its boundary @xmath1 , and gives their basic properties . throughout the paper we provide proofs of all statements that we make . section [ sec : fun ] gives some basic results on which we build the rest of the paper . the arguments are purely geometrical and based on the interplay between ( causal ) vector fields and surfaces with special properties of their mean curvature vectors . many of these results are already known @xcite but we do offer some sharpenings , such as theorem [ th : no - min ] and corollary [ cor : noposk ] . section [ sec : rw ] is an interlude dealing with robertson - walker spacetimes . in this case the general results of section 4 are sufficient to pin down the boundary @xmath1 exactly . section [ sec : general ] introduces spherically symmetric spacetimes and their unique spherically symmetric apparent 3-horizon . in section [ sec : notah ] we discuss perturbations of the resulting round marginally trapped tubes . we use the stability operator that describes how the outer expansion varies when a marginally outer trapped surface is deformed @xcite , and give a proof that one can always find trapped surfaces that extend to both sides of the spherically symmetric apparent 3-horizon . we also show that the region of the perturbed trapped sphere that is inside the apparent 3-horizon can be made arbitrarily small . section [ sec : imploding ] discusses imploding and asymptotically flat spherically symmetric spacetimes in general . in section [ sec : kodama ] we identify and discuss a past barrier through which future trapped surfaces can not pass . it is based on the presence of the hypersurface forming kodama vector field @xcite , and our restriction is a definite improvement on previous results @xcite . we also prove that any trapped surface must lie at least partly inside the spherically symmetric apparent 3-horizon . in section [ sec : converse ] we discuss the precise location and the properties of the boundary @xmath1 in a generic spherically symmetric and imploding spacetime . we demonstrate that the boundary @xmath1 of the trapped region can not be a marginally trapped tube . section [ sec : new ] raises a new issue : given that trapped surfaces must be confined at least partly within the spherically symmetric apparent horizon , what is the minimal region that must be removed from spacetime in order for it to have no closed trapped surfaces at all ? we call this the core of the trapped region . we show that the spherically symmetric apparent 3-horizon is the boundary of one such core . finally , the important example provided by the vaidya spacetime is treated in an appendix . trapped surfaces are the basic objects to be studied in this paper . thus , we start by providing their definition and their types , and by fixing our notation . standard references are @xcite . let @xmath3 be a 4-dimensional causally orientable spacetime with metric @xmath4 of signature @xmath5 . let @xmath6 denote a connected 2-dimensional surface with local intrinsic coordinates @xmath7 ( @xmath8 ) imbedded in @xmath9 by the @xmath10 parametric equations @xmath11 where @xmath12 are local coordinates for @xmath9 . the tangent vectors @xmath13 of @xmath6 are locally given by _ a e^_a . _ s ._s so that the first fundamental form of @xmath6 in @xmath9 is @xmath14 which collects the scalar products @xmath15 . from now on , we shall assume that @xmath16 is positive definite so that @xmath6 is a _ spacelike _ surface . then , the two linearly independent normal one - forms @xmath17 to @xmath6 can be chosen to be null and future directed everywhere on @xmath6 , so they satisfy @xmath18 where the last equality incorporates a condition of normalization . obviously , there still remains the freedom @xmath19 where @xmath20 is a positive function defined only on @xmath6 . the standard splitting into tangential and normal directions to @xmath6 leads to a formula relating the covariant derivatives on @xmath3 and on @xmath21 @xcite : @xmath22 where @xmath23 are the coefficients of the levi - civita connection @xmath24 of @xmath25 ( i.e. @xmath26 ) , and @xmath27 is the shape tensor ( also called second fundamental form vector , and extrinsic curvature vector ) of @xmath6 in @xmath3 . note that @xmath28 is symmetric , and orthogonal to @xmath6 , from where we deduce @xmath29 here , @xmath30 are two symmetric covariant tensor fields defined on @xmath6 and called the two null ( future ) second fundamental forms of @xmath6 in @xmath3 . they are explicitly defined by @xmath31 the shape tensor gives the difference between the projection to @xmath6 of the covariant derivative and the intrinsic derivative on @xmath6 by means of the fundamental relation e^_ae^_b_v_|_s=_a _ k^_ab [ nablas2 ] where , for all @xmath32 we denote by @xmath33 its projection to @xmath6 . the mean curvature vector of @xmath6 in @xmath3 @xcite is the trace of the shape tensor @xmath34 where @xmath35 is the contravariant metric on @xmath6 : @xmath36 . observe that @xmath37 is orthogonal to @xmath6 and that @xmath38 where @xmath39 are the traces of the null second fundamental forms , usually called the ( future ) null expansions . clearly , @xmath37 is invariant under transformations ( [ free ] ) . the class of _ weakly future - trapped _ ( f - trapped from now on ) surfaces are characterized by having @xmath37 pointing to the future everywhere on @xmath6 , and similarly for weakly past trapped . there are three important subcases that deserve their own name : ( i ) the traditional f - trapped surfaces have @xmath40 timelike everywhere on @xmath6 ; ( ii ) marginally f - trapped surfaces have @xmath41 null everywhere on @xmath6 ; and ( iii ) minimal surfaces have @xmath42 on @xmath6 . these conditions can be equivalently expressed in terms of the signs of the expansions as follows : [ cols="^,^,^,<",options="header " , ] this is to be compared with @xcite , as sometimes different names are given to the same objects , and vice versa . in particular , weakly f - trapped surfaces were called nearly f - trapped in @xcite . here we will follow the previous nomenclature which pretends to respect standard names as much as possible . see @xcite for further details . in the case that @xmath37 is proportional to one of the null normals but realizing both causal orientations the surface is said to be null dual , or marginally @xmath43-trapped , where the @xmath44 refers to the direction with vanishing expansion as the definition is equivalent to having either @xmath45 or @xmath46 vanishing . in the literature they are usually referred as mots ( marginally outer trapped surfaces " ) , by declaring the direction with vanishing expansion to be outer " . for completeness and future reference , we also mention that a surface is called _ untrapped _ if the mean curvature vector is spacelike everywhere , or equivalently , if both expansions have opposite signs . in this paper , we will be concerned with the following sets in @xmath3 , see also @xcite . the future - trapped region @xmath0 is defined as the set of points @xmath47 such that @xmath48 lies on a closed future - trapped surface . since the characterization of f - trapped surfaces is the negativity of both null expansions @xmath49 as defined in ( [ tr ] ) , the following general property follows easily . [ pr : topen ] the future - trapped region @xmath0 is an open set . take any @xmath50 and let @xmath51 be a closed f - trapped surface . we can perturb any such @xmath6 along an arbitrary direction @xmath52 defined on @xmath6 , by moving any point @xmath53 along @xmath52 a distance @xmath54 . the deformed surface @xmath55 has future null expansions @xmath56 , where @xmath49 are the null expansions on @xmath6 and the variations @xmath57 are given by precise formulas involving @xmath52 , the future null normals to @xmath6 and the geometric properties of @xmath6 , see e.g. @xcite . as @xmath58 and @xmath57 are continuous on @xmath6 , we can always choose @xmath59 small enough such that @xmath60 for _ any _ given direction @xmath52 . this implies that there exists a small neighborhood @xmath61 of @xmath62 such that @xmath63 . in general , @xmath0 does not have to be connected . any connected component of @xmath0 will thus be termed as a `` connected f - trapped region '' . it is clear that @xmath0 can be empty , or it can be the whole spacetime . an example of the former case @xmath64 is provided by any globally static spacetime @xcite . examples of the latter case @xmath65 are de sitter spacetime or some contracting robertson - walker geometries , see section [ sec : rw ] . however , in asymptotically flat black - hole type spacetimes @xcite neither of these cases will happen , because there are no trapped surfaces near spatial infinity , and there will appear future - trapped surfaces in the black hole region . in these cases @xmath0 will have a boundary on @xmath3 . [ def : b ] we denote by @xmath1 the boundary of the future - trapped region @xmath0 : @xmath66 [ pr : bclosed ] @xmath1 being the boundary of an open set , it is itself a _ closed _ set without boundary . moreover @xmath67 . [ pr:2sides ] @xmath1 is also the boundary of the untrapped region defined by the set of points @xmath68 , that is , such that @xmath48 does not lie on any closed f - trapped surface . we remark that @xmath1 is not necessarily connected . @xmath0 and @xmath1 are genuine spacetime objects , independent of any foliations or initial cauchy data sets . therefore , @xmath0 and @xmath1 are different in nature from the trapped regions and their boundaries contained in given slices , as recently studied in @xcite . the symmetries of the spacetimes respect @xmath0 and @xmath1 . more precisely : [ res : bandsym ] if @xmath69 is the group of isometries of the spacetime @xmath3 , then @xmath0 is invariant under the action of @xmath69 , and the transitivity surfaces of @xmath69 , relative to points of @xmath1 , remain in @xmath1 . take any point @xmath50 . then there is a closed f - trapped surface @xmath6 passing through @xmath48 . by moving @xmath6 via the motion group @xmath69 , and as f - trapped surfaces are moved to f - trapped surfaces by isometries , it follows that the whole transitivity surface of the group passing through @xmath48 lies in @xmath0 . similarly , let @xmath70 , so that any small neighborhood of @xmath71 intersects @xmath0 . by moving one such small neighborhood using @xmath69 one similarly deduces that the transitivity surface of @xmath69 relative to @xmath71 is part of @xmath1 . an implication of this result is that no globally defined killing vector can be transversal to @xmath72 . actually , this also holds for homothetic killing vectors ( vector fields @xmath73 satisfying @xmath74 ) as they also respect f - trapped surfaces ( if @xmath75 ) @xcite . [ cor : sym ] let @xmath76 be the ( global ) continuous group of isometries of @xmath3 where @xmath77 is its dimension ( i.e. , the number of linearly independent killing vectors ) and let @xmath78 be the dimension of its surfaces of transitivity . 1 . if @xmath79 and @xmath80 , that is , if the spacetime is homogeneous , then @xmath81 and either @xmath64 or @xmath65 . 2 . if @xmath82 and @xmath83 , then either @xmath81 or each connected component of @xmath1 is one of the 3-dimensional surfaces of transitivity . 3 . if @xmath84 and @xmath85 , then either @xmath81 or each connected component of @xmath1 is a hypersurface without boundary foliated by the 2-dimensional surfaces of transitivity . in particular , in arbitrary spherically symmetric spacetimes , @xmath1 ( if not empty ) is a spherically symmetric hypersurface without boundary . point 1 is immediate . points 2 and 3 follow because any connected component of @xmath1 can not have a boundary and can not be given by isolated 2-dimensional surfaces of transitivity , as these would contradict its basic properties [ pr : bclosed ] and [ pr:2sides ] . what is the possible relevance of @xmath1 ? apart from answering natural questions such as `` where can there be closed f - trapped surfaces ? '' or `` is this event part of a closed f - trapped surface ? '' , the location of @xmath1 provides important physical information due to the fundamental relevance of closed trapped surfaces in the development of black holes and singularities @xcite . more importantly , it provides a precise limit as to where dynamical horizons or marginally trapped tubes can develop . one could also hope that @xmath1 is related to the surface of a dynamical black hole . we are going to see that this suffers from the same problems as other candidates . in this section we present the main results that will allow us to put restrictions on the location of the region @xmath0 containing the closed future - trapped surfaces of a spacetime and its boundary @xmath1 . these results are fully general , and can be obtained within the framework of the interplay between generalized symmetries and submanifolds with special properties of their mean curvature vector . the underlying ideas come from @xcite . we start with the main formula to be used in what follows . let @xmath73 be an arbitrary @xmath86 vector field on @xmath9 defined on a neighbourhood of @xmath6 . recalling the identity @xmath87 where @xmath88 denotes the lie derivative with respect to @xmath73 , one gets on using ( [ nablas2 ] ) @xmath89 contracting now with @xmath35 we arrive at [ main ] where @xmath90 is the orthogonal projector that projects any object to its part tangent to @xmath6 . the elementary formula ( [ main ] ) is very useful and permits one to obtain many interesting results about the existence of weakly trapped closed surfaces in the presence of generalized symmetries , see @xcite . for instance : 1 . if @xmath6 is minimal ( @xmath91 ) , integrating ( [ main ] ) and using gauss theorem one finds that the divergence term does not contribute _ whenever _ @xmath6 is closed ( that is , compact without boundary ) , ergo @xmath92 observe that this relation must be satisfied for _ all _ possible vector fields @xmath73 . therefore , closed minimal surfaces are very rare . 2 . if @xmath73 is a killing vector , then the left hand side of ( [ main ] ) vanishes . integrating the righthand side on @xmath6 we get for closed @xmath6 @xmath93 therefore , if the killing vector @xmath73 is timelike on @xmath6 , then @xmath6 can not be weakly trapped ( neither future nor past ) , unless it is minimal . the following is an important consequence of formula ( [ main ] ) . [ lem : basic ] let @xmath73 be a vector field which is future - pointing on a region @xmath94 , and let @xmath6 be a surface contained in @xmath95 such that @xmath96 . then , @xmath6 can not be closed and weakly f - trapped unless @xmath97 and @xmath98 . integrating ( [ main ] ) on the closed @xmath6 , the divergence term integrates to zero and we get @xmath99 which implies that @xmath37 can not be future pointing all over @xmath6 , unless @xmath100 . * remarks : * * for non - minimal @xmath6 , the exceptional case @xmath97 implies that @xmath6 is marginally f - trapped and that @xmath101 is null and proportional to @xmath37 . * notice that only the averaged condition @xmath102 is needed here , so that @xmath103 can be negative somewhere on @xmath6 as long as the averaged formula holds . important instances where lemma [ lem : basic ] can be applied are given by the _ conformal killing vectors _ ( see e.g. @xcite ) and the _ kerr - schild vector fields _ @xcite . the former satisfy ( _ g)_=2 g _ [ ckv ] for some function @xmath104 , so that @xmath105 . thus , the condition on lemma [ lem : basic ] requires simply that @xmath106 . on the other hand , kerr - schild vector fields are characterized by ( _ g)_=2h _ _ , ( _ ) _ = b_[ksvf ] for some functions @xmath107 and @xmath108 , where @xmath109 is a fixed _ null _ one - form field ( @xmath110 ) . therefore @xmath111 and the condition on the lemma [ lem : basic ] requires now @xmath112 . stronger results can be found for the case where @xmath73 is a hypersurface - orthogonal vector field , that is to say , @xmath113}=0 \hspace{2 mm } \longleftrightarrow \hspace{2 mm } \xi_{\mu}=-f \partial_{\mu } \tau\ ] ] for some local functions @xmath114 and @xmath115 . then , @xmath73 is orthogonal to the hypersurfaces @xmath116const . , which are called the level hypersurfaces . in this case we have the following fundamental result . [ th : no - min ] let @xmath73 be a vector field which is future - pointing and hypersurface - orthogonal on a region @xmath94 with level hypersurfaces @xmath116const . and let @xmath6 be a f - trapped surface . then , @xmath6 can not have a local minimum of @xmath115 at any point @xmath117 where @xmath118 . in the case that @xmath6 is a weakly f - trapped surface the conclusion still holds unless .|_q=0 p^ ( _ g)_|_q=0 ._h^|_q=0 .[nueva ] let @xmath119 be a point where @xmath6 has a local extreme of @xmath115 . then , @xmath73 is orthogonal to @xmath6 at @xmath120 , that is to say , @xmath121 . another way of stating the same is that @xmath122 where @xmath123 is the local parametric expression of @xmath115 in terms of the local coordinates @xmath7 of @xmath6 ( note that @xmath124 with @xmath125 ) . using the previous expression , we can now compute the divergence @xmath126 at @xmath120 : @xmath127 introducing this in formula ( [ main ] ) we get at @xmath120 @xmath128 so that , if @xmath6 has a future pointing ( possibly vanishing ) @xmath129 we deduce @xmath130 which , given that @xmath35 is positive definite , implies that @xmath131 can not be positive ( semi)-definite . therefore , @xmath132 can not have a local minimum at @xmath120 . * remarks : * 1 . observe that @xmath6 does not need to be compact , nor contained in @xmath95 . 2 . notice that it is enough to assume @xmath118 only at the points @xmath120 that are local extrema of @xmath115 on @xmath6 . the sign of @xmath103 can thus be left arbitrary everywhere else on @xmath6 where @xmath133 . 3 . let us stress that the possibility with a positive semi - definite @xmath131 is also excluded , so that @xmath132 can not even be constant along one direction at @xmath120 . the only exceptional possibility is given by the case identified in the theorem satisfying ( [ nueva ] ) . if @xmath134 is timelike , the last in ( [ nueva ] ) implies that @xmath135 . + therefore , letting aside this exceptional possibility , @xmath115 will always decrease at least along one tangent direction in @xmath136 . it follows that , under the conditions of the theorem , starting from any point @xmath137 one can always follow a connected path along @xmath138 with decreasing @xmath115 . 4 . theorem [ th : no - min ] applies in particular ( but not only ! ) to ( i ) _ static _ killing vectors of course , ( ii ) hypersurface - orthogonal conformal killing vectors ( [ ckv ] ) with @xmath139 , and ( iii ) hypersurface - orthogonal kerr - schild vector fields ( [ ksvf ] ) with @xmath140 . 5 . the results in section iv of @xcite that any f - trapped surface penetrating a flat portion of the spacetime can not have a minimum of inertial time " there , so that they have to bend down in time" are simple consequences of the more general theorem [ th : no - min ] , which applies not only to flat spacetimes but in general to any static region , and to much more general cases as remarked above . another important result for hypersurface - orthogonal vector fields is ( see also @xcite ) : [ th : untrapped ] let @xmath73 be a vector field which is future - pointing and hypersurface - orthogonal on a region @xmath94 . then , all spacelike surfaces @xmath6 ( compact or not ) contained in one of the level hypersurfaces @xmath141const . within @xmath95 have 2_h^=p^ ( _ g|_s ) _ .[tau = c ] in particular , at any point @xmath62 such that @xmath142 , @xmath6 has a mean curvature vector which is not timelike future - pointing , and it can be future - pointing null or zero only if @xmath143 . let @xmath138 be a ( portion of a ) surface contained in one of the hypersurfaces @xmath141const . then @xmath144 all over @xmath138 so that from ( [ main ] ) we deduce ( [ tau = c ] ) . this immediately implies the rest of results . from theorems [ th : untrapped ] and [ th : no - min ] we deduce the following general property . [ cor : noposk ] no f - trapped surface ( closed or not ) can touch a spacelike hypersurface to its past at a single point , or have a 2-dimensional portion contained in the hypersurface , if the latter has a positive semi - definite second fundamental form . as the hypersurface is spacelike its normal vector , say @xmath73 , is timelike , can be extended to be hypersurface - orthogonal with level function @xmath115 and can be chosen to be future - pointing . then , the projection of @xmath145 to the hypersurface is proportional to its second fundamental form . thus , if this were positive semi - definite it would follow that @xmath96 where @xmath146 is the projector to any such surface @xmath6 , and the result follows . an intuitive explanation of this result is presented in figure [ fig : intuition ] . take the robertson - walker spacetimes , with line - element given by @xcite ds^2=-dt^2+a^2(t ) d^2_k[rw ] where @xmath147 is the standard metric on a maximally symmetric 3-dimensional space with normalized sectional curvature @xmath148 , and @xmath149 is a function of the preferred time @xmath150 called the scale factor . the future is defined by increasing values of @xmath150 . the above line - element possesses a timelike conformal killing vector given by @xmath151 such that @xmath104 in ( [ ckv ] ) is simply @xmath152 . furthermore , @xmath73 is obviously hypersurface - orthogonal @xmath153 with level surfaces given by the preferred hypersurfaces @xmath154const . @xmath73 has been used to derive results on closed weakly f - trapped surfaces in @xcite and more recently on mots in @xcite . [ adotnotpos ] no closed weakly f - trapped surface @xmath6 can be fully contained in the region @xmath155 of a robertson - walker spacetime , unless @xmath156 and @xmath6 is a minimal surface imbedded in a 3-sphere @xmath157 with @xmath158 . direct application of lemma [ lem : basic ] implies @xcite that the only open possibility is given by @xmath91 and @xmath159 . given that the hypersurface @xmath157 has vanishing second fundamental form , any such minimal surface must be minimal _ within _ the hypersurface @xmath157 . this immediately rules out the cases @xmath160 , as there are no compact minimal surfaces imbedded in euclidean or hyperbolic spaces @xcite . similarly , from theorem [ th : no - min ] follows that [ nominadotpos ] no closed weakly f - trapped surface can have a local minimum of @xmath150 at the region with @xmath155 . thus , the minimum of @xmath150 must be attained at a hypersurface @xmath161 with @xmath162 , if they exist . as a consequence , f - trapped closed surfaces are absent in generic _ expanding _ robertson - walker models . they can only be present in models with contracting phases . actually , it is well - known e.g . @xcite that there are closed trapped round spheres at any @xmath154constant slice with @xmath163 and @xmath164 . this last condition is simply the positivity of @xmath165 , where @xmath166 is the energy density relative to the preferred observer @xmath167 and @xmath168 the cosmological constant . given the homogeneity of the maximally symmetric slices , this implies that there are closed trapped round spheres through _ any _ point of the the slices with @xmath163 and @xmath164 . these spheres are , of course , past - trapped if @xmath169 and f - trapped if @xmath170 . the only remaining possibility keeping @xmath164 is that of 3-sphere slices ( @xmath156 ) with @xmath171 , in which case all round spheres in the slice are untrapped except for the equatorial one which is a minimal surface . from corollary [ cor : sym ] we already know that the boundary @xmath1 must be constituted by @xmath154constant slices . thus , @xmath1 splits into two different disconnected sets , @xmath172 and @xmath173 , according to whether the closed f - trapped surfaces lie locally to the future or past of @xmath1 , respectively . combining this with the arguments in the previous paragraph we can deduce the following . on a general robertson - walker spacetime with line - element ( [ rw ] ) and @xmath174 * if @xmath175 everywhere , then @xmath176 and thus @xmath81 . * if @xmath177 everywhere , then @xmath178 and thus @xmath81 . * in the case that @xmath179 changes sign , the past boundary @xmath172 , if non - empty , is necessarily contained in the region with @xmath180 and @xmath181 : @xmath182 the first two points are direct consequences of results [ adotnotpos ] and [ nominadotpos ] . to prove the third point , suppose that a f - trapped surface @xmath6 enters into a region with @xmath183 . then , @xmath6 can not have a minimum of @xmath150 there due to result [ nominadotpos ] . if @xmath6 is closed , @xmath6 must attain the minimum of @xmath150 , hence @xmath6 has to extend to the past for all values of @xmath150 until it crosses the first slice with @xmath184 and @xmath185 , entering into a region with @xmath170 . but we know that there are closed f - trapped round spheres through every point of such a region , so the boundary @xmath172 can not be there . proceeding towards the past , we either encounter another slice with @xmath171 and @xmath186 or not . in the second possiblility , there is no boundary @xmath172 for the connected component of @xmath187 . in the first case , either that slice is such a boundary @xmath172 , or else there are closed f - trapped surfaces crossing the slice towards the past , that is , entering into another region with @xmath169 . we can then repeat the argument from the beginning until one of the slices with @xmath171 and @xmath186 is the past boundary of the mentioned connected f - trapped region or there is no such a boundary . the last point in the previous result can be re - phrased as saying that the past boundary @xmath172 is constituted by preferred slices with @xmath184 , which have a vanishing second fundamental form and therefore are maximal and totally geodesic , and such that the spacetime starts to re - collapse there . if the robertson - walker geometry ( with @xmath174 ) has an initial expanding epoch , then either @xmath81 if it is expanding forever , or the first re - collapsing time is always part of @xmath172 . there is the open question whether the future boundary @xmath173 can be non - empty . if so , one easily obtains that it is necessarily contained in the region with @xmath188 . ( actually , one can further prove that @xmath189 , but this will not be necessary in what follows . ) the question of whether there can be weakly f - trapped surfaces intersecting both expanding and contracting regions is answered in the positive , as follows from de sitter spacetime or in more general cases from the results in @xcite . an important consequence of all the above is the following [ res : nowtsinb ] on a general robertson - walker spacetime with line - element ( [ rw ] ) and @xmath174 , the boundary @xmath1 of the future - trapped region @xmath0 does not contain any non - minimal weakly f - trapped surface ( closed or not ) . take any slice @xmath157 which is part of @xmath1 , so that @xmath190 , and choose any surface @xmath6 , closed or not , imbedded into @xmath157 . then @xmath73 is normal to @xmath6 so that @xmath191 and theorem [ th : untrapped ] implies that @xmath192 . thus , @xmath37 has to be spacelike , possibly zero , or past - pointing everywhere on @xmath6 ( the past - pointing possibility is forbidden if @xmath158 , i.e. , within @xmath172 . ) we will see that this turns out to be a rather general property in spherical symmetry , so that @xmath1 will never contain closed weakly f - trapped surfaces . observe that , in the case @xmath193 ( say ; a similar reasoning works for the case @xmath194 ) , if @xmath172 is non - empty , there are f - trapped round spheres as close to @xmath172 as we like . what happens if we try to approach @xmath172 following a sequence of such f - trapped round spheres ? the answer is simply that they get larger and larger , and if we go to the limit approaching @xmath172 , they actually break and become non - compact minimal planes . as we have seen , not all the @xmath184 hypersurfaces will be part of the boundary in general . an illustrative example is given by de sitter spacetime , which has @xmath156 and @xmath195 , so that @xmath179 is negative , zero , or positive for @xmath196 , @xmath197 or @xmath198 , respectively . thus , de sitter spacetime has f - trapped round spheres in any slice with @xmath196 . however , de sitter spacetime is maximally symmetric , and therefore all of its points are fully equivalent , so that there are f - trapped round spheres through every point . hence , @xmath81 and @xmath178 in de sitter spacetime . a different , more standard example , is provided by the closed friedman model for dust and @xmath199 , defined by @xmath156 with the following parametric form for @xmath149 : @xmath200 where @xmath201 is a constant . obviously @xmath202 . now there are closed f - trapped round spheres through all points of any slice with @xmath203 ( @xmath204 ) , as then @xmath170 . however , no closed f - trapped surface can enter the region with @xmath205 ( @xmath206 ) , as otherwise they would reach their minimum value of @xmath150 at a slice with @xmath183 . thus , in this case @xmath207 . the penrose diagram of this situation is depicted in figure [ fig : rw ] . obviously , the same holds for arbitrary models , with any value of @xmath208 ( but keeping @xmath209 ) , such that @xmath149 has a unique maximum value , that is , for models with one expanding phase , a unique re - collapsing time , and one contracting phase . observe that , in accordance with our general results , the slice @xmath197 in de sitter spacetime has @xmath184 but @xmath185 , while in the previous models with a unique re - collapsing time , such as the closed robertson - walker dust model of figure [ fig : rw ] , @xmath181 everywhere . a spacetime is spherically symmetric if it admits an so(3 ) group of isometries . this group acts transitively on round spheres embedded in space - time . let @xmath210 denote the standard metric on the unit round spheres , then the line - element can always be written as @xmath211 for some lorentzian 2-dimensional metric @xmath212 and where @xmath213 is constant on each round sphere . we assume from now on that @xmath214 , hence @xmath213 can be chosen as one of the coordinates for the metric @xmath212 . it is called the area coordinate because the preferred round spheres have an area equal to @xmath215 . we shall use the invariantly defined _ mass function _ @xmath216 by choosing @xmath217 as a coordinate labeling the incoming radial null geodesics , the spherically symmetric line - element can then be written as ds^2=-e^2(1-)dv^2 + 2e^dvdr+r^2d^2 [ gds2 ] with @xmath218 . the future is defined by increasing values of @xmath217 . these are called the advanced eddington - finkelstein coordinates . they may not be globally defined in some spacetimes , but they are well adapted to our purposes : describing the cases with incoming matter and radiation . the future - directed radial null geodesic vector fields are given by = -e^-_r , k=_v + ( 1-)e^_r[ks ] so that @xmath219 they satisfy @xmath220 . the mean curvature vector for each round sphere @xmath221 can be easily computed h_sph=(e^-_v+(1-)_r ) . [ hspheres ] taking into account that for each round sphere the two future - pointing null normals are @xmath222 and @xmath223 , the null expansions can be read off from ( [ hspheres ] ) : _ sph^+ = ( 1- ) , _ sph^-=- .[thetaspheres ] thus , these round spheres are f - trapped if and only if @xmath224 , and they are marginally f - trapped if @xmath225 . the set defined by @xmath226 is formed by hypersurfaces foliated by marginally f - trapped round spheres . thus , each of these hypersurfaces is a marginally trapped tube " @xcite . they are called the spherically symmetric apparent 3-horizons " , as each of its marginally f - trapped round spheres is an apparent horizon in the sense of @xcite , see also @xcite for a time slice that respects the symmetry . and it is unique with these properties : [ res : ahunique ] ah are the unique spherically symmetric hypersurfaces with respect to the given so(3 ) group foliated by marginally f - trapped surfaces . * remarks : * * there are lorentzian manifolds with several so(3 ) groups of isometries , such as flat , de sitter , or robertson - walker spacetimes ( simply change the origin of coordinates ) . this is why we have to fix the group defining the spherical symmetry . however , in generic situations the isometry group so(3 ) is unique , and then the ah is unique in absolute sense . * the _ foliation _ hypothesis is crucial here , as there can be marginally trapped surfaces different from round spheres , and actually of any genus , imbedded in spherically symmetric hypersurfaces . explicit examples are given in @xcite for robertson - walker spacetimes with @xmath156 , where marginally trapped surfaces of any genus are imbedded in @xmath154 const . the assumption of spherical symmetry for the hypersurface is essential here , as there are non - spherically - symmetric marginally trapped tubes in spherically symmetric spacetimes . this is a general property , as we will show in subsection [ subsec : pertnotiso ] . explicit examples for closed ( @xmath156 ) robertson - walker spacetimes are given in @xcite . that ah is the unique spherically symmetric set constituted by marginally f - trapped round spheres is obvious by construction . suppose then that there is a spherically symmetric hypersurface @xmath227 foliated by marginally f - trapped closed surfaces @xmath228 . pick up any such surface , say @xmath229 . by using the so(3 ) group of isometries , move @xmath229 to obtain a new , necessarily marginally f - trapped , closed surface @xmath230 in @xmath227 . this surface must be tangent at some point @xmath48 to one of the foliating surfaces , say @xmath231 . but then , using the maximum principle as in the proof of proposition 3.1 of @xcite one can deduce that @xmath232 , and a fortiori , that @xmath233 , so that @xmath229 must be tangent to the so(3 ) killing vectors . the proof in @xcite assumed that @xmath227 was spacelike , however this is not necessary and the reasoning works equally well for timelike @xmath227 . actually , it can be seen that the result holds for null @xmath227 as long as its null generator does not point along the null direction with vanishing expansion of @xmath234 . in the remaining case in which the null generator of @xmath227 points along the direction of vanishing null expansion of the @xmath234 , all possible cross - sections of @xmath227 have the corresponding null expansion vanishing in particular , the round spheres in @xmath227 are marginally f - trapped , and therefore the result holds true too . the exceptional situation that has arisen in the previous proof is only possible in cases where ah happens to have portions of isolated - horizon type @xcite , that is , null portions such that their null generators point along @xmath222 . to understand when this occurs , note that the normal one - form to ah is @xmath235 whose norm is @xmath236 so that ah is null at any point @xmath237 with @xmath238 . moreover , ah can be null at points where @xmath239 , and it can also be timelike or spacelike . therefore , ah is a dynamical horizon @xcite ( a spacelike marginally trapped tube ) on the region where it is spacelike , and an isolated horizon @xcite on any open region where @xmath240 . this isolated horizon portion of ah , if non - empty , is characterized also by : @xmath241 where @xmath242 is the einstein tensor of the spacetime . this will be relevant for the perturbations of ah to be studied in the next section . on the other hand , the dynamical horizon portion of ah is _ generic _ in the sense of @xcite , and therefore it is actually an outer trapping horizon in the sense of hayward @xcite . in an open region with @xmath243 the ah is null , but it is _ not _ an isolated horizon as the null normal to ah is not the null normal with vanishing expansion . a simple illustrative example of these different possibilities is provided by the robertson - walker spacetimes : fixing an origin of coordinates arbitrarily , the corresponding spherically symmetric apparent 3-horizon is a marginally trapped tube @xcite with the property of being spacelike , null or timelike if @xmath244 is less , equal or greater than zero , see pp.779 - 780 in @xcite see also @xcite , where @xmath166 and @xmath245 are the energy density and pressure , respectively ( @xmath246 , @xmath247 . ) a timelike ah is explicitly shown in figure [ fig : rw ] . an example of a null ah which is not an isolated horizon is provided by the robertson - walker spacetime with @xmath248 and @xmath199 @xcite . these results are obviously related to the instability of mots in robertson - walker geometries proven in @xcite if @xmath249 , @xmath250 and @xmath251 . observe that ah can be empty ( e.g. in flat spacetime ) , but this will only happen if there are no marginally f - trapped round spheres on the entire spacetime . as our aim is to study the region with closed f - trapped surfaces and its boundary , we will assume that they certainly exist . in this situation , ah can not be empty . under reasonable hypotheses , and for general asymptotically flat initial data sets ( so that the cosmological constant @xmath252 ) , one then knows @xcite that there is a regular complete future null infinity @xmath253 ( so that close to infinity the round spheres are untrapped ) . the event horizon ( eh ) is defined as the boundary of the causal past of future null infinity : @xmath254 @xcite , hence it is a null hypersurface by definition . in our case , it is also spherically symmetric . the apparent 3-horizon ah does not need to be connected , and it can have as many connected components as desired , even if @xmath255 is bounded everywhere by a finite positive mass @xmath256 . as an elementary example , take the case with @xmath240 , @xmath257 , so that ah is null everywhere . the connected components of ah are given by the null hypersurfaces @xmath258 where @xmath259 are the positive roots of the equation @xmath260 . in general , we will only be interested in the particular connected component of ah which is related to the event horizon ( eh ) of an asymptotically flat end ( in the example above , the one with the largest @xmath259 ) . this particular connected component of @xmath261 will be denoted by @xmath262 : @xmath263 the region where the round spheres are untrapped will be denoted by @xmath264 and we also use the notation @xmath265 note that @xmath266 can actually be empty . for instance , in cases where all the round spheres are f - trapped ( this happens for example in the kantowski - sachs models @xcite ) . however , @xmath266 can never be empty in the asymptotically flat cases considered herein , because then the round spheres close to spacelike infinity must be untrapped . notice , similarly , that the whole spacetime may sometimes coincide with @xmath95 , so that no round sphere is f - trapped , but they still can be marginally f - trapped . an example is provided by the extreme reissner - nordstrm solution which has a degenerate horizon @xcite . we avoid this situation due to the assumption of the existence of f - trapped spheres . we are going to use the stability operator @xcite to probe the possible perturbations of marginally f - trapped round spheres with the aim , in particular , of ascertaining if there can be closed f - trapped surfaces traversing ah @xcite . we will also find a characterization of the ah in terms of the einstein tensor , as well as other interesting results . related conclusions were derived in full generality , by the same method of perturbation , in @xcite . we will not restrict the causal character of ah , though , and we will also prove that the deformed surfaces can be made genuinely f - trapped while extending to both sides of @xmath267 . choose any connected component of ah , so that this is a spherically symmetric marginally f - trapped tube : a hypersurface foliated by marginally f - trapped round spheres ( defined by constant values of @xmath213 and @xmath217 ) . as explained in the previous section , the future null normals are given by ( [ ks ] ) with @xmath268 and @xmath269 , so that the corresponding expansions were presented in ( [ thetaspheres ] ) , which restricted to ah are : @xmath270 we can now perturb any such marginally f - trapped round sphere , say @xmath271 , along a direction @xmath272 defined on @xmath271 and orthogonal to it . following @xcite , @xmath273 is any function defined on @xmath271 and we describe @xmath274 by means of the outward vector field n = - . + k | _ [ n ] which is normalized with respect to the fixed null directions such that @xmath275 observe that the causal character of @xmath274 is unrestricted . deform the round sphere @xmath271 by going orthogonally to @xmath271 along @xmath272 a distance @xmath276 . the formula for the variation of the expansion @xmath45 can be found in many references , e.g. in @xcite , but the version better adapted to our purposes is that appearing in @xcite because it keeps the norm of @xmath274 free . given the marginal character of @xmath271 and its spherical symmetry most of the terms in the variation formula vanish and the outer expansion of the perturbed surface @xmath277 is given by ^+_= _ fn ^+ + o(^2 ) , _ fn ^+=-_f+f(-.g_k^u^| _ ) [ deltatheta ] where @xmath278 is the constant value of @xmath213 on @xmath271 , @xmath279 is the laplacian on @xmath271 , and @xmath280 is the following vector field orthogonal to @xmath271 and @xmath274 : @xmath281 observe , by the way , that selecting @xmath282constant ( [ deltatheta ] ) informs us that the vector @xmath280 such that @xmath283 produces no variation on @xmath45 , so that the corresponding @xmath274 is tangent to the ah simply leading to other marginally f - trapped round spheres on ah . let us call such a vector field @xmath284 , so that -.g_k^^|_- .g_k^k^|_=0 [ m ] together with @xmath285 characterizes @xmath286 , since @xmath284 is the unique spherically symmetric direction tangent to it . the exceptional isolated - horizon portion @xmath287 has the null @xmath288 as the tangent vector field . observe that the only forbidden direction due to the normalization used for @xmath274 ( and @xmath284 ) is that defined by @xmath288 ( which would correspond to @xmath289 ) . the situation is depicted in figure [ fig : pert ] . whether @xmath284 is spacelike , null or timelike ( and accordingly for ah ) depends on the magnitude of @xmath290 and the sign of @xmath291 . this sign is non - negative if the null energy condition is assumed . in the next section we will actually assume the stronger dominant energy condition , so that for most of our purposes the positive sign must be kept in mind . recall , though , that the condition @xmath292 defines the portions of ah which are isolated horizons . we have to consider both cases separately . assume that @xmath292 holds on a region so that we are dealing with @xmath287 . this can be seen equivalent to the condition @xmath293 . from the variation formula ( [ deltatheta ] ) we deduce that @xmath294 so that the perturbed expansion is _ independent of the direction _ of deformation @xmath274 . one can check that @xmath295 and the previous relation can be rewritten as _ f - f(1 - 2.|_)=- _ fn ^+ . [ deltathetaiso ] notice that the term between round brackets will generally be positive for instance if @xmath296 const. , and it will certainly be so for any part of @xmath287 related to an asymptotically flat end , because then @xmath297 changes from negative to positive values . eq.([deltathetaiso ] ) can be seen as an equation @xmath298 where @xmath299 is an elliptic operator on @xmath271 , and thus it is adapted for direct application of the maximum principle , see e.g. @xcite or appendix 3 in @xcite . in particular , if @xmath300 is non - positive everywhere it follows that @xmath273 must be negative everywhere on @xmath271 . combining this with the known fact that arbitrary perturbations along the null generator @xmath288 of the isolated horizon @xmath287 produce marginally f - trapped surfaces @xcite we obtain the following theorem . [ th : pertiso ] on any isolated - horizon portion @xmath287 of @xmath261 arbitrary deformations of its round spheres within @xmath287 lead to marginally f - trapped surfaces . moreover , if @xmath287 is such that @xmath301 , any other possible perturbation leading to weakly f - trapped surfaces has @xmath302 , so that the deformed surfaces lie strictly outside the region @xmath95 . observe that from ( [ deltathetaiso ] ) @xmath303 and , given that the righthand side can be chosen as small as desired , the minimum of the non - positive function @xmath273 can be made as small in magnitude as needed . in concrete situations , one can even use the freedom on choosing the variation vector @xmath274 if this helps . let us now consider the parts of @xmath261 with @xmath304 . from figure [ fig : pert ] we deduce that the perturbation along @xmath272 will enter into the region with f - trapped round spheres ( which is trivially part of @xmath0 and can be identified because @xmath288 always points into it ) at points with @xmath305 for easy control of these signs we note that , according to ( [ deltatheta]-[m ] ) , ( g_k^k^| _ ) f(n_n^ - m_m^ ) = -2(_f+_fn^+).[deltatheta1 ] an interesting conclusion arises by integrating this equality on @xmath271 @xmath306 from where we deduce the following facts : * spherically symmetric deformations , defined by having constant @xmath273 and @xmath307 , are uninteresting because they only lead to untrapped round spheres in the region @xmath266 if @xmath308 and to f - trapped round spheres outside @xmath95 if @xmath309 . * the deformed surface can be f - trapped with a negative definite sign of the variation only if @xmath310 is positive somewhere . hence , a f - trapped surface ( obtained in this way ) must lie at least partially in the region where the round spheres are f - trapped . this will turn out to be a fully general result ( corollary [ cor : nor>2 m ] ) . * the deformed surface can be untrapped only if @xmath310 is somewhere negative . * if the deformed surface lies entirely within @xmath266 so that @xmath311 everywhere , then @xmath312 must be positive somewhere . * if the deformed surface lies entirely outside @xmath95 , then @xmath312 must be negative somewhere . note that ( [ deltatheta ] ) is also adapted for direct application of the maximum principle , as it takes the form @xmath313 where now the elliptic operator @xmath314 . thus we also have * all possible perturbations with @xmath315 and leading to @xmath316 everywhere are such that @xmath273 is negative everywhere . thus , all perturbed weakly f - trapped surfaces with @xmath317 are strictly outside @xmath266 . in order to construct examples of f - trapped deformed surfaces which lie partly in @xmath266 we choose to consider perturbations such that @xmath318 for this choice the deformed surface enters the region with f - trapped round spheres if @xmath319 , and it enters @xmath266 if @xmath320 . we introduce a constant @xmath321 . we will aim for f - trapped surfaces for which @xmath322 by our assumptions this implies that @xmath323 if @xmath324 , so that the deformed surface is f - trapped . next we set @xmath325 for some as yet undetermined function @xmath326 . equation ( [ deltatheta1 ] ) becomes ( g_k^k^|_)(n_n^ - m_m^)f + 2_f = 0 . [ deltatheta2 ] we conclude that our assumptions require that ( g_k^k^|_)(n_n^ - m_m^ ) = - > 0 . [ finequal ] this is a ( mild ) restriction on the function @xmath326 . a simple solution is to choose @xmath326 to be an eigenfunction of the laplacian @xmath279 , say @xmath327 for a fixed @xmath328 and constants @xmath329 , where @xmath330 are the spherical harmonics . then , on using @xmath331 the deformation direction @xmath274 is determined by @xmath332 and the variation of the expansion then reads @xmath333 which is negative ( resp . positive ) for all @xmath334 ( resp . @xmath335 . ) as the other expansion was initally negative , by choosing @xmath276 very small we can always achieve that @xmath336 is also negative and therefore the deformed @xmath277 is f - trapped ( resp . untrapped ) . throughout we assume that the deformation is small enough so that the we can rely on the first order perturbation . it only remains to check that @xmath273 realizes all signs , so that the deformed surface criss - crosses ah . given that @xmath337 it is enough to adjust the constants @xmath329 to achieve this goal . for instance , the choice @xmath338 for @xmath339 and @xmath340 if @xmath324 ( or @xmath341 if @xmath335 ) will do , so that @xmath273 has the sign of @xmath342 at the region where @xmath343 , and the opposite sign around the north pole of @xmath271 where @xmath344 ( @xmath345 are the legendre polynomials ) . thus , we have proven the following theorem . [ th : ahnotb ] in arbitrary spherically symmetric spacetimes there are closed f - trapped , as well as untrapped , surfaces ( topological spheres ) penetrating both sides of the apparent 3-horizon ah at any region where @xmath346 . therefore , any part of ah which is not an isolated horizon belongs to the f - trapped region @xmath0 , so that these parts of ah never belong to the boundary @xmath1 . we remark that the previous reasoning is independent of the causal character of ah , which can be spacelike , null or timelike . the only restriction is that @xmath347 . observe also that the original round sphere has a positive gaussian curvature , and thus the deformed f - trapped surfaces penetrating both sides of ah will also have , for sufficiently small @xmath276 , positive gaussian curvature . this disproves a conjecture by hayward @xcite . actually , explicit examples of the same kind but going far away from ah were presented in @xcite . we can now address the non - uniqueness of dynamical horizons . the perturbation argument tells us that there are f - trapped surfaces penetrating into both sides of @xmath267 . we also know that there are untrapped round spheres lying just outside it . if @xmath261 is spacelike this means that we can find a spacelike hypersurface having such an outer trapped sphere as its inner boundary and an untrapped round sphere as its outer boundary , and such that it contains a path connecting the boundaries and lying entirely outside @xmath261 ( that is , inside @xmath266 ) . there is a theorem that ensures that such a spacelike hypersurface necessarily contains a marginally ( outer ) trapped surface @xcite . by construction it has a part lying inside @xmath266 , and we know that it must penetrate outside @xmath95 . moreover , generically such a surface ` evolves ' into a marginally outer trapped tube @xcite . as long as we stay sufficiently close to @xmath261 all the marginally outer trapped surfaces in the argument will be inner trapped as well . thus we have obtained : [ cor : lars ] in arbitrary spherically symmetric spacetimes there are marginally trapped tubes penetrating both sides of the apparent 3-horizon @xmath348 at any region where @xmath346 . explicit examples in robertson - walker spacetimes can be found in @xcite , and in the vaidya spacetime in @xcite . as a final question , we wonder how small the fraction of the closed f - trapped surface that extends outside @xmath95 can be made . this will be relevant in section [ sec : new ] , when we will ask the question of whether or not the complement of @xmath95 is the optimal set to be removed from spacetime in order to get rid of all closed f - trapped surfaces . with the assumptions used in the proof of theorem [ th : ahnotb ] we see that this means that we must produce a @xmath349 function @xmath326 defined on the sphere and obeying the inequality ( [ finequal ] ) , and which is positive only in a region that we can make arbitrarily small . if we choose a sufficiently small constant @xmath321 the last requirement implies that the region where the surface extends outside @xmath95 can be made arbitrarily small . to find such a function it is convenient to introduce stereographic coordinates @xmath350 on the sphere , so that the laplacian takes the form @xmath351 a solution to the problem as stated is f ( ) = \ { lll c_1 ( e^(2a-^2 ) - 1 ) & & ^2 < 4a + + -c_1(1+e^-1 ) & & ^2 > 4a . . this function is @xmath349 ( and can be further smoothed if necessary ) , and it is positive only if @xmath352 , that is on a disk surrounding the origin whose size can be chosen at will . the function obeys @xmath353 this is always larger than zero . thus we have proven the following important result . [ th : tiny ] in spherically symmetric spacetimes , there are closed f - trapped surfaces ( topological spheres ) penetrating both sides of the apparent 3-horizon @xmath354 with arbitrarily small portions outside the region @xmath95 . in this section we present the restrictions on the mass function in order to describe the case of inflow of matter and radiation , satisfying the dominant energy condition , entering into an initially flat spacetime and leading to the formation of a black hole . if the einstein field equations hold ( with vanishing cosmological constant ) , the dominant energy condition @xcite requires , among other restrictions , that the following inequalities hold , e.g. @xcite : 0 , [ dec1 ] + ( 1- ) , [ dec2 ] + -e^(1-)^2[dec3 ] . from ( [ dec1]-[dec2 ] ) one can deduce ( see , e.g. , @xcite ) @xmath355 which implies that , at any null hypersurface @xmath356const . , the mass function satisfies @xmath357 so that , if the mass function happens to be positive at any round sphere @xmath358 , then it is positive for all round spheres @xmath359 with @xmath360 . in particular , if the mass function is non - negative at @xmath361 , then it is non - negative everywhere . note also that , using ( [ dec1 ] ) , ( [ dec2 ] ) implies that @xmath362 similarly , from ( [ dec1][dec3 ] ) we deduce @xmath363 at the same region @xmath95 . these last two expressions can be physically reinterpreted if we note that they are equivalent to ( m)0 , k(m)0 [ dec4 ] where the null vector fields @xmath364 and @xmath288 are given in ( [ ks ] ) . in other words , the dominant energy condition implies that the mass function must be non - increasing ( respectively , non - decreasing ) along any future - pointing ingoing ( resp . outgoing ) radial null geodesic . observe also that the mass function must be non - decreasing along any spacelike outward direction on @xmath95 ( see e.g. @xcite ) , as follows from @xmath365 yet another implication of the above conditions is that the hypersurfaces @xmath366 const . are non - spacelike everywhere on @xmath95 . observe also that , on ah , the dominant energy condition ( [ dec3 ] ) implies .|_ah0 .[dotmah ] only continuous piecewise differentiable mass functions will be considered , so that distributional singularities on the curvature tensor such as shells of matter or radiation are avoided . we will restrict ourselves to mass functions bounded by a finite least upper bound @xmath367 , so that @xmath368 for all @xmath369 , and @xcite there is a regular complete future null infinity @xmath253 for an asymptotically flat end . we shall also restrict ourselves to the physical case where the mass - energy starts flowing in from past infinity at a given advanced time , so that previous to that instant the spacetime has no mass - energy and is flat . the value of @xmath217 when the mass inflow starts will be chosen as @xmath370 . then , the mass function satisfies m(v , r)=0 v<0 ; v>0 , 0 < m(v , r)m < [ massg ] we will not assume in general , however , that the energy - mass travels at the speed of light , so that the infalling mass can be composed of massive dust particles or more general matter . therefore , the hypersurface @xmath371 separating the flat portion and the rest of the spacetime can be timelike or null . the mass function can actually reach the value @xmath256 or not . in the former case , given that we are assuming that there is a regular future null infinity @xmath253 for an asymptotically flat end , there exists a value @xmath372 of @xmath217 such that @xmath373 for all @xmath374 . this implies that @xmath375 for all @xmath213 . note that charged cases ( such as those with an asymptotic , and static , reissner - nordstrm region ) are included in the other case characterized by @xmath376 everywhere . whether or not the spacetime becomes singular when the incoming matter reaches the centre depends on the particular properties of the falling matter and energy . the intersection of @xmath371 with @xmath361 will _ not _ be a curvature singularity so that there will be a regular centre @xmath361 at a portion of the non - flat region if @xmath78 and @xmath377 satisfy there ( e.g. @xcite ) : ( v,0)=0 ; m(v,0)= ( v,0)= ( v,0)=0 .[regular ] in this case , some later singularities can develop . if ( [ regular ] ) do not hold , then the singularity appears already at @xmath378 . at this general level , and in any of the previous cases , one can not know if the singularity will be spacelike , timelike , or null . thus , we will not prejudge this , and leave the future evolution of the spacetime open , not showing it in some of the penrose diagrams . these are depicted , for the essentially different possibilities of interest herein , in figures [ fig : general1 ] , [ fig : vaidya ] and [ fig : general2 ] . summarizing , we consider spacetimes with line - element ( [ gds2 ] ) satisfying the dominant energy condition and subject to ( [ massg ] ) so that there is an asymptotically flat end with regular @xmath253 and a non - degenerate eh , and such that @xmath262 is the connected component of @xmath261 associated to this eh . the actual position of eh depends on the particular properties of the mass function @xmath255 . generically , @xmath262 separates the region @xmath379 , defined as the connected subset of @xmath266 which contains the flat region of the spacetime , from a region containing f - trapped round spheres . under these assumptions , @xmath262 will eventually be spacelike ( actually achronal ) and asymptotic ( probably merging ) to the eh , see @xcite . ( the recent counterexamples presented in @xcite violate some of our assumptions . ) thus , @xmath262 has a portion that is a spherically symmetric , regular , dynamical horizon . nevertheless , in general @xmath262 can have timelike and null portions , see e.g. @xcite . this has been represented in the penrose diagrams of figure [ fig : general2 ] . apart from the above , we will need a further assumption , given by 0 ( _ 1_1)j^+(eh ) [ dotm ] to justify this assumption , and to understand its reasonability , let us make the following considerations . the region @xmath380 may contain a portion of the flat region , where @xmath381 so that the assumption is trivial there , and it is bounded to the future by @xmath262 , where ( [ dotm ] ) holds due to ( [ dotmah ] ) . the rest of its boundary is given by a portion of eh , whose null generators are given by @xmath288 . observe also that @xmath382 . if there is part of the origin @xmath361 in @xmath380 , then from ( [ dec3 ] ) we have @xmath383 . and if there is a flat portion in @xmath380 , this is separated from the rest by the spherically symmetric hypersurface @xmath371 , where @xmath384 by continuity . let us denote by @xmath385 the vector field tangent to the hypersurfaces @xmath366const . and orthogonal to the round spheres . thus , we have @xmath386 note that @xmath387 is future - pointing on @xmath95 . the hypersurface @xmath371 is imbedded in flat spacetime , so that the mass - energy is flowing in only if @xmath388 . but this implies that @xmath389 all in all , the dominant energy condition always ensures that there is a region of @xmath380 where ( [ dotm ] ) is automatically satisfied , and this includes its future boundary and its frontier with the flat region if non - empty . in summary , the assumption ( [ dotm ] ) is equivalent to assuming that the mass function is non - decreasing to the future along any hypersurface @xmath390const . on @xmath380 . in this section , we are going to identify a past barrier for f - trapped surfaces in the spacetimes considered in the previous section . this barrier severely restricts the possible locations of marginally trapped tubes and dynamical horizons . consider the vector field @xmath391 which characterizes the spherically symmetric directions tangent to the hypersurfaces @xmath390const . these hypersurfaces are timelike everywhere on @xmath266 , and null at ah , while they are spacelike outside @xmath95 . @xmath73 is hypersurface orthogonal , with the level function @xmath115 defined by _ dx^=-fd= dr - e^ ( 1- ) dv [ gtau ] . the hypersurfaces @xmath116const . are orthogonal to the hypersurfaces @xmath390const . everywhere . put in another way , the expansion of the round spheres along @xmath73 vanishes , that is , the mean curvature vector defined in ( [ hspheres ] ) is such that @xmath392 hence @xmath73 provides the invariantly defined direction in which the area of the round spheres remains constant . we note that @xmath73 is the kodama vector field @xcite , which has been recently used in related investigations @xcite . we have @xmath393 so that @xmath73 is future - pointing timelike on the region @xmath266 , future - pointing null at @xmath261 , and spacelike outside @xmath95 . therefore , @xmath115 can considered as a time function the kodama time " @xcite in the whole region @xmath266 . observe that @xmath394 the deformation of the metric along @xmath73 can be easily computed ( _ g ) _ = e^__-(_^r _ + _ ^r _ ) . [ deforxi ] thus , @xmath73 is a kerr - schild vector field @xcite on any open region with @xmath395 , an example is the vaidya spacetime treated in the appendix . @xmath73 is killing vector in the situations with @xmath396 , such as the case of the schwarzschild spacetime . we can use the kodama time to restrict the location of f - trapped surfaces . [ th : no - ming ] assume that the spacetime ( [ gds2 ] ) satisfies ( [ dotm ] ) . then , no f - trapped surface @xmath6 can have a local minimum of @xmath115 on @xmath266 , nor they can have an open portion with @xmath141 const . there . let @xmath119 be a point where @xmath6 has a local minimum of @xmath115 , or belonging to an open portion of @xmath397 for some constant @xmath398 . due to theorems [ th : no - min ] and [ th : untrapped ] , and since @xmath73 is future - pointing on @xmath95 , it is enough to show that on any such point @xmath399 projecting ( [ deforxi ] ) onto @xmath6 we get @xmath400 in particular , given that @xmath121 we obtain .p^ ( _ g|_s ) _ |_q=.e^ |_a |^a |_q 0 [ poscon ] whose non - negativity follows from ( [ dotm ] ) . by taking into account the third remark to theorem [ th : no - min ] , the same conclusion holds for weakly f - trapped surfaces unless the exceptional situation ( [ nueva ] ) occurs . [ cor : nor>2 m ] if the spacetime ( [ gds2 ] ) satisfies ( [ dotm ] ) , then no _ closed _ f - trapped surface can be contained in any connected component of @xmath95 . in particular , no _ closed _ f - trapped surface can be fully contained in the region @xmath401 . the only weakly f - trapped surfaces contained in @xmath95 are the marginally f - trapped surfaces foliating ah . combining this corollary with the standard result @xcite see @xcite for a rigourous derivation that no closed weakly f - trapped surface can penetrate outside the eh ( actually , no _ outer _ f - trapped closed surface penetrates into this region @xcite ) , we arrive at the following conclusion . [ coro : nod- ] letting aside the marginally f - trapped surfaces in @xmath262 , no closed weakly f - trapped surface can be fully contained in the region @xmath402 , and thus they must penetrate outside @xmath401 . observe that , in the cases when @xmath403 is spacelike so that it is a dynamical horizon , and recalling that then eh is its past cauchy horizon @xcite , eh=@xmath404 , corollary [ coro : nod- ] can be rephrased as `` no closed weakly f - trapped surface can be fully contained in the past cauchy development @xmath405 '' , which is in agreement with theorem 4.1 in @xcite . suppose that @xmath406 somewhere to one side of @xmath403 . we are naturally led to the question of what is the extension of the connected f - trapped region @xmath407 containing the f - trapped round spheres to that side of @xmath403 . equivalently , the question is what is the exact location of the connected component @xmath408 of the boundary @xmath1 which is to the past of @xmath403 . at first , one is tempted to think that @xmath403 could actually be this boundary @xmath408 , so that all f - trapped closed surfaces remain outside of @xmath409 , but we already know that this is not the case in general , as follows from theorem [ th : ahnotb ] . actually , f - trapped surfaces with spherical topology were explicitly exhibited in @xcite for the self - similar vaidya spacetime ( see appendix ) such that they penetrate @xmath379 and even extend to the flat region of the spacetime . the example from @xcite is shown in figure [ fig : improvement ] below , other examples were constructed in @xcite . thus , the connected f - trapped region @xmath410 will enter into @xmath411 . one can then wonder if @xmath410 will actually extend all the way down to eh . this was shown to be impossible in a particular vaidya solution with a shell of null dust in @xcite . we are going to prove in the following that this is a general property by identifying a past barrier to the connected f - trapped region @xmath410 . put by definition @xmath412 observe that @xmath413 is the least upper bound of @xmath115 on the event horizon eh . in other words , @xmath413 is either ( i ) the constant value of @xmath115 which defines the portion of the eh in the schwarzschild region of the spacetime in the case when @xmath373 for all @xmath374 , or ( ii ) the common limit of @xmath115 on both ah and eh when @xmath414 in the case that @xmath415 everywhere . @xmath227 can be completely characterized as the _ last _ hypersurface orthogonal to @xmath73 which is non - timelike everywhere . [ th : abovebg ] assume that the spacetime ( [ gds2 ] ) satisfies the dominant energy condition , ( [ massg ] ) and ( [ dotm ] ) . then , no closed f - trapped surface can penetrate into @xmath416 ( i.e. the region with @xmath417 . ) consider the closed set @xmath418 this set is contained in the region @xmath419 where @xmath73 is future pointing . @xmath420 is bounded to the future by @xmath421 and to the past by eh@xmath422 ( see figure [ fig : sigmainvaidya ] ) . therefore , any compact surface @xmath6 such that @xmath423 int@xmath424 will reach a minimum on @xmath420 . this minimum can not be on @xmath421 , because this is the maximum value of @xmath115 on @xmath420 . thus it will have to be either a non - local minimum on eh@xmath422 or a local one attained on @xmath425 . however these two possibilities forbid that @xmath6 be f - trapped , because the latter would contradict theorem [ th : no - ming ] , while the former would contradict the standard result @xcite that closed f - trapped surfaces never touch eh . thus , the hypersurface @xmath227 is a limit , to the past , for f - trapped closed surfaces . in fact , they can not even touch @xmath227 . [ th : aboveb ] under the assumptions of the previous theorem all closed f - trapped surfaces must be contained in the region @xmath426 ( defined by @xmath427 ) and penetrate outside @xmath428 . the last part of the theorem states that all f - trapped closed surfaces have points outside @xmath95 , that is points with @xmath406 , but this is already known from corollary [ coro : nod- ] . to prove the first part , observe that theorem [ th : abovebg ] ensures that @xmath429 on any such f - trapped closed surface @xmath6 . thus , we only need to show that @xmath115 can not reach the value @xmath413 on the surface , so that @xmath6 can never touch @xmath227 . but @xmath6 can not touch the portion of @xmath227 which coincides with eh ( if any ) , so this could only happen on the part of @xmath227 within @xmath430 , that is , with @xmath431 . but if there were a point @xmath432 , then theorem [ th : no - min ] would imply that @xmath433 can not have a local minimum on @xmath48 , and that there can not be any 1-dimensional line @xmath434 with @xmath435 such that @xmath436 ; theorem [ th : untrapped ] would imply that no 2-dimensional piece of @xmath6 can be entirely contained in @xmath227 . in summary , the existence of @xmath48 would lead to the existence of points on @xmath6 with @xmath417 , in contradiction . as we have shown , the hypersurface @xmath227 provides a strict limitation to the extension , towards the past , of the connected f - trapped region @xmath407 , and thereby to the location of its boundary @xmath408 . therefore , it is important to know the exact location of @xmath227 . this is the goal of this subsection . recall that @xmath227 is non - timelike everywhere , and actually spacelike on the entire portion where @xmath227 does not coincide with the eh . the location of @xmath227 depends , as is to be expected , on the particular properties of the mass function @xmath255 . however , one can deduce general properties of the hypersurfaces @xmath141const . ( one may also consult the appendix where the specific case of the imploding vaidya spacetime is treated in full , as then the equations can be explicitly integrated for appropriate choices of the mass function . ) in particular , we are going to answer partially the question of whether or not @xmath227 can penetrate into the flat region of the spacetime . to that end , recall the definition ( [ gtau ] ) of @xmath115 , so that the @xmath141const . hypersurfaces are given by the solutions to the ode = . [ odeg ] let @xmath437 be the value of @xmath438 at @xmath361 . the ode ( [ odeg ] ) will not have any critical point at @xmath439 whenever _ r0 = 0 .[nocrit ] in particular , this will be the case when there is no curvature singularity at @xmath440 , due to ( [ regular ] ) . on the other hand , if ( [ nocrit ] ) does not hold , the ode ( [ odeg ] ) is equivalent to the autonomous system @xmath441 which has a critical point at @xmath442 . its linear stability is ruled by the eigenvalues @xmath443 and eigenvectors @xmath444 of the corresponding matrix @xmath445 where @xmath446 , @xmath447 and @xmath448 are the limits of @xmath377 , @xmath449 and @xmath450 when approaching @xmath451 , respectively . these eigenvalues and eigenvectors are @xmath452 the character of the critical point is different depending on the sign of @xmath453 . the different possibilities are * if @xmath454 , the critical point is a focus if @xmath455 unstable or stable depending on whether @xmath456 is positive or negative . in this case no solution reaches @xmath440 . the hypersurface @xmath227 always penetrates the flat region in this case . see the illustrative explicit solution for the vaidya case in the appendix . if on the other hand @xmath457 , then the critical point is a centre at the linear level , and can become a focus , a node or remain as a centre depending on the properties of @xmath255 . * if @xmath458 , the critical point is a node , unstable or stable depending on whether @xmath456 is positive or negative . all possible solutions except one emerge from or approach @xmath440 with the same tangent direction , given by the eigenvector @xmath459 in the unstable case or by @xmath460 in the stable one . the exception for each case is given by one solution emerging from or approaching @xmath440 with the tangent direction of the other eigenvector . there exist three qualitatively different possibilities ( with the same values of @xmath447 and @xmath448 ) , depending on whether or not the special solutions @xmath461 corresponding to the eigenvalues @xmath462 at @xmath440 eventually meet @xmath262 . this , in turn , depends on the specific properties of the mass function and on the total mass @xmath256 . if at least one of the special solutions does _ not _ meet @xmath262 , then the hypersurface @xmath227 can not penetrate the flat region . on the other hand , if both special solutions meet @xmath262 then @xmath227 will have a portion in the flat region . explicit illustrative cases are given in the appendix for the vaidya spacetime . * if @xmath463 , the critical point is a degenerate node ( unstable or stable depending on whether @xmath456 is positive or negative ) in the linear stability analysis , and it remains as such , or it may become an unstable focus or node , depending on the specific properties of the mass function . its properties are once more analogous to those of the vaidya example in the appendix . in this section we want to discuss the possible extension of the connected f - trapped region @xmath407 associated to @xmath262 , and the relation between its boundary @xmath408 as defined in definition [ def : b ] with marginally trapped tubes and closed weakly f - trapped surfaces . having identified the past barrier @xmath227 for the connected f - trapped region @xmath407 , we can ask whether closed f - trapped surfaces can actually extend all the way down to @xmath227 , in other words , if @xmath227 coincides with the connected component @xmath408 of the boundary . this turns out not to be the case ( corollary [ cor : sigmanotb ] . ) we already know that the region outside @xmath95 , such that @xmath406 , belongs to @xmath0 . now we collect some important properties of how closed f - trapped surfaces can cross @xmath403 penetrating into @xmath430 . [ th : taum ] assume that the spacetime ( [ gds2 ] ) satisfies the dominant energy condition , ( [ massg ] ) and ( [ dotm ] ) . any closed f - trapped surface @xmath6 crossing @xmath262 attains the minimum value @xmath464 of @xmath115 outside @xmath409 , and has @xmath465 where @xmath466 is the minimum value of @xmath467 on @xmath262 , and @xmath468 is the value of @xmath213 at the round sphere @xmath469 . ( let us note that @xmath470eh . ) as @xmath6 is compact , it must attain a minimum of @xmath115 which is also a local minimum unless @xmath471const . this last possibility is not feasible for @xmath6 entering into @xmath430 due to theorem [ th : untrapped ] and the non - negativity of @xmath472 ( if @xmath191 ) as follows from ( [ poscon ] ) . for the former possibility , theorem [ th : no - min ] ensures that the local minimum can not lie on @xmath95 , so that @xmath464 has to be attained outside @xmath409 . pick up any value @xmath473 . as in the proof of theorem [ th : abovebg ] consider the closed set @xmath474 @xmath475 is bounded to the future partly by @xmath476 and partly by @xmath477 , and to the past by eh@xmath478 , see figure [ fig : sigmainvaidya ] . therefore , if @xmath479 int@xmath480 , @xmath481 will reach a minimum @xmath482 on @xmath475 . as usual , this minimum can not be on eh@xmath478 due to the standard result @xcite that @xmath6 can never touch eh ; it can not be on @xmath476 because this is the maximum value of @xmath115 on @xmath475 ; and it can not be on @xmath483 either , due to theorem [ th : no - min ] . therefore , such a minimum has to be attained on @xmath403 . besides , there can not be any point @xmath484 such that @xmath485 , because this would contradict either theorem [ th : no - min ] or theorem [ th : untrapped ] . thus , @xmath486 . consider now @xmath468 , and observe that it is the maximum value of @xmath213 on @xmath487 , because @xmath488 is a monotonically decreasing function of @xmath213 on ah as follows from the definition ( [ gtau ] ) . to prove the result , recall that the hypersurfaces @xmath141 const . and are orthogonal everywhere , the former are spacelike and the latter are timelike on @xmath266 , and they both become null and tangent at @xmath261 . therefore , the hypersurface @xmath489 is to the future of all hypersurfaces @xmath490 everywhere on @xmath430 . thus , if @xmath6 reached a value of @xmath491 at a point @xmath484 , then @xmath48 would be to the past of @xmath492 , that is , @xmath493 which is impossible . in the case that @xmath403 is spacelike a dynamical horizon , theorem 4.3 in @xcite implies that no closed f - trapped @xmath6 can penetrate into the region @xmath494 . theorem [ th : taum ] provides a stricter restriction , independently of the causal character of @xmath403 , since @xmath6 can not penetrate @xmath495 . this is graphically explained in figure [ fig : improvement ] . [ prop : taubr=0 ] assume that the spacetime ( [ gds2 ] ) satisfies the dominant energy condition , ( [ dotm ] ) and ( [ massg ] ) , and it has f - trapped round spheres to one side of @xmath262 . then , the connected component @xmath496 can not have a positive minimum value of @xmath213 , and furthermore @xmath497 that @xmath498 is obvious , as the spacetime is flat for @xmath499 so that there can not be f - trapped surfaces penetrating the past of @xmath496 . to see that @xmath500 , we first note that @xmath501 is impossible , as otherwise there would be f - trapped closed surfaces penetrating the region to the past of @xmath227 contradicting theorem [ th : abovebg ] . if @xmath502 , then @xmath496 would be fully contained in the region @xmath503 . but this would mean , due to properties [ pr : bclosed ] and [ pr:2sides ] , that the region defined by @xmath504 would be either part of @xmath0 , or completely external to it . however , this is again impossible because the part of this region with @xmath505 has f - trapped round spheres outside @xmath506 , and no closed f - trapped surface can penetrate its part with @xmath507 . the same reasoning serves to prove that @xmath508 for , if this infimum were positive , say @xmath509 , it would follow that @xmath496 would be fully contained in the region @xmath510 . but there are f - trapped round spheres for _ all _ values of @xmath511 outside @xmath506 . [ prop : merge ] under the same assumptions @xmath512 and @xmath496 merges with , or approaches asymptotically , @xmath227 , @xmath262 and eh in such a way that @xmath513 at any portion of @xmath262 with @xmath514 . furthermore , @xmath496 can not be non - spacelike everywhere . * remark : * in the case that @xmath372 is finite and the mass function is constant for @xmath374 the eh and @xmath262 coincide for all @xmath374 , and so does @xmath496 . this portion of @xmath261 is an isolated horizon with @xmath515=constant , and @xmath516 on that portion . however , there can be other portions of @xmath262 with @xmath516 such that they are isolated horizons . this happens if @xmath517 ( possibly constant ) for some interval @xmath518 with @xmath519 . physically , this means that the inflow of matter and radiation stops between @xmath520 and @xmath521 , and then it starts again . in principle , @xmath496 may coincide with @xmath262 on these particular portions of @xmath262 of isolated - horizon type . this is represented in figure [ fig : ahiso ] . it will be useful to have a name for these portions , so that we set : @xmath522 note that eh will belong to @xmath523 if @xmath372 is finite and the mass function reaches the value @xmath256 . also , that @xmath115 is constant on @xmath523 , as @xmath523 is defined by portions of @xmath390constant hypersurfaces within @xmath262 which are null . observe that the condition @xmath524 required also in the perturbations of section [ sec : notah] becomes @xmath525 if the dominant energy condition holds . there are f - trapped closed surfaces outside @xmath506 , but due to theorem [ th : abovebg ] there are none penetrating @xmath526 . thus , @xmath527 . however , @xmath496 can not meet @xmath528 according to theorem [ th : ahnotb ] . as @xmath227 and @xmath262 merge together , or approach each other asymptotically , so does @xmath496 . finally , consider the domain of dependence @xmath529 of @xmath227 . from the previuous observations , @xmath530 . but @xmath529 is globally hyperbolic with @xmath227 as a cauchy hypersurface , therefore if @xmath496 were non - spacelike everywhere it would have to cross @xmath227 @xcite , in contradiction with the fact that @xmath531 . as in the case of robertson - walker spacetimes ( result [ res : nowtsinb ] , section [ sec : rw ] ) we derive the following important result . [ th : nodh ] under the same assumptions , @xmath532 can not be a marginally trapped tube , let alone a dynamical or trapping horizon . furthermore , @xmath532 does not contain any non - minimal closed weakly f - trapped surface . from the previous corollary @xmath532 is contained in the region @xmath379 . but there are no closed weakly f - trapped surfaces completely contained in @xmath379 due to lemma [ lem : basic ] . thus , the only closed marginally f - trapped surfaces that can be contained in @xmath496 are those which are actually on its part @xmath533 , if any . in fact , this property could have been deduced more easily from result [ res : ahunique ] , as @xmath496 is a spherically symmetric hypersurface . we have decided to include the alternative proof as it may probably be generalized to non - spherically - symmetric cases . theorem [ th : nodh ] implies that the notion of limit section " in @xcite ( a spacelike 2-surface in @xmath1 arising as the uniform limit of a sequence of trapped surfaces approaching @xmath1 , definition in p.6473 ) is generically non - existent or ill defined . thus the assumptions of theorem 7 in @xcite are very rarely met . [ prop : bdecreasing ] assume that the spacetime ( [ gds2 ] ) satisfies the dominant energy condition , ( [ dotm ] ) and ( [ massg ] ) , and has f - trapped round spheres to one side of @xmath262 . then , @xmath115 is a non - increasing function of @xmath213 on any portion of the connected component @xmath496 which is locally to the past of @xmath410 . and it is actually strictly decreasing at least somewhere @xmath532 . * remark : * @xmath496 is a connected component of the boundary @xmath1 and , due to property [ pr:2sides ] , @xmath496 has two sides , one with and another without closed f - trapped surfaces . thus , by locally to the past " we mean that the kodama vector field points towards @xmath410 at @xmath532 at @xmath534 it is tangent . if @xmath496 coincides partly with @xmath523 the result is trivial there , as @xmath115 and @xmath213 are constant on @xmath523 . suppose then that @xmath535 were a non - decreasing function of @xmath213 around a round sphere @xmath536 given by @xmath537 for some constants @xmath538 and @xmath539 , and assume that the f - trapped region @xmath410 is to the future of @xmath271 ( see next figure ) . in 1,2,3,4 ( , 0 ) ( , 4 ) ; ( 0,1 ) ( 4.2,1 ) ; ( 0.3,0.3 ) node[sloped , above , very near end ] @xmath496 ( 4.1,4.1 ) ; ( 2,2 ) circle ( 2pt ) ; ( 2.15,1.85 ) node @xmath271 ; ( 0.7,2 ) ( 4.2,2 ) ; ( 4.7,1.8 ) node @xmath540 ; ( 0.6,2.8 ) ( 4.1,2.8 ) ; ( 4.75,2.95 ) node @xmath541 ; ( 0.7,2.5 ) ( 4.2,2.5 ) ; ( 4.85,2.5 ) node @xmath492 ; ( 2,-0.2 ) node @xmath542 ; ( 2.25,0.3 ) ( 2.25,4 ) ; ( 2.25,0.25 ) node @xmath543 ; ( 2,2.8 ) circle ( 1pt ) ; ( 1.87,2.93 ) node @xmath48 ; ( 2,2.8 ) .. controls ( 0.5,2.6 ) and ( 1.5,2.6 ) .. ( 2.25,2.5 ) ; ( 1.35,2.7 ) node[anchor = east ] ; ( 2.25,2.5 ) circle ( 1pt ) ; ( 2.4,2.63 ) node@xmath544 ; ( 1.5,3.5 ) node @xmath410 ; then , any point @xmath545 lying on the round spheres with @xmath542 and @xmath546 but near enough @xmath547 would belong to at least one f - trapped closed surface . pick up any such @xmath48 and a f - trapped @xmath51 , and let @xmath466 and @xmath468 be the minimum and the maximum values of @xmath467 and @xmath548 on @xmath262 , respectively . from theorem [ th : taum ] we know that @xmath549 and @xmath550 . in particular , @xmath551 , @xmath552 . from theorem [ th : no - min ] ( see its third remark ) there should be a connected path @xmath553 lying entirely on @xmath6 starting at @xmath48 and finishing on @xmath554 , @xmath555 , such that @xmath556 is non - increasing and strictly decreasing somewhere . thus , @xmath557 would eventually cross all hypersurfaces @xmath390const . with @xmath558 $ ] , and each of the crossings would happen with a smaller value of @xmath115 . due to the results in corollary [ cor : sym ] and proposition [ prop : taubr=0 ] , and since @xmath6 , and hence @xmath557 , can not intersect @xmath496 , this would mean that @xmath466 and @xmath468 should be such that @xmath559 and @xmath560 . but this leads to a contradiction , because if such a result held for _ all _ @xmath561 , by taking an appropriate sequence @xmath562 of such @xmath48 approaching @xmath271 , the sequence would have a limit on @xmath271 , which would in turn produce a sequence of round spheres @xmath563 , all of them belonging to @xmath262 , and converging to @xmath564 . as @xmath262 is closed , @xmath271 would belong to @xmath262 . but this is impossible as @xmath565 due to proposition [ prop : merge ] . [ cor : anot0 ] under the same assumptions , @xmath532 can not be tangent to a @xmath141const . hypersurface everywhere . in particular , @xmath496 never touches @xmath227 before they merge together ( and together with @xmath566eh ) , that is to say , @xmath567 . [ cor : sigmanotb ] the past barrier @xmath568 is not part of the boundary @xmath496 . [ cor : bspacelike ] under the same assumptions , @xmath496 is spacelike close enough to its merging ( or asymptotic approaching ) to eh , @xmath227 and @xmath262 . we already know that @xmath496 has to be spacelike somewhere . in the region of this corollary the f - trapped closed surfaces are to the future of @xmath496 , so that @xmath115 has to be a non - increasing function of @xmath213 . the limitation to the past by @xmath227 and to the future by @xmath262 then implies the result . the combinations of all results obtained hitherto can be schematically represented as in figure [ fig : whereisb ] . we can see that the boundary @xmath496 is highly non - local too , and it can have portions in flat regions of spacetime whose whole past is also flat . this is surely not a good candidate for the surface of a dynamical black hole . nevertheless , it remains as an interesting puzzle to find the exact location and the defining properties of @xmath1 . some relevant results in this direction are collected in the remaining of this section . proposition [ prop : bdecreasing ] informs us that @xmath532 has to bend down in the kodama time @xmath115 . this has direct consequences on the extrinsic curvature of @xmath496 . observe that , as follows from ( [ deforxi ] ) and ( [ dotm ] ) , we know that the level hypersurfaces @xmath141const . have a non - negative semi - definite second fundamental form . actually , they have two vanishing eigenvalues and the other one is proportional to @xmath449 . hence , one can prove that the boundary @xmath496 must have a second fundamental form with a non - positive double eigenvalue . [ prop : knegative ] assume that the spacetime ( [ gds2 ] ) satisfies the dominant energy condition , ( [ dotm ] ) and ( [ massg ] ) . then , any portion of the connected component @xmath496 which is locally to the past of @xmath569 has a second fundamental form with a non - positive ( and strictly negative whenever @xmath496 is not tangent to a @xmath141const . hypersurface ) double eigenvalue at any point point where it is spacelike . in particular , it can not have a positive semi - definite second fundamental form there . as @xmath496 is part of the boundary , it has spherical symmetry . take a portion @xmath570 of @xmath496 where it is spacelike and to the past of @xmath410 , and include this portion in a local foliation by spherically symmetric spacelike hypersurfaces @xmath571const . locally , the future - pointing vector field orthogonal to the foliation can be given by = e^-_v+a _ r , + _ dx^=-f dt = dr -e^(1 - -a)dv for some function @xmath572 . the level hypersurfaces are spacelike ( @xmath573 is timelike ) so that 1 - -2a>0 .[spacelike ] observe that @xmath574 , and furthermore @xmath575 by using the kodama time @xmath115 on @xmath576 . on @xmath577 we have that @xmath150 is constant , so that @xmath578 however , proposition [ prop : bdecreasing ] implies that this is strictly negative , which requires necessarily ( if @xmath579 ) @xmath580 given that @xmath581 at @xmath582 , this together with ( [ spacelike ] ) implies that ( if @xmath579 ) @xmath583 we are now going to show that @xmath584 is essentially the double eigenvalue of the second fundamental form of @xmath577 . a straightforward calculation provides the deformation of the metric along @xmath573 ( _ g ) _ = -2 _ _ + e^ ( _ _ + _ _ ) + + _ _ [ deforeta ] + + 2ra d _ where @xmath585 represents the angular part of the metric , that is , the metric on the unit round sphere . of course , the projection to @xmath496 of @xmath586 is ( proportional to ) the second fundamental form @xmath587 of @xmath496 whenever it is spacelike , so that @xmath588 is proportional to the restriction to @xmath577 of the second and third lines in ( [ deforeta ] ) . in particular , the double eigenvalue is proportional to @xmath584 . as a matter of fact , one can try to do better and try to set restrictions also on the third eigenvalue of @xmath588 . this is given by the expression on the second line of ( [ deforeta ] ) ( projected to @xmath577 ) . the idea is based on theorem [ th : no - min ] and corollary [ cor : noposk ] . there are closed f - trapped surfaces passing through every point @xmath589 which is locally to the future of @xmath570 . choose a sufficiently smooth sequence of closed f - trapped surfaces approaching @xmath570 . the smoothness is assumed such that one can take the limit and obtain a piece of a surface @xmath590 touching @xmath570 . the set obtained as limit of the surfaces may fail to be compact , e.g. the robertson - walker example mentioned after result [ res : nowtsinb ] ; or to be spacelike everywhere , or even to be connected . however , all these problems will be irrelevant in what follows as long as @xmath590 exists . ( this is the main problem when trying to turn this reasoning into a formal proof , as proving the existence of @xmath590 encounters some technical difficulties from a mathematical viewpoint ) . given that all the surfaces in the sequence are f - trapped , they all have @xmath591 , where @xmath592 is the constant value of @xmath150 at @xmath570 . therefore , @xmath590 is such that @xmath593 , which implies that @xmath590 has a local minimum of @xmath150 at the intersection @xmath594 , or that @xmath590 has a two - dimensional portion within @xmath570 . in particular , @xmath590 is spacelike around that minimum because @xmath573 is orthogonal to @xmath590 there . the mean curvature vector of this spacelike portion of @xmath590 ( including @xmath594 ) must be future - pointing or zero , given that @xmath590 is limit of the sequence of f - trapped surfaces . therefore , we have @xmath595 and also @xmath596 . now , we can apply the same reasoning as in theorem [ th : no - min ] , for which the compactness is not necessary . formula ( [ main ] ) applied to the spacelike portion of @xmath590 containing @xmath594 gives @xmath597 where @xmath146 is the orthogonal projector to @xmath590 . thus , we deduce that @xmath598 observe that , provided this argument can be promoted into a rigourous proof , it may restrict the third eigenvalue severely , because it has to hold for _ all _ projectors which are limits of the projectors to closed f - trapped surfaces close enough to @xmath577 . if one could gain control on the variety of such projectors , then much more precise restrictions could be set on the boundary @xmath1 . at this point we know that the eh is teleological , and also that closed f - trapped surfaces are clairvoyant : they are aware " of things that happen elsewhere , with spacelike separation . for instance , they can have portions in a flat region of spacetime whose whole past is also flat in clairvoyance of energy that crosses them elsewhere to make their compactness and trapping feasible @xcite , see figure [ fig : improvement ] . this non - local property of trapped surfaces is inherited by everything which is based on them , such as marginally trapped tubes including dynamical horizons . in conjunction with the non - uniqueness of dynamical horizons , this poses a fundamental puzzle for the physics of black holes , a problem that has been recognized and discussed many times lately , see e.g. @xcite and references therein . four possible solutions have been put forward @xcite . first , one can rely on the old and well defined event horizon . this encounters very serious problems because one needs to know the whole future evolution of the spacetime . the event horizon is unreasonably global @xcite . actually , the whole construction of trapping and dynamical horizons was developed to solve this problem and to have nice , local , definitions of the surface of a black hole @xcite . second , one can treat all possible horizons on equal footing . the problem is how to associate unique physical properties to the corresponding black hole , because each dynamical horizon comes with its own set of magnitudes . and they do not agree . the third strategy consisted in finding the boundary @xmath1 as defined in this paper . however , as we have shown , not only @xmath1 will not be a marginally trapped tube in general , it also suffers from the non - local properties associated to f - trapped surfaces . for instance , we have seen that @xmath1 can enter the flat regions of spacetime . finally , the fourth approach consists in trying to define a preferred dynamical horizon . hitherto , there has been no good definition for that . in the following we are going to pursue a novel strategy . the idea is based on the simple question : what part of the spacetime is absolutely indispensable for the existence of the black hole ? we already know that , in the cases considered so far with flat regions and matter imploding so that a black hole eventually forms , the flat region is certainly not essential for the existence of the black hole . what is ? by answering this question we might actually get a bonus and provide a positive , constructive solution to the fourth strategy mentioned before : it may happen that a unique dynamical horizon is selected . from corollary [ coro : nod- ] and theorem [ th : aboveb ] it is clear that if the whole complement of @xmath95 is removed from the spacetime , then no closed f - trapped surfaces remain . the question arises of whether or not proper subsets of that removed region suffice to achieve the same , that is , to short - circuit all closed f - trapped surfaces . to be precise , we give the following definition . a region @xmath599 is called the _ core _ of a connected component @xmath410 of the f - trapped region @xmath0 if it is a minimal closed connected set that needs to be removed from the spacetime in order to get rid of all closed f - trapped surfaces in @xmath410 , and such that any point on the boundary @xmath600 is connected to @xmath601 in the closure of the remainder . here , minimal " means that there is no other set @xmath602 with the same properties and properly contained in @xmath603 . the final condition states that the excised spacetime @xmath604 , which no longer has a connected f - trapped region @xmath410 , has the property that furthermore each point in the closure @xmath605 can be joined to the original boundary @xmath496 by a continuous curve fully contained in @xmath605 . ( this curve may have zero length at points @xmath606 ) . this is needed because one could identify a particular removable region to eliminate the f - trapped surfaces , excise it , but then put back a tiny but central isolated portion to make it smaller . however , this is not what one wants to cover with the definition . obviously , @xmath607 , however , @xmath603 will be generally smaller than @xmath410 . as an example , take the robertson - walker spacetime of figure [ fig : rw ] . there , the future trapped region @xmath0 is the whole future of the recollapsing time , shown in red . however , one only needs to remove the triangle to the future of the ah in order to get rid of all f - trapped surfaces . ( note that the boundary of this region is an apparent 3-horizon ah ) . this example also proves that @xmath2 is not unique : one can choose any other region @xmath2 equivalent to the chosen one by moving all its points by the group of symmetries on each homogeneous slice . actually this kind of non - uniqueness is rather trivial , and is due to the existence of a high degree of symmetry . nevertheless , even in less symmetric cases the uniqueness of the cores @xmath2 can not be assumed beforehand . we are actually going to show that it does not hold in general . we can use the results found in section [ sec : notah ] , especially theorem [ th : tiny ] , to identify one core of the f - trapped region in spherically symmetric spacetimes . assume that @xmath608 for simplicity . then , the complement of @xmath266 ( i.e. the region @xmath609 ) is the disjoint union of core f - trapped regions . each of its connected components is the core of the corresponding connected components of @xmath0 . * remark : * the cases with @xmath610 are technically more involved , but one expects that the result will hold true too . first of all , it is clear that every closed f - trapped surface has points in @xmath611 , as follows from corollary [ coro : nod- ] or theorem [ th : aboveb ] . hence , if we remove @xmath2 from the spacetime all closed f - trapped surfaces disappear . to see that there is no proper subset of @xmath2 with this property , observe that its boundary is @xmath612 . take then any closed connected proper subset @xmath602 of @xmath2 such that all points of @xmath613 are connected to its nearest part of the boundary @xmath1 . this implies that there is a curve from every @xmath614 to such a part of the boundary @xmath1 , and all these curves must therefore cross ah . in summary , @xmath615 always contains an open region around @xmath261 and outside @xmath616 . but then , theorem [ th : tiny ] ensures that there are closed f - trapped surfaces fully contained in @xmath617 , so that @xmath602 can not be a core . as a bonus , we have obtained that the boundary of the identified core @xmath618 is formed by the marginally trapped tubes ah , in particular by their dynamical horizon portions . one can wonder if this property selects these marginally trapped tubes in spherically symmetric spacetimes . observe , however , that according to the results in @xcite , given any regular dynamical horizon @xmath619 , there can not be any closed weakly f - trapped surface fully contained in its past domain of dependence @xmath620 . therefore , if we remove the appropriate future part of @xmath619 we also remove all possible closed f - trapped surfaces . the question now is whether or not these alternative would - be cores are actually optimal , or if they remove more than is needed from the spacetime to get rid of the f - trapped region . independently of whether or not they are optimal , the result in @xcite allows us to prove that there are non - spherically symmetric cores in spherically symmetric spacetimes . first we show that @xmath2 are the unique spherically symmetric cores . [ prop : sphsymcore ] in spherically symmetric spacetimes , @xmath618 are the only cores of @xmath0 which are invariant under the action of the corresponding so(3 ) group of isometries . therefore , @xmath621 are the only spherically symmetric boundaries of a core . suppose there were another spherically symmetric core @xmath602 . obviously @xmath602 could not be a proper subset of @xmath2 , nor vice versa , because both are cores . thus @xmath622 and this set would be spherically symmetric . however , every round sphere in @xmath2 is f - trapped , and therefore there would be f - trapped round spheres having no intersection with @xmath602 , contradicting the hypothesis that @xmath602 was a core of the trapped region . [ prop : nonsphsymcore ] there exist non - spherically symmetric cores of the f - trapped region in spherically symmetric spacetimes . consider the case when the spacetime has only one connected component of ah , which is spacelike , such as for example the vaidya spacetime in the appendix . from corollary [ cor : lars ] or the general results in @xcite we know that there are non - spherically symmetric dynamical horizons interweaving the ah see also the explicit constructions in @xcite . take any of these , say @xmath619 , so that @xmath619 lies partly to the future of ah ( and partly to its past ) . from theorem 4.1 in @xcite , no weakly f - trapped surface can be fully contained in the past domain of dependence of @xmath619 . consider then the causal future @xmath623 of @xmath619 . removing @xmath623 from the spacetime eliminates all closed f - trapped surfaces . nevertheless , it may happen that @xmath623 is not a core , because it is not minimal . in any case , there is a subset of @xmath623 which is a core of the f - trapped region @xmath0 . this new core will never include those parts of the spacetime which are to the future of ah but to the past of @xmath619 . thus , this core of @xmath0 is not @xmath2 , and due to proposition [ prop : sphsymcore ] , it can not be spherically symmetric . still , the identified core @xmath618 may be unique in the sense that its boundary @xmath621 is a marginally trapped tube . this would happen if , for instance in the example of the previous proof , any dynamical horizon @xmath619 other than ah is such that @xmath623 is _ not _ a core of the f - trapped region , the core being a proper subset of @xmath623 . if this is the case , then ah would be selected as the unique dynamical horizon which is the boundary of a core of the f - trapped region @xmath0 . whether or not this happens is a very interesting open question . we thank the wenner - gren foundation for making this research possible . ib was supported by the swedish research council . jmms thanks the theoretical physics division at fysikum in albanova , stockholms universitet , for hospitality . he also acknowledges financial support from grants fis2004 - 01626 ( micinn ) and giu06/37 ( upv / ehu ) . finally we happily acknowledge the help we received from robert wald , greg galloway , an anonymous referee , and jan man . consider the important case of the vaidya spacetime with incoming radiation . the line - element reads @xcite ds^2=-(1-)dv^2 + 2dvdr+r^2d^2 [ ds2 ] so that this is the particular case of ( [ gds2 ] ) with @xmath624 and a mass function independent of @xmath213 , @xmath625 . the einstein tensor of ( [ ds2 ] ) is of pure radiation type @xmath626 and thus , if the einstein field equations are assumed , the null convergence condition ( which in this particular case implies the dominant energy condition ) @xcite requires that 0 [ mdot ] so that the mass function can not decrease as a function of @xmath217 . thus , in this case the condition ( [ dotm ] ) is guaranteed . there is only one connected component of ah defined by @xmath627 and unique associated regions @xmath628 and @xmath629 . ah is a spacelike hypersurface for all @xmath217 such that @xmath630 , and it is null where @xmath631 . therefore , ah is a dynamical horizon @xcite on the region where it is spacelike , and an isolated horizon @xcite on any open region where @xmath632const . the spherically symmetric collapse of null radiation may lead to the formation of a naked singularity @xcite . to avoid this possibility one has to assume that @xcite @xmath633 a hidden , non - naked , curvature singularity is always present at @xmath634 . this is a spacelike future singularity . under all the above conditions , the penrose diagrams for the imploding vaidya spacetime are depicted in figure [ fig : vaidya ] , the first of them for the case with @xmath635 everywhere , the second with @xmath636 from @xmath637 on . the kodama vector field @xmath638 defined in subsection [ sec : kodama ] is actually a proper kerr - schild vector field @xcite ( ksvf from now on ) of type ( [ ksvf ] ) relative to the null direction @xmath364 for the vaidya spacetime . it is immediate to get @xmath639 so that the function @xmath107 in ( [ ksvf ] ) is @xmath640 . this is one of the requirements of theorem [ th : no - min ] . the other requirements are satisfied as in the general case , so that @xmath73 is future pointing on the region @xmath641 , and the level function @xmath115 is defined by _ dx^=-fd= dr-(1- ) dv . [ tau ] besides , @xmath73 is timelike on @xmath266 and null at the ah : @xmath642 . notice that the ksvf @xmath643 coincides with the standard static killing vector on the regions with @xmath632 const . and in particular with a static killing vector in the flat region @xmath499 where @xmath644 . thus , all portions of ah that are isolated horizons are actually killing horizons , including the portion of the eh with @xmath374 in the case that @xmath636 for all @xmath374 . [ [ the-level-hypersurfaces-tau-const.and-the-past-barrier-sigma-in-the-vaidya-spacetime . ] ] the level hypersurfaces @xmath141 const . and the past barrier @xmath227 in the vaidya spacetime . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ now , we analyze the shape and position of the level hypersurfaces @xmath141const . for the ksvf @xmath643 in the vaidya spacetime ( [ ds2 ] ) subject to ( [ mdot],[mass ] ) . observe that these hypersurfaces are characterized by being spherically symmetric and orthogonal to the hypersurfaces @xmath390const . the definition of @xmath115 is ( [ tau ] ) , so that the sought hypersurfaces are defined by the solution to the differential equation = . [ ode ] this is equivalent to the autonomous system @xmath645 so that its orbits on the phase plane @xmath646 provide the required hypersurfaces . given that @xmath647 , the ode ( [ ode ] ) has a critical point at @xmath648 . its linear stability is ruled by the eigenvalues @xmath443 and eigenvectors @xmath444 of the corresponding matrix @xmath649 which are given by @xmath650 the character of the critical point is different depending on whether @xmath651 or not . if @xmath652 , the critical point is an unstable focus , so that no solution actually reaches @xmath653 . the schematic phase plane is represented in figure [ fig : focus ] . as we can check , the hypersurface @xmath227 always penetrates the flat region in this case . if @xmath651 the critical point is an unstable node , with @xmath654 real and positive . in fact , one can see that @xmath655 , \hspace{5 mm } \lambda_-\in \left[0,\frac{1}{2}\right ) , \hspace{1 cm } \lambda_+ + \lambda_- = 1\ , .\ ] ] in this case , all possible solutions except one emerge from @xmath653 with the same tangent direction given by the eigenvector @xmath459 . the exception is given by one solution emerging from @xmath653 with the tangent direction of the other eigenvector @xmath460 . recall that @xmath656 is necessary to avoid ( locally ) naked singularities @xcite , and thus the allowed intervals for @xmath462 can be further restricted if such singularities are to be avoided . there are three qualitatively different possibilities in this case ( with the same value of @xmath657 ) , depending on whether or not the special solutions @xmath461 corresponding to the eigenvalues @xmath462 at @xmath653 eventually meet the ah . this , in turn , will depend on the specific properties of the mass function @xmath658 for @xmath659 and on the total mass @xmath256 . loosely speaking , if there is a period when the mass function has a large derivative , then the chances are that the special solutions will meet the ah . if at least one of the special solutions does _ not _ meet the ah , then the hypersurface @xmath227 can not penetrate the flat region . on the other hand , if both special solutions meet ah then @xmath227 will have a portion in the flat region . these possibilities are schematically represented in the figures [ fig : node1]-[fig : node2 ] . the limit case with @xmath660 has the character of a degenerate unstable node ( only one universal direction through which all solutions emerge from the critical point ) in the linear stability analysis , and it remains as such , or it may become an unstable focus or node , depending on the specific properties of the function @xmath658 around @xmath370 . the schematic structure of the phase portraits for this case are thus analogous to those already shown , with the small difference that the solutions @xmath661 and @xmath662 coincide when @xmath653 is a degenerate node for the full , non - linear , system . to illustrate the above , we present a particular example where the solutions of ( [ ode ] ) can be given explicitly in full . this is given by the self - similar vaidya spacetime , with a linear mass function @xmath663 which admits the following homothetic killing vector field for all @xmath664 @xmath665 observe that @xmath666 in this case , so that @xmath667 . the solutions to the ode ( [ ode ] ) provide the level hypersurfaces for @xmath73 . they are given by @xmath668 for all @xmath669 . for @xmath670 we have : * @xmath671 . the critical point is an unstable focus and the solutions are @xmath672 * @xmath673 . the critical point is a degenerate unstable node whose particular special solution is simply @xmath674 and the rest of solutions , all of them emanating from the origin tangent to the special solution @xmath675 , are @xmath676 * @xmath677 . the critical point is an unstable node with the special solutions @xmath678 the first of them being the exceptional one . the rest of solutions , all of them emanating from the origin tangent to the second special solution @xmath679 , are @xmath680 in all three cases , the hypersurface @xmath227 is given by @xmath421 , where @xmath681 . observe that @xmath227 does not enter the flat region if @xmath682 , and therefore no closed f - trapped surface can penetrate the flat region in these cases @xcite . we note as a final remark that the ah in this particular vaidya spacetime is an intrinsically flat hypersurface , and that the trace of its second fundamental form ( its expansion ) is proportional to @xmath683 . therefore , the ah is non - expanding exactly for the limit case with @xmath673 . in the cases where the past barrier @xmath684 enters the flat portion of the spacetime , the ah is contracting . senovilla , novel results on trapped surfaces , in mathematics of gravitation ii " , ( warsaw , september 1 - 9 , a krlak and k borkowski eds , 2003 ) ; http://www.impan.gov.pl/gravitation/confproc/index.html ( gr - qc/03011005 ) .

we consider the region @xmath0 in spacetime containing future - trapped closed surfaces and its boundary @xmath1 , and derive some of their general properties . we then concentrate on the case of spherical symmetry , but the methods we use are general and applicable to other situations . we argue that closed trapped surfaces have a non - local property , clairvoyance " , which is inherited by @xmath1 . we prove that @xmath1 is not a marginally trapped tube in general , and that it can have portions in regions whose whole past is flat . for asymptotically flat black holes , we identify a general past barrier , well inside the event horizon , to the location of @xmath1 under physically reasonable conditions . we also define the core @xmath2 of the trapped region as that part of @xmath0 which is indispensable to sustain closed trapped surfaces . we prove that the unique spherically symmetric dynamical horizon is the boundary of such a core , and we argue that this may serve to single it out . to illustrate the results , some explicit examples are discussed , namely robertson - walker geometries and the imploding vaidya spacetime . pacs : 04.70.bw , 04.20.cv

1009.0225

nonzero neutrino mass is necessary to explain the well - established phenomenon of neutrino oscillations in many experiments . theoretically , neutrino masses are usually assumed to be majorana and come from physics at an energy scale higher than that of electroweak symmetry breaking of order 100 gev . as such , the starting point of any theoretical discussion of the underlying theory of neutrino mass is the effective dimension - five operator @xcite @xmath8 where @xmath9 are the three left - handed lepton doublets of the standard model ( sm ) and @xmath10 is the one higgs scalar doublet . as @xmath11 acquires a nonzero vacuum expectation value @xmath12 , the neutrino mass matrix is given by @xmath13 note that @xmath14 breaks lepton number @xmath5 by two units . it is evident from eq . ( 2 ) that neutrino mass is seesaw in character , because it is inversely proportional to the large effective scale @xmath15 . the three well - known tree - level seesaw realizations @xcite of @xmath14 may be categorized by the specific heavy particle used to obtain it : ( i ) neutral fermion singlet @xmath16 , ( ii ) scalar triplet @xmath0 , ( iii ) fermion triplet @xmath17 . it is also possible to realize @xmath14 radiatively in one loop @xcite with the particles in the loop belonging to the dark sector , the lightest neutral one being the dark matter of the universe . the simplest such example @xcite is the well - studied `` scotogenic '' model , from the greek scotos meaning darkness . the one - loop diagram is shown in fig . 1 . scotogenic neutrino mass . ] the new particles are a second scalar doublet @xmath18 and three neutral singlet fermions @xmath19 . the dark @xmath20 is odd for @xmath18 and @xmath19 , whereas all sm particles are even . this is thus a type i radiative seesaw model . it is of course possible to replace @xmath16 with @xmath21 , so it becomes a type iii radiative seesaw model @xcite . what then about type ii ? since @xmath14 is a dimension - five operator , any loop realization is guaranteed to be finite . on the other hand , if a higgs triplet @xmath0 is added to the sm , a dimension - four coupling @xmath22 is allowed . as @xmath23 obtains a small vacuum expectation value @xcite from its interaction with the sm higgs doublet , neutrinos acquire small majorana masses , i.e. type ii tree - level seesaw . if an exact symmetry is used to forbid this dimension - four coupling , it will also forbid any possible loop realization of it . hence a type ii radiative seesaw is only possible if the symmetry used to forbid the hard dimension - four coupling is softly broken in the loop , as recently proposed @xcite . the symmetry used to forbid the hard @xmath24 coupling is lepton number @xmath25 under which @xmath26 . the scalar trilinear @xmath27 term is allowed and induces a small @xmath28 , but @xmath4 remains massless . to connect @xmath23 to @xmath29 in one loop , we add a new dirac fermion doublet @xmath30 with @xmath1 , together with three complex neutral scalar singlets @xmath31 with @xmath32 . the resulting one - loop diagram is shown in fig . 2 . higgs triplet . ] note that the hard terms @xmath33 and @xmath34 are allowed by @xmath5 conservation , whereas the @xmath35 terms break @xmath5 softly by two units to @xmath6 . a dark @xmath20 parity , i.e. @xmath36 , exists under which @xmath37 are odd and @xmath38 are even . hence the lightest @xmath31 is a possible dark - matter candidate . the three @xmath31 scalars are the analogs of the three right - handed sneutrinos in supersymmetry , and @xmath39 are the analogs of the two higgsinos . however , their interactions are simpler here and less constrained . the usual understanding of the type ii seesaw mechanism is that the scalar trilinear term @xmath40 induces a small vacuum expectation value @xmath41 if either @xmath42 is small or @xmath43 is large or both . more precisely , consider the scalar potential of @xmath44 and @xmath3 . @xmath45 let @xmath12 , then the conditions for the minimum of @xmath46 are given by @xcite @xmath47 + \mu v^2 & = & 0.\end{aligned}\ ] ] for @xmath48 but small , @xmath49 is also naturally small because it is approximately given by @xmath50 where @xmath51 . the physical masses of the @xmath1 higgs triplet are then given by @xmath52 since the hard term @xmath24 is forbidden , @xmath49 by itself does not generate a neutrino mass . its value does not have to be extremely small compared to the electroweak breaking scale . for example @xmath53 gev is acceptable , because its contribution to the precisely measured @xmath54 parameter @xmath55 @xcite is only of order @xmath56 . with the soft breaking of @xmath5 to @xmath6 shown in fig . 2 , type ii radiative seesaw neutrino masses are obtained . let the relevant yukawa interactions be given by @xmath57 together with the allowed mass terms @xmath58 , @xmath59 , and the @xmath5 breaking soft term @xmath60 , then @xmath61,\ ] ] where @xmath62 and @xmath63 , with @xmath64 using for example @xmath65 , @xmath66 , we obtain @xmath67 ev for @xmath53 gev . this implies that @xmath3 may be as light as a few hundred gev and be observable , with @xmath68 gev . for @xmath69 and @xmath70 a few hundred gev , the new contributions to the anomalous muon magnetic moment and @xmath71 are negligible in this model . in the case of three neutrinos , there are of course three @xmath31 scalars . assuming that the @xmath5 breaking soft terms @xmath72 neutrino mass matrix is diagonal to a very good approximation in the basis where the @xmath31 mass - squared matrix is diagonal . this means that the dark scalars @xmath73 couples to @xmath74 , where @xmath75 is the neutrino mixing matrix linking @xmath76 to the neutrino mass eigenstates @xmath77 . the salient feature of any type ii seesaw model is the doubly charged higgs boson @xmath7 . if there is a tree - level @xmath78 coupling , then the dominant decay of @xmath7 is to @xmath79 . current experimental limits @xcite on the mass of @xmath7 into @xmath80 , @xmath81 , and @xmath82 final states are about 490 to 550 gev , assuming for each a 100% branching fraction . in the present model , since the effective @xmath78 coupling is one - loop suppressed , @xmath83 should be considered @xcite instead , for which the present limit on @xmath84 is only about 84 gev @xcite . a dedicated search of the @xmath85 mode in the future is clearly called for . if @xmath86 , then the decay channel @xmath87 opens up and will dominate . in that case , the subsequent decay @xmath88 , i.e. charged lepton plus missing energy , will be the signature . the present experimental limit @xcite on @xmath70 , assuming electroweak pair production , is about 260 gev if @xmath89 gev for a 100% branching fraction to @xmath90 or @xmath42 , and no limit if @xmath91 gev . there is also a lower threshold for @xmath7 decay , i.e. @xmath84 sufficiently greater than @xmath92 , for which @xmath7 decays through a virtual @xmath93 pair to @xmath94 , resulting in same - sign dileptons plus missing energy . at 13 tev . ] in fig . 3 we plot the pair production cross section of @xmath95 at the large hadron collider ( lhc ) at a center - of - mass energy of 13 tev . we assume that @xmath96 and @xmath23 are heavier than @xmath7 so that we can focus only on the decay products of @xmath97 . the @xmath98 mode is always possible and should be looked for experimentally in any case . however , as already noted , a much more interesting possibility is the case @xmath86 , with the subsequent decay @xmath88 . this would yield four charged leptons plus missing energy , and depending on the linear combination of charged leptons coupling to @xmath31 , there could be exotic final states which have very little sm background , becoming thus excellent signatures to search for . suppose @xmath99 is the lightest scalar , and @xmath100 are heavier than @xmath101 , then @xmath101 decays to @xmath102 . hence the decay of @xmath95 could yield for example @xmath103 plus four @xmath99 ( missing energy ) in the final state . recent lhc searches for multilepton signatures at 8 tev by cms @xcite and atlas @xcite are consistent with sm expectations , and are potential restrictions on our model . in particular , the cms study includes rare sm events such as @xmath104 and @xmath105 . due to the absence of opposite - sign , same - flavor ( ossf ) @xmath106 pairs , both events are classified as ossf@xmath107 where lepton @xmath108 refers to electron , muon , or hadronically decaying tau . leptonic tau decays contribute to the electron and muon counts , and this determines the ossf@xmath109 category . details from cms are shown in table 1 for @xmath110 leptons and @xmath111 . & & + @xmath112 leptons & @xmath113 ( gev ) & obs . & exp.(sm ) & obs . & exp.(sm ) + sr1 & @xmath114 & 0 & @xmath115 & 0 & @xmath116 + sr2 & @xmath117 & 0 & @xmath118 & 0 & @xmath115 + sr3 & @xmath119 & 0 & @xmath118 & 0 & @xmath120 + 3 leptons & @xmath113 ( gev ) & obs . & exp.(sm ) & obs . & exp.(sm ) + sr4 & @xmath114 & 5 & @xmath121 & 7 & @xmath122 + sr5 & @xmath117 & 3 & @xmath123 & 35 & @xmath124 + sr6 & @xmath119 & 4 & @xmath125 & 53 & @xmath126 + the cms study estimates a negligible sm background for sr1-sr3 , and in our simulation we use the same selection criteria . we impose the cuts on transverse momentum @xmath127 10 gev and psuedorapidity @xmath128 for each charged lepton , with at least one lepton @xmath127 20 gev . in order to be isolated , each lepton with @xmath129 must satisfy @xmath130 , where the sum is over all objects within a cone of radius @xmath131 around the lepton direction . we implement our model with feynrules 2.0 @xcite . using the cteq6l1 parton distribution functions , we generate events using madgraph5 @xcite , which includes the pythia package for hadronization and showering . madanalysis @xcite is then used with the delphes card designed for cms detector simulation . generated events intially have 4 leptons . about half are detected as 3 lepton events , but the constraints from signal regions sr4-sr6 are less restrictive than sr1-sr3 . the number of detected events in the ossf0 @xmath132 4 lepton category is almost the same as @xmath133 with very few additional leptons from showering or initial / final state radiation . to examine the production of @xmath134 we take the mass of @xmath99 to be 130 gev , which allows @xmath99 to be dark matter as discussed in the next section . we use the values @xmath135 and @xmath136 , although the results are not sensitive to the exact values due to on - shell production and decay . the effects due to @xmath53 gev may be neglected . for our model , we scan the mass range of @xmath7 and @xmath101 . in fig . 4 we plot contours showing the expected number of detected events in the ossf0 @xmath132 4 lepton category for 13 tev at luminosity 100 fb@xmath137 assuming a negligible background as for the 8 tev case . although the branching fractions of @xmath101 to @xmath138 or @xmath139 are comparable , we find that most of the contributions from @xmath140 decay to @xmath141 or @xmath142 in the @xmath132 4 lepton final state are not detected . a similar analysis performed for 8 tev at 19.5 fb@xmath137 has a maximum number of detected events of 0.4 in the plot analogous to fig . 4 , which corresponds to a small estimated exclusion at the 15% confidence level . events for 13 tev at luminosity 100 fb@xmath137 . ] the lightest @xmath31 , say @xmath99 , is dark matter . its interaction with leptons is too weak to provide a large enough annihilation cross section to explain the present dark matter relic density @xmath143 of the universe . however , it also interacts with the sm higgs boson through the usual quartic coupling @xmath144 . for a value of @xmath145 consistent with @xmath143 , the direct - detection cross section in underground experiments is determined as a function of @xmath146 . a recent analysis @xcite for a real @xmath31 claims that the resulting allowed parameter space is limited to a small region near @xmath147 . in our model , we can evade this constraint by evoking @xmath100 . the mass - squared matrix spanning @xmath148 is given by @xmath149 whereas the coupling matrix of the one higgs @xmath150 to @xmath148 is @xmath151 . upon diagonalizing @xmath152 , the coupling matrix will not be diagonal in general . in the physical basis , @xmath99 will interact with @xmath153 through @xmath150 . this allows the annihilation of @xmath154 to @xmath155 through @xmath153 exchange , and contributes to @xmath143 without affecting the @xmath99 scattering cross section off nuclei through @xmath150 . this mechanism restores @xmath99 as a dark - matter candidate for @xmath156 . plotted against @xmath157 from relic abundance assuming @xmath158 . ] to demonstrate the scale of the values involved , we consider the simplifying case when @xmath159 and @xmath160 . the additional choice @xmath161 ensures that @xmath100 are heavier than @xmath99 , and is convenient because then the relic abundance requirement no longer depends explicitly on @xmath162 . taking into account that @xmath99 is a complex scalar , we use @xmath163 = 4.4 @xmath164@xmath165s@xmath137 @xcite and in fig . 5 we plot the allowed values for @xmath166 and @xmath157 taking @xmath158 for simplicity to satisfy the lux data . another possible scenario is to add a light scalar @xmath167 with @xmath1 , which acts as a mediator for @xmath31 self - interactions . this has important astrophysical implications @xcite . in this case , @xmath154 annihilating to @xmath168 becomes possible . we have studied a new radiative type ii seesaw model of neutrino mass with dark matter @xcite , which predicts a doubly charged higgs boson @xmath7 with suppressed decay to @xmath169 , thereby evading the present lhc bounds of 490 to 550 gev on its mass . in this model , @xmath7 may decay to two charged heavy fermions @xmath93 , each with odd dark parity . the subsequent decay of @xmath101 is into a charged lepton @xmath170 and a scalar @xmath31 which is dark matter . hence there is the interesting possibility of four charged leptons , such as @xmath171 , plus large missing energy in the final state . we show that the lhc at 13 tev will be able to probe such a doubly charged higgs boson with a mass of the order 400 to 500 gev . 99 s. weinberg , phys . lett . * 43 * , 1566 ( 1979 ) . e. ma , phys . . lett . * 81 * , 1171 ( 1998 ) . e. ma , phys . * d73 * , 077301 ( 2006 ) . e. ma and d. suematsu , mod . a24 * , 583 ( 2009 ) . e. ma and u. sarkar , phys . lett . * 80 * , 5716 ( 1998 ) . e. ma , phys . * 115 * , 011801 ( 2015 ) . particle data group , k. a. olive _ et al . _ , . phys . * c38 * , 090001 ( 2014 ) . atlas collaboration , g. aad _ et al . _ , jhep * 1503 * , 041 ( 2015 ) . s. kanemura , m. kikuchi , k. yagyu , and h. yokoya , phys . rev . * d90 * , 115018 ( 2014 ) . s. kanemura , m. kikuchi , k. yagyu , and h. yokoya , prog . ( * 2015 * ) 051b02 . cms collaboration , v. khachatryan _ et al . _ , j. * c74 * , 3036 ( 2014 ) . cms collaboration , s. chatrchyan _ et al . _ , phys . rev . * d90 * , 032006 ( 2014 ) . atlas collaboration , g. aad _ _ , phys . rev . * d90 * , 052001 ( 2014 ) . alloul , adam _ et al . _ , comput.phys.commun . * 185 * , 2250 ( 2014 ) . j. alwall _ et al . _ , j. high energy phys . * 1106 * , 128 ( 2011 ) . b. dumont , b. fuks , s. kraml _ et al . _ , eur.phys.j . * c75 * ( 2015 ) 2,56 l. feng , s. profumo , and l. ubaldi , jhep * 1503 * , 045 ( 2015 ) . g. steigman , b. dasgupta , and j. f. beacom , phys . rev . * d86 * , 023506 ( 2012 ) . d. n. spergel and p. j. steinhardt , phys . * 84 * , 3760 ( 2000 ) . j. mcdonald , phys . lett . * 88 * , 091304 ( 2002 ) . m. markevitch _ et al . _ , j. * 606 * , 819 ( 2004 ) . c. kouvaris , phys . lett . * 108 * , 191301 ( 2012 ) . m. kaplinghat , t. linden , and h .- b . yu , phys . lett . * 114 * , 211303 ( 2015 ) . for a recent brief review , see for example s. tulin , aip conf . proc . * 1604 * , 121 ( 2014 ) .

we consider a model of neutrino mass with a scalar triplet @xmath0 assigned lepton number @xmath1 , so that the tree - level yukawa coupling @xmath2 is not allowed . it is generated instead through the interaction of @xmath3 and @xmath4 with dark matter and the soft breaking of @xmath5 to @xmath6 . we discuss the phenomenological implications of this model , including @xmath7 decay and the prognosis of discovering the dark sector at the large hadron collider . ucrhep - t557 + november 2015 * type ii radiative seesaw model of + neutrino mass with dark matter + *

1511.06375

the algorithm presented in this paper solves the problem of * numerically * determining the decomposition of a finite dimensional irreducible unitary linear representation ( ` irrep ' in what follows ) of a group with respect to the unitary irreducible representations ( irreps ) of a given subgroup . more precisely , let @xmath1 be a compact lie group and @xmath2 a finite dimensional irreducible unitary representation of it , i.e. , @xmath3 is a group homomorphism that satisfies the following three conditions : @xmath4 here , @xmath5 is a complex hilbert space with inner product @xmath6 , @xmath7 is the group of unitary operators on @xmath5 , and @xmath8 stands for the adjoint . conditions ( 1 ) ( 3 ) above define a * unitary representation * @xmath9 of the group @xmath1 . the representation is said to be * irreducible * if there are no proper invariant subspaces of @xmath5 , i.e. , if any linear subspace @xmath10 is such that @xmath11 for all @xmath12 , then @xmath13 is either @xmath14 or @xmath5 . since the group @xmath1 is compact , any irreducible representation of @xmath1 will be finite - dimensional with dimension say @xmath15 ( @xmath16 ) . consider a closed subgroup @xmath17 . the restriction of @xmath18 to @xmath19 will define a unitary representation of @xmath19 which is reducible in general , that is , it will possess invariant subspaces @xmath20 such that @xmath21 for all @xmath22 . if we denote by @xmath23 the family of equivalence classes of irreps of @xmath19 ( recall that two unitary representations of @xmath19 , @xmath24 and @xmath25 , are equivalent if there exists a unitary map @xmath26 such that @xmath27 for all @xmath28 ) , then @xmath29 where the @xmath30 are non - negative integers , @xmath31 denotes a subset in the class of irreps of the group @xmath19 , i.e. , each @xmath32 denotes a finite dimensional irrep of @xmath19 formed by the pair @xmath33 , and @xmath34 denotes the direct sum of the linear space @xmath35 with itself @xmath30 times . thus , the family of non - negative integer numbers @xmath30 denotes the multiplicity of the irreps @xmath36 in @xmath9 . the numbers @xmath30 satisfy @xmath37 where @xmath38 and the invariant subspaces @xmath20 have dimension @xmath39 . notice that the unitary operator @xmath40 will have the corresponding block structure : @xmath41 where @xmath42 . the problem of determining an orthonormal basis of @xmath5 adapted to the decomposition will be called the * clebsch gordan problem * of @xmath9 with respect to the subgroup @xmath19 . to be more precise , the clebsch gordan problem of the representation @xmath18 of @xmath1 in @xmath5 with respect to the subgroup @xmath19 consists in finding an orthonormal basis of @xmath5 , @xmath43 , such that each family @xmath44 , for a given @xmath32 , defines an orthonormal basis of @xmath35 . thus , given an arbitrary orthonormal basis @xmath45 , we compute the @xmath46 unitary matrix @xmath47 with entries @xmath48 such that @xmath49 the coefficients @xmath50 of the matrix @xmath47 are usually expressed as the symbol @xmath51 and are called the * clebsch gordan coefficients * of the decomposition . the original clebsch gordan problem has its origin in the composition of two quantum systems possessing the same symmetry group : let @xmath52 and @xmath53 denote hilbert spaces corresponding , respectively , to two quantum systems @xmath54 and @xmath55 , which support respective irreps @xmath56 and @xmath57 of a lie group @xmath1 . then , the composite system , whose hilbert space is @xmath58 , supports an irrep of the product group @xmath59 . the interaction between both systems makes that the composite system possesses just @xmath1 as a symmetry group by considering the diagonal subgroup @xmath60 of the product group . the tensor product representation @xmath61 will no longer be irreducible with respect to the subgroup @xmath62 and we will be compelled to consider its decomposition in irrep components . a considerable effort has been put in computing the clebsch gordan matrix for various situations of physical interest . for instance , the groups @xmath63 have been widely discussed ( see @xcite , @xcite and references therein ) since when considering the groups @xmath64 and @xmath0 , the clebsch gordan matrix provides the multiplet structure and the spin components of a composite system of particles ( see @xcite , @xcite ) . however , all these results depend critically on the algebraic structure of the underlying group @xmath1 ( and the subgroup @xmath19 ) and no algorithm was known so far to efficiently compute the clebsch gordan matrix for a general subgroup @xmath17 of an arbitrary compact group @xmath1 . on the other hand , the problem of determining the decomposition of an irreducible representation with respect to a given subgroup has not been addressed from a numerical point of view . the multiplicity of a given irreducible representation @xmath33 of the compact group @xmath1 in the finite - dimensional representation @xmath65 is given by the inner product @xmath66 where @xmath67 and @xmath68 , @xmath69 , denote the characters of the corresponding representations , and @xmath70 stands for the standard inner product of central functions with respect to the ( left - invariant ) haar measure on @xmath1 . hence if the characters @xmath71 of the irreducible representations of @xmath1 are known , the computation of the multiplicities becomes , in principle , a simple task . moreover , given the characters @xmath71 of the irreducible representations , the projector method would allow us to explicitly construct the clebsch gordan matrix ( * ? ? ? 4 ) . however , if the irreducible representations of @xmath19 are not known in advance ( or are not explicitly described ) , there is no an easy way of determining the multiplicities @xmath30 . again , at least in principle , the computation of the irreducible representations of a finite group could be achieved by constructing its character table , i.e. , a @xmath72 unitary matrix where @xmath73 is the number of conjugacy classes of the group , but again , there is no a general - purpose numerical algorithm for doing that . recent developments in quantum group tomography require dealing with a broad family of representations of a large class of groups , compact or not , and their subgroups ( see @xcite and references therein for a recent overview on the subject ) . quantum tomography allows to extend ideas from standard classical tomography to analyze states of quantum systems . one implementation of quantum tomography is quantum group tomography . quantum group tomography is based on quantum systems supporting representations of groups . such representations allow to construct the corresponding tomograms for given quantum states @xcite , @xcite , @xcite . hence it is becoming increasingly relevant to have new tools able to efficiently handle group representations and their decompositions . it turns out that it is precisely the ideas and methods from quantum tomography which provide the clue for the numerical algorithm presented in this work . more explicitly , * mixed adapted quantum states * , i.e. , density matrices * adapted * to a given representation , will be used to compute the clebsch gordan matrix . section [ sec : preliminaries ] will be devoted to introduce the problem we want to solve . section [ sec : general_outline ] presents several results which will help us to show the correctness of the algorithm . the details of the numerical algorithm are contained in section [ the_algorithm : sec ] , while section [ sec : examples ] covers various examples and applications of the algorithm , among them , the decomposition of regular representations of any finite group and the decomposition of multipartite systems of spin particles . it is remarkable that the algorithm proposed here does not require an _ a priori _ knowledge of the irreducible representations of the groups and the irreducible representations themselves are returned as outcomes of the algorithm . this makes the proposed algorithm an effective tool for computing the irreducible representations of , in principle , any finite or compact group . for the sake of clarity , most of the analysis will be done in the case of finite groups . however , it should be noted that all statements and proofs can be easily lifted to compact groups by replacing finite sums over group elements by the corresponding integrals over the group with respect to the normalized haar measure on it . some additional remarks and outcomes will be discussed at the end in section [ sec : discussion ] . a final contains numerical results for the examples addressed in section [ sec : examples ] . let @xmath1 be a finite group of order @xmath74 and let @xmath75 be a subgroup , not necessarily normal of @xmath1 , of order @xmath76 . we label the elements of @xmath1 as @xmath77 , where the first @xmath78 elements correspond to the elements of the subgroup @xmath19 , i.e. , @xmath79 . in what follows , a generic element in the group @xmath1 will be simply denoted by @xmath12 unless some specific indexing is required . let @xmath18 be a unitary irreducible representation of @xmath1 on the finite dimensional hilbert space @xmath5 , @xmath80 , and let @xmath81 , @xmath82 , be any given orthonormal basis of @xmath5 . we denote by @xmath83_{i , j=1}^{n}\ ] ] the unitary matrix associated with @xmath84 , @xmath69 , in the chosen basis , i.e. , @xmath85 for every @xmath86 . the restriction of the representation @xmath18 to the subgroup @xmath19 , sometimes denoted by @xmath87 and called the _ subduced representation _ of @xmath18 to @xmath19 , will be , in general , reducible even if @xmath18 is irreducible . notice that the unitary matrix associated with @xmath88 , @xmath28 , is just a submatrix of @xmath89 obtained by restricting ourselves to the elements of the subgroup @xmath19 . a mixed state on @xmath5 , also called _ density matrix _ , is a @xmath46 normalized hermitian positive semidefinite matrix @xmath90 , i.e. , @xmath91 if the unitary representation @xmath18 of @xmath1 is irreducible , then any state @xmath90 can be written as @xmath92 to prove this formula one may use schur s orthogonality relations : @xmath93 where @xmath94 stands for the complex conjugate , and @xmath95 and @xmath96 denote , respectively , the entries of the unitary matrices @xmath97 and @xmath98 associated with the irreducible representations @xmath33 and @xmath99 of the group @xmath1 with respect to given arbitrary orthonormal bases in @xmath35 and @xmath100 . let us now consider a state @xmath90 satisfying the orthogonality relations @xmath101 clearly , because of eq . , such state verifies @xmath102 [ adapted_def ] a state @xmath90 in the hilbert space @xmath5 supporting an irrep of the group @xmath1 is said to be * adapted * to a closed subgroup @xmath19 if @xmath103 for @xmath104 . in other words , a state @xmath90 adapted to the subgroup @xmath19 of the finite group @xmath1 must be of the form @xmath105 even if the subduced representation @xmath87 is reducible . in view of the prominent role they will play in the algorithm , let us now discuss briefly the role of the inner products @xmath106 in the realm of quantum theory : given a linear operator @xmath54 on @xmath5 and a state @xmath90 , the number @xmath107 is called the expected value of the operator @xmath54 in the state @xmath90 and is denoted consequently as @xmath108 . if the operator @xmath54 is self - adjoint , the expected value @xmath108 is a real number and it truly represents the expected value of measuring the observable described by the operator @xmath54 on a quantum system in the state @xmath90 . in the language of quantum tomography , the group function @xmath109 is defined by the coefficients in the expansion written in eq . , @xmath110 and is called the _ characteristic function _ of the state @xmath90 associated with the representation @xmath65 or , depending on the emphasis , the _ smeared character _ of the representation @xmath18 with respect to the state @xmath90 ( see @xcite ) . one can easily check that the characteristic function @xmath111 is always positive semidefinite , i.e. , @xmath112 for all @xmath113 , @xmath114 and @xmath115 , and @xmath116 and @xmath117 . notice that if the state @xmath90 is @xmath118 , then the characteristic function @xmath111 is the standard character @xmath119 of the representation @xmath120 . moreover , if the representation @xmath120 is the trivial one , then @xmath121 for all @xmath12 . we are now in the position to specify which is the ultimate goal of our algorithm : computing the so - called _ clebsch gordan matrix_. [ cg_matrix_def ] let @xmath1 be a group , @xmath9 an irreducible unitary representation of @xmath1 and @xmath19 a closed subgroup of @xmath1 . the * clebsch gordan matrix * associated with @xmath122 and @xmath9 is the @xmath46 matrix @xmath47 such that @xmath123{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \mbox{\huge{$0 $ } } & \\ & & & \hspace{-1.7 cm } { \rotatebox{-46}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \\ & \hspace{-1.8cm}\mbox{\huge{$0 $ } } & & & \hspace{-1.4 cm } { \rotatebox{-46}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \\ & & & & & \hspace{-1.65cm}\mathds{1}_{c_n}\otimes d^n(h ) \end{pmatrix},\ ] ] for every @xmath22 , where the @xmath124 are the matrices defined in @xmath125 , the @xmath126 , are the matrices associated with the irreps of the subgroup @xmath19 and @xmath127 stands for the matrix kronecker product defined as @xmath128{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & a_{1n}b \\ a_{21}b & \hspace{-0.3 cm } a_{22}b & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&a_{2n}b\\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{-0.6 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } \\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{0.8 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}\\ \vspace{0.1cm}\hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.1em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.1em}\hfill\kern0pt } } } } & & \hspace{0.3 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.10em}\hfill\kern0pt}}}}\\ a_{m1}b & \hspace{-0.3cm}a_{m2}b & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & a_{mn}b \end{pmatrix}\ ] ] for arbitrary matrices @xmath129 and @xmath55 . since the unitary representation is unique ( modulo unitary transformations within each proper invariant subspace @xmath130 or permutations among the @xmath130 ) , the clebsch gordan matrix is also unique ( except for such transformations ) , ( see @xcite for more detailed information about this ) . finally , let us specify the kind of adapted states we will be using in the algorithm . as we shall see , such states will have to satisfy certain nondegeneracy conditions : given any adapted state @xmath90 , we know that , according to ( [ adapted ] ) , @xmath90 is a linear combination of the representations @xmath131 , so the clebsch gordan matrix @xmath47 in definition [ cg_matrix_def ] will block - diagonalize @xmath90 in the form @xmath132{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \mbox{\huge{0 } } & \\ & & & \hspace{-0.4 cm } { \rotatebox{-46}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \\ & \hspace{-0.8cm}\mbox{\huge{0 } } & & & \hspace{-0.1 cm } { \rotatebox{-46}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \\ & & & & & \hspace{-1.05cm}\mathds{1}_{c_n}\otimes \sigma^n \end{pmatrix},\ ] ] where each block @xmath133 , is a hermitian positive semidefinite matrix of the same dimension as the corresponding @xmath134 . now , consider the spectral decomposition of the matrices @xmath135 , i.e. , @xmath136 where the @xmath137 are orthonormal eigenvectors of @xmath135 within each proper subspace @xmath138 [ generic_def ] an adapted state @xmath90 is said to be * generic * if its eigenvalues have the minimum possible degeneracy , that is , @xmath139 for all @xmath140 and for all @xmath141 , @xmath142 . notice that the eigenvalues can not have what we might call minimal degeneracy since each @xmath143 has by construction multiplicity @xmath144 ( recall eq.([structure_rho ] ) ) . in the construction of the algorithm , a further concept of pair - wise genericity will be needed : [ generic_pair_def ] a pair @xmath145 of adapted states is said to be * mutually generic * if they are both generic , in the sense of definition @xmath146 , and no eigenvector @xmath147 of the block @xmath148 of @xmath149 is an eigenvector of the corresponding @xmath150 of @xmath151 whenever @xmath152 , where @xmath153 of course , we exclude the case @xmath154 in which the proper invariant subspace has dimension one and therefore the eigenvectors must coincide . before we provide a detailed description of the decomposition algorithm we propose , let us first give a rough outline of how the algorithm is organized and , especially , why does it work . the final goal of the algorithm is to find the clebsch gordan matrix @xmath47 which , as shown in definition [ cg_matrix_def ] , block - diagonalizes all the elements of the representation @xmath124 , @xmath22 . in other words , the columns of @xmath47 provide orthonormal bases for all proper invariant subspaces @xmath155 _ which are common to all @xmath124 , @xmath22 _ ( and consequently , common to all adapted states ) . now , consider any fixed adapted state @xmath90 and any unitary matrix @xmath156 diagonalizing @xmath90 pointwise , i.e. , such that @xmath157 is diagonal . the idea underlying our algorithm is that since the columns of both @xmath156 and @xmath47 span the same proper invariant subspaces , they must be somehow related . this connection , which is crucial to our argument , will be made explicit in theorem [ triple_v ] below , and implies that , after appropriate reordering of the columns of @xmath156 , any other adapted state ( more generally , any matrix which is a linear combination of the @xmath124 ) will be _ block - diagonalized _ by @xmath156 ( see corollary [ transformation_any_state ] below ) . furthermore , the diagonal blocks one obtains have a very particular structure which , once identified in corollary [ transformation_any_state ] , will be the key to extract the clebsch gordan matrix @xmath47 out of @xmath156 via appropriate similarity transformations , described both in corollary [ r_tilde : corollary ] and lemma [ permutation : lemma ] . the following result is the foundation of the algorithm we describe in [ the_algorithm : sec ] below : [ triple_v ] let @xmath90 be any generic adapted state and let @xmath156 be any unitary matrix such that @xmath157 is diagonal . then @xmath158 where @xmath47 is the clebsch gordan matrix , defined as in definition @xmath159 , @xmath160 is any permutation matrix , and @xmath161 , with @xmath162 given by @xmath163 for any set of @xmath164 unitary matrices @xmath165 , where @xmath166 is a set of eigenvectors of the matrices @xmath135 , @xmath167 , given in . * proof * : it follows from that @xmath168 for any choice of @xmath169 orthonormal bases @xmath170 , @xmath141 . recall that @xmath169 is the dimension of the invariant subspace @xmath35 or , equivalently , the number of rows and columns of the hermitian positive semidefinite matrices @xmath135 . on the other hand , @xmath30 is the multiplicity of that subspace , i.e. , the global multiplicity of the eigenvalues @xmath171 in the total matrix @xmath90 ( see ) . if we now construct unitary matrices @xmath172 such that their columns are the orthonormal vectors of the basis @xmath173 , then the matrix @xmath174 will diagonalize the matrix @xmath175 with its eigenvalues sorted as follows : @xmath176{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & & \mbox{\huge{0 } } & \\ & & & \hspace{-0.26 cm } { \rotatebox{-47}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & & \\ & \mbox{\huge{0 } } & & & \hspace{-0 cm } { \rotatebox{-47}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \\ & & & & & \hspace{-0.6cm}\lambda_{n_\alpha}^\alpha\mathds{1}_{c_\alpha } \end{pmatrix}=\lambda^\alpha\,.\ ] ] therefore , in view of , the matrix @xmath161 diagonalizes the matrix @xmath177 , @xmath178 and any permutation @xmath160 of the columns of the matrix @xmath179 will still diagonalize @xmath90 , which shows that any unitary matrix @xmath156 diagonalizing @xmath90 can be written as a product @xmath180 . @xmath181 [ transformation_any_state ] let @xmath90 be any adapted state , let @xmath182 be the associated block - diagonal matrix with blocks , let @xmath183 with @xmath184 , where each @xmath185 , @xmath186 , is a @xmath187 permutation matrix , and let @xmath180 . then , for any linear combination @xmath188 , it is verified that @xmath189{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \\ & & & \hspace{-1.99 cm } { \rotatebox{-43}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \\ & & & & \hspace{-2.4cm}c_nn_n\left\{\begin{array}{|ccc|}\hline \phantom{a } & \phantom{a } & \phantom{a}\\ \phantom{a } & \phantom{a } & \phantom{a}\\ \phantom{a } & \phantom{a } & \phantom{a}\\\hline \end{array}\right . \end{array } \right),\begin{array}{c } \hspace{-6.5cm}\vspace{4.08cm}\sigma^1 \end{array } \phantom{a } \begin{array}{c } \hspace{-5cm}\vspace{1.3cm}\sigma^2 \end{array } \phantom{a } \begin{array}{c } \hspace{-2.35cm}\vspace{-3.8cm}\sigma^n \end{array}\ ] ] where @xmath190{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & r_{1n_\alpha}^\alpha \\ r_{21}^\alpha & \hspace{-0.3 cm } r_{22}^\alpha & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&r_{2n_\alpha}^\alpha\\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{-0.6 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } \\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{0.8 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}\\ \vspace{0.1cm}\hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.1em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.1em}\hfill\kern0pt } } } } & & \hspace{0.3 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.10em}\hfill\kern0pt}}}}\\ r_{n_\alpha 1}^\alpha & \hspace{-0.3cm}r_{n_\alpha 2}^\alpha & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & r_{n_\alpha n_\alpha}^\alpha \end{pmatrix},\ ] ] with @xmath191 square matrices of size @xmath30 defined as @xmath192 where @xmath193 , @xmath167 , are the matrices on the block diagonal of @xmath194 after being transformed by @xmath47 , i.e. , those matrices such that @xmath195 . * proof * : we just transform @xmath194 with @xmath156 , @xmath196{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & & \hspace{-0cm}\mbox{\huge{0 } } & \\ & & & \hspace{-1.15 cm } { \rotatebox{-40}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & & \\ & \hspace{-2.8cm}\mbox{\huge{0 } } & & & \hspace{-0.4 cm } { \rotatebox{-40}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \\ & & & & & \hspace{-1.85cm}\big(x^np^n\big)^\dagger(\mathds{1}_{c_n}\otimes\tau^n)x^np^n \end{pmatrix}.\ ] ] hence , the matrices @xmath197 in the statement are @xmath198 . finally , if we substitute in @xmath199 the definition of @xmath200 in eq . , and use the property @xmath201 of the kronecker product for matrices @xmath202 such that the products @xmath203 and @xmath204 are feasible , we get @xmath205 @xmath181 this corollary is key to the algorithm described in section [ the_algorithm : sec ] below because it means that any matrix diagonalizing one generic adapted state @xmath90 , with the eigenvectors appropriately reordered , will transform any linear combination of the representation @xmath124 ( in particular , any other adapted state ) into the specific form given by corollary [ transformation_any_state ] , which has a very special structure . our next step amounts to exploit this special structure in order to reveal a finer block structure within each @xmath197 for any linear combination of the representation . [ r_tilde : corollary ] let @xmath206 and @xmath207 , be as in corollary [ transformation_any_state ] . let @xmath208 for any matrix @xmath209 , and set @xmath210 for any fixed @xmath211 . if @xmath212 , are the diagonal blocks of @xmath213 for some other @xmath214 , then @xmath215{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \tilde{s}_{{k_\alpha}1n_\alpha}^\alpha \mathds{1}_{c_\alpha } \\ \tilde{s}_{{k_\alpha}21}^\alpha \mathds{1}_{c_\alpha } & \hspace{-0.3 cm } \tilde{s}_{{k_\alpha}22}^\alpha \mathds{1}_{c_\alpha } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \tilde{s}_{{k_\alpha}2n_\alpha}^\alpha \mathds{1}_{c_\alpha}\\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{-1.2 cm } { \rotatebox{-25}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } \\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{0.8 cm } { \rotatebox{-25}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}\\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & & \hspace{0.56 cm } { \rotatebox{-25}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}\\ \tilde{s}_{{k_\alpha}n_\alpha 1}^\alpha \mathds{1}_{c_\alpha } & \hspace{-0.3 cm } \tilde{s}_{{k_\alpha}n_\alpha 2}^\alpha \mathds{1}_{c_\alpha } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \tilde{s}_{{k_\alpha}n_\alpha n_\alpha}^\alpha \mathds{1}_{c_\alpha}. \end{pmatrix}\ ] ] * proof * : if we write @xmath216{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & t_{1n_\alpha}^\alpha \\ t_{21}^\alpha & \hspace{-0.3 cm } t_{22}^\alpha & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&t_{2n_\alpha}^\alpha\\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{-0.6 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } \\ \hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}&\hspace{0.8 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[1.2em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt}}}}\\ \vspace{0.1cm}\hspace{-0.2 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.1em}\hfill\kern0pt } } } } & \hspace{-0.5 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.1em}\hfill\kern0pt } } } } & & \hspace{0.3 cm } { \rotatebox{-32}{\makebox[0pt]{\makebox[1.6em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \hspace{-0.4 cm } { \rotatebox{90}{\makebox[0pt]{\makebox[0.85em]{\xleaders\hbox{$\cdot$\hskip.10em}\hfill\kern0pt}}}}\\ t_{n_\alpha 1}^\alpha & \hspace{-0.3cm}t_{n_\alpha 2}^\alpha & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & { \rotatebox{0}{\makebox[0pt]{\makebox[1.4em]{\xleaders\hbox{$\cdot$\hskip.12em}\hfill\kern0pt } } } } & t_{n_\alpha n_\alpha}^\alpha \end{pmatrix},\ ] ] where @xmath217 , then one can easily check that @xmath218 @xmath181 notice that this transformation leads to a matrix with almost the structure of ( [ structure_rho ] ) , with the difference that the entries in the blocks @xmath135 are scattered everywhere instead of being concentrated in the diagonal blocks . in other words , if we set @xmath219 for @xmath220 such that @xmath221 for all @xmath222 , then @xmath223 while we would like to have the kronecker products in reverse order . it is well known that for any pair of matrices @xmath54 and @xmath55 of arbitrary dimensions , the two kronecker products @xmath224 and @xmath225 are permutationally equivalent ( i.e. , @xmath226 for appropriate permutation matrices @xmath160 and @xmath227 ) . moreover , when both @xmath54 and @xmath55 are square , they are actually permutationally similar ( i.e. , one can take @xmath228 above : see , for instance , corollary 4.3.10 in @xcite or @xcite ) . [ permutation : lemma ] given two matrices @xmath54 and @xmath55 of arbitrary sizes , there exist two permutation matrices @xmath160 and @xmath227 , which only depend on the dimensions of the matrices @xmath54 and @xmath55 , such that @xmath229 in the case in which @xmath54 and @xmath55 are square matrices of sizes @xmath15 and @xmath73 respectively , the permutation matrices are related by @xmath230 , where @xmath231 and @xmath232 are the following matrices of dimensions @xmath233 and @xmath234 respectively : @xmath235{\makebox[1.5em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & & \\ & & \hspace{-0.05 cm } { \rotatebox{-45}{\makebox[0pt]{\makebox[1.5em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \hspace{-0.1 cm } { \rotatebox{-45}{\makebox[0pt]{\makebox[1.5em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \hspace{0.2cm}\mbox{\huge{$0 $ } } & \\ & & & \hspace{0.02 cm } { \rotatebox{-45}{\makebox[0pt]{\makebox[1.5em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \hspace{-0.6 cm } { \rotatebox{-45}{\makebox[0pt]{\makebox[1.5em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \\ & \mbox{\huge{$0 $ } } & & & \hspace{0.05 cm } { \rotatebox{-45}{\makebox[0pt]{\makebox[1.5em]{\xleaders\hbox{$\cdot$\hskip.11em}\hfill\kern0pt } } } } & \hspace{-0.2cm}1 \\ & & & & & \hspace{-0.2cm}0 \end{pmatrix},\quad\qquad f=\left(\begin{array}{c|c } 1&\begin{array}{c } \vspace{-0.3cm}\\ \large{\text{$0$}}_{1\times(n-1)}\\ \vspace{-0.4 cm } \end{array}\\\hline \begin{array}{c } \vspace{0.2cm}\\ \large{\text{$0$}}_{(cn-1)\times 1 } \end{array}&\hspace{-0.19cm}\begin{array}{c } \vspace{-0.3cm}\\ \mathds{1}_{(n-1)}\otimes\left(\begin{array}{c } 0\\ \vdots\\ 0\\ 1 \end{array}\right)_{c\times 1 } \vspace{0.1cm}\\\hline \end{array}\\ & \begin{array}{c } \vspace{-0.3cm}\\ \huge{\text{$0$}}_{(c-1)\times ( n-1 ) } \end{array } \end{array}\right).\ ] ] as a consequence of lemma [ permutation : lemma ] , if we compute the matrix @xmath236 such that @xmath237 if @xmath156 is the unitary matrix in corollary [ transformation_any_state ] and @xmath238 is given by ( [ def_r_tilde ] ) , we conclude that @xmath239 is the clebsch gordan matrix in definition [ cg_matrix_def ] . we are now in the position to give a detailed description , step by step , of the decomposition algorithm that we have named smily . we first specify input and output of the algorithm : * * input * : a unitary representation of any finite group or compact lie group @xmath19 . * * output * : the clebsch gordan matrix @xmath240 , in a basis of eigenvectors of an initial adapted state @xmath149 . we may organize the smily algorithm into eight steps : 1 . * generate two adapted states * : we start by creating two mutually generic states @xmath149 and @xmath151 ( see definition [ generic_pair_def ] ) . to create them , we generate two random vectors @xmath241 and @xmath242 of size @xmath243 , with no zero components , and use their respective entries as coefficients to construct two linear combinations of the matrices @xmath131 : @xmath244 next , we symmetrize , @xmath245 shift them by the spectral radius and divide by the trace , @xmath246 to obtain two hermitian normalized positive semidefinite matrices @xmath149 and @xmath151 . having been randomly generated , it is safe to assume that they are mutually generic . * diagonalize pointwise the first state * : compute a unitary matrix @xmath247 which diagonalizes pointwise the state @xmath149 , i.e. , such that @xmath248 is a diagonal matrix . such matrix exists since @xmath149 is hermitian . + [ step_3_ps ] 3 . * first sorting * : reorder the columns of @xmath247 by grouping together the eigenvectors corresponding to a same proper subspace @xmath20 . recall that , according to corollary [ transformation_any_state ] , there is a reordering of the columns of @xmath247 which block - diagonalizes @xmath151 , and the dimensions of the diagonal blocks are the dimensions of the @xmath20 . notice that if two columns @xmath249 and @xmath250 of @xmath247 correspond to the same proper subspace @xmath20 , then @xmath251 . this will be our test for rearranging the columns of @xmath247 . more precisely , we use the following routine based on a divide - and - conquer approach : 1 . choose one column of @xmath247 , rename it as @xmath252 and move it into a list of vectors we will call @xmath253 . + + [ step_3_2 ] 2 . compute @xmath254 for another column @xmath250 of @xmath247 , and if @xmath255 , move @xmath250 into the list @xmath253 and rename it as @xmath256 . repeat on all remaining columns of @xmath247 , move those @xmath250 with @xmath257 into the list @xmath253 and label them as @xmath258 , with the index @xmath259 reflecting the order in which they have been included in the list . + . ] 3 . compute @xmath260 for @xmath261 , for those columns @xmath250 of @xmath247 not yet moved into @xmath253 in step . this is a re - check since there might be some vector left not included in the list in step because it happened to be orthogonal to @xmath252 in the scalar product defined by @xmath151 . the mutual genericity condition ensures that no vector in @xmath253 can be orthogonal to all remaining vectors in the list . + . ] once we have finished verifying all eigenvectors in @xmath253 , we take a block whose columns are the eigenvectors in @xmath253 and denote it as @xmath262 , since it is a set of @xmath263 vectors constituting an orthonormal basis of @xmath264 . after that , we come back to step and repeat the process with the rest of vectors until all of them have been sorted . + at the end of this step , we obtain a matrix we may call @xmath265 whose columns form bases @xmath266 of the proper subspaces @xmath20 for @xmath167 , i.e. , @xmath267 this step also gives the dimensions @xmath268 by counting the number of vectors in each subspace . * second sorting * : reorder the columns within each @xmath269 grouping together the eigenvectors corresponding to the same eigenvalue of @xmath149 . to do it , we just reorder the eigenvectors in each @xmath266 in decreasing order corresponding to their eigenvalues . thus , we obtain @xmath270 where @xmath271 + counting the multiplicity of one eigenvalue in each @xmath32 will give the multiplicity @xmath30 . hence , since we already got the products @xmath268 in step , we can also get the dimensions of the irreps @xmath169 by dividing those numbers by @xmath30 . at this point , it is also possible , if needed , to obtain the characters of the irreps in the decomposition of @xmath124 by computing @xmath272 5 . * coarse block - diagonalization of @xmath273 * : compute the matrix @xmath274 to obtain the coarse block - diagonalization of @xmath151 in terms of the matrices @xmath275 , as shown in corollary [ transformation_any_state ] , and identify the square matrices @xmath276 , of size @xmath30 . compute a matrix @xmath238 * : according to corollary [ r_tilde : corollary ] , for each @xmath275 choose a column of matrices @xmath277 such that @xmath278 for all @xmath141 , compute the unitary matrices @xmath279 and finally compute the unitary matrix @xmath280 7 . * compute the permutation matrix @xmath227 * : for each @xmath32 , compute the permutation matrix @xmath281 , as described in lemma [ permutation : lemma ] , and collect them in the block diagonal matrix @xmath282 8 . * final rearrangement * : compute the clebsch gordan matrix @xmath283 . the algorithm we have presented decomposes any finite dimensional unitary representation of any compact lie group . in the case of finite groups , it is natural to apply it to the regular representation because it contains every irreducible representation with multiplicity equal to the dimension of its irreps , @xmath284 ( * ? ? ? * ch.2 ) , thus : @xmath285 the regular representation of a group @xmath1 is the unitary representation obtained from the action of the group @xmath1 on the hilbert space of square integrable functions on the group , @xmath286 , where @xmath287 denotes the left(right)-invariant haar measure by left ( right ) translations . as before , we will restrict the discussion to finite groups @xmath1 as in sect . [ sec : preliminaries ] . the space of square integrable functions on @xmath1 can be identified canonically with the @xmath288-dimensional complex space formally generated by the elements of the group , i.e. , we will denote by @xmath289 $ ] the linear space whose elements are given by @xmath290 , @xmath291 , @xmath69 , with the natural addition law @xmath292 . notice that @xmath289 $ ] carries also a natural associative algebra structure @xmath293 although we will not make use of such structure here . the left regular representation is defined as @xmath294 thus , the matrix elements of the regular representation are obtained by computing the action of the group on the orthonormal basis @xmath295 , @xmath296 , of the hilbert space @xmath297 $ ] : @xmath298 then , the matrix representation of the left regular representation of the element @xmath299 can be easily computed from the table of the group written below ( notice the inverse of the elements along the rows ) . the matrix @xmath300 is obtaiend by constructing a matrix with ones in the positions where @xmath299 appears in the table and zeros in the rest . .group table . [ cols="^,^,^,^,^,^,^",options="header " , ] in the case of the regular representation , the input of our program can be the matrix @xmath301 constructed out of the table * t * ( see table[group_table ] ) relabeled by identifying @xmath302 with @xmath303 and @xmath295 with @xmath304 , and whose entries are defined as @xmath305 once we have the group multiplication table in this form , we do not need to compute , explicitly , the regular representation for each element @xmath306 to create the adapted states @xmath149 and @xmath151 in step since we can simply evaluate the random vectors @xmath307 on the elements of the table , that is , @xmath308_{ij}=\varphi_a \big(t_{ij}\big)\ , , \qquad a = 1,2 \ , .\ ] ] in the final , we will show the results obtained using our algorithm for the decomposition of the regular representation in two simple cases : the permutation group @xmath309 and the alternating group @xmath310 . to verify the accuracy of the results , we will compare characters , since they are independent of the choice of basis . we shall compute the characters of the irreps obtained after applying the unitary transformation @xmath240 provided by our algorithm and we will compare them with the exact characters by defining the error as @xmath311 where @xmath23 is the family of equivalence classes of irreps of @xmath19 . let @xmath1 be a compact lie group and @xmath19 a closed subgroup ( hence compact too ) . states adapted to @xmath19 will have the form @xmath312 where @xmath313 is the normalization factor @xmath314 and @xmath315 denotes the invariant haar measure on @xmath19 . because our algorithm is numerical , we need to approximate the integral with a finite sum . choosing a quadrature rule to approximate the integral for a given @xmath90 is equivalent to using another @xmath316 such that @xmath317 only at a finite number of elements of the group . then , the integral for @xmath316 reduces to a finite sum and the approximation of @xmath316 is exact . it could happen that the generic adapted states thus obtained do not have enough degrees of freedom , i.e. , it might happen that the block diagonal matrices of the representation were not irreducible . however , we will see that this is not a problem because in the case of lie groups , the clebsch gordan matrix decomposing all the elements of its lie algebra @xmath318 will be the clebsch gordan matrix decomposing all the elements of the representation . a lie algebra @xmath318 is an algebra closed under the lie bracket @xmath319:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ ] . any element of the lie algebra can be written in terms of linearly independent elements , called _ generators _ , @xmath320 , @xmath321 , which satisfy @xmath322=c_{ij}^k\xi_k\,,\qquad i , j , k=1,\ldots , n_\mathfrak{g}\,,\ ] ] where the coefficients @xmath323 are called the _ structure constants _ of the lie algebra @xmath318 and @xmath324 is its dimension . notice that the generators of any representation of the same lie algebra will have the same structure constants except by a multiplication factor . for lie groups , a unitary representation can be obtained via the exponential map of any element of its lie algebra @xmath318 , @xmath325 : @xmath326 one can immediately see that the clebsch gordan matrix @xmath47 that decomposes all the elements of the lie algebra @xmath327 will decompose all the elements of the representation and viceversa : @xmath328 where @xmath329 , @xmath167 , is the set of generators of the irreducible representations of the lie algebra @xmath318 and @xmath330 , their corresponding representations via the exponential map . in the case of compact lie groups , since the set of generators of its lie algebra is finite , @xmath331 , the matrix @xmath47 that decomposes in irreps @xmath332 non trivial linearly independent elements of the lie algebra , or @xmath332 linearly independent elements of the representation @xmath84 , @xmath333 , will be the clebsch gordan matrix . the original clebsch gordan problem consists in reducing a tensor product representation @xmath334 , @xmath335 , of two representations of the same group @xmath1 restricted to the diagonal subgroup of the product group . by associativity , this problem can be generalized to any number of tensor products @xmath336 . here , its associated lie algebra is given by @xmath337 where @xmath322=c_{ij}^k\xi_k,\qquad[\xi_i^\alpha,\xi_i^\alpha]=c_{ij}^k\xi_i^\alpha,\qquad\alpha=1,\ldots , n.\ ] ] let us now study the @xmath0 group : the generators of the representation of its associated lie algebra are given by the hermitian traceless angular momentum operators @xmath338 satisfying the commutation relations @xmath339=i\epsilon_{ij}^kj_k\ , , \qquad i , j , k = x , y , z\,,\qquad n_\mathfrak{g}=3\,.\ ] ] its associated representation of @xmath0 can be written as @xmath340 the matrix representation of momentum @xmath259 of the angular momentum operators @xmath341 is usually written in a basis of eigenvectors of @xmath342 , @xmath343 and the representation of the operators @xmath344 and @xmath345 is usually obtained from the representation of the ladder operators @xmath346 , @xmath347 for instance , if @xmath348 : @xmath349 @xmath350 in the standard basis @xmath351 the standard clebsch gordan matrix is constructed with eigenvectors of the total angular momentum operator @xmath352 with respect to the @xmath353 component , @xmath354 where @xmath15 is the number of parts of the system . the eigenvectors of this operator are usually denoted by @xmath355 , where @xmath356 represent the total angular momentum and @xmath357 : @xmath358 the standard procedure to obtain this clebsch gordan matrix consists in applying successively the ladder operator @xmath359 starting from the state of maximum momentum @xmath360 . notice that since the action of the matrix elements of the ladder operators is real , the clebsch gordan coefficients are real too . recall that the clebsch gordan matrix provided by our algorithm is written in terms of the eigenvectors of the first adapted state @xmath149 . thus , if we want to compare the clebsch gordan coefficients obtained from our algorithm with the standard ones , we have to find a clebsch gordan matrix @xmath361 which is conformed by eigenvectors of the operator @xmath362 . to do that , we first create two real adapted states , using the fact that the operators @xmath363 , @xmath364 , verify @xmath365 where @xmath366 denotes the complex conjugate therefore , for any adapted state @xmath90 , its complex conjugate @xmath367 is an adapted state too . hence , to create real adapted states , we first add each matrix @xmath368 , @xmath369 , in step in section[the_algorithm : sec ] to its complex conjugate to obtain real symmetric matrices , and then we multiply the result by its transpose to make it positive definite . finally , we normalize them dividing by their trace , i.e. , @xmath370 once we have two real adapted states @xmath371 and @xmath372 , we apply our algorithm to get the real clebsch gordan matrix @xmath240 . after that , we transform the operator @xmath362 with @xmath240 to decompose it in irreducible representations , @xmath373 and we diagonalize each block of this matrix , transforming it with a block - diagonal matrix @xmath374 which reorders the eigenvalues as follows : @xmath375{\makebox[1em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } & & & & & & & \\ & & & \hspace{-0.3cm}-j_1 & & & & & & \\ & & & & j_2 & & & & & \\ & & & & & \hspace{-0.2cm}j_2 - 1 & & & & \\ & & & & & & \hspace{-1.1 cm } { \rotatebox{-40}{\makebox[0pt]{\makebox[1em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } & & & \\ & & & & & & & \hspace{-0.6cm}-j_2 & & \\ & & & & & & & & \hspace{-0.4 cm } { \rotatebox{-40}{\makebox[0pt]{\makebox[1em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } & \\ & & & & & & & & & \hspace{-2.9 cm } { \rotatebox{-40}{\makebox[0pt]{\makebox[1em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } \\ & & & & & & & & & \hspace{-1.4cm}j_n \\ & & & & & & & & & \hspace{0.2cm}j_n-1 \\ & & & & & & & & & \hspace{0.8 cm } { \rotatebox{-40}{\makebox[0pt]{\makebox[1em]{\xleaders\hbox{$\cdot$\hskip.09em}\hfill\kern0pt } } } } \\ & & & & & & & & & \hspace{2.2cm}-j_n \\ \end{pmatrix}\,.\ ] ] therefore , the clebsch gordan matrix whose columns are the eigenvectors of @xmath362 , reordered in this way , is given by @xmath376 in the , we will show the computation of the clebsch gordan coefficients for the bipartite spin system @xmath377 and for the tripartite spin system @xmath378 . again , we will verify the accuracy by comparing the exact characters with the ones computed after transforming with the clebsch gordan matrix obtained with smily . for any irreducible representation of the @xmath0 group , it can be shown that the characters have the following expression : @xmath379 where @xmath380 is the dimension of the irrep . therefore , we measure the accuracy through @xmath381 with @xmath382 the number of elements in the quadrature approximation . a numerical algorithm to compute the decomposition of a finite - dimensional unitary representation of a compact lie group has been presented . such algorithm uses the notion of generic adapted quantum mixed states to obtain the block structure and , eventually , the coefficients of the clebsch gordan matrix solving the decomposition problem . the numerical algorithm is stable and accurate since it combines nothing but stable routines involving diagonalization of hermitian matrices , sorting and recombination of matrix blocks and matrix products . the numerical examples presented confirm this . the algorithm has been used successfully to decompose the regular representation of finite groups and the direct product of two and three representations of @xmath0 . in the first case , the main computational task was to prepare the group table , a preliminary task before the algorithm is used . in the second case , this preliminary part was much easier , since explicit expressions of the representations of the lie algebra @xmath383 , for any value of spin , are well - known . the algorithm can be easily extended to finite - dimensional representations of non - compact groups . however , because the representations will cease to be unitary , the numerical stability of the algorithm could be compromised . further insights on these questions will be considered elsewhere . [ appendix_ps]appendix in this appendix , we present the results obtained for the decomposition of the @xmath309 and @xmath310 group , and the clebsch gordan coefficients of the spin systems @xmath377 and @xmath384 . all experiments were conducted using matlab r2012a ( version 7.14.0.739 ) . a.1.1.*the decomposition of the left regular representation of the permutation group @xmath309 . * the @xmath309 group is the group of permutations of three elements and it has order six . the elements of this group can be generated with the set of transpositions @xmath385 , @xmath386 : @xmath387 our algorithm decomposes the regular representation in two representations @xmath388 and @xmath389 of dimension one and multiplicity one , and another one @xmath390 of dimension two and multiplicity two , exactly as expected . the representation @xmath388 corresponds to the trivial one , @xmath391 , @xmath392 , and the rest of representations obtained after applying the transformation @xmath240 given by smily are the following : @xmath393 if we use the formula to compute the accuracy of the characters of the irreps , we obtain @xmath394 a.1.2.*the decomposition of the left regular representation of the alternating group @xmath310 . * the alternating group @xmath310 is the group of even permutations of four elements . this group has twelve elements and it can be generated with three generators satisfying the relations @xmath395 the left regular representation of this group has four irreducible representations : three of dimension one and one of dimension three . hence smily will decompose the regular representation of this group in the three representations of dimension one with multiplicity one and in the representation of dimension three with multiplicity three . again , @xmath388 is the trivial representation @xmath391 , @xmath396 , and the rest are given by : @xmath397 @xmath398 in this case , the accuracy of the characters of the irreps computed with is given by @xmath399 a.2.1.*clebsch gordan coefficients for the spin system @xmath377 . * suppose we have a system of two particles in which the first particle has momentum @xmath400 and the second momentum @xmath303 . it is well - known ( * ? ? ? * ch.5 ) that this system is decomposed in the direct sum of systems of momentum @xmath401 , @xmath400 and @xmath402 , each one with multiplicity one , @xmath403 or , in other words , that the representation of @xmath0 corresponding to the tensor product @xmath377 has irreducible representations of momentum @xmath401 , @xmath400 and @xmath402 with multiplicity one each other . to create the adapted states for step of the algorithm , we have chosen three random vectors @xmath404 , @xmath405 , @xmath406 , for each adapted state , to obtain the three linearly independent elements of the representation . obviously , we have also created two random vectors @xmath307 of length @xmath407 to construct the matrices @xmath368 , @xmath369 , in step : @xmath408 where @xmath409 is the exponential representation given by and @xmath410 denotes the momentum of the representation @xmath32 . to represent the clebsch gordan coefficients , we will use the following standard arrangement : the coefficients obtained for the system @xmath377 applying the smily algorithm are as follows : to assess the accuracy , we have approximated the integral in with @xmath411 . the result we obtained is @xmath412 a.2.2.*clebsch gordan coefficients for the spin system @xmath384 . * to test the capabilities of our algorithm , we will compute the clebsch gordan coefficients of a system of three spin particles . these coefficients can be obtained from suitable choices of coefficients of products of two spins , for that reason , there are no tables for systems with more than two spins . the standard procedure consists in first reducing the representation of the first two particles , then reducing the result with the next particle , and so on , until there are no particles left . in our case , the product of three particles with spin @xmath402 , @xmath402 and @xmath400 yields @xmath413 this is , two irreps of momentum @xmath402 and @xmath401 with multiplicity one and other of momentum @xmath400 with multiplicity two . in the first step , we block - diagonalize the first two spins : @xmath414 and then we diagonalize the result : @xmath415 therefore , the clebsch gordan matrix of this system is @xmath416 in this example , we see that for a multipartite system of spins the multiplicities of the representations can be larger than one . thus , several eigenvectors may exist with the same values of @xmath356 and @xmath417 . therefore , it is necessary to add another ` quantum number ' , which we will denote by @xmath73 , to tell them apart . this ` quantum number ' will be a label indicating to which copy of the representation of multiplicity larger than one each of the eigenvectors with the same @xmath356 and @xmath417 belongs . hence the choice of @xmath73 to denote it , since this is the letter we used to denote the multiplicity in above . using our algorithm , we do not need to group the system in groups of bipartite systems as before , it can be done in one step . again , in this case , we have chosen three random vectors @xmath418 , @xmath406 , to obtain three linearly independent elements of the representation of the group , and another random vector @xmath419 of length @xmath407 to compute each linear combination @xmath194 . the coefficients will be represented in arrangements similar to the case of two spins but now including the label @xmath73 : notice that the table below is not unique because there exists more than one linear combination providing a valid clebsch gordan matrix that diagonalizes @xmath420 with the eigenvalues reordered in the way given in . the coefficients obtained for the tripartite system @xmath384 are the following : [ tabla_5 ] again , to assess the accuracy , we have approximated the integral in with @xmath411 , and the result obtained was @xmath421 the authors would like to thank the financial support provided by the ministry of economy and competitivity of spain , under grant mtm2014 - 54692 , and by the region of madrid research project quitemad+ , s2013/ice-2801 . ou and h.j . kimble . _ probability distribution of photoelectric currents in photodetection processes and its connection to the measurement of a quantum state_. phys . a. * 52 * , 31263146 ( 1995 ) .

a numerical algorithm that computes the decomposition of a finite - dimensional unitary reducible representation of a compact lie group is presented . the algorithm , inspired by notions of quantum mechanics , generates two adapted states and , after appropriate algebraic manipulations , returns the block matrix structure of the representation in terms of its irreducible components . it also provides an adapted orthonormal basis . the algorithm can be used to compute the clebsch gordan coefficients of the tensor product of irreducible representations of a given compact lie group . the performance of the algorithm is tested on various examples : the decomposition of the regular representation of finite groups and the computation of the clebsch gordan coefficients of tensor products of representations of @xmath0 .

1610.01054

dynamical systems preserving a geometrical structure have been studied quite extensively . especially those systems preserving a symplectic form have attracted a lot of attention , due to their fundamental importance in all kinds of applications . dynamical systems preserving a contact form are also of interest , both in mathematical considerations ( for example , in classifying partial differential equations ) and in specific applications ( study of euler equations ) . the 1form of liouville may be associated both with a symplectic form ( by taking the exterior derivative of it ) and with a contact form ( by adding to it a simple 1form of a new variable ) . we wish here to study dynamical systems respecting the form of liouville . as we shall see , they are symplectic systems which may be extented to contact ones . to set up the notation , let m be a smooth ( which , in this work , means continuously differentiable the sufficient number of times ) manifold of dimension @xmath0 . a contact form on m is a 1-form @xmath1 such that @xmath2 . a strict contactomorphism is a diffeomorphism of m which preserves the contact form ( their group will be denoted as @xmath3 ) while a vector field on m is called strictly contact if its flow consists of strict contactomorphims ( we denote their algebra as @xmath4 ) . in terms of the defining contact form @xmath1 , we have @xmath5 for a strict contactomorphism f and @xmath6 for a strictly contact vector field x , where @xmath7 denotes the lie derivative of @xmath8 in the direction of the field x. the classical example of a strictly contact vector field associated to @xmath8 is the vector field of reeb , @xmath9 , uniquely defined by the equations @xmath10 and @xmath11 . associated to every contact vector field x is a smooth function @xmath12 , called the contact hamiltonian of x , which is given as @xmath13 . conversely , every smooth function @xmath14 gives rise to a unique contact vector field @xmath15 , such that @xmath16 and @xmath17 . usually we write @xmath18 to denote the dependence of vector field @xmath18 on its ( contact ) hamiltonian function @xmath14 . results conserning the local behavior for systems of this kind may be found in @xcite , where the authors provide explicit conditions for their linearization , in the neighborhood of a hyperbolic singularity . the study of degenerate zeros , and of their bifurcations , remains , however , far from complete . here , in section 1 , we recall the form of strictly contact vector fields of @xmath19 , and their relation with symplectic vector fields of the plane . we show that the albegra @xmath20 of plane fields preserving the form of liouville @xmath21 may be obtained by projecting on @xmath22 stictly contact fields with constant third component . we begin the classification of vector fields belonging in @xmath20 ( we shall call them liouville vector fields ) by introducing the natural equivalence relation , and by showing that the problem of their classification is equivalent to a classification of functions up to a specific equivalence relation . in section 2 , ( germs at the orign of ) univariate functions are classified up to this equivalence relation , which we name `` restricted contact equivalence '' , due to its similarity with the classical contact equivalence of functions . we provide a complete list of normal forms for function germs up to arbitrary ( finite ) codimension . in section 3 , based on the previous results , we give local models for liouville vector fields of the plane . we first prove that all such fields are conjugate at points where they do not vanish , then we prove that they can be linearized at hyperbolic singularities , and finally we state the result conserning their finite determinacy , which is based on the finite determinacy theorem obtaind in section 2 . in section 4 , we first show how to construct a transversal unfolding of a singularity class of liouville vector fields and then we present transversal unfoldings for singularity classes of codimension 1 and 2 . phase portraits for generic bifurcations of members of @xmath20 are also given . next , in section 5 , we see that there is only one polynomial member of the group of plane diffeomorphisms preserving the form of liouville ( @xmath23 stands for this group ) . this is the linear liouville diffeomorphism , and we show the linearization of plane diffeomorphisms of this kind at hyperbolic fixed points . in section 6 , we return to members of @xmath24 to observe that the models obtained above are members of a specific base of the vector space of homogeneous vector fields . their linearization is again shown , albeit using classical methods of normal form theory . last section contains some observations concerning future directions . for a classical introduction to symplectic and contact topology the reader should consult @xcite , while @xcite offers a more complete study of the contact case . singularities of mappings are treated in a number of textbooks ; we recommend @xcite and @xcite ( see @xcite for a recent application of singularity theory to problems of dynamics ) . let m be a closed smooth manifold of dimension 2n+1 equipped with a contact form @xmath8 . the contact form is called regular if its reeb vector field , @xmath25 , generates a free @xmath26 action on m. in this case , m is the total space of a principal @xmath26 bundle , the so called boothby - wang bundle ( see @xcite for more details ) : @xmath27 , where @xmath28 is the action of the reeb field and @xmath29 is the canonical projection on @xmath30 . b is a symplectic manifold with symplectic form @xmath31 . the projection @xmath32 induces an algebra isomorphism between functions on the base b and functions on m which are preserved under the flow of @xmath25 ( such functions are called basic ) . it also induces a surjective homomorphism between strictly contact vector fields x of @xmath33 and hamiltonian vector fields y of @xmath34 ( that is , fields y with @xmath35 ) , the kernel of which homomorphism is generated by the vector field of reeb . in our local , three dimensional , case , things are of course simpler . using a local darboux chart , consider the euclidean space @xmath19 equipped with the standard contact structure @xmath36 . its reeb vector fiel , @xmath37 , induces the action @xmath38 , and the quotient of @xmath19 by this action , that is , the plane @xmath22 with coordinates @xmath39 , inherits the symplectic form @xmath40 . strictly contact vector fields of @xmath19 project to hamiltonian fields on this plane ( for a direct analogy with the volume preserving case the reader should consult @xcite ) . basic functions now depend , as one may easily verify , only on the first two variables , while the kernel of the above mentioned projection contains the multiples of @xmath41 . studying equation @xmath42 we get the general expression of @xmath43 : @xmath44 its contact hamiltonian is of course @xmath45 ( recall that it does not depend on the third variable ) , thus : @xmath46 observe that all vector fields of the @xmath39plane , preserving the symplectic structure @xmath47 , may be obtained in this way . in this work we restrict our attention to those members of @xmath48 , which preserve the form of liouville @xmath21 ( we shall denote their set as @xmath49 ) . the reason for this choise will become clear in section 6 . in this case , equation @xmath50 becomes : @xmath51 while @xmath52 . thus , their general form is @xmath53 , for some univariate function @xmath54 and a constant @xmath55 . observe that all vector fields of the plane presrving the form of liouville may be obtained by projecting the members of @xmath49 on the @xmath56 plane . we have , therefore , the following : [ basiclemma ] to every @xmath57 , corresponds a unique @xmath58 , namely @xmath59 . members of @xmath49 are trivially obtained by adding constant multiples of @xmath41 to members of @xmath20 . this lemma provides the general form of the vector fields we are interested in . our goal is the classification of these vector fields according to the natural relation defined in the obvious way : two fields @xmath60 are liouville conjugate if there exists a diffeomorphism of the plane preserving the form of liouville , @xmath61 , such that @xmath62 , while two fields @xmath63 are strictly contact conjugate if a @xmath64 exists , such that @xmath65 . observe that classifying members of @xmath20 leads to a classification of fields belonging in @xmath66 ; one needs only to extend @xmath67 to @xmath19 as @xmath68 . to proceed with the classification of liouville vector fields of the plane , we shall exploit their dependence on real valued functions . [ transforming ] let @xmath69 be a univariate function and @xmath70 a diffeomorphism of @xmath71 . the liouville vector field corresponding to function @xmath69 may be transformed , via a diffeomorphism respecting the form @xmath21 , to the liouville vector field corresponding to the function @xmath72 . constructing the fields corresponding to these two functions , according to the recipe given in lemma [ basiclemma ] , we conclude that the diffeomorphism accomplishing the desired transformation is @xmath73 which also preserves the liouville form . this lemma ensures that the classification of liouville vector fields , up to diffeomorphisms belonging in @xmath23 , reduces to a classification of univariate real functions . in the next section , we turn our attention to this classification . let @xmath74 be the germ at the origin of a smooth function . their ring will be denoted as @xmath75 . we introduce the following equivalence relation . let @xmath76 . we shall call them restrictively contact equivalent ( @xmath77-equivalent ) if there exists a germ of a smooth diffeomorphism @xmath78 such that @xmath79 . let @xmath76 , with @xmath80 . define @xmath81 . it is easy to check that @xmath70 is a local diffeomorphism at @xmath82 and @xmath79 . let us recall here that two univariate function germs @xmath76 are called contact equivalent if @xmath83 , for some function germ @xmath84 and diffeomorphism @xmath70 . the equivalence relation we study here requires @xmath85 . this explains why we called the above defined equivalence relation restricted contact . suppose now that @xmath86 is a curve of @xmath77equivalent germs , depending on the real parameter @xmath87 , with @xmath88 . there exists thus a curve of local diffeomorphisms @xmath89 , with @xmath90 and @xmath91 , such that @xmath92 . differenting with respect to s and evaluating at @xmath93 we get : @xmath94 , where @xmath95 is defined by the relation @xmath96 . note that @xmath97 , thus @xmath98 , the ideal of @xmath75 generated by @xmath99 . let @xmath100 . the ideal generated from the germs @xmath101 , equals @xmath102 . it is obvious that , if @xmath98 , then @xmath103 is a member of @xmath102 . let us prove the opposite inclusion . let @xmath104 . germs @xmath105 and @xmath106 exist , such that @xmath107 . we wish to find a germ @xmath108 such that : @xmath109 . one may easily check that a solution of the last differential equation is : @xmath110 which is well defined and smooth in a neighborhood of the origin and , therefore , for every @xmath104 a @xmath108 exists , such that @xmath111 , hence the conclusion . under the light of the lemma above , we proceed to the following : the tangent space of @xmath100 , with respect to @xmath77equivalence , is defined to be @xmath112 . the codimension of @xmath69 is defined as @xmath113 we calculate that , if @xmath114 , then @xmath115 , thus @xmath116 , while if @xmath117 , @xmath118 and @xmath119 . as usual , the germ @xmath100 is called @xmath120determined , with @xmath121 , if every other @xmath105 having the same @xmath120jet with @xmath69 is @xmath77equivalent to f. if such a finite @xmath120 does not exist , we say that @xmath69 is not finitely determined . the germ @xmath100 is @xmath120determined , with respect to @xmath77-equivalence , if @xmath122 . we have to prove that if @xmath123 , the germs @xmath69 and @xmath124 are @xmath77equivalent . towards this end , define @xmath125 $ ] . we shall construct diffeomorphisms @xmath126 , defined in a neighborhood of the origin , such that @xmath127 . differentiating with respect to @xmath87 , we get : @xmath128 . note that , for @xmath93 , we get the relation @xmath129 , which , by the previous lemma , has a solution @xmath98 since @xmath122 . we need to show that a solution exists for all @xmath130 $ ] . consider @xmath131 $ ] , let @xmath132 be the ring of function germs at @xmath133 $ ] and denote by @xmath134 the ideal of @xmath132 consisting of those germs vanishing at @xmath133 $ ] . we have : @xmath135 + @xmath136 + @xmath137 + @xmath138 + @xmath139 + @xmath140 , where the inclusion in the last line holds due to the nakayama lemma . thus , for every @xmath130 $ ] , we have that @xmath141 . we can therefore find @xmath142 , defining the germ of diffeomorphism @xmath143 which , for @xmath144 , establishes an equivalence between @xmath69 and @xmath124 . the classification of the elements of @xmath75 now follows . we begin with germs that either do not vanish at the origin , or have a regular point there . let @xmath100 . if @xmath145 , it is @xmath77equivalent to 1 , while if @xmath146 and @xmath147 , @xmath69 is @xmath148 to @xmath149 . let @xmath100 , with @xmath145 . to show that it is @xmath77equivalent to @xmath150 , we must find a local diffeomorphism @xmath151 such that @xmath152 , which is the same as @xmath153 , which is a differential equation with smooth right hand side , at least in a neighborhood of the origin , thus , such a smooth @xmath151 exists . on the other hand , let @xmath146 and @xmath147 . it is 1determined , thus @xmath77equivalent to its linear part @xmath149 , while , as may be easily verified , the germ @xmath149 is @xmath77equivalent to @xmath154 only if @xmath155 . let us know proceed to germs with critical points . let @xmath100 , with @xmath156 and @xmath157 . then @xmath69 is @xmath77equivalent to @xmath158 , if @xmath120 is an even number and to @xmath159 or @xmath160 , if @xmath120 is an odd number . if @xmath69 is such a germ , then , in a neighborhood of the origin , we may write @xmath161 , with @xmath162 . thus @xmath163 , and @xmath69 is @xmath120determined . it is thus @xmath77equivalent to @xmath164 , while , as may easily be verified , the germ of a diffeomorphism @xmath165 exists such that @xmath166 , for every @xmath167 , if @xmath120 is even , while if @xmath120 is odd then @xmath164 is @xmath77equivalent to @xmath168 , for @xmath169 and to @xmath158 , for @xmath170 . combining all the above , we may now state the main theorem for the classification of members of @xmath75 . [ functions ] if a member of @xmath75 does not vanish at the origin it is @xmath77-equivalent to the constant function @xmath150 . members of @xmath75 having codimension @xmath171 are @xmath77-equivalent to @xmath149 ( @xmath172 being the value of their derivative there ) . a member of @xmath75 of odd codimension @xmath120 is @xmath77-equivalent to @xmath173 , while if it is of even codimension @xmath120 it is @xmath77-equivalent to @xmath174 , depending on the sign of the value of its first non vanishing derivative at the orgin . table 1 contains the local models of members of @xmath75 having codimension up to five . we note that there are differences with the classical classification list for right equivalence ( in which list the @xmath175 and @xmath176 models may have both negative and positive sign ) and for contact equivalence ( in which , for example , the @xmath177 model does not depend on the constant @xmath172 , see @xcite ) . the interested reader should consult @xcite for a relation of contact and right equivalence , while the equivalence relation studied here provides more models than right and contact equivalence since it is stricter than both . table 1 [ cols="<,<,<",options="header " , ] for the cases @xmath178 we have ommited writing the vector fields for the negative and the positive sign since one may be obtained from the other after a multiplication with @xmath179 ( which means that their phase portraits are identical up to a reversal of time ) . except from the hyperbolic model ( and the non vanishing one ) , they all have an infinity of equillibria ( the @xmath180axis ) . othen than that , topologically their behavior is quite simple to analyze , since fuction @xmath181 serves as a first integral . it remains to analyze the behavior of pertubations of these vector fields . at regular points , members of @xmath20 are all conjugate to each other , via a diffeomorphism preserving the form of liouville . at hyperbolic singularities all such vector fields may be transformed to their linear part ; these linear parts are not conjugate to each other , since the eigenvalues there are a conjugacy invariant . however , up to topological equivalence , they are all saddle points , thus hyperbolic singularities are structurally stable . this is no more the case when we analyze vector fields belonging to the classes @xmath182 . to describe their local bifurcations we should first compute their transversal unfoldings . let @xmath15 be the germ at the origin of a liouville vector field . denote by @xmath183 its singularity class ( that is , the set of all germs at the origin of vector fields of liouville which are liouville equivalent to @xmath15 ) . a transversal unfolding of @xmath15 consists of a set of germs at the origin of liouville vector fields , which set intersects @xmath183 transversally at @xmath15 . thus , to construct transversal unfoldings of liouville vector fields , we must first compute the tangent spaces of singularity classes . let @xmath184 ( where @xmath69 is the function defining @xmath185 ) and @xmath183 its singularity class . we have : @xmath186 . let @xmath187 be the germ at the origin of a liouville vector field and @xmath188 the germ at the origin of a family of diffeomorphisms preserving the liouville form , where @xmath189 , @xmath190 and @xmath191 . define : @xmath192 . it is a curve of liouville vector fields belonging to @xmath183 , and we have @xmath193 . to calculate the tangent space @xmath194 we need to evaluate at @xmath93 the derivative with respect to the parameter @xmath87 of @xmath195 . it is : @xmath196 . we have denoted as @xmath197 the vector field defined by @xmath198 . note that @xmath199 is a liouville vector field , corresponding to the function @xmath200 , which belongs to @xmath201 , since @xmath202 . thus , the tangent space of @xmath183 at @xmath185 consists of those liouville fields corresponding to functions belonging in the ideal @xmath201 . the theorem above allows us to study bifurcations of liouville vector fields . to illustrate this , we present here such bifurcations of low codimension . we begin with the singularity class @xmath203 . the members of this class form a subset of codimension @xmath150 in the set of those members of @xmath20 vanishing at the origin . to transversally unfold them , we only need to add to their local model , linear terms preserving the form of liouville . we arrive thus at the vector field @xmath204 , where @xmath172 a real parameter . we have the following : the set of @xmath205 with @xmath206 and @xmath207 has codimension 1 in the set of liouville vector fields vanishing at the origin . its members are all conjugate to the @xmath203 model given above . the curve of vector fields @xmath208 intersects at @xmath209 this set transversally . the codimension and the conjugacy to the @xmath203 model follows easily from the analysis given in the previous sections . note that @xmath210 is the @xmath203 model , corresponding to the function @xmath211 . the intersection is transversal , since : @xmath212 . this is a liouville vector field corresponding to the function @xmath213 which is the only function ( up to a constant ) which vanishes at the origin and belongs to @xmath214 . thus , @xmath208 is a transversal unfolding of the @xmath203 singularity . vector fields depending on a single parameter undergoe , for isolated values of this parameter , the bifurcation depicted in figure 1 ; this bifurcation is therefore the codimension 1 bifurcation occuring in vector fields of interest . . the dotted line in the center picture stands for the line of singularities.,title="fig : " ] . the dotted line in the center picture stands for the line of singularities.,title="fig : " ] . the dotted line in the center picture stands for the line of singularities.,title="fig : " ] we proceed to bifurcations of codimension two . consider the @xmath215 model , and add to it terms of lower degree . we arrive at @xmath216 , where @xmath217 real parameters . we have the following : the set of @xmath205 with @xmath218 and @xmath219 has codimension 2 in the set of liouville vector fields vanishing at the origin . its members are all conjugate to the @xmath215 model given above . the surface of vector fields @xmath220 intersects at @xmath221 this set transversally . its proof goes along the lines of the previous proposition , and it is therefore omitted . in figure 2 we present the bifurcations system @xmath222 system undergoes , for characteristic parameter values . before discussing the diffeomorphism case , let us note that we could study bifurcations of arbitrary , finite , codimension following the exact same approach . ( left ) , @xmath221 ( center ) , @xmath223 ( right ) . the dotted line in the center picture stands for the line of singularities.,title="fig : " ] ( left ) , @xmath221 ( center ) , @xmath223 ( right ) . the dotted line in the center picture stands for the line of singularities.,title="fig : " ] ( left ) , @xmath221 ( center ) , @xmath223 ( right ) . the dotted line in the center picture stands for the line of singularities.,title="fig : " ] let us now turn our attention to diffeomorphisms of the plane respecting the form of liouville . as we saw , they are of the general form @xmath224 . diffeomorphism @xmath54 of @xmath71 uniquely defines such a diffeomorphism . the unique linear diffeomorphism preserving the form of liouville ( and the origin ) is thus @xmath225 . aside from this , there are no other polynomial members of @xmath23 ; as a consequence , finite jets ( of any order ) of liouville diffeomorphisms studied here do not belong to the same group . the classification of strict contactomorphisms , according to the natural equivalence relation , is of course our purpose ; @xmath226 are liouville conjugate if there exists a third liouville diffeomorphism @xmath67 such that @xmath227 . to continue , and since we focus on fixed points , we impose the conditions @xmath228 . generically , such diffeomorphisms may be linearized in a neighborhood of the origin . there exists a codimension zero subset of those members of @xmath23 vanishing at the origin , every member of which may be transformed , via a change of coordinates preserving the liouville form , to its linear part . let us consider the set of liouville diffeomorphisms having linear part @xmath229 . its codimension is zero ( in the set of liouville diffeomorphisms vanishing at the origin ) and its members are of the form @xmath224 where @xmath230 a local diffeomorphism ( we use h.o.t . as an abbreviation for higher order terms " ) . we have supposed that @xmath231 ; therefore a local diffeomorphism @xmath232 of @xmath71 exists such that @xmath233 ( this is the content of the sternberg linearization theorem , see @xcite ) . using this diffeomorphism define @xmath234 and observe that it is a diffeomorphism , preserving the liouville form , with inverse @xmath235 . as is easy to confirm , @xmath236 . we have thus found the generic model for the mappings under study , that is @xmath225 . as already remarked , it is actually the unique polynomial model for members of @xmath23 ; thus liouville diffeomorphisms either may be linearized or are not finitely determined ( at least finitely determined under the relation of liouville conjugacy ) . having completed the study of vector fields of liouville we may now state results for strictly contact vector fields of @xmath19 . indeed , one needs only to add constant multiples of @xmath41 to the local models presented above , to obtain vector fields which preserve both the contact form @xmath172 and the form of liouville . our choise of restricting our study to members of @xmath66 stems from the fact that they are the only strictly contact vector fields which may have homogeneous components . indeed , recall from section 1 the general form of a strictly contact vector field : @xmath237 . assuming that @xmath238 ( remember it does not depend on @xmath239 ) is a homogenous polynomial of degree @xmath240 , vector field @xmath15 above is homogeneous of degree @xmath241 only in case its third component is constant , for @xmath242 , or zero , for @xmath243 . members of @xmath66 are therefore the only homogeneous members of @xmath24 . we shall elaborate in this observation in this section , to show , using classical normal form theory , the linearization of strictly contact vector fields respecting the form of liouville . consider members of @xmath24 vanishing at the origin . if @xmath15 is such a field , let @xmath244 be its @xmath120jet at zero , for some natural number @xmath120 , where each @xmath245 is a homogeneous field of degree @xmath120 . it is easy to see , equating terms of the same degree in equation @xmath246 , that each @xmath247 is itself a member of @xmath24 . we denote as @xmath248 the subset of @xmath24 , the components of which are homogeneous functions of degree d. we easily prove the following : the vector space @xmath248 is one dimensional . for each @xmath249 , its base consists of the field @xmath250 . the local models of table 2 constitute , therefore , the basis generating the fields of interest . linear fields ( belonging to @xmath251 ) are of the form @xmath252 , with @xmath172 arbitrary constant . in our case , therefore , the existence of hyperbolic singularities is excluded ( actually , @xmath253 is also the unique linear member of @xmath254 ; strictly contact vector fields do not possess hyperbolic singularities ) . despite this fact , fields having non zero linear part can be linearized , in a neighborhood of the origin . we shall prove it now using an approach different from the one indicated above . there are @xmath255 monomials depending on three variables , having degree @xmath240 , as simple counting arguments may assure . thereupon , the vector space @xmath256 of homogeneous vector fields of degree @xmath240 is of dimension @xmath257 , and one may easily verify that the fields appearing in table 3 , being @xmath257 independent vector fields of degree @xmath240 , constitute a basis of it . vector fields of interest here belong to this base ( to obtain them , just set @xmath258 to the first field of the second class ) . this base was presented , in the general n dimensional case , in @xcite , section 4 of which contains the arguments we shall use to prove the next proposition . the author wishes to thank prof . j d meiss for clarifying them to him . table 3 lll fields & condition & number + + @xmath259 + @xmath260 + @xmath261 & @xmath262 & @xmath263 + + @xmath264 & @xmath265&@xmath265 + @xmath266 & @xmath267 & @xmath268 + + @xmath269 & @xmath267 & @xmath270 + if @xmath271 , the vector field @xmath272 $ ] , where @xmath253 is the unique , linear and non zero , strictly contact vector field presented above , is also homogeneous of degree d ( the brackets @xmath273 $ ] denote the usual commutator of vector fields ) . we may define therefore the operator @xmath274 , @xmath275 $ ] . vector fields belonging to the base of @xmath256 are eigenvectors of this operator ; thus the subspaces generated by them are invariant under @xmath276 , ensuring the diagonal form of its matrix . there exists a codimension zero subset of @xmath277 every member of which may be transformed to its linear part . the linearizing diffeomorphism is close to the identity and preserves the contact form . the subset we refer to is the set of vector fields of interest with non zero linearization , and its codimension is easily obtained . classical normal form theory ensures that , by changing coordinates , we may discard all terms of @xmath278 which are not contained in the complement of the range of this operator ( an operator which leaves invariant the spaces @xmath279 , as well as the subspaces generated by the basic vector fields , the subspace of fields which interest us included ) . the matrix of @xmath253 is self adjoint , so a complement to the range of @xmath276 is the kernel of this operator . this kernel however , as may easily be verified , is trivial , providing us with a diffeomorphism which transforms to its ( non zero ) linear part every field of @xmath277 . this diffeomorphism preserves the 1form defining the contact structure ; this stems from the diagonal form of the matrix of @xmath276 . strictly contact vector fields project to symplectic fields of the plane ; homogeneous strictly contact vector fields project to fields of the plane preserving the form of liouville . we have studied here the local behavior of the later ; the local study of the first remains a challenging task . contact systems have a long history , and attract a lot of attention , since they form a valuable tool in topological constructions , in hamiltonian dynamics and in many physical applications ( see @xcite for a textbook account of these fields , and further references ) . almost all contact systems possess hyperbolic singularities , as transversality arguments show . in this case , conditions for linearization have been obtained ( @xcite ) . results are much more rare , however , if the singularities are degenerate . we chose here to consider the simpler case of homogeneous strictly contact systems . this led us to the study of plane systems , preserving the form of liouville , a subject which has an interest of its own . to study these fields we had to classify univariate functions according to the restricted contact equivalence relation . all these admit generalizations and deserve more study . indeed , extending the definition of restricted contact equivalence to arbitrary dimensions we get of course the differential conjugacy relation for vector fields . one could probably reobtain results of normal form theory , using this approach , which would potentially help the problem of classifying vector fiels preserving the form of liouville in any dimension . and , as already mentioned , the general problem of analyzing the behavior of contact dynamical systems stands , both interesting and difficult . the author hopes to further comment on these subjects in the future . this work is dedicated to my two professors , tassos bountis and spyros pnevmatikos , on the occasion of their 65th birthday . it is only a pleasure for the author to acknowledge the influence they had on him and to thank them for their constant support .

we study vector fields of the plane preserving the form of liouville . we present their local models up to the natural equivalence relation , and describe local bifurcations of low codimension . to achieve that , a classification of univariate functions is given , according to a relation stricter than contact equivalence . we discuss , in addition , their relation with strictly contact vector fields in dimension three . analogous results for diffeomorphisms are also given . * keywords * systems preserving the form of liouville , strictly contact systems , classification , bifurcations + * msc[2000 ] * primary 37c15 , 37j10 , secondary 58k45 , 53d10

1606.09515

in july 2012 , the cms and atlas experimental collaborations at the large hadron collider ( lhc ) announced the observation of a new boson @xcite , consistent with a higgs particle , the last undiscovered object in the standard model ( sm ) . the initial results were based on data corresponding to integrated luminosities of @xmath4fb@xmath5 taken at @xmath6tev and 5.3fb@xmath5 at @xmath7tev and the search was performed in six decay modes : @xmath8 , @xmath9 , @xmath10 , @xmath11 , @xmath12 and @xmath13 . a @xmath145@xmath15 excess of events with respect to the background was clearly observed in the first and second of these decay modes , while the remaining ones yielded exclusion limits well above the sm expectation . both collaborations have since been regularly updating their findings @xcite , improving the mass and ( so - called ) ` signal strength ' measurements . in these searches , the magnitude of a possible signal is characterized by the production cross section times the relevant branching ratios ( brs ) relative to the sm expectations in a given higgs boson decay channel @xmath16 , denoted by @xmath17 ( i.e. , the signal strength ) . according to the latest results released by the two collaborations after the collection of @xmath1420 fb@xmath5 of data @xcite , a broad resonance compatible with a 125gev signal is now also visible in the @xmath18 decay channel . the mass of the observed particle is still centered around 125 gev but the measured values of its signal strength in different channels have changed considerably compared to the earlier results . these values now read + @xmath19 , @xmath20 , @xmath21 + at cms , and + @xmath22 , @xmath23 , @xmath24 + at atlas . the bulk of the event rates comes from the gluon - gluon fusion channel @xcite . furthermore , the signal has also been corroborated by tevatron analyses @xcite , covering the @xmath25 decay mode only , with the higgs boson stemming from associated production with a @xmath26 boson @xcite . however , there the comparisons against the sm higgs boson rates are biased by much larger experimental errors . if the current properties of the observed particle are confirmed after an analysis of the full 7 and 8tev data samples from the lhc , they will not only be a clear signature of a higgs boson , but also a significant hint for possible physics beyond the sm . in fact , quite apart from noting that the current data are not entirely compatible with sm higgs boson production rates , while the most significant lhc measurements point to a mass for the new resonance around 125gev the tevatron excess in the @xmath25 channel points to a range between 115gev and 135gev . while the possibility that the sm higgs boson state has any of such masses would be merely a coincidence ( as its mass is a free parameter ) , in generic supersymmetry ( susy ) models the mass of the lightest higgs boson with sm - like behavior is naturally confined to be less than 180gev or so @xcite . the reason is that susy , in essence , relates trilinear higgs boson and gauge couplings , so that the former are of the same size as the latter , in turn implying such a small higgs boson mass value . therefore , the new lhc results could well be perceived as being in favor of some low energy susy realisation . several representations of the latter have recently been studied in connection with the aforementioned lhc and tevatron data , including the minimal supersymmetric standard model ( mssm ) @xcite ( also the constrained version @xcite of it , in fact ) , the next - to - minimal supersymmetric standard model ( nmssm ) @xcite , the e@xmath27-inspired supersymmetric standard model ( e@xmath27ssm ) @xcite and the ( b - l ) supersymmetric standard model ( ( b - l)ssm ) @xcite . all of these scenarios can yield a sm - like higgs boson with mass around 125gev and most of them can additionally explain the excesses in the signal strength measurements in the di - photon channel . another approach to adopt in order to test the viability of susy solutions to the lhc higgs boson data is to consider the possibility of having cp - violating ( cpv ) phases ( for a general review of cp violation , see ref . @xcite ) in ( some of ) the susy parameters . these phases can substantially modify higgs boson phenomenology in both the mass spectrum and production / decay rates at the lhc @xcite , while at the same time providing a solution to electroweak baryogenesis @xcite . in the context of the lhc , the impact of cpv phases was emphasized long ago in ref . @xcite and revisited recently in ref . @xcite following the higgs boson discovery . in all such papers though , cpv effects were studied in the case of the mssm . in this paper , we consider the case of similar cpv effects in the nmssm . in particular , we study the possibility to have higgs boson signals with mass around 125gev in the cpv nmssm , which are in agreement with the aforementioned lhc data as well as the direct search constraints on sparticle masses from lep and lhc . we also investigate the dependence of the feasible cpv nmssm signals on the mass of the higgs boson as well as its couplings to both the relevant particle and sparticle states entering the model spectrum , chiefly , through the decay of the former into a @xmath28 pair . we thus aim at a general understanding of how such observables are affected by the possible complex phases explicitly entering the higgs sector of the next - to - minimal susy lagrangian . the paper is organized as follows . in the next section , we will briefly review the possible explicit cpv phases in the higgs sector of the nmssm . in sec . [ sec : params ] we will outline the independent cpv nmssm parameters and the methodology adopted to confine our attention to the subset of them that can impinge on the lhc higgs boson data . in the same section , we further investigate the possible numerical values of the complex parameters after performing scans of the low energy cpv nmssm observables compatible with the lep and lhc constraints on higgs boson and susy masses . in sec . [ sec : results ] we present our results on the higgs boson mass spectrum as well as signal rates in connection with the lhc . finally , we conclude in sec . [ sec : summa ] . the cpv phases appearing in the higgs potential of the nmssm at tree - level @xcite can be divided into three categories : 1 . @xmath29 and @xmath30 : the spontaneous cpv phases of the vacuum expectation values ( vevs ) of the up - type higgs doublet @xmath31 and the higgs singlet @xmath32 , respectively , with respect to the down - type higgs doublet @xmath33 ; 2 . @xmath34 and @xmath35 : the phases of the higgs boson trilinear couplings @xmath36 and @xmath37 ; 3 . @xmath38 and @xmath39 : the phases of the trilinear soft terms @xmath40 and @xmath41 . as explained in @xcite the phases in category 3 . above are determined by the minimisation conditions of the higgs potential with respect to the three higgs fields . furthermore , assuming vanishing spontaneous cpv phases in category 1 . ( and real sm yukawa couplings ) , the only actual physical phases appearing in the tree - level higgs potential are those in category 2 as the difference @xmath42 . beyond the born approximation , the phases of the trilinear couplings @xmath43 , @xmath44 , and @xmath45 also enter the higgs sector through radiative corrections from the third generation squarks and stau ( assuming negligible corrections from the first two generations ) . also , in the one - loop effective potential , @xmath34 can contribute independently from @xmath35 . the complete one - loop higgs mass matrix can be found in refs . @xcite . here we only reproduce the tree - level higgs as well as sfermion mass matrices in appendix a. the @xmath46 higgs mass matrix @xmath47 , defined in the basis @xmath48 ( after @xmath49-rotating the @xmath50 matrix to isolate the goldstone mode ) , is diagonalized with a unitary matrix @xmath51 to yield five mass eigenstates as @xmath52 where @xmath53 in order of increasing mass . for a nonzero value of any of the phases listed above , these mass eigenstates become cp indefinite due to scalar - pseudoscalar mixing . moreover , these cpv phases not only affect the masses of the higgs states but also their decay widths , since the higgs boson couplings to various particles are proportional to the elements of the unitary matrix @xmath51 ( see , e.g. , refs . additionally , alterations in the masses of light neutralinos and charginos , in particular , due to the phases in category 1 above , can also have an indirect impact on the brs of the higgs bosons into sm particles . the decay widths and brs of the higgs boson in the nmssm with cpv phases can be calculated using the methodology implemented in ref . explicit expressions for higgs boson couplings and widths in the cpv nmssm can be found in ref . @xcite , which follows the notation of @xcite . these widths and brs can then be used to obtain the signal strength of the @xmath28 channel ( also called _ reduced _ di - photon cross section ) , @xmath54 , defined , for a given higgs boson , @xmath55 , as @xmath56 where @xmath57 implies a sm higgs boson with the same mass as @xmath55 . in terms of the _ reduced _ couplings , @xmath58 ( couplings of @xmath55 with respect to those of @xmath57 ) , eq.([eq : rxsct1 ] ) can be approximated by @xmath59 ^ 2[c_i(\gamma\gamma)]^2\sum_x \frac{\gamma^{\textrm{total}}_{h_\textrm{sm}}}{\gamma^{\textrm{total}}_{h_i } } \ ] ] where @xmath60 denotes the total width of @xmath57 . in light of the recent lhc discovery of a sm higgs boson - like particle we scan the parameter space of the cpv higgs sector of the nmssm using a newly developed fortran code . in our scans the lep constraints on the model higgs bosons are imposed in a modified fashion ; i.e. , they have to be satisfied by the scalar and pseudoscalar components of all the cp - mixed higgs bosons . also imposed are the constraints from the direct searches of the third - generation squarks , stau and the light chargino at the lep . we point out here that , in the cp - conserving ( cpc ) limit , the higgs boson mass and brs have been compared with those given by nmssmtools @xcite and have been found to differ from the latter by @xmath141% and @xmath145% at the most , respectively . although no limits from @xmath61-physics or from relic density measurements have been imposed we confine ourselves to the regions the parameter space regions which have been found to comply with such constraints ( see , e.g. , ref . @xcite ) . we study the effects of the cpv phases described in the previous section on the mass and di - photon signal rate of a higgs boson predicted by the model that is compatible with the higgs boson discovery data from the lhc . in particular , we consider the three most likely scenarios specific to the cpv nmssm that comply with the latter . in our analysis , we assume minimal supergravity ( msugra)-like unification of the soft parameters at the susy - breaking energy scale , such that @xmath62 , + @xmath63 , + @xmath64 , + where @xmath65 and @xmath66 are the soft susy - breaking squared masses of the third - generation squarks and sleptons , respectively . these parameters are then fixed to their optimal values based on earlier studies @xcite in order to minimize the set of scanned parameters . we then focus only on the effects of the higgs sector parameters , which include the dimensionless higgs boson couplings @xmath36 and @xmath37 along with their phases @xmath34 and @xmath35 , as well as the soft susy - breaking parameters @xmath40 and @xmath41 . from outside the higgs sector , we only analyze the effect of the variation of the unified cpv phase of the third - generation trilinear couplings , @xmath67 ( @xmath68 ) . before we discuss the three scenarios mentioned above , we note that the two heaviest higgs boson mass eigenstates @xmath69 and @xmath70 always correspond to the interaction eigenstates @xmath71 and @xmath72 in eq.([eq : h - states ] ) . sits in the position corresponding to @xmath71 in eq.([eq : h - states ] ) before ordering by mass , even though evidently it contains components of the other higgs fields also . in particular , a sm - like @xmath55 contains adequate components of both @xmath71 and @xmath73 to have sm - like couplings to fermions and gauge bosons . ] hence , a scenario is defined by the higgs state that conforms to the lhc observations , out of the three light mass eigenstates , @xmath74 , @xmath75 and @xmath76 , and by the correspondence between the latter and the interaction eigenstates @xmath73 , @xmath77 and @xmath78 . however , note that such a definition is adopted only so that a distinction between different scenarios can be made conveniently . evidently , the behavior of the ` observed ' higgs boson , @xmath79 , with the cpv phases in a given scenario is a combined result of the set of parameters yielding that scenario rather than of its position among the mass - ordered higgs states . the criteria for choosing the ranges of the scanned model parameters as well as the values of the non - higgs - sector susy parameters thus depend on the scenario under consideration and are explained in the following . + in this scenario the lightest higgs state , @xmath74 , is the sm - like one and corresponds to @xmath73 , while @xmath75 and @xmath76 correspond to @xmath77 and @xmath78 , respectively . the requirement of obtaining a down - type higgs state with mass close to 125 gev and with sm - like couplings necessitates large soft susy masses and @xmath80 . the values of @xmath81 ( @xmath82 , where @xmath83 is the vev of @xmath32 ) and the gaugino masses are found to be in best agreement with the relic density constraints @xcite , giving a neutralino with a large higgsino component as the lightest susy particle . further , @xmath36 and @xmath37 are chosen such that there is enough mixing of the doublet with the singlet higgs boson so as to allow an @xmath74 with the correct mass while keeping its couplings close to their sm values . we test two cases for this scenario , corresponding to two representative values of the parameter @xmath84 ( @xmath85 , where @xmath86 and @xmath87 are the vevs of @xmath31 and @xmath33 , respectively ) , which is fixed to 8 in case1 and to 15 in case2 . + this scenario is defined by the sm - like @xmath14125gev higgs boson being the second lightest higgs boson , @xmath75 , of the model . there are two possibility entailing such a scenario . it can be @xmath88-like with a large singlet component , in which case it has @xmath54 sm - like or bigger , as shown in @xcite . we refer to this possibility as case1 of this scenario . it requires relatively large values of @xmath36 and @xmath37 , small values of the parameters @xmath40 and @xmath41 and moderate values of soft susy - breaking parameters . for case2 of this scenario , we take a slightly different region of the parameter space which yields a @xmath75 that is again @xmath73-like but with a much smaller singlet component , so that it has @xmath89 around the sm expectation . therefore , heavy unified soft squark mass and/or trilinear coupling are required in this case , but a light soft gaugino mass is preferred . @xmath36 can be small to intermediate while @xmath37 is always small . finally , in this scenario @xmath74 and @xmath76 are @xmath77- and @xmath78-like , respectively . + there also exists the possibility that the observed @xmath14125gev higgs boson is the @xmath76 of the model which corresponds to @xmath73 , while both @xmath77- and @xmath78-like higgs bosons are lighter . such a scenario can be realized for very fine - tuned ranges of the parameters @xmath40 and @xmath41 for a given @xmath84 value , with large soft squark and gaugino mass parameters preferred . note that in this case the @xmath78-like @xmath76 of case2 of scenario2 turns into @xmath75 by becoming lighter than the @xmath73-like state which , consequently , turns into @xmath76 . these two cases thus overlap slightly in terms of the relevant parameter space of the model . + note here that we do not consider a scenario with the @xmath14125gev higgs boson corresponding to the @xmath78 interaction eigenstate , since the pure ( or nearly pure ) pseudoscalar hypothesis is disfavored by the cms higgs boson analyses @xcite . moreover , in scenarios 2 and 3 above the masses of @xmath75 and @xmath76 can lie very close to each other . in fact , these two higgs bosons can be almost degenerate in mass near 125gev , in which case the signal observed at the lhc should be interpreted as a superposition of individual peaks due to each of them . however , in the nmssm , particularly in the presence of cpv phases , more than one possibilies with mass degenerate higgs bosons may arise ( see , e.g. , @xcite and @xcite ) . such possibilities warrant a dedicated study of their own , which is currently underway , and in this article we have , therefore , not taken any of them into account . for correctness of our results , we have thus imposed the condition of nondegeneracy during our scans , so that only those points are passed for which no other higgs boson apart from the signal higgs boson under consideration lies inside the mass range of interest , defined in the next section . values of the fixed parameters as well as ranges of the variable parameters for all the above scenarios are given in table[tab : range_par ] . + @xmath90(tev ) & & 0.8 & 3 & 3 + @xmath91(tev ) & & 0.35 & 0.35 & 1.5 + @xmath92(tev ) & & 1 & 4 & 4 + @xmath81(tev ) & & 0.14 & 0.14 & 0.14 + @xmath84 & 8 & 15 & 1.9 & 20 & 10 + + @xmath36 & & 0.5 0.6 & 0.01 0.3 & 0.1 0.3 + @xmath37 & & 0.3 0.4 & 0.01 0.1 & 0.05 0.1 + @xmath40(tev ) & & 0.14 0.2 & 0.2 0.6 & 0.95 1.05 + @xmath93(tev ) & & 0.2 0.25 & 0.1 0.3 & 0.07 0.09 + we perform scans for each of the scenarios described earlier requiring the mass of @xmath79 ( i.e. , of @xmath74 in scenario1 , @xmath75 in scenario2 and @xmath76 in scenario3 ) to lie in the range 124gev@xmath94127gev . we additionally impose the condition @xmath95 on the signal higgs boson . furthermore , to each case in a given scenario corresponds a set of three scans , such that in each of the scans only one of the following cpv phases is varied : + ( i ) @xmath96 , ( ii ) @xmath97 , ( iii ) @xmath67 , + while fixing the others to @xmath98 . each scan thus checks the effect of a different cpv source at the tree - level and/or beyond . in each scan we vary the relevant phase in steps of 1@xmath99 between @xmath98 and @xmath100 . the measurements of the electric dipole moment ( edm ) of the electron , neutron , and various atoms @xcite put constraints on the allowed values of @xmath34 and @xmath67 . however , the trilinear couplings of squarks and sleptons contribute to the edms only at the two - loop level , and their phases are thus rather weakly constrained . one can , furthermore , assign very heavy soft masses to the sfermions of first two generations in order to minimize the effect of @xmath67 on the edms , as pointed out in earlier studies for the mssm @xcite . in fact , such constraints can be neglected altogether by arguing that the phase combinations occurring in the edms can be different from the ones inducing higgs boson mixing @xcite . the phase of @xmath37 , in contrast , has been found to be virtually unconstrained by the edm measurements @xcite . below we present our results separately for each of the five cases investigated . for evaluating the effect of the phases on @xmath101 and @xmath102qualitatively , we choose a set of four representative points ( rps ) , referred to as rp1 , rp2 , rp3 , and rp4 in the following , for every case . as explained in sect.[sec : cpv ] , at the tree - level the only independent cpv phase entering the higgs mass matrix is the difference @xmath103 . at one - loop level , although @xmath34 can appear separately from @xmath35 , the contribution from the corresponding terms is much smaller than the tree - level dependence on @xmath103 . furthermore , since only @xmath104 appears in the diagonal cp - even and cp - odd blocks and only @xmath105 in the cp - mixing block ( see appendix a ) of the higgs mass matrix , the mass eigenstates show a very identical behavior when either of these two phases is varied while fixing the other to 0@xmath99 . only small differences arise for very large values of @xmath34 and @xmath35 due to the higher - order corrections . therefore , our rp1 for a given case corresponds to a point for which the effect of @xmath34 ( and equivalently @xmath35 ) on @xmath101 is maximized for that case . similarly , rp3 is chosen such that the variation in @xmath101 is maximal with @xmath67 , since this phase only appears at the one - loop level and can potentially cause a behavior different from that due to the tree - level cpv phase . the dependence on @xmath67 can , however , be expected to show an identical behavior across all cases , as it is largely independent of other higgs sector parameters ( except @xmath84 ) , which is indeed what we will observe in our results below . as already noted , the cpv phases also affect the higgs boson decay widths into fermions and gauge bosons , through the elements of the higgs mixing matrix @xmath51 . on the other hand , in the decays of a higgs boson into two lighter higgs bosons , the tree - level phase , @xmath103 , enters directly while the phase @xmath67 also enters through the one - loop cp - odd tadpole conditions at one - loop @xcite . rp2 and rp4 are , therefore , points with the largest effect on @xmath102 due to the variation in @xmath34/@xmath35 and @xmath67 , respectively , observed in our scans . note that in the discussion below the description of the behavior of a given rp may not be equally applicable to all other good points , since it is chosen only so as to understand the maximum possible impact of a given phase and to highlight some potentially distinguishing features of different cases and scenarios . in fig.[fig : s1c1]a we show , for the small @xmath84 case of this scenario , the variation in the number of good points , i.e. , points surviving the conditions imposed on @xmath101 and @xmath102 , with varying @xmath34 and @xmath35 . the number of surviving points first falls slowly with increasing @xmath34/@xmath35 and then abruptly for @xmath106 after which it remains almost constant for a while before falling further . the number of surviving points reduces to 0 for @xmath34 and @xmath35 larger than 75@xmath99 and 77@xmath99 , respectively . however , very few , @xmath1410 , surviving points reemerge for @xmath34 and @xmath35 larger than 155@xmath99 and 150@xmath99 , respectively , although it is not apparent from the figure . the drop in the number of points is not continuous since there are other parameters , @xmath36 , @xmath37 , @xmath40 and @xmath41 , which are also scanned over for every value of a given phase . moreover , the number of good points evidently depends on the conditions on @xmath101 and @xmath102 so that while both of these may be satisfied for one value of a phase one of these may be violated for the next . although , as we shall see below , @xmath101 is almost always mainly responsible for the drop in the number of good points . note that the cpc case is also subject to the conditions on @xmath101 and @xmath102 and , on account of being defined relative to this case , the number of good points does not represent all possible solutions for all values of the phases . thus , it is likely that the cpc case for a given parameter set falls outside the defined ranges of @xmath101 and/or @xmath102 , but the conditions on these are satisfied for a different value of a particular phase . such a value of the phase can thus result in a considerable number of good points which would be absent in the cpc case . nevertheless , the aim here is to give an estimate of the effect of the cpv phases on the number of good points relative to the cpc case , rather than to present a truly holistic picture . fig.[fig : s1c1]b shows the variation in the number of surviving points with that in @xmath67 . contrary to the case of @xmath34/@xmath35 , for this phase the number of surviving points falls abruptly to 0 when @xmath67 crosses @xmath107 and then rises again when @xmath67 reaches @xmath108 . the values of other parameters corresponding to each rp for this case are given below . [ cols="^,^,^,^,^",options="header " , ] in summary , we have demonstrated that the cpv nmssm offers some interesting solutions to the lhc higgs boson data , which differ substantially from well - known configurations of the cpc nmssm , thereby augmenting the regions of parameter space which can be scrutinized at the cern collider . we have concentrated on the case in which only three cpv phases , @xmath35 , @xmath34 and @xmath67 , enter the higgs sector . we have then checked the twofold impact of these phases , always varied independently from each other , on the mass as well as signal strength of the assumed signal higgs boson in the @xmath28 decay mode , in different model configurations . the overall picture that emerges is that any of the three lightest higgs states of the cpv nmssm can be the one discovered at the lhc . we have illustrated this by using five benchmark cases in the parameter space of the model that can easily be adopted for experimental analyses . our analysis also proves that the possibility of explicitly invoking cpv - phases is not ruled out by the current lhc higgs boson data in any of our tested plausible nmssm scenarios . finally , a numerical tool for analyzing the higgs sector of the cpv nmssm has also been produced and is available upon request . the obvious outlook of this analysis will be to consider the possibility that companion higgs boson signals to the one extracted at the lhc may emerge in the cpv nmssm , so as to put the lhc collaborations in the position of confirming or disproving this susy hypothesis . an investigation on these lines is now in progress . s. moretti is supported in part through the next institute . s. munir is funded in part by the welcome programme of the foundation for polish science . the work of p. poulose is partly supported by a serc , dst ( india ) project , project no . sr / s2/hep-41/2009 . detailed expressions for the one - loop higgs boson mass matrices can be found in refs . @xcite . here we only reproduce the tree - level mass matrix to show the dependence on @xmath34 and @xmath35 since the dominant contributions from these phases arise at this level . note that the tree - level sfermion , neutralino and chargino mass matrices given below are complex by definition . the one - loop effective higgs potential receives further contributions from @xmath34 and @xmath67 through the squark and stau sectors and from @xmath34 and @xmath35 through the neutralino and chargino sectors . * the neutral higgs boson mass matrix may be written as : + @xmath109 + using the minimization conditions of the higgs potential , one can define some convenient parameters , @xmath110 with @xmath111 + in terms of these parameters , the entries of the top left @xmath112 cp - even block in eq.([eq : mhiggs ] ) are given as @xmath113 + where @xmath114 and @xmath115 are the @xmath116 and @xmath117 gauge couplings , respectively , and the bottom right @xmath118 cp - odd block in reads @xmath119 + finally , the entries of the off - diagonal block in eq.([eq : mhiggs ] ) , which are responsible for mixing between cp - even and cp - odd states , are given as @xmath120 * the chargino mass matrix , in the @xmath121 basis , using the convention @xmath122 , can be written as @xmath123 \sqrt{2 } m_w \sin\beta & \frac{|\lambda| v_s}{\sqrt{2}}\,e^{i{\ensuremath{\phi_\lambda } } ' } \end{array}\right)\ , , \end{aligned}\ ] ] where @xmath124 and @xmath125 are the soft gaugino masses and @xmath126 is the mass of the @xmath26 boson . the above matrix is diagonalized by two different unitary matrices as @xmath127 , where @xmath128 . * the neutralino mass matrix , in the @xmath129 basis , can be written as @xmath130 & m_2 & m_z \cos\beta c_w & -m_z \sin\beta c_w & 0\\[2 mm ] & & 0 & -\frac{|\lambda| v_s}{\sqrt{2}}\,e^{i{\ensuremath{\phi_\lambda } } ' } & -\frac{|\lambda| v s_\beta}{\sqrt{2}}\,e^{i{\ensuremath{\phi_\lambda } } ' } \\[2 mm ] & & & 0 & -\frac{|\lambda| v \cos\beta}{\sqrt{2}}\,e^{i{\ensuremath{\phi_\lambda}}'}\\[2 mm ] & & & & \sqrt{2 } |\kappa| v_s \,e^{i{\ensuremath{\phi_\kappa } } ' } \end{array}\right).\,\end{aligned}\ ] ] with @xmath131 being the @xmath132 boson mass , @xmath133 and @xmath134 . this matrix is diagonalized as @xmath135 , where @xmath136 is a unitary matrix and @xmath137 . * for the stop , sbottom and stau matrices , in the @xmath138 basis , we have + @xmath139 + @xmath140 + @xmath141 + where @xmath142 , @xmath143 and @xmath144 are the masses of @xmath145 quarks and @xmath146 lepton , respectively , and @xmath147 , @xmath148 and @xmath149 are the corresponding yukawa couplings . @xmath150 and @xmath151 are the respective electric charges of the @xmath152 and @xmath61 quarks . the mass eigenstates of top and bottom squarks and stau are obtained by diagonalizing the above mass matrices as @xmath153 , such that @xmath154 , for @xmath155 . atlas collaboration , phys . b * 716 * , 1 ( 2012 ) . cms collaboration , phys . b * 716 * , 30 ( 2012 ) . cms collaboration , cms - 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while the properties of the 125 gev higgs boson - like particle observed by the atlas and cms collaborations are largely compatible with those predicted for the standard model state , significant deviations are present in some cases . we , therefore , test the viability of a beyond the standard model scenario based on supersymmetry , the cp - violating next - to - minimal supersymmetric standard model , against the corresponding experimental observations . namely , we identify possible model configurations in which one of its higgs bosons is consistent with the lhc observation and evaluate the role of the explicit complex phases in both the mass and di - photon decay of such a higgs boson . through a detailed analysis of some benchmark points corresponding to each of these configurations , we highlight the impact of the cp - violating phases on the model predictions compared to the cp - conserving case . shep-13 - 07 + april 2013 * 125 gev higgs boson signal within the complex nmssm * + + _ @xmath0 school of physics & astronomy , + university of southampton , southampton so17 1bj , uk . _ + _ @xmath1 national centre for nuclear research , ho . za 69 , 00 - 681 warsaw , poland . _ + _ @xmath2 department of physics , + iit guwahati , assam 781039 , india . _ + @xmath3e - mails : + , + smunir@fuw.edu.pl , + poulose@iitg.ernet.in .

1305.0166

one of the most important problems in mathematics is the proof of the riemann hypothesis ( rh ) which states that the non trivial zeros of the classical zeta function all have real part equal to 1/2 @xcite . the importance of this conjecture lies in its connection with the prime numbers . if the rh is true then the statistical distribution of the primes will be constrained in the most favorable way . according to michael berry the truth of the rh would mean that `` there is music in the primes '' @xcite . otherwise , in the words of bombieri , the failure of the rh would create havoc in the distribution of the prime numbers @xcite . in so far , the proof of the rh has resisted the attempts of many and most prominent mathematicians and physicist for more than a century , which explains in part its popularity @xcite . for these and other reasons the rh stands as one of the most fundamental problems in mathematics for the xxi century with possible implications in physics . in fact , physical ideas and techniques could probably be essential for a proof of the rh @xcite . this suggestion goes back to polya and hilbert which , according to the standard lore , proposed that the imaginary part of the non trivial riemann zeros are the eigenvalues of a self - adjoint operator @xmath5 and hence real numbers . in the language of quantum mechanics the operator @xmath5 would be nothing but a hamiltonian whose spectrum contains the riemann zeros . the polya - hilbert conjecture was for a long time regarded as a wild speculation until the works of selberg in the 50 s and those of montgomery in the 70 s . selberg found a remarkable duality between the length of geodesics on a riemann surface and the eigenvalues of the laplacian operator defined on it @xcite . this duality is encapsulated in the so called selberg trace formula , which has a strong similarity with the riemann explicit formula relating the zeros and the prime numbers . the riemann zeros would correspond to the eigenvalues , and the primes to the geodesics . this classical versus quantum version of the primes and the zeros was also at the heart of the so called quantum chaos approach to the rh ( see later ) . quite independently of the selberg work , montgomery showed that the riemann zeros are distributed randomly and obeying locally the statistical law of the random matrix models ( rmm ) @xcite . the rmm were originally proposed to explain the chaotic behaviour of the spectra of nuclei but it has applications in another branches of physics , specially in condensed matter @xcite . there are several universality classes of random matrices , and it turns out that the one related to the riemann zeros is the gaussian unitary ensemble ( gue ) associated to random hermitean matrices . montgomery analytical results found an impressive numerical confirmation in the works of odlyzko in the 80 s , so that the gue law , as applied to the riemann zeros is nowadays called the montgomery - odlyzko law @xcite . it is worth to mention that the prime numbers , unlike the riemann zeros , are distributed almost at random over the set of integers . indeed , it is believed that one can find arbitrary pairs of nearby odd numbers @xmath6 , as well as pairs arbitrarily separated . the only thing known about the distribution of the primes is the gauss law according to which the n@xmath7 prime @xmath8 behaves asymptotically as @xmath9 @xcite . this statement is called the prime number theorem proved by hadamard and de la valle - poussin in 1896 . if the rh is true then the deviation from the gauss law is of order @xmath10 . the analogue of the gauss law for the imaginary part of the riemann zeros ( called it @xmath11 ) is given by the riemann law where the n@xmath7-zero behaves as @xmath12 . hence , large prime numbers are progressively scarced , while large riemann zeros abound . an important hint suggested by the montgomery - odlyzko law is that the polya - hilbert hamiltonian @xmath5 must break the time reversal symmetry . the reason being that the gue statistics describes random hamiltonians where this symmetry is broken . a simple example is provided by materials with impurities subject to an external magnetic field , as in the quantum hall effect . a further step in the polya - hilbert - montgomery - odlyzko pathway was taken by berry @xcite . who noticed a similarity between the formula yielding the fluctuations of the number of zeros , around its average position @xmath13 , and a formula giving the fluctuations of the energy levels of a hamiltonian obtained by the quantization of a classical chaotic system @xcite . the comparison between these two formulas suggests that the prime numbers @xmath14 correspond to the isolated periodic orbits whose period is @xmath15 . in the quantum chaos scenario the prime numbers appear as classical objects , while the riemann zeros are quantal . this classical / quantum interpretation of the primes / zeros is certainly reminiscent of the one underlying the selberg trace formula mentioned earlier . one success of the quantum chaos approach is that it explains the deviations from the gue law of the zeros found numerically by odlyzko . the similarity between the fluctuation formulas described above , while rather appealing , has a serious drawback observed by connes which has to do with an overall sign difference between them @xcite . it is as if the periodic orbits were missing in the underlying classical chaotic dynamics , a fact that is difficult to understand physically . this and other observations lead connes to propose a rather abstract approach to the rh based on discrete mathematical objects known as adeles @xcite . the final outcome of connes work is a trace formula whose proof , not yet found , amounts to that of a generalized version of the rh . in connes approach there is an operator , which plays the role of the hamiltonian , whose spectrum is a continuum with missing spectral lines corresponding to the riemann zeros . we are thus confronted with two possible physical realizations of the riemann zeros , either as point like spectra or as missing spectra in a continuum . later on we shall see that both pictures can be reconciled in a qm model having a discrete spectra embedded in a continuum . in 1999 berry and keating on one hand @xcite , and connes on the other @xcite , proposed that the classical hamiltonian @xmath16 , where @xmath17 and @xmath14 are the position and momenta of a 1d particle , is closely related to the riemann zeros . the classical trayectories of the particle are hyperbolas in the phase space @xmath18 , hence one should not expect a discrete spectrum even at the semiclassical level ( see fig . [ semiclassical ] . to overcome this difficulty , berry and keating proposed to restrict the phase space to those points where @xmath19 and @xmath20 , with @xmath21 . the number of semiclassical states , @xmath22 , with energy between 0 and @xmath11 is given by the allowed area in phase space divided by @xmath23 ( see fig . [ semiclassical ] ) \(e ) = ( -1 ) + 1 [ 1 ] eq.([1 ] ) coincides asympotically with the smooth part of the formula that gives the number of riemann zeros in the same interval . this result is really striking given the simplicity of the hamiltonian and the asumptions made . on the other hand connes started on the same classical hamiltonian @xmath24 but constrained the phase space to those trayectories satisfying @xmath25 , with @xmath26 a cutoff which is sent to infinity at the end of the calculation . the number of semiclassical states is given by \(e ) = - ( -1 ) [ 2 ] the first term describes a continuum of states in the limit @xmath27 , while the second term coincides with minus the average position of the riemann zeros ( [ 1 ] ) . this result lead connes to propose the missing spectral interpretation of the riemann zeros described earlier . a third possible regularization of the xp model , proposed in references @xcite , is that @xmath28 , which leads to the following counting of semiclassical states , \(e ) = [ 3 ] this result agrees with the asymptotic part of ( [ 2 ] ) , meaning that there is a continuum spectrum but the possible connection with the riemann zeros is lost . the main advantage of the latter regularization is that it arises from a consistent quantization of @xmath0 unlike the two previous semiclassical regularizations . the classical hamiltonian @xmath0 can be consistently quantized in two cases depending on the choice of the domain in the @xmath17 coordinate : 1 ) @xmath31 and 2 ) @xmath28 @xcite . in the first case @xmath5 is essentially self - adjoint , while in the second it admits a one parameter self - adjoint extension . we shall consider the latter case . to do so one first define the normal ordered operator @xcite h_0 = ( x p + p x ) = -i ( + ) [ 4 ] where @xmath32 . the formal eigenfunctions of ( [ 4 ] ) are _ e(x ) = , 1 < x < n [ 5 ] where we have normalized @xmath33 and @xmath34 . one can show that ( [ 4 ] ) is self - adjoint if the wave functions satisfy the boundary condition e^i _ e(1 ) = n^1/2 _ e(n ) [ 6 ] where the angle @xmath35 parameterizes the self - adjoint extension of @xmath5 . imposing ( [ 6 ] ) on ( [ 5 ] ) yields n^i e = e^ i [ 7 ] which determines the eigenvalues of @xmath5 e_n = ( n + ) , n = 0 , 1 , [ 8 ] this equation agrees with the semiclassiclassical result ( [ 3 ] ) . in the particular case where @xmath36 , the spectrum ( [ 8 ] ) becomes symmetric around 0 . notice that the zero eigenvalue is excluded . another way to derive this result is by considering the inverse of the operator ( [ 4 ] ) . this can be done as follows , h_0 = ( x p + p x ) = x^ p x^ h^-1 = x^- p^-1 x^- [ 9 ] where @xmath37 is well defined since @xmath38 . the inverse of the momenta operator @xmath14 is given by the 1d green function p^-1 = , [ 10 ] where @xmath39 is the sign function . the schroedinger equation associated to ( [ 9 ] ) is _ e(x ) = _ e(x ) . [ 11 ] which in terms of the new wave function _ e(x ) = x^- _ e(x ) [ 12 ] becomes x _ e(x ) - _ 1^n dx ( x - x ) _ e(x ) = 0 [ 13 ] whose solution is n^i e = -1 , ( x ) = [ 14 ] hence we recover eqs.([5 ] ) and ( [ 7 ] ) in the particular case where @xmath36 . on the other hand , eq.([13 ] ) looks as the eigenvalue equation of yet another hamiltonian that we shall discuss next . in reference @xcite it was defined an extension of the bcs model of superconductivity , called the russian doll model , whose hamiltonian , when restricted to the one body case becomes h_rd(x , x ) = ( x ) ( x - x ) - ( g + i h ( x - x ) ) [ 15 ] where @xmath40 represents the energies of pairs of electrons occupying time reversed states in the band @xmath41 , @xmath42 is the standard bcs coupling constant and @xmath43 a coupling that breaks the time reversal symmetry . the eigentates and eigenfunctions of ( [ 15 ] ) are given by ( ) ^i h = , ( x ) = [ 16 ] comparing ( [ 16 ] ) with ( [ 14 ] ) , one obtains the following map between the eigenstates of the xp model and the rd model & e_rd = 0 e 0 & [ 17 ] + & h e & + & ( /2 ) & + & x^-1/2 _ e & in the particular case where @xmath44 and then @xmath36 . one can add a @xmath42 coupling term in the definition ( [ 9 ] ) of the operator @xmath45 , in which case the correspondence between the rd model and the xp model will cover all the self - adjoint extensions of @xmath46 . the rd model provides an example where the renormalization group , instead of ending at fixed points , run in cycles @xcite . in the rd model , the coupling that runs periodically under the rg is @xmath42 , with a period equal to @xmath47 , while the coupling @xmath43 remains invariant . this fact in turn implies the existence of several bound states whose number is given by @xmath48 . if we replace @xmath43 by @xmath11 , the latter equation becomes @xmath49 which coincides with the number of eigentates of the xp model . incidentally , we would like to mention that the field theory realizations of the cyclic renormalization group , of references @xcite , are at the origin of leclair s approach to the rh @xcite . in this reference the zeta function on the critical strip is related to the quantum statistical mechanics of non - relativistic , interacting fermionic gases in 1d with a quasi - periodic two - body potential . this quasi - periodicity is reminiscent of the zero temperature cyclic rg of the quantum mechanical hamiltonian of @xcite , but the general framework of both works is different . the previous results establish an interesting correspondence between two apparently different models which also suggests a way to add interactions to the xp model . indeed , the interacting term of the rd hamiltonian ( [ 15 ] ) that is proportional to the coupling constant @xmath42 is basically a proyector operator @xmath51 , with wave function @xmath52 @xmath53 . as we said above , adding that term to the inverse hamiltonian ( [ 9 ] ) would give rise to a @xmath54 term associated to the self - adjoint extensions of xp . instead we want to add an interacting term that reflects the existence of two boundaries in the bk regularization of the xp model . the simplest possibility is to define h^-1 = h_0 ^ -1 + ( |_a _ b | - |_b _ a | ) [ 18 ] where @xmath55 are two wave functions whose properties will be specified below . the matrix elements of ( [ 18 ] ) read h^-1(x , x ) = , 1 < x , x < n [ 19 ] where _ a(x ) = , _ b(x ) = [ 20 ] are real functions , which guarantee that @xmath45 is a hermitean and antisymmetric matrix , so that its eigenvalues are pairs of real numbers @xmath56 . a simplified version of ( [ 19 ] ) is obtained by choosing @xmath57 . the latter model will be denoted as type i , while the former as type ii . for these models to be well defined in the limit @xmath58 we impose the following normalization conditions f = \ { ll a & type + a , b & type + . a nice feature of the hamiltonian ( [ 19 ] ) is that the scroedinger equation is exactly solvable , the reason being that it is equivalent to a first order differential equation supplemented with a boundary condition . we shall next present the results obtained in reference @xcite . first of all , the eigenenergies @xmath11 of ( [ 19 ] ) satisfy the equation _ n(e ) + _ n(-e ) n^i e = 0 [ 22 ] where @xmath59 is a jost like function whose expression will given below in the limit @xmath58 . in that limit the eigenfunctions of the type ii model are given by _ e(x ) = [ 23 ] where @xmath60 are integration constants that depend on @xmath11 . in the limit @xmath61 the functions @xmath62 vanish sufficiently fast so that the wave function ( [ 23 ] ) is dominated by the first term , i.e. _ x > > 1 _ e(x ) ~ [ 24 ] it turns out that @xmath63 is given by the jost function ( up to a constant which can be taken as 1 ) , c_(e ) = ( e ) [ 25 ] hencefore the energies where @xmath64 is non zero correspond to delocalized states which behave asymptotically as the eigenfunctions of the unperturbed hamiltonian @xmath65 . for these states the ratio @xmath66 gives the scattering phase shift . on the other hand , @xmath63 vanishes whenever @xmath64 does . these energies correspond to localized states with a finite norm . in summary , the spectrum of @xmath5 consists of a continuum formed by those energies where @xmath67 , plus a discrete part given by the real zeros of @xmath64 ( see fig . [ bound - states ] ) . moreover , using the hermiticity of @xmath5 one can show from eq.([21 ] ) that @xmath68 does not have zeros with @xmath69 . \(e ) = 0 e 0 [ 26 ] the real zeros of @xmath70 correspond , as explained above , to localized states , while the complex zeros below the real axis correspond to resonances . these results are summarized in table 1 . [ cols="^,^,^,^",options="header " , ] table 1.- classification of eigenstates of the model . before giving the expression of @xmath64 for the type i and type ii model we shall introduce some definitions . first of all let us define the mellin transform of @xmath71 ( similarly for @xmath72 ) . \(t ) = _ 1^dx x^-1 + i t a(x ) , a(x ) = _ - ^ x^i t ( t ) [ 27 ] the reality of @xmath73 implies ^*(t ) = ( -t ) , t [ 28 ] condition @xmath74 in eq.([21 ] ) amounts to while condition @xmath75 in eq.([21 ] ) is equivalent to _ -^ | ( t ) |^2 < , [ 30 ] the function @xmath76 is in fact the fourier transform of @xmath73 in the variable @xmath77 , which takes values in the interval @xmath78 . in terms of @xmath79 , @xmath73 is a square normalizable and causal function , in which case @xmath76 has interesting analytic properties by a theorem due to titchmarsh @xcite . this theorem states that under the previous conditions @xmath76 is analytic in the complex upper - half plane and satisfies the formula \(z ) = p _ - ^ , z [ 31 ] where p denotes the cauchy principal value of the integral . to prove this formula one uses the fact that @xmath76 has no singularities in the upper - half plane and that the contour on integration on the circle @xmath80 vanishes since @xmath81 . for later purposes let us define the new function \(t ) = ( t ) , [ 32 ] and similarly @xmath82 , whose properties follow from those of @xmath76 namely : * reality : + ^*(t ) = ( -t ) , t [ 33 ] * regularity : + _ t 0 < |(0)| = 0 [ 34 ] * normalizability + _ -^ < , [ 35 ] * analiticity + \(z ) = p _ - ^ - p _ - ^ , z [ 36 ] + this eq . implies that @xmath83 is an analytic function in the upper - half plane which converges towards a constant value in a circle of infinite radius given by the last term of ( [ 36 ] ) . the next definition involves the product of two analytic functions @xmath84 and @xmath85 in the upper half - plane : ( ) ( z ) = ( z ) ( -z ) + _ - ^ , z [ 37 ] where the integration is understood in the cauchy sense as in eqs.([31 ] ) and ( [ 36 ] ) . one can show that @xmath86 is an analytic function in the upper half - plane provided @xmath87 is well behaved , which seems to be the case in all the examples we have analized . the analytic extension of @xmath86 to the lower half - plane will have in general singularities . in terms of this product we shall define the function @xcite : s_,(z ) = ( ) ( z ) - ( ) ( 0 ) [ 38 ] which satisfies the following conditions * reality : if @xmath88 and @xmath89 verify ( [ 33 ] ) then + s_,^*(z ) = s_,(-z ) , z [ 39 ] * regularity + s_,(0 ) = 1 [ 40 ] * shuffle relation + s_,(z ) + s_,(-z ) = 2 ( z ) ( -z ) [ 41 ] + the notation `` shuffle '' is borrowed from the theory of multiple zeta functions , as explained later on . after these definitions we can finally give the expression of the jost function @xmath64 in terms of the potentials @xmath90 and @xmath82 . for the type i model it reads \(t ) = 1 + 2 ( t ) + s _ , ( t ) [ 42 ] while for the type ii model it is \(t ) = 1 - s _ , ( t ) + s _ , ( t ) + s _ , ( t ) s _ , ( t ) - s _ , ( t ) s _ , ( t ) [ 43 ] from the properties of the @xmath91-functions one can easily derived : * reality : + ^*(t ) = ( -t ) z [ 44 ] + this condition guarantees that the ratio @xmath66 appearing in the eigenvalue eq.([22 ] ) is indeed a phase . * regularity + \(0 ) = 1 [ 45 ] + this condition implies that the zero energy is not an eigenvalue of the hamiltonian @xmath5 , which was the assumptions made by defining it in terms of its inverse . let us next consider the two models separately . the jost function ( [ 42 ] ) can be expressed as \(t ) = 1 + 2 ( t ) + ( ) ( t ) [ 46 ] where we used that @xmath92 , which implies ( ) ( 0 ) = 0 . [ 47 ] an important consequence of eq.([46 ] ) is the positivity of the real part of @xmath93 , \(t ) = |1 + ( t)|^2 0 [ 48 ] which imposes a strong contraint on the functions allowing a qm interpretation as jost functions of the type i model . in particular eq.([48 ] ) excludes the zeta function @xmath1 for @xmath94 , but not the case where @xmath2 as we shall see later on . the @xmath95-product defined in eq.([37 ] ) is non commutative and non associative . nevertheless it behaves nicely respect to the identity function @xmath96 , namely 1 1 = 1 [ 49 ] and 1 + 1 = 2 [ 50 ] where @xmath97 is an analytic function in the upper half - plane satisfying eq.([36 ] ) . using these two equations , the jost function ( [ 46 ] ) can be expressed as the _ square _ , with respect to the @xmath95-product , of a single function , i.e. = , = 1 + [ 51 ] so that @xmath88 is the @xmath95-_square root _ , of @xmath98 . using eq.([51 ] ) one can easily prove that @xmath99 does not have zeros in the upper half - plane . indeed , write @xmath100 as ( ) ( z ) = _ - ^ , z > 0 [ 52 ] then ( ) ( z ) = _ - ^ > 0 , z = x + i y [ 53 ] a simple but illustrative example of the theory developped so far is provided by the step potential , a(x ) = \ { lcl a_1 , & & 1 < x < x_1 + 0 , & & x_1 < x < + . [ 54 ] which yields = ( x_1^i t -1 ) , = ( 1- x_1^i e ) , [ 55 ] and the jost function \(t ) = 1 + ( x_1^ie -1 ) . [ 56 ] fig . [ step1 ] shows an argand plane representation of the real and imaginary parts of ( [ 56 ] ) for several values of @xmath101 . for @xmath102 the function @xmath99 vanishes at the values e_n = , n = 0 , 1 , [ 57 ] describing an infinite number of bound states . the remaining values of @xmath11 correspond to delocalized states . all the zeros of @xmath64 lie on the real axis for @xmath102 and below it for @xmath103 . the next problem we address is : given a function @xmath99 , satisfying the analyticity , reality , regularity and positivity conditions described above , which is the potential , or potentials , @xmath90 , verifying eq.([46 ] ) ? in this paragraph we shall give a perturbative method to construct one of those potential in terms of a series which converges under certain conditions placed on the function @xmath99 . let us first make the change @xmath104 in eq.([46 ] ) which becomes = + , = [ 58 ] iterating ( [ 58 ] ) generates the series expansion = + + ( ) + ( ) + [ 59 ] at order @xmath105 one gets all admissible bracketings , whose number is given by @xmath106 , where @xmath107 is the catalan number c_n = ( l 2 n + n ) [ 60 ] whose asymptotic behaviour is also described . to investigate the conditions for convergence of ( [ 59 ] ) , let us suppose that @xmath89 is given by the absolute convergent series \(t ) = _ n=1^g_n n^i t , |(t)| _ n=1^|g_n| < [ 61 ] using ( [ 37 ] ) one finds ( ) ( t ) = _ n=1^g_n^2 + 2 _ n > m g_n g_m ( n / m)^i t [ 62 ] so that similarly one finds that which , according to ( [ 60 ] ) , converges provided _ m=1^|g_m| [ 65 ] this condition is sufficient for convergence of the series ( [ 59 ] ) but it is not necessary . eq.([65 ] ) implies however the converse is not true ( i.e. eq.([66 ] ) does not imply ( [ 65 ] ) ) . we believe that ( [ 66 ] ) also guarantees the convergence of ( [ 59 ] ) , but this guess needs to be proved . observe also that ( [ 66 ] ) implies that @xmath108 , which is also a necessary condition for the existence of the potential @xmath90 . an application of these results is : for @xmath2 let us consider the function \(t ) = c ( - i t ) , c = [ 67 ] where the constant @xmath109 guarantees the normalization condition @xmath110 . the values of the constants @xmath111 appearing in ( [ 61 ] ) are given by g_1 = , g_n>1 = - [ 68 ] and the convergence criteria ( [ 65 ] ) yields _ m=1^|g_m| = ( ) 2 [ 69 ] where the latter conditions is satisfied if _ c = 1.72865 , ( _ c ) = 2 [ 70 ] moreover condition ( [ 66 ] ) can be checked numerically i.e. so we expect that the series ( [ 59 ] ) will also converge for any value @xmath2 [ re - im - s2 ] displays the real and imaginary parts of @xmath90 for the case @xmath112 obtained by the sum of the first terms of eq.([59 ] ) . the convergence towards a finite value is clear . the series ( [ 59 ] ) contains also an interesting analytical structure , which can be seen from the star product of two zeta functions , ( - i t ) = _ n=1^ ( ) ( t ) = ( 2 ) + 2 _ m 1^ [ 71 - 1 ] the double sum series of the rhs is equal to a euler - zagier zeta function for two variables which is a generalization of the zeta function . multivariable versions of this function have attracted much attention in various fields , as knot theory , perturbative quantum field theory , etc ( see @xcite , @xcite and references therein ) . the results obtained so far suggests that the riemann zeta function on the critical line @xmath113 could perhaps be realized as the jost function of the model . this idea is motivated by the scattering approach pionered by faddeev and pavlov in 1975 , and has been followed by many authors @xcite . an important result is that the phase of @xmath114 is related to the scattering phase shift of a particle moving on a surface with constant negative curvature . the chaotic nature of that phase is a well known feature . along this line of thoughts , bhaduri , khare and law ( bkl ) maded in 1994 an analogy between resonant quantum scattering amplitudes and the argand diagram of the zeta function @xmath115 , where the real part of @xmath116 ( along the @xmath17-axis ) is plotted against the imaginary part ( @xmath117-axis ) @xcite . the diagram consists of an infinite series of closed loops passing through the origin every time @xmath115 vanishes ( see fig . [ zeta1 ] ) . this loop structure is similar to the argand plots of partial wave amplitudes of some physical models with the two axis being interchanged . however the analogy is flawed since the real part of @xmath118 is negative in small regions of @xmath119 , a circumstance which never occurs in those physical systems . in fact , the loop structure of the models proposed by bkl is identical , up to a scale factor of 2 , to the model with the potential ( [ 54 ] ) ( see fig . ( [ step1 ] ) ) , where the loops representing @xmath64 , for @xmath102 , are circles of radius 1/2 , centered at @xmath120 . for general models of type i , the loops are not circles but the real part of @xmath121 is always positive ( see eq.([48 ] ) ) , and therefore they can never represent @xmath113 . incidentally , this constraint does not apply to the models of type ii , where @xmath122 may become negative . this suggests that @xmath113 could indeed be the jost function @xmath64 of a type ii model for a particular choice of @xmath71 and @xmath123 . if this were the case then the riemann zeros would become eigenenergies of the hamiltonian realizing in that manner the polya and hilbert conjecture which may also give hints into the solution of the riemann hypothesis . a complete answer to this problem is not yet known but we shall present below some encouraging results along this direction . the first step is to recover quantum mechanically the smooth approximation to the riemann zeros . this approximation is equivalent to the following condition 1 + e^ 2 i ( e ) = 0 [ 72 ] where e^ 2 i ( e ) = ^-i e [ 73 ] the function @xmath124 gives the phase of the zeta function , e.g. ( 1/2 - i t ) = z(t ) e^i ( t ) [ 74 ] while @xmath125 is the riemann - siegel zeta function which is real and even due to the duality relation satisfied by the zeta function ( e.g. @xmath126 ) . the reason that ( [ 72 ] ) is a good approximation to the location of the zeros , can be seen in fig . [ zeta1 ] which plots the real and imaginary parts of @xmath113 . observe that in the vicinity of a zero , the curves cut the imaginary axis where @xmath127 so that @xmath128 , which is nothing but eq.([72 ] ) . the value of @xmath129 satisfies n = + [ 75 ] which asymptotically coincides with eq . ( [ 1 ] ) up to a constant term . this result can be obtained choosing the following potential in the type i model a(x ) = - 2 [ 76 ] whose mellin transform ( [ 27 ] ) gives \(t ) = - e^2 i ( t ) + o(1/t ) [ 77 ] up to @xmath130 terms . the corresponding jost function is \(e ) = 2 ( 1 + e^2 i ( e ) ) + o(1/e ) [ 78 ] whose zeros agrees with ( [ 72 ] ) asymptotically . hence the smooth approximation to the riemann zeros can be obtained asymptotically by the potential ( [ 76 ] ) . why this potential is able to yield this result ? one may suspect that it must implement at the quantum level the bk boundary conditions of the semiclassical approach . indeed , let us show how this works in detail . the potential ( [ 76 ] ) corresponds to the wave function _ a(x ) = - 2 [ 79 ] which satisfies the equation h_0 ^ 2 _ a ( x ) = ( ( 2 x ) ^2 - ) _ a(x ) , [ 80 ] where @xmath131 is the hamiltonian ( [ 9 ] ) . dropping @xmath132 in both sides and replacing @xmath133 by @xmath134 , one obtains a classical version of ( [ 80 ] ) , ( x p ) ^2 = ( 2 x)^2 - p = 2 , [ 81 ] which describes a curve in phase space that approaches asymptotically the lines @xmath135 . we shall identify these asymptotes with the bk boundary in the momenta @xmath136 . recall on the other hand the boundary condition @xmath137 , which combined with the previous identification reproduces the planck cell quantization condition , l_p = 2 , l_x = 1 l_p l_x = 2 [ 82 ] the bk condition @xmath138 has already built in into the model and this is also reflected in the state @xmath139 which is concentrated near the position @xmath140 . in the model we have proposed , the two bk boundary conditions are realized asymmetrically , as opposed to the semiclassical model . it would be desirable to have a more symmetric treament of them . this indeed can be done and the results will be presented elsewhere . the next natural step is to see wether the zeta function @xmath113 can be realized as the jost function of the model . the analyticity properties of the jost functions implies that @xmath70 must be of the form \(e ) = c ( 1/2 - i e ) [ 83 ] which does not have poles in the upper - half plane ( @xmath109 and @xmath141 are normalization constants ) . in fig . [ zeta1 ] we plot an example with @xmath142 . since the real part of @xmath113 , and thus of @xmath70 , become negative in small regions of @xmath11 one is forced to consider the type ii model with two non trivial functions @xmath132 and @xmath143 . the problem of finding @xmath55 is rather non trivial . one can in principle fix one of them , say @xmath143 , and try to solve for @xmath132 as a function of the jost function ( [ 83 ] ) and @xmath143 . in the case of the zeta function @xmath144 , with @xmath2 , we were able to solve this problem perturbatively thanks to the fact that the zeta function is bounded . however for @xmath145 the zeta function is unbounded which may lead to problems of convergence . in any case , it seems clear that one needs further physical insights to make progress into this difficult problem . as we said above , one needs a more symmetric treatment of the coordinates @xmath17 and @xmath14 , and a clearer physical interpretation of the wave functions @xmath55 . another important ingredient to be implemented is the duality symmetry of the zeta function which in the present formulation of the model is not manifest but which is expected to play a central role . in summary , we have presented in this work an interacting version of the xp hamiltonian which may ultimately lead to a spectral realization of the riemann zeros , as suggested long ago by polya and hilbert . the main ingredient of the model is the non local character of the interaction in terms of two potentials which are the quantum version of the semiclassical phase space constraints of berry and keating . the generic spectrum of the model consists of a continuum of eigenstates in the thermodynamic limit which may also contain bound states embedded in it . we conjecture the existence of potentials giving rise to the riemann zeros as the discrete spectrum embedded in the continuum . if this were the case that would resolve the emission versus absortion spectral interpretation of the riemann zeros . this would also open the way to a better understanding of the riemann hypothesis . we have also pointed out the need to implement in an explicit way the duality properties of the zeta function , which implies a more symmetric treatment of the @xmath17 and @xmath14 coordinates as in the semiclassical model . i would like to thanks the organizers of the `` 5th international symposium on quantum theory and symmetries '' held at university of valladolid , spain , and specially mariano del olmo , for the invitation to present the results of the present work . i wish also to thank andre leclair for the many discussions we had on our joint work on the russian doll renormalization group and its relation to the riemann hypothesis . i also thank m. asorey , m. berry , l.j . boya , j. garca - esteve , j. keating , m.a . martn - delgado , g. mussardo , j. rodrguez - laguna and j. links for our conversations . this work was supported by the cicyt of spain under the contract fis2004 - 04885 . i also acknowledge esf science programme instans 2005 - 2010 . s. d. glazek and k. g. wilson , `` limit cycles in quantum theories '' , phys . * 89 * ( 2002 ) 230401 , hep - th/0203088 ; `` universality , marginal operators , and limit cycles '' , phys . rev . * b69 * , 094304 ( 2004 ) ; cond - mat/0303297 .

we discuss a possible spectral realization of the riemann zeros based on the hamiltonian @xmath0 perturbed by a term that depends on two potentials , which are related to the berry - keating semiclassical constraints . we find perturbatively the potentials whose jost function is given by the zeta function @xmath1 for @xmath2 . for @xmath3 we find the potentials that yield the smooth approximation to the zeros . we show that the existence of potentials realizing the zeta function at @xmath3 , as a jost function , would imply that the riemann zeros are point like spectrum embedded in the continuum , resolving in that way the emission / spectral interpretation . = cmss12 = cmu10 scaled1 h 0.2 cm \def\ ] ] ] @xmath4 ) # 1#2#1 # 2 # 11 # 1 # 1#1 # 1#2#1 # 2 # 1#1 # 1 | # 1 pi # 1e^^#1 # 1_#1 # 1#2 # 1 , # 2 pi # 1e^^#1 # 1_#1 = cmss12 = cmu10 scaled1 l

0711.1063

the _ kepler _ mission provides an unprecedented set of data on the photometric variability of stars @xcite . it is unparalleled in its combination of photometric precision and time coverage . it observes a large sample of stars , which consist primarily of main sequence stars in the solar neighborhood . although the primary mission is to discover exoplanets , clearly the stellar lightcurves are themselves a rich source of astrophysical information . stars can vary their luminosity for a variety of reasons . our orientation in this paper is to concentrate on stellar magnetic activity ( primarily seen through starspot modulation ) . questions of interest in this area include the stellar rotation periods , the differential rotation as a function of latitude , the distribution of spot areal coverages , the distributions in longitude and latitude on different stars , the presence and distribution of active longitudes , the timescale for evolution of spots of different sizes , spot contrasts , and the behavior of all these as a function of stellar mass , age , and rotation . of particular interest to the mission is the translation of photometric periods into stellar ages for planet - bearing stars . this paper is not intended to provide answers to any of the above questions , but rather to present the characteristics of the _ kepler _ dataset and its potential for making progress on these questions . starspots have long been studied with either photometry or spectroscopy @xcite . ground - based coverage , however , tends to have many gaps , and the spot areal coverage has to be many times solar for a signal to be robust . the promise of solving these deficiencies was well - illustrated first by the most @xcite and corot @xcite missions , but with less precision and length of coverage than will be available over the _ kepler _ mission . we have discussed the general level of variability in the first quarter of _ kepler _ data in our first paper @xcite ; hereafter paper i ) , where there is also a lengthier introduction to stellar variability in the context we are interested in here . the _ kepler _ quarter 1 ( q1 ) observations took place over @xmath133.5 days between may 13 and june 15 2009 . this paper refers to data gathered in q1 , which was almost entirely released to the public on june 15 , 2010 . we use only the so - called long cadence data which is formed on - board the spacecraft by summing 270 6 second integrations into successive 29.42 minute blocks ( see @xcite for additional details ) . q1 consists of 1639 such intervals from 13 may 2009 to 15 june 2009 , which provide continuous coverage . 156,097 stars were observed , almost all of which belong to the sample of stars that are identified as the core exoplanet target list . the construction of this target list has been described by @xcite . as described in the release notes , the data product which is relevant to this paper is the so - called raw data ( identified by the keyword ap_raw_flux in the fits files ) . this is not really raw pixel data - it has been added up in each target aperture , corrected for background , flat fielding , cosmic ray removal , non - linearities , and there is some treatment of selected instrumental effects and data gaps . what remains in the raw photometry are other instrumental artifacts @xcite , secular instrumental drifts @xcite , and intrinsic astrophysical variability . subsequent steps in the _ kepler _ pipeline attempt to render the data more suitable for the primary mission : the detection of transits . currently the methodology employed @xcite is not suitable for a study of stellar variability ; one can not be sure what components of variability have been removed , and the effects of reduction are different for different amounts of stellar variability . by comparing raw and corrected lightcurves , we determined that only raw lightcurves are suitable as a starting point for work that concentrates in detail on the variability due to stars ( that may change with future versions of the pipeline ) . we elect to use a form of differential photometry , produced by dividing each lightcurve by its median value then subtracting unity . we next investigated whether the secular trends seen in almost all raw lightcurves have an instrumental component that can be easily distinguished from true stellar variability . we made a linear fit to each curve , and examined the slopes as a function of position within a given detector or across the focal plane . it is not surprising to find these slopes ; the stars drift on the pixels due to pointing errors , positioning of the field , differential velocity aberration , and focal plane distortion due to thermal and other effects . the response of each pixel can vary . the psf of the _ kepler _ photometer has been intentionally designed to spread a star over several pixels , and apertures are intentionally larger than a target psf ( both help with photometric precision ) . stars at the edge of each target aperture can contribute more or less light over time depending on the direction of motion . because of the great precision of _ kepler _ photometry , all these effects may be important , depending on the target star and neighboring star field being measured . unfortunately , we conclude that to first order the overall effects on the measured intensity of a star are very local , and one can not use the behavior of nearby stars to reliably remove trends in a given target . of course , the linear trend in our fitting procedure could have a truly stellar component in some cases , and we have to acknowledge that in such cases it will be removed . after removal ( by subtraction ) of linear trends , it is clear that for many quiet stars , there remain curvatures in the lightcurve over the month of q1 ( there is no reason that instrumental secular effects should be purely linear ) . these also are not correlated spatially with each other , but it is clear that there is an instrumental component to them , because it is very common for such lightcurves with a linear fit removed to be high in the middle and low at the ends . with a quadratic fit removed , there was also some ( though not as much ) consistency in the slope of the curves at the ends . in most cases , therefore , we subtracted a third order polynomial from the lightcurve . this resulted in flat lightcurves for many of the quiet stars ; again with the danger that in some cases an astrophysical signal with a timescale of a couple of weeks or longer has been removed . this means that we are likely missing some slower low - amplitude rotators . we also tested the effect of removing a third - order fit from pure sine curves of 1 to 4 cycles , constructed using the same cadence and timespan as the real data . for a single cycle ( a period of 33.5 days ) , the third order fit crosses the sine wave 5 times and remains close to it . after subtraction one is left with a much smaller amplitude , and the periodogram has its maximum peak at about half the original period . the corrected lightcurve is much less affected for a 2-cycle curve ( with an intrinsic period of 16.75 days ) . the amplitude after subtraction of a polynomial fit remains the same ( with a slightly altered shape for the corrected lightcurve ) , and the inferred primary period is closer to 16 rather than 17 days . the effects for more cycles are even smaller . we conclude that our reduction procedure only allows valid discussion of primary periodicities of 16 days or less . obviously , as kepler continues to collect data this issue will be re - visited , along with development of methods for joining data across time intervals when the spacecraft has left and returned to the target field . a caveat to the removal of third order fits must be made in the case when a star is clearly quite variable primarily on a timescale of 10 days or more . that is to say , the corrections on quiet stars were never large in amplitude ( not larger than about 10 times the rms fluctuations on short timescales ) , and essentially all of them were flattened with a third order polynomial . this indicates that the spacecraft does not generally produce signals with higher amplitude and higher frequency than the polynomial . we therefore made an exception to the removal of a third order fit in the following circumstance . after removal of the linear fit , the differential lightcurve can be analyzed for zero crossings ( the number of times it goes from positive to negative or vice versa ) . for a quiet star with a flat lightcurve there will be many of these ( albeit with small excursions from zero ) . for a star with substantial low frequency variability the number of zero crossings gives an indicator of what its timescale of variability is . we used this to avoid removing a third order polynomial in lightcurves first smoothed over 10 hours which then exhibit 10 or fewer zero crossings . this is clearly somewhat arbitrary , and could be refined in the future . it is worth noting that in the sine wave test described above , even the 4-cycle test would not have had a third - order polynomial removed , since it has too few zero crossings . thus we believe that our procedure retains almost all of the variability we are primarily interested in . finally , there are a few cases ( usually less than 5 per 1000 ) where the raw lightcurve contains a sudden large discontinuity . these are removed in the pipeline after the raw stage , but because we can not use that version , we had to remove them ourselves . this was done by filtering all lightcurves with a routine that looked for large abrupt changes ( a version of the routine we also use to look for eclipses and flares ) . this yielded few enough cases that they could be dealt with manually . it is still true , however , the best way to treat them is not always obvious , and we employed three different strategies . in most cases , the discontinuity occurred somewhere in the middle of a not too variable lightcurve . in that case we removed a third order polynomial on each side of the discontinuity ; these tended to match across the gap , but if they did not we forced a match . this procedure removes variability on a shorter timescale than one polynomial fit across the whole lightcurve . in other cases with more variable stars , it was clearly damaging to use the first procedure ( the polynomials removed variations that are clearly stellar ) , so we simply removed the discontinuity by forcing a match across the gap , always adjusting the latter part of the curve to agree with the earlier part . this often resulted in a new residual linear slope , which we removed again . finally , in some cases the discontinuity occurred near one end or the other , and we selected which of the above procedures gave a more plausible solution . fortunately there were rather few discontinuities , although there remain some that are below our chosen detection threshold ; these do not matter in the current analysis . this manual intervention consumes most of the reduction time ; the rest of it ( including making the measurements described in the next section ) can be done in a few hours on a personal computer . clearly all of these corrections have the potential to alter an astrophysical signal . for now we must accept that the photometry is not perfect ( nor will it ever be ) , and look forward to the much longer measurement of each star that _ kepler _ will obtain , which should clarify in many cases what the right thing to do is . the purpose of this paper , in any case , is to produce a large statistical overview of the stellar photometry , rather than to model in detail particular stars . we are confident that the conclusions we draw do not depend too much on the details of the preliminary methodology we have employed in this paper . we are capturing most of the amplitude and characteristics of stellar variability on timescales less than about 2 weeks , and can distinguish between periodic and non - periodic behavior . we can see whether the observed behavior is repetitious over a month , and can detect flares and other short timescale phenomena . once the data were reduced as described in section ii , we proceeded to collect certain measurements on each lightcurve . here we describe only those which proved useful to the aims of this paper . as in paper i , one useful quantity is the variability range v@xmath2 ( sometimes just called range " ) , which is intended to represent the basic level of photometric variability . we slightly changed the means of computing this quantity from paper i , taking the unsmoothed differential lightcurve , sorting it by differential intensity , then computing the range between the 5@xmath3 and 95@xmath3 percentile of intensity ( this tends to avoid including really anomalous excursions up or down ) . we express v@xmath2 in mmag for convenience ( actually the units are differential intensity times 1085.84 , which is nearly correct but not logarithmic like the actual magnitude scale ) . we determined the zero crossings in both the reduced lightcurve itself and when it is smoothed by 10 hours ( 20 points ) , tabulating both the number of crossings and the mean and median separation of crossings . for all segments between crossings ( either above or below zero ) we saved the mean and median of absolute maximum excursions and integrated fluxes . the high frequency noise hf@xmath4 in the lightcurve was calculated by subtracting a four - point boxcar smoothed version of itself from the lightcurve and computing the standard deviation of the result . we also computed a lomb - scargle periodogram for each lightcurve , using 400 points spanning a range of periods between one and 100 days . these parameters were chosen because the procedure uses a logarithmic period spacing that is denser at short periods , so we extend the period range beyond the data length to force sufficient sampling in the periods of interest . we do not make use of results for periods beyond half the data length ( 16 days ) or for periods below 3 days ( to concentrate on solar - type rotating stars ) . this is the most computationally intensive part of our analysis , and we rebinned the lightcurve to 2 hour bins to save time . it then only requires a few hours on a small computer to obtain periodograms for all 156,000 stars . we saved the position , height , and width of all the peaks in them . we collected the height and integrated strength of the highest two peaks , the shortest and longest significant periods , and the number of peaks at least 10% as strong as the strongest one . our first division of the data was made on the basis of the strength of the strongest peak in the periodograms . after extensive manual examination , we found that strengths p@xmath5 of 60 or more showed by eye a clear believable periodicity , those between 35 and 60 were marginal , and those below 35 did not appear to be period in an obvious way . the values of p@xmath5 depend on the ratio of the periodic signal to the noise and on the number of data samples and periods tested . the formal false alarm probability for our case is 1% at p@xmath5 of 13 , and absurdly low for p@xmath5 of 35 , so we are being subjective but conservative in our judgment of what is a significant periodicity . the value of p@xmath5 for a pure sine wave is around 800 . in figure [ f1persample ] we show the kepler input catalog ( kic ) stellar parameters for the group of stars with p@xmath5 above 60 and below 35 ( we will refer to these as the periodic and non - periodic samples ) . there are features in the log(g)-t@xmath6 plane that are due to artifacts in the way the kic was constructed and in target selection criteria for the exoplanet search @xcite . we checked that the distribution of crowding factors ( the fraction of light in target apertures that does nt come from the intended target star ) is very similar for the two samples ; the median crowding factor is 0.135 . there are some clear differences between the two samples . giants are much more populous in the non - periodic sample . as described by @xcite they tend to have complex variability ; periodograms do not reach the level of p@xmath5 we have set as our minimum threshold for significant periodicity ( and exhibit multiple period peaks of comparable strength ) . by contrast , there are many solar - type stars in both the periodic and non - periodic samples . cool dwarfs tend to show up preferentially as periodic . hot stars also show up in both cases ; part of the reason for this is that we have not tested for very short periods . many of the hot pulsators are not flagged in our diagnostics but would be if we extended the search to shorter periods . there are more than 60,000 stars in our periodic sample , and nearly that many in the non - periodic sample . the marginal stars make up the roughly 35,000 remainder . the large number of stars with detected periods means that _ kepler _ will be quite powerful in measuring stellar rotation . we are only considering slightly over a month s data here and more than a year has already been collected . on one hand we should be able to detect spot rotations both longer in period and perhaps lower in amplitude as more data is available . on the other hand substantial issues remain to be resolved about how well stellar variability can be distinguished from instrumental effects , and how to patch across data segments where the stars are on different pixels . figure [ f2tempvarange ] shows v@xmath2 in the two samples as a function of effective temperature . for comparison , v@xmath2 for the active sun is about 1 mmag . the periodic sample has a cloud of points of high variability at all temperatures , and the non - periodic sample shows a two concentrations with lower variability . one is at temperatures corresponding to red giants ( between 4500k@xmath7t@xmath85200k ) and the other corresponds to less variable solar - type stars . most of the higher amplitude objects in the non - periodic sample are hot pulsators with periods less than a day , which are not flagged by us as periodic because we have filtered for longer periods to focus on stellar rotation . another way to look at the two samples is to examine the range of variability v@xmath2 against the high frequency noise hf@xmath4 in the lightcurve due to faintness of the stars . figure [ f3rmsrange ] shows the comparison between them . because of the way they are defined , v@xmath2 tends to be at least 4 times hf@xmath4 in almost all cases . this is the cause of the diagonal line along which many of the stars lie in both samples . these low - lying stars are ones for which the amplitude of variability is not much greater than the noise , although in the periodic sample a low - amplitude periodicity has been detected . conversely , the stars lying well above the line show high - amplitude variability , whether it is periodic or not . many of the giants show up rather clearly in the non - periodic sample at log(v@xmath2 ) between 0.2 to 0.5 , and hf@xmath4 between -1.2 and -0.7 . as a reminder , in the units of v@xmath2 the depth of an earth - sized transit is about -1.1 in the log . the objects which show up below the main diagonal are almost all cases where there are a few deep transit / eclipse dips . they do nt have enough points to generate high periodogram power , and are essentially ignored by the range calculation ( which drops the 5% extremes at each end ) , but have an influence on hf@xmath4 ( pushing it to the right off the main line ) . the periodic sample shows a much more robust cloud of higher amplitudes , at all values of hf@xmath4 . particularly for points above the main diagonal ridge , the lightcurves show obvious periodic variability , and in the upper ranges they almost all appear to be starspots rotating in and out of view ( with a few eclipsing systems with ellipsoidal lightcurves also present ) . in figure [ f4rangehist ] we show the distribution of the ranges for the periodic ( thicker line ) and non - periodic samples . this conveys some of the same information as in fig . [ f3rmsrange ] , but shows more clearly that the periodic sample has many more stars with high - amplitude variability ( as would be expected if starspots were the source of the variability ) . the distributions are much more similar at the low - amplitude end . here the stars are quiet ( do nt show much variability ) ; a little more than half of those are non - periodic . we have used spot modeling to convert an amplitude into a spot covering fraction , assuming spot contrast of about 0.6 ( which may not hold over all spectral classes ) . this yields the result that log ( v@xmath2)=0.3 corresponds to a spot coverage of 1% of the stellar surface . a simple estimation using half - black spots gives about the same answer . the active solar coverage is about 0.5% , the peak of the distribution here is at about 1% , and more than half of the stars flagged as periodic have coverages well above 1% , extending up to 20% or more in the largest cases . some of those extreme cases are not spots but ellipsoidal binaries ; others are indeed very spotted stars ( judging from the variations in the lightcurve from cycle to cycle , which binaries should not exhibit as readily ) . we now present some typical lightcurves . in fig . [ f5giants]a are some obviously periodic g dwarfs . the parameter space from which they are randomly chosen is 5800k@xmath7t@xmath95850 , log(g)@xmath104.2 , 8@xmath7v@xmath1112,100@xmath7 p@xmath12120 , and period ( days ) between 4 and 12 . [ f5giants]b shows a set of m dwarfs with the same parameters ( except 3900k@xmath7t@xmath94000k ) . one can see from the noise which stars are fainter ( the top middle m dwarf is substantially brighter than the others ) . there is not an obvious difference between the g and m stars , but of course we ve chosen them from the same relevant parameter spaces ( although it is also true that the g and m dwarfs do nt populate these spaces very differently ) . in fig . [ f5giants]c we return to the g dwarfs , but now look in the region 2@xmath7v@xmath114 and 65@xmath7p@xmath1270 . the intensity scale has been decreased by a factor of 2 to better display the decreased range . in most cases the periodicity is still obvious . in a couple of instances it is harder to pick out a periodicity by eye . finally , we present some lightcurves of stars in the giant domain in fig . [ f5giants]d . these have 4700k@xmath7t@xmath94750k , 2.5@xmath7log(g)@xmath73 , 2@xmath7v@xmath114 , and 20@xmath7p@xmath1225 . they look different from the g dwarfs in that the variability is aperiodic and fairly consistent in amplitude . we investigated whether one can isolate giants purely by their photometric behavior . it has been well established that they have a characteristic photometric signature @xcite ; the question we are asking here is how well one can eliminate dwarfs which might also show similar behavior ( even though the vast majority of them do not ) . we provide one reasonably successful attempt to do this without using kic gravities as priors for g , k , m stars ( we use them to check afterward ) . we choose stars with v@xmath2 between 1 and 5 ( motivated by where they are seen in fig . [ f3rmsrange ] ) , and with the number of peaks within 10% of the primary peak power greater than 10 ( based on the periodograms of known giants ) . finally we pick the number of smoothed zero crossings to be greater than 40 ( to pick out the right timescale for giant photometric excursions ) but the number of unsmoothed zero crossings to be less than 200 ( to eliminate noise as the primary source of zero crossings ) . figure [ f6photgiant ] shows the result ( with the unsmoothed zero crossings as the ordinate and kic gravity as the abscissa ) . most of the giants pass through our filter while very few dwarfs do . the remaining dwarfs have lightcurves that indeed closely resemble giant lightcurves , and perhaps some of them are actually misclassified giants . this can only be resolved through ground - based spectroscopy , or if the promise of detecting all the dwarf parallaxes from _ kepler _ data itself is realized . we now take a look at the periodic sample in more detail . figure [ f7perpowhist ] shows the distribution of primary periods within a sample that has been further restricted by choosing only stars with kic gravities greater than log(g)=4 ( which still leaves over 50,000 stars in the sample ) . the distribution of periods rises towards 2 weeks . beyond that we do not trust the results because the time coverage is only 34 days , and because removing a third order fit can definitely have an influence on whether the lightcurve shows features timescales longer than 2 weeks . the small peak at about 3.5 days might possibly be due to the fact that one of the guide stars was a spotted star with this period ( see the kepler data release 5 notes ; it was removed after q1 ) which caused the pointing to vary periodically . the structure at longer periods is due primarily to the fact that the sampling of periods tested is becoming sparser ( the actual sampling is given by the row of cyan plusses in the upper histogram ) . we also investigated whether the range of variability is a function of the primary period . generally , shorter periods indicate more active stars @xcite , and these might be both more obviously periodic or show larger amplitudes of variation ( unless the activity were too uniformly distributed ) . we see that the bulk of the stars have v@xmath2 a little greater than the active sun , and there is modest evidence of greater variability at shorter periods ( the lowest v@xmath2 are found mostly at the long periods ) . of course , by restricting this study to stars rotating fast enough to show periodicity in two weeks , we have biased the result against the slower rotators ( many of which presumably currently lie in our non - periodic sample ) . it is known that activity saturates at short periods and that might be playing a role here . still , it is perhaps a little surprising that there is not a more obvious effect , especially since we are sampling rotation periods up to solar in the case of a half - period of two weeks . we remind the reader that we did not analyze periods below 2 days ; there is indeed a population of short period cases but we expect that those are not due to rotation ( except perhaps in close binaries ) . it is important to note that one can not assume that the dominant period is the rotation period of the star , even when the signal looks very much like starspots . in many cases it is more likely to be the half - period ; when a star has an asymmetric spot distribution each face produces a dip of a different depth so that there could be two major dips per rotation . if there is essentially one active location , or a continuous enough distribution around the star , two discrete dips may not appear and then the periodogram will find the actual period . to produce an active sun sample for comparison , 20 segments of soho diarad @xcite data from 2001 were recast to have similar cadence and length as the _ kepler _ data ( as in paper i ) . periods are found between 10 - 50 days with a mean and median of about 20 days ( and mean p@xmath5 of 200 ) . of course , most of these periods ( even those matching the actual solar period ) would not be considered valid in our current analysis ( since they are derived from 33 day data segments and we do nt trust periods over 16 days ) . we composed a similar sample of quiet sun data from 1996 soho measurements , and actually obtained a similar period distribution with a mean p@xmath5 of about 150 . the high significance of p@xmath5 is primarily due to the very low noise in the soho data . we performed the additional experiment of removing a third order fit from the quiet sun data ; this reduced p@xmath5 to a little below 100 but left the dominant period distribution as rather similar . in order to better study this , we selected a subsample out of the periodic sample by restricting it to kic gravities greater than log(g ) of 4.2 , temperatures between 3800k and 6200k , periods between 3 and 16 days , p@xmath1370 , and v@xmath142 . the upper panel of figure [ f9nonpwrrng ] shows some trend in this sample of these nearly 14,000 clear dwarf rotators for p@xmath5 to increase with v@xmath2 , but there is a great deal of scatter along both axes . in this periodic dwarf sample there is a general decline in population to higher periodogram strengths , but all were chosen to show very clear periods . the analogous plot for the non - periodic sample ( lower panel ) is quite different , showing again that periodic stars tend to be more variable ( although there is a region of low variability in common ) . even these non - periodic stars lie primarily above the formal criterion for a significant period ( which is at p@xmath1515 ) . for solar - type main sequence stars , high variability very likely arises from starspots ( which then exhibit periodicity through stellar rotation ) , and objects which do nt exhibit periods ( or half - periods ) as short as 2 weeks are not as variable . we often see possible signs of differential rotation in the variable stars ( even with such a short span of data ) . such signs are typically that there are two dips of different depth , and they can be seen to move out of phase with respect to each other . an excellent longer span example of this is shown by corot - exo-2b @xcite . the dominant period can change as the spot distribution changes ( although we do nt have long enough data segments here to really see that ) . we find that the amplitude of variability is generally substantially larger for stars that are clearly periodic than for those that are not . among the non - periodic variables , giants are the dominant constituent . we are able to select for giants based on their photometric properties alone with reasonable efficiency . the high amplitude non - periodic stars are almost all high frequency pulsators ( too high for our tailored periodogram test to pick them up ) . for the periodic variables , the dominant period distribution rises to the long end of what we consider well - determined ( up to two weeks ) . since a substantial fraction of stars with dominant periods of two weeks may actually have rotation periods of a month , we are sampling into the pool of main sequence stars with solar rotation . some of the stars with even longer periods must wait for more time coverage , and some of them are quiet enough to have been classified as non - periodic in the current study . it is too early to give a real rotational period distribution or make statements about the age distribution in this sample . perhaps the main conclusion of this preliminary overview of stellar variability as seen by the mission is that this will be a very rich and unique dataset for studying the surfaces of stars . this study is preliminary both because the data reduction will require further refinement and mostly because we have only treated the first month of what is already more than a year ( expected to become at least 3.5 years ) of nearly continuous measurements at very high precision of roughly 150,000 stars . well over half of the sample of main sequence solar - type stars show variability at a level beyond the noise , and many of them show apparently periodic behavior ( 60,000 more clearly and 34,000 somewhat marginally ) . even restricting the sample to solar - type main sequence stars with strong amplitudes and periodicities produces a population of 14,000 . we have only looked by eye at a small minority of them so far . the stars whose variability is periodic but clearly not due to spots ( pulsators and eclipsers are the primary other variables ) are both far less numerous and can be filtered out ( or selected ) with good efficiency . while it is quite obvious in many cases that we are looking at starspots rotating in and out of view , evolving in strength , and differentially rotating at different latitudes , more work is needed to distinguish between these and other sources of variability in more ambiguous cases . we believe that the ambiguity of some of the lightcurves will be reduced or eliminated as longer time series become available . a lot of additional work is needed to model the spotted cases sufficiently well to extract all the information that is present . this will allow a qualitative leap in our understanding of magnetic activity on stars . the authors wish to thank the entire kepler mission team , including the engineers and managers who were so pivotal in the ultimate success of the mission . lw is grateful for the support of the _ kepler fellowship for the study of planet - bearing stars_. gb thanks the nsf through grant ast-0606748 for partial support of this work . funding for this discovery mission is provided by nasa s science mission directorate . affer , l. , micela , g. , favata , r. , flaccomio , e. 2009 , aip conf . ser . 1094 , cool stars , stellar systems and the sun ( melville , ny : aip ) , 341 alonso , r. et al . 2008 , _ a&a _ , 482 , l21 . basri , g. et al . 2010 , _ apj _ , 713 , l155 . batalha , n.m . et al . 2010 , _ apj _ , 713 , l109 . borucki , w. j. et al . _ science _ , 327 , 977 . batalha , n. b. et al . 2010 , _ apj _ , 713 , l103 . bedding , t.r . , et al . 2010 , _ apj _ , 713 , l176 . caldwell , d. a. et al . 2010 , _ apj _ , 713 , l120 . dewitte , s. , crommelynck , d. , joukoff , a. 2004 , j. geophys . 109 , a02102 . gilliland , r. l. , 2008 , _ aj _ , 136 , 566 . guenther , d. b. et al . 2008 , _ apj _ , 687 , 1448 . jenkins , j. m. et al . 2010 , _ apj _ , 713 , l87 . jenkins , j. m. et al . 2010 , _ apj _ , 713 , l120 . koch , d. et al . _ apjl _ , in press . pizzolato , n. , maggio , a. , micela , g. , sciortino , s. , ventura , p. 2003 , _ a&a _ , 397 , 147 . strassmeier , k. g. 2009 , , 17 , 251 0.4 in the non - periodic sample is the locus of post - main sequence stars.,title="fig:",scaledwidth=75.0% ] 0.4 in the non - periodic sample is the locus of post - main sequence stars.,title="fig:",scaledwidth=75.0% ]

we provide an overview of stellar variability in the first quarter of data from the _ kepler _ mission . the intent of this paper is to examine the entire sample of over 150,000 target stars for periodic behavior in their lightcurves , and relate this to stellar characteristics . these data constitute an unprecedented study of stellar variability given its great precision and complete time coverage ( with a half hour cadence ) . because the full kepler pipeline is not currently suitable for a study of stellar variability of this sort , we describe our procedures for treating the raw " pipeline data . about half of the total sample exhibits convincing periodic variability up to two weeks , with amplitudes ranging from differential intensity changes less than 10@xmath0 up to more than 10 percent . k and m dwarfs have a greater fraction of period behavior than g dwarfs . the giants in the sample have distinctive quasi - periodic behavior , but are not periodic in the way we define it . not all periodicities are due to rotation , and the most significant period is not necessarily the rotation period . we discuss properties of the lightcurves , and in particular look at a sample of very clearly periodic g dwarfs . it is clear that a large number of them do vary because of rotation and starspots , but it will take further analysis to fully exploit this . = 1

1008.1092

the swendsen - wang ( sw ) algorithm and related cluster methods @xcite have greatly improved the efficiency of simulating the critical region of a variety of spin models . the original sw algorithm can be modified to work for spin systems with internal symmetry breaking fields @xcite . spin models of this kind include the ising antiferromagnet in a uniform field , the random field ising model and lattice gas models of adsorption in porous media @xcite . the modification proposed in ref . is to assign boltzmann weights depending on the net field acting on the cluster to decide whether the cluster should be flipped . unfortunately , the modified sw algorithm is not efficient . the problem is that large clusters of spins usually have a large net field acting on them and are prevented from flipping by these fields . an algorithm for ising systems with fields that avoids this problem was introduced by redner , machta , and chayes@xcite . in this _ two - replica _ cluster algorithm large clusters are constructed from two replicas of the same system and have no net field acting on them so that they may be freely flipped . the two - replica cluster algorithm has been applied to study the phase transition of benzene adsorbed in zeolites @xcite and is more efficient than the conventional metropolis algorithm for locating and simulating the critical point and the phase coexistence line . combined with the replica exchange method of swendsen and wang @xcite , the two - replica method has been applied to the random field ising model @xcite . the two - replica method is closely related to the geometric cluster monte carlo method @xcite . in this paper , we report on a detailed investigation of the dynamics of the two - replica cluster ( trc ) algorithm as applied to the two - dimensional ising ferromagnetic in a staggered field ( equivalently , the ising antiferromagnet in a uniform field ) . the algorithm introduced in ref . has two components that are not required for detailed balance and ergodicity . we studied the contribution to the performance of the algorithm of these optional components . we find that the complete algorithm has a very small dynamic exponent @xmath0 . however , we also find that this small value of @xmath1 requires one of the optional components and that this component depends on a special symmetry of ising model in a staggered field . this observation leads to the question of whether cluster methods exist for efficiently simulating more general ising models with fields . we investigated other optional components for the algorithm but these do not lead to acceleration when fields are present . this paper is organized as follows . in sec . [ sec : ma ] we introduce the ising model in a staggered field and describe the algorithm . in sec . [ sec : methods ] we define the quantities to be measured and how errors are computed . in sec . [ sec : results ] we present the results . the paper closes in sec . [ sec : disc ] with a discussion . the hamiltonian for the ising model in a staggered field is @xmath2 = - k\sum_{<i , j>}\sigma_{i}\sigma_{j } -\sum_{i}h_{i}\sigma_{i}\ ] ] where the spin variables , @xmath3 take the values @xmath4 . @xmath5 is the coupling strength and @xmath6 is the magnetic field at site @xmath7 . the summation in the first term of eq . ( [ eq : h ] ) is over nearest neighbors on an @xmath8 square lattice with periodic boundary conditions and @xmath9 even . the second summation is over the sites of the lattice . the staggered field is obtained by setting @xmath10 if @xmath7 is in the even sublattice and @xmath11 if @xmath7 is in the odd sublattice . the staggered field breaks the up - down symmetry(@xmath12 ) of the zero field ising model , however two symmetries remain . the hamiltonian is invariant under even translations : @xmath13 with @xmath14 any vector in the even sublattice . the hamiltonian is also invariant under odd translations together with a global flip : @xmath15 with @xmath16 any vector in the odd sublattice . figure [ fig : phase ] shows the line of critical points , @xmath17 for this model . we carried out simulations at three points on the critical line taken from the high precision results of ref . , @xmath18 @xmath19 @xmath20 the basic idea of the two - replica cluster algorithm is to simultaneously simulate two independent ising systems , @xmath21 and @xmath22 , on the same lattice and in the same field . clusters of pairs of spins in this two - replica system are identified and flipped . in order to construct clusters , auxilliary bond variables are introduced . the bond variables \{@xmath23 } are defined for each bond @xmath24 and take values 0 and 1 . we say that @xmath25 is _ occupied _ if @xmath26 . a bond @xmath25 is _ satisfied _ if @xmath27 and @xmath28 . only satisfied bonds may be occupied . the two - replica algorithm simulates a joint distribution of the edwards - sokal @xcite type for \{@xmath29 } and \{@xmath30 } , and \{@xmath23}. the statistical weight @xmath31 $ ] for the joint distribution is @xmath32=e^{-g[\sigma,\tau ] } \delta[\sigma , \tau , \eta ] b_p[\eta]\ ] ] where @xmath33 @xmath34 is the standard bernoulli factor , @xmath35 = p^{|\eta| } ( 1-p)^{n_b-|\eta|}\ ] ] @xmath36 = # @xmath37 is the number of occupied bonds and @xmath38 is the total number of bonds of the lattice . the @xmath39 factor enforces the rule that only satisfied bonds are occupied : if for every bond @xmath24 such that @xmath40 the spins agree in both replicas ( @xmath41 and @xmath42 ) then @xmath43=1 $ ] ; otherwise @xmath43=0 $ ] . it is straightforward to show that integrating @xmath31 $ ] over the bond variables , @xmath44 yields the statistical weight for two independent ising model in the same field , @xmath45-\beta\mathcal{h}[\tau ] } = const\sum_{\{\eta\}}x[\sigma , \tau , \eta]\ ] ] if the identification is made that @xmath46 . the idea of the two - replica cluster algorithm is to carry out moves on the spin and bond variables that satisfy detailed balance and are ergodic with respect to the joint distribution of eq.([eq : x ] ) . the occupied bonds @xmath44 define connected clusters of sites . we call site @xmath7 an _ active site _ if @xmath47 and clusters are composed either entirely of active or inactive sites . if a cluster of active sites is flipped so that @xmath48 and @xmath49 the factor @xmath50 is unchanged . a single monte carlo sweep of the trc algorithm is composed of the following three steps : 1 . occupy satisfied bond connecting active sites with probability @xmath46 . identify clusters of active sites connected by occupied bond ( including single active sites ) . for each cluster @xmath51 , randomly and independently assign a spin value @xmath52 . if site @xmath7 is in cluster @xmath51 then the new spin values are @xmath53 and @xmath54 . in this way all active sites are updated . 2 . update each replica separately with one sweep of the metropolis algorithm . 3 . translate the @xmath22 replica by a random amount relative to the @xmath21 replica . if the translation is by an odd vector , all @xmath22 spins are flipped . step 1 of the is similar to a sweep of the sw algorithm except that clusters are grown in a two - replica system rather than in a single replica and only active clusters are flipped . note also that the bond occupation probability is @xmath46 for the algorithm and @xmath55 for the sw algorithm . it is straightforward to show that step 1 of the algorithm satisfies detailed balance with respect to the joint distribution eq . ( [ eq : x ] ) . since only active sites participate in step 1 of the algorithm , the metropolis sweep , step 2 , is required for ergodicity . step 3 contains the optional components of the algorithm : an even translation or an odd translation plus flip of one replica relative to the other . these moves are justified by the symmetries of the ising model in a staggered field stated in eqs . ( [ eq : even ] ) and ( [ eq : odd ] ) . when we refer to the algorithm without further specification , we mean the algorithm described by the steps 1 - 3 above . in the foregoing we also study the with only even translations or with only odd translations . in the algorithm we flip only active clusters but it is also possible to flip inactive clusters if a weight factor associated with the change in @xmath50 is used . we call a flip of an active cluster to an active cluster ( @xmath56 to @xmath57 or @xmath57 to @xmath56 ) an _ active flip_. the algorithm _ with inactive flips _ is obtained by replacing step 1 with the following : 1 . occupy satisfied bonds with probability @xmath46 . identify clusters connected by occupied bonds ( including single sites ) . for each cluster @xmath51 , taken one at a time , randomly propose two new spin values values , @xmath52 and @xmath58 for the @xmath21 and @xmath22 spins respectively . compute @xmath59 , the change in @xmath50 that would occur if the spins in the @xmath60 cluster are changed to the proposed values leaving spins in other clusters fixed . if @xmath61 accept the proposed spin values ( set @xmath62 and @xmath63 for all sites @xmath7 in cluster @xmath51 ) , otherwise , if @xmath64 accept the proposed spin values with probability @xmath65 . step 1@xmath66 is by itself ergodic however it may be useful to add metropolis sweeps and translations . we measured three observables using the algorithm : the absolute value of the magnetization of a single replica , _ m _ ; the energy of a single replica , @xmath67 ; and the absolute value of the net staggered magnetization for both replicas , _ s _ , where the definition of _ s _ is @xmath68 note that the staggered magnetization is conserved by all components of the algorithm except metropolis sweeps and inactive flips . for each of these observables we computed expectation values of the integrated autocorrelation time , @xmath69 and the exponential autocorrelation time , @xmath70 . from @xmath69 , we estimated the dynamic exponent @xmath1 . the autocorrelation function for @xmath71 , @xmath72 is given by , @xmath73 the integrated autocorrelation time for observable @xmath71 is defined by @xmath74 and the exponential autocorrelation time for an observable @xmath71 is defined by @xcite @xmath75 in practice the limits in eqs . ( [ eq : auto - func ] ) , ( [ eq : in - auto1 ] ) and ( [ eq : ex - auto ] ) must be evaluated at finite values . the length of the monte carlo runs determine @xmath76 and are discussed below . following ref . , we define @xmath77 and choose the cutoff @xmath78 to be the smallest integer such that @xmath79 , where @xmath80 = 6 . we used the least - squares method to fit @xmath81 as a function of @xmath82 to obtain the ratio of @xmath83 and chose a cut - off at @xmath84 . we used the blocking method @xcite to estimate errors . the whole sample of @xmath85 mc measurements was divided into @xmath86 blocks of equal length @xmath87 . for each block @xmath7 and each measured quantity @xmath88 , we computed the mean @xmath89 . our estimates of @xmath90 and its error @xmath91 are obtained from : @xmath92 @xmath93 in our simulations , we divided the whole sample into @xmath86 blocks where @xmath86 is between 10 and 30 . for the data collected using the algorithm , each block has a length @xmath94 . for the data collected using modifications of the algorithm , each block has a length @xmath95 . data were collected for @xmath96 , 2 and 4 and for size @xmath9 in the range 16 to 256 . table [ tab : m&s&e ] gives the integrated autocorrelation time using the algorithm for the magnetization , energy and staggered magnetization . table [ tab : m&s&e ] is comparable to the table in ref . but the present numbers are systematically larger , especially at the larger system sizes . this discrepancy may be due to the sliding cut - off @xmath78 used here instead of a fixed cut - off at 200 employed in ref . . table [ tab : odd&even ] gives the integrated autocorrelation times for magnetization using the with only even or only odd translations . the comparison of algorithm with only even translations and with only odd translations in tables [ tab : odd&even ] shows that odd translations together with global flips of one replica relative to another are essential to achieve small and slowly growing autocorrelation times when the staggered field is present . table [ tab : compare ] shows the magnetization autocorrelation times using different algorithms for system size @xmath97 . the swendsen - wang ( sw ) algorithm has the smallest @xmath98 in the absence of fields . however , when fields are present and the sw algorithm is then modified according to the method of ref . the performance is worse even than that of the metropolis algorithm . the slow equilibration of the sw algorithm in the presence of the staggered field is due to small acceptance probabilities for flipping large clusters . on the other hand , the presence of staggered fields does not significantly change the performance the two - replica algorithm so long as odd translations are present . inactive flips are helpful when there is no staggered field but when the staggered field is turned on , the autocorrelation time is not substantially improved by inactive flips . the explanation for the ineffectiveness of inactive flips when the staggered field is present is that the probability of accepting an inactive flip is small . for example , this probability is @xmath99 for @xmath97 and @xmath100 . the cpu time per spin on a pentium iii 450 mhz machine was also measured for the various algorithms and is listed in table [ tab : compare ] for @xmath97 . by considering a range of system sizes we found that the cpu time for one mc sweep of the algorithm increases nearly linearly with the number of spins . the algorithm is a factor of 3 slower than the metropolis algorithm but this difference is more than compensated for by system size @xmath101 by the much faster equilibration of the algorithm . even without odd translations , the algorithm outperforms metropolis for size 80 . the ratio of the integrated to exponential autocorrelation times was found to be nearly independent of system size over the range @xmath102 to @xmath103 . we found that over this size range @xmath104 varied from @xmath105 to @xmath106 for @xmath96 ; from @xmath107 to @xmath108 for @xmath109 ; and from @xmath110 to @xmath111 for @xmath100 . the ratio @xmath112 is also nearly independent of @xmath9 and @xmath113 and is about 0.45 . the ratio @xmath114 is nearly independent of @xmath9 but decreases slowly with @xmath113 ranging from 0.29 to 0.25 as @xmath113 ranges from 0 to 4 . the almost constant @xmath115 for different sizes suggests that the integrated and exponential autocorrelation times are governed by the same dynamic critical exponent . figures [ fig : m - bilog ] and [ fig : m - log ] show the magnetization integrated autocorrelation time for the plotted on log - log and log - linear scales , respectively . figures [ fig : e - bilog ] and [ fig : e - log ] show the energy integrated autocorrelation time for the plotted on log - log and log - linear scales , respectively . figures [ fig : s - bilog ] and [ fig : s - log ] show the staggered magnetization integrated autocorrelation time for the plotted on log - log and log - linear scales , respectively . for the whole range of @xmath9 , logarithmic growth appears to give a somewhat better fit than a simple power law , particularly for the magnetization . therefore , our results are consistent with @xmath116 for the algorithm . under the assumption that the dynamic exponent is not zero , we also carried out weighted least - squares fits to the form @xmath117 and varied @xmath118 , the minimum system size included in the fit . figures [ fig : m - z ] , [ fig : e - z ] and [ fig : s - z ] show the dynamic exponent @xmath1 for the magnetization , energy and staggered magnetization , respectively , as a function of @xmath118 using the algorithm . figures [ fig : even - z ] and [ fig : odd - z ] , show the dynamic exponent as a function of @xmath118 for the magnetization for the with only even translations and only odd translations , respectively . in all cases except @xmath119 , the dynamic exponent is a decreasing function of @xmath118 . for the magnetization , @xmath120 appears to extrapolate to a value between 0.1 and 0.2 as @xmath121 while for the energy and staggered magnetization , the dynamic exponent appears to extrapolate to a value between 0.3 and 0.4 . the small value of the dynamic exponent requires that odd translations and flips are included in the algorithm . from fig . [ fig : even - z ] it is clear that the dynamic exponent is near 2 for the algorithm with only even translations . table [ tab : dyn - exp ] gives results of the weighted least squares fits for @xmath1 for the smallest values of @xmath118 for which there is a reasonable confidence level . since there is a general downward curvature in the log - log graphs , these numbers are likely to be overestimates of the asymptotic values . thus , we can conclude that the asymptotic dynamic exponent for the algorithm is likely to be less than @xmath122 and is perhaps exactly zero . the dynamic exponent is apparently independent of the strength of the staggered field . for the case of the sw algorithm applied to the two - dimensional ising with no staggered field the best estimate is @xmath123@xcite but the results are also consistent with logarithmic growth of relaxation times . the numbers for dynamic exponent for the sw appear to be smaller than for the algorithm but this difference may simply reflect larger corrections to scaling in the case of the . we studied the dynamics of the two - replica cluster algorithm applied to the two - dimensional ising model in a staggered field . we found that the dynamic exponent of the algorithm is either very small ( @xmath124 ) or zero ( @xmath125 ) and that the dynamic exponent does not depend on the strength of the staggered field . a precise value of @xmath1 could not be determined because of large corrections to scaling . we tested the importance of various optional components of the algorithm and found that an odd translation and global flip of one replica relative to another is essential for achieving rapid equilibration . without this component , @xmath1 is near 2 so there is no qualitative improvement over the metropolis algorithm . an odd translation and global flip of one replica relative to the other allows for a large change of the total magnetization of the system with an acceptance fraction of @xmath126 . large changes in the global magnetization may also occur in the swendsen - wang algorithm in a field or via inactive flips in the algorithm but these flips have a small acceptance fraction due to the staggered field . unfortunately , the odd translation and flip move is allowed because of a special symmetry of the ising model in a staggered field . for more general ising systems with translationally invariant fields , we expect performance similar to the with even translations only . in this case , the autocorrelation time is significantly less than for the metropolis algorithm but the dynamic exponent is about the same . while the two - replica approach is useful for these more general problems of ising systems with fields , it does not constitute a method that overcomes critical slowing down except when additional symmetries are present that allow one replica to be flipped relative to the other . development of general methods for efficiently simulating critical spin systems with fields remains an open problem . .integrated autocorrelation times for the algorithm for the magnetization of a single replica @xmath127 , the net staggered magnetization of both replicas @xmath128 and the energy of a single replica @xmath129 . [ cols="^,^,^,^,^,^,^,^,^ " , ]

the dynamic critical behavior of the two - replica cluster algorithm is studied . several versions of the algorithm are applied to the two - dimensional , square lattice ising model with a staggered field . the dynamic exponent for the full algorithm is found to be less than 0.4 . it is found that odd translations of one replica with respect to the other together with global flips are essential for obtaining a small value of the dynamic exponent .

physics0007091

there has been recent progress in understanding physics of strongly correlated electronic systems and their electronic structure near a localization delocalization transition through the development of dynamical mean field theory ( dmft ) @xcite . merging this computationally tractable many body technique with realistic local density approximation ( lda ) @xcite based electronic structure calculations of strongly correlated solids is promising due to its simplicity and correctness in both band and atomic limits . at present , much effort is being made in this direction including the developments of a lda+dmft method anisimovkotliar , lda++ approach @xcite , combined gw and dmft theory @xcite , spectral density functional theory @xcite as well as applications to various systems such as la@xmath0sr@xmath1tio@xmath2 latio3 , v@xmath3o@xmath2 @xcite , fe and ni @xcite , ce @xcite , pu @xcite , transition metal oxides@xcite , and many others . for a review , see ref . . such _ ab initio _ dmft based self consistent electronic structure algorithms should be able to explore all space of parameters where neither dopings nor even degeneracy itself is kept fixed as different states may appear close to the fermi level during iterations towards self consistency . this is crucial if one would like to calculate properties of realistic solid state system where bandwidth and the strength of the interaction is not known at the beginning . it is very different from the ideology of model hamiltonians where the input set of parameters defines the regime of correlations , and the corresponding many body techniques may be applied afterwards . realistic dmft simulations of material properties require fast scans of the entire parameter space to determine the interaction for a given doping , degeneracy and bandwidth via the solution of the general multiorbital anderson impurity model ( aim ) @xcite . unfortunately , present approaches based on either non crossing approximation ( nca ) or iterative perturbation theory ( ipt ) are unable to provide the solution to that problem due to a limited number of regimes where these methods can be applied @xcite . the quantum monte carlo ( qmc ) technique dmft , jarrell is very accurate and can cope with multiorbital situation but not with multiplet interactions . also its applicability so far has been limited either to a small number of orbitals or to unphysically large temperatures due to its computational cost . recently some progress has been achieved using impurity solvers that improve upon the nca approximation rotors , jeschke , haule:2001 , but it has not been possible to retrieve fermi liquid behavior at very low temperatures with these methods in the orbitally degenerate case . as universal impurity solvers have not yet being designed in the past we need to explore other possibilities , and this paper proposes interpolative approach for the self energy in general multiorbital situation . we stress that this is not an attempt to develop an alternative method for solving the impurity problem , but follow up of the ideology of lda theory where approximations were designed by analytical fits @xcite to the quantum monte carlo simulations for homogeneous electron gas @xcite . numerically very expensive qmc calculations for the impurity model display smooth self energies at imaginary frequencies for a wide range of interactions and dopings , and it is therefore tempting to design such an interpolation . we also keep in mind that for many applications a high precision in reproducing the self energies may not be required . one of such applications is , for example , the calculation of the total energy ce , nature , science , nioprl which , as well known from lda based experience , may not be so sensitive to the details of the one electron spectra . as a result , we expect that even crude evaluations of the self energy shapes on imaginary axis may be sufficient for solving many realistic total energy problems , some of which have appeared already @xcite . another point is a computational efficiency and numerical stability . bringing full self consistent loops with respect to charge densities nature and other spectral functions require many iterations towards the convergency which may not need too accurate frequency resolutions at every step . however , the procedure which solves the impurity model should smoothly connect various regions of the parameter space . this is a crucial point if one would like to have a numerically stable algorithm and our new interpolational approach ideally solves this problem . in the calculations of properties such as the low energy spectroscopy and especially transport more delicate distribution of spectral weight is taken place at low energies , and the imaginary part of the analytically continued self energy needs to be computed with a greater precision . here we expect that our obtained spectral functions should be used with care . also , in a few well distinct regimes , such , e.g. , as very near the mott transition , the behavior maybe much more complicated and more difficult to interpolate . for the cases mentioned above extensions of the interpolative methods should be implemented and its beyond the scope of the present work . we can achieve a fast interpolative algorithm for the self energy by utilizing a rational representation . the coefficients in this interpolation can be found by forcing the self energy to obey several limits and constrains . for example , if infinite frequency ( hartree fock ) limit , positions of the hubbard bands , low frequency mass renormalization @xmath4 , mean number of particles @xmath5 as well as the value of the self energy at zero frequency @xmath6 are known from independent calculation , the set of interpolating coefficients is well defined . in this work , we explore the slave boson mean field ( sbmf ) approach @xcite and the hubbard i approximation @xcite to determine the functional dependence of these coefficients upon doping , degeneracy and the strength of the interaction @xmath7 . we verify all trends produced by this interpolative procedure in the regimes of weak , intermediate and strong interactions and at various dopings conditions . these trends are compared with known analytical limits as well as against calculations using the quantum monte carlo method . also , compared with qmc are self energies and spectral functions on both imaginary and real axes for selective values of dopings . they indicate that the sbmf approach can predict such parameters of interpolation as @xmath8and @xmath4 with a good accuracy while the hubbard i method fails in a number of regimes . however , the functional form of the atomic green function which appears within hubbard i can be used to determine positions of atomic satellites which helps to impose additional constraints on our procedure . giving an extraordinary computational speed of this approach we generally find a very satisfactory accuracy in comparisons with the numerically more accurate qmc calculations . if an increased accuracy is desired our method can be naturally extended by imposing more constraints and by implementing more refined impurity solvers other than the ones explored in this work . the paper is organized as follows . in section ii we discuss rational interpolation for the self energy and list the constraints . in section iii we discuss methods for solving anderson impurity model such as the slave boson mean field and the hubbard i approximations which can be used to find these constraints . a brief survey of the qmc method used to benchmark our algorithm is also given . we present numerical comparisons of sbmf and hubbard i techniques against the qmc simulations for such quantities as quasiparticle residue and multiple occupancies . in section iv we report the results of the interpolative method and compare the obtained spectral functions with the qmc . in section v we discuss possible improvements of the algorithm . section vi is the conclusion . to be specific , we concentrate on the anderson impurity hamiltonian @xmath9 , \label{ham}\end{aligned}\]]describing the interaction of the impurity level @xmath10 with bands of conduction electrons @xmath11 via hybridization @xmath12 . @xmath7 is the coulomb repulsion between different orbitals in the @xmath13band . inspired by the success of the iterative perturbation theory @xcite , in order to solve the anderson impurity model in general multiorbital case , we use a rational interpolative formula for the self energy . this can be encoded into a form@xmath14}{\prod\limits_{m=1}^{m}[\omega -p_{m}^{(\sigma ) } ] } , \label{rat}\]]the coefficients @xmath15 , @xmath16 , or , alternatively , the poles @xmath17 , zeroes @xmath18 and @xmath19 in this equation are to be determined . the form ( [ rat ] ) can be also viewed as a continuous fraction expansion but continuous fraction representation will not be necessary for the description of the method . our basic assumption is that only a well distinct set of poles in the rational representation ( [ rat ] ) is necessary to reproduce an overall frequency dependence of the self energy . extensive experience gained from solving hubbard and periodic anderson model within dmft at various ratios of the on site coulomb interaction @xmath7 to the bandwidth @xmath20 shows the appearance of lower and upper hubbard bands as well as renormalized quasiparticle peak in the spectrum of one electron excitations @xcite . it is clear that the hubbard bands are damped atomic excitations and to the lowest order approximation appear as the positions of the poles of the atomic green function . in the @xmath21 symmetry case which is described by the hamiltonian ( [ ham ] ) , these energies are numerated by the number of electrons occupying impurity level , i.e. @xmath22 and the atomic green function takes a simple functional form @xmath23where @xmath24 are the probabilities to find an atom in configuration with @xmath25 electrons while combinatorial factor @xmath26 arrives due to equivalence of all states with @xmath25 electrons in @xmath21 . we can represent the atomic green function ( [ leh ] ) using the rational representation ( [ rat ] ) , i.e. @xmath27}{\prod\limits_{n=1}^{n}[\omega -p_{n}^{(g ) } ] } , \label{gzp}\]]where @xmath28 are all @xmath29 atomic poles , while @xmath30 denote @xmath31 zeroes with @xmath29 being the total degeneracy . the centers of the hubbard bands are thus located at the atomic excitations @xmath32 . using standard expression for the atomic green function @xmath33 $ ] , we arrive to a desired representation for the atomic self energy @xmath34}{\prod\limits_{n=1}^{n-1}[\omega -z_{n}^{(g ) } ] } . \label{sat}\ ] ] using this functional form for finite @xmath35 modifies the positions of poles and zeroes via recalculating probabilities @xmath24 which is equivalent to the famous hubbard i approximation ( discussed in more detail in the next section ) . we now concentrate on the description of the quasiparticle peak which is present in metallic state of the system . for this an extra pole and zero have to be added in eq ( [ sat ] ) . to see this , let us consider the hubbard model for the @xmath21 case where the local green function can be written by the following hilbert transform @xmath36.$ ] if self energy lifetime effects are ignored , the local spectral function becomes @xmath37 $ ] where @xmath38 is the non interacting density of states . the peaks of the spectral functions thus appear as zeroes in eq . ( [ sat ] ) and in order to add the quasiparticle peak , one needs to add one extra zero ( denoted hereafter as @xmath39 to the numerator in eq . ( [ sat ] ) . to make the self energy finite at @xmath40 one has to also add one more pole ( denoted hereafter as @xmath41 ) which should appear in the denominator or eq . ( [ sat ] ) . furthermore , frequently the hartree fock value for the self energy can be computed separately and it is desirable to have a parameter in the functional form ( [ sat ] ) which will allow us to fix @xmath19 . an obvious candidate to be changed is that self energy pole in ( [ sat ] ) which is closest to @xmath42 equal zero . let us denote this parameter as @xmath43 and rewrite the denominator of ( [ sat ] ) as @xmath44 $ ] where the product is now extended over all zeroes of the atomic green functions except the one closest to zero and two extra poles @xmath41 and @xmath43 can control the width of the quasiparticle peak and @xmath45 thus , we arrive to the functional form for the self energy @xmath46}{(\omega -p_{1}^{(\sigma ) } ) ( \omega -p_{2}^{(\sigma ) } ) \prod\limits_{n=1}^{n-2}[\omega -z_{n}^{(g)}]}. \label{szp}\ ] ] we are now ready to list all constrains of our interpolative scheme . to fix the unknown coefficients @xmath47 @xmath48 @xmath49 @xmath50 in eq . ( [ szp ] ) and to write down the linear set of equations for the coefficients @xmath15 , @xmath16 in eq . ( [ rat ] ) . we use the following set of conditions . _ a ) _ _ hartree fock value _ @xmath45 in the limit @xmath40 the self energy takes its hartree fock form @xmath51mean level occupancy _ _ _ _ @xmath52is defined as a sum over all matsubara frequencies for the green s function , i.e.@xmath53where @xmath54defines the impurity green function and @xmath35 is the hybridization function . _ b ) _ _ zero frequency value _ @xmath55 the so called friedel sum rule establishes the relation between the total density and the real part of the self energy at zero frequency@xmath56 _ c ) quasiparticle mass renormalization value _ @xmath57 the slope of the self energy at zero frequency controls the quasiparticle residue , @xmath4 using the following relationship @xmath58formally , constraints ( _ b _ ) and ( _ c _ ) hold for zero temperature only but we expect no significant deviations in many regions of parameters as long as we stay at low enough temperatures . the behavior may be more elaborated in the vicinity of mott transition @xcite . _ d ) _ _ positions of hubbard bands . _ as we discussed , in order that the self energy obeys the atomic limit and places the centers of the hubbard bands at the positions of the atomic excitations , we demand that @xmath59 this condition fixes almost all self energy zeroes @xmath18 in eq.([rat ] ) to the poles @xmath60 however , it alone does not ensure that the weight is correctly distributed among the hubbard bands and that the very distant hubbard bands disappear . for this to occur , distant poles of green function have to be canceled out by nearby zeroes of the green function . it is clear that each pole @xmath28 far from the fermi level has to be accompanied by a nearby zero @xmath30 in order the weight of the pole be small . thus , the self energy has poles at the position of green s function zeroes which can be encoded into the constrain @xmath61^{-1}=0 . \label{zer}\]]we want to keep this property of the self energy for finite @xmath62 and thus demand that self energy diverges ( when lifetime effects are kept , it only reaches a local maximum ) at the zeros of the functional form ( [ leh ] ) of @xmath63 note that the relationship ( [ zer ] ) holds ( approximately ) for frequency @xmath42 larger than the renormalized bandwidth @xmath64 therefore the information about one @xmath65 which lies close to @xmath66 is omitted and replaced by the information about @xmath67 @xmath6 and @xmath4 as it is done by separating @xmath41 and @xmath43 in the denominator of eq . ( szp ) . we can now write down a set of linear equations for all unknown coefficients in the expression ( [ rat ] ) . there is total @xmath68 of parameters @xmath15 and @xmath69 where we can always set @xmath70 the conditions _ a)_,_b),c ) _ give @xmath71according to condition _ d ) _ we can use all @xmath29 poles @xmath28 and @xmath72 zeroes @xmath73 the zero @xmath30 closest to @xmath66 is dropped out . this brings additional @xmath74 equations for the coefficients and makes @xmath75 as the degree of the rational interpolation which are written below @xmath76^{m}-(p_{n}^{(g)}+\mu -\epsilon _ { f})\sum_{m=0}^{n}b_{m}[p_{n}^{(g)}]^{m } & = & 0\text { for } n=1 ... n , \label{co5 } \\ \sum_{m=0}^{n}b_{m}[z_{n}^{(g)}]^{m } & = & 0\text { for } n=1 ... n-2 .\label{co6}\end{aligned}\ ] ] note that while @xmath77 may be rather large , the actual number of poles contributing to the self energy behavior is indeed very small . we can directly see this from eq . ( [ sat ] ) which uses all @xmath29 poles @xmath28 fulfilling eq . ( [ pol ] ) and uses @xmath72 zeroes @xmath30 directly related to @xmath72 poles @xmath78 clearly , when the spectral weight of the atomic excitation becomes small , the corresponding @xmath79 becomes close to @xmath30 and the cancellation occurs . therefore in realistic situations when only the upper and lower hubbard bands have significant spectral weight along with the quasiparticle peak , the actual degree of the polynomial expansion is either two or three . however , it is advantageous numerically and cheap computationally to keep all poles and zeroes in eq . ( [ szp ] ) because the formula automatically distributes spectral weight over all existing hubbard bands . in the limit when @xmath80 the self energy automatically translates to the non interacting one . the atomic poles get close to each other but , most importantly , their spectral weight goes rapidly to zero as it gets accumulated within the quasiparticle band . in the mott insulating regime , the conditions _ b ) _ and _ c ) _ drop out while all poles @xmath28 and zeroes @xmath30 can be used to determine the interpolation . however , in this regime it does not matter whether one of @xmath30 closest to @xmath66 is dropped out or kept , since we can always replace this information by information about @xmath81 therefore the mott transition can be studied without changing the constraints . we thus see that in the insulating case the self energy correctly reproduces the well known result of the hubbard i method where the green function is computed after eq . ( [ imp ] ) with atomic self energy . if the lifetime effects are computed , the parameters @xmath28 and @xmath30 become complex and the hubbard bands will acquire an additional bandwidth . this effect is evident from the simulations using various perturbative or qmc impurity solvers and can be naturally incorporated into the interpolative formulas ( [ rat ] ) or ( [ szp ] ) . however , in practical implementation below we will omit it for illustrative purposes . let us now discuss the quality of interpolation from the perspective of the high frequency behavior for the self energy . the latter can be viewed moments as expansion in terms of the moments @xmath82 , i.e. , @xmath83 . most important for us is to look at highest moments which are given by the hartree fock value , eq . ( [ inf ] ) involving single occupancy matrix @xmath84 , as well as the first moment @xmath85u^{2 } , \label{ms1}\ ] ] containing a double occupancy matrix @xmath86 we see that the interpolation in part relies on the accuracy in computing multiple occupancies which are the functionals of both atomic excitations and the hybridization function . in this regard , using exact atomic green function to find poles @xmath28 and zeroes @xmath30 as part of the constrained procedure may not be as accurate since it would assume the use of _ _ atomic _ _ multiple occupancies which do _ not _ carry information about @xmath87on the other hand , we can also use only a functional form of the atomic green function where the multiple occupancies are computed in a more rigorous manner . in the next section we will show how this can be implemented using the sbmf multiple occupancies which will be found to be in better agreement with the quantum monte carlo data . note that the moments @xmath82 themselves can be used in establishing the constraints for interpolation coefficients . this would involve independent evaluations of @xmath88 , etc . as well as various integrals involving hybridization function @xmath35 . however , we may run into ill defined numerical problem since high frequency information will be used to extract the low frequency behavior . therefore , it is more advantageous numerically to use some poles and zeros of @xmath89 as given by condition _ d ) _ above . we thus see that the interpolational scheme is defined completely once a prescription for obtaining parameters such as @xmath90 @xmath91 as well as poles @xmath28 , and zeroes @xmath30 is given . for this purpose we will test two commonly used methods : sbmf method due to gutzwiller @xcite as described by kotliar and ruckenstein ruck and the well known hubbard i approximation @xcite . we compare these results against more accurate but computationally demanding quantum monte carlo calculations and establish the procedure to extract all necessary parameters . note that once the constraints such as @xmath4 are computed from a given approximate method , some of the quantities such as the total number of particles , @xmath84 , and the value of the self energy at zero frequency , @xmath92 can be computed fully self consistently . they can be compared with their non self consistent values . if the approximate scheme already provides a good approximation for @xmath84 and satisfies the friedel sum rule , the self consistency can be avoided hence accelerating the calculation . indeed we found that inclusion of the self consistency improves the results only marginally except when we are in a close vicinity to the mott transition but here we do not expect that our simple interpolative algorithm is very accurate . we now give the description of the approximate methods for solving the impurity model and then present the comparisons of our interpolative procedure with the qmc calculations . the quantum monte carlo method is a powerful and manifestly not perturbative approach in either interaction @xmath7 or the bandwidth @xmath20 . in the qmc method one introduces a hubbard stratonovich field and averages over it using the monte carlo sampling . this is a controlled approximation using different expansion parameter , the size of the mesh for the imaginary time discretization . unfortunately it is computationally very expensive as the number of time slices and the number of hubbard stratonovich fields increases . also the method works best at imaginary axis while analytical continuation is less accurate and has to be done with a great care . extensive description of this method can be found in ref . . we will use this method to benchmark our calculations using approximate algorithms described later in this section . a fast approach to solve a general impurity problem is the slave boson method @xcite . at the mean field level , it gives the results similar to the famous gutzwiller approximation @xcite . however , it is improvable by performing fluctuations around the saddle point . this approach is accurate as it has been shown recently to give the exact critical value of @xmath7 in the large degeneracy limit at half filling florens . the main idea is to rewrite atomic states consisting of @xmath25 electrons @xmath93 , @xmath94 with help of a set of slave bosons @xmath95 . in the following , we assume @xmath21 symmetric case , i.e. , equivalence between different states @xmath96 for fixed @xmath25 . formulae corresponding to a more general crystal field case are given in appendix b. the creation operator of a physical electron is expressed via slave particles in the standard manner @xcite . in order to recover the correct non interacting limit at the mean field level , the bose fields @xmath97 can be considered as classical values found from minimizing a lagrangian @xmath98 corresponding to the hamiltonian ( [ ham ] ) . two lagrange multipliers @xmath99 and @xmath100 should be introduced in this way , which correspond to the following two constrains : @xmath101 the numbers @xmath102 are similar to the probabilities @xmath24 discussed in connection to the formula for the atomic green function ( leh ) . we thus see the physical meaning of the first constrain which is the sum of probabilities to find atom in any state is equal to one , and the second constrain gives the mean number of electrons coinciding with that found from @xmath103 . a combinatorial factor @xmath104 arrives due to assumed equivalence of all states with @xmath25 electrons . minimization of @xmath98 with respect to @xmath105 leads us to the following set of equations to determine the quantities @xmath105 : @xmath106\psi _ { n}+nbt\sum_{i\omega } \delta ( i\omega ) g_{g}(i\omega ) [ lr\psi _ { n-1}+\psi _ { n}bl^{2}]+\]]@xmath107=0 , \label{guz}\]]where @xmath108 , determines the mass renormalization , and the coefficients @xmath109 , @xmath110 are normalization constants as in refs . . @xmath111 is the total energy of the atom with @xmath25 electrons in @xmath21 approximation . ( [ guz ] ) , along with the constrains ( [ one ] ) , ( [ two ] ) constitute a set of non linear equations which have to be solved iteratively . in practice , we consider eq . ( [ guz ] ) as an eigenvalue problem with @xmath100 being the eigenvalue and @xmath105 being the eigenvectors of the matrix . the physical root corresponds to the lowest eigenvalue of @xmath100 which gives a set of @xmath105 determining the mass renormalization @xmath112 since the matrix to be diagonalized depends non linearly on @xmath105 via the parameters @xmath113 and @xmath114 and also on @xmath99 , the solution of the whole problem assumes the self consistency : i ) we build an initial approximation to @xmath105 ( for example the hartree fock solution ) and fix some @xmath99 , ii ) we solve eigenvalue problem and find new normalized @xmath105 , iii ) we mix new @xmath105 with the old ones using the broyden method @xcite and build new @xmath115 and @xmath114 . steps ii ) and iii ) are repeated until the self consistency with respect to @xmath105 is reached . during the iterations we also vary @xmath99 to obey the constrains . the described procedure provides a stable computational algorithm for solving aim and gives us an access to the low frequency green s function and the self energy of the problem via knowledge of the slope of @xmath116 and the value @xmath117 at zero frequency . , ( b ) dependence of the spectral weight @xmath118 on concentration , and ( c ) density density correlation function , @xmath119 versus filling , @xmath5 , in the two band hubbard model in @xmath120 and @xmath121,title="fig:",scaledwidth=50.0% ] + the described slave boson method gives the following expression for the self energy : @xmath122the impurity green function @xmath123 in this limit is given by the expression @xmath124 as an illustration , we now give the solution of eq . ( [ guz ] ) for non degenerate case ( @xmath125 and at the particle hole symmetry point with @xmath126 . consider a dynamical mean field theory for the hubbard model which reduces the problem to solving the impurity model subject to the self consistency condition with respect to @xmath35 . starting with the semicircular density of states ( dos ) , the self consistency condition is given by eq . ( [ guz])@xmath127 we obtain the following simplifications : @xmath128 , @xmath129 , @xmath130 and @xmath131^{-1}.$ ] the sum @xmath132 appeared in eq . ( [ guz ] ) scales as @xmath133 with the constant @xmath134 being the characteristic of a particular density of states and approximately equal to 0.2 in the semicircular dos case . the self consistent solution of eq . ( [ guz ] ) is therefore possible and simply gives @xmath135 . the mott transition occurs when no sites with double occupancies can be found , i.e. when @xmath136 a critical value of @xmath137 . for @xmath138 , this gives @xmath139 and reproduces the result for @xmath140 known from the qmc calculation within a few percent accuracy . as degeneracy increases , critical @xmath7 is shifted towards higher values @xcite . from numerical calculations we obtained the following values of the critical interactions in the half filled case @xmath141 for @xmath142 ( @xmath143level)@xmath144 @xmath145 for @xmath146 ( @xmath147level)@xmath144 and @xmath148 for @xmath149 ( @xmath13level ) . density density correlation function @xmath119 for local states with @xmath25 electrons is proportional to the number of pairs formed by @xmath25 particles @xmath150 . since the probability for @xmath25 electrons to be occupied is given by : @xmath151 , the physical density density correlator can be deduced from : @xmath152 . similarly , the triple occupancy can be calculated from @xmath153 . let us now check the accuracy of this method by comparing its results with the qmc data . we consider the two band hubbard model in @xmath154 orbitally degenerate case . hybridization @xmath155 satisfies the dmft self consistency condition of the hubbard model on a bethe lattice @xmath156the coulomb interaction is chosen to be @xmath157 which is sufficiently large to open the mott gap at integer fillings . all calculations are done for the temperature @xmath158 . we first compare the average number of electrons vs. chemical potential determined from the slave bosons which is plotted in fig . [ figsbmf](a ) . this quantity is sensitive to the low frequency part of the green function which should be described well by the present method . we see that it reproduces the qmc data with a very high accuracy and only differs by 20 per cent very near the jump of @xmath159 at @xmath160 . the quasiparticle residue @xmath4 versus filling @xmath5 is plotted in fig . [ figsbmf](b ) . the slave boson method gives the fermi liquid and provides estimates for the quasiparticle residue with the overall discrepancy of the order of 30% . in fact , we have performed several additional calculations for other degeneracies ( @xmath161 and @xmath162 ) and for various parameters regimes . the trend to overestimate mass renormalization can be seen in many cases . it disappears only when @xmath7 approaches zero . we need to point out , however , that i ) the extractions of zero frequency self energy slopes from the high temperature qmc is by itself numerically not well grounded procedure , as information for the self energy is known at the matsubara points only , which is then extrapolated to @xmath66 , ii ) other methods for solving impurity model , such as nca or ipt display similar discrepancies and iii ) recent findings @xcite suggest that at least at half filling quasiparticle residues deduced from slave bosons becomes exact when @xmath163 . most importantly for our interpolative method is that the entire functional dependence of @xmath4 vs. filling , interaction and degeneracy is correctly captured . its overall accuracy is acceptable as it is evident from our comparisons of the spectral functions presented in the next section and well within the main goal of our work to provide fast scans of the entire parameter space necessary for simulating real materials . this is important as , for example , for general @xmath13electron material , the qmc method is prohibitively time consuming , but we expect from the sbmf method the results for mass renormalization not worse than 50% for such delicate regime as the vicinity of the mott transition . [ figsbmf](c ) shows the density density correlation function @xmath119 as a function of average occupation @xmath5 . the discrepancy is most pronounced for fillings @xmath164 [ see the inset of fig . [ figsbmf](c ) ] where the absolute values of @xmath119 are rather small . although our slave boson technique captures only the quasiparticle peak , it gives the correlation function in reasonable agreement with the qmc for dopings not too close to the mott transition . now we turn to the hubbard i approximation @xcite which is closely related to the moments expansion method @xcite . consider many body atomic states @xmath165 which in @xmath21 are all degenerate with index @xmath166 numerating these states for a given number of electrons @xmath167 the impurity green function is defined as the average@xmath168and becomes diagonal with all equal elements in @xmath169it is convenient to introduce the hubbard operators @xmath170and represent the one electron creation and destruction operators as follows@xmath171the impurity green function ( [ gim ] ) is given by @xmath172 where the matrix @xmath173 is defined as @xmath174 establishing the equations for @xmath173 can be performed using the method of equations of motion for the @xmath175 operators . performing their decoupling due to hubbard @xcite , carrying out the fourier transformation and analytical continuation to the real frequency axis , and summing over @xmath25 and @xmath176 after ( [ hub ] ) we arrive to the net result @xmath177the @xmath178 can be viewed in the matrix form ( [ hub ] ) with the following definition of a diagonal atomic green function@xmath179with @xmath111 being the total energies of the atom with @xmath25 electrons in @xmath169 the coefficients @xmath24 are the probabilities to find atom with @xmath25 electrons and were already discussed in connection to the formula ( [ leh ] ) for the atomic green function . they are similar to the coefficients @xmath102 introduced within the sbmf method but now found from different set of equations . these numbers are normalized to unity , @xmath180 and are expressed via diagonal elements of @xmath181 as follows : @xmath182their determination in principle assumes solving a non linear set of equations while determining @xmath183 the mean number of electrons can be measured as follows : @xmath184 or as follows @xmath185 the numbers @xmath24 can be also used to find the averages @xmath186 @xmath187 in the way similar to what has been done in the sbmf approach . , ( b ) dependence of the spectral weight @xmath118 on concentration , and ( c ) density density correlation function , @xmath188 versus filling , @xmath5 , in the two band hubbard model in @xmath120 and for @xmath189 . , title="fig:",scaledwidth=50.0% ] + if we neglect by the hybridization @xmath35 in eq . ( [ h1e ] ) , the probabilities @xmath24 become simply statistical weights : @xmath190we thus see that in principle there are several different ways to determine the coefficients @xmath24 , either via self consistent determination ( scw ) , or using statistical formula ( [ stw ] ) , or taking them from sbmf equation ( [ guz ] ) , i.e. setting @xmath191 while still utilizing the functional dependence provided by the hubbard i method . to determine the best procedure let us first consider limits of large and small @xmath7 s . when @xmath192 , @xmath123 is reduced to @xmath193 i.e. the hubbard i method reproduces the atomic limit . setting @xmath194 gives @xmath195^{-1}$ ] , which is the correct band limit . unfortunately , at half filling this limit has a pathology connected to the instability towards mott transition at any interaction strength @xmath7 . to see this , we consider a dynamical mean field theory for the hubbard model . using semicircular density of states , we obtain @xmath196^{-1}g_{at}(\omega ) $ ] and conclude that for any small @xmath7 the system opens a pathological gap in the spectrum . clearly , using hubbard i only , the behavior of the green function at @xmath197 can not be reproduced and the quality of the numbers @xmath24 is at question . this already emphasizes the importance of using the slave boson treatment at small frequencies . ultimately , making the comparisons with the qmc calculations is the best option in picking the most accurate procedure to compute the probabilities @xmath24 . to check the accuracy against the qmc we again consider the two band hubbard model in @xmath120 symmetry as above . the chemical potential@xmath144 mass renormalization and double occupancy are plotted versus filling in fig . [ fighub1 ] . all quantities here were computed with statistical weights after eq . ( [ stw ] ) but we found a similar agreement while utilizing the self consistent determination of @xmath24 after eq . ( [ scw ] ) . we first see that the hubbard i approximation does not give satisfactory agreement with the qmc data for @xmath198 because it misses the correct behavior at low frequencies . the comparisons for @xmath199 plotted in fig . [ fighub1](b ) surprisingly show a relatively good behavior . however , the pathology of this approximation at half filling would predict @xmath200 for any @xmath7 , which is a serious warning not to use it for extracting the quasiparticle weight . [ fighub1](c ) shows @xmath119 as a function of average occupation @xmath5 . as this quantity is directly related to the high frequency expansion one may expect a better accuracy here . however , comparing fig . [ fighub1](c ) and fig . [ figsbmf](c ) , it is clear that the slave boson method gives more accurate double occupancy . this is due to the fact that the density matrix obtained by the slave boson method is of higher quality than the one obtained from the hubbard i approximation . the results of these comparisons suggest that the probabilities @xmath201 provided by the slave boson method is a better way in determining the coefficients @xmath24 in the metallic region of parameters . therefore it is preferable to use these numbers while establishing the equations for the unknown coefficients in the interpolational form ( [ rat ] ) . however , the functional form ( [ atg ] ) of the hubbard i approximation with @xmath202 can still be used as it provides the positions of the poles @xmath28 and zeroes @xmath30 of the atomic green function necessary for the condition _ d ) _ in the previous section . this also ensures accurate high frequency behavior of the interpolated self energy since its moments expressed via multiple occupancies are directly related to @xmath203 interestingly , while more sophisticated qmc approach captures both the quasiparticle peak and the hubbard bands this is not the case for the slave boson mean field method . to obtain the hubbard bands in this method fluctuations need to be computed , which would be very tedious in a general multiorbital situation . however the slave boson method delivers many parameters in a good agreement with the qmc results , and , hence , it can be used to give a functional dependence of the coefficients of the rational approximation . by now the procedure to determine the coefficients is well established . we use the sbmf method to determine @xmath204 @xmath205 as well as poles and zeroes of the atomic green function provided by the sbmf probabilities @xmath102 and by the bare atomic energy levels @xmath206 ( we omit the lifetime effects for simplicity)@xmath127 this generates a set of linear equations for coefficients @xmath207 , @xmath208 in the rational interpolation formula ( [ rat ] ) . in the present section we show the trends our interpolative algorithm gives for the spectral functions in various regions of parameters as well as provide detailed comparisons for some values of doping for both imaginary and real axis spectral functions . the two band hubbard model with semicircular density of states and dmftself consistency condition after ( [ scf ] ) is utilized in @xmath154 symmetry in all cases using the bandwidth @xmath209 and temperature @xmath210 . [ figu2trend ] shows the behavior of the density of states @xmath211 for @xmath212 as a function of the chemical potential @xmath159 computed with respect to the particle hole symmetry point @xmath213 and as a function of frequency @xmath214 the semicircular quasiparticle band is seen at the central part of the figure . its bandwidth is only weakly renormalized by the interactions in this regime . it is half filled for @xmath215 ( i.e. when @xmath216 ) and gets fully emptied when chemical potential is shifted to negative values . several weak satellites can be also seen on this figure which are due to atomic poles . their spectral weight is extremely small in this case and any sizable lifetime effect ( which is not included while plotting this figure ) will smear these satellites out almost completely . while approaching fully emptied ( or fully filled situation ) the spectral weight of the hubbard bands disappears completely and only unrenormalized quasiparticle band remains . it is clear that even without shifting the atomic poles to the complex axis , the numerical procedure of generating the self energy is absolutely stable . and frequency for the two band hubbard model in @xmath120 and at @xmath217.,title="fig:",scaledwidth=50.0% ] + and frequency for the two band hubbard model in @xmath120 and at @xmath217 . , title="fig:",scaledwidth=50.0% ] + and frequency for the two band hubbard model in @xmath120 and at @xmath189.,title="fig:",scaledwidth=50.0% ] + and frequency for the two band hubbard model in @xmath120 and at @xmath189 . , title="fig:",scaledwidth=50.0% ] + this trend can be directly compared with the simulations using a more accurate qmc impurity solver . we present this in fig . [ figu2trendqmc ] for @xmath212 , which shows calculated density of states in the same region of parameters . remarkably that again we can distinguish the renormalized quasiparticle band and very weak hubbard satellites . the hubbard bands appear to be much more diffuse in this figure mainly due to the lifetime effects and partially due to maximum entropy method using for analytical continuation from imaginary to real axis . otherwise the entire picture looks very much like the one on fig . [ figu2trend ] , generated with much less computational effort . [ figu4trend ] gives the same behavior of the density of states for the strongly correlated regime @xmath218 in this case the situation at integer filling is totally different as the system undergoes metal insulator transition . this is seen around the dopings levels with @xmath159 between 0 and -1 and between -3 and -5 where the wight of the quasiparticle band collapses while lower and upper hubbard bands acquire all spectral weight . in the remaining region of parameters both strongly renormalized quasiparticle band and hubbard satellites remain . again , once full filling or full emptying is approached the quasiparticle bands restores its original bandwidth while the hubbard bands disappear . the qmc result for the same region of parameters is given in fig . [ figu4trendqmc ] . again we can distinguish the renormalized quasiparticle band and hubbard satellites as well as the areas of mott insulator and of strongly correlated metal . the hubbard bands appear to be more sharp in this figure which signals on approaching the atomic limit . we now turn to the comparison of the green functions and the self energies obtained using the formulae ( [ imp ] ) , and ( [ rat ] ) respectively against the predictions of the quantum monte carlo method . we will report our comparisons for the two band hubbard model and sets of dopings corresponding to @xmath219 using the value of @xmath220 other tests for different degeneracies , doping levels and the interactions have been performed which display similar accuracy . and @xmath220,scaledwidth=65.0% ] fig . [ figgrn ] shows the comparison between the real and imaginary parts of the green function obtained by the interpolative method with the results of the qmc calculations . as one can see almost complete agreement has been obtained for a wide regime of dopings . the agreement gets less accurate once the half filling is approached , but still very good giving an extraordinary computational speed of the given method compared to qmc . and @xmath220,title="fig:",scaledwidth=65.0% ] + fig . [ figsig ] shows similar comparison between the real and imaginary parts of the self energies obtained by the interpolative and the qmc method . we can see that the self energies exhibit some noise which is intrinsic to stochastic qmc procedure . the values of the self - energies near @xmath221 and @xmath222 are correctly captured with some residual discrepancies are attributed to slightly different chemical potentials used to reproduce given filling within every method . the results at the imaginary axis show slightly underestimated slopes of the self energies within the interpolative algorithm which is attributed to the underestimated values of @xmath4 obtained from the sbmf calculation . ultimately improving these numbers by inclusions of fluctuations beyond mean field will further improve the comparisons . however , even at the present stage of the accuracy all functional dependence brought by the sbmf method quantitatively captures the behavior of the self energy seen from time consuming qmc simulation . . and @xmath220,title="fig:",scaledwidth=65.0% ] + we also made detailed comparisons between calculated densities of states obtained at the real axis using the interpolative method and the qmcalgorithm . the qmc densities of states require an analytical continuation from the imaginary to real axis and were generated using the maximum entropy method . by itself this procedure introduces some errors within the qmc especially at higher frequencies . in fig . [ figdos ] , we show our results for the fillings corresponding to @xmath219 using the value of @xmath189 . one can see the appearance of the quasiparticle band and two hubbard bands distanced by the value of @xmath7 . it can be seen that the interpolative method remarkably reproduces the trend in shifting the hubbard bands upon changing the doping . it automatically holds the distance between them to the value of @xmath7 while this is not always true in the quantum monte carlo method . despite this result , the overall agreement between both methods is very satisfactory . here we would like to discuss possible ways to further improve the accuracy of the method . the inaccuracies are mainly seen in three different quantities : i ) the width of the hubbard bands , ii ) the mass renormalization @xmath223 which is borrowed from the sbmf method , and iii ) the number of electrons @xmath224 extracted from the interpolated impurity green function ( [ imp ] ) . the inaccuracy in the width of the hubbard band is mainly connected to neglecting the lifetime effect . provided it is computed , this will shift the positions of atomic poles onto the complex plane which is in principle trivial to account for within our interpolative algorithm . to improve the accuracy of @xmath223 one can , for example , work out a modified slave boson scheme which will account for the fluctuations around mean field solution . the inaccuracy in @xmath224 is small in many regions of parameters and typically amounts to 510 per cent . we can try to improve this agreement by requirement that @xmath224 obtained via interpolation matches with @xmath225 obtained by the sbmf method . the latter agrees very well with the qmc for a wide region of parameters as it is evident from fig . [ figsbmf](a ) . in reality , our analysis shows that in many cases the discrepancy in @xmath224 is connected with the overestimation of @xmath226 . therefore , points ii ) and iii ) mentioned above are interrelated . the requirement that @xmath227 can be enforced by adjusting the width of the quasiparticle band , and in many regions of parameters this is controlled by @xmath228 however , there are situations when the hubbard band appears in the vicinity of @xmath66 , and changing @xmath223 does not affect the bandwidth@xmath127 to gain a control in those cases it is better to replace the constraints @xmath229 by constraints of fixing the self energies at two frequencies@xmath144 say @xmath230 and @xmath231 where @xmath232 is the frequency of the order of renormalized bandwidth@xmath127 we have found that this scheme brings mass renormalizations which are about 30% smaller than the sbmf ones , and the agreement with the qmc is significantly improved . thus , inaccuracies ii ) and iii ) can be avoided with this very cheap trick . however , we also would like to point out that the condition @xmath227 is essentially non linear as the solution may not exist for all regions of parameters . it is , for example , evident that in such points where @xmath224 is given by a symmetry ( as , e.g. , particle hole symmetry point @xmath233 in the case considered above ) the mass renormalization does not affect the number of electrons . as the philosophy of our approach is to get the best possible fit we are also open to implementing any kinds of _ ad hoc _ renormalizations constants . one of such possibility could be the use of a quasiparticle residue 30 per cent smaller than @xmath234 as @xmath223 should go to unity when @xmath235 , the correction can , for example , be encoded into the formula @xmath236.$ ] we finally would like to remark that the scheme defined by a set of linear equations for the coefficients ( [ co1])([co6 ] ) is absolutely robust as solutions exist for all regimes of parameters such as strength of the interaction , doping and degeneracy . in general , bringing any information on the self energy @xmath237 at some frequency point @xmath238 or its derivative @xmath239 would generate a linear relationship between the interpolation coefficients , thus keeping robustness of the method . on the other hand , fixing such relationships as numbers of electrons brings non linearity to the problem which could lead to multiplicity or non existence of the solutions . it is also clear that by narrowing the regime of parameters , the accuracy of the interpolative algorithm can be systematically increased . to summarize , this paper shows the possibility to interpolate the self energies for a whole range of dopings , degeneracies and the interactions using a computationally efficient algorithm . the parameters of the interpolation are obtained from a set of constraints in the slave boson mean field method combined with the functional form of the atomic green function . the interpolative method reproduces all trends in remarkable agreement with such sophisticated and numerically accurate impurity solver as the qmc method . we also obtain a very good quantitative agreement in a whole range of parameters for such quantities as mean level occupancies , spectral functions and self energies . some residual discrepancies remain which can be corrected provided better algorithms delivering the constraints will be utilized . nevertheless , given the superior speed of the present approach , we have obtained a truly exceptional accuracy times efficiency of the proposed procedure . the work was supported by nsf dmr grants 0096462 , 02382188 , 0312478 , 0342290 and us doe grant no de the authors also acknowledge the financial support from the computational material science network operated by us doe and from the ministry of education , science and sport of slovenia . in the crystal field case we assume that @xmath29fold degenerate impurity level @xmath10 is split by a crystal field onto @xmath240 sublevels @xmath241 . we assume that for each sublevel there is still some partial degeneracy @xmath242 so that @xmath243 in limiting case of @xmath21 degeneracy , @xmath244 , and in non degenerate case , @xmath245 . we need to discuss how a number of electrons @xmath25 can be accommodated over different sublevels @xmath246 . introducing numbers of electrons on each sublevel , @xmath247 , we obtain @xmath248 note the restrictions : @xmath249 @xmath250 and @xmath251 . in @xmath21 case , @xmath252 , and in non degenerate case , @xmath253 , @xmath247 is either 0 or 1 . total energy for the shell with @xmath25 electrons depends on particular configuration @xmath254 @xmath255 . \label{slacrflev}\ ] ] many body wave function is also characterized by a set of numbers @xmath254 , i.e. @xmath256 energy @xmath257 remains degenerate , which can be calculated as product of how many combinations exists to accommodate electrons in each sublevel , i.e. @xmath258 let us further introduce probabilities @xmath259 to find a shell in a given state with energy @xmath260 sum of all probabilities should be equal to @xmath261 , i.e.@xmath262 there are two green functions in gutzwiller method : impurity green function @xmath263 and quasiparticle green function @xmath264 where matrix coefficients @xmath265 represent generalized mass renormalizations parameters . all matrices are assumed to be diagonal and have diagonal elements enumerated as follows : @xmath266 each element in the green function is represented as follows@xmath267@xmath268and determines a mean number of electrons in each sublevel@xmath269the total mean number of electrons is thus : @xmath270 hybridization function @xmath271 is the matrix which is assumed to be diagonal , and it has diagonal elements enumerated as follows : @xmath272 mass renormalizations @xmath273 are determined in each sublevel . the generalization of the non linear equations ( [ guz ] ) has the form@xmath276 \psi _ { n_{1} ... n_{g}}+ \notag \\ & & \sum_{\alpha = 1}^{g}n_{\alpha } [ t\sigma _ { i\omega } \delta _ { \alpha } ( i\omega ) g_{g\alpha } ( i\omega ) ] b_{\alpha } \left [ r_{\alpha } l_{\alpha } \psi _ { n_{1} ... n_{\alpha } -1 ... n_{g}}+b_{\alpha } l_{\alpha } ^{2}\psi _ { n_{1} ... n_{\alpha } ... n_{g}}\right ] + \notag \\ & & \sum_{\alpha = 1}^{g}(d_{\alpha } -n_{\alpha } ) [ t\sigma _ { i\omega } \delta _ { \alpha } ( i\omega ) g_{g\alpha } ( i\omega ) ] b_{\alpha } \left [ r_{\alpha } l_{\alpha } \psi _ { n_{1} ... n_{\alpha } + 1 ... n_{g}}+b_{\alpha } r_{\alpha } ^{2}\psi _ { n_{1} ... n_{\alpha } ... n_{g}}\right ] . \label{slacrfscf}\end{aligned}\ ] ] g. kotliar and s. y. savrasov , in _ new theoretical approaches to strongly correlated systems _ , edited by a. m. tsvelik , ( kluwer academic publishers , the netherlands , 2001 ) , p. 259 , ( available in cond mat/020824 ) ; s. biermann , f. aryasetiawan , and a. georges , phys . lett . * 90 * , 086402 ( 2003 ) . k. held , i. a. nekrasov , g. keller , v. eyert , n. bluemer , a. k. mcmahan , r. t. scalettar , th . pruschke , v. i. anisimov , and d. vollhardt , psi k newsletter # * 56 * ( april 2003 ) , p. 65 ; a. i. lichtenstein , m. i. katsnelson , and g. kotliar , in _ electron correlations and materials properties _ , ed . by a. gonis , n. kioussis and m. ciftan ( kluwer academic , plenum publishers , 2002 ) p. 428

a rational representation for the self energy is explored to interpolate the solution of the anderson impurity model in general orbitally degenerate case . several constrains such as the friedel s sum rule , positions of the hubbard bands as well as the value of quasiparticle residue are used to establish the equations for the coefficients of the interpolation . we employ two fast techniques , the slave boson mean field and the hubbard i approximations to determine the functional dependence of the coefficients on doping , degeneracy and the strength of the interaction . the obtained spectral functions and self energies are in good agreement with the results of numerically exact quantum monte carlo method .

cond-mat0410410

in type - i seesaw mechanism the lightness of the observed neutrinos are attributed to a seesaw scale around the gut scale incorporated in the theory . in this mechanism , right - handed neutrinos @xmath6 incorporated in the seesaw scale are usually identified with the mass of the @xmath7 : @xmath8 lightest of which is constrained from leptogenesis as @xmath9 gev@xcite . probing the new physics at such a high scale is far beyond the reach of ongoing collider experiments . moreover , apart from experimental accessibility , a theoretical analysis based on naturalness for a hierarchical @xmath7 masses @xmath10 put constraints on them as@xcite : @xmath11 where @xmath12 is the mass of the lightest neutrino . on the other hand , a seesaw scale in the tev range can be realized in some other variants , such as inverse seesaw , linear seesaw etc . by paying the price in terms of addition of extra singlet neutral fermions into these mechanisms which can explain the smallness of neutrino mass by a small lepton - number breaking mass matrix . the ingredients of these two models incorporate , in addition to the standard model singlet right - handed neutrinos @xmath13 , a set of singlet fermions @xmath14 , where @xmath15 ( = 1,2,3 ) are the flavour indices . the yukawa sector of such low energy seesaw mechanism is described by the lagrangian@xcite -_mass = m_d^_r+m^_r_r+m^_l(_l)^c+_s^s_r + + m^s_r+m^s_r+h.c . = & & m_l & m_d & m + m_d^t & m_r & m + m^t & m & _ s ( _ l)^c + _ r + s_r + h.c . where @xmath16 , @xmath17 , @xmath3 ( since it is due to combination of two different fields ) are the dirac type and the rest are the majorana type mass matrices . usually the linear seesaw mechanism is facilitated with the exclusion of all other lepton number violating mass terms expect ` @xmath3 ' whereas in inverse seesaw mechanism both @xmath2 and @xmath3 contain lepton number violating mass terms . thus for linear seesaw , we consider diagonal entries @xmath18 and for inverse seesaw , @xmath19 . therefore , the low energy effective neutrino mass matrix in linear seesaw@xcite can be written as m _ & & -m(m^-1m_d^t)-[m(m^-1m_d^t)]^t and accordingly in inverse seesaw it turns out as m _ & & m_d m^-1_s ( m_d m^-1)^t . [ inv ] now as there are fewer number of experimental constraints , a fruitful approach is to minimize the number of parameters in the lagrangian . popular paradigm is to consider some symmetry in the lagrangian that reduces the number of parameters or to assume texture zeros ( which are also dictated by some underlying symmetry ) in the fundamental mass matrices . 0.1 in in our present work we investigate both the low energy seesaw mechanisms mentioned earlier , incorporating the idea of maximal zero textures@xcite subjected to the criterion of non - zero eigenvalues of the charged lepton ( @xmath1 ) and effective neutrino mass matrix ( @xmath0 ) . we investigate the viable textures of @xmath0 with maximum number of zeros that can be accommodated with the current data . our methodology is as follows : + i ) first we explore to find out a minimal texture of charged lepton mass matrix ( @xmath1 ) which gives rise to three distinct nonzero eigenvalues , i.e , minimum number of parameters necessary to obtain det(@xmath20)@xmath21 0 . the textures obtained are such that they do not contribute to @xmath22 . + ii ) next we assume all the three light neutrino eigenvalues of @xmath0 are non - zero i.e. , det@xmath23 . the linear seesaw formula implies that @xmath16 , @xmath3 and @xmath17 are also non - singular . this fact unambiguously determines the possible minimal textures of @xmath16 , @xmath3 and @xmath17 . in the inverse seesaw , the same criterion fixes the minimal textures of @xmath16 , @xmath2 and @xmath3 . + iii ) fixing a particular minimal structure of @xmath16 and @xmath17 in linear seesaw ( or @xmath16 and @xmath3 in inverse seesaw ) , we systematically explore to obtain the minimal texture of the matrix @xmath3 ( in linear seesaw ) and @xmath2 in inverse seesaw by putting zeros in different entries , for the case of linear ( inverse ) seesaw . + iv ) following , we utilize the frampton and glashow and marfatia condition@xcite to eliminate emerged unphysical effective neutrino matrices ( @xmath0 ) . + v ) finally , we explore numerically the parameter space of the survived matrices utilizing the neutrino oscillation global fit data and predict @xmath24 , @xmath25 , @xmath26 , @xmath4 along with the hierarchical structure of neutrino masses . + the paper is organized as follows : sec . [ s2 ] contains minimally parametrized charged lepton mass matrices @xmath1 and it is obtained that they do not contribute to @xmath22 . effective neutrino mass matrices arising from texture zeros in linear seesaw is discussed in sec . [ s3 ] . the same analysis for inverse seesaw in presented in sec . [ s5 ] contains the summary of the present work . in general , the charged lepton mass matrix has the form @xmath27 we look for maximum zero textures ( minimum number of parameters ) of @xmath1 such that @xmath28 ( or non - zero eigenvalues for @xmath1 ) . a careful inspection of the determinant @xmath29 reveals six stringent possibilities and are presented accordingly in table [ t1 ] . .minimal textures of the charged lepton mass matrix @xmath1 [ cols="<,<,<",options="header " , ] * class ii : * parameter ranges of the matrices with @xmath30 unlike the previous case , this class of matrices ( @xmath31 and @xmath32 ) allow a sizable parameter space compatible with the experimental data . however , the matrices also predict constraint ranges of @xmath4 phase and @xmath24 . we present plots of these parameters in figure [ f1 ] and figure [ f2 ] respectively . from the first two plots of figure [ f1 ] the ranges of the parameters read as @xmath33 , @xmath34 and @xmath35 . + matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] + matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] the dirac cp phase is constrained as @xmath36 and the sum of the light neutrino masses ( @xmath24 ) is obtained within the range 0.094 ev @xmath37 0.18 ev which is well below the present experimental upper bound . in figure [ f2 ] we present the parameter ranges for @xmath32 . the matrix @xmath32 also allow a sizable parameter space and are depicted in first two plots of figure [ f2 ] . the ranges of @xmath38 , @xmath39 and @xmath40 can be read as @xmath41 , @xmath42 and @xmath43 . similar to the previous case , for this matrix also the ranges for @xmath4 and @xmath24 are constrained in a very narrow range as @xmath44 , 0.09 ev @xmath37 0.16 ev . the hierarchy is normal and is depicted in the extreme right plot of the bottom row of figure [ f2 ] . + matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] + matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] matrix . left plot of the bottom row is the variation of @xmath26 with @xmath4 and the right figure shows the hierarchy ( normal ) of the model.,title="fig : " ] we analyze two low energy seesaw ( linear seesaw and inverse seesaw ) mechanisms with the assumption of a minimal non - singular structure of the charged lepton mass matrix @xmath1 with three distinct eigenvalues and non - zero eigenvalues for the effective neutrino mass matrix . non - singular nature of @xmath1 and @xmath0 dictates certain possible textures for the constituent matrices . in the linear seesaw , in our minimalistic approach , it is seen that 5 is the maximal number of zeros that can be accommodated in matrix ` @xmath3 ' to obtain phenomenologically viable @xmath0 . on the other hand , in the inverse seesaw , all the allowed two - zero textures can be explicitly realized in terms of the minimally parametrized constituent matrices . we have numerically explored the allowed parameter ranges using neutrino oscillation global fit data and predict @xmath45 , @xmath25 , @xmath26 and @xmath4 along with the hierarchical structure of neutrino masses . one of the important prediction of this scheme is the vanishingly small value of @xmath4 which could be tested by the ongoing t2k experiment . all the matrices predict nonvanishing and highly constrained range of @xmath4 along with the normal hierarchical spectrum of neutrino masses . numerical analyses shows that two - zero textures can not give rise to large cp violation , and therefore if @xmath5 is established , this minimal scheme will be ruled out . however , we can possibly continue to have the same scheme in the neutrino sector but with other nontrivial charged lepton mass matrices such that @xmath46 is not diagonal to obtain large cp - violating phase . s. davidson and a. ibarra , phys . lett . b * 535 * , 25 ( 2002 ) [ hep - ph/0202239 ] . s. blanchet , p. s. b. dev and r. n. mohapatra , phys . d * 82 * , 115025 ( 2010 ) [ arxiv:1010.1471 [ hep - ph ] ] . j. d. clarke , r. foot and r. r. volkas , phys . d * 91 * , no . 7 , 073009 ( 2015 ) [ arxiv:1502.01352 [ hep - ph ] ] . r. n. mohapatra and j. w. f. valle , phys . d * 34 * , 1642 ( 1986 ) . j. bernabeu , a. santamaria , j. vidal , a. mendez and j. w. f. valle , phys . b * 187 * , 303 ( 1987 ) . r. n. mohapatra , phys . * 56 * , 561 ( 1986 ) . j. schechter and j. w. f. valle , phys . d * 25 * , 774 ( 1982 ) . j. schechter and j. w. f. valle , phys . d * 22 * , 2227 ( 1980 ) . s. fraser , e. ma and o. popov , phys . b * 737 * , 280 ( 2014 ) [ arxiv:1408.4785 [ hep - ph ] ] . a. ghosal and r. samanta , jhep * 1505 * , 077 ( 2015 ) [ arxiv:1501.00916 [ hep - ph ] ] . b. adhikary , a. ghosal and p. roy , indian j. phys . * 88 * , 979 ( 2014 ) [ arxiv:1311.6746 [ hep - ph ] ] . s. s. c. law and k. l. mcdonald , phys . d * 87 * , no . 11 , 113003 ( 2013 ) [ arxiv:1303.4887 [ hep - ph ] ] . p. s. b. dev and r. n. mohapatra , phys . d * 81 * , 013001 ( 2010 ) [ arxiv:0910.3924 [ hep - ph ] ] . a. e. crcamo hernndez and r. martinez , arxiv:1501.05937 [ hep - ph ] . a. e. crcamo hernndez , r. martnez and f. ochoa , arxiv:1309.6567 [ hep - ph ] . a. e. c. hernndez and i. d. m. varzielas , j. phys . g * 42 * , no . 6 , 065002 ( 2015 ) doi:10.1088/0954 - 3899/42/6/065002 [ arxiv:1410.2481 [ hep - ph ] ] . s. m. barr , phys . lett . * 92 * , 101601 ( 2004 ) [ hep - ph/0309152 ] . h. hettmansperger , m. lindner and w. rodejohann , jhep * 1104 * , 123 ( 2011 ) [ arxiv:1102.3432 [ hep - ph ] ] . m. hirsch , s. morisi and j. w. f. valle , phys . b * 679 * , 454 ( 2009 ) [ arxiv:0905.3056 [ hep - ph ] ] . m. chakraborty , h. z. devi and a. ghosal , phys . b * 741 * , 210 ( 2015 ) [ arxiv:1410.3276 [ hep - ph ] ] . p. h. frampton , s. l. glashow and d. marfatia , phys . b * 536 * , 79 ( 2002 ) [ hep - ph/0201008 ] . k. whisnant , j. liao and d. marfatia , aip conf . proc . * 1604 * , 273 ( 2014 ) . o. ludl and w. grimus , jhep * 1407 * , 090 ( 2014 ) [ arxiv:1406.3546 [ hep - ph ] ] . w. grimus and p. o. ludl , pos eps * -hep2013 * , 075 ( 2013 ) [ arxiv:1309.7883 [ hep - ph ] ] . j. liao , d. marfatia and k. whisnant , jhep * 1409 * , 013 ( 2014 ) [ arxiv:1311.2639 [ hep - ph ] ] . h. fritzsch , z. z. xing and s. zhou , jhep * 1109 * , 083 ( 2011 ) [ arxiv:1108.4534 [ hep - ph ] ] . a. merle and w. rodejohann , phys . d * 73 * , 073012 ( 2006 ) [ hep - ph/0603111 ] . w. wang , phys . b * 733 * , 320 ( 2014 ) [ erratum - ibid . b * 738 * , 524 ( 2014 ) ] [ arxiv:1401.3949 [ hep - ph ] ] . l. lavoura , phys . b * 609 * , 317 ( 2005 ) [ hep - ph/0411232 ] . a. kageyama , s. kaneko , n. shimoyama and m. tanimoto , phys . b * 538 * , 96 ( 2002 ) [ hep - ph/0204291 ] . w. wang , phys . d * 90 * , 033014 ( 2014 ) [ arxiv:1402.6808 [ hep - ph ] ] . s. baek , h. okada and k. yagyu , jhep * 1504 * , 049 ( 2015 ) [ arxiv:1501.01530 [ hep - ph ] ] . g. c. branco , d. emmanuel - costa , m. n. rebelo and p. roy , phys . d * 77 * ( 2008 ) 053011 [ arxiv:0712.0774 [ hep - ph ] ] . s. choubey , w. rodejohann and p. roy , nucl . b * 808 * , 272 ( 2009 ) [ erratum - ibid . * 818 * , 136 ( 2009 ) ] [ arxiv:0807.4289 [ hep - ph ] ] . b. adhikary , a. ghosal and p. roy , jhep * 0910 * ( 2009 ) 040 [ arxiv:0908.2686 [ hep - ph ] ] . b. adhikary , a. ghosal and p. roy , jcap * 1101 * ( 2011 ) 025 [ arxiv:1009.2635 [ hep - ph ] ] . b. adhikary , a. ghosal and p. roy , mod . a * 26 * ( 2011 ) 2427 [ arxiv:1103.0665 [ hep - ph ] ] . r. samanta , m. chakraborty and a. ghosal , arxiv:1502.06508 [ hep - ph ] . r. samanta and a. ghosal , arxiv:1507.02582 [ hep - ph ] . b. adhikary , m. chakraborty and a. ghosal , phys . d * 86 * , 013015 ( 2012 ) [ arxiv:1205.1355 [ hep - ph ] ] . d. v. forero , m. tortola and j. w. f. valle , phys . rev . d * 90 * , no . 9 , 093006 ( 2014 ) [ arxiv:1405.7540 [ hep - ph ] ] . p. a. r. ade _ et al . _ [ planck collaboration ] , arxiv:1502.01589 [ astro-ph.co ] .

we investigate linear and inverse seesaw mechanisms with maximal zero textures of the constituent matrices subjected to the assumption of non - zero eigenvalues for the neutrino mass matrix @xmath0 and charged lepton mass matrix @xmath1 . if we restrict to the minimally parametrized non - singular ` @xmath1 ' ( i.e. , with maximum number of zeros ) it gives rise to only 6 possible textures of @xmath1 . non - zero determinant of @xmath0 dictates six possible textures of the constituent matrices . we ask in this minimalistic approach , what are the phenomenologically allowed maximum zero textures are possible . it turns out that inverse seesaw leads to 7 allowed two - zero textures while the linear seesaw leads to only one . in inverse seesaw , we show that 2 is the maximum number of independent zeros that can be inserted into @xmath2 to obtain all 7 viable two - zero textures of @xmath0 . on the other hand , in linear seesaw mechanism , the minimal scheme allows maximum 5 zeros to be accommodated in ` @xmath3 ' so as to obtain viable effective neutrino mass matrices ( @xmath0 ) . interestingly , we find that our minimalistic approach in inverse seesaw leads to a realization of all the phenomenologically allowed two - zero textures whereas in linear seesaw only one such texture is viable . next our numerical analysis shows that none of the two - zero textures give rise to enough cp violation or significant @xmath4 . therefore , if @xmath5 is established , our minimalistic scheme may still be viable provided we allow more number of parameters in ` @xmath1 ' . = 1

1508.05227

in order to calculate physical observables from first principles in quantum chromodynamics ( qcd ) @xcite it is not enough to know its lagrangian . it is also necessary and important to know the true structure of its ground state . it is the response of the qcd vacuum which substantially modifies all the qcd green s functions from their free counterparts . these full ( `` dressed '' ) green s functions are needed for the above - mentioned calculations . the vacuum of qcd is a very complicated confining medium and its dynamical and topological complexity means that its structure can be organized at various levels : classical and quantum @xcite ( and references therein ) . it is mainly non - perturbative ( np ) by origin , character and magnitude , since the corresponding coupling constant is large . however , the virtual gluon field configurations and excitations of the perturbative ( pt ) origin , character and magnitude , due to asymptotic freedom ( af ) @xcite , are also present there . one of the main dynamical characteristics of the qcd ground state is the bag constant . its name comes from the famous bag models for hadrons @xcite , but its present understanding ( and thus modern definition ) not connecting to hadron properties . it is defined as the difference between the pt and the np vacuum energy densities ( veds ) @xcite . so , we can symbolically put @xmath1 , where @xmath2 is the np but contaminated by the pt contributions ( i.e. , this is a full @xmath2 like the full gluon propagator , see below ) . at the same time , we can continue as follows : @xmath3 = ved^{pt } - [ ved^{tnp } + ved^{pt } ] = - ved^{tnp } > 0 $ ] , since the ved is always negative . the bag constant is nothing but the truly np ( tnp ) ved , apart from the sign , by definition , and thus is free of the pt contributions ( contaminations ) . the symbolic subtraction presented here includes the subtraction at the fundamental gluon level , and two others at the hadronic level , i.e. , when the gluon degrees of freedom should be integrated out ( see section [ sec : tnpved ] below ) . in order to consider it also as a physical characteristic of the qcd ground state , the bag constant correctly calculated should satisfy some other necessary requirements such as colorlessness , finiteness , gauge - independence , no imaginary part ( stable vacuum ) , etc . the main purpose of this paper is to formulate a formalism how to calculate correctly the quantum part of the bag constant , using the effective potential approach for composite operators @xcite . in particular , to show how the above - mentioned subtractions are to be analytically made . on account of the confining effective charge , the bag constant has been numerically evaluated , satisfying all the necessary requirements mentioned above . using further the trace anomaly relation @xcite , we also develop a general formalism which makes it possible to relate the bag constant to another important np characteristic of the qcd ground state - the gluon condensate @xcite . here , we do not use the weak coupling solution for the corresponding @xmath0 function . finally we present our numerical result for the bag constant , which is in a good agreement with other phenomenological estimates of the gluon condensate @xcite . the quantum part of the ved is determined by the effective potential approach for composite operators @xcite . in the absence of external sources the effective potential is nothing but the ved . it is given in the form of the skeleton loop expansion containing all the types of the qcd full propagators and vertices , see fig . [ fig:1 ] . so each vacuum skeleton loop itself is a sum of an infinite number of the corresponding pt vacuum loops ( i.e. , containing the point - like vertices and free propagators , see fig . [ fig:2 ] , where one term in each lower order is shown , for simplicity ) . the number of the vacuum skeleton loops goes with the power of the planck constant , @xmath4 . . the solid lines describe the full quark propagators @xmath5 . @xmath6 is the full quark - gluon vertex , while @xmath7 and @xmath8 are the full three- and four - gluon vertices , respectively . ] ) . ] here we are going to formulate a general method of numerical calculation of the quantum part of the tnp yang - mills ( ym ) ved in the covariant gauge qcd . the gluon part of the ved to leading order ( the so - called log - loop level @xmath9 , the first skeleton loop diagram in fig . [ fig:1 ] , and which pt expansion is shown explicitly in fig . [ fig:2 ] ) is analytically given by the effective potential for composite operators as follows @xcite : @xmath10 where @xmath11 is the full gluon propagator and @xmath12 is its free counterpart ( see below ) . the traces over space - time and color group indices are assumed . evidently , the effective potential is normalized to @xmath13 , i.e. , the free pt vacuum is normalized to zero , as usual . next - to - leading and higher contributions ( two and more vacuum skeleton loops ) are suppressed at least by one order of magnitude in powers of @xmath4 . they generate very small numerical corrections to the log - loop terms , and thus are not important for the numerical calculation of the bag constant to leading order . the two - point green s function , describing the full gluon propagator , is @xmath14 where @xmath15 is the gluon invariant function ( dimensionless ) , the so - called lorentz structure ( sometimes , we will call it as the full gluon form factor or , equivalently , the effective charge ( `` running '' ) , see below ) , while @xmath16 is the gauge - fixing parameter and @xmath17 its free pt counterpart @xmath18 is obtained by putting the full gluon form factor @xmath15 in eq . ( [ eq:2 ] ) simply to one , i.e. , @xmath19 in order to evaluate the effective potential ( [ eq:1 ] ) , on account of eq . ( [ eq:2 ] ) , we use the well - known expression @xmath20,\ ] ] which becomes zero indeed when setting @xmath21 . going over to four - dimensional euclidean space in eq . ( [ eq:1 ] ) , one obtains ( @xmath22 ) @xmath23 - { 3 \over 4}d(q^2 ; \xi ) + a \right],\ ] ] where constant @xmath24 and the integration from zero to infinity over @xmath25 is assumed . the ved @xmath26 derived in eq . ( [ eq:6 ] ) is already a colorless quantity , since it has been summed over color indices . also it does not depend explicitly on the unphysical ( longitudinal ) part of the full gluon propagator due to the product @xmath27 , which , in its turn , comes from the above - mentioned normalization to zero . thus it is worth emphasizing that the transversal ( `` physical '' ) degrees of freedom only of gauge bosons contribute to this equation . note , in the effective potential approach to leading order there is no need for ghost degrees of freedom from the very beginning in order to cancel the longitudinal ( `` unphysical '' ) component of the full gluon propagator . this role is played by the normalization condition ( that is why the ghost skeleton loops are not shown in fig . [ fig:1 ] ) . furthermore , overall numerical factor @xmath28 has been introduced into eq . ( 1 ) in order to make the gluon degrees of freedom to be equal @xmath29 , where @xmath30 color of gluons times @xmath31 helicity ( transversal ) degrees of freedom , see eqs . ( [ eq:5 ] ) and ( [ eq:6 ] ) . in the connection with the above - mentioned normalization condition a few remarks are in order . it does not work for the higher order vacuum loops . as explained in ref . @xcite , for consistency with them in the pt qcd green s functions , for example in the hartree - fock approximation , the landau gauge should be used . in ref . @xcite the effective potential has been used to the two - loop order for the investigation of qcd chiral - symmetry breaking just in the landau gauge and in the hartree - fock approximatiion . in the general case ( i.e. , beyond the pt and at any gauge ) , however , the cancelation of unphysical gluon modes should proceed with the help of ghosts as it is described in more detail in appendix a. the derived expression ( [ eq:6 ] ) remains rather formal , since it suffers from the two serious problems : the coefficient of the transversal lorentz structure @xmath32 may still depend explicitly on @xmath16 . furthermore , it is divergent at least as the fourth power of the ultraviolet ( uv ) cutoff , and therefore suffers from different types of the pt contributions . in order to define the ved free of the above - mentioned pt contributions ( contaminations ) , let us make first the subtraction at the fundamental gluon level , namely @xmath33 where @xmath34 correctly describes the pt structure of the full effective charge @xmath32 , including its behavior in the uv limit ( af , @xcite ) , otherwise remaining arbitrary . on the other hand , @xmath35 defined by the above - made subtraction , is assumed to reproduce correctly the tnp structure of the full effective charge , including its asymptotic in the deep infrared ( ir ) limit . this underlines the strong intrinsic influence of the ir properties of the theory on its tnp dynamics . evidently , both terms are valid in the whole energy / momentum range , i.e , they are not asymptotics . let us also emphasize the principle difference between @xmath32 and @xmath35 . the former is the np quantity `` contaminated '' by the pt contributions , while the latter one being also np , nevertheless , is free of them . thus the formal separation between the tnp effective charge @xmath35 and its pt counterpart @xmath34 is achieved . for example , if the full effective charge explicitly depends on the scale responsible for the tnp dynamics in qcd , say @xmath36 - the so - called mass gap ( see section [ sec : effq ] below ) , then one can define the subtraction as follows : @xmath37 , which is , obviously , equivalent to the decomposition ( [ eq:7 ] ) . in this way the separation between the tnp effective charge and its pt counterpart becomes exact , but not unique let us emphasize that the dependence of the full effective charge @xmath38 on @xmath36 can be only regular . otherwise it is impossible to assign to it the above - mentioned physical meaning , since @xmath36 can be only zero ( the formal pt limit ) or finite , i.e. , it can not be infinitely large . in principle , in some special models of the qcd vacuum , such as the abelian higgs model @xcite , the np scale is to be identified with the mass of the dual gauge boson . there is also another serious reason for the subtraction in eq . ( [ eq:7 ] ) . the problem is that the above - mentioned uv asymptotic of the full effective charge may depend on the gauge - fixing parameter @xmath16 explicitly , namely to leading order @xmath39 , where the exponent @xmath40 explicitly depends on the gauge - fixing parameter @xmath16 via the coefficient @xmath41 based on ref . @xcite , and @xmath42 is the qcd asymptotic scale parameter . in this connection let us note that af being a physical phenomenon does not depend on the gauge choice ( it takes place at any gauge ) , while the uv asymptotic of the corresponding green s function may be still gauge - dependent . this is just explicitly shown above . evidently , in the decomposition ( [ eq:7 ] ) just the pt part of the full effective charge will be responsible for this explicit dependence on the gauge choice . subtracting it , we will be guaranteed that the remaining part will not depend explicitly on the gauge - fixing parameter ( that is why the dependence on @xmath16 is not explicitly shown in @xmath35 ) . let us note that if there is no exact criterion how to distinguish between the tnp and the pt parts in the full effective charge in eq . ( [ eq:7 ] ) as described above , then it is possible from the full effective charge to subtract its uv asymptotic only . however , in this case the separation between the tnp and the pt parts will be neither exact nor unique . for how to make this separation exact and unique at the same time see section [ sec : effq ] . substituting the decomposition ( [ eq:7 ] ) into eq . ( [ eq:6 ] ) and doing some simple rearrangements , one obtains @xmath43 - { 3 \over 4}d^{tnp}(q^2 ) \right ] + \epsilon_{pt},\ ] ] where the trivial integration over the angular variables in eq . ( 6 ) has been already done . here @xmath44 is @xmath45 - { 3 \over 4}d^{pt}(q^2 ; \xi ) + a \right].\ ] ] it contains the contribution which is mainly determined by the pt part of the full effective charge , @xmath46 . the constant @xmath47 should be also included , since it comes from the normalization of the free pt vacuum to zero . however , this is not the whole story yet . the first term in eq . ( 8) , depending only on the tnp effective charge , nevertheless , assumes the integration over the pt region up to infinity . it also represents the type of the pt contribution , which should be subtracted as well . if we separate the np region from the pt one , by introducing the so - called effective scale @xmath48 explicitly , then we get @xmath49 - { 3 \over 4}d^{tnp}(q^2 ) \right ] + \epsilon_{pt } + \epsilon'_{pt}\ , \ \ \ \ ] ] where @xmath50 - { 3 \over 4}d^{tnp}(q^2 ) \right].\ ] ] this integral represents the contribution to the ved which is determined by the tnp part of the full gluon propagator but integrated out over the pt region . along with @xmath44 given in eq . ( [ eq:9 ] ) it also represents a type of the pt contribution into the gluon part of the ved ( [ eq:8 ] ) , as mentioned above . this means that the two remaining terms in eq . ( [ eq:10 ] ) should be subtracted by introducing the tnp ym ved @xmath51 as follows : @xmath52 where the explicit expression for @xmath51 is given by the integral in eq . ( [ eq:10 ] ) . concluding , let us emphasize that both subtracted terms @xmath44 and @xmath53 , strictly speaking , are not the purely pt , since along with the nontrivial pt effective charge @xmath54 they contain the tnp effective charge @xmath35 as well . so to call them the pt contributions is a convention . more precisely it is better to say that these terms are `` contaminated '' by the pt contributions . the above - mentioned necessary subtractions can be made in a more sophisticated way by introducing explicitly the ghost degrees of freedom ( see appendix a ) . the bag constant ( the so - called bag pressure ) is defined as the difference between the pt and the np veds @xcite . so in our notations for the ym fields , and as it follows from the definition by eq . ( [ eq:12 ] ) , it is nothing but the tnp ym ved apart from the sign , i.e. , @xmath55 - { 3 \over 4 } \alpha_s^{tnp}(q^2 ) \right],\end{aligned}\ ] ] where from now on we introduce the notation @xmath56 since @xmath35 is the tnp effective charge @xmath57 , as noted above . this is a general expression for any model effective charge in order to calculate the bag constant , or the tnp ym ved apart from the sign , from first principles . it is our definition of the tnp ym ved and thus of the bag constant . so it is defined as the special function of the tnp effective charge integrated out over the np region ( soft momentum region , @xmath58 ) . it is free of the pt contributions , by construction . in this connection , let us recall that @xmath26 is also np , but contaminated by the pt contributions , which just to be subtracted in order to get eq . ( [ eq:13 ] ) from eq . ( 10 ) . comparing expressions ( [ eq:6 ] ) and ( [ eq:13 ] ) , one comes to the following _ prescription _ to get eq . ( [ eq:13 ] ) directly from eq . ( [ eq:6 ] ) : 1 . replacing @xmath59 or equivalently , @xmath60 . 2 . omitting the constant @xmath47 which normalizes the free pt vacuum to zero . 3 . introducing the effective scale @xmath61 which separates the np region from the pt one in the @xmath25-momentum space . 4 . omitting the minus sign for the bag constant . at this stage the bag constant defined by eq . ( [ eq:13 ] ) is definitely colorless ( color - singlet ) and free of the pt contributions ( `` contaminations '' ) . let us remind that it also depends on only transversal degrees of freedom of gauge bosons ( gluons ) . all its other properties mentioned above ( finiteness , positivity , no imaginary part , etc . ) depend on the chosen effective charge , more precisely on its tnp counterpart . it is worth emphasizing once more that in defining correctly the bag constant , three types of the corresponding subtractions have been introduced . the first one - in eq . ( [ eq:7 ] ) at the fundamental gluon level and the two others - in eq . ( [ eq:12 ] ) , when the gluon degrees of freedom were to be integrated out . for actual numerical calculations of the bag constant via the expression ( [ eq:13 ] ) it is always convenient to factorize its scale dependence . for this purpose , let us introduce the dimensionless variable and the tnp effective charge as follows : @xmath62 from the general expression for the bag constant ( [ eq:13 ] ) in these terms one then gets @xmath63 where we introduce the dimensionless tnp ym effective potential @xmath64 , for convenience . its explicit expression is @xmath65 - { 3 \over 4 } \alpha_s^{tnp}(z ) \right].\ ] ] let us emphasize that in order to factorize the scale dependence in the effective potential it is necessary to choose the fixed scale , like @xmath61 , and not the scale which can be varied , for example like the mass gap which can go to zero in order to recover the pt limit ( see section below ) . ( [ eq:16 ] ) and ( [ eq:17 ] ) are the main subject of our consideration in what follows . it is worth emphasizing once more that these expressions are general ones in order to correctly calculate the bag constant from first principles in any model gluon propagator . the only problem remaining to solve is to choose such tnp effective charge @xmath66 which , first of all should not _ explicitly _ depend on the gauge - fixing parameter @xmath16 . at the same time , the implicit gauge dependence is not a problem . such kind of the dependence is unavoidable in quantum or classical gauge theories , since the fields themselves are gauge - dependent @xcite . for the different tnp effective charges @xmath66 one gets different numerical results . that is why the choice for its explicit expression ( ansatz ) should be physically and mathematically well justified ( see below ) . in this connection , let us remind that the gluon schwinger dyson ( sd ) equation is highly non - linear one , and it has a very complicated mathematical structure , so there is no hope for an exact solutions , the number of which is not even fixed @xcite . this means that the number of independent solutions , obtained under specific truncation / approximation schemes and gauges , is not fixed @xmath67 as well . from the very beginning they should be considered on equal footing . let us choose the tnp effective charge as follows : @xmath68 where the superscript `` inp '' stands for the intrinsically np effective charge ( for a such replacement see remarks below ) . here @xmath69 is the so - called jaffe - witten ( jw ) mass gap , which is responsible for the large - scale structure of the qcd vacuum , and thus for its inp dynamics @xcite . let us note , that how the mass gap appears in qcd has been explicitly shown in our recent work in ref . @xcite . * the gauge independence is obvious , i.e. , it does not depend explicitly on the gauge choice , since the mass gap is already renormalized , and hence it is a finite quantity . * it satisfies the wilson criterion of confinement area law for heavy quarks @xcite or , equivalently , leads to the linear rising potential between heavy quarks @xcite in continuous qcd , `` seen '' also by lattice qcd @xcite . in this connection a few remarks are in order . in the case of heavy quarks the response of the vacuum can be neglected , and therefore the interaction between them and gluons effectively becomes point - pike . just this makes it possible to describe confinement of heavy quarks in terms of the linear rising potential , derived on the basis of the expression ( [ eq:18 ] ) . for the light quarks the response of the vacuum can not be neglected . the corresponding quark - gluon vertex is not point - like , and therefore there is no way to analyze confinement of light quarks in terms of the linear rising potential . however , the expression ( [ eq:18 ] ) can be still used for the solution of the sd equation for the quark propagator together with the corresponding slavnov - taylor ( st ) identity for the vertex @xcite . confinement of light quarks is due to the analytical properties of the corresponding green s functions ( unlike the electron propagator , the quark propagator should have no imaginary part ) . this is a principle difference in the description of confinement for light and heavy quarks . * the functional dependence in the confining expression ( [ eq:18 ] ) is , of course , the same for the ym fields and the full qcd . the dependence on the number of flavors can appear only in the mass gap . * it is exactly defined , since in the formal pt limit ( @xmath70 ) the inp effective charge ( [ eq:18 ] ) vanishes , and hence the bag constant itself . * it is uniquely defined as well . in order to show this explicitly , let us assume that it can be replaced by some arbitrary function as follows : @xmath71 where @xmath72 is the dimensionless arbitrary function , which is regular at zero in order not to change confining properties of the inp effective charge ( 18 ) . in this case it can be expand in taylor series around small @xmath25 , so one obtains @xmath73 , where @xmath74 is some auxiliary mass squared . then the inp effective charge in eq . ( [ eq:19 ] ) becomes @xmath75 and substituting this into the general decomposition ( [ eq:7 ] ) , one finally obtains @xmath76 where not loosing generality we include the finite number @xmath77 into the mass gap , and retaining the same notation , for simplicity . the uniqueness is achieved at the expense of the pt effective charge , which now becomes regularly dependent on the mass gap ( compare with the expression ( [ eq:9 ] ) ) . evidently , the uniqueness is due to the singular at origin structure of the inp effective charge in eq . ( [ eq:18 ] ) . in ref . @xcite it has been explicitly shown that the tnp part of the full gluon propagator as a function of the mass gap contains a regular at origin term as well . that is why it is not uniquely separated from the pt gluon propagator which effective charge is always regular at origin . we distinguish between the inp and the pt effective charges not only by the explicit presence of the mass gap , but by the character of the ir singularities as well @xcite . so only after the replacement of eq . ( [ eq:18 ] ) the obtained expression for the bag constant ( [ eq:13 ] ) _ becomes free of all the types of the pt contributions ( contaminations ) , indeed_. * in our recent work @xcite we have shown that the so - called inp gluon propagator is the purely transversal in a gauge invariant way , by construction . it exactly converges to the gluon propagator , which effective charge is in eq . ( [ eq:18 ] ) , after the renormalization of the mass gap is completed . for preliminary analytical investigation of such behavior see refs . @xcite as well ( and references therein ) . thus , we consider the expression ( [ eq:18 ] ) not only as physically and mathematically well confirmed but as uniquely justified within the confining inp qcd @xcite with its own mass gap identified with the jw mass gap for the pure ym fields ( see above ) . * there also exist direct lattice evidences that the zero momentum modes are enhanced in the full gluon propagator ( and hence in its effective charge ) @xcite ( and references therein ) . a np finite - size scaling technique was used in ref . @xcite to study the evolution of the running coupling in the su(3 ) ym lattice theory . at low energies it is shown to grow . the chosen analytical ansatz ( 18 ) can be considered as useful functional parametrization of these lattice results , indeed , while the scale of the enhancement is taken into account by the mass gap . * it is worth noting in advance that one of the attractive additional features of eq . ( 18 ) is that it allows one to perform an analytical summation over the matsubara frequencies in the generalization of the expression for the bag constant to non - zero temperatures . in this case one obtains the curve of the gluon matter pressure as a function of temperature . it and all other its derivatives ( entropy and energy densities , etc . ) then can be directly compared with the corresponding thermal lattice qcd curves @xcite . this will make it possible for better understanding of the thermodynamical structure of the gluon matter ( work in progress and preliminary numerical results are very encouraging ) . in conclusion , one may consider the expression ( [ eq:18 ] ) as the confining ansatz , for simplicity . however , it is worth emphasizing that only it satisfies all the necessary conditions discussed above . let us also note that for the theoretical and numerical results , depending on the confining effective charge , see discussion in section [ sec : concl ] . in terms of the variable in eq . ( [ eq:15 ] ) for the inp effective charge ( [ eq:18 ] ) , one gets : @xmath78 so that the dimensionless effective potential ( [ eq:17 ] ) becomes , @xmath79 - { 3\over 4 } { z_c \over z } \right].\ ] ] performing an almost trivial integration in this integral , one obtains @xmath80.\ ] ] it is easy to see now that as a function of @xmath81 , the effective potential ( [ eq:24 ] ) approaches zero from above as @xmath82 at @xmath83 limit . at infinity @xmath84 it diverges as @xmath85 . at a fixed effective scale @xmath61 and from eq . ( [ eq:22 ] ) it follows that @xmath86 is a correct pt regime , while @xmath84 is not a physical regime , since the mass gap @xmath36 is either finite or zero ( the pt limit ) , i.e. , it can not be infinitely large . in other words , at a fixed effective scale one recovers the correct pt limit for the bag constant , i.e. , the above - mentioned normalization condition is maintained for the bag constant , as it should be . the nontrivial second zero of the effective potential ( [ eq:24 ] ) follows obviously from the condition , @xmath87 which numerical solution is @xmath88 evidently , through the relation ( [ eq:22 ] ) this value determines a possible upper bound for @xmath36 and lower bound for @xmath48 , since @xmath89 is always positive / negative ( see figs . [ fig:3 ] and [ fig:4 ] ) . effective potential vs. @xmath81 . the non - physical region is @xmath90 , since @xmath91 should be always positive . at @xmath92 the effective potential is also zero . ] effective potential vs. @xmath81 . the non - physical region is @xmath90 , since @xmath51 should be always negative . at @xmath93 the effective potential is also zero . ] at @xmath92 , i.e. , @xmath94 the effective potential ( [ eq:24 ] ) vanishes identically , as it should be . from the above one can conclude that this effective potential as a function of @xmath81 has a maximum at some finite point , see fig . [ fig:3 ] . in the way how it has been introduced @xmath81 plays the role of the constant of integration of the effective potential though being formally a parameter of the theory . in general , by taking the first derivative of the effective potential with respect to the constant of integration one recovers the corresponding equations of motion @xcite . requiring thus @xmath95 , one obtains : @xmath96,\ ] ] which makes it possible to fix the constant of integration of the corresponding equation of motion at maximum . its numerical solution is @xmath97 so at maximum the ratio @xmath98 is always less than one . at this point the numerical value of the effective potential ( [ eq:24 ] ) is @xmath99 = 0.0263.\ ] ] the bag constant defined in eq . ( [ eq:16 ] ) , and hence the corresponding inp ved ( [ eq:13 ] ) , as a function of @xmath100 or , equivalently , of the mass gap @xmath101 thus becomes , @xmath102 where the relation @xmath103 has been already used . it is worth noting that a maximum for the bag constant corresponds to a minimum for the inp ym ved @xmath51 ( the so - called `` stationary '' state , see fig . [ fig:4 ] ) . so , we have explicitly demonstrated that in the considered case the bag constant ( [ eq:30 ] ) is finite , positive , and it has no imaginary part , indeed . it depends only on the mass gap responsible for the inp dynamics in the qcd ground state or , equivalently , on the effective scale squared separating the np region from the pt one . in order to complete the numerical calculation of the above defined bag constant all we need now is the value for the effective scale @xmath61 , which separates the np region from the pt one . similarly , the value for a scale at which the np effects become important , that is the mass gap @xmath36 , also allows one to achieve the same goal . if the pt regime for gluons ( as well as for quarks ) starts conventionally from @xmath104 , then this number is a natural choice for the effective scale . it makes it also possible to directly compare our values with the values of many phenomenological parameters calculated just at this scale ( see below ) . we consider this value as well justified and realistic upper limit for the effective scale defined above . thus , using further the relation ( [ eq:31 ] ) , one gets @xmath105 similarly , the numerical value of the mass gap @xmath36 has been obtained from the experimental value for the pion decay constant , @xmath106 , by implementing a physically well - motivated scale - setting scheme @xcite . in fact , we approximate the pion decay constant in the chiral limit @xmath107 by its experimental value , since the difference between them can be a few mev only . this is due to smallness of the corresponding light quark current masses . the pion decay constant is a good experimental number , since it is directly measured quantity , contrary to , for example the quark condensate or the dynamically generated quark mass . for the mass gap we have obtained the following numerical result @xmath108 , so similarly to the relations ( [ eq:32 ] ) , one yields @xmath109 in what follows we will consider this value as a realistic lower limit for the effective scale . one has to conclude that we have obtained rather close numerical results for the effective scale and the mass gap , by implementing rather different scale - setting schemes . it is worth emphasizing that the effective scale ( [ eq:33 ] ) covers quite well not only the deep ir region but the substantial part of the intermediate one as well . for the above - mentioned possible upper bounds for @xmath36 and lower bounds for @xmath61 our numerical results are for the scale - setting scheme ( [ eq:32 ] ) : @xmath110 then similarly , based on the scale - setting scheme ( [ eq:33 ] ) : @xmath111 evidently , their calculated values in each scale - setting scheme satisfy the corresponding bounds . for the bag constant ( and hence for the inp ym ved ) from eq . ( [ eq:30 ] ) , one obtains @xmath112 where the first and second numbers in brackets correspond to the numerical values given in eqs . ( [ eq:32 ] ) and ( [ eq:33 ] ) , respectively . in conclusion , let us note that in the pure ym theory there is no way to calculate the mass gap independently of the well - motivated scale - setting scheme , that s the effective scale in this case , i.e. , relations ( [ eq:32 ] ) . the scale - setting scheme ( [ eq:33 ] ) is based on the numerical value of the pion decay constant in the chiral limit . so this scheme is legitimated to use here as well , since the chiral quark condensates do not contribute to the ved in this limit , as it follows from the trace anomaly relation ( see next section ) . for further discussion on the numerical value of @xmath91 in different units see appendix b. the tnp ved ( and hence the bag constant ) is important by itself as the main dynamical characteristic of the qcd ground state . furthermore it assists in calculating such an important phenomenological parameter as the gluon condensate , introduced in the qcd sum rules approach to the physics of resonances @xcite . the famous trace anomaly relation @xcite in the general case of non - zero current quark masses @xmath113 is @xmath114 where @xmath115 is the trace of the energy - momentum tensor and @xmath116 being the gluon field strength tensor , while for the ratio @xmath117 see discussion below . the trace anomaly relation which includes the anomalous dimension for the quark mass has been derived in ref . @xcite , however , in our case of the pure gluon fields we can use the standard form of the trace anomaly relation ( [ eq:37 ] ) . sandwiching it between vacuum states and taking into account the obvious relation @xmath118 , one obtains @xmath119 is the chiral quark condensate . from this equation in the case of the pure ym fields ( i.e. , when the number of quark fields is zero @xmath120 ) , one can get @xmath121 where , evidently we saturate the total ved , @xmath122 by the tnp ym ved , @xmath123 defined in eq . ( [ eq:13 ] ) , i.e. , putting @xmath124 . let us note that the same result , i.e. , eq . ( [ eq:39 ] ) , will be obtained in the chiral limit for light quarks @xmath125 , for @xmath126 as well . if confinement happens then the @xmath0 function is always in the domain of attraction ( i.e. , always negative ) without ir stable fixed point @xcite . therefore , it is convenient to introduce the general definition of the gluon condensate not using the weak coupling limit solution to the @xmath0 function as follows : @xmath127 thus , the above defined general gluon condensate will be always positive , as it should be . the importance of this relation is that it gives the value of the gluon condensate as a function of the bag constant whatever solution of the @xmath0 function in terms of @xmath128 is . however , let us remind that there is a correlation between the two sides of this equation . the bag constant , correctly defined in eq . ( [ eq:13 ] ) , depends , in general , on the tnp effective charge @xmath129 . on the other hand , the renormalization group equation @xmath130 for the @xmath0 function gives it in terms of the corresponding effective charge . this makes it possible to determine the ratio @xmath131 , which appears in the left - hand - side of eq . ( [ eq:40 ] ) . of course , this equation should be solved for the chosen tnp effective charge ( see subsection 7.1 ) . concluding , let us only note that the quantum part of the total tnp ved at log - loop level is : @xmath132 where @xmath133 is the tnp quark skeleton loop contribution , see the corresponding skeleton loop diagram in fig . [ fig:1 ] . it is an order of magnitude less than @xmath51 because of much less quark degrees of freedom in the vacuum , and it is positive because of overall minus due to the quark loop . evidently , in terms of the ym bag constant , one obtains @xmath134,\ ] ] where we introduce @xmath135 and @xmath136 . so the replacement of the total bag constant by its ym counterpart only is a rather good approximation from the numerical point of view . in this connection , let us remind that in the large @xmath137-limit the pure gluon contribution scales as @xmath138 , while the quark contribution scales only as @xmath137 @xcite . however , in order to correctly calculate the bag constant in full qcd the quark part of the tnp ved @xmath133 is also important . let us note that it is non - zero even in the chiral limit . this part will be investigated and calculated in our subsequent paper . let us show explicitly now that our numerical values for the bag constant calculated in ( [ eq:36 ] ) are in rather good agreement with the phenomenological values of the gluon condensate . above we have already developed a general formalism which allows one to express the gluon condensate as a function of the bag constant . so substituting the numerical value of the bag constant into the eq . ( 40 ) , one obtains : @xmath139 on the other hand , the renormalization group equation for the @xmath0 function ( [ eq:41 ] ) after substitution of our solution for the inp effective charge ( [ eq:18 ] ) yields : @xmath140 as it is required for the confining theory where the @xmath0 function should be always in the domain of attraction , i.e. , negative ( see in ref . @xcite ) . the corresponding ratio as it appears in the left - hand - side of eq . ( [ eq:44 ] ) is @xmath141 substituting further this solution into the eq . ( [ eq:44 ] ) , it becomes @xmath142 which means that both sides of this relation between the bag constant and the gluon condensate have been calculated by using the same expression for the inp effective charge , and hence for the corresponding @xmath0 function . so from the numerical point of view the bag constant and the gluon condensate are in a self - consistent dependence from each other , making thus the latter one free of all the types of the pt contributions . our expression for the gluon condensate ( [ eq:47 ] ) allows one to recalculate any gluon condensate at any scale and any ratio , @xmath117 . to the gluon condensate a physical meaning can be indeed assigned as the global ( average ) vacuum characteristic which measures a density of the tnp gluon fields configurations in the qcd vacuum . however , it can not be directly compared with the phenomenological values for the standard gluon condensate estimated within different approaches @xcite . the problem is that it is necessary to remember that any value at the scale as in eq . ( [ eq:33 ] ) ( lower bound in the right - hand - side of eq . ( [ eq:47 ] ) ) is to be recalculated at the @xmath104 scale . moreover , both values explicitly shown in eq . ( [ eq:47 ] ) should be recalculated at the same ratio , as mentioned above . in phenomenology the standard ratio of the gluon condensate and its numerical value is : @xmath143 which can be changed within a factor of @xmath144 @xcite ( let us recall that this ratio comes from the weak coupling solution for the @xmath0 function , see for example in ref . @xcite ) . thus in order to achieve the same ratio the both sides of eq . ( [ eq:47 ] ) should be multiplied by @xmath145 . for the numerical value of the strong fine structure constant we use @xmath146 from the particle data group @xcite . in addition , the lower bound should be multiplied by the factor @xmath147 , coming form the numerical value by eq . ( [ eq:33 ] ) . then the recalculated gluon condensate in ( [ eq:47 ] ) , which is denoted as @xmath148 , finally becomes ( i.e. , both numbers in eq . ( [ eq:47 ] ) coincides , as it should be ) @xmath149 this numerical value for the gluon condensate should be compared with the numerical value coming from the phenomenology , see eq . ( [ eq:48 ] ) above . this shows that all our numerical results are in good agreement with various phenomenological estimates @xcite , taking into account that the quark contributions are approximately an order of magnitude less than the pure ym one to the full bag constant ( see remarks in this section just before subsection 7.1 ) . this confirms that our numerical values for the bag constant and hence for the gluon condensate are rather realistic ones . in summary , we have formulated a general method how to calculate numerically the quantum part of the tnp ym ved ( the ym bag constant , apart from the sign , by definition ) in the covariant gauge qcd from first principles . for this purpose we have used the effective potential approach for composite operators to leading order @xcite . it has an advantage to be directly the ved ( the pressure ) in the absence of external sources . the bag constant is defined as a special function of the tnp effective charge integrated out over the np region ( soft momentum region ) , see eq . ( [ eq:13 ] ) . at this stage the bag constant is colorless ( color - singlet ) and depends only on the transversal ( `` physical '' ) degrees of freedom of gauge bosons . it is also free of the pt contributions by its construction . this has been achieved due to the subtractions at the fundamental level as given by eg . ( [ eq:7 ] ) , as well as due to all other subtractions explicitly shown in eq . ( [ eq:12 ] ) , when the gluon degrees of freedom were to be integrated out . thus , our equations ( [ eq:16 ] ) and ( [ eq:17 ] ) are general ones in order to correctly calculate the bag constant as a function of any properly defined tnp effective charge . for the concrete calculation of the bag constant we replace the tnp effective charge by its confining inp counterpart in eq . ( [ eq:18 ] ) , since it is exactly and uniquely separated from the pt effective charge . the inp effective charge depends regularly on the mass gap , which is responsible for the large - scale structure of the qcd ground state @xcite . the scale - setting schemes have been chosen by the two different ways , leading , nevertheless , to a rather close numerical results for the mass gap and hence for the effective scale . the calculated bag constant in addition , is : finite , positive , and it has no imaginary part ( stable vacuum ) . it is also a manifestly gauge - invariant quantity ( i.e. , does not explicitly depend on the gauge - fixing parameter as it is required ) . the separation of `` soft versus hard '' gluon momenta is also exact because of the maximization / minimization procedure . it becomes possible since the effective potential ( [ eq:24 ] ) as a function of the constant of the integration @xmath81 and hence of the mass gap @xmath36 has a local maximum , see fig . [ fig:3 ] . this also makes it possible that in the above - mentioned scale - setting schemes either the mass gap or the effective scale is only independent , since the other one is to be determined via the relation ( [ eq:31 ] ) . in the scale - setting scheme ( [ eq:32 ] ) the effective scale is independent , while in the second scale - setting scheme ( [ eq:33 ] ) the mass gap is independent . _ it is worth emphasizing that the bag constant in our approach is not simply the difference between the pt and np veds , which is finite , colorless and manifestly gauge - invariant , etc . it is the energy density ( apart from the sign ) of the system of stable configurations of the purely transversal quantum virtual fields with the enhanced low - frequency components / large scale amplitudes due to the nl interaction of massless gluon modes , and which is being at `` stationary state '' , i.e. , being in the state with the minimum of energy , see fig . [ fig:4]_. in order to compare our numerical results with phenomenology we develop a general formalism which makes it possible to relate the bag constant to the gluon condensate in a unique and self - consistent way . in other words , the gluon condensate is defined and calculated at the same effective charge , which has been chosen for the calculation of the bag constant . for this purpose we use the trace anomaly relation without applying to the weak coupling solution for the corresponding @xmath0 function . in its turn , it is a solution of the corresponding renormalization group equation for the effective charge eq . ( our numerical results turned out to be in good agreement with phenomenological values of the gluon condensate calculated and estimated within different approaches and methods @xcite . it is instructive to briefly summarize our theoretical and numerical results for the bag constant in general and our specific ways : general properties of the bag constant determined by eqs . ( 16)-(17 ) are : : : * colorless ( color - singlet ) ; * electrically neutral ; * transversal , i.e. , depending only on `` physical '' degrees of freedom of gauge bosons ; * free of the pt contributions ( contaminations ) . results , depending on the confining effective charge eq . ( 18 ) are : : : * the explicit gauge invariance ; * uniqueness , i.e. , it is free of all the types of the pt contributions now ; * finiteness ; * positiveness ; * no imaginary part ( stable vacuum ) ; * existence of the stationary state for the corresponding ym energy density ( negative pressure , see fig . 4 ) ; * the final dependence on the mass gap only ; * a good numerical agreement with phenomenology . the above remarkable features all together are unique . apparently , it is due to the confining expression ( [ eq:18 ] ) and the correct determination of the bag constant itself in this investigation . it has been made in accordance with its modern definition as the difference between the pt and the np veds @xcite . our method can be generalized on the multi - loop skeleton contributions to the effective potential approach for composite operators , as well as to take into account the quark degrees of freedom , as plotted in fig . [ fig:1 ] . these terms , however , will produce numerical contributions an order of magnitude less , at least , in comparison with the leading log - loop level gluon term given by eq . ( [ eq:1 ] ) . what is necessary indeed , is to be able to extract the finite part of the tnp ved in a self - consistent and manifestly gauge - invariant ways . this is provided by our method which thus can be applied to any qcd vacuum quantum and classical models at any gauge ( covariant or non - covariant ) . it may serve as a test of them , providing an exact criterion for the separation `` stable versus unstable '' vacua . using our method we have already shown that the vacuum of classical dual abelian higgs model with string and without string contributions is unstable against quantum corrections @xcite . it would be also interesting to apply our general equations ( [ eq:16 ] ) and ( [ eq:17 ] ) in order to calculate the bag constant within the recently obtained analytical results for the gluon propagator in refs . the general formalism developed in our paper is aimed first of all at the analytical calculations of the bag constant ( or the vacuum energy density ) in any model gluon propagator in continuous qcd . however , as mentioned above the chosen ansatz ( 18 ) can be considered as useful parametrization of the corresponding lattice results . in this way our formalism can be extended to the lattice calculations as well . choosing an appropriate parametrization of any lattice result for the gluon propagator ( there is a lot of recent lattice data @xcite and references therein ) , one then can substitute it into our analytical expressions ( 16)-(17 ) . such a combination of the lattice and analytical calculations can be rather effective indeed , in order to understand what is the physics behind the lattice numbers and curves . on the other hand , all the analytical expressions and calculations will be put on solid numerical ground provided by the lattice simulations . so there is no doubt that the analytical and lattice calculations should not exclude each other , but contrary they should complement each other . all these possible developments are , of course , beyond the scope of the present investigation , and they have to be done elsewhere . let us begin with recalling that due to the above - mentioned normalization condition in the initial eq . ( 1 ) , its elaborated counterpart in eq . ( 6 ) depends only on the transversal ( `` physical '' ) component of the full gluon propagator . so there is no need for ghosts to cancel its longitudinal ( unphysical ) component , indeed . however , it is instructive to discuss the role of ghosts in general , and to clearly show that their explicit introduction leads to the same result for the bag constant , in particular . following ref . @xcite , the effective potential at the same log - loop order for the ghost degrees of freedom analytically is : @xmath150 where @xmath151 is the full ghost propagator , where @xmath152 is the ghost self - energy , while @xmath153 is its free pt counterpart . trace over color group indices is assumed . evidently , the effective potential is normalized to @xmath154 in the same way as the gluon part in eq . ( [ eq:1 ] ) . substituting these expressions into the ghost term ( [ eq : a.1 ] ) and again doing some algebra in four - dimensional euclidean space , one formally obtains that @xmath155 . this , in general , divergent constant contribution should be of course , regularized in order to assign to it a mathematical meaning . so the explicit functional dependence of the ghost propagator / self - energy on its argument is not important , since within the effective potential approach to calculate the ved it is always only constant . we have to sum up all the contributions for the pure ym fields at the same skeleton log - loop order . the relation given by eq . ( [ eq:10 ] ) then should look like as : @xmath156 - { 3 \over 4}d^{tnp}(q^2 ) \right ] + \nonumber \\ & + & \epsilon_{pt } + \epsilon'_{pt } + \epsilon_{gh}.\end{aligned}\ ] ] it is worth emphasizing that , the right - hand - side of this relation may still suffer from unphysical singularities by the integral in eq ( [ eq:9 ] ) , defining @xmath44 . the problem is that the pt effective charge , @xmath54 , which is responsible for af in qcd at large @xmath25 ( see , for example our paper @xcite ) , may have , in general , unphysical singularities below the scale @xmath42 , since in the integral ( [ eq:9 ] ) the integration is from zero to infinity . in addition , as mentioned above the integral ( [ eq:11 ] ) , defining @xmath53 , may be still divergent . thus the left - hand - side of the relation ( [ eq : a.2 ] ) is formal one , indeed . it suffers from various types of unphysical singularities which may appear in its right - hand - side . in order to get a physically meaningful expression , one has to remove the two integrals ( [ eq:9 ] ) and ( [ eq:11 ] ) from eq . ( [ eq:6 ] ) . this is to be done with the help of a ghost term by imposing the following condition of cancelation of unwanted terms in the most general form : @xmath157 . this condition can be always fulfilled , since it is a relation between three different ( unknown in general ) regularized constants . then the relation ( [ eq : a.2 ] ) thus becomes : @xmath158 - { 3 \over 4}d^{tnp}(q^2 ) \right],\end{aligned}\ ] ] in complete agreement with the relation ( [ eq:12 ] ) , and hence with the definition of the bag constant ( [ eq:13 ] ) , as it should be . so the tnp gluon contribution to the ved has been determined by subtracting unwanted terms by means of the ghost contribution . evidently , the subtracted terms are of no importance , while a ghost term plays no explicit role for further consideration . in qcd the general role of ghost degrees of freedom is to cancel all the unphysical degrees of freedom of gauge bosons @xcite , maintaining thus unitarity of the @xmath5-matrix . this is the main reason why they should be taken into account together with gluons always . this means that nothing should _ explicitly _ depend on them after the above - mentioned cancelation is performed . one of the main purposes of their introduction is to exclude the longitudinal ( unphysical ) component of the gluon propagator in every order of the pt , thus going beyond it and thus being a general one , indeed . if there is no need to cancel the longitudinal component of gauge boson propagators , then they should be used to eliminate the unphysical singularities of gauge bosons below the qcd asymptotic scale ( as it was described above ) , or some other ones which may be inevitably present in any solution / ansatz for the full gluon propagator . if one knows the ghost propagator exactly , then the above - mentioned cancelation of unphysical singularities of gauge bosons should proceed automatically , as usual in the pt calculus ( if , of course , all calculations are correct ) . for such an exact cancelation of the longitudinal part of the gluon propagator by the free pt ghost propagator in lower order of the pt see , for example ref . but if it is not known exactly or known approximately ( depending on the truncation / approximation scheme ) , as usual in the np calculus then nevertheless , one has to impose the corresponding condition of cancelation in order to fulfill their general role . this just has been done above . thus our subtraction scheme is in agreement with the general interpretation of ghosts to cancel all the unphysical degrees of freedom of gauge bosons @xcite . so by themselves the ghosts can not change the truly np dynamics of qcd , associated with the transversal component of the full gluon propagator in eq . ( 2 ) and described by its lorentz structure or , equivalently , by its effective charge ( see ref . @xcite as well ) . whatever solution(s ) for the full gluon propagator obtained by lattice qcd @xcite ( and references therein ) and by the analytical approach based on the corresponding sd system of equations @xcite ( and references therein ) might be ( smooth , singular , massive , etc . ) , it , however , should not undermine the above - mentioned general job of ghosts . it is worth emphasizing that by no coincidence in all the papers cited above the transversal landau gauge has been chosen by hand from the very beginning . so there is _ no and can not be the explicit _ dependence on the ghost degrees of freedom in any expressions for the physical quantities , in general , and in the expression for the bag constant , in particular . in this connection , let us remind that the confining effective charge ( [ eq:18 ] ) is the effective charge of the relevant gluon propagator , which becomes the purely transversal in a gauge invariant way , by construction @xcite . nevertheless , this does not mean that we need no ghosts at all . first of all , we need them in the higher orders of the two - particle irreducible vacuum graphs in the skeleton loop expansion of the effective potential @xcite , since for them the simple normalization of the free pt vacuum to zero does not work . so the cancelation of unphysical gluon modes should proceed with the help of the ghost degrees of freedom , as it was described in this appendix above . it is necessary to understand that the transversality of the gluon propagator in the landau gauge in order to correctly treat the pt qcd green s functions without ghosts is not enough to insure unitarity of the @xmath5-matrix in qcd . the whole machinery of all the st identities and the corresponding sd equations is still needed in order to insure the unitarity cancelations even in the landau gauge . for example , the quark st identity , contains the so - called ghost - quark scattering kernel explicitly @xcite . this kernel still makes an important contribution to the identity even if the gluon propagator is transversal @xcite . omitting ghosts at all in this identity , one will lose an important piece of information on the quark degrees of freedom themselves . as a result , any solution of the quark sd equation will suffer from unphysical singularities in the complex momentum plane . the problem is that via the quark - gluon vertex this equation will crucially depend on the term which comes from the identity even if the gluon propagator is transversal . the completely np analysis of this identity on the basis of the double pole structure of the full gluon propagator in the ir , eq . ( 18 ) , has been made in our earlier papers @xcite . we have derived the corresponding expression for the quark - gluon vertex following ref . @xcite only in more sophisticated way ( see ref . @xcite as well ) . we will take this result into account when we will directly calculate the confining quark contribution to the bag constant , as mentioned in section 7 just before subsection 7.1 . in order to show explicitly what magnitude of numbers we are dealing with , let us present our numerical value for the bag constant given by eq . ( [ eq:36 ] ) in different units , namely : @xmath159 this is a huge amount of energy stored in one @xmath160 of the qcd vacuum even in `` god - given '' units @xmath161 . using the number of different conversion factors ( see , for example ref . @xcite or the particle data group @xcite ) the bag constant can be expressed in different systems of units ( si , cgs , etc . ) . taking further into account that @xmath162 from eq . ( [ eq : b.1 ] ) one finally gets ( @xmath163 w @xmath164 kw @xmath165 mw @xmath166 gw ) @xmath167 or , equivalently , @xmath168 in familiar units of watt - hour ( wh ) . let us note that if one puts the effective scale squared as small as realistically possible @xmath169 ( see eq . ( [ eq:35 ] ) ) , then the previous number will be only slightly changed , namely @xmath170 . so both numbers still indicate a huge amount of the bag energy @xmath171 stored in one @xmath160 of the qcd vacuum . it is especially interesting to compare these numbers with the total production of primary energy of the 25 eu countries in year 2004 which was @xcite ( see also ref . @xcite ) @xmath172 where @xmath173 . approximately @xmath174 of this energy was produced by nuclear power plants @xcite . the huge difference between the numbers in eqs . ( [ eq : b.4 ] ) and ( [ eq : b.5 ] ) is very impressive and leads to some interesting still speculative but already possible discussion in appendix d below and in our preliminary work @xcite . apparently , our bag constant ( [ eq : b.1 ] ) may also contribute to the so - called dark energy density @xcite . at least , from the qualitative point of view it satisfies almost all the criteria necessary for the dark energy / matter candidate ( see here section 8 and discussions in refs . @xcite ) . from the quantitative numerical point of view it is also much better than the estimate from the higgs field s contribution to the ved , which is about @xcite @xmath175 in this notation our value ( b.1 ) is about @xmath176 the observed ved is very small indeed , namely @xmath177 see refs . so relatively to the value inferred from the cosmological constant ( i.e. , the above - mentioned observed ved ) @xmath178 while our is @xmath179 i.e , some @xmath180 orders of magnitude better , which is expected from the direct comparison of the estimate ( c.1 ) with our value ( c.2 ) . let us note that calculating at the plank length scale @xcite , we will obtain the same ratio , as it should be . from eq . ( b.1 ) it follows that @xmath181 where we used @xmath182 and @xmath183 denotes the above - mentioned plank length @xcite . in this units the observed ved is @xmath184 so that the ratio between ( c.6 ) and ( c.7 ) becomes again ( c.5 ) , indeed . of course , the ratio ( c.5 ) still remains very large , but it is much better than the ratio ( c.4 ) , as emphasized above . other possibility how qcd can be related to the dark energy puzzle has been described in ref . @xcite ( and references therein ) . concluding , the vacuum for which the value ( c.3 ) has been measured should not be mixed with the vacuum of any quantum field gauge theory . for the former one its energy is always positive ( i.e. , above zero ) , so the vacuum is simply treated as empty space . the energy of the latter one is always negative ( i.e. , below zero ) , and it is full of any kind of quantum excitations , fluctuations , etc . however , the qcd bag constant is always positive , finite , gauge - invariant , etc . ( if it has been correctly defined and calculated like in this work ) . that is the primary reason why we can compare our value ( c.2 ) and the estimate ( c.1 ) with ( c.3 ) . the lamb shift and the casimir effect are probably the two most famous experimental evidences of zero - point energy fluctuations in the vacuum of quantum electrodynamics ( qed ) @xcite . both effects are rather weak , since the qed vacuum is mainly pt by origin , character and magnitude ( the corresponding fine structure constant is weak ) . however , even in this case attempts have been already made to exploit the casimir effect in order to `` observe '' the negative energy and related affects @xcite and even to release energy from the vacuum ( see , for example refs . @xcite and references in the above - mentioned reviews @xcite ) . in ref . @xcite by investigating the thermodynamical properties of the quantum vacuum it has been concluded that no energy can be extracted cyclically from the vacuum ( see , however ref . @xcite and references therein ) . let us also note that in qed the photon propagator always remains pt even `` dressed '' @xcite . so formally we can define the bag constant in this theory as @xmath185 , since @xmath186 in the effective potential approach to leading order @xcite . it would be interesting to perform such a calculation , which will give one a correct finite value of the ved in qed , if , or course , the above proposed definition of the qed bag constant makes sense . but it is beyond the scope of the present investigation , and should be done elsewhere . since the qcd fine structure constant is strong , the idea to exploit the qcd vacuum in order to extract energy from it seems to be more attractive . however , before discussing the ways how to extract , it is necessary to discuss which minimum / maximum amount of energy at all can be released in a single cycle . who thinks that it is too early to discuss such kind of topic ( though we do not think so ) may entirely skip this appendix . the bag constant calculated here is a manifestly gauge - invariant , real and colorless ( color - singlet ) quantity , i.e. , it can be considered as a physical quantity . in fact , in this paper we have formulated a renormalization program to make the bag constant or , equivalently , the bag pressure finite and satisfying all other necessary requirements ( see section 8 above ) . the key elements of this program were the necessary subtractions at all levels . moreover , one of its attractive features , as emphasized above , is that it is the energy density of the purely transversal virtual gluon field configurations which are not only stable ( no imaginary part ) , but are being in the stationary state as well , i.e. , in the state with the minimum of energy ( see fig . that is why it makes sense to discuss the `` releasing '' of the bag constant from the vacuum , more precisely the bag energy ( b.4 ) . from the quantum statistical mechanics point of view , the energy is nothing but the pressure multiplied by the volume @xmath187 in the infinite - volume limit @xcite . so the vacuum energy @xmath188 in terms of the bag constant is and in @xmath189 units it diverges as follows : @xmath190 since @xmath191 always when the dimensionless uv cutoff @xmath192 goes to infinity . evidently , in deriving eq . ( d.1 ) we use the general relation @xmath193 , which is valid in any units for energy ( see appendix b above ) . let us imagine now that we can release the finite portion @xmath171 ( b.4 ) from the vacuum in @xmath194 different places ( different `` vacuum energy releasing facilities '' ( verf ) ) . it can be done by @xmath195 times in each place , where @xmath196 . then the releasing energy @xmath197 becomes @xmath198 the ideal case ( which , however , will never be achieved ) is when we could extract a finite portion of the energy an infinite number of times and in an infinite number of places . so the releasing energy ( d.2 ) might be divergent as follows : @xmath199 since the sum over @xmath200 diverges quadratically in the @xmath201 limit , and @xmath202 in this case . the difference between the vacuum energy ( d.1 ) and the releasing energy ( d.3 ) which is nothing but the remaining in the vacuum energy @xmath203 becomes @xmath204 , \quad \lambda \rightarrow \infty,\ ] ] i.e. , the qcd vacuum is an infinite and permanent reservoir of energy . the situation is even `` better '' if one takes into account the pt contributions to the vacuum energy ( in this case the convergence becomes of the order @xmath205 in eq . 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using the effective potential approach for composite operators , we have formulated a general method of calculation of the truly non - perturbative yang - mills vacuum energy density ( this is , by definition , the bag constant apart from the sign ) . it is the main dynamical characteristic of the qcd ground state . our method allows one to make it free of the perturbative contributions ( contaminations ) , by construction . we also perform an actual numerical calculation of the bag constant for the confining effective charge . its choice uniquely defines the bag constant , which becomes free of all the types of the perturbative contributions now , as well as possessing many other desirable properties as colorless , gauge independence , etc . using further the trace anomaly relation , we develop a general formalism which makes it possible to relate the bag constant to the gluon condensate defined at the same @xmath0 function ( or , equivalently , effective charge ) which has been chosen for the calculation of the bag constant itself . our numerical result for it shows a good agreement with other phenomenological estimates of the gluon condensate . we have argued that the calculated bag constant may contribute to the dark energy density . its contribution is by 10 orders of magnitude better than the estimate from the higgs field s contribution . we also propose to consider the bag energy as a possible amount of energy which can be released from the qcd ground state by a single cycle . the qcd ground state is shown to be an infinite and hence a permanent reservoir of energy .

0708.0163

understanding quantum correlations and entanglement in many - body systems is one of the main purposes of fundamental physics @xcite . although a general strategy for this task currently lacks , in the last decades many advances have been made . in particular , one - dimensional ( 1d ) systems play a very special role in this scenario , for two reasons : the first is of physical nature , and resides in the enhancement of the importance of quantum fluctuations , due to dimensionality @xcite ; the second is the existence of extremely powerful analytical and numerical techniques , such as exact solutions @xcite , bosonization @xcite , bethe ansatz @xcite , and matrix - product - states algorithms @xcite , allowing for the extraction of accurate information about low - lying excitations in a non - perturbative way . within the bosonization framework , the strategy is to identify the relevant degrees of freedom of the considered 1d model and , starting from them , to build up an effective field theory capturing its low - energy physics @xcite . if the ground state ( gs ) of the original model is gapless , the obtained field theory is usually conformally invariant , and is therefore called conformal field theory ( cft ) @xcite ( we remark that bosonization is not the only approach leading to effective cft s ; see , e.g. , ref . @xcite ) . due to their solvability in @xmath0d , cft s are particularly useful , allowing for the exact computation of all correlation functions , and therefore providing access to many interesting quantities in a controllable way @xcite . in the present work we deal with a particular class of entanglement measures , the rnyi entanglement entropies ( ree s ) @xcite , defined in the following way . let us consider a pure state of an extended quantum system , associated with a density matrix @xmath1 , and a spatial bipartition of the system itself , say @xmath2 . if we are interested in computing quantities that are spatially restricted to @xmath3 , we can employ , instead of the full density matrix , the reduced density matrix @xmath4 . the @xmath5-th ree , defined as @xmath6 describes the reduced density matrix . in the limit @xmath7 it reproduces the von neumann entanglement entropy ( vnee ) @xmath8 $ ] , the most common entanglement measure @xcite . in the last decade , the @xmath9 ree s have also become quite popular , for several reasons . analytical methods proved to be more suitable to calculate ree s than vnee , especially in field theory : e.g. , in cft , ree s have a clear interpretation as partition functions , while the vnee does not @xcite . from a fundamental point of view , the knowledge of the ree s @xmath10 is equivalent to the knowledge of @xmath11 itself , since they are proportional to the momenta of @xmath11 . at the same time , from a physical point of view , many of the important properties of the vnee , e.g. , the area law for gapped states @xcite and the proportionality to the central charge for critical systems @xcite , carry over to ree s as well . in fact , ree s are easily computable by matrix - product - states algorithms @xcite , and they allow for a precise estimation of the central charge from numerical data regarding the gs of the system ( see , e.g. , ref . finally , and very importantly , measurements of the @xmath12 ree in an ultracold - atoms setup have been recently performed @xcite , paving the way to the experimental study of entanglement measures in many - body systems . the computations of refs . @xcite , that deal with the gs of cft s , have been extended , in more recent times , to excited states @xcite . in cft , excitations are in one - to - one correspondence with fields , and can be organized in conformal towers @xcite . the lowest - energy state of each tower is in correspondence with a so - called _ primary _ field , while the remaining ones , called _ descendant _ , are in correspondence with _ secondary _ fields , obtained from the primaries by the application of conformal generators . while the properties of ree s for primary states are now quite well understood @xcite , much less is known about the descendants . in the pioneering work of ref . @xcite , a unifying picture for the computation of ree s of primary and descendant states was developed and both analytical and numerical computations for the scaling limit of simple spin chains with periodic boundary conditions ( pbc ) were performed . a related but physically different problem was considered in refs . @xcite , where the effect on the time evolution of inserting secondary operators at finite time was studied . in this paper we continue the work in ref . @xcite and extend its framework to the case of open boundary conditions ( obc ) that preserve conformal invariance . this is an important step , for several reasons . first , impurity problems @xcite and certain problems in string theory @xcite map to boundary cft ; not secondarily , the experimentally achievable setups often involve obc . the importance of descendant states stems from novel applications , including non - trivial checks of the universality class of critical lattice models , and understanding the behavior of degenerate multiplets . we will provide a general strategy for the computation of ree s , derive cft predictions for descendant states , and compare them to the numerical data obtained from lattice realizations of the considered cft s . we will also discuss some interesting complementary aspects arising naturally when considering descendant states : e.g. , we will need to consider the ree s of linear combinations on the cft side , in order to study degenerate states in the xx chain . the paper is structured as follows . in section [ cft_sec ] we will review the approach of ref . @xcite for systems with pbc , derive the procedure to compute the ree s for descendant states in cft s with ( and without ) boundaries , and provide some general results for states related to the tower of the identity . in section [ ising_sec ] we will compute explicitly the ree s for the @xmath13 minimal cft , and compare the scalings to the numerical data obtained for the spin-1/2 ising model in a transverse field ; in section [ potts ] we will perform the same study for the @xmath14 minimal cft and the three - state potts model , while in section [ xx ] for the compactified free boson and the spin-1/2 xx chain . in section [ conc ] we will draw our conclusions and some directions for future work . in the appendixes we will provide useful technical details of our discussion . in the scaling limit , critical 1d lattice models can be described by cft s @xcite . since cft s are exactly solvable , explicit universal expressions can be derived for the ree s . in this section we describe a calculation scheme for the ree s of arbitrary excited states in cft unifying the cases of pbc and obc . we consider here only unitary theories ; see ref . @xcite for a generalization to the ground states of non - unitary models . the case of a finite system with pbc has been widely studied in the past ( see , e.g. , refs . @xcite ) . in this section , we review the computation of the ree s for excited states in this case , following the approach of refs . @xcite and modifying it slightly , having in mind the generalization to the obc case . we consider an euclidean space - time manifold of an infinite cylinder : it is characterized by the complex variable @xmath15 , being @xmath16 a space coordinate , and @xmath17 a time one ; @xmath15 and @xmath18 , where @xmath19 is the size of the system ( acting as an ir cutoff for the field theory ) , are identified . the physical support is the circle at @xmath20 ; the subsystem @xmath3 is chosen to be the interval @xmath21 $ ] . the zero - temperature density matrix of the system is pure , i.e. , @xmath22 , where @xmath23 is a generic eigenstate of the hamiltonian . the starting point of our computation is the following identity : @xmath24 where we repeatedly inserted resolutions of the identity for the hilbert spaces relative to the segments @xmath3 and @xmath25 . because of the state - operator correspondence in cft , each eigenstate is generated by an operator @xmath26 , acting on the vacuum and placed at the infinite past @xcite . adopting this representation , the overlaps above are nothing but path integrals on half of the infinite complex cylinder , with the insertions of @xmath26 and @xmath27 in the far past and future respectively , and with boundary states @xmath28 and @xmath29 along the segments @xmath3 and @xmath25 . when the sums in eq . ( [ replica_eq ] ) are performed we obtain a correlation function of @xmath26 and @xmath27 operators on the so - called _ replica manifold _ , consisting of @xmath5 copies of the cylinder , glued together cyclically across cuts along @xmath3 @xcite . each of the @xmath5 copies can be transformed to the complex plane by the conformal mapping @xmath30 @xcite : the resulting @xmath5-sheeted manifold , that will be denoted by @xmath31 , is a collection of @xmath5 planes glued across the boundaries @xmath32 according to @xmath33 , where @xmath34 is the arc of the unit circle @xmath35 , @xmath36 $ ] ( @xmath37 is the relative subsystem size ) , and @xmath38 is a radial infinitesimal vector ( related to , e.g. , the lattice spacing in the case of a lattice model ) , introduced in order to uv - regularize the theory . assuming the usual normalization for the state , i.e. , @xmath39 , @xmath40 reduces to a @xmath41-point function on the replica manifold , that we then transform into a @xmath41-point function on a single plane ( the following relation contains a non - trivial statement : see the end of this section for a clarification of this point ) : @xmath42 where the constant @xmath43 ( @xmath44 is the partition function over @xmath31 ) sets the correct normalization @xmath45 , and takes , in the present case , the form @xmath46^{\frac{c}{6}\left(\frac{1}{n}-n\right)},\ ] ] where @xmath47 is the central charge of the considered cft @xcite . the second expression in eq . ( [ path - int ] ) is obtained by means of the conformal mapping @xmath48 from @xmath31 to the complex plane , i.e. , a composition of a mbius transformation and the @xmath5-th root : @xmath49 the mbius transformation brings the cut along the arc @xmath34 to the half line @xmath50 $ ] ; then , the @xmath5-th root transforms each replica sheet to a slice of the complex plane . we prescribe the @xmath51-th sheet ( @xmath52 ) to be transformed by the @xmath51-th branch of the @xmath5-th root . the mapping for @xmath12 is represented graphically in figure [ transf ] . finally , the operator @xmath53 is transformed into @xmath54 under the mapping ( [ ct ] ) . the symbol @xmath55 is not to be confused with the twist operator appearing when the rnyi entropy is expressed as a quantity in a local theory @xcite . the transformed operator is inserted at the point @xmath56 ; for details see [ appa ] . regularizing the replica manifold for both pbc and obc the mesh represent lines of equal time and space coordinates . in the obc case only the solid lines are in the physical domain . the purple ( red ) line represents the left ( right ) edge of the obc strip . left panel : pre - transformed space - time . right panel : transformed space - time.,title="fig : " ] @xmath57 regularizing the replica manifold for both pbc and obc . the mesh represent lines of equal time and space coordinates . in the obc case only the solid lines are in the physical domain . the purple ( red ) line represents the left ( right ) edge of the obc strip . left panel : pre - transformed space - time . right panel : transformed space - time.,title="fig : " ] the normalization constant can be identified ( apart from a non - universal additive constant ; see eq . ( [ ccree ] ) ) with the gs contribution @xmath58 then if we rewrite the ree as @xmath59 the @xmath5-th ( exponentiated ) _ excess entanglement entropy _ ( eee ) @xmath60 corresponds to the @xmath41-point function @xmath61 interestingly , from the cft point of view , @xmath62 is intrinsically regular , since the cutoffs @xmath38 and @xmath19 only appear in multiplicative state - independent factors in the normalization @xmath63 . the eee is related to the relative rnyi entropy of the excited state compared to the vacuum ( for some recent general results on relative entropies in 2d cft see ref . the relative entropy is defined as @xmath64 the @xmath5-th excess and relative entropies compared to the vacuum ( i.e. @xmath65 ) are related by the following : @xmath66 . the appearing two - point functions can be calculated straightforwardly once the transformation laws are known . the best - known result that can be obtained from eq . ( [ path - int ] ) is the following formula for the ree s of the gs of a finite system with pbc @xcite . such state is associated with the identity operator , for which the correlation function in eq . ( [ path - int ] ) is @xmath67 and only the constant @xmath63 plays a role . explicitly , eq . ( [ lognn ] ) looks : @xmath68+c_n,\ ] ] where @xmath69 is a non - universal constant accounting for regularization . this formula has become , in the years , the most used tool in order to extract the central charge of a cft from finite - size numerical data : for most of the available algorithms , ree s are among the most natural quantities to compute @xcite . moreover , traditional numerical methods need , in order to compute the central charge , the knowledge of information about both gs and the first excitations @xcite , while eq . ( [ ccree ] ) just involves data from the former . if @xmath26 is not the identity , corrections can arise , their precise shape depending on the considered cft and the operator itself . such corrections can be computed using some results from meromorphic cft @xcite . first , the adjoint of a field can be obtained by transforming it according to the map @xmath70 @xcite ; moreover , a sequence of mappings @xmath71 has the same effect on any operator as the single transformation @xmath72 , since only derivatives , logarithmic derivatives and derivatives of the schwarzian derivative @xcite occur in the transformation laws ( see [ appa ] ) . therefore , using the relations @xmath73 ( we supposed the field to be real ) , the @xmath5-th eee reads @xmath74 the operators @xmath75 can be determined straightforwardly from @xmath26 . the generic transformation rule for secondary fields @xcite yields a sum of lower descendants in the same tower ( see [ appa ] for details ) : the resulting @xmath41-point functions of secondary fields are thus evaluated by relating them to @xmath41-point functions of primaries . the two are in general connected by the action of a complicated differential operator , which however can be determined ( case by case ) by means of a systematic approach ( see ref . @xcite and [ appb ] ) . following this program , the @xmath5-th eee finally looks @xmath76 where @xmath77 is the cited differential operator , and @xmath78 is the primary field in the tower of @xmath26 . in this work , we will need the explicit form of a number of such @xmath41-point functions of primary fields . in some cases , we will use , when available , known results from the literature ; in the remaining ones , we will compute the correlations by means of the coulomb - gas approach @xcite , or different representations of the considered cft allowing for their derivation . finally , we remark that the strategy described in this section and adopted in ref . @xcite is not exactly the one used in ref . @xcite for primary states of periodic systems . in particular , we treated the problem starting directly from a replica manifold of planes , instead of `` physical '' cylinders . descendant states , @xmath79 , are more naturally defined on the plane , where the physical hamiltonian takes the form @xmath80 and descendants are obtained by acting with strings of virasoro generators on primary states @xcite . furthermore , we are allowed to start from planes since a sequence of two conformal mappings using two given holomorphic functions is equivalent to one map under the composition of these functions . coming back to the physical manifold would be redundant , and more importantly , we observed that starting from the planes simplifies the actual computations substantially by removing uncomfortable infinities , that would need careful regularization . we will use this approach also for the case of obc . in cft , obc reduce the operator content of the theory , and for minimal models the obc preserving conformal invariance are in one - to - one correspondence with the primary fields of the theory on the plane @xcite . the partition function of a cft on the upper half plane , that is the prototypical boundary manifold , looks @xcite @xmath81 with @xmath82 and @xmath83 being conformal weights of primary fields , indexing the possible conformal obc , @xmath84 the fusion coefficients , and @xmath85 the character corresponding to the primary operator of chiral dimension @xmath86 . by expanding eq . ( [ z_obc ] ) around @xmath87 , a series is obtained , whose integer coefficients are the degeneracies of the energy levels . for obc , the physical space - time is an infinite strip of finite width @xmath19 , described , again , by the complex variable @xmath15 ; the subsystem @xmath3 is taken as the @xmath20 interval @xmath88 $ ] . the infinite strip can be mapped to the upper half plane by the transformation @xmath89 @xcite ; @xmath3 is then mapped to the unit arc @xmath34 connecting @xmath90 and @xmath91 , and the operators associated with the excitation are placed at the origin and at infinity . opportunistically , we set up our framework for the ree in the previous subsection in a way that it needs no modification for obc . we use the same mapping ( [ ct ] ) unifying in this case the @xmath5 half - planes to a single unit disk ( see figure [ transf ] ) . after the transformation , the operators lie on the boundaries of the disk separating arcs with different conformal obc . the resulting @xmath41-point functions can be evaluated using boundary cft : compared to the pbc case now one of the chiralities is suppressed and the chiral building blocks ( conformal blocks ) combine with different , boundary - dependent , coefficients . this can be understood considering that with obc present in the system , some fusion channels , open in the pbc case , are now closed and the operator product expansion coefficients @xcite determining the weight of the conformal blocks in the @xmath41-point functions change . in particular , in the fusion of the boundary operators @xmath92 and @xmath93 , changing the boundary condition at the insertion points from @xmath82 to @xmath83 and @xmath83 to @xmath94 , respectively , will only involve fields whose towers are present in the partition function @xmath95 ( see eq . ( [ z_obc ] ) ) . this fact will be very useful in the situations where we will have to decide which fusion channels are to be considered in order to compute the desired correlators . we now discuss the eee for states associated to fields in the tower of the identity . such states are present in any bulk cft , and are also very common for theories living on manifolds with boundaries . moreover , they have the property of depending explicitly on the central charge of the cft : they thus allow , in principle , the determination of @xmath47 from numerical data , generalizing what is usually done for the gs to the whole tower . as an example , we compute the @xmath12 eee for the first descendant in the tower , i.e. , the state associated with the stress - energy tensor @xmath96 ( @xmath97 form the virasoro algebra @xcite ) . its transformation under a conformal map @xmath98 is @xmath99^{2}\left(l _ { -2}\mathbb{i}\right)\left(f(\xi)\right)+\frac{c}{12}\left\{f,\xi\right\}\mathbb { i},\ ] ] being @xmath100 ^ 2/2 $ ] the schwarzian derivative of @xmath98 . we can thus write @xmath101 where @xmath102 is the collection of complex - plane four - point functions of the operators @xmath103 and @xmath104 inserted at the appropriate points ; the matrix @xmath105 contains the ( potentially vanishing ) coefficients of the different correlations , and it can be guessed from the transformation relations @xmath106\left(+e^{+i\pi x/2}\right),\\ \mathcal{t}_{f_{2,x}}\left(l_{-2}\mathbb{i}\right)(0_{2 } ) & = \sin^2(\pi x)\left[\frac{c}{8}\,\mathbb{i}-e^{i\pi x}\left(l_{-2}\mathbb{i}\right)\right]\left(-e^{+i\pi x/2}\right).}\end{aligned}\ ] ] to obtain @xmath107 , we employ the recipe described in [ appb ] and derive the two- , three- and four - point functions of the stress - energy tensor ( @xmath108 is self - adjoint ) : @xmath109 being @xmath110 . after substituting @xmath111 , performing the sums in eq . ( [ l2i4p ] ) with the appropriate coefficients @xmath105 , determined from ( [ ttr ] ) , and multiplying by the normalization @xmath112 for the state @xmath113 , we obtain @xmath114}{16}c^{-1}+ \cr & + \frac{16200\cos(2\pi x)-228\cos(4\pi x)+120\cos(6\pi x)+\cos(8\pi x)+16675}{32768}+ \cr & + \frac{\sin^{4}(\pi x)\left[\cos(2\pi x)+7\right]^{2}}{1024}c+\frac{\sin^{8}(\pi x)}{1024}c^{2},}\ ] ] that is what we will compare to numerical data in the next sections . before doing it , it is worth to analyze eq . ( [ f_t ] ) both as a function of the relative subsystem size @xmath115 and of the central charge @xmath47 . in fig . [ figt](a ) we show @xmath116 for different values of @xmath47 . we observe that the small - block eee is independent of @xmath47 , which can also be seen by expanding @xmath116 around @xmath117 : @xmath118 this behavior is in agreement with the holographic result of ref . @xcite , where it was established that the excess vnee is proportional to the excitation energy , i.e. , @xmath119 , and in particular it is independent of @xmath47 . in the opposite limit , the half - block eee is given by @xmath120 interestingly , this function features a minimum located at @xmath121 . for small @xmath47 , the whole function @xmath116 diverges , signaling that in a @xmath122 unitary cft only the vacuum state exists @xcite . when @xmath47 is large , we see from the curves of fig . [ figt](a ) that the leading term @xmath123 starts to dominate . this leading term emerges when taking only the contributions of the identity in the transformation laws ( [ transf_t ] ) . this is the regime where the ads / cft correspondence should play a role ( see , for a review , e.g. , ref . @xcite ) ; however , to our knowledge , this result is not yet available in that context ( again , we remark that results for the small - block limit are already available @xcite . ) while in the present work we will only use the result for the stress - energy tensor , we emphasize that by means of the general transformation rule described in [ appa ] a similar analysis can be carried out for the whole identity tower . e.g. , for the state related to @xmath124 we obtain @xmath125}{16384}c^{-1}+ \cr & + \frac{3032808 \cos ( 2 \pi x)+819919 \cos ( 4 \pi x)-27612 \cos ( 6 \pi x)}{8388608}+ \cr & + \frac{386 \cos ( 8 \pi x)+8436 \cos ( 10 \pi x)+289 \cos ( 12 \pi x)+4554382}{8388608}+\cr & + \frac{\sin ^4(\pi x ) \cos ^2(\pi x ) \left[255 \cos ( 2 \pi x)+90 \cos ( 4 \pi x)+17 \cos ( 6 \pi x)+1686\right]}{8192}c\cr & + \frac { \sin ^8(\pi x ) \cos ^4(\pi x)}{64}c^{2}.}\ ] ] in figs . [ figt](b ) and ( c ) we show the @xmath12 eee s for @xmath126 and @xmath127 and several values of the central charge . similarly to the case of @xmath128 , the higher descendants also show a @xmath47-independence for small blocks , in agreement with ref . @xcite . comparing the three panels of fig . 2 , we observe that , increasing the level of the excitation , the shape of the curves for @xmath129 ( these values of course include minimal models ) becomes less and less dependent on the actual value of the central charge . in this section , we present analytical predictions for the @xmath13 minimal cft , and numerical data relative to the @xmath12 ree for the descendant states of the spin-1/2 ising chain in a transverse field . the ree s for the primary states were already discussed in ref . @xcite ; here , we just focus on descendant states . the @xmath13 minimal model is one of the simplest cft s @xcite . the operator content of the model on the plane is the following : the primary fields are the identity and the fields @xmath130 and @xmath131 , of chiral dimension @xmath90 , @xmath132 , and @xmath133 , respectively . for minimal models on the upper half plane , the bc preserving conformal invariance are in one - to - one correspondence with such primary fields , leading , as we shall see , to four possible pairs of obc . in this section we will consider , in any case , the first descendant state in each conformal tower . a 1d lattice realization of the @xmath13 minimal cft is the spin-1/2 ising chain in a transverse field at the critical point @xcite . the hamiltonian of such chain is given by @xmath134 where @xmath135 is a pauli matrix ( @xmath136 ) at the site @xmath137 ; the critical point is located at @xmath138 . the three possible conformal obc are the following @xcite : the @xmath115-component of the boundary spins must be fixed to @xmath133 ( @xmath139 ) , @xmath140 ( @xmath141 ) or let free ( @xmath142 ) , in correspondence , respectively , with the @xmath143 , @xmath131 and @xmath130 primary fields . once combined , because of the @xmath144 symmetry of hamiltonian ( [ ising_h ] ) , there are just four independent situations , namely @xmath145 , @xmath146 , @xmath147 and @xmath148 , where the first symbol indicates the boundary condition chosen for the first spin and the second for the last . for @xmath148bc , the hamiltonian can be written in terms of free spinless fermions by means of a jordan - wigner transformation @xcite . in such case , it can be diagonalized in an exact way , exploiting the properties of free fermions ; consequently , the ree s can be computed exactly , following the recipe of refs . @xcite . in the remaining cases , because of the presence of the fixed obc , the jordan - wigner - transformed hamiltonian is not anymore quadratic at the boundaries , and the use of approximated techniques is a necessity . we perform the computations by means of the density - matrix renormalization group ( dmrg ) technique @xcite in its multi - target version @xcite : it allows for a straightforward computation of the ree s of the first excited states of the chain , as well as the implementation of the obc in an exact way @xcite . in any case , we consider systems up to @xmath149 sites ; in the dmrg calculations , we employ 7 finite - size sweeps and keep up to 64 states , achieving , in the last steps , a maximum truncation error of the order of @xmath150 . what stated in this paragraph also holds for the results of sections [ potts ] and [ xx ] . the obtained numerical data has , in any case , been tested in two independent ways : by comparing the numerical degeneracies of energy multiplets and the ones predicted by cft ( see , e.g. , eq . ( [ zising ] ) ) ; by computing the conformal weight of the considered states from the finite - size scaling ( fss ) of their energies , according to the cft formula @xcite @xmath151 being @xmath152 the energy of the state of weight @xmath86 at size @xmath19 , and @xmath153 the sound velocity of the system , known to be @xmath67 for the spin-1/2 ising chain in a transverse field and the spin-1/2 xx chain @xcite . for the three - state potts chain , the sound velocity has been extracted numerically from the finite - size scaling of the gs energy density with @xmath154 boundary conditions ( see section [ potts ] ) , i.e. , from @xcite @xmath155 where @xmath156 and @xmath157 are , respectively , the bulk and surface component of the energy density . for the three - state potts chain , we obtain @xmath158 ( not shown ) . to compare our results for model ( [ ising_h ] ) with the cft predictions we need to identify the low - energy states in the two frameworks . as pointed out by cardy @xcite , the operator content of the low - energy effective field theory is affected by the bc in the following way : @xmath159 we can expand these partition functions in powers of @xmath160 in order to determine the degeneracies of the excited states @xcite : @xmath161 the first descendant states are , in any case , non degenerate : this makes our task easier , since , in general , the dmrg algorithm , when dealing with degeneracies , considers a non - trivial linear combination in the multiplet . the analysis of such case will be unavoidable when we will study , in section [ xx ] , the spin-1/2 xx chain . we start the analysis with the @xmath145 case , where the only present tower is the one of the identity . therefore , the first descendant state is the one associated with the stress - energy tensor , and its @xmath12 eee takes the form in eq . ( [ f_t ] ) , with @xmath13 : @xmath162 we show , in figure [ ising_fig](a ) , the difference between the @xmath12 ree for the first excited state and for the gs : as shown in figure [ ising_fig](b ) and table [ table : ising_fit ] , a fss analysis confirms that , in the thermodynamic limit , the dmrg data approaches the cft prediction with great accuracy . .fitting result of the finite - size data for the transverse - field ising chain with obc . first column : considered bc ; second column : position in the energy spectrum ( e.g. , @xmath163 : first excited state ) ; third column : conformal weight of the considered state ; fourth column : value of @xmath115 at which the finite - size fite is performed ; fifth , sixth and seventh columns : estimated fit parameters ( by means of the formula : @xmath164 ) ; eighth column : relative deviation @xmath165 of @xmath166 from the cft prediction ; nineth column : corresponding figure . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] the agreement between the numerical data and the cft prediction is remarkable . we then consider the @xmath167bc case : the operator content of the theory consists in the tower of the primary field with @xmath168 , that we dub @xmath131 . in order to compute the @xmath12 eee we need the four - point function of @xmath131 operators : we compute it relying on the parafermionic representation of the potts model @xcite . in this representation , the field @xmath131 is the parafermionic current itself , and its four - point function can be inferred from a recurrence relation . here , importantly , @xmath131 is different from its adjoint . the final result reads @xmath169.\ ] ] by means of this correlation function , the resulting eee for the gs is easily computed : @xmath170 for the first descendant state , applying the technique of [ appb ] , we obtain @xmath171 such predictions are compared with the dmrg data in figs . [ pottsprimaries_fig](c ) and [ pottsdescendants_fig](c ) ( for the results of the fss analysis , see tables [ table : pottsprimaries_fit ] and [ table : pottsdescendants_fit ] ) . again , the agreement between the cft prediction and the numerical data is remarkable . we point out that in the fusion rule @xmath172 only the identity channel is present : accordingly , a different result for the same excited states but with different bc is not possible . we conclude the section with the @xmath173 case . in order to compute the @xmath12 eee , we need the four - point function of the exotic primary field with scaling dimension @xmath174 , that we call @xmath175 . this field occupies the @xmath176 position in the kac table and its four - point function can be calculated by the standard coulomb - gas approach @xcite . for the considered obc the result is @xmath177 where @xmath178 is the hypergeometric function . the conformal block above corresponds to the identity channel , that is the only one permitted by the fusion rule @xmath179 which we read off from the partition function @xmath180 , eq . ( [ pottsaa ] ) . the eee for the gs is then @xmath181 and for the first descendant state we obtain , using eq . ( [ eq : l-1 ] ) , @xmath182,}\ ] ] with @xmath183 . we compare the analytical predictions and the dmrg data in figs . [ pottsprimaries_fig](d ) and [ pottsdescendants_fig](d ) : after a fss analysis ( see tables [ table : pottsprimaries_fit ] and [ table : pottsdescendants_fit ] for quantitative details ) we find , again , a nice agreement between them . summarizing , we were able to find quantitative agreement between the cft analytical predictions for the @xmath12 eee s and the dmrg data for all of the considered conformal obc . the last cft we consider is the free boson @xcite , described by the lagrangian @xmath184,\ ] ] compactified on a circle , i.e. , @xmath185 , @xmath186 being the compactification radius . this cft is characterized by a central charge @xmath187 and is not minimal ; the one - to - one correspondence between primary fields and conformal obc is therefore not available . indeed , it is possible to show that the conformal obc are of two types , named dirichlet ( @xmath188 ) and neumann ( @xmath189 ) @xcite . a simple lattice realization of the compactified free boson is the spin-1/2 xx chain in the vanishing - magnetization ( half - filling ) sector @xcite . in order for it to realize the upper - half - plane cft , the hamiltonian to be considered is @xcite @xmath190 where @xmath191 ; for @xmath192 ( @xmath193 ) , @xmath188bc ( @xmath189bc ) are realized . there are three possible combinations of conformal bc : @xmath194bc , @xmath195bc and @xmath196bc , respectively associated to the partition functions @xcite @xmath197,}\end{aligned}\ ] ] where the last equality , proved in ref . @xcite , exploits the naive intuition of additivity of central charges @xcite . once expanded around @xmath87 , they look @xmath198 in the theories described by eqs . ( [ z_dd ] ) and ( [ z_nn ] ) , the primary fields are the vertex operators @xmath199 , @xmath200 at the compactification radii @xmath201 , @xmath202 and the derivative of the field , @xmath203 . the contributions to the ree s originating from them have already been studied in ref . @xcite . in both cases , the first secondary operators have weight @xmath204 , and come in multiplets ; therefore , the problem of degeneracies can not be avoided anymore . the case of eq . ( [ z_nd ] ) has to be treated in a different way , i.e. , by opportunely multiplying @xmath13 corrections , that have already been studied in ref . @xcite and in section [ ising_sec ] . the numerical data is produced by means of exact diagonalization for @xmath194bc ( in this case , the hamiltonian can be reduced to a free spinless - fermions one by means of a jordan - wigner transformation @xcite ) , and by means of the dmrg technique in the remaining cases . a chain of @xmath205 sites has been considered ( the fss analysis for the ree s is unnecessary in this case , for reasons that will be clear soon ) ; 7 finite - size sweeps and up to @xmath206 schmidt states have been employed ; a maximum truncation error of @xmath150 has been achieved in the last steps of the finite - system algorithm . we start by considering the case of @xmath194bc : as shown by eq . ( [ z_dd ] ) , the excitations at levels @xmath207 and @xmath204 appear in multiplets . first , we consider the fourth and the fifth excited states , corresponding to the operators @xmath208 . the method of [ appb ] allows to compute the corresponding @xmath12 eee , starting from the well - known four - point correlations of vertex operators . the final result is @xmath209 and is plotted , together with the numerical data , in figure [ xx_fig](a ) ( the numerically - computed ree is also the same for both states ; however , for linear combinations , it would be different ) . the agreement between the two approaches is manifest , since the finite - size corrections oscillate as a function of @xmath115 . this situation is typical of luttinger liquids , i.e. , free bosonic theories , as proven in many different situations ( see , e.g. , refs . @xcite ) . the sixth , seventh , eigth and nineth excitations are also degenerate , and possess the same conformal weight . however , the first and the second two display different behaviors , as shown in figs . [ xx_fig](b ) and [ xx_fig](c ) . in the cft picture , there are four operators with conformal dimension @xmath204 : @xmath210 , @xmath108 and @xmath211 . in particular , the first two lead to vanishing eee s @xcite . comparing the prediction for them to the numerical data in figure [ xx_fig](b ) , we see that these two operators identify the sixth and the seventh excitation of the chain . instead , the profiles of figure [ xx_fig](c ) are well reproduced by the linear combinations of operators @xmath212 , that lead to the @xmath12 eee @xmath213 in the case of @xmath195bc , the degenerate descendant quadruplet is at level 2 , and it is formed by the operators @xmath108 , @xmath211 and @xmath214 . the linear combinations of operators that correctly interpret the dmrg data are @xmath215 and @xmath216 . remarkably , the first linear combinations correspond to a bosonization of the stress - energy tensor of the @xmath13 free massless majorana theory @xcite : indeed , we were able to check exactly , by means of the general @xmath5-point function of vertex operators , that such combinations lead to the same functional form , eq . ( [ f_t_ising ] ) , for the @xmath12 eee as the stress - energy tensor in the @xmath13 minimal cft . the cft prediction is displayed , together with the dmrg data , in figure [ xx_fig](d ) ( the difference between the numerically - computed eee s for the two states just resides in a relative minus sign in the coefficient of the oscillating part ) , showing remarkable agreement . instead , the eee for the other two states originates from a four - point function involving both @xmath203 and vertex operators . once explicitly computed , it looks @xmath217 and the resulting eee s are @xmath218 \left[1558 + 439 \cos ( 2 \pi x)+26 \cos ( 4 \pi x)+25 \cos ( 6 \pi x)\right]}{16384}.\ ] ] the successful comparison with the dmrg results is performed in figure [ xx_fig](e ) . finally , we consider @xmath219bc . this case is different from the previous two , since the partition function can be written as a product of characters of the @xmath13 minimal cft @xcite . as shown by eq . ( [ z_nd ] ) , the first descendant state arises with a conformal weight @xmath220 , and stems from the operator @xmath221 . the @xmath12 eee is therefore @xmath222 as computed in section [ ising_sec ] and in ref . the comparison with the dmrg data is performed in figure [ xx_fig](f ) : the theoretical prediction and the numerical data match , up to the usual oscillating corrections and to an additive affleck - ludwig contribution , with @xmath223 , that is known to be associated to @xmath189bc @xcite . concluding , even in this case we were able to show that the cft low - energy picture captures the main features of the numerically computed eee s . in this work we extended the results of ref . @xcite on rnyi entanglement entropies of descendant states in conformal systems , to the case of conformal systems with boundaries . we provided a unifying approach applicable for both periodic and open boundary conditions and described the computation of the corrections to eq . ( [ ccree ] ) for excited states . we also proved that for any choice of boundary conditions the rnyi entanglement entropies are formally given by the same @xmath41-point functions ; the only difference stems from the suppression of one of the chiralities and the different ope coefficients realized with different boundary conditions ( e.g. , different conformal blocks solving the same differential equation will play a role with different boundary conditions ) . using this framework we explicitly computed the @xmath12 rnyi entanglement entropies for the first few excitation for three lattice models , belonging to different universality classes ( ising , xx or xxz , and three - state potts ) , and compared them with numerical data , obtained by means of the density - matrix - renormalization - group algorithm ( and , where available , of free - fermions representations ) . in all the considered cases , the agreement between analytical predictions and numerical calculations was found to be excellent . moreover , we were able to solve , for the first time , the problem of the rnyi entanglement entropies of degenerate energy multiplets , where different linear combinations are observed on the lattice and in the field theory . the study of rnyi entanglement entropies in many - body systems offers several directions for future work . for instance , the understanding of finite - size corrections to the conformal scalings , that we have seen to arise in all of the analyzed situation , although with very different features , is still very incomplete , and deserves further investigation @xcite . in addition , the appearance of the affleck - ludwig constant contributions as part of the excess rnyi entanglement entropies is a phenomenon that has been observed since many years @xcite , and its complete theoretical understanding is still lacking . on the other side , rnyi entanglement entropies are among the most natural quantities that can be computed by means of the currently most popular numerical methods @xcite . the present study could help in order to numerically identify the lattice realization of conformal boundary conditions : for the majority of critical lattice systems , the boundary conditions preserving conformal invariance are unknown . moreover , our approach allows , in pronciple , for the numerical identification of the conformal fields corresponding to specific lattice states , especially for degenerate energy multiplets . we therefore think that our study can stimulate future activity in this fertile research field . we thank g. sierra for important discussions , for stimulating our interest in the project and for a careful reading of the manuscript ; we thank m. i. berganza and g. takcs for valuable discussions . we acknowledge infn - cnaf for providing computational resources and support , and d. cesini in particular ; we thank s. sinigardi for technical support . l. t. acknowledges financial support from the eu integrated project siqs , while t. p. from the hungarian academy of sciences through grant no . lp2012 - 50 and a postdoctoral fellowship . we review in this appendix the recipe , introduced in ref . @xcite , needed in order to perform the conformal transformation of a generic field . for sake of clarity , only the chiral part of the field is considered . primary operators transform in a particularly simple way : the conformal mapping @xmath224 takes the field @xmath225 , with conformal weight @xmath86 , to @xmath226^h\phi(w).\label{eq : primary}\ ] ] the transformation rule for secondary fields is much less known , and much more complicated . it is given by @xmath227(f(z)),\ ] ] where @xmath228 is the usual @xmath51-th virasoro generator ( relative to the origin ) , while @xmath229 is the operator corresponding to the state @xmath230 , inserted at @xmath231 . the coefficients @xmath232 were identified , in ref . @xcite , to be generated by the expression @xmath233 \mathcal{o}(0)\vert0\rangle\cr&=\sum_{(p)}h_{(p)}[f , z)l_{p_{1}}\ldots l_{p_{k}}\mathcal{o}(0)\vert0\rangle,}\ ] ] where , in turn , the coefficients @xmath234 are defined recursively as @xmath235 with @xmath236 given in ref . @xcite , the first few being @xmath237 it is easy to check that @xmath238 is the schwarzian derivative of @xmath98 multiplied by @xmath239 , reproducing the familiar transformation law of the stress - energy tensor ( eq . ( [ transf_t ] ) ) : @xmath240^{2}t(f(z))+\frac{c}{12}\{f , z\}.\ ] ] another example is the transformation law of the field @xmath241 , with @xmath225 a primary field of weigth @xmath86 : @xmath242^{h+1 } \partial\phi(f(z))+hf''(z)\left[f'(z)\right]^{h-1}\phi(f(z)).\ ] ] the relation above can independently be deduced from eq . ( [ eq : primary ] ) using the chain rule of differentiation . we review in this appendix the strategy , developed in ref . @xcite , for evaluating a generic @xmath189-point correlation function of chiral secondary fields . the basic object that is considered is @xmath243 where @xmath244 is a generic secondary field , generated from a primary by the application of virasoro generators . ultimately , as shown below , such @xmath189-point function can be transformed into a sum of derivatives of the corresponding @xmath189-point function of primary fields . in the first part of the procedure the generator @xmath245 is removed from the @xmath246 operator alone . in order to perform this task , the contour - integral representation of the virasoro generators is used @xcite : @xmath247 where the subscript @xmath231 indicates that the contour encircles @xmath231 . after inserting this expression in eq . ( [ basicobject ] ) , the contour of integration can be deformed in order to enclose all the other poles of this integral , i.e. , the @xmath248 insertion points . this gives @xmath249\dots a_{n}(z_{n})\right\rangle- \cr & \qquad\phantom{= } -\left\langle a_{1}(z_{1})a_{2}(z_{2})\left[\oint_{z_{3}}\frac{d\zeta}{2\pi i}(\zeta - z_{1})^{n+1}t(\zeta)a_{3}(z_{3})\right]\dots a_{n}(z_{n})\right\rangle- \ldots } \ ] ] now , using the relation @xmath250 and so on , the integrals can be removed and replaced by virasoro generators acting on the operators @xmath251 : @xmath252 iterating this procedure , all the virasoro generators can be removed from @xmath246 , paying the price of complicating the remaining operators in the correlator : indeed , what is obtained is a finite sum of correlators , with the operator inserted at @xmath253 being primary . the next step consists in repeating the procedure above for @xmath254 , reducing it to a primary . however , at this point , virasoro generators with @xmath255 will appear again in front of the first operator , potentially followed by other virasoro generator of any order : this apparently could make all the previous efforts useless . actually , this is not the case , because of the following identity : @xmath256 @xmath225 is a primary field , indicating that the addition of virasoro generators in front of powers of @xmath257 results in a sequence of @xmath257 generators , i.e. , a partial derivative . the final result is then @xmath258 with proper @xmath259 coefficients and @xmath260 all primaries . therefore , everything is reduced to the computation of sums of derivatives of @xmath189-point functions of primaries , computable by means of , e.g. , the coulomb gas formalism @xcite in the case of minimal models . 99 i. affleck , _ conformal field theory approach to quantum impurity problems _ in g. morandi , p. sodano , a. tagliacozzo , and v. tognetti ( eds . ) , _ field theories for low - dimensional condensed matter systems _ ( springer , berlin , 2000 ) .

we discuss the rnyi entanglement entropies of descendant states in critical one - dimensional systems with boundaries , that map to boundary conformal field theories in the scaling limit . we unify the previous conformal - field - theory approaches to describe primary and descendant states in systems with both open and closed boundaries . we provide universal expressions for the first two descendants in the identity family . we apply our technique to critical systems belonging to different universality classes with non - trivial boundary conditions that preserve conformal invariance , and find excellent agreement with numerical results obtained for finite spin chains . we also demonstrate that entanglement entropies are a powerful tool to resolve degeneracy of higher excited states in critical lattice models .

1606.02667

the majority of low - mass stars emerge from their parental clouds surrounded by disks of 0.0010.3 m@xmath2 @xcite . at ages of a few myr , these disks appear to evolve rapidly from optically thick at near- and mid - infrared and detectable at ( sub)millimeter wavelengths , to undetectable at all wavelengths @xcite . the physics behind this transition and its timescale holds clues about the planet formation process @xcite . the previous references focus on inner disk material traced by infrared excess . much less is known about colder material further from the star , even though this encompasses the bulk of the mass . this letter investigates the presence of cold material around several members of the mbm 12 young association . judging from the relative occurrences of k- ( @xmath6% ) and l- and n - band ( @xmath7% ) infrared excess , the mbm 12 ( l1457 ) young association is suspected to be at the very stage where disks start to disappear @xcite . after several unsuccessful attempts ( @xcite , @xcite ) , @xcite recently reported detection at 1 and 2 mm of continuum emission around two or possibly three classical t tauri stars in this 15 myr old association @xcite , indicating the presence of @xmath8 m@xmath2 of cold material around each object . this letter increases the number of detections of cold dust to four objects ( [ s : results ] ) , including the recently identified edge - on disk source lkh@xmath3 263 c @xcite . by extending the wavelength coverage into the submillimeter we can fit the spectral energy distributions ( seds ) , and gain more robust disk - mass estimates and insight into grain growth ( [ s : models ] ) . the letter concludes with a discussion of the inferred mass range in terms of multiplicity @xcite and disk - dispersal models ( [ s : naturenurture ] ) . the observations were obtained with the _ submillimeter common user bolometer array _ ( scuba ) @xcite on the james clerk maxwell telescope ( jcmt ) on 2002 december 12 under excellent weather conditions . typical opacities at 225 ghz were 0.040.06 . we obtained photometry of four systems : lkh@xmath3 262 , the triple lkh@xmath3 263 ( abc ) , lkh@xmath3 264 ( a ) , and the triple s18 ( abab ) ; table [ t : obs ] lists coordinates and observing details . the employed two - bolometer photometry technique allowed for increased observing efficiency , with a chop throw of @xmath9 in naysmith coordinates . individual integrations of 30 s were repeated for the totals listed in the table . to confirm the unresolved nature of the emission we obtained a 64-point jiggle map centered on lkh@xmath3 262 also containing lkh@xmath3 263 ( abc ) . while providing spatial information , the jiggle map is less sensitive than the single - point photometry data . the standards hl tau , crl 618 , and uranus provided focus checks and flux calibrations ; the nearby quasar 0235 + 164 served as pointing source every @xmath10 hours . pointing accuracy was good with excursions of less than a few arcsec . however , uncorrected pointing errors can still affect the photometry in the @xmath11 beam at 450 @xmath1 m . in spite of a @xmath12 dither included in the photometry , 450 @xmath1 m fluxes of lkh@xmath3 262 from photometry are lower by 60% than those from the jiggle map ( [ s : results ] ) . we conclude that uncorrected pointing offsets and calibration uncertainties at short wavelengths due to the imperfect beam shape of the jcmt resulted in 450 @xmath1 m photometry results that are strict lower limits to the actual source flux . we include a + 60% error in the uncertainty of the reported 450 @xmath1 m results . the 850 @xmath1 m photometry and the jiggle maps are unaffected . all four objects , containing a total of eight ( known ) stars , show emission at 850 and 450 @xmath1 m ( table [ t : flux ] ) . the jiggle map ( fig . [ f : map ] ) indicates that the emission is unresolved and confined to the source position , and is not structure in the cloud . because of the higher noise level , only lkh@xmath3 262 is detected in fig . [ f : map ] while lkh@xmath3 263 ( abc ) remain undetected . the separation of @xmath13 between lkh@xmath3 262 and 263 ( abc ) is large enough that @xmath142 mjy spill over at both wavelengths is expected , based on archival beam profiles . high signal - to - noise photometry and jiggle - map data of lkh@xmath3 262 are consistent at 850 @xmath1 m but discrepant at 450 @xmath1 m with respective fluxes of 166.2 mjy and 263.1 mjy . uncorrected pointing and calibration errors at 450 @xmath1 m are likely to blame ( [ s : obs ] ) ; the jiggle - map flux is extracted after gaussian profile fitting and does not suffer from pointing offsets . the reconcile the measurements , we include a + 60% error bar in table [ t : flux ] and stress that the 450 @xmath1 m photometry values are strict lower limits . the 850450 @xmath1 m spectral indices @xmath3 consequently contain a large uncertainty ( table [ t : flux ] ) . at their high end , corresponding to the high end of allowed 450 @xmath1 m fluxes , the indices ( @xmath151.52.5 ) are consistent with emission from cool ( @xmath16 k ) and coagulated dust grains . the spectral index @xmath17 , where the index of the dust emissivity @xmath18 decreases when grains coagulate ( e.g. , @xcite ) and where the slope of the planck function @xmath19 falls below the value of 2.0 outside the rayleigh - jeans limit when @xmath20 k at 450 @xmath1 m . [ f : seds ] plots our data and values from @xcite , @xcite , and @xcite ; the + 60% error bars to the 450 @xmath1 m photometry are explicitly included . from these seds we can infer the mass of cold material . the 850 @xmath1 m fluxes trace the absolute amount of material , while the seds help to constrain important model parameters . we choose the flared - disk model of @xcite to describe the disks . we use the disk mass as our only free parameter . the disk temperature distribution is fixed by scaling to the stellar luminosity @xmath21 , where @xmath22 l@xmath2 @xcite . source luminosities @xmath23 are from @xcite . we neglect any changes in disk structure due to changes in mass and temperature . following recent reassessments of the distance to mbm 12 @xcite we adopt @xmath24 pc . we assume an average inclination of @xmath25 . because most of the flux comes from optically thin regions , only near edge - on orientations would change our results significantly . the final model parameter is the dust emissivity @xmath26 : its absolute value and its variation with wavelength . often , @xmath26 is parameterized as @xmath27 with @xmath28 @xmath29 g@xmath30 ( dust ) and @xmath31 hz @xcite . from disk sed fitting , low values of @xmath18 are commonly found , 01 @xcite and interpreted as evidence for grain growth ; @xcite point out that grain mineralogy can affect @xmath18 and explain the low values . more accurate descriptions of @xmath32 follow from calculations of grain growth in astrophysical environments employing realistic mineralogies ( e.g. , @xcite ) . these models circumvent having to choose a value for @xmath33 , but still show show significant variation . we adopt a dual approach : we fit the seds with a parameterized @xmath26 to constrain @xmath18 and learn about possible grain growth , and then fit the seds with several physical grain - growth models to derive accurate disk masses with realistic uncertainties . with @xmath34 , @xmath35 minimization of the sed fit yields @xmath36 for lkh@xmath3 262 and @xmath37 for lkh@xmath3 264 ; the error bars for lkh@xmath3 263 and s18 are too large to constrain @xmath18 . for this fit we have used all detections and upper limits at wavelengths @xmath38 @xmath1 m . at the wavelengths of the infrared n - band and shorter , the flux is dominated by very small amounts of hot material and may contain contributions from silicate emission ; both depend strongly on the model parameters . this does not affect our results : @xmath14% of the 450 @xmath1 m flux originates from the @xmath39 au region that contributes @xmath40% of the 10 @xmath1 m emission . the inferred values @xmath36 and @xmath41 are similar to those commonly found in t tauri disks , and we interpret them as indications for grain growth in the mbm 12 disks . few of the grain - growth models of @xcite and @xcite have @xmath42 . of the models by @xcite , only strongly coagulated grains without ice mantles show such low @xmath18 . the authors note that such grains are unlikely in disk environments and we follow their advice and avoid this particular class . of the models by @xcite , @xmath42 is only found for average grain sizes @xmath43 cm . table [ t : flux ] lists the range of disk masses found by @xmath35 minimization . these vary from 0.0014 - 0.012 m@xmath2 for lkh@xmath3 263 to 0.0230.23 m@xmath2 for lkh@xmath3 262 . the allowed range for each source reflects the uncertainty in the dust opacity models . these these ranges overlap , but differences in disk mass between sources are robust if we assume similar dust properties in all objects . our modeling makes no assumptions about the opacity . we find that @xmath44% of the 450 @xmath1 m flux and @xmath45% at 1.3 mm originates from opaque regions ( @xmath46 . only much smaller disks ( @xmath47 au ) have @xmath48% of the 450 @xmath1 m flux coming from optically thick regions ( 45% at 1.3 mm ) . for such disks the sed slope can be explained by opacity instead of grain growth , and submillimeter fluxes no longer trace mass . only spatially resolved data can settle this issue . @xcite infer 0.0018 m@xmath2 for lkh@xmath3 263 c from modeling the scattered light , depending on the assumed dust properties . our values of 0.00140.012 m@xmath2 are consistent with their findings . an unknown fraction of our inferred mass may reside in disks around the companions a and b. spatially resolved data and detailed modeling are required to further characterize the disk of lkh@xmath3 263 c. @xmath4co 21 measurements @xcite indicate that co is depleted with respect to the dark cloud value of co / h@xmath49=@xmath50 by factors up to 300 . large depletion levels are found in disks around t tauri stars ( e.g. , @xcite ) and expected theoretically ( e.g. , @xcite ) . although our sample is small , with only eight stars in four unresolved systems , interesting conclusions can be reached about the disk evolution . our detections show that at least the classical t tauri stars in mbm 12 ( 15 myr ) have disks in the same mass range as the younger ( @xmath51 myr ) taurus and @xmath52 ophiuchus regions ( 0.0010.3 m@xmath2 ; @xcite ) . while @xcite shows that 0/95 stars in ic 348 ( @xmath10 myr ) have disk masses @xmath53 m@xmath2 , two out of four of our mbm 12 systems do ( or @xmath54% of all twelve mbm 12 systems ) ; in taurus , a comparable fraction of 14% of stars have disks @xmath53 m@xmath2 . @xcite argues that the k- and l - band excess suggest significant disk dispersal in mbm 12 . our sample does not directly address this issue because 3/4 objects have k - band excess ( s18 does not ) . @xcite showed that binaries with separations @xmath55100 au have reduced disk mass or no disks , compared to wider binaries and single stars . our data follow that trend , taking into account that the relative disk masses are robust ( [ s : models ] ) : lkh@xmath3 262 : @xmath56 au and 0.0230.23 m@xmath2 ; lkh@xmath3 264 a : @xmath57 au and 0.030.18 m@xmath2 ; s18 abab : @xmath58 au ( a b ) , @xmath59 au ( ba bb ) and 0.0050.025 m@xmath2 ; and lkh@xmath3 263 abc : @xmath60 au ( ab c ) , @xmath61 au ( a b ) and 0.00140.012 m@xmath2 ( binary separations from @xcite , @xcite and @xcite ) . this suggests that the presence of a close binary ( environment ) determines the total amount of disk material . if environment determines ( initial ) disk mass , can we still infer anything about disk dispersal from the amount of material ? in a viscouly evolving disk , disk mass and the accretion rate onto the central star are linked @xcite . [ f : evol ] plots ` evolutionary tracks ' of disks with initial masses of 0.001 , 0.01 , and 0.1 m@xmath2 around a 0.5 m@xmath2 star , following @xcite . this model assumes a standard @xmath3-disk description with @xmath62 , a central star with @xmath63 m@xmath2 and @xmath64 r@xmath2 , and a constant ionizing flux from the star with time - dependent contribution from accretion . the disk removal rate only depends weakly on the stellar parameters , and any uncertainties are insignificant compared to our observational error bars . muzerolle et al . ( in prep . ) report estimates of mass accretion rates onto three of our objects from br@xmath19 line measurements , using the calibration of @xcite . they find upper limits of ( 45)@xmath65 m@xmath2 yr@xmath30 for lkh@xmath3 263 and 264 , and a rate of @xmath66 m@xmath2 yr@xmath30 for s18 . these rates are comparable to those inferred for ic 348 @xcite and at the low end of the range found in taurus , which has a median of @xmath67 m@xmath2 yr@xmath30 . the errors include uncertainties in the observations and the stellar mass , assumed to be 0.5 m@xmath2 , but not the distance . because both accretion rate and disk mass depend on the distance squared , its uncertainty does not enter into the values of fig . [ f : evol ] ; @xmath68 also depends on @xmath69 , which we assume constant . the disk mass and accretion rate of s18 are consistent with the dispersal model for the age range of mbm 12 ( 15 myr ; shaded in fig . [ f : evol ] ) , suggesting that the object has lost one - third of its initial disk mass of @xmath70 m@xmath2 . the upper limit on the accretion rate of lkh@xmath3 264 also agrees with the model at 15 myr . the agreement improves if the disk mass is decreased somewhat , which is expected when dust grains have indeed grown to a few hundred @xmath1 m as suggested by the sed . grain sizes comparable to observing wavelength are efficient emitters , reducing the required mass to fit the observed flux . the limits on lkh@xmath3 263 do not provide useful constraints . in summary , we conclude that the classical t tauri stars in mbm 12 still have significant reservoirs of cold dust in circumstellar disk . there are indications for grain growth up to several hundred @xmath1 m in these disks , but spatially resolved observations are required to rule out opacity as an explanation for the flat spectral slopes . and while differences in disk mass are likely dominated by environment ( binary separation ; ` nature ' ) , available accretion rates and detected disk masses are consistent with a disk dispersal scenario ( ` nurture ' ) with one - third of the mass already lost . we thank our tsss j. kemp and j. hoge for excellent support during our observations . the staff of the jcmt and the jac are thanked for their hospitality . rj acknowledges support in part through nasa origins grant nag5 - 11905 . i.m . acknowledges support from the national reserach council , canada . the referee is thanked for a careful reading of our manuscript and insightful comments . , w. s. , robson , e. i. , gear , w. k. , cunningham , c. r. , lightfoot , j. f. , jenness , t. , ivison , r. j. , stevens , j. a. , ade , p. a. r. , griffin , m. j. , duncan , w. d. , murphy , j. a. , & naylor , d. a. 1999 , , 303 , 659 lrrrr lkh@xmath3262 & 02 56 07.9 & 20 03 25 & photom & 1200 + & & & jiggle & 4800 + lkh@xmath3263 abc & 02 56 08.7 & 20 03 41 & photom & 4800 + lkh@xmath3264 a & 02 56 37.5 & 20 05 38 & photom & 1200 + s18 abab & 03 02 21.1 & 17 10 35 & photom & 1200 + lrrrr lkh@xmath3 262 ( photom ) & @xmath71 & @xmath72 & 0.671.64 & 0.0230.23 + lkh@xmath3 262 ( jiggle ) & @xmath73 & @xmath74 & 1.351.84 & + lkh@xmath3 263abc & @xmath75 & @xmath76 & 1.163.17 & 0.00140.012 + lkh@xmath3 264a & @xmath77 & @xmath78 & 0.581.48 & 0.030.18 + s18 abab & @xmath79 & @xmath80 & 1.202.74 & 0.0050.025 +

we report detection of continuum emission at @xmath0 and 450 @xmath1 m from disks around four classical t tauri stars in the mbm 12 ( l1457 ) young association . using a simple model we infer masses of 0.00140.012 m@xmath2 for the disk of lkh@xmath3 263 abc , 0.0050.021 m@xmath2 for s18 abab , 0.030.18 m@xmath2 for lkh@xmath3 264 a , and 0.0230.23 m@xmath2 for lkh@xmath3 262 . the disk mass found for lkh@xmath3 263 abc is consistent with the 0.0018 m@xmath2 inferred from the scattered light image of the edge - on disk around component c. comparison to earlier @xmath4co line observations indicates co depletion by up to a factor 300 with respect to dark - cloud values . the spectral energy distributions ( sed ) suggest grain growth , possibly to sizes of a few hundred @xmath1 m , but our spatially unresolved data can not rule out opacity as an explanation for the sed shape . our observations show that these t tauri stars are still surrounded by significant reservoirs of cold material at an age of 15 myr . we conclude that the observed differences in disk mass are likely explained by binary separation affecting the initial value . with available accretion rate estimates we find that our data are consistent with theoretical expectations for viscously evolving disks having decreased their masses by @xmath5% .

astro-ph0307483

a relativistic collimated outflow ( known as a jet ) emerging from a black hole ( bh ) is ubiquitously observed in various spatial scales from stellar mass ( @xmath5 ) to supermassive ( @xmath6 ) bhs . polarimetry in the optical and near - infrared ( nir ) bands is a powerful method to unveil the emission mechanism and investigate the magnetic field structure inside the jet ( e.g. , * ? ? ? * ) . in active galactic nuclei ( agn ) and gamma - ray burst ( grb ) jets , it is well accepted that the optical and nir lights are produced by high - energy electrons through optically - thin synchrotron emission . high degrees of linear polarization in the optical / nir band observed from agn and grb jets ( @xmath7% up to 3040% ; see e.g. , * ? ? ? * ; * ? ? ? * ) indicate that the synchrotron process is operating , with highly ordered magnetic fields in the emission regions . measurements of polarization position angle ( pa ) are also useful to determine the magnetic field direction at the emission region . in addition , the detection of a polarization pa swing may manifest from the presence of a helical magnetic field along a jet or curved structure indicating the global jet geometry ( e.g. * ? ? ? * ; * ? ? ? * ) . it is also known that the jet emission in stellar - mass bhs appear in the nir band . while emission in the optical - band is dominated by the accretion disk , an excess with respect to the rayleigh - jeans tail of the disk blackbody component is often found in the nir - band . hence , if the nir emission is due to _ optically - thin _ synchrotron emission , a high polarization degree ( pd ) is theoretically expected . however , reliable optical / nir polarimetric measurements for galactic bh binary jets are still very limited mainly due to the difficulty in eliminating the interstellar polarization caused by large amounts of dust clouds in our galaxy in the target directions . another reason for the paucity of intrinsic polarization measurements is that the objects are often not sufficiently bright to perform polarimetry . in this regard , we note that clear nir polarization was detected from cyg x-2 and sco x-1 based on spectro - polarimetry @xcite . an opportunity to study the bh binary jet through optical / nir polarimetry was presented when v404 cygni ( a.k.a . , gs 2023 + 338 ; hereafter v404 cyg ) produced an exceptionally bright outburst in june 2015 . this object is one of the famous low - mass x - ray binaries ( lmxbs ) because a similar huge outburst was detected in 1989 with intensive multi - wavelength observations performed at that time ( e.g. , * ? ? ? the distance is accurately determined as @xmath8 kpc from parallax measurement using astrometric vlbi observations @xcite . in this paper , aimed at studying the non - thermal jet emission and constraining the physical parameters in a microquasar jet , we present results of linear polarization measurements in the optical and nir bands for v404 cyg during the brightest outburst in june 2015 performed by the _ 1.5 m and _ pirka _ 1.6 m telescopes in japan . observations and data reductions are described in [ sec - obs ] . we show the results in [ sec - results ] , and the implications of our findings are presented in [ sec - dis ] . we performed simultaneous optical and nir imaging polarimetry for v404 cyg using the hiroshima optical and near ir camera ( honir ; * ? ? ? * ) mounted on the _ kanata _ 1.5-m telescope in higashi - hiroshima , japan . the data presented here were taken on mjd 57193 and 57194 . the honir polarization measurements utilize a rotatable half - wave plate and a wollaston prism . we selected @xmath9 and @xmath1 bands as the two simultaneous observing filters . to study the wavelength dependence of polarization properties for the target , we also took @xmath10 imaging polarimetric data on mjd 57194.5057194.55 ( the `` c '' subscripts are herein suppressed ) . each observation consisted of a set of four exposures at half - wave plate position angles of 00 , 225 , 450 , and 675 . typical exposures in each frame were 30 s and 15 s for the @xmath0- and @xmath1-bands , respectively , but these exposures were sometimes modified depending on weather conditions ( decreased when seeing became good , and increased when cirrus clouds passed over the target ) . @xmath11- and @xmath12-band exposures on mjd 57194 were 30 s and 15 s , respectively . to calibrate these data , we observed standard stars on mjd 57195 that are known to be unpolarized ( hd 154892 ) and strongly - polarized ( hd 154445 and hd 155197 ; * ? ? ? * ; * ? ? ? we thereby confirmed that the instrumental pd is less than 0.2% and determined the instrumental polarization pa against the celestial coordinate grid . absolute flux calibration , which is needed to construct the optical and nir seds , was performed by observing standard stars on the photometric night mjd 57200 . we also performed optical @xmath0-band imaging polarimetry monitoring for v404 cyg using the multi - spectral imager ( msi ; watanabe et al . 2012 ) mounted on the 1.6-m pirka telescope located in hokkaido , japan , from mjd 5719057193 . the msi observations were performed in a similar manner to the _ kanata_/honir ones , namely the msi utilizes a rotatable half - wave plate and a wollaston prism , and a series of four exposures were taken for each polarization measurement . the typical exposure time of each frame was 15 s. to remove the instrumental polarization ( @xmath13% ) and to calibrate the polarization pa , we used past msi data of the two unpolarized stars ( bd+32 3739 and hd 212311 ; * ? ? ? * ) and three strongly - polarized stars ( hd 154445 , hd 155197 , and hd 204827 ; * ? ? ? * ) obtained on mjd 57167 and 57169 . we also confirmed the polarization efficiency of @xmath14% using a polarizer and flat - field lamp . we analyzed x - ray data for v404 cyg taken with xrt onboard the _ swift _ satellite using heasoft version 6.16 . swift_/xrt data analyzed here were taken on mjd 57193 and 57194 ( observation ids : 00031403040 and 00031403046 ) , which were almost simultaneous ( within less than 1 hour ) with the _ kanata_/honir multi - band photo - polarimetry data . clean events of grade 012 within the source rectangle region ( because the observation was in window - timing mode ) were selected . after subtracting background counts selected from both sides of the source with rectangle shapes , the 0.510 kev events were utilized for spectroscopy . we generated ancillary response files with the ` xrtmkarf ` tool . using xspec version 12.8.2 , we roughly fit the data by assuming a disk blackbody plus power - law model , both modulated by galactic absorption ( i.e. , ` wabs*(diskbb+pow ` ) ) , and converted the deabsorbed spectra to @xmath15 fluxes . [ fig : lc ] ( _ top _ panel ) shows the @xmath16-band light curve of v404 cyg during the bright outburst @xcite after the detection of burst - like activities by _ swift_/bat , _ fermi_/gbm , and _ maxi_/gsc on 2015 june 15 ( mjd 57188 ) @xcite . the optical flux increased by @xmath17 mag compared to the quiescent level ( @xmath18 mag , * ? ? ? * ) with a maximum around mjd 57194 and the highest flux level continued for about one week . during this brightest phase , the source showed large - amplitude ( as much as 3 mag ) and short - time variability . [ fig : lc ] ( two _ bottom _ panels ) illustrate intra - night variations of the @xmath0- and @xmath1-band fluxes , polarization degrees ( pds ) , and polarization position angles ( pas ) measured by _ kanata_/honir and _ pirka_/msi on mjd 57193 and 57194 ( corresponding to 2015 june 19 and 20 ) . note that the _ kanata_/honir @xmath0- and @xmath1-band photometric and polarimetric observations are strictly simultaneous . on mjd 57193 , the honir observations were interrupted by cloudy weather and stopped around mjd 57193.64 , while _ pirka_/msi continuously obtained @xmath0-band photometric and polarimetric data over @xmath19 hours . on the whole , the simultaneous @xmath0- and @xmath1-band light curves showed similar temporal variations . however , around mjd 57193.54 , a flux increase is evident only in the @xmath1 band , while no corresponding enhancement was observed in the @xmath0-band . during this nir flare , the @xmath1-band pd and pa did not show any significant variation despite the pronounced flux change . the @xmath1-band pd and pa were constant at @xmath20% and @xmath21 , respectively , throughout the honir observations that night . that same night , the @xmath0-band light curve showed a rapid and large - amplitude decrease around mjd 57193.64 and then gradually recovered . during the optical dip , the pd and pa were constant and did not show any significant variations . similarly , the @xmath0-band pds and pas measured by _ pirka_/msi on this night remained constant at @xmath22% and @xmath23 , respectively . note that there is a small discrepancy between the _ pirka_/msi and _ kanata_/honir @xmath0-band polarimetric results ( see table [ tab : pol ] ) . this could be due to a lack of cross - calibration but our subsequent discussion is unaffected by this small difference . on the next night , the observing conditions were relatively good until mjd 57194.7 and we obtained @xmath24-hours of continuous , simultaneous @xmath0- and @xmath1-band photometric and polarimetric data for v404 cyg . as shown in fig . [ fig : lc ] ( _ bottom - right _ panel ) , the honir photometric light curves in @xmath0- and @xmath1 bands exhibited quite similar temporal profiles , including the two ` dips ' around mjd 57194.68 . the pds and pas in both bands were again constant over the @xmath24-hour duration , even over the course of two observed flux dips . in addition , the pds and pas in each band were unchanged from those measured on the previous night ( mjd 57193 ) . we note the sporadic nature of the @xmath1-band polarimetric data points ( namely , pd and pa ) after mjd 57194.64 were due to the passage of cirrus clouds ( nir observations are more heavily affected by clouds compared to the optical ) . finally , we note that in addition to the photo - polarimetric data presented here we also obtained _ pirka_/msi @xmath0-band data on mjd 57190 , 57191 , and 57192 . as shown in table [ tab : pol ] , these data showed that the polarization parameters of the object remained constant at @xmath25 and @xmath26 over the three nights despite dramatic variability of the total flux of @xmath27 mag . to investigate the polarization properties of the sky region in the direction of v404 cyg , we also analyzed the _ kanata_/honir @xmath0-band data taken on mjd 57194 ( selected because the observing condition was much better compared to mjd 57193 ) and determined the pds and pas for the brightest field stars within the honir field - of - view ( fov ) . the results are displayed in fig . [ fig : syuui ] . we found that the pas of not only v404 cyg , but also the surrounding objects , showed almost the same direction . moreover , the measured pds were also observed at similar levels . these findings clearly indicate that , despite the relatively large pd of @xmath28% for v404 cyg , local dust clouds located between v404 cyg and the earth are the likely cause of the polarized emission in this sky direction and about half of the surrounding objects ( including v404 cyg ) are located beyond the dust cloud . this suggests the observed pd and pa for v404 cyg is _ not _ intrinsic _ but _ interstellar origin . the hypothesis is supported by the non - variable pds and pas for v404 cyg observed even during the large flux variations ( see two bottom panels in fig . [ fig : lc ] ) . we also plot in figure [ fig : versus ] the measured pa as a function of pd for each object within the fov . two clusterings of the data are clearly visible : objects with very small pd and a wide pa range over 180 are likely located in front of local dust clouds , while those with relatively large pds of typically @xmath29 and broadly similar pas of 030 are positioned beyond the dust clouds . we note that the slightly greater pd of v404 cyg ( @xmath30 ) with respect to the surrounding objects ( @xmath29 ) implies a small level of intrinsic polarization for v404 cyg of at most a few percent . furthermore , we show the @xmath31-band pds and pas of v404 cyg in fig . [ fig : multiband ] indicating a steep pd decrease toward longer wavelengths with constant pas over the six observation bands . this polarization behavior is similar to that of a highly reddened star , suggesting that the polarization is interstellar origin . detailed study of the interstellar dust based on these multi band polarimetric data will be reported in a forthcoming paper ( itoh et al . in preparation ) . from these observational results , we consider the measured polarization of v404 cyg is predominantly contaminated by interstellar dust between the object and the earth . the low intrinsic pd ( less than a few percent ) implies that the optical and nir emissions are dominated by either disk or optically - thick synchrotron emission , or both . the simultaneous _ kanata_/honir @xmath0- and @xmath1-band light curves showed almost the same temporal evolution , except the orphan @xmath1-band flare which peaked around mjd 57193.54 and lasted for @xmath2 mins ( see fig . [ fig : lc ] ) . apart from this orphan flare ( which is discussed in detail later ) , the quite similar @xmath0- and @xmath1-band light curves naturally leads to an interpretation that the nir and optical emissions come from the same component ( or have the same origin ) . there are two possible options to explain the optical and nir emissions : one is a disk origin and the other is from a jet . to gauge which is the most plausible , we constructed a broadband spectrum of v404 cyg from the radio to x - ray bands in @xmath32-@xmath33 representation ( fig.[fig : fnu ] ) . note that the radio fluxes are not simultaneous , while the _ kanata _ and _ swift_/xrt data were obtained within a one - hour timespan . the _ kanata _ optical and nir fluxes were dereddened by assuming @xmath34 and @xmath35 . the @xmath36 value of 4.0 was derived by @xcite from the spectral type of companion star and @xmath37 colors , which was also confirmed by subsequent studies ( e.g. , * ? ? ? * ; * ? ? ? after we corrected the observed fluxes for extinction using @xmath34 , we found a slightly rising , but almost flat ( @xmath38 ) shape in the optical and nir spectrum . this implies optically - thick synchrotron emission from an outer jet . indeed , the @xmath39 ghz radio spectrum obtained about 1.5 days before our observation ( on mjd 57191.95 ) can be smoothly extrapolated to the _ kanata_/honir spectral data assuming a @xmath40 form ( see fig [ fig : fnu ] ) . however , the flat optical / nir spectral shape can also be interpreted in a disk model ( see e.g. , * ? ? ? * extended data figure 6 therein ) . thus , it is difficult to determine the optical / nir emission mechanism solely from its spectral shape . no evidence of intrinsic linear polarization in the @xmath0- and @xmath1-bands are allowed in both scenarios because both optically - thick synchrotron and blackbody radiation only generate weak linear polarization of order @xmath41% . here , we focus on the orphan @xmath1-band flare which lasted for only @xmath2 mins at mjd 57193.54 . the observed red color and short duration imply synchrotron emission from a jet as the most plausible origin of the flare . indeed , the @xmath1-band peak flux of the flare reached @xmath27 jy , which was the same level measured during the giant radio and sub - mm flares observed by ratan-600 and sub millimeter array on mjd 57198.933 and mjd 57195.55 @xcite , respectively . as shown in fig . [ fig : fnu ] ( _ left _ panel ) , an extrapolation of the radio spectrum observed during the giant flare at ghz - frequencies on mjd 57198.933 nicely connects to the @xmath1-band peak flux by assuming a flat spectral shape ( i.e. , @xmath42 ) . more interestingly , even during the orphan flare , the @xmath1-band pd remains showed no significant temporal variation , indicating the nir emission is not strongly polarized . this result would be reasonably understood if the jet synchrotron emission in the @xmath1 band is still in the optically - thick regime . we therefore conjecture that this orphan @xmath1-band flare is produced by optically - thick synchrotron emission from an outer jet . if the optically - thick synchrotron emission extends up to the @xmath0-band with a flat spectral shape and the baseline @xmath0-band flux is not as high as the flaring component of @xmath27 jy ( after extinction correction , assuming @xmath34 ) , it significantly contributes to the @xmath0-band flux as well , making the flare visible also in the @xmath0-band light curve . a spectral break between the @xmath1 and @xmath0 bands of the flaring emission component caused by the transition of synchrotron emission from optically - thick to optically - thin regimes , if present , would make the contribution of the flaring component negligible with respect to the baseline flux , as observed . the quasi - simultaneous @xmath33 _ kanata_/honir and _ swift_/xrt spectra measured on mjd 57194.52 , together with the ratan-600 non - simultaneous radio fluxes are also shown . the radio , optical / nir , and x - ray spectra would be reasonably understood as optically - thick synchrotron emission from the outer jet , disk emission , and disk plus corona emissions , respectively . to constrain the jet parameters and physical quantities in the emission region , we attempted to model the spectral energy distribution ( sed ) of v404 cyg by using a one - zone synchrotron plus synchrotron self - compton ( ssc ) model @xcite , which is widely used for blazar sed modeling ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? we show in fig . [ fig : nufnu ] ( _ left _ panel ) the quasi - simultaneous broadband sed of v404 cyg during the @xmath1-band flare . this modeling assumes that a single emission region is located at the inner - most part of the jet . hence , the optically - thick synchrotron emission from an outer jet , which has a flat spectrum of @xmath43 observed in the radio up to nir band is not modeled , while the optically - thin synchrotron and ssc emissions at optical frequencies and higher are fitted . however , in the current case , we now know that the optical and x - ray emission are from a disk and disk plus corona , respectively . we therefore regard the _ kanata_/honir , _ swift_/xrt , and _ integral _ data points ( taken from fig . 3 of * ? ? ? * ) as upper limits for the jet emission . another constraint comes from the _ kanata_/honir observation that the @xmath1-band emission is not significantly polarized even during the orphan flare . this indicates that the @xmath1-band emission is still in the optically - thick regime and that the break frequency ( defined as @xmath44 ) is due to synchrotron self absorption ( ssa ) . the transition from the optically - thick to optically - thin regime is above the @xmath1-frequency band , thus @xmath45 hz , and we adopt a value of @xmath46 hz . we also assume that the synchrotron peak flux is 2 jy as observed by _ kanata_/honir ( and also by ratan-600 on mjd 57198.933 ) . we can then derive the magnetic field @xmath47 and the size of the emission region @xmath0 by using the standard formulae for synchrotron absorption coefficient and emissitivity ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , @xmath48 where @xmath49 is the power - law index of electron distribution ( see below ) and @xmath50 is the distance to v404 cyg . here we assumed almost equipartition between magnetic field and electron energy density , which was confirmed by the following sed modeling . the electron energy distribution is assumed to have a single power - law shape with exponential cutoff as @xmath51 for @xmath52 , where @xmath53 is the electron lorentz factor , @xmath54 is the electron normalization , @xmath49 is the power - law index , @xmath55 is the cutoff energy , @xmath56 and @xmath57 are the minimum and maximum electron energies and are respectively set to 1 and @xmath58 . the jet inclination angle of v404 cyg is estimated as @xmath59 ( e.g. , * ? ? ? * ; * ? ? ? * ) and assuming a jet velocity of @xmath60 , the corresponding doppler beaming factor is @xmath61^{-1 } \sim 0.9 $ ] . we can therefore safely neglect relativistic beaming effects . by changing the parameters of the electron energy distribution , we calculated the resultant synchrotron and ssc emissions . the calculated model curves are shown in fig . [ fig : nufnu ] ( _ left _ panel ) and all the model parameters are tabulated in table [ tab : model ] . the derived parameters for the electron energy distribution are @xmath62 , @xmath63 , and @xmath64 . importantly , the _ swift_/xrt data allowed us to constrain the cutoff energy as @xmath65 because larger @xmath55 violates these upper limits in the soft x - ray band . this implies that particle acceleration in this microquasar jet is not very efficient . the electron energy distribution cutoff energy is determined by the balance between acceleration and cooling times , where the acceleration time is defined as @xmath66 by using an electron energy @xmath67 and parameter @xmath68 , the number of gyrations an electron makes while doubling its energy ( e.g. , * ? ? ? * ; * ? ? ? since dominant cooling processes for electrons of @xmath69 are both synchrotron and ssc , the cooling time is estimated as @xmath70 , where @xmath71 and @xmath72 are the energy densities of magnetic field and synchrotron photons , respectively ( e.g. , * ? ? ? thus , we obtain @xmath73 by setting @xmath74 and @xmath75 g. this is much larger than @xmath76 in blazar jets ( e.g. , * ? ? ? * ) , indicating much longer acceleration times and inefficient acceleration in this microquasar jet . note that the electron power - law index of 2.2 we obtained from sed modeling has already been modified by rapid synchrotron and ssc cooling . in the current situation , because the magnetic field is strong ( @xmath77 g ) and emission region is small ( @xmath78 cm ) , we need to consider the following three energy loss processes : adiabatic cooling ( this is also equivalent to particle escape from emission region ) , synchrotron cooling , and ssc cooling . note here that we can neglect synchrotron cooling for electrons of @xmath79 because the optically - thick regime is below @xmath80 ( e.g. , * ? ? ? cooling timescales for these processes are estimated as @xmath81 and @xmath82 , respectively ( e.g. , * ? ? ? * ; * ? ? ? therefore , high - energy electrons of @xmath83 rapidly lose their energy via ssc emission and hence the electron power - law index becomes steeper by one power of @xmath67 , if injection of high - energy emitting electrons continued over a few tens of minutes ( which corresponds to the flare duration observed by _ kanata_/honir ) . namely , the electron energy distribution at @xmath83 is already in a fast - cooling regime , which indicates that the original ( or injected ) electron power - law index is 1.2 . this is much smaller than the standard power - law index of 2.0 derived by the first - order fermi acceleration theory ( e.g. , * ? ? ? we can now derive the total energy in electrons and magnetic field as @xmath84 erg and @xmath85 erg , respectively , thus the jet is poynting - flux dominated by a factor of @xmath19 . the jet power in electrons ( @xmath86 ) and magnetic field ( @xmath87 ) is calculated as @xmath88 , where @xmath89 is the electron energy density , @xmath90 is assumed , and the factor of 2 is due to the assumption of a two - sided jet ( e.g. , * ? ? ? then , we obtain @xmath91 erg s@xmath92 and @xmath93 erg s@xmath92 , and the summed power ( @xmath94 ) amounts to @xmath95 erg s@xmath92 . on the other hand , we can also calculate the total radiated power using the sed modeling result as @xmath96 erg s@xmath92 , which is larger than the summed @xmath97 erg s@xmath92 . this indicates that the poynting flux ( @xmath87 ) and @xmath86 are not sufficient to explain @xmath98 and another form of power is required . the simplest and most probable solution is to assume that the jet contains enough protons which have larger power than @xmath98 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this is another ( though indirect ) evidence of a baryon component in a microquasar jet . note that we reached the above conclusion based on jet energetics argument , but the baryonic jet in a microquasar has already been claimed by a different , independent method based on the detection of blue - shifted emission lines in the x - ray spectra for ss 433 and 4u 1630@xmath9947 @xcite . by assuming that the jet contains one cold proton per one relativistic ( emitting ) electron , we can derive the total energy of cold protons as @xmath100 erg , where @xmath101 is the proton mass . this corresponds to the cold proton power ( @xmath102 ) of @xmath103 erg s@xmath92 by using the relation of @xmath104 ( e.g. , * ? ? ? * ) , where @xmath105 is the energy density of cold protons and @xmath90 is assumed . we therefore obtain the total jet power @xmath106 and radiative efficiency of the jet as @xmath107 ( see also table [ tab : model ] ) . during the moderate and steady state on mjd 57194 , there were no observational constraints on @xmath44 due to the dominance of the disk component in the optical and nir bands , hence we assume that _ it remained the same as that during the bright flare , _ @xmath46 hz . we estimated the synchrotron peak flux as 0.2 jy , because such a flux level was observed in the ghz band during the high state @xcite and an extrapolation to the nir band with a flat shape ( @xmath108 ) seems reasonable . we thereby obtained the following estimates of @xmath109 g and @xmath110 cm ( see equations ( [ eq : mag ] ) and ( [ eq : size ] ) ) . important information about the non - thermal jet emission , which should be included in the sed modeling , comes from the _ integral _ detection of an additional power - law component of @xmath111 in the hard x - ray band @xcite . the _ kanata_/honir , _ swift_/xrt , and _ integral _ ( exponential cutoff power - law component dominant up to @xmath112 kev ) data points are treated as upper limits . based on these assumptions and the multi - wavelength data , we calculated the broadband non - thermal jet emission by accelerated electrons using the one - zone synchrotron and ssc model . the result is shown by solid lines in fig . [ fig : nufnu ] ( _ right _ panel ) and the model parameters are tabulated in table [ tab : model ] . we obtained the same parameter values of @xmath63 and @xmath64 for the electron energy distribution , but the electron normalization @xmath113 is smaller due to the fainter jet flux , as was observed in the radio band . we also found @xmath114 , suggesting again the jet is slightly poynting - flux dominated . more interestingly , the total radiated power is again larger than the summed electron and magnetic field powers in the jet ( see table [ tab : model ] ) , implying the presence of a baryonic component even during the fainter state . 1 . the ssa frequency and peak flux density enable us to estimate the magnetic field strength and size of the emission region by assuming equipartition between magnetic field and relativistic electrons . the derived magnetic field of @xmath115 gauss is much stronger , and size of the emission region of @xmath116 cm much smaller , compared to agn jets ( typically @xmath117 gauss and @xmath118 cm , see e.g. , @xcite ) . based on modeling of the broadband spectrum of v404 cyg , we found an upper limit to the cutoff lorentz factor of electrons of @xmath112 . because the cutoff is determined by the balance of the acceleration and cooling times , this result implies a longer acceleration time of @xmath73 in this microquasar jet , suggesting electron acceleration is much less efficient compared to agn jets ( that typically show @xmath76 ) . the original ( or injected ) power - law index of the electron energy distribution was derived as @xmath119 . in the sed modeling of agn jets , electrons are assumed to have a broken power - law shape . the power - law index below the break lorentz factor @xmath120 ( typically @xmath121 ) is estimated as @xmath122 ( e.g. , * ? ? ? hence , @xmath119 derived here in v404 cyg jet is comparable to that derived in agn jets , implying that same acceleration mechanism operates in these different systems . 4 . to account for the total radiated power of the jet of v404 cyg , a cold proton component is required inside the jet . this is the same situation as in agn jets . 5 . during the bright flare on mjd 57193 , the jet radiative efficiency ( @xmath123 ) is derived as @xmath124 . this is roughly comparable to that of agn and grb jets ( @xmath125 , see e.g. , @xcite ) . we appreciate the anonymous referee s constructive comments that helped to improve the manuscript . we thank dr . trushkin and dr . rodriguez for providing us with their ratan-600 and _ integral _ data , respectively . this work is supported by the optical & near - infrared astronomy inter - university cooperation program , and the mext of japan . we acknowledge with thanks the variable star observations from the aavso international database contributed by observers worldwide and used in this research . ytt is supported by kakenhi 15k17652 . mu is supported by kakenhi 25120007 . ccc is supported at nrl by nasa dpr s-15633-y . mw is supported by kakenhi 25707007 .

we present simultaneous optical and near - infrared ( nir ) polarimetric results for the black hole binary v404 cyg spanning the duration of its 7-day long optically - brightest phase of its 2015 june outburst . the simultaneous @xmath0 and @xmath1-band light curves showed almost the same temporal variation except for the isolated ( @xmath2 min duration ) orphan @xmath1-band flare observed at mjd 57193.54 . we did not find any significant temporal variation of polarization degree ( pd ) and position angle ( pa ) in both @xmath0 and @xmath1 bands throughout our observations , including the duration of the orphan nir flare . we show that the observed pd and pa are predominantly interstellar in origin by comparing the v404 cyg polarimetric results with those of the surrounding sources within the @xmath3 field - of - view . the low intrinsic pd ( less than a few percent ) implies that the optical and nir emissions are dominated by either disk or optically - thick synchrotron emission , or both . we also present the broadband spectra of v404 cyg during the orphan nir flare and a relatively faint and steady state by including quasi - simultaneous _ swift_/xrt and _ integral _ fluxes . by adopting a single - zone synchrotron plus inverse - compton model as widely used in modeling of blazars , we constrained the parameters of a putative jet . because the jet synchrotron component can not exceed the _ swift_/xrt disk / corona flux , the cutoff lorentz factor in the electron energy distribution is constrained to be @xmath4 , suggesting particle acceleration is less efficient in this microquasar jet outburst compared to agn jets . we also suggest that the loading of the baryon component inside the jet is inevitable based on energetic arguments .

1601.01312

a compact , nonempty set @xmath0 is called `` self - affine '' if it is the attractor of an iterated function system of affine maps . self - affine sets represent a natural class of fractal sets . on one hand , they are a natural generalization of self - similar sets . on the other hand , they appear in other areas , like theory of tilings and dynamical systems . they also represent a prototype for attractors of general , ( smooth ) non - linear i.f.s . despite these facts , they remain rather mysterious , and the study of their dimensional and topological properties is fraught with difficulties . one of the most important results available is falconer s theorem from 1988 @xcite . it states that for every collection @xmath1 of linear endomorphisms of @xmath2 such that @xmath3 for all @xmath4 , there exists a number @xmath5 -called the `` falconer dimension '' of @xmath6- such that for almost every @xmath7 ( in the sense of @xmath8 dimensional lebesgue measure ) , the attractor of the i.f.s . @xmath9 has hausdorff and box - counting dimensions equal to @xmath10 . there is an explicit , albeit difficult to compute , formula for @xmath10 . ( falconer s original version goes with @xmath11 as the bound for the norms ; later solomyak @xcite pointed out that @xmath12 works too ) . the hypothesis on the norms can be somewhat relaxed , but is essential . the simplest counterexample comes from the following family of self - affine sets , studied by f.przytycki and m.urbaski @xcite : for @xmath13 , let @xmath14 be the linear map given by @xmath15 let @xmath16 be the attractor of the i.f.s . @xmath17 . when @xmath18 these sets are easy to analyze , but the situation becomes much more complicated when @xmath19 . recall that the bernoulli convolution @xmath20 is defined as the distribution measure of the random sum @xmath21 , where the signs are chosen indepently with probabilty @xmath12 ; see @xcite , @xcite for further information on bernoulli convolutions . for us , the main feature of bernoulli convolutions is the following theorem of solomyak @xcite : for almost all @xmath22 , @xmath20 is an abolustely continuous measure with an @xmath23 density . the result of przytycki and urbaski is the following : assume that @xmath16 is totally disconnected , and @xmath24 . if @xmath20 has hausdorff dimension @xmath25 ( which , by solomyak s theorem , is the case for almost every @xmath26 ) then @xmath27 however , if @xmath28 is a pisot number ( i.e. , an algebraic number greater than @xmath25 all of whose algebraic conjugates have modulus less than @xmath25 ) , then @xmath29 . it is well known that the set of pisot number accumulates to @xmath30 ; thus this implies that the norm bound in falconer s theorem is sharp . the result of przytycki and urbaski suggests the following question : what can we say about the dimension and topological properties of @xmath16 in general ; i.e. allowing @xmath16 to be connected ? we will show that the same formula for the dimension holds for almost every @xmath31 in the natural region . more precisely , we have : [ th : main ] for almost @xmath31 such that @xmath32 , @xmath33 where @xmath34 . note that a direct connection between absolute continuity of @xmath35 and dimension of @xmath16 is lost . this is natural since for countably many values of @xmath31 there is an exact coincidence of cylinders which produces a dimension drop . the condition @xmath36 is also a natural one ; for @xmath37 one would expect @xmath16 to have positive lebesgue measure , and even non - empty interior . unfortunately , since transversality holds only in a small region inside @xmath38 , our results here are rather limited . it is convenient to state the result in terms of measures . let @xmath39 be the natural self - affine measure supported on @xmath16 ; it can be defined in several ways , for instance as the distribution of the random sum @xmath40 where signs are chosen independently with probability @xmath12 . note the close analogy with bernoulli convolutions ; we think of @xmath39 as _ self - affine _ bernoulli convolutions . [ th : mainbis ] there is an open set @xmath41 containing a neighborhood of @xmath42 and of the curve @xmath43 such that 1 . for almost all @xmath44 , @xmath39 is absolutely continuous with an @xmath23 density . in particular , @xmath45 ( we will denote @xmath46-dimensional lebesgue measure by @xmath47 ) . 2 . for almost all @xmath31 such that @xmath48 for some @xmath49 , @xmath39 is absolutely continuous with a continuous density . in particular , @xmath16 has nonempty interior . see corollary [ coro : abscontregion ] for the precise definition of @xmath50 . we remark that our results apply to more general families of self - affine sets , although the theorems above illustrate our main motivation and example . see theorems [ th : generalth ] and [ th : generalthbis ] for the general versions , as well as the extensions and further generalizations presented in section [ sec : extensions ] . we stress , however , that we are considering only families of self - affine sets where all the defining maps share the same linear part , which is a diagonalizable map . the method used to prove theorems [ th : main ] and [ th : mainbis ] is based on the transversality ideas that were successfully applied to many families of self - similar sets , starting with @xcite ; however , some new ideas are needed as well . in particular , we emphasize that the powerful projection scheme developed in @xcite does not seem to apply in this context , for two reasons . first , we have to deal with two different hlder exponents simultaneously ; second , and more important , the standard notion of transversality does not hold in a large enough region . to overcome the second problem we use transversality concurrently with absolute continuity of bernoulli convolutions , rather than transversality alone . finally , in section [ sec : extensions ] we consider some additional questions suggested by what is known in the self - similar case . we study some families of exceptions to the almost - everywhere results ; most of them are closely related to pisot numbers . this is to be expected , since for classical bernoulli convolutions reciprocals of pisot numbers are the only known parameters that yield a singular measure . according to theorem [ th : main ] , overlaps do not produce a dimension drop in the region @xmath51 except for a set of zero measure . we show that they do produce a measure drop ( i.e. the hausdorff measure in the critical dimension is @xmath52 ) in a big chunk of the overlapping region . this phenomenon was first observed for families of self - similar sets . it is well known that if a measure is the attractor of a general i.f.s . then it is either singular or absolutely continuous with respect to lebesgue measure . a more delicate question is whether , when absolutely continuous , it is actually equivalent to lebesgue measure on the attractor set . this is known to be true for self - similar sets @xcite , and here we show that the proof in the self - similar case can be adapted to cover many self - affine measures including the natural measures on @xmath16 . in this section we obtain our main results on hausdorff and box - counting dimension of certain certain families of self - affine sets , of which the class discussed in the introduction is a particular example . let @xmath53 be a set of real numbers which we call `` digits '' . we will normalize @xmath54 so that @xmath55 . associated to @xmath54 is a family of linear self - similar sets @xmath56 for @xmath57 , where @xmath58 is the attractor of the i.f.s . @xmath59 . furthermore , let @xmath20 be the natural self - similar measure supported on @xmath58 ; i.e. the probability measure defined by the relation @xmath60 where @xmath61 . now consider a two - dimensional version of this construction , where the i.f.s . is @xmath62 let @xmath16 be the attractor set , and @xmath39 the natural self - affine measure supported on it . these will be our main object of study in this paper . see figure 1 for some examples . * figure 1*. both figures correspond to the digit set @xmath63 . although both pictures look similar , the one on the left actually corresponds to an attractor of dimension strictly less than @xmath30 , while for parameters close to the one on the right we know that the attractor typically has positive lebesgue measure . in general , self - affine sets are harder to visualize than self - similar sets due to the fact that , after some iterations , cylinders sets have a very large excentricity . our most important example is the set of digits @xmath64 . in this case , @xmath65 and @xmath66 are the classical and self - affine bernoulli convolutions respectively . more precisely , the measure defined by ( [ eq : randomsum ] ) corresponds to the digit set @xmath67 , but the digits @xmath63 yield the same measures up to rescaling and translating . for the sake of simplicity we will use the digits @xmath63 at all places except where we are dealing with the fourier transform , for which the digits @xmath67 yield a slightly simpler formula . next we define a relevant class of power series . let @xmath68 let also @xmath69 be the subset of @xmath70 of power series with non - zero constant term . moreover , define @xmath71 as @xmath72 ; in other words , @xmath71 is the set of power series whose first non - zero coefficient is @xmath73 . we will identify ( subsets of ) @xmath70 with ( subsets of ) the symbolic space @xmath74 . as a matter of notational convenience we will often write @xmath75 instead of @xmath76 , and @xmath77 instead of @xmath78 . when we do so , we will use greek letters such as @xmath79 instead of @xmath80 . we will need a notion of transversality for power series . our definition is very close to the standard one ; however , we need to consider subsets of the class @xmath69 of power series rather than all of @xmath69 . [ def : transv ] let @xmath81 be a subset of @xmath69 . we say that @xmath82 is a set of transversality for @xmath81 if there exists a constant @xmath83 such that @xmath84 for every @xmath85 and @xmath86 . if a set @xmath87 can be expressed as @xmath88 for some @xmath89 , we say that @xmath90 is a set of transversality for @xmath91 if it is a set of transversality for @xmath81 . one easy but important observation is that to prove transversality it is enough to show if @xmath92 then @xmath93 for all @xmath94 and some @xmath95 . let @xmath96 be the uniform bernoulli measure on @xmath97 ; i.e. @xmath98 the projection map @xmath99 from @xmath97 to @xmath100 maps @xmath96 onto the bernoulli convolution @xmath20 . the measure @xmath96 induces a measure @xmath101 on @xmath102 via the difference map . in other words , @xmath101 is the bernoulli measure on @xmath70 where the symbol @xmath103 has weight @xmath104 in particular , @xmath105 . let @xmath106 be the shift operator on @xmath70 . finally , let us define some relevant sets . let @xmath107 throughout the paper we will assume that @xmath108 and @xmath26 have the same signs , although actually all the results and proofs are valid regardless of signs ( the conditions on @xmath108 and @xmath26 have to be replaced by conditions on their absolute values ; for example @xmath109 becomes @xmath110 ) . we are now in a position to state a technical proposition that contains our key estimate ; in particular , it is only here that transversality gets used . [ prop : main ] let @xmath111 be closed intervals such that @xmath112 . 1 . fix @xmath113 . suppose that for some @xmath114 there is a constant @xmath115 such that @xmath116 for all @xmath86 . assume also that @xmath117 is an interval of transversality for @xmath118 . let @xmath119 , and @xmath120 then @xmath121 2 . analogously , assume that ( [ eq : corrbound ] ) holds with @xmath122 replaced by @xmath123 for some @xmath124 , and that @xmath125 is an interval of transversality for @xmath118 . let @xmath126 and @xmath54 as in ( [ eq : dimformula ] ) . then @xmath127 _ proof_. we will prove only the first part of the proposition ; the second is just a restatement with @xmath108 and @xmath26 interchanged . let @xmath128 denote the inner integral in ( [ eq : keyestimate ] ) . using fubini s theorem and performing the change of variables @xmath129 we get @xmath130 where we used the identity @xmath131 . suppose that @xmath132 , and observe that @xmath133 whence @xmath134 therefore we obtain from ( [ eq : tech1 ] ) that @xmath135 write @xmath136 for the integrand in the right hand side of ( [ eq : tech2 ] ) , and define @xmath137 note that the integral in ( [ eq : keyestimate ] ) is equal to @xmath138 so our problem is reduced to estimating the series @xmath139 and @xmath140 . recall that @xmath141 . using this , ( [ eq : corrbound ] ) and the definition of @xmath101 we see that @xmath142 furthermore , since @xmath101 is a product measure , @xmath143 from here and fubini s theorem we compute @xmath144 where in the last step we used that @xmath145 and @xmath146 . therefore @xmath147 . it remains to show that @xmath148 . at this point we make use of the transversality hypothesis . let @xmath132 . we use ( [ eq : transv ] ) applied to the map @xmath149 to estimate @xmath150 using this , fubini s theorem and ( [ eq : tech3 ] ) we obtain @xmath151 where we used that @xmath152 and , for the convergence of the last integral , that @xmath153 . this completes the proof . @xmath154 in order to apply the previous lemma to obtain information about hausdorff dimension we need to establish transversality . the following easy lemma reduces this problem to the estimation of roots of power series . [ lemma : transv ] let @xmath155 be closed intervals such that the following holds : if @xmath156 and @xmath157 , then @xmath108 and @xmath26 are not both double roots of @xmath80 . then @xmath69 can be partitioned into two disjoint subsets @xmath158 such that @xmath159 are sets of transversality for @xmath160 respectively . _ proof_. since @xmath69 is a normal family and @xmath111 are closed , the following number is well defined and , by hypothesis , positive : @xmath161 let @xmath162 let also @xmath163 , and note that @xmath164 the lemma is now clear from the definition of transversality . @xmath154 of course , in order to effectively use the previous lemma we need some information on the location of the roots of the power series in @xmath69 . this is provided by the following result , which is a simple modification of theorem 2 in @xcite ( or rather the more general version stated after theorem 4 ) . [ th : transvest ] let @xmath165 if @xmath166 are roots of @xmath157 , counted with multiplicity , then @xmath167 in particular , if @xmath168 then the hypothesis of lemma [ lemma : transv ] holds . _ proof_. the core of the proof follows closely the proof of theorem 2 in @xcite . let @xmath169 , and let @xmath170 ; note that @xmath171 is a monic power series with coefficients bounded by @xmath172 . let @xmath173 a straightforward calculation shows that @xmath174 , where @xmath175 denotes @xmath23 norm on the unit circle . fix @xmath176 and let @xmath177 . then , using jensen s formula and jensen s inequality we obtain @xmath178 therefore @xmath179 taking @xmath180 yields ( [ eq : transvest1 ] ) , while setting @xmath181 and @xmath182 immediately gives ( [ eq : transvest2 ] ) . @xmath154 we will now state the main result of this section . theorem [ th : main ] ( in fact a more general version ) will be obtained as a corollary . we start by recalling the definition of _ lower correlation dimension _ of a measure @xmath183 on @xmath2 : @xmath184 if the above limit exists , we say that the correlation dimension exists and is given by the limiting value . [ th : generalth ] assume that : 1 . there is an open interval @xmath185 such that @xmath20 has lower correlation dimension @xmath25 for almost every @xmath186 ; 2 . there is an open interval @xmath187 such that if @xmath157 , @xmath80 has no double roots on @xmath90 ; 3 . there exists an open region @xmath188 such that if @xmath189 , @xmath157 , then @xmath108 and @xmath26 are not both double roots of @xmath80 . let @xmath190 then for almost all @xmath191 , @xmath192 before proving this theorem , some remarks are in order : 1 . correlation dimension is known to exist for arbitrary self - similar measures , see @xcite . 2 . one can only hope for this theorem to be valid in @xmath193 . outside this region falconer s dimension has another expression and one would expect falconer s dimension to coincide with hausdorff dimension for almost every parameter . therefore , one would like to make @xmath194 and @xmath188 as large as possible ; we will see that the case were the digits are equally spaced we do have @xmath195 ( note that for this to hold it is necessary that @xmath196 ) . it is well known that in the self - similar case , if @xmath90 is as the statement of the theorem , then @xmath20 is absolutely continuous for almost all @xmath197 ( see @xcite , theorem 4.3 for a proof ) ; in particular , @xmath198 . however , in order to obtain non - trivial results , @xmath199 has to be larger than @xmath90 . in the case of bernoulli convolutions , for example , solomyak s theorem implies that @xmath200 while @xmath90 is a much smaller interval . _ proof of theorem [ th : generalth]_. let us denote the right hand side of ( [ eq : dimform ] ) by @xmath201 . the inequality @xmath202 is standard for _ all _ @xmath203 ( this is a particular case of the upper bound in falconer s theorem ) . therefore it is enough to show that for every @xmath204 and for all @xmath205 there are closed ( non - degenerate ) intervals @xmath206 such that @xmath207 for almost all @xmath208 . we henceforth fix @xmath204 , @xmath209 and choose @xmath210 such that : * @xmath211 , @xmath212 . * @xmath213 for every @xmath214 . * @xmath215 or @xmath216 . if @xmath217 , let @xmath218 be the partition of @xmath70 given by lemma [ lemma : transv ] . otherwise , let @xmath219 . note that @xmath70 is a set of transversality for @xmath90 ( and so is trivially @xmath220 ) . this follows from the observation after definition [ def : transv ] and the fact that @xmath70 is a normal family . let @xmath221 if @xmath222 , define @xmath223 analogously ; otherwise , let @xmath224 . note that by definition of correlation dimension , @xmath225 therefore it suffices to verify ( [ eq : generalth1 ] ) for almost every @xmath226 and then let @xmath227 . we fix @xmath228 for the rest of the proof . we recall the well - known frostman s lemma : if there exists a radon measure @xmath183 supported on a compact set @xmath229 such that @xmath230 then @xmath231 . recall that @xmath96 is the uniform bernoulli measure on @xmath97 and @xmath39 the natural self - affine measure on @xmath16 ; it is easy to verify that @xmath232 . let @xmath233 by frostman s lemma , it is enough to show that @xmath234 . let @xmath235 be the inner double integral in ( [ eq : generalth2 ] ) . passing to the symbolic space and recalling the definition of @xmath101 we obtain @xmath236 where for the last inequality we used condition @xmath237 . let @xmath238 . use proposition [ prop : main ] to get @xmath239 ( note that in the case @xmath216 , ( [ eq : generalth3b ] ) holds trivially because @xmath240 ) . integrating ( [ eq : generalth3a ] ) over @xmath223 and ( [ eq : generalth3b ] ) over @xmath241 and applying fubini we get @xmath242 for @xmath243 . adding , interchanging the order of integration again and recalling ( [ eq : generalth2 ] ) we conclude @xmath244 this completes the proof . @xmath154 if @xmath245 ( @xmath246 ) then @xmath247 for almost all @xmath248 . proof_. in the case @xmath249 , @xmath20 is absolutely continuous for almost all @xmath250 . this was proved by solomyak @xcite in the case @xmath251 and by simon and tth @xcite in the case @xmath252 . therefore we can take @xmath253 in theorem [ th : generalth ] . it is an elementary exercise to verify that @xmath254 for all @xmath246 . therefore , theorem [ th : transvest ] tells us that @xmath255 is an appropriate region in theorem [ th : generalth ] , and @xmath256 on the other hand , we can take @xmath257 ( if @xmath157 then @xmath80 has no zeros at all in @xmath90 , let alone double zeros ) . since @xmath253 we have @xmath258 the corollary is now clear . @xmath154 we remark that in the region @xmath259 a more precise result can be obtained using the method of przytycki and urbaski ; this is due to the fact that the i.f.s . verifies the strong separation condition in that region . however , our proof has the advantage of being more elementary . it is not difficult to use the method of proof of absolute continuity of bernoulli convolutions to obtain intervals @xmath199 for more general digit sets , although in general it may difficult to show that @xmath253 . we remark , however , that theorem [ th : generalth ] actually holds for all sufficiently small perturbations of @xmath260 . in the region @xmath261 the falconer dimension of @xmath16 is @xmath30 and we expect @xmath16 to have positive lebesgue measure and even non - empty interior . our main objective in this section is to obtain results analogous to theorem [ th : generalth ] in the this region . however , it will be convenient to state our results in terms of measures supported on @xmath16 rather than the sets themselves . moreover , we will need to allow for more general measures than the uniform ones considered so far . let @xmath262 denote the set of borel probability measures on @xmath97 . for @xmath263 , let @xmath264 be the length of the longest common initial subsequence of @xmath79 and @xmath265 . we define the ( lower ) correlation dimension of @xmath266 as @xmath267 one important class of examples are the bernoulli measures @xmath268 , where @xmath269 is a probability vector . an inspection of the definitions shows that @xmath270 let us fix for the moment @xmath266 . the projection map @xmath271 induces a family of measures @xmath39 supported on @xmath16 , namely @xmath272 . we will also need to redefine @xmath101 to reflect the fact that @xmath96 is now allowed to be a more general measure : @xmath273 we can express the correlation dimension in terms of @xmath101 as follows : @xmath274 we now state a technical proposition which will play an analogous role to that of proposition [ prop : main ] . [ prop : mainbis ] assume that @xmath96 is a product measure . let @xmath111 be closed intervals such that @xmath275 . 1 . assume that @xmath117 is an interval of transversality for @xmath87 . then there is a constant @xmath276 such that @xmath277 for all @xmath278 , where @xmath279 if @xmath20 does not have a density in @xmath23 . analogously , if @xmath125 is an interval of transversality for @xmath118 then there exists @xmath276 such that @xmath280 for all @xmath281 . before proving the proposition , we state a simple lemma we will need . let @xmath282 be a measure on @xmath100 with an @xmath23 density , which we denote by @xmath80 . then @xmath283 _ proof of the lemma_. this is standard but we were not able to find a reference , so a proof is provided for the convenience of the reader . let @xmath284 let also @xmath285 , where @xmath286 since @xmath287 is an approximate identity , @xmath288 in @xmath23 . therefore @xmath289 the lemma is proved . @xmath154 _ proof of proposition [ prop : mainbis]_. we will prove only the first part , since the second is just a restatement with the parameters interchanged . let @xmath290 . let us write @xmath291 set @xmath292 and @xmath293 . we have that @xmath294 for some @xmath95 , since @xmath117 is an interval of transversality . from this and the fact that @xmath96 is a product measure we obtain @xmath295 note however that @xmath296 from this , ( [ eq : mainbis1 ] ) and ( [ eq : mainbis2 ] ) we obtain @xmath297 since @xmath298 , we deduce from ( [ eq : corrdim ] ) that @xmath299 this completes the proof . we can now state our first result giving regions where @xmath39 is absolutely continuous and , in consequence , @xmath45 . roughly speaking , this theorem follows from proposition [ prop : mainbis ] in the same way that theorem [ th : generalth ] follows from proposition [ prop : main ] . however , when @xmath64 and @xmath300 , the theorem gives only a small region where absolute continuity holds . we later extend this region somewhat ( in particular , we show it contains a neighborhood of @xmath42 ) , although , as mentioned in the introduction , results are far for complete here . [ th : generalthbis ] let @xmath96 be a product measure on @xmath301 , and let @xmath302 . assume that : 1 . there is an open interval @xmath185 such that @xmath303 whenever @xmath304 is compactly contained in @xmath199 . 2 . there is an open interval @xmath305 such that if @xmath157 , @xmath80 has no double roots on @xmath90 ; 3 . there exists an open region @xmath188 such that if @xmath189 and @xmath157 , then @xmath108 and @xmath26 are not both double roots of @xmath80 . let @xmath306 then for almost all @xmath307 , @xmath39 is absolutely continuous with a density in @xmath23 . we make some remarks before the proof . 1 . note that @xmath308 when @xmath96 is the uniform bernoulli measure @xmath301 , but in general there is a gap between the regions given by theorems [ th : generalth ] and [ th : generalthbis ] , even when @xmath64 . if the analogy with bernoulli convolutions holds , @xmath309 is the treshold for @xmath23 density when @xmath96 is not uniform , but @xmath39 could still be absolutely continuous even when @xmath310 . 2 . of course , if @xmath39 is absolutely continuous then @xmath45 . 3 . the hypothesis on @xmath199 may appear too strong , but in practice it does in fact follow from the same proof that shows absolute continuity of @xmath20 for a.e.@xmath186 . _ proof of theorem [ th : generalthbis]_. the proof follows the scheme of theorem 4.3 in @xcite . however , we need to introduce the variants we have used before in the proof of theorem [ th : generalth ] . let @xmath111 be closed intervals such that @xmath217 or @xmath216 . in the first case , let us choose @xmath311 such that @xmath312 is an interval of transversality for @xmath313 ( @xmath243 ) . otherwise , let @xmath314 . the first steps mimic the proof @xcite , theorem 4.3 , so we only sketch them . let @xmath315 be the lower density of @xmath39 at @xmath316 . if we can show that @xmath317 then the criterion for absolute continuity in @xcite , section 2.12 , will imply that @xmath39 is absolutely continuous for almost all @xmath318 and , moreover , @xmath319 . we can proceed like in @xcite , theorem 4.3 to estimate @xmath320 therefore it is enough to show that the integral in the right hand side above is bounded by @xmath321 when restricted to @xmath91 or @xmath322 . we consider only the restriction to @xmath91 since the other case is similar . using fubini , proposition [ prop : mainbis ] and ( [ eq : condnorm ] ) , we can estimate the integral in ( [ eq : generalthbis1 ] ) restricted to @xmath323 as @xmath324 the proof is now complete . we now give a concrete region where we can show that @xmath39 is absolutely continuous in the case @xmath251 . other values of @xmath325 can be handled analogously . [ coro : abscontregion ] let @xmath326 further , let @xmath327 then @xmath39 is absolutely continuous for almost every @xmath44 . furthermore , @xmath39 has a continuous density for almost every @xmath328 . _ proof_. it was proved in @xcite that we can take @xmath329 for any @xmath330 and @xmath331 . on the other hand , theorem [ th : transvest ] applied with @xmath332 and @xmath333 shows that we can take @xmath334 . therefore theorem [ th : mainbis ] shows that @xmath39 is absolutely continuous for almost every @xmath307 . for the rest of the corollary it suffices to show that if @xmath335 is absolutely continuous with a density in @xmath23 , then @xmath39 is absolutely continuous with a continuous density . this is standard but we include a proof for completeness . by decomposing the measure @xmath39 as an infinite convolution of bernoulli measures we obtain @xmath336 ( recall that we are using the digits @xmath67 rather than @xmath63 . therefore , for all @xmath49 , @xmath337 assume @xmath338 has an @xmath23 density ; then so does its fourier transform . now ( [ eq : fourierdecomposition ] ) shows that @xmath339 ; on the other hand , @xmath340 is bounded since @xmath39 is a finite measure . therefore @xmath341 and , after taking inverse fourier transform , we conclude that @xmath39 is absolutely continuous with a continuous density , as desired @xmath154 the regions @xmath342 are pictured in figure 2 . * figure 2*. the region @xmath343 in corollary [ coro : abscontregion ] , and the first @xmath344 pieces or @xmath345 . the shaded region is @xmath346 , which is also contained in @xmath345 . several remarks are in order . 1 . @xmath345 contains a neighborhood of @xmath42 ; more precisely , @xmath347 . indeed , one can see that @xmath348 as each interval in the union in the left hand side overlaps with the next . 2 . using tricks such as those in @xcite it is possible to extend the region @xmath50 , but not significantly . it is also possible to use a computer - assisted rigorous estimation of the region @xmath188 to show that @xmath50 can be enlarged to cover more than @xmath349 of @xmath350 ; details will be discussed in a forthcoming paper . in this section we discuss several natural questions . some of them can be easily answered with the current techniques , while others appear to be harder . for the most part we restrict ourselves to the case @xmath64 or @xmath245 . an exception is subsection [ sec : puretype ] , where we consider more general self - affine sets and measures . we introduce some notation we will be using repeatedly . recall that @xmath351 if @xmath352 , let @xmath353 $ ] be the associated cylinder set in @xmath97 , let @xmath354 be the cylinder set @xmath355)$ ] and , finally , let @xmath356 note that @xmath357 . we will often omit the subscripts @xmath31 whenever they are fixed in a context . in this subsection we consider the case @xmath64 . a natural question is for what values of @xmath31 is @xmath16 connected ; this parameter set is called the `` mandelbrot set '' or `` connectedness locus '' associated to the family ; it will be denoted by @xmath262 . b. solomyak @xcite has some interesting results on this topic . without going into details , let us state some of the known basic facts which will be useful later . _ 3_. assume @xmath365 is disconnected . denote by @xmath366 the @xmath367 neighborhood of @xmath368 . let @xmath369 be the cylinders of step @xmath25 , and pick @xmath367 such that @xmath370 and @xmath371 are disjoint . since @xmath372 because @xmath373 is contractive , we must have @xmath374 hence @xmath375 . @xmath154 for bernoulli convolutions , the only values of @xmath26 for which is known that @xmath20 is singular are reciprocals of pisot numbers . this is proved by showing that the fourier transform @xmath376 does not go to zero as @xmath377 . in fact , it is known that @xmath378 does not converge to @xmath52 at infinity if and only if @xmath28 is pisot ( @xcite , proposition 15.3.2 ) . the problem of determining all @xmath26 such that @xmath20 is singular is currently open and seems to be very hard . in @xcite it is proved that for @xmath379 and @xmath26 the reciprocal of a pisot number , it is verified that @xmath380 ( here @xmath16 is the attractor for the digit set @xmath63 ) . their proof extends readily to arbitrary @xmath108 such that @xmath16 is totally disconnected . although they do not write down the details , they remark that a simpler proof of the same fact can be obtained using the technique of mcmullen @xcite . here we investigate the set of exceptions to the almost everywhere results obtained in the earlier sections . in particular , we write down the details of the proof suggested by przytycki and urbaski , which extends to the overlapping case , and in fact it provides examples of @xmath381 such that @xmath382 . throughout the section we assume that @xmath251 , although for the most part analogous considerations are valid for @xmath252 as well . let @xmath383 be the set of all @xmath384 such that @xmath108 and @xmath26 are roots of a _ polynomial _ with coefficients in @xmath385 . for all @xmath386 , @xmath387 moreover , the set @xmath383 is dense in @xmath388 , where @xmath262 is the connectedness locus for the family @xmath389 . _ _ _ proof_. if @xmath386 we can find two words @xmath390 for some @xmath391 such that @xmath392 and therefore @xmath393 . it follows that @xmath394 can be covered by @xmath395 rectangles of size @xmath396 , and more generally , by @xmath397 rectangles of size @xmath398 . by subdividing each of those rectangles into @xmath399 squares of side @xmath400 we conclude that @xmath16 can be covered by @xmath401 balls of radius @xmath402 . therefore @xmath403 and the first assertion follows . the fact that @xmath383 is dense in @xmath262 is an immediate consequence of lemma [ lemma : conneclocus ] and rouch s theorem . @xmath154 we will say that @xmath404 is a _ pisot pair _ if @xmath405 and there exists a monic irreducible polynomial @xmath406 $ ] such that @xmath407 and @xmath408 are the only roots of @xmath409 with modulus greater or equal than @xmath25 . pisot pairs ( or rather more general `` pisot families '' ) have been studied by several authors ; see for instance @xcite . _ _ _ proof_. we know that the vertical and horizontal projections of @xmath39 are @xmath35 and @xmath415 respectively . therefore the restrictions of @xmath340 are @xmath416 and @xmath417 . hence whenever one of @xmath416 or @xmath417 does not converge to @xmath52 at infinity , the same happens to @xmath340 . in particular , this is the case if either @xmath410 or @xmath411 are pisot . if @xmath412 is a pisot pair then we can apply the technique used to prove that @xmath376 does not converge to @xmath52 at infinity . an even closer example is the family of complex bernoulli convolutions studied in @xcite . we indicate the idea , but refer to the proof of theorem 2.3 in @xcite for the details . let @xmath409 be the polynomial in the definition of pisot pair , and let @xmath418 be the roots of @xmath409 ( so @xmath419 ) . since for all integers @xmath46 , @xmath420 we have that @xmath421 for some @xmath422 . this fact combined with the expression ( [ eq : fourier ] ) for the fourier transform imply ( after some technical considerations ) that @xmath423 for all positive integers @xmath424 and some @xmath367 independent of @xmath424 . @xmath154 it is not difficult to obtain many examples of pisot pairs using a computer . for instance , the polynomial @xmath425 has exactly two positive roots @xmath426 and all the other roots are complex and of absolute value less than @xmath25 . the approximate values are @xmath427 and @xmath428 . note that in this example @xmath37 . _ proof_. the proof consists of two parts . in the first part we obtain an upper bound for the dimension of @xmath16 using mcmullen s technique ; in the second part we show that this upper bound verifies the inequality in the theorem . in the course of the proof , @xmath432 etc will denote constants independent of @xmath433 and @xmath108 ( they may depend on the fixed number @xmath26 ) . let @xmath434 we recall garsia s lemma @xcite : the distance between any two different elements of @xmath435 is bounded below by @xmath436 . on the other hand , since @xmath19 one can easily see , using for instance the greedy algorithm , that the distance between consecutive elements of @xmath435 is at most @xmath437 . since @xmath438 for all @xmath439 , it follows that @xmath440 let @xmath441 be the elements of @xmath435 ( so that @xmath442 ) , and let @xmath443 note that @xmath444 we will show the following : @xmath445 where @xmath446 . in the course of the proof of ( [ eq : mcmullenest ] ) all numbers @xmath26 , @xmath108 and @xmath433 will remain fixed . let @xmath447 be the symbolic space @xmath97 , where @xmath448 . further let @xmath449 let @xmath450 . define the @xmath46-th symbolic approximate square @xmath451 as @xmath452 , where @xmath453 let @xmath454 be the projection map given by @xmath455 ( so in other words @xmath456 , where @xmath265 is the sequence obtained by concatenating all the @xmath457 ) . note that @xmath458 is surjective but not necessarily injective . observe that if @xmath459 then @xmath460 represent the first and second coordinates . therefore @xmath461 is contained in a ball of center @xmath462 and radius comparable to @xmath463 . let @xmath96 be the bernoulli measure on @xmath447 giving weight @xmath464 to all @xmath465 such that @xmath466 ( see @xcite for a motivation for this choice of weights ) . our next step is to show that for all @xmath450 , @xmath467 we will do so by using a clever trick due to mcmullen . for @xmath468 write @xmath469 if @xmath470 . note from the definition of @xmath471 that @xmath472 therefore @xmath473 write the right hand side above as @xmath474 , where @xmath475 note that @xmath476 is bounded over all positive @xmath407 . therefore , by telescoping over the sequence @xmath477 we deduce that @xmath478 is also bounded over all @xmath424 . observe that since @xmath479 is bounded , @xmath480 as @xmath481 . therefore we must have @xmath482 this together with ( [ eq : estmm1 ] ) and ( [ eq : estmm2 ] ) show that ( [ eq : snest ] ) is verified . recall that @xmath483 is contained in a ball @xmath484 . it follows from @xmath485 that if @xmath183 is the projection of @xmath96 under @xmath458 then @xmath486 since @xmath458 is surjective , the mass distribution principle ( @xcite , proposition 2.2 ) shows that ( [ eq : mcmullenest ] ) is satisfied . this concludes the first part of the proof . from now on , @xmath26 will remain fixed , but we will consider both @xmath433 and @xmath108 ( or rather @xmath487 ) as variables . let @xmath488 . recalling that @xmath446 , write the upper bound in ( [ eq : mcmullenest ] ) as @xmath489 let @xmath490 . notice that @xmath491 therefore in order to establish the theorem it is enough to show that the right hand side above defines a continuous function of @xmath487 which is strictly negative on @xmath492 . we claim that for fixed @xmath487 the sequence @xmath493 is submultiplicative . indeed , @xmath494 on the other hand , each @xmath495 is the sum of one or more numbers of the form @xmath496 , and each pair @xmath497 appears in exactly one of the @xmath495 . the submultiplicativity is then consequence of the inequality @xmath498 for a finite collection of positive numbers @xmath499 , which holds since @xmath500 . by taking logarithms and using subadditivity we obtain @xmath501 denote the limiting function by @xmath502 . since @xmath503 and @xmath504 , we deduce from ( [ eq : boundnk ] ) and ( [ eq : sumakj ] ) that @xmath505 . also , notice that @xmath506 is a convex function . since , because of subadditivity , @xmath502 is then a pointwise limit of decreasing convex functions , @xmath502 must itself be convex . in particular , @xmath507 is continuous and , since it agrees with the linear function @xmath508 at @xmath52 and @xmath25 , we must have @xmath509 for all @xmath500 . moreover , if we can show that @xmath502 is strictly convex on @xmath510 $ ] , this will imply that @xmath511 for all @xmath500 and , as noted before ( see ( [ eq : dimdiff ] ) and the associated remark ) , this will yield the theorem . a straightforward calculation shows that if we let @xmath512 then @xmath513 denote the sum in the numerator above by @xmath514 . in the course of the proof that @xmath515 ( which essentially goes back to garsia @xcite ) , it is shown that @xmath516 see @xcite , pp . 179 - 180 for a proof of this fact . we remark that the proof uses both garsia s lemma and the singularity of @xmath20 , but is otherwise elementary . recalling that @xmath517 is bounded by a constant multiple of @xmath518 , we deduce from ( [ eq : derivtau ] ) and ( [ eq : entropyineq ] ) that @xmath519 for some sufficiently large @xmath520 . since , on the other hand , @xmath521 and @xmath522 for all @xmath523 , we conclude that @xmath524 , where @xmath525 denotes the left derivative . this shows that @xmath502 can not agree with @xmath526 on @xmath492 and therefore must be strictly convex , completing the proof @xmath154 we make some remarks about the above proof . first , the proof is about the hausdorff dimension of @xmath16 . under strong separation , the box dimension does not drop when @xmath411 is pisot . when there are overlaps , it is no longer so clear what happens to the box dimension , but in principle there is no reason to believe it will also drop . second , the function @xmath502 that appeared in the course of the proof is actually ( the negative of ) the @xmath527-spectrum of @xmath20 . the fact that @xmath502 is strictly convex corresponds , under the multifractal formalism , to the fact that @xmath20 is a multifractal measure ; i.e. it has a range of local dimensions . in general , the left and right derivatives of the @xmath527-spectrum at @xmath25 give substantial information about the measure ; see for example @xcite . thus the proof is another indication of the delicate relationship between bernoulli convolutions and the sets @xmath16 . _ proof_. the first part is clear from theorem [ th : dimdrop ] . for the second part , note that when @xmath533 , we have that @xmath534 ( since @xmath530 ) and the falconer dimension is exactly @xmath30 . therefore the second part follows from the continuity of the drop in theorem [ th : dimdrop ] . @xmath154 interestingly , there are exactly two pisot numbers whose reciprocals are greater than @xmath535 . the smallest pisot number is the real root of @xmath536 , which is about @xmath537 . the second smallest pisot number is the positive root of the polynomial @xmath538 ; it is about @xmath539 . the next pisot number is the positive root of @xmath540 , which is already greater than @xmath541 . see @xcite , theorem 7.2.1 . for a proof of these facts . in this subsection we assume @xmath245 . we showed that when the falconer dimension is less than @xmath30 , the hausdorff dimension of @xmath16 is almost everywhere equal to the falconer dimension . hence it is natural to ask what is the hausdorff measure in the critical dimension @xmath542 . unlike the self - similar case , this is a non - trivial question even in the strong separation case , provided @xmath543 ( otherwise @xmath16 can be seen to be bi - lipschitz equivalent to the cantor set @xmath58 ) . it is in fact very easy to show that @xmath544 for _ all _ @xmath31 , by considering the natural cover . the following result was communicated to us by m. rams , but seems to be folklore . [ th : hmnooverlaps ] assume the strong separation is verified for @xmath16 , where @xmath545 , and let @xmath546 be the falconer dimension of @xmath16 . then @xmath547 if and only if @xmath20 is absolutely continuous with a bounded density . [ lemma : equivalence ] assume that @xmath543 , @xmath109 and @xmath548 , where @xmath546 is the falconer dimension of @xmath16 . then @xmath549 assigns the same mass to all cylinders of level @xmath433 , where @xmath550 is the measure of hausdorff type obtained by considering covers by open squares only . ( here we do not assume a separation condition ) . _ proof of lemma_. for notational convenience we will omit the subscripts @xmath551 . let @xmath552 . we will argue by contradiction . since @xmath553 it follows that some @xmath554 has measure _ larger _ than @xmath555 ; fix such a @xmath465 and choose @xmath204 such that @xmath556 by decomposing @xmath465 into sub - cylinders and using subadditivity again , it follows that for all sufficiently large @xmath433 there is @xmath352 verifying ( [ eq : lemmaequiv1 ] ) . now fix @xmath557 and choose a cover @xmath558 of @xmath394 by squares such that @xmath559 also fix @xmath433 such that @xmath560 and ( [ eq : lemmaequiv1 ] ) holds for some @xmath352 . consider a cover @xmath561 of @xmath554 defined as follows : for each @xmath562 , cover the rectangle @xmath563 by @xmath564 squares of side @xmath565 , and take the union of those squares over all @xmath562 . therefore we have @xmath566 where we used that @xmath567 . letting @xmath568 and recalling ( [ eq : lemmaequiv2 ] ) we conclude that @xmath569 . this contradicts ( [ eq : lemmaequiv1 ] ) , as desired . @xmath154 _ proof of theorem [ th : hmnooverlaps]_. we use the same notation as in the lemma . for @xmath570 let @xmath571 be the open square centered at @xmath572 and half - side @xmath573 . one consequence of strong separation , the self - affine relation and the fact that @xmath183 projects onto the bernoulli convolution @xmath20 is that @xmath574 from this it immediately follows that if @xmath282 has a bounded density then @xmath575 where @xmath576 is independent of @xmath433 . hence @xmath183 is an @xmath546-dimensional frostman measure and it follows that @xmath577 . now suppose that @xmath578 is not a bounded function ( or @xmath282 is not absolutely continuous at all ) , and fix @xmath83 . let @xmath579 if @xmath580 is small enough then @xmath581 has positive measure . but in this case , the ergodic theorem implies that for almost every @xmath79 , @xmath582 visits @xmath581 with positive frequency , and therefore we deduce from ( [ eq : hmnooverlaps1 ] ) that @xmath583 under strong separation , @xmath183 assigns the same mass to all cylinders of the same level ; therefore @xmath183 is a constant multiple of @xmath584 by the lemma , and thus equivalent to @xmath585 . but by the density theorems ( see @xcite , theorem 6.2 . ( 1 ) ) , @xmath586 therefore @xmath587 as desired . ( note that we do need the lemma ; otherwise @xmath588 might be concentrated in the exceptional set of @xmath589 ) . @xmath154 we now consider the overlapping case . in the self - similar setting , solomyak @xcite in a particular situation and later peres , simon and solomyak @xcite in greater generality showed that , assuming transversality , self - similar sets with overlap have typically @xmath52 measure . it turns out that the proof in @xcite extends to our setting . unfortunately , we are not able to check the needed concept of transversality ( which is different from the one used in the first sections ) in all of the relevant region @xmath193 , although it does hold for a large chunk of the overlapping region by results of solomyak @xcite . _ proof_. our proof follows the idea of bandt and graf ; details are provided for completeness . assume by way of contradiction that @xmath577 . we know from lemma [ lemma : equivalence ] that in this case all cylinders of the same level are disjoint in @xmath550-measure . let @xmath552 , and choose a cover @xmath598 of @xmath394 by open squares such that @xmath599 where @xmath367 is to be chosen later . let @xmath600 be the union of the @xmath601 , and pick @xmath204 such that if @xmath602 for some map @xmath368 then @xmath603 ; this is possible since @xmath600 is open . next take @xmath596 such that ( [ eq : epsilonclose ] ) holds with this @xmath330 . note that if @xmath330 is small enough then @xmath465 and @xmath604 must have the same length , say @xmath433 ; note also that @xmath433 can be made arbitrarily large by taking @xmath330 small . because of the way @xmath330 was chosen , we have @xmath605 . now adapt the covering @xmath598 to @xmath597 as in lemma [ lemma : equivalence ] ( mapping the @xmath601 by @xmath606 and then dividing the resulting rectangles into squares ) . the union of this covering clearly contains @xmath607 , whence it is also a covering of @xmath608 ; this is a contradiction since @xmath608 and @xmath609 are measure disjoint and this covering is almost optimal . more precisely , use ( [ eq : lemmaquiv3 ] ) to get @xmath610 taking @xmath611 and @xmath433 large enough so that @xmath612 yields the desired contradiction . @xmath154 in applying the previous lemma we will need to prove that certain sets have zero measure , and following @xcite we will do so by showing that those sets have no lebesgue density points . however , we will need a different notion of density points , defined using averages over rectangles of size @xmath613 rather than balls . the following lemma shows that this makes no difference to us . [ lemma : densityrect ] let @xmath614 be the rectangle centered at @xmath615 having dimensions @xmath616 ( @xmath576 is allowed to depend on @xmath617 but not on @xmath433 ) . let @xmath368 be a measurable subset of @xmath618 . if for all @xmath619 we have that @xmath620 then @xmath621 . the requirement that @xmath108 and @xmath26 are zeros of @xmath80 is natural since we want @xmath16 to have overlaps ( recall lemma [ lemma : conneclocus ] ; also note that @xmath16 is not necessarily connected if @xmath627 ) . note that in the previous definition of transversality one of @xmath108 , @xmath26 was allowed to be a double zero as long as the other parameter was not ; this is not the case here . on the other hand , here we need only the _ existence _ of such an @xmath80 ; it is therefore natural to conjecture that @xmath628 where @xmath629 denote the set of parameters where there is an overlap ( @xmath630 stands for `` intersection parameters '' ; this terminology was introduced in @xcite ) . however , we were not able to take advantage of the fact that only existence of an @xmath80 is needed . solomyak @xcite has obtained a large region of @xmath624-transversality in the case @xmath251 . although it does not cover all of @xmath631 , it does contain a big chunk of it . we refer to his paper for details . _ proof_. the proof follows closely the proof of theorem 2.1 in @xcite ( in fact , the easier homogeneous case ) . to begin , observe that for @xmath204 small , two cylinders @xmath635 are @xmath330-relatively close if and only if @xmath636 for some large @xmath433 and @xmath637 be the set of all @xmath632 such that ( [ eq : zeromeas1 ] ) holds for some @xmath596 of the same length . invoking lemma [ lemma : bandtgraf ] , it is enough to show that @xmath638 has zero lebesgue measure for all @xmath330 sufficiently small . to this end , we will show the following : for all @xmath639 , some @xmath640 , @xmath641 , and all sufficiently large @xmath433 ( all depending on @xmath642 , @xmath643 where @xmath644 is as defined in lemma [ lemma : densityrect ] . once we have shown this , the same lemma will give that @xmath645 has zero lebesgue measure . from now on fix @xmath204 and @xmath646 . since @xmath647 is a region of @xmath624-transversality , there exists @xmath648 such that @xmath649 . write @xmath650 , where @xmath651 . it follows that @xmath652 for all @xmath653 ( where @xmath654 denotes the initial word of length @xmath433 of @xmath457 ) . let @xmath655 be a small open square centered at @xmath656 and compactly contained in @xmath618 such that @xmath657 since the closure of @xmath655 is contained in @xmath618 , @xmath658 also , since @xmath659 , @xmath660 . for this choice of @xmath664 , the upper bound in ( [ eq : derbounds ] ) implies that @xmath668 whence @xmath669 note however that if @xmath433 is large enough @xmath670 where @xmath667 or @xmath26 . a similar argument shows that @xmath671 for @xmath667 or @xmath26 and large enough @xmath433 . on the other hand , @xmath673 therefore if we take @xmath674 and @xmath675 , we obtain that @xmath676 together with ( [ eq : t ] ) this implies ( [ eq : conditionzeromeas ] ) , and the proof is complete . @xmath154 it is known that self - similar measures are either singular or mutually absolutely continuous with respect to lebesgue measure on the attractor . this was proved , in increasing levels of generality , in @xcite , @xcite and @xcite . all of those papers use the lebesgue density theorem and assume that the maps are at least conformal ; however , by using density bases more general than balls it is possible to adapt those proofs to our setting . recall the a density basis @xmath345 for a borel set @xmath394 is a family of open sets such that the following holds : for all @xmath677 there are arbitrarily small sets @xmath678 containing @xmath589 and if @xmath679 is a borel set , then @xmath680 ( here @xmath681 means that @xmath682 ) . let @xmath394 be the attractor of an affine i.f.s . @xmath683 on @xmath2 , and assume that @xmath684 . we will say that @xmath394 is _ differentiation - regular _ if there exists a density basis @xmath345 for @xmath394 and a constant @xmath685 such that the following holds : for every @xmath677 there is a sequence @xmath686 in @xmath345 with @xmath687 , such that if @xmath688 for some @xmath562 , there exists a finite word @xmath689 verifying @xmath690 _ _ very roughly speaking , a self - affine set is differentiation - regular if we can pick a differentiation basis for @xmath394 consisting of open sets which look like some cylinder in the construction of @xmath394 . the next proposition shows that some important classes of self - affine sets , including those studied in this paper , are indeed differentiation - regular . [ prop : diffregular ] let @xmath691 be contracting affine maps on @xmath2 , and let @xmath692 denote the linear part of @xmath373 . let @xmath394 be the attractor of @xmath683 . assume that @xmath684 and any one of the following conditions hold : 1 . @xmath693 and all the maps @xmath692 are equal . 2 . there exists a finite generating system @xmath694 of @xmath2 , such that @xmath695 for all @xmath696 . 3 . all the maps @xmath692 are simultaneously diagonalizable . _ proof_. assume first that @xmath693 and all the linear parts are equal to @xmath368 . let @xmath697 be a ball centered at the origin and containing @xmath394 , and let @xmath698 be the direction of the major axis of the ellipse @xmath699 . then we can pick a subsequence @xmath700 which is either constant or lacunary ( a sequence @xmath701 is lacunary if it converges to some @xmath702 and there is @xmath703 such that @xmath704 for all @xmath705 ) . in either case , it is well - known that the family @xmath706 is a density basis of @xmath622 ( this is originally due to r.fromberg , a proof can be found in @xcite ) . now if @xmath465 is a word of length @xmath707 we have that @xmath708 and and @xmath709 where @xmath710 . this shows that @xmath394 is differentiation - regular . now we consider the second case . fix a generating set @xmath694 as in the statement , and let @xmath409 be a convex polyhedra containing @xmath394 and whose sides are parallel to elements of @xmath694 ( since @xmath694 contains a basis , we can take @xmath409 to be a suitable parallelepiped ) . the set of all convex polyhedra with sides parallel to some element of @xmath694 is known to be a density basis of @xmath2 ( @xcite , p.137 ) ; let us denote this basis by @xmath345 . by hypothesis , @xmath711 for all @xmath465 , and we can conclude that @xmath394 is differentiation - regular as in ( [ eq : diffregular ] ) . [ prop : puretype ] let @xmath394 be the attractor of an affine i.f.s . @xmath683 on @xmath2 , and assume that @xmath394 is differentiation - regular . then for any self - affine measure @xmath183 supported on @xmath394 , @xmath712 is either singular or mutually absolutely continuous with respect to @xmath183 . _ proof_. it is well known that @xmath183 is either singular or absolutely continuous with respect to lebesgue measure ( this can be seen for instance by decomposing @xmath183 into absolutely continuous and singular parts , and showing that each of them also verifies the self - affine relation , hence one of them must be trivial ) . therefore it is enough to show that if @xmath183 is the attractor of the weighted i.f.s . @xmath713 and @xmath714 , then @xmath715 . following @xcite , proposition 3.1 , let @xmath716 since we are assuming that @xmath183 is not singular , @xmath717 ; we want to show that in fact @xmath718 . fix a borel subset @xmath719 of @xmath394 such that @xmath720 . let @xmath677 and @xmath688 for some @xmath589 . pick a word @xmath465 such that ( [ eq : densitybasecondition ] ) holds . the borel set @xmath721 is contained in @xmath394 ; moreover it has zero @xmath183-measure since @xmath183 is self - affine ( and thus @xmath722 is dominated by a multiple of @xmath183 ) . we finish this section with some remarks . first , we did not really use that @xmath183 is self - affine ; just that @xmath727 for all borel sets @xmath368 . we may call measures verifying ( [ eq : weaksa ] ) _ weakly self - affine _ . _ _ second , although one can check the differentiation - regular condition in many interesting cases , there are many natural instances where this property appears to fail ( although proving it rigorously looks difficult ) . for instance , let @xmath728 be two closed angular sectors in the open first quadrant , disjoint except at the origin ( for simplicity we consider only the case @xmath693 ) . if the linear part of @xmath373 maps @xmath692 into @xmath692 , then the i.f.s . @xmath729 induces another i.f.s . in the circle @xmath730 , whose attractor is a cantor set of directions . however , it is known that for many cantor sets @xmath576 , the set of rectangles with sides parallel to @xmath576 is _ not _ a density basis @xcite , while no cantor set is known for which the opposite is true . thus checking out differentiation - regularity appears unlikely . of course , it could be that proposition [ prop : puretype ] is true for arbitrary self - affine sets , using a different method of proof . however we were not able to verify this and we believe that there may be a counterexample . so far we have only considered self - affine sets where the digits lie all on the line @xmath731 . when @xmath64 this is of course not a restriction , but when @xmath627 it is certainly a strong assumption . in fact , almost all the results in this paper can be generalized , in theory , to a more general setting described below ; the problem is that the needed notion of transversality can be very hard to check , or simply false . on the plus side , most of the results can be shown to hold under a perturbation of the digits . in particular , transversality ( in all of its various forms ) is an open condition . if anything , this shows that collinearity of the digits is not a necessary condition for the type of results obtained in this paper . for @xmath735 let @xmath736 we say that @xmath737 is a _ region of transversality _ if whenever @xmath189 and @xmath738 , either @xmath108 is not a double root of @xmath739 or @xmath26 is not a double root of @xmath740 . _ _ in this framework , suitable versions of theorems [ th : generalth ] and [ th : generalthbis ] apply . in general , obtaining regions of transversality looks very difficult unless the digits are almost collinear or have some very special form . however , the regions @xmath259 in the aforementioned theorems can still be efficiently estimated , since @xmath199 and @xmath90 are in this case independent from one another . as an example , we have the following result , whose proof works exactly as in theorem [ th : generalth ] . let @xmath312 be open intervals such that @xmath741 has no double roots in @xmath742 for all @xmath743 . let @xmath742 be open intervals such that @xmath744 has correlation dimension @xmath25 for almost every @xmath745 , @xmath243 ( here @xmath746 is the b.c . associated to @xmath747 ) . finally , let @xmath188 be a region of transversality . let @xmath748 then @xmath749 for almost all @xmath750 . @xmath154 note that we are somehow in the reverse situation with respect to falconer s theorem , which holds for _ all _ families of linear maps satisfying certain conditions and for _ almost every _ digit . here , the result holds for _ all _ digits in some open set and _ almost every _ parameter . another possible variant is to allow the digits to depend on the parameters . this is standard in the self - similar setting and present no additional complications here ; most of the results , when stated appropriately , hold also in this case . yuval peres , wilhelm schlag , and boris solomyak . sixty years of bernoulli convolutions . in _ fractal geometry and stochastics , ii ( greifswald / koserow , 1998 ) _ , volume 46 of _ progr . _ , pages 3965 . birkhuser , basel , 2000 .

we study families of possibly overlapping self - affine sets . our main example is a family that can be considered the self - affine version of bernoulli convolutions and was studied , in the non - overlapping case , by f.przytycki and m.urbaski @xcite . we extend their results to the overlapping region and also consider some extensions and generalizations .

math0408203

coordinated networks of mobile robots are already in use for environmental monitoring and warehouse logistics . in the near future , autonomous robotic teams will revolutionize transportation of passengers and goods , search and rescue operations , and other applications . these tasks share a common feature : the robots are asked to provide service over a space . one question which arises is : when a group of robots is waiting for a task request to come in , how can they best position themselves to be ready to respond ? the distributed _ environment partitioning problem _ for robotic networks consists of designing individual control and communication laws such that the team divides a large space into regions . typically , partitioning is done so as to optimize a cost function which measures the quality of service provided over all of the regions . _ coverage control _ additionally optimizes the positioning of robots inside a region as shown in fig . [ fig : cover_example ] . this paper describes a distributed partitioning and coverage control algorithm for a network of robots to minimize the expected distance between the closest robot and spatially distributed events which will appear at discrete points in a non - convex environment . optimality is defined with reference to a relevant `` multicenter '' cost function . as with all multirobot coordination applications , the challenge comes from reducing the communication requirements : the proposed algorithm requires only short - range gossip " communication , i.e. , asynchronous and unreliable communication between nearby robots . territory partitioning and coverage control have applications in many fields . in cyber - physical systems , applications include automated environmental monitoring @xcite , fetching and delivery @xcite , construction @xcite , and other vehicle routing scenarios @xcite . more generally , coverage of discrete sets is also closely related to the literature on data clustering and @xmath0-means @xcite , as well as the facility location or @xmath0-center problem @xcite . partitioning of graphs is its own field of research , see @xcite for a survey . territory partitioning through local interactions is also studied for animal groups , see for example @xcite . a broad discussion of algorithms for partitioning and coverage control in robotic networks is presented in @xcite which builds on the classic work of lloyd @xcite on optimal quantizer selection through centering and partitioning . " the lloyd approach was first adapted for distributed coverage control in @xcite . since this beginning , similar algorithms have been applied to non - convex environments @xcite , unknown density functions @xcite , equitable partitioning @xcite , and construction of truss - like objects @xcite . there are also multi - agent partitioning algorithms built on market principles or auctions , see @xcite for a survey . while lloyd iterative optimization algorithms are popular and work well in simulation , they require synchronous and reliable communication among neighboring robots . as robots with adjacent regions may be arbitrarily far apart , these communication requirements are burdensome and unrealistic for deployed robotic networks . in response to this issue , in @xcite the authors have shown how a group of robotic agents can optimize the partition of a convex bounded set using a lloyd algorithm with gossip communication . a lloyd algorithm with gossip communication has also been applied to optimizing partitions of non - convex environments in @xcite , the key idea being to transform the coverage problem in euclidean space into a coverage problem on a graph with geodesic distances . distributed lloyd methods are built around separate partitioning and centering steps , and they are attractive because there are known ways to characterize their equilibrium sets ( the so - called centroidal voronoi partitions ) and prove convergence . unfortunately , even for very simple environments ( both continuous and discrete ) the set of centroidal voronoi partitions may contain several sub - optimal configurations . we are thus interested in studying ( discrete ) gossip coverage algorithms for two reasons : ( 1 ) they apply to more realistic robot network models featuring very limited communication in large non - convex environments , and ( 2 ) they are more flexible than typical lloyd algorithms meaning they can avoid poor suboptimal configurations and improve performance . there are three main contributions in this paper . first , we present a discrete partitioning and coverage optimization algorithm for mobile robots with unreliable , asynchronous , and short - range communication . our algorithm has two components : a _ motion protocol _ which drives the robots to meet their neighbors , and a _ pairwise partitioning rule _ to update territories when two robots meet . the partitioning rule optimizes coverage of a set of points connected by edges to form a graph . the flexibility of graphs allows the algorithm to operate in non - convex , non - polygonal environments with holes . our graph partition optimization approach can also be applied to non - planar problems , existing transportation or logistics networks , or more general data sets . second , we provide an analysis of both the convergence properties and computational requirements of the algorithm . by studying a dynamical system of partitions of the graph s vertices , we prove that almost surely the algorithm converges to a pairwise - optimal partition in finite time . the set of pairwise - optimal partitions is shown to be a proper subset of the well - studied set of centroidal voronoi partitions . we further describe how our pairwise partitioning rule can be implemented to run in anytime and how the computational requirements of the algorithm can scale up for large domains and large teams . third , we detail experimental results from our implementation of the algorithm in the player / stage robot control system . we present a simulation of 30 robots providing coverage of a portion of a college campus to demonstrate that our algorithm can handle large robot teams , and a hardware - in - the - loop experiment conducted in our lab which incorporates sensor noise and uncertainty in robot position . through numerical analysis we also show how our new approach to partitioning represents a significant performance improvement over both common lloyd - type methods and the recent results in @xcite . the present work differs from the gossip lloyd method @xcite in three respects . first , while @xcite focuses on territory partitioning in a convex continuous domain , here we operate on a graph which allows our approach to consider geodesic distances , work in non - convex environments , and maintain connected territories . second , instead of a pairwise lloyd - like update , we use an iterative optimal two - partitioning approach which yields better final solutions . third , we also present a motion protocol to produce the sporadic pairwise communications required for our gossip algorithm and characterize the computational complexity of our proposal . preliminary versions of this paper appeared in @xcite and @xcite . compared to these , the new content here includes : ( 1 ) a motion protocol ; ( 2 ) a simplified and improved pairwise partitioning rule ; ( 3 ) proofs of the convergence results ; and ( 4 ) a description of our implementation and a hardware - in - the - loop experiment . in section [ sec : prelim ] we review and adapt coverage and geometric concepts ( e.g. , centroids , voronoi partitions ) to a discrete environment like a graph . we formally describe our robot network model and the discrete partitioning problem in section [ sec : algorithm ] , and then state our coverage algorithm and its properties . section [ sec : convergence ] contains proofs of the main convergence results . in section [ sec : results ] we detail our implementation of the algorithm and present experiments and comparative analysis . some conclusions are given in section [ sec : conclusion ] . in our notation , @xmath1 denotes the set of non - negative real numbers and @xmath2 the set of non - negative integers . given a set @xmath3 , @xmath4 denotes the number of elements in @xmath3 . given sets @xmath5 , their difference is @xmath6 . a set - valued map , denoted by @xmath7 , associates to an element of @xmath3 a subset of @xmath8 . we are given a team of @xmath9 robots tasked with providing coverage of a finite set of points in a non - convex and non - polygonal environment . in this section we translate concepts used in coverage of continuous environments to graphs . let @xmath10 be a finite set of points in a continuous environment . these points represent locations of interest , and are assumed to be connected by weighted edges . let @xmath11 be an ( undirected ) weighted graph with edge set @xmath12 and weight map @xmath13 ; we let @xmath14 be the weight of edge @xmath15 . we assume that @xmath16 is connected and think of the edge weights as distances between locations . [ rem : discretization ] for the examples in this paper we will use a coarse _ occupancy grid map _ as a representation of a continuous environment . in an occupancy grid @xcite , each grid cell is either free space or an obstacle ( occupied ) . to form a weighted graph , each free cell becomes a vertex and free cells are connected with edges if they border each other in the grid . edge weights are the distances between the centers of the cells , i.e. , the grid resolution . there are many other methods to discretize a space , including triangularization and other approaches from computational geometry @xcite , which could also be used . in any weighted graph @xmath16 there is a standard notion of distance between vertices defined as follows . a _ path _ in @xmath17 is an ordered sequence of vertices such that any consecutive pair of vertices is an edge of @xmath17 . the _ weight of a path _ is the sum of the weights of the edges in the path . given vertices @xmath18 and @xmath0 in @xmath17 , the _ distance _ between @xmath18 and @xmath0 , denoted @xmath19 , is the weight of the lowest weight path between them , or @xmath20 if there is no path . if @xmath17 is connected , then the distance between any two vertices in @xmath17 is finite . by convention , @xmath21 if @xmath22 . note that @xmath23 , for any @xmath24 . we will be partitioning @xmath10 into @xmath9 connected subsets or regions which will each be covered by an individual robot . to do so we need to define distances on induced subgraphs of @xmath16 . given @xmath25 , the _ subgraph induced by the restriction of @xmath17 to @xmath26 _ , denoted by @xmath27 , is the graph with vertex set equal to @xmath26 and edge set containing all weighted edges of @xmath17 where both vertices belong to @xmath26 . in other words , we set @xmath28 . the induced subgraph is a weighted graph with a notion of distance between vertices : given @xmath29 , we write @xmath30 note that @xmath31 we define a _ connected subset of @xmath10 _ as a subset @xmath32 such that @xmath33 and @xmath34 is connected . we can then partition @xmath10 into connected subsets as follows . [ def : conpartitions ] given the graph @xmath35 we define a _ connected @xmath36partition of @xmath10 _ as a collection @xmath37 of @xmath9 subsets of @xmath10 such that 1 . @xmath38 ; 2 . @xmath39 if @xmath40 ; 3 . @xmath41 for all @xmath42 ; and 4 . @xmath43 is connected for all @xmath42 . let @xmath44 to be the set of connected @xmath36partitions of @xmath10 . property ( ii ) implies that each element of @xmath10 belongs to just one @xmath43 , i.e. , each location in the environment is covered by just one robot . notice that each @xmath45 induces a connected subgraph in @xmath16 . in subsequent references to @xmath43 we will often mean @xmath46 , and in fact we refer to @xmath47 as the _ dominance subgraph _ or _ region _ of the @xmath48-th robot at time @xmath49 . among the ways of partitioning @xmath10 , there are some which are worth special attention . given a vector of distinct points @xmath50 , the partition @xmath51 is said to be a _ voronoi partition of q generated by c _ if , for each @xmath43 and all @xmath52 , we have @xmath53 and @xmath54 , @xmath55 . note that the voronoi partition generated by @xmath56 is not unique since how to apportion tied vertices is unspecified . for our gossip algorithms we need to introduce the notion of adjacent subgraphs . two distinct connected subgraphs @xmath43 , @xmath57 are said to be _ adjacent _ if there are two vertices @xmath58 , @xmath59 belonging , respectively , to @xmath43 and @xmath57 such that @xmath60 . observe that if @xmath43 and @xmath57 are adjacent then @xmath61 is connected . similarly , we say that robots @xmath48 and @xmath62 are adjacent or are neighbors if their subgraphs @xmath43 and @xmath57 are adjacent . accordingly , we introduce the following useful notion . for @xmath63 , we define the _ adjacency graph _ between regions of partition @xmath64 as @xmath65 , where @xmath66 if @xmath43 and @xmath57 are adjacent . note that @xmath67 is always connected since @xmath16 is . we define three coverage cost functions for graphs : @xmath68 , @xmath69 , and @xmath70 . let the _ weight function _ @xmath71 assign a relative weight to each element of @xmath10 . the _ one - center function _ @xmath68 gives the cost for a robot to cover a connected subset @xmath32 from a vertex @xmath72 with relative prioritization set by @xmath73 : @xmath74 a technical assumption is needed to solve the problem of minimizing @xmath75 : we assume from now on that a _ total order _ relation , @xmath76 , is defined on @xmath10 , i.e. , that @xmath77 . with this assumption we can deterministically pick a vertex in @xmath3 which minimizes @xmath68 as follows . [ def : centroid ] let @xmath10 be a totally ordered set , and let @xmath32 . we define the set of generalized centroids of @xmath3 as the set of vertices in @xmath3 which minimize @xmath68 , i.e. , @xmath78 further , we define the map @xmath79 as @xmath80 . we call @xmath81 the _ generalized centroid _ of @xmath3 . in subsequent use we drop the word generalized " for brevity . note that with this definition the centroid is well - defined , and also that the centroid of a region always belongs to the region . with a slight notational abuse , we define @xmath82 as the map which associates to a partition the vector of the centroids of its elements . we define the _ multicenter function _ @xmath69 to measure the cost for @xmath9 robots to cover a connected @xmath9-partition @xmath64 from the vertex set @xmath83 : @xmath84 we aim to minimize the performance function @xmath69 with respect to both the vertices @xmath56 and the partition @xmath64 . we can now state the coverage cost function we will be concerned with for the rest of this paper . let @xmath85 be defined by @xmath86 in the motivational scenario we are considering , each robot will periodically be asked to perform a task somewhere in its region with tasks appearing according to distribution @xmath73 . when idle , the robots would position themselves at the centroid of their region . by partitioning @xmath17 so as to minimize @xmath70 , the robot team would minimize the expected distance between a task and the robot which will service it . we introduce two notions of optimal partitions : centroidal voronoi and pairwise - optimal . our discussion starts with the following simple result about the multicenter cost function . [ prop : optimal - for - hgeneric ] let @xmath63 and @xmath50 . if @xmath87 is a voronoi partition generated by @xmath56 and @xmath88 is such that @xmath89 , then @xmath90 the second inequality is strict if any @xmath91 . proposition [ prop : optimal - for - hgeneric ] implies the following necessary condition : if @xmath92 minimizes @xmath69 , then @xmath93 and @xmath64 must be a voronoi partition generated by @xmath56 . thus , @xmath70 has the following property as an immediate consequence of proposition [ prop : optimal - for - hgeneric ] : given @xmath63 , if @xmath94 is a voronoi partition generated by @xmath95 then @xmath96 this fact motivates the following definition . @xmath63 is a _ centroidal voronoi partition _ of @xmath10 if there exists a @xmath97 such that @xmath64 is a voronoi partition generated by @xmath56 and @xmath98 . the set of _ pairwise - optimal partitions _ provides an alternative definition for the optimality of a partition : a partition is pairwise - optimal if , for every pair of adjacent regions , one can not find a better two - partition of the union of the two regions . this condition is formally stated as follows . @xmath63 is a _ pairwise - optimal partition _ if for every @xmath66 , @xmath99 the following proposition states that the set pairwise - optimal partitions is in fact a subset of the set of centroidal voronoi partitions . the proof is involved and is deferred to appendix [ sec : appendix_c ] . see fig . [ fig : voronoi ] for an example which demonstrates that the inclusion is strict . [ prop : optpair ] let @xmath51 be a _ pairwise - optimal partition_. then @xmath64 is also a _ centroidal voronoi partition_. for a given environment @xmath10 , a pair made of a centroidal voronoi partition @xmath64 and the corresponding vector of centroids @xmath56 is locally optimal in the following sense : @xmath70 can not be reduced by changing either @xmath64 or @xmath56 independently . a pairwise - optimal partition achieves this property and adds that for every pair of neighboring robots @xmath100 , there does not exist a two - partition of @xmath101 with a lower coverage cost . in other words , positioning the robots at the centroids of a centroidal voronoi partition ( locally ) minimizes the expected distance between a task appearing randomly in @xmath10 according to relative weights @xmath73 and the robot who owns the vertex where the task appears . positioning at the centroids of a pairwise - optimal partition improves performance by reducing the number of sub - optimal solutions which the team might converge to . we aim to partition @xmath10 among @xmath9 robotic agents using only asynchronous , unreliable , short - range communication . in section [ sec : model ] we describe the computation , motion , and communication capabilities required of the team of robots , and in section [ sec : problemformulation ] we formally state the problem we are addressing . in section [ sec : algorithm ] we propose our solution , the _ discrete gossip coverage algorithm _ , and in [ sec : illustrative ] we provide an illustration . in sections [ sec : convprop ] and [ sec : computation ] we state the algorithm s convergence and complexity properties . our discrete gossip coverage algorithm requires a team of @xmath9 robotic agents where each agent @xmath42 has the following basic computation and motion capabilities : 1 . agent @xmath48 knows its unique identifier @xmath48 ; 2 . agent @xmath48 has a processor with the ability to store @xmath16 and perform operations on subgraphs of @xmath102 ; and 3 . agent @xmath48 can determine which vertex in @xmath10 it occupies and can move at speed @xmath103 along the edges of @xmath16 to any other vertex in @xmath10 . the localization requirement in ( c3 ) is actually quite loose . localization is only used for navigation and not for updating partitions , thus limited duration localization errors are not a problem . the robotic agents are assumed to be able to communicate with each other according to the _ range - limited gossip communication model _ which is described as follows : 1 . given a communication range @xmath104 , when any two agents reside for some positive duration at a distance @xmath105 , they communicate at the sample times of a poisson process with intensity @xmath106 . recall that an homogeneous poisson process is a widely - used stochastic model for events which occur randomly and independently in time , where the expected number of events in a period @xmath107 is @xmath108 . [ rem : comm ] ( 1 ) this communication capability is the minimum necessary for our algorithm , any additional capability can only reduce the time required for convergence . for example , it would be acceptable to have intensity @xmath109 depend upon the pairwise robot distance in such a way that @xmath110 for @xmath105 . ( 2)we use distances in the graph to model limited range communication . these graph distances are assumed to approximate geodesic distances in the underlying continuous environment and thus path distances for a diffracting wave or moving robot . assume that , for all @xmath111 , each agent @xmath42 maintains in memory a connected subset @xmath47 of environment @xmath10 . our goal is to design a distributed algorithm that iteratively updates the partition @xmath112 while solving the following optimization problem : @xmath113 subject to the constraints imposed by the robot network model with range - limited gossip communication from section [ sec : model ] . in the design of an algorithm for the minimization problem there are two main questions which must be addressed . first , given the limited communication capabilities in ( c4 ) , how should the robots move inside @xmath10 to guarantee frequent enough meetings between pairs of robots ? second , when two robots are communicating , what information should they exchange and how should they update their regions ? in this section we introduce the _ discrete gossip coverage algorithm _ which , following these two questions , consists of two components : 1 . the _ random destination & wait motion protocol _ ; and 2 . the _ pairwise partitioning rule_. the concurrent implementation of the random destination & wait motion protocol and the pairwise partitioning rule determines the evolution of the positions and dominance subgraphs of the agents as we now formally describe . we start with the random destination & wait motion protocol . ' '' '' width height .4pt * random destination & wait motion protocol * ' '' '' width height .4pt each agent @xmath114 determines its motion by repeatedly performing the following actions : agent @xmath48 samples a _ destination vertex _ @xmath58 from a uniform distribution over its dominance subgraph @xmath43 ; agent @xmath48 moves to vertex @xmath58 through the shortest path in @xmath43 connecting the vertex it currently occupies and @xmath58 ; and agent @xmath48 waits at @xmath58 for a duration @xmath115 . ' '' '' width height .4pt if agent @xmath48 is moving from one vertex to another we say that agent @xmath48 is in the _ moving _ state while if agent @xmath48 is waiting at some vertex we say that it is in the _ waiting _ state . the motion protocol is designed to ensure frequent enough communication between pairs of robots . in general , any motion protocol can be used which meets this requirement , so @xmath48 could select @xmath58 from the boundary of @xmath43 or use some heuristic non - uniform distribution over @xmath43 . if any two agents @xmath48 and @xmath62 reside in two vertices at a graphical distance smaller that @xmath116 for some positive duration , then at the sample times of the corresponding communication poisson process the two agents exchange sufficient information to update their respective dominance subgraphs @xmath43 and @xmath57 via the pairwise partitioning rule . ' '' '' width height .4pt * pairwise partitioning rule * ' '' '' width height .4pt assume that at time @xmath111 , agent @xmath48 and agent @xmath62 communicate . without loss of generality assume that @xmath117 . let @xmath47 and @xmath118 denote the current dominance subgraphs of @xmath48 and @xmath62 , respectively . moreover , let @xmath119 denote the time instant just after @xmath49 . then , agents @xmath48 and @xmath62 perform the following tasks : agent @xmath48 transmits @xmath47 to agent @xmath62 and vice - versa initialize @xmath120 , @xmath121 , @xmath122 , @xmath123 compute @xmath124 and an ordered list @xmath125 of all pairs of vertices in @xmath126 compute the sets + @xmath127 + @xmath128 * if * @xmath129 + @xmath130 * then * @xmath131 @xmath132 ' '' '' width height .4pt some remarks are now in order . ( 1 ) the pairwise partitioning rule is designed to find a minimum cost two - partition of @xmath126 . more formally , if list @xmath125 and sets @xmath133 and @xmath134 for @xmath135 are defined as in the pairwise partitioning rule , then @xmath133 and @xmath134 are an optimal two - partition of @xmath126 . \(2 ) while the loop in steps 4 - 7 must run to completion to guarantee that @xmath133 and @xmath134 are an optimal two - partition of @xmath126 , the loop is designed to return an intermediate sub - optimal result if need be . if @xmath43 and @xmath57 change , then @xmath70 will decrease and this is enough to ensure eventual convergence . ( 3)we make a simplifying assumption in the pairwise partitioning rule that , once two agents communicate , the application of the partitioning rule is instantaneous . we discuss the actual computation time required in section [ sec : computation ] and some implementation details in section [ sec : results ] . ( 4)notice that simply assigning @xmath133 to @xmath48 and @xmath134 to @xmath62 can cause the robots to `` switch sides '' in @xmath126 . while convergence is guaranteed regardless , switching may be undesirable in some applications . in that case , any smart matching of @xmath133 and @xmath134 to @xmath48 and @xmath62 may be inserted . ( 5)agents who are not adjacent may communicate but the partitioning rule will not change their regions . indeed , in this case @xmath133 and @xmath134 will not change from @xmath47 and @xmath118 . some possible modifications and extensions to the algorithm are worth mentioning . in case the robots have heterogeneous dynamics , line 5 can be modified to consider per - robot travel times between vertices . for example , @xmath136 could be replaced by the expected time for robot @xmath48 to travel from @xmath137 to @xmath138 while @xmath139 would consider robot @xmath62 . here we focus on partitioning territory , but this algorithm can easily be combined with methods to provide a service in @xmath10 as in @xcite . the agents could split their time between moving to meet their neighbors and update territory , and performing requested tasks in their region . the simulation in fig . [ fig : sim_four ] shows four robots partitioning a square environment with obstacles where the free space is represented by a @xmath140 grid . in the initial partition shown in the left panel , the robot in the top right controls most of the environment while the robot in the bottom left controls very little . the robots then move according to the random destination & wait motion protocol , and communicate according to range - limited gossip communication model with @xmath141 ( four edges in the graph ) . the first pairwise territory exchange is shown in the second panel , where the bottom left robot claims some territory from the robot on the top left . a later exchange between the two robots on the top is shown in the next two panels . notice that the cyan robot in the top right gives away the vertex it currently occupies . in such a scenario , we direct the robot to follow the shortest path in @xmath16 to its updated territory before continuing on to a random destination . after 9 pairwise territory exchanges , the robots reach the pairwise - optimal partition shown at right in fig . [ fig : sim_four ] . the expected distance between a random vertex and the closest robot decreases from @xmath142 down to @xmath143 . the strength of the discrete gossip coverage algorithm is the possibility of enforcing that a partition will converge to a pairwise - optimal partition through pairwise territory exchange . in theorem [ th : main ] we summarize this convergence property , with proofs given in section [ sec : convergence ] . [ th : main ] consider a network of @xmath9 robotic agents endowed with computation and motion capacities ( c1 ) , ( c2 ) , ( c3 ) , and communication capacities ( c4 ) . assume the agents implement the _ discrete gossip coverage algorithm _ consisting of the concurrent implementation of the _ random destination & wait motion protocol _ and the _ pairwise partitioning rule_. then , a. [ item : well - posedness ] the partition @xmath144 remains connected and is described by @xmath145 and b. [ item : convergence ] @xmath144 converges almost surely in finite time to a pairwise - optimal partition . by definition , a pairwise - optimal partition is optimal in that @xmath70 can not be improved by changing only two regions in the partition . for simplicity we assume uniform robot speeds , communication processes , and waiting times . an extension to non - uniform processes would be straightforward . in this subsection we explore the computational requirements of the discrete gossip coverage algorithm , and make some comments on implementation . cost function @xmath146 is the sum of the distances between @xmath18 and all other vertices in @xmath43 . this computation of one - to - all distances is the core computation of the algorithm . for most graphs of interest the total number of edges @xmath147 is proportional to @xmath148 , so we will state bounds on this computation in terms of @xmath149 . computing one - to - all distances requires one of the following : * if all edge weights in @xmath16 are the same ( e.g. , for a graph from an occupancy grid ) , a breadth - first search approach can be used which requires @xmath150 in time and memory ; * otherwise , dijkstra s algorithm must be used which requires @xmath151 in time and @xmath150 in memory . let @xmath152 be the time to compute one - to - all distances in @xmath43 , then computing @xmath146 requires @xmath153 in time . [ prop : computation ] the motion protocol requires @xmath150 in memory , and @xmath153 in computation time . the partitioning rule requires @xmath154 in communication bandwidth between robots @xmath48 and @xmath62 , @xmath154 in memory , and can run in any time . we first prove the claims for the motion protocol . step 2 is the only non - trivial step and requires finding a shortest path in @xmath43 , which is equivalent to computing one - to - all distances from the robot s current vertex . hence , it requires @xmath153 in time and @xmath155 in memory . we now prove the claims for the partitioning rule . in step 1 , robots @xmath48 and @xmath62 transmit their subgraphs to each other , which requires @xmath154 in communication bandwidth . for step 3 , the robots determine @xmath156 , which requires @xmath154 in memory to store . step 4 is the start of a loop which executes @xmath157 times , affecting the time complexity of steps 5 , 6 and 7 . step 5 requires two computations of one - to - all distances in @xmath126 which each take @xmath158 . step 6 involves four computations of @xmath68 over different subsets of @xmath126 , however those for @xmath133 and @xmath134 can be stored from previous computation . since @xmath159 and @xmath160 are strict subsets of @xmath126 , step 5 takes longer than step 6 . step 7 is trivial , as is step 8 . the total time complexity of the loop is thus @xmath161 . however , the loop in steps 4 - 7 can be truncated after any number of iterations . while it must run to completion to guarantee that @xmath133 and @xmath134 are an optimal two - partition of @xmath126 , the loop is designed to return an intermediate sub - optimal result if need be . if @xmath43 and @xmath57 change , then @xmath70 will decrease . our convergence result will hold provided that all elements of @xmath125 are eventually checked if @xmath43 and @xmath57 do not change . thus , the partitioning rule can run in any time with each iteration requiring @xmath158 . all of the computation and communication requirements in proposition [ prop : computation ] are independent of the number of robots and scale with the size of a robot s partition , meaning the discrete gossip coverage algorithm can easily scale up for large teams of robots in large environments . this section is devoted to proving the two statements in theorem [ th : main ] . the proof that the pairwise partitioning rule maps a connected @xmath9-partition into a connected @xmath9-partition is straightforward . the proof of convergence is more involved and is based on the application of lemma [ lem : finite - lasalle ] in appendix [ sec : appendix_a ] to the discrete gossip coverage algorithm . lemma [ lem : finite - lasalle ] establishes strong convergence properties for a particular class of set valued maps ( set - valued maps are briefly reviewed in appendix [ sec : appendix_a ] ) . we start by proving that the pairwise partitioning rule is well - posed in the sense that it maintains a connected partition . to prove the statement we need to show that @xmath162 satisfies points ( i ) through ( iv ) of definition [ def : conpartitions ] . from the definition of the pairwise partitioning rule , we have that @xmath163 and @xmath164 . moreover , since @xmath165 and @xmath166 , it follows that @xmath167 and @xmath168 . these observations imply the validity of points ( i ) , ( ii ) , and ( iii ) for @xmath162 . finally , we must show that @xmath169 and @xmath170 are connected , i.e. , @xmath162 also satisfies point ( iv ) . to do so we show that , given @xmath171 , any shortest path in @xmath172 connecting @xmath138 to @xmath173 completely belongs to @xmath133 . we proceed by contradiction . let @xmath174 denote a shortest path in @xmath175 connecting @xmath138 to @xmath173 and let us assume that there exists @xmath176 such that @xmath177 . for @xmath178 to be in @xmath134 means that @xmath179 . this implies that @xmath180 this is a contradiction for @xmath171 . similar considerations hold for @xmath134 . the rest of this section is dedicated to proving convergence . our first step is to show that the evolution determined by the discrete gossip coverage algorithm can be seen as a set - valued map . to this end , for any pair of robots @xmath181 , @xmath182 , we define the map @xmath183 by @xmath184 where @xmath185 and @xmath186 . if at time @xmath187 the pair @xmath100 and no other pair of robots perform an iteration of the pairwise partitioning rule , then the dynamical system on the space of partitions is described by @xmath188 we define the set - valued map @xmath189 as @xmath190 observe that can then be rewritten as @xmath191 . the next two propositions state facts whose validity is ensured by lemma [ lemma : onmotionprotocol ] of appendix [ sec : appendix_b ] which states a key property of the random destination & wait motion protocol . [ prop : tk ] consider @xmath9 robots implementing the discrete gossip coverage algorithm . then , there almost surely exists an increasing sequence of time instants @xmath192 such that @xmath193 for some @xmath194 . the proof follows directly from lemma [ lemma : onmotionprotocol ] which implies that the time between two consecutive pairwise communications is almost surely finite . the existence of time sequence @xmath192 allows us to to express the evolution generate by the discrete gossip coverage algorithm as a discrete time process . let @xmath195 and @xmath196 , then @xmath197 where @xmath189 is defined as in . given @xmath198 , let @xmath199 denote the information which completely characterizes the state of discrete gossip coverage algorithm just after the @xmath0-th iteration of the partitioning rule , i.e. , at time @xmath200 . specifically , @xmath199 contains the information related to the partition @xmath201 , the positions of the robots at @xmath200 , and whether each robot is in the _ waiting _ or _ moving _ state at @xmath200 . the following result characterizes the probability that , given @xmath199 , the @xmath202-th iteration of the partitioning rule is governed by any of the maps @xmath203 , @xmath204 . [ prop : pi ] consider a team of @xmath9 robots with capacities ( c1 ) , ( c2 ) , ( c3 ) , and ( c4 ) implementing the discrete gossip coverage algorithm . then , there exists a real number @xmath205 , such that , for any @xmath206 and @xmath204 @xmath207\geq \bar{\pi}.\ ] ] assume that at time @xmath208 one pair of robots communicates . given a pair @xmath209 , we must find a lower bound for the probability that @xmath210 is the communicating pair . since all the poisson communication processes have the same intensity , the distribution of the chance of communication is uniform over the pairs which are `` able to communicate , '' i.e. , closer than @xmath116 to each other . thus , we must only show that @xmath210 has a positive probability of being able to communicate at time @xmath208 , which is equivalent to showing that @xmath210 is able to communicate for a positive fraction of time with positive probability . the proof of lemma [ lemma : onmotionprotocol ] implies that with probability at least @xmath211 any pair in @xmath212 is able to communicate for a fraction of time not smaller than @xmath213 where @xmath214 and @xmath107 are defined in the proof of lemma [ lemma : onmotionprotocol ] . hence the result follows . the property in proposition [ prop : pi ] can also be formulated as follows . let @xmath215 be the stochastic process such that @xmath216 is the communicating pair at time @xmath0 . then , the sequence of pairs of robots performing the partitioning rule at time instants @xmath192 can be seen as a realization of the process @xmath217 , which satisfies @xmath218 \geq \bar{\pi}\ ] ] for all @xmath204 . next we show that the cost function decreases whenever the application of @xmath219 from changes the territory partition . this fact is a key ingredient to apply lemma [ lem : finite - lasalle ] . [ lemma : tdecr ] let @xmath220 and let @xmath221 . if @xmath222 , then @xmath223 . without loss of generality assume that @xmath100 is the pair executing the pairwise partitioning rule . then @xmath224 according to the definition of the pairwise partitioning rule we have that if @xmath225 , @xmath226 , then @xmath227 from which the statement follows . we now complete the proof of the main result , theorem [ th : main ] . note that the algorithm evolves in a finite space of partitions , and by theorem [ th : main ] statement ( [ item : well - posedness ] ) , the set @xmath44 is strongly positively invariant . this fact implies that assumption ( i ) of lemma [ lem : finite - lasalle ] is satisfied . from lemma [ lemma : tdecr ] it follows that assumption ( ii ) is also satisfied , with @xmath70 playing the role of the function @xmath126 . finally , the property in is equivalent to the property of _ persistent random switches _ stated in assumption ( iii ) of lemma [ lem : finite - lasalle ] , for the special case @xmath228 . hence , we are in the position to apply lemma [ lem : finite - lasalle ] and conclude convergence in finite - time to an element of the intersection of the equilibria of the maps @xmath203 , which by definition is the set of the pairwise - optimal partitions . to demonstrate the utility and study practical issues of the discrete gossip coverage algorithm , we implemented it using the open - source player / stage robot control system @xcite and the boost graph library ( bgl ) @xcite . all results presented here were generated using player 2.1.1 , stage 2.1.1 , and bgl 1.34.1 . to compute distances in uniform edge weight graphs we extended the bgl breadth - first search routine with a distance recorder event visitor . to evaluate the performance of our gossip coverage algorithm with larger teams , we tested 30 simulated robots partitioning a map representing a @xmath229 portion of campus at the university of california at santa barbara . as shown in fig . [ fig : large_sim ] , the robots are tasked with providing coverage of the open space around some of the buildings on campus , a space which includes a couple open quads , some narrower passages between buildings , and a few dead - end spurs . for this large environment the simulated robots are @xmath230 on a side and can move at @xmath231 . each territory cell is @xmath232 . in this simulation we handle communication and partitioning as follows . the communication range is set to @xmath233 ( 10 edges in the graph ) with @xmath234 . the robots wait at their destination vertices for @xmath235 . this value for @xmath236 was chosen so that on average one quarter of the robots are waiting at any moment . lower values of @xmath236 mean the robots are moving more of the time and as a result more frequently miss connections , while for higher @xmath236 the robots spend more time stationary which also reduces the rate of convergence . with the goal of improving communication , we implemented a minor modification to the motion protocol : each robot picks its random destination from the cells forming the open boundary is the set of vertices in @xmath43 which are adjacent to at least one vertex owned by another agent . ] of its territory . in our implementation , the full partitioning loop may take @xmath237 seconds for the largest initial territories in fig . [ fig : large_sim ] . we chose to stop the loop after a quarter second for this simulation to verify the anytime computation claim the 30 robots start clustered in the center of the map between engineering ii and broida hall , and an initial voronoi partition is generated from these starting positions . this initial partition is shown on the left in fig . [ fig : large_sim ] with the robots positioned at the centroids of their starting regions . the initial partition has a cost of @xmath238 . the team spends about 27 minutes moving and communicating according to the discrete gossip coverage algorithm before settling on the final partition on the right of fig . [ fig : large_sim ] . the coverage cost of the final equilibrium improved by @xmath239 to @xmath240 . visually , the final partition is also dramatically more uniform than the initial condition . this result demonstrates that the algorithm is effective for large teams in large non - convex environments . over time for the simulation in fig . [ fig : large_sim].,height=125 ] fig . [ fig : large_sim_cost ] shows the evolution of @xmath70 during the simulation . the largest cost improvements happen early when the robots that own the large territories on the left and right of the map communicate with neighbors with much smaller territories . these big territory changes then propagate through the network as the robots meet and are pushed and pulled towards a lower cost partition . we conducted an experiment to test the algorithm using three physical robots in our lab , augmented by six simulated robots in a synthetic environment extending beyond the lab . our lab space is @xmath241 on a side and is represented by the upper left portion of the territory maps in fig . [ fig : experiment ] . the territory graph loops around a center island of desks . we extended the lab space through three connections into a simulated environment around the lab , producing a @xmath242 environment . the map of the environment was specified with a @xmath243 bitmap which we overlayed with a @xmath244 resolution occupancy grid representing the free territory for the robots to cover . the result is a lattice - like graph with all edge weights equal to @xmath244 . the @xmath244 resolution was chosen so that our physical robots would fit easily inside a cell . additional details of our implementation are as follows . we use erratic mobile robots from videre design , as shown in fig . [ fig : robot ] . the vehicle platform has a roughly square footprint @xmath245 , with two differential drive wheels and a single rear caster . each robot carries an onboard computer with a 1.8ghz core 2 duo processor , 1 gb of memory , and 802.11 g wireless communication . for navigation and localization , each robot is equipped with a hokuyo urg-04lx laser rangefinder . the rangefinder scans @xmath246 points over @xmath247 at @xmath248 with a range of @xmath249 meters . our mixed physical and virtual robot experiments are run from a central computer which is attached to a wireless router so it can communicate with the physical robots . the central computer creates a simulated world using stage which mirrors and extends the real space in which the physical robots operate . the central computer also simulates the virtual members of the robot team . these virtual robots are modeled off of our hardware : they are differential drive with the same geometry as the erratic platform and use simulated hokuyo urg-04lx rangefinders . we use the ` amcl ` driver in player which implements adaptive monte - carlo localization @xcite . the physical robots are provided with a map of our lab with a @xmath250 resolution and told their starting pose within the map . we set an initial pose standard deviation of @xmath251 in position and @xmath252 in orientation , and request localization updates using @xmath253 of the sensor s range measurements for each change of @xmath254 in position or @xmath255 in orientation reported by the robot s odometry system . we then use the most likely pose estimate output by ` amcl ` as the location of the robot . for simplicity and reduced computational demand , we allow the virtual robots access to perfect localization information . each robot continuously executes the random destination & wait motion protocol , with navigation handled by the ` snd ` driver in player which implements smooth nearness diagram navigation @xcite . for ` snd ` we set the robot radius parameter to @xmath256 , obstacle avoidance distance to @xmath257 , and maximum speeds to @xmath258 and @xmath259 . the ` snd ` driver is a local obstacle avoidance planner , so we feed it a series of waypoints every couple meters along paths found in @xmath16 . we consider a robot to have achieved its target location when it is within @xmath260 and it will then wait for @xmath235 . for the physical robots the motion protocol and navigation processes run on board , while there are separate threads for each virtual robot on the central computer . as the robots move , a central process monitors their positions and simulates the range - limited gossip communication model between both real and virtual robots . we set @xmath141 and @xmath261 . these parameters were chosen so that the robots would be likely to communicate when separated by at most four edges , but would also sometimes not connect despite being close . when this process determines two robots should communicate , it informs the robots who then perform the pairwise partitioning rule . our pairwise communication implementation is blocking : if robot @xmath48 is exchanging territory with @xmath62 , then it informs the match making process that it is unavailable until the exchange is complete . the results of our experiment with three physical robots and six simulated robots are shown in figs . [ fig : experiment ] and [ fig : exp_cost ] . the left column in fig . [ fig : experiment ] shows the starting positions of the team of robots , with the physical robots , labeled 1 , 2 , and 3 , lined up in a corner of the lab and the simulated robots arrayed around them . the starting positions are used to generate the initial voronoi partition of the environment . the physical robots own the orange , blue , and lime green territories in the upper left quadrant . we chose this initial configuration to have a high coverage cost , while ensuring that the physical robots will remain in the lab as the partition evolves . in the middle column , robots 1 and 2 have met along their shared border and are exchanging territory . in the territory map , the solid red line indicates 1 and 2 are communicating and their updated territories are drawn with solid orange and blue , respectively . the camera view confirms that the two robots have met on the near side of the center island of desks . the final partition at right in fig . [ fig : experiment ] is reached after @xmath262 minutes . all of the robots are positioned at the centroids of their final territories . the three physical robots have gone from a cluster in one corner of the lab to a more even spread around the space . . the total cost @xmath70 is shown above in black , while @xmath68 for each robot is shown below in the robot s color . ] [ fig : exp_cost ] shows the evolution of the cost function @xmath70 as the experiment progresses , including the costs for each robot . as expected , the total cost never increases and the disparity of costs for the individual robots shrinks over time until settling at a pairwise - optimal partition . in this experiment the hardware challenges of sensor noise , navigation , and uncertainty in position were efficiently handled by the ` amcl ` and ` snd ` drivers . the coverage algorithm assumed the role of a higher - level planner , taking in position data from ` amcl ` and directing ` snd ` . by far the most computationally demanding component was ` amcl ` , but the position hypotheses from ` amcl ` are actually unnecessary : our coverage algorithm only requires knowledge of the vertex a robot occupies . if a less intensive localization method is available , the algorithm could run on robots with significantly lower compute power . in this subsection we present a numerical comparison of the performance of the discrete gossip coverage algorithm and the following two lloyd - type algorithms . this method is from @xcite and @xcite , we describe it here for convenience . at each discrete time instant @xmath263 , each robot @xmath48 performs the following tasks : ( 1 ) @xmath48 transmits its position and receives the positions of all adjacent robots ; ( 2 ) @xmath48 computes its voronoi region @xmath43 based on the information received ; and ( 3 ) @xmath48 moves to @xmath264 . this method is from @xcite . it is a gossip algorithm , and so we have used the same communication model and the random destination & wait motion protocol to create meetings between robots . say robots @xmath48 and @xmath62 meet at time @xmath49 , then the pairwise lloyd partitioning rule works as follows : ( 1 ) robot @xmath48 transmits @xmath47 to @xmath62 and vice versa ; ( 2 ) both robots determine @xmath265 ; ( 3 ) robot @xmath48 sets @xmath169 to be its voronoi region of @xmath126 based on @xmath266 and @xmath267 , and @xmath62 does the equivalent . for both lloyd algorithms we use the same tie breaking rule when creating voronoi regions as is present in the pairwise partitioning rule : ties go to the robot with the lowest index . our first numerical result uses a monte carlo probability estimation method from @xcite to place probabilistic bounds on the performance of the two gossip algorithms . recall that the chernoff bound describes the minimum number of random samples @xmath268 required to reach a certain level of accuracy in a probability estimate from independent bernoulli tests . for an accuracy @xmath269 and confidence @xmath270 , the number of samples is given by @xmath271 for @xmath272 and @xmath273 , at least 116 samples are required . figure [ fig : bad_start ] shows both the initial territory partition of the extended laboratory environment used and also a histogram of the final results for the following monte carlo test . the environment and robot motion models used are described in section [ sec : implementation ] . starting from the indicated initial condition , we ran 116 simulations of both gossip algorithms . the randomness in the test comes from the sequence of pairwise communications . these sequences were generated using : ( 1 ) the random destination & wait motion protocol with @xmath58 sampled uniformly from the open boundary of @xmath43 and @xmath235 ; and ( 2 ) the range - limited gossip communication model with @xmath141 and @xmath261 . the cost of the initial partition in fig . [ fig : bad_start ] is @xmath274 , while the best known partition for this environment has a cost of just under @xmath275 . the histogram in fig . [ fig : bad_start ] shows the final equilibrium costs for 116 simulations of the discrete gossip coverage algorithm ( black ) and the gossip lloyd algorithm ( gray ) . it also shows the final cost using the decentralized lloyd algorithm ( red dashed line ) , which is deterministic from a given initial condition . the histogram bins have a width of @xmath276 and start from @xmath277 . for the discrete gossip coverage algorithm , @xmath278 out of @xmath279 trials reach the bin containing the best known partition and the mean final cost is @xmath280 . the gossip lloyd algorithm reaches the lowest bin in only @xmath237 of @xmath279 trials and has a mean final cost of @xmath281 . the decentralized lloyd algorithm settles at @xmath282 . our new gossip algorithm requires an average of @xmath283 pairwise communications to reach an equilibrium , whereas gossip lloyd requires @xmath284 . based on these results , we can conclude with @xmath285 confidence that there is at least an @xmath286 probability that 9 robots executing the discrete gossip coverage algorithm starting from the initial partition shown in fig . [ fig : bad_start ] will reach a pairwise - optimal partition which has a cost within @xmath287 of the best known cost . we can further conclude with @xmath285 confidence that the gossip lloyd algorithm will settle more than @xmath287 above the best known cost at least @xmath288 of the time starting from this initial condition . comparing discrete gossip coverage algorithm ( black bars ) , gossip lloyd algorithm ( gray bars ) , and decentralized lloyd algorithm ( red dashed line ) . for the gossip algorithms , 116 simulations were performed with different sequences of pairwise communications . the decentralized lloyd algorithm is deterministic given an initial condition so only one final cost is shown . the initial cost for each test is drawn with the green dashed line . ] figure [ fig : multi_compare ] compares final cost histograms for @xmath289 different initial conditions for the same environment and parameters as described above . each initial condition was created by selecting unique starting locations for the robots uniformly at random and using these locations to generate an initial voronoi partition . the initial cost for each test is shown with the green dashed line . in 9 out of 10 tests the discrete gossip coverage algorithm reaches the histogram bin with the best known partition in at least @xmath290 of @xmath279 trials . the two lloyd methods get stuck in sub - optimal centroidal voronoi partitions more than @xmath287 away from the best known partition in more than half the trials in 7 of 10 tests . we have presented a novel distributed partitioning and coverage control algorithm which requires only unreliable short - range communication between pairs of robots and works in non - convex environments . the classic lloyd approach to coverage optimization involves iteration of separate centering and voronoi partitioning steps . for gossip algorithms , however , this separation is unnecessary computationally and we have shown that improved performance can be achieved without it . our new discrete gossip coverage algorithm provably converges to a subset of the set of centroidal voronoi partitions which we labeled pairwise - optimal partitions . through numerical comparisons we demonstrated that this new subset of solutions avoids many of the local minima in which lloyd - type algorithms can get stuck . our vision is that this partitioning and coverage algorithm will form the foundation of a distributed task servicing setup for teams of mobile robots . the robots would split their time between servicing tasks in their territory and moving to contact their neighbors and improve the coverage of the space . our convergence results only require sporadic improvements to the cost function , affording flexibility in robot behaviors and capacities , and offering the ability to handle heterogeneous robotic networks . in the bigger picture , this paper demonstrates the potential of gossip communication in distributed coordination algorithms . there appear to be many other problems where this realistic and minimal communication model could be fruitfully applied . given a set @xmath291 , a set - valued map @xmath292 is a map which associates to an element @xmath293 a subset @xmath294 a set - valued map is non - empty if @xmath295 for all @xmath296 . given a non - empty set - valued map @xmath219 , an evolution of the dynamical system associated to @xmath219 is a sequence @xmath297 where @xmath298 for all @xmath299 a set @xmath300 is _ strongly positively invariant _ for @xmath219 if @xmath301 for all @xmath302 . [ lem : finite - lasalle ] let @xmath303 be a finite metric space . given a collection of maps @xmath304 , define the set - valued map @xmath292 by @xmath305 . given a stochastic process @xmath306 , consider an evolution @xmath307 of @xmath219 satisfying @xmath308 assume that : 1 . there exists a set @xmath309 that is strongly positively invariant for @xmath219 ; 2 . there exists a function @xmath310 such that @xmath311 , for all @xmath312 and @xmath313 ; and 3 . there exist @xmath314 and @xmath315 such that , for all @xmath316 and @xmath317 , there exists @xmath318 such that @xmath319 \geq p. $ ] for @xmath320 , let @xmath321 be the set of fixed points of @xmath322 in @xmath323 , i.e. , @xmath324 . if @xmath325 , then the evolution @xmath307 converges almost surely in finite time to an element of the set @xmath326 , i.e. , there exists almost surely @xmath327 such that , for some @xmath328 , @xmath329 for @xmath330 [ lemma : onmotionprotocol ] consider @xmath9 robots implementing the discrete gossip coverage algorithm starting from an arbitrary @xmath51 . consider @xmath187 and let @xmath144 denote the partition at time @xmath49 . assume that at time @xmath49 no two robots are communicating . then , there exist @xmath331 and @xmath332 , independent of @xmath144 and the positions and states of the robots at time @xmath49 , such that , for every @xmath333 , @xmath334\ge \alpha.$ ] our goal is to lower bound the probability that @xmath48 and @xmath62 will communicate within the interval @xmath340 . to do so we construct _ one _ sequence of events of positive probability which enables such communication . consider the following situation : @xmath48 is in the _ moving _ state and needs time @xmath341 to reach its destination @xmath58 , whereas robot @xmath62 is in the _ waiting _ state at vertex @xmath59 and must wait there for time @xmath342 . we denote by @xmath343 ( resp . @xmath344 ) the time needed for @xmath48 ( resp . @xmath62 ) to travel from @xmath58 ( resp . @xmath59 ) to @xmath137 ( resp . @xmath345 ) . let @xmath346 be the event such that @xmath48 performs the following actions in @xmath347 without communicating with any robot @xmath348 : next , we lower bound the probability that event @xmath355 occurs . recall the definition of @xmath356 from sec . [ sec : model ] . since a robot can have at most @xmath357 neighbors , the probability that ( i ) of @xmath355 happens is lower bounded by @xmath358 for ( ii ) , the probability that @xmath48 chooses @xmath137 is @xmath359 , which is lower bounded by @xmath360 . then , in order to spend at least @xmath361 at @xmath137 , @xmath48 must choose @xmath137 for @xmath362 consecutive times . finally , the probability that during this interval @xmath48 will not communicate with any robot other than @xmath62 is lower bounded by @xmath363 the probability that ( ii ) occurs is thus lower bounded by @xmath364 combining the bounds for ( i ) and ( ii ) , it follows that @xmath365\geq \bigl(\tfrac{1}{{\left|q\right|}}\bigr)^{\lceil \frac{\delta}{\tau } \rceil } e^{-{{\lambda_{\textup{comm}}}}(\delta+\tau ) n}.\ ] ] the same lower bound holds for @xmath366 $ ] , meaning that @xmath367&={\mathbb{p}}\left[e_{i}\right]\ , { \mathbb{p}}\left[e_{j}\right ] \geq \bigl(\tfrac{1}{{\left|q\right|}}\bigr)^{2 \lceil \frac{\delta}{\tau } \rceil } e^{-2 { { \lambda_{\textup{comm}}}}(\delta+\tau ) n}.\end{aligned}\ ] ] if event @xmath368 occurs , then robots @xmath48 and @xmath62 will be at adjacent vertices for an amount of time during the interval @xmath347 equal to @xmath369 since @xmath343 and @xmath344 are no more than @xmath370 , we can conclude that @xmath48 and @xmath62 will be within @xmath116 for at least @xmath236 . conditioned on @xmath368 occurring , the probability that @xmath48 and @xmath62 communicate in @xmath347 is lower bounded by @xmath371 . a suitable choice for @xmath214 from the statement of the lemma is thus @xmath372 it can be shown that this also constitutes a lower bound for the other possible combinations of initial states : robot @xmath48 is _ waiting _ and robot @xmath62 is _ moving _ ; robots @xmath48 and @xmath62 are both _ moving _ ; and robots @xmath48 and @xmath62 are both _ waiting_. to create a contradiction , assume that @xmath51 is a pairwise - optimal partition but not a centroidal voronoi partition . in other words , there exist components @xmath43 and @xmath57 in @xmath64 and an element @xmath138 of one component , say @xmath373 , such that @xmath374 choose @xmath57 such that for all @xmath375 @xmath376 let @xmath377 be a shortest path in @xmath17 connecting @xmath137 to @xmath345 and let @xmath378 be the first element of the path starting from @xmath379 which is not in @xmath57 . let @xmath380 be such that @xmath381 . in the first case , we again have a contradiction using the same logic above with @xmath178 in place of @xmath138 . in the second case , we must further consider whether there exists a @xmath389 such that every vertex in @xmath389 is also in @xmath390 . if there is not such a path , then @xmath391 and we again have a contradiction as above . if there is such a path , then we can instead repeat this analysis using using @xmath380 in place of @xmath62 and considering the path formed by this @xmath389 and the vertices in @xmath383 after @xmath178 . since the next vertex playing the role of @xmath178 must be closer to @xmath138 , we will eventually find a vertex which creates a contradiction . r. smith , j. das , h. heidarsson , a. pereira , f. arrichiello , i. cetnic , l. darjany , m .- e . garneau , m. howard , c. oberg , m. ragan , e. seubert , e. smith , b. stauffer , a. schnetzer , g. toro - farmer , d. caron , b. jones , and g. sukhatme , `` usc cinaps builds bridges , '' _ ieee robotics & automation magazine _ , vol . 17 , no . 1 , pp . 2030 , 2010 . s. yun , m. schwager , and d. rus , `` coordinating construction of truss structures using distributed equal - mass partitioning , '' in _ international symposium on robotics research _ , ( lucerne , switzerland ) , aug . 2009 . l. c. a. pimenta , v. kumar , r. c. mesquita , and g. a. s. pereira , `` sensing and coverage for a network of heterogeneous robots , '' in _ ieee conf . on decision and control _ , ( cancn , mxico ) , pp . 39473952 , dec . 2008 . r. cortez , h. tanner , and r. lumia , `` distributed robotic radiation mapping , '' in _ experimental robotics _ ( o. khatib , v. kumar , and g. pappas , eds . ) , vol . 54 of _ springer tracts in advanced robotics _ , pp . 147156 , springer , 2009 . f. bullo , r. carli , and p. frasca , `` gossip coverage control for robotic networks : dynamical systems on the the space of partitions , '' _ siam journal on control and optimization _ , aug available at http://motion.me.ucsb.edu/pdf/2008u-bcf.pdf . j. w. durham , r. carli , p. frasca , and f. bullo , `` discrete partitioning and coverage control with gossip communication , '' in _ asme dynamic systems and control conference _ , ( hollywood , ca , usa ) , pp . 225232 , j. w. durham , r. carli , and f. bullo , `` pairwise optimal coverage control for robotic networks in discretized environments , '' in _ ieee conf . on decision and control _ , ( atlanta , ga , usa ) , pp . 72867291 , dec . 2010 . b. gerkey , r. t. vaughan , and a. howard , `` the player / stage project : tools for multi - robot and distributed sensor systems , '' in _ int . conference on advanced robotics _ , ( coimbra , portugal ) , pp . 317323 , june 2003 .

we propose distributed algorithms to automatically deploy a team of mobile robots to partition and provide coverage of a non - convex environment . to handle arbitrary non - convex environments , we represent them as graphs . our partitioning and coverage algorithm requires only short - range , unreliable pairwise `` gossip '' communication . the algorithm has two components : ( 1 ) a motion protocol to ensure that neighboring robots communicate at least sporadically , and ( 2 ) a pairwise partitioning rule to update territory ownership when two robots communicate . by studying an appropriate dynamical system on the space of partitions of the graph vertices , we prove that territory ownership converges to a pairwise - optimal partition in finite time . this new equilibrium set represents improved performance over common lloyd - type algorithms . additionally , we detail how our algorithm scales well for large teams in large environments and how the computation can run in anytime with limited resources . finally , we report on large - scale simulations in complex environments and hardware experiments using the player / stage robot control system .

1011.1939

reachability for continuous and hybrid systems has been an important topic of research in the dynamics and control literature . numerous problems regarding safety of air traffic management systems @xcite , @xcite , flight control @xcite , @xcite , @xcite ground transportation systems @xcite , @xcite , etc . have been formulated in the framework of reachability theory . in most of these applications the main aim was to design suitable controllers to steer or keep the state of the system in a `` safe '' part of the state space . the synthesis of such safe controllers for hybrid systems relies on the ability to solve target problems for the case where state constraints are also present . the sets that represent the solution to those problems are known as capture basins @xcite . one direct way of computing these sets was proposed in @xcite , @xcite , and was formulated in the context of viability theory @xcite . following the same approach , the authors of @xcite , @xcite formulated viability , invariance and pursuit - evasion gaming problems for hybrid systems and used non - smooth analysis tools to characterize their solutions . computational tools to support this approach have been already developed by @xcite . an alternative , indirect way of characterizing such problems is through the level sets of the value function of an appropriate optimal control problem . by using dynamic programming , for reachability / invariant / viability problems without state constraints , the value function can be characterized as the viscosity solution to a first order partial differential equation in the standard hamilton - jacobi form @xcite , @xcite , and @xcite . numerical algorithms based on level set methods have been developed by @xcite , @xcite , have been coded in efficient computational tools by @xcite , @xcite and can be directly applied to reachability computations . in the case where state constraints are also present , this target hitting problem is the solution to a reach - avoid problem in the sense of @xcite . the authors of @xcite , @xcite developed a reach - avoid computation , whose value function was characterized as a solution to a pair of coupled variational inequalities . in @xcite , @xcite , @xcite the authors proposed another characterization , which involved only one hamilton - jacobi type partial differential equation together with an inequality constraint . these methods are hampered from a numerical computation point of view by the fact that the hamiltonian of the system is in general discontinuous @xcite . in @xcite , a scheme based on ellipsoidal techniques so as to compute reachable sets for control systems with constraints on the state was proposed . this approach was restricted to the class of linear systems . in @xcite , this approach was extended to a list of interesting target problems with state constraints . the calculation of a solution to the equations proposed in @xcite , @xcite is in general not easy apart from the case of linear systems , where duality techniques of convex analysis can be used . in this paper we propose a new framework of characterizing reach - avoid sets of nonlinear control systems as the solution to an optimal control problem . we consider the case where we have competing inputs and hence adopt the gaming formulation proposed in @xcite . we first restrict our attention to a specific reach - avoid scenario , where the objective of the control input is to make the states of the system hit the target at the end of our time horizon and without violating the state constraints , while the disturbance input tries to steer the trajectories of the system away from the target . we then generalize our approach to the case where the controller aims to steer the system towards the target not necessarily at the terminal , but at some time within the specified time horizon . both problems could be treated as pursuit - evasion games , and for a worst case setting we define a value function similar to @xcite and prove that it is the unique continuous viscosity solution to a quasi - variational inequality of a form similar to @xcite , @xcite . the advantage of this approach is that the properties of the value function and the hamiltonian ( both of them are continuous ) enable us using existing tools to compute the solution of the problem numerically . to illustrate our approach , we consider a reach - avoid problem that arises in the area of air traffic management , in particular the problem of collision avoidance in the presence of 4d constraints , called target windows . target windows ( tw ) are spatial and temporal constraints and form the basis of the cats research project @xcite , whose aim is to increase punctuality and predictability during the flight . in @xcite a reachability approach of encoding tw constraints was proposed . we adopt this framework and consider a multi - agent setting , where each aircraft should respect its tw constraints while avoiding conflict with other aircraft in the presence of wind . since both control and disturbance inputs ( in our case the wind ) are present , this problem can be treated as a pursuit - evasion differential game with state constraints , which are determined dynamically by performing conflict detection . in section ii we pose two reach - avoid problems for continuous systems with competing inputs and state constraints , and formulate them in the optimal control framework . section iii provides the characterization of the value functions of these problems as the viscosity solution to two variational inequalities . in section iv we present an application of this approach to a two aircraft collision avoidance scenario with realistic data . finally , in section v we provide some concluding remarks and directions for future work . consider the continuous time control system @xmath0 , and an arbitrary time horizon @xmath1 . with @xmath2 , @xmath3 , @xmath4 , and @xmath5 . let @xmath6}$ ] , @xmath7}$ ] denote the set of lebesgue measurable functions from the interval @xmath8 $ ] to @xmath9 , and @xmath10 respectively . consider also two functions @xmath11 , @xmath12 to be used to encode the target and state constraints respectively , + * assumption 1 . * _ @xmath13 and @xmath14 are compact . @xmath15 , @xmath16 and @xmath17 are bounded and lipschitz continuous in x and continuous in u and v. _ under assumption @xmath18 the system admits a unique solution @xmath19 \rightarrow \mathbb{r}^n$ ] for all @xmath20 $ ] , @xmath21}$ ] and @xmath22}$ ] . for @xmath23 $ ] this solution will be denoted as @xmath24 let @xmath25 be a bound such that for all @xmath26 and @xmath21}$ ] and for all @xmath27 , @xmath28 let also @xmath29 and @xmath30 be such that @xmath31 } \rightarrow \mathcal{u}_{[0,t]}$ ] such that for all @xmath32 $ ] and for all @xmath33 , if @xmath34 for almost every @xmath35 $ ] , then @xmath36(\tau)=\gamma[\hat{v}](\tau)$ ] for almost every @xmath35 $ ] . we then use @xmath37}$ ] to denote the class of non - anticipative strategies . consider the sets @xmath38 , @xmath39 related to the level sets of the two bounded , lipschitz continuous functions @xmath40 and @xmath41 respectively . for technical purposes assume that @xmath38 is closed whereas @xmath39 is open . then @xmath38 and @xmath39 could be characterized as @xmath42 consider now a closed set @xmath43 that we would like to reach while avoiding an open set @xmath44 . one would like to characterize the set of the initial states from which trajectories can start and reach the set @xmath38 at the terminal time @xmath45 without passing through the set @xmath39 over the time horizon @xmath46 $ ] . to answer this question on needs to determine whether there exists a choice of @xmath47}$ ] such that for all @xmath22}$ ] , the trajectory @xmath48 satisfies @xmath49 and @xmath50 for all @xmath23 $ ] . the set of initial conditions that have this property is then @xmath51 } , ~\forall v(\cdot ) \in \mathcal{v}_{[t , t ] } , \\ & ( \phi(t , t , x,\gamma(\cdot),v(\cdot ) ) \in r ) \land ( \forall \tau \in [ t , t ] , ~ \phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) \notin a ) \}. \nonumber\end{aligned}\ ] ] now introduce the value function @xmath52 \rightarrow \mathbb{r}$ ] @xmath53 } } \sup_{v(\cdot ) \in \mathcal{v}_{[t , t ] } } \max \{l(\phi(t , t , x , u(\cdot),v(\cdot ) ) ) , \max_{\tau \in [ t , t ] } h(\phi(\tau , t , x , u(\cdot),v(\cdot ) ) ) \}.\ ] ] @xmath54 can be thought of as the value function of a differential game , where @xmath55 is trying to minimize , whereas @xmath56 is trying to maximize the maximum between the value attained by @xmath16 at the end @xmath45 of the time horizon and the maximum value attained by @xmath17 along the state trajectory over the horizon @xmath46 $ ] . based on @xcite , @xcite and @xcite , we will show that the value function defined by @xmath57 is the unique viscosity solution of the following quasi - variational inequality . @xmath58 with terminal condition @xmath59 . it is then easy to link the set @xmath60 of @xmath61 to the level set of the value function @xmath62 defined in @xmath57 . + * proposition 1 . * @xmath63 . + @xmath64 if and only if @xmath65 } } \sup_{v(\cdot ) \in \mathcal{v}_{[t , t ] } } \max \{l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , \\ \max_{\tau \in [ t , t ] } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \ } \leq 0 $ ] . equivalently , there exists a strategy @xmath66}$ ] such that for all @xmath67}$ ] , @xmath68 } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \ } \leq 0 $ ] . the last statement is equivalent to there exists a @xmath66}$ ] such that for all @xmath67}$ ] , @xmath69 and @xmath70 } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \leq 0 $ ] . or in other words , there exists a @xmath66}$ ] such that for all @xmath67}$ ] , @xmath71 and for all @xmath23~ \phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) \notin a$ ] . another related problem that one might need to characterize is the set of initial states from which trajectories can start , and for any disturbance input can reach the set @xmath38 not at the terminal , but at some time within the time horizon @xmath46 $ ] , and without passing through the set @xmath39 until they hit @xmath38 . in other words , we would like to determine the set @xmath72 } , ~\forall v(\cdot ) \in \mathcal{v}_{[t , t ] } , \\ & \exists \tau_1 \in [ t , t],~ ( \phi(\tau_1,t , x,\gamma(\cdot),v(\cdot ) ) \in r ) \land ( \forall \tau_2 \in [ t,\tau_1 ] , ~ \phi(\tau_2,t , x,\gamma(\cdot),v(\cdot ) ) \notin a ) \}. \nonumber\end{aligned}\ ] ] based on @xcite , define the augmented input as @xmath73 \in \textit{u } \times [ 0,1]$ ] and consider the dynamics @xmath74 let @xmath75 denote the solution of the augmented system , and define @xmath76 , @xmath77 and @xmath78 similarly to the previous case . following @xcite for every @xmath79}$ ] the pseudo - time variable @xmath80 \rightarrow [ t , t]$ ] is given by @xmath81 consider @xmath82 to be almost an inverse of @xmath83 in the sense that @xmath84 . in @xcite , @xmath82 was defined as the limit of a convergent sequence of functions , and it was shown that @xmath85 for any @xmath23 $ ] . based on the analysis of @xcite , equation @xmath86 implies that the trajectory @xmath87 of the augmented system visits only the subset of the states visited by the trajectory @xmath88 of the original system in the time interval @xmath89 $ ] . define now the value function @xmath90 } } \sup_{v(\cdot ) \in \mathcal{v}_{[t , t ] } } \max \{l(\tilde{\phi}(t , t , x,\tilde{\gamma}[v](\cdot),v(\cdot ) ) ) , \max_{\tau \in [ t , t ] } h(\tilde{\phi}(\tau , t , x,\tilde{\gamma}[v](\cdot),v(\cdot ) ) ) \}.\ ] ] one can then show that @xmath91 is related to the set @xmath92 . + * proposition 2 . * _ for @xmath93 $ ] , @xmath94 . _ the proof of this proposition is given in appendix a. we first establish the consequences of the principle of optimality for @xmath54 . + * lemma 1 . * _ for all @xmath95 $ ] and all @xmath96 $ ] : _ @xmath97 } } \sup_{v(\cdot)\in \mathcal{v}_{[t , t+\alpha ] } } \big [ \max \big \ { \max_{\tau \in [ t , t+\alpha ] } h(\phi(\tau , t , x , u(\cdot ) ) ) , v(\phi(t+\alpha , t , x , u(\cdot)),t+\alpha ) \big \ } \big].\ ] ] _ moreover , for all @xmath95 , ~ v(x , t ) \geq h(x)$ ] . _ the proof for the second part is straightforward and follows from the definition of @xmath54 . the proof for the first part is given in appendix b. we now show that @xmath54 is a bounded , lipschitz continuous function . + * lemma 2 . * _ there exists a constant @xmath98 such that for all @xmath99 $ ] : _ @xmath100 the proof of this lemma is given in appendix b. we now introduce the hamiltonian @xmath101 , defined by @xmath102 * lemma 3 . * _ there exists a constant @xmath98 such that for all @xmath103 , and all @xmath104 _ : @xmath105 the proof of this fact is straightforward ( see @xcite , @xcite for details ) . we are now in a position to state and prove the following theorem , which is the main result of this section . + * theorem 1 . * _ @xmath54 is the unique viscosity solution over @xmath95 $ ] of the variational inequality _ @xmath58 _ with terminal condition @xmath59 . _ uniqueness follows from lemma 2 , lemma 3 and @xcite . note also that by definition of the value function we have @xmath106 . therefore it suffices to show that 1 . for all @xmath107 and for all smooth @xmath108 : @xmath109 , if @xmath110 attains a local maximum at @xmath111 , then @xmath112 2 . for all @xmath107 and for all smooth @xmath108 : @xmath109 , if @xmath110 attains a local minimum at @xmath111 , then @xmath113 the case @xmath114 is automatically captured by @xcite . + * part 1 . * consider an arbitrary @xmath107 and a smooth @xmath115 such that @xmath110 has a local maximum at @xmath111 . then , there exists @xmath116 such that for all @xmath117 with @xmath118 @xmath119 we would like to show that @xmath120 since by lemma 1 @xmath121 , either @xmath122 or , @xmath123 . for the former the claim holds , whereas for the latter it suffices to show that there exists @xmath124 such that for all @xmath125 @xmath126 for the sake of contradiction assume that for all @xmath124 there exists @xmath125 such that for some @xmath127 @xmath128 since @xmath108 is smooth and @xmath15 is continuous , then based on @xcite we have that @xmath129 for all @xmath130 and some @xmath131 , where @xmath132 denotes a ball centered at @xmath56 with radius @xmath133 . because @xmath10 is compact there exist finitely many distinct points @xmath134 , and @xmath135 such that @xmath136 and for @xmath137 @xmath138 define @xmath139 by setting for @xmath140 , @xmath141 if @xmath142 . then @xmath143 since @xmath108 is smooth and @xmath15 is continuous , there exists @xmath144 such that for all @xmath117 with @xmath145 @xmath146 finally , define @xmath147 } \rightarrow \mathcal{u}_{[t_0,t]}$ ] by @xmath36(\tau ) = g(v(\tau))$ ] for all @xmath148 $ ] . it is easy to see that @xmath149 is now non - anticipative and hence @xmath150}$ ] . so for all @xmath151}$ ] and all @xmath117 such that @xmath145 , @xmath152(\cdot),v(\cdot ) ) < -\theta < 0.\ ] ] by continuity , there exists @xmath153 such that @xmath154(\cdot),v(\cdot))-x_0|^2 + ( t - t_0)^2 < \delta_2 $ ] for all @xmath155 $ ] . therefore , for all @xmath151}$ ] @xmath156 let @xmath157 $ ] be such that @xmath158 } h(\phi(\tau , t_0,x_0,\gamma(\cdot),v(\cdot))).\ ] ] * case 1.1 : * if @xmath159 $ ] , then for @xmath160 we have @xmath161 then by the dynamic programming argument of lemma @xmath18 we have : @xmath162 } } \big [ \max \big \ { & \max_{\tau \in [ t_0,t_0+\delta_3 ] } h(\phi(\tau , t_0,x_0,\gamma(\cdot),v(\cdot ) ) ) , \\&v(\phi(\tau_0,t_0,x_0,\gamma(\cdot),v(\cdot)),\tau_0 ) \big \ } \big].\end{aligned}\ ] ] we can choose @xmath163}$ ] such that @xmath164 } h(\phi(\tau , t_0,x_0,\gamma(\cdot),v(\cdot ) ) ) , v(\phi(\tau_0,t_0,x_0,\gamma(\cdot),v(\cdot)),\tau_0 ) \big \ } + \epsilon,\ ] ] and set @xmath165 . since @xmath121 for all @xmath117 we have that @xmath166 } h(\phi(\tau , t_0,x_0,\gamma(\cdot),\hat{v}(\cdot ) ) ) = h(\phi(\tau_0,t_0,x_0,\gamma(\cdot),\hat{v}(\cdot ) ) ) \leq v(\phi(\tau_0,t_0,x_0,\gamma(\cdot),\hat{v}(\cdot)),\tau_0).\ ] ] hence @xmath167 since @xmath168 holds for all @xmath151}$ ] , it will also hold for @xmath169 , and hence the last argument establishes a contradiction . + * case 1.2 : * if @xmath170 then for @xmath171 we have that for all @xmath151}$ ] @xmath172 since by lemma 1 @xmath162 } } \max \big \ { & \max_{\tau \in [ t_0,t_0+\delta_3 ] } h(\phi(\tau , t_0,x_0,\gamma(\cdot),v(\cdot ) ) ) , \\&v(\phi(t_0 + \delta_3,t_0,x_0,\gamma(\cdot),v(\cdot)),t_0 + \delta_3 ) \big \ } , \end{aligned}\ ] ] then if @xmath173 } } v(\phi(t_0 + \delta_3,t_0,x_0,\gamma(\cdot),v(\cdot)),t_0 + \delta_3 ) , \ ] ] we can choose @xmath163}$ ] such that @xmath174 which establishes a contradiction . + if @xmath173 } } \max_{\tau \in [ t_0,t_0+\delta_3 ] } h(\phi(\tau , t_0,x_0,\gamma(\cdot),v(\cdot))),\ ] ] then we can choose @xmath163}$ ] such that @xmath175 } h(\phi(\tau , t_0,x_0,\gamma(\cdot),\hat{v}(\cdot ) ) ) + \epsilon,\ ] ] or equivalently @xmath176 , since @xmath170 . based on our initial hypothesis that @xmath177 , there exists a @xmath178 such that @xmath179 . if we take @xmath180 we establish a contradiction . + * consider an arbitrary @xmath107 and a smooth @xmath115 such that @xmath110 has a local minimum at @xmath111 . then , there exists @xmath116 such that for all @xmath117 with @xmath118 @xmath181 we would like to show that @xmath182 since @xmath183 it suffices to show that @xmath184 . this implies that for all @xmath124 there exists a @xmath125 such that @xmath185 for the sake of contradiction assume that there exists @xmath186 such that for all @xmath125 there exists @xmath127 such that @xmath187 since @xmath108 is smooth , there exists @xmath144 such that for all @xmath117 with @xmath145 @xmath188 hence , following @xcite , for @xmath189 and any @xmath150}$ ] @xmath190 by continuity , there exists @xmath153 such that @xmath191 for all @xmath155 $ ] . therefore , for all @xmath150}$ ] @xmath192 but by the dynamic programming argument of lemma @xmath18 we can choose a @xmath193}$ ] such that @xmath194 } } \big [ \max \big \ { \max_{\tau \in [ t_0,t_0+\delta_3 ] } h(\phi(\tau , t_0,x_0,\hat{\gamma}(\cdot),v(\cdot ) ) ) , \\&v(\phi(t_0 + \delta_3,t_0,x_0,\hat{\gamma}(\cdot),v(\cdot)),t_0 + \delta_3 ) \big \ } \big ] - \frac{\delta_3 \theta}{2 } \\ & \geq \max \big \ { \max_{\tau \in [ t_0,t_0+\delta_3 ] } h(\phi(\tau , t_0,x_0,\hat{\gamma}(\cdot),v(\cdot ) ) ) , v(\phi(t_0 + \delta_3,t_0,x_0,\hat{\gamma}(\cdot),v(\cdot)),t_0 + \delta_3 ) \big \ } - \frac{\delta_3 \theta}{2 } \\ & \geq v(\phi(t_0 + \delta_3,t_0,x_0,\hat{\gamma}(\cdot),v(\cdot)),t_0 + \delta_3 ) - \frac{\delta_3 \theta}{2}.\end{aligned}\ ] ] the last statement establishes a contradiction , and completes the proof . consider the value function @xmath91 defined in the previous section . the following theorem proposes that @xmath91 is the unique viscosity solution of another variational inequality . + * theorem 2 . * _ @xmath195 \rightarrow \mathbb{r}$ ] is the unique viscosity solution of the variational inequality _ @xmath196 _ with terminal condition @xmath197 . _ by theorem 1 , @xmath198 is the unique viscosity solution of @xmath199 , subject to @xmath200 . if we let @xmath201 then , following the proof of theorem 2 of @xcite , we have that @xmath202 consequently , the two variational inequalities @xmath199 and @xmath203 are equivalent , and so @xmath198 is the viscosity solution of @xmath203 . since the solution to @xmath203 is unique @xcite , one could easily show that @xmath204 } } \sup_{v(\cdot ) \in \mathcal{v}_{[t , t ] } } \min_{\tau_1 \in [ t , t ] } \max \{l(\phi(\tau_1,t , x , u(\cdot),v(\cdot ) ) ) , \max_{\tau_2 \in [ t,\tau_1 ] } h(\phi(\tau_2,t , x , u(\cdot),v(\cdot ) ) ) \}.\ ] ] to illustrate the approach described in the previous sections , we consider a problem from the air traffic management area . the increase in air traffic is bound to lead to further en - route delays and potentially safety problems in the immediate future @xcite , @xcite . a major difficulty with accommodating this expected increase in air traffic is uncertainty about the future evolution of flights . therefore , the cats research project has proposed a novel concept of operations , which aims to increase punctuality and safety during the flight . this concept is mainly based on imposing spatial and temporal constraints at different parts of the flight plan of each aircraft . these 4d constraints are known as target windows ( tw ) @xcite , and represent the commitment from each actor ( air traffic controllers , airports , airlines , air navigation service providers ) to deliver a particular aircraft within the tw constraint . this commitment is known as the contract of objectives ( coo ) @xcite , and can be viewed as a first step towards the implementation of the reference business trajectory envisioned by the sesar joint undertaking @xcite . in this section we follow the approach proposed in @xcite to code the tw constraints , and use the reach - avoid formulation of sections ii and iii , to investigate collision avoidance in the presence of tw constraints . for this purpose we consider a two - aircraft scenario , where each aircraft should respect its tw constraints , while avoiding conflict with other aircraft . each aircraft @xmath205 is assumed to have a predetermined flight plan , which comprises a series of way points @xmath206^t \in \mathbb{r}_+^3 $ ] , where @xmath207 . the angle @xmath208 that each segment forms with the @xmath209 axis and the flight path angle @xmath210 that it forms with the horizontal plane are shown in fig . 1 . the discrete state @xmath211 stores the segment of the flight plan that the aircraft is currently in , and for @xmath212 we can define @xmath213 where @xmath214 is the length of the projection of its segment on the horizontal plane . assume perfect lateral tracking and set @xmath215 denote the the part of each segment covered on the horizontal plane ( see fig . 1 ) . based on our assumption that each aircraft has constant heading angle @xmath208 at each segment , its @xmath209 and @xmath216 coordinates can be computed by : @xmath217 [ fig : subfigureexample ] to approximate accurately the physical model , the flight path angle @xmath218 , which is the angle that the aircraft forms with the horizontal plane , is a control input fixed according to the angle @xmath210 that the segment forms with the horizontal plane . if @xmath219 the aircraft will be cruising @xmath220 at that segment , whereas if it is positive or negative it will be climbing @xmath221)$ ] or descending @xmath222)$ ] respectively . the speed of each aircraft apart from its type depends also on the altitude @xmath223 . at each flight level there is a nominal airspeed that aircraft tend to track , giving rise to a function @xmath224 . the dependence on the flight path angle indicates the discrete mode i.e. cruise , climb , descent , that an aircraft could be . for our simulations , we have assumed that at every level the airspeed could vary within @xmath225 of the nominal one ; this is restricted by the control input @xmath226 $ ] . figure 2 shows the speed - altitude profiles of the a320 ( the simulated aircraft ) , based on the bada database @xcite , for the different phases of flight . these curves have been computed by linear interpolation between the predetermined @xmath227 points . [ fig : subfigureexample ] most of the reachability numerical methods are based on gridding the state space , so the memory and time necessary for the computation grow exponentially in the state dimension . therefore , using a full , five- or six - state , point mass model of the aircraft , like the one described in @xcite , would be computationally expensive to analyze using the existing computational tools . in @xcite , the full point mass model for the aircraft , was abstracted to a simplified one to make the reachability computation tractable . motivated by the fact that aircraft track laterally very well , it was assumed that the heading angle remains constant at each segment . the dynamics of each aircraft are modeled by a hybrid automaton @xmath228 ( in the notation of @xcite ) , with : * continuous states @xmath229^t \in \mathbb{r}_+^3 = x_j$ ] . * discrete states @xmath230 . * initial states @xmath231 . * control inputs @xmath232^t\in \left[-1,1\right ] \times \left[-\overline{\gamma_j}^p,\overline{\gamma_j}^p\right]$ ] . * disturbance inputs @xmath233^t\in\mathbb{r}^3 $ ] * vector field @xmath234 . @xmath235 * domain @xmath236 . * guards @xmath237 . * reset map @xmath238 . apart from @xmath239 , the other two continuous states are the altitude @xmath240 , and the time @xmath241 . the last equation was included in order to track the tw temporal constraints . as stated above , @xmath242 is the flight path angle and @xmath243 is the wind speed , which acts as a bounded disturbance with @xmath244 , and for our simulations we used @xmath245 . since the flight path angle @xmath218 does not exceed 5@xmath246 , for simplification we can assume that @xmath247 and @xmath248 . target windows represent spatial and temporal constraints that aircraft should respect . following @xcite , we assume that tw are located on the surface area between two air traffic control sectors . based on the structure of those sectors , the tw are either adjacent or superimposed ( fig . 3 ) , and for simplicity we assume that there is a way point centered in the middle of each tw . our objective is to compute the set @xmath249 of all initial states at time @xmath241 for which there exists a non - anticipative control strategy @xmath149 , that despite the wind input @xmath56 can lead the aircraft @xmath250 inside the tw constraint set at least once within its time and space window , while avoiding conflict with the other aircraft . in air traffic , conflict refers to the loss of minimum separation between two aircraft . each aircraft is surrounded by a protected zone , which is generally thought of as a cylinder of radius 5nmi and height 2000 ft centered at the aircraft . if this zone is violated by another aircraft , then a conflict is said to have occurred . to achieve this goal , we adopt another simplification introduced in @xcite ; we eliminate time from the state equations , and perform a two - stage calculation . we define the spatial constraints of a tw centered at the way point @xmath211 as @xmath251)$ ] if the tw is adjacent , and @xmath252,z_{(i , j)})$ ] if the tw is superimposed . let also @xmath253 $ ] denote the time window of @xmath254 . then @xmath249 could be computed as : + * stage 1 : * compute for each aircraft @xmath250 the set @xmath255 of states @xmath256 at time @xmath257 ( beginning of target window ) from which there exist a control trajectory that despite the wind can lead the aircraft inside @xmath254 at least once within the time interval @xmath253 $ ] . but this set is the @xmath258 set , which was shown in section iii to be the zero sublevel set of @xmath91 , which is the solution to the following partial differential equation @xmath259 the terminal condition @xmath260 was chosen to be the signed distance to the set @xmath261 , and the avoid set @xmath262 is characterized by @xmath263 . this function represents the area where a conflict might occur , and it is computed online by performing conflict detection ( see appendix c ) . + * stage 2 : * compute the set @xmath249 of all states that start at time @xmath264 and for every wind can reach the set @xmath255 at time @xmath257 , while avoiding conflict with other aircraft . based on the analysis of section ii , this is the @xmath265 set , that can be computed by solving @xmath266 with terminal condition @xmath267 . the set @xmath255 is defined as @xmath268 whereas @xmath262 depends once again on the obstacle function @xmath263 . [ fig : subfigureexample ] the simulations for each aircraft are running in parallel , so at every instance @xmath241 , we have full knowledge of the backward reachable sets of each aircraft . based on that , algorithm 1 of appendix c describes the implemented steps for the reach - avoid computation . consider now the case where we have two aircraft each one with a tw , whose flight plans intersect , and they enter the same air traffic sector with a @xmath269 difference . 4a depicts the two flight plans and fig . 4b the projection of the flight plans on the horizontal plane . the target windows are centered at the last way point of each flight plan . the result of the two - stage backward reachability computation with tw as terminal sets is depicted in fig . the tubes at this figure include all the states that each aircraft could be , and reach its tw . we should also note that the tubes are the union of the corresponding sets . these sets at a specific time instance , would include all the states that could start at that time and reach the tw at the end of the horizon . 5b is the projection of these tubes on the horizontal plane . as it was expected , the @xmath209-@xmath216 projection coincides with the projection of the flight plans on the horizontal plane . this is reasonable , since in the hybrid model we assumed constant heading angle at each segment . moreover , based on the speed - altitude profiles , aircraft fly faster at higher levels , so at those altitudes there are more states that can reach the target . [ fig : subfigureexample ] we can repeat the previous computation , but now checking at every time if the sets , in the sense described before , satisfy the minimum separation standards . that way , the time and the points of each set where a conflict might occur , can be detected . the result of this calculation is illustrated in fig . the `` hole '' that is now around the intersection area of fig . 5a represents the area where the two aircraft might be in conflict . [ fig : subfigureexample ] now that we managed to perform conflict detection , we are in a position to compute at every instance the obstacle function @xmath263 . since the conflict does not occur within the time interval of the tw , the set of the initial states that an aircraft @xmath250 could start and reach the set @xmath255 at time @xmath257 , while avoiding conflict with the other aircraft , should be computed . once the aircraft hits @xmath270 , it can also reach the tw within its time constraints . to obtain the solution to this reach - avoid problem , the variational inequality @xmath199 should be solved . if the conflict had occurred in @xmath271 $ ] equation @xmath203 should be solved instead . one could either use numerical methods developed by @xcite , or the level set method toolbox @xcite , whose authors propose a way to code obstacles on the value function . the latter was used in this paper , and the obstacle function @xmath263 was dynamically determined , since at every time it is the result of the conflict detection . 6b shows the reach - avoid tubes at @xmath272 . as it was expected , the set of states that could reach the target while avoiding conflict with the other aircraft , does not include the conflict zone of fig . 6a , but also some more states that would end up in this zone . this is a `` centralized '' solution , since for the safe sets shown in fig . 6b , there exists a combination of inputs for the two aircraft that could satisfy all constraints , i.e. reaching their tw while avoiding conflict . it should be also noted that all simulations where performed on the same grid , but were running in parallel . hence he have two 2d computations ( one for each aircraft ) instead of a 4d one , and @xmath263 was determined at each step by comparing the obtained sets as described in algorithm 1 . a new framework of solving nonlinear systems with state constraints and competing inputs was presented . this formulation was based on reachability and game theory , has the advantage of maintaining the continuity in the hamiltonian of the system , and hence it has very good properties in terms of the numerical solution . the problem of reaching a desired set , in this case the tw , while avoiding conflict with other aircraft was formulated as a reach - avoid problem , and was computed numerically by using the existing tools . in future work , we plan to use these reach - avoid bounds in order to perform conflict resolution by optimizing some cost criterion . another issue would be to extend the proposed approach to formulate games in the case where the obstacle function is time and/or control dependant . finally , we intend to validate our approach with fast time simulation studies using realistic aircraft and flight management system models , flight plans and wind uncertainty . [ [ section ] ] * part 1 . * following @xcite we first show that @xmath273 . consider @xmath274 and for the sake of contradiction assume that @xmath275 . then there exists @xmath276 such that @xmath277 } } \nonumber \max \{l(\tilde{\phi}(t , t , x,\tilde{\gamma}[v](\cdot),v(\cdot ) ) ) , \max_{\tau \in [ t , t ] } h(\tilde{\phi}(\tau , t , x,\tilde{\gamma}[v](\cdot),\\v(\cdot ) ) ) \ } > 2\epsilon > 0 $ ] . this in turn implies that there exists @xmath278}$ ] such that either @xmath279\\(\cdot),\hat{v}(\cdot)))>\epsilon>0 $ ] or there exists @xmath23 $ ] such that @xmath280(\cdot),\hat{v}(\cdot ) ) ) > \epsilon>0 $ ] . consider now the implications of @xmath274 . equation @xmath281 implies that there exists a @xmath66}$ ] such that for all @xmath22}$ ] , and so also for @xmath169 , we can define @xmath282(\cdot)$ ] . then , for this @xmath283 and @xmath169 there exists @xmath284 $ ] such that @xmath285 and for all @xmath286 ~ \phi(\tau_2,t , x , u(\cdot),\hat{v}(\cdot ) ) \notin a$ ] . choose the freezing input signal as @xmath287\\ 0 & \text{for } s \in [ \tau_1,t ] \end{array } \right.\ ] ] if we combine @xmath288 with @xmath283 , we can get the input @xmath289 which will generate a trajectory @xmath290\\ \phi(\tau_1 , x , t , u(\cdot),\hat{v}(\cdot ) ) & \text{for } \tau \in [ \tau_1,t ] \end{array } \right.\ ] ] * case 1.1 : * consider first the case where for all @xmath291 } ~l(\tilde{\phi}(t , t , x , \tilde{\gamma}[\hat{v}](\cdot),\hat{v}(\cdot ) ) ) > \epsilon>0 $ ] . for @xmath292 we have that @xmath293 since @xmath274 , we showed before that @xmath285 , i.e. @xmath294 . so from we have that @xmath295 . since @xmath282(\cdot)$ ] is already non - anticipative , and a non - anticipative strategy for @xmath288 can be designed , @xmath289 will also be non - anticipative . therefore , the previous statement establishes a contradiction . + * case 1.2 : * consider now the case where for all @xmath291}$ ] there exists @xmath23 $ ] such that @xmath280(\cdot),\hat{v}(\cdot ) ) ) > \epsilon>0 $ ] . since we showed that for all @xmath296 , ~ \phi(\tau , t , x , u(\cdot),\hat{v}(\cdot ) ) \notin a$ ] , we can conclude that for all @xmath296 $ ] @xmath297 if @xmath298 $ ] , then we have that @xmath299 so @xmath300 . hence , for all @xmath23 $ ] we have that @xmath301 . since in case 1.1 , @xmath289 was shown to be non - anticipative , we have a contradiction . + * part 2 . * next , we show that @xmath302 . consider @xmath303 such that @xmath304 and assume for the sake of contradiction that @xmath305 . then for all for all @xmath66}$ ] there exists @xmath306}$ ] such that for all @xmath284 $ ] either @xmath307 or there exists @xmath286 $ ] such that @xmath308 . following the analysis of @xcite , consider that the strategy @xmath66}$ ] is extracted from @xmath291}$ ] , and choose the @xmath169 that corresponds to that strategy . in @xcite , was proven that the set of states visited by the augmented trajectory is a subset of the states visited by the original one . we therefore have that for all @xmath284 $ ] @xmath309(\cdot),\hat{v}(\cdot ) ) \notin r \longrightarrow \tilde{\phi}(\tau_1 , x , t,\tilde{\gamma}[\hat{v}](\cdot),\hat{v}(\cdot ) ) \notin r,\ ] ] and also @xmath310 $ ] @xmath311(\cdot),\hat{v}(\cdot ) ) \in a \longrightarrow \tilde{\phi}(\tau_2 , x , t,\tilde{\gamma}[\hat{v}](\cdot),\hat{v}(\cdot ) ) \in a.\ ] ] by @xmath312 we conclude that there exists a @xmath313 such that either for all @xmath284 $ ] @xmath314(\cdot),\hat{v}(\cdot ) ) ) > \delta > 0,\ ] ] or for some @xmath315 $ ] @xmath316(\cdot),\hat{v}(\cdot ) ) ) > \delta > 0.\ ] ] since @xmath304 , then for all @xmath276 there exists a non - anticipative strategy @xmath291}$ ] such that @xmath277}},~ \max \{l(\tilde{\phi}(t , t , x,\tilde{\gamma}[v](\cdot),v(\cdot ) ) ) , \max_{\tau \in [ t , t ] } h(\tilde{\phi}(\tau , t , x,\tilde{\gamma}[v](\cdot),v(\cdot ) ) ) \ } \leq \epsilon$ ] . hence for all @xmath67}$ ] , @xmath317(\cdot),v(\cdot)))\leq \epsilon$ ] and for all @xmath23 $ ] , @xmath318(\cdot),v(\cdot ) ) ) \leq \epsilon$ ] . for @xmath319 the last argument implies that @xmath320(\cdot),\hat{v}(\cdot ) ) ) \leq \epsilon,\ ] ] and there exists @xmath286 $ ] @xmath321(\cdot),\hat{v}(\cdot ) ) ) \leq \epsilon.\ ] ] if we choose @xmath322 , the last statements contradict @xmath323 and complete the proof . [ [ section-1 ] ] following @xcite we can define @xmath324 } } \sup_{v(\cdot)\in \mathcal{v}_{[t , t+\alpha ] } } \big [ \max \big \ { \max_{\tau \in [ t , t+\alpha ] } h(\phi(\tau , t , x , u(\cdot ) ) ) , v(\phi(t+\alpha , t , x , u(\cdot)),t+\alpha ) \big \ } \big].\ ] ] we will then show that for all @xmath325 , @xmath326 and @xmath327 . then since @xmath325 is arbitrary , @xmath328 . + * case 1 : * @xmath326 . fix @xmath325 and choose @xmath329}$ ] such that @xmath330 } } \big [ \max \big \ { & \max_{\tau \in [ t , t+\alpha ] } h(\phi(\tau , t , x,\gamma_1(\cdot),v_1(\cdot ) ) ) , \\&v(\phi(t+\alpha , t , x,\gamma_1(\cdot),v_1(\cdot)),t+\alpha ) \big \ } \big ] - \epsilon , \nonumber\end{aligned}\ ] ] similarly , choose @xmath331}$ ] such that @xmath332 } } \max \big \ { l(\phi(t , t+\alpha,\phi(t+\alpha , t , x,\gamma_1(\cdot),v_1(\cdot)),\gamma_2(\cdot),v_2(\cdot ) ) ) , \\ & \max_{\tau \in [ t+\alpha , t ] } h(\phi(\tau , t+\alpha,\phi(t+\alpha , t , x,\gamma_1(\cdot),v_1(\cdot)),\gamma_2(\cdot),v_2(\cdot ) ) ) \big \ } - \epsilon.\end{aligned}\ ] ] for any @xmath333}$ ] we can define @xmath334}$ ] and @xmath335}$ ] such that @xmath336 for all @xmath337 and @xmath338 for all @xmath339 $ ] . define also @xmath340}$ ] by @xmath341(\tau)= \left\ { \begin{array}{rl } \gamma_1[v_1](\tau ) & \text{if } \tau \in [ t , t+\alpha)\\ \gamma_2[v_2](\tau ) & \text{if } \tau \in [ t+\alpha , t ] . \end{array } \right.\ ] ] it easy to see that @xmath342 } \rightarrow \mathcal{u}_{[t , t]}$ ] is non - anticipative . by uniqueness , @xmath343 if @xmath337 , and @xmath344 if @xmath339 $ ] . + hence , @xmath345 } } \sup_{v_2(\cdot)\in \mathcal{v}_{[t+\alpha , t ] } } \max \big \ { \max_{\tau \in [ t , t+\alpha ] } h(\phi(\tau , t , x,\gamma_1(\cdot),v_1(\cdot ) ) ) , \\&l(\phi(t , t+\alpha,\phi(t+\alpha , t , x,\gamma_1(\cdot),v_1(\cdot)),\gamma_2(\cdot),v_2(\cdot))),\\ & \max_{\tau \in [ t+\alpha , t ] } h(\phi(\tau , t+\alpha,\phi(t+\alpha , t , x,\gamma_1(\cdot),v_1(\cdot)),\gamma_2(\cdot),v_2(\cdot ) ) ) \big \ } - 2\epsilon\\ & \geq \sup_{v(\cdot)\in \mathcal{v}_{[t , t]}}\max \big \ { l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , \max_{\tau \in [ t , t ] } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \big \ } - 2\epsilon \\ & \geq v(x , t)-2\epsilon.\end{aligned}\ ] ] therefore , @xmath326 . + * case 2 : * @xmath327 . fix @xmath325 and choose now @xmath340}$ ] such that @xmath346 } } \max \big \ { l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , \max_{\tau \in [ t , t ] } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \big \}-\epsilon.\ ] ] by the definition of @xmath347 @xmath348 } } \big [ \max \big \ { \max_{\tau \in [ t , t+\alpha ] } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) , v(\phi(t+\alpha , t , x,\gamma(\cdot),v(\cdot)),t+\alpha ) \big \ } \big].\ ] ] hence there exists a @xmath349}$ ] such that @xmath350 } h(\phi(\tau , t , x,\gamma(\cdot),v_1(\cdot ) ) ) , v(\phi(t+\alpha , t , x,\gamma(\cdot),v_1(\cdot)),t+\alpha ) \big \ } + \epsilon.\ ] ] let @xmath351 for all @xmath337 and @xmath352 for all @xmath339 $ ] . let also @xmath353}$ ] to be the restriction of the non - anticipative strategy @xmath354 over @xmath355 $ ] . then , for all @xmath339 $ ] , we define @xmath356(\tau ) = \gamma[\hat{v}](\tau)$ ] . hence @xmath357 } } \max \big \ { l(\phi(t , t+\alpha,\phi(t+\alpha , t , x,\gamma(\cdot),v_1(\cdot)),\gamma'(\cdot),v'(\cdot ) ) ) , \\ & \max_{\tau \in [ t+\alpha , t ] } h(\phi(\tau , t+\alpha,\phi(t+\alpha , t , x,\gamma(\cdot),v_1(\cdot)),\gamma'(\cdot),v'(\cdot ) ) ) \big \},\end{aligned}\ ] ] and so there exists a @xmath358}$ ] such that @xmath359 } h(\phi(\tau , t+\alpha,\phi(t+\alpha , t , x,\gamma(\cdot),v_1(\cdot)),\gamma'(\cdot),v_2(\cdot ) ) ) \big \ } + \epsilon . \nonumber\end{aligned}\ ] ] we can define @xmath360 \end{array } \right.\ ] ] therefore , from ( [ eq:8 ] ) and ( [ eq:9 ] ) @xmath361 } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \big \ } + 2\epsilon , \nonumber\ ] ] which together with ( [ eq:7 ] ) implies @xmath362 . since @xmath16 , and @xmath17 are bounded , @xmath54 is also bounded . for the second part fix @xmath363 and @xmath20 $ ] . let @xmath325 and choose @xmath364}$ ] such that @xmath365 } } \max_{\tau \in [ t , t ] } \max \{l(\phi(t , t,\hat{x},\hat{\gamma}(\cdot),v(\cdot ) ) ) , h(\phi(\tau , t,\hat{x},\hat{\gamma}(\cdot),v(\cdot ) ) ) \ } - \epsilon.\ ] ] by definition @xmath366 } } \max_{\tau \in [ t , t ] } \max \{l(\phi(t , t , x,\hat{\gamma}(\cdot),v(\cdot ) ) ) , h(\phi(\tau , t , x,\hat{\gamma}(\cdot),v(\cdot ) ) ) \}.\ ] ] we can choose @xmath306}$ ] such that @xmath367 } \max \{l(\phi(t , t , x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) , h(\phi(\tau , t , x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) \ } + \epsilon,\ ] ] and hence @xmath368 } \max \{l(\phi(t , t , x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) , h(\phi(\tau , t , x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) \ } \\ & - \max_{\tau \in [ t , t ] } \max \{l(\phi(t , t,\hat{x},\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) , h(\phi(\tau , t,\hat{x},\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) \ } + 2\epsilon.\end{aligned}\ ] ] for all @xmath23 $ ] : @xmath369 where @xmath370 is the lipschitz constant of @xmath15 . by the gronwall - bellman lemma @xcite , there exists a constant @xmath371 such that for all @xmath23 $ ] @xmath372 let @xmath373 $ ] be such that @xmath374 } h(\phi(\tau , t , x,\hat{\gamma}(\cdot),\hat{v}(\cdot))).\ ] ] then @xmath375 * case 1 . * @xmath376 @xmath377 * case 2 . * @xmath378 @xmath379 so in any case @xmath380 . the same argument with the roles of @xmath209 , @xmath381 reversed establishes that @xmath382 . since @xmath383 is arbitrary , @xmath384 finally consider @xmath2 and @xmath385 $ ] . without loss of generality assume that @xmath386 . let @xmath325 and choose @xmath66}$ ] such that @xmath387 } } \max_{\tau \in [ t , t ] } \max \{l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \ } - \epsilon \\ & \geq \max_{\tau \in [ t , t ] } \max \{l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \ } - \epsilon\end{aligned}\ ] ] by definition , @xmath388 } } \max_{\tau \in [ \hat{t},t ] } \max \{l(\phi(t,\hat{t},x,\hat{\gamma}(\cdot),v(\cdot ) ) ) , h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),v(\cdot ) ) ) \}.\ ] ] so we can choose @xmath389}$ ] such that @xmath390 } \max \{l(\phi(t,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) , h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) \ } + \epsilon,\ ] ] where @xmath391}$ ] is the restriction of @xmath354 over @xmath392 $ ] . then , for all @xmath393 $ ] , we define @xmath394(\tau ) = \gamma[v](\tau)$ ] , and @xmath395 . by uniqueness , for all @xmath393 $ ] we have that @xmath396 . @xmath397 } \max \{l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \}\\ & - \max_{\tau \in [ \hat{t},t ] } \max \{l(\phi(t,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) , h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) \ } - 2\epsilon.\end{aligned}\ ] ] * case 1 . * @xmath398 } h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot)))$ ] @xmath397 } \max \{l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \ } \\&-l(\phi(t,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) - 2\epsilon \\ & \geq l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) - l(\phi(t,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) - 2\epsilon \\ & = l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) - l(\phi(t+t-\hat{t},t , x,\gamma(\cdot),v(\cdot ) ) ) - 2\epsilon \\ & \geq -c_l c_f |t - t - t+\hat{t}|-2\epsilon \\ & = -c_l c_f |\hat{t}-t| - 2\epsilon,\end{aligned}\ ] ] where @xmath399 is the lipschitz constant of @xmath16 . + * case 2 . * @xmath400 } h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot)))$ ] @xmath397 } \max \{l(\phi(t , t , x,\gamma(\cdot),v(\cdot ) ) ) , h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) \ } \\&-\max_{\tau \in [ \hat{t},t ] } h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) - 2\epsilon \\ & \geq \max_{\tau \in [ t , t ] } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) -\max_{\tau \in [ \hat{t},t ] } h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) - 2\epsilon.\end{aligned}\ ] ] let @xmath401 $ ] be such that @xmath402 } h(\phi(\tau,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot))).\ ] ] then @xmath403 } h(\phi(\tau , t , x,\gamma(\cdot),v(\cdot ) ) ) -h(\phi(\tau_0,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) - 2\epsilon \\ & \geq h(\phi(\tau_0,t , x,\gamma(\cdot),v(\cdot ) ) ) -h(\phi(\tau_0,\hat{t},x,\hat{\gamma}(\cdot),\hat{v}(\cdot ) ) ) - 2\epsilon \\ & = h(\phi(\tau_0,t , x,\gamma(\cdot),v(\cdot ) ) ) -h(\phi(\tau_0+t-\hat{t},t , x,\gamma(\cdot),v(\cdot ) ) ) - 2\epsilon \\ & \geq -c_h c_f |\tau_0-\tau_0-t+\hat{t}|-2\epsilon \\ & = -c_h c_f |\hat{t}-t|-2\epsilon,\end{aligned}\ ] ] where @xmath404 is the lipschitz constant of @xmath17 . in any case we have that @xmath405 a symmetric argument shows that @xmath406 , and since @xmath383 is arbitrary this concludes the proof . [ [ section-2 ] ] the following algorithm summarizes the steps of the reach - avoid computation described in section iv . for simplicity , we have assumed that the tw do not overlap . [ alg : scpf ] * initialization*. set : + @xmath407 for @xmath205 , + @xmath408 for @xmath205 , + * @xmath409 * , + * @xmath410 * . * while @xmath250 is in the sector * * if @xmath411 * + solve @xmath412 . * for all @xmath413 in the sector * + @xmath414 . * for all @xmath415 * + define @xmath416 such that @xmath417 is a box . + let @xmath418 , and @xmath419 , + @xmath420 , + @xmath421 . + * else * + @xmath422 . + * end for * + * end for * + * @xmath423*. + * else if @xmath264 * + solve @xmath424 . + repeat steps @xmath425 with @xmath54 instead of @xmath91 . + * end if * + * end while * research was supported by the european commission under the project cats , fp6-tren-036889 . c. tomlin , g. pappas , and s. sastry , `` conflict resolution for air traffic management : a study in multiagent hybrid systems , '' _ ieee transactions on automatic control _ , vol . 43 , no . 4 , pp . 509521 , 1998 . j. p. aubin , j. lygeros , m. quincampoix , s. sastry , and n. seube , `` impulse differential inclusions : a viability approach to hybrid systems , '' _ ieee transactions on automatic control _ , vol . 47 , no . 1 , pp . 220 , 2002 . p. cardaliaguet , m. quincampoix , and p. saint - pierre , `` set valued numerical analysis for optimal control and differential games , '' _ in m.bardi , t.raghaven , and t.papasarathy ( eds . ) annals of the international society of dynamic games _ , pp . 177247 , 1999 . l. evans and p. souganidis , `` differential games and representation formulas for solutions of hamilton - jacobi - isaacs equations , '' _ indiana university of mathematics journal _ , vol . 33 , no . 5 , pp . 773797 , 1984 . i. mitchell , a. m. bayen , and c. tomlin , `` validating a hamilton jacobi approximation to hybrid reachable sets , '' _ in m.di.benedetto and a.sangiovanni-vincentelli ( eds . ) hybrid systems : computation and control springer verlag _ , pp . 418432 , 2001 . i. mitchell and c. tomlin , `` level set methods for computations in hybrid systems , '' _ in m.di.benedetto and a.sangiovanni-vincentelli ( eds . ) hybrid systems : computation and control springer verlag _ , pp . 310323 , 2000 . c. tomlin , _ hybrid control of air traffic management systems_.1em plus 0.5em minus 0.4emuniversity of california , berkeley : ph.d . dissertation , department of electrical engineering and computer sciences , 1998 . `` sesar definition phase - deliverable d1 - air transport framework - the current situation , '' july 2006 . [ online ] . available : http://www.eurocontrol.int / sesar / public / standard_page / documentation.htm% l[http://www.eurocontrol.int / sesar / public / standard_page / documentation.htm% l ] i. lymperopoulos , j. lygeros , a. lecchini , w. glover , and j. maciejowski , `` a stochastic hybrid model for air traffic management processes , '' _ university of cambridge , department of engineering , technical report _ , vol . aut07 - 15 , 2007 .

a new framework for formulating reachability problems with competing inputs , nonlinear dynamics and state constraints as optimal control problems is developed . such reach - avoid problems arise in , among others , the study of safety problems in hybrid systems . earlier approaches to reach - avoid computations are either restricted to linear systems , or face numerical difficulties due to possible discontinuities in the hamiltonian of the optimal control problem . the main advantage of the approach proposed in this paper is that it can be applied to a general class of target hitting continuous dynamic games with nonlinear dynamics , and has very good properties in terms of its numerical solution , since the value function and the hamiltonian of the system are both continuous . the performance of the proposed method is demonstrated by applying it to a two aircraft collision avoidance scenario under target window constraints and in the presence of wind disturbance . target windows are a novel concept in air traffic management , and represent spatial and temporal constraints , that the aircraft have to respect to meet their schedule .

0911.4625

there have been many experimental @xcite , computational @xcite , and theoretical @xcite studies of the structural and mechanical properties of disordered static packings of frictionless disks in 2d and spheres in 3d . in these systems , counting arguments , which assume that all particle contacts give rise to linearly independent impenetrability constraints on the particle positions , predict that the minimum number of contacts required for the system to be collectively jammed is @xmath12 , where @xmath13 for fixed boundary conditions and @xmath14 for periodic boundary conditions @xcite , where @xmath15 is the spatial dimension and @xmath9 is the number of particles @xcite . the additional contact is required because contacts between hard particles provide only inequality constraints on particle separations @xcite . in the large - system limit , this relation for the minimum number of contacts reduces to @xmath16 , where @xmath17 is the average contact number . disordered packings of frictionless spheres are typically isostatic at jamming onset with @xmath18 , and possess the minimal number of contacts required to be collectively jammed @xcite . further , it has been shown in numerical simulations that collectively jammed hard - sphere packings correspond to mechanically stable soft - sphere packings in the limit of vanishing particle overlaps @xcite . in contrast , several numerical @xcite and experimental studies @xcite have found that disordered packings of ellipsoidal particles possess fewer contacts ( @xmath3 ) than predicted by naive counting arguments , which assume that all contacts give rise to linearly independent constraints on the interparticle separations . despite this , these packings were found to be mechanically stable ( ms ) @xcite . in a recent manuscript @xcite by donev , _ , the authors explained this apparent contradiction that static packings of ellipsoidal particles are mechanically stable , yet possess @xmath3 . the main points of the argument are included here . the set of @xmath19 interparticle contacts imposes @xmath19 constraints , @xmath20 , where @xmath21 is the center - to - center separation and @xmath22 is the contact distance along @xmath23 between particles @xmath24 and @xmath25 . in disordered ms sphere packings with @xmath26 , each of the @xmath19 interparticle contacts represents a linearly independent constraint . in contrast , some of the @xmath19 contacts for ms packings of ellipsoidal particles give rise to linearly _ dependent _ constraints . linearly dependent constraints do not block the degrees of freedom that appear in the constraint equations for sphere packings , whereas they can block multiple degrees of freedom in packings of ellipsoidal particles because they have convex particle shape with a varying radius of curvature @xcite . for static packings of spherical particles , interparticle contacts give rise to only convex " constraints ( fig . [ convex_concave ] ( a ) ) , while contacts can yield convex " or concave " constraints in packings of ellipsoidal particles ( figs . [ convex_concave ] ( a ) and ( b ) ) . note that the distinction between convex and concave constraints is different than the distinction between convex and concave particles . for example , ellipsoids have a convex particle shape , but static ellipsoid packings can possess concave interparticle constraints . ) are shaded blue . the axes labeled @xmath27 and @xmath28 indicate two representative orthogonal directions in configuration space.,scaledwidth=45.0% ] in jammed packings of ellipsoidal particles , there are @xmath29 special directions @xmath30 in configuration space along which perturbations give rise to interparticle overlaps that scale quadratically with the perturbation amplitude , @xmath31 @xcite . displacements in all other directions yield overlaps that scale linearly with @xmath11 , @xmath32 , as found for jammed sphere packings . this novel scaling behavior for packings of ellipsoidal particles can be explained by decomposing the dynamical matrix @xmath4 for these packings into two important components : the stiffness matrix @xmath5 that contains all second - order derivatives of the total potential energy @xmath33 with respect to the configurational degrees of freedom , and the stress matrix @xmath6 that includes all first - order derivatives of @xmath33 with respect to the particle coordinates . the directions @xmath30 are the eigenvectors of the stiffness matrix @xmath5 with zero eigenvalues . for static packings of ellipsoidal particles at the jamming threshold ( @xmath34 ) that interact via purely repulsive linear spring potentials ( _ i.e. _ @xmath35 ) , we find that the total potential energy increases quartically when the system is perturbed by @xmath11 along the @xmath30 directions , @xmath36 , where the constant @xmath37 . also , at the jamming threshold , the stress matrix @xmath38 and zero modes of the stiffness matrix are zero modes of the dynamical matrix . in this manuscript , we will investigate how the mechanical stability of packings of ellipsoidal particles is modified at finite compression ( @xmath39 ) . for example , when a system at finite @xmath0 is perturbed by amplitude @xmath11 along @xmath30 , do quadratic terms in @xmath11 arise in the total potential energy or do the contributions remain zero to second order ? if quadratic terms are present , do they stabilize or destabilize the packings , and how do the lowest frequency modes of the dynamical matrix scale with @xmath0 and aspect ratio ? we find a number of key results for static packings of ellipsoidal particles at finite compression ( @xmath39 ) including : 1 ) packings of ellipsoidal particles generically satisfy @xmath40 @xcite ; 2 ) the stiffness matrix @xmath5 possesses @xmath41 eigenmodes @xmath42 with zero eigenvalues even at finite compression @xmath43 ; and 3 ) the modes @xmath8 are nearly eigenvectors of the dynamical matrix ( and the stress matrix @xmath44 ) with eigenvalues that scale as @xmath45 , with @xmath37 , and thus finite compression stabilizes packings of ellipsoidal particles @xcite . in contrast , for static packings of spherical particles , the stiffness matrix contributions to the dynamical matrix stabilize all modes near jamming onset . at jamming onset ( @xmath46 ) , the harmonic response of packings of ellipsoidal particles vanishes , and the total potential energy scales as @xmath10 for perturbations by amplitude @xmath11 along these ` quartic ' modes , @xmath8 . our findings illustrate the significant differences between amorphous packings of spherical and ellipsoidal particles . the remainder of the manuscript will be organized as follows . in sec . [ methods ] , we describe the numerical methods that we employed to measure interparticle overlaps , generate static packings , and assess the mechanical stability of packings of ellipsoidal particles . in sec . [ vibration ] , we describe results from measurements of the density of vibrational modes in the harmonic approximation , the decomposition of the dynamical matrix eigenvalues into contributions from the stiffness and stress matrices , and the relative contributions of the translational and rotational degrees of freedom to the vibrational modes as a function of overcompression and aspect ratio using several packing - generation protocols . in sec . [ conclusion ] , we summarize our conclusions and provide promising directions for future research . we also include two appendices . in appendix [ contact_formation ] , we show that the formation of new interparticle contacts affects the scaling behavior of the potential energy with the amplitude of small perturbations along eigenmodes of the dynamical matrix . in appendix [ appendixb ] , we provide analytical expressions for the elements of the dynamical matrix for ellipse - shaped particles in 2d that interact via a purely repulsive linear spring potential . defined as the ratio of the major to minor axis and ( b ) prolate ellipsoids in 3d where @xmath47 is the ratio of the polar to equatorial lengths.,scaledwidth=30.0% ] in this section , we describe the computational methods employed to generate static packings of convex , anisotropic particles , _ i.e. _ ellipses in 2d and prolate ellipsoids in 3d with aspect ratio @xmath48 of the major to minor axes ( fig . [ intro_fig1 ] ) , and analyze their mechanical properties . to inhibit ordering in 2d , we studied bidisperse mixtures ( 2-to-1 relative number density ) , where the ratio of the major ( and minor ) axes of the large and small particles is @xmath49 . in 3d , we focused on a monodisperse size distribution of prolate ellipsoids . we employed periodic boundaries conditions in unit square ( 2d ) and cubic ( 3d ) cells and studied systems sizes in the range from @xmath50 to @xmath51 particles to address finite - size effects . for ellipsoidal particles @xmath24 and @xmath25 with unit vectors @xmath52 and @xmath53 that characterize the orientations of their major axes . @xmath22 is the center - to - center separation at which ellipsoidal particles first touch when they are brought together along @xmath23 at fixed orientation.,scaledwidth=30.0% ] in both 2d and 3d , we assume that particles interact via the following pairwise , purely repulsive linear spring potential @xmath54 where @xmath55 is the characteristic energy of the interaction , @xmath21 is the center - to - center separation between particles @xmath24 and @xmath25 , and @xmath22 is the orientation - dependent center - to - center separation at which particles @xmath24 and @xmath25 come into contact as shown in fig . [ fig : fig10_chapter ] . below , energies , lengths , and time scales will be expressed in units of @xmath55 , @xmath56 , and @xmath57 , respectively , where @xmath58 and @xmath59 are the mass and moment of inertia of the ellipsoidal particles . perram and wertheim developed an efficient method for calculating the exact contact distance between ellipsoidal particles with any aspect ratio and size distribution in 2d and 3d @xcite . in their formulation , the contact distance is obtained from @xmath60 the approximation @xmath61 is equivalent to the commonly used gay - berne approximation for the contact distance @xcite . the accuracy of the gay - berne approximation depends on the relative orientation of the two ellipsoidal particles , and in general is more accurate for monodisperse systems . for example , in fig . [ fig : fig12_chapter ] , we show @xmath62 for several relative orientations of both monodisperse and bidisperse systems . the relative deviation from the true contact distance can be as large as @xmath63 for @xmath64 and @xmath65 . thus , the gay - berne approximation should be used with caution when studying polydisperse packings of ellipsoidal particles . for monodisperse ellipses with @xmath65 , @xmath66 . we find similar results for 3d systems . unless stated otherwise , we employ the exact expression for contact distance , and thus @xmath67 , @xmath68 , @xmath69 , and @xmath70 , where @xmath71 is the minimum obtained from eq . [ sigma_functional ] . positioned at the gay - berne contact distance @xmath72 . for two ellipses with the same size , the ( a ) end - to - end configuration is exact , while the ( b ) side - to - end configuration has a @xmath73 relative error . for two ellipses with @xmath64 , the ( c ) end - to - end configuration has a relative error of @xmath74 , while the ( d ) side - to - end configuration has a relative error of @xmath75.,scaledwidth=45.0% ] we employ a frequently used isotropic compression method for soft , purely repulsive particles @xcite to generate static packings of ellipsoidal particles at jamming onset ( @xmath1 ) . static packings at jamming onset are characterized by infinitesimal but nonzero total potential energy and pressure . this isotropic compression method consists of the following steps . we begin by randomly placing particles at low packing fraction ( @xmath76 ) with random orientations and zero velocities in the simulation cell . we successively compress the system by small packing fraction increments @xmath77 , with each compression followed by conjugate gradient ( cg ) energy minimization until the total potential energy per particle drops below a small threshold , @xmath78 , or the total potential energy per particle between successive iterations of the minimization routine is @xmath79 . the algorithm switches from compression to decompression if the minimized energy is greater than @xmath80 . each time the algorithm toggles from compression to decompression or vice versa , the packing fraction increment is halved . the packing - generation algorithm is terminated when the total potential energy per particle satisfies @xmath81 . thus , using this method we can locate the jammed packing fraction @xmath82 and particle positions at jamming onset for each initial condition to within @xmath83 . after jamming onset is identified , we also generate configurations at specified @xmath84 over six orders of magnitude from @xmath83 to @xmath85 by applying a prescribed set of compressions with each followed by energy minimization . to determine whether the accuracy of the energy minimization algorithm affects our results ( see sec . [ protocol ] ) , we calculate the eigenvalues of the dynamical matrix as a function of the total kinetic energy ( or deviation from zero in force and torque balance on each particle ) at each @xmath0 . to do this , we initialize the system with ms packings from the above packing - generation algorithm and use molecular dynamics ( md ) simulations with damping terms proportional to the translational and rotational velocities of the ellipsoidal particles to remove excess kinetic energy from the system @xcite . the damped md simulations are terminated when the total kinetic energy per particle is below @xmath86 , where @xmath87 is varied from @xmath88 to @xmath89 . this provides accuracy in the particle positions of the energy minimized states in the range from @xmath83 to @xmath88 . for the damped md simulations , we solve newton s equations of motion ( using fifth - order gear predictor - corrector methods @xcite ) for the center of mass position and angles that characterize the orientation of the long axis of the ellipsoidal particles . in 2d , we solve @xmath90 where @xmath91 is the angle the long axis of ellipse @xmath24 makes with the horizontal axis , @xmath92 is the translational velocity of particle @xmath24 , @xmath93 is the rotational speed of particle @xmath24 , @xmath94 and @xmath95 are the damping coefficients for the position and angle degrees of freedom , and the moment of inertia @xmath96 . the force @xmath97 on ellipse @xmath24 arising from an overlap with ellipse @xmath25 is @xmath98 where @xmath99+\nonumber\\ & & ( \beta^{-2}-\chi)\sin\left[2(\psi_{ij}-\theta_j)\right]\big ] \times \nonumber\\ & & \left(1-\chi^2\cos^2\left[\theta_i-\theta_j\right]\right)^{-1 } , \label{eq : force2}\end{aligned}\ ] ] @xmath100 for the purely repulsive linear spring potential in eq . [ eq : ellipse_energy ] , and @xmath101 and @xmath102 are illustrated in fig . [ torque_schematic ] . to calculate the torque @xmath103 \cdot { \hat z}$ ] in eq . [ angle ] , we must identify the point of contact between particles @xmath24 and @xmath25 , @xmath104,\end{aligned}\ ] ] where @xmath105 and @xmath106 , @xmath107 , and @xmath108 are depicted in fig . [ torque_schematic ] . from eqs . [ eq : force ] and [ moment ] , we find @xmath109 and @xmath110 characterize the orientation of particles @xmath24 and @xmath25 relative to the horizontal axis , _ i.e. _ @xmath111 . @xmath107 gives the angle between the center - to - center separation vector @xmath112 and the horizontal axis and @xmath113 is the angle unit vector in polar coordinates . the unit vector @xmath114 points in the direction of the force on particle @xmath24 due to particle @xmath25 at the point of contact . @xmath115 points from the center of particle @xmath24 to the point of contact with particle @xmath25 , and @xmath108 is the angle between @xmath52 and @xmath116.,scaledwidth=30.0% ] to investigate the mechanical properties of static packings of ellipsoidal particles , we will calculate the eigenvalues of the dynamical matrix and the resulting density of vibrational modes in the harmonic approximation @xcite . the dynamical matrix is defined as @xmath117 where @xmath118 ( with @xmath119 ) represent the @xmath120 degrees of freedom in the system and @xmath121 is the number of degrees of freedom per particle . in 2d @xmath122 with @xmath123,@xmath124,@xmath125,@xmath126,@xmath127,@xmath128,@xmath125,@xmath129,@xmath130,@xmath131,@xmath125,@xmath132 } and in 3d for prolate ellipsoids @xmath133 with @xmath123,@xmath124,@xmath125,@xmath126,@xmath127 , @xmath128,@xmath125,@xmath129,@xmath134,@xmath135,@xmath125,@xmath136,@xmath137,@xmath138,@xmath125,@xmath139,@xmath140,@xmath141,@xmath125 , @xmath142 } , where @xmath91 is the polar angle and @xmath143 is the azimuthal angle in spherical coordinates , @xmath144 , @xmath145 , and @xmath146 . the dynamical matrix requires calculations of the first and second derivatives of the total potential energy @xmath33 with respect to all positional and angular degrees of freedom in the system . the first derivatives of @xmath33 with respect to the positions of the centers of mass of the particles @xmath147 can be obtained from eq . [ fijvec ] . in 2d , there is only one first derivative involving angles , @xmath148 , where @xmath149)r_{ij}},\nonumber \\ c&=&\cos^2(\theta_i-\theta_j ) . \nonumber\end{aligned}\ ] ] complete expressions for the matrix elements of the dynamical matrix for ellipses in 2d are provided in appendix [ appendixb ] . in 3d , we calculated the first derivatives of @xmath33 with respect to the particle coordinates analytically , and then evaluated the second derivatives for the dynamical matrix numerically . the vibrational frequencies in the harmonic approximation can be obtained from the @xmath150 nontrivial eigenvalues @xmath151 of the dynamical matrix , @xmath152 . @xmath15 of the eigenvalues are zero due to periodic boundary conditions . for all static packings , we have verified that the smallest nontrivial eigenvalue satisfies @xmath153 . below , we will study the density of vibrational frequencies @xmath154 as a function of compression @xmath0 and aspect ratio @xmath47 , where @xmath155 is the number of vibrational frequencies less than @xmath156 . we will also investigate the relative contributions of the translational and rotational degrees of freedom to the nontrivial eigenvectors of the dynamical matrix , @xmath157 for ellipses in 2d and @xmath158 @xmath159 for prolate ellipsoids in 3d , where @xmath24 labels the eigenvector and runs from @xmath160 to @xmath150 . the eigenvectors are normalized such that @xmath161 . the dynamical matrix ( eq . [ eq : dm_eq ] ) can be decomposed into two component matrices @xmath4 : 1 ) the stiffness matrix @xmath5 that includes only second - order derivatives of the total potential energy @xmath33 with respect to the configurational degrees of freedom and 2 ) the stress matrix @xmath6 that includes only first - order derivatives of @xmath33 . the @xmath162 elements of @xmath5 and @xmath6 are given by @xmath163 where the sums are over distinct pairs of overlapping particles @xmath24 and @xmath25 . since @xmath164 for the purely repulsive linear spring potential ( eq . [ eq : ellipse_energy ] ) , the stiffness matrix depends only on the geometry of the packing ( _ i.e. _ @xmath165 . also , at zero compression @xmath34 , @xmath38 , @xmath166 , and only the stiffness matrix contributes to the dynamical matrix . the frequencies associated with the eigenvalues @xmath167 of the stiffness matrix ( at any @xmath0 ) are denoted by @xmath168 , and the stiffness matrix eigenvectors are normalized such that @xmath169 . when counting the number of interparticle contacts @xmath19 , we remove all rattler particles ( defined as those with fewer than @xmath170 contacts ) and do not include the contacts that rattler particles make with non - rattler particles @xcite . removing these contacts may cause non - rattler particles to become rattlers , and thus this process is performed recursively . note that for ellipsoidal particles with @xmath170 contacts , the lines normal to the points ( or planes in 3d ) of contact must all intersect , otherwise the system is not mechanically stable . the number of contacts per particle is defined as @xmath171 , where @xmath172 is the number of rattlers . we find that the number of rattler particles decreases with aspect ratio from approximately @xmath73 of the system at @xmath173 to zero for @xmath174 in both 2d and 3d . static packings of ellipsoidal particles at jamming onset typically possess fewer contacts than predicted by isostatic counting arguments @xcite , @xmath3 , over a wide range of aspect ratio as shown in fig . this finding raises a number of important questions . for example , are static packings of ellipsoidal particles mechanically stable at finite @xmath39 , _ i.e. _ does the dynamical matrix for these systems possess nontrivial zero - frequency modes at @xmath39 ? in this section , we will show that packings of ellipsoidal particles are indeed mechanically stable ( with no nontrivial zero - frequency modes ) by calculating the dynamical , stress , and stiffness matrices for these systems as a function of compression @xmath0 , aspect ratio @xmath47 , and packing - generation protocol . further , we will show that the density of vibrational modes for these systems possesses three characteristic frequency regimes and determine the scaling of these characteristic frequencies with @xmath0 and @xmath47 . versus aspect ratio @xmath47 for static packings of ( a ) bidisperse ellipses in 2d and ( b ) prolate ellipsoids in 3d at jamming onset . the isostatic values @xmath175 ( 2d ) and @xmath176 ( 3d ) are indicated by dashed lines.,scaledwidth=45.0% ] a number of studies have shown that amorphous sphere packings are fragile solids in the sense that the density of vibrational frequencies ( in the harmonic approximation ) @xmath177 for these systems possesses an excess of low - frequency modes over debye solids near jamming onset , _ i.e. _ a plateau forms and extends to lower frequencies as @xmath178 @xcite . in this work , we will calculate @xmath177 as a function of @xmath0 and aspect ratio @xmath47 for amorphous packings of ellipsoidal particles and show that the density of vibrational modes for these systems shows significant qualitative differences from that for spherical particles . for @xmath179 ellipse - shaped particles at @xmath180 with aspect ratio @xmath181 ( solid ) , @xmath182 ( dotted ) , @xmath183 ( dashed ) , and @xmath184 ( dot - dashed ) . @xmath177 for @xmath185 has been scaled by @xmath186 relative to those with @xmath187 to achieve collapse at low aspect ratios . ( b ) @xmath177 for the same aspect ratios in ( a ) on a @xmath188-@xmath188 scale . the inset illustrates the three characteristic frequencies @xmath189 , @xmath190 , and @xmath191 in @xmath177 for @xmath192.,scaledwidth=40.0% ] for @xmath193 prolate ellipsoids at @xmath194 for @xmath181 ( solid ) , @xmath182 ( dotted ) , @xmath195 ( dashed ) , and @xmath196 ( dot - dashed ) . @xmath177 for @xmath185 has been scaled by @xmath197 relative to those with @xmath187 to achieve collapse at low aspect ratios . ( b ) @xmath177 for the same aspect ratios in ( a ) on a @xmath188-@xmath188 scale . the inset illustrates the three characteristic frequencies @xmath189 , @xmath190 , and @xmath191 in @xmath177 for @xmath192.,scaledwidth=45.0% ] in figs . [ fig_dm_2d ] ( a ) and ( b ) , we show @xmath177 on linear and @xmath188-@xmath188 scales , respectively , for ellipse - shaped particles in 2d at @xmath198 over a range of aspect ratios from @xmath173 to @xmath199 . we find several key features in @xmath177 : 1 ) for low aspect ratios @xmath200 , @xmath177 collapses with that for disks ( @xmath185 ) at intermediate and large frequencies @xmath201 ; 2 ) for large aspect ratios @xmath202 , @xmath177 is qualitatively different for ellipses than for disks over the entire frequency range ; and 3 ) a strong peak near @xmath203 and a smaller secondary peak at intermediate frequencies ( evident on the log - log scale in fig . [ fig_dm_2d ] ( b ) ) occur in @xmath177 for @xmath204 . note that at finite compression @xmath39 , we do not find any nontrivial zero - frequency modes of the dynamical matrix in static packings of ellipses and ellipsoids . the only zero - frequency modes in these systems correspond to the @xmath15 constant translations that arise from periodic boundary conditions and zero - frequency modes associated with ` rattler ' particles with fewer than @xmath170 interparticle contacts . for @xmath179 ellipses as a function of compression @xmath205 ( solid ) , @xmath206 ( dashed ) , @xmath207 ( dotted ) , and @xmath85 ( dot - dashed ) for ( a ) @xmath208 and ( b ) @xmath199.,scaledwidth=40.0% ] ( circles ) , @xmath190 ( squares ) , and @xmath191 ( diamonds ) from @xmath177 as a function of aspect ratio @xmath209 for @xmath179 ellipses in 2d at @xmath180 . the solid ( dashed ) lines have slope @xmath210 ( @xmath211 . ( b ) @xmath212 for systems with @xmath179 ellipses in 2d at @xmath205 ( circles ) , @xmath213 ( squares ) , @xmath206 ( diamonds ) , @xmath214 ( upward triangles ) , @xmath207 , ( downward triangles ) , and @xmath85 ( crosses ) . the solid line has slope @xmath210.,scaledwidth=40.0% ] to monitor the key features of @xmath177 as a function of @xmath215 and @xmath47 , we define three characteristic frequencies as shown in the inset to fig . [ fig_dm_2d ] ( b ) . @xmath189 and @xmath190 identify the locations of the small and intermediate frequency peaks in @xmath177 , and @xmath191 marks the onset of the high - frequency plateau regime in @xmath177 . for our analysis , we define @xmath191 as the largest frequency ( @xmath216 ) with @xmath217 , which is approximately half of the height of the plateau in @xmath177 at large frequencies . all three characteristic frequencies increase with aspect ratio . note that we only track @xmath190 and @xmath191 for aspect ratios where @xmath218 . for example , the intermediate and high - frequency bands characterized by @xmath190 and @xmath191 merge for @xmath219 . as shown in fig . [ fig_dm_3d ] , @xmath177 for 3d prolate ellipsoids displays similar behavior to that for ellipses in 2d ( fig . [ fig_dm_2d ] ) for aspect ratios @xmath220 . for example , @xmath177 for ellipsoids possesses low , intermediate , and high frequency regimes , whose characteristic frequencies @xmath189 , @xmath190 , and @xmath191 increase with aspect ratio . note that the intermediate and high - frequency bands @xmath190 and @xmath191 merge for @xmath221 , which occurs at lower aspect ratio than the merging of the bands in 2d . another significant difference is that in 3d @xmath177 extends to higher frequencies at large aspect ratios ( @xmath222 ) than @xmath177 for ellipses . ( circles ) , @xmath190 ( squares ) , and @xmath191 ( diamonds ) from @xmath177 as a function of aspect ratio @xmath209 for @xmath179 prolate ellipsoids in 3d at @xmath194 . the solid ( dashed ) lines have slope @xmath210 ( @xmath211 . ( b ) @xmath212 for systems with @xmath179 prolate ellipsoids at @xmath223 ( circles ) , @xmath206 ( squares ) , and @xmath214 ( diamonds ) . the solid line has slope @xmath210.,scaledwidth=40.0% ] we note the qualitative similarity between the @xmath177 for @xmath224 ellipsoids shown in fig . [ fig_dm_3d ] ( b ) and @xmath177 for @xmath225 presented in fig . 1 ( c ) of ref . @xcite for @xmath226 . however , zeravcic , _ et al . _ suggest that there is no weight in @xmath177 for @xmath227 except at @xmath228 for both oblate and prolate ellipsoids , in contrast to our results in fig . [ fig_dm_3d ] . in fig . [ fig_dm_2d_comp ] , we show the behavior of @xmath177 for ellipse packings as a function of compression @xmath0 for two aspect ratios , @xmath208 and @xmath199 . we find that the low - frequency band ( characterized by @xmath229 depends on @xmath0 , while the intermediate and high frequency bands do not . the intermediate and high frequencies bands do not change significantly until the low - frequency band centered at @xmath189 merges with them at @xmath230 and @xmath231 for @xmath208 and @xmath199 , respectively . we plot the characteristic frequencies @xmath189 , @xmath190 , and @xmath191 versus aspect ratio @xmath232 for ellipse packings in fig . [ fig_char_freq ] and ellipsoid packings in fig . [ fig_char_freq_3d ] . the characteristic frequencies obey the following scaling laws over at least two orders of magnitude in @xmath232 and five orders of magnitude in @xmath0 : @xmath233 similar results for the scaling of @xmath190 and @xmath191 with @xmath232 were found in ref . . we will refer to the modes in the low - frequency band in @xmath177 ( with characteristic frequency @xmath189 ) as ` quartic modes ' , and these will be discussed in detail sec . [ quartic_modes ] . the scaling of the quartic mode frequencies with compression , @xmath234 , has important consequences for the linear response behavior of ellipsoidal particles to applied stress @xcite . associated with the eigenvalues of the stiffness matrix @xmath5 for @xmath179 ellipse packings as a function of compression @xmath235 ( dotted ) , @xmath207 ( dashed ) , and @xmath85 ( dot - dashed ) for @xmath208 . the vertical solid line indicates the ` zero - frequency ' tolerance @xmath236 , which is the lowest frequency obtained for the dynamical matrix for packings at @xmath237 and the smallest compression ( @xmath198 ) in fig . [ fig_dm_2d].,scaledwidth=40.0% ] as shown in fig . [ z ] , static packings of ellipsoidal particles can possess @xmath3 over a wide range of aspect ratio , yet as described in sec . [ dos ] , the dynamical matrix @xmath238 contains a complete spectrum of @xmath239 nonzero eigenvalues @xmath151 near jamming . to investigate this intriguing property , we first calculate the eigenvalues of the stiffness matrix @xmath5 , show that it possesses @xmath240 ` zero'-frequency modes whose number matches the deviation in the contact number from the isostatic value , and then identify the separate contributions from the stiffness and stress matrices to the dynamical matrix eigenvalues . in fig . [ fig_dm_stiffness_comp ] , we show the distribution of frequencies @xmath241 associated with the eigenvalues of the stiffness matrix for ellipse packings at @xmath208 as a function of compression @xmath0 . we find three striking features in fig . [ fig_dm_stiffness_comp ] : 1 ) many modes of the stiffness matrix exist near and below the zero - frequency threshold ( determined by the vibrational frequencies of the dynamical matrix at @xmath237 and @xmath242 ) , 2 ) frequencies that correspond to the low - frequency band characterized by @xmath189 are absent , and 3 ) the nonzero frequency modes ( with @xmath243 ) do not scale with @xmath0 as pointed out for the dynamical matrix eigenvalues in eqs . [ scaling2 ] and [ scaling3 ] . further , we find that the number of zero - frequency modes @xmath240 of the stiffness matrix matches the deviation in the number of contacts from the isostatic value ( @xmath244 ) for each @xmath0 and aspect ratio . specifically , @xmath245 over the full range of @xmath0 for @xmath246 of the more than @xmath247 packings for aspect ratio @xmath248 and for @xmath249 of the more than @xmath247 packings for @xmath250 . and ( b ) @xmath251 , plotted versus @xmath252 for ellipse packings with @xmath208 and @xmath253 ( circles ) , @xmath206 ( squares ) , @xmath214 ( diamonds ) , and @xmath207 ( triangles ) . in ( a ) and ( b ) , the solid lines correspond to @xmath254 and @xmath255 . in the main panel and inset of ( a ) , only modes corresponding the intermediate and high frequency bands are included . in the main panel and inset of ( b ) , only modes corresponding the low frequency band are included . the insets to ( a ) and ( b ) , which plot @xmath256 versus @xmath257 and @xmath258 versus @xmath259 , show the deviations @xmath260 for high- and intermediate - frequency modes and @xmath261 for low - frequency modes.,scaledwidth=40.0% ] in fig . [ stiffness_matrix ] , we calculate the projection of the dynamical matrix eigenvectors @xmath262 onto the stiffness and stress matrices , @xmath263 and @xmath264 , where @xmath265 is the transpose of @xmath262 and @xmath266 . fig . [ stiffness_matrix ] ( a ) shows that for large eigenvalues @xmath267 of the dynamical matrix ( _ i.e. _ within the intermediate and high frequency bands characterized by @xmath190 and @xmath191 in fig . [ fig_dm_2d ] ) , the eigenvalues of the stiffness and dynamical matrices are approximately the same , @xmath268 . the deviation @xmath269 , shown in the inset to fig . [ stiffness_matrix ] ( a ) , scales linearly with @xmath270 . thus , we find that the intermediate and high frequency modes for packings of ellipsoidal particles are stabilized by the stiffness matrix @xmath5 . in the main panel of fig . [ stiffness_matrix ] ( b ) , we show that for frequencies in the lowest frequency band ( characterized by @xmath189 ) the eigenvalues of the stress and dynamical matrices are approximately the same , @xmath271 . in the inset to fig . [ stiffness_matrix ] ( b ) , we show that the deviation @xmath272 scales as @xmath273 . thus , we find that the lowest frequency modes for packings of ellipsoidal particles are stabilized by the stress matrix @xmath44 over a wide range of compression @xmath0 . similar results were found previously for packings of hard ellipsoidal particles @xcite . in contrast , for static packings of spherical particles , the stress matrix contributions to the dynamical matrix are destabilizing with @xmath274 for all frequencies near jamming , and @xmath275 stabilizes the packings as shown in fig . [ decompose2d ] . and ( b ) @xmath276 , plotted versus the eigenvalues of the dynamical matrix @xmath257 for bidisperse disk packings at @xmath223 ( circles ) , @xmath206 ( squares ) , @xmath214 ( diamonds ) , and @xmath207 ( triangles ) . in ( a ) the solid line corresponds to @xmath254 . note that @xmath274 over the entire range of frequencies.,scaledwidth=40.0% ] we showed in sec . [ dos ] that the dynamical matrix @xmath238 for packings of ellipsoidal particles contains a complete spectrum of @xmath277 nonzero eigenvalues @xmath151 for @xmath278 despite that fact that @xmath3 . further , we showed that the modes in the lowest frequency band scale as @xmath279 in the @xmath280 limit . what happens at jamming onset when @xmath281 , _ i.e. _ are these low - frequency modes that become true zero - frequency modes at @xmath1 stabilized or destabilized by higher - order terms in the expansion of the potential energy in powers of the perturbation amplitude ? to investigate this question , we apply the following deformation to static packings of ellipsoidal particles : @xmath282 where @xmath11 is the amplitude of the perturbation , @xmath262 is an eigenvector of the dynamical matrix , and @xmath283 is the point in configuration space corresponding to the original static packing , followed by conjugate gradient energy minimization . we then measure the change in the total potential energy before and after the perturbation , @xmath284 . we plot @xmath285 versus @xmath11 in fig . [ fig7b ] for perturbations along eigenvectors that correspond to the smallest nontrivial eigenvalue @xmath286 of the dynamical matrix for static packings of ( a ) disks and ellipses and ( b ) spheres and prolate ellipsoids at @xmath287 . as expected , for disks and spheres , we find that @xmath288 over a wide range of @xmath11 in response to perturbations along eigenvectors that correspond to the smallest nontrivial eigenvalue . in contrast , we find novel behavior for @xmath289 when we apply perturbations along the eigendirection that corresponds the lowest nonzero eigenvalue of the dynamical matrix for packings of ellipsoidal particles . in fig . [ fig15 ] , we show that @xmath285 obeys @xmath290 where @xmath291 and the constants @xmath292 , for perturbations along all modes @xmath293 in the lowest frequency band of @xmath177 for packings of ellipsoidal particles when we do not include changes in the contact network following the perturbation and relaxation . ( see appendix [ contact_formation ] for measurements of @xmath285 when we include changes in the contact network . ) eigenmodes in the lowest frequency band are termed ` quartic ' because at @xmath281 they are stabilized by quartic terms in the expansion of the total potential energy with respect to small displacements @xcite . for @xmath294 , the change in potential energy scales as @xmath295 , whereas @xmath296 for @xmath297 , where the characteristic perturbation amplitude @xmath298 . in the insets to fig . [ fig15 ] ( a ) and ( b ) , we show that the characteristic perturbation amplitude averaged over modes in the lowest frequency band scale as @xmath299 for static packings of ellipses in 2d and prolate ellipsoids in 3d , which indicates that the @xmath300 possess nontrivial dependence on aspect ratio @xmath47 . the quartic modes have additional interesting features . for example , quartic modes are dominated by the rotational rather than translational degrees of freedom . we identify the relative contributions of the translational and rotational degrees of freedom to the eigenvectors of the dynamical matrix in figs . [ fig_comp_2d ] and [ fig_comp_3d ] . the contribution of the translational degrees of freedom to eigenvector @xmath262 is defined as @xmath301 where the sum over @xmath302 includes @xmath303 and @xmath304 in 2d and @xmath303 , @xmath304 , and @xmath305 in 3d and the eigenvectors are indexed in increasing order of the corresponding eigenvalues . since the eigenvectors are normalized , the rotational contribution to each eigenvector is @xmath306 . for both ellipses in 2d and prolate ellipsoids in 3d , we find that at low aspect ratios ( @xmath307 ) , the first @xmath9 ( @xmath308 ) modes in 2d ( 3d ) are predominately rotational and the remaining @xmath308 ( @xmath309 ) modes in 2d ( 3d ) are predominately translational . in the inset to figs . [ fig_comp_2d](b ) and [ fig_comp_3d ] , we show that @xmath310 increases as @xmath311 , where @xmath312 ( @xmath313 ) for ellipses ( prolate ellipsoids ) , for both the low and intermediate frequency modes . for @xmath314 , we find mode - mixing , especially at intermediate frequencies , where modes have finite contributions from both the rotational and translational degrees of freedom . for @xmath315 , the modes become increasingly more translational with increasing frequency . for @xmath174 , the modes become more rotational in character at the highest frequencies . our results show that the modes with significant rotational content at low @xmath47 correspond to modes in the low and intermediate frequency bands of @xmath177 , while the modes with significant translational content at low @xmath47 correspond to modes in the high frequency band of @xmath177 . to each eigenvector @xmath316 of the dynamical matrix versus frequency @xmath156 in packings of ellipses in 2d at @xmath317 . ( a ) shows data for aspect ratios @xmath208 ( black solid ) , @xmath196 ( red dashed ) , @xmath318 ( green dash - dash - dot ) , @xmath184 ( blue dash - dot ) , and @xmath319 ( purple dot - dot - dash ) and ( b ) shows data for aspect ratios @xmath192 ( black solid ) , @xmath320 ( red dashed ) , @xmath195 ( green dot - dot - dash ) , @xmath321 ( blue dash - dot ) , @xmath322 ( purple dot - dot - dash ) , and @xmath183 ( cyan dotted ) . the inset to ( b ) shows @xmath323 averaged over modes in the lowest ( squares ) and intermediate ( circles ) frequency regimes . the solid line has slope @xmath318.,scaledwidth=40.0% ] to each eigenvector @xmath316 of the dynamical matrix versus frequency @xmath156 in packings of prolate ellipsoids in 3d at @xmath324 . ( a ) shows data for aspect ratios @xmath325 ( black solid ) , @xmath183 ( red dashed ) , @xmath326 ( green dash - dash - dot ) , @xmath196 ( blue dash - dot ) , and @xmath318 ( purple dot - dot - dash ) and ( b ) shows data for aspect ratios @xmath192 ( black solid ) , @xmath320 ( red dashed ) , @xmath195 ( green dash - dash - dot ) , @xmath321 ( blue dash - dot ) , @xmath322 ( purple dot - dot - dash ) , and @xmath183 ( cyan dotted ) . the inset to ( b ) shows @xmath327 averaged over modes in the lowest ( squares ) and intermediate ( circles ) frequency regimes . the solid line has slope @xmath313.,scaledwidth=40.0% ] we performed several checks to test the robustness and accuracy of our calculations of the density of vibrational modes in the harmonic approximation @xmath177 for static packings of ellipsoidal particles : 1 ) we compared @xmath177 obtained from static packings of ellipsoidal particles using perram and wertheim s exact expression ( eq . [ sigma_functional ] ) for the contact distance between pairs of ellipsoidal particles and the gay - berne approximation described in sec . [ contact ] ; 2 ) we calculated @xmath177 for static packings as a function of the tolerance used to terminate energy minimization for both the md and cg methods ; and 3 ) we studied the system - size dependence of @xmath177 in systems ranging from @xmath50 to @xmath51 particles . in fig . [ fig11 ] , we show that the density of vibrational modes @xmath177 is nearly the same when we use the perram and wertheim exact expression and the gay - berne approximation to the contact distance for ellipse - shaped particles . @xmath177 for static packings of ellipse - shaped particles is also not dependent on @xmath328 , which controls the accuracy of the conjugate gradient energy minimization ( sec . [ packing ] ) , for sufficiently small values . our calculations in fig . [ fig11 ] ( b ) also show that @xmath177 is not sensitive to the energy minimization procedure ( _ i.e. _ md vs. cg ) for small values of the minimization tolerance @xmath87 . in addition , key features of the density of vibrational modes are not strongly dependent on system size . for example , in fig . [ system_size ] , we show @xmath177 for ellipses in 2d at aspect ratio @xmath237 and compression @xmath287 over a range of system sizes from @xmath50 to @xmath51 . ( for reference , @xmath177 at fixed system size @xmath179 and @xmath198 over a range of aspect ratios is shown in fig . [ fig_dm_2d ] . ) @xmath177 in the low and intermediate frequency bands and plateau region overlap for all system sizes . the only feature of @xmath177 that changes with system size is that successively lower frequency , long wavelength translational modes extend from the plateau region as system size increases . in the large system - size limit @xmath329 , which we do not reach in these studies , the lowest frequency modes will scale as @xmath330 . we performed extensive numerical simulations of static packings of frictionless , purely repulsive ellipses in 2d and prolate ellipsoids in 3d as a function of aspect ratio @xmath47 and compression from jamming onset @xmath0 . we found several important results that highlight the significant differences between amorphous packings of spherical and ellipsoidal particles near jamming . first , as found previously , static packings of ellipsoidal particles generically satisfy @xmath40 @xcite ; _ i.e. _ they possess fewer contacts than the minimum required for mechanical stability as predicted by counting arguments that assume all contacts give rise to linearly independent constraints on particle positions . second , we decomposed the dynamical matrix @xmath4 into the stiffness @xmath5 and stress @xmath6 matrices . we found that the stiffness matrix possesses @xmath41 eigenmodes @xmath8 with zero eigenvalues over a wide range of compressions @xmath43 . third , we found that the modes @xmath8 are nearly eigenvectors of the dynamical matrix ( and the stress matrix @xmath44 ) with eigenvalues that scale as @xmath45 , with @xmath37 , and thus finite compression stabilizes packings of ellipsoidal particles @xcite . at jamming onset , the harmonic response of packings of ellipsoidal particles vanishes , and the total potential energy scales as @xmath10 for perturbations by amplitude @xmath11 along these ` quartic ' modes , @xmath8 . in addition , we have shown that these results are robust ; for example , the density of vibrational modes @xmath177 ( in the harmonic approximation ) is not sensitive to the error tolerance of the energy minimization procedure , the system size , and the accuracy of the determination of the interparticle contacts over the range of parameters employed in the simulations . these results raise several fundamental questions for static granular packings : 1 ) which classes of particle shapes give rise to quartic modes ? ; 2 ) is there a more general isostatic counting argument that can predict the number of quartic modes at jamming onset ( for a given packing - generation protocol ) ? ; and 3 ) are systems with quartic modes even more anharmonic @xcite than packings of spherical particles in the presence of thermal and other sources of fluctuations ? we will address these important questions in our future studies . support from nsf grant numbers dmr-0905880 ( bc and mm ) and dms-0835742 ( cs and co ) is acknowledged . we also thank t. bertrand , m. bi , and m. shattuck for helpful discussions . the scaling behavior of @xmath285 ( shown in figs . [ fig7b ] and [ fig15 ] ) as a function of the amplitude @xmath11 of the perturbation along the eigenmodes of the dynamical matrix is valid only when the original contact network of the perturbed static packing does not change . as shown in fig . [ contact_breaking_fig ] , @xmath289 does not obey the power - law scaling described in eq . [ quartic_equation ] when new interparticle contacts form . note that changes in the contact network are more likely for systems with @xmath331 as shown previously in ref . @xcite . in a future publication , we will measure the critical perturbation amplitude @xmath332 below which new contacts do not form and existing contacts do not change for each mode @xmath293 . this work is closely related to determining the nonlinear vibrational response of packings of ellipsoidal and other anisotropic particles . in this appendix , we provide explicit expressions for the dynamical matrix elements ( eq . [ eq : dm_eq ] ) for ellipse - shaped particles that interact via the purely repulsive linear spring potential ( eq . [ eq : ellipse_energy ] ) . the nine dynamical matrix elements for @xmath333 are @xmath334 and the nine dynamical matrix elements for @xmath335 are @xmath336 where @xmath107 is the polar angle defined in fig . [ torque_schematic ] , @xmath56 , @xmath337 is given in eq . [ fij ] , @xmath338 @xmath339 and @xmath340 for systems with ` rattler ' particles that possess fewer than @xmath170 contacts , @xmath341 ( @xmath342 ) for fixed ( periodic ) boundary conditions , where @xmath343 and @xmath172 is the number of rattler particles . c. f. schreck and c. s. ohern , computational methods to study jammed systems " , in _ experimental and computational techniques in soft condensed matter physics _ , ed . by j. s. olafsen , ( cambridge university press , new york , 2010 ) . the cg gradient energy minimization technique we implement relies on numerous evaluations of the total potential energy to identify local minima . however , when we integrate newton s equations of motion with damping forces proportional to particle velocities ( eqs . [ newton ] and [ angle ] ) , energy minimization and the minimization stopping criteria are based on the evaluation of forces and torques , which allows increased accuracy compared to the cg technique .

we numerically investigate the mechanical properties of static packings of ellipsoidal particles in 2d and 3d over a range of aspect ratio and compression @xmath0 . while amorphous packings of spherical particles at jamming onset ( @xmath1 ) are isostatic and possess the minimum contact number @xmath2 required for them to be collectively jammed , amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming ( @xmath3 ) from naive counting arguments , which assume that all contacts give rise to linearly independent constraints on interparticle separations . to understand this behavior , we decompose the dynamical matrix @xmath4 for static packings of ellipsoidal particles into two important components : the stiffness @xmath5 and stress @xmath6 matrices . we find that the stiffness matrix possesses @xmath7 eigenmodes @xmath8 with zero eigenvalues even at finite compression , where @xmath9 is the number of particles . in addition , these modes @xmath8 are nearly eigenvectors of the dynamical matrix with eigenvalues that scale as @xmath0 , and thus finite compression stabilizes packings of ellipsoidal particles . at jamming onset , the harmonic response of static packings of ellipsoidal particles vanishes , and the total potential energy scales as @xmath10 for perturbations by amplitude @xmath11 along these ` quartic ' modes , @xmath8 . these findings illustrate the significant differences between static packings of spherical and ellipsoidal particles .

1201.3965

type ia supernovae ( sne ia ) are important cosmological probes that first revealed the accelerating expansion of the universe @xcite . the cosmological results rely on the normal sne ia whose brightness correlates with their light curve shapes and colors @xcite , allowing them to be used as standardizable candles . observations of similar but peculiar objects are useful for understanding the nature of the progenitor systems and the physics of the explosion , particularly how they might differ between objects . it is also important to understand objects which may be found in cosmological samples but do not follow the relationships between the luminosity and the light curve shape . the similar peak luminosities of sne ia suggested explosions of similar mass and energy . the widely - held theory is that a sn ia results from the thermonuclear disruption of a carbon - oxygen white dwarf ( co - wd ) as it approaches the chandrasekhar limit . this could be due to accretion from a non - degenerate companion ( also called the single degenerate scenario ; @xcite ) or the disruption of a wd companion ( also called the double degenerate scenario ; @xcite ) . the nature of an sn ia progenitor as a c - o wd ( and admittedly for a single case ) has only recently been confirmed by very early time observations of sn 2011fe @xcite . the wd mass at explosion might not need approach the chandrasekhar limit , as helium shell detonations can trigger a core detonation in sub - chandrasekhar mass progenitors @xcite . the nature of the companion remains unknown , and recent results suggest that sne ia may result from both single degenerate and double degnerate systems . early observations of many sne ia do not show the interaction expected @xcite if the sn explosion were to interact with a red giant ( rg ) companion @xcite . x - ray limits also rule out red giants due to the lack of shock interaction @xcite . pre - explosion , multi - wavelength , and extremely early observations of sn 2011fe rule out a rg @xcite and even a main sequence ( ms ) companion @xcite for that object . searches for the leftover companion in snr 0509 - 67.5 rule out a non - degenerate companion @xcite . on the other hand , high resolution spectroscopy of nearby sne has found a preference for blue shifted sodium absorption in about 20 - 25% of sne ia @xcite and even variable absorption @xcite suggestive of a local csm wind from a non - degenerate companion . ptf11kx observations showed signatures of a recurrent nova progenitor in a single degenerate system @xcite . thus , multiple channels might be required to create the explosions classified as sne ia . the idea that the accreting progenitor explodes as it approaches the chandrasekhar mass has been challenged by a class of sne that appear spectroscopically similar to sne ia but are overluminous for their light curve shape . detailed modeling of the light curves appears to require more than a chandrasekhar mass of ejected material . sn 2003fg was the first discovered @xcite with sne 2006gz @xcite , 2007if @xcite and 2009dc @xcite showing similarities . @xcite discovered five additional , similar objects in sn factory observations , though only one was conclusively above the chandrasekhar limit . association with this subclass is sometimes based on spectroscopic similarity to others of the class , to a high inferred luminosity , or to actually modeling the light curve and determining a high ejecta mass . variations exist amongst candidates of his subclass , which is not surprising given our limited understanding of their origin and relationship to normal sne ia . @xcite highlight the observational differences between sne 2003fg and 2006gz , two probable super - chandarasekhar mass candidates . the most common means of estimating the mass from sne ia comes from the application of `` arnett s law '' @xcite . at maximum light the luminosity output is approximately equal to the instantaneous rate of energy release from radioactive decay . thus the peak bolometric luminosity is proportional to the mass of @xmath0ni synthesized in the explosion . the @xmath0ni can also be estimated from the late light curve @xcite or nebular spectra @xcite . the total mass can be estimated based on energetics using the observed luminosities and expansion velocities and assumptions on the density profile ( e.g. @xcite ) . the mass can also be estimated by constructing models of various masses and explosion scenarios and comparing to the observed light curves @xcite and spectra @xcite . not all of the luminosity necessarily comes from radioactive decay . excess luminosity could also come from circumstellar interaction @xcite or result from asymmetric explosions viewed at a favorable angle @xcite . asymmetric explosions can not explain the brightest of sc sne , and spectropolarimetry of sn 2009dc implies no large scale asymmetries in the plane of the sky @xcite ) . @xcite find that the late time observations of sn 2006gz require less radioactive ni than suggested from peak optical observations , drawing into question the overluminous nature of the event . they suggest that the luminosity is overestimated due to an over - correction for extinction . sc sne are hot , high - energy explosions , so ultraviolet ( uv ) coverage is important to better measure the total luminosity and determine its origin , in particular whether it originates from shocks or simply a hot photosphere . the ultraviolet / optical telescope ( uvot ; @xcite ) on the swift satellite @xcite presents an excellent opportunity to obtain unique , early - time uv data . this paper will focus on three objects : sn 2009dc a well - studied member of the super - chandrasekhar mass sn class and sne 2011aa and 2012dn which share some characteristics . we will refer to these candidate super - chandrasekhar sne ia as sc sne below , though a firm mass determination will require more data and is beyond the scope of this work . comparisons will focus on the differences and similarities between sn 2009dc and the less studied sne 2011aa and 2012dn , and the differences of these three sc sne compared to other sne ia . in section [ obs ] we discuss these three sc sne and present uv / optical photometry and spectra from uvot . in section [ results ] we compare the colors , absolute magnitudes , spectra , and integrated luminosities , comparing sne 2011aa and 2012dn to 2009dc and the three to a larger sample of `` normal '' sne ia . in section [ discussion ] we discuss the results and summarize . sn 2009dc was discovered by @xcite on 2009 april 9.31 ( all dates ut ) . @xcite reported spectroscopic similarities to sc sne on april 22 . swift observations began april 25.5 . swift / uvot photometry has been published by @xcite and also referred to by @xcite . an epoch of uv grism spectroscopy was performed may 1.0 ( 4.9 days after the time of maximum light in the b - band ) . sn 2009dc has been extensively studied @xcite including theoretical modeling of the light curves @xcite and spectra @xcite . assuming that its luminosity is powered by radioactive decay , sn 2009dc likely had a @xmath0ni yield between 1.2 and 1.8 m@xmath1 depending on the assumed extinction ( though @xcite also calculate a @xmath0ni mass of 3.7 m@xmath1 for their largest plausible reddening ) . sn 2009dc exploded outside of ugc 10064 toward the disrupted companion ugc 10063 ( see @xcite and @xcite for further discussion of the host environment ) . the redshift of ugc 10064 is 0.021391 @xmath2 0.000070 @xcite . the foreground galactic extinction along the line of sight is a@xmath3=0.191 @xcite . sn 2011aa was discovered by @xcite on 2011 february 6.3 . it was also independently discovered by master on 2011 february 13.54 @xcite . from optical spectra taken february 8.9 it was spectroscopically identified as a young sn ia by @xcite who found a best match to the normal sn ia 1998aq a week before maximum light . observations with the swift spacecraft began on feb 11.6 and continued for 16 epochs of uv and optical photometry ( every other day around maximum light and then more spread out at later times ) . one epoch of spectroscopy with the uvot s uv grism was performed on february 28.0 ( 8.1 days after maximum light in the b - band ) , but overlap with a field star significantly contaminates the spectrum . @xcite found photometric similarities between sn 2011aa optical observations and sc sne ia models . sn 2011aa is located at the intersection of two galaxies designated ugc3906 at a redshift of 0.012355 + /- 0.000087 @xcite . the foreground galactic extinction along the line of sight is a@xmath3=0.078 @xcite . sn 2012dn was discovered by @xcite on 2012 july 8.5 . @xcite spectroscopically classified it as a sn ia before maximum light from a spectrum obtained july 10.2 . they noted strong cii absorption and similarities to three sne ia described as sc sne ia . @xcite also noticed similarities to sc sne ia spectra . swift observations began july 13.1 . one epoch of uv grism spectroscopy was obtained on july 22.6 ( 2.2 days before maximum light in the b band ) . we also use an optical spectrum obtained by the south african large telescope ( salt ) on july 24 from parrent et al . ( 2013 , in prep ) . the uvot and salt spectra were combined by normalizing to the same b - band magnitude and splicing together at 4750 . sn 2012dn is located in the galaxy eso 462-g016 at a redshift of 0.010187 @xmath2 0.000020 @xcite . the foreground galactic extinction along the line of sight is a@xmath3=0.167 @xcite . the swift / uvot observations used the following six broadband filters with the corresponding central wavelengths @xcite : uvw2 ( 1928 ) , uvm2 ( 2246 ) , uvw1 ( 2600 ) , u ( 3465 ) , b ( 4392 ) , and v ( 5468 ) . those filters are sometimes referred to as w2 , m2 , w1 , uu , bb , and vv , respectively . swift / uvot data were analyzed following the methods of @xcite and @xcite but incorporating the revised uv zeropoints and time - dependent sensitivity from @xcite . the photometry is given in table [ table_photometry ] . the light curves of the three sc sne are displayed in figure [ fig_lightcurves ] . the uvot data for sn 2009dc were originally published in @xcite and here we rereduce the data with the new zeropoints , sensitivity corrections , and subtraction of the underlying galaxy flux . the difference is small typically less than 0.05 mag . the photometry for sn 2011aa also includes galaxy subtraction , so the late time flattening in the uv filters appears to be real . sn 2012dn does not have galaxy template images , but the amount of galaxy contamination is likely small . the uvot b and v bands are similar to johnson @xmath4 and @xmath5 while the swift u - band is extends to much shorter wavelengths than johnson @xmath6 or sloan @xmath7 ( and does not suffer from atmospheric attenuation ) , of particular importance for sne such as these with different uv spectral shapes than normal sne . while we have obtained photometry in six bands , for simplicity we will focus on three filters with which to measure colors and absolute magnitudes . we use uvm2 for the mid - uv ( or muv ) , uvw1 for the near - uv ( or nuv ) , and the v - band for the optical . the uvot grism data was extracted and calibrated using the default parameters of the uvotpy package ( kuin et al . 2014 , in preparation ) . the nominal wavelength accuracy is 20 and the flux calibration is accurate to about 10% . individual exposures were extracted and the spectra combined using a weighted mean . for comparison , we use previously published photometry ( updated to the latest calibration as described above ) from spectroscopically normal sne ia with @xmath8@xmath9 1.4 with detections in all three uv filters @xcite . for sn 2011fe we use spectrophotometry of sn 2011fe using the spectra from @xcite due to the uvot data s optical saturation near peak @xcite . further comparisons are made with sne spectroscopically similar to sn 2002cx ( sne 2005hk and 2012z ) and sn 1991 t ( sne 2007s , 2007cq , and 2011dn ) . the photometry from 2011dn is presented here for the first time while sn 2012z will be presented in stritzinger et al . ( 2014 , in preparation ) and the others were previously published in @xcite . the light curves of sne 2009dc , 2011aa and 2012dn are shown in figure [ fig_lightcurves ] . swift observations of sn 2009dc began near maximum light so the curves monotonically fade , but all appear qualitatively very similar , including the crossing points of the different filters . the evolution of sn 2011aa , though , is much slower . we show in figure [ fig_11ay01ay ] that it has a similar decay rate in b and v as sn 2001ay , `` the most slowly declining type ia supernova '' @xcite . the uv light curves are also broader than those of normal sne ia , most of which have very similar post - maximum light curve shapes in the nuv @xcite . we parameterize the light curves of sne 2011aa and 2012dn by their peak magnitudes and by @xmath10m@xmath11 , the number of magnitudes they fade in the 15 days following maximum light in that same band . the peak is determined by stretching a template light curve to the data between 5 days before and 5 days after maximum light . @xmath10m@xmath11 is determined by stretching a template light curve to the data between 2 days before and 15 days after maximum light to the data and interpolating from the stretched template . the uv templates ( with uvw1 template also being used for u band due to its similar light curve shape ) come from sn 2011fe @xcite and b and v from mlcs2k2 @xcite . we note that for very broad sne such as some of these , the measurement of light curve parameters depend heavily on how the fitting and determination of the peak time is done . the light curve parameters are tabulated in table [ table_parameters ] . we also list the difference in time between when the sn peaks in the b band compared to the other filters . the difference in peak times between b and uvw1 are much larger than the normal sne analyzed by @xcite with a mean of 2.22 days and the largest being 3.7 . sn 2009dc is excluded since observations began near the optical peak , while the uv was already fading . the b - band light curves are found to peak at mjd 54947.1 ( 2009 april 26.1 ) , 55611.9 ( 2011 february 19.9 ) , and 56132.8 ( 2012 july 24.8 ) for sne 2009dc , 2011aa , and 2012dn , respectively . these values are used as the reference epochs for the light curves and spectra displayed . llrrrrr sn2012dn & uvw2 & 56121.1333 & 16.393 & 0.100 & 2.482 & 0.230 + sn2012dn & uvm2 & 56124.7118 & 15.857 & 0.089 & 2.495 & 0.204 + sn2012dn & uvw1 & 56121.1419 & 15.262 & 0.056 & 7.434 & 0.383 + sn2012dn & u & 56121.1303 & 14.324 & 0.031 & 40.397 & 1.152 + sn2012dn & b & 56121.1313 & 15.344 & 0.036 & 32.096 & 1.059 + sn2012dn & v & 56124.7077 & 14.720 & 0.048 & 18.529 & 0.822 + lrr m@xmath12(peak ) & 16.15 @xmath13 0.04 & 15.86 @xmath13 0.04 + @xmath10m@xmath11(w2 ) & 0.97 @xmath13 0.10 & 1.03 @xmath13 0.06 + t@xmath14(w2)-t@xmath14(b ) & -4.4 @xmath13 1.4 & -7.1 @xmath13 0.5 + m@xmath15(peak ) & 15.70 @xmath13 0.04 & 15.84 @xmath13 0.15 + @xmath10m@xmath11(m2 ) & 1.15 @xmath13 0.08 & 1.02 @xmath13 0.05 + t@xmath14(m2)-t@xmath14(b ) & -3.7 @xmath13 1.2 & -7.3 @xmath13 1.6 + m@xmath16(peak ) & 15.01 @xmath13 0.02 & 14.71 @xmath13 0.02 + @xmath10m@xmath11(w1 ) & 0.78 @xmath13 0.06 & 1.21 @xmath13 0.05 + t@xmath14(w1)-t@xmath14(b ) & -4.9 @xmath13 0.8 & -7.0 @xmath13 0.4 + m@xmath17(peak ) & 14.14 @xmath13 0.01 & 13.69 @xmath13 0.01 + @xmath10m@xmath11(u ) & 0.67 @xmath13 0.02 & 1.14 @xmath13 0.03 + t@xmath14(u)-t@xmath14(b ) & -4.3 @xmath13 0.5 & -5.0 @xmath13 0.3 + m@xmath18(peak ) & 14.80 @xmath13 0.01 & 14.38 @xmath13 0.07 + @xmath10m@xmath11(b ) & 0.59 @xmath13 0.07 & 1.08 @xmath13 0.03 + m@xmath19(peak ) & 14.73 @xmath13 0.02 & 14.36 @xmath13 0.10 + @xmath10m@xmath11(v ) & 0.30 @xmath13 0.07 & 0.44 @xmath13 0.04 + t@xmath14(v)-t@xmath14(b ) & 2.4 @xmath13 0.7 & 0.6 @xmath13 1.8 + figure [ fig_colors ] shows the color evolution in uvm2-uvw1 and uvw1-v of the three sc sne ia compared to spectroscopically normal swift sne with @xmath10m@xmath11(b ) @xmath201.4 . the left panel shows that the uvw1-v colors of normal sne evolve from red to blue , reaching a minimum color a few days before the optical maximum and then becoming redder again . the uvm2-uvw1 colors of sne ia have a large dispersion and tend to become slowly bluer . the range of normal sn colors is compared to our sc sample in the right panel . sn 2009dc , which was first observed near the optical maximum , is at the blue end of both colors but not extremely so . sn 2012dn has similar colors at similar epochs but was also observed at earlier epochs . at those pre - maximum epochs sn 2012dn was bluer than the normal sne . sn 2011aa had similar premaximum colors to sn 2012dn but did not redden as quickly as the others due to the slower light curve evolution shown above . @xcite suggest that the spread in the nuv colors of normal sne ia can be viewed as two separate subclasses . in addition to their bluer colors , the nuv - blue subclass also shares a spectroscopic trait with sc sne in showing cii in their optical spectra @xcite . sne 2003fg @xcite , 2006gz @xcite , 2007if @xcite , 2009dc @xcite , and 2012dn @xcite all noted cii , often very strong . two slow - decliners that are nt considered sc candidates , 2001ay and 2009ig , may have had weak cii features @xcite . while sn 2009dc is actually included in the nuv - blue subclass ( as the bluest member ) in @xcite , the early phase observations of sne 2011aa and 2012dn presented here show that the sc sne are much bluer than normal sne ia at earlier times . sc sne may not have the early red to blue evolution of normal sne ia , or it happens earlier than ten days before optical maximum . their slower evolution , on the other hand , might make the time of optical maximum a poor reference point for giving physical meaning to their behavior compared to normal sne ia . nevertheless , it is clear that the sc sne ia extend the diversity in the uv more than that already found @xcite . early uv observations appear to be a way of photometrically separating sc sne from normal sne . this could be quantified as the magnitude or timing of the minimum color ( i.e. the color at its bluest epoch ) or the time difference between maximum light in the uv and optical bands . two additional classes of sne also warrant further comparison . spectroscopic similarity to sn 1991 t was used as a follow up criterion to discover new sc candidates by @xcite . sne similar to sn 2002cx ( also called sne iax ; @xcite ) also show hot , highly - ionized photospheres . we are not making a physical connection between the groups , but they warrant further comparison because of how their similar physical conditions have a strong effect on their uv flux and because of possible confusion in spectroscopic classification @xcite . several examples of each are displayed in the right hand panel of figure [ fig_colors ] . similar to the sc sne , 1991t - like and 2002cx - like sne showed a monotonic reddening in uvw1-v from the onset of swift observations . with the exception of sn 2007s ( whose optical colors suggest reddening from the host galaxy ; @xcite ) , all could have been as blue ( in uvw1-v ) as the sc sne if they were observed early enough but the colors became redder at a much faster rate than the sc sne . in the uvm2-uvw1 color , one of each class had a comparable color . the 91t - like sn 2007cq was classified by @xcite as a muv - blue , because it follows the nuv - red group in uvw1-v but is relatively blue in uvm2-uvw1 . thus multi - epoch multi - wavelength uv photometry reveals complicated similarities and differences amongst sne of different subclasses and within the same subclass . further observations of members of these classes , including uv spectroscopy and even earlier uv photometry , will help explain the physical origins of the uv flux . since one common characteristic of the strongest sc sn candidates is their high luminosity , we now examine the absolute magnitudes of sne 2009dc , 2011aa , and 2012dn . as in @xcite , most distance moduli are computed from the host galaxy recessional velocity , corrected for local velocity flows @xcite , and a hubble constant of 72 km / s / mpc @xcite . distances from cephieds , the tully - fisher relation , or surface brightness fluctuations are used when available , as listed in @xcite and @xcite . for sne 2009dc , 2011aa , and 2012dn the adopted hubble flow distances are 34.94 @xmath13 0.16 , 33.894 @xmath13 0.18 , and 33.324 @xmath13 0.20 , respectively . the sne in this sample are at relatively low redshifts ( mostly with recessional velocities less than 6000 km / s ) , so thermal velocities of the galaxies can add a significant dispersion to distances calculated assuming a hubble flow . thus the scatter of absolute magnitudes in the optical is much larger than found for larger samples of sne ( see @xcite for more details on this sample ) . @xcite suggest that the extinction and thus the luminosity of sn 2006gz could be overestimated . to avoid such overcorrections , we do not correct any of the sne ia for host extinction . we do correct for line sight extinction in the milky way ( mw ) by converting the @xcite v - band extinction from ned to an e(b - v ) reddening ( by dividing by 3.1 ) and then multiplying by the extinction coefficients calculated for the uv - optical spectrum of sn 1992a @xcite . figure [ fig_abscurves ] shows the absolute magnitudes in the optical ( v band ) , nuv ( uvw1 ) , and muv ( uvm2 ) . while sc sne ia are brighter than most ( but not all ; see below ) in the optical , they are almost one magnitude brighter than the brightest in the nuv and about two magnitudes brighter in the muv . the light curve shapes are similar in shape , but sn 2012dn fades a little faster while sne 2009dc and 2011aa remain brighter than normal sne ia for a month after peak . for comparison , sne similar to sne 1991 t and 2002cx are also plotted . sc and 1991t - like sne ia are at the bright end of the normal distribution in the optical and distinctly brighter in the uv . the 2002cx - like sne are at the faint end of the normal distribution in the optical but become relatively brighter ( and peak much earlier ) at shorter wavelengths . figure [ fig_dmb_abs ] shows the peak absolute magnitudes compared to @xmath8 for the normal and sc sne ia . the sc sne candidates all have broad optical light curves ( i.e. low values of @xmath8 ) but not uniquely broad . in the uv , all three are noticeably brighter . the peak optical luminosities of sne 2011aa and 2012dn are comparable to those of the normal sne ia . sn 2009dc is significantly brighter in the optical . figure [ fig_dmb_vabs ] zooms in on the absolute v - band magnitudes of the broad sne , including ground based observations of other broad sne . sn 2009dc lies clearly amongst the other sc sne while sne 2011aa and 2012dn have v - band absolute magnitudes consistent with the other sne ia . the absolute magnitudes , especially in the uv , are very sensitive to the extinction . one concern for the analysis of sn 2006gz based on the luminosity is that it is very sensitive to the assumed extinction sn 2006gz could be fainter if there is less host extinction or if the extinction coefficient is smaller . in multiple analyses sn 2009dc is bright even if no host galaxy extinction is assumed but could be even brighter . in this plot we have assumed no host dust extinction , yet they could be extinguished by dust in the host galaxy , and thus intrinsically brighter . the extremely blue colors would suggest that the host reddening is minimal , but the intrinsic colors of these objects are not actually known . a larger sample is needed to determine observationally what the range of colors might be and how blue the unreddened color could be . the degeneracy between reddening and luminosity means that the sne 2011aa and 2012dn could have low reddening and be optically underluminous compared to sn 2009dc . alternatively , significant reddening would mean they are intrinsically bluer and even more overluminous in the uv . either way , these three sne show similarities but are not identical . figure [ fig_uvotspectra ] shows the uvot grism spectra of sne 2009dc , 2011aa , and 2012dn . the signal to noise ratio is much lower than usual for optical sn spectroscopy , but comparable to other swift / uvot spectra @xcite . sn 2011aa was contaminated by an overlapping stellar spectrum , but sne 2009dc and 2012dn exhibit similar continuum shapes and features . absorption from mgii appears in the nuv , while shortward of that the spectrum is blanketed by overlapping lines of iron - peak elements . the bottom panel of figure [ fig_uvotspectra ] compares the combined uvot - salt spectrum of sn2012dn to an hst uv / optical spectrum of sn 2011fe taken 2011 september 7 ( 2.9 days before maximum light in the b band ; @xcite ) and a swift / uvot grism spectrum of the broad but normal sn 2009ig @xcite taken 2009 september 3.7 ( 2.3 days before maximum light in the b band ) . all three spectra have been normalized to the same b - band magnitude to compare the relative flux in the uv . sn 2011fe , classified as a nuv - blue sn @xcite , and sn 1009ig are not dissimilar to sn 2012dn above 4000 . the caii h&k lines of sn 2009ig are very broad and deep @xcite , reducing its nuv flux . in the muv , sne 2009ig and 2011fe have a much lower flux and a smoother pseudocontinuum . while we do not want to overinterpret the grism spectrum by studying individual features at this time , the strong undulations in the muv of sne 2009dc and 2012dn suggest a lower opacity ( and thus gaps in the line blanketing ) , rather than a hot blackbody from a shock interaction , as the source of the increased uv luminosity . to determine how much flux is observed , we need to convert from the observed magnitudes . flux conversion factors are very spectrum dependent in the uv @xcite , differing by source type and phase for objects ( like sne ) with time - variable spectra . simpler seds based on the photometry have a problem reproducing the multi - filter photometry . to estimate the flux contributions of different wavelength regions , we use the combined uv / optical spectrum we have for sn 2012dn and warp it to match the photometry as follows . first , the spectrum was smoothed with a running average over 10 . the spectrum is extrapolated beyond the uvot filter range using the mean of the spectrum in the shortest 50 ( between 2200 and 2250 ) . then the whole spectrum was scaled by a constant value to match the observed b - band magnitude at that epoch . a warping function is created from linear segments from 1500 to 8100 ( just beyond the uvot bounds ) with pivot points near where the uvot filter curves intersect each other ( after normalizing by the integral of the effective area of the curve to deweight the broader filters ) . these points are at 2030 , 2460 , 3050 , 3870 , and 4960 . these seven points are iteratively adjusted to minimize the magnitude differences between the observed photometry and that of the warped spectrum . this method better reproduces the spectral shape than converting the observed photometry to independent flux density points and simply connecting the dots ( brown et al . 2014 , in preparation ) . the sn2012dn spectrum is used for sne 2012dn and 2011aa . for sn 2009dc we combine the uvot grism spectrum with a comparable epoch spectrum from @xcite obtained from the wiserep database @xcite . from these warped spectra we calculate the amount of flux coming from the full uvot range ( 1600 - 6000 ) and three regions muv ( 1600 - 2800 ) , nuv ( 2800 - 4000 ) , and optical ( 4000 - 6000 ) at each epoch with photometry in all six uvot filters . the top panel of figure [ fig_flux ] shows the integrated flux of the three sc sne compared to a direct integration of the uv / optical spectra of @xcite for the normal nuv - blue sn 2011fe . despite their bright uv luminosity , sne 2011aa and 2012dn which have about the same integrated luminosity as the normal sn 2011fe . sn 2009dc is about twice as bright . the evolution of the muv and nuv flux fractions ( compared to the total 1600 - 6000 flux ) are displayed in the middle and lower panel of figure [ fig_flux ] . the nuv fractions for the sc sne and sn2011fe all peak between 44 and 48% . sne 2012dn and 2011aa have 10 and 9% , respectively , of their flux in the muv in their earliest observation . by b - band maximum light , the fraction for sn 2012dn has dropped down to 4% , only modestly above sn 2011fe . sn 2011aa has a significantly larger fraction of its flux ( compared to the others ) for at least ten days after the b - band maximum . as a simple check on the effect of the template spectrum we perform the same color matching for all three sne using the sne 2009dc and 2012dn spectra , uv / optical spectra of sn 2011fe from @xcite near maximum light and 24 days after maximum , the near maximum spectrum of the sn ia 1992a from @xcite and a spectrum of vega from @xcite . the total flux changes by only a few percent and the muv fraction changes by up to 15% ( in a relative sense ) , indicating that the luminosity measurement is dependent on , but not dominated by , the spectral template inputed . the integrated luminosity curves allow us to compare in a relative sense the bolometric luminosity of sne 2011aa and 2012dn to the well - studied sn 2009dc , and thus the inferred @xmath0ni mass . since the colors of sne 2011aa and 2012dn are similar to or bluer in color than sn 2009dc , we assume for now that they do not suffer significantly more dust extinction than sn 2009dc and that the same fractions of the bolometric luminosities lie outside of our 1600 - 6000 range for all three sne . we also assume the rise time is similar for the three sne and that the ratio of the bolometric luminosity to the radioactive luminosity is the same . under these ( many ) assumptions , the mass of @xmath0ni is proportional to the integrated luminosity l@xmath21 . for a range of host galaxy reddening values , @xcite determined a @xmath0ni between 1.2 and 3.7 @xmath22 , with a most likely value of 1.7 @xmath2 0.4 @xmath22 . since sne 2011aa and 2012dn have about half the integrated luminosity , their @xmath0ni would likely be around 0.9 @xmath22 . while smaller than sn 2009dc , it is close to the the amount of @xmath0ni produced ( 0.92 @xmath22 ) in a chandrasekhar - mass detonation where the entire mass is converted into iron group elements @xcite . one could also make the comparison with sn 2011fe . because of its smaller luminosity and shorter rise time ( 16.58 days for sn 2011fe compared to at least 21.1 days for sn 2009dc ) , the @xmath0ni mass is estimated to be 0.53 @xmath22 @xcite . if sne 2011aa and 2012dn were assumed to have similar rise times and radiative efficiencies as sn 2011fe , the @xmath0ni masses would also be similar . so the estimate relies in part on assumptions about an unobserved property the rise time . the observed properties of sne 2011aa and 2012dn are more similar to sn 2009dc than sn 2011fe , but this highlights the need for a better understanding these objects and limiting the assumptions that must be made . further analysis is needed to determine more accurately the @xmath0ni mass required and what progenitor / explosion scenarios might result in these observables . one suggestion for the increased luminosity in sc sne is shock interaction @xcite . @xcite suggest uv observations as a means to probe the influence of shock interactions on the early luminosity . @xcite performed numerical calculations of the spectra and simulated uvot light curves resulting from a double - degnerate sn ia exploding within a shell of unaccreted material . while our candidate sc sne ia have peak luminosities comparable to those studied by @xcite , the light curves shapes are much different . the light curve shapes can vary based on the amount and spatial distribution of the surrounding material , but the smoothness of the uv light curves and their qualitative similarity to the optical light curves suggest a photospheric origin . a photospheric origin for the emission is supported by the uv spectra of sne 2009dc and 2012dn , which show stronger features in the muv ( below 2700 ) than seen in normal sne ia , while the flux from a hot shock would be relatively smooth and would dilute the photospheric features @xcite . on the other hand , the optical features are also much stronger than for sn 2007if , for which the top lighting of a shock was invoked as one explanation for its diluted features and high luminosity @xcite . the uv spectra of sne 2009dc and 2012dn do not allow a smooth blackbody source for the excess flux . such a spectrum might be expected from a high temperature shock with a hydrogen - rich circumstellar medium , as was used to explain the diluted features of sn 2002ic ( @xcite , see also @xcite ) . a structured spectrum with emission and absorption , due to reprocessing of the shock emission or originating from a different composition , can not be excluded . @xcite found adding a spectrum of the ibn sn 2006jc to their theoretical sn ia spectrum gave reasonable matches to the observed spectra of sn 2009dc . higher quality uv spectra of ibn and sc sne are needed to perform similar tests in the uv . nevertheless , photometric observations may already contain enough information to further constrain photometric ( e.g. @xcite ) or spectroscopic modeling ( e.g. @xcite ) . while the optical light curves of sne 2011aa and 2012dn are not dissimilar to normal sne ia ( though extremely broad in the case of sn 2011aa ) , the nuv - optical and especially the muv - nuv ( or muv - optical ) colors are markedly different . the rest - frame uv also peaks earlier for sc than normal sne ia . early rest - frame uv photometry might allow optically overluminous sne such as sn 2009dc to be excluded from cosmological analysis . @xcite estimate the rate of sc sne to be a few percent of all sne ia locally , but a bias could result from an evolutionary shift @xcite if these are more common in the early universe than they are locally . @xcite show that the relative fractions of nuv - blue and nuv - red normal sne ia change with redshift . the origin of the uv diversity amongst normal and sc candidate sne ia may point to ways to reduce the dispersion at longer wavelengths and understand potential biases in sn ia standardization at different epochs in the history of the universe . bolometric light curve comparisons between models and observations serve as an important diagnostic of allowed models and parameters . the creation of bolometric light curves , however , especially the treatment of missing wavelength ranges , varies greatly . sometimes the nuv ( or at least the ground based u band ) is included , and the muv may or may not be included . often the uv portion of the flux is considered to be negligible ( a reasonable assumption in some cases ) . if it is included , it is often set at a constant percentage of the flux . as shown here , there is also a lot of variation in the nuv and muv flux fractions between various sne ia , and the fractions evolve quite significantly with time . the data given here will allow the bolometric light curves of these objects to be more accurately determined . for example , the falling uv fraction means that inclusion of the uv flux will broaden the pre - maximum rise of the bolometric flux . this could lead to a longer implied rise time if fit with a light curve template . this longer rise time may not be accurate , however , if the stretched light curve template did not include the uv in its construction . @xcite use multi - wavelength modeling to show the difference between a bvri , uv - optical - ir ( uvoir ) , and true bolometric light curve . the distinction between these is important . uv data will allow more constraints on the modelling . while we have pushed the knowledge of the uv behavior for sc sne ia @xmath23 days earlier , the very earliest epochs would also be important for looking for the effects of shock interaction with a non - degenerate companion @xcite or differences in the uv - optical flux evolution at the earliest times @xcite . as the uv - optical colors are still bluest at the first epochs observed , the bolometric contribution before then may be larger still and are in any case uncertain . higher quality uv spectra at the earliest possible epochs will better probe the mechanism responsible for the excess uv emission and how to account for it in mass determinations . in summary , we have presented uv / optical photometry and spectroscopy for three sne ia , 2009dc , 2011aa and 2012dn , which have been suggested as candidate super - chandrasekhar mass sne ia . while their optical properties are not dissimilar to normal sne ia , they are significantly bluer and more luminous in the uv than normal sne ia , with muv luminosities about a factor of @xmath24 higher . uv spectra of sne 2009dc and 2012dn feature structure not expected for shock interaction , suggesting a photospheric origin of the excess uv luminosity . the uv is shown to contribute significantly ( but still smaller than the optical ) to the bolometric luminosity , especially at early times . the integrated luminosities of sne 2011aa and 2012dn are much lower than 2009dc , however . this suggests a larger diversity in the class , if they are indeed in the same class , when considering uv and optical photometric and spectroscopic characteristics . a more detailed study of these sne is required to determine if they were above the chandrasekhar mass . is supported by the mitchell postdoctoral fellowship and nsf grant ast-0708873 . p.a.m acknowledges support from nasa adap grant nnx10ad58 g . we are grateful to j. vinko and j. parrent for sharing the spectra of sn 2012dn . we are also grateful to h. marion and x. wang for looking at optical spectra of sn 2011aa . this work made use of public data in the _ swift _ data archive and the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , california institute of technology , under contract with nasa . , a. a. , landsman , w. , holland , s. t. , et al . 2011 , in american institute of physics conference series , vol . 1358 , american institute of physics conference series , ed . j. e. mcenery , j. l. racusin , & n. gehrels , 373376 , g. , de vaucouleurs , a. , corwin , jr . , h. g. , et al . 1991 , third reference catalogue of bright galaxies . volume i : explanations and references . volume ii : data for galaxies between 0@xmath25 and 12@xmath25 . volume iii : data for galaxies between 12@xmath25 and 24@xmath25 .

among type ia supernovae ( sne ia ) exist a class of overluminous objects whose ejecta mass is inferred to be larger than the canonical chandrasekhar mass . we present and discuss the uv / optical photometric light curves , colors , absolute magnitudes , and spectra of three candidate super - chandrasekhar mass sne2009dc , 2011aa , and 2012dn observed with the _ swift _ ultraviolet / optical telescope . the light curves are at the broad end for sne ia , with the light curves of sn 2011aa being amongst the broadest ever observed . we find all three to have very blue colors which may provide a means of excluding these overluminous sne from cosmological analysis , though there is some overlap with the bluest of `` normal '' sne ia . all three are overluminous in their uv absolute magnitudes compared to normal and broad sne ia , but sne 2011aa and 2012dn are not optically overluminous compared to normal sne ia . the integrated luminosity curves of sne 2011aa and 2012dn in the uvot range ( 1600 - 6000 ) are only half as bright as sn 2009dc , implying a smaller @xmath0ni yield . while not enough to strongly affect the bolometric flux , the early time mid - uv flux makes a significant contribution at early times . the strong spectral features in the mid - uv spectra of sne 2009dc and 2012dn suggest a higher temperature and lower opacity to be the cause of the uv excess rather than a hot , smooth blackbody from shock interaction . further work is needed to determine the ejecta and @xmath0ni masses of sne 2011aa and 2012dn and fully explain their high uv luminosities .

1404.0650

let @xmath2 be the cubic surface defined by @xmath3 then @xmath4 is a singular del pezzo surface with a unique singularity @xmath5 of type @xmath1 and three lines , each of which is defined over @xmath6 . on the @xmath1 cubic surface.,width=491 ] let @xmath7 be the zariski open subset formed by deleting the lines from @xmath4 . our principal object of study in this paper is the cardinality @xmath8 for any @xmath9 . here @xmath10 is the usual height on @xmath0 , in which @xmath11 is defined as @xmath12 , provided that the point @xmath13 is represented by integral coordinates @xmath14 that are relatively coprime . in figure [ fig : d5 ] we have plotted an affine model of @xmath4 , together with all of the rational points of low height that it contains . the following is our principal result . we have @xmath15 where the leading constant is @xmath16 with @xmath17 it is straightforward to check that the surface @xmath4 is neither toric nor an equivariant compactification of @xmath18 . thus this result does not follow from the work of tschinkel and his collaborators @xcite . our theorem confirms the conjecture of manin @xcite since the picard group of the minimal desingularisation @xmath19 of the split del pezzo surface @xmath4 has rank @xmath20 . furthermore , the leading constant @xmath21 coincides with peyre s prediction @xcite . to check this we begin by observing that @xmath22 by ( * ? ? ? * theorem 4 ) and ( * ? ? ? * theorem 1.3 ) , where @xmath23 is a split smooth cubic surface and @xmath24 is the order of the weyl group of the root system @xmath1 . next one easily verifies that the constant @xmath25 in the theorem is the real density , which is computed by writing @xmath26 as a function of @xmath27 and using the leray form @xmath28 . finally , it is straightforward to compute the @xmath29-adic densities as being equal to @xmath30 . our work is the latest in a sequence of attacks upon the manin conjecture for del pezzo surfaces , a comprehensive survey of which can be found in @xcite . a number of authors have established the conjecture for the surface @xmath31 which has singularity type @xmath32 . the sharpest unconditional result available is due to la bretche @xcite . furthermore , in joint work with la bretche @xcite , the authors have recently resolved the conjecture for the surface @xmath33 which has singularity type @xmath34 . our main result signifies only the third example of a cubic surface for which the manin conjecture has been resolved . the proof of the theorem draws upon the expanding store of technical machinery that has been developed to study the growth rate of rational points on singular del pezzo surfaces . in particular , we will take advantage of the estimates involving exponential sums that featured in @xcite . in the latter setting these tools were required to get an asymptotic formula for the relevant counting function with error term of the shape @xmath35 . however , in their present form , they are not even enough to establish an asymptotic formula in the @xmath1 setting . instead we will need to revisit the proofs of these results in order to sharpen the estimates to an extent that they can be used to establish the theorem . in addition to these refined estimates , we will often be in a position to abbreviate our argument by taking advantage of @xcite , where several useful auxiliary results are framed in a more general context . in keeping with current thinking on the arithmetic of split del pezzo surfaces , the proof of our theorem relies on passing to a universal torsor , which in the present setting is an open subset of the hypersurface @xmath36 embedded in @xmath37 $ ] . furthermore , as with most proofs of the manin conjecture for singular del pezzo surfaces of low degree , the shape of the cone of effective divisors of the corresponding minimal desingularisation plays an important role in our work . for the surfaces treated in @xcite , @xcite , @xcite , the fact that the effective cone is simplicial streamlines the proofs considerably . for the surface studied in @xcite , this was not the case , but it was nonetheless possible to exploit the fact that the dual of the effective cone is the difference of two simplicial cones . for the cubic surface , the dual of the effective cone is again the difference of two simplicial cones . however , we choose to ignore this fact and rely on a more general strategy instead . while working on this paper the first author was supported by epsrc grant number ` ep / e053262/1 ` . the second author was partially supported by a feodor lynen research fellowship of the alexander von humboldt foundation . the authors are grateful to the referee for a number of useful comments that have improved the exposition of this paper . define the multiplicative arithmetic functions @xmath38 for any @xmath39 , where @xmath40 denotes the number of distinct prime factors of @xmath41 . these functions will feature quite heavily in our work and we will need to know the average order of the latter . [ lem : sum_h_k ] for any @xmath42 we have @xmath43 let @xmath42 be given and let @xmath44 . then we have @xmath45}2^{\omega(u)}\\ & \ll_{\varepsilon}q\log q \sum_{d_1,\ldots , d_k=1}^{\infty}\frac{(d_1\cdots d_k)^{{\varepsilon}-1/2}}{[d_1,\ldots , d_k]},\end{aligned}\ ] ] where @xmath46 $ ] denotes the least common multiple of @xmath47 . we easily check that the final sum is absolutely convergent by considering the corresponding euler product , which has local factors of the shape @xmath48 . given integers @xmath49 , with @xmath50 , we will be led to consider the quadratic exponential sum @xmath51 our study of this should be compared with the corresponding sum studied in ( * ? ? ? * eq . ( 3.1 ) ) , involving instead a cubic phase @xmath52 . in ( * ? ? * lemma 4 ) an upper bound of the shape @xmath53 is established for the cubic sum . the following result shows that we can do better in the quadratic setting . [ lem : lemma4 ] for any @xmath54 with @xmath55 , we have @xmath56 writing @xmath57 in the second step , we find that @xmath58 the inner sum is @xmath41 if @xmath59 and @xmath60 otherwise . let @xmath61 and write @xmath62 , @xmath63 with @xmath64 . then @xmath65 and the result follows . our next results concern the function @xmath66 , where @xmath67 is the fractional part of @xmath68 . the following estimate improves upon ( * ? ? ? * lemma 5 ) . [ lem : sum_mod_q ] for any @xmath69 , @xmath70 , @xmath71 with @xmath72 , we have @xmath73 let @xmath74 denote the sum that is to be estimated . by mbius inversion it follows that @xmath75 we claim that @xmath76 for any @xmath69 , @xmath70 and @xmath71 with @xmath72 . under this assumption , it therefore follows that @xmath77 this is satisfactory for the lemma , since @xmath78 . to establish we follow the proof of ( * lemma 4 ) , finding that @xmath79 where @xmath80 rather than applying weyl s inequality as in ( * ? ? ? * lemma 4 ) , we simply break into @xmath81 residue classes modulo @xmath82 and apply lemma [ lem : lemma4 ] to deduce that @xmath83 now @xmath84 for any @xmath85 . hence @xmath86 which thereby concludes the proof of . for positive integers @xmath87 , we define the function @xmath88 we combine lemma [ lem : sum_mod_q ] with the proof of ( * ? ? ? * lemma 1 ) to obtain the following result . [ lem : sum_mod_q_2 ] let @xmath89 and @xmath90 . we have @xmath91 in the proof of ( * ? ? ? * lemma 1 ) , @xmath92 is estimated as @xmath93 , for given @xmath94 coprime to @xmath41 . using ( * ? ? ? * lemma 7 ) , we make this precise as @xmath95 where @xmath96 is chosen such that @xmath97 . our task is to compute @xmath98 for the main term , we may extend the summation over @xmath94 to all positive integers , since @xmath99 as in ( * ? ? ? * lemma 1 ) , we see that the sum over @xmath100 is @xmath101 , with @xmath102 summing this over @xmath103 , we get @xmath104 . it is easy to see that @xmath105 agrees with the leading constant in the statement of the lemma . for the error term , we exchange the summations over @xmath94 and @xmath103 . applying lemma [ lem : sum_mod_q ] , we obtain the contribution @xmath106 with @xmath107 . this completes the proof of the lemma . given @xmath108 such that @xmath50 and a real - valued function @xmath109 defined on an interval @xmath110 , let @xmath111 it is interesting to compare this sum with the sort of sums that featured in our corresponding investigation of the @xmath34 cubic surface . the sole difference between ( * ? ? ? * eq . ( 4.1 ) ) and @xmath112 is that the argument involves @xmath113 , rather than @xmath114 . we will be interested in studying @xmath112 when @xmath115 . here , if @xmath116 $ ] and @xmath117 , then @xmath118 is defined to be the set of real - valued differentiable functions @xmath109 , such that @xmath119 is monotonic and of constant sign on @xmath120 , with @xmath121 . it will be convenient to define @xmath122 we will need a version of ( * ? ? ? * lemma 10 ) , in which the factor @xmath123 is made more explicit . this is achieved in the following result . [ lem : lemma10 ] let @xmath124 . assume @xmath125 and @xmath115 . for any @xmath44 , we have @xmath126 where @xmath127 is the divisor function . in comparing this with ( * ? ? ? * lemma 10 ) , one sees that the first and third term in both results share the same approximate order of magnitude . however , the middle term is improved from @xmath128 to @xmath129 . this saving is crucial in our work . it arises from the fact that the current set - up leads us to estimate the quadratic exponential sums with @xmath130 , rather than the corresponding cubic sums with phase @xmath52 and @xmath131 . in the former case we are dealing with linear exponential sums , for which we have very good control , and in the latter case we only have the bound @xmath132 available . let @xmath133 . replacing the bound @xmath134 by @xmath135 in the application of vaaler s trigonometric formula in the proof of ( * ? ? ? * lemma 10 ) , we obtain @xmath136 for any @xmath137 , where @xmath138 as in ( * ? ? ? * lemma 10 ) , we rewrite this as @xmath139 with @xmath140 and @xmath141 where @xmath142 is the multiplicative inverse of @xmath143 modulo @xmath41 . since @xmath144 , we have ( with @xmath145 ) @xmath146 write each @xmath147 modulo @xmath41 uniquely as @xmath148 with @xmath149 and @xmath150 . then @xmath151 with @xmath152 , just as in ( * ? ? ? * lemma 10 ) . therefore , @xmath153 for the contribution from the case @xmath154 , note that @xmath155 . we have @xmath156 trivially , and @xmath157 the inner sum is @xmath158 if @xmath159 ( which is possible only in the case @xmath160 since @xmath161 ) and @xmath60 otherwise . thus @xmath162 , whence the total contribution to @xmath163 from the case @xmath154 is @xmath164 where @xmath165 is the sum of divisors function . for the total contribution to @xmath163 from the case @xmath166 , we note that @xmath167 by ( * ? ? ? * lemma 5 ) for @xmath115 . also @xmath168 where @xmath169 . by lemma [ lem : lemma4 ] , @xmath170 the contribution from the case @xmath171 is therefore @xmath172 plugging the contribution from @xmath173 and @xmath166 to @xmath163 into @xmath112 , we deduce , for any @xmath137 , that @xmath112 is @xmath174 observing that @xmath175 we therefore deduce that @xmath176 let @xmath177 if @xmath137 , we may use this @xmath10 in the estimate above , together with @xmath178 , in order to obtain the lemma . if @xmath179 , so that @xmath180 , we deduce from the trivial estimate @xmath181 that the lemma holds in this case too . let @xmath4 be the @xmath1 cubic surface , let @xmath182 be the open subset formed by deleting the lines from @xmath4 and let @xmath19 be the minimal desingularisation of @xmath4 . in this section we will establish an explicit bijection between @xmath183 and the integral points on the universal torsor above @xmath19 , subject to a number of coprimality conditions . for this we will follow the strategy explained in @xcite . to establish the bijection we will introduce new variables @xmath184 and @xmath185 . it will be convenient to henceforth write @xmath186 and @xmath187 for any @xmath188 . let us recall some information concerning the geometry of @xmath4 from ( * ? ? ? * section 8) . blowing up the singularity @xmath5 on @xmath4 results in the exceptional divisors @xmath189 in a @xmath1-configuration on the minimal desingularisation @xmath190 . let @xmath191 resp . @xmath192 on @xmath19 be the strict transforms under @xmath193 of the three lines @xmath194 , @xmath195 , @xmath196 resp . the curves @xmath197 and @xmath198 on @xmath4 . the extended dynkin diagram in figure [ fig : dynkin ] is the dual graph of the configuration of the curves @xmath199 on @xmath19 . @xmath200 \ar@{-}[dr ] \ar@{-}[dd ] & & e_6 \ar@{-}[rr ] & & { * + < 10pt>[o][f]{e_2 } } \ar@{-}[dr]\\ & e_8 \ar@{-}[r ] & e_7 \ar@{-}[r ] & { * + < 10pt>[o][f]{e_5 } } \ar@{-}[r ] & { * + < 10pt>[o][f]{e_4 } } \ar@{-}[r ] & { * + < 10pt>[o][f]{e_1}}\\ a_1 \ar@{-}[rrrr ] \ar@{-}[ur ] & & & & { * + < 10pt>[o][f]{e_3 } } \ar@{-}[ur]}\ ] ] by ( * ? ? ? * section 8) , non - zero global sections @xmath201 corresponding to @xmath199 form a generating set of the cox ring of @xmath19 . the ideal of relations in @xmath202 is generated by @xmath203 . we express the sections @xmath204 , for @xmath205 , of the anticanonical class @xmath206 in terms of the generators of @xmath202 as follows : @xmath207 the general strategy of @xcite suggests that @xmath183 should be parametrised by certain integral points on the variety @xmath208 . this is confirmed in the the following result . [ lem : bijection ] we have @xmath209 where @xmath210 is the set of @xmath211 such that holds , with @xmath212 and @xmath213 the coprimality conditions in are achieved by taking @xmath214 and @xmath215 to be coprime if and only if the divisors @xmath216 and @xmath217 are not adjacent in the diagram . the reader is invited to consider the correspondence between * the variables of the parametrisation and the generators of @xmath202 , * the torsor equation ( [ eq : torsor ] ) and the relation in @xmath202 , * the height conditions ( [ eq : height ] ) and the expressions of @xmath204 in terms of the generators of @xmath202 , * the coprimality conditions ( [ eq : cpa2]) ( [ eq : cpe ] ) and the configuration of the curves associated to the generators of @xmath202 encoded in figure [ fig : dynkin ] . the proof of lemma [ lem : bijection ] is elementary , but modelled according to the geometry of @xmath4 . the following additional geometric information is relevant . contracting @xmath218 in this order leads to a map @xmath219 that is the blow - up of six points in the projective plane . we may choose @xmath220 as the coordinate lines in @xmath221 . then @xmath222 is the quadric @xmath223 . the morphisms @xmath224 and the projection @xmath225{cccc } \phi_2 : & s & { \dashrightarrow } & { { { \mathbb{p}}^2}},\\ & { \mathbf{x } } & \mapsto & ( x_0:x_1:x_2 ) \end{array}\ ] ] from the singularity @xmath5 , form a commutative diagram of rational maps between @xmath226 and @xmath227 . the inverse map of @xmath228 is @xmath225{cccc } \phi_3 : & { { { \mathbb{p}}^2}}&{\dashrightarrow}&s,\\ & ( { \eta}_4':{\alpha}_1':{\eta}_8')&\mapsto&({\eta}_4'^3:{\eta}_4'^2{\alpha}_1':{\eta}_4'^2{\eta}_8':{\eta}_8'{\alpha}_2 ' ) \end{array}\ ] ] where @xmath229 . the maps @xmath230 give a bijection between the complement @xmath7 of the lines on @xmath4 and @xmath231 , and furthermore , induces a bijection between @xmath183 and the integral points @xmath232 motivated by the way the curves @xmath233 occur in @xmath234 as the blow - ups of intersection points of @xmath235 , one introduces the following further variables @xmath225{lll } { \eta}_5=\gcd({\eta}_4,{\eta}_8),&{\eta}_7=\gcd({\eta}_5,{\eta}_8 ) , & { \eta}_3=\gcd({\eta}_4 , { \alpha}_1 , { \alpha}_2),\\ { \eta}_1=\gcd({\eta}_3,{\eta}_4,{\alpha}_2),&{\eta}_2=\gcd({\eta}_1,{\alpha}_2)&{\eta}_6=\gcd({\eta}_2,{\alpha}_2 ) . \end{array}\ ] ] although we omit the details here , it is now straightforward to derive the bijection described in the statement of lemma [ lem : bijection ] using elementary number theory . in analysing the height conditions apparent in we will meet a number of real - valued functions , whose size it will be crucial to understand . we begin with the observation that is equivalent to @xmath236 , where @xmath237 in what follows we will need to work with the regions @xmath238 in keeping with the philosophy of @xcite , the definitions of these regions is dictated by the polytope whose volume is defined to be the constant @xmath239 , as computed using an alternative method in the introduction . in fact one has @xmath240 to which @xmath241 is closely related . perhaps a few more words are in order concerning the role of the cone of effective divisors in our work . the parametrisation of @xmath183 in lemma [ lem : bijection ] suggests that @xmath242 should be comparable to the volume of @xmath243 . on the other hand , the factors @xmath239 and @xmath25 of the conjectured leading constant in our theorem suggest the appearance of @xmath244 instead . the latter is constructed from @xmath241 , which comes from the dual of the effective cone , and from @xmath245 , which is obtained from the region whose volume is @xmath25 . at some point we will therefore need to make a transition from @xmath243 to @xmath244 . rather than distributing this procedure over the entire proof , as in our previous investigation @xcite , we will save this transition until lemma [ lem : final_step ] , where it signifies the final step in our argument . we are now ready to record the various integrals that will feature in our work , together with some basic estimates for them . all of the bounds are simple enough to deduce in themselves , but readily follow from applications of ( * ? ? ? * lemma 5.1 ) . bearing this in mind , we have @xmath246 and @xmath247 and finally @xmath248 we now have everything in place to start the proof of the theorem . for fixed @xmath249 , let @xmath250 be the number of @xmath251 that contribute to @xmath242 . let @xmath252 be the set of @xmath253 satisfying @xmath254 . by definition , @xmath255 . we would like to begin by applying ( * ? ? ? * proposition 2.4 ) , which is concerned with a much more general setting . in order to facilitate our use of this result , table [ t : dict ] presents a dictionary between the notation adopted in @xcite and the special case considered here . .dictionary for applying ( * ? ? ? * proposition 2.4 ) [ cols=">,<,>,<",options="header " , ] we may now apply ( * ? ? ? * proposition 2.4 ) to deduce that @xmath256 where @xmath257 and the error term @xmath258 is the sum of terms of the form @xmath259 with @xmath260 one for each of the intervals that form @xmath261 , with start and end points @xmath262 and @xmath263 . here , @xmath264 denotes the multiplicative inverse of an integer @xmath265 . our first task is to show that the overall contribution from @xmath266 makes a satisfactory contribution to @xmath242 . [ lem : d5_first_sum ] we have @xmath267 we must show that once summed over @xmath268 such that ( [ eq : cpe8 ] ) , ( [ eq : cpe7 ] ) and ( [ eq : cpe ] ) hold , the term @xmath258 contributes @xmath269 . let @xmath270 . we remove ( [ eq : cpe8 ] ) by a mbius inversion . this leads us to estimate @xmath271 where @xmath272 is defined to be @xmath273 with @xmath274 where @xmath275 is the allowed interval for @xmath276 and @xmath277 as above depend on @xmath278 and @xmath279 . we may split the summation over @xmath280 into subintervals @xmath281 where we have @xmath282 as functions of @xmath276 . in view of the bounds for @xmath283 and @xmath284 that follow from the inequalities in the definition of @xmath285 , it follows that @xmath286 since @xmath287 , we may restrict the summation over @xmath288 to @xmath289 such that @xmath290 . then @xmath291 and @xmath292 , so that we can apply lemma [ lem : lemma10 ] to obtain @xmath293 note that @xmath294 for any @xmath42 . writing , temporarily , @xmath295 we deduce that the total contribution from the first term is @xmath296 by lemma [ lem : sum_h_k ] . the total contribution from the second term is @xmath297 finally , the total contribution from the third term is @xmath298 this therefore completes the proof of the lemma . let @xmath299 be the number of @xmath300 subject to @xmath301 , and let @xmath302 be the remaining number of elements of @xmath210 . lemma [ lem : d5_first_sum ] can be modified in an obvious way to give estimates for @xmath299 and @xmath302 . for @xmath299 , we sum over @xmath303 first and over @xmath304 afterwards , and for @xmath302 , we do the reverse . we rewrite the result of lemma [ lem : d5_first_sum ] as follows . removing ( [ eq : cpe8 ] ) by a mbius inversion , and adding @xmath306 to prevent that @xmath307 , we arrive at the formula @xmath308 where @xmath309 [ lem:8 ] we have @xmath310 where @xmath311 let @xmath312 and @xmath313 where @xmath314 \mid { { { \eta}_4{\eta}_7k_8{\eta}_8 ' } \equiv { -{\varrho}^2{\eta}_3}\ ( \mathrm{mod}\ { q})}\}.\ ] ] as in ( * ? ? ? * section 8.3 ) , we have @xmath315 where @xmath316 is the unique integer modulo @xmath41 with @xmath317 clearly @xmath318 is equivalent to @xmath319 for any such @xmath316 . using lemma [ lem : sum_mod_q ] , we deduce that @xmath320 is @xmath321 a straightforward application of partial summation therefore reveals the total error as being @xmath322 here , in the second step , we have used @xmath323 and @xmath324 and the bound ( [ eq : estimate_v1 ] ) for @xmath325 . the final step uses lemma [ lem : sum_h_k ] . we rewrite the result of lemma [ lem : d5_first_sum ] . recall the definition of the function @xmath327 for positive integers @xmath87 . noting that we may replace ( [ eq : cpe7 ] ) by @xmath328 , it follows that @xmath329 where @xmath330 here we automatically have @xmath331 . thus the congruence involving @xmath103 in @xmath332 determines @xmath303 uniquely modulo @xmath333 . [ lem:9 ] we have @xmath334 where @xmath335 let @xmath312 , @xmath336 and @xmath337 it follows from lemma [ lem : sum_mod_q_2 ] that @xmath320 is @xmath338 a little thought reveals that the main term here is @xmath339 . using partial summation , we estimate the total error as @xmath340 using @xmath341 and ( [ eq : estimate_v1 ] ) in the second step and lemma [ lem : sum_mod_q ] in the final step . throughout the remainder of the paper we set @xmath342 for the total error term that appears in our main result . in this section and the next we will need to compute the average order of certain complicated multi - variable arithmetic functions , sometimes weighted by piecewise continuous functions . as previously , we will place ourselves in the more general investigation carried out in @xcite . here , given @xmath343 and @xmath344 , a number of rather general sets of functions are introduced : @xmath345 ( * ? ? ? * definition 3.8 ) , @xmath346 ( * ? ? ? * definition 4.2 ) , @xmath347 ( * ? ? ? * definition 7.7 ) and @xmath348 ( * ? ? ? * definition 7.8 ) . we will not redefine these sets here , but content ourselves with recording the inclusions @xmath349 ( * ? ? ? * corollary 7.9 ) . in the notation of ( * definition 7.7 ) , our manipulations will involve the function @xmath350 for any @xmath351 , where @xmath352 and @xmath353 [ lem : d5_third_sum_b ] we have @xmath354 where @xmath355 is given by . our proof of the lemma is based on combining ( * ? ? ? * proposition 3.9 ) with lemma [ lem:8 ] . we will apply the former to @xmath356 summed over @xmath357 , with @xmath358 and @xmath359 there are a number of preliminary hypotheses that need to be checked in using ( * ? ? ? * proposition 3.9 ) . local factors of @xmath360 are given by @xmath361 , equal to @xmath362 we see that @xmath363 , for an appropriate @xmath364 . for @xmath365 , we observe that ( [ eq : estimate_v2b ] ) implies @xmath366 and that @xmath367 unless @xmath323 . thus everything is in place for an application of ( * ? ? ? * proposition 3.9 ) , giving @xmath368 where we check @xmath369 by ( * ? ? ? * corollary 7.10 ) . [ lem : d5_third_sum_a ] we have @xmath370 with @xmath355 given by . this time our argument is based on combining ( * ? ? ? * proposition 3.10 ) with lemma [ lem:9 ] , the former being applied with @xmath371 . as previously , there are a number of preliminary hypotheses that need to be checked in order to use this result . for the first of these , we define @xmath372 as in the proof of lemma [ lem : d5_third_sum_b ] , we have @xmath373 , for some @xmath364 . next , ( [ eq : estimate_v2a ] ) implies that @xmath374 an application of ( * ? ? ? * proposition 3.10 ) now gives the expected main term , together with a total error term @xmath375 . we put back together our estimates for @xmath302 and @xmath299 that were obtained in lemmas [ lem : d5_third_sum_b ] and [ lem : d5_third_sum_a ] , respectively . this yields the following result . since @xmath379 , there is a @xmath364 such that @xmath380 . this and the bound ( [ eq : estimate_v3 ] ) for @xmath381 show that we are able to apply ( * ? ? ? * proposition 4.3 ) with @xmath382 to conclude that @xmath383 here , @xmath384 and @xmath385 is the `` average '' of @xmath386 over @xmath387 , which is computed as @xmath388 using ( * ? ? ? * corollary 7.10 ) . it turns out that in applying ( * ? ? ? * lemma 5.1 ) to obtain , only the inequality @xmath396 is used in the definition of @xmath243 . hence the same bounds hold if we replace @xmath243 by @xmath244 in the definitions of @xmath397 . finally , for @xmath415 we use ( [ eq : estimate_v2b ] ) for the integration over @xmath416 . then , integrating over @xmath417 , @xmath399 and @xmath418 we obtain @xmath419 this completes the proof of the lemma . substituting @xmath420 into @xmath25 , for fixed @xmath421 , we obtain @xmath422 finally , by substituting @xmath423 for @xmath424 into ( [ eq : alpha_volume ] ) , written as an integral , we deduce that @xmath425 this completes the proof of the theorem . t. d. browning , an overview of manin s conjecture for del pezzo surfaces . _ analytic number theory a tribute to gauss and dirichlet ( gttingen , 20th june 24th june , 2005 ) _ , 3956 , clay mathematics proceedings * 7 * , ams , 2007 . u. derenthal and yu . tschinkel , universal torsors over del pezzo surfaces and rational points . _ equidistribution in number theory , an introduction _ , 169196 , nato sci . phys . chem . * 237 * , springer , 2006 .

the manin conjecture is established for a split singular cubic surface in @xmath0 , with singularity type @xmath1 .

0807.4733

since the discovery of increased pinning in heavy - ion irradiated samples , the interaction between flux - lines and columnar defects in high - t@xmath5 superconductors has been the subject of intense experimental and theoretical investigations.@xcite an angle dependent critical current enhancement has been put into evidence both in the moderately anisotropic material yba@xmath0cu@xmath1o@xmath2 @xcite as in highly anisotropic materials such as bi@xmath0sr@xmath0cacu@xmath0o@xmath6.@xcite the influence of correlated disorder on the equilibrium properties of the flux - line lattice has been the subject of fewer experimental investigations . torque and magnetization experiments on bi@xmath0sr@xmath0cacu@xmath0o@xmath6 have revealed a pinning energy contribution to the equilibrium magnetization arising from the presence of the defects , but , in contrast to the irreversible magnetic moment , the equilibrium magnetization did not show any other angle dependent contribution than the one arising from the layering of the material.@xcite here , we investigate the less anisotropic compound yba@xmath0cu@xmath1o@xmath2 and show that there exists a narrow domain in the ( h - t ) diagram where a reversible angular dependent contribution to the torque arises due to the interaction of flux - lines with the linear defects . this constitutes a direct demonstration that vortex lines in the liquid phase distort in order to accomodate to the linear irradiation defects . the experiments were performed on a single crystal of dimensions @xmath7 @xmath8m@xmath9 ; the shortest dimension was along the @xmath10axis . the transition temperature after irradiation was @xmath11 k. a single domain of parallel twin planes was observed , the planes running at 45@xmath12 with respect to the crystal s longest edge . the sample was irradiated with 5.8 gev pb ions to a dose @xmath13 @xmath14 , equivalent to a matching ( dose equivalent ) field @xmath15 kg . the ion beam was directed perpendicular to the longer crystal edge , at an angle of 30 degrees with respect to the @xmath10-axis . the irradiation created continuous linear amorphous defects of radius @xmath16 , oriented along the direction of the ion beam,@xcite with density @xmath17 . after the irradiation , the sample was characterized using the magneto - optic flux visualization technique at 65 , 77 and 82 k ; no evidence of any remaining influence of the twin planes on the flux penetration could be observed . the torque was measured using a piezoresistive microlever from park scientific instruments , as described in ref . . the microlever formed part of a low temperature wheatstone resistive bridge , in which a second lever with no sample was inserted in order to compensate for the background signal originating from the magnetoresistance of the levers . the measuring lever was fed with a current of 300 @xmath8a and thermalized to better than 0.01 k using he@xmath18 exchange gas . the torque setup was calibrated from the meissner slope of the reversible magnetization as a function of field at a fixed angle , as described elsewhere@xcite . in torque experiments with a single rotation axis , the plane in which the applied field @xmath19 is rotated is always at a misorientation angle @xmath20 with respect to the plane enclosing the @xmath10-axis and the irradiation direction ( fig . [ lever ] ) . this angle is not known _ a priori _ : it results from the uncertainty in both the irradiation direction and the sample positioning . we estimate that it is less than a few degrees . the result of the misorientation is that the applied field is never strictly aligned with the irradiation direction ; @xmath20 is therefore the minimum angle between @xmath19 and the ion tracks when the field direction is varied . in a separate experiment , the irreversibility line was measured using squid _ ac_-susceptometry . it was located as the onset of the in phase ( reactive ) component of the _ ac _ susceptibility measured in an oscillatory field of amplitude 0.1 oe and frequency 13 hz , oriented parallel to the dc field . these measurements were performed for two orientations of the static field , applied parallel to the direction of the tracks ( _ i.e . at 30@xmath12 with respect to the @xmath10axis ) , and applied in the symmetric direction with respect to the @xmath10-axis . the irreversibility fields for both orientations were found to be linear with temperature ; the line obtained with the field applied parallel to the tracks clearly lies above the one for the symmetric orientation ( fig . [ irrline ] ) . in contrast to what is observed for bi@xmath0sr@xmath0cacu@xmath0o@xmath6 , and more recently , in heavy - ion irradiated yba@xmath0cu@xmath1o@xmath2 thick films,@xcite there is no change in the behavior at @xmath21 up to our maximum measuring field of @xmath22 koe , and the lines do not merge above the irradiation field . _ typical torque signals are displayed in fig . [ data ] . below the irreversibility line determined by squid _ ac_-susceptometry with the field along the tracks , the torque measurements reveal a hysteretic behavior when the field is aligned with the irradiation direction . above the line , the system is in the so called vortex liquid phase and the torque signal is reversible ; however , in a narrow region typically 1 to 2 k wide , a kink is found , roughly symmetric with respect to the orientation of the columnar defects ( fig . [ data ] ) . this behavior is similar to what is observed for conventional torque on a layered superconductor when the field is rotated across the plane of the layers , and indicates that the vortex lines deform in order to have their direction coincide with that of the linear defects . in other words , the free energy of the vortex liquid phase is lowered by flux line pinning onto the columnar tracks . at low temperature , where thermal fluctuations are not important , theory@xcite predicts that when the external field is applied sufficiently close to the layer / track direction ( _ i.e . the angle between applied field and the tracks @xmath23 where @xmath24 is the lock - in angle ) , the equilibrium configuration of a single flux - line is that in which the whole length of the line is aligned with the defect . at larger angles @xmath25 , one expects a staircase configuration in which line segments aligned with the defects alternate with segments wandering between defects.@xcite for @xmath26 larger than the accomodation angle @xmath27 , the vortices do not readjust to the columnar defects at all . in our experiment , it is unlikely that we achieve the locked configuration , as this would require the alignment of the external field with the track direction to within some angle @xmath28 . in the locked configuration , one should observe a linear variation of the torque signal with angle , with a slope @xmath29 erg @xmath30 rad@xmath31 ( neglecting the anisotropy in the demagnetizing factors ) i.e. @xmath32 erg deg@xmath31 in a 10 koe field in our case . this is three orders of magnitude larger than the highest of the slopes in fig . _ the predicted contribution @xmath33 to the torque signal@xcite is shown in fig . [ theory ] . as the field angle is increased from the irradiation direction , the torque first increases linearly , reaching a maximum at the lock - in angle , and then decreases linearly beyond this . for angles larger than @xmath34 , the torque contribution arising from the interaction between vortices and ion tracks should be zero . in practice , the lock - in angle is quite small , therefore the torque signal should be quasi discontinuous when the field and track direction coincide . in the single vortex regime , the magnitude of the torque signal close to the irradiation direction may be obtained in terms of the lock - in angle,@xcite @xmath35 with @xmath36 the pinning energy per unit length . the accomodation angle @xmath37 , where @xmath38 is the vortex line tension , the energy scale @xmath39 , and @xmath40 is the typical wavevector of the vortex distortion induced by the columns . in optimally doped yba@xmath41cu@xmath42o@xmath2 , the penetration depth @xmath43 , the @xmath44plane coherence length @xmath45 , and the anisotropy parameter @xmath46.@xcite > from eq . ( [ eq : torque ] ) , one sees that one can _ directly obtain an estimate of the lock in angle from the torque jump observed when the field is aligned with the columns and eq . ( [ eq : torque ] ) . a good estimate of the pinning energy @xmath47 is equally obtained from the torque jump : _ @xmath48 ( @xmath49 is the mean separation between vortices ) . this method to obtain the pinning energy@xcite is more direct than estimates based on the angular dependence of the resistivity @xmath50 . those rely on the identification of a shallow maximum of @xmath50 at @xmath27 , or , alternatively , with a `` depinning angle''@xcite determined by the rate at which vortices can liberate themselves from a track ; the relation of the latter with the accomodation angle is not certain . since the pinning energy is predicted to be proportional to @xmath51 , the method based on transport measurements can result in a large uncertainty in @xmath47 . the present approach has the advantage that there is only one assumption , which concerns the precise form of @xmath52 . taking the curve at @xmath53 koe and @xmath54 k ( @xmath55 ) , _ i.e. at the onset of magnetic irreversibility , one has a typical value of the torque jump @xmath56 400 erg @xmath30 ( fig . [ data ] ) ; consequently , @xmath57 deg . the parameter values @xmath58 , @xmath59 , @xmath60 , and @xmath61 , yield the pinning energy per unit length @xmath62 erg cm@xmath31 , and @xmath64 . the obtained value of the accomodation angle seems reasonable : the difference between the extrapolation of the torque from large positive and negative angles to @xmath65 , at which the field and the ion track are nearly aligned , is a good indication that @xmath27 lies beyond the angular range depicted in fig . clearly , @xmath27 greatly exceeds the angular width of the irreversible regime just below the irreversibility line , which is about 8@xmath12 at 88.5 k and @xmath53 koe . the accomodation angle is comparable to the low temperature limit of the `` depinning angle '' measured on an untwinned yba@xmath41cu@xmath42o@xmath2 single crystal irradiated with 1.0 gev u ions to the same nominal dose.@xcite _ returning to the experimental data in fig . [ data ] , one observes that , in spite of the fact that a clear jump in the torque signal can be defined , the discontinuity at the irradiation angle is rather smooth . the smoothness of the curve is possibly due to the non - zero misalignment angle @xmath20 . the effect of misalignment can be quantitatively accounted for using simple trigonometric considerations . projecting the torque as given in ref . on the experimental torque axis * u * ( fig . [ lever ] ) , one obtains the magnitude of the measured torque signal : @xmath66 where @xmath67^{1/2}.\ ] ] @xmath68 is the field rotation angle in the laboratory frame , and @xmath26 , as before , is the real angle between the direction of the magnetic field and that of the ion tracks . the curves plotted in fig . [ theory ] shows that the effect of the misalignment is both to widen the angular interval between the torque maxima ( now @xmath69 from one another ) and to decrease the torque value at the maximum . using @xmath70 , @xmath71 , and @xmath72 we find that the maximum torque is only about 0.6 @xmath73 so that the pinning potential estimated from the apparent torque jump is in this case only about 40@xmath74 of the actual value , which , at @xmath53 koe and @xmath75 k , would amount to @xmath76 erg cm@xmath31 . the absolute value of the pinning energy per unit length is in reasonable agreement with the estimate for core pinning of individual vortices,@xcite @xmath77 ( with @xmath78 erg cm@xmath31 ) in the temperature regime of interest , this mechanism is more relevant than electromagnetic pinning@xcite because @xmath79 greatly exceeds the track radius . using the same parameter values as above , the model yields the theoretical value @xmath80 erg cm@xmath31 ( at @xmath81 ) . recent measurements on heavy ion irradiated bi@xmath41sr@xmath41cacu@xmath41o@xmath6@xcite showed that in that material , the dependence of pinning energy on track diameter and temperature is in agreement with the core pinning model , although the magnitude of the pinning energy exceeded the theoretical expectation ( [ eq : core ] ) by a factor 5 . in the present case , the strong temperature and field dependence of the experimentally obtained pinning energy , displayed in fig . [ fig : upin ] , show that a simple `` zero temperature '' single vortex pinning approach is inadequate . the reasons for this are that ( i ) the fields under consideration are not small with respect to @xmath3 , so that only a fraction of vortices can be expected to be actually trapped on a columnar track , and ( ii ) the proximity to @xmath4 possibly necessitates the inclusion of the effect of strong thermal fluctuations.@xcite a theoretical description of the effect of a field rotation , or even of the total pinning energy , in the case where the vortex density is comparable to the density of a system of strong linear pins has , to our knowledge , not been developped at present . although the decrease of the pinning energy per unit volume as field is increased , and the eventual disappearance of the torque jump at @xmath82 , is the straightforward consequence of the averaging of the pinning energy gain obtained from the restricted number of vortices trapped on an ion track and the ever increasing number of those that are not , there are few predictions about the resulting field dependence of the equilibrium magnetization . extensive numerical calculations of the vortex energy distribution in the presence of columnar pins were carried out by wengel and tuber;@xcite however , they did not make any specific predictions as to the precise temperature or field dependence of the magnetization . the effect of thermal fluctuations must also be considered . in resistivity measurements , such fluctuations are usually accounted for by stating that , at the angle at which depinning occurs , _ i.e. at which the probability to find a pinned vortex segment becomes exponentially small , the thermal energy and the pinning energy of a single trapped vortex segment are equal.@xcite as a consequence , the `` depinning angle '' measured by the angular dependence of the resistivity is smaller than the accomodation angle and is given by:@xcite _ @xmath83 with the parameters values as used above , we obtain @xmath84 15@xmath12 , which is comparable to the angular width of the irreversible regime in fig . [ data ] . the same type of argument leads one to conclude that at the same temperature , thermal fluctuations are much less efficient when the field is aligned with the track direction , because the length and the trapping energy of the pinned line segments are large . nevertheless , the fact that the magnetization is _ reversible and that the resistivity measured under similar condictions is _ linear @xcite implies that although it may be small , the thermal depinning rate is non zero . since the measured pinning energy is proportional to the average vortex length trapped on a columnar defect at any one moment , the rapid decrease of @xmath47 with temperature , which is in agreement with estimates obtained from resistivity data,@xcite does not seem to be an artefact of the method used to analyze torque or resistivity data , but reflects the increasing efficiency of thermal fluctuations in liberating vortices from the tracks . strong vortex wandering above a `` depinning temperature '' @xmath85 , such as proposed in refs . , and would lead to a torque jump that follows an exponential temperature dependence . such a dependence was observed in ref . , where the rapid decrease of the accomodation angle measured in heavy - ion irradiated bi@xmath0sr@xmath0cacu@xmath0o@xmath6 was attributed to the effect of thermal fluctuations . although the reduced range of temperatures over which @xmath47 could be determined in the present experiments makes a direct comparison very difficult , thermal wandering of flux lines could be responsible for the disappearance of the pinning energy and the torque jump at temperatures _ below @xmath86 ( see fig . [ irrline ] ) . we have , from thermodynamic torque measurements , obtained the first evidence for an angle dependent contribution of amorphous columnar defects to the equilibrium magnetization in yba@xmath41cu@xmath42o@xmath2 . the analysis of the torque signal allowed us to directly determine the lock - in angle and the pinning energy of the linear defects . the magnitude of the pinning energy is in qualitative agreement with the core pinning mechanism by columnar defects ; however , the observed strong field dependence means that the interactions between flux lines are not negligible in the range of magnetic fields investigated here . the strong temperature dependence of the torque jump , and the disappearance of the pinning energy below @xmath87 are the consequence of thermal fluctuations in the vortex liquid state , which are increasingly efficient in liberating vortex segments from the tracks as temperature increases . the work of stj is funded by the ec , tmr grant nr . we thank f. holtzberg ( emeritus , i.b.m . thomas j. watson research center , yorktown heights ) for providing the yba@xmath41cu@xmath42o@xmath2 single crystal . this feature is also valid in the presence of weak irreversibility . for a method how to estimate the equilibrium torque in such a situation , see l. fruchter and i.a . campbell , phys . b * 40 * , 5158 ( 1989 )

we have measured an angle dependent contribution to the equilibrium magnetization of a yba@xmath0cu@xmath1o@xmath2 single crystal with columnar defects created by irradiation with 5.8 gev pb ions . this contribution manifests itself as a jump in the equilibrium torque signal , when the magnetic field direction crosses that of the defects . the magnitude of the jump , which is observed in a narrow temperature interval of less than 2 k wide , for fields up to about twice the dose equivalent field @xmath3 , is used to estimate the energy gained by vortex pinning on the defects . the vanishing of the effective pinning energy at a temperature below @xmath4 is attributed to its renormalization by thermal fluctuations . 2

cond-mat9903407

the molecular beam magnetic resonance ( mbmr ) technique has significantly contributed , as is well known , to the development of atomic and molecular physics @xcite . and it makes possible to measure the larmor frequency of an atom or molecule in the presence of a magnetic field . in the original technique , developed by i.i . rabi and others @xcite , @xcite the molecular beam is forced to pass through four different fields : a non - homogeneous polarizer field ( a ) where the molecules are prepared . a resonant unit ( c ) that consists of two , a static and an oscillating , fields . a non - homogeneous analyzer field ( b ) . only molecules in the prepared state reach the detector . the two non - homogeneous magnetic fields a and b have opposite directions . the molecular beam describes a sigmoidal trajectory and , finally , is collected in a detector ( see fig . [ fig:1 ] ) . typical path of molecules in a m.b.m.r . the two solid curves show the paths of the molecules whose moments do not change when passing through the resonant cell . ] rabi explained this effect in terms of spatial reorientation of the angular moment due to a change of state when the transition occurs . in this case the depletion explanation is based on the interaction between the molecular magnetic dipole moment and the non - homogeneous fields . @xmath2 the force is provided by the field gradient interacting with the molecular dipolar moment ( electric or magnetic ) . on the resonant unit the molecular dipole interacts with both , homogeneous and oscillating , fields . when the oscillating field is tuned to a transition resonant frequency between two sub states , a fraction of the molecular beam molecules is removed from the initial prepared state . the dipolar moment changes in this fraction and as a consequence , the interaction force with the non - homogeneous analyzer field ( b ) . as only molecules in the initial prepared state reach the detector the signal in the detector diminishes . during the last years some interesting experimental results have been reported for n@xmath0o , no , no dimer , h@xmath0 and bafch@xmath1 cluster @xcite - @xcite . the main result consists in the observation of molecular beam depletion when the molecules of a pulsed beam interact with a static electric or magnetic field and an oscillating field ( rf ) as in the rabi s experiments . but , in these cases , instead of using four fields , only two fields , those which configure the resonant unit ( c ) , are used , that is , without using the non - homogeneous magnetic , a and b , fields . see fig.[fig:2 ] the dotted line path show the trajectory change of the fraction of the molecular beam that is removed from the initial prepared state when passing through the resonant cell . ] in a similar way , when the oscillating field is tuned to a transition resonant frequency between two sub states , the fraction of the molecular beam that is removed from the initial prepared state does not reach the detector . but the important thing is : differently to the previous method , it happens without using non - homogeneous fields . obviously , the trajectory change has to be explained without considering the force provided by the field gradient . there must be another molecular feature that explains the depletion . it looks as though the linear momentum conservation principle were not satisfied . these experiments suggest that a force depending on other fundamental magnitude of the particle , different from mass and charge must be taken into account . in order to find out an explanation , let s consider the following case : an electron is moving , with speed , @xmath3 constant in modulus , in a homogeneous magnetic field @xmath4 where @xmath3 is perpendicular to @xmath4 . its kinetic energy will be : @xmath5 the electron , as is well known , describes a circular trajectory ( in general case an helix ) with a radius @xmath6 , being : @xmath7 and : @xmath8 due to the lorentz force : @xmath9 on the other hand , as the electron has a magnetic moment , @xmath10 , and spin @xmath11 , the presence of the magnetic field @xmath4 produces a torque when interacting with the electron magnetic moment @xmath10 . the angle between @xmath11 and o@xmath12 ( the direction of the magnetic field @xmath4 ) remains constant but the spin @xmath11 revolves about o@xmath12 with angular velocity @xmath13 . this phenomenon bears the name of larmor precession . the electron kinetic energy must increase with the energy due to spin precession . but it should be considered that the forces producing the torque are perpendicular to the precession motion and , as a consequence , do not modify the energy of the system . it looks like if the principle of energy conservation be violated . if the rotation around an axis is considered as origin of the spin , in a classic ( and impossible ) interpretation , one could imagine the electron rotating in a slowly way and offsetting the increase in energy due to the precession movement . but , as it is well known , the spin is a quantized quantity ; its modulus is constant and immutable . this option is , as a consequence , not acceptable . let us consider now that the helicity is a constant of motion . helicity , @xmath14 , is defined as the scalar product of linear momentum and the spin : @xmath15 is this hypothesis consistent with quantum mechanics ? let us consider an electron in a uniform magnetic field @xmath4 , and let us choose the o@xmath12 axis along @xmath4 . the classical potential energy due to electron magnetic moment @xmath10 is then @xmath16 where @xmath17 is the modulus of the magnetic field . let us set : @xmath18 @xmath19 being the classical angular precession velocity . ( as is well known , @xmath20 has dimensions of the inverse of a time , that is , of an angular velocity . ) if we replace @xmath21 by the operator @xmath21 the classic energy becomes an operator : the hamiltonian @xmath22 which describes the evolution of the spin of the electron in the field @xmath4 is : @xmath23 since this operator is time independent , solving the corresponding schr@xmath24dinger equation amounts to solving the eigenvalue equation of h. we immediately see that the eigenvectors of h are those of @xmath21 ( see ref.@xcite ) : @xmath25 @xmath26 there are therefore two energy levels , @xmath27 and @xmath28 their separation @xmath29 is proportional to the magnetic field and define a _ single bohr frequency _ is it possible to distinguish , in a uniform magnetic field @xmath4 , which electrons are the state @xmath31 and which are the state @xmath32 ? the answer is no . their behavior inside the field is exactly the same . but , nevertheless , if we introduce a oscillating magnetic field h@xmath33 with a frequency resonant with the transition @xmath34 , then it will be possible to distinguish both states by the difference in their trajectories , see ref.@xcite . let us assume that , at time @xmath35 , the spin is in the state : @xmath36 to calculate the state @xmath37 in an arbitrary state @xmath38 and as @xmath39 is already expanded in terms of the eigenstates of the hamiltonian we will obtain : @xmath40 or , using the values of e@xmath41 and e@xmath42 : @xmath43 the presence of the magnetic field @xmath4 therefore introduces a phase shift , proportional to time , between the coefficients of the kets @xmath31 and @xmath32 . comparing the expression ( [ eq:14 ] ) for @xmath37 with that for the eigenket @xmath44 for the observable @xmath45 @xmath46 we see that the direction @xmath47 along which the component is @xmath48,with certainty , is defined by the polar angles : @xmath49 the angle between @xmath47 and o@xmath12 ( the direction of the magnetic field @xmath4 ) therefore remains constant , but @xmath47 revolves around o@xmath12 with angular velocity @xmath50 proportional to the magnetic field . thus , we find in quantum mechanics the phenomenon equivalent to that described for a particle with classic magnetic moment and spin and which bears the name of larmor precession . we redefine now the helicity , @xmath14 , in order that its eigenvalues be @xmath511 , as @xmath52 , where @xmath53 and @xmath54 . the initial velocity of the electron is @xmath55 , and we assume the initial spin state of the electron to be an eigenstate of the helicity with eigenvalue + 1 , which is given in ( [ eq12 ] ) , with @xmath56 , that is : @xmath57 at the time @xmath58 the velocity of the electron is , as it is known , @xmath59 \label{eq:18}\ ] ] where @xmath20 is given in ( [ eq:7 ] ) . according to ( [ eq:15 ] ) and ( [ eq:16 ] ) , with @xmath56 , at time @xmath58 the spin state is : @xmath60 and the helicity at time @xmath58 , @xmath61 , @xmath62 \label{eq:20}\ ] ] now , taking into account that , @xmath63 we easily obtain : @xmath64 this shows that @xmath65 is an eigenstate of the helicity of eigenvalue + 1 ; in other words , helicity is conserved along the electron s ( classical ) trajectory . it has been proven that helicity is a constant . as a consequence of this result , the linear momentum @xmath66 must have the same precession angular velocity ( larmor angular velocity ) @xmath13 than the spin @xmath11 . the equation of motion describing the linear momentum evolution must be then equivalent of the equation of motion which describe the evolution of the spin @xmath11 . this means that : @xmath67 @xmath68 it is concluded the particle will be under a central acceleration , @xmath69 perpendicular to @xmath3 . the particle is then under a central force : @xmath70 this kind of forces related with the spin will be designed as lorentz - like forces . in this case , the trajectory will be a circular one . the radius will be : @xmath71 and its kinetic energy : @xmath72 which is equal to the initial one shown in ( [ eq:2 ] ) . the force ( [ eq:24 ] ) is the responsible of the electron circular trajectory inside the field @xmath4 and should be related to the spin @xmath11 of the electron . if the case of an electron in a magnetic field is considered , then the force due to the spin of the electron will be : @xmath73 where @xmath13 is the spin larmor precession velocity around o@xmath12 . but is known that : @xmath74 substituting in ( [ eq:24 ] ) the expression for the force acting on the particle is obtained . this force has its origin on the spin . this expression is : @xmath75 as for an electron @xmath76 , the final result is : @xmath77 surprisingly this expression for the lorentz - like force , related to the spin , coincide with that known as lorentz force related to the charge . considering the spin as responsible of the lorentz - like force , a new deflection mechanism has been proposed ( see ref . the equations of motion for a system with intrinsic angular momentum when applying torques are described and , according with the theory , when the frequency of the oscillating field coincides with a transition resonant frequency ( larmor frequency ) , the molecules that change their state from the original one are removed from their trajectories and , as a consequence , do not reach the detector and the corresponding signal decreases . in 1939 alvarez and bloch @xcite measured the neutron magnetic moment by using a neutron beam passing through a resonant unit . neutrons from the be + d reactions were slowed to thermal velocities and diffused down a cadmium lined tube through the water tank to the polarizer magnet , b@xmath78 . after passing through the resonant unit that consists of two , a static and an oscillating , fields and the analyzer magnet , b@xmath79 they were detected in a bf@xmath1 chamber . the polarizer b@xmath78 and analyzer b@xmath79 are strongly magnetized iron pieces . a neutron resonant dip is observed in the signal of the neutron beam when the oscillating resonant frequency corresponding to the transition between the two states up and down is reached . according to the previous theoretical description and recently results obtained for no@xmath0 , no , no dimer , h@xmath0 and bafch@xmath1 cluster , if the alvarez and bloch experiment is carried out without using analyzer magnet b@xmath79 , we anticipate that the experimental results will be the same as those obtained by alvarez and bloch in the experiment of 1939 . according to the new explanation , the trajectory change takes place when neutrons pass through the resonant unit and the oscillating field is tuned to a transition resonant frequency between two states , up and down , of the spin of the neutron . in case of alvarez and bloch experiment , they used a magnetic field for the neutron resonance of 622 gauss and a resonant frequency of oscillator of 1843 kilocycles . the author is very grateful to prof . jos l. snchez gmez , universidad autnoma de madrid for useful discussions .

during the last years some interesting experimental results have been reported for experiments in n@xmath0o , no , no dimer , h@xmath0 , toluene and bafch@xmath1 cluster . the main result consists in the observation of molecular beam depletion when the molecules of a pulsed beam interact with a static electric or magnetic field and an oscillating field ( rf ) . in these cases , and as a main difference , instead of using four fields as in the original technique developed by i.i . rabi and others , only two fields , those which configure the resonant unit , are used . that is , without using the non - homogeneous magnetic fields . the depletion explanation for i.i . rabi and others is based in the interaction between the molecular electric or magnetic dipole moment and the non - homogeneous fields . but , obviously , the change in the molecules trajectories observed on these new experiments has to be explained without considering the force provided by the field gradient because it happens without using non - homogeneous fields . in this paper a theoretical way for the explanation of these new experimental results is presented . one important point emerges as a result of this development , namely , the existence of an until now unknown , spin - dependent force , which would be responsible of the aforementioned deviation of the molecules .

1404.2625

the most common kind of superconductivity ( sc ) is based on bound electron pairs coupled by deformation of the lattice . however , sc of more subtle origins is rife in strongly correlated electron systems including many heavy - fermion ( hf ) , cuprate and organic superconductors . in particular , a number of studies on @xmath13-electron compounds revealed that unconventional sc arises at or close to a quantum critical point ( qcp ) , where magnetic order disappears at low temperature ( @xmath14 ) as a function of lattice density via application of hydrostatic pressure ( @xmath3 ) @xcite . these findings suggest that the mechanism forming cooper pairs can be magnetic in origin . namely , on the verge of magnetic order , the magnetically soft electron liquid can mediate spin - dependent attractive interactions between the charge carriers @xcite . however , the nature of sc and magnetism is still unclear when sc appears very close to the antiferromagnetism ( afm ) . therefore , in light of an exotic interplay between these phases , unconventional electronic and magnetic properties around qcp have attracted much attention and a lot of experimental and theoretical works are being extensively made . si@xmath0 @xcite , cein@xmath2 @xcite and cerh@xmath0si@xmath0 @xcite : ( b ) cecu@xmath0si@xmath0 @xcite and cerhin@xmath1 @xcite . dotted and solid lines indicate the @xmath3 dependence of @xmath10 and @xmath9 , respectively.,width=268 ] the phase diagram , schematically shown in figure 1(a ) , has been observed in antiferromagnetic hf compounds such as cepd@xmath0si@xmath0 @xcite , cein@xmath2 @xcite , and cerh@xmath0si@xmath0 @xcite . markedly different behavior , schematically shown in figure 1(b ) , has been found in the archetypal hf superconductor cecu@xmath0si@xmath0 @xcite and the more recently discovered cerhin@xmath1 @xcite . although an analogous behavior relevant to a magnetic qcp has been demonstrated in these compounds , it is noteworthy that the associated superconducting region extends to higher densities than in the other compounds ; their value of @xmath9 reaches its maximum away from the verge of afm @xcite . in this article , we review the recent studies under @xmath3 on cecu@xmath0si@xmath0 , cerhin@xmath1 and cein@xmath2 via nuclear - quadrupole - resonance ( nqr ) measurements . these systematic works have revealed the homogeneous mixed phase of sc and afm and that its novel superconducting nature exhibits the gapless nature in the low - lying excitations below @xmath9 , which differ from the superconducting characteristics for the hf superconductors reported to possess the line - node gap @xcite . the firstly - discovered hf superconductor cecu@xmath0si@xmath0 is located just at the border to the afm at @xmath15 @xcite . this was evidenced by various magnetic anomalies observed above @xmath9 @xcite and by the fact that the magnetic _ a - phase _ appears when sc is suppressed by a magnetic field @xmath16 @xcite . furthermore , the transport , thermodynamic and nqr measurements consistently indicated that nominally off - tuned ce@xmath17cu@xmath18si@xmath0 is located just at @xmath19 and crosses its qcp by applying a minute pressure of @xmath20 gpa @xcite . the magnetic and superconducting properties in cecu@xmath0si@xmath0 were investigated around the qcp as the functions of @xmath3 for ce@xmath17cu@xmath18si@xmath0 just at the border to afm and of ge content @xmath21 for cecu@xmath0(si@xmath5ge@xmath6)@xmath0 by cu - nqr measurements @xcite . figure 2 shows the phase diagram referred from the literature @xcite . here , @xmath22 is an effective fermi temperature below which the nuclear - spin - lattice - relaxation rate divided by temperature ( @xmath23 ) stays constant and @xmath24 is a temperature below which the slowly fluctuating antiferromagnetic waves start to develop . note that a primary effect of ge doping expands the lattice @xcite and that its chemical pressure is @xmath25 gpa per 1% ge doping as suggested from the @xmath3 variation of cu - nqr frequency @xmath26 in cecu@xmath0ge@xmath0 and cecu@xmath0si@xmath0 @xcite . ( si@xmath5ge@xmath27)@xmath0 and for ce@xmath17cu@xmath18si@xmath0 under @xmath3 . @xmath10 and @xmath9 are the respective transition temperature of afm and sc . also shown are @xmath24 below which the slowly fluctuating afm waves develop and @xmath22 below which @xmath23 becomes const . , width=313 ] in the normal state , the slowly fluctuating antiferromagnetic waves propagate over a long - range distance without any trace of afm below @xmath28 k. the exotic sc emerges in ce@xmath17cu@xmath18si@xmath0 below @xmath29 k , where low - lying magnetic excitations remain active even below @xmath9 . a rapid decrease below @xmath9 in @xmath30 evidences the opening of superconducting energy gap , whereas the large enhancement in @xmath23 well below @xmath9 reveals the gapless nature in the low - lying excitations in its superconducting state . with increasing @xmath3 , as a result of the marked suppression of antiferromagnetic critical fluctuations , the exotic sc evolves into a typical hf - sc with the line - node gap that is characterized by the relation of @xmath31 above @xmath32 gpa . markedly by substituting only 1% ge , afm emerges at @xmath33 0.7 k , followed by the sc at @xmath34 0.5 k . unexpectedly , @xmath30 does not show any significant reduction at @xmath9 , but follows a @xmath23 = const . behavior well below @xmath9 as observed in ce@xmath17cu@xmath18si@xmath0 as presented in fig.3 . it was revealed that the uniform mixed phase of sc and afm is unconventional , exhibiting that low - lying magnetic excitations remain active even below @xmath9 as shown later on fig.4 . as ge content increases , @xmath10 is progressively increased , while @xmath9 is steeply decreased . as a result of the suppression of antiferromagnetic critical fluctuations for the samples at more than @xmath35 , the magnetic properties above @xmath10 progressively change to those in a localized regime as observed in cecu@xmath0ge@xmath0 @xcite . further insight into the exotic sc is obtained on cecu@xmath0(si@xmath36ge@xmath37)@xmath0 that reveals the uniform mixed phase of afm ( @xmath33 0.75 k ) and sc ( @xmath34 0.4 k ) under @xmath38 . figure 3 shows the @xmath14 dependence of @xmath30 at @xmath3 = 0 gpa ( closed circles ) , 0.56 gpa ( open circles ) and 0.91 gpa ( closed squares ) . here , the data at @xmath3 = 0.19 gpa are not shown , since they are nearly equivalent to those at @xmath3 = 0.56 gpa . in the entire @xmath14 range , @xmath30 is suppressed with increasing @xmath3 , evidencing that the low - energy component of spin fluctuations is forced to shift to a high - energy range . as expected from the fact that the afm is already suppressed at pressures exceeding @xmath39 gpa , any trace of anomaly associated with it is not observed at all down to @xmath34 0.45 k at @xmath3 = 0.56 gpa and 0.91 gpa . it is , therefore , considered that the afm in cecu@xmath0(si@xmath36ge@xmath37)@xmath0 is not triggered by some disorder effect but by the intrinsic lattice expansion due to the ge doping . dependence of @xmath30 of cecu@xmath0(si@xmath36ge@xmath37)@xmath0 at several pressures . inset shows the @xmath14 dependence of @xmath23 of cecu@xmath0(si@xmath36ge@xmath37)@xmath0 at @xmath3 = 0 gpa ( closed circles ) and 0.91 gpa ( closed squares ) and those of ce@xmath17cu@xmath18si@xmath0 at @xmath3 = 0 ( open squares ) and 0.85 gpa ( open triangles ) . arrows indicate @xmath40 k and @xmath34 0.4 k at @xmath38 gpa for cecu@xmath0(si@xmath36ge@xmath37)@xmath0 . , width=302 ] in order to demonstrate a systematic evolution of low - energy magnetic excitations at the paramagnetic state , the inset of figure 3 shows the @xmath14 dependence of @xmath23 in cecu@xmath0(si@xmath36ge@xmath37)@xmath0 at @xmath3 = 0 gpa ( closed circles ) and 0.91 gpa ( closed squares ) along with the results in ce@xmath17cu@xmath18si@xmath0 at @xmath3 = 0 gpa ( open squares ) and 0.85 gpa ( open triangles ) @xcite . the result of @xmath23 in cecu@xmath0(si@xmath36ge@xmath37)@xmath0 at @xmath38 gpa is well explained by the spin - fluctuations theory for weakly itinerant afm in @xmath41 k around @xmath33 0.75 k @xcite . the good agreement between the experiment and the calculation indicates that a long - range nature of the afm is in the itinerant regime . at @xmath42 gpa , @xmath23 , which probes the development of magnetic excitations , is suppressed and resembles a behavior that would be expected at an intermediate pressure between @xmath38 gpa and 0.85 gpa for ce@xmath17cu@xmath18si@xmath0 . next , we discuss an intimate @xmath3-induced evolution of low - lying magnetic excitations in the superconducting state . as seen in figure 3 and its inset , the @xmath30 and @xmath23 at @xmath43 0 gpa do not show a distinct reduction below @xmath9 , but instead , a @xmath44 const . behavior emerges well below @xmath9 . at @xmath3 = 0.56 gpa , the afm is depressed , but anftiferromagnetic critical fluctuations develop in the normal state . it is noteworthy that the relation of @xmath45 is still valid below @xmath9 , resembling the behavior for ce@xmath17cu@xmath18si@xmath0 at @xmath38 gpa @xcite . by contrast at @xmath42 gpa , @xmath30 follows a relation of @xmath46 below @xmath34 0.45 k , consistent with the line - node gap at the fermi surface . this typical hf - sc in @xmath30 was observed in ce@xmath17cu@xmath18si@xmath0 at pressures exceeding @xmath47 0.85 gpa as well @xcite . a small deviation from @xmath31 behavior at @xmath42 gpa far below @xmath9 may be associated with an inevitable ge - impurity effect for @xmath48-wave superconductors in general @xcite . therefore , it is considered that the unconventional sc at @xmath38 gpa and 0.56 gpa evolves into the typical hf - sc with the line - node gap at pressures exceeding @xmath490.91 gpa . apparently , these results exclude a possible impurity effect as a primary cause for the @xmath50 = const . behavior below @xmath9 at @xmath43 0 gpa . we stress that a reason why the @xmath30 at @xmath38 gpa is deviated from @xmath31 below @xmath9 is ascribed not to the impurity effect but to the persistence of low - lying magnetic excitations well below @xmath9 . figure 4 indicates the @xmath14 dependence of nqr intensity multiplied by temperature @xmath51 in cecu@xmath0(si@xmath36ge@xmath37)@xmath0 at @xmath43 0 , 0.19 , 0.56 , and 0.91 gpa . here , the @xmath52 normalized by the value at 4.2 k is an integrated intensity over frequencies where nqr spectrum was observed . note that @xmath51 stays constant generally , if @xmath53 and/or @xmath54 range in the observable time window that is typically more than several microseconds . therefore , the distinct reduction in @xmath51 upon cooling is ascribed to the development of antiferromagnetic critical fluctuations , since it leads to an extraordinary short relaxation time of @xmath55 0.14 @xmath56sec @xcite . dependence of @xmath51 at several pressures , where @xmath52 is an nqr intensity normalized by the value at 4.2 k. , width=302 ] the @xmath57 at @xmath38 gpa decreases down to about @xmath58 at @xmath40 k upon cooling below @xmath59 1.2 k. its reduction stops around @xmath10 , but does no longer recover with further decreasing @xmath14 . note that its reduction below @xmath60 k is due to the superconducting diamagnetic shielding of rf field for the nqr measurement . as @xmath3 increases , @xmath24 becomes smaller , in agreement with the result presented in the phase diagram of fig . 2 , and the reduction in @xmath51 becomes moderate in the normal state . with further increasing @xmath3 up to 0.91 gpa , eventually , @xmath51 remains nearly constant down to @xmath61 k , indicative of no anomaly related to antiferromagnetic critical fluctuations . this behavior resembles the result observed at pressures exceeding @xmath62 gpa in ce@xmath17cu@xmath18si@xmath0 . these results also assure that the ge substitution expands the lattice of ce@xmath17cu@xmath18si@xmath0 . thus , the exotic sc in ce@xmath17cu@xmath18si@xmath0 and cecu@xmath0(si@xmath36ge@xmath37)@xmath0 at @xmath38 gpa is characterized by the persistence of low - lying antiferromagnetic critical excitations . it was argued that these excitations may be related to a collective mode in the uniform mixed phase of afm and sc @xcite . it seems , therefore , that these exotic sc could be rather robust against the appearance of afm . ) vs @xmath14 phase diagram of cecu@xmath0si@xmath0.@xcite , width=253 ] here , we deal with the @xmath16 vs @xmath14 phase diagram of ce@xmath17cu@xmath0si@xmath0 shown in figure 5 . when @xmath16 suppresses its sc , exceeding an upper critical field @xmath63 , various measurements revealed an evolution from the sc into some magnetic phase that was called as _ a - phase _ this phase emerges below @xmath64 close to the @xmath9 at @xmath65 . recent neutron diffraction experiment has revealed that a single crystal exhibiting _ a - phase _ anomalies below @xmath66 k undergoes a long - range incommensurate afm at @xmath65 . it was suggested that a spin - density - wave ( sdw ) instability is the origin of the magnetic qcp in cecu@xmath0si@xmath0 @xcite . markedly , the recent cu - nmr measurement is consistent with the onset of static magnetic order as the nature of @xmath16 induced _ a - phase _ @xcite , although _ b - phase _ is not yet under detailed investigations . upon accepting the existence of a second order quantum critical point @xmath67 between the uniform mixed phase of sc+sdw and the phase of sc , a promising theoretical approach is available for the analysis of experiments in the study of the influence of an applied magnetic field on such the critical point @xcite . in order to address an origin of _ a - phase _ , further extensive works are required and now in progress . antiferromagnetic critical fluctuations develop below @xmath68 in @xmath69 in cecu@xmath0(si@xmath5ge@xmath6)@xmath0 and @xmath70 gpa in cecu@xmath0si@xmath0 . remarkably this marginal afm emerges closely to the border between afm and sc . once slight ge is substituted for si to expand its lattice , the afm suddenly sets in . by contrast , afm is not observed down to 0.012 k at @xmath71 . with increasing @xmath21 , @xmath72 is progressively increased , while @xmath73 is steeply decreased . correspondingly , antiferromagnetic critical fluctuations are suppressed for the samples at more than @xmath21 = 0.06 . it is noteworthy that the afm seems to suddenly disappear at @xmath21 = 0 as if @xmath72 was replaced by @xmath73 . eventually , the sc coexists with the marginal afm at @xmath71 @xcite . this fact suggests that afm and sc have a common background . in @xmath70 gpa , the marginal afm is expelled by the onset of the sc below @xmath73 at @xmath16 = 0 . however , when the application of @xmath16 turns on to suppress the sc , the first - order like transition from the sc to the magnetic _ a - phase _ takes place @xcite . in @xmath74 0.2 gpa , as a result of the complete suppression of the marginal afm , the typical hf sc takes place with the line - node gap . in cecu@xmath0(si@xmath5ge@xmath6)@xmath0 , one @xmath75 electron per ce ion plays vital role for both the sc and afm in @xmath69 leading to the novel states of matter . we have proposed that the uniform mixed phase of afm and sc in the slightly ge substituted compounds , and the magnetic - field induced _ a - phase _ for the homogeneous cecu@xmath0si@xmath0 in @xmath70 gpa are accounted for on the basis of an so(5 ) theory @xcite . it is considered that the marginal afm in @xmath69 below @xmath73 may be identified as a collective magnetic mode in the uniform mixed phase of afm and sc and that it turns out to be competitive with the onset of sc in @xmath70 gpa at @xmath16 = 0 . the latter is relevant with the @xmath16-induced first - order transition from the sc to the _ a - phase_. concerning the interplay between afm and sc , we would propose that the marginal afm in the sc at @xmath21 = 0 may correspond to a pseudo goldstone mode due to the broken u(1 ) symmetry . due to the closeness to the magnetic qcp , however , such the gapped mode in the sc should be characterized by an extremely tiny excitation ( resonance ) energy . the intimate interplay between sc and afm found in uniform cecu@xmath0si@xmath0 has been a long - standing problem - unresolved for over a decade . we have proposed that the so(5 ) theory constructed on the basis of quantum - field theory may give a coherent interpretation for these exotic phases found in the hf superconductor cecu@xmath0si@xmath0 @xcite . in this context , we would suggest that the sc in cecu@xmath0si@xmath0 could be mediated by the same magnetic interaction as leads to the afm in cecu@xmath0(si@xmath5ge@xmath6)@xmath0 . this is in marked contrast to the bcs superconductors in which the pair binding is mediated by phonons @xmath76 vibrations of the lattice density . a new antiferromagnetic hf compound cerhin@xmath1 undergoes the helical magnetic order at a nel temperature @xmath77 k with an incommensurate wave vector @xmath78 @xcite . a neutron experiment revealed the reduced ce magnetic moments @xmath79 0.8@xmath80 @xcite . the @xmath3-induced transition from afm to sc takes place at a relatively lower critical pressure @xmath81 gpa and higher @xmath82 k than in previous examples @xcite . figure 6 indicates the @xmath3 vs @xmath14 phase diagram of cerhin@xmath1 for afm and sc that was determined by the in - nqr measurements under @xmath3 . vs @xmath14 phase diagram for cerhin@xmath1 . the respective marks denoted by solid square , open triangle and cross correspond to the pseudogap temperature @xmath83 , the antiferromagnetic ordering temperature @xmath10 and the internal field @xmath84 at the in site . the open and solid circles correspond to the onset temperature @xmath85 and @xmath86 of superconducting transition ( see text ) . dotted line denotes the position for @xmath87 gpa . shaded region indicates the coexistent @xmath3 region of afm and sc.,width=291 ] the nqr study showed that @xmath10 gradually increases up to 4 k as @xmath3 increases up to @xmath88 gpa and decreases with further increasing @xmath3 @xcite . in addition , the @xmath14 dependence of @xmath30 probed the pseudogap behavior at @xmath89 and 1.6 gpa @xcite . this suggests that cerhin@xmath1 may resemble other strongly correlated electron systems @xcite . note that the value of bulk superconducting transition temperature @xmath86 is progressively reduced as shown by closed circle in figure 6 . apart from the afm at @xmath90 gpa exceeding @xmath19 , _ @xmath30 decreases obeying a @xmath91 law _ without the coherence peak just below @xmath9 . this indicates that the sc of cerhin@xmath1 is unconventional with the line - node gap @xcite . vs @xmath92 at @xmath3 = 0 and 1.6 gpa ( see text ) . inset shows the @xmath93in - nqr spectra of 1@xmath26 at @xmath3 = 1.6 gpa above and below @xmath10 = 2.8 k. , width=245 ] we present microscopic evidence for the exotic sc at the uniform mixed phase of afm and sc in cerhin@xmath1 at @xmath94 gpa . the inset of figure 7 displays the nqr spectra above and below @xmath10 at @xmath94 gpa . below @xmath95 k , the nqr spectrum splits into two peaks due to the appearance of @xmath84 at the in site . this is clear evidence for the occurrence of afm at @xmath94 gpa . the plots of @xmath96 vs @xmath97 at @xmath38 and 1.6 gpa are compared in figure 7 , showing nearly the same behavior . here @xmath98 is the value extrapolated to zero at @xmath67 k and @xmath99 is the @xmath14 dependence of spontaneous staggered magnetic moment . the character of afm at @xmath94 gpa is expected to be not so much different from that at @xmath38 . figure 8 indicates the @xmath14 dependence of @xmath30 at @xmath94 gpa . a clear peak in @xmath30 is due to antiferromagnetic critical fluctuations at @xmath95 k. below @xmath95 k , @xmath30 continues to decrease moderately down to @xmath100 k even though passing across @xmath101 k. this relaxation behavior suggests that sc does not develop following the mean - field approximation below @xmath85 . markedly , @xmath30 decreases below @xmath86 , exhibiting a faint @xmath91 behavior in a narrow @xmath14 range . with further decreasing @xmath14 , @xmath30 becomes proportional to the temperature , indicative of the gapless nature in low - lying excitation spectrum in the coexistent state of sc and afm on a microscopic level . thus the @xmath53 measurement has revealed that the intimate interplay between afm and sc gives rise to an _ amplitude fluctuation of superconducting order parameter _ between @xmath85 and @xmath86 . such fluctuations may be responsible for the broad transition in resistance and ac - susceptibility ( @xmath102 ) measurements . furthermore , the @xmath44 const behavior well below @xmath86 evidences the gapless nature in low - lying excitations at the uniform mixed phase of afm and sc . this result is consistent with those in cecu@xmath0si@xmath0 at the border to afm @xcite and a series of cecu@xmath0(si@xmath5ge@xmath0)@xmath0 compounds that show the uniform mixed phase of afm and sc @xcite . the specific - heat result under @xmath3 , that probed a finite value of its @xmath14-linear contribution , @xmath103 @xmath55 100 mj / molk@xmath104 at @xmath105 gpa is now understood due not to a first - order like transition from afm to sc @xcite , but to the gapless nature in the uniform mixed phase of afm and sc . dependence of @xmath30 at @xmath3 = 1.6 gpa . both dotted lines correspond to @xmath45 and @xmath31 . inset indicates the @xmath14 dependence of @xmath106al - nqr @xmath30 of upd@xmath0al@xmath2 cited from the literature @xcite . dotted line corresponds to @xmath31.,width=264 ] it is noteworthy that such @xmath50 = const . behavior is not observed below @xmath9 at @xmath3 = 2.1 gpa @xcite , consistent with the specific - heat result under @xmath3 as well @xcite . this means that the origin for the @xmath44 const . behavior below @xmath86 at @xmath94 gpa is not associated with some impurity effect . if it were the case , the residual density of states below @xmath9 should not depend on @xmath3 . this novel feature differs from the uranium(u)-based hf antiferromagnetic superconductor upd@xmath0al@xmath2 which has multiple 5@xmath13 electrons . in upd@xmath0al@xmath2 , a superconducting transition occurs at @xmath9 = 1.8 k well below @xmath10 = 14.3 k @xcite . as indicated in the inset of figure 8 @xcite , in upd@xmath0al@xmath2 , _ @xmath30 decreases obeying a @xmath91 law over three orders of magnitude _ below the onset of @xmath9 without any trace for the @xmath50 = const . behavior . this is consistent with the line - node gap even in the uniform mixed phase of afm and sc . in order to highlight the novel sc on a microscopic level , the @xmath14 dependence of @xmath23 is shown in figure 9(a ) at @xmath94 gpa in @xmath107 k and is compared with the @xmath14 dependence of the resistance @xmath108 at @xmath109 gpa referred from the literature @xcite . although each value of @xmath3 is not exactly the same , they only differ by 2% . we remark that the @xmath14 dependence of @xmath23 points to the pseudogap behavior around @xmath83 = 4.2 k , the afm at @xmath10 = 2.8 k , and the sc at @xmath86 = 0.9 k at which @xmath110 has a peak as seen in figure 9(b ) . this result itself evidences the uniform mixed state of afm and sc . a comparison of @xmath23 with the @xmath108 at @xmath109 gpa in figure 9(b ) is informative in shedding light on the uniqueness of afm and sc . below @xmath83 , @xmath108 starts to decrease more rapidly than a @xmath14-linear variation extrapolated from a high @xmath14 side . it continues to decrease across @xmath10 = 2.8 k , reaching the zero resistance at @xmath111 1.5 k. the resistive superconducting transition width becomes broader . unexpectedly , @xmath112 2 k , that is defined as the temperature below which the diamagnetism starts to appear , is higher than @xmath111 1.5 k. any signature for the onset of sc from the @xmath30 measurement is not evident in between @xmath85 and @xmath86 , demonstrating that the mean - field type of gap does not grow up down to @xmath113 0.9 k. _ the existence of fluctuations due to the interplay of afm and sc is responsible for the broad transition towards the uniform mixed phase of sc and afm . _ dependence of @xmath23 at @xmath94 gpa . ( b ) the @xmath14 dependencies of @xmath110 at @xmath94 gpa and resistance at @xmath109 gpa cited from the literature @xcite . @xmath86 and @xmath85 correspond to the respective temperatures at which @xmath110 has a peak and below which @xmath102 starts to decrease . @xmath10 corresponds to the antiferromagnetic ordering temperature at which @xmath23 exhibits a peak and @xmath83 to the pseudogap temperature below which it starts to decrease . a solid line is an eye guide for the @xmath14- linear variation in resistance at temperatures higher than @xmath83.,width=291 ] recent neutron - diffraction experiment suggests that the size of staggered moment @xmath114 in the afm is almost independent of @xmath3 @xcite . its relatively large size of moment with @xmath115 seems to support such a picture that the _ same @xmath13-electron _ exhibits simultaneously itinerant and localized dual nature , because there is only one @xmath75-electron per ce@xmath116 ion . in this context , it is natural to consider that the superconducting nature in the uniform mixed phase of afm and sc belongs to a novel class of phase which differs from the _ unconventional @xmath48-wave sc with the line - node gap_. as a matter of fact , a theoretical model has been recently put forth to address the underlying issue in the uniform mixed phase of afm and sc @xcite . figure 10 indicates the @xmath3 vs @xmath14 phase diagram in cein@xmath2 around @xmath19 . this work has deepened the understanding of the physical properties on the verge of afm in cein@xmath2 that exhibits the archetypal phase diagram shown in figure 1(a ) @xcite . the localized magnetic character is robust up to @xmath117 gpa . the characteristic temperature @xmath118 , below which the system crosses over to an itinerant magnetic regime , increases dramatically with further increase of @xmath3 . as a result , the measurements of @xmath30 and @xmath102 at @xmath119 gpa down to @xmath120 mk provided the first evidence of unconventional sc at @xmath121 mk in cein@xmath2 , which arises in the hf state fully established below @xmath122 k @xcite . by contrast , the phase separation into afm and paramagnetism ( pm ) is evidenced in cein@xmath2 from the observation of two kinds of nqr spectra in @xmath123 gpa . nevertheless , it is highlighted that the sc in cein@xmath2 occurs in both the afm and pm at @xmath124 gpa . the maximum value of @xmath125 mk is observed for the sc in pm . markedly , the sc coexisting with afm emerges below @xmath126 mk . the present results indicate the occurrence of the first - order phase transition from the uniform mixed phase of sc and afm to the single phase of sc under the hf state of pm around @xmath19 . therefore , a qcp is absent in cein@xmath2@xcite . vs @xmath14 phase diagram of cein@xmath2 determined from the present experiment . the @xmath3 and @xmath14 ranges where the phase separation of afm and pm occurs are shaded in the figure.,width=291 ] cein@xmath2 forms in the cubic aucu@xmath2 structure and orders antiferromagnetically below @xmath127 k at @xmath38 with an ordering vector * q * = ( 1/2,1/2,1/2 ) and ce magnetic moment @xmath128 , which were determined by nqr measurements@xcite and the neutron - diffraction experiment on single crystals@xcite , respectively . the resistivity measurements of cein@xmath2 have clarified the @xmath129 phase diagram of afm and sc : @xmath10 decreases with increasing @xmath3 . on the verge of afm , the sc emerges in a narrow @xmath3 range of about 0.5 gpa , exhibiting a maximum value of @xmath130 k at @xmath131 gpa where afm disappears @xcite . dependence of @xmath93 in nqr spectrum for cein@xmath2 at @xmath132 gpa above @xmath10 ( a ) and @xmath3 = 2.37 gpa ( b ) , 2.43 gpa ( c ) and 2.50 gpa ( d ) at temperatures lower than the @xmath10 and @xmath9 . the dotted line indicates the peak position at which the nqr spectrum is observed for pm.,width=283 ] figure 11 shows the nqr spectra of 1@xmath133 transition for the pm at ( a ) @xmath134 gpa and for temperatures lower than @xmath10 and @xmath9 at ( b ) @xmath134 gpa , ( c ) @xmath124 gpa and ( d ) @xmath135 gpa . note that the 1@xmath26 transition can sensitively probe the appearance of internal field @xmath136 associated with even tiny ce ordered moments on the verge of afm . as a matter of fact , as seen in figures 11(a ) and ( b ) , a drastic change in the nqr spectral shape is observed due to the occurrence of @xmath136 at the in nuclei below @xmath10 . by contrast , the spectra at ( c ) @xmath124 gpa and ( d ) @xmath135 gpa include two kinds of spectra arising from afm and pm provides firm microscopic evidence for the emergence of magnetic phase separation . the volume fraction of afm at @xmath137 mk at each pressure is plotted as function of @xmath3 , as seen in figure 12(a ) ( solid triangles ) . it should be noted that , as shown in figure 10 , the phase separations at @xmath138 and 2.37 gpa are observed only between @xmath139 k and @xmath140 k and between @xmath141 k and @xmath142 k , respectively . dependence of the volume fraction of afm(solid triangles ) and sc in pm(open circles ) . the volume fraction of afm is evaluated from the integration of the nqr spectrum over the frequency at @xmath14 = 100 mk at each pressure . the superconducting volume fraction at each pressure is estimated by comparing the value of @xmath102 at the lowest temperature for cein@xmath2 with that for the hf superconductor ceirin@xmath1@xcite . the phase separation of afm and pm takes place at @xmath143 gpa . note that the uniform mixed phase of sc and afm emerges in the shaded region . ( b ) the @xmath14 dependence of @xmath102 in @xmath144 gpa . the solid arrow indicates the onset @xmath9 for sc . , width=283 ] figure 12(b ) shows the @xmath14 dependence of @xmath102 for cein@xmath2 under various values of @xmath3 . even though the magnetic phase separation takes place , as shown by shading in figure 10 , the clear decrease in @xmath102 points to the bulk nature of sc at @xmath145 gpa . it is , however , noteworthy that there is no indication of sc down to @xmath14 = 30 mk at @xmath3 = 2.17 gpa where no magnetic phase separation is observed at all against @xmath3 and @xmath14 . when taking into account the fact that the value of @xmath9 reaches a maximum at @xmath146 gpa where the volume fraction of afm and pm remains comparable , it is expected that the uniform mixed phase of sc and afm takes place only in the @xmath3 region where the phases of afm and pm are separated . the results of the @xmath3 dependence of nqr spectrum and @xmath147 strongly suggested that the uniform mixed phase of afm and sc is separated from the phase of hf sc under pm in the shaded region in figure 12(a ) . the present results revealed that there emerges the first - order transition from the uniform mixed phase of sc and afm to the phase of hf sc under pm . note that the sc takes place under both the backgrounds of afm and pm , although the superconducting characteristics are significantly different . the uniform mixed phase of afm and sc is corroborated by direct evidence from the @xmath14 dependence of @xmath23 that can probe the low - lying excitations due to quasiparticles in sc and the magnetic excitations in afm . figure 13 shows the drastic evolution in the @xmath3 and @xmath14 dependencies of @xmath23 for afm ( solid symbols ) and pm ( open symbols ) at @xmath3 = 2.17 , 2.28 , 2.43 and 2.50 gpa . here , @xmath10 was determined as the temperature below which the nqr intensity for pm decreases due to the emergence of afm associated with the magnetic phase separation . the @xmath53 for afm and pm is separately measured at respective nqr peaks which are clearly distinguished from each other , as shown in figures 11(b ) , 11(c ) and 11(d ) . thus , the respective @xmath148 and @xmath149 for afm and pm are determined as the temperature below which @xmath23 decreases markedly due to the opening of superconducting gap . these results verify that the uniform mixed phase of afm and sc takes place on a microscopic level in @xmath123 gpa . dependence of @xmath150 in cein@xmath2 at @xmath151 and 2.28 gpa ( a ) , 2.43 gpa ( b ) and 2.50 gpa ( c ) . open and solid symbols indicate the data for pm and afm measured at @xmath55 9.8 mhz and @xmath55 8.2 mhz , respectively . the solid arrow indicates the respective superconducting transition temperatures @xmath149 and @xmath148 for pm and afm . the dotted and dashed arrows indicate , respectively , the @xmath10 and the characteristic temperature @xmath152 below which the @xmath153const . law ( dotted line ) is valid , which is characteristic of the fermi - liquid state.,width=321 ] in the pm at @xmath124 and 2.50 gpa , the @xmath23 = const . relation is valid below @xmath154 k and @xmath155 k , respectively , which indicates that the fermi - liquid state is realized . this result is in good agreement with that of the previous resistivity measurement from which the @xmath156 dependence in resistance was confirmed @xcite . note that the @xmath30 for the sc in the pm at @xmath135 gpa follows a @xmath91 dependence below @xmath126 mk , consistent with the line - node gap model characteristic for unconventional hf sc . as shown in figure 13(a ) , no phase separation occurs below @xmath157 k at @xmath151 gpa , and @xmath158 const . behavior is observed well below @xmath10 . it is also observed in the previous nqr measurements at @xmath38 @xcite . the @xmath14 dependence of @xmath23 at @xmath138 gpa resembles the behavior at @xmath151 gpa above @xmath159 1 k. however , at @xmath138 gpa , the phase separation of afm and pm occurs in the small @xmath14 window between @xmath140 k and 3 k. in contrast to @xmath158 const . behavior well below @xmath10 at @xmath151 gpa , the @xmath23 at @xmath138 gpa continues to increase upon cooling below @xmath10 and exceeds the values of around @xmath10 in spite of antiferromagnetic spin polarization being induced . these results suggest that the occurrence of magnetic phase separation is closely responsible for the onset of sc , and yields the large enhancement of the low - lying magnetic excitations for afm . this feature is also seen for the afm at @xmath124 gpa where @xmath23 is larger than the value for pm , as shown in figure 13(b ) . it is still unknown why such low - lying magnetic excitations continue to be enhanced well below @xmath10 . some spin - density fluctuations may be responsible for this feature in association with the phase separation of afm and pm . in this context , cein@xmath2 is not in a magnetically soft electron liquid state @xcite , but instead , the relevant magnetic excitations , such as spin - density fluctuations , induced by the first - order transition from afm to pm might mediate attractive interaction . whatever its pairing mechanism is at @xmath138 gpa where afm is realized over the whole sample below @xmath139 k , the clear decrease in @xmath23 and @xmath102 provide convincing evidence for the uniform mixed phase of afm and sc in cein@xmath2 at @xmath138 gpa . further evidence for the new type of sc uniformly mixed with afm was obtained from the results at @xmath124 gpa , as indicated in figure 13(b ) . at temperatures lower than the respective values of @xmath149 = 230 mk and @xmath148 = 190 mk for pm and afm , unexpectedly , the magnitude of @xmath158 const . coincide with one another for both the phases that are magnetically separated into afm and pm with the respective different values of @xmath9 . this means that the quasi - particle excitations for the uniform mixed phase of sc and afm may be the same in origin as for the phase of sc in pm . how does this happen ? it may be possible that both the phases are in a dynamically separated regime with time scales smaller than the inverse of nqr frequency so as to make each superconducting phase for the sc under afm and for the pm uniform . in this context , the observed magnetically separated phases and the relevant phases of sc may belong to new phases of matter . we have provided evidence for the phase separation of afm and pm ( shaded area in figure 10 ) and the new type of sc uniformly mixed with the afm near @xmath19 in cein@xmath2 . it has been found that the highest value of @xmath160 mk in cein@xmath2 is observed for the pm at @xmath124 gpa where the volume fraction of afm and pm becomes almost the same . the present experiments have revealed that this new type of sc is mediated by a novel pairing interaction associated with the magnetic phase separation . we propose that _ the magnetic excitations , such as spin - density fluctuations , induced by the first - order magnetic phase transition might mediate attractive interaction to form cooper pairs in cein@xmath2 ; this is indeed a new type of pairing mechanism_. the sc in hf compounds has not yet been explained from the microscopic point of view , mainly due to the strong correlation effect and the complicated band structures . an essential task seems to identify the residual interaction between quasi - particles through analyzing the effective @xmath13-band model by choosing dominant bands @xcite . here , we have demonstrated that hf superconductors possess a great variety of ground states at the boundary between sc and afm with anomalous magnetic and superconducting properties . a genuine uniform mixed phase of afm and sc has been observed in cecu@xmath0si@xmath0 and cerhin@xmath1 in the pressure ( @xmath3 ) versus temperature ( @xmath14 ) phase diagram through the extensive and precise nqr measurements under @xmath3 . in other strongly correlated electron systems , the sc appears near the boundary to the afm . even though the underlying solid state chemistries are rather different , the resulting phase diagrams are strikingly similar and robust . this similarity suggests that the overall feature of all these phase diagrams is controlled by a single energy scale . in order to gain an insight into the interplay between afm and sc , here , we try to focus on a particular theory , which unifies the afm and sc of the heavy - fermion systems based on an so(5 ) theory , because symmetry unifies apparently different physical phenomena into a common framework as all fundamental laws of nature @xcite . the uniform mixed phase diagram of afm and sc and the exotic sc affected by strong antiferromagnetic fluctuations could be understood in terms of an so(5 ) superspin picture @xcite . by contrast , cein@xmath2 has revealed the @xmath3 induced first - order transition from afm to pm as functions of pressure and temperature near their boundary . unexpectedly , however , the sc is robust under the phase separation into afm and pm , even though the superconducting characteristics in the afm strikingly differ from those in the pm . the uniform mixed phase of afm and sc has been also observed , however , neither superconducting fluctuations nor the development of low - lying magnetic excitations have been revealed . these features differ significantly from those observed in cecu@xmath0si@xmath0 and cerhin@xmath1 . instead , the spin - density fluctuations develop as temperature goes down far below @xmath10 , leading to the onset of the sc under the background of afm which is separated from the pm . these new phenomena observed in cein@xmath2 should be understood in terms of a quantum phase separation because these new phases of matter are induced by applying pressure . in fermion systems , if the magnetic critical temperature at the termination point of the first - order transition is suppressed at @xmath19 , the diverging magnetic density fluctuations inherent at the critical point from the magnetic to paramagnetic transition become involved in the quantum fermi degeneracy region . the fermi degeneracy by itself generates various instabilities called as the fermi surface effects , one of which is a superconducting transition . on the basis of a general argument on quantum criticality , it is shown that the coexistence of the fermi degeneracy and the critical density fluctuations yield a new type of quantum criticality @xcite . in this context , the results on cein@xmath2 deserve further theoretical investigations . we believe that the results presented here on cecu@xmath0si@xmath0 , cerhin@xmath1 and cein@xmath2 provide vital clue to unravel the essential interplay between the afm and the sc , and to extend the universality of the understanding on the sc in strongly correlated electron systems . these works are currently stimulated in collaboration with c. geibel and f. steglich on the study of cecu@xmath0si@xmath0 and with prof . y. nuki and his many coworkers on the studies of cerhin@xmath1 and cein@xmath2 . these works were supported by the coe research grant from mext of japan ( grant no . 10ce2004 ) . + steglich f. @xmath193 @xmath194 . , in _ proceedings of physical phenomena at high magnetic fields - ii _ , edited by fisk z , gorkov l , meltzer d and schrieffer r ( world scientific , singapore , 1996 ) , p. 125 .

we report the discovery of exotic superconductivity ( sc ) and novel magnetism in heavy - fermion ( hf ) compounds , cecu@xmath0si@xmath0 , cerhin@xmath1 and cein@xmath2 on the verge of antiferromagnetism ( afm ) through nuclear - quadrupole - resonance ( nqr ) measurements under pressure ( @xmath3 ) . the exotic sc in a homogeneous cecu@xmath0si@xmath0 ( @xmath4 k ) revealed _ antiferromagnetic critical fluctuations _ at the border to afm or marginal afm . remarkably , it has been found that the application of magnetic field induces an spin - density - wave ( sdw ) transition by suppressing the sc near the upper critical field . furthermore , the uniform mixed phase of sc and afm in cecu@xmath0(si@xmath5ge@xmath6)@xmath0 emerges on a microscopic level , once a tiny amount of 1%ge(@xmath7 ) is substituted for si to expand its lattice . the application of minute pressure ( @xmath8 gpa ) suppresses the sudden emergence of the afm caused by doping ge . the persistence of the low - lying magnetic excitations at temperatures lower than @xmath9 and @xmath10 is ascribed due to the uniform mixed phase of sc and afm . likewise , the @xmath3-induced hf superconductor cerhin@xmath1 coexists with afm on a microscopic level in @xmath11 - 1.9 gpa . it is demonstrated that sc does not yield any trace of gap opening in low - lying excitations below the onset temperature , presumably associated with _ an amplitude fluctuation of superconducting order parameter_. the unconventional gapless nature of sc in the low - lying excitation spectrum emerges due to the uniform mixed phase of afm and sc . by contrast , in cein@xmath2 , the @xmath3-induced _ phase separation _ into afm and paramagnetism ( pm ) takes place without any trace for a quantum phase transition . the outstanding finding is that sc sets in at both the phases magnetically separated into afm and pm in @xmath12 gpa . a new type of sc forms the uniform mixed phase with the afm and the hf sc takes place in the pm . we propose that the magnetic excitations such as spin - density fluctuations induced by the first - order phase transition from the afm to the pm might mediate attractive interaction to form the cooper pairs in the novel phase of afm .

cond-mat0405348

twitter and other social media have become important communication channels for the general public . it is thus not surprising that various stakeholder groups in science also participate on these platforms . scientists , for instance , use twitter for generating research ideas and disseminating and discussing scientific results @xcite . many biomedical practitioners use twitter for engaging in continuing education ( e.g. , journal clubs on twitter ) and other community - based purposes @xcite . policy makers are active on twitter , opening lines of discourse between scientists and those making policy on science @xcite . quantitative investigations of scholarly activities on social media often called altmetrics can now be done at scale , given the availability of apis on several platforms , most notably twitter @xcite . much of the extant literature has focused on the comparison between the amount of online attention and traditional citations collected by publications , showing low levels of correlation . such low correlation has been used to argue that altmetrics provide alternative measures of impact , particularly the broader impact on the society @xcite , given that social media provide open platforms where people with diverse backgrounds can engage in direct conversations without any barriers . however , this argument has not been empirically grounded , impeding further understanding of the validity of altmetrics and the broader impact of articles . a crucial step towards empirical validation of the broader impact claim of altmetrics is to identify scientists on twitter , because altmetric activities are often assumed to be generated by the public " rather than scientists , although it is not necessarily the case . to verify this , we need to be able to identify scientists and non - scientists . although there have been some attempts , they suffer from a narrow disciplinary focus @xcite and/or small scale @xcite . moreover , most studies use purposive sampling techniques , pre - selecting candidate scientists based on their success in other sources ( e.g. , highly cited in web of science ) , instead of organically finding scientists from the twitter platform itself . such reliance on bibliographic databases binds these studies to traditional citation indicators and thus introduces bias . for instance , this approach overlooks early - career scientists and favors certain disciplines . here we present the first large - scale and systematic study of scientists across many disciplines on twitter . as our method does not rely on external bibliographic databases and is capable of identifying any user types that are captured in twitter list , it can be adapted to identify other types of stakeholders , occupations , and entities . we study the demographics of the set of scientists in terms of discipline and gender , finding over - representation of social scientists , under - representation of mathematical and physical scientists , and a better representation of women compared to the statistics from scholarly publishing . we then analyze the sharing behaviors of scientists , reporting that only a small portion of shared urls are science - related . finally , we find an assortative mixing with respect to disciplines in the follower , retweet , and mention networks between scientists . our study serves as a basic building block to study scholarly communication on twitter and the broader impact of altmetrics . we classify current literature into two main categories , namely _ product_- vs. _ _ producer-__centric perspectives . the former examines the sharing of scholarly papers in social media and its impact , the latter focuses on who generates the attention . * product - centric perspective . * priem and costello formally defined twitter citations as direct or indirect links from a tweet to a peer - reviewed scholarly article online " and distinguished between first- and second - order citations based on whether there is an intermediate web page mentioning the article @xcite . the accumulation of these links , they argued , would provide a new type of metric , coined as altmetrics , " which could measure the broader impact beyond academia of diverse scholarly products @xcite . many studies argued that only a small portion of research papers are mentioned on twitter @xcite . for instance , a systematic study covering @xmath0 million papers indexed by both pubmed and web of science found that only @xmath1 of them have mentions on twitter @xcite , yet this is much higher than other social media metrics except mendeley . the coverages vary across disciplines ; medical and social sciences papers that may be more likely to appeal to a wider public are more likely to be covered on twitter @xcite . mixed results have been reported regarding the correlation between altmetrics and citations @xcite . a recent meta - analysis showed that the correlation is negligible ( @xmath2 ) @xcite ; however , there is dramatic differences across studies depending on disciplines , journals , and time window . * producer - centric perspective . * survey - based studies examined how scholars present themselves on social media @xcite . a large - scale survey with more than @xmath3 responses conducted by _ nature _ in @xmath4 revealed that more than @xmath5 were aware of twitter , yet only @xmath6 were regular users @xcite . a handful of studies analyzed how twitter is used by scientists . priem and costello examined @xmath7 scholars to study how and why they share scholarly papers on twitter @xcite . an analysis of @xmath8 emergency physicians concluded that many users do not connect to their colleagues while a small number of users are tightly interconnected @xcite . holmberg and thelwall selected researchers in @xmath9 disciplines and found clear disciplinary differences in twitter usages , such as more retweets by biochemists and more sharing of links for economists @xcite . note that these studies first selected scientists outside of twitter and then manually searched their twitter profiles . two limitations thus exist for these studies . first , the sample size is small due to the nature of manual searching @xcite . second , the samples are biased towards more well - known scientists . one notable exception is a study by hadgu and jschke , who presented a supervised learning based approach to identifying researchers on twitter , where the training set contains users who were related to some computer science conference handles @xcite . although this study used a more systematic method , it still relied on the dblp , an external bibliographic dataset for computer science , and is confined in a single discipline . defining science and scientists is a herculean task and beyond the scope of this paper . we thus adopt a practical definition , turning to the @xmath10 standard occupational classification ( soc ) system ( http://www.bls.gov/soc/ ) released by the bureau of labor statistics , united states department of labor . we use soc because not only it is a practical and authoritative guidance for the definition of scientists but also many official statistics ( e.g. , total employment of social scientists ) are released according to this classification system . soc is a hierarchical system that classifies workers into @xmath11 major occupational groups , among which we are interested in two , namely ( 1 ) computer and mathematical occupations ( code 15 - 0000 ) and ( 2 ) life , physical , and social science occupations ( code 19 - 0000 ) . from the two groups , we compile @xmath7 scientist occupations ( supporting table s1 ) . although authoritative , the soc does not always meet our intuitive classifications of scientists . for instance , biologists " is not presented in the classification . we therefore consider another source wikipedia to augment the set of scientist occupations . in particular , we add the occupations listed at http://en.wikipedia.org/wiki/scientist#by_field . we then compile a list of scientist titles from the two sources . this is done by combining titles from soc , wikipedia , and illustrative examples under each soc occupation . we also add two general titles : scientists " and researchers . " for each title , we consider its singular form and the core disciplinary term . for instance , for the title clinical psychologists , " we also consider clinical psychologist , " psychologists , " and psychologist . " we assemble a set of @xmath12 scientist titles using this method . our method of identifying scientists is inspired by a previous study that used twitter _ lists _ to identify user expertise @xcite . a twitter _ list _ is a set of twitter users that can be created by any twitter user . the creator of a list needs to provide a name and optional description . although the purpose of lists is to help users organize their subscriptions , the names and descriptions of lists can be leveraged to infer attributes of users in the lists . imagine a user creating a list called economist " and putting http://twitter.com/betseystevenson[@betseystevenson ] in it ; this signals that @betseystevenson may be an economist . if @betseystevenson is included in numerous lists all named economist , " which means that many independent twitter users classify her as an economist , it is highly likely that @betseystevenson is indeed an economist . this is illustrated in fig [ fig : list - name - wordcloud ] where the word cloud of the names of twitter lists containing @betseystevenson is shown . we can see that economist " is a top word frequently appeared in the titles , signaling the occupation of this user . in other words , we crowdsource " the identity of each twitter user . . ] in principle , we could use twitter s ` memberships ` api ( https://dev.twitter.com/rest/reference/get/lists/memberships ) , for each user , to get all the lists containing this user , and then infer whether this user is a scientist by analyzing the names and descriptions of these lists . however , this method is highly infeasible , because ( 1 ) most users are not scientists , ( 2 ) the distribution of listed counts is right - skewed : lady gaga , for example , is listed more than @xmath13 times ( https://www.electoralhq.com/twitter-users/most-listed ) , and ( 3 ) twitter api has rate limits . we instead employ a previously introduced list - based snowball sampling method @xcite that starts from a given initial set of users and expands to discover more . we improve this approach by more systematically obtaining the job title lexicon and the seed user set ( supporting text ) . we use the snowball sampling ( breadth - first search ) on twitter lists . we first identify seed users ( supporting text ) and put them into a queue . for each public user in the queue , we get all the lists in which the user appears , using the twitter ` memberships ` api . then , for each public list in the subset resulting lists whose name contains at least one scientist title , we get its members using the twitter ` members ` api ( https://dev.twitter.com/rest/reference/get/lists/members ) and put those who have not been visited into the queue . the two steps are repeated until the queue is empty , which completes the sampling process . note that to remove many organizations and anonymous users as well as to speed up the sampling , we only consider users whose names contain spaces . we acknowledge that this may drop many users with non - english names or the ones who do not disclose their names in a standard way . also note that this procedure is inherently blind towards those scientists who are not listed . from the sampling procedure , we get @xmath14 users appearing in @xmath15 lists whose names contain scientist titles . to increase the precision of our method , the final dataset contains those users whose profile descriptions also contain scientist titles . a total number of @xmath16 users are found . table [ tab : top - listed ] shows the top @xmath17 scientists based on number of lists whose names contain scientist titles , suggesting that our sampling method can identify scientists in diverse disciplines . these top scientists have very different levels of popularities , with the most followed scientist neil degrasse tyson attracting more than @xmath18 million followers while miles kimball has @xmath19 thousand followers . -2.25in0 in .*top 30 scientists based on the number of twitter lists whose names contain scientist titles . * [ cols="<,>,>,>,<,>,>,>",options="header " , ] [ tab : community - top - nodes ] 1 m. gabielkov , a. rao , and a. legout . studying social networks at scale : macroscopic anatomy of the twitter social graph . in _ proc . of the 2014 acm international conference on measurement and modeling of computer systems _ , pages 277288 , 2014 . n. k. sharma , s. ghosh , f. benevenuto , n. ganguly , and k. gummadi . inferring who - is - who in the twitter social network . in _ proc . of the 2012 acm workshop on online social networks _ , pages 5560 , 2012 .

metrics derived from twitter and other social media often referred to as altmetrics are increasingly used to estimate the broader social impacts of scholarship . such efforts , however , may produce highly misleading results , as the entities that participate in conversations about science on these platforms are largely unknown . for instance , if altmetric activities are generated mainly by scientists , does it really capture broader social impacts of science ? here we present a systematic approach to identifying and analyzing scientists on twitter . our method can be easily adapted to identify other stakeholder groups in science . we investigate the demographics , sharing behaviors , and interconnectivity of the identified scientists . our work contributes to the literature both methodologically and conceptually we provide new methods for disambiguating and identifying particular actors on social media and describing the behaviors of scientists , thus providing foundational information for the construction and use of indicators on the basis of social media metrics .

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* power - law fit for overall flight . * first , we fit the flight length distribution of the geolife and nokia mdc datasets regardless of transportation modes ( see methods section ) . we fit truncated power - law , lognormal , power - law and exponential distribution ( see supplementary table s1 ) . we find that the overall flight length ( @xmath0 ) distributions fit a truncated power - law @xmath1 with exponent @xmath2 as 1.57 in the geolife dataset ( @xmath3 ) and 1.39 in the nokia mdc dataset ( @xmath4 ) ( fig . [ fig : all ] ) , better than other alternatives such as power - law , lognormal or exponential . figure . [ fig : all ] illustrates the pdfs and their best fitted distributions according to akaike weights . the best fitted distribution ( truncated power - law ) is represented as a solid line and the rest are dotted lines . we use logarithm bins to remove tail noises @xcite . our result is consistent with previous research ( @xcite ) , and the exponent @xmath2 is close to their results . we show the akaike weights for all fitted distributions in the supplementary table s2 . the akaike weight is a value between 0 and 1 . the larger it is , the better the distribution is fitted @xcite . the akaike weights of the power - law distributions regardless of transportation modes are 1.0000 in both datasets . the p - value is less than 0.01 in all our tests , which means that our results are very strong in terms of statistical significance . note that here the differences between fitted distributions are not remarkable as shown in the fig . [ fig : all ] , especially between the truncated power - law and the lognormal distribution . we use the loglikelihood ratio to further compare these two candidate distributions . the loglikelihood ratio is positive if the data is more likely in the power - law distribution , and negative if the data is more likely in the lognormal distribution . the loglikelihood ratio is 1279.98 and 3279.82 ( with the significance value @xmath5 ) in the geolife and the nokiamdc datasets respectively , indicating that the data is much better fitted with the truncated power - law distribution . * lognormal fit for single transportation mode . * however , the distribution of flight lengths in each single transportation mode is not well fitted with the power - law distribution . instead , they are better fitted with the lognormal distribution ( see supplementary table s2 ) . all the segments of each transportation flight length are best approximated by the lognormal distribution with different parameters . in fig . [ fig : geolife ] and supplementary fig . s1 , we represent the flight length distributions of walk / run , bike , subway / train and car / taxi / bus in the geolife and the nokia mdc dataset correspondingly . the best fitted distribution ( lognormal ) is represented as a solid line and the rest are dotted lines . table [ tab : parameters ] shows the fitted parameter for all the distributions ( @xmath2 in the truncated power - law , @xmath6 and @xmath7 in the lognormal ) . we can easily find that the @xmath6 is increasing over these transportation modes ( walk / run , bike , car / taxi / bus and subway / train ) , identifying an increasing average distance . compared to walk / run , bike or car / taxi / bus , the flight distribution in subway / train mode is more right - skewed , which means that people usually travel to a more distant location by subway / train . it must be noted that our findings for the car / taxi / bus mode are different from these recent research results @xcite , which also investigated the case of a single transportation mode , and found that the scaling of human mobility is exponential by examining taxi gps datasets . the differences are mainly because few people tend to travel a long distance by taxi due to economic considerations . so the displacements in their results decay faster than those measured in our car / taxi / bus mode cases . * mechanisms behind the power - law pattern . * we characterize the mechanism of the power - law pattern with lvy flights by mixing the lognormal distributions of the transportation modes . previous research has shown that a mixture of lognormal distributions based on an exponential distribution is a power - law distribution @xcite . based on their findings , we demonstrate that the reason that human movement follows the lvy walk pattern is due to the mixture of the transportation modes they take . we demonstrate that the mixture of the lognormal distributions of different transportation modes ( walk / run , bike , train / subway or car / taxi / bus ) is a power - law distribution given two new findings : first , we define the change rate as the relative change of length between two consecutive flights with the same transport mode . the change rate in the same transportation mode is small over time . second , the elapsed time between different transportation modes is exponentially distributed . * lognormal in the same transportation mode . * let us consider a generic flight @xmath8 . the flight length at next interval of time @xmath9 , given the change rate @xmath10 , is @xmath11 it has been found that the change rate @xmath12 in the same transportation mode is small over time @xcite . the change rate @xmath12 reflects the correlation between two consecutive displacements in one trip . to obtain the pattern of correlation between consecutive displacements in each transportation mode , we plot the flight length point ( @xmath8 , @xmath9 ) from the geolife dataset ( fig . [ fig : rate ] ) . here @xmath8 represents the @xmath13-th flight length and @xmath9 represents the @xmath14-th flight length in a consecutive trajectory in one transportation mode @xcite . figure . [ fig : rate ] shows the density of flight lengths correlation in car / taxi / bus , walk / run , subway / train and bike correspondingly . ( @xmath8 , @xmath9 ) are posited near the diagonal line , which identifies a clear positive correlation . similar results are also found in the nokia mdc dataset ( see supplementary fig . we use the pearson correlation coefficient to quantify the strength of the correlation between two consecutive flights in one transportation mode @xcite . the value of pearson correlation coefficient @xmath15 is shown in the supplementary table s3 . the @xmath16 value is less than 0.01 in all the cases , identifying very strong statistical significances . @xmath15 is positive in each transportation mode and ranges from 0.3640 to 0.6445 , which means that there is a significant positive correlation between consecutive flights in the same transportation mode , and the change rate @xmath12 in the same transportation mode between two time steps is small . the difference @xmath12 in the same transportation mode between two time steps is small due to a small difference @xmath17 in consecutive flights . we sum all the contributions as follows : @xmath18 we plot the change rate samples @xmath12 of the car / taxi / bus mode from the geolife dataset as an example in supplementary fig . we observe that the change rate @xmath12 fluctuates in an uncorrelated fashion from one time interval to the other in one transportation mode due to the unpredictable character of the change rate . the pearson correlation coefficient accepts the findings at the 0.03 - 0.13 level with p - value less than 0.05 ( see supplementary table s4 ) . by the central limit theorem , the sum of the change rate @xmath12 is normally distributed with the mean @xmath19 and the variance @xmath20 , where @xmath6 and @xmath21 are the mean and variance of the change rate @xmath12 and @xmath22 is the elapsed time . then we can assert that for every time step @xmath13 , the logarithm of @xmath0 is also normally distributed with a mean @xmath23 and variance @xmath24 @xcite . note here that @xmath25 is the length of the flight at the time @xmath22 after @xmath22 intervals of elapsed time . in the same transportation mode , the distribution of the flight length with the same change rate mean is lognormal , its density is given by @xmath26,\ ] ] which corresponds to our findings that in each single transportation mode the flight length is lognormal distributed . * transportation mode elapsed time . * we define elapsed time as the time spent in a particular transportation mode ; we found that it is exponentially distributed . for example , the trajectory samples shown in fig . [ fig : trajectory ] contain six trajectories with three different transportation modes , ( taxi , walk , subway , walk , taxi , walk ) . thus the elapsed time also consists of six samples ( @xmath27 , @xmath28 , @xmath29 , @xmath30 , @xmath31 , @xmath32 ) . the elapsed time @xmath13 is weighted exponentially between the different transportation modes ( see supplementary fig . similar results are also reported in @xcite . the exponentially weighted time interval is mainly due to a large portion of walk / run flight intervals . walk / run is usually a connecting mode between different transportation modes ( e.g. , the trajectory samples shown in fig . [ fig : trajectory ] ) , and walk / run usually takes much shorter time than any other modes . thus the elapsed time decays exponentially . for example , 87.93@xmath33 of the walk distance connecting other transportation modes is within 500 meters and the travelling time is within 5 minutes in the geolife dataset . * mixture of the transportation modes . * given these lognormal distributions @xmath34 in each transportation mode and the exponential elapsed time @xmath13 between different modes , we make use of mixtures of distributions . we obtain the overall human mobility probability by considering that the distribution of flight length is determined by the time @xmath13 , the transportation mode change rate @xmath12 mean @xmath6 and variance @xmath21 . we obtain the distribution of single transportation mode distribution with the time @xmath13 , the change rate mean @xmath6 and variance @xmath21 fixed . we then compute the mixture over the distribution of @xmath13 since @xmath13 is exponentially distributed over different transportation modes with an exponential parameter @xmath35 . if the distribution of @xmath0 , @xmath36 , depends on the parameter @xmath13 . @xmath13 is also distributed according to its own distribution @xmath37 . then the distribution of @xmath0 , @xmath38 is given by @xmath39 . here the @xmath13 in @xmath36 is the same as the @xmath13 in the @xmath37 . @xmath37 is the exponential distribution of elapsed time @xmath13 with an exponential parameter @xmath35 . so the mixture ( overall flight length @xmath40 ) of these lognormal distributions in one transportation mode given an exponential elapsed time ( with an exponent @xmath35 ) between each transportation mode is @xmath41 d t , \ ] ] which can be calculated to give @xmath42 where the power law exponent @xmath43 is determined by @xmath44 @xcite . the calculation to obtain @xmath43 is given in supplementary note 1 . if we substitute the parameters presented in table [ tab : parameters ] , we will get the @xmath45 in the geolife dataset , which is close to the original parameter @xmath46 , and @xmath47 in the nokia mdc dataset , which is close to the original parameter @xmath48 . the result verifies that the mixture of these correlated lognormal distributed flights in one transportation mode given an exponential elapsed time between different modes is a truncated power - law distribution . previous research suggests that it might be the underlying road network that governs the lvy flight human mobility , by exploring the human mobility and examining taxi traces in one city in sweden @xcite . to verify their hypothesis , we use a road network dataset of beijing containing 433,391 roads with 171,504 conjunctions and plot the road length distribution @xcite . as shown in supplementary fig . s5 , the road length distribution is very different to our power - law fit in flights distribution regardless of transportation modes . the @xmath2 in road length distribution is 3.4 , much larger than our previous findings @xmath46 in the geolife and @xmath48 in the nokia mdc . thus the underlying street network can not fully explain the lvy flight in human mobility . this is mainly due to the fact that it does not consider many long flights caused by metro or train , and people do not always turn even if they arrive at a conjunction of a road . thus the flight length tails in the human mobility should be much larger than those in the road networks . * data sets . * we use two large real - life gps trajectory datasets in our work , the geolife dataset @xcite and the nokia mdc dataset @xcite . the key information provided by these two datasets is summarized in table [ tab : mobilitydataset ] . we extract the following information from the dataset : flight lengths and their corresponding transportation modes . geolife @xcite is a public dataset with 182 users gps trajectory over five years ( from april 2007 to august 2012 ) gathered mainly in beijing , china . this dataset contains over 24 million gps samples with a total distance of 1,292,951 kilometers and a total of 50,176 hours . it includes not only daily life routines such as going to work and back home in beijing , but also some leisure and sports activities , such as sightseeing , and walking in other cities . the transportation mode information in this dataset is manually logged by the participants . the nokia mdc dataset @xcite is a public dataset from nokia research switzerland that aims to study smartphone user behaviour . the dataset contains extensive smartphone data of two hundred volunteers in the lake geneva region over one and a half years ( from september 2009 to april 2011 ) . this dataset contains 11 million data points and the corresponding transportation modes . * obtaining transportation mode and the corresponding flight length . * we categorize human mobility into four different kinds of transportation modality : walk / run , car / bus / taxi , subway / train and bike . the four transportation modes cover the most frequently used human mobility cases . to the best of our knowledge , this article is the first work that examines the flight distribution with all kinds of transportation modes in both urban and inter - city environments . in the geolife dataset , users have labelled their trajectories with transportation modes , such as driving , taking a bus or a train , riding a bike and walking . there is a label file storing the transportation mode labels in each user s folder , from which we can obtain the ground truth transportation mode each user is taking and the corresponding timestamps . similar to the geolife dataset , there is also a file storing the transportation mode with an activity i d in the nokia mdc dataset . we treat the transportation mode information in these two datasets as the ground truth . in order to obtain the flight distribution in each transportation mode , we need to extract the flights . we define a flight as the longest straight - line trip from one point to another without change of direction @xcite . one trail from an original to a destination may include several different flights ( fig . [ fig : trajectory ] ) . in order to mitigate gps errors , we recompute a position by averaging samples ( latitude , longitude ) every minute . since people do not necessarily move in perfect straight lines , we need to allow some margin of error in defining the ` straight ' line . we use a rectangular model to simplify the trajectory and obtain the flight length : when we draw a straight line between the first point and the last point , the sampled positions between these two endpoints are at a distance less than 10 meters from the line . the same trajectory simplification mechanism has been used in other articles which investigates the lvy walk nature of human mobility @xcite . we map the flight length with transportation modes according to timestamp in the geolife dataset and activity i d in the nokia mdc dataset and obtain the final ( transportation mode , flight length ) patterns . we obtain 202,702 and 224,723 flights with transportation mode knowledge in the geolife and nokia mdc dataset , respectively . * identifying the scale range . * to fit a heavy tailed distribution such as a power - law distribution , we need to determine what portion of the data to fit ( @xmath49 ) and the scaling parameter ( @xmath2 ) . we use the methods from @xcite to determine @xmath49 and @xmath2 . we create a power - law fit starting from each value in the dataset . then we select the one that results in the minimal kolmogorov - smirnov distance , between the data and the fit , as the optimal value of @xmath49 . after that , the scaling parameter @xmath2 in the power - law distribution is given by @xmath50 where @xmath51 are the observed values of @xmath52 and @xmath53 is number of samples . * akaike weights . * we use akaike weights to choose the best fitted distribution . an akaike weight is a normalized distribution selection criterion @xcite . its value is between 0 and 1 . the larger the value is , the better the distribution is fitted . akaike s information criterion ( aic ) is used in combination with maximum likelihood estimation ( mle ) . mle finds an estimator of @xmath54 that maximizes the likelihood function @xmath55 of one distribution . aic is used to describe the best fitting one among all fitted distributions , @xmath56 here k is the number of estimable parameters in the approximating model . after determining the aic value of each fitted distribution , we normalize these values as follows . first of all , we extract the difference between different aic values called @xmath57 , @xmath58 then akaike weights @xmath59 are calculated as follows , @xmath60 10 url # 1`#1`urlprefix[2]#2 [ 2][]#2 & _ _ * * , ( ) . & _ _ * * , ( ) . , & _ _ * * , ( ) . , , , & _ _ * * , ( ) . , , & in _ _ ( ) . , & in _ _ ( ) . , & _ _ * * , ( ) . , , & _ _ * * , ( ) . , & in _ _ ( ) . , , , & _ _ * * ( ) . , , & in _ _ & in _ _ ( ) . , , , & _ _ * * , ( ) . , & _ _ * * , ( ) . , & _ _ * * , ( ) . _ et al . _ _ _ * * , ( ) . , , & _ _ * * , ; _ _ ( ) . , , & _ _ * * , ( ) . , & _ _ * * , ( ) . , & _ _ * * , ( ) . , , & in _ _ ( ) . , , , & in _ _ ( ) . & _ _ ( , ) . , , , & _ _ * * , ; _ _ ( ) . , & _ _ * * , ; _ _ ( ) . , , & _ _ * * , ; _ _ ( ) . , , , & _ _ * * , ( ) . & _ _ ( ) . . , , & . in _ _ _ _ * * , ( ) . & _ _ _ _ ( ) . & _ _ ( ) . _ _ * * , ( ) . , & _ _ * * , ; _ _ ( ) . _ _ ( , ) . _ _ ( , ) . _ et al . _ in _ _ ( ) . , , , & in _ _ ( ) . , & _ _ * * , ( ) . k.z . , m.m . , p.h . , w.r . and s.t . designed the research based on the initial idea by k.z . and s.t .. k.z . executed the experiments guided by m.m . , w.r . and s.t . k.z . and s.t . performed statistical analyses , and prepared the figures . w.r . and s.t . wrote the manuscript . all authors reviewed the manuscript . * competing financial interests : * the authors declare no competing financial interests . .the geolife and the nokia mdc human mobility datasets . [ cols="^,^,^",options="header " , ] given @xmath61 d t .\ ] ] @xmath62 d t \\ & = \frac{\lambda}{\sigma}\frac{1}{\sqrt{2\pi } } x^{-1 } \\ & \int_{t=0}^{\infty } exp(-\lambda t ) exp[-\frac{(\ln(x)-\mu t)^2}{2t\sigma ^2 } ] \frac{1}{\sqrt{t}}]dt \\ & = \frac{\lambda}{\sigma}\frac{1}{\sqrt{2\pi } } x^{-1 } \\ & \int_{t=0}^{\infty } exp[\frac{-(\ln(x)-\mu t)^2 - 2\lambda\sigma^2 t } { 2t\sigma ^2 } ] \frac{1}{\sqrt{t}}]d t \\ & = \frac{\lambda}{\sigma}\frac{1}{\sqrt{2\pi } } x^{-1 } exp(\frac{\ln{x } \mu}{\sigma^2 } ) \\ & \int_{t=0}^{\infty } exp[-(\frac{\mu^2 + 2\lambda\sigma^2}{2\sigma^2})t - \frac{(\ln x)^2}{2\sigma^2}\frac{1}{t } ] \frac{1}{\sqrt{t}}]d t .\end{aligned}\ ] ] using the substitution @xmath63 gives @xmath64 \frac{1}{\sqrt{u^2 } } ] 2 u du .\end{aligned}\ ] ] let @xmath65 and @xmath66 , from the integral table we get @xmath67 which helps us to get the expression for @xmath68 , @xmath69 the expression for @xmath43 is @xmath70 here the @xmath6 and the @xmath21 are the normalized mean and variance of the change rate , while the @xmath35 is the exponential parameter of elapsed time between different transportation modes . we normalize the @xmath6 and @xmath21 of different transportation modes following @xmath71 and @xmath72 . note here @xmath73 , @xmath74 , @xmath75 , @xmath76 and @xmath77 , @xmath78 , @xmath79 , @xmath80 represent the mean and standard deviation of the change rate in each transportation modes in both datasets , as shown in the table . the mean value @xmath6 is 5.54 and 6.05 and the variance @xmath21 is 0.5954 and 1.0165 in the geolife dataset and in the nokia mdc dataset respectively . combining the fitted exponential parameter @xmath81 in the geolife dataset and @xmath82 in the nokia mdc dataset , we obtain the final @xmath45 in the geolife dataset , which is close to the original parameter @xmath46 , and @xmath47 in the nokia mdc dataset , which is close to the original parameter @xmath48 .

human mobility has been empirically observed to exhibit lvy flight characteristics and behaviour with power - law distributed jump size . the fundamental mechanisms behind this behaviour has not yet been fully explained . in this paper , we propose to explain the lvy walk behaviour observed in human mobility patterns by decomposing them into different classes according to the different transportation modes , such as walk / run , bike , train / subway or car / taxi / bus . our analysis is based on two real - life gps datasets containing approximately 10 and 20 million gps samples with transportation mode information . we show that human mobility can be modelled as a mixture of different transportation modes , and that these single movement patterns can be approximated by a lognormal distribution rather than a power - law distribution . then , we demonstrate that the mixture of the decomposed lognormal flight distributions associated with each modality is a power - law distribution , providing an explanation to the emergence of lvy walk patterns that characterize human mobility patterns . understanding human mobility is crucial for epidemic control @xcite , urban planning @xcite , traffic forecasting systems @xcite and , more recently , various mobile and network applications @xcite . previous research has shown that trajectories in human mobility have statistically similar features as lvy walks by studying the traces of bank notes @xcite , cell phone users locations @xcite and gps @xcite . according to the this model , human movement contains many short flights and some long flights , and these flights follow a power - law distribution . intuitively , these long flights and short flights reflect different transportation modalities . figure . [ fig : trajectory ] shows a person s one - day trip with three transportation modalities in beijing based on the geolife dataset ( table [ tab : mobilitydataset ] ) @xcite . starting from the bottom right corner of the figure , the person takes a taxi and then walks to the destination in the top left part . after two hours , the person takes the subway to another location ( bottom left ) and spends five hours there . then the journey continues and the person takes a taxi back to the original location ( bottom right ) . the short flights are associated with walking and the second short - distance taxi trip , whereas the long flights are associated with the subway and the initial taxi trip . based on this simple example , we observe that the flight distribution of each transportation mode is different . in this paper , we study human mobility with two large gps datasets , the geolife and nokia mdc datasets ( approximately 10 million and 20 million gps samples respectively ) , both containing transportation mode information such as walk / run , bike , train / subway or car / taxi / bus . the four transportation modes ( walk / run , bike , train / subway and car / taxi / bus ) cover the most frequently used human mobility cases . first , we simplify the trajectories obtained from the datasets using a rectangular model , from which we obtain the flight length @xcite . here a flight is the longest straight - line trip from one point to another without change of direction @xcite . one trail from an origin to a destination may include several different flights ( fig . [ fig : trajectory ] ) . then , we determine the flight length distributions for different transportation modes . we fit the flight distribution of each transportation mode according to the akaike information criteria @xcite in order to find the best fit distribution . we show that human movement exhibiting different transportation modalities is better fitted with the lognormal distribution rather than the power - law distribution . finally , we demonstrate that the mixture of these transportation mode distributions is a power - law distribution based on two new findings : first , there is a significant positive correlation between consecutive flights in the same transportation mode , and second , the elapsed time in each transportation mode is exponentially distributed . the contribution of this paper is twofold . first , we extract the distribution function of displacement with different transportation modes . this is important for many applications @xcite . for example , a population - weighted opportunities ( pwo ) model @xcite has been developed to predict human mobility patterns in cities . they find that there is a relatively high mobility at the city scale due to highly developed traffic systems inside cities . our results significantly deepen the understanding of urban human mobilities with different transportation modes . second , we demonstrate that the mixture of different transportations can be approximated as a truncated lvy walk . this result is a step towards explaining the emergence of lvy walk patterns in human mobility .

1408.4910

quantum bits , or qubits @xcite , have been realized using , for example , superconducting circuits @xcite , quantum dots @xcite , trapped ions @xcite , single dopants in silicon @xcite , and nitrogen vacancy centres @xcite . the state of a qubit is affected by various sources of error such as finite qubit lifetime , measurement imperfections , non - ideal initialization , and imprecise external control . provided that these errors are below a certain threshold , they can be corrected with quantum error correction codes @xcite which encode the information of a logical qubit into an ensemble of physical qubits . surface codes @xcite , error correction codes with the highest known thresholds , may require thousands of physical qubits for each fault - tolerant logical qubit . controlling such a large ensemble of qubits consumes a great amount of power , rendering heat management at the qubit register an important challenge . the power consumption of a quantum processor can be decreased by implementing more accurate physical qubits , thus leading to smaller ensembles forming the logical qubits . however , it is known that gate errors also arise from the quantum - mechanical uncertainties in the control pulse @xcite . in the case of a resonant disposable control pulse , this type of error is inversely proportional to the pulse energy , and hence poses a trade - off in the power management of the quantum computer . even in the absence of all other types of error , this result implies such a high level of dissipated power at the chip temperature that it challenges the commercially available cryogenic equipment , as we estimate in appendix [ appa ] for a typical superconducting quantum computer running a surface code to factorize a 2000-bit integer . in this work , we derive the greatest lower bound for the gate error within the resonant jaynes cummings model @xcite . the inevitable error originates from the quantum nature of the driving mode and becomes dominant in the regime of low driving powers . in contrast to previous work @xcite , our constructive derivation does not need to assume any particular state of the system and is applicable to qubit rotations of arbitrary angles . in addition to the lower bound itself , our method naturally finds the bosonic quantum states of the pulse that reach the bound . we explicitly show that single - qubit rotations are optimally realized by applying a certain amount of squeezing to coherent states . the optimal states do not alone solve the above - mentioned heat dissipation problem , but we additionally find that back - action - induced correlations between the control pulse and the controlled qubit can be transferred to auxiliary qubits ( see also refs . thus we propose a control protocol where multiple gates are generated with a single control pulse which is frequently refreshed using auxiliary qubits . whereas previous studies suggest that it is not possible to save energy by reusing control pulses without sacrificing the minimum gate fidelity @xcite , our method exhibits orders of magnitude smaller energy consumption with no drop in the average gate fidelity . this paper is organized as follows . in sec . [ sec : semiclassical ] , we briefly summarize the formalism used to describe qubit rotations and discuss gate errors in the semiclassical model . in sec . [ sec : optimization ] , we derive the quantum limit of gate error . the refreshing protocol is constructed and studied in sec . [ sec : protocol ] and the key results are summarized and discussed further in sec . [ sec : discussion ] . let us first review the semiclassical formalism of single - qubit control and the resulting gate errors . the state of a qubit can be represented as a bloch vector constrained inside a unit sphere , see fig . single - qubit logic gates @xmath0 , realized using , e.g. , microwave pulses , rotate the bloch vector by @xmath1 about the axis @xmath2 . assuming that the control pulse is a classical waveform in resonance with the qubit transition energy @xmath3 , the system may be described in the rotating frame using a semiclassical interaction hamiltonian of the form @xcite @xmath4 where @xmath5 and @xmath6 denote the ground and excited states of the qubit , respectively , @xmath7 represents the classical amplitude @xmath8 and phase @xmath9 of the control field , @xmath10 is the coupling constant including the pulse envelope , and @xmath11 is the reduced planck constant . the gate @xmath0 is implemented by choosing the interaction time @xmath12 and the pulse envelope such that they satisfy @xmath13 . for example , setting @xmath14 and @xmath2 along the @xmath15-axis , the temporal evolution operator @xmath16 $ ] becomes @xmath17 , where @xmath18 is the pauli @xmath19-operator . thus , up to a redundant global phase factor , the interaction implements a perfect not gate @xmath20 . model system . ( a ) ideal two - level system ( bottom ) interacting with a harmonic oscillator ( top ) . ( b ) bloch vector representation of the qubit state @xmath21 and an example @xmath20 rotation . ] we assess gate errors by utilizing the state transformation error @xmath22 where the initial qubit state is given by @xmath23 and @xmath24 is the desired gate . in general , the qubit state is unknown during the computation , and therefore we choose not to restrict our analysis to any specific state . instead , we study the average of a given error measure @xmath25 over a uniform state distribution on the bloch sphere , generally given by @xmath26 semiclassically , a source of gate error arises from uncertainties in the phase and the photon number @xmath27 , which are , for small phase fluctuations , fundamentally bounded by quantum mechanics through the minimal uncertainty relation @xcite @xmath28 . thus we consider a control pulse with an average of @xmath29 photons and minimal uncertainties @xmath30 and @xmath31 , where @xmath32 is a free squeezing parameter . these uncertainties carry on to the temporal evolution operator @xmath33 , and we find from eq . that the average gate error becomes inversely proportional to the photon number . for the @xmath20 gate for example , we obtain the average gate error @xmath34 in the limit @xmath35 . interestingly , the error is minimized with a non - zero squeezing parameter @xmath36 , a result also obtained in the full quantum treatment in sec . [ minimizationmethod ] . an alternative qubit - independent error quantity is the maximum gate error given by @xmath37 , which obeys a similar @xmath38-dependence @xcite . let us proceed to the full quantum treatment , where the gate operation arises from the quantum - mechanical interaction between the qubit and a single bosonic mode referred to as the drive . utilization of such quantum drive @xcite allows us to account for the changes in its state arising from the interaction with the qubit . in practice , qubits are also driven by propagating photons described by a continuum of modes , but such arrangements do not save energy in comparison to a well - controlled single mode . hence our description below is expected to yield a fundamental lower bound for the energy needed for controlling a single qubit at a given fidelity . in contrast to the semiclassical model , the evolution of the qubit is not unitary . after the interaction , the qubit state is extracted by taking a partial trace over the drive degrees of freedom as @xmath39,\ ] ] where @xmath40 and @xmath41 denote the arbitrary initial density operator and the evolution operator of the qubit drive system , respectively . the error , or infidelity , between the target and the resulting qubit state is here defined as @xmath42=1-\m{tr}\left [ \hat{\chi}(t)\hat{k}\hat{\chi}_{0}\hat{k}^{\dagger}\right ] , \label{eq : errortraces}\ ] ] which can be regarded as a generalization of eq . . the dynamics of the qubit drive system is generally described by the jaynes cummings model @xcite , which includes the rotating - wave approximation . assuming resonant interaction , the system is governed by the interaction hamiltonian @xmath43 where @xmath44 is the bosonic annihilation operator of the drive mode . without loss of generality , we assume an on - off envelope such that @xmath45 for @xmath46 and @xmath47 otherwise . most features of the semiclassical model are reobtained if @xmath48 and the drive is in the coherent state @xmath49 , where @xmath50 is the @xmath27th fock state . for example , taking the expectation value of @xmath51 in the state @xmath52 yields the semiclassical hamiltonian in eq . ( [ eq : classicalh ] ) . thus the coherent state approximately induces a gate @xmath0 if the timing condition @xmath53 is satisfied . if the initial state of the joint system is separable , @xmath54 , where @xmath55 and @xmath56 denote the initial qubit and drive states , respectively , the gate error of eq . induced by the jaynes cummings interaction can be written in the general form @xmath57 here , @xmath58 denotes either the transformation error @xmath59 of a particular qubit state , the average gate error @xmath60 [ eq . ] , or the maximum gate error @xmath61 . the information about the desired gate and chosen interaction time is contained in the corresponding operator @xmath62 which is denoted either by @xmath63 , @xmath64 , or @xmath65 , respectively . an analytical expression for @xmath63 and @xmath64 can be found for any gate , whereas an expression for @xmath65 exists for at least rotations @xmath66 , where the rotation axis @xmath67 is restricted to the @xmath68-plane of the bloch sphere . see appendix [ appb ] for derivations and detailed expressions . optimal drive states and the resulting error . ( a ) numerically solved initial drive states @xmath69 that minimize the average error of rotations @xmath70 and @xmath20 as wigner distributions above and below the dashed line , respectively . the wigner function is defined as @xmath71 $ ] , where @xmath72 is the displacement operator . the interaction time for each operation @xmath0 is @xmath73 , which is expected to yield states with @xmath74 . ( b ) gate error for an @xmath20 operation as a function of the average photon number @xmath75 of the driving pulse which is initialized either in the coherent state ( red color ) or the squeezed cat state ( blue color ) . the highlighted areas indicate the range of error , depending on the initial state of the qubit , and the solid lines show the error averaged over qubit states distributed uniformly on the bloch sphere , @xmath60 . the inset shows the difference @xmath76 between the numerically calculated errors and their analytical first - order approximations ( table [ tab_1 ] ) , with dashed lines indicating the difference in maximum errors . ] [ bt ! ] [ cols="^,^,^,^,^,^,^,^,^ " , ] we solve the drive states that minimize the average or maximum gate error for a given interaction time and a desired rotation @xmath77 . to this end , it is sufficient to consider only pure states @xcite , and hence we may employ the forms given by eq . . the error - minimizing states are the eigenstates of operators @xmath62 that correspond to the largest eigenvalue @xmath78 , @xmath79 by definition , the optimal states @xmath80 provide a fundamental lower bound for the error @xmath58 . we solve this eigenvalue equation numerically . examples of fidelity - optimal solutions are shown in fig . [ fig2]a using the wigner pseudo - probability function @xcite . the numerically obtained states can be accurately described using the squeezed coherent states @xmath81 , where @xmath82 and @xmath83 are the displacement and squeezing operators , respectively @xcite . importantly , the numerical solutions possess the correct amplitude and phase to satisfy the timing condition @xmath53 and to set the desired direction of the rotation axis , without imposing them explicitly . furthermore , the average errors , as well as the optimal squeezing parameters , are equal to those obtained in the semiclassical approach in sec . [ sec : semiclassical ] . in the specific case of @xmath84-rotations , a sum of two eigenvectors , i.e. , the squeezed cat state @xcite @xmath85 where the positive constant @xmath86 ensures normalization , is a state that minimizes both the average and the maximum error simultaneously ( see appendix [ appb ] ) . comparison of errors produced by such a state and a coherent state is presented in fig . [ fig2]b . the numerical approach for solving the eigenstates of @xmath62 has the disadvantage of truncating the infinite - dimensional state vector to a finite vector of length @xmath87 , which might distort or exclude some of the possible solutions . however , the obtained gaussian - like solutions are not affected by changes in the cut - off for @xmath88 . raising the cut - off reveals more energetic solutions , but these correspond to pulses that implement the chosen gate after an integer number of unnecessary @xmath89 rotations . generally for @xmath90 gates , we find solutions with errors that vanish as @xmath38 in the limit @xmath35 , as shown in appendix [ appc ] . the lower bounds together with errors induced by non - squeezed coherent states are shown in table [ tab_1 ] . other gates , such as the pauli - z gate and the hadamard gate , can be constructed as sequences of @xmath90 gates . recently , it was shown that squeezing also improves the fidelity of the phase gate in the dispersive regime @xcite . schematic diagram of the drive - refreshing protocol . during one cycle , the circulating drive pulse ( red ) induces a chosen rotation @xmath91 on one of the qubits @xmath92 in the register and is then refreshed by sequential @xmath20 interactions with each ancillary qubit \{@xmath93}. in an ideal setting , each ancilla is prepared precisely into the state @xmath94 and reset after each cycle . in practice , the ancilla qubits are initially in their ground states and their preparation and reset is implemented by a circulating corrector pulse ( green ) . ] all of the fundamental lower bounds derived above are inversely proportional to the average photon number . intuitively , a drive with a large photon number should be capable of inducing multiple gates without changing substantially , thus decreasing the required amount of energy per gate for nearly equal error level . we show below that reusing a drive effectively decreases the energy consumption well below the lower bound of average gate error for disposable pulses . furthermore , the drive can be corrected between successive gates such that the consumption drops without essential decrease of the average gate fidelity . in our protocol , an itinerant control drive cyclically interacts with a register of resonant qubits and ancillary qubits , see fig . a cycle begins with the drive , initially in a suitable squeezed coherent state , applying a chosen gate operation with minimal error on a register qubit . consequently , the drive state changes due to the quantum back - action . to undo this , the drive is set to sequentially interact with corrective ancilla qubits , initialized in a superposition of ground and excited states , for a time corresponding to a @xmath84-rotation . as a result , the purity , energy , and phase of the drive are restored in successive interactions ( see appendix [ appd ] ) . at the end of the cycle , the ancilla qubits are reset and the refreshed drive is usable for another high - fidelity gate . with increasing number of ancilla qubits , the execution time of a full cycle increases and thus one itinerant pulse applies a gate on the register less frequently . to compensate for this , one could add another drive pulse for each ancilla in the array , and synchronize their travel times such that each qubit would interact with one of the pulses at a given time . such a system would apply as many gates on the register per cycle as there are itinerant pulses in circulation . however , we restrict our analysis to a single pulse . evolution of an ancilla state during a refreshing cycle : ( i ) preparation from the ground state into the state @xmath95 , ( ii ) drive refresh as a result the primary rotation ( red ) , and ( iii ) ancilla reset . we either assume that the preparative steps ( i ) and ( iii ) are ideal or induced by a corrector pulse as shown in fig . [ fig3 ] . ] the refreshing by the ancillary interactions is understood by considering the path traversed by the bloch vector of the ancillary qubit , as illustrated in fig . a drive lacking energy rotates the vector with smaller angular frequency , leaving the ancilla slightly biased towards the ground state and gaining energy in the process . similarly , excessive energy in the drive is transferred to the ancilla due to rotating it closer to the excited state . the hilbert space of this system is formally a composite space of the fock space @xmath96 of the drive and the two - level spaces @xmath97 of the register and ancilla qubits , @xmath98 the drive only interacts with one qubit at a time and therefore each interaction can be calculated in the subspace of the relevant qubit and the drive , assuming the qubits are not correlated . after the interaction , the drive state is extracted by tracing over the associated qubit space . namely , the @xmath99th iteration of the drive state is given by @xmath100 , \label{eq : nextstate}\ ] ] where @xmath101 acts in the subspace of the drive and the @xmath99th qubit in the protocol sequences described in the following sections . consider first the case where the ancilla qubits are perfectly reset during each cycle , and the gate we wish to apply on each register qubit is @xmath102 . the protocol is executed with the following steps : 1 . the drive state is initialized to the @xmath103-minimizing state @xmath104 . a new register qubit is initialized in a random pure state , chosen uniformly from the bloch sphere . the drive interacts with the register qubit for interaction time @xmath105 [ eq . . the @xmath106 ancilla qubits are initialized to @xmath95 . the drive interacts with an ancilla qubit for interaction time @xmath105 . repeat for all ancillas . evaluate the average error @xmath60 of a hypothetical @xmath20 gate with eqs . and using the current drive state . continue from step ( ii ) . for gates other than @xmath102 , the phases of the drive and ancillas , as well as the interaction time in step ( iii ) , but not step ( v ) , would be shifted accordingly . average error @xmath60 of @xmath20 gates generated by an itinerant drive pulse which initially had an average photon number @xmath75 and has reached the steady state due to ancilla refreshing . the drive is set to interact with @xmath106 ideal ancillas ( @xmath95 ) per cycle as indicated , leading to effective refreshing of the drive state . the dashed line indicates the lower bound of error which is achieved either with a disposable optimal pulse or with a pulse refreshed by infinitely many ideal ancillas . the inset shows the average gate error as a function of @xmath106 for @xmath107 . ] we numerically simulate the evolution of the drive and evaluate the average error of the gate @xmath20 for a register qubit after each cycle . during the protocol , the average error @xmath60 will increase from its initial lower - bound value at varying rates depending on the randomized states of the register qubits . we find that after many cycles , the drive reaches a steady state that generates the desired gates with a predictable average error . with 13 ancillas per cycle , the average error saturates after a hundred cycles ; with ten or more ancillas , the saturation takes less than ten cycles . if no corrective ancillas are used , the average error eventually reaches @xmath108 . figure [ fig5 ] shows how the eventual error level depends on the number of photons and ancillas . the average gate error approaches its theoretical lower bound , in the limit of many drive - refreshing ancilla qubits . for smaller rotation angles , qualitatively similar results are obtained with more slowly accumulating error . thus a single itinerant drive pulse supplied with ideal ancilla states can generate an infinite number of high - fidelity gates . in the previous section , the qubits in the register were assumed to be essentially uncorrelated to justify the partial tracing over each qubit after the respective interaction . here we demonstrate the beneficial performance of our method in the case where the register qubits are maximally entangled . we initialize the register of @xmath109 qubits in the greenberger horne zeilinger ( ghz ) state @xmath110 . the control protocol is physically the same as in the previous section : the drive interacts with only one qubit at a time to implement a single - qubit gate @xmath24 and is refreshed by @xmath106 ideally prepared ancillas between each such gate . the target operation on the register is thus @xmath111 . due to the entangled register , the temporal evolution operators must be calculated in the hilbert space @xmath112 or @xmath113 for interactions between the drive and a register qubit , or drive and the @xmath114th ancilla , respectively . no partial trace over any register qubit is taken . after the drive has interacted with every register qubit once , the state of the register has transformed into @xmath115 and the total transformation error is computed as @xmath116.\ ] ] we divide this error by the number of qubits to obtain the effective error per gate , @xmath117 . state preparation error per qubit @xmath118 for a register of @xmath109 qubits initially in a ghz state . the target gate is an @xmath119 rotation for all qubits individually , implemented by a squeezed state of @xmath120 photons ( @xmath121 ) that is refreshed by @xmath106 ideal ancillas per cycle . the circles represent the data , whereas the coloured lines extend the line segments between the first two data points , to distinguish deviations from linear behaviour . the black dashed line represents the error obtained using @xmath109 disposable pulses of constant photon number @xmath120 . the dotted line is the error due to disposable pulses of constant total energy @xmath122 . ] results of a simulation for an @xmath119 gate with the initial drive state @xmath123 are shown in fig . a behaviour similar to fig . [ fig5 ] is observed : with enough ancillary corrections between the register gates , the error produced by an itinerant drive can be reduced to the level given by individual pulses . the figure also suggests that even without corrections , reusing a drive of certain energy is more beneficial in practice than dividing the same amount of photons into individual , weaker disposable pulses . thus we conclude that regardless of the state of the register , refreshment of a drive pulse likely serves to improve the trade - off between gate error and required energy . the above case of entangled qubits also provides a way to compare our results to the previous work by gea - banacloche and ozawa @xcite , where they studied a register in a ghz state that was operated by a drive of @xmath75 photons on average . they showed that the maximum error of the @xmath119 gate in this system scales as @xmath124 per qubit . this scaling was used to argue that a pulse of average photon number @xmath125 can not outperform @xmath109 individual pulses of @xmath126 average photons , although their performance was not compared explicitly . the key differences here are that ref . @xcite does not consider the possibility of using ancillary qubits , and that it employs a definition of error which also accounts for the infidelity of the drive state . our results suggest that even though the errors due to both reused and disposable pulses of equal total energy increase almost linearly with @xmath109 , the prefactor of the former is much smaller and can be greatly improved by the refreshing protocol . average gate error as a function of the total mean number of initial photons , @xmath127 for @xmath128 and @xmath75 for @xmath129 , divided by the number of @xmath20 register gates generated . the ancilla states are non - ideally prepared by a corrector pulse initially in state @xmath130 . during the protocol , the curve advances from right to left and the results are averaged over multiple simulations . the dashed line indicates the lower bound of error which is achieved either with a disposable optimal pulse or with a pulse refreshed by infinitely many ideal ancillas . ] the total energy consumption of the protocol can be meaningfully estimated only if the method and energy cost of the ancilla preparation is specified . to this end , we propose to prepare the ancillas by a circulating corrector pulse shown in fig . [ fig3 ] . in the full protocol , the ancilla qubits are first prepared in their ground state and then controlled by the corrector pulse from cycle to cycle . with opposite phase and half the interaction time compared with the drive , the corrector pulse applies an @xmath131 gate on the ancilla before and after a @xmath20 gate introduced by the drive pulse . for simplicity , we assume that the state of the register is separable . the full protocol is given by the following steps : 1 . the drive state is initialized to @xmath132 , the corrector pulse to @xmath133 and all @xmath106 ancillas to the ground state . a new register qubit is initialized in a random pure state . the drive interacts with the register qubit with interaction time @xmath105 [ eq . . 4 . an ancilla qubit interacts sequentially with the corrector , the drive , and the corrector again , with interaction times @xmath134 , @xmath12 , and @xmath134 , respectively . repeat for all other ancillas . 5 . evaluate the average error @xmath60 of a hypothetical @xmath20 gate with eqs . and using the current drive state . continue from step ( ii ) . in addition to computing the drive state after each interaction , the state of the interacting qubit is also extracted for subsequent use by a partial trace over the drive degrees of freedom . this is justified if the ancilla qubits do not become strongly correlated during the evolution . this approximation is more accurate the closer the control pulses are to classical pulses which do not induce entanglement . since all ancilla qubits are prepared to the ground state , the energy consumption fully arises from the drive and corrector pulses , both of which have the initial average energy @xmath135 . thus , the average energy consumption per register gate is @xmath136 , where @xmath109 is the number of elapsed cycles , or equally gates generated . in the case where the drive - refreshing protocol is not used , @xmath129 , we have @xmath137 . results from multiple simulations are averaged and shown in fig . in contrast to the ideal case , the system accumulates error over repeated cycles and the average gate error does not saturate . nevertheless , we find that with a sufficient number of ancillary qubit interactions between the register gates , the average error remains nearly constant for a large number of successive gates . the protocol can be stopped before the error reaches a desired threshold . this shows that the total energy cost per register gate is effectively reduced to orders of magnitude below the lower bound for disposable pulses . in fact , fig . [ fig7 ] suggests that the gate error may be , in theory , reduced indefinitely without increasing the power consumption by using more energetic pulses . in this work , we derived the greatest lower bound for the error of a single - qubit gate implemented with a single resonant control mode of certain mean energy . in contrast to previous work , our method for obtaining the bound is not restricted to any particular gate or state of the qubit drive system . the method can also be used to find the quantum state of the drive mode that minimizes the average gate error , or alternatively the transformation error for a chosen initial qubit state . specifically , we found that the lower bounds for rotations about axes in the @xmath68-plane are achieved by squeezing the quantum state of a coherent drive pulse by an amount that depends on the target gate . together with the recent result that squeezing also significantly improves the phase gate in the dispersive regime @xcite , our results suggest that squeezing may generally yield useful improvements in different control schemes . this calls for experimental studies on outperforming the widely - used coherent state . importantly , our results also impose a lower bound on the energy consumption of individually driven qubits . delivering the required power to the qubit level , possibly through a series of attenuators , implies heat management challenges that must be addressed in future large - scale quantum computers . as a solution , we introduced a concrete protocol where an itinerant control pulse is used to generate multiple gates and is refreshed between them to avoid loss of gate fidelity . the refreshing process may also prove useful in correcting the phase and amplitude errors of a noisy control pulse . our protocol can possibly be realized in some form with future low - loss microwave components such as photon routers @xcite , circulators , and nanoelectromechanical systems @xcite . technical limitations in the quality of these devices will set in practice the trade - off between the achievable gate fidelity and the dissipated power . in the future , our work can be extended to error bounds for 2-qubit gates , state preservation , pulse amplification , and propagating control pulses composed of a continuum of bosonic modes . we thank paolo solinas and benjamin huard for useful discussions . this work was supported by the european research council under starting independent researcher grant no . 278117 ( singleout ) and under consolidator grant no . 681311 ( quess ) . we also acknowledge funding from the academy of finland through its centres of excellence program ( grant nos 251748 and 284621 ) and grant ( no . 286215 ) and from the finnish cultural foundation . we estimate the power required by a superconductor - based quantum computer solving a 2000-bit factorization problem , stabilized by a surface code . for this particular computation , the needed number of physical qubits has been estimated by fowler @xcite to be @xmath138 . we assume that the physical qubits are controlled with typical coherent microwave pulses and that @xmath139 gates are completed in equal time and with lower power than @xmath84 gates . the average power needed during one surface code cycle is calculated by counting the frequency of measurements , @xmath140 , @xmath141 , and cnot operations , and by taking a duration - weighted average of the corresponding powers . the operation times depend on implementation . using operation times achieved in ref . @xcite , @xmath142 ns , @xmath143 ns , and @xmath144 ns , for @xmath84-rotations , controlled phase gates , and measurements , respectively , and assuming that our code executes as many operations in parallel as possible , the average power per physical qubit is approximately @xmath145 where the @xmath146 s denote the average drive powers for the the above - mentioned operations . for simplicity , we neglect the two - qubit gates and measurements and use @xmath147 . typical powers at the chip are of the order of @xmath148 w , after being generated in the room temperature and attenuated by tens of decibels on their way to roughly 10-mk base temperature . using only @xmath149 db of attenuation at the base temperature , the total power dissipation here becomes @xmath150 mw . such power level is much higher than the typical cooling power of @xmath151w in state - of - the - art dilution refrigerators at 10 mk . note that using an open transmission line is expected to consume more power than required in the single - mode case considered in sec . [ sec : optimization ] . the average energy density in a transmission line is given by @xmath152 , where @xmath153 is the capacitance per unit length and @xmath154 is the root mean square of the voltage . in a time interval @xmath155 , a propagating drive pulse advances a distance @xmath156 , effectively transporting a power of @xmath157 , where @xmath158 is the photon wavelength . in comparison , consider a @xmath159 resonator which is used to apply @xmath154 to the qubit for an equal operation time . the resonator requires a power @xmath160 , and with a typical qubit frequency of @xmath161 ghz , the ratio between the powers is @xmath162 . thus qubit control using propagating photons in a transmission line seems to lead to orders of magnitude higher power consumption than our single - mode case . however , a more comprehensive study employing the quantization of the transmission line is required to reach accurate estimates . we leave such study for future research . finally , let us consider the lower bound for the power to drive the qubits using disposable pulses . the minimum amount of photons ( see sec . [ minimizationmethod ] ) to produce the gate error @xmath163 used by fowler in ref . @xcite is @xmath164 photons at the qubit level . with @xmath161 ghz , the corresponding powers are @xmath165 w and @xmath166w . this suggests that the lower bound for our example problem size is at the border where current refrigeration equipment fail to deliver the required cooling power , and hence significant increments in the problem size or non - ideal implementation of the suggested driving techniques call for inventive solutions to the emerging heat management problem . a way to avoid the attenuation at the base temperature would be to generate the control pulses at the chip level . to our knowledge , however , no present chip - level photon source is capable of producing pulses that are accurate and intense enough to induce quantum gates of high fidelity . furthermore , the operation efficiency of such devices needs to be sufficiently high to be a considerable alternative . typically microwave sources internally dissipate much more power than their maximum output . assuming the qubit drive system is initially in a pure state , @xmath167 , eq . reduces to @xmath168\hat{k}\c_{0}\hat{k}^{\dagger}\right\ } \nonumber \\ \quad\;=1-\sum_{k=0}^{\infty}\left|\b{\chi_{0},k}(\hat{k}^{\dagger}\otimes\hat{\mathbb{i}})\hat{u}(t)\k{\chi_{0},\sigma_{0}}\right|^{2 } , \label{eq : err1}\end{gathered}\ ] ] where @xmath24 is the desired gate , @xmath12 is the interaction time , @xmath41 is the temporal evolution operator , and @xmath169 are the photon number states . we represent the basis of the qubit space using vectors @xmath170 and @xmath171 , and explicitly write @xmath172 in this basis , @xmath173 , and @xmath41 is given by @xmath174 with the shorthand notations @xmath175 , @xmath176 , and @xmath177 . using the expressions above , the matrix element in eq . can be structured as @xmath178 where @xmath179,\nonumber \\ \gamma_{01}^{k}(\vartheta,\varphi ) & = & -is_{k}\left[k_{12}^{*}\sin^{2}\left(\frac{\vartheta}{2}\right)+k_{11}^{*}\frac{1}{2}\sin\left(\vartheta\right)\e{i\varphi}\right],\nonumber \\ \gamma_{10}^{k}(\vartheta,\varphi ) & = & -is_{k+1}\left[k_{21}^{*}\cos^{2}\left(\frac{\vartheta}{2}\right)+k_{22}^{*}\frac{1}{2}\sin\left(\vartheta\right)\e{-i\varphi}\right],\nonumber \\ \gamma_{11}^{k}(\vartheta,\varphi ) & = & c_{k+1}\left[k_{22}^{*}\sin^{2}\left(\frac{\vartheta}{2}\right)+k_{21}^{*}\frac{1}{2}\sin\left(\vartheta\right)\e{i\varphi}\right].\nonumber\end{aligned}\ ] ] the error is thus given by @xmath180 we can define the transformation operator through its matrix elements in the photon number basis as @xmath181 , where @xmath182 and @xmath183 is a kronecker delta that is zero for any negative index . equation is thus reduced to the form given in eq . , that is , @xmath184 using eq . , the average error and its corresponding operator can be structured in a similar manner . defining the matrix elements of the operator @xmath185 as @xmath186 the average error also assumes the form of eq . . as shown in ref . @xcite , the average error integrated over the bloch sphere is equal to the arithmetic mean of the error of six so - called axial states . this provides an alternative expression for the operator @xmath185 , namely , @xmath187.\end{aligned}\ ] ] we can optimize the maximum error if there exists an initial qubit state which produces the highest error regardless of the drive state , i.e. , @xmath188 . specifically for gates @xmath66 , computing the gradients of @xmath189 with respect to @xmath190 and @xmath191 shows that the maximum point is virtually independent of the drive state , and that the maximum error is obtained with @xmath192 and @xmath193 , or equivalently @xmath194 , where @xmath9 is the angle between the horizontal rotation axis @xmath67 and the @xmath15-axis . due to symmetry , the initial drive state that optimizes @xmath61 is an eigenvector of @xmath195 which corresponds to the mean error of these two states . the elements of the commutator @xmath196 $ ] turn out to decrease as @xmath197 . thus it follows that the eigenvectors of @xmath185 , i.e. , the squeezed cat states given by eq . , simultaneously minimize both @xmath61 and @xmath60 in the limit @xmath35 . this section shows how the gate error can be analytically approximated for a specific gate . as an example , we choose @xmath60 for the gate @xmath20 . using @xmath198 , it is straightforward to evaluate the integrals in eq . , and hence the average error [ eq . becomes @xmath199\right . \nonumber \\ & & \left . + 2s_{n}(t)s_{n+1}(t)\m{re}\left(c_{n+1}c_{n-1}^{*}\right)\right\}. \label{eq : exactaverageerror}\end{aligned}\ ] ] the error is then obtained by inserting the coefficients @xmath200 of the desired drive state : coherent , squeezed cat , or some other state . we choose a squeezed cat state @xmath201 where @xmath32 is an unknown squeezing parameter and the amplitude satisfying @xmath202 is real . in the high energy limit , the occupation numbers in a squeezed state @xmath203 obey the normal distribution @xmath204,\ ] ] with mean @xmath205 and standard deviation @xmath206 . the amplitudes of a squeezed cat state @xmath207 , are obtained by allowing either even ( or odd ) states to be occupied , such that @xmath208,\ ] ] if @xmath27 is even ( odd ) and @xmath209 otherwise . insertion of these coefficients into eq . yields @xmath210 where @xmath211 \left[\sin^{2}\left(\frac{\pi}{2}\sqrt{2m\alpha^{-2}}\right)+\sin^{2}\left(\frac{\pi}{2}\sqrt{\left(2m+1\right)\alpha^{-2}}\right)\right],\ ] ] and @xmath212^{2}+\left(2m\alpha^{-1}-\alpha\right)^{2}\right\ } \right ) \sin\left[\frac{\pi}{2}\sqrt{\left(2m+1\right)\alpha^{-2}}\right]\sin\left[\frac{\pi}{2}\sqrt{\left(2m+2\right)\alpha^{-2}}\right].\ ] ] the sums can be computed by a change of variables @xmath213 and treating the infinite sum @xmath214 as an integral @xmath215 , which justified in the limit @xmath216 , where the functions are rather smooth and have support on a region much wider than unity . approximating the result to the lowest order in @xmath217 eventually yields @xmath218 this expression is minimized with the choice of the squeezing parameter @xmath36 , independently of @xmath219 . thus we have @xmath220 . approximate average errors for gates @xmath119 , @xmath221 , and @xmath222 implemented by different drive states , such as the coherent state , can be computed in a similar fashion . this approach also works with the maximum error for rotations of @xmath84 . expressions obtained this way are listed in table [ tab_1 ] . to build our protocol , we first search for initial states of the ancillas and the drive , such that each ancilla drive interaction would steer the drive towards a stable state . for the protocol to work , the following conditions must be satisfied : 1 . in the vicinity of the stable state , the drive is able to induce high - fidelity gates on a register qubit . an ancilla drive interaction increases the purity of the drive state , defined here as @xcite @xmath223 . interactions with register qubits in randomized states tend to decrease the purity of a drive , rendering it less useful for subsequent gates . thus increasing the purity effectively transfers entropy from the drive to the ancilla qubits . an ancilla drive interaction steers the amplitude , or equally energy , of the drive towards its steady state value . this is needed since we want to generate gates with a fixed interaction time . the interaction steers the relative phase of the drive towards its initial value . purity change of a drive state due to the interaction with an ancilla qubit initialized in state @xmath224 . the shown change is @xmath225 averaged over multiple simulations , where @xmath226 is the result of the evolution of @xmath227 . the impure initial drive state @xmath228 was obtained by letting a squeezed coherent state @xmath229 with @xmath230 interact with 10 qubits in randomly chosen initial states . the interaction time corresponds to a @xmath84 rotation . ] condition ( i ) suggests that the most promising candidates for the initial drive state are the coherent state @xmath231 and its squeezed variant @xmath232 . we study condition ( ii ) by computing the change in purity @xmath233 where @xmath234 is given by eq . . figure [ fig8 ] shows that the purity of the drive state increases if the ancilla is initialized close to the state @xmath95 . the figure also implies that the protocol works even if there is some error when preparing the ancilla in this state . change of the average photon number @xmath235 in an initial drive state @xmath236 as a function of the deviation @xmath237 . the change results from an interaction of a fixed time @xmath238 with an ancilla qubit prepared in the state @xmath239 . ] furthermore , changes in photon occupation , @xmath240 with different initial energies of the drive shows in fig . [ fig9 ] that this ancilla state also enforces negative feedback on the average photon number , satisfying condition ( iii ) . our study of the wigner representation of the drive state after successive ancilla interactions shows that condition ( iv ) is also satisfied . for a protocol generating only @xmath140 gates , the squeezed cat state @xmath241 was also tested as an initial drive state . unfortunately , in this case conditions ( ii ) and ( iii ) do not hold for any choice of the ancilla state since the cat state does not rotate the bloch vector of the ancilla in a specific direction , which is essential for the feedback mechanism depicted in fig . [ fig4 ] . 33ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1098/rspa.1998.0167 [ * * , ( ) ] http://dx.doi.org/10.1038/19718 [ * * , ( ) ] http://dx.doi.org/10.1038/nature13171 [ * * , ( ) ] http://dx.doi.org/10.1038/nature14270 [ * * , ( ) ] link:\doibase 10.1126/science.282.5393.1473 [ * * , ( ) ] http://dx.doi.org/10.1038/nature15263 [ * * , ( ) ] http://dx.doi.org/10.1038/35005011 [ * * , ( ) ] link:\doibase 10.1038/nature18648 [ * * , ( ) ] http://dx.doi.org/10.1038/nature11449 [ * * , ( ) ] http://dx.doi.org/10.1038/nature09256 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.87.307 [ * * , ( ) ] link:\doibase 10.1103/physreva.86.032324 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.89.057902 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.89.217901 [ * * , ( ) ] http://stacks.iop.org/1464-4266/7/i=10/a=017 [ * * , ( ) ] link:\doibase 10.1103/physreva.74.060301 [ * * , ( ) ] link:\doibase 10.1103/physreva.78.032331 [ * * , ( ) ] http://stacks.iop.org/1751-8121/42/i=22/a=225303 [ * * , ( ) ] link:\doibase 10.1103/physreva.87.022321 [ * * , ( ) ] link:\doibase 10.1109/proc.1963.1664 [ * * , ( ) ] link:\doibase 10.1080/09500349314551321 [ * * , ( ) ] link:\doibase 10.1103/physreva.93.040301 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.63.934 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.150402 [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1103/physreva.39.1665 [ * * , ( ) ] link:\doibase 10.1103/physreve.89.052128 [ * * , ( ) ] @noop _ _ ( , ) http://stacks.iop.org/1464-4266/4/i=1/a=201 [ * * , ( ) ] http://stacks.iop.org/1367-2630/16/i=4/a=045011 [ * * , ( ) ] link:\doibase 10.1126/science.1243289 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.116.180501 [ * * , ( ) ] link:\doibase 10.1103/physrevapplied.6.024009 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.073601 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys2527 [ * * , ( ) ] link:\doibase 10.1016/s0375 - 9601(02)00069 - 5 [ * * , ( ) ]

in the near future , a major challenge in quantum computing is to scale up robust qubit prototypes to practical problem sizes and to implement comprehensive error correction for computational precision . due to inevitable quantum uncertainties in resonant control pulses , increasing the precision of quantum gates comes with the expense of increased energy consumption . consequently , the power dissipated in the vicinity of the processor in a well - working large - scale quantum computer seems unacceptably large in typical systems requiring low operation temperatures . here , we introduce a method for qubit driving and show that it serves to decrease the single - qubit gate error without increasing the average power dissipated per gate . previously , single - qubit gate error induced by a bosonic drive mode has been considered to be inversely proportional to the energy of the control pulse , but we circumvent this bound by reusing and correcting itinerant control pulses . thus our work suggests that heat dissipation does not pose a fundamental limitation , but a necessary practical challenge in future implementations of large - scale quantum computers .

1609.02732

a hgte / cdte quantum well is a system where the dirac fermions appear only in a single valley , at the @xmath1 point of the brillouin zone , unlike graphene where there are two valleys of the dirac fermions with a strong inter - valley scattering . the energies of spatially quantized sub - bands at the quasimomentum @xmath2 and energy spectrum @xmath3 for different widths of the quantum well ( @xmath4 ) were calculated within _ kp _ method in numerous papers @xcite . as seen from fig . [ f1 ] , various types of energy spectrum are realized upon increasing the hgte quantum - well width ; namely , `` normal '' , when @xmath4 is less than a critical width @xmath5 nm , dirac - like at small quasimomenta for @xmath6 , inverted when @xmath7 , and finally , semimetallic when @xmath8 nm . to interpret experimental data , these calculations of the energy spectrum are used practically always . they well describe the width dependence of the energies of both electron and hole subbands at @xmath2 and the energy dependence of the electron effective mass ( @xmath9 ) . however , quite a lot of differences between the experimental data and the results of these calculations on the energy spectrum of the carriers have been accumulated to date . first of all , they refer to the spectrum of the valence band . the hole effective mass ( @xmath10 ) at @xmath11 nm within the wide hole density range @xmath12 @xmath13 is substantially less than the calculated one : @xmath14 @xcite instead of @xmath15 @xcite . the top of the valence band in the nominally symmetric structures with @xmath16 ( @xmath17 nm ) was found to be very strongly split by spin - orbit ( so ) interaction @xcite . therewith , the so splitting of the conduction band in the same structures does not reveal itself @xcite . it is surprising that such so splitting is observed in structures both with inverted and normal spectrum despite the fact that at @xmath18 and @xmath7 the conduction band is formed from different terms ( see fig . [ f1 ] ) . at @xmath18 , the conduction band is formed from electron states and states of light hole , while at @xmath7 , it is formed from heavy - hole states . such so splitting was not described by byckov - rashba effect taken into account within _ kp _ method . it was assumed @xcite that such a surprising behavior of so splitting is a result of the interface inversion asymmetry ( iia ) in the hgte quantum well , which was not taken into account in _ kp _ calculations in @xcite . the question arises : how other spin - dependent effects , for example , the zeeman splitting , depend on the spectrum type normal or inverted . we found only two papers where the zeeman splitting of electron spectrum was measured in the hgte quantum wells with the width @xmath4 which is more or less close to @xmath19 @xcite . in ref . @xcite , the zeeman splitting was determined in a structure with normal spectrum , @xmath20 nm , at very large electron density @xmath21 @xmath13 . in ref . @xcite , it was determined in a structure with inverted spectrum with @xmath22 nm that is noticeably larger than @xmath19 , at @xmath23 @xmath13 . so , up to now a systematic study of the zeeman splitting and a comparison of it with theoretical calculations are absent . in this paper , we present the results of the investigation of the shubnikov - de haas ( sdh ) oscillations in tilt magnetic fields in the hgte quantum wells with normal and inverted spectra . to find the ratio of the zeeman splitting to the orbital one , we have used a modified coincidence method which consists in measuring the angle dependence of amplitudes of the sdh oscillations in low magnetic fields . the simultaneous analysis of this dependence and the shape of oscillations of @xmath24 made it possible to determine both the ratio of the zeeman splitting to the cyclotron one and the anisotropy of @xmath0-factor ( @xmath25 ) over a wide electron - density range , where @xmath26 and @xmath27 are the in - plane and transverse @xmath0-factor , respectively . . the dependences @xmath3 of the conduction and valence bands for @xmath18 ( b ) and @xmath7 ( c ) . the marked area in ( a ) shows the range of quantum well widths under study . ] our samples with the hgte quantum wells were realized on the basis of hgte / hg@xmath28cd@xmath29te ( @xmath30 ) heterostructures grown by the molecular beam epitaxy on a gaas substrate with the ( 013 ) surface orientation @xcite . the samples were mesa etched into standard hall bars of @xmath31 mm width and the distance between the potential probes was @xmath31 mm . to change and control the carrier density in the quantum well , the field - effect transistors were fabricated with parylene as an insulator and aluminium as a gate electrode . for each heterostructure , several samples were fabricated and studied . the zeeman splitting of the conduction band has been obtained from measurements of the sdh effect in a tilted magnetic field , i.e. we used the so - called coincidence method . this method is based on the fact that the spin splitting , @xmath32 , depends on the total magnetic field ( @xmath33 ) whereas the orbital splitting of the landau levels ( lls ) in 2d systems , @xmath34 , is proportional to the component of the magnetic field which is perpendicular to the 2d plane ( @xmath35 ) : @xmath36 , where @xmath37 . .the parameters of heterostructures under study [ cols="^,^,^,^,^ " , ] thus , the ratio of the zeeman splitting to the orbital one , @xmath38 , will change upon varying the tilt angle as @xmath39 . it is clear that there are particular angles @xmath40 when @xmath41 . at integer @xmath42 values , the energies of the lls with different numbers and opposite spin coincide with each other and the distances between the pairs of such degenerate lls are equal to @xmath34 . when @xmath42 is half - integer , the energy distances between nearest lls are twice as low , @xmath43 . as a result , the oscillation periods will differ twice for these cases . knowing values of @xmath40 , one can find the ratio @xmath44 . for example , when @xmath45 , @xmath46 . in this paper , we used the modified coincidence method @xcite . we have measured the oscillations at low magnetic field when : ( i ) @xmath35 is significantly less than the field of the onset of the quantum hall effect ( qhe ) ; ( ii ) amplitude of the oscillation is small so that oscillations of the fermi energy are negligible ; ( iii ) the sdh oscillations are spin - unsplit . in this case , the study of the angular dependence of the oscillation amplitude @xmath47 at a given @xmath35 value within the entire range of angles ( rather than the determination of critical angles ) allows one not only to determine the ratio of the zeeman splitting to the orbital one , but estimate the @xmath0-factor anisotropy . to find analytic expression for the tilt - angle dependence of the oscillation amplitude @xmath47 , it is convenient to represent the oscillations as the sum of the contributions from the two series of the landau spin sublevels . at low magnetic field , when the sdh oscillations are unsplit , the main contribution to the oscillations of @xmath24 comes from the first harmonic , , and the amplitude of the first harmonic is zero . ] and the well known lifshits - kosevitch ( lk ) formula for the sdh oscillations reduces to the following expression @xmath48 \nonumber \\ & + & a_\downarrow \cos\left[2\pi\left(\frac{e_f}{\hbar\omega _ c}+\frac{1}{2}-\frac{x(b)}{2}\right)\right ] . \label{eq1}\end{aligned}\ ] ] here , the factors @xmath49 and @xmath50 depend on the dingle factor , temperature , and magnetic field . when @xmath51 , eq . ( [ eq1 ] ) is @xmath52}\cos{\left[2\pi\left(\frac{e_f}{\hbar\omega _ c}+\frac{1}{2}\right)\right]}. \label{eq2}\ ] ] thus , over this magnetic field range , the values of @xmath35 corresponding to the extremes of @xmath24 should not depend on the tilt angle , while the amplitudes of oscillations @xmath53 $ ] should periodically change with @xmath54 and the angular dependence of the relative amplitude is @xmath55}{\cos[\pi x(1)]}. \label{eq3}\ ] ] let us begin our analysis with the results obtained in the structures with a normal spectrum ( @xmath18 ) ( see table [ tab1 ] ) . as an example , consider the results for the structure 1520 . before discussing the oscillations in the tilt magnetic field , it is necessary to estimate the magnetic field range where eq . ( [ eq1 ] ) is valid for this structure . to this end , let us inspect the magnetic field dependences of @xmath24 and @xmath56 in the normal field , which are presented in fig . [ f2](a ) for the electron density @xmath57 @xmath13 . it is seen that at @xmath58 t , the amplitude of oscillations of @xmath24 is @xmath59 percent less , and the steps in @xmath60 ( with even filling factors @xmath61 ) appear only at @xmath62 t ; therefore one can neglect the oscillations of the fermi energy within this range of @xmath33 . the electron density found from the period of oscillations under assumption that the landau levels are two - fold degenerate , coincides with the hall density . , @xmath63 and @xmath64 , within experimental error , coincides with @xmath65 , where @xmath66 is the capacitance measured in the same sample ; @xmath67 is the gate area . ] thus , at @xmath58 t , the conditions of applicability of eq . ( [ eq1 ] ) are met . now let us inspect the sdh oscillations in the tilt magnetic field . to remove the monotonic part we plotted in fig . [ f2](b ) the @xmath70 versus @xmath35 dependences , measured at different tilt angles . to make it easier to trace the position of oscillations at different angles , we mark the position of one of the maxima @xmath71 at the normal field @xmath72 t by a dashed line . it is clearly seen that the positions of extremes of @xmath70 do not change with tilt angle but the maxima are transformed to the minima at @xmath73 , and upon further rotation they are transformed to the maxima again at @xmath74 . note , a noticeable difference was not observed when the parallel component of the field was along or perpendicular to the current . at @xmath57 @xmath13 . points are experimental data found at @xmath75 and @xmath76 t. the inversion of the amplitude sign corresponds to the change of the oscillation phase by @xmath77 . solid and dash lines are the dependences eq.([eq3 ] ) with @xmath78 and @xmath79 , respectively . other lines are the calculated dependences with taking into account @xmath0-factor anisotropy . ] for the quantitative analysis , the amplitude of oscillations @xmath80 at a given @xmath54 was found by fitting of the oscillatory part of @xmath80 versus @xmath35 curves to the oscillating function corresponding to the electron density @xmath81 measured in the normal field , with the amplitude @xmath82 . the relative amplitudes of the oscillations found in this way @xmath83 as a function of @xmath54 are plotted for some values of @xmath35 in fig . the inversion of the amplitude sign corresponds to the change in the oscillations phase by @xmath77 . note , that @xmath83 does not depend practically on @xmath35 when the magnetic field is sufficiently small so that eq . ( [ eq1 ] ) is valid . in the same figure we have shown the dependences which are given by eq . ( [ eq3 ] ) for some values of the ratio @xmath84=@xmath85 . one can see that eq . ( [ eq3 ] ) well describes the experimental data with @xmath86 . some deviation at @xmath87 can result from the @xmath0-factor anisotropy . indeed , taking this possibility into account in simplest form @xmath88 with @xmath89 we obtain the exact coincidence over all range of tilt angles with @xmath90 ( see dot line in fig . [ f3 ] ) . in fig . [ f3 ] , we have plotted also the angular dependences of @xmath83 with close pairs of parameters . this makes it possible to assess how uniquely these parameters are determined . note that the value of @xmath91 is consistent with the fact that the onset of qhe is observed with even numbers [ see fig . [ f2](a ) ] . such data treatment was carried out for other electron densities and all the results for @xmath92 versus the electron density are plotted in fig . [ f7 ] together with the results obtained for another structure with @xmath18 , 1122 ( see table [ tab1 ] ) . as an example , in fig . [ f4 ] , we have presented the data for structure 1023 at @xmath94 @xmath13 . one can see that at @xmath95 t , the oscillations of @xmath56 are practically absent and the spin splitting of the oscillations of @xmath24 is not observed . so , at @xmath95 t , the conditions for applicability of eq . ( [ eq1 ] ) are met . the derivatives @xmath70 , measured at different tilt angles as a function of @xmath35 are presented in fig . [ f4](b ) for different @xmath54 . for clarity , we have plotted the dashed line at @xmath96 which corresponds to the position of one of maxima . it is clearly seen that with the tilt angle increase , the positions of the extremes of @xmath71 in @xmath97 , similarly to in structures with @xmath98 ( see . fig . [ f2]b ) , does not change but the amplitude of the oscillations decreases significantly slower than in the structure with @xmath18 , and the inversion of the oscillations phase is not observed up to @xmath99 [ compare fig . [ f2](b ) and [ f4](b ) ] . the dependence of the amplitude of oscillations on @xmath54 together with the calculated dependence , eq . ( [ eq3 ] ) , with the isotropic @xmath0-factor is presented in fig . one can see that this simple dependence describes well the data over @xmath54 range from @xmath100 to @xmath101 with @xmath102 . note , this value is three - four times as low as that for the structures with normal spectrum ( see fig . let us check how unambiguously the value of @xmath103 is determined for this case . to this end , we have plotted in fig . [ f5 ] the @xmath80 versus @xmath54 dependences which were calculated using two free parameters , namely @xmath25 and @xmath103 . one can see that the experimental data are equally well described with very different pairs of @xmath103 and @xmath25 . to avoid such a large ambiguity , let us consider oscillations of @xmath104 in a larger magnetic field , where the zeeman splitting starts to be observed but lower than the onset of qhe . such experimental dependence in the normal field is presented in fig . [ f6 ] together with the curves calculated using the lk formula with different @xmath103 values . we assumed the lorentz broadening of lls with parameter @xmath105 mev which was found from the magnetic field dependence of @xmath104 at @xmath106 t. the inset shows that the calculated curve with @xmath102 radically differs from the experimental curve , it does not demonstrate the zeeman splitting of the oscillations up to @xmath107 t. the calculated curves with @xmath108 are significantly closer to the experimental dependence @xmath104 , therewith the curve with @xmath109 practically reproduces the experimental dependence . thus , the comparison of the data with the calculated curves in fig . [ f5 ] and fig . [ f6 ] gives the possibility to find unambiguously the values of @xmath103 and @xmath25 as @xmath110 and @xmath111 , respectively . as seen from fig . [ f5 ] , some discrepancy between the @xmath83 data and calculated curves remains at @xmath112 . the reasons for this discrepancy are not clear . tthey may be : ( i ) effect of sufficiently large in - plane component of @xmath33 because at @xmath113 , the in - plane component of @xmath33 is about @xmath114 t , which corresponds to the magnetic length @xmath115 nm so that @xmath116 became comparable to the width of the quantum well width @xmath117 nm . that can change the energy spectrum noticeably . ( ii ) a difference in the broadening ( or difference in contribution to the oscillations ) of different spin sub - levels ; ( iii ) imperfect flatness of the quantum well and so on . nevertheless , we believe that the value of @xmath118 corresponds to the ratio of the zeeman splitting to the cyclotron energy in the normal magnetic field and @xmath0-factor anisotropy is @xmath119 . . points are experimental data found at @xmath120 , and @xmath121 t at the electron density @xmath94 @xmath13 . the lines are the dependences eq . ( [ eq3 ] ) with different pairs of the parameters @xmath103 and @xmath25 shown in figure . ] the described above measurements and data treatment were carried out for all structures from table [ tab1 ] over the wide electron density range . all obtained values of @xmath103 and @xmath25 versus the electron density are summarized in fig . [ f7 ] . and @xmath56 for structure 1023 in the normal magnetic field at @xmath93 @xmath13 for magnetic field range larger than in fig . [ f4 ] ( points ) . the solid lines are the calculated @xmath24 curves with different values of @xmath103 . these curves are shifted for clarity . the inset shows comparison of the data with the calculation with @xmath102 up to @xmath122 t ( see text ) . ] -factor anisotropy ( b ) plotted against the electron density . the solid and open symbols are the experimental data for the structures with normal and inverted spectra , respectively . the structures numbers are presented in ( a ) . the diagonal crosses are the result of @xcite and straight cross are the result of @xcite . the lines are the calculated dependences for @xmath123 nm and @xmath124 nm ( see text ) . ] one can see that the ratio of the zeeman splitting to the orbital one is close to each other for both types of the structures , with the normal and inverted spectrum . this ratio decreases slightly from @xmath125 to @xmath126 , as the electron density increases from @xmath127 @xmath13 to @xmath128 @xmath13 . the values of the @xmath0-factor anisotropy in the structures with normal and inverted spectra differ significantly [ see fig . [ f7](b ) ] . the values of @xmath25 in the structures with the normal spectrum are in the range of @xmath129 , while in the structures with inverted spectrum they are in the range of @xmath130 . for both types of the structures the values of @xmath25 increase with increasing electron density . let us compare our data with the results of previous studies . we have found only two articles @xcite where the zeeman splitting was studied in the structures with @xmath16 and we plotted them in fig . [ f7 ] . in paper @xcite , the zeeman splitting was determined for the normal magnetic field only in the structure with @xmath22 nm at the electron density @xmath131 @xmath13 . the value of @xmath103 agrees well with our data ( see fig . [ f7 ] ) . in ref . @xcite , both the zeeman splitting and @xmath0-factor anisotropy were found for the structure with @xmath20 nm for the very high electron density , @xmath132 @xmath13 . the value of the @xmath0-factor anisotropy is found to be close to our data for the structure with @xmath133 nm [ see fig . [ f7](b ) ] , while the ratio @xmath38 is significantly larger than our data : @xmath134 instead of @xmath135 . such difference is unclear . one of possible reasons is role of spin - orbit interaction , which can be large for so high electron density and was not taken into account in the analysis of the data . now let us compare the obtained results with the theoretical ones . to find @xmath103=@xmath85 , the positions of the landau levels have been calculated in framework of the @xmath136-band _ @xmath137 _ model @xcite . since there are different notations of the landau levels in various papers , we have numbered the levels in a row , starting from unity for the lowest ll of the conduction band . the zeeman splitting was found as the energy distance between the levels @xmath138 and @xmath139 with odd @xmath138 , while the orbital splitting was found as the distance between the levels @xmath138 and @xmath140 . ] the calculated @xmath92 versus @xmath138 dependences are plotted in fig . [ f7](a ) by solid lines . it is seen that the experimental values are slightly ower than the calculated ones for the structures with the normal and inverted spectrum . it is instructive to compare the results of the calculations performed in the framework of the 8-band _ kp _ model with those obtained within the framework of the bernevig - hughes - zhang ( bhz ) model @xcite , which is often used to analysis various effects . we have used the parameters of the bhz model which give the dependence @xmath3 very close to that calculated in framework of the @xmath136-band _ kp _ model . however , the zeeman splitting in this case appears to be @xmath141 percent larger [ see dashed lines in fig . [ f7](a ) ] . to compare the data for the @xmath0-factor anisotropy with the theory , one needs to know the values of @xmath26 together with @xmath27 calculated just above . the dependences of @xmath26 on electron density were calculated using the results of the paper @xcite where the energy spectrum of the hgte quantum wells in the in - plane magnetic field was studied . the calculated dependences of @xmath25 versus electron density are shown in fig . [ f7](b ) . it is seen that for both types of the structures , with the normal and inverted spectrum , the theoretical values of @xmath25 are small , they are close to each other , and increase with the electron density increase . the calculated values of @xmath25 significantly differ from the experimental data for both types of the structures . for the structures with the inverted spectrum ( @xmath7 ) , the experimental values are to @xmath142 times lower . for the structures with the normal spectrum , the difference is larger and the experimental data are to @xmath114 times higher than the calculated ones . to discuss possible reasons for the discrepancy , let us remind the results of our previous paper @xcite . we have shown ( see introduction ) that for @xmath16 in nominally symmetric structures the top of the valence band is very strongly split by so interaction @xcite . therewith , the so splitting of the conduction band in the same structures does not reveal itself @xcite . it is surprising that such so splitting is observed in structures with the inverted and normal spectrum despite the fact that at @xmath18 and @xmath7 the conduction band is formed from different terms ( see fig . it was assumed in @xcite that such surprising behavior of the so splitting is a result of the interface inversion asymmetry in the hgte quantum well , which is not taken into account in _ kp _ calculations @xcite . we believe that the disagreement between the experimental data on @xmath0-factor anisotropy and calculations is also a result of the interface inversion asymmetry in the hgte quantum well , which is not taken into account in _ kp _ calculations . in summary , the ratio of the zeeman splitting to the orbital one and anisotropy of the @xmath0-factor in the hgte quantum wells both with normal and inverted spectrum have been studied experimentally within a wide electron density range . to obtain two these parameters unambiguously , we have analyzed both the tilt angle dependence of the sdh oscillations in low magnetic fields and the shape of the oscillations in moderate magnetic fields . it has been shown that the ratios of the zeeman splitting to the orbital one are close to each other in the structures with normal and inverted spectra , these ratios decrease when the electron density increases and they are quite close to the values calculated within the _ kp _ method . in contrast , the anisotropy of @xmath0-factor in the structures with the normal and inverted spectrum is strongly different and for both cases differ significantly from the calculated ones . we believe that such disagreement with the calculations is a result of the interface inversion asymmetry in the hgte quantum well , which is not taken into account in the _ kp _ calculations . we are grateful to s. studenikin , v. aleshkin , a.v . germanenko and o.e . raichev for useful discussions and m. zholudev for calculations of lls in framework of 8-band @xmath137 model . the work has been supported in part by the russian foundation for basic research ( grants no . 16 - 02 - 00516 and no . 15 - 02 - 02072 ) and by act 211 government of the russian federation , agreement no . and o.e.r . gratefully acknowledge financial support from the ministry of education and science of the russian federation under projects no . 3.571.2014/k and no . 19ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) link:\doibase 10.1103/physrevb.63.245305 [ * * , ( ) ] link:\doibase 10.1103/physrevb.72.035321 [ * * , ( ) ] link:\doibase 10.1126/science.1133734 [ * * , ( ) ] @noop ph.d . thesis , ( ) @noop * * , ( ) , link:\doibase 10.1103/physrevb.88.155306 [ * * , ( ) ] link:\doibase 10.1103/physrevb.89.165311 [ * * , ( ) ] link:\doibase 10.1103/physrevb.93.155304 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.235414 [ * * , ( ) ] link:\doibase 10.1103/physrevb.69.115340 [ * * , ( ) ] link:\doibase 10.1504/ijnt.2006.008725 [ * * , ( ) ] link:\doibase 10.1103/physrev.174.823 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.84.121407 [ * * , ( ) ] link:\doibase 10.1103/physrevb.85.045310 [ * * , ( ) ]

the zeeman splitting of the conduction band in the hgte quantum wells both with normal and inverted spectrum has been studied experimentally in a wide electron density range . the simultaneous analysis of the sdh oscillations in low magnetic fields at different tilt angles and of the shape of the oscillations in moderate magnetic fields gives a possibility to find the ratio of the zeeman splitting to the orbital one and anisotropy of @xmath0-factor . it is shown that the ratios of the zeeman splitting to the orbital one are close to each other for both types of structures , with a normal and inverted spectrum and they are close enough to the values calculated within _ kp _ method . in contrast , the values of @xmath0-factor anisotropy in the structures with normal and inverted spectra is strongly different and for both cases differs significantly from the calculated ones . we believe that such disagreement with calculations is a result of the interface inversion asymmetry in the hgte quantum well , which is not taken into account in the _ kp _ calculations .

1606.08614

whenever the agenda is about wormholes exotic matter ( i.e. matter violating the energy conditions ) continues to occupy a major issue in general relativity @xcite . it is a fact that einstein s equations admit wormhole solutions that require such matter for its maintenance . in quantum theory temporary violation of energy conditions is permissible but in classical physics this can hardly be justified . one way to minimize such exotic matter , even if we can not ignore it completely , is to concentrate it on a thin - shell . this seemed feasible , because general relativity admits such thin - shell solutions and by employing these shells at the throat region may provide the necessary repulsion to support the wormhole against collapse . the ultimate aim of course , is to get rid of exotic matter completely , no matter how small . in the 4-dimensional ( 4d ) general relativity with a cosmological term , however , such a dream never turned into reality . for this reason the next search should naturally cover extensions of general relativity to higher dimensions and with additional structures . one such possibility that received a great deal of attention in recent years , for a number of reasons , is the gauss - bonnet ( gb ) extension of general relativity @xcite . in the brane - world scenario our universe is modelled as a brane in a 5d bulk universe in which the higher order curvature terms , and therefore the gb gravity comes in naturally . einstein - gauss - bonnet ( egb ) gravity , with additional sources such as maxwell , yang - mills , dilaton etc . has already been investigated extensively in the literature @xcite . not to mention , all these theories admit black hole , wormhole @xcite and other physically interesting solutions . as it is the usual trend in theoretical physics , each new parameter invokes new hopes and from that token , the gb parameter @xmath2 does the same . although the case @xmath3 has been exalted much more than the case @xmath4 in egb gravity so far @xcite ( and references cited therein ) , it turns out here in the stable , normal matter thin - shell wormholes that the latter comes first time to the fore . construction and maintenance of thin - shell wormholes has been the subject of a large literature , so that we shall provide only a brief review here . instead , one class @xcite that made use of non - exotic matter for its maintenance attracted our interest and we intend to analyze its stability in this paper . this is the 5d thin - shell solution of einstein - maxwell - gauss - bonnet ( emgb ) gravity , whose radius is identified with the minimum radius of the wormhole . for this purpose we employ radial , linear perturbations to cast the motion into a potential - well problem in the background . in doing this , a reasonable assumption employed routinely , which is adopted here also , is to relate pressure and energy density by a linear expression @xcite . for special choices of parameters we obtain islands of stability for such wormholes . to this end , we make use of numerical computation and plotting since the problem involves highly intricate functions for an analytical treatment . the paper is organized as follows . in sec . ii the 5d emgb thin - shell wormhole formalism has been reviewed briefly . we perturb the wormhole through radial linear perturbation and cast the problem into a potential - well problem in sec . iii . in sec . iv we impose constraint conditions on parameters to determine possible stable regions through numerical analysis . the paper ends with conclusion which appears in sec . the action of emgb gravity in 5d ( without cosmological constant , i.e. @xmath5 ) is@xmath6 in which @xmath7 is related to the 5d newton constant and @xmath8 is the gb parameter . beside the maxwell lagrangian the gb lagrangian @xmath9 consists of the quadratic scalar invariants in the combination @xmath10 in which @xmath11scalar curvature , @xmath12ricci tensor and @xmath13riemann tensor . variational principle of @xmath14 with respect to @xmath15 yields @xmath16where the lovelock ( @xmath17 ) and maxwell ( @xmath18 ) tensors respectively are @xmath19 @xmath20 the einstein tensor @xmath21 is to be found from our metric ansatz@xmath22 in which @xmath23 will be determined from ( 3 ) . a thin - shell wormhole is constructed in emgb theory as follows . two copies of the spacetime are chosen from which the regions@xmath24are removed . we note that @xmath25 will be identified in the sequel as the radius of the thin - shell and @xmath26 stands for the event horizon radius . ( note that our notation @xmath25 corresponds to @xmath27 in ref . other notations all agree with those in ref . @xcite ) . the boundary , time - like surface @xmath28of each @xmath29 , accordingly will be@xmath30next , these surfaces are identified on @xmath31 with a surface energy - momentum of a thin - shell such that geodesic completeness holds . following the darmois - israel formalism @xcite in terms of the original coordinates @xmath32 we define @xmath33 , with @xmath34 the proper time . the gb extension of the thin - shell em theory requires further modifications . this entails the generalized darmois - israel boundary conditions @xcite , where the surface energy - momentum tensor is expressed by @xmath35diag@xmath36 . we are interested in the thin - shell geometry whose radius is assumed a function of @xmath34 , so that the hypersurface becomes@xmath37the generalized darmois - israel conditions on @xmath38 take the form @xmath39where a bracket implies a jump across @xmath38 , and @xmath40 is the induced metric on @xmath38 with normal vector @xmath41 @xmath42 is the extrinsic curvature ( with trace @xmath43 ) , defined by @xmath44the remaining expressions are as follows . the divergence - free part of the riemann tensor @xmath45 and the tensor @xmath46 ( with trace @xmath47 ) are given by@xmath48 .\end{aligned}\]]the emgb solution that will be employed as a thin - shell solution with a normal matter @xcite is given by ( with @xmath49)@xmath50with constants , @xmath51mass and @xmath52charge . for a black hole solution the inner ( @xmath53 ) and event horizons ( @xmath54 ) are @xmath55 ^{1/2}}.\]]by employing the solution ( 14 ) we determine the surface energy - momentum on the thin - shell , which will play the major role in the perturbation . we shall address this problem in the next section . in order to study the radial perturbations of the wormhole we take the throat radius as a function of the proper time , i.e. , @xmath56 . based on the generalized birkhoff theorem , for @xmath57 the geometry will be given still by ( 6 ) . for the metric function @xmath58 given in ( 14 ) one finds the energy density and pressures as @xcite @xmath59 , \\ s_{\theta } ^{\theta } & = & s_{\phi } ^{\phi } = s_{\psi } ^{\psi } = p=\frac{1}{4\pi } \left [ \frac{2\delta } { a}+\frac{\ell } { \delta } -\frac{4\alpha } { a^{2}}% \left ( \ell \delta -\frac{\ell } { \delta } \left ( 1+\dot{a}^{2}\right ) -2\ddot{% a}\delta \right ) \right ] , \end{aligned}\]]where @xmath60 and @xmath61 in which @xmath62we note that in our notation a dot denotes derivative with respect to the proper time @xmath34 and a prime implies differentiation with respect to the argument of the function . by a simple substitution one can show that , the conservation equation @xmath63is satisfied . the static configuration of radius @xmath64 has the following density and pressures @xmath65 , \\ p_{0 } & = & \frac{\sqrt{f\left ( a_{0}\right ) } } { 4\pi } \left [ \frac{2}{a_{0}}+% \frac{f^{\prime } \left ( a_{0}\right ) } { 2f\left ( a_{0}\right ) } -\frac{2\alpha } { a_{0}^{2}}\frac{f^{\prime } \left ( a_{0}\right ) } { f\left ( a_{0}\right ) } % \left ( f\left ( a_{0}\right ) -1\right ) \right ] .\end{aligned}\ ] ] in what follows we shall study small radial perturbations around the radius of equilibrium @xmath66 to this end we adapt a linear relation between @xmath67 and @xmath68 as @xcite @xmath69here since we are interested in the wormholes which are supported by normal matter , @xmath70 is the speed of sound . by virtue of eq.s ( 19 ) and ( 22 ) we find the energy density in the form @xmath71this , together with ( 16 ) lead us to the equation of motion for the radius of the throat , which reads@xmath72 = \left ( \frac{\sigma _ { 0+}p_{0}}{\beta ^{2}+1}\right ) \left ( \frac{a_{0}}{a}% \right ) ^{3\left ( \beta ^{2}+1\right ) } + \frac{\beta ^{2}\sigma _ { 0-}p_{0}}{% \beta ^{2}+1}.\]]after some manipulation this can be cast into @xmath73where @xmath74 ^{1/3}-\frac{b}{\left [ \sqrt{a^{2}+b^{3}}-a\right ] ^{1/3}}\right ) ^{2}\]]in which the functions @xmath75 and @xmath76 are @xmath77 , \\ b & = & \frac{a^{2}}{8\alpha } + \frac{1-f\left ( a\right ) } { 2}.\end{aligned}\]]we notice that @xmath78 and more tediously @xmath79 both vanish at @xmath80 the stability requirement for equilibrium reduces therefore to the determination of @xmath81 and it is needless to add that , @xmath82 is complicated enough for an immediate analytical result . for this reason we shall proceed through numerical calculation to see whether stability regions/ islands develop or not . since the hopes for obtaining thin - shell wormholes with normal matter when @xmath3 have already been dashed 5 , we shall investigate here only the case for @xmath83 in order to analyze the behavior of @xmath82 ( and its second derivative ) we introduce new parameterization as follows@xmath84accordingly , our new variables @xmath85 @xmath86 @xmath87 @xmath75 and @xmath76 take the following forms @xmath88and @xmath89 , \\ \tilde{p}_{0 } & = & \frac{\sqrt{f\left ( \tilde{a}_{0}\right ) } } { 4\pi } \left [ \frac{2}{\tilde{a}_{0}}+\frac{f^{\prime } \left ( \tilde{a}_{0}\right ) } { % 2f\left ( \tilde{a}_{0}\right ) } + \frac{2}{\tilde{a}_{0}^{2}}\frac{f^{\prime } \left ( \tilde{a}_{0}\right ) } { f\left ( \tilde{a}_{0}\right ) } \left ( f\left ( \tilde{a}_{0}\right ) -1\right ) \right ] , \end{aligned}\]]@xmath90 , \\ b & = & -\frac{\tilde{a}^{2}}{8}+\frac{1-f\left ( \tilde{a}\right ) } { 2}.\end{aligned}\]]following this parametrization our eq . ( 25 ) takes the form @xmath91where@xmath92 in the next section we explore all possible constraints on our parameters that must satisfy to materialize a stable , normal matter wormhole through the requirement @xmath93 @xmath94 ) starting from the metric function we must have @xmath95 @xmath96 ) in the potential , the reality condition requires also that @xmath97at the location of the throat this amounts to @xmath98or after some manipulation it yields @xmath99this is equivalent to@xmath100 @xmath101 ) our last constraint condition concerns , the positivity of the energy density , which means that @xmath102this implies , from ( 31 ) that @xmath103 < 0\]]or equivalently@xmath104it is remarkable to observe now that the foregoing constraints ( @xmath105 ) on our parameters can all be expressed as a single constraint condition , namely @xmath100 we plot @xmath106 from ( 26 ) for various fixed values of mass and charge , as a projection into the plane with coordinates @xmath107 and @xmath108 in other words , we search and identify the regions for which @xmath109 , in @xmath110dimensional figures considered as a projection in the @xmath111 plane . the metric function @xmath112 and energy density @xmath113 behavior also are given in fig.s 1 - 4 . it is evident from fig.s 1 - 4 that for increasing charge the stability regions shrink to smaller domains and tends ultimately to disappear completely . for smaller @xmath114 bounds we obtain fluctuations in @xmath115 which is smooth otherwise . in each plot it is observed that the maximum of @xmath116 occurs at the right - below corner ( say , at @xmath117 ) which decreases to the left ( with @xmath114 ) and in the upward direction ( with @xmath107 ) . beyond certain limit ( say @xmath118 ) , the region of instability takes the start . the proper time domain of stability can be computed from ( 35 ) as @xmath119from a distant observer s point of view the timespan @xmath120 can be found by using the radial geodesics lagrangian which admits the energy integral @xmath121this gives the lifetime of each stability region determined by @xmath122once @xmath118 ( @xmath123 ) are found numerically , assuming that no zeros of @xmath124 and @xmath125 occurs for @xmath126 the lifespan of each stability island can be determined . we must admit that the mathematical complexity discouraged us to search for possible metastable region that may be triggered by employing a semi - classical treatment . our numerical analysis shows that for @xmath0 and specific ranges of mass and charge the 5d emgb thin - shell wormholes with normal matter can be made stable against linear , radial perturbations . the fact that for @xmath127 there is no such wormholes is well - known . the magnitude of @xmath8 is irrelevant to the stability analysis . this reflects the universality of wormholes in parallel with black holes , i.e. , the fact that they arise at each scale . stable regions develop for each set of finely - tuned parameters which determine the lifespan of each such region . beyond those regions instability takes the start . our study concerns entirely the exact emgb gravity solution given in ref . it is our belief that beside emgb theory in different theories also such stable , normal - matter wormholes are abound , which will be our next venture in this line of research . 1 : @xmath128 region ( @xmath129 , @xmath130 ) for various ranges of @xmath107 and @xmath108 the lower and upper limits of the parameters are evident in the figure . the metric function @xmath131 and @xmath132 , are also indicated in the smaller figures .

recently in ( phys . rev . d 76 , 087502 ( 2007 ) and phys . rev . d 77 , 089903(e ) ( 2008 ) ) a thin - shell wormhole has been introduced in 5-dimensional einstein - maxwell - gauss - bonnet ( emgb ) gravity which was supported by normal matter . we wish to consider this solution and investigate its stability . our analysis shows that for the gauss - bonnet ( gb ) parameter @xmath0 stability regions form for a narrow band of finely - tuned mass and charge . for the case @xmath1 , we iterate once more that no stable , normal matter thin - shell wormhole exists .

1001.4384

it is widely accepted that quantum chromodynamics ( qcd ) is the fundamental theory for the strong interaction . the high - energy behavior of qcd is well described by the perturbed qcd because of its asymptotic freedom . it explains the experimental data such as those in the deep inelastic scattering process . on the other hand , it is difficult to describe the low energy properties of qcd because the effective coupling constant increases with the decreasing momentum as the renormalization group analysis suggests . thooft proposed to use the inverse number of colors @xmath2 as an expansion parameter by generalizing qcd to the su(@xmath3 ) gauge theory@xcite . in the large @xmath3 limit , it becomes the theory of the weakly interacting meson . witten pointed out that the baryons should appear as topological solitons@xcite . the skyrme model is recognized as an effective theory of qcd in the large @xmath4 limit although the skyrme lagrangian is not derived from qcd directly@xcite . the baryon number is introduced into the skyrme model from a topological point of view . skyrme proposed a stable configuration called the hedgehog ansatz . the quantization is performed by introducing the collective coordinates , the flavor rotation of the hedgehog configuration@xcite . the skyrme model explains the static properties of the baryon such as the charge radius and the magnetic moment with 30% accuracy . the product ansatz is a two - skyrmion configuration proposed by skyrme@xcite . it is a good approximation so far the two skyrmions are separated in the long distance . the numerical simulation is a direct method to obtain the exact two - baryon configuration@xcite . there is another way to describe the skyrmion configuration with a few parameters . the atiyah - manton ansatz is constructed from the instanton configuration in the su(2 ) gauge theory@xcite . the stable configuration with the torus shape can be described by this ansatz . even if the skyrmion configuration is obtained , there is a problem that the attraction in the central potential at the intermediate range is absent . it is being solved by considering the n-@xmath5 mixing through the intermediate state@xcite , the finite-@xmath3 effect@xcite , the higher - order terms generated by @xmath6-meson , and the radial excitation . by diagonalizing the potential between the nn and n@xmath5 states for each channel , one can construct the better eigenstate . it amounts to the n-@xmath5 mixing . the finite @xmath3 effects are often considered together with the nn - n@xmath5 mixing . since the skyrme model is recognized as an effective theory in the large @xmath4 limit , the finite @xmath3 correction is required@xcite . the skyrme model is extended into the su(3 ) flavor symmetry . there are two approaches to deal with the extra strange degrees of freedom . one is the bound state approach@xcite in which the symmetry breaking is regarded as large . the k - meson is introduced as a small fluctuation from the su(2 ) symmetry . another is the collective coordinate method which is based on the su(3 ) symmetry . in this method , the symmetry breaking is taken to be small . the symmetry breaking is treated perturbatively@xcite . yabu and ando unified these two approaches by the exact treatment of the symmetry breaking@xcite . yabu - ando approach reproduces the mass splitting of the baryons in the same multiplet . in the two - baryon case , only the product ansatz has been investigated because of the complexity of the numerical simulation of the su(3 ) skyrme model@xcite . in this paper , we investigate the interaction between the hyperon and the nucleon in the su(3 ) skyrme model . the atiyah - manton ansatz extended to the su(3 ) symmetry is adopted as the two - baryon configuration . the static potential is expanded in the modified su(3 ) rotational matrices . we obtain the interaction between the baryons by integrating the static potential with the initial and final wave functions over the euler angles . to obtain the attractive force in the central channel of the @xmath0-n interaction , we take account of the @xmath0n-@xmath1n mixing through the intermediate state . in sec . ii , we construct the two - baryon configuration by the atiyah - manton ansatz . in sec . iii , we express the potential in the modified su(3 ) rotational matrices and obtain its matrix element between the baryons . in sec . iv , we consider the @xmath0n-@xmath1n mixing through the intermediate state together with the finite @xmath3 effects . in sec . v , we discuss our results . let us consider the non - linear field of the pseudo - scalar meson @xmath7 within the flavor su(3 ) symmetry . the action of the su(3 ) skyrme model is given by @xmath8 where @xmath9 ^ 2 , \\ l_{\rm sb } & = & \int d^3x\biggl\ { \frac{f_\pi^2}{32}(m_\pi^2+m_\eta^2 ) { \rm tr}\left(u+u^\dagger-2\right ) \nonumber \\ & & + \frac{\sqrt{3}f_\pi^2}{24}(m_\pi^2-m_{{\rm k}}^2 ) { \rm tr}\left(\lambda_8(u+u^\dagger)\right)\biggr\ } , \label{eq : lsb } \\ \gamma & = & -\frac{i}{240\pi^2}\int_q d^5x\epsilon^{ijklm } { \rm tr}\left(u^\dagger(\partial_i u)u^\dagger(\partial_j u)u^\dagger(\partial_k u ) u^\dagger(\partial_l u)u^\dagger(\partial_m u)\right ) . \label{eq : swz}\end{aligned}\ ] ] the summation over the repeated indices is assumed and @xmath10 denote the gell - mann matrices . the symmetry breaking part of the lagrangian ( [ eq : lsb ] ) reproduces the mass terms expanded in the pseudo - scalar meson fields with the gell - mann - okubo relation @xmath11 . in the wess - zumino - witten term ( [ eq : swz ] ) , the integration is taken over the 5-dimensional disc @xmath12 the boundary of which is the usual spacetime . the length and the meson mass are often measured in the unit @xmath13 , and the energy in @xmath14 , called the skyrme units . the hedgehog configuration is also stable in the su(3 ) skyrme model . the quantization is done with respect to the collective coordinates expressed as the rotation of the hedgehog configuration , @xmath15 it is difficult to construct the two - baryon configuration in the general form . the product ansatz is used as the first approximation . the product ansatz holds when the two skyrmions are separated in the long distance . the atiyah - manton ansatz is another method to construct the two - baryon configuration , which has been used in the su(2 ) skyrme model . it is obtained from the instanton , a topological configuration of the gauge field defined in the euclidean spacetime , @xmath16 the instanton configuration given by thooft@xcite is expressed as @xmath17 where @xmath18 and @xmath19 are the instanton coordinate and the spreading of the @xmath20-th instanton respectively . jackiw - nohl - rebbi ( jnr ) proposed the more general form of the instanton configuration@xcite . the two - instanton superpotential is expressed as @xmath21 from the skyrmion point of view , it can describe the stable configuration with the torus shape . however , it is difficult to apply the jnr form to the su(3 ) skyrme model because of the complex relation between the instanton parameters @xmath22 and the skyrmion position . therefore , we concentrate our efforts on the thooft form . to apply the above method to our problem , we extend the atiyah - manton ansatz to the su(3 ) symmetry . we can change eq . ( [ eq : inst ] ) into the form @xmath23 because the differentiation with respect to the spatial variables can be separated into that with the instanton coordinates . now , we extend the gauge group from su(2 ) to su(3 ) by replacing the su(2 ) @xmath24-matrices with the generators of the su(3 ) group @xmath25 and @xmath26 different for each instanton coordinate , @xmath27 where we have used the notations @xmath28 the atiyah - manton configuration does not have the exponential damping behavior of the massive meson in the long distance . we introduce the additional parameters for the exponential damping by the substitution in eq . ( [ eq : modam ] ) , @xmath29 this substitution improves the long - distance behavior of the atiyah - manton configuration for the massive case . it corresponds to the hedgehog solution with the profile function @xmath30 the long - distance behavior of the profile function leads to @xmath31 in spite of the above modifications , the baryon number of the atiyah - manton configuration is still conserved . indeed , the baryon number is confirmed to be two within 1% discrepancy by the numerical simulation . we use the third parameter set in ref . @xcite ( @xmath32=82.9mev , @xmath33=4.87 , @xmath34=769mev ) throughout this paper . the subtraction of the vacuum - like energy does not matter because we use the energy difference between the two configurations . there is ambiguity in determining the separation of the two skyrmions for the generated configuration . we adopt the separation between the two baryons as @xmath35^{1/2},\ ] ] where @xmath36 is the baryon number density@xcite . we perform the numerical simulation by taking the orientations of the individual skyrmions as @xmath37 and @xmath38 which ensures the symmetry under the exchange between the two skyrmions . we determine the instanton parameters @xmath39 . from the symmetry under the exchange of the two skyrmions , we require that they should be equal for the both skyrmions . it turns out that @xmath40 for the relative orientation @xmath41 and @xmath42 for @xmath43 by minimizing the static energy . these parameters give the classical mass @xmath44 in the skyrme unit which is consistent with the exact value 39.849 estimated by the numerical simulation in ref . the static potentials @xmath45 , @xmath46 , @xmath47 , and @xmath48 for the orientations @xmath41 , @xmath49 , @xmath50 , and @xmath51 respectively are shown in fig . 1 . ( 6.5,3.5 ) ( 0,0.5)(6.5,3 ) ( 0,0 ) ( 6.5,0 ) fig . 1 static potential in mev as a function of the separation between the two baryons @xmath52(fm ) @xmath53 , @xmath54 , @xmath55 , @xmath56 for the relative orientation @xmath57 , @xmath58 , @xmath59 , @xmath60 . the static potential is generally expanded in the su(3 ) rotation matrices@xcite . from the symmetry of the solution , the static potential is reduced to the form @xmath61 with the condition @xmath62 . in the symmetry - breaking case , it should be expanded in the modified rotation matrices rather than the non - breaking ones because the mixing of a certain representation with its higher ones is not so small as to be neglected . the su(3 ) rotation is parameterized by the su(3 ) euler angles , @xmath63 where the matrices @xmath64 and @xmath65 are expressed in the usual su(2 ) euler angles@xcite . the wave function for the baryon belonging to the multiplet @xmath66 is given by yabu and ando as the modified su(3 ) rotation matrix , @xmath67 @xmath68 where @xmath69 and @xmath70 stand for the usual su(2 ) rotation matrices and @xmath71 is the multiplicity of the representation @xmath72 . the properties of the symmetry breaking are contained in the strange - mixing function @xmath73 . a subsidiary condition derived from the wess - zumino term is imposed on the physical states , @xmath74 we can determine the coefficients @xmath75 in the static potential ( [ eq : vexp ] ) by observing it for several relative orientations . once the static potential is given , the interaction between the hyperon and the nucleon is obtained by integrating the static potential between the initial and the final wave functions over all orientations , @xmath76 where we adopt the direct product of the yabu - ando wave function as the two - baryon state for the first approximation . the interaction is obtained in the form @xmath77 for the nn - interaction , @xmath78 for the @xmath0n - interaction , and @xmath79 for the @xmath1n - interaction . since the baryons stay in the @xmath80-axis , the tensor operator is defined by @xmath81 . the potentials @xmath82 for the relative orientations determine the coefficients @xmath83 , @xmath84 , @xmath85 , @xmath86 in the static potential ( [ eq : vexp ] ) expanded up to the octet representation . the graphs of the interaction @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 , @xmath92 , and @xmath93 are shown in figs . 2 - 8 . ( 6.5,6 ) ( 0,4)(6.5,2 ) ( 0,3 ) ( 6.5,0.5 ) fig . 2 central part of the nn - potential @xmath94 in mev as a function of the separation @xmath95(fm ) , dashed curve denotes the product ansatz and dot - dashed one the one - boson exchange model . ( 0,0.5)(3.25,2 ) ( 0,0 ) ( 3.25,0.5 ) fig . 3 spin - isospin part of the nn - potential @xmath96 . ( 3.25,0.5)(3.25,2 ) ( 3.25,0 ) ( 3.25,0.5 ) fig . 4 tensor - isospin part of the nn - potential @xmath97 . ( 6.5,5 ) ( 0,3)(3.25,2 ) ( 0,2.5 ) ( 3.25,0.5 ) fig . 5 central part of the @xmath98n - potential @xmath94 . ( 3.25,3)(3.25,2 ) ( 3.25,2.5 ) ( 3.25,0.5 ) fig . 6 central part of the @xmath99n - potential @xmath94 . ( 0,0.5)(3.25,2 ) ( 0,0 ) ( 3.25,0.5 ) fig . 7 spin - isospin part of @xmath99n - potential @xmath96 . ( 3.25,0.5)(3.25,2 ) ( 3.25,0 ) ( 3.25,0.5 ) fig . 8 tensor - isospin part of the @xmath99n - potential @xmath97 . the central potential @xmath94 of the nn , @xmath0n , @xmath1n systems is still repulsive , while the atiyah - manton ansatz tends to show the less repulsive force than the product ansatz @xcite . for the spin - isospin part @xmath96 , the results show a good agreement with the nijmegen potential ( model d ) , the one - boson - exchange potential developed by the nijmegen group@xcite at the range @xmath100 . the behavior of the tensor - isospin part @xmath97 is consistent with the nijmegen model . it is well known that the naive estimation of the interaction is insufficient to give the central attraction at the intermediate range . in the su(2 ) case , the @xmath5-n mixing is taken into account . the direct product of the wave functions as the two - baryon eigenstate becomes worse when the separation between the skyrmions decreases . the candidate for the su(3 ) symmetry is the @xmath0-@xmath1 mixing . we consider the @xmath0n-@xmath1n mixing in the intermediate state together with the finite @xmath3 effects . it is shown by the fact that the off - diagonal element of the potential survives , @xmath101 where @xmath102 it is found that the isospin is conserved under the baryon - baryon interaction . the total spin is not an invariant of the hyperon - nucleon system whereas the projection of the spin in the @xmath80-direction is still conserved . the potentials with respect to the total spin and isospin states are written as @xmath103 for the @xmath104 channel , @xmath105 for the @xmath106 channel , and @xmath107 for the @xmath108 channel . the non - zero off - diagonal matrix element @xmath109 shows that the direct product of the single - baryon wave functions is not a good eigenstate for the two - baryon system . one can obtain the better two - baryon state by diagonalizing the matrix @xmath110v_@xmath111v_@xmath112v_@xmath111v_@xmath113 for each channel . after the diagonalization , the lowest eigenvalue is adopted for the @xmath0n central potential . the finite @xmath3 effects should be taken into consideration together with the @xmath0-@xmath1 mixing because the skyrme model is recognized as an effective theory in the large @xmath3 limit . the spin - isospin matrix elements are enhanced in the @xmath114 case compared with those in the large @xmath3 limit by the factors 20/9 for @xmath115 , @xmath116 and @xmath117 for @xmath118 , @xmath119 , from the analysis of the quark hedgehog model as in ref . the graph of the @xmath0-n interaction in the central channel @xmath90 with the @xmath0n-@xmath1n mixing is shown in fig ( 6.5,4 ) ( 0,1)(6.5,3 ) ( 0,0 ) ( 6.5,0.5 ) fig . 9 central part of the @xmath98n - potential @xmath94 with @xmath0n-@xmath1n mixing , solid curve stands for that with the @xmath0-@xmath1 mixing , dashed one for the naive estimation , and dot - dashed one for the one - boson - exchange model . the attractive force at the intermediate range appears by taking account of the @xmath0n-@xmath1n mixing through the intermediate state together with the finite @xmath3 effects . we discuss the results in this section . the static potential is obtained from the atiyah - manton ansatz extended to the su(3 ) symmetry . the atiyah - manton ansatz gives a lower energy than the product ansatz at the intermediate range . since the symmetry breaking is not small , the static potential is expanded in the modified su(3 ) rotation matrices up to the octet representation . we have obtained the baryon - baryon interaction by integrating the static potential between the two - baryon states over the euler angles . in the naive estimation , we have not obtained the intermediate attraction of the central force although the result from the atiyah - manton ansatz is less repulsive than the product ansatz . to improve the estimation of the central potential , we have to take account of the several effects as in the su(2 ) case . one of such effects is introduced by considering the mixing with the higher excitations , the @xmath5-n mixing in the su(2 ) case , through the intermediate state . the candidate within the su(3 ) symmetry is the mixing between @xmath120 and @xmath121@xcite . the effects from the mixing of these particles are expected to play a significant role in the hyperon - nucleon interaction because the mass difference between @xmath0 and @xmath1 , @xmath122 , is smaller than that between n and @xmath123 , @xmath124 . the central potential between @xmath0 and n with the @xmath0n-@xmath1n mixing shows the attraction at the intermediate range . this result is consistent with the one - boson - exchange model . the direct product of the two single - baryon states is not a good eigenstate when the two skyrmions close together . it is suggested by the non - vanishing off - diagonal matrix element between the @xmath0-n and @xmath1-n states . by diagonalizing this matrix for each channel , one can obtain the better eigenstate of the two - baryon system . the lowest eigenvalue is adpoted for the @xmath0n potential . this procedure amounts to taking account of the @xmath0n-@xmath1n mixing through the intermediate state . the finite @xmath3 effects is considered together with the @xmath0-@xmath1 mixing . the finite @xmath3 correction is estimated from the analysis of quark hedgehog model . the spin - isospin part @xmath96 and the tensor - isospin part @xmath97 shows a consistent behavior with the one - boson - exchange model at the range @xmath100 . this implies that the long - range force which is dominated by the @xmath125-exchange reproduces the one - boson - exchange potential well . in the present paper , we have observed that the naive estimation of the interaction between the hyperon and the nucleon does not show the attractive central force at the intermediate range . the atiyah - manton ansatz is adopted to improve the medium - range behavior of the skyrmion configuration rather than the product ansatz . the @xmath0n-@xmath1n mixing in the intermediate state is taken into account . the finite @xmath3 effects are included from the quark hedgehog model . after these treatments are taken , the hyperon - nucleon interaction shows the central attraction which is consistent with the one boson exchange model . therefore , we conclude that the configuration with a certain accuracy , the @xmath0n-@xmath1n mixing , and the finite @xmath3 effects are required for the attractive force in the @xmath0-n interaction . finally , we discuss the validity of the atiyah - manton configuration based on the thooft instanton . in this paper , we have adopted the thooft form as a starting point of the su(3 ) skyrmion configuration . on the other hand , the jackiw - nohl - rebbi form gives the more general configuration . it can describe the stable configuration with the torus shape . indeed , it is significant to reproduce such a configuration in the su(3 ) model as well . in this field , the stable point in the manifold of the atiyah - manton configuration is investigated . by quantizing the fluctuation around it , one constructs the quantum state which has the same quantum number as the deuteron@xcite . however , the jnr form is difficult to handle the modification of the long - distance behavior caused by the mass of the pseudo - scalar meson . at this point , it is convenient to take the thooft form owing to the transparent relation between the instanton coordinate and that of the skyrmion . furthermore , our configuration based on the thooft form is still valid in the region where the individual skyrmions are identified . the author thanks professor k. ohta for his suggestion to investigate this subject . g. thooft , nucl . * b72 * , 461 ( 1974 ) ; nucl . * b75 * , 461 ( 1974 ) . e. witten , nucl . phys . * b160 * , 57 ( 1979 ) . t. h. r. skyrme , proc . * a260 * , 127 ( 1961 ) . g. s. adkins , c. r. nappi , and e. witten , nucl . * b228 * , 552 ( 1983 ) ; g. s. adkins and c. r. nappi , nucl . * b233 * , 109 ( 1984 ) . j. j. m. verbaarschot , t. s. walhout , j. wambach , and h. w. wyld , nucl . * a461 * , 603 ( 1986 ) . m. f. atiyah and n. s. manton , phys . * 222b * , 438 ( 1989 ) ; commun . phys . * 153 * , 422 ( 1993 ) . a. hosaka , m. oka , and r. d. amado , nucl . * a530 * , 507 ( 1991 ) . t. otofuji , s. saito , m. yasuno , t. kurihara , and h. kanada , phys . rev . c * 34 * , 1559 ( 1986 ) . h. depace , h. mther , and a. faessler , nucl . phys . * a457 * , 541 ( 1986 ) ; phys . * 188b * , 307 ( 1987 ) . a. jackson , a. d. jackson , and v. pasquier , nucl . * a432 * , 567 ( 1985 ) . t. s. walhout and j. wambach , phys . lett . * 67 * , 314 ( 1991 ) ; n. r. walet and r. d. amado , phys . lett . * 68 * , 3849 ( 1992 ) . c. g. callan and i. klebanov , nucl . * b262 * , 365 ( 1985 ) . p. o. mazur , m. a. nowak , and m. praszalowicz , phys . 147b * 137 ( 1984 ) ; m. praszalowicz , phys . * 158b * , 264 ( 1985 ) . m. chemtob , nucl . phys . * b256 * , 600 ( 1985 ) . h. yabu and k. ando , nucl . * b301 * , 601 ( 1988 ) . g. klbermann and j. m. eisenberg , phys . * 235b * , 6 ( 1990 ) . b. schwesinger , f. g. scholtz , and h. b. geyer , phys . d * 51 * , 1228 ( 1995 ) . g. thooft , phys . lett . * 37 * , 8 ( 1976 ) . r. jackiw , c. nohl , and c. rebbi , phys . d * 15 * , 1642 ( 1977 ) . d. f. holland , j. math . * 10 * , 531 ( 1969 ) . m. m. nagels , t. a. rijken , and j. j. de swart , phys . d * 12 * , 744 ( 1975 ) ; phys . d * 15 * , 2547 ( 1977 ) ; phys . d * 20 * , 1633 ( 1979 ) . a. gal , adv . * 8 * , 1 ( 1975 ) . r. a. leese , n. s. manton , and b. j. schroers , nucl . * b442 * , 228 ( 1995 ) ; n. r. walet , nucl . * a606 * , 429 ( 1996 ) ; * a586 * , 649 ( 1995 ) .

the interaction between the hyperon and the nucleon is investigated in the su(3 ) skyrme model . the static potential , which is expanded in terms of the modified su(3 ) rotation matrices , is obtained for several orientations with the atiyah - manton ansatz . the interaction is calculated for the nn , @xmath0n , and @xmath1n systems . the medium - range attraction of the central potential between @xmath0 and n is obtained by considering the @xmath0-@xmath1 mixing through the intermediate state .

hep-ph9710469

density functional theory@xcite ( dft ) is nowadays the most widely used method for electronic structure calculations , in both condensed matter physics and quantum chemistry , thanks to the combination of low computational cost and remarkable accuracy for a wide variety of chemical bonds and solid - state systems . there are , however , notable exceptions to such an accuracy . for example , even the best available approximations of the exchange - correlation functional , the key ingredient of the dft , fail to recover long - range van der waals interactions,@xcite are not completely safe for the description of the hydrogen bond @xcite and have intrinsic problems with situations of near degeneracy ( when two sets of orbitals happen to have very close energies ) . @xcite more generally , the `` chemical accuracy '' ( the accuracy needed to predict the rates of chemical reactions ) has not yet been reached . for all these reasons the search for new approximate functionals , or even new ways of exploiting the basic ideas and advantages of the dft , is very active.@xcite in this context several authors@xcite have suggested to split the electron - electron interaction into a short - range part , to be treated within the dft , and a long - range part , to be handled by other techniques . the motivation behind these mixed schemes " is that the dft , even in the simplest local - density approximation ( lda ) , provides an accurate description of the short - range electron - electron repulsion,@xcite while other techniques which give a poor description of short - range properties , like the configuration interaction ( ci ) method or the random - phase approximation ( rpa),@xcite can , instead , accurately capture long - range correlation effects . of course there is no unique way to split the coulomb potential @xmath2 into a short - range ( sr ) and a long - range ( lr ) part . the error function and its complement @xmath3 have been already used for this purpose @xcite ( see fig . [ fig_erf ] ) , and we stick to this choice , which yields analytic matrix elements for both gaussians and plane waves , i.e. , the most common basis functions in quantum chemistry and solid - state physics , respectively . this form still leaves room for some arbitrariness : the choice of the most convenient cutoff parameter @xmath1 , which may be different for different mixed schemes " . into a short - range ( sr ) part and a long - range ( lr ) part , according to eqs . ( [ eq_srpotential])-([eq_lrpotential ] ) , with @xmath4.,width=249 ] the combination of a short - range dft calculation and a different treatment of the long - range part of the electron - electron interaction can be founded on a rigorous basis through the adiabatic connection formalism.@xcite depending on the specific problem addressed ( van der waals forces , near - degeneracy , ... ) , and thus on the particular approach to the long - range part of the electron - electron interaction , different mixed schemes " have been proposed.@xcite but in all of them , as in standard dft , a crucial role is played by the exchange - correlation functional , which now must be built for a modified electron - electron interaction . the schemes of refs . need a pure short - range functional , @xmath5 $ ] , whose lda version is given by @xmath6=\int n(\rv)\ , \exc(n(\rv),\mu)\,d\rv , \label{eq_ldasr}\ ] ] where @xmath7 is the exchange - correlation energy per electron of a uniform gas of density @xmath8 interacting with a short - range potential like eq . ( [ eq_srpotential ] ) . the value of @xmath1 in eq . ( [ eq_ldasr ] ) can be either a constant , or , possibly , a convenient function of the density , @xmath9.@xcite the local functional @xmath7 is the quantity which we provide in this paper . we start from a jellium - like hamiltonian ( in hartree atomic units used throughout this work ) @xmath10 where @xmath11 is the modified electron - electron interaction @xmath12 @xmath13 is , accordingly , the interaction between the electrons and a rigid , positive , uniform background of density @xmath14 @xmath15 and @xmath16 is the corresponding background - background interaction @xmath17 first we calculate the ground - state energy per electron of this model hamiltonian , as a function of the density parameter @xmath18 and of the parameter @xmath1 , with a diffusion monte carlo method ( sec . [ sec_dmc ] ) . then we derive the asymptotic behaviors of the correlation energy @xmath19 ( sec . [ sec_limits ] ) . on these grounds we finally ( sec . [ sec_param ] ) present a convenient analytic parametrization of the correlation energy , thus following in the footsteps from quantum simulations of the regular jellium model to the best available lda functionals.@xcite a local density functional for the short - range potential of eqs . ( [ eq_vee])-([eq_vbb ] ) should recover the ceperley - alder@xcite ( ca ) correlation energy for @xmath20 . in this section we outline the implications of this condition on the technical aspects of our calculation , which is in all respects a standard application of the diffusion monte carlo method in the fixed node approximation ( fn dmc).@xcite the fn dmc method gives the energy @xmath21 of the lowest lying fermionic eigenstate of the hamiltonian which has the same nodes as the chosen trial function @xmath22 . the error in @xmath21 is variational , and it vanishes as the nodal structure of @xmath22 approaches the ( unknown ) nodal structure of the exact ground state . the simplest choice for the trial function of a homogeneous fluid@xcite is the jastrow slater form , @xmath23 , where the symmetric jastrow factor @xmath24 $ ] describes pair correlations , and @xmath25 is the product of one slater determinant of plane waves ( pw ) for each spin component ( @xmath26 denotes the coordinates of all the particles ) . a better nodal structure is provided by the so called backflow ( bf ) wave function.@xcite the method used in ref . is in principle exact : it starts from the fn solution and then it performs a `` nodal relaxation '' , whereby the energy converges to the exact ground state result . however , this second process is accompanied by an increasing statistical noise , which may hinder full convergence of the results . in practice , the results of ref . are between the fn energies recently calculated with pw and bf nodes@xcite , and actually somewhat closer to the former . since , on one hand , bf calculations are considerably more demanding , and , on the other , the most widely used local - density functionals are constructed to fit the quantum monte carlo results of ref . , we choose to stick to the simple trial function with slater determinants of plane waves . in this way our short - range local - density functional " will continuously merge into the ceperley - alder@xcite - based local - density functionals as @xmath27 . all the other errors in the simulation can be controlled and eliminated . it is easy to ensure that the biases due to a finite time step and a finite population of walkers@xcite are much smaller than the statistical uncertainty of the ca results , which we set as our target precision . the number extrapolation is more delicate . we simulate @xmath28 particles in a cubic box with periodic boundary conditions , interacting via the potential of eq . ( [ eq_srpotential ] ) . since for very small values of @xmath1 we rely on the analytic asymptotic behavior described in sec . [ sec_limits ] , the only simulations we need to do will deal with really short - range potentials , which we may safely treat using the minimum image convention.@xcite the dependence of the energy on the number of particles is determined with the variational monte carlo ( vmc ) method , which calculates the expectation value of the hamiltonian operator on the trial wave function and is cheaper than dmc . for several values of @xmath28 ( namely 38 , 54 , 66 , 114 , 162 ) , we use vmc to calculate ( i ) the variational energy @xmath29 ( after optimization of the jastrow factor ) , and ( ii ) the hartree fock energy @xmath30 , which corresponds to @xmath31 . for each value of @xmath18 and @xmath1 , the resulting estimate of the correlation energy per electron , @xmath32 , is fitted to the following form : @xmath33 \nonumber \\ & & + b(r_s,\mu)/n . \label{eq_size}\end{aligned}\ ] ] here @xmath34 is the kinetic energy of @xmath28 non interacting electrons at @xmath35 , and @xmath36 , @xmath37 and the correlation energy in the thermodynamic limit , @xmath38 are fitting parameters . the size dependence of the vmc result for the correlation energy is shown in fig . [ fig_size ] for the case where it is largest ( small @xmath18 and small @xmath1 ) . we point out that the simple functional guess of eq . ( [ eq_size ] ) ( solid line ) accurately models the size dependence of the vmc data which , although on a small energy scale , are still far from a smooth dependence ( dots with error bars ) . our final result for the correlation energy is obtained by adding the infinite - size extrapolation obtained from eq . ( [ eq_size ] ) to the result of a single dmc simulation with @xmath39 . , for different numbers of particles @xmath28 . the fitting function of eq . ( [ eq_size ] ) ( line ) favorably compares with the vmc data ( dots ) . , width=249 ] in this section we derive some limiting behaviors of the correlation energy @xmath19 , which will be used for its parametrization in sec . [ sec_param ] . the detailed study carried out here can be also of interest for the choice of a density - dependent @xmath1 parameter in the mixed schemes of refs . and . we consider two different regimes : when our system approaches the standard jellium model ( i.e. , full interaction @xmath2 ) , and when it approaches the noninteracting fermi gas . in the first case ( secs . [ picmu ] and [ picrs ] ) we find that the correlation energy is a function of the scaled variable @xmath40 , while in the second case ( sec . [ granmurs ] ) the relevant scaled variable is @xmath41 . since for small @xmath1 @xmath42 if we fix the density and let the parameter @xmath1 approach zero , we can write @xmath43 where @xmath44 in eq . ( [ eq_hpert ] ) , and in the rest of this paper , the suffix `` coul '' indicates quantities of the standard uniform electron gas ( jellium ) , with coulomb interaction @xmath2 . thus , for small @xmath1 we are perturbing the jellium model , @xmath45 since @xmath46 is a constant , we immediately find @xmath47 and @xmath48 , which , combined with @xmath49 , also gives @xmath50 and @xmath51 . then @xmath52 is simply @xmath53 and can be easily evaluated , since it is related to the plasma oscillation,@xcite @xmath54 eqs . ( [ eq_epert])-([eq_e3 ] ) hold because the expectation values of @xmath46 and @xmath55 on @xmath56 exist , as it will be more explicitly shown in eqs . ( [ eq_hef])-([eq_prova ] ) . taking the energy per particle @xmath57 , and dividing it into the non - interacting kinetic part @xmath58 and the exchange - correlation contribution @xmath59 , we then have the small-@xmath1 expansion @xmath60 the same result can be obtained by differentiation of @xmath61 with respect to @xmath1 and by using the helmann - feynmann theorem , which leads to the exact expression ( see also ref . ): @xmath62 , \label{eq_hef}\ ] ] where @xmath63 , and @xmath64 is the pair - distribution function@xcite corresponding to the hamiltonian of eq . ( [ eq_ham ] ) . the evaluation of eq . ( [ eq_hef ] ) at @xmath65 , immediately gives the first - order result , @xmath66 . higher - order derivatives of @xmath59 at @xmath65 can be obtained by further differentiating eq . ( [ eq_hef ] ) , provided that the conditions for differentiation under the integral sign are fulfilled . since @xmath67 implies @xmath68 and @xmath69 , the possibility of extracting the second and third derivatives of @xmath59 at @xmath65 from eq . ( [ eq_hef ] ) depends on whether the integrals @xmath70\ ] ] with @xmath71 and @xmath72 exist . this is the case , since @xmath73 is a well - behaved function whose oscillation - averaged part@xcite goes to zero as@xcite @xmath74 when @xmath75 . we thus obtain from eq . ( [ eq_hef ] ) @xmath76 \nonumber \\ & = & \frac{6}{\sqrt{\pi}}r_s^2\left(-\frac{1}{r_s^2\omega_p}\right)= -\frac{6}{\sqrt{3\pi}}r_s^{3/2 } , \label{eq_prova}\end{aligned}\ ] ] in agreement with eq . ( [ eq_expexc ] ) . we see that since @xmath77 no further information can be extracted from eq . ( [ eq_hef ] ) , or , equivalently , by going further with the expansion of eq . ( [ eq_hpert ] ) . one can divide @xmath59 into its exchange and correlation parts , @xmath78 . the exchange energy @xmath79 has been calculated by savin,@xcite and is reported in appendix [ app_ex ] . its small-@xmath1 expansion is @xmath80 where @xmath81 . the @xmath20 behavior of @xmath82 , is then @xmath83 notice that if we divide the pair - distribution function @xmath84 into its exchange and correlation parts , @xmath85 , we have @xmath86 , \label{eq_hex}\\ \frac{\partial\ec}{\partial\mu } & = & -\frac{3}{\sqrt{\pi}}\int_0^{\infty } ds\,s^2\,e^{-\mu^2r_s^2s^2}g_c(s , r_s,\mu ) . \label{eq_hec}\end{aligned}\ ] ] ( this follows directly from the hellmann - feynmann theorem and from the fact that @xmath87 corresponds to the noninteracting gas and thus does not depend on @xmath1 . ) if we take the limit @xmath20 of eqs . ( [ eq_hex ] ) and ( [ eq_hec ] ) we recover the first - order result in eqs . ( [ eq_exexp ] ) and ( [ eq_ecexp ] ) . however , higher - order derivatives at @xmath65 of @xmath79 and @xmath88 can not be obtained by differentiating eqs . ( [ eq_hex ] ) and ( [ eq_hec ] ) . this is due to the long - range tail of @xmath89 and @xmath90 : when @xmath91 they both approach zero as@xcite @xmath92 . thus , integrals of the kind @xmath93 $ ] and @xmath94 diverge . the long - range tails of @xmath89 and @xmath90 exactly cancel@xcite in @xmath73 . this is why , at small @xmath1 , both @xmath79 and @xmath88 have terms @xmath95 which cancel out in @xmath59 . if we use the relevant scaled units @xmath97 and we let @xmath18 approach zero , the potential has the expansion @xmath98 which has the coulomb interaction as leading term . we are thus approaching again the jellium model , so that eq . ( [ eq_ecexp ] ) is also valid for finite @xmath1 and @xmath96 . in eq . ( [ eq_ecexp ] ) the relevant scaled variable is @xmath40 . this can be understood in the following way . the coulomb gas presents screening effects at lenghts @xmath99 , where @xmath100 is the thomas - fermi wave vector . since the @xmath101 function amounts to some sort of artificial screening at lenghts @xmath102 , the thomas - fermi screening appears , exactly as in the coulomb gas , when @xmath103 . when @xmath105 , the potential terms of eqs . ( [ eq_vee])-([eq_vbb ] ) rapidly vanish ( @xmath106 ) . in this regime we can treat the whole potential as a perturbation to the non - interacting fermi gas . the first - order ( in the potential ) correction to the non - interacting energy @xmath107 gives @xmath79 of appendix [ app_ex ] . the second - order term can be computed by standard rayleigh - schrdinger perturbation theory @xmath108 as in the case of jellium , @xmath109 is the sum of a direct term and of a second - order exchange term,@xcite which in fourier space read @xmath110 here all the momenta are expressed in units of @xmath111 , and @xmath112 is the heaviside step function . now , consider the case @xmath113 and divide the integral over @xmath114 in eqs . ( [ eq_ec2dir ] ) and ( [ eq_ec2ex ] ) into two parts : @xmath115 in the first part , when @xmath116 $ ] , we can write @xmath117 ( since @xmath118 , and the integrals of eqs . ( [ eq_ec2dir ] ) and ( [ eq_ec2ex ] ) are restricted to the domain @xmath119 , @xmath120 ) . equations ( [ eq_ec2dir ] ) and ( [ eq_ec2ex ] ) then reduce to integrals of the same kind , which can be summed to yield @xmath121 i.e. , they give a term which vanishes as @xmath122 . in the second part , @xmath123 , having chosen @xmath124 , we can write @xmath125 equations ( [ eq_ec2dir ] ) and ( [ eq_ec2ex ] ) again reduce to integrals of the same kind , which can be summed to yield @xmath126 the right - hand side of eq . ( [ eq_eclargemurs ] ) can be evaluated analytically and then expanded for @xmath113 . its leading term is ( correctly ) independent of @xmath127 and equals @xmath128 . we thus have @xmath129 with @xmath130 hartree . since the perturbation series expansion whose second - order term corresponds to eq . ( [ eq_secondorder ] ) is done with respect to the whole potential @xmath131 and not with respect to the parameter @xmath1 , higher - order terms could also contribute to the value of @xmath132 . for this reason , in our parametrization of @xmath133 @xmath132 is left as a free parameter , to be optimized with a fit on the dmc data . we expect to find a value of @xmath132 of the same order of the one estimated with eq . ( [ eq_secondorder ] ) , since the potential @xmath134 vanishes very rapidly as @xmath105 , so that the higher - order - term contribution to @xmath132 should be small . , for different densities . our fitting function ( lines ) is compared with our dmc data ( dots ) . the error bars are comparable with the symbol sizes.,width=249 ] an accurate and simple analytic representation of the correlation energy @xmath19 can be obtained by a pad form which interpolates between the limiting behaviors given by our eqs . ( [ eq_ecexp ] ) and ( [ eq_mugrandi ] ) , and contains some free parameters to fit our dmc data . we write @xmath135}{1+b_1(r_s)\mu + b_2(r_s)\mu^2+b_3(r_s)\mu^3+b_4(r_s)\mu^4},\ ] ] where @xmath136 and @xmath137 is one of the standard parametrizations@xcite of the correlation energy of the unpolarized jellium . here we used the parametrization of perdew and wang.@xcite the two parameters @xmath138 and @xmath132 are fixed by a two - dimensional ( @xmath139 ) best fit to our dmc data . we find : @xmath140 this fit yields a reduced @xmath141 of 2.7 . in fig . [ fig_allrs ] we show our dmc data together with the fitting function for different values of @xmath18 . notice that our analytic @xmath19 does not break down at high ( @xmath96 ) or low ( @xmath142 ) densities , being constrained by exact behaviors . we have presented a comprehensive numerical and analytic study of the ground - state energy of a ( spin unpolarized ) uniform electron gas with modified , short - range - only electron - electron interaction @xmath143 , as a function of the cutoff parameter @xmath1 and of the electronic density . our chief goal has been the publication , in a convenient form for application , of a reliable local density functional for the correlation energy of this model system , which ( i ) fits the results of our quantum monte carlo simulations and ( ii ) automatically incorporates exact limits . such a functional is a crucial ingredient for some recently proposed mixed schemes " , which exploit the dft only for the short - range part of the electron - electron interaction . in this context the natural extension of this study will be the generalization of our functional to the spin - polarized case . what we obtained in this paper is not the only possible short - range local - density functional of interest to mixed schemes " . in some of them @xcite the dft treatment of the short - range part is handled through another functional @xmath144 $ ] , defined as the difference between the standard exchange - correlation energy functional ( corresponding to the coulomb interaction ) and a long - range - only functional @xmath145=e_{xc}[n]-{e}_{xc}^{\rm lr}[n].\ ] ] another direction of future work will thus be the study of the uniform electron gas with a long - range - only interaction of the form of eq . ( [ eq_lrpotential ] ) , and , possibly , other modified interactions proposed in the same spirit.@xcite we thank s. baroni , a. savin , and j. toulouse for useful discussions , and gratefully acknowledge financial support from the italian ministry of education , university and research ( miur ) through cofin 2003 - 2004 and the allocation of computer resources from infm iniziativa calcolo parallelo . the exchange energy corresponding to the hamiltonian ( [ eq_ham ] ) has been calculated by savin in ref . , and is equal to @xmath146\biggr\},\end{aligned}\ ] ] with @xmath147 . the exchange energy thus satisfies @xmath148 99 w. kohn , rev . phys . * 71 * , 1253 ( 1999 ) . mattsson , science * 298 * , 759 ( 2002 ) . c. fiolhais , f. nogueira , and m. marques ( eds . ) , _ a primer in density functional theory _ ( springer - verlag , berlin , 2003 ) . h. rydberg , m. dion , n. jacobson , e. schrder , p. hyldgaard , s. i. simak , d.c . langreth , and b.i . lundqvist , phys . lett . * 91 * , 126402 ( 2003 ) . j. dobson and j. wang , phys . b * 69 * , 235104 ( 2004 ) . j. tao , j.p . perdew , v.n . staroverov , and g.e . scuseria , phys . . lett . * 91 * , 146401 ( 2003 ) ; j.p . perdew , j. tao , v.n . staroverov , and g.e . scuseria , j. chem . phys . * 120 * , 6898 ( 2004 ) . j. ireta , j. neugebauer , and m. scheffler , _ on the accuracy of dft for describing hydrogen bonds : dependence on the bond directionality _ , a. savin , in _ recent developments and applications of modern density functional theory _ , edited by j.m . seminario ( elsevier , amsterdam , 1996 ) . r. pollet , a. savin , t. leininger , and h. stoll , j. chem . phys . * 116 * , 1250 ( 2002 ) ; t. leininger , h. stoll , h .- j . werner , and a. savin , chem . * 275 * , 151 ( 1997 ) . w. kohn , y. meir , d.e . makarov , phys . . lett . * 80 * , 4153 ( 1998 ) . j. heyd , g.e . scuseria , and m. ernzerhof , j. chem . phys . * 118 * , 8207 ( 2003 ) ; j. heyd and g.e . scuseria , j. chem . 120 , 7274 ( 2004 ) . perdew , a. savin , and k. burke , phys . a * 51 * , 4531 ( 1995 ) ; k. burke , j.p . perdew , and m. ernzerhof , j. chem . phys . * 109 * , 3760 ( 1998 ) . m. fuchs and x. gonze , phys . b * 65 * , 235109 ( 2002 ) . see also the comment of m. fuchs , k. burke , y .- m . niquet , and x. gonze , phys . lett . * 90 * , 189701 ( 2003 ) . a. savin , f. colonna , and r. pollet , int . j. quantum chem . * 93 * , 166 ( 2003 ) . see also refs . j. toulouse , f. colonna , and a. savin , in preparation . ceperley and b.j . alder , phys . * 45 * , 566 ( 1980 ) . vosko , l. wilk , and m. nusair , can . j. phys . * 58 * , 1200 ( 1980 ) . perdew and a. zunger , phys . b * 23 * , 5048 ( 1981 ) . j.p . perdew and y. wang , phys . b * 45 * , 13244 ( 1992 ) . for a recent review on fixed - node diffusion monte carlo and further references , see m. foulkes , l. mitas , r. needs and g. rajagopal , rev . phys . * 73 * , 33 - 83 ( 2001 ) . y. kwon , d. m. ceperley and r. m. martin , phys . b * 58 * 6800 ( 1998 ) . allen and d.j . tildesley , _ computer simulation of liquids _ ( clarendon press , oxford , 1987 ) . see , e.g. , d. pines and p. nozires , _ theory of quantum liquids _ ( benjamin , new york , 1966 ) . r. pollet , f. colonna , t. leininger , h. stoll , h .- j . werner , and a. savin , int . j. quantum chem . * 91 * , 84 ( 2003 ) . gori - giorgi , f. sacchetti , and g.b . bachelet , phys . b * 61 * , 7353 ( 2000 ) ; * 66 * , 159901(e ) ( 2002 ) . p. gori - giorgi and j.p . perdew , phys . b * 66 * , 165118 ( 2002 ) . perdew and y. wang , phys . b * 46 * , 12947 ( 1992 ) ; * 56 * , 7018(e ) ( 1997 ) . energetically unimportant oscillating terms are dealt with the evaluation of integrals of the kind : @xmath149 . a. savin and h .- j . flad , int . j. quantum chem . * 56 * , 327 ( 1995 ) ; j. toulouse , a. savin , and h .- j . flad , int . j. quantum chem . , to appear .

motivated by recent suggestions to split the electron - electron interaction into a short - range part , to be treated within the density functional theory , and a long - range part , to be handled by other techniques we compute , with a diffusion monte carlo method , the ground - state energy of a uniform electron gas with a modified , short - range - only electron - electron interaction @xmath0 , for different values of the cutoff parameter @xmath1 and of the electron density . after deriving some exact limits , we propose an analytic representation of the correlation energy which accurately fits our monte carlo data and also includes , by construction , these exact limits , thus providing a reliable `` short - range local - density functional '' .

cond-mat0406375

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

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

gr-qc0703019

mid - infrared imaging surveys underway with the spitzer space telescope demonstrate the presence of large numbers of sources whose luminosity must arise primarily from dust re - emission and which are significantly obscured at optical wavelengths ( e.g. @xcite ) . the source counts are consistent with expectations derived from efforts to account for the total infrared background and have been modeled as showing the evolution of luminous , star - forming galaxies ( e.g. @xcite , @xcite ) . spectroscopic and photometric redshifts also indicate that the majority of such sources have infrared luminosity powered primarily by star formation @xcite . confirming the nature of these sources and their redshift distribution is crucial to understanding the evolution of star formation and agn activity in the universe , especially at early epochs . a similar situation has existed for much longer in efforts to understand the nature of optically faint radio sources ( e.g. @xcite , @xcite ) . the observation that many sub - mjy radio sources are identified with a population of faint , blue galaxies initially indicated that most optically - faint radio sources are powered primarily by star formation @xcite . this had also been indicated by the correlation between infrared and radio fluxes for starburst systems ( @xcite , @xcite , @xcite ) . however , recent studies have shown that many faint ( @xmath5 100 @xmath3jy ) radio sources do not show the infrared fluxes @xcite or submillimeter detections @xcite expected for starburst systems . it is important to determine , therefore , what fraction of the faint radio source population does have firm indications of starbursts . a major observational challenge is to obtain redshifts or spectral diagnostics for infrared and radio sources too faint for optical spectroscopy , sources having optical magnitudes @xmath1 24mag . for sources that are sufficiently bright in the infrared ( f@xmath2 ( 24@xmath3 m ) @xmath4 0.75mjy ) , redshifts can be determined to z @xmath5 2.8 using the infrared spectrograph on spitzer ( irs ) , based on strong spectral features from silicate absorption or pah emission . results for the first set of sources selected only on the basis of extreme ir / optical flux ratios indicated that such sources were typically at z @xmath5 2 and were usually similar to the absorption spectra of local sources powered by agn ( @xcite ; hereinafter h05 ) . however , some sources selected using colors characterising starbursts @xcite or using submillimeter detections @xcite showed pah emission characteristic of starbursts . submillimeter observations of spitzer 24@xmath3 m sources selected because of extreme ir / optical ratios did not show the submillimeter fluxes expected for a starburst - dominated sample @xcite ; instead , most of these sources typically have stronger mid - infrared fluxes , consistent with the presence of hotter dust powered by an agn . it is clear , therefore , that the optically faint , infrared selected population has both agn and starburst constituents , and further samples are needed to define their redshift distributions and relative fractions in the overall luminosity function of dusty sources . the spitzer first look survey has imaged 4.4 deg@xmath11 at 24@xmath3 m with the multiband imaging photometer ( mips ) instrument @xcite ; initial results are described by @xcite although no catalogs are as yet available . the survey also has accompanying deep imaging with the very large array ( vla ) for which a source catalog is available @xcite and imaging in @xmath0 band from the national optical astronomy observatory . the initial comparison of 24@xmath3 m flux densities with 20 cm flux densities ( a ratio parameterized by q = log@xmath6f@xmath2(24 @xmath3m)@xmath7f@xmath2(20 cm)@xmath8 $ ] ) indicated that sources which are detected in both infrared and radio cluster around a median q value of 0.8 @xcite , which is the ratio expected from the previously known radio - infrared correlations for starbursts . this apparent agreement with expectations is misleading , because it does not include the large numbers of sources detected in the radio but not in the infrared , or vice - versa . a comparison of spitzer and vla 20 cm surveys to similar detection limits , although in a much smaller sky area , indicated that only 9% of the 24@xmath3 m sources are detected in the radio , and only 33% of the radio sources detected in the infrared @xcite . taking into account these limits , radio sources have a median q that is negative whereas infrared sources have a median q of about unity . this result was used by @xcite to conclude that the majority of radio sources in these vla surveys are powered by agn , whereas the majority of infrared sources are powered by starbursts . the result of starburst dominance for the spitzer 24@xmath3 m sources based on various survey comparisons and modeling of counts is inconsistent with the conclusion of h05 based on irs spectroscopy that most optically faint sources of @xmath5 1mjy at 24@xmath3 m are distant ultraluminous galaxies powered by agn . they observed 31 sources in the bootes field of the noao deep wide - field survey ( ndwfs , @xcite ) having @xmath0 @xmath1 25mag and f@xmath2 ( 24@xmath3 m ) @xmath4 0.75mjy . of the 17 sources with determinable redshifts , 16 were best fit with heavily absorbed templates and have median z of 2.2 . because the absorbed spectra for the local templates arise from objects with agn ( the prototype being markarian 231 ) , and because of the radio properties and luminosities of the sources , h05 interpret the heavily absorbed sources as bring primarily powered by obscured agn . spectroscopy of 8 sources in the spitzer fls selected by @xcite using color criteria targeted to select starbursts showed that 2 of the 6 sources with measurable spectral features show strong pah emission features . two sources observed by @xcite selected because of previous scuba detections also show pah emission . it is clear , therefore , that the optically faint infrared population contains a variety of sources , and selection criteria other than simply the infrared to optical ratio need to be invoked in efforts to sort and classify this population . to extend our samples and to begin investigation of the nature of infrared / radio sources , we undertook spectroscopic observations of sources in the fls area selected with infrared and optical criteria similar to those in our bootes survey but with the additional criterion of a radio detection at 20 cm . our selection criteria for spectroscopic targets were to use the mips and vla surveys with the @xmath0 band optical image to inspect all sources in the vla 20 cm catalog of the fls area @xcite having f@xmath2 ( 24@xmath3 m ) @xmath4 0.75mjy . to do this , we obtained the 24@xmath3 m images from the spitzer archive and produced an fls catalog of 24@xmath3 m sources , reduced with the mips data analysis tool @xcite . point source extraction was performed using an empirical point spread function ( psf ) constructed from the brightest objects found in the 24@xmath3 m image and fitted to all the sources detected in the data . the flux of each object is derived using the scaled fitted psf after applying a slight correction to account for the finite size of the modeled point spread function . for the present study , we are only interested in examining sources with f@xmath2 ( 24@xmath3 m ) @xmath4 0.75mjy whereas the fls 5 sigma limit is about 0.15mjy ; our sample is complete , therefore , for these bright sources . our 24@xmath3 m catalog was compared to the vla 20 cm catalog , and all sources having vla detections ( brighter than 115@xmath3jy ) were selected for comparison to the noao @xmath0 band catalog , also available in the spitzer archive , and all of the radio / infrared sources having @xmath0 @xmath4 23.9mag or being optically unidentifed were examined . ( our criteria for source association were that cataloged source coordinates agree to within 2 . ) our final sample consists of the 26 sources in the fls having f@xmath2 ( 24@xmath3 m ) @xmath4 1.0mjy which are in the 20 cm catalog and have @xmath0 @xmath12 23.9mag . we have obtained spectra for 18 of these sources with the irs ; the remaining 8 sources meeting these criteria were already within the program described by @xcite and not accessible to us . observations and results for the sources discussed in this paper are summarized in table 1 . coordinates listed are the 24@xmath3 m coordinates , which were used for the irs targets . spectroscopic observations were made with the irs short low module in order 1 only ( sl1 ) and with the long low module in orders 1 and 2 ( ll1 and ll2 ) , described in @xcite . these give low resolution spectral coverage from @xmath58@xmath3 m to @xmath535@xmath3 m . sources were normally placed on the slits by offsetting from nearby 2mass stars ; in a few cases with no sufficiently nearby 2mass stars , direct pointing without offsets was used successfully . sources which are observed are only a few percent of the brightness of the background flux , so the background is the dominant source of noise , and accurate background subtraction is essential before extracting a source spectrum . background subtraction for ll1 and ll2 modules can utilize the background observed in the off - source order ; e.g. , the ll1 slit provides an observation only of background when the source is in the ll2 slit , and this background can be subtracted from the observation when the source is in the ll1 slit . all images when the source was in one of the two nod positions on each slit were coadded to obtain the image containing the source spectrum for that nod position . the background which was subtracted from this coadded source - spectrum image included coadded background images of both nod positions when the source was in the other slit and including images from the other nod position in the same slit . this means that the background observation subtracted from a source observation includes three times the integration time as for the source . this improves signal to noise in the subtracted background . we experimented with the addition of background images from observations of different sources in an attempt to produce a standard background ( `` supersky '' ) for application to all sources , but the background observed from source to source was not sufficiently stable to produce a reliable supersky . the differenced source minus background image for each nod position was used for the spectral extraction , giving two independent extractions of the spectrum for each ll order . these two were compared to reject any highly outlying pixels in either spectrum , and a final mean spectrum was produced . for sl1 , there was no separate background observation with the source in the sl2 slit , so background subtraction was done by differencing coadded images of the two nod positions in sl1 . data were processed with version 11.0 of the ssc pipeline , and the bcd.fits files were used for our spectral extractions . extraction of source spectra was done with the smart analysis package @xcite ; location of spectra on the slit for extraction was done by individual examination because in several cases serendipitous sources also fell on the slit , and adopted backgrounds had to exclude these sources . typical extraction widths shown in the cross - dispersion profile displayed by smart were 3 to 3.5 pixels , to minimize noise from pixels containing little source flux . while use of an extraction window this narrow reduces the source flux somewhat compared to the calibration observations that utilize a wider extraction , the 24@xmath3 m mips fluxes are available for determining normalization of extracted spectral fluxes . it can be seen from comparison of the spectra shown in the figures with the mips fluxes in table 1 that the extracted spectra typically agree at 24@xmath3 m to within 10 % of the mips flux . final spectra for analysis of redshifts and features were smoothed to approximately a resolution element , applying boxcar smoothing of 0.2@xmath3 m for sl1 , 0.3@xmath3 m for ll2 , and 0.4@xmath3 m for ll1 . these resulting spectra are shown in figures 1 , 2 and 3 . even in spectra of faint sources with poor s / n , spectral features with sufficiently large equivalent widths exist in the mid - infrared spectra for redshift determination . at one extreme , these features are the strong pah emission features characteristic of starburst galaxies . the other extreme shows strong absorption features , the strongest being silicate absorption , with no pah emission . examples of pah spectra for galaxies observed with the irs are in @xcite for ngc 7714 , in @xcite for the mean of starbursts , and in @xcite for ngc 3079 . the prototype absorbed agn is markarian 231 , which shows an apparent excess above the continuum centered at 8@xmath3 m ( rest frame ) , caused by absorption on either side , followed by a deep trough of silicate absorption centered at 9.7@xmath3 m ; the irs spectrum is discussed in @xcite . an even more extremely absorbed source is iras f00183 - 7111 , with irs spectra in @xcite . an intermediate template with both absorption and emission is arp 220 , for which we utilize unpublished irs spectra . this is characterized primarily by absorption , but the excess at 8@xmath3 m is accentuated by the presence of pah 7.7@xmath3 m emission . redshifts can be determined by seeking either the 8@xmath3 m `` excess '' and following absorption , or the set of strong pah emission features . the strongest pah feature is at 7.7@xmath3 m ( rest frame ) , so a similar redshift would be derived even if a spectrum is ambiguous as to whether the strongest feature is the 8@xmath3 m apparent excess or true pah emission ; the interpretation of the source , however , would be very different with the two alternatives . pah features of about half the strength of 7.7@xmath3 m are at 6.2@xmath3 m and 11.2@xmath3 m . in order to interpret a feature as pah , we require an indication that at least one of the other pah features is present with the correct shape and relative flux as scaled to the 7.7@xmath3 m feature , although the 11.2@xmath3 m feature is often redshifted out of our observed spectral range . redshifts are estimated in two ways : in the first , we simply seek the features described above and assign a redshift based on the peaks in the spectrum . in the second , we fit a selection of templates that attempt a formal chi - squared fit to the full spectrum so that strengths of the features relative to the continuum are also considered . these templates are a pure pah emission spectrum such as m82 or ngc 3079 , arp 220 ( with both pah emission and deep silicate absorption ) , markarian 231 , and f00183 - 7111 . the redshifts derived with both techniques are listed in table 1 . based on the differences in derived redshift depending on the template used , and on the differences in redshift measures from fitting templates and from seeking only the individual features , we estimate that the uncertainty in redshift is typically 0.1 . the final classification of an object as characterized by pah emission ( em ) or silicate absorption ( abs ) is adopted as listed in table 1 . spectra with a straightforward choice of best - fitting template and spectra for which no features could be found are shown in figure 1 . spectra having definite features but with a more uncertain assignment of template fits are in figures 2 and 3 . because we are attempting to use the presence of pah emission or silicate absorption as an indicator of starburst - derived luminosity or agn - derived luminosity , respectively , it is important to determine which characterizes the spectrum even if the redshift is not affected by this choice . this is not always an unambiguous decision , and we present in figures 2 and 3 examples of the 7 sources in which either interpretation ( all emission and no absorption , or all absorption and no emission ) can provide a possible fit to the spectral features seen . we present these examples primarily as illustrations of the cautions that must be used in interpreting these irs spectra with poor s / n ; it is important not to assign high confidence to features that may not be real . some of this ambiguity may arise because sources actually are composite and would formally best be fit with varied combinations of pah emission and silicate absorption spectrum . we do not feel , however , that the s / n is adequate for this to improve our results regarding the classification of absorption dominance or emission dominance , but believe instead that the ambiguities arise primarily because of the difficulty in judging if a particular spectral peak is a real feature or is noise . the ambiguous sources which we illustrate are numbers 1 , 5 , 9 , 11 , 14 , 17 , and 18 ; their spectra and the alternative fits are shown in figures 2 and 3 . the choice of fit ( absorption or emission ) for these objects depends on deciding whether seemingly strong spectral features are real rather than accidents of noise . we display all spectra to the longest wavelength , 35@xmath3 m , at which some signal may be meaningfully recorded for these faint sources . we caution , however , that any large features beyond 33@xmath3 m should receive very little weight , because noise spikes are often found at these longer wavelengths where the detector sensitivity is falling rapidly . as examples of the uncertainties introduced in the fits , sources 1 and 5 in figure 2 could be well fit by pah features if the spike at @xmath5 33@xmath3 m is a real feature , but if it is not real , the fit can not be pah . conversely , source 11 in figure 2 can not be pah if the collection of spikes between 29@xmath3 m and 33@xmath3 m represents real signal . our final decision on which fit to accept from these ambiguous cases is based on our overall judgment of which spectral features are real , but we illustrate the alternative fits so that the reader may make their own judgment and use these examples for comparison to future ambiguous spectra . there is only one case where the ambiguity in fit would result in a significant difference in redshift , because of ambiguity regarding identification of features . this is for source 17 in figure 3 . this is the poorest fit we have , and assigning a redshift based on the markarian 231 template requires ignoring as noise an apparent excess of flux near 31@xmath3 m . if this is considered as a real feature , it is consistent with a 7.7@xmath3 m feature at a higher redshift than any source we have yet found , and which poorly explains the continuum flux at shorter wavelengths . we illustrate source 17 as the example of the one source for which we can not decide if it has a measurable redshift or not . in our final classifications summarized in table 1 , we do not include a redshift for source 17 and provide only the best fit power law for this source . the fits we finally adopt for the remaining ambiguous sources are given in table 1 ; these fits are judged by whether the pure absorption or pure emission templates shown in figures 2 and 3 provide the better overall fit to the observed spectra . only two of the 7 ambiguous cases in figures 2 and 3 are judged to be better fit by pah emission whereas the remainder are assigned to the absorption fit , except for number 17 with an indeterminate redshift . for the total set of 18 sources , we are able to assign redshifts to 14 . the median redshift of our sample is 2.1 and that of h05 is 2.2 . the most notable difference in the results for the present sample compared to h05 is that we find proportionally a few more sources which are dominated by pah emission spectra . two of 14 sources ( numbers 3 and 9 ) are well fit with a pure pah spectrum , whereas only one of the 17 sources in h05 was fit by the pah - dominated spectrum of ngc 7714 . we have 3 of 14 sources best fit by arp 220 , which contains pah emission in the template , and there were also 3 of these in h04 . this slight excess of pah sources is the only difference between the present sample and that of h05 which might be attributable to the radio selection , presumed to favor starbursts . the majority of sources with redshifts ( 12 of 14 ) are characterized by templates dominated by absorption ( including the arp 220 template ) , as were 16 of the 17 in h05 . for further discussion and as a working hypothesis , we assume a simple classification into agn - powered sources or starburst - powered sources determined only by whether the source shows strong absorption or strong pah emission , respectively . there are 4 sources without detectable features ; for these , the index @xmath13 of the power law which best fits the observed spectrum is given in table 1 . the classification we assume for these objects is that they are also agn characterised by hot dust but at unknown redshift ; the absorption may be too weak to detect , or may be at z @xmath1 2.8 , where it would be out of the detectable range of the irs spectra . it is important to note an important caveat regarding the classification of agn sources . this arises because of the metal - poor , compact starburst sbs 0335 - 052 , this has proven to have a starburst spectrum in the mid - infrared which is unique among all of the starbursts observed with the irs . unlike all other galaxies observed because they were previously classified as starbursts @xcite , it has no indication of pah features ( @xcite,@xcite ) and is the most extreme in this respect of all blue compact dwarf starbursts so far observed with the irs @xcite . it also shows weak silicate absorption . as a result , its observed mid - infrared spectral characteristics are very similar to the characteristics which we assign to agn - powered sources . despite its unusual spectral characteristics compared to other starbursts , @xcite and @xcite suggest that sbs 0335 - 052 is the best local example for the spectral shape of a primordial starburst . if this is so , and if such objects exist at luminosities and redshifts comparable to the sources we have found , the fraction of sources which we classify as agn - powered is smaller than we have estimated . adopting the agn classifications as described , it means that our sample has either 16 of 18 agn , if arp 220 absorption templates are assigned as agn , or 13 of 18 agn , if arp 220 templates are considered as starbursts . with either interpretation , the majority of sources are agn . the comparable numbers for the h05 sample were 30 of 31 agn , or 27 of 31 agn . the median 24@xmath3 m flux density of the h05 sample is 1.1mjy and of our present sample is 1.3mjy , so both samples are very similar in this criterion . the current sample has median @xmath0 @xmath4 24mag ( because 11 of 18 do not have measurable @xmath0 , and the faintest measured is @xmath0 = 24.5 ) , whereas the h05 sample has median @xmath0 @xmath4 25mag ( because the optical survey limit in bootes is fainter than in the fls ) , so both samples are also similar in optical magnitude selection . the primary difference is the use of radio flux densities to define the current sample . because the sample was initially defined by the infrared limit , there was no constraint on the radio flux density as long as there was a radio detection . the radio sample reaches sufficiently faint to find sources with q as large as 1.0 . the mean q for the 18 sources is 0.56 . only one source has a negative q ; this is one of the four power law sources so is consistent with being an agn . the median q value is a consequence of the fact that the great majority of sources in the fls detected both at 24@xmath3 m and at 20 cm have approximately this value of q , a value consistent with that from starbursts @xcite . there are numerous radio sources in the fls with more negative q , as expected from agn , but these sources are not found in large numbers in a sample requiring detection at 24@xmath3 m above flux densities of about one mjy . because of the flux density limits used for our selection , our criteria lead , therefore , to a sample of objects which should be dominated by dusty , obscured starbursts according to our current understanding of the infrared and radio characteristics of such starbursts . as a result , we expected starburst spectral classifications to dominate our sample . in this context , it is significant that only a minority of sources show the starburst classification determined from the presence of pah features . the mean q for the 13 agn ( counting the power law sources ) , as classified in the irs spectra , is 0.5 ; the mean for the 5 starbursts ( counting the arp 220 templates ) is 0.6 . though obviously based on limited statistics , this result indicates that the q value is not a reliable diagnostic of whether a dusty source will show the pah features expected from starbursts or the absorption expected from agn . this is not a surprising result if we use markarian 231 as the prototype dusty , absorbed agn . at z = 2 , it would have observed 24@xmath3 m flux density of 0.6mjy , based on the rest - frame flux density of 1.5 jy at 8@xmath3 m in @xcite , so this absorbed agn would be comparable to the flux densities of the infrared sources in our sample . more importantly , markarian 231 would have a q value of 0.56 , based on the ratio in the rest frame of f@xmath2(8 @xmath3 m ) compared to the f@xmath2(6 cm ) of 0.41 jy from @xcite ; this value of q agrees very well with the median of the sources in our sample . we have observed a new sample of 18 spitzer 24@xmath3 m sources with the spitzer irs , chosen to be optically faint ( @xmath0 @xmath1 24mag ) and also to have radio detections at 20 cm . the most definitive result of this study is confirmation of the conclusions of h05 that optically - faint spitzer sources of f@xmath2(24@xmath3 m ) @xmath5 1mjy are systematically at high redshift . of the 18 sources in our sample , 12 ( 67 % ) have confident redshifts z @xmath12 1.8 . within h05 , at least 15 of 31 ( 48 % ) have z @xmath12 1.8 . this is extraordinarily different from the redshift distribution expected from a flux limited 24@xmath3 m sample as predicted using current models or observations for starburst galaxies . for example , the redshift distribution by @xcite for spitzer sources to f@xmath2(24@xmath3 m ) @xmath5 0.1mjy , based primarily on photometric redshifts for starburst templates , has @xmath9 10% of sources with z @xmath4 1.8 . the earlier models of @xcite predicted that for 24@xmath3 m samples to 1.5mjy , less than 1% would have z @xmath4 1.8 . the redshift distribution of our optically - faint samples is also very different from the results for optically - bright samples of similar infrared flux . @xcite use a sample similar to ours in infrared flux limit ( f@xmath2(24@xmath3 m ) @xmath4 1mjy ) but much brighter optically ( @xmath0 @xmath9 21.7 ) . they have optically - determined redshifts of 255 sources in the bootes field for which 61 , or 24% , have z @xmath4 1.8 , compared to the 67% which we find . their selection is deliberately directed toward finding type 1 quasars by selecting objects with compact optical morphologies , and the resulting luminosity functions are similar to those derived from optically - selected quasar surveys to similar optical limits . it is clear , therefore , that an extreme infrared to optical flux ratio is a simple but powerful criterion for preferentially selecting dusty sources at high redshift . more challenging than determining the redshift is the question of classifying these dusty , high redshift sources as having their mid - infrared luminosities derived primarily from starbursts or agn . this classification is essential for utilizing counts and redshifts of mid - infrared sources to determine the correct relative contribution of these two fundamental sources of luminosity in the early universe . our working hypothesis for this classification , based on analogies to local ultraluminous infrared galaxies , is that sources showing strong absorption features or showing power - law continua without spectral features are classified as agn , and those sources with conspicuous pah features are classified as starbursts . using this initial hypothesis , we can examine some of the consequences of our results and make predictions regarding future results . considering the results in table 1 , 16 of the 18 sources show absorption features or power law spectra . for h05 , the fraction is 30 of 31 . in either sample , this implies that over 90% of the sources are agn . we can determine a lower limit to the agn fraction by allowing that sources fit by the arp 220 template are powered primarily by starbursts , because this template does include pah emission even though it is dominated by absorption . in this case , 13 of the 18 sources in table 1 are agn , and 27 of the 31 in h05 . these diagnostics using irs spectra indicate , therefore , that a large fraction of optically - faint sources in samples chosen at 24@xmath3 m are powered by agn . furthermore , these diagnostics indicate that at least 75% of those faint radio sources which would be attributed to star formation based on their infrared to radio flux ratios are actually powered primarily by agn . these results indicate that the agn population at high redshift is higher than currently realized , in both radio and infrared samples , primarily because of a large fraction of optically obscured agn which were previously unknown . there is a straightforward interpretation of these results that explains comparisons to existing models of 24@xmath3 m source counts and makes some testable predictions . the population of sources we have observed has f@xmath2(24@xmath3 m ) @xmath1 1mjy and has extreme infrared to optical ratios such that @xmath0 @xmath1 24mag . if we conclude , as explained above , that this population is primarily obscured agn , it means that the population of high redshift , obscured agn is much larger in number than the population of starbursts at mid - infrared luminosities corresponding to this flux density limit . it is not inconsistent , therefore , that the models of counts and redshift distributions that accomodate starbursts have not predicted this population of high luminosity , high redshift sources , because such models do not include the agn population . this conclusion is also consistent with observations of the only known population of luminous , high redshift , dusty starbursts , which is the scuba or mambo submillimeter population @xcite . these submillimeter sources are known to have f@xmath2(24@xmath3 m ) @xmath10 0.5mjy ( @xcite,@xcite ) . mambo observations of optically obscured sources selected to have f@xmath2(24@xmath3 m ) @xmath4 1mjy also indicate that these sources do not have the submillimeter fluxes expected for starburst galaxies , but have relatively larger mid - infrared fluxes @xcite . these various observations indicate that the luminous , dusty starburst population at z @xmath5 2 is not sufficiently luminous in the mid - infrared to dominate the samples we selected for irs observation . the important conclusion is that the high end of the infrared luminosity function for optically - obscured , dusty sources at z @xmath5 2 is dominated by agn but that the relative fraction of starbursts increases for f@xmath2(24@xmath3 m ) @xmath10 0.5mjy . this leads to a prediction about expected results from future observations of optically - faint sources selected only on the basis of f@xmath2(24@xmath3 m ) : targets observed at fainter 24@xmath3 m flux density levels should show a higher fraction of pah - dominated spectra than in the samples we have presented . the one such source so far observed with f@xmath2(24@xmath3 m ) @xmath5 0.5mjy , although chosen because of its submillimeter detection , does show pah emission @xcite . this conclusion also predicts that source characteristics as determined from overall spectral templates which fit a wide range of wavelengths should indicate an increasing proportion of starburst sources with decreasing f@xmath2(24@xmath3 m ) , but populations with f@xmath2(24@xmath3 m ) @xmath1 1 mjy should be dominated by agn templates . if we predict that the nature of optically faint 24@xmath3 m sources changes for f@xmath2(24@xmath3 m ) @xmath10 0.5mjy ( primarily starbursts ) compared to f@xmath2(24@xmath3 m ) @xmath1 1.0mjy ( primarily agn ) , this also implies consequences regarding x - ray characteristics . in particular , comparison of x - ray and infrared detections should show a difference in the typical infrared to x - ray flux ratio at these different infrared flux limits . the mean infrared to x - ray flux ratio should be smaller for the brighter infrared sources than for the fainter infrared sources , because strong x - ray sources correspond to agn . while comparisons of chandra and spitzer surveys have been made @xcite , these have been deep surveys in small areas of the sky . as a result , there are insufficient sources in common at the relevant flux levels to test this prediction meaningfully . for example , there are only 10 sources with x - ray detections and f@xmath2(24@xmath3 m ) @xmath4 1.0mjy in these samples as displayed by @xcite , and we do not know how many of those are optically faint or at high redshift . our classification has defined sources with an absorbed infrared spectrum or power - law infrared spectrum as agn . if this classification is correct , optically faint sources having f@xmath2(24@xmath3 m ) @xmath1 1mjy chosen using x - ray selection as an additional criterion should always show power - law or absorbed spectra and never show pah spectra , because the x - ray criterion is a firm agn indicator . so far , no irs results are available for sources chosen from combined spitzer and chandra surveys , but we have programs underway to obtain these . we can be optimistic , therefore , that as spectroscopic samples of spitzer sources accumulate , we will be able to produce quantitative luminosity functions for both dusty starbursts and dusty agn at the crucial epoch of z @xmath5 2 . we thank t. herter for assistance in developing the template - fitting routines used in this paper , and we thank d. devost , g. sloan , k. uchida , and p. hall for help in improving our irs spectral analysis . this work is based in part on observations made with the spitzer space telescope , which is operated by the jet propulsion laboratory , california institute of technology under nasa contract 1407 . support for this work by the irs gto team at cornell university was provided by nasa through contract number 1257184 issued by jpl / caltech . alonso - 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gonzalez , p.g . , et al . 2005 , , in press richards , e. a. , fomalont , e. b. , kellermann , k. i. , windhorst , r. a. , partridge , r. b. , cowie , l. l. , and barger , a. j. 1999 , , 526 , l73 rieke , g.h . et al . , 2004 , , 154 , 25 rigby , j.r . , et al . 2004 , , 154 , 160 spoon , h. w. w. , et al . 2004 , , 154 , 184 thuan , t.x . , sauvage , m. , and madden , s. 1999 , , 516 , 783 weedman , d.w . , et al . 2005 , , in press wu , y. , charmandaris , v. , hao , l. , brandl , b. , and houck , j.r . 2005 , , submitted yan , l. , et al . 2005 , , 628 , 604 yun , m. , reddy , n. , and condon , j. 2001 , , 554 , 803 \1 . & sst24 j171440.57 + 584104.9 & 1.30 & & 0.5 & 1200 & 1.98(abs))&2.00(231 ) & + 2 . & sst24 j172210.27 + 584617.2 & 2.28 & 23.9 & 1.0 & 720 & 1.16(abs ) & 1.20(231 ) & + 3 . & sst24 j172540.26 + 584953.4 & 1.47 & 24.2 & 1.0 & 720 & 1.94(em ) & 1.80(m82 ) & + 4 . & sst24 j172249.03 + 585027.6 & 1.12 & 24.5 & 0.7 & 1440 & & & 1.5 + 5 . & sst24 j171151.01 + 585104.1 & 1.05 & 24.4 & 0.5 & 1440 & 1.94(abs ) & 1.92(183 ) & + 6 . & sst24 j171144.26 + 585225.8 & 1.29 & 23.9 & 0.9 & 720 & 1.25(abs ) & 1.22(231 ) & + 7 . & sst24 j171527.08 + 585802.2 & 1.04 & & -0.6 & 1440 & & & 2.4 + 8 . & sst24 j171205.09 + 591508.6 & 1.10 & & 1.0 & 1440 & 2.31(abs ) & 2.40(231 ) & + 9 . & sst24 j171927.28 + 591536.5 & 2.01 & & 0.4 & 720 & 2.25(em ) & 2.20(m82 ) & + 10 . & sst24 j170939.20 + 592728.3 & 1.20 & & 0.7 & 1440 & 2.50(abs)&2.52(220 ) & + 11 . & sst24 j172708.53 + 593727.7 & 1.11 & & 0.8 & 1440 & 1.96(abs)&2.00(231 ) & + 12 . & sst24 j171342.79 + 593920.3 & 1.89 & & 0.3 & 720 & & & 1.1 + 13 . & sst24 j172048.02 + 594320.6 & 1.54 & & 0.3 & 1200 & 2.21(abs ) & 2.29(183 ) & + 14 . & sst24 j171343.94 + 595714.4 & 1.03 & 24.0 & 0.4 & 1440 & 1.70(abs ) & 1.81(183 ) & + 15 . & sst24 j171057.45 + 600745.2 & 2.05 & 24.5 & 0.7 & 720 & 2.38(abs ) & 2.40(220 ) & + 16 . & sst24 j172448.65 + 601439.1 & 1.68 & & 0.4 & 720 & 2.22(abs ) & 2.40(220 ) & + 17 . & sst24 j172001.44 + 602544.9 & 1.55 & & 0.2 & 1200 & & & 0.6 + 18 . & sst24 j171434.56 + 602828.6 & 1.51 & 24.2 & 0.1 & 1200 & 2.13(abs ) & 2.35(183 ) & +

spectra have been obtained with the infrared spectrograph on the spitzer space telescope for 18 optically faint sources ( @xmath0 @xmath1 23.9mag ) having f@xmath2 ( 24@xmath3 m ) @xmath4 1.0mjy and having radio detections at 20 cm to a limit of 115@xmath3jy . sources are within the spitzer first look survey . redshifts are determined for 14 sources from strong silicate absorption features ( 12 sources ) or strong pah emission features ( 2 sources ) , with median redshift of 2.1 . results confirm that optically faint sources of @xmath51 mjy at 24@xmath3 m are typically at redshifts z @xmath5 2 , verifying the high efficiency in selecting high redshift sources based on extreme infrared to optical flux ratio , and indicate that 24@xmath3 m sources which also have radio counterparts are not systematically different than samples chosen only by their infrared to optical flux ratios . using the parameter q = log@xmath6f@xmath2(24 @xmath3m)@xmath7f@xmath2(20 cm)@xmath8 $ ] , 17 of the 18 sources observed have values of 0@xmath9q@xmath91 , in the range expected for starburst - powered sources , but only a few of these show strong pah emission as expected from starbursts , with the remainder showing absorbed or power - law spectra consistent with an agn luminosity source . this confirms previous indications that optically faint spitzer sources with f@xmath2(24@xmath3 m ) @xmath1 1.0mjy are predominately agn and represent the upper end of the luminosity function of dusty sources at z @xmath5 2 . based on the characteristics of the sources observed so far , we predict that the nature of sources selected at 24@xmath3 m will change for f@xmath2(24@xmath3 m ) @xmath10 0.5mjy to sources dominated primarily by starbursts .

astro-ph0510621

loop quantum gravity had never been considered a candidate of the unification of matter and gravity until a remarkable series of discoveries emerged recently . first , markopoulou and kribs@xcite discovered that loop quantum gravity and many related theories of dynamical quantum geometry have emergent excitations which carry conserved quantum numbers not associated with geometry or the gravitational field . around the same time , bilson - thompson@xcite found that a composite or preon " model of the quarks , leptons and vector bosons could be coded in the possible ways that three ribbons can be braided and twisted . this suggested that the particles of the standard model could be discovered amidst the emergent braid states and their conserved quantum numbers associated with those of the standard model . one realization of this was then given in @xcite , for a particular class of dynamic quantum geometry models based on 3-valent quantum spin - networks obtained by gluing trinions together . these are coded in the knotting and braiding of the edges of the spin network ; they are degrees of freedom because of the basic result that quantum gravity or the quantization of any diffeomorphism invariant gauge theory has a basis of states given by embeddings up to diffeomorphisms of a set of labeled graphs in a spatial manifold . indeed , the role of the braiding of the edges of the graphs had been a mystery for many years . however , spin foam models in @xmath0 dimensions involve embedded 4-valent spin networks@xcite . it is then natural to ask if there are conservation laws associated with braids in 4-valent spin - networks . besides , quantum gravity with a positive cosmological cons- tant@xcite and quantum deformation of quantum gravity@xcite suggest the framing of embedded spin - networks . in this paper we extend the investigation of the braid excitations from the 3-valent case to the 4-valent case . we study ( framed ) 4-valent spin - networks embedded in 3d . due to the complexity of embedded 4-valent spin - networks , to deal with the braid excitations of them we need a consistent and convenient mathematical formalism . in this paper , which is the first of a series of papers on the subject , we first propose a new notation of the embedded ( framed ) 4-valent spin - networks and define what we mean by braids , then discuss equivalence moves with the help of our notation , which relate all diffeomorphic embedded 4-valent graphs and form the graphical calculus of the kinematics of these graphs , and at the end present a classification of the braids . these results are key to our subsequent papers . we focus on 3-strand braids , which are the simplest non - trivial and interesting braid excitations living on embedded 4-valent spin - networks . firstly , we fix the notation , namely a tube - sphere notation . we work in the category of framed graphs , in particular the two dimentional projections representing embedded framed 4-valent spin networks up to diffeomorphisms . there is a single diffeomorphism class of nodes . we therefore represent nodes by rigid 2-spheres and edges by tubes . such a node can be considered locally dual to a tetrahedron , as shown in fig . [ notation](a ) . if the spin - nets are not framed , we simply reduce tubes to lines but still keep spheres as nodes . to fully characterize the embedding of a spin - net in a 3-manifold , we assume that not only the nodes are rigid , i.e. they can only be rotated or translated , but also the positions on the node where the edges are attached are fixed . this requirement and the local duality ensures the non - degeneracy of the nodes , i.e. no more than two edges of a node are co - planar . for the convenience of calculation , we simplify the tube - sphere notation in fig . [ notation](a ) to fig . [ notation](b ) , in which 1 ) the sphere is replaced by a solid circle ; 2 ) the two tubes in the front , @xmath1 and @xmath2 in ( a ) , are replaced by a solid line piercing through the circle in ( b ) ; and 3 ) the two tubes in the back , @xmath3 and @xmath4 in ( a ) are substituted by @xmath3 and @xmath4 in ( b ) with a dashed line connecting them through the circle . there is no loss of generality in taking this simplified notation , because one can always arrange a node in the two states like fig . [ notation](b ) & ( c ) by diffeomorphism before taking a projection . due to the local duality between a node and a tetrahedron and the fact that all the four edges of a node are on an equal footing , if we choose one of the four edges of a node at a time , the other three edges are still on an equal footing , in respect to a rotation symmetry with the specially chosen edge as the rotation axis , e.g. the edge @xmath3 in fig . [ notation](b ) & ( c ) . this rotation symmetry will be discussed in detail in the next section . there could exist twists on embedded tubes , e.g. the @xmath5-twist on the edge @xmath3 with respect to the solid red dot , shown in fig . [ notationtwist](a ) . note that we put twists in the unit of @xmath5 for two reasons . the first reason is that the possible states by which a node may be represented in a projection can be taken into each other by @xmath5 rotations around one of the edges of the node ( this will become clear in section [ subsecrot ] ) . by the local duality of a node to a tetrahedron , these correspond to the @xmath5 rotations that relate the different ways that two tetrahedra may be glued together on a triangular face . these rotations create twists in the edges and , as a result of the restriction on projections of nodes we impose , set the twists in a projection of an edge of a spin network in units of @xmath5 . the other reason is that the least twist distinguishable from zero of a piece of tube in a projection is @xmath5 and all higher twists distinguishable from each other in the projection must then be multiples of @xmath5 . because an edge is always between two nodes and a rotation of a node creates / annihilates twists on its edges , one usually needs to specify the fixed point on an edge with respect to which a twist is counted , as shown in fig . [ notationtwist](a ) . in this manner , the 1 unit of twist in fig . [ notationtwist](a ) is obviously equivalent to that in [ notationtwist](b ) , which is the same amount of twist in the opposite direction on the other side of the fixed point . interestingly , both twists in fig . [ notationtwist](a ) and ( b ) are right - handed twists if one point his / her right thumb to the node on the same sides of the fixed point as that of the twists ; therefore , we can unambiguously assign the same value to them , namely @xmath6 ( unit of @xmath5 ) . this provides a way of simplifying the notation of twist , i.e. we can simply label an edge with a ( left- ) right - handed twist a ( negative ) positive integer . for example , fig . [ notationtwist](a ) and ( b ) can be replaced by [ notationtwist](c ) without ambiguity . recalling the rotation axis mentioned before , one can assign states to a node with respect to its rotation axis . if the rotation axis is an edge in the back , we say the node is in state @xmath7 , or is simply called a @xmath7-node , e.g. fig . [ notation](b ) with edge @xmath3 as the rotation axis . otherwise , if the rotation axis is an edge in the back , the node is called a @xmath8-node or in the state @xmath8 , e.g. fig . [ notation](c ) with edge @xmath4 . the results of this paper will refer to the case of framed spin networks , defined above . however , unframed graphs are used in loop quantum gravity and it is useful to have results then for that case as well . the particular notation of unframed graphs is obtained from the framed case discussed here by dropping information about twists of the edges ( which thus represent curves rather than tubes ) , but keeping the nodes as rigid spheres , locally dual to tetrahedra . this is necessary so that the evolution moves are well defined for unframed embedded graphs , which will be explained in the second of this series of papers . in the rest of this paper we refer always to the framed case . results for the unframed case will be understood from those for the framed case by neglecting the twists of the edges , unless we explicitly describe them . equipped with the notation defined above , we are interested in a type of topological structures as sub - structures of embedded 4-valent spin networks , namely 3-strand braids , which are defined as follows . [ defbraid]a * * 3-strand braid ( * * or a * braid * for simplicity * * ) * * is a sub - spinnet of an embedded 4-valent spin network , which is a three dimensional object formed by two nodes with three common edges , now named * * strand**s ; the two nodes are called * * end - node**s , each of which has one and only one free edge , called an * external edge*. the two dimensional projections of these braids denoted in our notation are called braid diagrams , a typical example of which is shown in fig . [ braid ] . the following conditions should be satisfied : 1 . if braids are arranged horizontally , then the ( left ) right external edge of a braid can always be the ( left- ) right - most edge of the ( left ) right end - node , and always stretches to the ( left ) right , which has no tangles with the strands , for the left part of the braid diagram in fig . [ notbraid](a ) as an example ; 2 . what is captured between the two end - nodes , e.g. the region between the two dashed lines in fig . [ braid ] , should meet the definition of braid in the ordinary braid theory , for the braid diagram in fig . [ notbraid](b ) as an example ; 3 . the three strands of a braid are never tangled with any other edge of the spin - net , as illustrated in right side of the braid diagram in fig . [ notbraid](a ) , for example . + [ p ] we would like to emphasize that the braids defined above are 3d structures , each of which has many diffeomorphic embeddings that are represented by their 2d projections in our notation . as a result , the 2d projections of many braids , i.e. their braid diagrams , which appear to be different are actually equivalent to each other in the sense of diffeomorphism . the precise set of equivalence relations will be the topic of the next section . bearing this in mind , in the rest of the paper we are not going to distinguish braids from their braid diagrams , unless an ambiguity arises . these kinds of braids are different from the braids in the context of ordinary braid theory , since the two end - nodes of such a braid are topologically significant to the state of the braid . these braid are stable under a certain stability condition regarding the evolution of spin - nets , which will be brought up in the companion paper . however , in this paper , we focus only on the intrinsic properties of these braids , or in other words the pure topological properties of the braids up to diffeomorphism , i.e. without dynamic evolution . to do so , we need to first describe the non - dynamical operations that can be applied to the embedded 4-valent spin networks . we can assign a number to a crossing according to its chirality , viz @xmath6 for a right - handed crossing , @xmath9 for a left - handed crossing , and @xmath10 otherwise . [ assignment ] shows this assignment . [ p ] such a scheme of assignment will become useful in the subsequent discussions . as aforementioned , the tube diagrams of an embedded spin network belong to different equivalence classes . it is therefore obligatory to characterize these equivalence classes by equivalence relations . to do so , one needs to find the full set of local moves , operating on the nodes and edges , which do nt change the diffeomorphism class of the embedding of a diagram . in the discussion below we work in the framed case . in the unframed case , one just ignores the twists . an obvious set of equivalence moves consists of the usual three reidemeister moves , framed or unframed , whose details are not repeated here ; these moves will be applied without further notice . more importantly , there are two kinds of equivalence moves that can be peculiarly defined on an embedded 4-valent spin - net , under which two diagrams , in particular two braids , that are related by a sequence of equivalence moves are thought to be equivalent . the first kind composes of translation moves . the second type of equivalence moves are rotations defined on the nodes . we discuss translation moves first . translation moves , which are in fact extended reidemeister type moves , involve not only the edges but also the nodes of an embedded spin - net ; they reflect the translation symmetry of the embedded spin - nets . let us look at the simplest example first . [ translation](a ) shows a node @xmath11 connected to other places of the network via its four edges ; red points represent attached points on other nodes . one can slide the node @xmath11 along its edge @xmath12 to the left , which leads to fig . [ translation](b ) ; this does not change anything of the topology of the embedded spin - net . [ translationx ] illustrates more complicated cases where a crossing is taken into account . [ p ] in fig . [ translationx](a1 ) there is a node @xmath11 and a crossing ; however , since the crossing is between the edge @xmath12 of node @xmath11 and the edge @xmath13 of some other node and node @xmath11 together with all its edges are above edge @xmath13 , one can safely translate node @xmath11 along edge @xmath12 to the left passing the crossing , which results in fig . [ translationx](a2 ) , in which the crossing turns out to be between edge @xmath13 and edge @xmath14 . this , which is obviously a symmetry , may be understood as a reidemeister move ii . [ p ] apart from the translation symmetry , there is also a rotation symmetry that gives rise to rotations defined on a node , with respect to one of its four edges , of an embedded spin - net . these rotations are not those with rigid metric but only the ones that change projections of an embedding , without affecting diffeomorphisms . as mentioned before , @xmath5 rotations take states representing a node in a projection into each other . it is time to see in detail how these rotations affect a subgraph consisting of a node and its four edges . [ pi3rot+ ] shows such a rotation in the case where the node is in a @xmath7-state with respect to the chosen rotation axis before imposing the rotation , while fig . [ pi3rot- ] illustrates the opposite case . [ p ] [ p ] a @xmath5 rotation creates a crossing of two edges of the node and causes twists , which are explicitly labeled , on all edges of the node . the twist number on the rotation axis of a node is always opposite to that of the other three edges of the node . note that a @xmath5-rotation changes the state of a node , as shown in figures [ pi3rot+ ] and [ pi3rot- ] , i.e. if the node is in state @xmath7 before the rotation , it becomes a @xmath8-node after the rotation . this is the key to the first reason that we put the twists in an edge in units of @xmath5 . a @xmath5 rotation relates two projections of an embedded spin network , which belong to the same diffeomorphism class . two consecutive @xmath5 rotations certainly give rise to a @xmath15 rotation . however , it is intuitive to understand @xmath15 rotations in a more topological way . obviously , rotating a tetrahedron by an angle of @xmath15 with respect to the normal of any of the four faces of the tetrahedron does not change the view of it . therefore , by the local duality between a node and a tetrahedron , as long as an edge of a node is chosen , the other three edges of the node are on an equal footing . if we rotate a node with respect to any of its four edges by @xmath15 , the resulting diagram should be diffeomorphic to , or in our context equivalent to , the original one . in fig . [ 2pi3rot+ ] and fig . [ 2pi3rot- ] we list all the @xmath15-rotations . [ p ] [ p ] each of such rotations generate two crossings and twists on all four edges . the twist number on the rotation axis of a node is always opposite to that of the other three edges of the node . note that a @xmath15 rotation does not change the state of a node with respect to the rotation axis , i.e. if a node is in state @xmath7 with respect to @xmath16 before the rotation , it is still a @xmath7-node after the rotation . the @xmath5 and @xmath15 rotations can be used to construct larger rotations , for example the @xmath17 rotations , which also certainly do not change the diffeomorphism class a projection belongs to . for the convenience of future use , we depict these four possible rotations in fig . [ pirot+ ] and fig . [ pirot- ] . [ p ] [ p ] note that a @xmath17 rotation changes the state of a node , i.e. if a node is in state @xmath7 with respect to its rotation axis before the rotation , it becomes a @xmath8-node with respect to the same axis after the rotation . @xmath5 rotations are the smallest building blocks of all possible rotations ; they are thus the generators of all rotations . this is illustrated in fig . [ pi3rot+ ] through fig . [ pirot- ] , each of which can be directly used in a graphic calculation . recall that all the equivalence moves defined above are diffeomorphic operations on the embedded graphs . as an example , [ equibraids ] depicts two braids that can be deformed into each other by a @xmath5 rotation of node 2 with respect to its external edge @xmath16 . we say these two braids are equivalent to each other . [ ph ] note that for an end - node of a braid , only its external edge is allowed to be the rotation axis , with respect to which the equivalence rotation moves are applied . otherwise , one may end up with a situation similar to fig . [ badbraid ] , which does not satisfy definition [ defbraid ] . therefore , although sub - spinnets like fig . [ badbraid ] are equivalent to well - defined braids by rotation moves , they are not to be investigated because they complicate the clear structure of braids and do not have any new interesting property . thus for simplicity we only allow the external edge of an end - node of a braid to be the rotation axis . if a node is not an end - node of a braid , any of its four edges can be chosen as a rotation axis . [ ph ] by looking carefully at the rotations and the crossings and twists generated accordingly one can find that the assignment of values to crossings , shown in fig . [ assignment ] , is consistent with the assignment of values to twists , shown in fig . [ notationtwist ] . given that the rotations and translations are well - defined equivalence moves , there should be a conserved quantity , which is the same before and after the moves . rotations create or annihilate twist and crossings simultaneously , we thus define a composite quantity , christened * effective twist number of a rotation * , @xmath18 where @xmath19 is the twist number created by the rotation on an edge of the node , @xmath20 is the crossing number of a crossing created by the rotation between any two edges of the node , and the factor of 2 comes from the fact that a crossing always involve two edges . one can easily check that the rotations in fig . [ pi3rot+ ] through fig . [ pirot- ] satisfy @xmath21 . that is , rotations have a zero effective twist number . therefore we can enlarge @xmath22 to a more general quantity @xmath23 , the * effective twist number of subdiagrams * of an embedded spin - net , which are related by rotations of nodes , by taking into account all the edges that are affected by rotations . we define @xmath24 where @xmath19 is the twist number on an edge of the subdiagram , @xmath20 is the crossing number of a crossing in the subdiagram . since @xmath21 , @xmath23 is indeed a conserved quantity under rotations . important examples of subdiagrams are braids , which will become clear when we talk about propagation and interactions . the effective twist number @xmath23 in eq . [ theta0 ] is also found to be preserved by translation moves . note that the effective twist number is not defined in the unframed case , simply because the unframed case has no notion of twists . with the help of equivalence moves , in particular the rotation moves , we can classify all possible 3-strand braids into two major types , namely reducible braids and irreducible braids , whose definitions are given below . the aforementioned restriction that only the external edge of an end - node of a braid can be the rotation axis of the node ensures the unambiguous assignments of states to the end nodes of a braid and keeps the classification of braids simple . note that twists on edges are irrelevant to the calculation in the section ; they are thus neglected throughout the discussion . nevertheless , the results are valid for both framed and unframed cases . for the purpose of classifying the braids , we also consider braids as if they are isolated regions in a graph . [ defredubraid]a braid is called a * reducible braid * , if it is equivalent to a braid with fewer crossings ; otherwise , it is * irreducible*. the braid on the top part of fig . [ equibraids ] is an example of a reducible braid , whereas the one at the bottom of the figure is an irreducible braid . to classify the braids in a convenient way , we need a new notation and some auxiliary definitions . since we have a way of assigning crossings integers @xmath6 or @xmath9 , as in fig . [ assignment ] , we can use @xmath25 matrices with two end - nodes in either state @xmath7 or @xmath8 to denote a 3-strand braid with @xmath26 crossings , as shown in fig . [ matrixbraid ] and its caption , keeping in mind that the state of an end - node is and can only be with respect to its external edge . for the purpose of calculation , it is also convenient to associate crossings with one of the two end - nodes of a braid . for example , in fig . [ matrixbraid ] the left end - node with its nearest crossing can be denoted by @xmath27{c}-1\\ 0 \end{array } \right ) $ ] , and the right end - node with its nearest two crossings can be written as @xmath28{cc}0 & 0\\ + 1 & + 1 \end{array } \right ) \ominus$ ] , which has @xmath28{c}0\\ + 1 \end{array } \right ) \ominus$ ] as its 1-crossing * sub end - node*. end - nodes represented in this way are called 1-crossing end - nodes , 2-crossing end - nodes , etc . an end - node without any crossing is christened a * bare end - node*. [ ph ] a braid can be decomposed into or recombined from a left end - node , a right end - node , and a bunch of crossings represented by matrices . for instance,@xmath29{ccc}-1 & 0 & -1\\ 0 & + 1 & 0 \end{array } \right ) \oplus\longleftrightarrow\oplus\left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) + \left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) + \left ( \begin{array } [ c]{ccc}-1 & 0 & -1\\ 0 & + 1 & 0 \end{array } \right ) \oplus,\ ] ] where the `` @xmath30 '' between two matrices on the rhs means direct sum or concatenation of two pieces of braids . one can see from the above equation that the first two crossings or the second and third crossings on the rhs are cancelled . given this , we have the following definition . [ defredunode]an @xmath26-crossing end - node is said to be a * reducible end - node * , if it is equivalent to a @xmath31-crossing end - node with @xmath32 , by equivalence moves done on the node ; otherwise , it is irreducible . the definition above gives rise to another definition of reducible braid , which is equivalent to definition [ defredubraid ] . [ defredubraid2]a braid is said to be * reducible * if it has a reducible end - node . if a braid has a reducible left or right end - node , or both , it is called * left- * , or * right- , * or * two - way - reducible . * for consistency we may also symbolize the rotation moves . because rotations are exerted only on the end - nodes of a braid , we can denote all possible moves by rotation operators @xmath33 , @xmath34 , @xmath35 , and @xmath36 , where the superscript @xmath37 is the angle of rotation , the first subscript @xmath38 ( @xmath39 ) reads that the operation is on the left ( right ) end - node of a braid , and the second subscript @xmath38 ( @xmath39 ) indicates that the direction of rotation is left- ( right- ) handed . the left- ( right- ) handedness of the rotation is defined in such a way that if you grab the rotation axis of a node in your left ( right ) hand , with the thumb pointing to the node , your hand wraps up in the direction of rotation . results of the rotation operators have been shown graphically in fig . [ pi3rot+ ] through fig . [ pirot- ] . here we show an example of the algebra in the following equation.@xmath40{ccc}-1 & 0 & 0\\ 0 & + 1 & + 1 \end{array } \right ) \ominus\right ] \\ & = \oplus\left ( \begin{array } [ c]{ccc}-1 & 0 & 0\\ 0 & + 1 & + 1 \end{array } \right ) r_{rr}^{\pi/3}\left ( \ominus\right ) \\ & = \oplus\left ( \begin{array } [ c]{ccc}-1 & 0 & 0\\ 0 & + 1 & + 1 \end{array } \right ) + \left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) \oplus\\ & = \oplus\left ( \begin{array } [ c]{cccc}-1 & 0 & 0 & -1\\ 0 & + 1 & + 1 & 0 \end{array } \right ) \oplus.\end{aligned}\ ] ] because a braid can be reduced only from its end - nodes , we first classify the ( ir)reducible end - nodes . we start from 1-crossing end - nodes ; all the possible ones are listed in table [ tb : all1xnodes ] [ c]|l|l|l|left end - nodes & & right end - nodes + & & + & & + & & + & & + the following equations then show how all the reducible 1-crossing end - nodes are reduced to bare nodes.@xmath41{c}+1\\ 0 \end{array } \right ) \right ] & = \ominus\left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) + \left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) = \ominus\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \label{reduce1x}\\ r_{ll}^{\pi/3}\left [ \oplus\left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) \right ] & = \ominus\left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) + \left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) = \ominus\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \nonumber\\ r_{ll}^{\pi/3}\left [ \ominus\left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) \right ] & = \oplus\left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) + \left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) = \oplus\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \nonumber\\ r_{lr}^{\pi/3}\left [ \ominus\left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) \right ] & = \oplus\left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) + \left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) = \oplus\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \nonumber\\ r_{rl}^{\pi/3}\left [ \left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) \oplus\right ] & = \left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) + \left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) \ominus=\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \ominus\nonumber\\ r_{rr}^{\pi/3}\left [ \left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) \oplus\right ] & = \left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) + \left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) \ominus=\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \ominus\nonumber\\ r_{rr}^{\pi/3}\left [ \left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) \ominus\right ] & = \left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) + \left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) \oplus=\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \oplus\nonumber\\ r_{rl}^{\pi/3}\left [ \left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) \ominus\right ] & = \left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) + \left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) \oplus=\left ( \begin{array } [ c]{c}0\\ 0 \end{array } \right ) \oplus\nonumber\end{aligned}\ ] ] with the help of the above calculations , we can easily list all the irreducible 1-crossing end - nodes in table [ tb : irred1xnodes ] . [ c]|l|l|l|left end - nodes & & right end - nodes + & & + & & + & & + & & + now we consider 2-crossing end - nodes ; there are a total of 48 of this kind , including left and right end - nodes . to find all the irreducible 2-crossing end - nodes , we need only to think about those whose sub 1-crossing nodes are irreducible , since otherwise a 2-crossing end - node is already reducible ; this excludes 24 2-crossing end - nodes . if a 2-crossing node has an irreducible sub 1-crossing node , its crossings can definitely not be reduced by @xmath42-rotations , because a @xmath15-rotation is made of two consecutive @xmath43-rotations that do not reduce any irreducible 1-crossing node , and it does not flip the state of a bare node . interestingly , however , a 2-crossing end - node with an irreducible sub 1-crossing end - node may still be reduced to an irreducible 1-crossing end - node by @xmath17-rotations , which can be seen from the following equations.@xmath44{cc}-1 & 0\\ 0 & -1 \end{array } \right ) \right ] & = \ominus\left ( \begin{array } [ c]{ccc}+1 & 0 & + 1\\ 0 & + 1 & 0 \end{array } \right ) + \left ( \begin{array } [ c]{cc}-1 & 0\\ 0 & -1 \end{array } \right ) = \ominus\left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) \label{reduce2x}\\ r_{lr}^{\pi}\left [ \oplus\left ( \begin{array } [ c]{cc}0 & + 1\\ + 1 & 0 \end{array } \right ) \right ] & = \ominus\left ( \begin{array } [ c]{ccc}0 & -1 & 0\\ -1 & 0 & -1 \end{array } \right ) + \left ( \begin{array } [ c]{cc}0 & + 1\\ + 1 & 0 \end{array } \right ) = \ominus\left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) \nonumber\\ r_{lr}^{\pi}\left [ \ominus\left ( \begin{array } [ c]{cc}+1 & 0\\ 0 & + 1 \end{array } \right ) \right ] & = \oplus\left ( \begin{array } [ c]{ccc}-1 & 0 & -1\\ 0 & -1 & 0 \end{array } \right ) + \left ( \begin{array } [ c]{cc}+1 & 0\\ 0 & + 1 \end{array } \right ) = \oplus\left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) \nonumber\\ r_{ll}^{\pi}\left [ \ominus\left ( \begin{array } [ c]{cc}0 & -1\\ -1 & 0 \end{array } \right ) \right ] & = \oplus\left ( \begin{array } [ c]{ccc}0 & + 1 & 0\\ + 1 & 0 & + 1 \end{array } \right ) + \left ( \begin{array } [ c]{cc}0 & -1\\ -1 & 0 \end{array } \right ) = \oplus\left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) \nonumber\\ r_{rr}^{\pi}\left [ \left ( \begin{array } [ c]{cc}0 & + 1\\ + 1 & 0 \end{array } \right ) \oplus\right ] & = \left ( \begin{array } [ c]{cc}0 & + 1\\ + 1 & 0 \end{array } \right ) + \left ( \begin{array } [ c]{ccc}-1 & 0 & -1\\ 0 & -1 & 0 \end{array } \right ) \ominus=\left ( \begin{array } [ c]{c}-1\\ 0 \end{array } \right ) \ominus\nonumber\\ r_{rl}^{\pi}\left [ \left ( \begin{array } [ c]{cc}-1 & 0\\ 0 & -1 \end{array } \right ) \oplus\right ] & = \left ( \begin{array } [ c]{cc}-1 & 0\\ 0 & -1 \end{array } \right ) + \left ( \begin{array } [ c]{ccc}0 & + 1 & 0\\ + 1 & 0 & + 1 \end{array } \right ) \ominus=\left ( \begin{array } [ c]{c}0\\ + 1 \end{array } \right ) \ominus\nonumber\\ r_{ll}^{\pi}\left [ \left ( \begin{array } [ c]{cc}0 & -1\\ -1 & 0 \end{array } \right ) \ominus\right ] & = \left ( \begin{array } [ c]{cc}0 & -1\\ -1 & 0 \end{array } \right ) + \left ( \begin{array } [ c]{ccc}+1 & 0 & + 1\\ 0 & + 1 & 0 \end{array } \right ) \oplus=\left ( \begin{array } [ c]{c}+1\\ 0 \end{array } \right ) \oplus\nonumber\\ r_{lr}^{\pi}\left [ \left ( \begin{array } [ c]{cc}+1 & 0\\ 0 & + 1 \end{array } \right ) \ominus\right ] & = \left ( \begin{array } [ c]{cc}+1 & 0\\ 0 & + 1 \end{array } \right ) + \left ( \begin{array } [ c]{ccc}0 & -1 & 0\\ -1 & 0 & -1 \end{array } \right ) \oplus=\left ( \begin{array } [ c]{c}0\\ -1 \end{array } \right ) \oplus\nonumber\end{aligned}\ ] ] consequently , we can list all the irreducible 2-crossing end - nodes in table [ tb : irred2xnodes ] [ c]|c|l|c|c|l| & & + & @xmath27{cc}-1 & -1\\ 0 & 0 \end{array } \right ) $ ] & & @xmath28{cc}0 & + 1\\ -1 & 0 \end{array } \right ) \oplus$ ] & @xmath28{cc}+1 & + 1\\ 0 & 0 \end{array } \right ) \oplus$ ] + & @xmath27{cc}0 & 0\\ + 1 & + 1 \end{array } \right ) $ ] & & @xmath28{cc}+1 & 0\\ 0 & -1 \end{array } \right ) \oplus$ ] & @xmath28{cc}0 & 0\\ -1 & -1 \end{array } \right ) \oplus$ ] + & @xmath45{cc}+1 & + 1\\ 0 & 0 \end{array } \right ) $ ] & & @xmath28{cc}0 & -1\\ + 1 & 0 \end{array } \right ) \ominus$ ] & @xmath28{cc}-1 & -1\\ 0 & 0 \end{array } \right ) \ominus$ ] + & @xmath45{cc}0 & 0\\ -1 & -1 \end{array } \right ) $ ] & & @xmath28{cc}-1 & 0\\ 0 & + 1 \end{array } \right ) \ominus$ ] & @xmath28{cc}0 & 0\\ + 1 & + 1 \end{array } \right ) \ominus$ ] + the following theorem states that there is no need to investigate end - nodes with more crossings to see if they are irreducible . [ theonxnode]an @xmath26-crossing end - node , @xmath46 , which has an irreducible 2-crossing sub end - node , is irreducible . if the @xmath26-crossing end - node has a irreducible 2-crossing sub end - node , the two crossings nearest to the node are not reducible by either a single @xmath43- or a single @xmath15-rotation on the node . we may consider @xmath47-rotations on its 3-crossing sub end - node . however , if a 3-crossing end - node is reducible by a @xmath17-rotation , it must contain a reducible 2-crossing sub end - node according to fig . [ pirot+ ] , fig . [ pirot- ] and eq.[reduce2x ] , which is contradictory to the condition given in the theorem . this is then true for all cases where @xmath48 by simple induction . therefore , the theorem holds . equipped with the knowledge of ( ir)reducible end - nodes , we are ready to classify braids . the two end - nodes of a braid are either in the same states or in opposite states , we first take a look at braids whose end nodes are in the same states . [ theo123xbraids]all @xmath26-crossing braids in the form @xmath27{ccc } & \cdots & \\ & \cdots & \end{array } \right ) \oplus$ ] and @xmath45{ccc } & \cdots & \\ & \cdots & \end{array } \right ) \ominus$ ] are reducible for @xmath49 . it suffices to prove the @xmath50 case , the case of @xmath51 follows similarly or by symmetry . \1 ) @xmath52 . there are only four possibilities , namely @xmath27{c}\pm1\\ 0 \end{array } \right ) \oplus$ ] and @xmath27{c}0\\ \pm1 \end{array } \right ) \oplus$ ] ; however , they are all reducible because they all contain one reducible 1-crossing end - node according to eq . [ reduce1x ] . \2 ) @xmath53 . we first consider the braids formed by an irreducible 2-crossing end - node @xmath54 or @xmath55 , and a bare end - node @xmath56 . we do the following decomposition@xmath57 then from table [ tb : irred2xnodes ] , it is readily seen that @xmath58 and @xmath59 are always reducible end - nodes for any choice of @xmath54 and @xmath55respectively . that is , the braids formed this way are reducible . we then consider braids formed by two irreducible 1-crossing end - nodes . the first two rows in table [ tb : irred1xnodes ] and eq . [ reduce2x ] clearly shows that the result is either an unbraid or one with a reducible 2-crossing end - node . \3 ) @xmath60 . we need only consider braids , each of which is formed by the direct sum of a 2-crossing irreducible end - node and a 1-crossing irreducible end - node . this can be done by taking the direct sum between the ( right ) left end - nodes in the first two rows of table [ tb : irred1xnodes ] , and the ( left ) right end - nodes in the first two rows of table [ tb : irred2xnodes ] . it is not hard to see that any resultant braid has merely two possibilities : i ) two neighboring crossings are cancelled by the direct sum , which leads to 1-crossing braids that are proven to be reducible in the case of @xmath52 ; and ii ) a crossing in the irreducible 2-crossing end - node is combined with the irreducible 1-crossing end - node to form a reducible 2-crossing end - node , i.e. the braid is reducible . theorem [ theo123xbraids ] does not cover the case where @xmath61 , which will be included in another theorem soon . before that , let us consider the braids whose end - nodes are in opposite states . @xmath26-crossing braids in the form @xmath27{ccc } & \cdots & \\ & \cdots & \end{array } \right ) \ominus$ ] and @xmath45{ccc } & \cdots & \\ & \cdots & \end{array } \right ) \oplus$ ] , for @xmath49 . note that due to theorem[theo123xbraids ] , the set of irreducible 1-crossing braids to be found here represents the full set of irreducible braids for @xmath49 , regardless of the states of the end - nodes . 1 . @xmath52 . an irreducible braid can only be made by an irreducible 1-crossing end - node and a bare node . from table [ tb : irred1xnodes ] , there are only four options , which are indeed all irreducible ; they are now listed in table [ tb : irred1xbraids ] . + [ c]|l|l|l| & & + & & + 2 . it is sufficient to consider the braids formed by an irreducible 2-crossing end - node and a bare end - node in the opposite state . the reason is that if a 2-crossing braid is irreducible , its two 2-crossing end - nodes must be irreducible as well ; moreover , if a 2-crossing end - node is irreducible , its 1-crossing sub end - node is already irreducible . therefore , one can simply add to each irreducible end - node in table [ tb : irred2xnodes ] a bare end - node in the opposite state to create an irreducible 2-crossing braid . being a bit redundant , we list all the 16 irreducible 2-crossing braids in table [ tb : irred2xbraids ] . + [ c]|c|l|c|c|l| & @xmath27{cc}-1 & -1\\ 0 & 0 \end{array } \right ) \ominus$ ] & & @xmath45{cc}0 & + 1\\ -1 & 0 \end{array } \right ) \oplus$ ] & @xmath45{cc}+1 & + 1\\ 0 & 0 \end{array } \right ) \oplus$ ] + & @xmath27{cc}0 & 0\\ + 1 & + 1 \end{array } \right ) \ominus$ ] & & @xmath45{cc}+1 & 0\\ 0 & -1 \end{array } \right ) \oplus$ ] & @xmath45{cc}0 & 0\\ -1 & -1 \end{array } \right ) \oplus$ ] + & @xmath45{cc}+1 & + 1\\ 0 & 0 \end{array } \right ) \oplus$ ] & & @xmath27{cc}0 & -1\\ + 1 & 0 \end{array } \right ) \ominus$ ] & @xmath27{cc}-1 & -1\\ 0 & 0 \end{array } \right ) \ominus$ ] + & @xmath45{cc}0 & 0\\ -1 & -1 \end{array } \right ) \oplus$ ] & & @xmath27{cc}-1 & 0\\ 0 & + 1 \end{array } \right ) \ominus$ ] & @xmath27{cc}0 & 0\\ + 1 & + 1 \end{array } \right ) \ominus$ ] + 3 . @xmath60 . a 3-crossing braid in this case is irreducible if and only if it admits the following two decompositions.@xmath62 where `` @xmath30 '' is understood as the direct sum . the proof of this claim follows immediately from theorem [ theonxnode ] . it is time to summarize the case of @xmath61 for @xmath26-crossing braids , regardless of the states of the end - nodes , by the following theorem . [ theonxbraids]a @xmath26-crossing braid for @xmath61 is irreducible , if and only if it admits the decomposition@xmath63 where `` @xmath30 '' is understood as the direct sum . the only constraint of the arbitrary sequence of crossings is that its last crossing on each side does not cancel the neighboring crossing associated with the end - node on the same side . an irreducible 2-crossing end - node contains an irreducible 1-crossing end - node . by theorem [ theonxnode ] , if the above decomposition is admitted , the braid is not reducible on either end - node whatever the arbitrary sequence of crossings is up to the constraint . therefore , the theorem holds . the braids that are interesting to us are those reducible ones , which is shown in the companion paper . thus we may make more detailed divisions in the type of reducible braids by the definition below . given a reducible braid @xmath3 , a braid @xmath64 obtained from @xmath3 by doing as much reduction as possible is called an * extremum * of @xmath3 ; @xmath3 may have more than one * extrema * , but all the extrema have the same number of crossings . we then have the following . 1 . if all extrema of @xmath3 are unbraids , i.e. braids with no crossing , @xmath3 is said to be * completely reducible . * 2 . if an extremum of @xmath3 can be reached by equivalence moves exerted only on its ( left)right end - node , @xmath3 is called * extremely * ( * left-)right - reducible ; * if @xmath3 is also completely reducible , @xmath3 is then said to be * completely ( left-)right - reducible . * note that completely ( left- ) right - reducible implies extremely ( left- ) right - reducible , but not vice versa in general . in this paper , we proposed a new notation , namely the tube - sphere notation , for embedded ( framed ) 4-valent spin - networks . by means of this notation , we discovered a type of topological structures , the 3-strand braids , as sub - diagrams of an embedded spin - net . equivalence moves , including translations and rotations , which divide projections of embeddings of spin - networks into different equivalence classes , are defined and discussed in detail . the equivalence moves are important and useful in two aspects . firstly , by rotations , we classify 3-strand braids into two major types : reducible braids and irreducible braids , the former of which are further classified for the purpose of subsequent works . secondly , by equivalence moves one is able to carry out the calculation of braid propagation and interactions of embedded 4-valent spin - nets . these results serve as foundations for the work in the companion paper and all our future work dealing with braid - like excitations of embedded 4-valent spin - networks . in another paper , we will propose the evolution moves of embedded 4-valent spin - networks , by which some of the ( reducible ) 3-strand braids are able to propagate on the spin - nets and interact with each other and provide a possible formulation of the dynamics of these local excitations . the author is in debt to lee smolin , the author s advisor , for his great insight and heuristic discussion . he is grateful to fotini markopoulou for her critical comments . he would appreciate the helpful discussions with isabeau premont - schwarz , aristide baratin , and thomasz konopka . gratitude must also go to sundance bilson - thompson for his proof - reading of the manuscript . research at perimeter institute is supported in part by the government of canada through nserc and by the province of ontario through medt .

we propose a new notation for the states in some models of quantum gravity , namely 4-valent spin networks embedded in a topological three manifold . with the help of this notation , equivalence moves , namely translations and rotations , can be defined , which relate the projections of diffeomorphic embeddings of a spin network . certain types of topological structures , viz 3-strand braids as local excitations of embedded spin networks , are defined and classified by means of the equivalence moves . this paper formulates a mathematical approach to the further research of particle - like excitations in quantum gravity . = 1

0710.1312

the past decades have seen an accelerating miniaturization of both mechanical and electrical devices , so that a better understanding of properties of ultrasmall systems is required in increasing detail . the first measurements of conductance quantization in the late 1980s @xcite in constrictions of two - dimensional electron gases formed by means of gates , have demonstrated the importance of quantum confinement effects in these systems and opened a wide field of research . a major step has been the discovery of conductance quantization in metallic nanocontacts @xcite : the conductance measured during the elongation of a metal nanowire is a steplike function where the typical step height is frequently near a multiple of the conductance quantum @xmath0 , where @xmath1 is the electron charge and @xmath2 planck s constant . surprisingly , this was initially not interpreted as a quantum effect but rather as a consequence of abrupt atomic rearrangements and elastic deformation stages . this interpretation , supported by a series of molecular dynamics simulations @xcite , was claimed to be confirmed by another pioneering experiment @xcite measuring simultaneously the conductance and the cohesive force of gold nanowires with diameters ranging from several ngstroms to several nanometers . as the contact was pulled apart , oscillations in the force of order 1nn were observed in perfect correlation with the conductance steps . it came as a surprise when stafford _ et al . _ ( 1997 ) introduced the free - electron model of a nanocontact referred to as the _ nanoscale free - electron model _ ( nfem ) henceforth and showed that this comparatively simple model , which emphasizes the quantum confinement effects of the metallic electrons , is able to reproduce quantitatively the main features of the experimental observations . in this approach , the nanowire is understood to act as a quantum waveguide for the conduction electrons ( which are responsible for both conduction and cohesion in simple metals ) : each quantized mode transmitted through the contact contributes @xmath3 to the conductance and a force of order @xmath4 to the cohesion , where @xmath5 and @xmath6 are the fermi energy and wavelength , respectively . conductance channels act as delocalized bonds whose stretching and breaking is responsible for the observed force oscillations , thus explaining straightforwardly their correlations with the conductance steps . since then , free - standing metal nanowires , suspended from electrical contacts at their ends , have been fabricated by a number of different techniques . metal wires down to a single atom thick were extruded using a scanning - tunneling microscope tip @xcite . metal nanobridges were shown to `` self - assemble '' under electron - beam irradiation of thin metal films ( kondo _ et al . _ 1997 , 2000 , rodrigues _ et al . _ 2000 ) , leading to nearly perfect cylinders down to four atoms in diameter , with lengths up to fifteen nanometers . in particular , the mechanically - controllable break junction technique , introduced by moreland and ekin ( 1985 ) and refined by ruitenbeek and coworkers @xcite , has allowed for systematic studies of nanowire properties for a variety of materials . for a survey see the review by agrat _ et al . _ ( 2003 ) . a remarkable feature of metal nanowires is that they are stable at all . most atoms in such a thin wire are at the surface , with small coordination numbers , so that _ surface effects _ play a key role in their energetics . indeed , macroscopic arguments comparing the surface - induced stress to the yield strength indicate a minimum radius for solidity of order ten nanometers @xcite . below this critical radius and absent some other stabilizing mechanism , plastic flow would lead to a rayleigh instability @xcite breaking the wire apart into clusters . already in the 19th century plateau ( 1873 ) realized that this surface - tension - driven instability is unavoidable if cohesion is due solely to classical pairwise interactions between atoms . the experimental evidence accumulated over the past decade on the remarkable stability of nanowires considerably thinner than the above estimate clearly shows that electronic effects emphasized by the nfem dominate over atomistic effects for sufficiently small radii . a series of experiments on alkali metal nanocontacts ( yanson _ et al . _ 1999 , 2001 ) identified _ electron - shell effects _ , which represent the semiclassical limit of the quantum - size effects discussed above , as a key mechanism influencing nanowire stability . energetically - favorable structures were revealed as peaks in conductance histograms , periodic in the nanowire radius , analogous to the electron - shell structure previously observed in metal clusters @xcite . a supershell structure was also observed @xcite , in the form of a periodic modulation of the peak heights . more recently , such electron - shell effects have also been observed , even at room temperature , for the noble metals gold , copper , and silver ( diaz _ et al . _ 2003 , mares _ et al . _ 2004 , 2005 ) as well as for aluminum @xcite . soon after the first experimental evidence for electron shell effects in metal nanowires , a theoretical analysis using the nfem found that nanowire stability can be explained by a competition of the two key factors , surface tension and electron - shell effects @xcite . both linear @xcite and nonlinear ( brki _ et al . _ 2003 , 2005a ) stability analyses of axially symmetric nanowires found that the surface - tension driven instability can be completely suppressed in the vicinity of certain `` magic radii . '' however , the restriction to axial symmetry implies characteristic gaps in the sequence of stable nanowires , which is not fully consistent with the experimentally observed nearly perfect periodicity of the conductance peak positions . a jahn - teller deformation breaking the symmetry can lead to more stable deformed configurations . recently , the linear stability analysis was extended to wires with arbitrary cross - section ( urban _ et al . _ 2004a , 2006 ) . this general analysis confirms the existence of a sequence of magic cylindrical wires of exceptional stability which represent roughly 75% of the main structures observed in conductance histograms . the remaining 25% are deformed and predominantly of elliptical or quadrupolar shapes . this result allows for a consistent interpretation of experimental conductance histograms for alkali and noble metals , including both the electronic shell and supershell structures @xcite . this chapter is intended to give an introduction to the nfem . section [ sec : model ] summarizes the assumptions and features of the model while the general formalism is described in sec.[sec : formalism ] . in the following sections , two applications of the nfem will be discussed : first , we give a unified explanation of electrical transport and cohesion in metal nanocontacts ( sec . [ sec : forcecond ] ) and second , the linear stability analysis for straight metal nanowires will be presented ( sec . [ sec : stabana ] ) . the latter will include cylindrical wires as well as wires with broken axial symmetry , thereby discussing the jahn - teller - effect . guided by the importance of conduction electrons in the cohesion of metals , and by the success of the jellium model in describing metal clusters @xcite , the nfem replaces the metal ions by a uniform , positively charged background that provides a confining potential for the electrons . the electron motion is free along the wire , and confined in the transverse directions . usually an infinite confinement potential ( hard - wall boundary conditions ) for the electrons is chosen . this is motivated by the fact that the effective potential confining the electrons to the wire will be short ranged due to the strong screening in good metals . in a first approximation electron - electron interactions are neglected , which is reasonable due to the excellent screening @xcite in metal wires with @xmath7 . it is known from cluster physics that a free electron model gives qualitative agreement and certainly describes the essential physics involved . interaction , exchange and correlation effects as well as a realistic confinement potential have to be taken into account , however , for quantitative agreement . from this we infer that the same is true for metal nanowires , where similar confinement effects are important . remarkably , the electron - shell effects crucial to the stabilization of long wires are described with quantitative accuracy by the simple free - electron model , as discussed below . in addition , the nfem assumes that the positive background behaves like an incompressible fluid when deforming the nanowire . this takes into account , to lowest order , the hard - core repulsion of core electrons as well as the exchange energy of conduction electrons . when using a hard - wall confinement , the fermi energy @xmath5 ( or equivalently the fermi wavelength @xmath6 ) is the only parameter entering the nfem . as @xmath5 is material dependent and experimentally accessible , there is no adjustable parameter . this pleasant feature needs to be abandoned in order to model different materials more realistically . different kinds of appropriate surface boundary conditions are imaginable in order to model the behaviour of an incompressible fluid and to fit the surface properties of various metals . this will be discussed in detail in sec . [ sec : generalconstraint ] . a more refined model of a nanocontact would consider effects of scattering from disorder ( brki _ et al . _ 1999a , b ) and electron - electron interaction via a hartree approximation @xcite . the inclusion of disorder in particular leads to a better quantitative agreement with transport measurements , but does not change the cohesive properties qualitatively in any significant way , while electron - electron interactions are found to be a small correction in most cases . as a result , efforts to make the nfem more realistic do not improve it significantly , while removing one of its main strengths , the absence of any adjustable parameters . the major shortcoming of the nfem is that its applicability is limited to good metals having a nearly spherical fermi surface . it is best suited for the ( highly reactive ) s - orbital alkali metals , providing a theoretical understanding of the important physics in nanowires . the nfem has also been proven to qualitatively ( and often semi - quantitatively ) describe noble metal nanowires , and in particular , gold . lately , it has been shown that the nfem can even be applied ( within a certain parameter range ) to describe the multivalent metal aluminum , since al shows an almost spherical fermi surface in the extended - zone scheme . the nfem is especially suitable to describe shell effects due to the conduction - band s - electrons , and the experimental observation of a crossover from atomic - shell to electron - shell effects with decreasing radius in both metal clusters @xcite and nanowires @xcite justifies _ a posteriori _ the use of the nfem in the later regime . naturally , the nfem does not capture effects originating from the directionality of bonding , such as the effect of surface reconstruction observed for au . for this reason it can not be used to model atomic chains of au atoms , which are currently extensively studied experimentally . keeping these limitations in mind , the nfem is applicable within a certain range of radius , capturing nanowires with only very few atoms in cross - section up to wires of several nanometers in thickness , depending on the material under consideration . a metal nanowire represents an open system connected to metallic electrodes at each end . these macroscopic electrodes act as ideal electron reservoirs in thermal equilibrium with a well - defined temperature and chemical potential . when treating an open system , the schrdinger equation is most naturally formulated as a scattering problem . the basic idea of the scattering approach is to relate physical properties of the wire with transmission and reflection amplitudes for electrons being injected from the leads . the fundamental quantity describing the properties of the system is the energy - dependent unitary scattering matrix @xmath8 connecting incoming and outgoing asymptotic states of conduction electrons in the electrodes . for a quantum wire , @xmath8 can be decomposed into four submatrices @xmath9 , @xmath10 , @xmath11 , where 1 ( 2 ) indicates the left ( right ) lead . each submatrix @xmath12 determines how an incoming eigenmode of lead @xmath13 is scattered into a linear combination of outgoing eigenmodes of lead @xmath10 . the eigenmodes of the leads are also referred to as scattering channels . the formulation of electrical transport in terms of the scattering matrix was developed by landauer and bttiker : the ( linear response ) electrical conductance @xmath14 can be expressed as a function of the submatrix @xmath15 which describes transmission from the source electrode 1 to the drain electrode 2 and is given by @xcite @xmath16 here @xmath17 + 1\}^{-1}$ ] is the fermi distribution function for electrons in the reservoirs , @xmath18 is the inverse temperature and @xmath19 is the electron chemical potential , specified by the macroscopic electrodes . the trace @xmath20 sums over all eigenmodes of the source . the appropriate thermodynamic potential to describe the energetics of an open system is the grand canonical potential @xmath21,\ ] ] where @xmath22 is the electronic density of states ( dos ) of the nanowire . notably , the dos of an open system may also be expressed in terms of the scattering matrix as @xcite @xmath23 where tr sums over the states of both electrodes . this formula is also known as wigner delay . note that eqs . ( [ condformula ] ) , ( [ eq : omegavond ] ) , and ( [ eq : dauss ] ) include a factor of 2 for spin degeneracy . thus , once the electronic scattering problem for the nanowire is solved , both transport and energetic quantities can be readily calculated . ) ] . lower - left part : sketch of transverse energies for different transverse channels @xmath24 , @xmath25 , and @xmath26 as a function of the @xmath27-coordinate . channel @xmath24 is transmitted through the constriction as its maximum transverse energy is smaller than the fermi energy , channel @xmath25 is partly transmitted , and channel @xmath26 is almost totally reflected . right part : density plots of @xmath28 for the three eigenmodes depicted on the lower - left part , corresponding to five states due to degeneracies of energies @xmath29 and @xmath30 . ] for an axially symmetric constriction aligned along the @xmath27-axis , as depicted in fig . [ fig : sketchwkb ] , its geometry is characterized by the @xmath27-dependent radius @xmath31 . outside the constriction , the solutions of the schrdinger equation decompose into plane waves along the wire and discrete eigenmodes of a circular billiard in transverse direction . the eigenenergies @xmath32 of a circular billiard are given by @xmath33 where the quantum number @xmath34 is the @xmath35-th root of the bessel function @xmath36 of order @xmath19 and @xmath37 is the radius of the wire outside the constriction . in cylindrical coordinates @xmath38 , @xmath39 , and @xmath27 , the asymptotic scattering states read @xmath40 where @xmath41 is the longitudinal wavevector . in the following , we use multi - indices @xmath42 in order to simplify the notation . if the constriction is smooth , i.e. @xmath43 , one may use an adiabatic approximation . in the adiabatic limit , the transverse motion is separable from the motion parallel to the @xmath27-axis even in the region of the constriction , and the channel index @xmath44 of an incoming electron is preserved throughout the wire . accordingly , eqs . ( [ e.nu ] ) and ( [ inoutstates ] ) remain valid in the region of the constriction , with @xmath37 replaced by @xmath31 . the channel energies become functions of @xmath27 , @xmath45 , as is sketched in the lower part of fig . [ fig : sketchwkb ] , and act as a potential barrier for the effective one - dimensional scattering problem in channel @xmath44 . the corresponding schrdinger equation for the longitudinal part @xmath46 of the wave function reads @xmath47\phi_n(z)&=&0\;,\end{aligned}\ ] ] and is solved within the wkb approximation ( see , e.g. , @xcite ) by @xmath48\;.\end{aligned}\ ] ] for a constriction of length @xmath49 the transmission amplitude in channel @xmath44 is then given by the familiar wkb barrier transmission factor @xmath50 \;\equiv\ ; \sqrt{{{\cal t}}_n(e)}\,e^{i \theta_n(e)}\,.\end{aligned}\ ] ] here @xmath51 is the transmission coefficient of channel @xmath44 and @xmath52 is the corresponding phase shift . the transmission amplitude gets exponentially damped in regions where the transverse energy is larger than the state total energy . the full s - matrix is now found to be of the form @xmath53 where for simplicity of notation we have suppressed the channel indices and each of the entries is understood to be a diagonal matrix in the channels . using the formulas of sec . [ sec : smatrix ] , we may proceed to determine physical quantities . from eq . ( [ condformula ] ) we deduce that the electrical conductance at zero temperature reads , @xmath54,\nonumber\end{aligned}\ ] ] where the second line is obtained by using eq . ( [ eq : tn ] ) . here @xmath55 denotes the heaviside step function ( @xmath56 for @xmath57 , @xmath58 otherwise . ) the density of states is found to be connected with the phase shift @xmath52 , @xmath59 from the dos , one gets the grand canonical potential in the limit of zero temperature as @xmath60 which can then be used to calculate the tensile force and stability of the nanowire , as discussed in the following sections . the formalism presented in the previous subsection can be readily extended to non - axisymmetric wires . in general , the surface of the wire is given by the radius function @xmath61 , which may be decomposed into a multipole expansion @xmath62\right\},\ ] ] where the sums run over positive integers . the parameterization is chosen in such a way that @xmath63 is the cross - sectional area at position @xmath27 . the parameter functions @xmath64 and @xmath65 compose a vector @xmath66 , characterizing the cross - sectional shape of the wire . the transverse problem at fixed longitudinal position @xmath27 now takes the form @xmath67 with boundary condition @xmath68 for all @xmath69 $ ] . this determines the transverse eigenenergies @xmath70 which now depend on the cross - sectional shape through the boundary condition . with the cross - section parametrization ( [ eq : deformation ] ) , their dependence on geometry can be written as @xmath71 where the shape - dependent functions @xmath72 remain to be determined . in general , and in particular for non - integrable cross - sections , this has to be done by solving eq.([eq : seq : transverse ] ) numerically @xcite . the adiabatic approximation ( long - wavelength limit ) implies the decoupling of transverse and longitudinal motions . one starts with the ansatz @xmath73 and neglects all @xmath27-derivatives of the transverse wavefunction @xmath74 . again one is left with a series of effective one - dimensional scattering problems ( eq.[eq : wkb : schroedingerc ] ) for the longitudinal wave functions @xmath75 , in which the transverse eigenenergies @xmath76 act as additional potentials for the motion along the wire . these scattering problems can again be solved using the wkb approximation and eqs . ( [ eq : dos : wkb ] ) and ( [ eq : omegawkb ] ) apply . semiclassical approximations often give an intuitive picture of the important physics and , due to their simplicity , allow for a better understanding of some general features . a very early analysis of the density of eigenmodes of a cavity with reflecting walls goes back to weyl ( 1911 ) who proposed an expression in terms of the volume and surface area of the cavity . his formula was later rigorously proved and further terms in the expansion were calculated . quite generally , we can express any extensive thermodynamic quantity as the sum of such a semiclassical weyl expansion , which depends on geometrical quantities such as the system volume @xmath77 , surface area @xmath78 , and integrated mean curvature @xmath79 , as well as an oscillatory shell - correction due to quantum - size effects @xcite . in particular , the grand - canonical potential ( [ eq : omegavond ] ) can be written as @xmath80 where the energy density @xmath81 , surface tension coefficient @xmath82 , and curvature energy @xmath83 are , in general , material- and temperature - dependent coefficients . on the other hand , the shell correction @xmath84 can be shown , based on very general arguments @xcite , to be a single - particle effect , which is well described by the nfem . within the nfem there is only one parameter entering the calculation apart from the contact geometry : the fermi energy @xmath5 , which is material dependent and in general well known ( see tab . [ tab : sigma.gamma ] ) . nevertheless , the energy cost of a deformation due to surface and curvature energy , which can vary significantly for different materials , plays a crucial role when determining the stability of a nanowire . obviously , when working with a free - electron model , contributions of correlation and exchange energy are not included , while they are known to play an essential role in a correct treatment of the surface energy @xcite . using the nfem _ a priori _ implies the macroscopic free energy density @xmath85 , the macroscopic surface energy @xmath86 , and the macroscopic curvature energy @xmath87 . when drawing conclusions for metals having surface tensions and curvature energies that are rather different from these values , one has to think of an appropriate way to include these material - specific properties in the calculation . a convenient way of modeling the material properties without losing the pleasant features of the nfem is via the implementation of an appropriate surface boundary condition . any atom - conserving deformation of the structure is subject to a constraint of the form @xmath88 this constraint on deformations of the nanowire interpolates between incompressibility and electroneutrality as side conditions , that is between volume conservation ( @xmath89 ) and treating the semiclassical expectation value for the charge @xmath90 @xcite as an invariant ( @xmath91 , @xmath92 ) . .material parameters @xcite of several monovalent metals : fermi energy @xmath5 , fermi wavevector @xmath93 , surface tension @xmath82 , and curvature energy @xmath83 , along with the corresponding values of @xmath94 and @xmath95 . the last column gives the corresponding values for the multivalent metal al ( see discussion in sec . [ sec : materialdependence ] . ) adapted from @xcite . [ cols="<,^,^,^,^,^,^,^",options="header " , ] table [ tab : deformedwires ] lists the most stable deformed sodium wires with quadrupolar cross section , obtained by the procedure described above . the deformation of the stable structures is characterized by the parameter @xmath96 or equivalently by the aspect ratio @xmath97 clearly , nanowires with highly - deformed cross sections are only stable at small conductance . the maximum temperature up to which the wires remain stable , given in the last column of tab.[tab : deformedwires ] , is expressed in units of @xmath98 . the use of this characteristic temperature reflects the temperature dependence of the shell correction to the wire energy @xcite . deformations with higher @xmath99 cost more and more surface energy . compared to the quadrupolar wires , the number of stable configurations with 3- , 4- , 5- , and 6-fold symmetry , their maximum temperature of stability , and their size of the deformations involved , all decrease rapidly with increasing order @xmath99 of the deformation . for @xmath100 no stable geometries are known . all this reflects the increase in surface energy with increasing order @xmath99 of the deformation . it is possible to derive the complete stability diagram for cylinders , i.e. , to determine the radii of cylindrical wires that are stable with respect to _ arbitrary _ small , long - wavelength deformations @xcite . at first sight , considering arbitrary deformations , and therefore theoretically an infinite number of perturbation parameters seems a formidable task . fortunately , the stability matrix @xmath101 for cylinders is found to be diagonal , and therefore the different fourier contributions of the deformation decouple . this simplifies the problem considerably , since it allows to determine the stability of cylindrical wires with respect to arbitrary deformations through the study of a set of pure @xmath99-deformations , i.e. deformations as given by eq.([eq : deformation ] ) with only one non - zero @xmath102 . figure [ fig : stabcylinders ] shows the stable cylindrical wires in dark gray as a function of temperature . the surface tension was fixed at the value for na , see tab . [ tab : sigma.gamma ] . the stability diagram was obtained by intersecting a set of individual stability diagrams allowing @xmath103-deformations with @xmath104 . this analysis confirms the extraordinary stability of a set of wires with so called `` magic radii '' . they exhibit conductance values @xmath105 1 , 3 , 6 , 12 , 17 , 23 , 34 , 42 , 51 , ... it is noteworthy that some wires that are stable at low temperatures when considering only axisymmetric perturbations , e.g. , @xmath105 5 , 10 , 14 , are found to be unstable when allowing more general , symmetry - breaking deformations . the heights of the dominant stability peaks in fig.[fig : stabcylinders ] exhibit a periodic modulation , with minima occurring near @xmath105 9 , 29 , 59 , 117 , ... the positions of these minima are in perfect agreement with the observed supershell structure in conductance histograms of alkali metal nanowires @xcite . interestingly , the nodes of the supershell structure , where the shell effect for a cylinder is suppressed , are precisely where the most stable deformed nanowires are predicted to occur ( see discussion above ) . thus symmetry breaking distortions and the supershell effect are inextricably linked . linear stability is a necessary but not a sufficient condition for a nanostructure to be observed experimentally . the linearly stable nanocylinders revealed in the above analysis are in fact _ metastable _ structures , and an analysis of their lifetime has been carried out within an axisymmetric stochastic field theory by brki _ et al . _ ( 2005a ) . there is a strong correlation between the height of the stable fingers in the linear stability analysis and the size of the activation barriers @xmath106 , which determines the nanowire lifetime @xmath107 through the kramers formula @xmath108 . this suggests that the linear stability analysis , with temperature expressed in units of @xmath109 , provides a good measure of the total stability of metal nanowires . in particular , the `` universal '' stability of the most stable cylinders is reproduced , wherein the absolute stability of the magic cylinders is essentially independent of radius ( aside from the small supershell oscillations ) . a detailed comparison between the theoretically most stable structures and experimental data for sodium is provided in fig.[fig : compare ] . for each stable finger in the linear stability analysis its mean conductance is extracted and plotted as a function of its index number , together with experimental data by yanson _ et al . _ ( 1999 ) . this comparison shows that there is a one - to - one relation between observed conductance peaks and theoretically stable geometries which in particular allows for a prediction of the cross - sectional shape of the wires . this striking fit is only possible when including non - axisymmetric wires , which represent roughly 25% of the most stable structures and which are labeled by the corresponding aspect ratios @xmath110 in fig . [ fig : compare ] . the remaining @xmath111 of the principal structures correspond to the magic cylinders . the role of symmetry in the stability of metal nanowires is thus fundamentally different from the case of atomic nuclei or metal clusters , where the vast majority of stable structures have broken symmetry . the crucial difference between the stability of metal nanowires and metal clusters is not the shell effect , which is similar in both cases , but rather the surface energy , which favors the sphere , but abhors the cylinder . besides the geometries entering the comparison above , the stability analysis also reveals two highly deformed quadrupolar nanowires with conductance values of @xmath112 and @xmath113 , cf.tab . [ tab : deformedwires ] . they are expected to appear more rarely due to their reduced stability relative to the neighboring peaks , and their large aspect ratio @xmath110 that renders them rather isolated in configuration space . nevertheless they can be identified by a detailed analysis of conductance histograms of the alkali metals @xcite . results for different metals are similar in respect to the number of stable configurations and the conductance of the wires . on the other hand , the deviations from axial symmetry and the relative stability of jahn - teller deformed wires is sensitive to the material - specific surface tension and fermi temperature . the relative stability of the highly deformed wires decreases with increasing surface tension @xmath114 , measured in intrinsic units , and this decrease becomes stronger with increasing order @xmath99 of the deformation . therefore , for the simple s - orbital metals under consideration ( tab.[tab : sigma.gamma ] ) , deformed li wires have the highest , and au wires have the lowest relative stability compared to cylinders of `` magic radii . '' notable in this respect is aluminum with @xmath115 , some five times smaller than the value for au . aluminum is a trivalent metal , but the fermi surface of bulk al resembles a free - electron fermi sphere in the extended - zone scheme . this suggests the applicability of the nfem to al nanowires , although the continuum approximation is more severe than for monovalent metals . . thick lines mark stable wires in the configuration space of rms radius @xmath116 and deformation parameter @xmath96 . the dashed box emphasizes a series of very stable superdeformed wires , whose peanut - shaped cross section is shown as an inset . this sequence was recently identified experimentally @xcite . [ fig : alstabdia],scaledwidth=70.0% ] recent experiments @xcite have found evidence for the fact that the stability of aluminum nanowires also is governed by shell filling effects . two magic series of stable structures have been observed with a crossover at @xmath117 and the exceptionally stable structures have been related to electronic and atomic shell effects , respectively . concerning the former , the nfem can quantitatively explain the conductance and geometry of the stable structures for wires with @xmath118 and there is a perfect one - to - one correspondence of the predicted stable al nanowires and the experimental electron - shell structure . moreover , an experimentally observed third sequence of stable structures with conductance @xmath119 provides intriguing evidence for the existence of `` superdeformed '' nanowires whose cross sections have an aspect ratio near 2:1 . theoretically , these wires are quite stable compared to other highly deformed structures and , more importantly , are very isolated in configuration space , as illustrated in the stability diagram shown in fig.[fig : alstabdia ] . this favors their experimental detection if the initial structure of the nanocontact formed in the break junction is rather planar with a large aspect ratio since then it is likely that the aspect ratio is maintained as the wire necks down elastically . aluminum is unique in this respect and evidence of superdeformation has not been reported in any of the previous experiments on alkali and noble metals , presumably because highly - deformed structures are intrinsically less stable than nearly axisymmetric structures , due to their larger surface energy . in this chapter we have given an overview on the nanoscale free - electron model , treating a metal nanowire as a non - interacting electron gas confined to a given geometry by hard - wall boundary conditions . at first sight , the nfem seems to be an overly simple model , but closer study reveals that it contains very rich and complex features . since its first introduction in 1997 , it has repeatedly shown that it captures the important physics and is able to explain qualitatively , when not quantitatively , many of the experimentally observed properties of alkali and noble metal nanowires . its strengths compared to other approaches are , in particular , the absence of any free parameters and the treatment of electrical and mechanical properties on an equal footing . moreover , the advantage of obtaining analytical results allows the possibility to gain some detailed understanding of the underlying mechanisms governing the stability and structural dynamics of metal nanowires . the nfem correctly describes electronic quantum size effects , which play an essential role in the stability of nanowires . a linear stability analysis shows that the classical rayleigh instability of a long wire under surface tension can be completely suppressed by electronic shell effects , leading to a sequence of certain stable `` magic '' wire geometries . the derived sequence of stable cylindrical and quadrupolar wires explains the experimentally observed shell and supershell structures for the alkali and noble metals as well as for aluminum . the most stable wires with broken axial symmetry are found at the nodes of the supershell structure , indicating that the jahn - teller distortions and the supershell effect are inextricably linked . in addition , a series of superdeformed aluminum nanowires with an aspect ratio near 2:1 is found which has lately been identified experimentally . a more elaborate fully quantum mechanical analysis within the nfem reveals an interplay between the rayleigh and a peierls - type instability . the latter is length - dependent and limits the maximal length of stable nanowires but other than that confirms the results obtained by the long wavelength expansion discussed above . remarkably , certain gold nanowires are predicted to remain stable even at room temperature up to a maximal length in the micrometer range , sufficient for future nanotechnological applications . the nfem can be expanded by including the structural dynamics of the wire in terms of a continuum model of the surface diffusion of the ions . furthermore , defects and structural fluctuations may also be accounted for . these extensions improve the agreement with experiments but do not alter the main conclusions . however , the nfem does not address the discrete atomic structure of metal nanowires . with increasing thickness of the wire the effects of surface tension decrease and there is a crossover from plastic flow of ions to crystalline order , the latter implying atomic shell effects observed for thicker nanowires . therefore , the nfem applies to a window of conductance values between a few @xmath3 and about @xmath120 , depending on the material under consideration . promising extensions of the nfem in view of current research activities are directed , e.g. , towards the study of metal nanowires in nanoelectromechanical systems ( nems ) which couple nanoscale mechanical resonators to electronic devices of similar dimensions . the nfem is ideally suited for the investigation of such systems since it naturally comprises electrical as well as mechanical properties . it is hoped that the generic behaviour of metal nanostructures elucidated by the nfem can guide the exploration of more elaborate , material - specific models , in the same way that the free - electron model provides an important theoretical reference point from which to understand the complex properties of real bulk metals .

a brief review of the nanoscale free - electron model of metal nanowires is presented . this continuum description of metal nanostructures allows for a unified treatment of cohesive and conducting properties . conductance channels act as delocalized chemical bonds whose breaking is responsible for jumps in the conductance and force oscillations . it is argued that surface and quantum - size effects are the two dominant factors in the energetics of a nanowire , and much of the phenomenology of nanowire stability and structural dynamics can be understood based on the interplay of these two competing factors . a linear stability analysis reveals a sequence of `` magic '' conductance values for which the underlying nanowire geometry is exceptionally stable . the stable configurations include jahn - teller deformed wires of broken axial symmetry . the model naturally explains the experimentally observed shell and supershell structures .

1006.3910

the advent of modern computers has made it possible to study many - dimensional dynamical systems in detail and to ferret out previously unsuspected characteristics . phenomena such as relaxation to equilibrium @xcite , lack of ergodicity @xcite , and connection of dynamical to statistical properties @xcite have been explored numerically . a category of models that continue to enjoy great attention is that of interacting - electron systems . in this paper we consider a particular member of this class : the classical dynamics of the confined two - dimensional electron gas ( 2deg ) . while various aspects of 2deg have been well - studied , it remains a convenient and relatively tractable system to understand dynamical properties of complex systems . it is well - known that at low temperatures the electron system freezes into an hexagonal wigner lattice @xcite . extensive studies have been made of the possible conformations that result when this simple system is perturbed @xcite . a simple example is provided by the inter - layer coupling between vertically separated 2degs which enhances the stability of the square lattice relative to the hexagonal @xcite . the melting and structural transitions have also been well - studied @xcite . the confinement is also a possible perturbation ideal hexagonal lattice is obtained only for infinite system . however , commonly used confining potentials preserve the hexagonal character with only slight edge distortions . perturbed 2deg models have relevance beyond electron systems : experiments on dusty plasmas @xcite and ion plasmas @xcite have been usefully explained in terms of formation of few layered and bi - layered coulomb lattices . the classical behavior of a confined 2deg subject to an external magnetic field has been of some recent theoretical use @xcite . in particular the magnetotransport phenomena in 2deg have lead to important physical insights . new systems have been proposed in which the electron motion in 2deg is nontrivially altered by a ( possibly non - homogeneous ) magnetic field @xcite . some possibilities have been realized by advances in the semiconductor technology @xcite . such systems are studied quantum mechanically for a full understanding but many features can already be appreciated within the classical theory . for example , the classical chaos was found to control the low - field transport in systems with competing magnetic and electrostatic modulation @xcite . in this paper we analyze the structure and dynamics of 2deg on a novel geometry . the electrons are constrained to the surface of a cylinder of radius @xmath0 and confined along the @xmath1 direction to a strip of width @xmath2 by the potential of the uniform static positive background . we chose this geometry because it allows the electrons to carry a current along the @xmath3 direction . we apply the external electric and magnetic fields and observe the time - dependent response . we have been motivated to this problem by certain reflections on usual textbook derivations of the classical 3-dimensional hall effect @xcite . these derivations are based on free - electron drude theory and consider independent electrons drifting with a common and constant drift velocity @xmath4 . it is then easy to show that a constant transverse electric field , generated by excess boundary charge , can balance the magnetic force and bring a steady state . in the context of a classical 2-dimensional many degree - of - freedom interacting system this simple picture could be dynamically unstable . here we conduct numerical experiments to see if a regime of interacting electrons drifting with a common uniform velocity can be attained in a 2d many - body system . a peculiarity of 2deg is that the coulomb field escapes from the surface ; in 3d a constant transverse hall field can be nicely produced by the boundary electrons only , while in 2d , because of gauss s law , these boundary electrons can only produce a @xmath5 field ( the field of a charged wire ) . we see then that a global charge redistribution is called upon for to produce a constant field . the lagrangian for @xmath6 interacting electrons confined by the background potential @xmath7 and subject to an external radial magnetic field @xmath8 is @xmath9 where @xmath10 is the coordinate of the @xmath11th electron and @xmath12 is the distance between @xmath11th and @xmath13th electrons and @xmath14 is the speed of light . the electronic mass and charge are @xmath15 and @xmath16 respectively . to remove any ambiguity we clarify that distances are not calculated along the surface but are ordinary three - dimensional distances : @xmath17 . the rotation symmetry possessed by @xmath18 implies the noether s constant : @xmath19 the resulting equations of motion are @xmath20 where the confining potential @xmath7 of the positive background is taken to be either @xmath21 or @xmath22 the stripe being symmetrical about @xmath23 and extending from @xmath24 to @xmath25 . the first potential is flat but rises steeply near the edges . the second potential has more parabolic character and is an approximation to the exact potential of a positive background : we found analytical solution for the potential of a positively charged flat rectangle with the same width and length of our cylindrical background , and the above second potential approximates this analytic form . the use of potentials with differing characters helps in distinguishing any peculiar effect of confinement from overall observations . these potentials are sketched in fig.([confine ] ) . the form of equation ( 1 ) is simplified by using scaled units : lengths may be scaled by the average interelectronic distance @xmath26 : @xmath27 and time is scaled as @xmath28 where @xmath29 . in these units the equations of motion are @xmath30 where @xmath31 and @xmath32 is the two - dimensional number density of the electron gas . for example at the experimentally attainable @xmath33 @xmath34 the above formula gives @xmath35 tesla . because of the scale - invariance of the coulomb interaction , @xmath36 is independent of @xmath37 equation ( 4 ) has two components for each particle corresponding to @xmath1 and @xmath3 motions . we deliberately add an additionally force along the @xmath3 direction represented by a small electric field @xmath38 and a weak damping so that the final equation for @xmath3 is @xmath39 we chose this form of damping such that if all electrons cycle with a constant common velocity @xmath4 , the damping force is balanced by @xmath40 the last term in the above equation is just the coulomb repulsion between electrons . the equations of motion are integrated numerically using a 7/8 order embedded runge - kutta pair with self - adjusted step . the initial states for the numerical integrations are generated in the following way : the energy of the confined coulomb system is @xmath41 a naive candidate for a minimal - energy initial condition would be one which minimizes the above energy and with all electrons cycling with the same velocity @xmath42 it is straightforward to find this condition by steepest descent quenching and one obtains a translating lattice which approximates an ideal triangular lattice with some edge distortions . the problem with this condition is that in the presence of the external magnetic field @xmath8 ( which does not appear explicitly in the energy ) , the translating electrons deflect upwards causing a lattice deformation . we observe , by integrating the equations of motion numerically from this condition with inclusion of the above defined weak dissipation , that the lattice shifts to a different configuration consistent with the presence of magnetic field and with a slightly higher steady state energy . ( notice that we are driving the system , so energy does not have to be constant ) . an equivalent way to accomplish this same final state is to seek minimum energy configurations with a certain property : they must describe a uniform motion of the entire system along the @xmath3 direction under a radial magnetic field @xmath8 ( @xmath43 what we are looking for is that the total force acting on each electron be zero @xmath44 this configuration can be obtained by a steepest descent procedure : we integrate the ( modified ) quenching equation @xmath45 @xmath46 being the parameter along the quenching path . the quenched configurations are obtained for various values of the parameter @xmath47 and then used as initial conditions for subsequent molecular dynamical runs . fig.(2 ) shows a representative configuration obtained with the confinement of eq.(2 ) . the electrons arrange themselves in an hexagonal lattice slightly distorted by the confinement and magnetic field ( notice that this is a global distortion , not a simple boundary perturbation of a perfect lattice ) . electrons are projected out with an initial @xmath3 velocity @xmath48 plus small random components along @xmath3 and @xmath1 directions . various quantities are calculated along the trajectory : the instantaneous rotation rate @xmath49 which is related to the current @xmath50 ; the @xmath3 averaged potential difference between the top and bottom edges : @xmath51 ( hall voltage ) ; and the hall resistance @xmath52 . all these quantities are function of time and hence their time - development is likely to be informative . we also observe snapshots of the system at regular intervals . these snapshots could be folded to @xmath53 or left unfolded to preserve information about angular motion . we report the results of the simulations of @xmath54 and 484 particles performed with the confining potential of eqs ( 2 ) and ( 3 ) . we distinguish between runs carried out with @xmath55 zero and nonzero . simulations with nonzero @xmath55 reach a steady state after a brief transient which , is not attained by zero @xmath55 runs . we henceforth call this state a perfectly translating lattice ( ptl ) . in the ptl state the electrons cycle with a common constant @xmath3 velocity @xmath48 ; motion along @xmath1 being rapidly damped out . the resulting hall resistance @xmath56 should be a constant in such circumstances but we observe small fluctuations in @xmath57 . the amplitude of these fluctuations is an irregular function of @xmath58 generally varying between 0.1 - 1.0 percent . the presence of fluctuating @xmath59 is not disquieting however and can be understood as an artifact of the method employed to calculate hall resistance of small finite number of electrons : because we used a finite number of points to average the potential difference , small instantaneous fluctuations are generated if the number of electrons is small and we verify that the fluctuations become smaller for larger @xmath6 ( number of electrons ) . two cases can be further differentiated with regard to the initial phase of the dynamics . initial configuration for certain values of the external magnetic field @xmath8 turns out to be dynamically unstable . the system makes a transition to a new configuration through coordinated row - jumping of many electrons simultaneously . such a transition is made possible by the existence of numerous local minima in the energy surface whose presence has been confirmed numerically by extensive quenching runs starting from distinct initial conditions . the instability only appears for scaled magnetic fields greater than 7.0 for @xmath54 and 8.5 for @xmath60 ( for electron density of @xmath61 @xmath34 these fields correspond to 1.9 t and 2.3 t respectively ) . in a way , this result shows that the initial state found by minimization ceased to be stable and another extremum ( not the minimum anymore ) of the functional became an stable fixed point , a bifurcation . this instability is accompanied by slower relaxation to a ptl state and can be seen in fig.(3 ) which plots the spread in the instantaneous rotation rate vs. time . even more dramatically , velocity inhomogeneities are developed if all electrons are released with a common velocity @xmath4 with no random components . these inhomogeneities are not long lasting and ultimately a ptl is attained . the resultant of this temporarily existing velocity profile can be visualized most easily in fig.(4 ) where the @xmath3 coordinates have not been folded to @xmath62 . this striking profile is not apparent after folding and the finally established ptls do not differ from ptls in cases where this instability is absent . we call this instability a _ shearing _ instability since it is visible as a shear in the velocity profile of the electrons . this shear is seen as a dispersion in unfolded lattice along @xmath63 coordinate . we have carried out runs without forcing and dissipation i.e. @xmath64 and these do not achieve a ptl state . they may or may not display the initial shearing instability but in all cases the initial randomness in velocity distribution is magnified and the @xmath1 dynamics is not damped . an irregular velocity distribution develops even from a perfectly homogeneous initial velocity distribution . the instantaneous hall resistance @xmath65 is unsteady with large amplitude fluctuations and the system can not be said to be in a hall regime . from these simulations a plot of the hall resistance @xmath56 vs. @xmath8 can be drawn . only converged values of @xmath56 are used which rules out our undriven simulations ( @xmath64 ) . in fig.(5 ) results from unsheared and sheared states are displayed for @xmath54 and 484 . data from the simulations employing confining potential of equation ( 3 ) has been plotted for @xmath54 also . all are ptl configurations but the sheared states have experienced the shearing instability referred to earlier . we observe that @xmath56 from the unsheared states lie on straight lines though different slopes are obtained for the two confining potentials used : 1.07 and 1.05 for potential ( 1 ) and ( 2 ) respectively . data for @xmath54 and 484 overlap . points from the sheared states are scattered haphazardly about this straight line . how far do these data match our expectations ? a plausible argument can be made for reasonableness of our simulation results : the hall voltage @xmath51 is the difference between the top and the bottom edges of the stripe and may be expressed as @xmath66 where @xmath67 and the integral is taken along a straight line from bottom to the top edge ; @xmath68 is the distance of the @xmath69th electron from integration element . now the integrand may be split as @xmath70 where @xmath32 labels the electron nearest to the integration element and @xmath71 is the unit vector along @xmath1 direction . this follows from the assuming that the force - balance condition ( eq . 9 ) holds in a neighborhood of the electron nearest to 0 : @xmath72 where @xmath73 label electrons . this equation holds only at the position of an electron and we use as an approximate equality in a neighborhood of the nearest point @xmath74 . from the above it follows @xmath75 the first term on the right yields the straight line dependence of @xmath56 vs. @xmath8 with a unit slope but the second term provides a correction . to evaluate this correction , we take an integration path passing through a column of electrons , avoiding each electrons by making a small semi - circle around it . if the distance between the @xmath11th and @xmath76th electrons along this path is @xmath77 , this correction evaluates to @xmath78 hence the correction to the unit slope depends on the degree of compression that the lattice undergoes under the external magnetic field . a 3d lattice does not suffer this kind of bulk squishing but only edge distortions . if the lattice rearrangement is global in the sense that lattice distances are affected throughout the bulk and not just at the edges then we can fairly expect a significant alteration of the slope of hall resistance plot . this expectation is realized in our simulations as we have seen that the slopes of @xmath56 vs @xmath8 plot indeed differ from unity by a few percent . lattices which are dynamically unstable and undergo rearrangement via the shearing instability would seem to require more correction according to this picture and in fact provide a needed check for the theory . in conclusion we have performed dynamical simulations on a 2deg constrained to a cylindrical stripe and subject to crossed electric and magnetic fields . the classical confined electron gas has a natural non - trivial minimal energy state and provides a convenient test bed to study many - body dynamics and long - range effects . in this work we included many - body effects on the dynamical picture used in the classical derivations of the hall effect in two dimensional systems . we have analyzed the formation of the steady state presupposed in classical derivations . this state , which we refer to as a perfectly translating lattice ( ptl ) , in which all electrons cycle with a common constant velocity , is formed by a relaxation process . the initial configurations are obtained by a generalized quenching procedure . this initial configuration is liable to be dynamically unstable for magnetic fields above a threshold . the quenching procedure sometimes yields rather shallow local minima which readily allow further rearrangements to nearby wide basins . the inter - basin like motion is manifested in simultaneous jumping of many electrons and a slower relaxation to ptl state . the hall resistance @xmath56 can be calculated and plotted as a function of external magnetic field @xmath79 appropriately for ptl states @xmath56 is a linear function of @xmath8 except for sheared states . an explanation has been put forward based upon force - balance condition as it obtains for ptl states . if one repeated the same study for a confined 3d electron - gas , the minimal energy state would be a tridimensional wigner lattice and inclusion of a drift velocity in the presence of an external magnetic field would produce only boundary charge rearrangement . the obtained ptl would be the same ideal wigner lattice and there would be no many - body correction to the hall coefficient . our results are specific to 2d classical systems . the present work evolved from our earlier attempts to study the same problem but with the electrons interacting via the darwin lagrangian , which is the first relativistic correction to the coulomb interaction . we find that the relativistic corrections break the scale - invariance of the coulomb interaction , even in the absence of a magnetic field , only by requiring a critical density of the electron gas@xcite . even though these is a much richer dynamical system , we did not continue the studies because the equations of motion are algebraic - differential and become impossible to integrate above the critical density even by use of the modern specialized integrators radau @xciteand dassl @xcite . for some recent computational papers on freezing transition of 2deg and winger lattice , v. m. bedanov and f. m. peeters , phys . b * 49 * , 2667 ( 1994 ) ; l. candido , j. p. rino , n. studart , and f. peeters , j. phys . condens . matter * 10 * , 11627 ( 1998 ) .

we perform molecular dynamics simulations on an interacting electron gas confined to a cylindrical surface and subject to a radial magnetic field and the field of the positive background . in order to study the system at lowest energy states that still carry a current , initial configurations are obtained by a special quenching procedure . we observe the formation of a steady state in which the entire electron - lattice cycles with a common uniform velocity . certain runs show an intermediate instability leading to lattice rearrangements . a hall resistance can be defined and depends linearly on the magnetic field with an anomalous coefficient reflecting the manybody contributions peculiar to two dimensions .

cond-mat0103353

bekenstein and hawking have showed that the entropy of black holes is proportional to the area of their event horizon @xcite . in units of @xmath0 and @xmath1 , the black hole entropy is given as @xmath2 where @xmath3 is the area of event horizon of the black hole . hawking have shown that the black hole can evaporate by emitting radiation , consequently it s event horizon area decreases . he had also shown that the event horizon of the black hole posses temperature , which is inversely proportional to it s mass or proportional to it s surface gravity . during the process of evaporation the entropy of the black hole will decrease . but due to the emitted radiation , the entropy of the surrounding universe will increase . hence the second law of thermodynamics was modified in such a way that , the entropy of the black hole plus the entropy of the exterior environment of the black hole will never decrease , this is called as the generalized second law(gsl ) , which can be represented as , @xmath4 where @xmath5 is the entropy of environment exterior to the black hole and @xmath6 is the entropy of the black hole . the thermodynamic properties of the event horizon , was shown to exist in a more basic level@xcite , by recasting the einstein s field equation for a spherically symmetric space time as in the form of the first law of thermodynamics . in references @xcite one can find investigations on the applicability of the first law of thermodynamics to cosmological event horizon . jacobson @xcite showed that , einstein s field equations are equivalent to the thermodynamical equation of state of the space time . in cosmology the counter part of black hole horizon is the cosmological event horizon . gibbons and hawking @xcite proposed that analogous to black hole horizon , the cosmological event horizon also do possess entropy , proportional to their area . they have proved it particularly for de sitter universe for which an event horizon is existing . for cosmological horizon , gsl implies that , the entropy of the horizon together with the matter enclosed by the event horizon of the universe will never decrease . that is the rate of change of entropy of the cosmological event horizon together with that of material contents within the horizon of the universe , must be greater than or equal to zero , @xmath7 where @xmath8 is the entropy of the cosmological event horizon and @xmath9 represents the entropy of the matter or radiation ( or both together ) of the universe . the validity of gsl for cosmological horizon was confirmed and extended to universe consisting of radiation by numerical analyses by davies @xcite and others @xcite . in reference @xcite , the authors analyzed the gsl with some variable models of f(t ) gravity . in reference @xcite gsl was analyzed with reference to brane scenario . ujjal debnath et . @xcite have analyzed the gsl for frw cosmology with power - law entropy correction . there are investigations connecting the entropy and hidden information . in the case of black hole horizon , the observer is outside the horizon , and the entropy of the black hole is considered as measure of the information hidden within the black hole . while regarding cosmological horizon , the observer is inside the horizon . this will cause problems in explaining the entropy of the cosmological horizon as the measure of hidden information as in the case of black hole . in the case of black hole the hidden region is finite , while in the case of cosmological horizon , there may be infinite region beyond the event horizon of the universe , which causing problems in explaining the cosmological horizon entropy as the hidden information . another important fact is regarding the existence of dominant energy condition for the non decreasing horizon area . in the case of black hole , hawking proved an area theorem , that the area of the black hole will never decrease if it is not radiating @xcite . davies @xcite proved an analogous theorem for cosmological event horizon that the area of the cosmological event horizon will never decrease , provided it satisfies the dominant energy condition , @xmath10 where @xmath11 is the density of the cosmic fluid and @xmath12 is its pressure . regarding the applicability of the generalized second law to the friedmann universe , analysis were done by considering the friedmann universe as a small deviation form the de sitter phase@xcite . in these works the authors calculated the horizon entropy through a numerical computation of the comoving distance to the event horizon . in the present work we obtained an analytical equation for the hubble parameter and proceeded to the calculation of the entropy of the cosmological event horizon in an analytical way . we also checked the validity of dominant energy condition by using the derived hubble parameter . our analysis is for a flat universe which consists of ( i ) radiation and positive cosmological constant and ( ii ) non - relativistic matter and positive cosmological constant . we have considered the flat universe because of the fact that , the inflationary cosmological models predicts flat universe and more over the flatness of the space is confirmed by observations , for example , the current value of the curvature parameter is @xmath13 @xcite . the paper is arranged as follows . in section two , we consider the flat friedmann universe with radiation and a positive cosmological constant . we are presenting the calculation of the entropy of radiation , event horizon and the total entropy of universe and the respective time evolutions . we have also checked the validity of the generalized second law in this section . in section three we present the analogous calculations for the flat friedmann universe with non - relativistic matter and a positive cosmological constant . in section four we present the particular behaviour of the radiation entropy in the friedmann universe with reference to the development of the event horizon . in section five we present the discussion followed by conclusions . for a flat friedmann universe with frw metric , the dynamics are governed by the friedmann equations(by choosing @xmath14 ) , @xmath15 and @xmath16 where @xmath17 is the radiation density , @xmath18 is the radiation pressure , @xmath19 is the constant cosmological constant , @xmath20 is the hubble parameter and the dot over the density represents derivative with respect to time . for radiation , the pressure is , @xmath21 . from equations ( [ eqn : friedmann1 ] ) and ( [ eqn : friedmann2 ] ) the scale factor of this universe can be obtained as , @xmath22 where @xmath23 , @xmath24 and @xmath25 is the present value of the hubble parameter . this equation shows that as @xmath26 the scale factor @xmath27 the radiation dominated phase of the friedmann universe and as @xmath28 the scale factor @xmath29 , with time @xmath30 in gyrs , the blue line ( lower line ) corresponds to the friedmann universe and red line ( upper line ) for de sitter universe . _ _ ] the de sitter phase . the behaviour of the scale factor with time is shown in figure [ fig : af1 ] , in comparison with the scale factor of the de sitter phase . from the plot it is evident that the scale factor of the friedmann universe tends to the de sitter phase at large times . so at smaller times the universe is in the radiation dominated phase and it is decelerating , consequently it does nt have event horizon . at larger times the universe enters the accelerated expansion phase , where it posses an event horizon . the co - moving distance to the event horizon , can be obtained by using the relation , @xmath31 thus the proper distance to the event horizon is @xmath32 for the existence of the event horizon , the integral has to converge . with the scale factor in equation ( [ eqn : afact1 ] ) , the integral in the equation for comoving distance to the event horizon can not be solved analytically . so as a first step we made a numerical computation of the comoving distance to the event horizon , as it is necessary to understand the time evolution of the comoving distance and the result is shown in figure [ fig : codist1 ] . [ fig : codist1 ] to the event horizon with time in gyrs for friedmann universe with radiation and cosmological constant , title="fig : " ] the plot shows that the comoving distance to the event horizon is decreasing with time . since the comoving horizon distance is decreasing , the comoving volume of the universe within the horizon also decreases . the radiation density behaves as @xmath33 , therefore the radiation content within the horizon is decreasing with time . which nevertheless implies that the radiation is crossing the horizon , hence the radiation entropy within the horizon is decreasing . this method and conclusion is in line with the result of t m davies et . one can also find investigations of the same spirit regarding the heat flow through the cosmic horizon in references @xcite . in fact this result is true for any model of the universe having an event horizon . the horizon entropy can be obtained as per the bekenstein equation ( [ eqn : hentro ] ) . for that the area of the event horizon can be taken as @xmath34 in the work of davies et , al . the entropy was calculated in a numerical way , but we are obtaining the entropy of the horizon using the hubble parameter obtained form the scale factor . we are substituting @xmath35 in terms of the hubble parameter . the scale factor in equation ( [ eqn : afact1 ] ) shows that , at large time the scale factor is approaching to that of de sitter phase . for de sitter phase , it can be shown that , @xmath36 since the friedmann universe considered here is approaching the de sitter phase at large times , it will not be unfair in taking , the comoving distance @xmath37 for the friedmann universe in consideration . for the scale factor in equation ( [ eqn : afact1 ] ) , the hubble parameter is , @xmath38 before going for a calculation of the entropy of the event horizon , we will check here the validity of the area theorem proposed by davies , with the obtained hubble parameter . from equation ( [ eqn : friedmann1 ] ) and ( [ eqn : h1 ] ) , the condition for non - decreasing horizon area , equation ( [ eqn : condition1 ] ) , leads to @xmath39 using equation ( [ eqn : h1 ] ) we have plotted @xmath40 versus time in figure [ fig : cond1 ] . we have used the parameter values , @xmath41 @xcite and a standard value @xmath42 through out for our calculations . the plot shows that the area of the event horizon of the friedmann universe with radiation and a positive cosmological constant will never decrease , hence the entropy of horizon will never decrease . ) ] on the other hand , the entropy of the radiation is decreasing with time as we have argued earlier . in oder to satisfy the gsl , the decrease in the entropy of the radiation is to be balanced by the increase in the horizon entropy . the horizon area is always increasing , implies that there exist some kind of trading of the entropy between the horizon and the radiation content of the universe . the entropy of the event horizon is @xmath43 as we have argued earlier , taking @xmath44 , the horizon entropy become , @xmath45 the entropy of the radiation can be obtained using the relation , @xmath46 where @xmath47 is the volume of the event horizon and @xmath48 is the temperature of the radiation . taking @xmath49 , substituting temperature form radiation energy density , @xmath50 , @xmath51 which after substituting @xmath20 parameter form equation ( [ eqn : h1 ] ) and @xmath17 in terms of @xmath20 , using the friedmann equations , become @xmath52 where @xmath53 the radiation constant . we have plotted the time variation of @xmath8 , @xmath54 and @xmath55 in figure 4 . the figure shows that , at sufficiently large times the radiation entropy is decreasing , while the horizon entropy is increasing . the increase in the horizon entropy is more than required to compensate for the decrease in the radiation entropy because of that , total entropy comprising the entropy of the radiation and horizon will increase . this is confirming the validity of the generalized second law for the cosmological horizon , that the entropy of the horizon plus the entropy of the fluid within the horizon will never decrease . this result is in confirmation with the earlier works of davies and others , but they have arrived at the conclusion through straight numerical work , on the other hand our work is more of an analytical way . the conditions for satisfying the generalized second law can be obtained by analysing the validity of the exact statement of the law as given equation ( [ eqn : gsl ] ) . the time rate of the horizon entropy is , @xmath56 where the dot over @xmath20 represents the derivative with time given as @xmath57 , leads to @xmath58 the time rate of radiation entropy can be given from ( [ eqn : sgamma1 ] ) as , @xmath59 the above two equations reveal that the time rate of horizon entropy is positive hence the horizon entropy is at the increase , while the time rate of radiation entropy is negative hence the radiation entropy is at the decrease . the generalized second law , can then be represented as , @xmath60 [ eqn : gslcond3 ] replacing @xmath20 and @xmath61 using the equation ( [ eqn : h1 ] ) , the above condition become , @xmath62 expressing @xmath17 , in terms of @xmath20 , using the friedmann equation , we have evaluated the time evolution of the left hand side of the above inequality condition and the result is shown in figure [ fig : gslcon1 ] . ] the figure shows that the condition for the gsl is always satisfied . from the gsl condition in equation([eqn : gslcon3 ] ) , we can obtain a condition regarding the temperature of the radiation within the horizon . with @xmath63 , the condition ( [ eqn : gslcon3 ] ) , become , @xmath64 taking @xmath65 , then an inequality condition constraints the present value of the temperature of the radiation can be obtained as , @xmath66 the above condition leads to a numerical value , @xmath67k . compared to the present temperature of the radiation @xmath68 , this is very much in favour of the validity of second law in the friedmann universe . this result is agreeing with the result obtained by davies et . al , that @xmath69 , where @xmath20 is now taken as the temperature of the horizon . by using the fundamental constants , the temeprature of the horizon , is @xmath70 where @xmath71 is the boltzmann constant , which implies a present value , @xmath72k . so the temperature of the horizon is less than that of the event horizon , which indicates that , the radiation can aproach the horizon . in this section we are analysing the friedmann universe with matter and a positive cosmological constant , regarding the horizon entropy and the generalized second law . the friedmann equation , in this case is , @xmath73 the scale factor can then be obtained as , @xmath74 as in the previous case , here also the solution will tends to the de sitter phase , @xmath75 as @xmath76 which means that the model posses an event horizon . the hubble parameter corresponds to the scale factor is , @xmath77 it can be seen that the dominant energy condition is being satisfied , as in the case of friedmann universe with radiation , such that @xmath78 at all time . the comoving distance to the horizon is evaluated using the scale factor in the equation ( [ eqn : a2 ] ) , and is deceasing with time as shown is shown in figure [ fig : dist1 ] . so the matter entropy within the horizon ] will decrease and hence the matter will crosses the event horizon . entropy of the event horizon is calculated as , @xmath79 entropy of matter can calculated using an analogous relation corresponds to equation ( [ eqn : radentro ] ) , and taking temperature of matter approximately as @xmath80 , @xmath81 which after substitution of @xmath20 parameter become , @xmath82 the behaviour of @xmath83 and @xmath84 with time is shown in figure [ fig : totent2 ] . and @xmath84 with time in gyrs for friedman universe with matter and cosmological constant . the continuous line representing entropy of horizon plus that of matter , dashed line representing the entropy of horizon and dash - dot line is for entropy of matter ] the figure shows that the total entropy of the universe is increasing and the increase in the entropy of the horizon is more than that required for compensating the decrease in the matter entropy . the general behaviour is the same as that of friedmann universe with radiation , that universe with matter also will satisfy the generalized second law . the condition for satisfying the generalized second law for this universe can be obtained by incorporating the time derivatives of the corresponding entropies into the second law , as @xmath85 using the hubble parameter equation ( [ eqn : h2 ] ) , the above condition become , @xmath86 substituting @xmath87 in terms of the hubble parameter using the friedmann equation , we have plotted the time evolution of the left hand side of the above equation and is shown in figure [ fig : secndlaw2 ] . as in the case of the friedmann universe with radiation , here also the plot shows that the inequality condition corresponds to the generalized second law is satisfied . as in the previous section , the generalized second law can leads to constraint on the temperature of matter . equation ( [ eqn : gslma ] ) can be recast , by taking @xmath88 , as @xmath89 from this it can be shown that , the present temperature of matter in the universe satisfies , @xmath90 for the standard value @xmath91 the above condition also gives , @xmath92 the temperature of the horizon is @xmath93 @xcite , and with proper parameters , @xmath94 so the present temperature of the matter is greater than the temperature of the horizon , which supports the conclusion that the matter can cross the event horizon . in this section we will restrict our analysis to firedmann universe with radiation and cosmological constant only . however one can easily see that the conclusions made are in general true for firedmann universe with matter also , but with different numerical values . our aim here is to show that the entropy of the contend of the universe does have a small increase before the development of event horizon . in the last two sections we have discussed the behaviour of horizon entropy and entropy of the material within the horizon . we have concentrated on checking the validity of the generalized second law . we have shown that the total entropy of the universe is always increasing , and the cosmological event horizon is satisfying the generalized second law . however it is to be noted from the figure [ fig : entrorad ] ( from figure [ fig : totent2 ] ) that the entropy of the radiation ( matter ) is increasing first , attaining a maximum , then after it is decreasing . and the plot on the right represents the time evolution of the @xmath95factor of the same friedmann universe with time in giga years.,title="fig : " ] and the plot on the right represents the time evolution of the @xmath95factor of the same friedmann universe with time in giga years.,title="fig : " ] for clarity regarding this we will show in figure [ fig : radentro ] , the time evolution of the radiation entropy for a friedmann universe having radiation and a positive cosmological constant . the figure shows that the radiation entropy first increases and then decreases to zero at very large times . in the previous section we have concluded that the decrease in the entropy of radiation ( or matter ) is due to the escape of the radiation ( or matter ) from within the horizon . the horizon will exist only when the universe is accelerating . from the bahaviour of the scale factor we have noted that , the universe will be in the radiation dominated ( or matter dominated ) phase as time @xmath96 in the radiation dominated or matter dominated phase the expansion of the universe is decelerating , hence no horizon . the horizon will develop only when the universe enters the @xmath19 dominated phase . a clear demarcation between the deceleration and acceleration phases during the evolution of the universe can be obtained by calculating the deceleration parameter @xmath97 , which is defined as @xmath98 we have calculated the @xmath95factor using the hubble parameter in equation ( [ eqn : h1 ] ) for the friedmann universe with radiation and @xmath19 and the time evolution of which is shown in figure [ fig : radentro ] along with the time evolution of the radiation entropy for an easy comparison . the universe enters the accelerating phase , corresponding to the time at which the @xmath95factor starts to have negative values and as per the figure that is around a time @xmath99 as per the above analysis the universe enters the accelerating phase at around @xmath100 , and at around this time the event horizon starts developing . at this transition time the horizon was tiny . even at this time the difference in the entropy of the event horizon and radiation was very high . entropy of the radiation given in the equation ( [ eqn : sgamma2 ] ) , gives a value for @xmath101 , @xmath102 . while the entropy of the event horizon , as in equation ( [ eqn : ceh2 ] ) , leads to value of @xmath103 , for the same time . these shows that , even at the formation of the event horizon , the entropy of it is eight orders of magnitude greater than the radiation entropy . so even at the tiny stage of the event horizon the entropy of radiation is not so significant . a comparison of the plots in figure [ fig : radentro ] shows that the entropy of radiation is increasing at first and is start decreasing at the same time when @xmath95factor become negative , as the universe entering the accelerating phase . in the decelerating phase , corresponds to positive values of @xmath95factor the radiation entropy is increasing as there is no horizon for the radiation to escape . since the radiation entropy is increasing during the initial stages , the generalized second law is still valid such that @xmath104 will become the gsl as there in no event horizon . when the universe enters the accelerated expanding phase , where it has event horizon , the radiation entropy is decreasing , because now the radiation is crossing the event horizon . but nevertheless , in the accelerating phase , the horizon entropy is increasing at a faster rate compensated to the decrease in the radiation entropy , which in turn leads to the increase in the total entropy of the universe , guaranteeing the validity of the generalized second law . the time rate of radiation entropy is given in equation ( [ eqn : radentrot ] ) . substituting for @xmath105 for friedmann universe with radiation , the equation can be reduced to , @xmath106 when the radiation entropy is maximum , the time rate is zero , then the above equation leads to the condition , @xmath107 where we have used the friedmann equation to substitute for the @xmath108 from which the corresponding time can be obtained as @xmath109 for the standard parameters , the value of the above time , corresponds to the decreasing of radiation entropy , is around @xmath110 which is in confirmation with the figure [ fig : radentro ] . in the case friedmann universe with matter and a positive cosmological constant also , it is evident form the figure [ fig : totent2 ] , that the entropy of matter too have an increase before the formation of the event horizon . so one can easily see that in the case matter also , the above conclusions are true in general . gibbons and hawking have conjectured that cosmological event horizon of the de sitter universe have entropy like black hole event horizon , and the total entropy of such a universe will never decrease , that is it satisfies the gsl . later davies and others have extended this conjecture to friedmann universe with radiation and dust such that the friedmann universe satisfies the gsl . however their work is mainly based on the numerical computation . in this paper we have presented an analytical analysis of the entropy of the event horizon and fluid within the horizon and also the constraints followed from the validity of the gsl . we have considered two types of friedmann universes . type one is the friedmann universe with radiation and a positive cosmological constant . the other type is the friedmann universe with non - relativistic matter and a positive cosmological constant . we have obtained the expansion scale factor and the hubble parameter for the friedmann universe with radiation ( and matter ) and cosmological constant . the time evolution of the scale factor is plotted and have found that at sufficiently small times the friedmann universe is radiation(or matter ) dominated and is in the decelerating phase . but at large times , the universe become dominated by the cosmological constant , hence in the accelerated expansion and will approach de sitter phase at very large times . during the accelerated expansion phase , the universe has got an event horizon . we have numerically evaluated the time evolution of the comoving distance to the event horizon and verified that the comoving distance is decreasing with time in both types of the universes . as a result the comoving volume of the event horizon decreases , subsequently the radiation ( matter ) can cross the event horizon . this implies that the entropy of the radiation ( or matter ) is decreasing consequent to the escaping of radiation ( or matter ) through the horizon . analogous to the area theorem in black hole , davies proposed a corresponding theorem for the cosmological event horizon which implies a dominant energy condition as given in equation ( [ eqn : condition1 ] ) . in the present case of the friedmann universe , the dominant energy condition implies that , @xmath111 the plot in figure [ fig : cond1 ] conclusively proves this . so once the event horizon is formed it s area will never decrease . so unlike in the case of the black holes , where the area of the event horizon decreases when it is radiating , the area of the cosmological event horizon increases when radiation ( matter ) crosses the horizon . we have obtained the analytical relations for the entropy of the event horizon and radiation ( matter ) for the friedman universe . the entropy of the event horizon is given in equation ( [ eqn : ceh2 ] ) , according to which the present value of the event horizon entropy will be around , @xmath112 we have plotted the time evolution of these entropies and found that the net entropy of the radiation ( or matter ) is decreasing but the entropy of the event horizon is increasing at faster rate as the universe expands . this implies that the total entropy of the friedmann universe , that is the sum of the entropy of the radiation ( or matter ) and event horizon , is increasing . this indicate the validity of the gsl for both types of the friedmann universes . the constraints imposed by the gsl is obtained . for the friedmann universe with radiation and cosmological constant , the gsl constraint the present temperature of the radiation as , @xmath67k in standard units . compared to the latest value of the radiation temperature form cobe , 2.725@xmath1130.002 k @xcite , the above constraint implies the friedmann universe in consideration is very well in the purview of the gsl . the temperature of the horizon , is @xmath114 in the standard units . for the the present case , this temperature is around @xmath115k . the comparison of the above temepratures shows that there is a radiation drain form within the horizon . at this point one should note the result of davies et . al.@xcite that , the temperature of the radiation is higher than that of the horizon . so there is natural flow direction towards the horizon . therefore in the present context , we can conclude that the temperature of the horizon of the friedmann universe with radiation and cosmological constant at present is less than @xmath116 k. for the universe with non - relativistic matter and cosmological constant , the gsl constraint the matter temperature as @xmath117k , which in turn implies that the horizon temperature of the friedmann universe with matter and cosmological constant is less than @xmath116k , so that there is matter flow towards the event horizon . the time evolution of the radiation ( matter ) entropy shows that , it increases first , attains a maximum and then decreases as shown in the figure [ fig : radentro ] . it is seen that the increase in the radiation ( matter ) entropy is during the deceleration phase of the universe , when the radiation ( matter ) is dominating the cosmological constant . it is to be noted that there is no event horizon when the universe is decelerating . so the corresponding increase in the entropy of the radiation is due to the non - existence of the of the event horizon . if the event horizon is absent , there is no crossing of the radiation over the horizon , and it retained within causal region of our universe , which facilitate the small increase in the radiation entropy . we made this point clear by comparing the time evolution of the radiation entropy and @xmath95factor , such that the entropy of the radiation is start decreasing when the @xmath95factor become negative , consequently the expansion is accelerating at which condition the universe posses an event horizon . we have computed the the time corresponding to the maximum of the radiation entropy at which the @xmath95factor is critically become negative , as @xmath118 , and is evident form figure [ fig : radentro ] . bekenstein j d , phys . d * 7 * , 2333 ( 1973 ) . hawking s w _ commun . phys . _ * 43 * 199 ( 1975 ) j m bardeen , b carter and s w hawking , commun . math . phys . * 31 * , 161 ( 1973 ) t padmanabhan , phys . rept . * 49 * , 406 ( 2005 ) t padmanabhan , phys . rept . * 73 * , 046901 ( 2010 ) t jacobson , phys . lett . * 75 * , 1260 ( 1995 ) r g cai , phys . b * 525 * , 331 ( 2002 ) r g cai and s p kim , jhep . textbf0502 , 050 ( 2005 ) gibbons g w and hawking s w , phys . d * 15 * , 2738 ( 1977 ) davies p c w , class . quantum grav . * 4 * , l225 ( 1987 ) t m davies , p c w davies and c h lineweaver , class . quantum grav . * 20 * , 2753 ( 2003 ) davies p c w , class . quantum grav . * 5 * , 1349 ( 1988 ) r amanullah et . j. * 716 * , 712 ( 2010 ) m d pollock , t p singh , class . quantum grav . * 6 * , 901 ( 19 ) r brustein , phys . lett . * 84 * , 2072 ( 2000 ) diego pavan , class . quantum grav . * 7 * , 487 ( 1990 ) r bousso , phys . d * 71 * , 064024 ( 2005 ) b wang , y gong and e abdalla , phys . d * 74 * , 083520 ( 2006 ) k karami and a abdomaleki , jcap * 04 * , 007 ( 2012 ) rahul ghosh , antonio pasqua and surajit chattopadhyay , eur . j. plus * 128 * , 12 ( 2013 ) nairwita mazunder and subenoy chakraborthy , eur . j. c * 70 * , 329 ( 2010 ) ujjal debnath , surajith chattopadhyay , ibrar hussain , mubasher jamil and ratbay myrzakulov , eur . j. c * 72 * , 1875 ( 2012 ) sergio del campo , ramon herrera and diego pavon , phys . b * 707 * , 8 ( 2012 ) a frolov and lev kofman jcap , * 0305 * , 009 ( 2003 ) hawking s w and ellis g f r , _ the large scale structure of space - time _ cambridge university press , ( 1973 ) e komatsu et . ( wmap collaborations ) , astrophys . . ser . * 192 * 18 ( 2011 ) c l bennet et . al . astrophys . * 464 * , l1 ( 1996 )

we discuss the generalized second law ( gsl ) and the constraints imposed by it for two types of friedmann universes . the first one is the friedmann universe with radiation and a positive cosmological constant , and the second one consists of non - relativistic matter and a positive cosmological constant . the time evolution of the event horizon entropy and the entropy of the contents within the horizon are analyses in an analytical way by obtaining the hubble parameter . it is shown that the gsl constraint the temperature of both the radiation and matter of the friedmann universe . it is also shown that , even though the net entropy of the radiation ( or matter ) is decreasing at sufficiently large times as the universe expand , it exhibit an increase during the early times when universe is decelerating . that is the entropy of the radiation within the comoving volume is decreasing only when the universe has got an event horizon . keywords : friedmann universe , entropy , generalised second law . pacs : 04.70.dy,97.60lf,98.80jk,02.60jh,04.20cv

1306.2714

the lisa observatory @xcite has incredible science potential , but that potential can only be fully realized by employing advanced data analysis techniques . lisa will explore the low frequency portion of the gravitational wave spectrum , which is thought to be home to a vast number of sources . since gravitational wave sources typically evolve on timescales that are long compared to the gravitational wave period , individual low frequency sources will be `` on '' for large fractions of the nominal three year lisa mission lifetime . moreover , unlike a traditional telescope , lisa can not be pointed at a particular point on the sky . the upshot is that the lisa data stream will contain the signals from tens of thousands of individual sources , and ways must be found to isolate individual voices from the crowd . this `` cocktail party problem '' is the central issue in lisa data analysis . the types of sources lisa is expected to detect include galactic and extra - galactic compact stellar binaries , super massive black hole binaries , and extreme mass ratio inspirals of compact stars into supermassive black holes ( emris ) . other potential sources include intermediate mass black hole binaries , cosmic strings , and a cosmic gravitational wave background produced by processes in the early universe . in the case of compact stellar binaries @xcite and emris @xcite , the number of sources is likely to be so large that it will be impossible to resolve all the sources individually , so that there will be a residual signal that is variously referred to as a confusion limited background or confusion noise . it is important that this confusion noise be made as small as possible so as not to hinder the detection of other high value targets . several estimates of the confusion noise level have been made @xcite , and they all suggest that unresolved signals will be the dominant source of low frequency noise for lisa . however , these estimates are based on assumptions about the efficacy of the data analysis algorithms that will be used to identify and regress sources from the lisa data stream , and it is unclear at present how reasonable these assumptions might be . indeed , the very notion that one can first clean the data stream of one type of signal before moving on to search for other targets is suspect as the gravitational wave signals from different sources are not orthogonal . for example , when the signal from a supermassive black hole binary sweeps past the signal from a white dwarf binary of period @xmath0 , the two signals will have significant overlap for a time interval equal to the geometric mean of @xmath0 and @xmath1 , where @xmath1 is the time remaining before the black holes merge . thus , by a process dubbed `` the white dwarf transform , '' it is possible to decompose the signal from a supermassive black hole binary into signals from a collection of white dwarf binaries . as described in [ cocktail ] , optimal filtering of the lisa data would require the construction of a filter bank that described the signals from every source that contributes to the data stream . in principle one could construct a vast template bank describing all possible sources and look for the best match with the data . in practice the enormous size of the search space and the presence of unmodeled sources renders this direct approach impractical . possible alternatives to a full template based search include iterative refinement of a source - by - source search , ergodic exploration of the parameter space using markov chain monte carlo ( mcmc ) algorithms , darwinian optimization by genetic algorithms , and global iterative refinement using the maximum entropy method ( mem ) . each approach has its strengths and weakness , and at this stage it is not obvious which approach will prove superior . here we apply the popular markov chain monte carlo @xcite method to simulated lisa data . this is not the first time that mcmc methods have been applied to gravitational wave data analysis , but it is first outing with realistic simulated lisa data . our simulated data streams contain the signals from multiple galactic binaries . previously , mcmc methods have been used to study the extraction of coalescing binary @xcite and spinning neutron star @xcite signals from terrestrial interferometers . more recently , mcmc methods have been applied to a simplified toy problem @xcite that shares some of the features of the lisa cocktail party problem . these studies have shown that mcmc methods hold considerable promise for gravitational wave data analysis , and offer many advantages over the standard template grid searches . for example , the emri data analysis problem @xcite is often cited as the greatest challenge facing lisa science . neglecting the spin of the smaller body yields a 14 dimensional parameter space , which would require @xmath2 templates to explore in a grid based search @xcite . this huge computational cost arises because grid based searches scale geometrically with the parameter space dimension @xmath3 . in contrast , the computational cost of mcmc based searches scale linearly with the @xmath3 . in fields such as finance , mcmc methods are routinely applied to problems with @xmath4 , making the lisa emri problem seem trivial in comparison . a _ google _ search on `` markov chain monte carlo '' returns almost 250,000 results , and a quick scan of these pages demonstrates the wide range of fields where mcmc methods are routinely used . we found it amusing that one of the _ google _ search results is a link to the _ pagerank _ @xcite mcmc algorithm that powers the _ google _ search engine . the structure of the paper follows the development sequence we took to arrive at a fast and robust mcmc algorithm . in [ cocktail ] we outline the lisa data analysis problem and the particular challenges posed by the galactic background . a basic mcmc algorithm is introduced in [ mcmc7 ] and applied to a full 7 parameter search for a single galactic binary . a generalized multi - channel , multi - source f - statistic for reducing the search space from @xmath5 to @xmath6 is described in [ fstat ] . the performance of a basic mcmc algorithm that uses the f - statistic is studied in [ mcmc_f ] and a number of problems with this simple approach are identified . a more advanced mixed mcmc algorithm that incorporates simulated annealing is introduced in [ mcmc_mix ] and is successfully applied to multi - source searches . the issue of model selection is addressed in [ bayes ] , and approximate bayes factor are calculated by super - cooling the markov chains to extract maximum likelihood estimates . we conclude with a discussion of future refinements and extensions of our approach in [ conclude ] . space based detectors such as lisa are able to return several interferometer outputs @xcite . the strains registered in the interferometer in response to a gravitational wave pick up modulations due to the motion of the detector . the orbital motion introduces amplitude , frequency , and phase modulation into the observed gravitational wave signal . the amplitude modulation results from the detector s antenna pattern being swept across the sky , the frequency modulation is due to the doppler shift from the relative motion of the detector and source , and the phase modulation results from the detector s varying response to the two gravitational wave polarizations @xcite . these modulations encode information about the location of the source . the modulations spread a monochromatic signal over a bandwidth @xmath7 , where @xmath8 is the co - latitude of the source and @xmath9 is the modulation frequency . in the low frequency limit , where the wavelengths are large compared to the armlengths of the detector , the interferometer outputs @xmath10 can be combined to simulate the response of two independent 90 degree interferometers , @xmath11 and @xmath12 , rotated by 45 degrees with respect to each other @xcite . this allows lisa to measure both polarizations of the gravitational wave simultaneously . a third combination of signals in the low frequency limit yields the symmetric sagnac variable @xcite , which is insensitive to gravitational waves and can be used to monitor the instrument noise . when the wavelengths of the gravitational waves become comparable to the size of the detector , which for lisa corresponds to frequencies above 10 mhz , the interferometry signals can be combined to give three independent time series with comparable sensitivities @xcite . the output of each lisa data stream can be written as @xmath13 here @xmath14 describes the response registered in detector channel @xmath15 to a source with parameters @xmath16 . the quantity @xmath17 denotes the combined response to a collection of @xmath18 sources with total parameter vector @xmath19 and @xmath20 denotes the instrument noise in channel @xmath15 . extracting the parameters of each individual source from the combined response to all sources defines the lisa cocktail party problem . in practice it will be impossible to resolve all of the millions of signals that contribute to the lisa data streams . for one , there will not be enough bits of information in the entire lisa data archive to describe all @xmath18 sources in the universe with signals that fall within the lisa band . moreover , most sources will produce signals that are well below the instrument noise level , and even after optimal filtering most of these sources will have signal to noise ratios below one . a more reasonable goal might be to provide estimates for the parameters describing each of the @xmath21 sources that have integrated signal to noise ratios ( snr ) above some threshold ( such as @xmath22 ) , where it is now understood that the noise includes the instrument noise , residuals from the regression of bright sources , and the signals from unresolved sources . while the noise will be neither stationary nor gaussian , it is not unreasonable to hope that the departures from gaussianity and stationarity will be mild . it is well know that matched filtering is the optimal linear signal processing technique for signals with stationary gaussian noise @xcite . matched filtering is used extensively in all fields of science , and is a popular data analysis technique in ground based gravitational wave astronomy @xcite . switching to the fourier domain , the signal can be written as @xmath23 , where @xmath24 includes instrument noise and confusion noise , and the signals are described by parameters @xmath25 . using the standard noise weighted inner product for the independent data channels over a finite observation time @xmath0 , @xmath26 a wiener filter statistic can be defined : @xmath27 the noise spectral density @xmath28 is given in terms of the autocorrelation of the noise @xmath29 here and elsewhere angle brackets @xmath30 denote an expectation value . an estimate for the source parameters @xmath25 can be found by maximizing @xmath31 . if the noise is gaussian and a signal is present , @xmath31 will be gaussian distributed with unit variance and mean equal to the integrated signal to noise ratio @xmath32 the optimal filter for the lisa signal ( [ lisa_sig ] ) is a matched template describing all @xmath21 resolvable sources . the number of parameters @xmath33 required to describe a source ranges from 7 for a slowly evolving circular galactic binary to 17 for a massive black hole binary . a reasonable estimate @xcite for @xmath21 is around @xmath34 , so the full parameter space has dimension @xmath35 . since the number of templates required to uniformly cover a parameter space grows exponentially with @xmath3 , a grid based search using the full optimal filter is out of the question . clearly an alternative approach has to be found . moreover , the number of resolvable sources @xmath21 is not known a priori , so some stopping criteria must be found to avoid over - fitting the data . existing approaches to the lisa cocktail party problem employ iterative schemes . the first such approach was dubbed `` gclean '' @xcite due to its similarity with the `` clean '' @xcite algorithm that is used for astronomical image reconstruction . the `` gclean '' procedure identifies and records the brightest source that remains in the data stream , then subtracts a small amount of this source . the procedure is iterated until a prescribed residual is reached , at which time the individual sources are reconstructed from the subtraction record . a much faster iterative approach dubbed `` slice & dice '' @xcite was recently proposed that proceeds by identifying and fully subtracting the brightest source that remains in the data stream . a global least squares re - fit to all the current list of sources is then performed , and the new parameter record is used to produce a regressed data stream for the next iteration . bayes factors are used to provide a stopping criteria . there is always the danger with iterative approaches that the procedure `` gets off on the wrong foot , '' and is unable to find its way back to the optimal solution . this can happen when two signals have a high degree of overlap . a very different approach to the lisa source confusion problem is to solve for all sources simultaneously using ergodic sampling techniques . markov chain monte carlo ( mcmc ) @xcite is a method for estimating the posterior distribution , @xmath36 , that can be used with very large parameter spaces . the method is now in widespread use in many fields , and is starting to be used by astronomers and cosmologists . one of the advantages of mcmc is that it combines detection , parameter estimation , and the calculation of confidence intervals in one procedure , as everything one can ask about a model is contained in @xmath36 . another nice feature of mcmc is that there are implementations that allow the number of parameters in the model to be variable , with built in penalties for using too many parameters in the fit . in an mcmc approach , parameter estimates from wiener matched filtering are replaced by the bayes estimator @xcite @xmath37 which requires knowledge of @xmath36 - the posterior distribution of @xmath38 ( _ i.e. _ the distribution of @xmath38 conditioned on the data @xmath39 ) . by bayes theorem , the posterior distribution is related to the prior distribution @xmath40 and the likelihood @xmath41 by @xmath42 until recently the bayes estimator was little used in practical applications as the integrals appearing in ( [ be ] ) and ( [ post ] ) are often analytically intractable . the traditional solution has been to use approximations to the bayes estimator , such as the maximum likelihood estimator described below , however advances in the markov chain monte carlo technique allow direct numerical estimates to be made . when the noise @xmath43 is a normal process with zero mean , the likelihood is given by @xcite @xmath44\ , , \ ] ] where the normalization constant @xmath45 is independent of @xmath39 . in the large snr limit the bayes estimator can be approximated by finding the dominant mode of the posterior distribution , @xmath36 , which finn @xcite and cutler & flannagan@xcite refer to as a maximum likelihood estimator . other authors @xcite define the maximum likelihood estimator to be the value of @xmath38 that maximizes the likelihood , @xmath41 . the former has the advantage of incorporating prior information , but the disadvantage of not being invariant under parameter space coordinate transformations . the latter definition corresponds to the standard definition used by most statisticians , and while it does not take into account prior information , it is coordinate invariant . the two definitions give the same result for uniform priors , and very similar results in most cases ( the exception being where the priors have a large gradient at maximum likelihood ) . the standard definition of the likelihood yields an estimator that is identical to wiener matched filtering@xcite . absorbing normalization factors by adopting the inverted relative likelihood @xmath46 , we have @xmath47 in the gravitational wave literature the quantity @xmath48 is usually referred to as the log likelihood , despite the inversion and rescaling . note that @xmath49 the maximum likelihood estimator ( mle ) , @xmath50 , is found by solving the coupled set of equations @xmath51 . parameter uncertainties can be estimated from the negative hessian of @xmath52 , which yields the fisher information matrix @xmath53 in the large snr limit the mle can be found by writing @xmath54 and taylor expanding ( [ ml ] ) . setting @xmath55 yields the lowest order solution @xmath56 the expectation value of the maximum of the log likelihood is then @xmath57 this value exceeds that found in ( [ comp ] ) by an amount that depends on the total number of parameters used in the fit , @xmath3 , reflecting the fact that models with more parameters generally give better fits to the data . deciding how many parameters to allow in the fit is an important issue in lisa data analysis as the number of resolvable sources is not known a priori . this issue does not usually arise for ground based gravitational wave detectors as most high frequency gravitational wave sources are transient . the relevant question there is whether or not a gravitational wave signal is present in a section of the data stream , and this question can be dealt with by the neyman - pearson test or other similar tests that use thresholds on the likelihood @xmath58 that are related to the false alarm and false dismissal rates . demanding that @xmath59 - so it is more likely that a signal is present than not - and setting a detection threshold of @xmath60 yields a false alarm probability of 0.006 and a detection probability of 0.994 ( if the noise is stationary and gaussian ) . a simple acceptance threshold of @xmath60 for each individual signal used to fit the lisa data would help restrict the total number of parameters in the fit , however there are better criteria that can be employed . the simplest is related to the neyman - pearson test and compares the likelihoods of models with different numbers of parameters . for nested models this ratio has an approximately chi squared distribution which allows the significance of adding extra parameters to be determined from standard statistical tables . a better approach is to compute the bayes factor , @xmath61 which gives the relative weight of evidence for models @xmath62 and @xmath63 in terms of the ratio of marginal likelihoods @xmath64 here @xmath65 is the likelihood distribution for model @xmath62 and @xmath66 is the prior distribution for model @xmath62 . the difficulty with this approach is that the integral in ( [ marginal ] ) is hard to calculate , though estimates can be made using the laplace approximation or the bayesian information criterion ( bic ) @xcite . the laplace approximation is based on the method of steepest descents , and for uniform priors yields @xmath67 where @xmath68 is the maximum likelihood for the model , @xmath69 is the volume of the model s parameter space , and @xmath70 is the volume of the uncertainty ellipsoid ( estimated using the fisher matrix ) . models with more parameters generally provide a better fit to the data and a higher maximum likelihood , but they get penalized by the @xmath71 term which acts as a built in occam s razor . we begin by implementing a basic mcmc search for galactic binaries that searches over the full @xmath72 dimensional parameter space using the metropolis - hastings @xcite algorithm . the idea is to generate a set of samples , @xmath73 , that correspond to draws from the posterior distribution , @xmath36 . to do this we start at a randomly chosen point @xmath74 and generate a markov chain according to the following algorithm : using a proposal distribution @xmath75 , draw a new point @xmath76 . evaluate the hastings ratio @xmath77 accept the candidate point @xmath76 with probability @xmath78 , otherwise remain at the current state @xmath74 ( metropolis rejection @xcite ) . remarkably , this sampling scheme produces a markov chain with a stationary distribution equal to the posterior distribution of interest , @xmath36 , regardless of the choice of proposal distribution @xcite . a concise introduction to mcmc methods can be found in the review paper by andrieu _ et al _ @xcite . on the other hand , a poor choice of the proposal distribution will result in the algorithm taking a very long time to converge to the stationary distribution ( known as the burn - in time ) . elements of the markov chain produced during the burn - in phase have to be discarded as they do not represent the stationary distribution . when dealing with large parameter spaces the burn - in time can be very long if poor techniques are used . for example , the metropolis sampler , which uses symmetric proposal distributions , explores the parameter space with an efficiency of at most @xmath79 , making it a poor choice for high dimension searches . regardless of the sampling scheme , the mixing of the markov chain can be inhibited by the presence of strongly correlated parameters . correlated parameters can be dealt with by making a local coordinate transformation at @xmath74 to a new set of coordinates that diagonalises the fisher matrix , @xmath80 . we tried a number of proposal distributions and update schemes to search for a single galactic binary . the results were very disappointing . bold proposals that attempted large jumps had a very poor acceptance rate , while timid proposals that attempted small jumps had a good acceptance rate , but they explored the parameter space very slowly , and got stuck at local modes of the posterior . lorentzian proposal distributions fared the best as their heavy tails and concentrated peaks lead to a mixture of bold and timid jumps , but the burn in times were still very long and the subsequent mixing of the chain was torpid . the mcmc literature is full of similar examples of slow exploration of large parameter spaces , and a host of schemes have been suggested to speed up the burn - in . many of the accelerated algorithms use adaptation to tune the proposal distribution . this violates the markov nature of the chain as the updates depend on the history of the chain . more complicated adaptive algorithms have been invented that restore the markov property by using additional metropolis rejection steps . the popular delayed rejection method @xcite and reversible jump method @xcite are examples of adaptive mcmc algorithms . a simpler approach is to use a non - markov scheme during burn - in , such as adaptation or simulated annealing , then transition to a markov scheme after burn - in . since the burn - in portion of the chain is discarded , it does not matter if the mcmc rules are broken ( the burn - in phase is more like las vegas than monte carlo ) . before resorting to complex acceleration schemes we tried a much simpler approach that proved to be very successful . when using the metropolis - hastings algorithm there is no reason to restrict the updates to a single proposal distribution . for example , every update could use a different proposal distribution so long as the choice of distribution is not based on the history of the chain . the proposal distributions to be used at each update can be chosen at random , or they can be applied in a fixed sequence . our experience with single proposal distributions suggested that a scheme that combined a very bold proposal with a very timid proposal would lead to fast burn - in and efficient mixing . for the bold proposal we chose a uniform distribution for each of the source parameters @xmath81 . here @xmath82 is the amplitude , @xmath83 is the gravitational wave frequency , @xmath8 and @xmath84 are the ecliptic co - latitude and longitude , @xmath85 is the polarization angle , @xmath86 is the inclination of the orbital plane , and @xmath87 is the orbital phase at some fiducial time . the amplitudes were restricted to the range @xmath88 $ ] and the frequencies were restricted to lie within the range of the data snippet @xmath89 $ ] mhz ( the data snippet contained 100 frequency bins of width @xmath90 ) . a better choice would have been to use a cosine distribution for the co - latitude @xmath8 and inclination @xmath86 , but the choice is not particularly important . when multiple sources were present each source was updated separately during the bold proposal stage . for the timid proposal we used a normal distribution for each eigendirection of the fisher matrix , @xmath80 . the standard deviation @xmath91 for each eigendirection @xmath92 was set equal to @xmath93 , where @xmath94 is the corresponding eigenvalue of @xmath80 , and @xmath72 is the search dimension . the factor of @xmath95 ensures a healthy acceptance rate as the typical total jump is then @xmath96 . all @xmath18 sources were updated simultaneously during the timid proposal stage . note that the timid proposal distributions are not symmetric since @xmath97 . one set of bold proposals ( one for each source ) was followed by ten timid proposals in a repeating cycle . the ratio of the number of bold to timid proposals impacted the burn - in times and the final mixing rate , but ratios anywhere from 1:1 to 1:100 worked well . we used uniform priors , @xmath98 , for all the parameters , though once again a cosine distribution would have been better for @xmath8 and @xmath86 . two independent lisa data channels were simulated directly in the frequency domain using the method described in ref . @xcite , with the sources chosen at random using the same uniform distributions employed by the bold proposal . the data covers 1 year of observations , and the data snippet contains 100 frequency bins ( of width @xmath99 ) . the instrument noise was assumed to be stationary and gaussian , with position noise spectral density @xmath100 and acceleration noise spectral density @xmath101 . .7 parameter mcmc search for a single galactic binary [ cols="<,^,^,^,^,^,^,^",options="header " , ] [ tab9 ] it is also interesting to compare the output of the 10 source mcmc search to the maximum likelihood one gets by starting at the true source parameters then applying the super cooling procedure ( in other words , cheat by starting in the neighborhood of the true solution ) . we found @xmath102 , and @xmath103 , which tells us that the mcmc solution , while getting two of the source parameters wrong , provides an equally good fit to the data . in other words , there is _ no _ data analysis algorithm that can fully deblend the two highly overlapping sources . our first pass at applying the mcmc method to lisa data analysis has shown the method to have considerable promise . the next step is to push the existing algorithm until it breaks . simulations of the galactic background suggest that bright galactic sources reach a peak density of one source per five @xmath99 frequency bins @xcite . we have shown that our current f - mcmc algorithm can handle a source density of one source per ten frequency bins across a one hundred bin snippet . we have yet to try larger numbers of sources as the current version of the algorithm employs the full @xmath72 dimensional fisher matrix in many of the updates , which leads to a large computational overhead . we are in the process of modifying the algorithm so that sources are first grouped into blocks that have strong overlap . each block is effectively independent of the others . this allows each block to be updated separately , while still taking care of any strongly correlated parameters that might impede mixing of the chain . we have already seen some evidence that high local source densities pose a challenge to the current algorithm . the lesson so far has been that adding new , specially tailored proposal distributions to the mix helps to keep the chain from sticking at secondary modes of the posterior ( it takes a cocktail to solve the cocktail party problem ) . on the other hand , we have also seen evidence of strong multi - modality whereby the secondary modes have likelihoods within a few percent of the global maximum . in those cases the chain tends to jump back and forth between modes before being forced into a decision by the super - cooling process that follows the main mcmc run . indeed , we may already be pushing the limits of what is possible using any data analysis method . for example , the 10 source search used a model with 70 parameters to fit 400 pieces of data ( 2 channels @xmath104 2 fourier components @xmath104 100 bins ) . one of our goals is to better understand the theoretical limits of what can be achieved so that we know when to stop trying to improve the algorithm ! it would be interesting to compare the performance of the different methods that have been proposed to solve the lisa cocktail party problem . do iterative methods like gclean and slice & dice or global maximization methods like maximum entropy have different strengths and weakness compared to mcmc methods , or do they all fail in the same way as they approach the confusion limit ? it may well be that methods that perform better with idealized , stationary , gaussian instrument noise will not prove to be the best when faced with real instrumental noise . p. bender _ _ , _ lisa pre - phase a report _ , ( 1998 ) . c. r. evans , i. iben & l. smarr , apj * 323 * , 129 ( 1987 ) . v. m. lipunov , k. a. postnov & m. e. prokhorov , a&a * 176 * , l1 ( 1987 ) . d. hils , p. l. bender & r. f. webbink , apj * 360 * , 75 ( 1990 ) . d. hils & p. l. bender , apj * 537 * , 334 ( 2000 ) . g. nelemans , l. r. yungelson & s. f. portegies zwart , a&a * 375 * , 890 ( 2001 ) . l. barack & c. cutler , phys . rev . d*69 * , 082005 ( 2004 ) . j. r. gair , l. barack , t. creighton , c. cutler , s. l. larson , e. s. phinney & m. vallisneri , class . * 21 * , s1595 ( 2004 ) . a. j. farmer & e. s. phinney , mon . not . . soc . * 346 * , 1197 ( 2003 ) . s. timpano , l. j. rubbo & n. j. cornish , gr - qc/0504071 ( 2005 ) . l. barack & c. cutler , phys . rev . d*70 * , 122002 ( 2004 ) . n. christensen & r. meyer , phys . rev . d*58 * , 082001 ( 1998 ) ; n. christensen & r. meyer , phys . rev . d*64 * , 022001 ( 2001 ) ; n. christensen , r. meyer & a. libson , class . * 21 * , 317 ( 2004 ) . n. christensen , r. j. dupuis , g. woan & r. meyer , phys . rev . d*70 * , 022001 ( 2004 ) ; r. umstatter , r. meyer , r. j. dupuis , j. veitch , g. woan & n. christensen , gr - qc/0404025 ( 2004 ) . r. umstatter , n. christensen , m. hendry , r. meyer , v. simha , j. veitch , s. viegland & g. woan , gr - qc/0503121 ( 2005 ) . l. page , s. brin , r. motwani & t. winograd , stanford digital libraries working paper ( 1998 ) . m. tinto , j. w. armstrong & f. b. estabrook , phys . rev . d*63 * , 021101(r ) ( 2001 ) . c. cutler , phys . d * 57 * , 7089 ( 1998 ) . n. j. cornish & l. j. rubbo , phys . rev . d*67 * , 022001 ( 2003 ) . t. a. prince , m. tinto , s. l. larson & j. w. armstrong , phys . d*66 * , 122002 ( 2002 ) . c. w. helstrom , _ statistical theory of signal detection _ , 2nd edition ( pergamon press , london , 1968 ) . wainstein and v.d . zubakov , _ extraction of signals from noise _ ( prentice - hall , englewood cliffs , 1962 ) . thorne , in _ 300 years of gravitation _ , edited by s.w . hawking and w. israel ( cambridge university press , cambridge , england , 1987 ) , p. 330 . schutz , in _ the detection of gravitational waves _ , edited by d.g . blair ( cambridge university press , cambridge , england , 1991 ) , p. 406 . sathyaprakash and s.v . dhurandhar , phys . d * 44 * , 3819 ( 1991 ) . s.v . dhurandhar and b.s . sathyaprakash , phys . d * 49 * , 1707 ( 1994 ) . c. cutler and . . flanagan , phys . d * 49 * , 2658 ( 1994 ) . r. balasubramanian and s.v . dhurandhar , phys . d * 50 * , 6080 ( 1994 ) . sathyaprakash , phys . rev . d * 50 * , 7111 ( 1994 ) . apostolatos , phys . d * 52 * , 605 ( 1996 ) . e. poisson and c.m . will , phys . d * 52 * , 848 ( 1995 ) . r. balasubramanian , b.s . sathyaprakash , and s.v . dhurandhar , phys . d * 53 * , 3033 ( 1996 ) . owen , phys . d * 53 * , 6749 ( 1996 ) . b.j . owen and b.s . sathyaprakash , phys . d * 60 * , 022002 ( 1999 ) . l. s. finn , phys . d * 46 * 5236 ( 1992 ) . p. jaranowski & a. krolak , phys . rev . d*49 * , 1723 ( 1994 ) . p. jaranowski , a. krolak & b. f. schutz , phys . rev . d*58 * 063001 ( 1998 ) . m. h. a. davis , in _ gravitational wave data analysis _ , edited by b. f. schutz , ( kluwer academic , boston , 1989 ) . f. echeverria , phys . d * 40 * , 3194 ( 1989 ) . n.j . cornish & s.l . larson , phys . d*67 * , 103001 ( 2003 ) . j. hgbom , astr . * 15 * , 417 ( 1974 ) . cornish , _ talk given at gr17 , dublin , july ( 2004 ) _ ; n.j . cornish , l.j . rubbo & r. hellings , _ in preparation _ ( 2005 ) . g. schwarz , ann . stats . * 5 * , 461 ( 1978 ) . , w. r. gilks , s. richardson & d. j. spiegelhalter , ( chapman & hall , london , 1996 ) . d. gamerman , _ markov chain monte carlo : stochastic simulation of bayesian inference _ , ( chapman & hall , london , 1997 ) . c. andrieu , n. de freitas , a. doucet & m. jordan , machine learning * 50 * , 5 ( 2003 ) .

the laser interferometer space antenna ( lisa ) is expected to simultaneously detect many thousands of low frequency gravitational wave signals . this presents a data analysis challenge that is very different to the one encountered in ground based gravitational wave astronomy . lisa data analysis requires the identification of individual signals from a data stream containing an unknown number of overlapping signals . because of the signal overlaps , a global fit to all the signals has to be performed in order to avoid biasing the solution . however , performing such a global fit requires the exploration of an enormous parameter space with a dimension upwards of 50,000 . markov chain monte carlo ( mcmc ) methods offer a very promising solution to the lisa data analysis problem . mcmc algorithms are able to efficiently explore large parameter spaces , simultaneously providing parameter estimates , error analysis , and even model selection . here we present the first application of mcmc methods to simulated lisa data and demonstrate the great potential of the mcmc approach . our implementation uses a generalized f - statistic to evaluate the likelihoods , and simulated annealing to speed convergence of the markov chains . as a final step we super - cool the chains to extract maximum likelihood estimates , and estimates of the bayes factors for competing models . we find that the mcmc approach is able to correctly identify the number of signals present , extract the source parameters , and return error estimates consistent with fisher information matrix predictions .

gr-qc0506059

type ia supernova ( sn ia ) is thought to originate from an accreting carbon - oxygen white dwarf ( wd ) in a binary system @xcite . however , it is uncertain what is the companion star supplying its mass . there are two scenarios to explain sne ia . one is the single degenerate ( sd ) scenario , in which the companion star is a main sequence or a red giant star @xcite . the envelope of the companion star overflows the roche lobe , and transfers to the surface of the wd . the other is the double degenerate ( dd ) scenario , in which two wds merge and end up with explosion @xcite . to distinguish the two scenarios , we need to find observable differences between them . in the sd scenario , sn ejecta collide with its companion . @xcite simulated this collision and found that high energy photons are emitted from the shock - heated region into a certain limited solid angle and that the emission becomes prominent before the peak of the light curve due to @xmath1 decay . this emission is strong especially in a wd + red giant system . @xcite investigated early phases of observed light curves gathered by the sloan digital sky survey ( sdss ) to see whether they have this prompt emission . by comparing its early light curve with the observations , they found that there is no obvious emission feature in the light curves , which constrained the event rate of each model . as a result , a progenitor system with a main sequence companion more massive than @xmath2 or a red giant is ruled out as the primary source of sne ia . @xcite assumed the lte condition and further that whole matter is composed of radiation dominated gases with @xmath3 . we suspect that they overestimated the radiation temperature of the emission regions . in fact , based on their detailed calculations of spectra from the same hydrodynamical models presented here , @xcite argued that hydrogen rich matter filling in the hole excavated by the companion star prevents the photosphere from quickly receding to the ni - rich region and makes the sn redder than expected by the previous study @xcite . in addition , @xcite focused only on the impacts on light curves . we investigate also the influence on spectra . if the material stripped from the companion does not spread in a wide solid angle , hydrogen lines from the companion might appear in the spectrum , depending on the line of sight . @xcite investigated nebula spectra of some sne ia and found the upper limit of @xmath4 for solar abundance material . several authors @xcite calculated the mass of the material stripped from the companion star by the collision . @xcite performed hydrodynamical simulations to estimate the stripped mass . according to their results , the stripped masses are @xmath5 for the @xmath6 main sequence companion , @xmath7 for the @xmath6 red giant companion . on the other hand , there is no spectral feature of the stripped material in observed spectra . recently , @xcite adopted a more realistic profile for a main sequence companion from a binary evolution theory and found that the stripped mass decreases to @xmath8 . @xcite investigated signatures of hydrogen lines in the phase later than the maximum light when the shock heating due to the collision has been already consumed by adiabatic expansion . in this study , we relax the assumption of the instantaneous coupling between gas and radiation to examine whether or not the decoupling of gas and radiation changes the influence on light curves found by @xcite . in addition , we investigate whether hydrogen in the stripped material leaves its trace in spectra especially before maximum light . in later phases , @xcite already studied spectra for the same model presented here . at first , we simulated the collision of a sn ia with its companion with a radiation hydrodynamical code . secondly , we calculated spectra by ray - tracing snapshots of the distributions of density , temperatures , and velocity obtained from the simulation . section 2 describes our models . section 3 and 4 present the numerical methods for the simulation and ray - tracing . in section 5 , we describe our results . section 6 concludes this study . as progenitor systems , we consider three models named ms , rga , and rgb ( table.[model.tbl ] ) . model ms is a close binary system with a main sequence companion . we use a polytropic star with the index @xmath9 to model the structure of the companion star . we assume the companion star composed of a solar abundance material . its mass @xmath10 is equal to @xmath11 . the binary separation @xmath12 is equal to @xmath13 . this model is the same as the @xmath11 main sequence model in @xcite . model rga is a close binary system with a red giant companion . in this model , the companion star consists of the helium core and the hydrogen - rich envelope with the solar abundance . its core mass and total mass @xmath10 are equal to @xmath14 and @xmath6 , respectively . the binary separation @xmath12 is equal to @xmath15 . we assume a fully convective envelope with @xmath16 . in the hydrodynamical simulation , the core is treated as a point source of gravity . this model is the same as the @xmath11 red giant model in @xcite . model rgb is the same as model rga but with a longer separation , i.e. , @xmath17 . the radii @xmath18 of the companion stars in the above models are obtained by an empirical law for close binary systems assuming that the companion stars fill the roche lobe . @xmath19 here , @xmath20 is the mass of the progenitor wd , which is equal to @xmath21 . thus @xmath22 . .models in our calculation . [ cols="<,^,^,^",options="header " , ] when neutral hydrogen is located outside the photosphere where the temperature is lower than the photospheric temperature , h@xmath0 line might appear as absorption in the spectrum . results from the simulation and ray - tracing explained in the previous sections are presented in the following subsections . from the above simulation , we obtain spatial structures of the ejecta and the stripped material . the stripped mass from the companion star is calculated by adding up the mass of hydrogen rich gas with the fluid velocity exceeding the escape velocity . the masses are @xmath23 in model ms , @xmath24 in model rga , and @xmath25 in model rgb . in model ms , the amount is overestimated , compared to the previous studies ( about @xmath8 in @xcite ) . this is because we do not have enough resolution in the low velocity region around the companion star due to the remapping process . on the other hand , in models rga and rgb , the amount is slightly smaller than the previous studies @xcite . nevertheless , the trend that the entire envelope is stripped by the impact is consistent with these studies . figures [ pict2d-ms.eps]-[pict2d-rgb.eps ] show distributions of density @xmath26 and total number abundance @xmath27 at days 1 and 20 after the explosion in each model . as shown in the figures , the hydrogen - rich material is stripped along the line @xmath28 . at day 20 , the ejecta and the stripped material are already expanding homologously . a cavity is created around the axis @xmath28 with the spread angle of @xmath29 . if we view this sn with a viewing angle in this range , the stripped hydrogen is not hidden by the surrounding ejecta . figure [ pict1d.eps ] shows one - dimensional structures of the density @xmath26 , the gas temperature @xmath30 , and the radiation temperature @xmath31 along the line @xmath28 at day 1 . we calculate the position of the photosphere where thermalization length is equal to unity , which is indicated by the dotted vertical line . the photospheric temperature is equal to @xmath32 for model ms , @xmath33 for model rga , and @xmath34 for model rgb . @xcite analytically estimated the effective temperature of the prompt emission as @xmath35 where @xmath36 , @xmath37 is the electron scattering opacity , and @xmath38 is the time in days . at @xmath39 , @xmath40 is estimated as @xmath41 for model ms , @xmath42 for model rga , and @xmath43 for model rgb , in comparison , the temperatures become lower in our simulation . this is because the collision heats the gas first in our simulations , which separately treat the temporal evolutions of the radiation and gas energy densities . the subsequent emission from the gas is not enough to raise the photospheric temperature as high as this formula due to the low densities . bolometric light curves with viewing angles @xmath44 , @xmath45 , @xmath46 , @xmath47 and @xmath48 are shown in figure [ lcurve.eps ] . the collision radiates prompt emission especially in the range of @xmath49 . @xcite analytically estimated bolometric luminosities due to the collision at day 1 to be @xmath50 for the @xmath6 main sequence model , @xmath51 for the @xmath2 main sequence model , and @xmath52 for the @xmath6 red giant model . in comparison with our results , the luminosity is equal to @xmath53 for model ms , @xmath54 for model rga , and @xmath55 for model rgb . the luminosities in models ms , rga are dimmer than the previous study . this is because the temperature of emission region becomes lower than their results . in the analysis of @xcite , the luminosity in the @xmath2 main sequence model is the threshold above which a progenitor system is ruled out from the primary source of sne ia . the luminosity in model rga is on this threshold . a model with a slightly longer separation as model rgb is to be ruled out because the emission can yet be detected . here one should note that our models may underestimate the strength of the coupling between the gas and radiation because the free - free emission and the energy exchange by compton scattering , which our models do not include , accelerate the equilibration as pointed out by @xcite . as a result , we may significantly underestimate the luminosity due to the shock heating . more elaborate numerical studies including the comptonization are needed to assess this effect . in figure [ rvcurve.eps ] , we compare our results of model light curves in the r and v bands viewing from the side of the companion star ( @xmath56 ) with two well observed type ia sne 2011fe and 2014j . though the early light curve of sn 2014j is of unfiltered observations or of approximate @xmath57 magnitudes estimated from narrowband detections using h@xmath58 filters@xcite , the shape indicates some influence from the collision between the sn ejecta and the companion star . therefore sn 2014j might have a red giant as the ex - companion star and prefers the sd scenario . note that we do not intend to fit the observed light curve of sn 2014j with our current models . a detailed comparison will be discussed elsewhere . we calculate spectra with several viewing angles @xmath59 every 5 days up to 40 days since explosion . figures [ spectra-ms-2.eps]-[spectra-rgb-2.eps ] show spectrum of each model . here the monochromatic luminosity @xmath60 is normalized by the continuum luminosity @xmath61 . in the early spectra , strong absorption features come from c ii line ( @xmath62 ) and si ii line ( @xmath63 ) . on the other hand , h@xmath0 feature is quite weak or hidden by the c ii absorption , because w7 model retains a large amount of carbon in the outer ejecta . in reality , c ii absorption features are weak and rarely detected in observed sn ia spectra @xcite . we overestimate their absorption feature and therefore there is a possibility to detect h@xmath0 feature in some sne ia depending on the carbon content and the viewing angle . especially , model ms has a strong h@xmath0 feature on its spectra before day 10 and after day 30 . as was mentioned before , the overestimate of the stripped mass in this model enhances the feature as compared with reality . in models rga and rgb , h@xmath0 absorption becomes prominent only after day 30 . on the other hand , c ii feature already becomes weak in this epoch and does not disturb the h@xmath0 feature . the s / n ratio needed for the 3@xmath64 detection of these features is written as @xmath65 at day 35 , the s / n ratio to detect the most strong feature in the spectrum is calculated as 59 for model rga , and 125 for model rgb . figure [ ew.eps ] shows the time evolution of the equivalent width of h@xmath0 line . the peak value of the equivalent width with the viewing angle @xmath28 is 6 for model ms , 5 for model rga , and 4 for model rgb . with @xmath66 , the peak value decreases by a factor of about two compared with the above value in each model . thus , in models rga and rgb , the h@xmath0 feature can be detected when the viewing angle is in the range of @xmath67 , or within the solid angle of @xmath68 . the event ratio in each model is estimated as @xmath69 . however , this h@xmath0 feature has never been observed in sn ia spectra . thus , these systems are ruled out from the progenitor candidates . here , it should be noted that these calculations assume that the ionization states are in thermal equilibrium and that we need to know the continuum spectra with sufficient accuracies in advance . @xcite already discussed hydrogen lines in this late phase in more detail including the paschen lines . as another spectral feature to be investigated , figure [ vel-si.eps ] shows time evolution of silicon velocity with different viewing angles . the differences of velocities between @xmath44 and @xmath48 are about @xmath70 for all models . a spectral resolution of about 20 could distinguish the differences . this variation occurs because the edge of the ejecta is decelerated and masked by the companion star . we usually observe a sn from a single direction and it is difficult to distinguish this effect from the variety of expansion speeds of individual sne ia . one possibility to extract this effect is a sn ia for which a several light echoes are discovered . they will have different silicon velocities depending on the viewing angles , and can be compared with our results . @xcite obtained a light echo spectrum of snr tycho with the spectral resolution of 24 , observed by the focas camera on the subaru telescope . the resolution we require is higher . furthermore , the light echo spectrum is constructed by a mixture of sn spectra for about 10 days . thus , it is not feasible to detect the variation at present . these spectral features confirmed the results obtained by more sophisticated treatment of radiative transfer in the same models @xcite except that the temporal evolution of silicon line . @xcite obtained the silicon velocities independent of viewing angles at day @xmath71 while we always see lower velocities when viewing from the companion side . this difference might be partly due to different treatments of the shock heating . @xcite redetermined the temperature by imposing the radiative equilibrium condition ignoring the shock heating . higher temperature due to the shock heating reduces the number of si ii ions and lowers the velocities at the absorption minima . in later phases , the adiabatic cooling diminishes contributions of the shock heating and the results of the two different simulations converge . in the sd scenario of sne ia , the collision between the ejecta and its companion occurs . due to this collision , their light curve and spectra are changed from the case of sn ia without the companion . these characteristics are expected to constrain progenitor systems . in this study , we perform radiation hydrodynamical simulations of this collision for three binary systems in the sd scenario . using the obtained snapshots , we solve the radiation transport equation , and obtain their light curves and spectra with several viewing angles . then , we determine whether they can constrain the event rate of the sd progenitor . the results are as follows . in comparison to @xcite , the prompt emission due to the collision becomes weaker , since the photospheric temperature becomes lower by the weaker coupling of gas and radiation . in models ms and rga , the luminosity does not exceed that of the @xmath2 main sequence model in @xcite , above which the emission can be detected in observed sne ia . on the contrary , in model rgb , the luminosity still exceeds the threshold . thus , we conclude that binary systems with the separation @xmath72 is allowed as a progenitor of sne ia . as exemplified by sn 2014j , photometric observations during the first few days for a large sample of sne are crucial for distinguishing the two scenarios . in all models , the equivalent widths of h@xmath0 absorption become a few 30 days after the explosion . however , in model ms , the stripped mass is overestimated and the absorption will become weaker in reality . on the other hand , in models rga and rgb , the h@xmath0 feature can be detected if the viewing angle @xmath67 . then , 2% of sne ia in these models should have the feature on their spectra , while there is no sign of h@xmath0 absorption in the observed spectra . if a sufficient number of spectra were obtained with good s / n ratios and were not showed this feature , a model with a red giant companion filling the roche lobe could be ruled out from sne ia progenitors . the central wavelength of si ii absorption varies depending on viewing angles . if more than one light echoes are discovered in a single sn , we obtain its spectra with different viewing angles . since the difference of silicon velocities depending on the viewing angles is about @xmath70 , the required resolution is about 20 . the current observation has not reached this resolution for light echo spectra . we need higher resolution which will be reached by future observations . as shown above , our models indicate that the prompt emission can not constrain binary systems with a short separation . one should note here that our models may significantly underestimate the luminosity due to the shock heating because we do not take into account the coupling of radiation and gas through compton scattering . thus we need to address the effects of compton scattering before reaching a conclusion on this issue . at the same time , we also found that the spectral feature might constrain the occurrence of thermalization through compton scattering . furthermore , h@xmath0 feature brings us additional information of the progenitor of an individual sn ia . because we can not predict where and when a sn will occur , the prompt emission is difficult to be found . it will be detected only when a monitoring telescope is occasionally viewing around its site . in contrast , some hydrogen features become prominent after the maximum light , which relaxes the required observing cadence . if we continuously observe the sn and take spectra with a right viewing angle , we will never miss it . this is a great advantage .

in the single degenerate ( sd ) scenario of type ia supernovae ( sne ia ) , the collision of the ejecta with its companion results in stripping hydrogen rich matter from the companion star . this hydrogen rich matter might leave its trace in the light curves and/or spectra . in this paper , we perform radiation hydrodynamical simulations of this collision for three binary systems . as a result , we find that the emission from the shock - heated region is not as strong as in the previous study . this weak emission , however , may be a result of our underestimate of the coupling between the gas and radiation in the shock interaction . therefore , though our results suggest that the observed early light curves of sne ia can not rule out binary systems with a short separation as the progenitor system , more elaborate numerical studies will be needed to reach a fair conclusion . alternatively , our results indicate that the feature observed in the early phase of a recent type ia sn 2014j might result from interaction of the ejecta with a red giant companion star . we also discuss the dependence of spectral features of h@xmath0 and si ii absorption lines on viewing angles and investigate whether they can constrain the event rate of the sd progenitor .

1504.01234

properties of heavy - flavored hadrons such as masses and decay widths can , in principle , be described in the theoretical framework of quantum chromodynamics ( qcd ) . however , they are difficult to calculate in practice with the perturbative qcd technique due to the fact that the strong coupling constant @xmath11 is large in this low energy regime . to overcome this difficulty , other methods such as lattice qcd @xcite , heavy quark effective theory @xcite , quark model @xcite , qcd sum rule @xcite , and bag model @xcite are deployed . the properties of the @xmath12 baryons have been measured by many experiments @xcite , but the total uncertainties of the world averages remain large @xcite . for example , the relative uncertainties of the decay widths are around 10% of their central values . furthermore , the relative uncertainty of the mass splitting @xmath13 is about 40% , and there is no significant measurement for the mass splitting @xmath14 @xcite . due to the mass hierarchy between the @xmath15 and @xmath16 quarks , one may expect that the @xmath17 @xmath18 baryon is heavier than the @xmath19 @xmath20 baryon ; however , many experimental results contradict this naive expectation @xcite . to explain the discrepancy , various models have been introduced @xcite that predict positive mass splittings . precise measurements of the mass splittings are necessary to test these models . in this paper , we present precise measurements of the masses and decay widths of the @xmath0 and @xmath1 baryons , and of their mass splittings . throughout this paper , the charge - conjugate decay modes are implied . this study uses a data sample corresponding to an integrated luminosity of 711 fb@xmath2 collected with the belle detector at the kekb @xmath3 asymmetric - energy collider @xcite operating at the @xmath4 resonance . the belle detector is a large solid angle magnetic spectrometer that consists of a silicon vertex detector ( svd ) , a 50-layer central drift chamber ( cdc ) , an array of aerogel threshold cherenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter comprising csi(tl ) crystals located inside a superconducting solenoid coil that provides a 1.5 t magnetic field . an iron flux return located outside the coil is instrumented to detect @xmath21 mesons and to identify muons . a detailed description of the belle detector can be found in ref . @xcite . the @xmath12 baryons are reconstructed via their @xmath22 decays , where @xmath23 is a low - momentum ( `` slow '' ) pion . charged tracks are required to have an impact parameter with respect to the interaction point of less than 3 cm along the beam direction ( the @xmath24 axis ) and less than 1 cm in the plane transverse to the beam direction . in addition , each track is required to have at least two associated vertex detector hits each in the @xmath24 and azimuthal strips of the svd . the particles are identified using likelihood @xcite criteria that have efficiencies of 84% , 91% , 93% , and 99% for @xmath25 , @xmath26 , @xmath27 , and @xmath23 , respectively . @xmath5 candidates are reconstructed as combinations of @xmath25 , @xmath28 , and @xmath29 candidates with an invariant mass between 2278.07 and 2295.27 mev/@xmath9 , corresponding to @xmath30 around the nominal @xmath5 mass , where @xmath31 represents the @xmath5 invariant mass resolution . @xmath5 daughter tracks are refit assuming they originate from a common vertex . the @xmath5 production vertex is defined by the intersection of its trajectory with the @xmath3 interaction region . @xmath5 candidates are combined with @xmath23 candidates to form @xmath12 candidates . @xmath23 candidates are required to originate from the @xmath5 production vertex in order to improve their momentum resolution , which results in an enhanced signal - to - background ratio . signal candidates retained for further analysis are required to have a confidence level greater than 0.1% for the @xmath23 vertex fit constrained to the @xmath5 production vertex . to suppress combinatorial backgrounds , we also require the momentum of @xmath12 baryons in the center - of - mass frame to be greater than 2.0 gev/@xmath32 . the distributions of the mass difference @xmath33 for all reconstructed @xmath12 candidates are shown in fig . [ fig : feeddown ] . we also use a monte carlo ( mc ) simulation sample for various purposes in this study , where events are generated with pythia @xcite , decays of unstable particles are modeled with evtgen @xcite , and the detector response is simulated with geant3 @xcite . the sample of selected @xmath12 candidates includes two types of backgrounds : partially reconstructed decays of excited @xmath5 baryons ( referred to as `` feed - down backgrounds '' ) and random combinations of the final state particles . the procedures used to parameterize these backgrounds are described in this section . from the tracks of a @xmath35 decay , a @xmath36 candidate can be reconstructed if one of the slow pions is left out . this can be either a signal ( from a @xmath12 resonant decay of an excited @xmath5 state ) or a feed - down background event . the feed - down backgrounds from the @xmath37 and @xmath38 states appear in the @xmath0 mass region . in order to remove these backgrounds , we tag events that have a mass difference @xmath39 ( @xmath40 being a charged track ) that falls either in the [ 302 , 312 ] mev/@xmath9 or the [ 336 , 347 ] mev/@xmath9 mass interval , corresponding to the @xmath37 and @xmath38 signals , respectively ( see fig . [ fig : excitedlambdac ] ) . the tagged events are subtracted from the @xmath41 distributions as shown in fig . [ fig : feeddown ] . to prevent a possible bias in the subtraction , we estimate the backgrounds under the @xmath42 peaks from mc simulations and subtract them from the tagged feed - down backgrounds . furthermore , we take into account the charged track detection efficiency of 74% on average to correct for the feed - down backgrounds . since the shape of the feed - down backgrounds depends on the @xmath23 momentum , we obtain and apply the efficiency correction as a function of this quantity . mass difference of @xmath43 . signal regions of the @xmath37 ( filled ) and @xmath38 ( hatched ) are defined in the text . ] the remaining background consists of random combinations , with or without a true @xmath5 baryon . in the latter case , the background level is estimated from the @xmath5 mass sidebands , defined as @xmath44 @xmath45 [ 2259.16 , 2267.76 ] mev/@xmath9 or @xmath44 @xmath45 [ 2305.58 , 2314.18 ] mev/@xmath9 . the treatment of the random backgrounds in the fit is discussed in sec . [ sec : fitprocedure ] . the parameters of the @xmath0 and @xmath1 signals , namely the decay widths and the mass differences with respect to the @xmath5 mass , are determined by performing binned maximum likelihood fits . due to the small fraction of the weighted events in the region where the feed - down background is subtracted , a correction to the covariance matrix of the fit parameters is applied to obtain the proper errors the @xmath0 and @xmath1 baryons are described by a relativistic breit - wigner probability density function ( pdf ) convolved with the detector response function as @xmath46 where @xmath47 is a relativistic breit - wigner with the nominal mass difference @xmath48 and the decay width @xmath49 as fit parameters , and @xmath50 is the detector response function . the resolution function @xmath50 is parameterized as the sum of three gaussian functions centered at zero . the parameters are obtained from an mc simulation separately for the @xmath51 and @xmath52 signals . the detector resolutions for the @xmath51 and @xmath52 baryons are found to be @xmath53 and @xmath54 mev/@xmath9 , respectively , from the weighted variances of the three gaussian distributions where the errors are statistical . the random backgrounds without true @xmath5 baryons are modeled as histogram pdfs with shape and normalization taken from the @xmath5 baryon data sidebands . the random backgrounds with true @xmath5 baryons are described with a threshold function : @xmath55 where @xmath56 , @xmath57 are fit parameters and @xmath58 is the known charged pion mass @xcite . in the neutral channel , we find a small peak near @xmath59 mev/@xmath9 . based on studies performed using mc and data samples , we confirm the origin of this peak to be the as - of - yet unobserved decay of @xmath60 . we describe this peak with a gaussian function . the mean and width of the gaussian from the fit are found to be @xmath61 and @xmath62 mev/@xmath9 , respectively ; the former is consistent with that from the world average ( @xmath63 mev/@xmath9 ) @xcite and the latter is consistent with that from mc . the fit results to @xmath41 are shown in fig . [ fig : globalfit ] . the goodness - of - fit values are @xmath64 with 347 degrees of freedom for @xmath17 and @xmath65 with 350 degrees of freedom for @xmath19 . to estimate systematic uncertainties , three sources are studied : momentum scale , resolution and fit model , and background parameterization . these are summarized in table [ table : systematics ] . mass measurements are sensitive to the momentum scale of the detector . because there is a possible bias in the measurements of the charged track momenta , which may be due to the energy loss of the charged particles in materials , one should consider the precision of the momentum calibration . to minimize the possible bias , we calibrate the momentum scale using the copious @xmath66 sample . charged tracks are iteratively calibrated as functions of the curvature , polar angle , and momentum of each track in the laboratory frame by comparing the reconstructed and world average @xcite masses of @xmath67 meson as a function of the @xmath67 momentum . the obtained corrections are applied to the data sets used in this study . to estimate the accuracy , we choose a control sample of @xmath68 decay , and compare the mass difference of @xmath69 over the @xmath23 momentum bins with the world average @xcite as shown in fig . [ fig : systresolution ] . we observe the largest difference to be 0.02 mev/@xmath9 , which we assign as the systematic uncertainty on the mass difference measurements due to the momentum calibration . mass difference @xmath69 obtained from mc ( red triangle ) and data ( black circle ) using the @xmath68 decay as a function of the @xmath23 momentum . the uncertainties of each points are too small to be displayed . the world average with its total uncertainty @xcite is also shown as a hatched area . ] since our detector resolution model is evaluated from the mc as discussed in sec . [ sec : fitprocedure ] , the discrepancy between the mc and data is considered as a source of systematic uncertainty . to estimate the discrepancy , we compare the detector resolution in data and mc using the same control sample of @xmath68 decay . since the decay width of the @xmath70 meson is small , one can assume that the distribution of the mass difference @xmath69 is dominated by the detector resolution . we vary the widths of the detector response functions from + 1.7% to + 11.8% in the fits to @xmath41 by choosing the largest and smallest differences between the mc and data obtained by comparing @xmath69 as a function of the @xmath23 momentum . the uncertainties are found to be 0.19 , 0.25 , and 0.24 mev/@xmath9 for the widths of the @xmath0 , @xmath71 , and @xmath72 baryons , respectively . we also vary the detector response functions by @xmath73 deviation from the fitted resolution parameters , where @xmath31 is the statistical error , and only small uncertainties are found for the decay widths of 0.01 and 0.04 mev/@xmath9 for the @xmath0 and @xmath1 baryons , respectively . we also check the internal consistency of the fitting procedure . in order to probe any bias from the fitter , we perform 10,000 pseudo - experiments for each of the mass differences , @xmath74 and @xmath75 , and the decay widths @xmath76 and @xmath77 . in the production of the pseudo - experiments , we set the input values to be those obtained from the data . from the study , we find negligible discrepancies . the effect of binning is studied by varying the bin size in the fits to @xmath41 from 0.1 mev/@xmath9 to 1.0 mev/@xmath9 . the uncertainties of @xmath78 are negligible , and we find small uncertainties for the widths of 0.09 , 0.06 , 0.04 , and 0.05 mev/@xmath9 for the @xmath79 , @xmath80 , @xmath71 , and the @xmath72 baryons , respectively . we also test the effect of various fit ranges . we choose several fit ranges , some of which include both the @xmath0 and @xmath1 signals and others only one of them . though the results from the various fit ranges are consistent within the statistical fluctuations , we conservatively assign the variations in the fit results , 0.03 and 0.01 mev/@xmath9 for @xmath81 and @xmath82 , respectively , and 0.19 and 0.17 mev/@xmath9 for the widths of the @xmath71 and @xmath72 baryons , respectively , as systematic uncertainties . [ cols="^,^,^,^,^,^,^,^,^ " , ] since we correct the feed - down backgrounds by taking into account the efficiency as discussed in sec . [ sec : backgrounds ] , the uncertainty of the efficiency should also be taken into account . the systematic uncertainty from the feed - down model is estimated as 1.87% from the error propagation of the statistical uncertainties of the feed - down backgrounds , the uncertainties of the tracking efficiency and the acceptance of the detector . we vary the yields of the feed - down background by @xmath83 without significant effect on the fit results compared with the statistical uncertainties . since we fix the yields of the random backgrounds without true @xmath5 baryons , as discussed in sec . [ sec : backgrounds ] , we also vary the yields of the random backgrounds by their uncertainties ; only negligible effects are obtained . finally , we test other threshold functions to describe the random backgrounds with true @xmath5 baryons , but again find only negligible effects . our measurements for the mass differences ( with respect to the @xmath5 mass ) and the decay widths of the @xmath0 and @xmath1 baryons are summarized in table [ table : result ] . we also calculate the mass splittings @xmath84 from @xmath85 and @xmath86 as @xmath87 mev/@xmath9 and @xmath88 mev/@xmath9 where the first error is statistical and the second is systematic . since the mass splittings are calculated from @xmath78 , most of the systematic uncertainties cancel , such as that from the momentum calibration . these measurements are the most precise to date . the mass splitting @xmath89 is found to be positive as expected by the models @xcite . we thank the kekb group for the excellent operation of the accelerator ; the kek cryogenics group for the efficient operation of the solenoid ; and the kek computer group , the national institute of informatics , and the pnnl / emsl computing group for valuable computing and sinet4 network support . we acknowledge support from the ministry of education , culture , sports , science , and technology ( mext ) of japan , the japan society for the promotion of science ( jsps ) , and the tau - lepton physics research center of nagoya university ; the australian research council and the australian department of industry , innovation , science and research ; austrian science fund under grant no . p 22742-n16 ; the national natural science foundation of china under contracts no . 10575109 , no . 10775142 , no . 10825524 , no . 10875115 , no . 10935008 and no . 11175187 ; the ministry of education , youth and sports of the czech republic under contract no . lg14034 ; the carl zeiss foundation , the deutsche forschungsgemeinschaft and the volkswagenstiftung ; the department of science and technology of india ; the istituto nazionale di fisica nucleare of italy ; the wcu program of the ministry education science and technology , national research foundation of korea grants no . 2011 - 0029457 , no . 2012 - 0008143 , no . 2012r1a1a2008330 , no . 2013r1a1a3007772 ; the brl program under nrf grant no . krf-2011 - 0020333 , no . krf-2011 - 0021196 , center for korean j - parc users , no . nrf-2013k1a3a7a06056592 ; the bk21 plus program and the gsdc of the korea institute of science and technology information ; the polish ministry of science and higher education and the national science center ; the ministry of education and science of the russian federation and the russian federal agency for atomic energy ; the slovenian research agency ; the basque foundation for science ( ikerbasque ) and the upv / ehu under program ufi 11/55 ; the swiss national science foundation ; the national science council and the ministry of education of taiwan ; and the u.s.department of energy and the national science foundation . this work is supported by a grant - in - aid from mext for science research in a priority area ( `` new development of flavor physics '' ) and from jsps for creative scientific research ( `` evolution of tau - lepton physics '' ) . e. won acknowledges support by nrf grant no . 2010 - 0021174 , b. r. ko by nrf grant no . 2010 - 0021279 . r. lewis , n. mathur , and r. m. woloshyn , phys . d * 64 * , 094509 ( 2001 ) . n. mathur , r. lewis , and r. m. woloshyn , phys . d * 66 * , 014502 ( 2002 ) . y. namekawa _ et al . _ , d * 87 * , 094512 ( 2013 ) . w. roberts and m. pervin , int a * 23 * , 2817 ( 2008 ) . d. ebert , r. n. faustov , and v. o. galkin , phys . b * 659 * , 612 ( 2008 ) . zhang and m. -q . huang , phys . d * 78 * , 094015 ( 2008 ) . a. bernotas and v. imonis , lith . * 49 * , 19 ( 2009 ) . e. m. aitala _ ( e791 collaboration ) , phys . b * 379 * , 292 ( 1996 ) . j. m. link _ et al . _ ( focus collaboration ) , phys . b * 488 * , 218 ( 2002 ) . j. m. link _ et al . _ ( focus collaboration ) , phys . b * 525 * , 205 ( 2002 ) . m. artuso _ ( cleo collaboration ) , phys . d * 65 * , 071101(r ) ( 2002 ) . s. b. athar _ ( cleo collaboration ) , phys . d * 71 * , 051101(r ) ( 2005 ) . t. aaltonen _ ( cdf collaboration ) , phys . d * 84 * , 012003 ( 2011 ) . b. aubert _ ( babar collaboration ) , phys . d * 78 * , 112003 ( 2008 ) . j. beringer _ ( particle data group ) , phys . d * 86 * , 010001 ( 2012 ) . g. brandenburg _ et al . _ ( cleo collaboration ) , phys . rev . lett . * 78 * , 2304 ( 1997 ) . chan , phys . d * 31 * , 204 ( 1985 ) . p. hwang and d. b. lichtenberg , phys . d * 35 * , 3526 ( 1987 ) . s. capstick , phys . d * 36 * , 2800 ( 1987 ) . r. c. verma and s. srivastava , phys . d * 38 * , 1623 ( 1988 ) . r. e. cutkosky and p. geiger , phys . d * 48 * , 1315 ( 1993 ) . m. genovese , j .- richard , and b. silvestre - brac , phys . d * 59 * , 014012 ( 1998 ) . b. silvestre - brac , f. brau , and c. semay , j. phys . g * 29 * , 2685 ( 2003 ) . s. kurokawa and e. kikutani , nucl . methods phys . a * 499 * , 1 ( 2003 ) , and other papers included in this volume ; t.abe _ et al . _ , prog . phys . * 2013 * , 03a001 ( 2013 ) and following articles up to 03a011 . a. abashian _ et al . _ ( belle collaboration ) , nucl . methods phys . a * 479 * , 117 ( 2002 ) ; also see detector section in j. brodzicka _ et al . _ , 2012 * , 04d001 ( 2012 ) . e. nakano , nucl . methods phys . a * 494 * , 402 ( 2002 ) . t. sjstrand _ et al . _ , * 135 * , 238 ( 2001 ) . d. j. lange , nucl . methods phys . sect . a * 462 * , 152 ( 2001 ) . r. brun _ et al . _ , cern report dd / ee/84 - 1 .

we present measurements of the masses and decay widths of the baryonic states @xmath0 and @xmath1 using a data sample corresponding to an integrated luminosity of 711 fb@xmath2 collected with the belle detector at the kekb @xmath3 asymmetric - energy collider operating at the @xmath4 resonance . we report the mass differences with respect to the @xmath5 baryon @xmath6 and the decay widths @xmath7 where the first uncertainties are statistical and the second are systematic . the isospin mass splittings are measured to be @xmath8 mev/@xmath9 and @xmath10 mev/@xmath9 . these results are the most precise to date .

1404.5389

the study of nonequilibrium phase transitions is a topic of growing interest due to its application to a variety of complex systems@xmath3 : contact process , domain growth , catalysis , phase separation and transport phenomena . although there is no general theory to account for nonequilibrium model systems , in recent years some progress has been achieved in understanding the stationary states of these systems employing approximate analytical methods and simulations . some rigorous mathematical questions concerning the phase transitions of these complex interacting particle systems can be appreciated in the books of liggett@xmath4 and konno@xmath5 . in this paper we focus our attention on the phase transitions observed in the surface reaction model proposed by ziff , gulari and barshad@xmath6 ( zgb ) , which describes some kinetic aspects of the oxidation of @xmath7 over a catalytic surface . in particular , here we consider a modified version of the zgb model , where we incorporate a random distribution of inert sites on the catalytic surface . the original zgb model is an irreversible lattice model for surface reactions based on the langmuir - hinshelwood mechanism , where the reactants must be adsorbed before reacting . the steps used to describe the zgb model ( a lattice markov process ) are the following : molecules of @xmath0 and @xmath1 from a gaseous phase can be adsorbed onto the sites of a regular square lattice of identical sites . these molecules arrive at the surface according to their partial pressures in the gas mixture , that is , the probability of a @xmath0 molecule arriving is @xmath8 and @xmath9 for the @xmath1 molecule . the @xmath0 molecule requires only a single vacant site to be adsorbed , while the @xmath1 is adsorbed if it finds a nearest - neighbor pair of empty sites . upon adsorption , the @xmath1 molecule dissociates and the two free @xmath10 atoms can react independently . if , after an adsorption step , a nearest - neighbor @xmath11 pair appears on the lattice , they immediately react , forming a @xmath2 molecule that goes to the gas phase , leaving two empty sites on the lattice . therefore , in this adsorption controlled limit , only a single parameter ( @xmath8 ) is sufficient to describe the dynamics of the model . the simulations performed by ziff and co - workers have shown that the system exhibits two phase transitions between active and poisoned states : for @xmath12 , an o - poisoned state is formed , while for @xmath13 the lattice is poisoned by @xmath0 . for @xmath14 a reactive steady - state is found , in which a nonzero number of vacant sites is present in the lattice . at @xmath15 the transition is continuous , whereas at @xmath16 the transition is of the first - order type . using a mean field theory , dickman@xmath17 qualitatively reproduced the phase diagram of the zgb model and showed that , at the level of site approximation , only the first - order transition appears . however , employing the pair approximation , both continuous and first - order transitions are obtained . we are interested on the effects of inert sites on the phase transitions of the zgb model . we have investigated in detail the dependence of the phase transitions on the concentration of inert sites . this problem presents some experimental interest in the automobile industry , where lead particles are deposited over the catalyst during the exhaust of the gases after combustion . this affects the efficiency of the catalytic surface due to the pinning of these lead particles on the surface , forbidding the adsorption of @xmath0 and @xmath1 molecules at the lead positions and reducing the reaction paths . hovi and co - workers@xmath18 , have studied by computer simulations the effect of preadsorbed poison and promoters on the irreversible zgb model . they calculated the coverage of species as a function of the concentration of inert sites for a wide range of values , finding the interesting result that the first - order transition changes to a continuous one at a critical value of the concentration . corts and valencia@xmath19 have also reported some results concerning random impurities distributed over the catalyst , in which they observed the change of the first - order transition into a continuous one as one increases the concentration of impurities . albano@xmath20 simulated the zgb model on incipient percolation clusters ( ipc s ) with a fractal dimension of 1.90 . he showed that both transitions , at @xmath15 and @xmath16 are continuous , and that for an infinite lattice , in which @xmath8 is larger than 0.408 , the reactions stop at finite times because the ipc s are poisoned by pure @xmath0 . casties et al.@xmath21 also performed a monte carlo simulation of the @xmath0 oxidation on probabilistic fractals . they observed a change in the character of the transition at @xmath16 from first order on regular lattices to second order on percolation clusters ( for @xmath22 larger than @xmath23 , which is the percolation threshold on the square lattice ) . in this work we have performed mean - field ( site and pair approximations ) calculations and monte carlo simulations for different values of the concentration of inert sites . the model studied here is a variant of the original zgb model , where inert sites are randomly distributed over the lattice . our approach is close related to that presented by vigil and willmore@xmath24 to study the effects of spatial correlations on the oscillatory behavior of a modified zgb model , where defects are continually added and desorbed from the surface . in their studies , they considered the mean - field site and pair approximations , as well as monte carlo simulations . in the present work we have determined the phase diagram for different concentrations , and the spinodal and transition lines as a function of the concentration of inert sites . we have constructed hysteresis curves to find the critical concentration at which the first - order transition changes into a continuous one . this paper is organized as follows : in sec . ii we present the results obtained within the site approximation ; in sec . iii we introduce the pair approximation equations and show the results obtained using this scheme ; sec . iv presents the results of simulations , and finally , in sec . v we present our conclusions . we take a square lattice as our catalytic surface . a fraction @xmath25 of the sites is randomly distributed over the lattice representing the pinned inert sites . the remaining sites of the lattice can be vacant , or occupied by either @xmath10 atoms or @xmath0 molecules . the zgb model is described by the following steps : @xmath26 where the labels @xmath27 and @xmath28 denote the gaseous phase and an adsorbed reactant on the surface , respectively , and @xmath29 indicates a vacant site . steps ( 1 ) and ( 2 ) indicate the adsorption of the species , whereas the third step is the proper reaction , between distinct species located at adjacent sites of the lattice . in the site approximation the time evolution equations of the concentrations are given by @xmath30 where @xmath31 , @xmath32 and @xmath33 represent , respectively , the coverages of @xmath10 , @xmath0 and blank sites in the lattice . @xmath8 gives the arrival probability of a @xmath0 molecule . in addition , there is the following constraint among the concentrations @xmath34 the steady - state solutions of the above system of equations are given by @xmath35 , that corresponds to a poisoned surface , and @xmath36 inserting eq . ( 7 ) into eq . ( 4 ) we obtain an expression for the steady - state values of the concentration @xmath32 : @xmath37 we exhibit in fig . 1 a typical diagram for the coverages of @xmath0 , @xmath10 and vacant sites obtained for @xmath38 . this diagram was obtained by integrating the equations of motion for the @xmath32 and @xmath31 concentrations , starting from an initial condition in which the number of empty sites is @xmath39 . the site approximation does not give any continuous transition for all values of the concentration of inert sites . this was already pointed out by dickman@xmath17 for the zgb model without inert sites . we observe in fig . 1 , that the limit of stability of the reactive phase is @xmath40 , which corresponds to the spinodal point . therefore , a reactive steady - state is found for all values of @xmath41 . for values of @xmath42 , the system becomes poisoned , with a large amount of @xmath0 and a small concentration of @xmath10 atoms . the presence of @xmath10 atoms in the region @xmath42 is due to the inert sites that can block some oxygen , and also to the simplicity of the site approximation , which does not forbid the formation of @xmath43 nearest - neighbor pairs in the lattice . the tolerance of these @xmath43 pairs also explains the absence of the continuous phase transition , which is observed in the simulations . 2 is a plot of the solutions @xmath32 of eq.(8 ) versus the parameter @xmath44 for different values of concentration @xmath25 of inert sites . we obtain two solutions , which we call @xmath45 and @xmath46 , that join together at the spinodal point . for instance , for @xmath47 the value we find is @xmath48 , which furnishes the value @xmath49 . we also note in fig . 2 that , at the spinodal point , the concentration of @xmath32 molecules remains the same irrespective of the value we choose for @xmath25 . this special value is @xmath50 . then , the net effect of adding @xmath25 is to shift the curves horizontally . in this site approximation , solutions are possible only for values of @xmath51 . this happens because above this value the solution would correspond to the non - physical value @xmath52 . so , the meaning of the two solutions in fig . 2 is the following : the branch @xmath46 represents the stable steady - state solutions whereas the @xmath45 branch gives the unstable solutions . these solutions were obtained after numerical integration of the equations of motion for @xmath32 and @xmath31 , starting from the state described by @xmath39 . for the initial condition @xmath53 and @xmath32 larger than @xmath45 the system evolves to the poisoned state . the initial condition @xmath53 and @xmath32 less than @xmath45 drives the system to the lower stable reactive solution @xmath46 . fig . 2 also shows that , as we approach the spinodal point for any value of @xmath25 , the region of stability becomes narrower . then , we expect that for some value of @xmath41 a first - order transition occurs , that is , the concentration @xmath32 must increase abruptally from a small value ( reactive phase ) to a large value ( poisoned phase ) . unfortunately , we can not use here the usual thermodynamic considerations based on the minimization of a suitable thermodynamic potential . in order to find this first - order transition we adopt the same kinetic criteria employed by dickman@xmath17 , which was borrowed from the work of ziff et al@xmath6 . the phase transition was determined by choosing an initial state where half of the lattice was empty and the other half was completely filled with @xmath0 . in this work we choose as our initial state , to solve the equations of motion for @xmath32 and @xmath31 , the values @xmath54 . it is clear that this choice is not the same as that considered by ziff et al . , because we can not discriminate which sites are empty or not . the phase boundary is defined at the special value @xmath16 where the solution of the equations of motion changes from the reactive to the poisoned state as we vary the value of @xmath8 for the same initial condition , as established above . for @xmath47 we obtain the same value found by dickman . we exhibit in fig . 3 the results obtained for the first - order transition and the spinodal points for @xmath38 . the spinodal was obtained from the initial condition @xmath39 , and the first - order transition from the condition @xmath54 . for this particular value of @xmath38 , we have @xmath55 and @xmath56 . we have considered all values of the concentration of inert sites , and fig . 4 shows the values of @xmath57 ( dashed line ) and @xmath16 ( full line ) as a function of the concentration of inert sites . at the particular value @xmath58 the two lines merge . for values of @xmath59 the transition still remains of the first - order type , although the number of vacant sites that stay in the active state is very small . for instance , for @xmath60 , at the transition point ( @xmath61 ) , the number of vacant sites changes from @xmath62 in the active state , to @xmath63 in the poisoned state . throughout our analysis we considered a given state to be active if the number of vacant sites is larger than @xmath64 . we also exhibit in fig . 5 the number of vacant sites @xmath33 at the active state as a function of the number of inert sites @xmath25 , at the transition and spinodal points . we observe that for all values of @xmath65 , the number of vacant sites at the spinodal point is always larger than that at the transition point . let us consider the application of the pair approximation procedure to this zgb model that includes inert sites . here we introduce the pair probability @xmath66 of a random nearest neighbor pair of sites being occupied by species @xmath67 and @xmath68 . we have the following types of species : @xmath29 , @xmath69 , @xmath70 , and @xmath10 , which represent , respectively , vacant , inert , carbon monoxide , and oxygen . as in the previous treatments @xmath71 we need to consider only the changes that occur at a particular central pair in the lattice . in the table below we display the allowed and forbbiden ( indicated by @xmath72 ) nearest - neighbor pairs in the present model . @xmath73 the next table also exhibits all the possible transitions among pairs . we obtain @xmath74 independent transitions , labelled by numbers in the range @xmath75 . in the table transitions indicated by @xmath72 are prohibited . @xmath76 then , we write the equations relating the probability of each element with the corresponding pair probabilities : @xmath77 the pair probabilities also satisfy the constraint @xmath78 next , we need to write the time evolution equations for the pair probabilities . examining the latter table we can construct the desired equations of evolution . we explicitly write the equations of motion for the pair probabilities @xmath79 . @xmath80 where @xmath81 to @xmath82 are the transition rates . the factors of two arising in the equation of motion for @xmath83 are due to the fact that the pair probabilities @xmath84 and @xmath85 are equal by symmetry . for instance , from the pair @xmath86 we can obtain , with the same probability , the different configurations @xmath87 and @xmath88 . in general , the expressions for the transition rates are lengthy , and we present these transition rates in the appendix . in this pair approximation we can not obtain analytical solutions as we have done in the site approximation . we solved the coupled set of eight nonlinear equations by the fourth - order runge - kutta method , searching for the stationary solutions . we considered the two different initial conditions as in the case of the site approximation . let us first consider the evolution from the initial state where @xmath89 , in which only the pairs @xmath86 , @xmath90 and @xmath91 are present in the lattice at @xmath92 . figure 6 shows the diagram of the model for @xmath93 . for @xmath94 the lattice poisons with oxygen . in the range @xmath95 there is an active region , and for @xmath96 the lattice poisons with @xmath0 . when @xmath97 , we found the same figures obtained by dickman in his pair approximation . for instance , the site and pair approximations give the same value for the spinodal point @xmath57 . however , when we consider some inert sites in the lattice , the spinodal point found in the site approximation is always smaller than that obtained within the pair approximation . for this particular value , @xmath93 , the site approximation yields @xmath98 , whereas @xmath56 is obtained by the pair approximation . the value of @xmath8 at the continuous transition , which now arises in this pair approximation , decreases slightly with increasing values of the concentration of inert sites . we also considered the solutions evolving from an initial condition where half of the free sites ( @xmath89 ) is filled with @xmath0 molecules and the other half left empty . in order to be close to the initial condition used in the simulation , we chose for the initial pair conditions @xmath99 , @xmath100 and @xmath101 , which mimics a division of the lattice into two parts : on one side of the lattice we would have inert sites and @xmath0 molecules and , on the other side , vacant and inert sites . if @xmath97 , we found for the transition between the active and co - poisoned states the value @xmath102 , which agrees with the value found in the simulations . 7 displays the concentration of @xmath0 molecules at the transition point for which @xmath93 . in this pair approximation , the values of @xmath57 and @xmath16 are very close . we also show in fig . 8 the concentration of vacant sites as a function of the concentration of inert sites , at the transition point , and also at the spinodal point . both curves join at @xmath103 , and for @xmath104 , we can not observe any active state . as in the site approximation , an active state is defined only if @xmath105 . then , the calculations performed within the pair approximation give results that are very similar to those obtained by the site approximation , concerning the spinodal and transition points . in addition , it was observed that initial conditions do not affect the point in which the continuous phase transition occurs . in fig . 9 we exhibit the phase diagram for this zgb model with inert sites . the size of the reactive window decreases as we increase the concentration of inert sites . we have plotted the transition line for the first - order transition and for the spinodal line , which gives the limit of stability of the reactive phase . the line separating the active and o - poisoned phases is a continuous transition line . we have performed monte carlo simulations in the zgb model with inert sites in order to check the results we have obtained in the site and pair approximations . the simulations were carried for different values of the concentration of inert sites @xmath106 . for small values of @xmath106 , we considered square lattices of linear size @xmath107 , but for large values of @xmath106 we have used lattices of linear size up to @xmath108 . the first step in the simulation is to randomly distribute the selected fraction @xmath106 of inert sites in the lattice . all simulations then started with a fraction of empty sites equal to @xmath89 . the @xmath0 molecules arrive at the surface with a probability @xmath8 and the @xmath1 molecules with probability @xmath109 . the rules for adsorption and reaction of the species are exactly the same as in the original zgb model@xmath6 . since adsorption of oxygen requires two nearest neighbor empty sites , the effect of the inert sites is to favour the adsorption of @xmath0 relatively to that of @xmath1 molecules . in general , we have taken @xmath110 monte carlo steps ( mcs ) to attain the stationary states , and @xmath110 more to calculate the concentration averages at the stationary states . one mcs is equal to @xmath111 trials of deposition of species , where @xmath112 is the linear size of the lattice . to speed up the simulations we worked with a suitable list of empty sites . we exhibit in fig . 10 the phase diagram of the model in the plane @xmath8 versus @xmath106 . it is similar to that obtained within the pair approximation . however , there is a fundamental difference between the transition line separating the active and co - poisoned phases in both approaches . in the pair approximation the transition line is always of the first - order type , whereas in the simulations there is a critical concentration above which the transition becomes continuous . we have done detailed simulations to find the critical concentration at which the transition becomes continuous . we have found for the critical concentration of inert sites the value @xmath113 . we arrived at this value by looking at the hysteresis loops in the curves of @xmath114 versus @xmath8 for different values of the concentration @xmath106 , as we can see in fig . we proceed as follows : in fig . 11a we fixed the concentration of inert sites at the value @xmath115 and the curve with circles , which is the proper transition curve , was obtained from an initial state where @xmath89 , that is , with a lattice almost empty . the curve with squares was determined from an initial state in which the lattice was almost covered by @xmath0 . we have taken a fraction of only @xmath116 of randomly empty sites over the lattice at the starting time . then , we clearly observe the hysteresis loop at the concentration @xmath106 , which implies that the transition is of first - order . on the other hand , fig . 11b , where the fraction of inert sites is @xmath117 , does not exhibit the hysteresis loop and the transition is clearly a continuous one . the critical value of @xmath113 was obtained analysing the behavior of these curves in the range @xmath118 . as we have pointed out in the introduction , hovi et al.@xmath18 had already observed the change in the nature of this transition as a function of the concentration . the phase boundary separating the active and the o - poisoned phases in fig . 10 is continuous for all values of @xmath106 . we have checked this fact by observing that no hysteresis loop was found for any value of @xmath106 . the width of the active phase decreases with increasing values of @xmath106 . for values of @xmath119 the lattice is poisoned ( absence of empty sites ) with different amounts of @xmath0 and @xmath10 species . due to finite size effects , this value is larger than the value 0.408 found by albano@xmath20 in the limit of very large ipc s . we have also noted that the production rate of @xmath2 molecules attains its maximum value exactly at the first - order transition , for values of @xmath120 . if @xmath121 the maximum production rate of @xmath2 molecules is located inside of the reactive window . this is seen in fig . 12 , where the circles indicate the points where the production rate of @xmath2 is maximum . in the site and pair approximations this maximum occurs always at the phase boundary , irrespective of the value of @xmath106 . 13 shows the production rate @xmath122 of @xmath2 molecules as a function of @xmath106 . as expected , the role of inert sites is also of blocking the reactions over the catalyst . the maximum production rate occurs at a surface free of impurities . we have studied the effects of a random distribution of inert sites on the phase diagram of the zgb model . we determined the time evolution equations for the concentrations of the different species over the catalytic surface within an effective field theory , at the level of site and pair approximations , and also performed monte carlo simulations on the model . we obtained the coverages of the species as function of the deposition rate of @xmath0 and of the concentration of inert sites . in the site and pair approximations we found the transition line and the limit of stability of the reactive phase . in the site approximation , the continuous transition between the o - poisoned and reactive states is absent for any values of the concentration of inert sites . the width of the reactive window exhibits the same behavior , as a function of concentration of inert sites , in both pair approximation and monte carlo simulations . however , the transition between the reactive and co - poisoned phase is always of first - order in the site and pair approximations , whereas monte carlo simulations give a critical point where the transition changes nature . for values of the concentration of inert sites less than the critical value , the transition is first - order and above this value , it changes to a continuous one . the determination of this critical concentration was possible through the analysis of the hysteresis curves for different values of the concentration of inert sites . the production rate of @xmath2 molecules is maximum at the first - order transition , in both site and pair approximations . this is the case in the simulations , but the transition is of the first - order type . when the concentration of inert sites is greater than the critical value , the maximum production rate of @xmath2 molecules moves towards the reactive window . the overall effect of inert sites is to reduce the production of @xmath2 molecules . we would like to thank ron dickman by his many valuable suggestions , and luis g. c. rego by the critical reading of the manuscript . this work was supported by the brazilian agencies capes , cnpq and finep . we present the transition rates in the pair approximation , which we used in section iii to solve the time evolution equations for the pair probabilities . the transition rates are @xmath81 to @xmath82 , which are given by @xmath123 @xmath132 + \frac 13\left ( \frac{p_{vo}}{p_v}\right ) ^2\left ( 1-\frac{p_{oo}}{p_o}\right ) \nonumber \\ & & + \left . \frac 23\frac{p_{vo}}{p_v}\left ( 1-\frac{p_{vo}}{p_v}\right ) \frac{% p_{oo}}{p_o}+\frac 14\left ( \frac{p_{vo}}{p_v}\right ) ^2\frac{p_{oo}}{p_o}% \right\ } \nonumber\end{aligned}\ ] ] @xmath135 + \frac{p_{vc}}{p_c}% \left [ \left ( 1-\frac{p_{vc}}{p_v}\right ) ^2\right . \nonumber \\ & & + \left . \frac{p_{vc}}{p_v}\left ( 1-\frac{p_{vc}}{p_v}\right ) + \frac 13% \left ( \frac{p_{vc}}{p_v}\right ) ^2\right ] \times \left . \left [ \frac{p_{vc}% } { p_v}\left ( 1-\frac{p_{vc}}{p_v}\right ) + \frac 23\left ( \frac{p_{vc}}{p_v}% \right ) ^2\right ] \right\ } \nonumber\end{aligned}\ ] ] @xmath136 \right\ } \nonumber \\ t_{14b } & = & 2(1-y_{co})p_{cd}\frac{p_{vc}}{p_c}\left\ { 2\frac{p_{vv}}{p_v}\left [ \left ( 1-\frac{p_{vc}}{p_v}\right ) \left ( 1-\frac{p_{cd}}{p_d}\right ) + \frac 12\frac{p_{vc}}{p_v}\left ( 1-\frac{p_{cd}}{p_d}\right ) \right . \right . \nonumber \\ & & + \left . \frac 12\left ( 1-\frac{p_{vc}}{p_v}\right ) \frac{p_{cd}}{p_d}+% \frac 13\frac{p_{vc}}{p_v}\frac{p_{cd}}{p_d}\right ] + \frac{p_{vd}}{p_d}% \left ( 1-\frac{p_{vc}}{p_v}\right ) ^2+\frac 22\frac{p_{vc}}{p_v}\frac{p_{vd}% } { p_d } \nonumber \\ & & + \left . \left . \frac 13\left ( \frac{p_{vc}}{p_v}\right ) ^2\right ] \right\ } \nonumber\end{aligned}\ ] ] j. marro and r. dickman , _ nonequilibrium phase transitions in lattice models _ ( cambridige university press , cambridge , 1999 ) . v. privman , ed . , _ nonequilibrium statistical mechanics in one dimension _ ( cambridge university press , cambridge , 1996 ) . liggett , _ interacting particle systems _ ( springer - verlag , new york , 1985 ) . n. konno , _ phase transitions of interacting particle systems _ ( world scientific , singapore , 1994 ) . r. m. ziff , e. gulari , and y. barshad , phys . 56 * , 2553 ( 1986 ) . r. dickman , phys . rev . a * 34 * , 4246 ( 1986 ) . hovi , j. vaari , h. -p . kaukonen and r.m . nieminen , compt . science , * 1 * , 33 ( 1992 ) . joaquin corts and eliana valencia , surf . sci . * 425 * , l357 ( 1999 ) . albano , phys . b * 42 * , 10818 ( 1990 ) . a. casties , j. mai , and w. von niessen , j. chem . phys . * 99 * , 3082 ( 1993 ) . r. dennis vigil and frank t. willmore , phys . e * 54 * , 1225 ( 1996 ) . dickman , b.c.s . grandi , w. figueiredo and r. dickman , phys . e * 59 * , 6361 ( 1999 ) . da costa and w. figueiredo , phys . e * 61 * , 1134 ( 2000 ) .

a random distribution of inert sites is introduced in the ziff - gulari - barshad model to study the phase transitions between active and poisoned states . the adsorption of @xmath0 and @xmath1 molecules is not possible at the position of the inert sites . this model is investigated in the site and pair approximations , as well as through monte carlo simulations . we determine the mean coverages of the elements as a function of the dilution and show that the continuous transition between the active and o - poisoned state is slightly affected by moderate values of dilution in the pair approximation and in the simulations . on the other hand , from the analysis of the hysteresis curves , the transition between the active and co - poisoned states changes from first - order to continuous as one increases the concentration of inactive sites . the observed transition in the site and pair approximations is always of first - order nature . we also found the lines of transition and spinodal points as a function of the concentration of inert sites . finally , the production rate of @xmath2 is calculated as a function of the dilution of sites . pacs number(s ) : 05.70.ln , 05.70.fh , 82.65.jv , 82.20.mj

cond-mat0008283

the point scatterer on a torus is a popular model to study the transition between integrable and chaotic dynamics in quantum systems . it rose to prominence in the quantum chaos literature in a famous paper of petr seba @xcite which dealt with the closely related case of rectangular billiards . the model first appeared in solid state physics @xcite in the 1930s to explain electronic band structure and conductivity in solid crystals . many applications arose in nuclear physics throughout the 1960s and 1970s , see for instance @xcite . the purpose of this article is to give an introduction to this important model which belongs to the class of pseudo - integrable systems and to report on some recent progress in this field . the reader will also be introduced to some important open problems . in 1931 kronig and penney @xcite studied the quantum mechanics of an electron in a periodic crystal lattice with the goal of understanding the conductivity properties of solid crystals . they introduced the periodic 1d hamiltonian @xmath0}(x - k ) , \quad v_0>0 , \quad 0<a\ll 1\ ] ] where @xmath1 denotes the characteristic function . according to bloch theory , we have the decomposition @xmath2 where @xmath3 is the space of quasiperiodic functions with quasimomentum @xmath4 : @xmath5 let us consider the special case of periodic boundary conditions @xmath6 . to simplify the hamiltonian @xmath7 it is convenient to take the limit @xmath8 let @xmath9 . the calculation @xmath10}(x)f(x)dx=\frac{\alpha}{a}\int_{-a/2}^{a/2}f(x)dx \to \alpha f(0 ) , \quad a\searrow 0\ ] ] shows that the hamiltonian @xmath7 converges in the distributional sense to a singular rank - one perturbation of the 1d laplacian @xmath11 } \to h_\alpha=-\frac{d^2}{dx^2}+\alpha\left\langle \delta_0,\cdot \right\rangle\delta_0 , \quad a\searrow 0.\ ] ] the operator @xmath12 can be realised rigorously by using von neumann s self - adjoint extension theory . we will be interested in studying the analogues of the operator @xmath12 on 2d and 3d tori . let @xmath13 be a rectangle with side lengths @xmath14 , @xmath15 . we define the aspect ratio of @xmath13 as the quotient @xmath16 . in a 1990 paper @xcite petr seba studied the operator @xmath17 on a rectangle with irrational aspect ratio and dirichlet boundary conditions . seba s motivation was to find a quantum system which displayed the features of quantised chaotic systems such as quantum ergodicity and level repulsion , yet whose classical dynamics was close to integrable . as was pointed out later by shigehara @xcite the energy levels obtained in seba s quantisation do not repell each other , in fact careful numerical experiments conducted by shigehara show that the spacing distribution coincides with that of the laplacian which is conjectured to be poissonian . we will discuss rigorous mathematical results in this direction in section 5 . shigehara suggested a different quantisation in his paper which should produce energy levels which display level repulsion . in the present paper we refer to seba s quantisation as `` weak coupling '' and to shigehara s as `` strong coupling '' . a detailled discussion of these two different quantisations is given in section 3 . in the present paper we will deal with a system closely related to the seba billiard a point scatterer on a flat torus ( which means periodic boundary conditions ) , however , the results which will be presented can probably be easily extended to rectangular domains with dirichlet or neumann boundary conditions . * acknowledgements : * i would like to thank zeev rudnick and stephane nonnenmacher for many helpful comments and suggestions that have led to the improvement of this paper . we consider a rectangle with side lengths @xmath18 , @xmath19 , where @xmath20 , and identify opposite sides to obtain the torus @xmath21 where @xmath22 . we want to study the formal operator @xmath23 to treat @xmath12 rigorously we will employ von neumann s theory of self - adjoint extensions . for an introduction to this standard machinery see @xcite . the main idea is to restrict @xmath12 to a domain where we understand how it acts functions which vanish at the position of the scatterer and therefore do not feel " its presence . we denote by @xmath24 the domain of @xmath25-functions which vanish in a neighbourhood of @xmath26 . clearly @xmath27 . we denote @xmath28 . the restricted laplacian @xmath29 is a symmetric operator , however it is not self - adjoint . by restricting @xmath30 to the domain @xmath31 we are enlarging the domain of its adjoint . therefore we have @xmath32 . a simple computation of the adjoint @xmath33 shows that its domain is given by @xmath34 we have the following definition . the deficiency spaces of a symmetric densely defined operator @xmath35 are given by the kernels @xmath36 the deficiency indices of @xmath35 are defined as @xmath37 and @xmath38 . if @xmath39 , then we say that @xmath35 is essentially self - adjoint . for @xmath40 denote by @xmath41 the corresponding green s function , namely the integral kernel of the resolvent @xmath42 and therefore we have the following distributional identity @xmath43 indeed , if we compute the deficiency elements of @xmath33 we have to solve @xmath44 for some @xmath45 . this shows that the deficiency spaces are spanned by the green s functions @xmath46 . we thus have @xmath47 where the orthogonal decomposition is with respect to the graph inner product @xmath48 and the closure is taken with respect to the associated graph norm @xmath49 . the following theorem is due to von neumann . let @xmath35 be a densely defined symmetric operator . if @xmath35 has deficiency indices @xmath50 , then there exists a family of self - adjoint extensions which is parametrised by @xmath51 , the group of unitary maps on @xmath52 . the domain of the extension @xmath53 is given by @xmath54 where @xmath55 , @xmath56 are the vectors whose entries are the deficiency elements and the closure of @xmath57 is taken with respect to the graph norm of @xmath35 . the operator @xmath58 is essentially self - adjoint . in our case we have for @xmath59 $ ] @xmath60 and @xmath61 is the restriction of @xmath33 to this domain . functions @xmath62 satisfy @xmath63 and near @xmath26 they have the asymptotic @xmath64 the extension associated with the choice @xmath65 is just the self - adjoint laplacian on @xmath66 . we will be interested in studying the extensions for @xmath67 . an orthonormal basis of eigenfunctions of the laplacian on @xmath21 is given by the complex exponentials @xmath68 and @xmath69 is the dual lattice of @xmath70 : @xmath71 the eigenvalue of @xmath72 is given by @xmath73 . we introduce the set of distinct norms @xmath74 and denote the multiplicity of @xmath75 by @xmath76 if @xmath69 is a rational lattice , i. e. if @xmath77 , then the multiplicities can be large and we have the bound ( see e.g. @xcite , lemma 7.2 ) @xmath78 we have the following lemma which can be found in the standard literature on point scatterers , or in the appendix to the paper @xcite . [ quantisation lemma ] let @xmath67 . we have that @xmath40 is an eigenvalue of @xmath61 on @xmath79 iff @xmath80 where @xmath81 and the corresponding eigenfunction is a multiple of the green s function @xmath82 for which we have the @xmath83-identity @xmath84 the eigenfunctions in the lemma are not the only eigenfunctions of @xmath61 . if @xmath85 ( which in our case happens for all @xmath86 ) , then @xmath87 appears in the spectrum of @xmath61 with multiplicity @xmath88 . the associated eigenfunctions are superpositions of laplacian eigenfunctions and vanish at @xmath26 and therefore do not feel the effect of the scatterer . in the present article we will only be interested in the new eigenvalues , which are solutions to , and the associated new eigenfunctions . two different quantisations appear in the literature on point scatterers , and we will follow the terminology used by shigehara et al . in the papers @xcite in referring to the two models as _ weak _ and _ strong _ coupling . the purpose of this section is to explain how these different quantisations arose in the literature and give a derivation of the strong coupling quantisation on a 2d torus . in his famous paper on wave chaos in a singular billiard @xcite seba considered a point scatterer on a rectangle with irrational aspect ratio and dirichlet boundary conditions . seba computed the spectrum of @xmath61 by solving the equation numerically and a plot of the level spacing distribution seemed to suggest level repulsion . in 1994 shigehara came to investigate this question and following careful numerical investigations he observed @xcite that the eigenvalue spacing distribution of the self - adjoint extension @xmath61 seemed to coincide with that of the laplacian , which according to the conjecture of berry and tabor @xcite is believed to be poissonian . shigehara observed that the apparent weakness " of the point scatterer in seba s quantisation could be corrected by adjusting the parameter @xmath89 in a suitable way as the eigenvalue @xmath90 tends to infinity . the operator can be realised by employing the self - adjoint extension theory discussed in the previous section . the formal operators @xmath91 are associated with the family of extensions @xmath92 it is well known that in 1d there is an exact relation which links the physical coupling constant @xmath93 and the parameter @xmath67 . the situation is more complicated in 2 or 3d . a relation can be derived ( at a physical level of rigour ) from the scattering problem for a spherical scatterer ( cf . for instance @xcite ) by shrinking its diameter to zero . one obtains a relation which in contrast to the 1d case contains a logarithmic divergence in the spectral parameter @xmath90 @xmath94 for certain real constants @xmath95 , @xmath96 which depend on the domain . so a fixed choice of @xmath89 corresponds to weak coupling , @xmath97 on the other hand a fixed physical coupling constant @xmath98 would require a renormalisation of the parameter @xmath89 , which means @xmath99 should be allowed to depend on @xmath90 as @xmath100 , @xmath101 the renormalisation condition is equivalent to a different quantisation condition for a point scatterer ( cf . for instance @xcite ) which only takes into account the physically relevant energies in the summation , @xmath102 where @xmath103 is the physical coupling constant . we require the following lemma which gives an asymptotic for the sum over the energies outside the interval @xmath104 $ ] . [ truncation ] consider a general torus @xmath79 . let @xmath105 , where @xmath106 . we have the asymptotic @xmath107 we have @xmath108 we will use the circle law @xmath109 where the best known exponent @xmath110 is due to huxley @xcite and the optimal exponent is expected to be @xmath111 ( gauss s circle problem ) . summation by parts allows us to compare a lattice sum with an integral @xmath112 where @xmath113 is some differentiable function . we obtain for the first sum @xmath114 similarly for the second sum @xmath115 with the help of the lemma above we can now show that the quantisation is in fact equivalent to the strong coupling quantisation given by the renormalisation condition . we obtain @xmath116 which forces the renormalisation @xmath117 let @xmath118 denote the solutions to the spectral equation ( where @xmath89 is fixed ) . the eigenvalues @xmath118 interlace with the norms @xmath119 in the following way @xmath120 we denote the solutions to the strong coupling quantisation condition by @xmath121 . similarly as above the eigenvalues @xmath121 interlace with the norms @xmath119 . the corresponding eigenfunctions are the green s functions @xmath122 . we are interested in the statistical behaviour of the eigenfunctions @xmath123 and @xmath122 in the limit as the eigenvalue tends to infinity . let @xmath124 where @xmath125 . let @xmath126 and expand @xmath127 into a fourier series @xmath128 in order to quantise the classical symbol @xmath14 it is convenient to realise the torus as the quotient @xmath129 $ ] , where @xmath130=\{x_1+{{\mathrm{i}}}x_2 \mid ( x_1,x_2)\in\scrl_0\}.\ ] ] for @xmath131 we define by @xmath132 the corresponding element of @xmath133.$ ] we associate with the symbol @xmath14 the zeroth order pseudo - differential operator @xmath134 defined on the fourier transform side by @xmath135 and @xmath136 where @xmath137 let @xmath138 . denote the @xmath83-normalised green s function by @xmath139 . we would like to study the behaviour of the matrix elements @xmath140 in the limit as @xmath141 . similarly we would like to study the matrix elements of @xmath142 in the strong coupling quantisation , @xmath143 we first study the behaviour of the eigenfunction in position space . if we take a classical symbol on position space @xmath144 , then the operator @xmath142 is simply given by multiplication @xmath145 we have the following theorem , which is proven in @xcite . in the paper the theorem is stated for the specific case of the eigenfunctions @xmath146 of the weakly coupled point scatterer . however , the result holds for any increasing sequence of numbers which interlaces with the norms @xmath119 . in particular it also applies to the eigenfunctions @xmath147 of a strongly coupled point scatterer . we state the theorem in full generality . [ wave function stats]*(rudnick - u . , 2012 ) * + let @xmath21 be a general flat torus . let @xmath144 . recall that @xmath139 denotes the @xmath83-normalised green s function . for any increasing sequence of numbers @xmath148 which interlaces with the norms @xmath149 there exists a density one subsequence @xmath150 such that @xmath151 as @xmath141 along @xmath152 . in particular this theorem implies that * both * in the weak * and * strong coupling regimes the eigenfunctions equidistribute in position space . also note that the result holds for both rational and irrational lattices . the analogous result was proved in @xcite for the standard 3d torus and general 3d tori which satisfy certain irrationality conditions . the proof of theorem [ wave function stats ] uses the explicit formula for the green s function and by approximating the green s function by a sum over a polynomial size interval the problem can be translated into a number theoretical problem about the well - spacedness of lattice points in thin annuli . in a recent paper @xcite marklof and rudnick have shown that for rational polygons a full density of eigenfunctions of the laplacian equidistributes in position space . in particular their result applies to the square torus . their proof uses egorov s theorem . quantum ergodicity does of course not hold for the laplacian eigenfunctions on the torus . if one chooses an eigenbasis of plane waves , the eigenfunctions are obviously localised in momentum . it is an interesting question to ask if the presence of a ( weakly or strongly coupled ) point scatterer can change this . curiously , the answer depends on the arithmetic properties of the lattice @xmath70 . for both a weakly and strongly coupled point scatterer it is possible @xcite to prove quantum ergodicity for @xmath153 and it is likely that one can generalise this to any rational lattice . on the other hand one can disprove quantum ergodicity in the irrational case . the failure of quantum ergodicity in the closely related case of the seba billiard ( which has an irrational aspect ratio ) was already conjectured by berkolaiko , keating and winn @xcite . in @xcite keating , marklof and winn prove under assumptions on the spectrum ( which are consistent with the berry - tabor conjecture ) that there exists a positive density subsequence of eigenfunctions for the seba billiard ( weak coupling ) which become localised around two laplacian eigenfunctions . in particular it follows that this subsequence becomes localised in the high energy limit , therefore disproving quantum ergodicity . this phenomenon is in some sense similar to scarring on unstable periodic orbits of the classical system . the authors construct a sequence of quasimodes which converge to the real eigenfunctions in order to obtain their results . it would be interesting to prove this result without any assumptions on the spectrum . in the closely related case of a weakly coupled point scatterer on an irrational 2d torus the failure of quantum ergodicity can be proved unconditionally @xcite . another interesting question is to see if one can classify quantum limits which are localised in momentum without any assumptions on the spectrum . * open problems : * _ it would be interesting to try to prove the analogous statement for the strong coupling limit : can one disprove quantum ergodicity for irrational lattices ? _ the microcolal lift @xmath154 of the measures @xmath155 is defined by the identity @xmath156 the quantum limits are the limit points of the sequence @xmath154 in the weak- * topology . in @xcite jakobson classified the quantum limits for the laplacian on the square torus . one can pose the same problem for a point scatterer . * open problems : * _ what are the quantum limits for a point scatterer on the square torus in the weak and strong coupling regimes ? how about general rational and irrational 2d tori ? how about 3d tori ? _ one of the main observations in seba s paper @xcite was level repulsion for a point scatterer in a rectangular billiard with dirichlet boundary conditions and irrational aspect ratio . seba computed the eigenvalues numerically and plotted the spacing distribution which did indeed reveal some form of level repulsion and he initially conjectured that the spacing distribution should coincide with that of the gaussian orthogonal ensemble ( goe ) in random matrix theory . as shigehara @xcite discovered later , level repulsion , as observed by seba , could only be expected in the strong coupling regime , where the extension parameter is normalised in a suitable way as the eigenvalue tends to infinity . it is likely that seba carried out a truncation of the spectral equation ( similar to the one in lemma [ truncation ] ) when performing the numerics , so effectively he calculated the eigenvalues for the strong coupling quantisation . based on careful numerics , shigehara predicted that one should recover poissonian level statistics in the weak coupling regime , as was expected for the unperturbed laplacian on an irrational rectangle . in other words , the effect of the scatterer in this regime was too weak to have an impact in the high energy limit . in 3d , however , the situation is very different . numerics suggest that the eigenvalues of a fixed self - adjoint extension obey intermediate level statistics , just as in the strong coupling regime in 2d @xcite . in a 1999 paper @xcite bogomolny , gerland and schmit argued that the spacing distribution was close to a semi - poissonian distribution . the seba billiard thus belongs to an intermediate class of systems which shows a weaker form of level repulsion than chaotic systems . another example of such intermediate systems are flat surfaces with conic singularities . intermediate statistics have also been observed for certain families of quantum maps @xcite and near the transition point in the anderson model in 3d ( cf . @xcite , section 6 . 1 . 3 . , p. 332 - 3 ) . there is also a recent paper by stckmann , kuhl and tudorovskiy @xcite which claims to revise the results of bogomolny , gerland and schmit in @xcite . the paper contains heuristic mathematical arguments and numerics which show a transition from level repulsion to poissonian spacing statistics for the seba biliard in the high energy limit . however , the quantisation condition used in @xcite can be shown to correspond to the weak coupling regime as opposed to the strong coupling regime considered by bogomolny , gerland and schmit . the poissonian statistics observed in @xcite are therefore hardly surprising , as they were already predicted by shigehara in @xcite . a rigorous mathematical proof that the spacing distribution of a weakly coupled point scatterer coincides with that of the laplacian on the 2d torus is given in the paper @xcite and we will discuss this result and others below . there are many open questions regarding the spectral statistics of point scatterers on flat tori . even in the case of the unperturbed laplacian , little can be said rigorously . let @xmath157 and define @xmath158 if @xmath153 , we have the following asymptotic which is due to landau @xcite @xmath159 if @xmath70 is an irrational lattice , then the multiplicity of the laplacian eigenvalues is on average @xmath160 ( for a generic vector @xmath161 $ ] the other three choices are @xmath162 , @xmath163 and @xmath164 ) . indeed we have in this case @xmath165 we define the mean spacing up to a threshold @xmath166 by @xmath167 and observe that @xmath168 we introduce the mean normalised spacings @xmath169 analogously let @xmath170 and we introduce the mean normalised spacings @xmath171 and it is easy to see that @xmath172 as @xmath173 ( recall that @xmath174 are the spacings of the new eigenvalues which interlace with the norms @xmath87 ) . we have the following conjecture which is due to berry and tabor @xcite and is expected to hold for a generic classically integrable quantum system . we will only state it in the special case of flat 2d tori . this conjecture remains out of reach , even a proof of the existence of the spacing distribution is not available . however , some progress has been made with regard to the pair correlation function @xcite , in cases where the aspect ratio satisfies certain diophantine properties . it is an interesting problem to study the effect of the scatterer on the distribution of the spacings @xmath178 . we introduce the difference between laplacian eigenvalues and the neighbouring eigenvalue of the point scatterer : @xmath179 and we denote its mean by @xmath180 one has the following bound on the mean of @xmath181 which shows that on average the eigenvalues @xmath87 and @xmath182 clump " together . the result is derived in @xcite and the main tool is an exact trace formula for a point scatterer , similar to the trace formula proved in @xcite . this implies that if either the limiting distribution of @xmath184 or @xmath178 exist , then they must coincide . by use of the trace formula proved in @xcite it is possible to extend this result to surfaces of constant negative curvature . the situation is very different in 3d . denote @xmath185 , @xmath186 and consider the torus @xmath187 . an orthonormal basis of laplacian eigenfunctions is given by the exponentials @xmath188 where @xmath189 is the dual of @xmath190 , and the eigenvalue is given by the norm @xmath73 . we denote the set of distinct norms by @xmath191 . let @xmath192 . in 3d the operator @xmath28 , where @xmath193 , has deficiency indices @xmath194 the analogous operator @xmath29 is essentially self - adjoint . ] and we denote the self - adjoint extensions by @xmath61 . the new eigenvalues of @xmath61 ( which interlace with the norms @xmath191 ) are denoted by @xmath195 and we define the differences @xmath196 and @xmath197 . as opposed to theorem [ clumpingthm ] the eigenvalues of the point scatterer on a 3d torus seem to lie on average in the middle of the neigbouring laplacian eigenvalues . we have the following result which is consistent with intermediate statistics . the proof is given in @xcite and the methods are the same as in the case of theorem [ clumpingthm ] . it would be desirable to obtain information about the variance of @xmath181 . theorems [ clumpingthm ] and [ intermediate ] are proved using a trace formula for the point scatterer of the type proved in @xcite . however , this approach fails for higher moments . + * open problems : * consider a 3d torus . can one obtain an asymptotic for @xmath199 or a useful bound such as @xmath200 for some @xmath201 ? what about higher moments ? f. a. berezin and l. d. faddeev , _ remark on the schrdinger equation with a singular potential _ , dokl nauk sssr 137 , 10111014 , 1961 ( russian ) ; english translation : soviet mathematics 2 , 372375 , 1961 .

this survey article deals with a delta potential - also known as a point scatterer - on flat 2d and 3d tori . we introduce the main conjectures regarding the spectral and wave function statistics of this model in the so - called weak and strong coupling regimes . we report on recent progress as well as a number of open problems in this field . [ section ] [ thm]lemma [ thm]proposition [ thm]corollary [ thm]conjecture [ thm]definition

1212.1086

connecting the puzzling disturbances in both the gas and stellar disk of the milky way ( mw ) with the dark matter distribution of our galaxy and its dwarf companions may become possible in the gaia era ( perryman et al . gaia will provide parallaxes and proper motions for a billion stars down to @xmath0 ( de bruijne et al . 2014 ) and radial velocities for stars with @xmath1 . by now , a plethora of stellar tidal streams have been discovered , including the sagittarius ( sgr ) tidal stream ( ibata et al . 1997 ) , the monoceros stream ( newberg et al . 2002 ) , and many others ( belokurov et al . a number of authors have attempted to infer the galactic potential by modeling stellar tidal streams ( e.g. johnston et al . 1999 ) , but the limitations of determining accurate phase space information for the stream and simplistic modeling ( for example static halos ) have led to large uncertainties in the reconstruction of the galactic potential . more recently , observations of an asymmetry in the number density and bulk velocity of solar neighborhood stars have been interpreted as arising from a dark sub - halo or dwarf galaxy passing through the galactic disk , exciting vertical waves ( widrow et al . 2012 ; carlin et al . 2013 ; xu et al . 2015 ) . this corroborates a similar previous suggestion that the disturbances in the outer hi disk of our galaxy may be due to a massive , perturbing satellite ( chakrabarti & blitz 2009 ; henceforth cb09 ) . there is some evidence now for this predicted satellite , which may mark the first success of galactoseismology ( chakrabarti et al . 2016 ) . galaxy outskirts hold particularly important clues to the past galactic accretion history and dynamical impacts . extended hi disks reach to several times the optical radius ( walter et al . 2008 ) , presenting the largest possible cross - section for interaction with sub - halos at large distances ( where theoretical models _ expect _ them to be , e.g. springel et al . the gas disk of our galaxy manifests large planar disturbances and is warped ( levine , blitz & heiles 2006 ) . chakrabarti & blitz ( 2009 ; 2011 ) found that these puzzling planar disturbances in the gas disk of our galaxy could be reproduced by an interaction with a sub - halo with a mass one - hundredth that of the milky way , with a pericenter distance of @xmath2 7 kpc , which is currently at @xmath2 90 kpc . this interaction also produces structures in the stellar disk that are similar to the monoceros stream at present day . chakrabarti et al . ( 2015 ) found an excess of faint variables at @xmath3 , and chakrabarti et al . ( 2016 ) obtained spectroscopic observations of three cepheid candidates that are part of this excess . the average radial velocities of these stars is @xmath2 163 km / s , which is large and distinct from the stellar disk of the galaxy ( which in the fourth quadrant is negative ) . using the period - luminosity relations for type i cepheids , we obtained an average distance of 73 kpc for these stars ( chakrabarti et al . 2016 ) . tidal interactions remain manifest in the stellar disk for many crossing times , but the gas is collisional and disturbances in the gas disk dissipate on the order of a dynamical time . therefore , an analysis of disturbances in the gas disk can provide a constraint on the time of encounter ( chakrabarti et al . 2011 ) . ultimately , a joint analysis of the gas ( a cold , responsive , dissipative component that is extended such as the hi disk ) _ and _ the stars ( that retain memory of the encounter for many crossing times ) holds the most promise for unearthing clues about recent _ and _ past encounters . 1:100 mass ratio perturber , ( right ) an image of the stellar density distribution . from chakrabarti & blitz ( 2009 ) . ] extended hi disks of local spirals have low sound speeds compared to their rotation velocity , and so are extremely sensitive to gravitational disturbances . furthermore , in the outskirts , atomic hydrogen traces the bulk of the ism ( bigiel et al . therefore , the outskirts of galaxies are less subject to the effects of feedback from supernovae and star formation that complicate the ism structure ( and the modeling thereof ) in the inner regions of galaxies ( christensen et al . 2013 ) . using the sensitivity of gaseous disks to disturbances , we constrained the mass and current radial distance of galactic satellites ( chakrabarti et al . 2011 ; cb11 ; cb09 ) and its azimuth to zeroth order by finding the best - fit to the low - order fourier modes ( i.e. , low m modes that trace large - scale structures , @xmath4 kpc- scale , in the disk ) of the projected gas surface density of an observed galaxy . we tested our ability to characterize the galactic satellites of spirals with optically visible companions , namely , m51 and ngc 1512 , which span the range from having a very low mass companion ( @xmath2 1:100 mass ratio ) to a fairly massive companion ( @xmath2 1:3 mass ratio ) . we accurately recover the masses and relative positions of the satellites in both these systems ( chakrabarti et al . 2011 ) . to facilitate a statistical study , we developed a simplified numerical approach along with a semi - analytic method to study the excitation of disturbances in galactic disks by passing satellites , and derived a simple scaling relation between the mass of the satellite and the sum of the fourier modes ( chang & chakrabarti 2011 ) . we later extended this method to also constrain the dark matter density profile of spiral galaxies ( chakrabarti 2013 ) . of particular interest now with the advent of gaia , is if we can detect the kinematical signature of this interaction in the stars that it perturbed at pericenter . if the stars for which radial velocities were obtained by chakrabarti et al . ( 2016 ) are indeed part of the dwarf galaxy predicted by cb09 , then such a detection would enable a constraint on the orbit and angular momentum of this dwarf galaxy . price - whelan et al . ( 2013 ) have noted that the gaia data can be complemented by measuring rr lyrae stars in the mid - infrared , which would allow for distances accurate to 2 % out to @xmath2 30 kpc , i.e. , this would give accurate distances for the outer hi disk . the puzzles of the milky way disk the large ripples in the gas disk , the many stellar streams , and the vertical waves in the galactic disk , need to be studied comprehensively . only for our galaxy , can we connect the dots between the orbits , the dynamical evolution of the satellites , the disk structure , and its place in the broader context of galaxy formation . a joint analysis of the data from gaia and hi surveys of the milky way should enable this effort .

the gaia satellite will provide unprecedented phase - space information for our galaxy and enable a new era of galactic dynamics . we may soon see successful realizations of galactoseismology , i.e. , inferring the characteristics of the galactic potential and sub - structure from a dynamical analysis of observed perturbations in the gas or stellar disk of the milky way . here , we argue that to maximally take advantage of the gaia data and other complementary surveys , it is necessary to build comprehensive models for both the stars and the gas . we outline several key morphological puzzles of the galactic disk and proposed solutions that may soon be tested .

1608.04150

the purpose of this paper is to discuss the hydrodynamic limit for interacting particle systems in the crystal lattice . problems of the hydrodynamic limit have been studied intensively in the case where the underlying space is the euclidean lattice . we extend problems to the case where the underlying space has geometric structures : the _ crystal lattice_. the crystal lattice is a generalization of classical lattice , the square lattice , the triangular lattice , the hexagonal lattice , the kagom lattice ( figure[crystals ] ) and the diamond lattice . before explaining difficulties for this extension and entering into details , we motivate to study these problems . there are many problems on the scaling limit of interacting particle systems , which have their origins in the statistical mechanics and the hydrodynamics . ( see @xcite , @xcite and references therein . ) the hydrodynamic limit for the exclusion process is one of the most studied models in this context . here we give only one example for exclusion processes in the integer lattice , which is a prototype of our results , due to kipnis , olla and varadhan ( @xcite ) . from the view point of physics and mathematics , it is natural to ask for the scaling limit of interacting particle systems evolving in more general spaces and to discuss the relationship between macroscopic behaviors of particles and geometric structures of the underlying spaces . in this paper , we deal with the crystal lattice , which is the simplest extension of the euclidean lattice @xmath0 . although the crystal lattice has periodic global structures , it has inhomogeneous local structures . on the other hand , crystal lattices have been studied in view of discrete geometric analysis by kotani and sunada ( @xcite , @xcite , @xcite , and the expository article @xcite ) . they formulate a crystal lattice as an abelian covering graph , and then they study random walks on crystal lattices and discuss the relationship between asymptotic behaviors of random walks and geometric structures of crystal lattices . in @xcite , they introduce the _ standard realization _ , which is a discrete harmonic map from a crystal lattice into a euclidean space to characterize an equilibrium configuration of crystals . in @xcite , they discuss the relationship between the _ albanese metric _ which is introduced into the euclidean space , associated with the standard realization and the central limit theorem for random walks on the crystal lattice . considering exclusion processes on the crystal lattice , one is interested to ask what geometric structures appear in the case where the interactions depend on the local structures . given a graph , the exclusion process on it describes the following dynamics : particles attempt to jump to nearest neighbor sites , however , they are forbidden to jump to sites which other particles have already occupied . so , particles are able to jump to nearest neighbor vacant sites . then , the problem of the hydrodynamic limit is to capture the collective behavior of particles via the scaling limit . if we take a suitable scaling limit of space and time , then we observe that the density of particles is governed by a partial differential equation as a macroscopic model . here it is necessary to construct a suitable scaling limit for a graph and to know some analytic properties of the limit space . a crystal lattice is defined as an infinite graph @xmath1 which admits a free action of a free abelian group @xmath2 with a finite quotient graph @xmath3 . we construct a scaling limit of a crystal lattice as follows : let @xmath4 be a positive integer . take a finite index subgroup @xmath5 of @xmath2 , which is isomorphic to @xmath6 when @xmath2 is isomorphic to @xmath0 . then we take the quotient of @xmath1 by @xmath5-action : @xmath7 . we call this finite quotient graph @xmath7 the _ @xmath4-scaling finite graph_. the quotient group @xmath8 acts freely on @xmath7 . here we consider exclusion processes on @xmath7 . to observe these processes in the continuous space , we embed @xmath7 into a torus . we construct an embedding map @xmath9 from @xmath7 into a torus by using a harmonic map @xmath10 in the discrete sense in order that the image @xmath11 converges to a torus as @xmath4 goes to the infinity . ( here the convergence of metric spaces is verified by using the gromov - hausdorff topology , however , we do not need this notion in this paper . ) then we obtain exclusion processes embedded by @xmath9 into the torus . in this paper , we deal with the simplest case among exclusion processes : _ the symmetric simple exclusion process _ and its perturbation : _ the weakly asymmetric simple exclusion process_. in the latter case , we obtain a heat equation with nonlinear drift terms on torus as the limit of process of empirical density ( theorem[main ] and examples below ) . we observe that the diffusion coefficient matrices and nonlinear drift terms can be computed by data of a finite quotient graph @xmath3 and a harmonic map @xmath10 . ( see also examples in section [ harmonic ] . ) the hydrodynamic limit for these processes on the crystal lattice is obtained as an extension of the one on @xmath0 . so , first , we review the outline of the proof for @xmath0 , following the method by guo , papanicolaou and varadhan in @xcite . since the lattice @xmath0 is naturally embedded into @xmath12 , the combinatorial laplacian on the scaled discrete torus converges to the laplacian on the torus according to this natural embedding . the local ergodic theorem is the key step of the proof since it enables us to replace local averages by global averages and to verify the derivation of the limit partial differential equation . it is formulated by using local functions on the configuration space and the shift action on the discrete torus . the proof of the local ergodic theorem is based on the one - block estimate and the two - blocks estimate . roughly speaking , the one - block estimate is interpreted as the local law of large numbers and the two - blocks estimate is interpreted as the asymptotic independence of two different local laws of large numbers . second , we look over the outline of the proof for the crystal lattice . there are two main points with regard to the difference between @xmath0 and the crystal lattice , that are the convergence of the laplacian and the local ergodic theorem . although the crystal lattice @xmath1 is embedded into an euclidean space by a harmonic map @xmath10 , the combinatorial laplacian on the image of the @xmath4-scaling finite graph @xmath11 does not converge to the laplacian on the torus straightforwardly . it is proved by averaging each fundamental domain by @xmath2-action because of the local inhomogeneity of the crystal lattice . thus , it is necessary to obtain the local ergodic theorem compatible with the convergence of the laplacian . furthermore , it is also necessary to obtain the local ergodic theorem compatible with the local inhomogeneity of the crystal lattice . for these reasons , we have to modify the local ergodic theorem in the case of crystal lattices . to formulate the local ergodic theorem in the crystal lattice , we introduce the notion of _ @xmath2-periodic local function bundles_. a @xmath2-periodic local function bundle is a family of local functions on the configuration space which is parametrized by vertices periodically . moreover , we introduce two different ways to take local averages of a @xmath2-periodic local function bundle . the first one is to take averages per each fundamental domain as a unit . the second one is to take averages on each @xmath2-orbit . the local ergodic theorem in the crystal lattice is formulated by using @xmath2-periodic local function bundles , two types of local averages and the @xmath13-action on the @xmath4-scaling finite graph @xmath7 . in fact , we use only special @xmath2-periodic local function bundles to handle the weakly asymmetric simple exclusion process . the proof of this local ergodic theorem is also based on the one - block estimate and the two - blocks estimate . proofs of these two estimates are analogous to the case of the discrete torus since we use the fact that the whole crystal lattice is covered by the @xmath2-action of a fundamental domain in the first type of the local average and we restrict to a @xmath2-orbit in the second type of the local average . in this paper , we call the local ergodic theorem the _ replacement theorem _ and prove it in the form of the super exponential estimate . the derivation of the hydrodynamic equation is the same manner as the case of the discrete torus . let us mention related works . interacting particle systems are categorized into the gradient system and the non - gradient system , according to types of interactions . we call the system the gradient system when the interaction term is represented by the difference of local functions . otherwise , we call the system the non - gradient system . we mention a recent work on the non - gradient system by sasada @xcite . the symmetric simple exclusion process is a model of the gradient system . our problems essentially correspond to problems for the gradient system since the hydrodynamic limit for the weakly asymmetric simple exclusion process is reduced to the one for the symmetric simple exclusion process , following @xcite . as for the hydrodynamic limit on spaces other than the euclidean lattice , jara investigates the hydrodynamic limit for zero - range processes in the sierpinski gasket ( @xcite ) . as for the crystal lattice , there is another type of the scaling limit . in @xcite , shubin and sunada study lattice vibrations of crystal lattices and calculate one of the thermodynamic quantities : the specific heat . they derive the equation of motion by taking the continuum limit of the crystal lattice . as a further problem , we mention the following problem : recently , attentions have been payed for interacting particle systems evolving in random environments ( e.g. , @xcite , @xcite and @xcite ) . for example , the quenched invariance principle for the random walk on the infinite cluster of supercritical percolation of @xmath0 with @xmath14 is proved by berger and biskup ( @xcite ) . their argument is based on a harmonic embedding of percolation cluster into @xmath12 . our use of the harmonic map @xmath10 and local function bundles will play a role in the systematic treatment of particle systems in more general random graphs . furthermore , the hydrodynamic limit on the inhomogeneous crystal lattice is considered as the case where the crystal lattice has topological defects . this problem would be interesting in connection with material sciences . this paper is organized as follows : in section [ crystal lattices ] , we introduce the crystal lattice and construct the scaling limit by using discrete harmonic maps . in section [ hydrodynamic limit ] , we formulate the weakly asymmetric simple exclusion process on the crystal lattice and state the main theorem ( theorem [ main ] ) . in section [ replacement theorem ] , we introduce @xmath2-periodic local function bundles and show the replacement theorem ( theorem [ super exponential estimate ] ) . we prove the one - block estimate and the two - blocks estimate . in section [ the proof of the main theorem ] , we derive the quasi - linear parabolic equation , applying the replacement theorem and complete the proof of theorem [ main ] . section [ appendixa ] is appendix;a . we prove some lemmas related to approximation by combinatorial metrics to complete the scaling limit argument . section [ appendixb ] is appendix;b . we refer an energy estimate of a weak solution and a uniqueness result for the partial differential equation to this appendix . _ landau asymptotic notation . _ throughout the paper , we use the notation @xmath15 to mean that @xmath16 as @xmath17 . we also use the notation @xmath18 to mean that @xmath16 as @xmath19 . in this section , we introduce the crystal lattice as an infinite graph and its realization into the euclidean space . let @xmath20 be a locally finite connected graph , where @xmath21 is a set of vertices and @xmath22 a set of all oriented edges . the graph @xmath1 may have loops and multiple edges . for an oriented edge @xmath23 , we denote by @xmath24 the origin of @xmath25 , by @xmath26 the terminus and by @xmath27 the inverse edge of @xmath25 . here we regard @xmath1 as a weighted graph , whose weight functions on @xmath21 and @xmath22 are all equal to one . we call a locally finite connected graph @xmath20 a _ @xmath2-crystal lattice _ if a free abelian group @xmath2 acts freely on @xmath1 and the quotient graph @xmath28 is a finite graph @xmath29 . more precisely , each @xmath30 defines a graph isomorphism @xmath31 and the graph isomorphism is fixed point - free except for @xmath32 . in other words , a @xmath2-crystal lattice @xmath1 is an abelian covering graph of a finite graph @xmath3 whose covering transformation group is @xmath2 . let us construct an embedding of a @xmath2-crystal lattice @xmath1 into the euclidean space @xmath12 of dimension @xmath33 . given an injective homomorphism @xmath34 such that there exits a basis @xmath35 , @xmath36 then we define a harmonic map associated with @xmath37 . fix an injective homomorphism @xmath37 as above . we call an embedding @xmath38 , a _ @xmath37-periodic harmonic map _ if @xmath10 satisfies the followings : _ @xmath10 is @xmath2-periodic _ , i.e. , for any @xmath39 and any @xmath30 , @xmath40 and _ @xmath10 is harmonic _ , i.e. , for any @xmath39 , @xmath41=0 $ ] , where @xmath42 . we note that a @xmath37-periodic harmonic map @xmath10 depends on @xmath37 and call @xmath10 a _ periodic harmonic map _ in short when we fix some @xmath37 . for @xmath43 , we take a lift @xmath44 of @xmath25 , and define @xmath45 . by the @xmath2-periodicity , @xmath46 does not depend on the choices of lifts . for @xmath47 , let us define a @xmath48-matrix by @xmath49 here the matrix is symmetric and positive definite . we call the matrix @xmath50 the _ diffusion coefficient matrix_. _ examples _ 1 . the one dimensional standard lattice . the one dimensional standard lattice @xmath1 which we identify the set of vertices @xmath21 with @xmath51 and the set of ( unoriented ) edges with the set of pairs of vertices @xmath52 . now @xmath51 acts freely on @xmath1 by the additive operation in @xmath51 and the quotient finite graph consists of one vertex and one loop as un oriented graph . when we regard @xmath1 as an oriented graph , we add both oriented edges to @xmath1 and the quotient graph consists of one vertex and two oriented loops ( figure [ one - dim ] ) . + let us choose a canonical injective homomorphism @xmath53 . in our formulation , choose a basis @xmath54 in @xmath55 and define @xmath53 by setting @xmath56 for @xmath57 so that @xmath58 . by identifying the set of vertices of @xmath1 with @xmath51 , we define an embedding map @xmath59 , @xmath60 . this embedding map @xmath10 is a @xmath51-periodic harmonic map . in this case , + 3 . let us give another example of periodic harmonic map for the one dimensional standard lattice @xmath1 . now we define a @xmath51-action on @xmath1 in the following way : for @xmath62 , @xmath39 , define @xmath63 . then this induces a free @xmath51-action on @xmath1 and the quotient graph consists of two vertices and two edges between them as an unoriented graph . let @xmath37 be the injective homomorphism as the same as in example 0a . we define an embedding map @xmath64 by setting @xmath65 , @xmath66 . then @xmath10 is a periodic harmonic map . the image of @xmath10 is not isomorphic to the previous one ( figure [ one - dim2 ] ) . in this case , + in example 0b.,width=487 ] 4 . the square lattice . the square lattice has the standard embedding in @xmath68 and this embedding is shown to be periodic and harmonic in our sense in the following . we identify the set of vertices of the square lattice @xmath1 with @xmath69 and the set of edges with the set of pairs of vertices @xmath70 . now @xmath69 acts freely on @xmath1 by the additive operation in @xmath69 and the quotient graph is the bouquet graph with one vertex and two unoriented loops . when we regard @xmath1 as an oriented graph , we add both oriented edges to @xmath1 and the quotient finite graph is the bouquet graph with one vertex and four oriented loops . + let us choose a canonical injective homomorphism @xmath71 . that is , choose a basis @xmath72 in @xmath68 and define @xmath71 by setting @xmath73 for @xmath74 so that @xmath75 . by identifying the set of vertices of @xmath1 with @xmath69 , we define an embedding map @xmath59 , @xmath76 . this embedding map @xmath10 is a @xmath69-periodic harmonic map . in this case , @xmath77 6 . let us give another example of periodic harmonic map for the square lattice @xmath1 . choose a basis @xmath78 in @xmath68 and define @xmath71 by setting @xmath79 for @xmath74 so that @xmath80 . in the same way as above example 1a , we define an embedding map @xmath59 , @xmath76 . ( figure [ square ] . ) this embedding map @xmath10 is a @xmath69-periodic harmonic map . in this case , @xmath81 + 7 . let us give an example an embedding map @xmath10 which is periodic but not harmonic . we choose an action of @xmath69 on the square lattice @xmath1 in the following way : again , we identify the set of vertices @xmath21 of @xmath1 with @xmath69 . for @xmath82 , @xmath83 , define @xmath84 . then this induces a free @xmath51-action and the quotient graph consists of two vertices , two edges between them and one loop on each vertex ( two loops ) as an unoriented graph . let @xmath37 be the same as in example 1a . we define an embedding map @xmath85 by setting @xmath86 , @xmath87 for @xmath88 . then @xmath89 is periodic but not harmonic since for @xmath90 , @xmath91 = ( 1 , 0)+(-1 , 0 ) + ( 1 , 1/2 ) + ( 1 , -1/2)=(2 , 0 ) \neq ( 0,0)$ ] . the hexagonal lattice . + the hexagonal lattice admits a free @xmath69-action with the quotient graph consisting of two vertices and three edges as an unoriented graph . we define a fundamental subgraph @xmath92 by setting the set of vertices @xmath93 and the set of ( unoriented ) edges @xmath94 . then the hexagonal lattice has a subgraph isomorphic to @xmath92 and is covered by copies of the subgraph translated by the @xmath69-action . choose a basis @xmath95 in @xmath68 and define @xmath71 by setting @xmath79 for @xmath74 so that @xmath80 . we define an embedding map @xmath10 by setting @xmath96 , @xmath97 , @xmath98 and @xmath99 for @xmath88 . ( figure [ hexagonal ] . ) then @xmath10 is a periodic harmonic map . in this case , @xmath100 + 9 . the kagom lattice . + the kagom lattice admits a free @xmath69-action with the quotient graph consisting of three vertices and six edges ( two edges between each pair of vertices ) as an unoriented graph . we define a fundamental subgraph @xmath92 by setting the set of vertices @xmath101 and the set of ( unoriented ) edges @xmath102 . then the kagom lattice has a subgraph isomorphic to @xmath92 and is covered by copies of the subgraph translated by the @xmath69-action . choose a basis @xmath95 in @xmath68 and define @xmath71 as the same as in example 2 . we define an embedding map @xmath10 by setting @xmath103 , @xmath104 , @xmath105 , @xmath106 , @xmath107 for @xmath88 . ( figure [ kagome ] . ) then @xmath10 is a periodic harmonic map . in this case , @xmath100 + _ remark_. the notion of periodic harmonic map on @xmath2-crystal lattice is studied by kotani and sunada and including the standard realization which they introduced in @xcite as a special case . they use harmonic maps to characterize _ equilibrium configurations _ of crystals . in fact , a periodic harmonic map is characterized by a critical map for some discrete analogue of energy functional . the standard realization is not only a critical map but also the map whose energy itself is minimized by changing flat metrics on torus with fixed volume . ( more precisely , see@xcite ) . the existence of periodic harmonic map for every injective homomorphism producing lattices in @xmath12 and the uniqueness up to translation is proved in theorem 2.3 and theorem 2.4 in @xcite . let us construct the scaling limit of the crystal lattice . suppose that @xmath2 is isomorphic to @xmath0 . let @xmath108 be an arbitrary positive integer and @xmath5 the subgroup isomorphic to @xmath6 . the subgroup @xmath5 acts also freely on @xmath1 and its quotient graph @xmath109 is also a finite graph @xmath110 . then @xmath111 acts freely on @xmath7 . we call @xmath7 the _ @xmath4-scaling finite graph_. the map @xmath112 satisfies that @xmath113 for all @xmath39 and all @xmath30 since @xmath10 is @xmath2-equivariant . we have the torus @xmath114 , equipped with the flat metric induced from the euclidean metric . the torus depends on @xmath115 , however , we do not specify it in the following . then the map @xmath116 induces the map @xmath117 we call @xmath9 the _ @xmath4-scaling map_. ( figure[scaling ] . ) @xmath118 -scaling finite graph by a harmonic map in the covering space , width=487 ] next , we observe convergence of the combinatorial laplacian on @xmath7 . since the degrees of @xmath119 might be different , depending on each @xmath120 , we consider average " of the combinatorial laplacian on a fundamental domain . let @xmath121 be the degree of a vertex @xmath119 , i.e. , the cardinality of the set @xmath122 . define the combinatorial laplacian @xmath123 associated with @xmath124 acting on the space of continuous functions @xmath125 by @xmath126,\ ] ] for @xmath127 and @xmath119 . we show that the combinatorial laplacian converges to the laplacian on @xmath128 in the following sense : for every twice continuous derivative functions @xmath129 , for every @xmath130 , for each @xmath131 , take an arbitrary sequence of vertices @xmath132 such that @xmath133 is a lift of @xmath120 and @xmath134 as @xmath17 , then by the taylor formula , @xmath135 since @xmath10 is harmonic , @xmath136 since @xmath137 , the last term is equal to @xmath138 , where @xmath50 is a diffusion coefficient matrix and @xmath139 and @xmath140 . we formulate the symmetric simple exclusion process and the weakly asymmetric simple exclusion process in crystal lattices . as we see below , the former is a particular case of the latter . let @xmath110 be the @xmath4-scaling finite graph of @xmath1 . we denote the configuration space by @xmath141 . we denote the configuration space for the whole crystal lattice @xmath142 by @xmath143 . each configuration is defined by @xmath144 with @xmath145 or @xmath146 and by @xmath147 in the same way . we consider the bernoulli measure @xmath148 and @xmath149 on @xmath150 , @xmath151 , respectively , for @xmath152 . they are defined as the product measures of the bernoulli measure @xmath153 on @xmath154 , where @xmath155 . let @xmath156 be the @xmath157-space of @xmath55-valued functions on @xmath150 . the action of @xmath13 on @xmath7 lifts on @xmath150 by setting @xmath158 for @xmath159 and @xmath119 . the group @xmath13 also acts on @xmath156 by @xmath160 for @xmath161 . for @xmath162 and @xmath163 , we denote by @xmath164 the configuration defined by exchanging the values of @xmath165 and @xmath166 , i.e. , @xmath167 for each @xmath162 , we define the operator @xmath168 by setting @xmath169 . we see that @xmath170 and @xmath171 for @xmath172 . the above notations also indicate corresponding ones for @xmath173 the configuration space on the whole crystal lattice . the symmetric simple exclusion process is defined by the generator @xmath174 acting on @xmath175 as @xmath176 the weakly asymmetric simple exclusion process is defined as a perturbation of the symmetric simple exclusion process . we denote by @xmath177\times \mathbb{t}^{d})$ ] the space of continuous functions with continuous derivatives in @xmath178 $ ] and the twice continuous derivatives in @xmath128 . for each function @xmath179\times \mathbb{t}^{d})$ ] , the weakly asymmetric simple exclusion process on @xmath7 is defined by the generator @xmath180 acting on @xmath175 as @xmath181 where @xmath182.\ ] ] here @xmath183 is the @xmath4-scaling map . the meaning of the perturbation is as follows : we introduce a `` small '' drift depending on space and time in particles . in the original process , a particle jumps with rate @xmath184 from @xmath24 to @xmath26 ( @xmath25 is an edge ) at time @xmath185 , while in the perturbed process , a particle jumps approximately with rate @xmath186 therefore , the external field which is now @xmath187 gives a small asymmetry of the order @xmath188 in the jump rate . notice that we obtain the symmetric simple exclusion process when @xmath189 is constant . _ remark _ the weakly asymmetric simple exclusion process which we introduced here does not include the well - studied case where for one dimensional lattice , the external field is @xmath190 for some constant @xmath191 and its limit equation produces the viscous burgers equation ( e.g. , @xcite ) . this process corresponds to the case with @xmath192 which we do not treat here . let @xmath193 , z_{n})$ ] be the space of paths which are right continuous and have left limits for some arbitrary fixed time @xmath194 . for a probability measure @xmath195 on @xmath150 , we denote by @xmath196 the distribution on @xmath193 , z_{n})$ ] of the continuous time markov chain @xmath197 generated by @xmath198 with the initial measure @xmath195 . the main theorem is stated as follows : [ main ] let @xmath199 $ ] be a measurable function . if a sequence of probability measures @xmath195 on @xmath150 satisfies that @xmath200=0,\ ] ] for every @xmath201 and for every continuous functions @xmath202 , then for every @xmath203 , @xmath204=0,\ ] ] for every @xmath201 and for every continuous functions @xmath202 , where @xmath205 is the unique weak solution of the following quasi - linear parabolic equation : @xmath206 here we define @xmath207 for @xmath43 . we give examples corresponding to ones in section [ harmonic ] . _ examples _ 1 . the one dimensional standard lattice . + for the embedding in example 0a , we recover the equation in theorem 3.1 in @xcite : @xmath208 + for the embedding in example 0b . , we have the following equation : @xmath209 2 . the square lattice . + for the square lattice and its embedding in example 1a . , we have the following equation : @xmath210 + for the square lattice and its embedding in example 1b . , we have the following equation : + @xmath211 3 . the hexagonal lattice , the kagom lattice . + for the hexagonal lattice , the kagom lattice and their embeddings in example 2 . and 3 , we have the following same equation : @xmath212 in this section , we formulate the replacement theorem and give its proof . the replacement theorem is given by the form of super exponential estimate and follows from the one - block estimate and the two blocks estimate . for our purpose , we introduce local function bundles which describe the local interactions of particles and the two types of local averages for local function bundles . a local function bundle @xmath213 on @xmath214 is a function @xmath215 , which satisfies that for each @xmath216 there exists @xmath217 such that @xmath218 depends only on @xmath219 . here @xmath220 is the graph distance in @xmath1 . we say that a local function bundle @xmath221 is @xmath2-periodic if it holds that @xmath222 for any @xmath30 , @xmath39 and @xmath147 . here we give examples of @xmath2-periodic local function bundles on @xmath223 . we use the first one and the third one later . _ examples _ * if we define @xmath221 by for @xmath39 and @xmath147 @xmath224 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . * if we define @xmath221 by for @xmath39 and @xmath147 @xmath225 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . * fix @xmath226 . if we define @xmath227 by @xmath228 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . note that a @xmath2-periodic local function bundle @xmath221 induces a map @xmath229 for large enough @xmath4 in the natural way . to abuse the notation , we indicate the induced map by the same character @xmath213 . first , for @xmath230 , we define the @xmath231-ball by @xmath232 here @xmath233 is the word metric appearing in section [ approximation ] . we regard that @xmath234 via the covering map when @xmath4 is large enough for @xmath231 . for a local function bundle @xmath215 , we define the local average of @xmath213 on blocks @xmath235 by for @xmath39 , @xmath236b(d_{x_{0}},r)\right|}\sum_{z \in [ x]b(d_{x_{0}},r ) } f_{z } : z \to \mathbb{r},\ ] ] where @xmath237 $ ] is a unique element @xmath30 such that @xmath238 and @xmath239b(d_{x_{0}},r)\right|$ ] denotes the cardinality of the set , which is equal to @xmath240 . note that @xmath241 for every @xmath242 . as a special case , we define for @xmath147 and @xmath39 , @xmath243b(d_{x_{0}},r)\right|}\sum_{z \in [ x]b(d_{x_{0}},r ) } \eta_{z } : z \to \mathbb{r}.\ ] ] second , we define the local average of @xmath213 on each @xmath2-orbit , @xmath244 by for @xmath39 , @xmath245 note that @xmath246 and @xmath247 are @xmath2-periodic when @xmath213 is @xmath2-periodic . if a local function bundle @xmath213 is @xmath2-periodic and @xmath4 is large enough , then @xmath248 induce the functions on @xmath150 in the natural way . to abuse the notation , we indicate the induced maps by the same characters @xmath248 . for a local function bundle @xmath215 , for @xmath39 , let us define @xmath249 $ ] , the expectation with respect to the bernoulli measure @xmath149 . the following estimate allows us to replace the local averages of the local function bundle by the global averages of the empirical density . we call the following theorem the replacement theorem . we prove it in the form of the super exponential estimate . [ super exponential estimate](super exponential estimate of the replacement theorem ) fix @xmath194 . for any @xmath2-periodic local function bundles @xmath215 , for every @xmath242 and for every @xmath201 , it holds that @xmath250 where @xmath251 note that for every @xmath242 , @xmath252 @xmath253 . we denote by @xmath254 the corresponding distribution on @xmath193 , z_{n})$ ] of continuous time markov chain @xmath197 generated by @xmath255 with the initial measure @xmath195 . furthermore , we denote by @xmath256 the corresponding distribution on @xmath193 , z_{n})$ ] of continuous time markov chain @xmath197 generated by @xmath255 with the initial measure @xmath257 , i.e. , an equilibrium measure . we denote by @xmath258 the expectation with respect to @xmath196 , by @xmath259 the one with respect to @xmath254 and by @xmath260 the one with respect to @xmath256 . for a probability measure @xmath261 on some probability space , we also denote by @xmath262 the expectation with respect to @xmath261 . by the following proposition , we reduce the super exponential estimate for @xmath196 to the one for @xmath254 . [ radon - nikodym ] there exists a constant @xmath263 such that @xmath264 to simplify the notation , put @xmath265 for @xmath119 . to calculate the radon - nikodym derivative : @xmath266 \\ & = \exp\bigg[\sum_{x \in v_{n}}h(t , x)\eta_{x}(t)-\sum_{x \in v_{n}}h(0 , x)\eta_{x}(0 ) -\int_{0}^{t}\bigg\{\sum_{x \in v_{n}}\frac{\partial h}{\partial t}(t , x)\eta_{x}(t ) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \frac{n^{2}}{4}\sum_{e \in e_{n}}\big(\eta_{oe}(1-\eta_{te})e^{h(t , te)-h(t , oe ) } + \eta_{te}(1-\eta_{oe})e^{h(t , oe)-h(t , te ) } \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\eta_{oe}(1-\eta_{te } ) - \eta_{te}(1-\eta_{oe})\big)\bigg\}dt\bigg ] \\ & = \exp\bigg[\sum_{x \in v_{n}}h(t , x)\eta_{x}(t)-\sum_{x \in v_{n}}h(0 , x)\eta_{x}(0 ) -\int_{0}^{t}\big\{\sum_{x \in v_{n}}\frac{\partial h}{\partial t}(t , x)\eta_{x}(t ) \\ & + \frac{n^{2}}{2}\sum_{x \in v_{n}}\sum_{e \in e_{n , x}}\eta_{x}\left(e^{h(t , te)-h(t , oe)}-1\right ) - \frac{n^{2}}{4}\sum_{e \in e_{n}}\left(e^{h(t , te)-h(t , oe)}+e^{h(t , oe)-h(t , te)}-2\right)\eta_{oe}\eta_{te}\big\}dt\bigg ] . \\\end{aligned}\ ] ] by the inequality @xmath267 for @xmath268 , we have that @xmath269 \\ & -n^{2}\sum_{e \in e_{n , x}}\left(h(t , te)- h(t , oe)\right ) - \frac{1}{2}n^{2}\sum_{e \in e_{n , x}}\left(h(t , te)- h(t , oe)\right)^{2}\big| \\ & \le \frac{1}{6}n^{2}\sum_{e \in e_{n , x}}\left|h(t , te)- h(t , oe)\right|^{3}e^{\left|h(t , te)- h(t , oe)\right|},\end{aligned}\ ] ] and thus , since @xmath10 is harmonic , @xmath270 \\ & = \frac{1}{2}\sum_{e \in e_{n , x}}\sum_{i , j=1}^{d}\frac{\partial^{2}h}{\partial x_{i } \partial x_{j}}(t , x)v_{i}(e)v_{j}(e ) + \frac{1}{2}\sum_{e \in e_{n , x}}\left(\nabla_{{\bf v}(e)}h(t , x)\right)^{2 } + o_{n}.\end{aligned}\ ] ] furthermore , @xmath271 hence , there exists a constant @xmath263 depending only on @xmath189 and @xmath272 such that @xmath273 is bounded from above by @xmath274 . it completes the proof . the super exponential estimate for @xmath254 induces the one for @xmath196 since it holds that for any borel sets @xmath275 , z_{n})$ ] , @xmath276 . furthermore , it is enough to prove the super exponential estimate for @xmath256 since for any borel sets @xmath275 , z_{n})$ ] , @xmath277 . let @xmath278 be the spaces of probability measures on @xmath279 , respectively . define @xmath280 , @xmath281 the @xmath282-bernoulli measure on @xmath150 , @xmath151 , respectively . here we introduce a functional on @xmath283 , which is the dirichlet form for the density function . for @xmath284 , put the density @xmath285 . the dirichlet form of @xmath286 is defined by @xmath287 note that @xmath288 . _ the functional @xmath289 is also called the @xmath290-functional in the different literatures . let us define the subset of the space of probability measures on @xmath150 by for @xmath291 , @xmath292 the proof of the super exponential estimate for @xmath256 is reduced to the following : [ local ergodic theorem ] for every @xmath291 and every @xmath242 , @xmath293 first , we prove theorem[super exponential estimate ] by using therem[local ergodic theorem ] . fix any @xmath294 . by the above argument , it is enough to show that for any @xmath2-periodic local function bundles @xmath215 , for every @xmath242 and for every @xmath201 , @xmath295 where @xmath251 by the chebychev inequality , for every @xmath296 and every @xmath201 , @xmath297 now , the operator @xmath298 acting on @xmath299 is self - adjoint for all @xmath296 and all @xmath242 . let @xmath300 be the largest eigenvalue of @xmath298 . by using the feynman - kac formula , @xmath301 therefore , it is suffice to show that for every @xmath296 , @xmath302 since ( [ eigenvalue ] ) implies that @xmath303 and we obtain the theorem by taking @xmath304 to the infinity . by the variational principle , the largest eigenvalue @xmath300 is represented by the following : @xmath305 see @xcite appendix 3 for more details . denote the average of @xmath261 by @xmath13-action by @xmath306 then @xmath307 is @xmath13-invariant so that @xmath308=\mathbb{e}_{\widetilde \mu}\left|\widetilde f_{x , k } - \langle f_{x}\rangle(\overline \eta_{x_{0 } , \epsilon n})\right|.\ ] ] the functional @xmath309 is also @xmath13-invariant , i.e. , @xmath310 for @xmath284 , @xmath311 . thus , it is suffice to consider @xmath13-invariant measures @xmath261 to estimate the largest eigenvalue @xmath300 . furthermore , it is suffice to consider the case where @xmath261 satisfies @xmath312 there exists a constant @xmath313 depending on @xmath213 such that @xmath314 , thus we reduce @xmath284 to every @xmath13-invariant measure satisfying that for every @xmath315 , @xmath316 this shows that we can reduce to @xmath317 for every @xmath318 , and thus it is enough to show for every @xmath315 and every @xmath242 , @xmath319 to obtain ( [ eigenvalue ] ) . this follows from theorem[local ergodic theorem ] . it completes the proof . in this section , we prove the one - block estimate . we regard a probability measure @xmath261 on @xmath150 as one on @xmath151 by periodic extension . let @xmath320 be the covering map by @xmath5-action , and define the periodic inclusion @xmath321 by @xmath322 . we identify @xmath261 on @xmath150 with its push forward by @xmath323 . on the other hand , we identify an @xmath5-invariant probability measure on @xmath151 with a probabiltiy measure on @xmath150 . first , for a finite subgraph @xmath324 of x , we define the restricted state space @xmath325 and the @xmath282-bernoulli measure by @xmath326 on @xmath327 . let us define the operator acting on @xmath328 by @xmath329 . for @xmath330 , @xmath331 stands for the restriction of @xmath261 on @xmath327 and @xmath332 its density . we also define the corresponding dirichlet form of @xmath333 by @xmath334 for large enough @xmath4 , we regard @xmath335 as a subgraph of @xmath7 by taking a suitable fundamental domain in @xmath21 for @xmath5-action . by the convexity of the dirichlet form , @xmath336 by putting @xmath337 . the one - block estimate is stated as follows : [ the one - block estimate](the one - block estimate . ) for every @xmath2-periodic local function bundles , @xmath215 , every @xmath242 and every @xmath315 , @xmath338 for any probability measure @xmath339 , we apply the above argument by setting @xmath335 as @xmath340 and @xmath341 . since @xmath342 and @xmath343 are @xmath13-invariant , @xmath344 since @xmath339 and @xmath345 , it holds that @xmath346 we note that @xmath347 is compact with respect to the weak topology , and thus @xmath348 has a subsequence which convergences to some @xmath261 in @xmath347 . let @xmath349 be the set of all limit points of @xmath350 in @xmath347 . by the above argument , @xmath351 for all @xmath352 . therefore , since @xmath353 for every @xmath352 by the definition of @xmath354 , we obtain that @xmath355 for every @xmath356 and every @xmath357 . this shows that random variables @xmath358 are exchangeable under @xmath261 . by the de finetti theorem , there exists a probability measure @xmath359 on @xmath360 $ ] such that @xmath361 . since @xmath362 it is enough to show that @xmath363}\mathbb{e}_{\nu_{\rho}}\left| \widetilde f_{x , k } - \langle f_{x } \rangle \left(\overline \eta_{x_{0 } , k}\right ) \right|=0,\ ] ] for every @xmath2-invariant local function bundles @xmath213 . by the definition of the @xmath2-periodic local function bundle , there exists a constant @xmath364 such that for every @xmath39 , @xmath365 depends on at most @xmath366b(d_{x_{0}},l)\}$ ] . therefore we obtain that there exists a constant @xmath313 depending only on @xmath213 such that @xmath367\right]^{2}\le c(f)\cdot \frac{l^{d}}{k^{d } } \to 0 \ \ \ \ \ \text{as $ k \to \infty$}.\ ] ] note that @xmath368 $ ] for every @xmath242 since the bernoulli measure @xmath149 is @xmath13-invariant . in addition , we also obtain that there exists a constant @xmath369 not depending on @xmath370 , @xmath371 finally , since @xmath372 is a polynomial with respect to @xmath370 , in particular , uniformly continuous on @xmath360 $ ] , we obtain that @xmath373 } \mathbb{e}_{\nu_{\rho}}\left| \widetilde f_{x , k } - \langle f_{x } \rangle(\overline \eta_{x_{0 } , k } ) \right|=0 \ \ \ \ \ \text{for every $ x \in d_{x_{0}}$}.\ ] ] this concludes the theorem @xmath374 in this section , we prove the two - blocks estimate . we identify a probability measure on @xmath150 with its periodic extension on @xmath151 in the same manner as section[section of the one - block estimate ] . ( the two - blocks estimate)[the two - blocks estimate ] for every @xmath315 , @xmath375 let us denote by @xmath376 the space of probability measures on @xmath377 . we define the map for @xmath30 , @xmath378 by @xmath379 , @xmath147 . for @xmath330 , we define the push forward of @xmath261 by @xmath380 via @xmath381 . let us denote by @xmath382 the set of all limit points of @xmath383 in @xmath384 as @xmath17 and by @xmath385 the set of all limit points of @xmath386 as @xmath387 . we put @xmath388 , where @xmath389 stands for a copy of @xmath390 . then , it holds that @xmath391 note that @xmath392 we introduce two types of generators acting on @xmath393 and the corresponding dirichlet forms . the first one is used for treating two different states at the same time independently . the second one is used for treating exchanges of particles between two different states . as in section[section of the one - block estimate ] , we define a subgraph @xmath394 of @xmath1 by setting @xmath395 and the operator acting on @xmath328 by @xmath396 for @xmath397 , we denote by @xmath398 the restriction of @xmath261 on @xmath399 . define the dirichlet form of @xmath400 by @xmath401 let us introduce the notation which describes exchanges of states for @xmath402 . for @xmath403 , @xmath404 is the configuration obtained by exchanging values @xmath405 and @xmath406 , i.e. , @xmath407 is defined by setting @xmath408 @xmath409 moreover , for @xmath410 , we define @xmath411 . we define the operator acting on @xmath412 by @xmath413 the corresponding dirichlet form of @xmath400 is defined by @xmath414 we prove two lemmas needed later . the first one is easy to show , so we omit the proof . [ quasi - iso ] there exist constants @xmath415 such that for all @xmath30 , @xmath416 where @xmath220 is the graph distance of @xmath1 . for @xmath417 and for @xmath163 , @xmath418 is the configuration obtained by exchanging two values @xmath405 and @xmath419 , i.e. , @xmath420 and moreover for @xmath417 , we define the operator @xmath421 for @xmath422 . these notations also indicates ones for @xmath151 . the second one is the following : [ path ] for every @xmath13-periodic functions @xmath422 and every @xmath423 , @xmath424 for @xmath425 , there exists a path @xmath426 such that @xmath427 and @xmath428 . define a sequence of edges @xmath429 by setting @xmath430 . for @xmath163 , let us define @xmath431 , inductively . we note that @xmath432 . then we have that @xmath433 if @xmath434 is @xmath13-periodic , then for each @xmath435 , @xmath436 therefore @xmath437 it completes the proof . let us define the subset of @xmath384 by for a constant @xmath438 , @xmath439 then we have the following lemma . [ inclusion ] there exists a constant @xmath438 such that @xmath440 as in section[section of the one - block estimate ] , we define a subgraph @xmath441 of @xmath1 by setting @xmath442 for @xmath443 . we take large enough @xmath444 for the diameter of @xmath335 , @xmath443 , so that @xmath445 for @xmath446 and @xmath447 for @xmath448 by taking a suitable fundamental domain in @xmath21 by @xmath5-action . take @xmath449 . for @xmath450 , if we define @xmath451 by @xmath452 and @xmath453 then @xmath454 . let us consider the operator acting on @xmath328 by @xmath455 for @xmath456 , we denote the density of @xmath457 by @xmath458 . then @xmath459 for any @xmath456 , we put @xmath337 . by the convexity of the dirichlet form and the @xmath13-invariance of @xmath37 , @xmath460 since @xmath461 , @xmath462 therefore @xmath463 for any @xmath464 . furthermore @xmath465 for any @xmath466 by the continuity of the functional @xmath467 . for @xmath456 , by the convexity of the dirichlet form , @xmath468 by the @xmath13-invariance of @xmath456 and by lemma[path ] , @xmath469 by lemma[quasi - iso ] and the definition of @xmath456 , for all @xmath30 such that @xmath448 , @xmath470 by setting @xmath471 and the continuity of the @xmath467 , for every @xmath466 , we have that @xmath472 . it concludes that @xmath473 . let us prove theorem[the two - blocks estimate ] . denote by @xmath474 the set of all limit points of @xmath475 as @xmath19 . for every @xmath476 , it holds that @xmath477 and @xmath478 by the continuity of the functionals @xmath467 and @xmath479 . these show that for any @xmath480 , @xmath481 and thus for any @xmath482 and for any @xmath483 , @xmath484 , @xmath485 and @xmath486 , i.e. , @xmath487 is exchangeable on @xmath377 . by the de finetti theorem there exists a probability measure @xmath359 on @xmath488 $ ] such that @xmath489}\nu_{\rho}\otimes \nu_{\rho}\lambda(d\rho)$ ] . as in the proof of theorem[the one - block estimate ] , @xmath490}\mathbb{e}_{\nu_{\rho}}\left|\overline \eta_{x_{0 } , k } - \rho \right|^{2}=0 $ ] , therefore , by the triangular inequality , @xmath491}\mathbb{e}_{\nu_{\rho}\otimes \nu_{\rho}}\left|\overline \eta_{x_{0 } , k}-\overline \eta_{x'_{0 } , k}\right| \\ & \le 2 \sup_{\rho \in [ 0 , 1]}\mathbb{e}_{\nu_{\rho}}\left|\overline \eta_{x_{0 } , k}- \rho \right| \to 0 \ \ \ \ \ \text{as $ k \to \infty$}.\end{aligned}\ ] ] finally , by lemma[inclusion ] , @xmath473 for some @xmath438 and thus , @xmath492 this completes the theorem @xmath493 let us prove theorem[local ergodic theorem ] by using the one - block estimate theorem[the one - block estimate ] and the two - blocks estimate theorem[the two - blocks estimate ] . first , we note that there exist positive constants @xmath494 and @xmath495 such that for any @xmath147 , @xmath496 and thus there exists a constant @xmath497 depending on @xmath498 and @xmath443 such that for any @xmath147 , @xmath499 uniformly . then , since @xmath261 is @xmath2-invariant as a probability measure on @xmath151 , @xmath500 where the last inequality comes from the fact that for any @xmath501 it holds that @xmath502 . applying theorem[the two - blocks estimate ] , we have that @xmath503 for every @xmath2-periodic local function bundles @xmath213 , @xmath504 is uniformly continuous on @xmath488 $ ] . therefore , @xmath505 furthermore , applying the one - block estimate theorem[the one - block estimate ] , for every @xmath242 , @xmath506 it completes the proof of theorem[local ergodic theorem ] . in this section , we prove theorem[main ] . let @xmath507 be the @xmath4-scaling map . we define the empirical density by @xmath508 where @xmath509 is the delta measure at @xmath510 . the empirical density is the measure valued process . we denote by @xmath511 \times \mathbb{t}^{d})$ ] the space of continuous functions with continuous derivatives in @xmath178 $ ] and twice continuous derivatives in @xmath128 . for every @xmath512 \times \mathbb{t}^{d})$ ] , we define @xmath513 to abuse the notation , we denote the inner product in @xmath514 by @xmath515 let us define the process as follows : @xmath516 where @xmath517 and @xmath518 where @xmath519 here @xmath520 are martingales with respect to the filtration @xmath521 , where @xmath522 and it holds that @xmath523 then we have the following lemma by applying the doob inequality . [ doob ] @xmath524=0.\ ] ] for @xmath525 \times \mathbb{t}^{d})$ ] and for each @xmath162 , we have that @xmath526\left[j_{s}\left(\phi_{n}(te)\right)-j_{s}\left(\phi_{n}(oe)\right)\right].\end{aligned}\ ] ] by the regularity of @xmath527 and the compactness of @xmath128 , we note that there exists a constant @xmath528 depending only on @xmath527 , such that uniformly , @xmath529 thus @xmath530^{2 } \le n^{2}|e_{n}|c'\frac{c(j)^{2}}{|v_{n}|^{2}n^{2 } } = \frac{c''}{|v_{n}|},\ ] ] where @xmath531 is a constant such that @xmath532 and @xmath533 . then we obtain that @xmath534 applying the doob inequality for the right continuous martingale @xmath535 , @xmath536 \le 4 \mathbb{e}_{n}^{h}\left|m_{n}(t)\right|^{2},\ ] ] we conclude that @xmath537=0 $ ] . we denote by @xmath538 the space of nonnegative borel measures with the total measure less than or equal to one on @xmath128 , endowed with the weak topology . since @xmath128 is a compact metric space , the space of continuous functions @xmath125 with the supremum norm is separable . fix a dense countable subset @xmath539 of @xmath125 , then the weak topology of @xmath540 is given by the distance @xmath541 by @xmath542 for @xmath543 where @xmath544 . we note that @xmath540 with the weak topology is compact . define the space of paths in @xmath540 by @xmath545 , \mathcal{m}):=\left\{\xi_{\cdot } : [ 0 , t ] \to \mathcal{m } \ \text{$\xi$ is right continuous with left limits.}\right\},\ ] ] equipped with the skorohod topology . for a given process @xmath546 \to \mathcal{m}$ ] such that @xmath547 , \mathcal{m})\right)=1 $ ] , we denote by @xmath548 the distribution of @xmath549 on @xmath550 , \mathcal{m})$ ] . then we show that the sequence @xmath551 has a subsequential limit . the following proposition gives a sufficient condition for this . see @xcite section 4 , theorem 1.3 for the proof . [ relative compactness ] if for every @xmath552 , @xmath553 , and every @xmath201 , @xmath554 then there exists a subsequence @xmath555 and a probability measure @xmath556 on @xmath550 , \mathcal{m})$ ] such that @xmath557 weakly converges to @xmath556 as @xmath558 . the next proposition claims that each subsequential limit @xmath556 in proposition[relative compactness ] is absolutely continuous with respect to lebesgue measure on the torus for each time @xmath185 , and its density has the value in @xmath488 $ ] a.e . the proof is the same as in @xcite section 4 , pp.57 , so we omit the proof . [ absolute continuity ] all limit points @xmath556 of @xmath551 are concentrated on trajectories of absolutely continuous measures with respect to the lebesgue measure for each time @xmath185 , i.e. , there exists a borel set @xmath559 , \mathcal{m})$ ] such that @xmath560 and for every @xmath561 and every @xmath562 $ ] , @xmath563 is absolutely continuous with respect to the lebesgue measure @xmath564 . moreover , the density @xmath565 satisfies that @xmath566 , @xmath564-a.e . to simplify the notation , we put @xmath567 for @xmath568 , \mathbb{t}^{d})$ ] , respectively . then we have : @xmath569 \cdot ( -\eta_{oe}\eta_{te}+\eta_{oe})(j(t , te)- j(t , oe ) ) \\ & = \frac{n^{2}}{2|v_{n}|}\sum_{x \in v_{n}}\sum_{e \in e_{n , x}}\left[\left\{\exp\left(h(t , te ) - h(t , oe)\right)-1\right\}\cdot \left(j(t , te ) - j(t , oe)\right)\eta_{oe}+\left(j(t , te)- j(t , oe)\right)\eta_{oe}\right ] \\ & -\frac{n^{2}}{4|v_{n}|}\sum_{e \in e_{n}}\left\{\exp\left(h(t , te ) - h(t , oe)\right)- \exp\left(h(t , oe)-h(t , te)\right)\right\}\cdot \eta_{oe}\eta_{te}(j(t , te)- j(t , oe)).\end{aligned}\ ] ] for @xmath23 , we denote the directional derivative along @xmath46 by @xmath570\times \mathbb{t}^{d}$}.\ ] ] applying the inequality @xmath571 for @xmath268 , by the regularity of @xmath189 and by the compactness @xmath128 , there exists a constant @xmath318 not depending on each point of @xmath178\times \mathbb{t}^{d}$ ] such that for every @xmath4 and for every @xmath162 , @xmath572 @xmath573 @xmath574 by the convergence of the combinatorial laplacian in section[scaling limit ] , we have that @xmath575 hence , @xmath576 we replace @xmath196 by @xmath548 regarding the empirical density @xmath549 as the measure on @xmath550,\mathcal{m})$ ] . let us prove the following lemma : [ equi - continuity ] for every @xmath127 and for every @xmath201 , @xmath577=0.\ ] ] for every continuous functions @xmath578 , it holds that @xmath579 since by ( [ n2l ] ) there exists a constant @xmath580 such that for large enough @xmath4 , @xmath581 uniformly , we obtain that by the chebychev inequality and by the triangular inequality , for every @xmath201 , for every @xmath582 and for large enough @xmath4 , @xmath583 \le ( 1/\delta)\mathbb{e}_{n}^{h}\left[c \gamma + 2 \sup_{0 \le t \le t } \left|m_{n}(t)\right|\right].\ ] ] then by lemma[doob ] , @xmath584\le c\frac{\gamma}{\delta}.\ ] ] therefore for every @xmath201 , @xmath585=0.\ ] ] it completes the proof . we have the following estimate : there exists a constant @xmath586 depending only on @xmath189 and @xmath527 such that for every @xmath162 and for every @xmath587 , @xmath588 thus , by the regularity of @xmath189 and @xmath527 , putting @xmath589 for @xmath162 , @xmath590 here we regard @xmath591 as a local function bundle independent of states and denote by @xmath592 the local average . by the uniform continuity of the twice derivative of @xmath527 , putting @xmath593 it holds that @xmath594 here we also regard @xmath595 as a local function bundle . by the above argument , we obtain that @xmath596 here @xmath597 is the local function bundle appearing in the third example in section[local function bundles ] and @xmath598 its local average . by applying theorem[super exponential estimate ] and by the continuity of @xmath599 , @xmath600 and @xmath601 on the compact space @xmath178 \times \mathbb{t}^{d}$ ] , it holds that for every @xmath562 $ ] and for every @xmath201 , @xmath602 @xmath603 and @xmath604 here we use @xmath605 for every @xmath119 in the third estimate above . by the triangular inequality , it holds that for every @xmath562 $ ] and for every @xmath201 , @xmath606 applying the convergence of the combinatorial laplacian in section [ scaling limit ] , we have that @xmath607 recall that @xmath608 by lemma[doob ] and by the chebychev inequality , for every @xmath201 , @xmath609 furthermore , by the triangular inequality , we have that for every @xmath610 and for every @xmath562 $ ] , @xmath611 by lemma[characteristic function ] , we replace @xmath612 by @xmath613 and the summation for @xmath311 by the integral . since by lemma[equi - continuity ] and by proposition[relative compactness ] , the sequence @xmath614 is relatively compact in the weak topology , for a limit point @xmath556 there exists a subsequence @xmath615 weakly converging to @xmath556 . by proposition[absolute continuity ] , the empirical density @xmath616 concentrates on an absolutely continuous trajectory @xmath617 as @xmath558 . by the assumption of theorem[main ] , we replace @xmath618 by @xmath619 , and then we have that for every @xmath201 and for every @xmath620 $ ] , @xmath621 here we replace @xmath46 for @xmath226 by @xmath46 for @xmath43 by the @xmath2-invariance of @xmath622 . by the lebesgue dominated convergence theorem as @xmath19 and by the triangular inequality , we have that for every @xmath201 and for every @xmath620 $ ] , @xmath623 this shows @xmath624 concentrates on @xmath370 , which is a weak solution of the quasi - linear parabolic equation ( [ pde1 ] ) . furthermore , @xmath370 has finite energy by lemma[energy ] . by the uniqueness result of the weak solution lemma[uniqueness ] in section[appendixb ] , we conclude that the limit point @xmath556 of @xmath551 is unique and @xmath549 concentrates on @xmath625 as @xmath4 goes to the infinity . that is , for every @xmath201 , @xmath626 where @xmath627 is the skorohod distance in @xmath193 , \mathcal{m})$ ] . in particular , since @xmath194 is arbitrary , it follows that for every @xmath203 , for every @xmath201 and for every continuous functions @xmath127 , @xmath628=0.\ ] ] it completes the proof of theorem[main ] . take a @xmath51-basis @xmath629 of @xmath2 and identify @xmath2 with @xmath0 . define the standard generator system of @xmath2 by @xmath630 . we introduce the length function associated with @xmath631 , @xmath632 by @xmath633 for @xmath30 . then the map @xmath634 induces the metric in @xmath2 , which is called the word metric associated with @xmath631 . let @xmath635 be the image of @xmath629 by the natural homomorphism @xmath636 . then @xmath637 generates @xmath13 . the length function associated with @xmath638 , @xmath639 is also defined in the same way . to abuse the notation , we denote the word metric in @xmath13 associated with @xmath638 by the same symbol @xmath233 . we define an @xmath640-norm in @xmath641 by @xmath642 for @xmath643 and the distance @xmath644 in @xmath12 by @xmath645 for @xmath646 . denote by @xmath647 the induced metric in @xmath128 from @xmath648 . fix @xmath649 and a fundamental domain @xmath650 such that @xmath651 and @xmath652 is connected in the following sense : for any @xmath653 there exist a path @xmath654 in @xmath22 such that @xmath655 and @xmath656 are all in @xmath652 . this kind of set @xmath652 always exists if we take a spanning tree in @xmath3 and its lift in @xmath1 . to abuse the notation , we denote by @xmath657 the images of @xmath658 by the covering map , respectively . we also fix a fundamental domain @xmath659 which is identified with @xmath660 . to abuse the notation , we denote by @xmath661 the image of @xmath662 by the covering map . we define the map @xmath663:v \to \gamma$ ] as follows : for @xmath39 , there exists a unique element @xmath30 such that @xmath238 since @xmath2 acts on @xmath1 freely . define @xmath237:=\sigma$ ] . since there exists a constant @xmath664x_{0})-\phi(x)\|_{1}$ ] by the @xmath2-perodicity , we have that @xmath665x_{0})-\phi([z]x_{0})\|_{1 } -2c_{0 } \le \|\phi(x)-\phi(z)\|_{1 } \le \|\phi([x]x_{0})-\phi([z]x_{0})\|_{1 } + 2c_{0}\ ] ] for any @xmath666 . furthermore , since @xmath667x_{0})- \phi([z]x_{0})\|_{1}=\left|[x]-[z]\right|$ ] , we have that @xmath668-[z]\right| -2c_{0 } \le \|\phi(x)-\phi(z)\|_{1 } \le \left|[x ] -[z]\right| + 2c_{0}.\ ] ] as in section[harmonic ] , suppose that we have an injective homomorphism @xmath669 such that @xmath670 take a fundamental parallelotope @xmath671 . in the similar way to the above , we define the map @xmath663:\gamma \otimes \mathbb{r } \to \gamma$ ] as follows : for @xmath672 , there exists a unique element @xmath30 such that @xmath673 . define @xmath674:=\sigma$ ] . since there exists a constant @xmath675x_{0})- { \bf x}\|_{1}$ ] by the @xmath2-periodicity , we have that @xmath676x_{0})-\phi([{\bf z}]x_{0})\|_{1 } -2c_{1 } \le \|{\bf x } - { \bf z}\|_{1 } \le \|\phi([{\bf x}]x_{0})-\phi([{\bf z}]x_{0})\|_{1 } + 2c_{1}\ ] ] for any @xmath677 . furthermore , we have that @xmath678-[{\bf z}]\right| -2c_{1 } \le \|{\bf x}-{\bf z}\|_{1 } \le \left|[{\bf x } ] -[{\bf z}]\right| + 2c_{1}.\ ] ] let us define an @xmath679-ball in @xmath128 of the center @xmath680 by @xmath681 . we show the following : [ ball approximation ] there exists a constant @xmath682 depending only on @xmath679 such that for any @xmath680 and for any @xmath108 , @xmath683 here @xmath684 stands for the volume of a borel set @xmath272 and @xmath685 the cardinality of a set @xmath686 . for any @xmath680 , take a lift @xmath687 then @xmath688x_{0})- \tilde { \bf z}\|_{1 } \le c_{1}$ ] . for sufficiently small @xmath689 , take a lift @xmath690 of @xmath691 . again , from the above argument , it holds that @xmath692p \subset \widetilde b^{1}_{\tilde{\bf z}}(n\epsilon ) \subset \cup_{|\sigma| \le \epsilon n + 2c_{1}}\sigma [ \tilde{\bf z}]p$ ] . note that @xmath693 , and thus @xmath694p\right)\right| \le vol\left(p \right)(2^{d}/d!)\left((\epsilon n + 2c_{1})^{d } -(\epsilon n -2c_{1})^{d}\right)$ ] . since @xmath695p\right ) / vol\left(p\right)= \left| \cup_{|\sigma| \le \epsilon n}\sigma [ \tilde{\bf z}]d_{x_{0}}\right| / |v_{0}|$ ] and @xmath696 , it concludes that there exists a constant @xmath697 depending only on @xmath679 such that @xmath698d_{x_{0 } } \right|}{n^{d}|v_{0}|}\right| \le \frac{c(\epsilon)}{n}.\ ] ] the cardinality of the set @xmath699 is invariant under translation . it completes the proof . let us define a measure @xmath261 on @xmath128 by @xmath700 . let @xmath701 be a characteristic function defined by @xmath702 for the empirical density @xmath703 on @xmath128 , @xmath163 , then we have the following lemma . [ characteristic function ] there exists a constant @xmath704 depending only on @xmath689 , such that for any @xmath163 and any @xmath705 , @xmath706d_{x_{0}}\right|}\sum_{x \in \cup_{|\sigma|\le \epsilon n}\sigma [ z]d_{x_{0 } } } \eta_{x}\right| \le \frac{c(\epsilon)}{n},\ ] ] where @xmath707 take a lift @xmath708 of @xmath705 and a lift @xmath709 of @xmath710 . in the similar way to the proof of lemma[ball approximation ] , we obtain that @xmath711d_{x_{0 } } \subset \left\{x \in v \ \bigg| \ \left\|\frac{1}{n}\phi(x ) - \frac{1}{n}\phi(\tilde z)\right\|_{1 } \le \epsilon \right\ } \subset \bigcup_{|\sigma|\le \epsilon n + 2c_{0}}\sigma [ \tilde z]d_{x_{0}}.\ ] ] furthermore , we take a lift @xmath712 of @xmath163 , then it holds that @xmath713 and @xmath714d_{x_{0 } } } \widetilde \eta_{x}\right| \le \frac{1}{|v_{n}|}\sum_{x \in \cup_{\epsilon n - 2c_{0 } \le |\sigma|\le \epsilon n + 2c_{0}}\sigma [ \tilde z]d_{x_{0}}}\widetilde \eta_{x}.\ ] ] the last term is bounded by @xmath715 , and thus there exists a constant @xmath716 depending only on @xmath679 such that @xmath714d_{x_{0 } } } \widetilde \eta_{x}\right| \le \frac{c_{1}(\epsilon)}{n}.\ ] ] by lemma[ball approximation ] and @xmath717d_{x_{0 } } \right|=\left|\cup_{|\sigma| \le \epsilon n}\sigma d_{x_{0 } } \right|=\left|\cup_{|\sigma| \le \epsilon n}\sigma[z]d_{x_{0 } } \right|$ ] , @xmath718d_{x_{0 } } \right|}\right| & \le \frac{1}{|v_{n}|\mu\left(b^{1}_{\phi_{n}(z)}(\epsilon)\right)}\frac{1}{\left|\cup_{|\sigma| \le \epsilon n}\sigma[z]d_{x_{0 } } \right|}\frac{c(\epsilon)}{n } |v_{n}| \\ & \le \frac{c_{2}(\epsilon)}{n^{d+1}},\end{aligned}\ ] ] where @xmath719 is a constant depending only on @xmath679 . finally , @xmath720d_{x_{0}}\right|}\sum_{x \in \cup_{|\sigma|\le \epsilon n}\sigma [ z]d_{x_{0 } } } \eta_{x}\right| & \le \frac{1}{\mu\left(b^{1}_{\phi_{n}(z)}(\epsilon)\right)}\frac{c_{1}(\epsilon)}{n } + \frac{c_{2}(\epsilon)}{n^{d+1}}|v_{0}|(\epsilon n)^{d } \\ & \le \frac{c_{3}(\epsilon)}{n},\end{aligned}\ ] ] where @xmath721 is a constant depending only on @xmath679 . it completes the proof . [ energy ] suppose that @xmath722 is a sequence of probability measures on @xmath193,\mathcal{m})$ ] . for any limit point @xmath723 of @xmath722 , @xmath723-a.s . there exists a measurable function @xmath205 such that @xmath724 , @xmath370 has @xmath725\times \mathbb{t}^{d}),\ ] ] for @xmath726 , and satisfies @xmath727 for every @xmath728\times \mathbb{t}^{d})$ ] and @xmath726 . for fixed @xmath729 , we define a lattice @xmath730 whose vertex set @xmath731 is the subset of @xmath732 in the following : @xmath731 is the orbit of @xmath390 by @xmath13 , i.e. , @xmath733 . define @xmath734 the set of oriented edges @xmath213 such that @xmath735 for some @xmath736 . then @xmath13 acts on @xmath737 naturally . a configuration @xmath738 on @xmath7 induces the one on @xmath737 by restriction . we use the same symbol @xmath738 for this restriction . for @xmath739\times \mathbb{t}^{d})$ ] and for @xmath740 , we define @xmath741,\mathbb{r})$ ] , @xmath742 and @xmath743 for any @xmath284 , we have that @xmath744 we use the cauchy - schwarz inequality in the last inequality . by using the argument in the proof of lemma[path ] , there exists a constant @xmath580 such that @xmath745 since @xmath746 and @xmath747 , we get @xmath748 note that @xmath749 for large enough @xmath4 . consider for @xmath296 the self - adjoint operator @xmath750 and suppose @xmath751 to be the largest eigenvalue of this operator . by the variational formula @xmath752 by the simple inequality for @xmath753 , @xmath754 , the last formula is bounded by @xmath755 . we put @xmath756 . on the other hand , @xmath757 where @xmath758 is the directional derivative along @xmath759 and @xmath760 . by the entropy inequality , @xmath761 since @xmath762 for some constant @xmath531 , by the feynman - kac formula , we obtain @xmath763 \times \mathbb{t}^{d})}^{2 } + |v_{0}|c'.\ ] ] for a limit point of @xmath764 , @xmath723 , we get @xmath765 \le 16ct \|j\|_{l^{2}([0,t ] \times \mathbb{t}^{d})}^{2}+|v_{0}|c'.\ ] ] denote a countable dense subset of @xmath766\times \mathbb{t}^{d})$ ] by @xmath767 , we also get the following estimate : @xmath768 \le |v_{0}|c'.\ ] ] see @xmath769 pp.107 , section 5.7 for details . therefore for almost all @xmath370 , there exists @xmath770 such that for every @xmath728\times \mathbb{t}^{d})$ ] , @xmath771 that is , @xmath772 this implies the linear functional @xmath773\times \mathbb{t}^{d } ) \to \mathbb{r}$ ] defined by @xmath774 is extended on @xmath775\times\mathbb{t}^{d})$ ] . by the riesz representation theorem , there exists @xmath776\times\mathbb{t}^{d})$ ] such that @xmath777 for every @xmath728\times \mathbb{t}^{d})$ ] and every @xmath778 . this yields lemma[energy ] . [ uniqueness ] for any @xmath779\times \mathbb{t}^{d})$ ] , a weak solution of the quasi - linear partial differential equation @xmath780 with the measurable initial value @xmath199 $ ] , of bounded energy , i.e , @xmath781 is unique . first , the author would like to thank professor tsuyoshi kato for his constant encouragement and helpful suggestions . he wishes to express his gratitude to professors motoko kotani , nobuaki sugimine , satoshi ishiwata and makiko sasada for their valuable advice and helpful discussions . second , he would like to thank professor yukio nagahata for his helpful comments on this subject and the background of the hydrodynamic limit . third , he would like to thank doctors sei - ichiro kusuoka and makoto nakashima for their advices on the presentation of this paper . fourth , he would like to thank professor kazumasa kuwada for his valuable comments on the earlier version of the manuscript and his encouragement . the author could not write up this paper without a great deal of his advice . fifth , the author partially carried out this work at riken center for developmental biology in kobe . he would like to thank professors hiroki r. ueda and yohei koyama for their interest in this topic and their encouragement . sixth , the author partially carried out this work at max planck institute for mathematics in the sciences in leipzig . he would like to thank professors jrgen jost and nihat ay for helpful discussions . the author is supported by the research fellowships of the japan society for the promotion of science for young scientists . 99 berger , n. , biskup , m. : quenched invariance principle for simple random walk on percolation clusters . prob . theory related fields * 137 * , 83 - 120 ( 2007 ) bertini , l. , landim , c. , mourragui , m. : dynamical large deviations for the boundary driven weakly asymmetric exclusion process . * 37 * ( 6 ) , 2357 - 2403 ( 2009 ) faggionato , a. : random walks and exclusion processes among random conductances on random infinite clusters : homogenization and hydrodynamic limit . electron . j. probab . * 13 * , no . 73 , 2217 - 2247 ( 2008 ) gonalves , p. , jara , m. : density fluctuations for a zero - range process on the percolation cluster . comm . in probab . * 14 * , 382 - 395 ( 2009 ) guo , m.z . , papanicolaou , g.c . , varadhan , s.r.s . : nonlinear diffusion limit for a system with nearest neighbor interactions . * 118 * , 31 - 59 ( 1988 ) jara , m. : hydrodynamic limit for a zero - range process in the sierpinski gasket . comm . math . * 288 * , 773 - 797 ( 2009 ) kipnis , c. , landim , c. : scaling limits of interacting particle systems . grundlehren der mathematischen wissenschaften * 320 * , springer - verlag , berlin ( 1999 ) kipnis , c. , olla , s. , varadhan , s. r. s. : hydrodynamics and large deviation for simple exclusion processes . comm . pure appl . xlii , 115 - 137 ( 1989 ) kotani , m. , sunada , t. : albanese maps and off diagonal long time asymptotics for the heat kernel . phys . * 209 * , 633 - 670 ( 2000 ) kotani , m. , sunada , t. : standard realizations of crystal lattices via harmonic maps . trans . * 353 * , 1 - 20 ( 2001 ) kotani , m. , sunada , t. : large deviation and the tangent cone at infinity of a crystal lattice . math . z. * 254 * , 837 - 870 ( 2006 ) koukkous , a. : hydrodynamic behavior of symmetric zero - range processes with random rates . stochastic process appl . * 84 * 297 - 312 ( 1999 ) sasada , m. : hydrodynamic limit for two - species exclusion processes . appl . * 120 * , issue 4 , 494 - 521 ( 2010 ) shubin , m. , sunada , t. : geometric theory of lattice vibrations and specific heat , pure appl . math . q. * 2 * no . 3 , 745 - 777 ( 2006 ) spohn , h. : large scale dynamics of interacting particles , texts and monograph in physics , springer verlag , heidelberg ( 1991 ) sunada , t. : crystals that nature might miss creating . notices of the ams , * 55 * no . 2 , 208 - 215 ( 2008 )

we investigate the hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices . we construct a suitable scaling limit by using a discrete harmonic map . as we shall observe , the quasi - linear parabolic equation in the limit is defined on a flat torus and depends on both the local structure of the crystal lattice and the discrete harmonic map . we formulate the local ergodic theorem on the crystal lattice by introducing the notion of local function bundle , which is a family of local functions on the configuration space . the ideas and methods are taken from the discrete geometric analysis to these problems . results we obtain are extensions of ones by kipnis , olla and varadhan to crystal lattices .

1105.6220

recently there has been a renewed interest in the finite temperature dynamics of the one dimensional spin 1/2 heisenberg model , especially on the question of diffusive spin transport@xcite . in particular , it was argued that the integrability of the model implies pathological spin dynamics and presumably the absence of spin diffusion@xcite . the role of conservation laws was pointed out in reference@xcite were it was shown that in several quantum integrable models the uniform ( @xmath0 ) current correlations do not decay to zero at long times . this result , established using the mazur inequality@xcite , suggests pathological finite temperature dynamics . as far as the heisenberg model is concerned , the analysis of conservation laws has shown that the energy current operator commutes with the hamiltonian , suggesting anomalous finite-(@xmath1 ) energy density correlations . however , for zero magnetic field , this method turned out to be insufficient for deciding about the decay of the uniform spin current correlations . this case is closely related to the behavior of the finite temperature conductivity in the one dimensional model of spinless fermions at half - filling interacting with a nearest neighbor interaction ( the t - v " model@xcite ) . in this work , we address the issues raised above by the numerical diagonalization of the hamiltonian matrix on finite size lattices . more precisely , we study the implications of the energy current conservation on the ( @xmath1 ) energy density correlations , and , as an alternative route to the analysis of spin diffusion , we investigate the decay of the uniform ( @xmath0 ) spin current correlations . the paper is organized as follows : in section 2 , we recall the heisenberg hamiltonian and define the various quantities studied below . in section 3 we briefly summarize the phenomenological picture of diffusion . there , we also argue that the decay of the uniform spin current correlations to a finite value is incompatible with a diffusive behavior , _ assuming _ continuity in the wave - vector @xmath2 of the correlations at @xmath0 . next , we test these ideas in section 4 in the xy limit , where results can be obtained analytically . turning to the numerical results , in section 5.1 we present the energy density correlations at infinite temperature for the case of the isotropic heisenberg model . a simple ansatz for the observed behavior suggests a logarithmic dependence at low frequencies for the energy autocorrelation function . as far as the spin dynamics is concerned , numerous studies of the @xmath3 spin density correlations exist@xcite . therefore , in section 5.2 , we restrict ourselves to the decay of the uniform spin current correlations for various values of the anisotropy parameter @xmath4 and temperatures interestingly , it turns out that these do not decay to zero for @xmath5 . according to the argument given in section 3 , this result implies non - diffusive spin transport . section 6 contains a short discussion on experimental relevance of these findings and open questions . the anisotropic heisenberg hamiltonian for a chain of @xmath6 sites with periodic boundary conditions is given by : @xmath7 where @xmath8 , @xmath9 are the pauli spin operators with components @xmath10 at site @xmath11 . for a conserved quantity @xmath12 , @xmath13=0 $ ] , the continuity equation in @xmath14space defines the current @xmath15 : @xmath16 with @xmath17 and @xmath18 . setting @xmath19 we find the following spin and energy currents respectively : @xmath20 @xmath21 for the discussion of dynamic correlations at finite temperatures , we chose to analyze the anticommutator form : @xmath22 where @xmath23 is the thermal average at temperature @xmath24 over a complete set of states . further , the frequency dependent correlation function defined by : @xmath25 is symmetric in frequency , @xmath26 . a central point in our approach is the relation between the dynamic correlations of a quantity a and its corresponding current correlations , which we obtain by using the continuity equation ( [ ce ] ) : @xmath27 in particular , we will discuss the asymptotic value of the current correlations @xmath28 a finite value of @xmath29 translates to a @xmath30 peak in @xmath31 and , as we will discuss below , implies restrictions in the behavior of @xmath32 . an important observation is that the energy current @xmath33 of the heisenberg model commutes with the hamiltonian@xcite , so that @xmath34 , whereas the spin current does not . however , it will turn out that @xmath35 for @xmath5 , meaning that the spin current and energy current correlations are similar in the sense that in their frequency representation , they both exhibit a finite weight @xmath30 function . when we consider the @xmath3-dependent correlations of a conserved quantity @xmath36 such as the magnetization , it is usually assumed , largely on phenomenological grounds , that they exhibit a diffusive behavior in the long - time @xmath37 , short wavelength @xmath38 regime@xcite : @xmath39 where @xmath40 is the corresponding diffusion constant , or @xmath41 for @xmath42 . this lorentzian form correctly reduces to a @xmath30 function in the limit @xmath38 , as implied by @xmath13=0 $ ] . further , using the continuity equation ( [ cs ] ) for @xmath38 , we obtain : @xmath43 which gives the diffusion constant @xmath40 when first , the limit @xmath38 and then , @xmath44 are taken . on the other hand , if the current correlations for @xmath0 do not decay to zero at long times , @xmath45 and @xmath46 has a finite weight @xmath30 component which is incompatible with the diffusive form ( [ djbqo ] ) . in this reasoning , we must assume a regular behavior of the correlation functions in the @xmath2 variable . to summarize the argument , if a quantity @xmath36 is conserved ( @xmath13=0 $ ] ) and its current @xmath47 is either conserved ( @xmath48=0 $ ] ) , or @xmath45 , then continuity in @xmath2 at @xmath0 excludes a diffusive form ( [ db ] ) for the corresponding correlation @xmath49 . a simple model for testing these ideas is the @xmath50 limit ( @xmath51 ) , of the heisenberg model . in this case , both the energy current @xmath33 and the spin current @xmath52 commute with the hamiltonian . the model can be mapped to a free spinless fermion model by using a jordan - wigner transformation which allows us also to evaluate explicitly the spin and energy dynamic correlations at @xmath53 . in the spin case , these are well known results@xcite : @xmath54 @xmath55 these forms are indeed consistent with the conservation of both spin ( energy ) and spin current ( energy current ) as they reduce to a @xmath30 function when the limits @xmath38 , @xmath44 are taken . further , the time decay of the autocorrelations at @xmath53 is not of the form @xmath56 , as predicted by the diffusion hypothesis . indeed , @xmath57 @xmath58 which both behave as @xmath59 for @xmath60 . as we mentioned earlier , the energy current @xmath33 associated with the anisotropic heisenberg model ( [ heis ] ) commutes with the hamiltonian for all values of the parameter @xmath4 . therefore , the time correlations do not decay at all ( @xmath34 ) and according to the argument explained in section 3 , no diffusive energy transport occurs . however , the conservation of @xmath33 does not provide us with any details about the shape of @xmath61 at finite @xmath2 . in the absence of an analytical solution , we investigate this quantity by numerical diagonalization of the hamiltonian matrix on a ring of 16 sites . in figure 1 , we show @xmath61 for @xmath62 , which is experimenally the most interesting point as it describes isotropic quasi one - dimensional antiferromagnets . we study the high temperature limit @xmath53 , which is the most convenient for a numerical study as it involves the full excitation spectrum , but is also relevant experimentally for spin systems as the magnitude of @xmath63 can be of the order of @xmath64 . the plot is represented as histograms of width @xmath65 , all the frequencies which fall into one interval are summed up . the inset shows the normalized , integrated ( prior to summing nearby frequencies ) quantity @xmath66 which has the advantage of smoothing out the finite size discontinuities . to point out the practically linear integrated behavior of the pure heisenberg model , we also show the same quantity for a more generic case obtained by adding a next - next neighbor ( non - integrable ) interaction @xmath67 . the simplest way to describe this behavior is by means of plateaus " given by the following ansatz : @xmath68 which satisfy the first @xmath69 and the second @xmath70 exact moments for @xmath53 . further , this ansatz is compatible with the limit @xmath71 as implied by the conservation of energy . using the continuity equation ( [ cs ] ) , we obtain for small @xmath2 and @xmath72 @xmath73 which correctly reduces to a @xmath30-function for @xmath38 , in agreement with the conservation of the energy current @xmath74=0 $ ] . using this ansatz we find for the energy autocorrelation function ( obtained by integration over @xmath2 ) : @xmath75 a logarithmic behavior at low frequencies , in contrast to the diffusion form @xmath76 . we should stress that these results are only _ indicative _ , as they are obtained from small size lattices which can provide reliable information only for correspondingly high frequencies and large wave - vectors . nevertheless , the consistency of these results with the arguments presented above against a diffusion form are encouraging . the spin density dynamic correlations @xmath77 have been the subject of many studies which have not been able to answer the question of spin diffusion unambiguously . here , we revisit this problem by investigating the compatibility between spin density and spin current correlations , which requires that we calculate @xmath78 . in contrast to the energy current , the spin current @xmath52 does not commute with the hamiltonian , so that @xmath79 is different from a pure @xmath30 function . nevertheless , if @xmath80 , which means that @xmath79 has a finite weight @xmath81function at @xmath82 , our previous arguments against diffusion still hold . in determining @xmath78 , we noticed a peculiar difference in the low frequency behavior of @xmath79 depending on the anisotropy parameter @xmath4 . in figure 2 , we show @xmath83 the corresponding integrated , normalized quantity . we see that for @xmath84 , @xmath85 ( @xmath86 in the figure ) all the low frequency weight of @xmath79 is concentrated in the @xmath87-function at @xmath82 . in contrast , for neighboring values such as @xmath88 or @xmath89 , we observe a shift of weight to a low frequency region whose size decreases as the system grows ( inset ) and eventually vanishes as @xmath90 . we believe that the behavior of this special @xmath4 points is related to the existence of finite length strings ( bound states ) as they appear in the formulation of the thermodynamics of the heisenberg model , within the bethe ansatz method@xcite . it seems that in order to determine @xmath78 from finite size systems for @xmath91 , we should include the weight from these low frequency regions . as an example , doing so for @xmath88 gives us a value of @xmath92 for l=16 ( figure 2 ) . having discussed this technical issue , we can then determine @xmath78 for different size systems , as a function of temperature and @xmath4 . by extrapolating our finite size results to the thermodynamic limit using second order polynomials in @xmath93 for @xmath94 , we obtain the results shown in figure 3 . their striking feature is that for @xmath95 , @xmath78 is finite in the @xmath5 region , and practically zero when @xmath96 . in this regime , according to our previous argument , we expect a non- diffusive behavior . deciding about the behavior of @xmath78 for @xmath96 at finite temperatures is rather subtle . the reason is that in the heisenberg model , @xmath62 corresponds to a point of change of symmetry , from easy plane to easy axis , accompanied by the opening of a gap . in the fermionic version of the model , the t - v " model , it corresponds to a metal - insulator mott - hubbard type transition , with the charge stiffness changing discontinuously@xcite at zero temperature . we should note that this discontinuity is difficult to reproduce by numerical simulations on small finite size lattices , as the transition corresponds to the divergence of the localization length . considering that at high temperature , @xmath78 behaves similarly to the charge stiffness in the t - v " model@xcite we understand why it is difficult to decide whether @xmath78 is greater than zero in the region @xmath97 and @xmath98 . for the same reason , we can not exclude that @xmath78 behaves discontinuously at @xmath62 . nevertheless , it seems unambiguous that @xmath99 for @xmath100 and @xmath95 . the results presented are of interest in recent experimental studies@xcite of spin dynamics in quasi - one dimensional materials such as @xmath101 and @xmath102 . particular attention should be paid to the unusually high value of the diffusion constant found in nmr experiments on @xmath102@xcite , perhaps related to the integrability of the heisenberg model as discussed above . furthermore , our results on the behavior of energy density correlations are of interest in the interpretation of the quasi - elastic raman scattering , related to magnetic energy fluctuations@xcite . we should emphasize that no diffusion form should be expected for the energy density correlations in the isotropic heisenberg model with only nearest neighbor interaction . an eventual diffusive behavior should be attributed to next - nearest neighbor coupling , interaction with phonons or deviations from one dimensionality . finally , the main unresolved issue in this work is a better understanding of the finite temperature spin dynamics at the isotropic point . we would like to thank p. prelovek for useful discussions . this work was supported by the swiss national science foundation grant no . 20 - 49486.96 , the university of fribourg and the university of neuchtel . 99 bhm m and leschke h 1993 _ physica a _ * 199 * 116 fabricius k , lw u and stolze j 1997 _ phys . b _ * 55 * 5833 starykh o a , sandvik a w and singh r r p 1997 _ phys . rev . b _ * 55 * 14953 damle k and sachdev s _ cond - mat/9711014 _ + narozhny b n 1996 _ phys . rev . b _ * 54 * 3311 + mccoy b m _ cond - mat/9706194 _ + sachdev s 1994 _ phys . b _ * 50 * 13006 zotos x and prelovek p 1996 _ phys . rev . b _ * 53 * 983 + castella h , zotos x and prelovek p 1995 _ phys . rev . * 74 * 972 mccoy b m 1995 _ statistical mechanics and field theory , proceedings of the seventh summer physics school _ ( world scientific : the australian national university ) zotos x , naef f and prelovek p 1997 _ phys b _ * 55 * 11029 mazur p 1969 _ physica _ * 43 * 533 kadanoff l p and martin p c 1963 _ ann . ph . _ * 24 * 419 niemeijer t 1967 _ physica _ * 36 * 377 takahashi m and suzuki m 1972 _ prog _ * 48 * 2187 shastry b s and sutherland b 1990 _ phys . lett . _ * 65 * 243 takigawa m , motoyama n , eisaki h and uchida s 1996 _ phys . * 24 * 4612 kuroe h 1997 _ phys . rev . b _ * 55 * 409 halley j w 1978 _ phys . rev . * 41 * 1065

in this paper , we study the spin and energy dynamic correlations of the one dimensional spin 1/2 heisenberg model , using mostly exact diagonalization numerical techniques . in particular , observing that the uniform spin and energy currents decay to finite values at long times , we argue for the absence of spin and energy diffusion in the easy plane anisotropic heisenberg model .

cond-mat9711264

non - gaussian quantum states , endowed with properly enhanced nonclassical properties , may constitute powerful resources for the efficient implementation of quantum information , communication , computation and metrology tasks @xcite . indeed , it has been shown that , at fixed first and second moments , gaussian states _ minimize _ various nonclassical properties @xcite . therefore , many theoretical and experimental efforts have been made towards engineering and controlling highly nonclassical , non - gaussian states of the radiation field ( for a review on quantum state engineering , see e.g. @xcite ) . in particular , several proposals for the generation of non - gaussian states have been presented @xcite , and some successful ground - breaking experimental realizations have been already performed @xcite . concerning continuous - variable ( cv ) quantum teleportation , to date the experimental demonstration of the vaidman - braunstein - kimble ( vbk ) teleportation protocol @xcite has been reported both for input coherent states @xcite , and for squeezed vacuum states @xcite . in particular , ref . @xcite has reported the teleportation of squeezing , and consequently of entanglement , between upper and lower sidebands of the same spatial mode . it is worth to remark that the efficient teleportation of squeezing , as well as of entanglement , is a necessary requirement for the realization of a quantum information network based on multi - step information processing @xcite . in this paper , adopting the vbk protocol , we study in full generality , e.g. including loss mechanisms and non - unity gain regimes , the teleportation of input single - mode coherent squeezed states using as non - gaussian entangled resources a class of non - gaussian entangled quantum states , the class of squeezed bell states @xcite . this class includes , for specific choices of the parameters , non - gaussian photon - added and photon - subtracted squeezed states . in tackling our goal , we use the formalism of the characteristic function introduced in ref . @xcite for an ideal protocol , and extended to the non - ideal instance in ref . @xcite . here , in analogy with the teleportation of coherent states , we first optimize the teleportation fidelity , that is , we look for the maximization of the overlap between the input and the output states . but the presence of squeezing in the unknown input state to be teleported prompts also an alternative procedure , depending on the physical quantities of interest . in fact , if one cares about reproducing in the most faithful way the initial state in phase - space , then the fidelity is the natural quantity that needs to be optimized . on the other hand , one can be interested in preserving as much as possible the squeezing degree at the output of the teleportation process , even at the expense of the condition of maximum similarity between input and output states . in this case , one aims at minimizing the difference between the output and input quadrature averages and the quadrature variances . it is important to observe that this distinction makes sense only if one exploits non - gaussian entangled resources endowed with tunable free parameters , so that enough flexibility is allowed to realize different optimization schemes . indeed , it is straightforward to verify that this is impossible using gaussian entangled resources . we will thus show that exploiting non - gaussian resources one can identify the best strategies for the optimization of different tasks in quantum teleportation , such as state teleportation vs teleportation of squeezing . comparison with the same protocols realized using gaussian resources will confirm the greater effectiveness of non - gaussian states vs gaussian ones as entangled resources in the teleportation of quantum states of continuous variable systems . the paper is organized as follows . in section [ secqtelep ] , we introduce the single - mode input states and the two - mode entangled resources , and we recall the basics of both the ideal and the imperfect vkb quantum teleportation protocols . with respect to the instance of gaussian resources ( twin beam ) , the further free parameters of the non - gaussian resource ( squeezed bell state ) allow one to undertake an optimization procedure to improve the efficiency of the protocols . in section [ sectelepfidelity ] we investigate the optimization procedure based on the maximization of the teleportation fidelity . we then analyze an alternative optimization procedure leading to the minimization of the difference between the quadrature variances of the output and input fields . this analysis is carried out in section [ secoptvar ] . we show that , unlike gaussian resources , in the instance of non - gaussian resources the two procedures lead to different results and , moreover , always allow one to improve on the optimization procedures that can be implemented with gaussian resources . finally , in section [ secconcl ] we draw our conclusions and discuss future outlooks . in this section , we briefly recall the basics of the ideal and imperfect vbk cv teleportation protocols ( for details see ref . the scheme of the ( cv ) teleportation protocol is the following . alice wishes to send to bob , who is at a remote location , a quantum state , drawn from a particular set according to a prior probability distribution . the set of input states and the prior distribution are known to alice and bob , however the specific state to be teleported that is prepared by alice remains unknown . alice and bob share a resource , e.g. a two - mode entangled state . the input state and one of the modes of the resource are available for alice , while the other mode of the resource is sent to bob . alice performs a suitable ( homodyne ) bell measurement , and communicates the result to bob exploiting a classical communication channel . then bob , depending on the result communicated by alice , performs a local unitary ( displacement ) transformation , and retrieves the output teleported state . the non - ideal ( realistic ) teleportation protocol includes mechanisms of loss and inefficiency : the photon losses occurring in the realistic bell measurements , and the noise arising in the propagation of optical fields in noisy channels ( fibers ) when the second mode of the resource is sent to bob . the photon losses occurring in the realistic bell measurements are modeled by placing in front of an ideal detector a fictitious beam splitter with non - unity transmissivity @xmath0 ( and corresponding non - zero reflectivity @xmath1 ) @xcite . the propagation in fiber is modeled by the interaction with a gaussian bath with an effective photon number @xmath2 , yielding a damping process with inverse - time rate @xmath3 @xcite . denoting by @xmath4 the input field mode , and by @xmath5 and @xmath6 , respectively , the first and the second mode of the entangled resource , the decoherence due to imperfect photo - detection in the homodyne measurement performed by alice involves the input field mode @xmath4 , and one mode of the resource , e.g. mode @xmath5 . throughout , we assume a pure entangled resource . indeed , it is simple to verify that considering mixed ( impure ) resources is equivalent to a consider a suitable nonvanishing detection inefficiency @xmath7 @xcite . the degradation due to propagation in fiber affects the other mode of the resource , e.g. mode @xmath6 , which has to reach bob s remote place at the output stage . denoting now by @xmath8 and @xmath9 the projectors corresponding , respectively , to a generic pure input single - mode state and a generic pure two - mode entangled resource , the characteristic function @xmath10 of the single - mode output field @xmath11 can be written as @xcite : @xmath12 \\&=e^{- \gamma_{\tau , r}|\alpha|^{2 } } \chi_{in}\left(g t \ , \alpha \right ) \chi_{res}\left(g t \ , \alpha^{*};e^{-\frac{\tau}{2 } } \ , \alpha\right ) , \end{split } \label{chioutfinale}\ ] ] where @xmath13 is the glauber displacement operator , @xmath14 $ ] is the characteristic function of the input state , @xmath15 $ ] is the characteristic function of the resource , @xmath16 is the gain factor of the protocol @xcite , @xmath17 is the scaled dimensionless time proportional to the fiber propagation length , and the function @xmath18 is defined as : @xmath19 we assume in principle to have some knowledge about the characteristics of the experimental apparatus : the inefficiency @xmath7 ( or @xmath20 ) of the photo - detectors , and the loss parameters @xmath21 and @xmath2 of the noisy communication channel . we consider as input state a single - mode coherent and squeezed ( cs ) state @xmath22 with unknown squeezing parameter @xmath23 and unknown coherent amplitude @xmath24 . we then consider as non - gaussian entangled resource the two - mode squeezed bell ( sb ) state @xmath25 , defined as @xcite : @xmath26 here @xmath27 is , as before , the displacement operator , @xmath28 is the single - mode squeezing operator , @xmath29 is the two - mode squeezing operator @xmath30 , with @xmath31 denoting the annihilation operator for mode @xmath32 @xmath33 , @xmath34 is the two - mode fock state ( of modes 1 and 2 ) with @xmath35 photons in the first mode and @xmath36 photons in the second mode , and @xmath37 and @xmath38 are two intrinsic free parameters of the resource entangled state , in addition to @xmath39 and @xmath40 , which can be exploited for optimization . note that particular choices of the angle @xmath38 in the class of squeezed bell states eq . ( [ squeezbell ] ) allow one to recover different instances of two - mode gaussian and non - gaussian entangled states : for @xmath41 the gaussian twin beam ( twb ) ; for @xmath42 $ ] and @xmath43 the two - mode photon - added squeezed ( pas ) state @xmath44 ; for @xmath45 $ ] and @xmath43 the two - mode photon - subtracted squeezed ( pss ) state @xmath46 . the last two non - gaussian states are defined as : @xmath47 and are already experimentally realizable with current technology @xcite . in the following section we study , in comparison with the instance of two - mode gaussian entangled resources , the performance of the optimized two - mode squeezed bell states when used as entangled resources for the teleportation of input single - mode coherent squeezed states . for completeness , in the same context we make also a comparison with the performance , as entangled resources , of the more specific realizations ( [ photaddsqueez ] ) , ( [ photsubsqueez ] ) . the characteristic functions of states ( [ cohsqueezst ] ) , ( [ squeezbell ] ) , ( [ photaddsqueez ] ) , and ( [ photsubsqueez ] ) are computed and their explicit expressions are given in appendix [ appendixstates ] . .summary of the notation employed throughout this work to describe the different parameters that characterize the input coherent squeezed ( cs ) states [ eq . ( [ cohsqueezst ] ) ] , the shared entangled two - mode squeezed bell ( sb ) resources [ eq . ( [ squeezbell ] ) ] , and the characteristics of non - ideal teleportation setups @xcite . see text for further details on the role of each parameter . [ cols="^,^,^",options="header " , ] for ease of reference , table [ tableparam ] provides a summary of the parameters associated with the input states , the shared resources , and the sources of noise in the teleportation protocol . the commonly used measure to quantify the performance of a quantum teleportation protocol is the fidelity of teleportation @xcite , @xmath48 $ ] , which amounts to the overlap between a pure input state @xmath49 and the ( generally mixed ) teleported state @xmath11 . in the formalism of the characteristic function the fidelity reads @xmath50 where @xmath51 is the characteristic function of the single - mode input state @xmath52 , eq . ( [ cohsqueezst ] ) , and @xmath53 is the characteristic function for the output teleported state , eq . ( [ chioutfinale ] ) . in this section , we will make use of eq . ( [ fidelitychi ] ) to analyze the efficiency of the cv teleportation protocol . in the instance of non - gaussian squeezed bell resources ( [ squeezbell ] ) , at fixed squeezing parameter , the optimization procedure amounts to the maximization of the teleportation fidelity ( [ fidelitychi ] ) over the free parameters of the entangled resource . it can be shown that the optimal choice for the phases @xmath40 and @xmath37 is @xmath54 and @xmath55 . the analytic expression for the fidelity @xmath56 of the non - ideal quantum teleportation of coherent squeezed states using squeezed bell resources reads @xmath57 \right . \nonumber \\ & & + \frac{1}{4}e^{-2(2r+\tau ) } \delta_{2}^{2}\sin^{2}\delta \left[\frac{1}{\lambda_{1}^{2}}\left(3+\frac{12 \omega_{1}^{2}}{\lambda_{1}}+\frac{4\omega_{1}^{4}}{\lambda_{1}^{2}}\right)+ \frac{1}{\lambda_{2}^{2}}\left(3-\frac{12 \omega_{2}^{2}}{\lambda_{2}}+\frac{4\omega_{2}^{4}}{\lambda_{2}^{2}}\right)\right . \\ & & \left . + \frac{2}{\lambda_{1}\lambda_{2}}\left(1+\frac{2\omega_{1}^{2}}{\lambda_{1}}-\frac{2\omega_{2}^{2}}{\lambda_{2 } } -\frac{4\omega_{1}^{2}\omega_{2}^{2}}{\lambda_{1}\lambda_{2}}\right)\right ] \right\ } \ , , \nonumber \label{telepfidelitycohsq}\end{aligned}\ ] ] where , introducing @xmath58 , the quantities @xmath59 , @xmath60 , @xmath61 , @xmath62 , @xmath63 , and @xmath64 are defined by the following relations : @xmath65 for different choices of @xmath38 in eq . ( [ telepfidelitycohsq ] ) , see section [ secqtelep ] , one obtains the teleportation fidelities associated to photon - added and photon - subtracted squeezed resource states . let us observe that the fidelity in eq . ( [ telepfidelitycohsq ] ) depends both on the input coherent amplitude @xmath24 , and on the input single - mode squeezing parameter @xmath66 , while it is independent of the input squeezing phase @xmath67 . once again , it is worth stressing that , in the teleportation paradigm , the input state is unknown and only partial ( probabilistic ) knowledge on the alphabet of input states is admitted . it is thus required , in principle , to assume teleportation protocols independent of the input parameters , as it turns out to be the case for the vbk protocol with gaussian entangled resources and input coherent states . however , in more general cases , one can study the behavior of the so - called one - shot fidelity , that is the teleportation fidelity at specific values of the input parameters . suitable averages of the one - shot fidelity over the set of input states and parameters , according to an assigned prior distribution , will then result in the average quantum teleportation fidelity . the latter quantity can then be confronted with so - called classical fidelity thresholds ( benchmarks ) that correspond to the maximum achievable average fidelity between the input state ( measured by alice in order to achieve an optimal estimation of it ) and the output state ( prepared by bob according to alice s measurement outcomes ) , without the use of any shared entanglement @xcite . while teleportation benchmarks are available for the cases of coherent input states ( with completely unknown @xmath24 ) @xcite , purely squeezed input states ( with @xmath68 and completely unknown @xmath66 ) @xcite , as well as for states with known squeezing degree and unknown displacement and phase @xcite , a benchmark for the case of input states with totally unknown displacement and squeezing has not yet been derived , and stands as a challenging problem in quantum estimation theory . henceforth , assuming _ a priori _ that the input parameters ( displacement and squeezing degree ) are completely random , we adopt then the following approach to optimize the quantum teleportation fidelity . we exploit a non - unity gain strategy to remove at least the @xmath24-dependence in the one - shot fidelity ; then , we study the behavior of the @xmath24-independent one - shot fidelity for specific values of the input squeezing parameter @xmath66 , in order to identify an effective , @xmath66-independent approximation . indeed , fixing the gain @xmath16 at the value @xmath69 @xmath70 in eq . ( [ telepfidelitycohsq ] ) yields the @xmath24-independent fidelity @xmath71 : @xmath72 where the quantities @xmath59 , @xmath60 , @xmath61 , @xmath62 , @xmath63 , and @xmath64 are defined in eq . ( [ relations ] ) . for different choices of @xmath38 ( see section [ secqtelep ] ) , one obtains the teleportation fidelities associated to the use of different gaussian and non - gaussian entangled resources : the twin beam , the photon - added , and the photon - subtracted squeezed states . for such resources no optimization procedure is possible as @xmath38 is a specific function of @xmath39 . instead , the optimization of the fidelity ( [ telepfidelitysqvac ] ) with respect to the free non - gaussian parameter @xmath38 identifies the optimal squeezed bell resource associated to the optimal value : @xmath73\!\end{array}\!\!. \label{deltaoptfid}\ ] ] let us notice that , for @xmath74 ( ideal protocol ) and @xmath75 ( input coherent states ) , eq . ( [ deltaoptfid ] ) reduces to @xcite : @xmath76 \ , . \label{deltaoptfid2}\ ] ] the displacement - independent one - shot fidelity @xmath71 and the optimal angle @xmath77 are still dependent on @xmath66 , the input squeezing . unfortunately , the optimization of the non - gaussian resource based on the choice ( [ deltaoptfid ] ) as optimal angle would be practically unfeasible because the input squeezing is not known . in order to circumvent this problem , we introduce a sub - optimal angle @xmath78 such that @xmath79 where @xmath80 is a fixed effective value of the input squeezing chosen , according to a suitable criterion that will be clarified below , in the range of possible values of the squeezing parameter @xmath66 . at fixed @xmath66 , and as a function of the angle @xmath38 parameterized by @xmath80 , expressed in db , i.e. @xmath81 , see eq . ( [ deltasubopt ] ) , both in the instance of the ideal protocol @xmath82 ( full lines ) , and of a non - ideal protocol , with @xmath83 , @xmath84 , and @xmath85 ( dashed lines ) . the one - shot fidelities are drawn for three different values of the input squeezing : @xmath86 db . the curves are ordered from top to bottom for increasing @xmath66.,width=321 ] in the following we will express the squeezing parameters @xmath39 and @xmath66 in decibels , according to the relation @xcite : @xmath87 the practical rationale for introducing a sub - optimal characterization in the maximization of the output fidelity is based on the observation that the assumption of a completely random degree of input squeezing @xmath66 is clearly unrealistic . it is instead very sensible to consider that the range of possible values of @xmath66 falls in a window @xmath88 $ ] db . indeed , to date , the experimentally reachable values of squeezing fall roughly in such a range with @xmath89 db @xcite . we can then study the behavior of @xmath71 corresponding to the angle @xmath78 as a function of the effective input squeezing parameter @xmath80 , at fixed squeezing parameters of the resource and of the input state , respectively @xmath39 and @xmath66 , and at fixed loss parameters @xmath21 , @xmath2 , and @xmath7 . [ fig1sfidsbar ] shows that @xmath71 is quite insensitive to the value of @xmath80 . assuming the realistic range @xmath90 $ ] db , the choice of a sub - optimal angle such that @xmath91 db ( average value of the interval ) , leads to a decrease of the optimized fidelity , compared to the choice of @xmath77 , of at most @xmath92 in ideal conditions , and even smaller in realistic conditions . in other words , the teleportation fidelity is essentially constant in the considered interval of variability for the angle @xmath38 . therefore , throughout in the following , we fix @xmath93 db in the expression eq . ( [ deltaoptfid ] ) to make it @xmath66-independent . in fig . [ fig1sfid ] , we plot the teleportation fidelity associated to the various considered resources ( gaussian twin beam , optimized two - mode squeezed bell - like state , two - mode squeezed photon - subtracted state ) both for the ideal protocol ( panel i ) and for the non - ideal protocol ( panel ii ) . we see that , at fixed ( finite ) squeezing @xmath39 of the resource , the gaussian twin beam is always outperformed by the optimal non - gaussian squeezed bell resource in the ideal protocol . it is worth to remark that for very high values of the squeezing @xmath39 , the advantage of the non - gaussian resources fades and gaussian twin beams perform in practice equally well for the teleportation of the considered input states . this reflects the well known fact that , using the ideal vbk protocol and an ideal einstein podolsky rosen resource ( corresponding , e.g. , to a twin beam in the limit @xmath94 ) , _ any _ quantum state can be unconditionally teleported with unit fidelity @xcite . all the one - shot fidelities decrease for increasing squeezing @xmath66 of the input and , interestingly , in the non - ideal protocol they achieve a maximum at a finite value @xmath95 of the squeezing @xmath39 of the resource . the optimal squeezed bell resource and the twin beam share the same @xmath96 db and coincide at that point . in fig . [ fig1sfid ] we also plot the one - shot fidelities associated with the two - mode photon - subtracted squeezed states , eq . ( [ photsubsqueez ] ) . the two - mode photon - subtracted squeezed state always outperforms the twin beam in the ideal protocol , and at low and intermediate values of the resource squeezing @xmath39 in the non - ideal case . it is always outperformed by the optimized squeezed bell resource . we note that , on the other hand , the two - mode photon - added squeezed states always exhibit a performance worse than the two - mode photon - subtracted squeezed states and the squeezed bell resources ( the corresponding fidelities are omitted in the plots for clarity ) . in a given range of the squeezing @xmath39 , @xmath46 and @xmath25 exhibit comparable levels in the fidelity of teleportation . in conclusion , properly optimized non - gaussian resources maximize the fidelity of teleportation of squeezed coherent states both in the ideal and imperfect vbk protocols , outperforming the corresponding gaussian resources . in the next section we carry out a similar analysis with the aim of identifying the optimal strategy that maximizes the reproduction at the output of the input squeezing . in this section , we introduce a different approach to the optimization of the teleportation protocol , aimed at retaining and faithfully reproducing at the output the variances and thus the squeezing of the input state . the strategy is to constrain the first and second order moments of the output field to reproduce the ones of the input field , by exploiting the free parameters of the non - gaussian resources . we introduce the mean values @xmath97 $ ] , with @xmath98 @xmath99 , and the variances @xmath100 - { \rm tr } [ z_j \rho_{j } ] ^{2}$ ] , and @xmath101 - 2{\rm tr } [ x_j \rho_{j } ] { \rm tr } [ p_j \rho_{j } ] $ ] ( the cross - quadrature variance , with @xmath102 denoting the symmetrization ) of the quadrature operators @xmath103 , @xmath104 , associated with the single - mode input state @xmath49 and the output state @xmath11 of the teleportation protocol . the explicit expressions for the quantities @xmath105 , @xmath106 , and @xmath107 are reported in the appendix [ appendixquadratures ] . the quantities measuring the deviation of the output from the input are the differences between the output and input first and second quadrature moments : @xmath108 with @xmath109 given by eq . ( [ eqc ] ) . from the above equations , we see that the assumption @xmath110 ( i.e. @xmath69 ) yields @xmath111 and @xmath112 . therefore , for @xmath110 , the input and output fields possess equal average position and momentum ( equal first moments),and equal cross - quadrature variance ; then , the optimization procedure reduces to the minimization of the quantity @xmath113 with respect to the free parameters of the non - gaussian squeezed bell resource , i.e. @xmath114 . moreover , as for the optimization procedure of section [ sectelepfidelity ] , it can be shown that the optimal choice for @xmath40 and @xmath37 is , once again , @xmath54 and @xmath55 . the optimization on the remaining free parameter @xmath38 yields the optimal value @xmath115 : @xmath116 \ , . \label{deltaoptvar}\ ] ] the optimal angle @xmath115 , corresponding to the minimization of the differences @xmath117 and @xmath118 between the output and input quadrature variances , is independent of @xmath7 , at variance with the optimal value @xmath77 , eq . ( [ deltaoptfid ] ) , corresponding to the maximization of the teleportation fidelity . it is also important to note that in this case there are no questions related to a dependence on the input squeezing @xmath66 . for @xmath119 eq . ( [ deltaoptvar ] ) reduces to @xmath120 . such a value is equal to the asymptotic value given by eq . ( [ deltaoptfid2 ] ) for @xmath94 , so that , in this extreme limit the two optimization procedures become equivalent . in the particular cases of photon - added and photon - subtracted resources , no optimization procedure can be carried out , and the parameter @xmath38 is simply a given specific function of @xmath39 ( see section [ secqtelep ] ) . we remark that , having automatically zero difference in the cross - quadrature variance at @xmath110 , finding the angles that minimize @xmath117 and @xmath118 precisely solves the problem of achieving the optimal teleportation of both the first moments and the full covariance matrix of the input state at once . in order to compare the performances of the gaussian and non - gaussian resources , and to emphasize the improvement of the efficiency of teleportation with squeezed bell - like states , we consider first the instance of ideal protocol ( @xmath119 , @xmath84 , @xmath121 ) , and compute , and explicitly report below , the output variances @xmath122 of the teleported state associated with non - gaussian resources ( i.e. optimized squeezed bell - like states @xmath123 , photon - added squeezed states @xmath124 , photon - subtracted squeezed states @xmath125 ) , and with gaussian resources , i.e. twin beams @xmath126 . from eqs . ( [ varxout])([eqc3 ] ) , we get : @xmath127 eq . ( [ dzsb ] ) is derived exploiting the optimal angle ( [ deltaoptvar ] ) , which reduces to eq . ( [ deltaoptfid2 ] ) in the ideal case . independently of the resource , the teleportation process will in general result in an amplification of the input variance . however , the use of non - gaussian optimized resources , compared to the gaussian ones , reduces sensibly the amplification of the variances at the output . looking at eq . ( [ dztwb ] ) , we see that the teleportation with the twin beam resource produces an excess , quantified by the exponential term @xmath128 , of the output variance with respect to the input one . on the other hand , the use of the non - gaussian squeezed bell resource eq . ( [ dzsb ] ) yields a reduction in the excess of the output variance with respect to the input one by a factor @xmath129 . let us now analyze the behaviors of the photon - added squeezed resources and of the photon - subtracted squeezed resources , eqs . ( [ dzpas ] ) and ( [ dzpss ] ) , respectively . we observe that , in analogy with the findings of the previous section , the photon - subtracted squeezed resources exhibit an intermediate behavior in the ideal protocol ; indeed for low values of @xmath39 they perform better than the gaussian twin beam , but worse than the optimized squeezed bell states . the photon - added squeezed resources perform worse than both the twin beam and the other non - gaussian resources . these considerations follow straightforwardly from a quantitative analysis of the terms associated with the excess of the output variance in eqs . ( [ dzpas ] ) and ( [ dzpss ] ) . moreover , again in analogy with the analysis of the optimal fidelity , for low values of @xmath39 , there exists a region in which the performance of photon - subtracted squeezed states and optimized squeezed bell states are comparable . finally , again in analogy with the case of the fidelity optimization , the output variance associated with the gaussian twin beam and with the optimized squeezed bell states coincide at a specific , large value of @xmath39 , at which the two resources become identical . the input variances @xmath130 ( [ varxin ] ) and ( [ varpin ] ) , and the output variances @xmath131 , are plotted in panels i and ii of fig . [ figvar ] for the ideal vkb protocol and in panels iii and iv of fig . [ figvar ] for the non - ideal protocol . in the instance of realistic protocol , for small resource squeezing degree @xmath39 , similar conclusions can be drawn , leading to the same hierarchy among the entangled resources . however , analogously to the behavior of the teleportation fidelity , for high values of @xmath39 the photon - subtracted squeezed resources are very sensitive to decoherence . in fact , such resources perform worse and worse than the gaussian twin beam for @xmath39 greater than a specific finite threshold value . rather than minimizing the differences between output and input quadrature variances , one might be naively tempted to consider minimizing the difference between the ratio of the output variances @xmath132 and the ratio of the input variances @xmath133 . this quantity might appear to be of some interest because it is a good measure of how well squeezing is teleported in all those cases in which the input and output quadrature variances are very different , that is those situations in which the statistical moments are teleported with very low efficiency . however , it is of little use to preserve formally a scale parameter if the noise on the quadrature averages grows out of control . the procedure of minimizing the difference between output and input quadrature statistical moments is the only one that guarantees the simultaneous preservation of the squeezing degree and the reduction of the excess noise on the output averages and statistical moments of the field observables . we have studied the efficiency of the vbk cv quantum teleportation protocol for the transmission of quantum states and averages of observables using optimized non - gaussian entangled resources . we have considered the problem of teleporting gaussian squeezed and coherent states , i.e. input states with two unknown parameters , the coherent amplitude and the squeezing . the non - gaussian resources ( squeezed bell states ) are endowed with free parameters that can be tuned to maximize the teleportation efficiency either of the state or of physical quantities such as squeezing , quadrature averages , and statistical moments . we have discussed two different optimization procedures : the maximization of the teleportation fidelity of the state , and the optimization of the teleportation of average values and variances of the field quadratures . the first procedure maximizes the similarity in phase space between the teleported and the input state , while the second one maximizes the preservation at the output of the displacement and squeezing contents of the input . we have shown that optimized non - gaussian entangled resources such as the squeezed bell states , as well as other more conventional non - gaussian entangled resources , such as the two - mode squeezed photon - subtracted states , outperform , in the realistic intervals of the squeezing parameter @xmath39 of the entangled resource achievable with the current technology , entangled gaussian resources both for the maximization of the teleportation fidelity and for the maximal preservation of the input squeezing and statistical moments . these findings are consistent and go in line with previous results on the improvement of various quantum information protocols replacing gaussian with suitably identified non - gaussian resources @xcite . in the process , we have found that the two optimal values of the resource angle @xmath38 associated with the two optimization procedures are different and identified , respectively , by eqs . ( [ deltaoptfid ] ) and ( [ deltaoptvar ] ) . this inequivalence is connected to the fact that , when using entangled non - gaussian resources with free parameters that are amenable to optimization , the fidelity is closely related to the form of the different input properties that one wishes to teleport , e.g. quasi - probability distribution in the phase space , squeezing , statistical moments of higher order , and so on . different quantities correspond to different optimal teleportation strategies . finally , regarding the vbk protocol , it is worth remarking that the maximization of the teleportation fidelity corresponds to the maximization of the squared modulus of the overlap between the input and the output ( teleported ) state , without taking into account the characteristics of the output with respect to the input state . therefore , part of the non - gaussian character of the entangled resource is unavoidably transferred to the output state . the latter then acquires unavoidably a certain degree of non - gaussianity , even if the presence of pure gaussian inputs . moreover , as verified in the case of non - ideal protocols , the output state is also strongly affected by decoherence . thus , in order to recover the purity and the gaussianity of the teleported state , purification and gaussification protocols should be implemented serially after transmission through the teleportation channel is completed @xcite . if the second ( squeezing preserving ) procedure is instead considered , the possible deformation of the gaussian character is not so relevant , because the shape reproduction is not the main goal , while purification procedures are again needed to correct for the extra noise added during teleportation when finite entanglement and realistic conditions are considered . an important open problem is determining a proper teleportation benchmark for the class of gaussian input states with unknown displacement and squeezing . such a benchmark is expected to be certainly smaller than @xmath134 in terms of teleportation fidelity , the latter being the benchmark for purely coherent input states with completely random displacement in phase space @xcite . our results indicate that optimized non - gaussian entangled resources will allow one to beat the classical benchmark , thus achieving unambiguous quantum state transmission via a truly quantum teleportation , with a smaller amount of nonclassical resources , such as squeezing and entanglement , compared to the case of shared gaussian twin beam resources . in this context , [ fig1sfid ] provides strong and encouraging evidence that suitable uses of non - gaussianity in tailored resources , feasible with current technology @xcite , may lead to a genuine demonstration of cv quantum teleportation of displaced squeezed states in realistic conditions of the experimental apparatus . this would constitute a crucial step forward after the successful recent experimental achievement of the quantum storage of a displaced squeezed thermal state of light into an atomic ensemble memory @xcite . we acknowledge financial support from the european union under the fp7 strep project hip ( hybrid information processing ) , grant agreement no . here we report the characteristic functions for the single - mode input states and for the two - mode entangled resources . the characteristic function for the coherent squeezed states ( [ cohsqueezst ] ) , i.e. @xmath135 reads : @xmath136 the characteristic function for the squeezed bell - like resource ( [ squeezbell ] ) , i.e. @xmath137 reads : @xmath138 \ , , \end{split } \label{charfuncsb}\ ] ] where the complex variables @xmath139 are defined as : @xmath140 it is worth noticing that , for @xmath41 , eq . ( [ charfuncsb ] ) reduces to the well - known gaussian characteristic function of the twin beam . given the characteristic functions for the single - mode the input state and for the two - mode entangled resource , eqs . ( [ chiinput ] ) and ( [ charfuncsb ] ) , respectively , it is straightforward to obtain the characteristic function for the single - mode output state of the teleportation protocol by using eq . ( [ chioutfinale ] ) and replacing @xmath141 with @xmath142 . in this appendix , we report the analytical expressions for the mean values @xmath97 $ ] , with @xmath98 @xmath99 , and the variances @xmath100 - { \rm tr } [ z_j \rho_{j } ] ^{2}$ ] of the quadrature operators @xmath103 , @xmath104 , associated with the single - mode input state @xmath49 and the output state @xmath11 of the teleportation protocol . we also compute the cross - quadrature variance @xmath101 - 2{\rm tr } [ x_j \rho_{j } ] { \rm tr } [ p_j \rho_{j } ] $ ] , associated with the non - diagonal term of the covariance matrix of the density operator , where the subscript @xmath102 denotes the symmetrization . the mean values and the variances associated with the input single - mode coherent squeezed state ( [ cohsqueezst ] ) can be easily computed : @xmath143 and @xmath144 the mean values and the variances associated with the output single - mode teleported state , described by the characteristic function ( [ chioutfinale ] ) read : @xmath145 and @xmath146 with in the instance of gaussian resource @xmath149 , such quantity simplifies to : @xmath150 for suitable choices of @xmath38 in eq . ( [ eqc2 ] ) , see section [ secqtelep ] , one can easily obtain the output variances associated with photon - added and photon - subtracted squeezed states . s. suzuki , h. yonezawa , f. kannari , m. sasaki , and a. furusawa , appl . 89 * , 061116 ( 2006 ) ; h. vahlbruch , m. mehmet , n. lastzka , b. hage , s. chelkowski , a. franzen , s. gossler , k. danzmann , and r. schnabel , phys . lett . * 100 * , 033602 ( 2008 ) . o. glckl , u. l. andersen , r. filip , w. p. bowen , and g. leuchs , phys . lett . * 97 * , 053601 ( 2006 ) ; j. heersink , ch . marquardt , r. dong , r. filip , s. lorenz , g. leuchs , and u. l. andersen , phys . lett . * 96 * , 253601 ( 2006 ) ; a. franzen , b. hage , j. diguglielmo , j. fiurasek , and r. schnabel , phys . lett . * 97 * , 150505 ( 2006 ) ; b. hage , a. samblowski , j. diguglielmo , a. franzen , j. fiurasek , and r. schnabel , nature phys . * 4 * , 915 ( 2008 ) ; r. dong , m. lassen , j. heersink , ch . marquardt , r. filip , g. leuchs , and u. l. andersen , nature phys . * 4 * , 919 ( 2008 ) . k. jensen , w. wasilewski , h. krauter , t. fernholz , b. m. nielsen , a. serafini , m. owari , m. b. plenio , m. m. wolf , and e. s. polzik , e print arxiv:1002.1920 ( 2010 ) , nature phys . ( advance online publication , doi:10.1038/nphys1819 ) .

we study the continuous - variable quantum teleportation of states , statistical moments of observables , and scale parameters such as squeezing . we investigate the problem both in ideal and imperfect vaidman - braunstein - kimble protocol setups . we show how the teleportation fidelity is maximized and the difference between output and input variances is minimized by using suitably optimized entangled resources . specifically , we consider the teleportation of coherent squeezed states , exploiting squeezed bell states as entangled resources . this class of non - gaussian states , introduced in references @xcite , includes photon - added and photon - subtracted squeezed states as special cases . at variance with the case of entangled gaussian resources , the use of entangled non - gaussian squeezed bell resources allows one to choose different optimization procedures that lead to inequivalent results . performing two independent optimization procedures one can either maximize the state teleportation fidelity , or minimize the difference between input and output quadrature variances the two different procedures are compared depending on the degrees of displacement and squeezing of the input states and on the working conditions in ideal and non - ideal setups .

1009.6155

in standard thin disk accretion theory @xcite , the angular momentum axis of the accretion flow is assumed to be aligned with the black hole spin axis . @xcite found that even if the initial angular momentum axis of the accretion flow is misaligned from the black hole spin axis , the inner part of the disk will still align on the viscous timescale . however , this so - called viscous " regime only operates when @xmath1 , where @xmath2 is the scale height of the accretion disk , and @xmath3 is the parameterized viscosity @xcite . this is applicable in active galactic nuclei ( agn ) and the high / soft or thermal state of black hole x - ray binaries . on the other hand , advection - dominated accretion flows ( adafs ) are expected in the low / hard state of black hole x - ray binaries @xcite and in low - luminosity agn . adafs are unable to cool through efficient radiation , and are geometrically thick . it is likely that the accretion flow in many of these sources is misaligned , or `` tilted . '' contemporary general relativistic mhd simulations ( grmhd , * ? ? ? * ; * ? ? ? * ) currently provide the most physically realistic description of the inner portion of accretion flows around spinning black holes . radiation can be calculated from these simulations in post - processing by assuming that it is dynamically and thermodynamically negligible . this method has been used to look for high frequency quasi - periodic oscillations ( hfqpos ) in simulated data @xcite and to create radiative models of sagittarius a * @xcite . all of this work assumed alignment between the angular momentum axis of the accretion flow and the black hole spin axis . @xcite were the first to do grmhd simulations of disks with a tilt between these two axes . these new simulations yielded a number of unexpected features . first , the main body of the disk remained tilted with respect to the symmetry plane of the black hole ; thus there was no indication of a bardeen - petterson effect in the disk at large . the torque of the black hole instead principally caused a global precession of the main disk body @xcite . the time - steady structure of the disk was also warped , with latitude - dependent radial epicyclic motion driven by pressure gradients attributable to the warp @xcite . the tilted disks also truncated at a larger radius than expected for an untilted disk . in fact , based on dynamical measures , the inner edge of these tilted disks was found to be independent of black hole spin @xcite , in sharp contrast to the expectation that accretion flows truncate at the marginally stable orbit of the black hole . finally , @xcite found evidence for trapped inertial waves in a simulation with a black spin @xmath4 , producing excess power at a frequency @xmath5 hz . in this work we use relativistic ray tracing to produce images and light curves of some of these numerically simulated tilted and untilted black - hole accretion disks . our goal in this paper is to discuss observable differences between the two types of accretion flows , and to identify observational signatures of tilted black hole accretion disks . the simulations used here are from @xcite . the parameters are given in table [ sims ] . all of the simulations used the cosmos++ grmhd code @xcite , with an effective resolution of @xmath6 for the spherical - polar grid ( except near the poles where the grid was purposefully underresolved ) and @xmath7 for the cubed - sphere grid . the simulations were initialized with an analytically solvable , time - steady , axisymmetric gas torus @xcite , threaded with a weak , purely poloidal magnetic field that follows the isodensity contours and has a minimum @xmath8 initially . the magnetorotational instability ( mri ) arose naturally from the initial conditions , and the disk quickly became fully turbulent . the simulations were all evolved for @xmath98000 m , or @xmath940 orbits at @xmath10 m in units with @xmath11 . only data from the final @xmath12 of the simulation are used in this analysis , once the disks are fully turbulent as measured by a peak in the accretion rate and in the mass inside of @xmath10 m . this is chosen to utilize as much of the simulation data as possible , and none of our results depend on which time interval in the simulation is used . cccc 0h & 0 & ... & spherical - polar + 315h & 0.3 & @xmath0 & spherical - polar + 50h & 0.5 & @xmath13 & cubed - sphere + 515h & 0.5 & @xmath0 & spherical - polar + 715h & 0.7 & @xmath0 & spherical - polar + 90h & 0.9 & @xmath13 & spherical - polar + 915h & 0.9 & @xmath0 & spherical - polar these simulations all evolved an internal energy equation , and injected entropy at shocks . such a formulation does not conserve energy , and produces a more slender , cooler torus than conservative formulations which capture the heat from numerical reconnection of magnetic fields @xcite . the scale height spanned the range @xmath14 in these simulations , with larger scale heights for higher spin simulations . relativistic radiative transfer is computed from simulation data via ray tracing . starting from an observer s camera , rays are traced backwards in time assuming they are null geodesics ( geometric optics approximation ) , using the public code described in @xcite . in the region where rays intersect the accretion flow , the radiative transfer equation is solved along the geodesic @xcite in the form given in @xcite , which then represents a pixel of the image . this procedure is repeated for many rays to produce an image , and at many time steps of the simulation to produce time - dependent images ( movies ) . light curves are computed by integrating over the individual images . sample images of two simulations are given in figure [ imgs ] . doppler beaming causes asymmetry in the intensity from approaching ( left ) and receding ( right ) fluid . photons emitted from the far side of the accretion flow are deflected toward the observer , causing it to appear above the black hole . the thick , central ring is due to gravitational lensing from material passing under the black hole , while the underresolved circular ring is caused by photons that orbit the black hole one or more times before escaping . these ring features are in excellent agreement with the predictions made by @xcite . to calculate fluid properties at each point on a ray , the spacetime coordinates of the geodesic are transformed from boyer - lindquist to the kerr - schild coordinates used in the simulation . since the accretion flow is dynamic , light travel time delays along the geodesic are taken into account . data from the sixteen nearest zone centers ( eight on the simulation grid over two time steps ) are interpolated to each point on the geodesic . between levels of resolution near the poles on the spherical - polar grid , data from the higher resolution layer are averaged to create synthetic lower resolution points , which are then interpolated . very little emission originates in the underresolved regions of the simulation . the simulations provide mass density , pressure , velocity and magnetic field in code units . these are converted into cgs units following the procedure described in @xcite and @xcite . the length- and time - scales are set by the black hole mass , taken to be @xmath15 throughout . we consider two emission models . the thin line emissivity from @xcite is a toy model that traces the mass density in the accretion flow . the thermal emission model from @xcite uses free - free emission and absorption coefficients , and is used as a model for the high / soft state . although we do not expect tilted disks to accurately represent the high / soft state , this model may be appropriate for sources radiating at an appreciable fraction of eddington , where the infall time is shorter than the radiative diffusion time and the accretion flow becomes geometrically slim . " when taking the temperature from the ideal gas law rather than the radiation - dominated equation of state used in @xcite , this model may be qualitatively appropriate for modeling the low / hard state in x - ray binaries or low - luminosity agn . in [ radedge ] , we consider emission from inside of @xmath16 m , while in [ lines ] and [ var ] fluid inside of @xmath17 m is used for the ray tracing . for all results here , we take the temperature from the ideal gas law rather than assuming a radiation - dominated equation of state . all of our results are qualitatively identical when using the radiation - dominated equation of state to calculate the temperature . inferring the inner edge of accretion flows is important for attempts to measure spin from broad iron lines ( e.g. * ? ? ? * ) or continuum fitting ( e.g. * ? ? ? * ; * ? ? ? such measurements assume that the disk has a sharp cutoff at the innermost stable circular orbit , which depends on spin @xcite . @xcite used four different dynamical measures from @xcite to compare the inner edges of simulated tilted and untilted accretion disks . here we use the ray traced models to locate the radiation edge , " the radius inside of which the contribution to the total flux is negligible . for each emission model , images are calculated for all simulations over a grid of observer inclination , observer time , and observer azimuth ( for the tilted simulations ) . we then compute images cutting out fluid inside of successive values of the radius @xmath18 . the radiation edge is functionally defined as the radius where the ratio of intensities , @xmath19 , drops below an arbitrary fraction @xmath20 , chosen so that the untilted radiation edge agrees as well as possible with @xmath21 , the marginally stable orbit . figure [ tfluxrline1 ] shows a plot of @xmath19 as a function of @xmath18 averaged over observer time , azimuth and inclination for the thin line emissivity . from these curves we extract values of the radiation edge , @xmath22 . results are shown in figure [ rinline1 ] , where the error bars are computed from the standard deviation of @xmath22 as a function of time , averaged over the other parameters . this result agrees well with the dynamical measures from @xcite . while the radiation edge moves in towards the black hole with increasing spin for untilted simulations , there is no such trend in the tilted simulations . instead , the radiation edge appears to be independent of spin . figures [ tfluxrline2 ] and [ rinline2 ] show the same plots for the thermal emission model with observed photon energy @xmath23 kev . the conclusions are identical with this emission model . the untilted simulations have radiation edges which agree quite well with @xmath21 , while the tilted simulations show no correlation between spin and @xmath22 . again , these results are consistent with @xcite , although we find no trend of _ increasing _ radiation edge with spin , as was found for a couple of the dynamical measures used in @xcite . plots from other observed photon energies are not shown ; although the relative flux falls off much more quickly with increasing @xmath18 at higher photon energies , the results for the radiation edge remain completely unchanged . spectra from agn and x - ray binaries typically include strong emission and absorption features . as the observed line shapes are sensitive to both the velocity of the emitting / absorbing fluid , and also to the local gravitational redshift , they can provide information about the dynamics of the accretion flow @xcite . untilted accretion flows have nearly keplerian velocity distributions outside the marginally stable orbit , where the velocities smoothly transition to plunging . simulated tilted accretion disks , on the other hand , show three major differences . the keplerian velocity structure is now tilted . secondly , the warped structure of the tilted disks leads to epicyclic motions with velocity magnitudes comparable to the local geodesic orbital velocity @xcite . finally , the larger radiation edge values of the tilted disks identified in [ radedge ] means that the transition to plunging orbits occurs at larger radius than in untilted disks . these effects indicate that we should expect a number of differences in line profiles from tilted accretion flows @xcite . the maximum blueshift should be larger for tilted accretion disks , except for edge - on viewing . for @xmath24 , where @xmath25 is the observer s inclination angle and @xmath26 is the initial tilt angle , both relative to the black hole spin axis , the tilted accretion flow should mimic an untilted one with a larger inclination . in contrast , the red wing should be less pronounced in the tilted disks due to their larger truncation radii . on the redshifted side , tilted disks behave similar to lower spin , untilted disks . producing a detailed reflection spectrum would require a significant number of assumptions to model the metallicity , ionization levels , and incident x - ray flux throughout the accretion flow . for simplicity , we instead use toy model emissivities of the form @xmath27 , where @xmath28 is the fluid mass density , @xmath29 is the photon - energy integrated emissivity and @xmath30,@xmath31 . the two values correspond to assuming the emitted line flux is proportional to the incident flux from an irradiating source on the spin axis and to local dissipation of heat , respectively ( e.g. , * ? ? ? this simple form allows us to focus on general features to be expected from emission lines from tilted black hole accretion disks . figure [ iline ] shows sample line profiles for an inclination of @xmath32 for four observer azimuths from the 915h simulation . only a single observer azimuth from the 90h simulation is shown , since the time - averaged emission line is independent of observer azimuth for untilted simulations . in all cases , the lines consist of a strong peak near the rest energy of the line ( @xmath33 ) , a smaller peak at lower energy and a red wing , " whose extent and strength depends on the amount of emission arising very close to the black hole ( small @xmath34 ) . the location of the blue " peak ( large @xmath34 ) depends on the maximum velocity along the line of sight in the accretion flow . for an untilted disk , this corresponds directly to the observer s inclination angle , since all fluid velocities are essentially in the equatorial plane . for the tilted model shown in figure [ iline ] , the location and strength of the blue peak changes significantly with observer azimuth . when the angular momentum axis of the accretion flow is in the plane of the sky ( @xmath35 , depending on the simulation time ) , its fluid velocities are maximally aligned with the observer s line of sight , leading to the largest blueshifts . this is the same condition as an untilted disk being viewed edge - on . for other orientations , the blue tail can extend to significantly higher photon energies in the tilted simulations because the largest effective inclination is approximately @xmath36 . when the accretion flow is not edge - on , there will exist orientations where @xmath37 , and the blue peak for a tilted simulation will occur at higher energy than possible for untilted accretion flows . the red wing , on the other hand , remains largely unchanged with observer azimuth , since it is caused by gravitational redshifts rather than doppler boosts . since the radiation edge for the 915h simulation was found to occur at significantly larger radius than that of 90h , it is expected that the red wing should extend further in the 90h simulation . the effect is subtle , but identifiable in figure [ iline ] . to quantify these trends , for all simulations we compute the extent of the line profile , as well as the strengths and locations of their red and blue peaks . most clear are the results for the line extents , shown for @xmath32 in figures [ ilinew1 ] and [ ilinew2 ] . as expected , the red wing extends to lower photon energies at higher spins for untilted simulations , while there is no similar trend for the tilted models . also as expected , the blue wing extends to systematically higher photon energies in the tilted simulations because of the difference between @xmath38 and @xmath25 noted above and the epicyclic motion in the tilted simulations . perhaps the most striking feature of the line profiles is the variation with observer azimuth seen in all tilted simulations . these changes in line shape between different observer azimuths are typically larger than the full range of changes seen between different spins for untilted simulations . this suggests that the most powerful means of recognizing a tilted accretion disk may be to measure changes in an emission line profile over time as the disk precesses . x - ray timing of black hole binaries has allowed the characterization of power spectra and the detection of transient qpos ( for a review , see * ? ? ? * ) . high - frequency qpos are seen in the steep power law state ( spl ) , while low - frequency qpos have been observed in both the hard state and the spl . the geometry of the accretion flow in both these states is uncertain , and there is no reason to assume complete alignment between the accretion flow angular momentum and black hole spin axes in these states . given the time - dependent nature of the ray tracing , we can analyze the variability of the simulated accretion flows for the simplistic emission models used here to analyze the shape of their power spectra and to look for possible qpos . the best time sampling of the simulations is in 90h and 915h , which are used here at 8 observer azimuths , 3 inclinations and 3 observed photon energies using the thermal emission model . each light curve captures roughly 6 ( 20 ) orbits at @xmath17 m ( 10 m ) , corresponding to a total observer time , @xmath39 s , where @xmath40 is the black hole mass in units of @xmath41 . this duration is about @xmath42 of the total precession period for the torus in the 915h simulation . figure [ lexample ] shows sample light curves and power spectra from the thermal emission model at @xmath43kev for an observer inclination of @xmath32 . the secular trend is removed by subtracting the linear best fit from the light curve before computing the power spectrum . all power spectra are well fit by broken power law models of the form : @xmath44 where @xmath45 , @xmath46 are power law indices and the break frequency , @xmath47 , lies near @xmath48hz @xmath49 in both simulations . the tilted disk power spectra tend to flatten out at the highest sampled frequencies , @xmath91000hz @xmath49 . figure [ allpsds ] shows median power spectra for the three different inclination angles from each simulation . the error bars are estimated from the standard deviation in @xmath50 over observer azimuths and photon energies . at higher inclinations , the peaks in the power around @xmath48hz grow , especially for the tilted simulations . this would be expected from a source of excess power in the inner radii , where the larger doppler shifts at higher inclination would enhance the signal . to quantitatively compare the power spectra between the two simulations , the ratio between median power spectra in untilted and tilted simulations is plotted for each inclination in figure [ relpower ] . the values are normalized to the combined uncertainties at each frequency . the overall plots are shifted according to the mean ratio between power spectra . at almost all frequencies , these ratios are within @xmath51 , and are unlikely to be observed as significant features . however , there are a few noteworthy features near @xmath52hz . these are particularly interesting given the finding by @xcite that the tilted simulation 915h contains excess power due to trapped inertial waves at @xmath53hz . to assess the significance of possible features in the psds , the power spectrum is fit with a broken power law model . the parameters from the best fit are used to simulate many random light curves with the same parameters , and which contain no significant features . the significance is determined by comparing the values for the power at each frequency for each model power spectrum with the distribution of random ones . an example is shown in figure [ fitexample ] , where a single power spectrum from the 915h simulation is shown , as well as the best fit broken power law model and upper and lower 99.9% confidence intervals from simulating random light curves . no obvious qpo features show up in this analysis . in several of the 915h light curves , the feature near @xmath54hz shows up as 99.9% significant . it appears at high significance in more of the light curves at high inclinations . in the 90h simulations , almost all significant features are found at very high frequencies @xmath91000@xmath55hz . these are spurious , caused by slight errors in the fit to the post - break slope incurred by ignoring all frequencies larger than @xmath56hz . including the highest frequencies in the fit can favor models with break frequencies @xmath9500 @xmath49hz , steep initial slopes and shallow post - break slopes . this occurs due to the denser sampling of the psd at high frequencies . simply ignoring the highest frequencies gives better results than a variety of more complicated weighting schemes . the features near @xmath52hz from figure [ fitexample ] never show up at more than 99% significance . in general , while the feature near @xmath54hz in the tilted simulations is more convincing than anything from the untilted simulations , it does not appear at high enough significance at enough observer frequencies and azimuths to be identified as a qpo . finally , fitting the sets of power spectra provides a general idea for the range of best fit values of the broken power law parameters . the median parameters found from the tilted and untilted simulation are listed in table [ vartable ] , where the quoted uncertainties are the standard deviations from light curves with different observer azimuths and frequencies . break frequencies have the units @xmath49hz . the break in slope becomes more pronounced at higher inclination as the initial slope becomes shallower while the post - break slope becomes steeper . the post - break slope is slightly shallower in the tilted simulations , while the initial slope is more strongly dependent on inclination in the untilted case . lcccccc @xmath45 & @xmath57 & @xmath58 & @xmath59 & @xmath60 & @xmath61 & @xmath62 + @xmath46 & @xmath63 & @xmath64 & @xmath65 & @xmath66 & @xmath67 & @xmath68 + @xmath47 & @xmath69 & @xmath70 & @xmath71 & @xmath72 & @xmath73 & @xmath74 the observable signatures of tilted disks discussed so far are , for the most part , due to two main differences between tilted and untilted disks : tilted disks precess , and they are truncated outside @xmath21 . @xcite already discussed why the simulated accretion flows precess . it is our interest to better understand the physical cause for the large truncation radius . the first thing to note is that rapidly rotating black holes provide more centrifugal support to an accretion disk than slowly rotating black holes . therefore , the angular momentum extraction mechanism at play in the tilted disks must be more effective at higher spin . this is confirmed in figure [ deltal ] , where we plot the difference in density - weighted , shell - averaged specific angular momentum for tilted and untilted simulations of comparable spin . the angular momentum is defined as @xmath75 , where @xmath76 is the fluid four - velocity and the shell - average of a quantity @xmath77 is given by , @xmath78 where @xmath79 is the coordinate solid angle , @xmath34 is the metric determinant and @xmath80 . the density - weighted shell - average of @xmath77 is defined as @xmath81 . the angular momentum profiles for the untilted simulations are nearly geodesic outside of @xmath82 m . inside of @xmath83 m , the tilted simulations become increasingly sub - geodesic , with the higher spin cases deviating more than the lower spin ones . the same trend holds when comparing the tilted simulations to the analytic result for the angular momentum profile of material on geodesic orbits in an equatorial disk inclined @xmath0 to the black hole spin axis . @xcite suggested that the non - axisymmetric standing shocks that occur in the inner radii above and below the midplane of the disk may enhance the outward transport of angular momentum , causing fluid to plunge from outside the marginally stable orbit . to connect the enhanced angular momentum loss of the tilted disks with the standing shocks , we next look at a plot of the density - weighted , shell - averaged entropy profiles in figure [ entropy ] . since these simulations conserve entropy except across shocks , the excess inside of @xmath84 m in the tilted simulations signifies the presence of extra shocks . the steepness of the entropy gradient gives some measure of the strength of these shocks . again , we see that the effect is greatest in the simulations with the fastest spinning black holes . further evidence linking the sub - geodesic angular momentum profiles of the tilted simulations with the standing shocks can be found from looking at the time - dependence of the shell - averaged angular momentum . while the untilted simulation remains nearly geodesic , the tilted simulations are continuously transporting angular momentum outward from @xmath83 m for the first @xmath85 m before reaching a steady state , as would be expected from a dynamical mechanism . finally , vertically integrated contour plots such as figure [ cplots ] show that the angular momentum in the tilted simulations is non - axisymmetrically distributed . the regions of depleted angular momentum correspond to the standing shocks , which appear as regions of excess entropy in the bottom panels of figure [ cplots ] . following @xcite , we postulate that the standing shocks are caused by deviations from circular orbits near the black hole . figure [ eccentricity ] shows the shell - averaged eccentricities of the orbits in each simulation , estimated at one scale - height in the disk using @xmath86 where @xmath26 is the tilt and @xmath87 is the precession of each orbital shell . differs from @xcite by a phase factor of @xmath88 in @xmath87 . @xcite used the formula from @xcite without modification . ] all quantities are calculated from fitting the shell - averaged disk tilt and twist ( eqs . 32 and 41 of @xcite ) with power laws , and using the resulting expressions in equation ( [ eq : ecc ] ) . the increase in eccentricity toward smaller radii leads to a crowding of orbits near their apocenters @xcite , which leads to the formation of the standing shocks . the eccentricity is larger for higher black hole spin , except inside the plunging region where the fits become poor and the eccentricity is ill - defined . equation ( [ eq : ecc ] ) may indicate how these results depend on the initial tilt of the simulations . if we assume that the strongest dependence of @xmath89 on tilt is through @xmath26 and that @xmath90 and @xmath91 remain unchanged for different tilts , then equation ( [ eq : ecc ] ) suggests that the eccentricity of the orbits should vary roughly linearly with the initial tilt , at least for small angles . this prediction is tentatively confirmed by a simulation we have done that started with an initial tilt of @xmath92 . tilted accretion flows will inevitably be present in a significant fraction of black hole sources with @xmath93 and possibly @xmath94 ( thick or slim disks ) . using relativistic ray tracing and a set of simple emissivities , we have compared the radiation edge , emission line profiles and power spectra of simulated black hole accretion flows with a tilt of @xmath0 to their untilted counterparts . we find the radiation edge is independent of black hole spin , while the untilted simulations agreed with the expected qualitative trend of decreasing inner radius with increasing spin . these results for the radiation edge confirm the work of @xcite , who used dynamical measures to locate the inner edge . due to the independence of inner edge on spin , the red wing of tilted accretion flow emission line profiles is also fairly independent of spin . this introduces a possible complication for attempts to measure black hole spin from sources which may be geometrically thick . in general , measurements of small spin ( large inner radius ) may be unreliable unless the disk is known to be untilted . a reliable estimate of a large black hole spin ( small inner radius ) , in contrast , could rule out the presence of a tilted disk . the blue wing can be much broader for tilted accretion flows , and the tilted - disk line profiles depend strongly on the observer azimuth as well as inclination . since a tilted disk is expected to precess @xcite , highly variable emission line profiles could signify the presence of a tilted accretion flow , as pointed out by @xcite for warped thin disks . since many llagn and x - ray binaries in the low / hard state should be tilted , time - variable emission lines should be quite common , and this effect is unlikely to significantly depend on accurate reflection spectrum modeling . although the simulations can only be run for a short time compared to the precession time scale , precession is a possible source of low frequency quasi - periodic oscillations when the accretion flow is optically thin due to the modulation of doppler shifts as the velocities in the accretion flow align and misalign with the observer s line of sight ( see * ? ? ? * for more discussion of qpos from precessing tilted disks ) . finally , we have studied power spectra for our simple models . we find broken power law spectra with break frequencies around @xmath52hz and power law indices in the range 0 - 2 ( 3 - 4 ) pre- ( post- ) break for both tilted and untilted simulations . previous studies @xcite found single power laws with index @xmath92 . @xcite found that power spectra from individual annuli are well described by broken power laws where the break frequency is close to the local orbital frequency the averaging of many annuli with an emissivity that falls with radius smooths the power spectrum into a single power law . we see the same behavior in our simulations ; the break frequencies from power spectra of individual radial shells agree with the local orbital frequency for both simulations 90h and 915h . a break frequency @xmath52hz then implies a radius of @xmath95 m . our broken power law spectra are therefore likely due to the fact that our emissivity peaks relatively near the outer radius used for the ray tracing , @xmath17 m . a larger radial domain would likely shift the break to smaller frequencies . observed break frequencies in the low / hard state are typically @xmath96hz , which may be caused by the transition from a thin disk to a thicker , adaf flow @xcite . that would imply a transition radius @xmath97 . our results for pre- and post - break slopes from both tilted and untilted simulations agree with those found in cygnus x-1 @xcite for an inclination @xmath98 . in gro j1655 - 40 @xcite our pre - break slopes agree for all inclinations . however , the psd for that source is well described by a single power law . there is no clear evidence in our work for high frequency qpos due to the trapped inertial waves identified by @xcite , although there are more features in power spectra from the 915h simulation at higher significance than in 90h . even when computing psds for sets of spherical shells from the simulations , there are no clear features in the tilted power spectra that are not also present in the untilted case . it is possible that this result could depend on the chosen emissivity . alternatively , the excess power in trapped inertial waves could be insufficient to rise above the red noise continuum . the independence of the inner radius of the tilted simulations on black hole spin is attributable to the extra angular momentum transport provided by the asymmetric standing shocks . these shocks are only present in the tilted simulations . their strength scales with black hole spin , which is a necessary condition for countering the greater centrifugal support at higher spins . the standing shocks , in turn , appear to be attributable to epicyclic motion within the disk driven by pressure gradients associated with the warped structure . again , this effect scales with the spin of the black hole , which contributes to the stronger shocks . for small tilt angles , the orbital eccentricity scales as @xmath99 . this suggests that significant deviations between the spin - dependence of the radiation edge and the marginally stable orbit should be present even at modest tilt angles @xmath100 . at larger tilts , it is unclear if the increasing eccentricity will lead to an inner edge that increases with spin . this is both due to the uncertainty in the radial tilt and twist profiles @xmath101 and @xmath102 at larger tilts , and to the lack of a quantitative connection between inner disk edge and eccentricity . the dynamical measures from @xcite place the location of the inner edge in a simulation with @xmath103 and @xmath104 closer to the location of 915h than 90h . this data point supports the idea that a noticeable departure between @xmath22 and @xmath105 should exist between tilted and untilted disks even for @xmath100 . it also suggests that at larger tilt angles , @xmath22 is likely to increase with spin unless the effect saturates at @xmath106 . simulations with larger tilt angles will be able to address this question with certainty . we thank omer blaes and eric agol for many stimulating discussions . this work was partially supported by nasa grants 05-atp05 - 96 and nnx08ax59h ; a graduate fellowship at the kavli institute for theoretical physics at the university of california , santa barbara under nsf grant phy05 - 51164 ; and nsf grant ast08 - 07385 .

geometrically thick accretion flows may be present in black hole x - ray binaries observed in the low / hard state and in low - luminosity active galactic nuclei . unlike in geometrically thin disks , the angular momentum axis in these sources is not expected to align with the black hole spin axis . we compute images from three - dimensional general relativistic magnetohydrodynamic simulations of misaligned ( tilted ) accretion flows using relativistic radiative transfer , and compare the estimated locations of the radiation edge with expectations from their aligned ( untilted ) counterparts . the radiation edge in the tilted simulations is independent of black hole spin for a tilt of @xmath0 , in stark contrast to the results for untilted simulations , which agree with the monotonic dependence on spin expected from thin accretion disk theory . synthetic emission line profiles from the tilted simulations depend strongly on the observer s azimuth , and exhibit unique features such as broad blue wings . " coupled with precession , the azimuthal variation could generate time fluctuations in observed emission lines , which would be a clear `` signature '' of a tilted accretion flow . finally , we evaluate the possibility that the observed low- and high - frequency quasi - periodic oscillations ( qpos ) from black hole binaries could be produced by misaligned accretion flows . although low - frequency qpos from precessing , tilted disks remains a viable option , we find little evidence for significant power in our light curves in the frequency range of high - frequency qpos .

1101.3783

until recently , little was known about the population of transient sources at low radio frequencies due to the lack of previous dedicated , sensitive surveys . many of the known target transient populations are synchrotron sources , hence predicted to be faint and vary on long timescales at low radio frequencies ( such as afterglows from gamma - ray bursts and tidal disruption events ; for a recent review see * ? ? ? however , there are a number of different populations of sources that are expected to emit short duration bursts of low frequency coherent radio emission and are anticipated to be detectable in short snapshot low radio frequency images ( e.g. giant pulses from pulsars and flares from brown dwarfs or exoplanets ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? one such coherently emitting target is the population of fast radio bursts ( frbs ; * ? ? ? * ; * ? ? ? frbs were discovered at 1.4 ghz using high time resolution observations from the parkes radio telescope . these sources constitute single , non - repeating , bright pulses of millisecond duration at 1.4 ghz that are highly dispersed , suggesting an extra - galactic origin . a number of theories have been proposed as the progenitors of frbs , including both extra - galactic ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and galactic origins ( e.g. * ? ? ? the scattering for frbs is highly dependent upon the observing frequency and is expected to smear out the pulse to much longer durations at low radio frequencies @xcite . the pulse durations at low radio frequencies make them more difficult to detect using standard search methods at high time resolution . instead , their durations are expected to be comparable to those attainable in short snapshot images . however , it is unclear what the rates of frbs at low frequencies will be because the rates are still being constrained at higher frequencies and little is known about their spectral shape ( e.g. * ? ? ? * ; * ? ? ? therefore , observations at low frequencies will aid in constraining both the rates and the spectral slopes of frbs . by more tightly constraining the rates , some progenitor mechanisms may be ruled out , including those associated with other populations with relatively low rates ( such as short gamma - ray bursts ; * ? ? ? additionally all frbs to date have been detected using single dish telescopes leading to large positional uncertainties ( e.g. 14 arcmin ; * ? ? ? * ) . by detecting frbs in short snapshot image plane data observed using a low frequency radio interferometer , their positions can be constrained to higher accuracy ( @xmath31 arcmin ) enabling host galaxy associations and deep constraints on multi - wavelength counterparts . additionally , an interferometer will obtain more reliable flux densities , as single dish observations are subject to flux density uncertainties as the position of the source within the primary beam is unknown . this provides better constraints on the flux density distribution of sources ( @xmath6@xmath7 distribution ) . over the past few years , the search for transient sources at low radio frequencies has intensified with the arrival of sensitive , wide - field instruments such as the murchison wide - field array ( mwa ; * ? ? ? * ; * ? ? ? * ) , the low frequency array ( lofar ; * ? ? ? * ) and the long wavelength array station 1 ( lwa1 ; * ? ? ? additionally , the automated processing of very large datasets is being enabled via the long - term source monitoring capabilities of specially developed pipelines , including the lofar transients pipeline ( trap ; * ? ? ? * ) and the pipeline for the askap survey for variables and slow transients ( vast ; * ? ? ? dedicated transient surveys are utilising the improvement in instrumentation and software to constrain the surface densities of transients at these low frequencies on a range of timescales and sensitivities ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? orders of magnitude improvement in sensitivity or search area will be required to more tightly constrain their rates . this can be attained by the next generation of radio telescopes , such as the square kilometre array ( ska ; e.g. * ? ? ? . however , obtaining the required observation time may be difficult on over - subscribed instruments and transient surveys will need to utilise commensal observations . this paper uses observations from one such dataset , the mwa observations for the campaign to detect the epoch of re - ionisation ( eor ) in which hundreds of hours of observing time are required on individual fields . this dataset can probe variability and transients on timescales ranging from seconds up to years , enabling constraints to be placed on both the long timescale incoherent emission mechanisms , e.g. synchrotron emission from active galactic nuclei ( agn ) , short timescale coherent emission mechanisms such as frbs and scintillation processes on a range of timescales . this paper describes a pilot transient and variability search using 78 hours of the mwa eor dataset , producing highly competitive transient rates . the 28 second snapshot timescale is chosen to specifically target the expected population of frbs . this work complements @xcite , a search for frbs using mwa observations imaged on a much shorter integration time ( 2 seconds ) and conducting an image plane de - dispersion to search for frbs . via this method , @xcite are able to attain an improvement in sensitivity for frbs in comparison to the standard processing strategies at the expense of processing speed and resolution . whereas a standard imaging strategy , such as that utilised in this paper , enables more observations to be processed in a comparable timescale and the use of the data products for additional science such as longer duration transient and variability studies . without de - dispersion , a dispersed frb will be detected at a lower flux density in the short snapshot images as the original signal is averaged over both time and frequency . therefore , these two approaches are complementary ; @xcite increases sensitivity by sacrificing surveyed area whereas the survey conducted in this paper sacrifices sensitivity to increase the amount of surveyed area . additionally , candidate frbs identified in this analysis can be independently confirmed as frbs by measuring their dispersed signal using the pipeline developed @xcite . section 2 of this paper describes the processing strategies used to make all the images and the analysis strategies implemented to conduct quality control and to search for transient sources . in section 3 , we present the limits on transients detected on a range of timescales and focus on the implications for the rates and spectral shapes of frbs by comparison to previous studies at other frequencies . finally , section 4 provides an initial analysis of variability of known sources within the field . the data used in this paper are obtained from a commensally observed dataset for the transients team and the eor team . the full dataset comprises of @xmath81000 hours targeting 3 specific fields , well off the galactic plane , centred on 2 different observing frequencies ( 154 and 182 mhz ) . processing the full dataset is a `` big data '' scale computational challenge due to the supercomputing time required and the data volume at all of the processing stages . by targeting a subsample of the dataset , we can develop automated strategies to make the data volume more manageable and quantify the supercomputing requirements for the full dataset . in this study , we choose observations of a single target field , centred on ra : 0.00 deg , dec : -27.00 deg ( 00:00:00 , @xmath927:00:00 ; j2000 ) , at the observing frequency of 182 mhz and at elevations of @xmath875 degrees . this field is centred on the galactic co - ordinates l : 30.636 deg , b : -78.553 deg ( 30:38:08.4 , -78:33:10.6 ) . the observations were conducted by taking multiple pointed observations as the field drifts through 5 different azimuth - elevation pointing directions centred on zenith . these observations were then phase centred to ra : 0.00 dec : @xmath927.00 deg ( j2000 ) , with a primary beam half width half maximum ( hwhm ) of 11.3 degrees . this leads to a sample size of 3010 individual observations of 2 minute integration times , or 100 hours , in the time range 2013 august 23 2014 september 14 . .the wsclean settings used to image all the observations presented in this analysis . all other settings were the default settings . [ cols="<,^",options="header " , ] for this section , we put all the images through trap utilising the monitoring strategy described in section 2.6 and monitor the variability of sources detected in the median image with flux densities in excess of 0.5 jy . as stated previously , there are residual primary beam issues when comparing images at different az - el pointing directions so we process each pointing direction separately . variable candidates are then compared between each of the pointing directions . trap measures two key variability parameters for every unique source in the dataset . the first parameter is the reduced weighted @xmath10 at a given observing frequency , @xmath11 , given by : @xmath12 where @xmath13 is the number of datapoints , @xmath14 is the flux density of a datapoint , @xmath15 , @xmath16 is the error on the @xmath17-th flux density measurement and over bars represent the mean values ( the full derivation of this from the standard reduced weighted @xmath10 is given in * ? ? ? * ) . the second parameter is the coefficient of variation ( also known as the modulation index ) at each observing frequency , @xmath18 , given by : @xmath19 where @xmath20 is the standard deviation of the observed flux densities . these parameters are measured for each time step that the source is observed . in the following analysis , we focus on the variability parameters for each unique source from the final time step . in figure [ variplt ] , we show these variability parameters from the end of each trap run for each of the 5 unique azimuth - elevation pointing directions . as the timescales probed by each of the different pointing directions is roughly the same , it is expected that the typical source parameters for the different pointing directions will be in good agreement and this is clearly the case in figure [ variplt ] . the @xmath18 distribution is well fitted with a gaussian distribution , with a typical value of @xmath21 , consistent with the typical flux density uncertainties measured in section 2.8 . we note that the absolute @xmath11 values should be a factor @xmath03 lower due to the correlated noise observed in these images ( see section 2.4 ) ; this does not affect the analysis in this section as we are only considering sources that are anomalous to the distribution . this factor will need to be considered when quantifying low level variability in future analysis . the @xmath11 distribution is clearly right - skewed with an excess of sources at higher values , suggesting that there may be variability in some of the source light curves . however , as the @xmath18 parameters of these sources are comparable to the rest of the population , this is unlikely ( variable sources have anomalously high values for both variability parameters and are expected reside in the top right corner of this plot ) . by visual inspection , we note that many sources show variation on specific nights , pointing to a possible ionospheric origin or residual calibration issues . further analysis is ongoing . using additional source parameters ( particularly the maximum flux density that a source attains and the ratio between that maximum flux density and the average flux density of the source ) can aid in understanding the population of sources and can be used to more clearly separate the variable sources from a stable population . in figure [ variplt2 ] , we plot these 4 parameters for each of the different pointing directions . we see a clear correlation between the maximum flux density and @xmath11 as expected ( this is caused by the measurement accuracy of flux densities for bright sources , which does not take into account systematic uncertainties ) . additionally , we observe a negative trend between the maximum flux densities and @xmath18 , however there is a clear diversion from this trend at flux densities @xmath82 jy . this diversion is caused by a large number of images , coincident in time , having a systematic flux density scale offset ( of order 10% ) from the rest of the images . this is likely caused by uncertainties in the primary beam model and is expected to be resolved in future work when we address the issues between different pointing directions . to avoid the region where the systematic uncertainties are dominating , we utilise a @xmath22 threshold on both the @xmath11 and @xmath18 parameters to identify variable sources . this corresponds to variable sources requiring @xmath23 and @xmath24 , equivalent to a flux density variation in excess of 20% . these thresholds will only identify sources which are significantly more variable than the typical population and does not address any intrinsic low level variability , such as the phenomenon of `` low frequency variability '' likely caused by refractive interstellar scintillation ( although this is not expected to occur in our dataset as it typically has timescales of @xmath51 year and is more prevalent at lower galactic latitudes ; e.g. * ? ? ? * ) . from the skewed @xmath11 distribution and visual inspection of a large number of light curves , we note that there is low - level variability on specific observing nights which may correspond to ionospheric activity . no significantly variable sources were identified via this method . two sources within this field have observed variability , with timescales of 2 seconds at @xmath0150 mhz , which has been interpreted as interplanetary scintillation ( ips ; * ? ? ? using the zenith observations , we identify these two sources within our dataset and determine their variability parameters . pks 2322 - 275 has @xmath25 and @xmath26 and pks 2318 - 195 has @xmath27 and @xmath28 . both of these sources are well below the variability thresholds and their parameters are highly typical for the source population . @xcite show that these sources have a typical variability timescale which is significantly shorter then the 30 second integration timescale used in this analysis . therefore , any ips events would be statistically averaged to the mean value in these images . additionally , there is one pulsar within the field , psr 2327 - 20 . this pulsar has a low dm and , hence , may undergo diffractive and refractive scintillation . in our median image , we note that this source is very faint , with a flux density of @xmath00.07 jy , and is unlikely to be detected in our images . we monitored the position of this pulsar to see if this pulsar scintillates above the detection threshold . psr 2327 - 20 is not detected in any of the snapshot images . although this analysis has not identified any significant variability , we note that the variability analysis needs a significant amount of further work to be able to identify variability of sources corresponding to @xmath2920% of their flux densities . for future analysis : * we intend to resolve remaining systematic primary beam uncertainties within the images . this will enable all the pointing directions to be processed at once , giving a much larger dataset for characterising the sources . additionally , it will resolve the deviation in @xmath18 at flux densities in excess of 2 jy , which will lead to an increase in sensitivity . * the variability parameters for sources can change significantly as the number of data points in the light curve increase . for instance , if a source emits a single flare at early times this variability may not be apparent using the variability parameters from the final time step in the dataset . this means that we may be missing interesting variability on short timescales due to processing large numbers of images at once . trap records these variability parameters as a function of snapshot and , in future , we aim to develop methods to study the variability of sources as a function of time . from our analysis of @xmath010,000 images , we note that the eor0 field is remarkably stable at 182 mhz . there are no convincing transient candidates and all sources have flux density variations of @xmath2920% . in future work , we will target remaining systematic flux density uncertainties to enable us explore low level variation within the field . the transient surface densities obtained are more constraining than previous surveys by orders of magnitude in timescale , sensitivity and snapshot rates ; although we note that this field is not sensitive to a galactic population of transient sources due to being well off the galactic plane ( galactic latitudes @xmath30 degrees ) . despite expecting to observe transients comparable to the source observed by @xcite , we instead place a constraining limit on their surface densities and/or spectral indices . on the shortest timescale , predictions scaled from the observed populations suggested that this survey would identify a small number of frbs . again , there are no detections , which are consistent with suggestions of lower rates and flat spectral indices . to further pursue these elusive transients at low radio frequencies , we need to conduct similar surveys at a range of frequencies , whilst also increasing the sensitivities and surveyed area by an order of magnitude or more . finally , there are a range of timescales that this survey does not explore , most notably the very short and @xmath81 year where a range of transient sources are anticipated . this scientific work makes use of the murchison radio - astronomy observatory , operated by csiro . we acknowledge the wajarri yamatji people as the traditional owners of the observatory site . support for the mwa comes from the u.s . national science foundation ( grants ast-0457585 , phy-0835713 , career-0847753 , and ast-0908884 ) , the australian research council ( lief grants le0775621 and le0882938 ) , the u.s . air force office of scientific research ( grant fa9550 - 0510247 ) , and the centre for all - sky astrophysics ( an australian research council centre of excellence funded by grant ce110001020 ) . support is also provided by the smithsonian astrophysical observatory , the mit school of science , the raman research institute , the australian national university , and the victoria university of wellington ( via grant med - e1799 from the new zealand ministry of economic development and an ibm shared university research grant ) . the australian federal government provides additional support via the commonwealth scientific and industrial research organisation ( csiro ) , national collaborative research infrastructure strategy , education investment fund , and the australia india strategic research fund , and astronomy australia limited , under contract to curtin university . we acknowledge the ivec petabyte data store , the initiative in innovative computing and the cuda center for excellence sponsored by nvidia at harvard university , and the international centre for radio astronomy research ( icrar ) , a joint venture of curtin university and the university of western australia , funded by the western australian state government . this research was undertaken with the assistance of resources from the national computational infrastructure ( nci ) , which is supported by the australian government . this work was supported by the flagship allocation scheme of the nci national facility at the anu . bannister k. w. , murphy t. , gaensler b. m. , hunstead r. w. , chatterjee s. , 2011 , mnras , 412 , 634 bastian t. s. , dulk g. a. , leblanc y. , 2000 , apj , 545 , 1058 bell m. e. , et al . , 2014 , mnras , 438 , 352 berger e. , et al . , 2001 , natur , 410 , 338 carbone d. , et al . , 2014 , arxiv , arxiv:1411.7928 cendes y. , et al . , 2014 , arxiv , arxiv:1412.3986 coenen t. , et al . , 2014 , a&a , 570 , a60 cohen a. s. , lane w. m. , cotton w. d. , kassim n. e. , lazio t. j. w. , perley r. a. , condon j. j. , erickson w. c. , 2007 , aj , 134 , 1245 condon j. j. , cotton w. d. , greisen e. w. , yin q. f. , perley r. a. , taylor g. b. , broderick j. j. , 1998 , aj , 115 , 1693 condon j. j. , 1974 , apj , 188 , 279 cordes j. m. , lazio t. j. w. , 2002 , astro , arxiv : astro - 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we present a survey for transient and variable sources , on timescales from 28 seconds to @xmath01 year , using the murchison widefield array ( mwa ) at 182 mhz . down to a detection threshold of 0.285 jy , no transient candidates were identified , making this the most constraining low - frequency survey to date and placing a limit on the surface density of transients of @xmath1 deg@xmath2 for the shortest timescale considered . at these frequencies , emission from fast radio bursts ( frbs ) is expected to be detectable in the shortest timescale images without any corrections for interstellar or intergalactic dispersion . at an frb limiting flux density of 7980 jy , we find a rate of @xmath382 frbs per sky per day for dispersion measures @xmath3700 pc @xmath4 . assuming a cosmological population of standard candles , our rate limits are consistent with the frb rates obtained by @xcite if they have a flat spectral slope . finally , we conduct an initial variability survey of sources in the field with flux densities @xmath50.5 jy and identify no sources with significant variability in their lightcurves . however , we note that substantial further work is required to fully characterise both the short term and low level variability within this field . [ firstpage ] instrumentation : interferometers - techniques : image processing - catalogues - radio continuum : general

1602.07544

[ [ general - assumptions . ] ] general assumptions . + + + + + + + + + + + + + + + + + + + + the luminosity distance of m87 is @xmath35 @xcite , so that an angle of @xmath36 milli - arc sec ( mas ) corresponds to @xmath37 . the mass of the central black hole has been recently determined using detailed modeling of long slit spectra of the central regions of m87 to be @xmath38 @xcite , corrected for the distance we use . as those authors point out , that mass is about @xmath39 higher than previous determinations @xcite , corrected for the different assumed distances . the reason for such a large change is not fully explained yet . the use of the new mass @xcite implies a higher resolution of our vlba observations in terms of @xmath6 , but does not substantially affect our conclusions . ] . the schwarzschild radius is @xmath40 ; @xmath41 therefore corresponds to approximately @xmath42 . the observed timing delay @xmath43 between the onset of the vhe flare and the radio peak is about @xmath44 ( @xmath45 ) . the following equations hold : the observed velocity in units of the speed of light @xmath9 is calculated as @xmath46 / [ 1 - \beta_{\rm{int } } \cos \theta]$ ] , the doppler boost factor is @xmath47^{-1}$ ] , the observed time delay is @xmath48 $ ] , and the distance along the jet is @xmath49 for @xmath50 ( see below ) . [ [ the - jet - orientation - angle . ] ] the jet orientation angle . + + + + + + + + + + + + + + + + + + + + + + + + + + the most rapid optical proper motions are observed along the jet with superluminal speeds @xcite of @xmath51 ( hst-1 ) , @xmath52 ( knot d ) , and @xmath53 ( knot e ) . on the other hand , the most rapid radio proper motions seen from features moving through hst-1 indicate superluminal speeds in the range of @xmath54 @xcite and through knot d indicate a superluminal speed of @xmath55 @xcite . assuming the jet plasma moves at the speed of light ( @xmath56 ) , one obtains a maximum viewing angle of @xmath57 / [ \beta_{\rm{obs}}^{2 } + 1 ] $ ] . the highest optically determined superluminal speed requires a jet viewing angle of @xmath58 . on the other hand , the highest radio determined superluminal speed requires @xmath59 for @xmath60 , respectively . jet angles of @xmath61 were derived @xcite based on radio observations at @xmath20 , assuming the component to the east of the core is a counter - jet , and with a velocity measurement based on only one pair of observations . in this paper we assume a likely range of the jet angle of @xmath62 . [ [ the - jet - opening - angle . ] ] the jet opening angle . + + + + + + + + + + + + + + + + + + + + + + the observations indicate the following jet full opening angles as a function of the distance to the core : @xmath63 @xcite , @xmath64 @xcite , and @xmath65 @xcite . assuming a jet viewing angle of @xmath50 the corresponding intrinsic full opening angles are : @xmath66 , @xmath67 , and @xmath68 . the observed pattern of opening angles suggest that the radio core corresponds to the formation region of the jet . [ [ the - counter - jet . ] ] the counter - jet . + + + + + + + + + + + + + + + + the optical identification of an emission feature observed at radio frequencies located @xmath69 away from the nucleus in the direction opposite to the jet resulted in a first indication of a counter - jet in m87 @xcite . this feature is also seen in observations at mid - infrared frequencies @xcite . radio observations at wavelengths of @xmath70 show clear indications of a counter - feature extending up to @xmath71 from the radio core in the direction opposite to the jet @xcite . while individual features move along the jet ( determined from 21 images taken over a time span of more than 10 years ) , the counter feature seems to move in the opposite direction with an apparent velocity of @xmath72 , strengthening the counter - jet interpretation . however , it can not be fully excluded that this apparent movement is a result of temporal under - sampling in the kinematic analysis , although the individual tracked jet features give consistent brightness temperatures across the observation epochs , which would be unlikely in case of incorrect cross - identifications of the components . the jet to counter - jet brightness ratio ( @xmath73 ) is @xmath74 . a counter - feature is also observed at @xmath75 and @xmath76 , extending roughly @xmath41 from the core in the direction opposite to the jet @xcite , see also fig . the jet to counter - jet brightness ratio is calculated to be @xmath77 . marginal indication for an apparent velocity of @xmath78 of the counter - jet away from the core is found from three @xmath25 radio images . the observations , however , suggest a temporal under - sampling of those data . [ [ position - of - the - black - hole . ] ] position of the black hole . + + + + + + + + + + + + + + + + + + + + + + + + + + + at radio frequencies of @xmath79 , the core is no larger than @xmath80 @xcite , and its jet morphology is consistent with a wide - opening angle jet base , as seen at @xmath20 , converging near the point of the black hole . farther out between about @xmath36 and @xmath81 the bright edges of the jet converge toward a point close to the maximum extent of the counter - feature . interpreting it as a part of the jet would require that the jet before the radio peak is better collimated than afterwards , although it appears to be as wide as the jet itself . the astrometric results relative to m84 show that the root mean square scatter of the position of the radio core is about @xmath82 along and across the beam with no clear systematic motion ( davies et al . , in preparation ) . if the radio core is farther down the jet and the jet power is going up fractionally like the flux density , this would imply a very stable position of the shock region . the spatial stability is a reasonable assumption for the radio core being located close to the black hole . theoretical modelling also supports the hypothesis that the @xmath20 radio peak emission results from the position of the black hole @xcite . [ [ vhe . ] ] vhe . + + + + the h.e.s.s . , magic and veritas collaborations operate imaging atmospheric che - renkov telescopes ( iacts ) located in namibia , the canary islands ( spain ) and arizona ( usa ) , respectively . the telescopes measure cosmic @xmath1-ray photons ( entering the atmosphere of the earth ) in an energy range of @xmath19 up to several 10 s of tev . m87 has been observed at those energies for the last ten years . except for the 2008 observation campaign , the observations were scheduled in advance and did not follow any external or internal triggers , leading to arbitrarily sampled light curves . during the observations of the past 10 years only two episodes of flaring activity have been measured : in 2005 @xcite and in 2008 ( reported in this paper ) . for the first time , m87 was observed by h.e.s.s . , magic and veritas in a joint campaign for more than @xmath83 in 2008 ( more than @xmath18 of data after quality selection ) . the integral fluxes presented in this paper ( fig . [ fig2 ] ) were calculated . ] under the assumption that the spectrum of m87 is described by a power - law function @xmath84 @xcite . any correlation between the spectral shape and the flux level has not yet been established for m87 . the relative frequency of flaring activity was estimated by fitting the night - by - night binned light curves as measured by h.e.s.s . , magic and veritas with a constant function ( using all available data partly archival from 2004 to 2008 ) . subsequently flux nights with the most significant deviation from the average were removed until the fit resulted in a reduced @xmath85 per degree of freedom of less than @xmath36 ; all removed points corresponded to flux values higher than the average . the light curves are compatible with constant emission for 49 out of 53 nights ( h.e.s.s . , 2004 - 2008 ) , 12 out of 21 nights ( magic , 2008 ) and 50 out of 51 nights ( veritas , 2007 - 2008 ) . combining these numbers one finds flaring activity in the so far recorded data in 14 out of 125 nights of observations , resulting in a relative frequency of flares on the order of @xmath24 of all observed nights . almost all data were recorded arbitrarily and except for four nights ( with a time difference of @xmath86 between the veritas and the h.e.s.s./magic observations ) all observations were separated in time by more than one day . therefore we assume that this number gives an estimate of the general chance to measuring a vhe @xmath1-ray flare from m87 . however , the relative frequency of flaring activity is overestimated by the fact that the 2008 observations were intensified for some nights during the high flux state following the vhe trigger by magic @xcite . [ [ x - ray . ] ] x - ray . + + + + + + m87 was regularly observed at x - ray energies with chandra , resulting in 61 measurements of the x - ray flux of the nucleus during the last ten years @xcite . three measurements exceed a flux level higher than 2 times the root mean square ( rms ) of the average flux of all data points ( relative occurrence of @xmath87 ) . only one measurement exceeds the level of @xmath88 which was taken during the radio flare with a deviation of @xmath89 ( fig . [ fig2 ] in the main text ) . [ [ radio . ] ] radio . + + + + + + throughout 2007 , m87 was observed with the vlba on a regular basis roughly every three weeks @xcite . the aim of this movie project was to study morphological changes of the plasma jet with time . preliminary analysis of the first 7 months showed a fast evolving structure , somewhat reminiscent of a smoke plume , with apparent velocities of about twice the speed of light . these motions were faster than expected so the movie project was extended from january to april 2008 with a sampling interval of 5 days . a full analysis of these data is in progress and details will be published elsewhere . the observed radio flux densities reached at the end of the 2008 observations , roughly 2 months after the vhe flare occurred , are larger than seen in any previous vlbi observations of m87 at this frequency , including during the preceding 12 months of intensive monitoring , in 6 observations in 2006 and in individual observations in 1999 , 2000 , 2001 , 2002 , and 2004 @xcite . the mojave project web site gives @xmath90 vlba flux densities at 27 epochs since 1995 , with the highest flux value measured on may 1 , 2008 flux reported in this paper . however , no further @xmath91 data for 2008 are listed on the mojave web page . ] ; most of the data ( except the last two epochs ) are published in @xcite . assuming a flare duration of @xmath234 months , a similar flare was not observed during a total period of @xmath92 months based on the 5 observations from 2004 and earlier , which are well separated , 8 months taking into account overlap based on the 2006 pilot observations spread over 4 months , and 14 months during the 2007/2008 monitoring including 2 months before the start but not including the time during the observed flare . that is 42 months total for which a similar flare was not in progress . by the same accounting , there are 4 months with a flare . so the probability of a radio flare being in progress at any given time is @xmath93 , suggesting that radio flares of the observed magnitude are uncommon . [ [ the - observed - radiox - rayvhe - pattern . ] ] the observed radio / x - ray / vhe - pattern . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the probability of observing @xmath94 out of @xmath95 nights exceeding a flux baseline ( for which the chance probability is @xmath96 ) is : @xmath97 . the probability of observing @xmath94 or more flare nights is @xmath98 . a joint estimated of the chance probability of the observed pattern is difficult , since the characteristic time scales of flux changes are different and the data are not sampled equally , or are partly sampled with higher frequencies as the characteristic time scale of the flux changes ( over - sampling ) or much less frequent ( under - sampling ) . however , if defining a time window covering the whole increase of the radio flux between end of january to mid of april one finds 8 out of 40 vhe measurements ( @xmath99 ) and 1 out of 2 x - ray measurements ( @xmath100 ) exceeded their baselines . the chance probability of observing this pattern during the radio flare is on the order of @xmath101 . the fluxes in the radio , x - ray , and vhe bands reached their highest archival level during the defined time window . [ [ the - model . ] ] the model . + + + + + + + + + + whereas the vhe @xmath1-ray emission from m87 varies on time scales of a few days , the @xmath20 radio emission from the nucleus steadily increases over a time period of two months ( fig . [ fig2 ] ) . a model calculation was performed to test if the slow variations of the radio flux can be explained with a self - absorbed synchrotron model of electrons injected into a `` slow outer sheath '' of jet plasma . the slow outer sheath has the geometry of a hollow cone , an assumption which is supported by the edge - brightened structure of the jet observed at radio frequencies ( fig . [ fig1]d ) . as the plasma travels down the jet , it expands , leading to a decline of the frozen - in magnetic field and to adiabatic cooling of the electrons . in the model , a @xmath1-ray flare leads to the injection of radio - emitting plasma at the base of the jet and at the base of the slow outer sheath . the model assumes that the vhe @xmath1-rays are produced very close to the black hole , e.g. in the black hole magnetosphere and does not attempt to describe the production mechanism . it merely uses the observed @xmath1-ray fluxes to normalize the energy spectrum of the electrons responsible for the radio emission . in the beginning , the radio - emitting plasma is optically thick , and the synchrotron emission can not escape . owing to the adiabatic expansion , the plasma eventually becomes optically thin leading to a radio flare . the radio flare dies down owing to the decline of the magnetic field and the adiabatic cooling of the electrons . following the injection of radio - emitting plasma at time @xmath102 , a ring of plasma with radius @xmath103 ( with the thickness of the radio bright sheath being 1/5th of the cone radius ) travels down the jet . the emission of the ring is computed in the frame of the moving plasma , assuming that the magnetic field scales as @xmath104 , and the electrons cool adiabatically . the calculation uses the standard equations for lorentz transforming the emission of different sections of the ring , see for example @xcite . taking into account light travel time effects , the received radio flux at @xmath20 is computed . the overall normalization of the electron energy spectrum is adjusted to reproduce the observed radio flux . the results of the model strongly depend on the choice of the minimal radius @xmath105 , and weakly depend on the choices of the magnetic field @xmath106 , the jet opening angle @xmath107 , and and the thickness of the sheath . [ [ results . ] ] results . + + + + + + + + we assume an intrinsic cone opening angle of @xmath108 , a jet angle of @xmath50 and a rather low magnetic field of @xmath109 at the base of the jet for which radiative cooling can be neglected . the simulated radio light curve produced by a single injection of radio - emitting plasma is shown in fig . [ figs1 ] for an assumed bulk lorentz factor of @xmath110 ( @xmath111 , giving the best fit result ) . the radio flux needs approximately 20 days to reach its maximum . figure [ figs2 ] shows the corresponding radio light curve obtained when choosing a time - dependent electron injection function proportional to the measured vhe @xmath1-ray fluxes , starting with the vhe data taken in january 2008 . the spatial extent of the predicted radio source after 50 days is @xmath112 light days and therefore still within the central resolution element of the vlba observations ( fig . [ fig3 ] ) . the model was chosen to minimize the number of free parameters and assumptions . other dependencies could affect the results as follows : ( i ) if the emitting plasma is more compact ( and the volume filling factor is @xmath113 ) , it stays synchrotron self - absorbed for a longer time , increasing the time lag between the @xmath1-ray flares and the rise of the radio flux . ( ii ) the emission volume may expand slower than proportional to @xmath114 , as assumed in the model . slower expansion would slow down the time scale for the rise and decay of the radio flux . ( iii ) a higher value of @xmath115 would result in a faster radio flare . ( iv ) all non - thermal particles are injected into the jet right at the base of the slow sheath . this model assumption may not be accurate . additional non - thermal particles may be accelerated further downstream which would lead to a longer duration of the radio flare . ( v ) if the magnetic field at the base of the jet is stronger , radiative cooling is not negligible any longer and fitting the data would require an assumption of continued acceleration as the outer shell flows down the jet . ( vi ) in a turbulent jet flow , efficient stochastic re - acceleration may occur , which could change the picture . however , the current model calculations show that synchrotron self - absorption may play a role in explaining the observed slower turn - on of the radio emission . the key questions for the understanding of the vhe @xmath1-ray emission measured from the radio galaxy m87 and from the more than 20 known vhe @xmath1-ray blazars are : ( i ) what is the underlying particle distribution which is accelerated , ( ii ) what are the mechanisms to generate the @xmath1-rays , and ( iii ) where is the region of the emission located . in the following paragraphs a selection of models discussed for m87 in the literature is investigated with a focus on the question whether they can explain the observed vhe / radio light curves . note , however , that some of the models have difficulties in explaining the observed hard vhe @xmath1-ray spectra @xcite , which will however not be discussed in more detail . [ [ models - of - vhe - emission - from - the - black - hole - magnetosphere . ] ] models of vhe emission from the black hole magnetosphere . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the electromagnetic mechanism of extraction of the rotational energy of black holes by the blandford & znajek scenario @xcite seems to be a viable mechanism for powering the relativistic jets of agn . particles can be accelerated by the electric field of vacuum gaps in the black hole magnetosphere @xcite ( the electric field component parallel to the ordered magnetic field is not screened out ) or due to centrifugal acceleration in an active plasma - rich environment , where the parallel electric field is screened @xcite . synchrotron and curvature radiation of the charged particles , and inverse compton scattering of thermal photons can produce vhe @xmath1-ray photons @xcite . an important question in this scenario is if the @xmath1-rays can escape the central region or if they are absorbed through pair creation processes with either photons from the accretion disk @xcite or infrared photons emitted by a potential dust torus for which no clear observational evidence is found so far @xcite . if m87 harbours a non - standard ( advection - dominated ) accretion disk , @xmath1-rays could escape without being absorbed @xcite . an alternative scenario could be that the primary photons create a pair cascade whose leakage produces the observed @xmath1-ray emission @xcite . the delayed radio emission could be explained by the effect of synchrotron self - absorption ( see sec . 3 ) or the time needed to cool the electrons before they dominantly emit synchrotron radiation in the radio regime . [ [ the - model - of - reimer - et - al.-2004 . ] ] the model of reimer et al . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in this model a primary relativistic electron population is injected together with high - energy protons into a highly magnetized emission region @xcite . the vhe emission is dominated by either @xmath116 synchrotron radiation or by proton synchrotron radiation . the low - energy component is explained by the synchrotron emission of the electron population . however , the radio flux is underestimated by the ( steady - state ) model ( explanations are discussed in the paper ) so that a discussion of the observed radio / vhe flare is beyond the scope of this particular model . [ [ the - model - of - lenain - et - al.-2008 . ] ] the model of lenain et al . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in this model the high - energy emission region ( x - rays up to vhe ) consists of small blobs ( @xmath117 ) travelling through the extended jet and radiating at distances just beyond the alfven surface @xcite . the emission takes place in the broadened jet formation region , in the innermost part of the jet ( corresponding to the central resolution element in the @xmath20 radio map ) . this multi - blob model is a two - flow model , where the fast , compact blobs contribute to x - rays and @xmath1-rays through the synchrotron self - compton mechanism , and are embedded in an extended , diluted and slower jet emitting synchrotron radiation from radio to optical frequencies . even though this model describes only steady state emission , the observed radio / vhe variability can be discussed qualitatively . for instance , a sudden rise of the density of the underlying leptonic population at the stationary shock ( i.e. in the blobs ) translates into a flare of x - rays and @xmath1-rays , but no immediate rise of radio emission is expected , because the emission volume is synchrotron self - absorbed at radio frequencies , see sec . 3 . however , as the flare propagates into the extended , less magnetized , neighbouring jet , the leptons in the jet are energized and could cool by emitting at radio frequencies with some delay , creating a diluted radio flare in response to the vhe flare . an alternative scenario by giannios et al . ( 2009 ) explains the fast variability of vhe @xmath1-ray radiation in blazars as a possible result from large lorentz factor ( 100 ) filaments within a more slowly moving jet flow @xcite . [ [ the - model - of - tavecchio - and - ghisellini-2008 . ] ] the model of tavecchio and ghisellini ( 2008 ) . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + according to this model @xcite the jet consists of a fast spine and slow sheath layer . the photons from the fast spine are external - compton boosted to vhe by the slower sheath . in this framework one may assume that the vhe flare comes from near to the core and the time lag to the radio maximum is entirely a result of propagation of some disturbance down the jet and the associated reduction in synchrotron self - absorption . the x - rays should be synchrotron emission that is not self - absorbed and flare at about the same time as the vhe @xmath1-rays . in the radio data we see significant edge brightening suggestive of a de - boosted spine . note , however , that edge brightening like that observed can also be produced by enhanced surface emissivity . in the model the radio emission originates from a region different from that producing the vhe emission , so that a strict flux correlation is not required . detailed modeling would be needed to explain the observed light curves in the framework of this model . [ [ the - model - of - georganopoulos - et - al.-2005 . ] ] the model of georganopoulos et al . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in this model the jet decelerates over a length of @xmath118 @xcite . the vhe emission is assumed to come from the fast moving part near the jet base by inverse - compton scattering of low - frequency photons from the slower moving part of the jet . the model calculations are steady state so that they are difficult to apply to an ejection event . however , one can assume that the vhe flare is directly associated with the ejection event . as the disturbance propagates down the jet and decelerates we see the radio rise later as a result of synchrotron self - absorption effects , similar to the model described in sec . 3 . here the x - ray flux can still rise and fall with the vhe if it comes directly from the disturbance ; the majority of the observed power comes from the slower part of the flow , but this is a more or less steady state result . a similar model by levinson ( 2007 ) of radiative decelerating blobs in the jets of vhe @xmath1-ray emitting blazars is described in @xcite . [ [ the - model - of - marscher - et - al.-2008 . ] ] the model of marscher et al . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the model is based on the observation of a double flare in bllac @xcite . the first flare is seen at x - ray / optical energies accompanied by polarization measurements at a date of 2005.82 . it indicates an injection event into the jet acceleration and collimation region near to the black hole . the flare was followed by the appearance of a new radio component in vlba images that approached and passed through the radio core , accompanied by a second x - ray / optical flare at 2005.92 , where the @xmath119 radio flux begins to increase and peaks at about 2006.0 . this is interpreted as the passage of the disturbance through a standing shock in the jet at a distance of @xmath120 , see fig . 3 in @xcite . vhe @xmath1-ray emission has been detected in 2005 , however , no evidence for flux variability has been found in the data . in this picture the radio core is located at the standing shock and the radio emission in the acceleration and collimation region may be ( i ) either intrinsically weak or ( ii ) synchrotron self - absorbed . the non - coincidence between the radio peak and the second peak in the x - ray / optical is proposed to arise from the longer lifetime of particles radiating at radio frequencies . since the disturbance passes down the expanding jet the radio emission might last longer than the second x - ray / optical flare but does not increase in strength after the disturbance passes through the shock . applying this model to m87 one can assume that the observed vhe flare corresponds to ( a ) the first or ( b ) the second flare . a : the vhe flare indicates the injection at the base of the jet and the increase of the radio flux corresponds to the passage through a standing shock that is located at the m87 radio core . in this case , the counter - feature would have to be interpreted as the jet before the standing shock . although the peak of the radio emission is delayed , the radio flux started to increase at about the same time as the vhe flare , which indicates that the two emission regions are not spatially separated , making this scenario unlikely . b : the vhe flare indicates the passage of the disturbance through the standing shock accompanied by a slow increase of the radio flux . in this interpretation the first flare related to the injection at the base of the jet would have been missed completely in any of the wavelengths . there would still be a problem with a non time coincident maximum in the vhe and radio emission . the vhe to radio lag could be explained by synchrotron self - absorption ( see sec . 3 . ) , which would seem to require a coincidental juxtaposition of the standing shock with just the right radio optical depth and subsequent optical depth decline down the expanding jet . an alternative scenario could be that the shock - accelerated particles causing the vhe emission cool rapidly until they later emit photons dominantly at radio frequencies . bllac is @xmath121 times farther away than m87 , and the jet angle is considerably smaller ( @xmath122 ) as compared to m87 . the black hole in m87 , on the other hand , is @xmath123 times more massive than the one in bllac . therefore , our data have a @xmath124 times higher spatial resolution ] and provide a more than two orders of magnitude more detailed insight into the jet physics on gravitational scales : in case of the m87 observations presented here , @xmath41 corresponds to @xmath42 , whereas in the case of the bllac observations , @xmath41 corresponds to @xmath125 . although the marscher et al . model makes use of the observed vhe emission in bllac , there is no experimental evidence that would constrain the spatial region of that emission . our observations connect the vhe emission with the radio emission from the nucleus in m87 and therefore contrain the vhe emission region to lie within the collimation region of the jet , at maximum a few hundred @xmath6 away from the black hole . 3 j.a . gaidos , _ et al . _ , _ nature _ * 383 * , 319 - 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890 ( 1979 ) . blandford , & r.l . znajek , _ mnras _ * 179 * , 433 - 456 ( 1977 ) . h. krawczynski , _ apj _ * 659 * , 1063 - 1073 ( 2007 ) . d. whysong , & r. antonucci , _ apj _ * 602 * , 116 - 122 ( 2004 ) . w. bednarek , _ mnras _ * 285 * , 69 - 81 ( 1997 ) . d. giannios , d. uzdensky , & m.c . begelman , _ mnras _ , * 395 * , l29-l33 ( 2009 ) . a. levinson , _ apj _ * 671 * , l29-l32 ( 2007 ) . * the veritas collaboration : * v. a. acciari@xmath126 , e. aliu@xmath127 , t. arlen@xmath128 , m. bautista@xmath129 , m. beilicke@xmath130 , w. benbow@xmath126 , s. m. bradbury@xmath131 , j. h. buckley@xmath130 , v. bugaev@xmath130 , y. butt@xmath132 , k. byrum@xmath133 , a. cannon@xmath134 , o. celik@xmath128 , a. cesarini@xmath135 , y. c. chow@xmath128 , l. ciupik@xmath136 , p. cogan@xmath129 , w. cui@xmath137 , r. dickherber@xmath130 , s. j. fegan@xmath128 , j. p. finley@xmath137 , p. fortin@xmath138 , l. fortson@xmath136 , a. furniss@xmath139 , d. gall@xmath137 , g. h. gillanders@xmath135 , j. grube@xmath134 , r. guenette@xmath129 , g. gyuk@xmath136 , d. hanna@xmath129 , j. holder@xmath127 , d. horan@xmath140 , c. m. hui@xmath141 , t. b. humensky@xmath142 , a. imran@xmath143 , p. kaaret@xmath144 , n. karlsson@xmath136 , d. kieda@xmath141 , j. kildea@xmath126 , a. konopelko@xmath145 , h. krawczynski@xmath130 , f. krennrich@xmath143 , m. j. lang@xmath135 , s. lebohec@xmath141 , g. maier@xmath129 , a. mccann@xmath129 , m. mccutcheon@xmath129 , j. millis@xmath146 , p. moriarty@xmath147 , r. a. ong@xmath128 , a. n. otte@xmath139 , d. pandel@xmath144 , j. s. perkins@xmath126 , d. petry@xmath148 , m. pohl@xmath143 , j. quinn@xmath134 , k. ragan@xmath129 , l. c. reyes@xmath149 , p. t. reynolds@xmath150 , e. roache@xmath126 , e. roache@xmath126 , h. j. rose@xmath131 , m. schroedter@xmath143 , g. h. sembroski@xmath137 , a. w. smith@xmath133 , s. p. swordy@xmath142 , m. theiling@xmath126 , j. a. toner@xmath135 , a. varlotta@xmath137 , s. vincent@xmath141 , s. p. wakely@xmath142 , j. e. ward@xmath134 , t. c. weekes@xmath126 , a. weinstein@xmath128 , d. a. williams@xmath139 , s. wissel@xmath142 , m. wood@xmath128 * the h.e.s.s . collaboration : * f. aharonian@xmath156 , a.g . akhperjanian@xmath157 , g. anton@xmath158 , u. barres de almeida@xmath159 , a.r . bazer - bachi@xmath160 , y. becherini@xmath161 , b. behera@xmath162 , k. bernlhr@xmath163 , a. bochow@xmath164 , c. boisson@xmath165 , j. bolmont@xmath166 , v. borrel@xmath160 , j. brucker@xmath158 , f. brun@xmath166 , p. brun@xmath167 , r. bhler@xmath164 , t. bulik@xmath168 , i. bsching@xmath169 , t. boutelier@xmath170 , p.m. chadwick@xmath171 , a. charbonnier@xmath166 , r.c.g . chaves@xmath164 , a. cheesebrough@xmath171 , l .- m . chounet@xmath172 , a.c . clapson@xmath164 , g. coignet@xmath173 , m. dalton@xmath174 , m.k . daniel@xmath171 , i.d . davids@xmath175 , b. degrange@xmath172 , c. deil@xmath164 , h.j . dickinson@xmath171 , a. djannati - ata@xmath161 , w. domainko@xmath164 , l.oc . drury@xmath176 , f. dubois@xmath173 , g. dubus@xmath170 , j. dyks@xmath168 , m. dyrda@xmath177 , k. egberts@xmath164 , d. emmanoulopoulos@xmath162 , p. espigat@xmath161 , c. farnier@xmath178 , f. feinstein@xmath178 , a. fiasson@xmath178 , a. frster@xmath164 , g. fontaine@xmath172 , m. fling@xmath174 , s. gabici@xmath176 , y.a . gallant@xmath178 , l. grard@xmath161 , d. gerbig@xmath179 , b. giebels@xmath172 , j.f . glicenstein@xmath167 , b. glck@xmath158 , p. goret@xmath167 , d. ghring@xmath158 , d. hauser@xmath162 , m. hauser@xmath162 , s. heinz@xmath158 , g. heinzelmann@xmath180 , g. henri@xmath170 , g. hermann@xmath164 , j.a . hinton@xmath181 , a. hoffmann@xmath182 , w. hofmann@xmath164 , m. holleran@xmath169 , s. hoppe@xmath164 , d. horns@xmath180 , a. jacholkowska@xmath166 , o.c . de jager@xmath169 , c. jahn@xmath158 , i. jung@xmath158 , k. katarzyski@xmath183 , u. katz@xmath158 , s. kaufmann@xmath162 , e. kendziorra@xmath182 , m. kerschhaggl@xmath174 , d. khangulyan@xmath164 , b. khlifi@xmath172 , d. keogh@xmath171 , w. kluniak@xmath168 , t. kneiske@xmath180 , nu . komin@xmath167 , k. kosack@xmath164 , g. lamanna@xmath173 , j .- p . lenain@xmath165 , t. lohse@xmath174 , v. marandon@xmath161 , j.m . martin@xmath165 , o. martineau - huynh@xmath166 , a. marcowith@xmath178 , d. maurin@xmath166 , t.j.l . mccomb@xmath171 , m.c . medina@xmath165 , r. moderski@xmath168 , e. moulin@xmath167 , m. naumann - godo@xmath172 , m. de naurois@xmath166 , d. nedbal@xmath184 , d. nekrassov@xmath164 , b. nicholas@xmath185 , j. niemiec@xmath177 , s.j . nolan@xmath171 , s. ohm@xmath164 , j - f . olive@xmath160 , e. de oa wilhelmi@xmath186 , k.j . orford@xmath171 , m. ostrowski@xmath187 , m. panter@xmath164 , m. paz arribas@xmath174 , g. pedaletti@xmath162 , g. pelletier@xmath170 , p .- o . petrucci@xmath170 , s. pita@xmath161 , g. phlhofer@xmath162 , m. punch@xmath161 , a. quirrenbach@xmath162 , b.c . raubenheimer@xmath169 , m. raue@xmath188 , s.m . rayner@xmath171 , m. renaud@xmath189 , f. rieger@xmath188 , j. ripken@xmath180 , l. rob@xmath184 , s. rosier - lees@xmath173 , g. rowell@xmath185 , b. rudak@xmath168 , c.b . rulten@xmath171 , j. ruppel@xmath179 , v. sahakian@xmath157 , a. santangelo@xmath182 , r. schlickeiser@xmath179 , f.m . schck@xmath158 , r. schrder@xmath179 , u. schwanke@xmath174 , s. schwarzburg@xmath182 , s. schwemmer@xmath162 , a. shalchi@xmath179 , m. sikora@xmath168 , j.l . skilton@xmath181 , h. sol@xmath165 , d. spangler@xmath171 , . stawarz@xmath187 , r. steenkamp@xmath190 , c. stegmann@xmath158 , f. stinzing@xmath158 , g. superina@xmath172 , a. szostek@xmath191 , p.h . tam@xmath162 , j .- tavernet@xmath166 , r. terrier@xmath161 , o. tibolla@xmath192 , m. tluczykont@xmath180 , c. van eldik@xmath164 , g. vasileiadis@xmath178 , c. venter@xmath169 , l. venter@xmath165 , j.p . vialle@xmath173 , p. vincent@xmath166 , m. vivier@xmath167 , h.j . vlk@xmath164 , f. volpe@xmath193 , s.j . wagner@xmath162 , m. ward@xmath171 , a.a . zdziarski@xmath168 , a. zech@xmath165 , * the magic collaboration : * h. anderhub@xmath194 , l. a. antonelli@xmath195 , p. antoranz@xmath196 , m. backes@xmath197 , c. baixeras@xmath198 , s. balestra@xmath196 , j. a. barrio@xmath196 , d. bastieri@xmath199 , j. becerra gonzlez@xmath200 , j. k. becker@xmath197 , w. bednarek@xmath201 , k. berger@xmath201 , e. bernardini@xmath202 , a. biland@xmath194 , r. k. bock@xmath203 , g. bonnoli@xmath204 , p. bordas@xmath205 , d. borla tridon@xmath206 , v. bosch - ramon@xmath205 , d. bose@xmath196 , i. braun@xmath194 , t. bretz@xmath207 , i. britvitch@xmath194 , m. camara@xmath196 , e. carmona@xmath206 , s. commichau@xmath194 , j. l. contreras@xmath196 , j. cortina@xmath208 , m. t. costado@xmath209 , s. covino@xmath195 , v. curtef@xmath197 , f. dazzi@xmath210 , a. de angelis@xmath211 , e. de cea del pozo@xmath212 , c. delgado mendez@xmath200 , r. de los reyes@xmath196 , b. de lotto@xmath211 , m. de maria@xmath211 , f. de sabata@xmath211 , a. dominguez@xmath213 , d. dorner@xmath194 , m. doro@xmath199 , d. elsaesser@xmath207 , m. errando@xmath208 , d. ferenc@xmath214 , e. fernndez@xmath208 , r. firpo@xmath208 , m. v. fonseca@xmath196 , l. font@xmath198 , n. galante@xmath206 , r. j. garca lpez@xmath209 , m. garczarczyk@xmath208 , m. gaug@xmath200 , f. goebel@xmath215 , d. hadasch@xmath198 , m. hayashida@xmath206 , a. herrero@xmath209 , d. hildebrand@xmath194 , d. hhne - mnch@xmath207 , j. hose@xmath206 , c. c. hsu@xmath206 , t. jogler@xmath206 , d. kranich@xmath194 , a. la barbera@xmath195 , a. laille@xmath214 , e. leonardo@xmath204 , e. lindfors@xmath216 , s. lombardi@xmath199 , f. longo@xmath211 , m. lpez@xmath199 , e. lorenz@xmath217 , p. majumdar@xmath202 , g. maneva@xmath218 , n. mankuzhiyil@xmath211 , k. mannheim@xmath207 , l. maraschi@xmath195 , m. mariotti@xmath199 , m. martnez@xmath208 , d. mazin@xmath208 , m. meucci@xmath204 , j. m. miranda@xmath196 , r. mirzoyan@xmath206 , h. miyamoto@xmath206 , j. moldn@xmath205 , m. moles@xmath213 , a. moralejo@xmath208 , d. nieto@xmath196 , k. nilsson@xmath216 , j. ninkovic@xmath206 , i. oya@xmath196 , r. paoletti@xmath204 , j. m. paredes@xmath205 , m. pasanen@xmath216 , d. pascoli@xmath199 , f. pauss@xmath194 , r. g. pegna@xmath204 , m. a. perez - torres@xmath213 , m. persic@xmath219 , l. peruzzo@xmath199 , f. prada@xmath213 , e. prandini@xmath199 , n. puchades@xmath208 , i. reichardt@xmath208 , w. rhode@xmath197 , m. rib@xmath205 , j. rico@xmath220 , m. rissi@xmath194 , a. robert@xmath198 , s. rgamer@xmath207 , a. saggion@xmath199 , t. y. saito@xmath206 , m. salvati@xmath195 , m. sanchez - conde@xmath213 , k. satalecka@xmath202 , v. scalzotto@xmath199 , v. scapin@xmath211 , t. schweizer@xmath206 , m. shayduk@xmath206 , s. n. shore@xmath221 , n. sidro@xmath208 , a. sierpowska - bartosik@xmath212 , a. sillanp@xmath216 , j. sitarek@xmath222 , d. sobczynska@xmath201 , f. spanier@xmath207 , a. stamerra@xmath204 , l. s. stark@xmath194 , l. takalo@xmath216 , f. tavecchio@xmath195 , p. temnikov@xmath218 , d. tescaro@xmath208 , m. teshima@xmath206 , d. f. torres@xmath223 , n. turini@xmath204 , h. vankov@xmath218 , r. m. wagner@xmath206 , v. zabalza@xmath205 , f. zandanel@xmath213 , r. zanin@xmath208 , j. zapatero@xmath198 . @xmath126fred lawrence whipple observatory , harvard - smithsonian center for astrophysics , amado , az 85645 , usa _ , _ @xmath127department of physics and astronomy and the bartol research institute , university of delaware , newark , de 19716 , usa _ , _ @xmath128department of physics and astronomy , university of california , los angeles , ca 90095 , usa _ , _ @xmath129physics department , mcgill university , montreal , qc h3a 2t8 , canada _ , _ @xmath130department of physics , washington university , st . louis , mo 63130 , usa _ , _ @xmath131school of physics and astronomy , university of leeds , leeds , ls2 9jt , uk _ , _ @xmath132harvard - smithsonian center for astrophysics , 60 garden street , cambridge , ma 02138 , usa _ , _ @xmath133argonne national laboratory , 9700 s. cass avenue , argonne , il 60439 , usa _ , _ @xmath134school of physics , university college dublin , belfield , dublin 4 , ireland _ , _ @xmath135school of physics , national university of ireland , galway , ireland _ , _ @xmath136astronomy department , adler planetarium and astronomy museum , chicago , il 60605 , usa _ , _ @xmath137department of physics , purdue university , west lafayette , in 47907 , usa _ , _ @xmath138department of physics and astronomy , barnard college , columbia university , ny 10027 , usa _ , _ @xmath139santa cruz institute for particle physics and department of physics , university of california , santa cruz , ca 95064 , usa _ , _ @xmath140laboratoire leprince - ringuet , ecole polytechnique , cnrs / in2p3 , f-91128 palaiseau , france _ , _ @xmath141department of physics and astronomy , university of utah , salt lake city , ut 84112 , usa _ , _ @xmath142enrico fermi institute , university of chicago , chicago , il 60637 , usa _ , _ @xmath143department of physics and astronomy , iowa state university , ames , ia 50011 , usa _ , _ @xmath144department of physics and astronomy , university of iowa , van allen hall , iowa city , ia 52242 , usa _ , _ @xmath145department of physics , pittsburg state university , 1701 south broadway , pittsburg , ks 66762 , usa _ , _ @xmath146department of physics , anderson university , 1100 east 5th street , anderson , in 46012 _ , _ @xmath147department of life and physical sciences , galway - mayo institute of technology , dublin road , galway , ireland _ , _ @xmath148european southern observatory , karl - schwarzschild - strasse 2 , 85748 garching , germany _ , _ @xmath149kavli institute for cosmological physics , university of chicago , chicago , il 60637 , usa _ , _ @xmath150department of applied physics and instrumentation , cork institute of technology , bishopstown , cork , ireland _ , _ @xmath151national radio astronomy observatory , socorro , nm 87801 , usa _ , _ @xmath224physics department , 333 workman center , new mexico institute of mining and technology , 801 leroy place , socorro , nm 87801 , usa _ , _ @xmath153department of physics and astronomy , university of alabama , tuscaloosa , al 35487 , usa _ , _ @xmath154isr-2 , ms - d436 , los alamos national laboratory , los alamos , nm 87545 , usa _ , _ @xmath155department of astronomy , university of california , los angeles , ca 90095 - 1547 , usa _ , _ @xmath164max - planck - institut fr kernphysik , p.o . box 103980 , d-69029 heidelberg , germany _ , _ @xmath157yerevan physics institute , 2 alikhanian brothers st . , 375036 yerevan , armenia _ , _ @xmath160centre detude spatiale des rayonnements , cnrs / ups , 9 av . du colonel roche , bp 4346 , f-31029 toulouse cedex 4 , france _ , _ @xmath180universitt hamburg , institut fr experimentalphysik , luruper chaussee 149 , d-22761 hamburg , germany _ , _ @xmath174institut fr physik , humboldt - universitt zu berlin , newtonstr . 15 , d-12489 berlin , germany _ , _ @xmath165luth , observatoire de paris , cnrs , universit paris diderot , 5 place jules janssen , 92190 meudon , france _ , _ @xmath167irfu / dsm / cea , ce saclay , f-91191 gif - sur - yvette , cedex , france _ , _ @xmath171university of durham , department of physics , south road , durham dh1 3le , u.k . @xmath169unit for space physics , north - west university , potchefstroom 2520 , south africa _ , _ @xmath172laboratoire leprince - ringuet , ecole polytechnique , cnrs / in2p3 , f-91128 palaiseau , france _ , _ @xmath173laboratoire dannecy - le - vieux de physique des particules , cnrs / in2p3 , 9 chemin de bellevue - bp 110 f-74941 annecy - le - vieux cedex , france _ , _ @xmath161astroparticule et cosmologie ( apc ) , cnrs , universite paris 7 denis diderot , 10 , rue alice domon et leonie duquet , f-75205 paris cedex 13 , france ; umr 7164 ( cnrs , universit paris vii , cea , observatoire de paris ) _ , _ @xmath176dublin institute for advanced studies , 5 merrion square , dublin 2 , ireland _ , _ @xmath162landessternwarte , universitt heidelberg , knigstuhl , d-69117 heidelberg , germany _ , _ @xmath178laboratoire de physique thorique et astroparticules , universit montpellier 2 , cnrs / in2p3 , cc 70 , place eugne bataillon , f-34095 montpellier cedex 5 , france _ , _ @xmath158universitt erlangen - nrnberg , physikalisches institut , erwin - rommel - str . 1,d-91058 erlangen , germany _ , _ @xmath170laboratoire dastrophysique de grenoble , insu / cnrs , universit joseph fourier , bp 53 , f-38041 grenoble cedex 9 , france _ , _ @xmath182institut fr astronomie und astrophysik , universitt tbingen , sand 1 , d-72076 tbingen , germany _ , _ @xmath166lpnhe , universit pierre et marie curie paris 6 , universit denis diderot paris 7 , cnrs / in2p3 , 4 place jussieu , f-75252 , paris cedex 5 , france _ , _ @xmath184charles university , faculty of mathematics and physics , institute of particle and nuclear physics , v holeovikch 2 , 180 00 _ , _ @xmath179institut fr theoretische physik , lehrstuhl iv : weltraum und astrophysik , ruhr - universitt bochum , d-44780 bochum , germany _ , _ @xmath190university of namibia , private bag 13301 , windhoek , namibia _ , _ @xmath187obserwatorium astronomiczne , uniwersytet jagielloski , ul . orla 171 , 30 - 244 krakw , poland _ , _ @xmath168nicolaus copernicus astronomical center , ul . bartycka 18 , 00 - 716 warsaw , poland _ , _ @xmath181school of physics & astronomy , university of leeds , leeds ls2 9jt , uk _ , _ @xmath185school of chemistry & physics , university of adelaide , adelaide 5005 , australia _ , _ @xmath183toru centre for astronomy , nicolaus copernicus university , ul . gagarina 11 , 87 - 100 toru , poland _ , _ @xmath177instytut fizyki jadrowej pan , ul . radzikowskiego 152 , 31 - 342 krakw , poland _ , _ @xmath225european associated laboratory for gamma - ray astronomy , jointly supported by cnrs and mpg _ , _ @xmath226supported by capes foundation , ministry of education of brazil _ , _ @xmath194eth zurich , ch-8093 switzerland _ , _ @xmath195inaf national institute for astrophysics , i-00136 rome , italy _ , _ @xmath196universidad complutense , e-28040 madrid , spain _ , _ @xmath197technische universitt dortmund , d-44221 dortmund , germany _ , _ @xmath198universitat autnoma de barcelona , e-08193 bellaterra , spain _ , _ @xmath199universit di padova and infn , i-35131 padova , italy _ , _ @xmath200inst . de astrofsica de canarias , e-38200 la laguna , tenerife , spain _ , _ @xmath201university of d , pl-90236 lodz , poland _ , _ @xmath202deutsches elektronen - synchrotron ( desy ) , d-15738 zeuthen , germany _ , _ @xmath206max - planck - institut fr physik , d-80805 mnchen , germany _ , _ @xmath204universit di siena , and infn pisa , i-53100 siena , italy _ , _ @xmath205universitat de barcelona ( icc / ieec ) , e-08028 barcelona , spain _ , _ @xmath207universitt wrzburg , d-97074 wrzburg , germany _ , _ @xmath208ifae , edifici cn . , campus uab , e-08193 bellaterra , spain _ , _ @xmath227depto . de astrofisica , universidad , e-38206 la laguna , tenerife , spain _ , _ @xmath211universit di udine , and infn trieste , i-33100 udine , italy _ , _ @xmath212institut de ciencis de lespai ( ieec - csic ) , e-08193 bellaterra , spain _ , _ @xmath213inst . de astrofsica de andalucia ( csic ) , e-18080 granada , spain _ , _ @xmath214university of california , davis , ca-95616 - 8677 , usa _ , _ @xmath216tuorla observatory , turku university , fi-21500 piikki , finland _ , _ @xmath218inst . for nucl . research and nucl . energy , bg-1784 sofia , bulgaria _ , _ @xmath228inaf / osservatorio astronomico and infn , i-34143 trieste , italy _ , _ @xmath229icrea , e-08010 barcelona , spain _ , _ @xmath221universit di pisa , and infn pisa , i-56126 pisa , italy _ , _ @xmath230supported by infn padova _ , _ @xmath231deceased_.

the accretion of matter onto a massive black hole is believed to feed the relativistic plasma jets found in many active galactic nuclei ( agn ) . although some agn accelerate particles to energies exceeding @xmath0 electron volts ( ev ) and are bright sources of very - high - energy ( vhe ) @xmath1-ray emission , it is not yet known where the vhe emission originates . here we report on radio and vhe observations of the radio galaxy m87 , revealing a period of extremely strong vhe @xmath1-ray flares accompanied by a strong increase of the radio flux from its nucleus . these results imply that charged particles are accelerated to very high energies in the immediate vicinity of the black hole . = 1 active galactic nuclei ( agn ) are extragalactic objects thought to be powered by massive black holes in their centres . they can show strong emission from the core , which is often dominated by broadband continuum radiation ranging from radio to x - rays and by substantial flux variability on different time scales . more than 20 agn have been established as vhe @xmath1-ray emitters with measured energies above @xmath2 tera electron volts ( tev ) ; the jets of most of these sources are believed to be aligned with the line - of - sight to within a few degrees . the size of the vhe @xmath1-ray emission region can generally be constrained by the time scale of the observed flux variability @xcite but its location remains unknown . we studied the inner structure of the jet of the giant radio galaxy m87 , a known vhe @xmath1-ray emitting agn @xcite with a @xmath3 black hole @xcite , scaled by distance , located @xmath4 ( 54 million light years ) away in the virgo cluster of galaxies . the angle between its plasma jet and the line - of - sight is estimated to lie between @xmath5 ( see supporting online text ) . the substructures of the jet , which are expected to scale with the schwarzschild radius @xmath6 of the black hole is defined as @xmath7 , @xmath8 is the gravitational constant , and @xmath9 is the speed of light . the schwarzschild radius defines the event horizon of the black hole . ] , are resolved in the x - ray , optical and radio wavebands @xcite ( fig . [ fig1 ] ) . high - frequency radio very long baseline interferometry ( vlbi ) observations with sub - milliarcsecond ( mas ) resolution are starting to probe the collimation region of the jet @xcite . with its proximity , bright and well - resolved jet , and very massive black hole , m87 provides a unique laboratory in which to study relativistic jet physics in connection with the mechanisms of vhe @xmath1-ray emission in agn . vlbi observations of the m87 inner jet show a well resolved , edge - brightened structure extending to within @xmath10 ( @xmath11 or @xmath12 ) of the core . closer to the core , the jet has a wide opening angle suggesting that this is the collimation region @xcite . generally , the core can be offset from the actual location of the black hole by an unknown amount @xcite , in which case it could mark the location of a shock structure or the region where the jet becomes optically thin . however , in the case of m87 a weak structure is seen on the opposite side of the core from the main jet , which may be the counter - jet , based on its morphology and length @xcite . together with the observed pattern in opening angles , this suggests that the black hole of m87 is located within the central resolution element of the vlbi images , at most a few tens of @xmath6 from the radio core ( see supporting online text ) . along the jet , previous monitoring observations show both near - stationary components @xcite ( pc - scale ) and features that move at apparent superluminal speeds @xcite ( 100 pc - scale ) . the presence of superluminal motions and the strong asymmetry of the jet brightness indicate that the jet flow is relativistic . the near - stationary components could be related to shocks or instabilities , that can be either stationary , for example if they are the result of interaction with the external medium , or slowly moving if they are the result of instabilities in the flow . a first indication of vhe @xmath1-ray emission from m87 was reported by the high energy gamma - ray astronomy ( hegra ) collaboration in 1998/99 @xcite . the emission was confirmed by the high energy stereoscopic system ( h.e.s.s . ) in 2003 - 2006 @xcite , with @xmath1-ray flux variability on time scales of days . m87 was detected again with the very energetic radiation imaging telescope array system ( veritas ) in 2007 @xcite and , recently , the short - term variability was confirmed with the major atmospheric gamma - ray imaging cherenkov ( magic ) telescope during a strong vhe @xmath1-ray outburst @xcite in february 2008 . causality arguments imply that the emission region should have a spatial extent of less than @xmath13 , where @xmath14 is the relativistic doppler factor . this rules out explanations for the vhe @xmath1-ray emission on the basis of ( i ) dark matter annihilation @xcite , ( ii ) cosmic - ray interactions with the matter in m87 @xcite , or ( iii ) the knots in the plasma jet ( fig . [ fig1]c ) . leptonic @xcite and hadronic @xcite vhe @xmath1-ray jet emission models have been proposed . however , the location of the emission region is still unknown . the nucleus @xcite , the inner jet @xcite or larger structures in the jet , such as the knot hst-1 ( fig . [ fig1]c ) , have been discussed as possible sites @xcite . because the angular resolution of vhe experiments is of the order of @xmath15 , the key to identifying the location of the vhe @xmath1-ray emission lies in connecting it to measurements at other wavebands with considerably higher spatial resolutions . an angular resolution more than six orders of magnitude better ( less than @xmath16 degrees , corresponding to approximately @xmath17 in case of m87 ) can be achieved with radio observations ( fig . [ fig1 ] ) . we used the h.e.s.s . @xcite , magic @xcite and veritas @xcite instruments to observe m87 during 50 nights between january and may 2008 , accumulating over @xmath18 of data ( corrected for the detector dead times ) in the energy range between @xmath19 and several 10 s of tev . simultaneously , we monitored m87 with the very long baseline array @xcite ( vlba ) at @xmath20 with a resolution of @xmath21 @xcite , corresponding to about @xmath22 , see @xcite . during the first half of 2008 , three x - ray pointings were performed with the chandra satellite @xcite . our light curves are shown in fig . [ fig2 ] . we detected multiple flares at vhe in february 2008 with denser sampling , following a trigger sent by magic [ @xmath2323 h of the data published in @xcite ] . the short - term vhe variability , first observed in 2005 @xcite , is clearly confirmed and the flux reached the highest level observed so far from m87 , amounting to more than @xmath24 of that of the crab nebula . at x - ray frequencies the innermost knot in the jet ( hst-1 ) is found in a low state , whereas in mid february 2008 the nucleus was found in its highest x - ray flux state since 2000 @xcite . this is in contrast to the 2005 vhe @xmath1-ray flares @xcite , which happened after an increase of the x - ray flux of hst-1 over several years @xcite , allowing speculation that hst-1 might be the source of the vhe @xmath1-ray emission @xcite ; no @xmath25 radio observations were obtained at that time . given its low x - ray flux in 2008 , hst-1 is an unlikely site of the 2008 vhe flaring activity . over at least the following two months , until the vlba monitoring project ended , the @xmath20 radio flux density from the region within @xmath26 of the core rose by @xmath27 as compared with its level at the time of the start of the vhe flare and by @xmath28 as compared with the average level in 2007 ( fig . [ fig2 ] ) . the resolution of the @xmath25 images corresponds to @xmath22 and the initial radio flux density increase was located in the unresolved core . the region around the core brightened as the flare progressed ( fig . [ fig3 ] ) , suggesting that new components were emerging from the core . at the end of the observations , the brightened region extended about @xmath29 from the peak of the core , implying an average apparent velocity of @xmath30 ( @xmath9 is the speed of light ) , well under the approximately @xmath31 seen just beyond that distance in the first half of 2007 . astrometric results obtained as part of the vlba monitoring program show that the position of the m87 radio peak , relative to m84 , did not move by more than @xmath236 @xmath6 during the flare , suggesting that the peak emission corresponds to the nucleus of m87 . because vhe , x - ray and radio flares of the observed magnitude are uncommon , the fact that they happen together ( chance probability of @xmath32 , supporting online text ) is good evidence that they are connected . this is supported by our joint modeling of the vhe and radio light curves : the observed pattern can be explained by an event in the central region causing the vhe flare . the plasma travels down the jet and the effect of synchrotron self - absorption causes a delay of the observed peak in radio emission because the region is not transparent at radio energies at the beginning of the injection ( supporting online text , sec . 3 ) . the vlbi structure of the flare along with the timing of the vhe activity , imply that the vhe emission occurred in a region that is small when compared with the vlba resolution . unless a source of infrared radiation is located very close to the central black hole , which is not supported by current observations @xcite , tev @xmath1-ray photons can escape the central region of m87 without being heavily absorbed through @xmath33 pair production @xcite . the light curve might indicate a rise in radio flux above the range of variations observed in the past , starting before the first vhe flare was detected . this could imply that the radio emission is coming from portions of the jet launched from further out in the accretion disk than that responsible for the vhe emission . however , it is difficult to derive a quantitive statement on this , because no vhe data were taken in the week previous to the flaring . thus , an earlier start of the vhe activity can not be excluded , either . a possible injection of plasma at the base of the jet observed at optical and x - ray energies with a delayed passage through the radio core @xmath23@xmath34 further down the jet interpreted as a standing shock and accompanied by an increase in radio emission has been discussed in the case of bllac @xcite ( with evidence for vhe emission , see supporting online text for more details ) . m87 is much closer than bllac and has a much more massive black hole , allowing the vlba to start resolving the jet collimation region whose size , from general relativistic magnetohydrodynamic simulations @xcite , is thought to extend over @xmath231000 @xmath6 . in case of m87 the radio core does not appear to be offset by more than the vlba resolution of @xmath2350 @xmath6 from the black hole ( see supporting online text ) and the jet has a larger angle to the line - of - sight than in bllac . thus the coincidence of the vhe and radio flares ( separated in photon frequency by 16 orders of magnitude ) , constrains the vhe emission to occur well within the jet collimation region . the support of the namibian authorities and of the university of namibia in facilitating the construction and operation of h.e.s.s . is gratefully acknowledged , as is the support by the german ministry for education and research ( bmbf ) , the max planck society , the french ministry for research , the cnrs - in2p3 and the astroparticle interdisciplinary programme of the cnrs , the u.k . science and technology facilities council ( stfc ) , the ipnp of the charles university , the polish ministry of science and higher education , the south african department of science and technology and national research foundation , and by the university of namibia . we appreciate the excellent work of the technical support staff in berlin , durham , hamburg , heidelberg , palaiseau , paris , saclay , and in namibia in the construction and operation of the equipment . _ magic : _ the collaboration thanks the instituto de astrofsica de canarias for the excellent working conditions at the observatorio del roque de los muchachos in la palma , as well as the german bmbf and mpg , the italian infn and spanish micinn . this work was also supported by eth research grant th 34/043 , by the polish mniszw grant n n203 390834 , and by the yip of the helmholtz gemeinschaft . _ veritas : _ this research is supported by grants from the u.s . department of energy , the u.s . national science foundation and the smithsonian institution , by nserc in canada , by science foundation ireland and by the stfc in the u.k . we acknowledge the excellent work of the technical support staff at the flwo and the collaborating institutions in the construction and operation of the instrument . _ vlba : _ the very long baseline array is operated by the national radio astronomy observatory , a facility of the u.s . national science foundation , operated under cooperative agreement by associated universities , inc .

0908.0511

in terms of the role in increasing the efficiency and aggregate network throughput , cognitive radio concept plays differently than the conventional spectrum allocation methods @xcite . in cognitive networks , unlicensed secondary users opportunistically access radio bandwidth owned by licensed primary users in order to maximize their performance , while limiting interference to primary users communications . previously , cognitive radio mostly focused on a white space approach @xcite , where the secondary users are allowed to access only those time / frequency slots left unused by the licensed users . white space approach is based on zero interference rationale . but , due to noise and fading in channel and mechanism of channel sensing , errors in measurement are inevitable @xcite . therefore , in practical scenarios , there is some probability of having collision between primary and secondary users , which can be measured and used as a constraint for the optimization problem . there are some works investigating the coexistence of primary / secondary signals in the same time / frequency band by focusing on physical layer methods for static scenarios , e.g. , @xcite . considering the dynamism while superimposition of primary and secondary users on the same time / frequency slot , a strategy of secondary user has been derived where the primary user operates in slotted arq based networks @xcite . we consider ieee 802.11 based networks where primary users follow dcf protocol in order to access the channel . unlike the work @xcite , in our contemporary work @xcite , we have developed a transmission strategy for the secondary user which picks a backoff counter intelligently or remains idle after having a transmission in a multiplexed manner . as the user needs to pass difs and backoff time period before flushing a packet into the air , the secondary user does not know the exact state of the primary user . therefore , the performace constraint of the primary user plays a great role in the decision making process of secondary user . our previous work revealed solution by formulating the problem as linear program being assumed that secondary user does know the traffic arrival distribution of primary user . as this approach assumes that the secondary transmitter has some knowledge of the current state and probabilistic model of the primary transmitter / receiver pair , limiting its applicability . for example , while it is likely that the secondary might read acks for the primary system , it is unlikely that the secondary will have knowledge of the pending workload of packets at the primary transmitter or will know the distribution of packet arrivals at the primary transmitter . therefore , we address this limitation by developing an on line learning approach that uses one feedback bit sent by the primary user and that approximately converges to the optimal secondary control policy . we will show that when the secondary user has access to such tiny knowledge , an online algorithm can obtain performance similar to an offline algorithm with some state information . rest of the paper is organized as follows , section [ sec : sysmodel ] illustrates system model of the network , which includes the detailed optimization problem and solution thereafter . results obtained from simulation have been shown in section [ sec : perfeval ] in order to verify the efficacy of the algorithm . finally section [ sec : concl ] concludes the paper . we consider interference mitigation scenario in ieee 802.11 based networks . the prime assumption on the interference mitigation strategy is that both users can decode their packets with some probability when they transmit together or individually . however , secondary user is constrained to cause no more than a fixed maximum degradation of the primary s performance . this approach is the other end of white space one . if primary user can not tolerate any loss , the optimal strategy for the secondary user is not to transmit at all . whereas in the work @xcite , secondary user can detect the slot occupancy and can only transmit in the slots which it finds empty and therefore incurs some throughput even if primary user can not tolerate any throughput loss . consider the network in figure [ fig : sysmodel ] with a primary and secondary source , namely @xmath0 and @xmath1 . destination of these source nodes are @xmath2 and @xmath3 respectively . we assume a quasi static channel , and time is divided into slots . before initiating a packet transmission , both users first undergo difs period and decrements the backoff counter which is as large as each single time slot . while decrementing backoff counter , if the station detects a busy channel , it halts its decrementing process and resumes until it detects idle channel for the length of difs period . when the counter reaches to zero , packet is flushed out into the air . packets have a fixed size of l - bits , and transmission of a packet plus its associated feedback message fits the duration of a slot . ideally , packet transmission time is variable , but in this work for the sake simplicity , it is constant i.e. multiple of some slots . we denote by @xmath4 , @xmath5 , @xmath6 and @xmath7 , the random variables corresponding to the channel coefficients respectively between @xmath0 and @xmath2 ; @xmath0 and @xmath3 ; @xmath1 and @xmath3 ; @xmath1 and @xmath2 with @xmath8 , @xmath9 , @xmath10 and @xmath11 their respective probability distribution . the average decoding failure probability at the primary destination @xmath2 associated with a silent secondary source is denoted by @xmath12 , while the same probability when the secondary source transmits is @xmath13 . analogously , the average decoding failure probability at the secondary destination @xmath3 when the primary source is silent and transmitting is denoted with @xmath14 and @xmath15 respectively . control protocols implemented by the primary user is greatly impacted by the secondary user s transmission as discussed in the above paragraph . thus , it degrades the primary user s performance and this manner is true for the secondary user as well . however , the goal of the system design is to optimize secondary user s performance without doing harm to the primary user in some extent . therefore , upon receiving the feedback from the primary user , secondary one adjusts its transmission policy . packet arrival at the primary user is designed as a poisson arrival process with the parameter @xmath16 . the state of the network can be modeled as a homogeneous markov process . two parameters ( backoff stage , counter value ) referred to as ( b , c ) describe the state of a user , where @xmath17 can take any value between 0 and @xmath18 . backoff stage b varies from 1 to maximum backoff stage @xmath19 . here , @xmath19 is the maximum retry limit . having a transmission failure , each packet is attempted by the primary user for retransmission at most @xmath19 times . at each backoff stage , if a station reaches state @xmath20 ( i.e. backoff counter value becomes 0 ) , the station will send out a packet . if the transmission failure occurs at this point with some probability , the primary user moves to higher backoff stage @xmath21 with probability @xmath22 . if successful packet transmission happens , the primary user goes to idle state @xmath23 ( if there is no outstanding packet in queue ) or in the initial backoff stage having picked some backoff counter with the probability of @xmath24 . markov chain model of primary user has been illustrated in figure [ fig : primmarkov ] . secondary user tries each packet only once , after having transmission , it goes to idle state with some probability or picks a backoff counter @xmath25 with probability @xmath26 for the transmission of new packet from the queue . note that , secondary user s packet is assumed as backlogged or there is always one packet in the queue . however , in order to meet the performance loss constraint of primary user , secondary user needs to keep silent and therefore we have introduced a fake variable @xmath27 i.e. secondary user s packet arrival rate . markov chain model for the secondary user has been shown in figure [ fig : secmarkov ] . in both figures , @xmath28 and @xmath29 are function of @xmath16 and @xmath27 . detailed state transitions and steady state distribution of the problem have been skipped in this work due to space constraint . goal of this work to find a optimal strategy for the secondary limiting the performance loss of primary user . let us define the cost functions @xmath30 as the average cost incurred by the markov process in state @xmath31 if action @xmath32 is chosen . note that , @xmath33 represents the secondary source keeps silent and @xmath34 represents the picking of a backoff counter from secondary backoff counter window i.e. @xmath35 $ ] . and , average generic cost function yields to @xmath36\ ] ] where @xmath37 is the sequence of actions of the secondary source and @xmath38 is an exogenous random variable which is not instantaneously obtained due to protocol specific behavior . for example , if secondary user picks a backoff counter @xmath25 , it has to go through first difs and @xmath25 times backoff slots before having transmission . while passing through the backoff slots , it might be halted by the transmission of primary user and reduces the overall throughput than the case of not being halted . this incidence is also true for the primary user as well . moreover , state variable @xmath39 is not explicit to the secondary user , because secondary user does not know if the primary user is in backoff stage or in idle slot . however , secondary user can sense the primary user s presence if the primary user transmits in a slot . considering all these issues , our high level cost functions have been derived below . @xmath40 @xmath41 and @xmath42 are the instantaneous calculated secondary user s throughput assuming the failure probability of transmitted packet is @xmath43 and @xmath44 respectively . as discussed previously , @xmath44 and @xmath43 are the failure probability of secondary user s transmitted packet when primary user transmits and does not transmit respectively . besides these two cases , @xmath45 is just the throughput of secondary user considering the current time slot as we know secondary user s queue is backlogged . sitting idle in other s transmission time and backoff slots are taken account into the calculation of throughput . @xmath46 and again , @xmath47 can be interpreted as the fraction of time slots in which the primary source fails the last allowed transmission and the packet would not be delivered and @xmath48 is the fraction of time slots where the primary begins the service of a new packet . in this paper we define the failure probability as the average ratio of dropped packets after @xmath19 retransmission , to the total number of new packets sent , one can see that @xmath49 is equivalent to the failure probability of the primary source s packets . the optimization problem is then given by @xmath50 it is shown in @xcite that the optimization problem in equation [ eq : opt - prob ] is solved by formulating the problem as a linear program . parameters in the formulation have been derived from the steady state distribution of the markov chain . finally , the obtained optimal strategy has been denoted by a vector @xmath51 . elements of this vector holds the proportion of time secondary user keeps silent or in which probability should it picks the backoff counter from the given contention window . solution needs a little brute force search with some standard policy that have been proven analytically . the offline solution of the optimization problem requires full knowledge of state @xmath39 , which corresponds to the transmission index and queue state of the primary source , as well as knowledge of the transition probabilities and cost functions . however , the full knowledge of @xmath39 requires an explicit exchange of information . we address this limitation in two steps . first , by assuming that the secondary only has information about what can be directly observed about the primary , and second , by using an on line learning technique that learns the necessary parameters without requiring knowledge of the transition probabilities . by sensing the channel , primary user can not instantaneously detect the channel condition as primary user follows dcf protocol . therefore , there is no way to get the information about the state of the primary user when it is in backoff state or in idle state i.e. primary user s queue is empty . secondary user can get to know if primary user transmits in a certain slot by sensing the channel . in some cases , secondary user can get knowledge if primary user s transmitted packet is new or old . the header includes the sequence number of the packet , which increases if the transmitted packet is a new one and remains the same if it is a retransmission . however , when the retransmitted packet reaches to its maximum limit , there is no way for the secondary user to know , in the next slot whether it will go through another backoff stage with fresh packet , since the buffer state is completely unknown to the secondary user . even though primary user can gather such little information , in the proposed solution , secondary user does not rely on these information . rather , the proposed solution depends on a simple bit which indicates whether the performance constraint of the primary user is satisfied in the current time slot or not . this information is sent by the primary user as piggy backed form in either ack or the actual packet s header . having this information , secondary user regulates its transmission strategy . it is shown in the following section via numerical results that this such partial knowledge is sufficient to implement a learning algorithm operating close to the limit provided by full state knowledge . note that , the state of the primary user is overlooked here , it does not help in the decision making process of the secondary user . rather the cost functions are most important driving factor of the proposed online algorithm . most approaches to optimal control require knowledge of an underlying probabilistic model of the system dynamics which requires certain assumptions to be made , and this entails a separate estimation step to estimate the parameters of the model . in particular , in our optimization paradigm @xcite , the optimal randomized stationary policy can be found if the failure probabilities @xmath52 , @xmath53 , @xmath43 , @xmath44 are known to the secondary user , together with some knowledge of state @xmath54 . in this section we describe how we can use an adaptive learning algorithm called q - learning @xcite to find the optimal policy without a priori knowledge about our probabilistic model . the q - learning algorithm is a long - term average reward reinforcement learning technique . it works by learning an action - value function @xmath55 that gives the expected utility of taking a given action @xmath56 in a given state @xmath54 and following a fixed policy thereafter . intuitively , the q - function captures the relative cost of the choice of a particular allocation for the next time - step at a given state , assuming that an optimal policy is used for all future time steps . q - learning is based on the adaptive iterative learning of q factors . however , as discussed previously , it is almost impossible to get to know about the information of primary user s current state and thus it ignores the current state @xmath39 while learning the system and behavioral parameters of primary user . since , secondary user overlooks current state @xmath39 while taking any action , we can call it as the variant of markov decision process(mdp ) . the original mdp means , the agent takes action based on the current state of the environment . no matter , secondary user follows mdp or variant of mdp , it needs to fix a cost function which is typically named as reward . ultimate reward of the secondary user is its own throughput i.e. @xmath57 which it wants to maximize . however , in order to maximize throughput , we have adopted some indirect approach to get the maximized value of @xmath57 . cost function is associated with the action of secondary user . proability of each action of secondary user is resided in the vector @xmath58 $ ] . length of this vector is @xmath59 ( @xmath60 is backoff window size of secondary user ) . index @xmath61 denotes the proportion of time secondary user keeps itself silent , subsequent indexes @xmath62 denote the portion of time backoff counter @xmath63 is chosen by the secondary user . as discussed previously , outcome of secondary user s action is not obtained instantaneously until the secondary user has its transmission . due to the interaction of secondary and primary user , the obtained throughput from each action vary and our cost function is the obtained average throughput ( added to the long term average throughput ) resultant from the taken action . let @xmath64 is the average throughput of secondary user while taking the action @xmath56 and @xmath65 is the average throughput when the secondary user really completes its packet transmission . then , the cost function at time @xmath66 is defined as follows : @xmath67 and our optimization problem thus stands to @xmath68\ ] ] and the q - learning algorithm for solving equation [ eq : opt ] is illustrated as follows . * * step 1 : * let the time step @xmath69 . initialize each element of reward vector @xmath70 as some small number , such as 0 . * * step 2 : * check if the constraint of the primary user is satisfied . if not , choose the action of @xmath33 . otherwise , choose the action @xmath56 with index @xmath71 $ ] that has the highest @xmath72 value with some probability say @xmath73 , else let @xmath56 be a random exploratory action . in other words , + @xmath74 * * step 3 : * carry out action @xmath75 . wait until secondary user completes its transmission if it picks any backoff counter . or secondary user may choose the option of being silent . in either case , calculate the cost function @xmath76 and update the reward variable for the corresponding action . if the current state is @xmath54 and the resultant state is @xmath77 after taking the action @xmath75 , reward is updated as follows . + @xmath78 * * step 4 : * set the current state as @xmath77 and repeat step 2 . when convergence is achieved , set @xmath79 . this is the typical q - learning algorithm . in our case , we do nt know the primary user s exact current state and also do nt know what the next state will be . therefore the equation [ eq : q - learning ] reduces to @xmath80 in order to obtain the optimal value of @xmath81 , we have found the following theorem . * theorem 1 : * step size parameter @xmath82 gives the convergence to the algorithm . * proof : * the choice @xmath82 results in the sample - average method , which is guaranteed to converge to the true action values by the law of large numbers . a well - known result in stochastic approximation theory gives us the conditions required to assure convergence with probability 1 : @xmath83 the first condition is required to guarantee that the steps are large enough to eventually overcome any initial conditions or random fluctuations . the second condition guarantees that eventually the steps become small enough to assure convergence . note that , both convergence conditions are met for the sample - average case , @xmath82 . in this section we will evaluate the performace of our online algorithm . in addition , we have compared performance of this algorithm with the algorithm which has some information of primary user as presented in our work @xcite . throughout the simulation , we assume that the buffer size of the primary source is @xmath84 and the maximum retransmission time is @xmath85 . backoff window size in each stage is 4 , 6 , 8 , and 10 respectively . secondary user s backoff window size is as @xmath86 . we set the failure probabilities for the transmission of the primary source @xmath87 , @xmath88 , depending on the fact that secondary is silent or not , respectively . similarly , the failure probabilities of the secondary source are set to be @xmath89 . note that these failure probabilities are not known at the secondary source and it has to learn the optimal policy without any assumption on these parameters in advance . once again , the goal of the algorithm is to maximize the throughput of the secondary source . and @xmath90 figure [ fig : thrput - convergence ] depicts the convergence of secondary and primary user s throughput from 0th iteration to some number of iterations . throughput loss is defined as the difference between maximum achievable throughput and instantaneous throughput at a particular slot . from the given parameters , maximum achievable throughput is calculated considering only a single user ( primary or secondary ) is acting on the channel . we see the convergence of throughput loss happens after a few iterations . and @xmath90 in order to extrapolate the cost functions of our algorithm , we also have shown convergence process of two actions picked up by the secondary user , i.e. probability of picking backoff counter 0 and 1 respectively in the figure [ fig : reward - convergence ] . we have initialzed cost of all actions at time slot zero . as the algorithm moves along with time , it updates its average reward according the formula presented in the algorithm . the algorithm is more prone to pick backoff counter with lower value that will be shown in the subsequent figures . however , in terms of general rule , algorithm does not pick the same action repeatedly . this is because , due to the interaction between primary and secondary users , the repeated action may cause to the degradation of the primary user s performance or it may degrade its own average reward value than the other actions . consequently , the algorithm moves to the other action and the average reward value over the time for different actions look similar . ] figure [ fig : secthrput - comparison ] shows the throughput of primary and secondary source with the increased packet arrival rate @xmath16 for a fixed tolerable primary source s throughput loss . as expected , throughput of the secondary source decreases as @xmath16 is increased gradually . a larger @xmath16 means that the primary source is accessing the channel more often . therefore , the number of slots in which the secondary source can transmit while meeting the constraint on the throughput loss of the primary source decreases . in addition , in this figure , we have projected the result obtained by our optimal algorithm @xcite . optimal algorithm though due to the protocol behavior is not fully aware of state of the system , has some better information than our proposed online algorithm . therefore , it incurs better performance in terms of achievable throughput for different @xmath16 value . whereas , our online algorithm though does not look like have similar performance , but gains better one than other blind generic algorithm . generic algorithm means , here secondary user picks its backoff counter uniformly . with this strategy , we see the performance for the secondary user is the worst . even worst news is that , this algorithm is completely blind about the performance constraint of primary user . ] figure [ fig : secstrategy - comparison ] compares the obtained secondary user s strategy for both our optimal and online algorithms . we have presented the proportion of idle slots and probability of picking backoff counter 0 . for the sake of page limit , we have skipped other results here . in this result apparently , we do nt see any match between two algorithms . however , we can explain the difference . in fact , online algorithm is mostly dependent on the primary user performance loss violation indicator and its own reward value for different actions . it tries to pick the action with maximum value , which is usually the backoff counter with lower value . otherwise , upon the signal of constraint violation , it keeps silent . therefore , we see that online algorithm puts more weights to the backoff of lower value and again backoff counter of lower value breaks the constraint more often and thus it keeps more silent than offline algorithm . whereas , optimal algorithm knows the arrival rate of primary user , it runs a near brute - force algorithm in order to find the optimal strategy of secondary user . we have proposed an on line learning approach in interference mitigation adopted ieee 802.11 based networks for the cognitive user . our approach relies only on the little performance violation feedback of the primary transmitter and uses q - learning to converge to nearly optimal secondary transmitter control policies . numerical simulations suggest that this approach offers performance that is close to the performance of the system when complete system state information is known . although , the strategy of both algorithms does not follow the exactly similar trend . q. zhao , l. tong , a. swami , and y. chen , `` decentralized cognitive mac for opportunistic spectrum access in ad hoc networks : a pomdp framework , '' _ ieee journal on selected areas in communications _ , vol . 25 , no . 3 , pp . 589 600 , apr . o. simeone , y. bar - ness , and u. spagnolini , `` stable throughput of cognitive radios with and without relaying capability , '' _ communications , ieee transactions on _ , vol . 55 , no . 12 , 2351 2360 , dec . h. su and x. zhang , `` cross - layer based opportunistic mac protocols for qos provisionings over cognitive radio wireless networks , '' _ selected areas in communications , ieee journal on _ , vol . 26 , no . 1 118 129 , jan . 2008 . y. chen , q. zhao , and a. swami , `` joint design and separation principle for opportunistic spectrum access in the presence of sensing errors , '' _ information theory , ieee transactions on _ , vol . 54 , no . 5 , pp . 2053 2071 , may 2008 . w. zhang and u. mitra , `` spectrum shaping : a new perspective on cognitive radio - part i : coexistence with coded legacy transmission , '' _ communications , ieee transactions on _ , vol . 58 , no . 6 , pp . 1857 1867 , june 2010 . y. xing , c. n. mathur , m. haleem , r. chandramouli , and k. subbalakshmi , `` dynamic spectrum access with qos and interference temperature constraints , '' _ mobile computing , ieee transactions on _ , vol . 6 , no . 4 , pp . 423 433 , april 2007 . , `` cognitive interference management in retransmission - based wireless networks , '' in _ communication , control , and computing , 2009 . allerton 2009 . 47th annual allerton conference on _ , 30 2009-oct . 2 2009 , pp . 94 101 . r. ruby and v. c. leung , `` determining the transmission strategy of congnive user in ieee 802.11 based networks , '' the university of british columbia , tech . rep . , 2011 . [ online ] . available : http://www.ece.ubc.ca/~rukhsana/files/icc2012.pdf s. mahadevan , `` average reward reinforcement learning : foundations , algorithms , and empirical results , '' _ mach . _ , vol . 22 , pp . 159195 , january 1996 . [ online ] . available : http://dl.acm.org/citation.cfm?id=225667.225681

traditional concept of cognitive radio is the coexistence of primary and secondary user in multiplexed manner . we consider the opportunistic channel access scheme in ieee 802.11 based networks subject to the interference mitigation scenario . according to the protocol rule and due to the constraint of message passing , secondary user is unaware of the exact state of the primary user . in this paper , we have proposed an online algorithm for the secondary which assist determining a backoff counter or the decision of being idle for utilizing the time / frequency slot unoccupied by the primary user . proposed algorithm is based on conventional reinforcement learning technique namely q - learning . simulation has been conducted in order to prove the strength of this algorithm and also results have been compared with our contemporary solution of this problem where secondary user is aware of some states of primary user . cognitive radio , ism band , reinforcement learning , optimization , q - learning

1510.04692

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

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

0809.4265

techniques in popular use in the nuclear three - body problem include the faddeev@xcite , green s function monte carlo ( gfmc ) , and correlated hyper - spherical harmonics expansion methods ( see , for instance , refs.@xcite ) . these methods have been used with realistic phenomenological potentials ( _ e.g. _ av18@xcite , , cd - bonn@xcite , etc . . . ) with strong short - range interactions , yielding binding energies accurate to within @xmath5%@xcite . all of these methods treat the bare @xmath6 interactions in hilbert spaces that are effectively infinite . because the complexity of the hilbert spaces grows very rapidly with nucleon number a , the extension of such methods to heavier systems becomes increasingly difficult . this motivates another approach : solving a - body problems in more tractable finite `` included spaces , '' while accounting for the missing physics of the `` excluded space '' through an effective interaction . for shell - model - inspired effective theories that define included spaces in terms of harmonic oscillator ( ho ) bases , the missing physics includes both high - momentum and long - wavelength interactions . the former are missing because the included space contains only low - energy oscillator shells , while the latter are connected with the finite extent of the basis states , which are overconfined in an ho potential . shell - model calculations often employ two - body effective interactions , but these are generally determined phenomenologically . in recent years there has been growing success in calculating these effective interactions directly from the underlying bare interaction in a precise and systematic way ( _ e.g. _ refs.@xcite and references within ) . the purpose of this paper is to extend upon this work by calculating the more complicated @xmath7 that could be used in a shell - model - inspired effective theory , through three - body order in the excluded space .. the inclusion of three - body contributions is generally believed essential in building in the density dependence crucial to saturation . the exact effective interaction for the a - body problem is an a - body operator . the calculation of this interaction is clearly as difficult as solving the original a - body problem in an infinite hilbert space . however , there are reasons to hope that such an a - body calculation might be unnecessary . this hope depends on the plausible notion that the effective interaction might converge as an expansion in the number of nucleons interacting in the excluded space . were such an expansion to converge at the three- or four - body cluster level , then standard three- or four - body methods could be used to treat the excluded space . these more complicated but still tractable effective interactions could then be diagonalized in the shell - model - inspired effective theories to produce accurate binding energies and wave functions . there are a couple of arguments that such a scheme might converge . one is the success of the shell model when phenomenological two - body effective interactions are used . this suggests that a good part of the excluded - space physics is two - body . another is the qualitative argument that short - range clustering is increasingly unlikely as the size of the nucleon cluster increases : such spacial clustering is necessary for strong , multi - particle , high - momentum interactions . finally , we know that the pauli principle forbids short - range s - wave interactions above a=4 . additional support for this notion comes from recent large - basis _ ab initio _ no - core shell - model ( ncsm ) calculations@xcite . these calculations use a large but finite ho hilbert space . a lee - suzuki ( ls ) similarity transformation@xcite coupled with a folded - diagram sum@xcite is performed on the bare nn interaction to yield an energy - independent effective interaction . early calculations utilized only the effective two - body interaction as derived from these transformations . hence calculations in large hilbert spaces , or included spaces , were usually needed to ensure convergence which , if judiciously chosen , could improve convergence greatly . ] of the binding energy ( _ i.e. _ the included space had to be large enough such that the contributions to the binding energy from the effective three- and higher - body terms were small ) . recent calculations that now include induced three - body interactions derived from the ls procedure show significant improvement in the convergence of binding energies for the alpha particle and some @xmath8-shell nuclei@xcite . an alternative approach to calculating the induced three - body effective interaction was developed in ref.@xcite using an excitation - energy-_dependent _ three - body interaction based on the self - consistent solution of the bloch - horowitz ( bh ) equation@xcite . this method was successfuly used to calculate the binding energy of @xmath0he . however , both of these works to date ( _ i.e. _ ls and bh procedures ) involve handling the excluded - space physics in a very large but still finite hilbert space . this approximation introduces an extra dependence to the three - body binding on the size of the excluded space . such dependence can only be removed by a true summation over all excluded high - energy modes ( i.e. excluded space of infinte size ) . though this dependence is small due to the fact that results of these previous works are quite accurate ( both groups examined the convergence of their results as a function of the size of the excluded space ) , the rate of convergence clearly depends in detail on how singular the underlying nn interaction is . here we develop a method for calculating the effective three - body interaction that introduces no such high - momentum cutoff , but instead truly integrates over all high - energy modes . the accuracy of our results is only limited by our numerical precision . thus it can be applied with equal confidence to any underlying nn reaction , regardless of that potential s description of short - range physics . this interaction is used in the simplest test cases , @xmath0he/@xmath0h , and the results are compared with those that would result from evaluating the effective interaction only at the two - body level . our treatment is based on solving the bh equation as well , which produces a state - dependent ( and thus excitation - energy - dependent ) hermitian interaction , @xmath9 where @xmath10 here the intrinsic hamiltonian @xmath11 is obtained by summing the relative kinetic energy and potential energy operators @xmath12 and @xmath13 over all nucleon pairs @xmath14 and @xmath15 . the relative kinetic energy is found by subtracting the center - of - mass ( cm ) kinetic energy from the total kinetic energy obtained by summing @xmath16 over all nucleons . @xmath17 is the total mass of the a - body system , @xmath18 is the excluded space projection operator , and @xmath19 is the included - space projection operator that defines the finite , low - energy space in which a direct diagonalization will be done , once @xmath7 is obtained . as @xmath20 is the desired eigenvalue , which is not known @xmath21 @xmath22 , the equations must be solved self - consistently . in the calculations reported here , the av18 potential is used as the bare @xmath6 interaction and no bare three - body interaction is included . because our application is to @xmath0he/@xmath0h , rather than to a heavier nucleus , we will have solved the effective interaction at the a - body level . thus we are not testing the assumption of a cluster expansion here . instead , our focus is on the technique for solving the effective interaction , and its relation to the approximate two - body result . using techniques familiar from faddeev calculations , we first determine an effective two - body - like interaction , @xmath23 , from the scattering @xmath24-matrix . we find that the induced effective three - body interaction can be evaluated perturbatively by iterating on @xmath23 . for a certain range of oscillator parameters @xmath1 , convergence is achieved even for very small included spaces , including the limiting @xmath25 space . we also develop a non - perturbative method . by rearranging the terms of the original expansion for the three - body interaction , the series can be summed exactly through numerical solution of an integral equation . this method is very accurate and efficient for any included space and any oscillator parameter . in section [ preliminaries ] we generalize the two - body formulation presented in ref.@xcite . section [ sect : perturbative ] describes our perturbative treatment of the effective interaction , while section [ sect : nonperturbative ] presents the nonperturbative solution via an integral equation . the associated discussions focus on qualitative issues , with derivations reserved for the appendices . we discuss the results and the need for a simple test of the cluster assumption ( by applying current results to @xmath26he ) in section [ sect : conclusion ] . to guarantee that @xmath7 , like the bare @xmath11 , is translationally invariant one can work in a complete basis of three - nucleon ho slater determinants that includes all configurations with quanta @xmath27 . such a basis , in traditional independent - particle shell - model coordinates , is overcomplete , consisting of subspaces characterized by the eigenvalues of the center - of - mass hamiltonian @xmath28 that do not interact via @xmath7 . for @xmath294 , it is convenient to avoid this overcompleteness by using a jacobi basis , which reduces the @xmath30-body intrinsic - hamiltonian problem to an ( @xmath30 - 1)-body problem . this is the choice we make here . the jacobi included - space ( @xmath31 ) basis corresponding to the ho shell - model basis described above is thus simple , consisting of all relative - coordinate configurations with @xmath27 . ( we note that for @xmath324 , the single - particle basis is the more efficient basis for many - body calculations . however , an @xmath7 developed in a jacobi basis can be easily transformed to single - particle coordinates for use in shell - model - inspired diagonalizations . ) such included spaces for the two - nucleon system are simple to construct . for example , a @xmath2=2 included space consists of spatial wave functions of the form @xmath33 , @xmath34 , and @xmath35 , coupled to spin and isospin wave functions to maintain antisymmetry . this requires @xmath36 to be odd . the corresponding construction of fully anti - symmetrized @xmath30-body wave functions is less trivial @xcite . here we use the codes described in ref.@xcite to generate these for @xmath30=3 . in table [ tab : dimension ] dimensions of various @xmath30=3 included spaces are given as a function of @xmath2 . for the three - body system one can obtain accurate results by diagonalizing the bare @xmath11 in a jacobi basis , provided @xmath2 is made sufficiently large . this was done in ref.@xcite for the av18 potential with @xmath2=60 , resulting in a binding energy accurate to @xmath4 20 kev . the motivation for solving this problem for small @xmath2 using effective interactions is to explore techniques that might be more feasible for larger @xmath30 , where the model - space growth analogous to table [ tab : dimension ] will be much steeper . one goal of the current work is to find techniques that might make the integration over the excluded space more tractable . in our earlier work on the two - body system@xcite we found that this integration is difficult at both long and short distances . we found it convenient in ref.@xcite to remove the long - distance difficulties , the pathologies associated with the overbinding of the ho , at the outset . as the overbinding of the ho becomes arbitrarily large as @xmath37 increases , part of the tail of the wave function remains unresolved in any finite - basis treatment though numerically the contribution of the tail becomes increasingly unimportant in direct diagonalizations as more shells are added . ( this contrasts with the short - range problem , as av18 and other modern potentials are regulated at short distance by some assumed functional form . such potentials can be fully resolved in finite bases , provided @xmath2 is larger than the scale implicit in that functional form . ) in ref.@xcite the long - range behavior was handled by the following rearrangement of eq . ( [ eqn : bh ] ) @xmath38\frac{e}{e - qt}\right\}p,\ ] ] where @xmath39 here @xmath40 and @xmath41 are shorthand for @xmath42 and @xmath43 , respectively . due to space restrictions eq . [ eqn : bhre ] was stated in ref.@xcite without derivation . hence we show its derivation for completeness in appendix [ app : bhfinal ] appearing in ref.@xcite differs from the one shown in eq . [ eqn : teff ] . however , both expressions , from an operator standpoint , are completely equivalent . ] . to calculate included - space matrix elements of eq . [ eqn : bhre ] , it was first convenient to define the states @xmath44 where @xmath45 is some state that resides in the included space ( _ i.e. _ @xmath46 ) . with this definition , matrix elements of eq . [ eqn : bhre ] became @xmath47 our expansion of @xmath7 was formed by expanding the resolvent of @xmath48 ( see eq . [ eqn : veff ] ) in powers of @xmath49 : @xmath50 an iterative procedure was then used to calculate included - space matrix elements of this expansion . at each order we solved for the self - consistent energy @xmath20 . the rearrangement of eq . ( [ eqn : bhre ] ) was applied to the deuteron in ref.@xcite and led to excellent results in perturbation theory for suitably chosen oscillator parameters @xmath1 ( only few orders of @xmath49 were needed to reproduce the deuteron binding energy to within one kev ) . figure [ fig : endshell ] shows that @xmath51 acting on the ho state @xmath45 drastically changes that state s large-@xmath37 behavior : the asymptotic behavior of @xmath52 is @xmath53 , where @xmath54 , @xmath55 being the mass of the particle@xcite . as this is the proper fall - off , the overconfining effects of the ho are thus repaired . the effects are important for the deuteron , which has an extended wave function because of its small binding energy . the operator @xmath51 modifies only those ho states that reside in the last shell of the included space . for all other states within the included space , the operator is identical to unity . the `` endshell - corrected '' states thus control all of the proper asymptotic behavior . the resummation of the kinetic energy operator allows one to adjust basis wave functions to absorb much of the short - range behavior of the @xmath6 interaction into the included space , leaving a weak residual interaction that can be handled perturbatively . by adjusting the oscillator parameter @xmath1 to low values ( @xmath56 fm ) , the residual q - space contributions to the binding energy due to @xmath57 can be evaluated to within @xmath4 one kev in third - order perturbation theory , even for very small included spaces . this is only possible because the kinetic energy has been summed to all orders independent of @xmath1 , thus guaranteeing that the choice of a small @xmath1 will not alter the wave function at large @xmath37 . the optimal @xmath1 presumably provides the best resolution of the hard core without substantially altering the ability of the basis to reconstruct the intermediate - range potential . in the next section we will explore a similar expansion for @xmath0h/@xmath0he . we find modifications are necessary to account for the disparate length scales that come about from the inclusion of a third particle . before starting this discussion , we end this section by presenting compact expressions for the operators @xmath58 and @xmath59 . such operators appear frequently in subsequent formulae . the operator @xmath60 closely resembles the free particle propagator . however , due to the projection operator q in the denominator , calculating matrix elements of @xmath58 requires a little ingenuity . here we show that the excluded - space green s function can be expanded in terms of the much simpler full green s function and operators within the included space that can be inverted easily . following ref.@xcite , we first consider @xmath61 where @xmath62 . collecting terms in this `` schwinger - dyson '' form and noting that @xmath63 gives @xmath64 the last term in eq . [ eqn : step2 ] , @xmath65 , can be rewritten @xmath66 plugging the above equation into eq . [ eqn : step2 ] finally gives @xmath67 similarly , the `` schwinger - dyson '' form of @xmath58 is @xmath68.\ ] ] using eq . ( [ eqn : step3 ] ) then yields @xmath69 this expression is relatively simple to use . the free particle propagator , @xmath70 , is known analytically . indeed , its form is diagonal in momentum space . the operator @xmath71 represents a matrix composed of included - space overlaps of the free particle propagator . since we work within small included spaces , inverting this matrix is not difficult . the included - space matrix elements are easy to calculate , as analytic expressions for @xmath72 for any @xmath30-body system exists . these expressions involve multiple sums over hypergeometric functions and gamma functions . in appendix [ sect:3bodyetme ] we give the relevant expression for the three - body system . to derive an analogous expression for @xmath59 , we first consider @xmath73 where we have substituted eq . [ eqn : finalstep ] in the second line above . with the use of eqs . [ eqn : finalstep ] and [ eqn : eminusqt ] , @xmath59 becomes @xmath74 the physical interpretation of eq . [ eqn : eminusqtq ] is simple : the term on the _ lhs _ represents free propagation in the excluded ( @xmath18 ) space , while the first term on the _ rhs _ ( bottom line ) represents free propagation in the full ( @xmath75 ) space and the second term subtracts off the contribution coming from included - space ( @xmath31 ) propagation . it is convenient to make the following definitions : @xmath76 with these definitions , eqs . [ eqn : finalstep ] and [ eqn : eminusqtq ] become @xmath77 finally , with the expressions above , we can express our perturbative expansion in a concise form , @xmath78 where @xmath79 in the expansion given by eq . [ eqn : bhfinal ] , @xmath80 corresponds to @xmath81 , while @xmath82 corresponds to @xmath83 , and so on . as we stressed in ref.@xcite , matrix elements of the operators shown in eq . [ eqn : gamma_n ] can be simply calculated using a recursive procedure . similarly , one might apply the expansion given by eqs . ( [ eqn : bhfinal ] ) and ( [ eqn : gamma_n ] ) directly to @xmath0he/@xmath0h . as was done in the two - body system , we construct endshell - corrected states to build in the correct aymptotic forms . figure [ fig:3bodyendshell ] shows momentum - space results for one @xmath84 endshell state . as expected , the modified wavefunction acquires a strong peak near zero - momentum , corresponding to large - r corrections . to determine an optimal @xmath1 for subsequent calculations , @xmath11 is minimized within the included space . the results are given in fig . [ fig : he3variational ] for several included spaces . as expected , the endshell states improve the binding of the three - body system and lower the optimal @xmath1 , relative to uncorrected ho results . however , the improved binding energies are still far away from the exact answers . this contrasts with the deuterium results ( see fig.2 of ref . @xcite ) , where 0th - order results using endshell states were accurate ( @xmath4 50 kev ) , and became nearly exact in second- or third - order perturbation theory . the 0th order a=3 results underbind by @xmath85 mev . as fig . [ fig:10hwhe3energy ] shows , these results are not corrected in perturbation theory . successive orders produce wildly oscillating values for the binding energy , in contrast to the rapid convergence found for the deuteron . the difficulty is that a single distance scale @xmath1 does not provide sufficient freedom to describe short - range behavior governed by two relative cordinates . extended states such as those of fig . [ fig : efimov ] are problematic . once one makes a choice of jacobi coordinates , the short - range behavior of some two - body cluster will be difficult to describe . for example , the choice @xmath86 and @xmath87 prevents one from adjusting @xmath1 to account for the short - range interactions of nucleons 1 and 2 . consequently , important hard - core interactions lie outside the included space , leading to nonperturbative behavior . because the deuteron was easily treated , it is possible to sum to all orders the repeated excluded space scatterings of any two - nucleon cluster . such a partial summation should account for the strong repulsive interaction between any two coupled nucleons , leaving only intermediate- and long - range interactions . such interactions can be treated reasonably accurately within the included space using our endshell - corrected states , leaving perturbative corrections . this partial summation corresponds to the faddeev decomposition@xcite . the starting point is again eq . [ eqn : heff_me ] , @xmath88 where we cast @xmath48 in its integral form , @xmath89 here the superscript @xmath0 denotes that this is a three - body effective interaction . now consider the state @xmath90 @xmath91 @xmath92 , which satisfies @xmath93 this can be decomposed into faddeev components @xmath94 , given by @xmath95 where @xmath96 equations [ eqn : psi_sum ] and [ eqn : psi_faddeev ] show that the faddeev components require the solution of three coupled integral equations . for example , @xmath97 satisfies @xmath98 equation [ eqn : psi12_relation ] can be inverted with respect to the @xmath97 faddeev component , giving @xmath99 where @xmath100 analogous equations can be found for the remaining faddeev components . the superscript @xmath101 indicates that the effective interaction given by eq . [ eqn : v12eff ] represents the sum of repeated potential scatterings between the same two nucleons ( in this case , nucleons 1 and 2 ) , while the third nucleon remains a spectator , as illustrated in fig . [ fig : v12eff ] . this expression is exactly the partial sum mentioned above . notice that the middle expression in eq . [ eqn : v12eff ] is very similar to eq . [ eqn : veff ] . furthermore , the integral form of @xmath102 is similar to that of the g - matrix@xcite . yet there are subtleties that differentiate the two . we will return to this point later . the integral equation given by eq . [ eqn : v12eff ] can also be solved exactly in terms of the ( 2 + 1)-body @xmath24-matrices@xcite . in appendix [ sect : exactv12eff ] we show @xmath103^{-1}g_0t_{12},\ ] ] where @xmath104 and @xmath105 is given by eq . [ eqn : invgamma0 ] . the physical interpretation of eq . [ eqn : exactv12eff ] is similar to the one given below eq . [ eqn : eminusqt ] : the first term on the _ rhs _ represents the effective interaction coming from repeated scatterings in the full ( @xmath75 ) space , while the second term subtracts off the contribution from the included ( @xmath31 ) space . the advantage in using the faddeev decompositions comes from exploiting particle exchange symmetries . as the states @xmath106 are fully anti - symmetric , the individual faddeev components can be related to each other by simple permutation operators@xcite . in particular , @xmath107 where the permutation operator @xmath108 is defined as @xmath109 . the last expression in eq . [ eqn : psi12_relation ] then becomes @xmath110 which can be formally inverted with respect to @xmath111 to give @xmath112 where @xmath113 equation [ eqn : v30eff ] defines the induced three - body interaction coming from repeated scatterings of @xmath102 ( see fig . [ fig : resolvent30 ] ) . the operator @xmath108 ensures that every insertion of @xmath102 represents scatterings coming from different pairs of nucleons ( i.e. the rungs of fig . [ fig : resolvent30 ] alternate ) , preventing any double couting . as there are no spectator nucleons in this expression , it is labeled with the superscript @xmath114 . matrix elements @xmath115 are given by @xmath116 where the factor of @xmath117 in the last expression comes from the anti - symmetry of the states . an expansion of eq . [ eqn : v3effme ] can now be found by expanding the resolvent of eq . [ eqn : v30eff ] in powers of @xmath118 , @xmath119 expressions analogous to those of eq . ( [ eqn : gamma_n ] ) can be constructed from @xmath108 and @xmath102 , @xmath120 combining eq . [ eqn : gamma_ns ] with the expansion of eq . [ eqn : v3effme ] gives the final result ( compare with eq . [ eqn : bhfinal ] ) , @xmath121 figure [ fig : threebodyenergies ] shows the convergence of the binding energies for @xmath0h and @xmath0he for several included - space and oscillator parameter choices . the expansion of eq . ( [ eqn : bhfinal3 ] ) converges even for very small included spaces , including @xmath122 ( which corresponds to a single matrix element ) . surprisingly , the most rapid convergence occurs for the smallest included spaces . we do not have a convincing physical explanation for this result . note that the zeroth order results ( i.e. @xmath123 ) all overbind in fig . [ fig : threebodyenergies ] . this is possible since the zeroth order calculation is no longer variational , as @xmath102 now depends on @xmath20 ( see eq . [ eqn : v12eff ] ) . hence there is no prescribed method for finding the optimal @xmath1 , other than by trial and error . ( we stress that fully converged results will be independent of @xmath1 , as we have executed the effective theory faithfully . by an optimal @xmath1 we mean one that will speed the convergence . ) the values of @xmath1 used in fig . [ fig : threebodyenergies ] should be near the optimal . note that these values are much larger than those found for the deuteron calculations of ref.@xcite . since @xmath102 is calculated exactly , all short range two - body correlations are correctly taken into account . hence @xmath1 is no longer forced to small values by the demands of short-@xmath37 physics , but instead can relax to values characteristic of the size of the three - body system . as mentioned earlier , @xmath102 is reminiscent of the @xmath124-matrix found in traditional nuclear many - body theory , which is also an infinite ladder sum in particle - particle propagation . however the differences are substantial . traditional shell - model calculations have always used g - matrix interactions at the two - body level , ignoring any dependence the operator may have on spectator nucleons ( including in general certain violations of the pauli principle ) . the operator @xmath102 , on the other hand , depends explicitly on the spectator nucleon kinetic energy , as well as on the kinetic energies of the interacting nucleons . this dependence is manifest in eq . [ eqn : v12eff ] , as the operator @xmath40 in the resolvent @xmath125 represents the sum of the kinetic energies of all nucleons . this dependence on the spectator nucleon is essential for calculating the correct two - body correlations within the three - body system , as @xmath18 is defined by the quanta carried by the three - body configurations . similarly , for @xmath30-body calculations , the analogous operator , @xmath126 , would carry the dependence of all spectator nucleons . it appears that a perturbative expansion for the binding energy of @xmath0he/@xmath0h results only if one first sums two - body ladder interactions to all orders ( _ i.e. _ @xmath102 ) . this summation softens the two - body interaction , which on iteration can then generate the effective three - body interaction ( _ i.e. _ @xmath127 ) perturbatively . a similar procedure was followed in refs.@xcite , where the triton binding energy was calculated by perturbing in di - baryon fields . the di - baryon fields themselves represent an infinite scattering of two - body interactions , in analogy with @xmath102 . a drawback of the perturbative approach of the previous section is that for each included space defined by @xmath2 , there is only a limited range of @xmath1s for which the expansion converges readily . furthermore , there is no definite procedure for estimating the optimal @xmath1 , as the zeroth - order calculation is no longer variational . thus a nonperturbative procedure would be attractive : any @xmath1 could be chosen , and physical observables would prove to be independent of @xmath1 , as they must for a rigorously executed effective theory . such a non - perturbative summation is indeed possible by reshuffling the terms of eq . ( [ eqn : bhfinal3 ] ) to form a summable geometric series and by numerically solving an integral equation . this approach provides an opportunity to directly compare the relative sizes of @xmath102 and @xmath127 as functions of both @xmath2 and @xmath1 . in this section we only show results for the triton system , as @xmath0he calculations produce similar resultshe results , the reader is referred to ref.@xcite ] . the starting point this time is eq . [ eqn : bhfinal3 ] , @xmath128 where the set @xmath129 is given by eq . [ eqn : gamma_ns ] . this expansion can be rearranged into @xmath130 equation [ eqn:3bhfinalnp ] can be verified by expanding its terms out and directly comparing with eq . [ eqn : bhfinal3repeat ] . the following definitions can be made to simplify eq . [ eqn:3bhfinalnp ] , @xmath131 where the above expressions represent infinite summations . substituting these expressions into eq . [ eqn:3bhfinalnp ] gives @xmath132 the two geometric series in the equation above can be summed to give the final desired result , @xmath133 where we have made the following identifications , @xmath134 the usefulness of eq . ( [ eqn:3bodynpert ] ) depends on having an efficient method for calculating the included - space matrix elements of @xmath135 and @xmath136 . this can be done by first considering the state @xmath137 , which , using eq . [ eqn : gamma_inftys ] , satisfies @xmath138 hence @xmath139 satisfies the fredholm integral equation of the second kind@xcite . there are numerous numerical methods available for solving this particular integral equation . we use a general projection algorithm@xcite . due to the large number of partial waves and the complicated structure of the operator @xmath108 , solving the integral equation by matrix inversion is impractical . once @xmath139 is found , the matrix elements @xmath140 can be evaluated . then the matrix elements @xmath141 are easily calculated since @xmath142 which can be verified using eqs . [ eqn : gamma_ns ] and [ eqn : gamma_inftys ] . hence @xmath143 tables [ tab : triton_results ] and [ tab : tritonnp3half ] show the calculated binding energies for the triton system as a function of the included - space size @xmath2 and oscillator parameter @xmath1 . in table [ tab : triton_results ] we have ignored the small isospin - violating parts of the av18 potential . hence isospin is conserved at @xmath144 . table [ tab : tritonnp3half ] includes isospin - violating contributions . this causes a small admixture of @xmath145 components into the ground state . at about the kev level , the results are independent of the choice of the included space , i.e. , of @xmath1 and @xmath2 . the @xmath4 few kev variations illustrate the level of our numerical precision . note that for the values @xmath1=.83 and 1.17 fm , our calculations of the previous section would give non - converging binding energies . this of course emphasizes the non - perturbative nature of these calculations . note that eq . [ eqn:3bodynpert ] has the correct limiting behavior as @xmath146 , @xmath147 . in this limit the projection operator @xmath148 , giving @xmath149 . thus @xmath150 it is also straightforward to show that @xmath151 where we have used eq . [ eqn : t12def ] to obtain the second line above . finally , it is obvious that @xmath127 vanishes in this limit since @xmath152 hence @xmath153 . tables [ tab:00tritonme]- [ tab:22tritonme ] show the variation of the matrix elements of @xmath102 and @xmath127 with @xmath2 and @xmath1 . isospin - violating effects have also been ignored in these results . as the number of included - space examples is small , it is difficult to extract any limiting behavior . yet it does seem that as @xmath2 increases , matrix elements of @xmath102 approach the bare @xmath154 ( indicated by the symbol @xmath155 in the tables ) . this is especially evident in table [ tab:02tritonme ] , where the effect of renormalization is small ( i.e. the renormalized matrix elements are not too different from their bare matrix elements ) . in the cases involving @xmath127 , limiting behavior is also difficult to extract . indeed , in tables . [ tab:00tritonme ] and [ tab:22tritonme ] matrix elements of @xmath127 tend to grow away from zero with increasing @xmath2 rather than tend toward zero . however , in these cases the renormalization is strong , so that limiting behavior would not be expected for such small @xmath2 . in table [ tab:02tritonme ] the renormalization is much weaker , and the limiting behavior @xmath156 is clearly seen for the @xmath157 and @xmath158 mev examples . it is also clear from tables . [ tab:00tritonme]- [ tab:22tritonme ] that the matrix elements of @xmath102 and @xmath127 have a strong dependence on @xmath2 and @xmath1 ( @xmath159 ) , a dependence often ignored in traditional shell - model calculations . generally the contribution of @xmath160 to the overall binding energy is small compared to @xmath161 ( @xmath162 of @xmath102 ) , as is evident from fig . [ fig : v21_vs_v30 ] . this graph gives the @xmath163 , @xmath164 , and @xmath165 contributions to the binding energy as functions of @xmath2 and @xmath1 . interestingly , for each included - space @xmath2 , there are two values of @xmath1 at which the contribution of @xmath160 completely vanishes . table [ tab : b_zeros ] lists specific values . if this result proves to be a generic property of the three - body @xmath7 in heavier nuclei ( that is , if zeros appear in those calculations at approximately the same value of @xmath1 ) , this could greatly simplify the more complicated included - space diagonalizations required in heavy systems . for example , the three - body contribution could be ignored or could be explored in low - order perturbation theory . this would require establishing that the a=3 zero for @xmath160 does persist at about the same @xmath1 in heavier systems . this seems plausible , as the three - body ladder in a heavy nucleus is quite similar to that in a=3 . we found in section [ sect : perturbative ] that the simple perturbative scheme found in ref.@xcite ( _ i.e. _ eq . ( [ eqn : bhfinal ] ) ) did not directly extend to the three - body system due to the additional length scale introduced by the third nucleon . for any choice of jacobi coordinates , this second length scale excludes a class of hard - core interactions from the included space regardless of the choice of @xmath1 . this problem was circumvented by invoking the faddeev decomposition on @xmath7 , which allowed us to sum to all orders the repeated potential scattering between the same two nucleons , generating @xmath102 . hence the strong repulsive two - nucleon force was treated non - perturbatively . the remaining contribution to @xmath7 , @xmath127 , can then be calculated by summing repeated insertions of @xmath102 on alternate pairs of nucleons . for certain ranges of @xmath1 , this expansion converged after several orders of perturbation even for the smallest allowed included spaces , as shown in fig . [ fig : threebodyenergies ] . in section [ sect : nonperturbative ] we described a non - perturbative method for calculating @xmath7 in the three - body system , summarized in eq . ( [ eqn:3bodynpert ] ) . this allowed a direct comparison of the contributions of @xmath163 , @xmath161 , and @xmath160 . because the calculation was non - perturbative , we were able to explore the dependence of the included - space matrix elements ( and establish the independence of the binding energy ) for a wider range of @xmath1s and @xmath2s . also , for each included space , we found two values of @xmath1 where the total binding was due to @xmath102 , a result that should be explored in heavier systems . the numerical methods included an efficient algorithm for solving integral equations ( _ i.e. _ eq . ( [ eqn:3bodyintegraleq ] ) ) . in the current applications to a=3 systems , the numerical effort was not substantial . work is in progress to apply a similar formalism to the alpha particle . as in the three - body case , one must account for the various length scales inherent to the four - body system . this can be done by invoking the faddeev - yakubovsky decompositions on @xmath7 , in direct analogy to the three - body decompositions presented in this paper . such a procedure will not only yield @xmath163 , @xmath166 and @xmath167 ( direct analogs of @xmath161 and @xmath160 ) , but also @xmath168 . this last expression represents the induced effective four - body interaction . it will be interesting to see if these calculations verify the commonly implicit assumption of the hierarchical sizes of these interactions ( _ i.e. _ @xmath169 ) . the four - body effective interaction is the most complicated lowest - order ( s - wave ) operator that can be constructed . furthermore , alpha - particle clustering is an important phenomenological feature of nuclear structure , apparent even in simple systems like @xmath170be . thus it will be interesting to determine the size of this contribution . the a=4 calculations will also check whether zeros in the three - body effective interaction persist . if they do , we may learn something about their `` trajectories '' in @xmath1 as a is inceased . for completeness , we show the derivation of eq . [ eqn : bhre ] , which was first used in ref.@xcite in deriving a perturbative expansion for the deuteron . we begin by explicitly expressing the bh equation in its various components , @xmath171 next consider the operator @xmath172qt\\ = & t\frac{1}{e - qt}\left[1+qv\frac{1}{e - qh}\right]qt\\ = & t\frac{1}{e - qt}\left[1+qv\left\{\frac{1}{e - qt}+\frac{1}{e - qh}qv\frac{1}{e - qt}\right\}\right]qt\\ = & tq\frac{1}{e - tq}\left[e - qt+qv+qv\frac{1}{e - qh}qv\right]\frac{1}{e - qt}qt , \end{aligned}\ ] ] where the relation @xmath173 was used . a similar calculation gives @xmath174,\\ v\frac{1}{e - qh}qt=&\left[v+v\frac{1}{e - qh}qv\right]\frac{1}{e - qt}qt . \end{aligned}\ ] ] substituting eqs . [ eqn : teffexpand ] and [ eqn : vteffexpand ] into eq . [ eqn : heffexpand ] and using the fact that @xmath175 and @xmath176 gives the desired result : @xmath177 hence @xmath178 where @xmath179 and @xmath180 note that the resolvent in eq . [ eqn : appveff ] is now sandwiched between @xmath41s only , and not @xmath40s . forming a perturbative expansion involves expanding this resolvent as was done in eq . [ eqn : resolvent_expansion ] . the operator @xmath51 acts non - trivially only on endshell states . that is , @xmath181 which can be easily verified by expanding the operator in powers of @xmath182 and noting the fact that the kinetic energy operator @xmath40 can change the principal quantum number @xmath183 by at most @xmath184 . these ` tilde ' states @xmath185 represent the culmination of the infinitely repeated scatterings of the operator @xmath182 on the original states @xmath186 . such scatterings persist at large-@xmath37 even though the potential @xmath41 has already vanished . iterating the last expression of eq . [ eqn : v12eff ] on @xmath23 gives @xmath187 replacing @xmath59 by the relevant expression below eq . [ eqn : invgamma0 ] , eq . [ eqn : v12effexpansion1 ] becomes @xmath188 the various terms of eq . [ eqn : v12effexpansion2 ] can be re - ordered to give @xmath189 where @xmath190 represents an infinite sum of two - body interactions between particles 1 and 2 . the solution to eq . [ eqn : v12sum ] is given by the operator @xmath191 , where @xmath192 as mentioned near the end of sect . [ sect : perturbative ] , the operator @xmath191 differs from the usual two - body @xmath24-matrix since it is imbedded within a three - body space . hence @xmath193 contains a dependence on the jacobi momentum @xmath194 of the third ` spectator ' nucleon . this dependence carries over to the operator @xmath191 . hence the designation @xmath195-body @xmath24-matrix . replacing the infinite sum of eq . [ eqn : v12sum ] by @xmath191 and inserting this into eq . [ eqn : v12effexpansion3 ] gives @xmath196 where @xmath197 is given by eq . [ eqn : gamma21def ] , i.e. @xmath198 equation [ eqn : v12effexpansion4 ] can now be summed geometrically , giving the desired result @xmath199^{-1}g_0t_{12}.\ ] ] the relevant matrix element is @xmath200 replacing the radial integrals by their series expansion and changing to dimensionless variables , eq . [ eqn:3mea ] becomes @xmath201 where @xmath202 and @xmath203 an analytic solution to the integral in eq . [ eqn:3meb ] can be found by first changing variables to cylindrical coordinates : @xmath204 hence the integral becomes @xmath205\\ \times \left[\int_0^{\pi/2}d\phi \ ( cos\phi)^{2(1+l+m+m')}(sin\phi)^{2(1+l+m+m')}\right].\end{gathered}\ ] ] the integrals in square brackets have analytic solutions that can be found in any book of integrals ( e.g. see ref.@xcite ) . they are written here for completeness : @xmath206 @xmath207 this procedure generalizes for matrix elements of @xmath70 for higher - body systems . for an a - body problem within jacobi coordinates , a change of variables to ( a-1)-dimensional spherical coordinates is needed so that the resulting ( a-1 ) integrals ` factorize ' ( e.g. those in square brackets of eq . [ eqn:3mec ] ) . we thank andreas nogga for his insightful discussions and contribution of codes for calculating the permutation operator @xmath108 . this work was supported in part by the division of nuclear physics , office of science , u.s . department of energy , and in part under the auspices of the u. s. department of energy by the university of california , lawrence livermore national laboratory under contract no . w-7405-eng-48 . ( 3217,246)(0,0)[l]@xmath37 ( fm)(2712,1021)(0,0)[l ] @xmath208 fm(2712,1111)(0,0)[l ] @xmath209(2712,1246)(0,0)[l]@xmath210(2743,23)(0,0)(d)(3533,90)(0,0 ) 25(3217,90)(0,0 ) 20(2901,90)(0,0 ) 15(2586,90)(0,0 ) 10(2270,90)(0,0 ) 5(1954,90)(0,0 ) 0(1932,1468)(0,0)[r ] 0.4(1932,1246)(0,0)[r ] 0.3(1932,1024)(0,0)[r ] 0.2(1932,801)(0,0)[r ] 0.1(1932,579)(0,0)[r ] 0(1932,357)(0,0)[r]-0.1(1932,135)(0,0)[r]-0.2(1417,246)(0,0)[l]@xmath37 ( fm)(912,1021)(0,0)[l ] @xmath208 fm(912,1111)(0,0)[l ] @xmath209(912,1246)(0,0)[l]@xmath211(943,23)(0,0)(c)(1733,90)(0,0 ) 25(1417,90)(0,0 ) 20(1101,90)(0,0 ) 15(786,90)(0,0 ) 10(470,90)(0,0 ) 5(154,90)(0,0 ) 0(132,1468)(0,0)[r ] 0.8(132,1246)(0,0)[r ] 0.6(132,1024)(0,0)[r ] 0.4(132,801)(0,0)[r ] 0.2(132,579)(0,0)[r ] 0(132,357)(0,0)[r]-0.2(132,135)(0,0)[r]-0.4(3217,1758)(0,0)[l]@xmath37 ( fm)(2712,2533)(0,0)[l ] @xmath208 fm(2712,2623)(0,0)[l ] @xmath209(2712,2758)(0,0)[l]@xmath212(2743,1535)(0,0)(b)(3533,1602)(0,0 ) 25(3217,1602)(0,0 ) 20(2901,1602)(0,0 ) 15(2586,1602)(0,0 ) 10(2270,1602)(0,0 ) 5(1954,1602)(0,0 ) 0(1932,2980)(0,0)[r ] 1.4(1932,2832)(0,0)[r ] 1.2(1932,2684)(0,0)[r ] 1(1932,2536)(0,0)[r ] 0.8(1932,2388)(0,0)[r ] 0.6(1932,2239)(0,0)[r ] 0.4(1932,2091)(0,0)[r ] 0.2(1932,1943)(0,0)[r ] 0(1932,1795)(0,0)[r]-0.2(1932,1647)(0,0)[r]-0.4(1417,1750)(0,0)[l]@xmath37 ( fm)(912,2550)(0,0)[l ] @xmath208 fm(912,2640)(0,0)[l ] @xmath209(912,2775)(0,0)[l]@xmath213(943,1535)(0,0)(a)(1733,1602)(0,0 ) 25(1417,1602)(0,0 ) 20(1101,1602)(0,0 ) 15(786,1602)(0,0 ) 10(470,1602)(0,0 ) 5(154,1602)(0,0 ) 0(132,2980)(0,0)[r ] 2.5(132,2775)(0,0)[r ] 2(132,2570)(0,0)[r ] 1.5(132,2365)(0,0)[r ] 1(132,2160)(0,0)[r ] 0.5(132,1955)(0,0)[r ] 0(132,1750)(0,0)[r]-0.5 ( 1952,1489)(0,0)[r ] 0.4(1952,1423)(0,0)[r ] 0.2(1952,1357)(0,0)[r ] 0(1952,1292)(0,0)[r]-0.2(1952,1226)(0,0)[r]-0.4(1952,1160)(0,0)[r]-0.6(1952,1094)(0,0)[r]-0.8(1952,1028)(0,0)[r]-1(1952,962)(0,0)[r]-1.2(1952,896)(0,0)[r]-1.4(1952,830)(0,0)[r]-1.6(3291,391)(0,0)@xmath194(3339,696)(0,0)[l ] 8(3282,634)(0,0)[l ] 7(3225,572)(0,0)[l ] 6(3168,511)(0,0)[l ] 5(3112,449)(0,0)[l ] 4(3055,387)(0,0)[l ] 3(2998,325)(0,0)[l ] 2(2941,264)(0,0)[l ] 1(2884,202)(0,0)[l ] 0(2358,115)(0,0)(d)(2358,235)(0,0)@xmath8(2852,186)(0,0 ) 8(2754,221)(0,0 ) 7(2655,257)(0,0 ) 6(2557,293)(0,0 ) 5(2458,328)(0,0 ) 4(2359,364)(0,0 ) 3(2261,400)(0,0 ) 2(2162,435)(0,0 ) 1(2064,471)(0,0 ) 0(152,1489)(0,0)[r ] 0.1(152,1416)(0,0)[r ] 0.08(152,1343)(0,0)[r ] 0.06(152,1270)(0,0)[r ] 0.04(152,1196)(0,0)[r ] 0.02(152,1123)(0,0)[r ] 0(152,1050)(0,0)[r]-0.02(152,977)(0,0)[r]-0.04(152,904)(0,0)[r]-0.06(152,830)(0,0)[r]-0.08(1491,391)(0,0)@xmath194(1539,696)(0,0)[l ] 8(1482,634)(0,0)[l ] 7(1425,572)(0,0)[l ] 6(1368,511)(0,0)[l ] 5(1312,449)(0,0)[l ] 4(1255,387)(0,0)[l ] 3(1198,325)(0,0)[l ] 2(1141,264)(0,0)[l ] 1(1084,202)(0,0)[l ] 0(558,115)(0,0)(c)(558,235)(0,0)@xmath8(1052,186)(0,0 ) 8(954,221)(0,0 ) 7(855,257)(0,0 ) 6(757,293)(0,0 ) 5(658,328)(0,0 ) 4(559,364)(0,0 ) 3(461,400)(0,0 ) 2(362,435)(0,0 ) 1(264,471)(0,0 ) 0(1952,3649)(0,0)[r ] 1.6(1952,3576)(0,0)[r ] 1.4(1952,3503)(0,0)[r ] 1.2(1952,3430)(0,0)[r ] 1(1952,3356)(0,0)[r ] 0.8(1952,3283)(0,0)[r ] 0.6(1952,3210)(0,0)[r ] 0.4(1952,3137)(0,0)[r ] 0.2(1952,3064)(0,0)[r ] 0(1952,2990)(0,0)[r]-0.2(3291,2551)(0,0)@xmath194(3339,2856)(0,0)[l ] 8(3282,2794)(0,0)[l ] 7(3225,2732)(0,0)[l ] 6(3168,2671)(0,0)[l ] 5(3112,2609)(0,0)[l ] 4(3055,2547)(0,0)[l ] 3(2998,2485)(0,0)[l ] 2(2941,2424)(0,0)[l ] 1(2884,2362)(0,0)[l ] 0(2358,2275)(0,0)(b)(2358,2395)(0,0)@xmath8(2852,2346)(0,0 ) 8(2754,2381)(0,0 ) 7(2655,2417)(0,0 ) 6(2557,2453)(0,0 ) 5(2458,2488)(0,0 ) 4(2359,2524)(0,0 ) 3(2261,2560)(0,0 ) 2(2162,2595)(0,0 ) 1(2064,2631)(0,0 ) 0(152,3649)(0,0)[r ] 0.08(152,3539)(0,0)[r ] 0.06(152,3430)(0,0)[r ] 0.04(152,3320)(0,0)[r ] 0.02(152,3210)(0,0)[r ] 0(152,3100)(0,0)[r]-0.02(152,2990)(0,0)[r]-0.04(1491,2551)(0,0)@xmath194(1539,2856)(0,0)[l ] 8(1482,2794)(0,0)[l ] 7(1425,2732)(0,0)[l ] 6(1368,2671)(0,0)[l ] 5(1312,2609)(0,0)[l ] 4(1255,2547)(0,0)[l ] 3(1198,2485)(0,0)[l ] 2(1141,2424)(0,0)[l ] 1(1084,2362)(0,0)[l ] 0(558,2275)(0,0)(a)(558,2395)(0,0)@xmath8(1052,2346)(0,0 ) 8(954,2381)(0,0 ) 7(855,2417)(0,0 ) 6(757,2453)(0,0 ) 5(658,2488)(0,0 ) 4(559,2524)(0,0 ) 3(461,2560)(0,0 ) 2(362,2595)(0,0 ) 1(264,2631)(0,0 ) 0 ( 3196,389)(0,0)[r]triton(3196,506)(0,0)[r]helium3(3196,584)(0,0)[r ] b=1.66 fm(3196,664)(0,0)[r]@xmath220(2730,20)(0,0)(d)(3540,80)(0,0 ) 10(3216,80)(0,0 ) 8(2892,80)(0,0 ) 6(2568,80)(0,0 ) 4(2244,80)(0,0 ) 2(1920,80)(0,0 ) 0(1900,1364)(0,0)[r]-2(1900,1209)(0,0)[r]-4(1900,1053)(0,0)[r]-6(1900,898)(0,0)[r]-8(1900,742)(0,0)[r]-10(1900,587)(0,0)[r]-12(1900,431)(0,0)[r]-14(1900,276)(0,0)[r]-16(1900,120)(0,0)[r]-18(3192,2394)(0,0)[r]triton(3192,2511)(0,0)[r]helium3(3192,2589)(0,0)[r ] b=2.04 fm(3192,2669)(0,0)[r]@xmath221(2720,1532)(0,0)(b)(3540,1592)(0,0 ) 10(3212,1592)(0,0 ) 8(2884,1592)(0,0 ) 6(2556,1592)(0,0 ) 4(2228,1592)(0,0 ) 2(1900,1592)(0,0 ) 0(1880,2876)(0,0)[r]-3(1880,2669)(0,0)[r]-4(1880,2461)(0,0)[r]-5(1880,2254)(0,0)[r]-6(1880,2047)(0,0)[r]-7(1880,1839)(0,0)[r]-8(1880,1632)(0,0)[r]-9(1404,951)(0,0)[r]triton(1404,1068)(0,0)[r]helium3(1404,1146)(0,0)[r ] b=2.04 fm(1404,1226)(0,0)[r]@xmath222(950,20)(0,0)(c)(40,742 ) ( 1740,80)(0,0 ) 10(1424,80)(0,0 ) 8(1108,80)(0,0 ) 6(792,80)(0,0 ) 4(476,80)(0,0 ) 2(160,80)(0,0 ) 0(140,1364)(0,0)[r]-2(140,1226)(0,0)[r]-3(140,1088)(0,0)[r]-4(140,949)(0,0)[r]-5(140,811)(0,0)[r]-6(140,673)(0,0)[r]-7(140,535)(0,0)[r]-8(140,396)(0,0)[r]-9(140,258)(0,0)[r]-10(140,120)(0,0)[r]-11(1408,2375)(0,0)[r]triton(1408,2492)(0,0)[r]helium3(1408,2570)(0,0)[r ] b=1.66 fm(1408,2650)(0,0)[r]@xmath3(960,1532)(0,0)(a)(40,2254 ) ( 1740,1592)(0,0 ) 10(1428,1592)(0,0 ) 8(1116,1592)(0,0 ) 6(804,1592)(0,0 ) 4(492,1592)(0,0 ) 2(180,1592)(0,0 ) 0(160,2876)(0,0)[r]-3(160,2763)(0,0)[r]-3.5(160,2650)(0,0)[r]-4(160,2537)(0,0)[r]-4.5(160,2424)(0,0)[r]-5(160,2311)(0,0)[r]-5.5(160,2197)(0,0)[r]-6(160,2084)(0,0)[r]-6.5(160,1971)(0,0)[r]-7(160,1858)(0,0)[r]-7.5(160,1745)(0,0)[r]-8(160,1632)(0,0)[r]-8.5 ( 0,0 ) , @xmath223 , and @xmath224 for @xmath25 and @xmath225 triton calculations as a function of @xmath1 . the upper pairs of lines refer to @xmath226 , the middle pair @xmath127,and the bottom two @xmath102 . note that the respective sums of all three terms gives the triton binding energy.,title="fig : " ] 0 & -7.6147 ( 218.533 ) & -7.6144 ( 74.113 ) & -7.6148 ( 26.480 ) + 2 & -7.6143 ( 65.260 ) & -7.6144 ( 26.732 ) & -7.6145 ( 12.556 ) + 4 & -7.6144 ( 26.226 ) & -7.6141 ( 15.840 ) & -7.6141 ( 9.068 ) + 6 & -7.6179 ( 13.268 ) & -7.6140 ( 9.899 ) & -7.6141 ( 6.524 ) + 8 & -7.6179 ( 5.985 ) & -7.6136 ( 6.725 ) & -7.6139 ( 5.257 ) + 10 & -7.6182 ( 1.606 ) & -7.6131 ( 4.156 ) & -7.6138 ( 4.093 ) + 8 & 10 & @xmath155 + 60 & 64.51 & 90.00 & 90.00 & 90.00 & 90.00 & 90.00 & 90.00 + & -68.36 & -38.41 & -32.54 & -26.40 & -21.03 & -16.52 & 128.50 + & -3.76 & 8.26 & 22.38 & 37.25 & 51.34 & 64.06 & 0.00 + 30 & 33.59 & 45.00 & 45.00 & 45.00 & 45.00 & 45.00 & 45.00 + & -45.11 & -47.20 & -45.61 & -44.05 & -42.50 & -40.99 & 29.10 + & 3.91 & 3.71 & 6.17 & 8.96 & 11.96 & 15.03 & 0.00 + 15 & 17.67 & 22.50 & 22.50 & 22.50 & 22.50 & 22.50 & 22.50 + & -26.37 & -31.54 & -30.43 & -29.73 & -29.16 & -28.66 & 3.98 + & 1.08 & 2.43 & 4.16 & 5.03 & 5.68 & 6.26 & 0.00 + 10 & @xmath155 + 60 & -33.149 & -47.107 & -46.936 & -46.873 & -46.837 & -45.375 + & 0.146 & -1.771 & -1.692 & -1.016 & -0.330 & 0.000 + 30 & -13.400 & -18.898 & -18.772 & -18.665 & -18.564 & -18.952 + & 0.641 & -0.197 & -0.655 & -0.920 & -1.065 & 0.000 + 15 & -5.053 & -6.964 & -6.911 & -6.884 & -6.857 & -6.769 + & 0.398 & 0.385 & 0.296 & 0.194 & 0.104 & 0.000 + 10 & @xmath155 + 60 & 122.210 & 150.000 & 150.000 & 150.000 & 150.000 & 150.000 + & -11.616 & 11.714 & 15.141 & 18.365 & 20.788 & 89.884 + & 10.381 & 18.639 & 25.302 & 31.312 & 36.794 & 0.000 + 30 & 61.986 & 75.000 & 75.000 & 75.000 & 75.000 & 75.000 + & -16.159 & -12.223 & -11.338 & -10.376 & -9.496 & 21.034 + & 4.571 & 7.211 & 8.547 & 9.790 & 11.033 & 0.000 + 15 & 31.679 & 37.500 & 37.500 & 37.500 & 37.500 & 37.500 + & -10.649 & -10.579 & -10.162 & -9.805 & -9.509 & 4.019 + & 2.144 & 3.365 & 3.782 & 4.071 & 4.335 & 0.000 +

the three - body energy - dependent effective interaction given by the bloch - horowitz ( bh ) equation is evaluated for various shell - model oscillator spaces . the results are applied to the test case of the three - body problem ( @xmath0h and @xmath0he ) , where it is shown that the interaction reproduces the exact binding energy , regardless of the parameterization ( number of oscillator quanta or value of the oscillator parameter @xmath1 ) of the low - energy included space . we demonstrate a non - perturbative technique for summing the excluded - space three - body ladder diagrams , but also show that accurate results can be obtained perturbatively by iterating the two - body ladders . we examine the evolution of the effective two - body and induced three - body terms as @xmath1 and the size of the included space @xmath2 are varied , including the case of a single included shell , @xmath3 . for typical ranges of @xmath1 , the induced effective three - body interaction , essential for giving the exact three - body binding , is found to contribute @xmath410% to the binding energy .

nucl-th0404028

the study of harmonic maps was initiated by f. b. fuller , j. nash and j. h. sampson @xcite while the first general result on the existence of harmonic maps is due to eells - sampson @xcite . harmonic maps are extrema ( critical points ) of the energy functional defined on the space of smooth maps between riemannian ( pseudo - riemannian ) manifolds . the trace of the second fundamental form of such maps vanishes . more precisely , let @xmath0 , @xmath1 be pseudo - riemannian manifolds and @xmath2 denotes the sections of the tangent bundle @xmath3 of @xmath4 , that is , the space of vector fields on @xmath4 . then _ @xmath5 of a smooth map @xmath6 is defined by the formula @xmath7 where @xmath8 is the volume measure associated to the metric @xmath9 and the _ energy density _ @xmath10 of @xmath11 is the smooth function @xmath12 given by @xmath13 for each @xmath14 . in the above equation @xmath15 is a linear map @xmath16 therefore it can be considered as a section of the bundle @xmath17 where @xmath18 is the pullback bundle having fibres @xmath19 , @xmath14 and @xmath20 is the pullback metric on @xmath21 . if we denote by @xmath22 and @xmath23 the levi - civita connections on @xmath24 and @xmath25 respectively , then the second fundamental form of @xmath11 is the symmetric map @xmath26 defined by @xmath27 for any @xmath28 . where @xmath29 is the pullback of the levi - civita connection @xmath23 of @xmath25 to the induced vector bundle @xmath30 . the section @xmath31 , defined by @xmath32 is called the _ tension field _ of @xmath11 and a map is said to be harmonic if its tension field vanishes identically ( see @xcite ) . if we consider @xmath33 a smooth two - parameter variation of @xmath11 such that @xmath34 and let @xmath35 be the corresponding variational vector fields then @xmath36 the _ hessian _ of a harmonic map @xmath11 is defined by : @xmath37 the index of a harmonic map @xmath38 is defined as the dimension of the tangent subspace of @xmath39 on which the hessian @xmath40 is negative definite . a harmonic map @xmath11 is said to be _ stable _ if morse index ( _ i.e. _ , the dimension of largest subspace of @xmath41 on which the hessian @xmath40 is negative definite ) of @xmath11 is zero and otherwise , it is said to be _ unstable _ ( see @xcite ) . for a non - degenerate point @xmath42 , we decompose the space @xmath43 into its _ vertical space _ @xmath44 and its _ horizontal space _ @xmath45 , that is , @xmath46 , so that @xmath47 . the map is said to be horizontally conformal if for each @xmath14 either the rank of @xmath48 is zero ( that is , @xmath49 is a critical point ) , or the restriction of @xmath48 to the horizontal space @xmath50 is surjective and conformal ( here @xmath49 is a regular point ) @xcite . the premise of harmonic maps has acknowledged several important contributions and has been successfully applied in computational fluid dynamics ( cfd ) , minimal surface theory , string theory , gravity and quantum field theory ( see @xcite ) . most of works on harmonic maps are between riemannian manifolds @xcite . the harmonic maps between pseudo - riemannian manifolds behave differently and their study must be subject to some restricted classes of pseudo - riemannian manifolds @xcite . this paper is organized as follows . in sect . [ prem ] , the basic definitions about almost para - hermitian manifolds , almost paracontact manifolds and normal almost paracontact manifolds are given . in sect . [ paraholo ] , we define and study paraholomorphic map . we prove that the tension field of any @xmath51-paraholomorphic map between almost para - hermitian manifold and para - sasakian manifold lies in @xmath52 . [ parap ] deals with parapluriharmonic map in which we obtain the necessary and sufficient condition for a @xmath53-paraholomorphic map between para - sasakian manifolds to be @xmath54-parapluriharmonic and give an example for its illustrations . a smooth manifold @xmath55 of dimension @xmath56 is said to be an almost product structure if it admits a tensor field @xmath57 of type @xmath58 satisfying : @xmath59 in this case the pair @xmath60 is called an almost product manifold . an almost para - complex manifold is an almost product manifold @xmath61 such that the eigenbundles @xmath62 associated with the eigenvalues @xmath63 of tensor field @xmath57 have the same rank @xcite . an almost para - hermitian manifold @xmath64 is a smooth manifold endowed with an almost para - complex structure @xmath57 and a pseudo - riemannian metric @xmath65 compatible in the sense that @xmath66 it follows that the metric @xmath65 has signature @xmath67 and the eigenbundles @xmath62 are totally isotropic with respect to @xmath65 . let @xmath68 be an orthonormal basis and denote @xmath69 : @xmath70 for @xmath71 and @xmath72 for @xmath73 . the fundamental @xmath74-form of almost para - hermitian manifold is defined by @xmath75 and the co - differential @xmath76 of @xmath77 is given as follows @xmath78 an almost para - hermitian manifold is called para - khler if @xmath79 @xcite . a @xmath80 smooth manifold @xmath81 of dimension @xmath82 is said to have a triplet @xmath83-structure if it admits an endomorphism @xmath84 , a unique vector field @xmath85 and a @xmath86-form @xmath87 satisfying : @xmath88 where @xmath89 is the identity transformation ; and the endomorphism @xmath84 induces an almost paracomplex structure on each fibre of @xmath90 the contact subbundle , _ i.e. _ , eigen distributions @xmath91 corresponding to the characteristic values @xmath63 of @xmath84 have equal dimension @xmath92 . + from the equation ( [ eta ] ) , it can be easily deduced that @xmath93 this triplet structure @xmath83 is called an almost paracontact structure and the manifold @xmath81 equipped with the @xmath83-structure is called an almost paracontact manifold ( see also @xcite ) . if an almost paracontact manifold admits a pseudo - riemannian metric @xmath94 satisfying : @xmath95 where signature of @xmath94 is necessarily @xmath96 for any vector fields @xmath97 and @xmath98 ; then the quadruple @xmath99 is called an almost paracontact metric structure and the manifold @xmath81 equipped with paracontact metric structure is called an almost paracontact metric manifold . with respect to @xmath94 , @xmath87 is metrically dual to @xmath85 , that is @xmath100 also , equation ( [ gphi ] ) implies that @xmath101 further , in addition to the above properties , if the structure-@xmath99 satisfies : @xmath102 for all vector fields @xmath103 on @xmath81 , then the manifold is called a paracontact metric manifold and the corresponding structure-@xmath99 is called a paracontact structure with the associated metric @xmath94 @xcite . for an almost paracontact metric manifold , there always exists a special kind of local pseudo - orthonormal basis @xmath104 ; where @xmath105 @xmath85 and @xmath106 s are space - like vector fields and @xmath107 s are time - like . such a basis is called a @xmath84-basis . hence , an almost paracontact metric manifold @xmath108 is an odd dimensional manifold with a structure group @xmath109 , where @xmath110 is the para - unitary group isomorphic to @xmath111 . an almost paracontact metric structure-@xmath99 is para - sasakian if and only if @xmath112 from eqs . , and , it can be easily deduced for a para - sasakian manifold that @xmath113 in particular , a para - sasakian manifold is @xmath114-paracontact @xcite . on an almost paracontact metric manifold , one defines the @xmath115-tensor field @xmath116 by @xmath117 - 2\,d\eta\otimes\xi,\end{aligned}\ ] ] where @xmath118 $ ] is the nijenhuis torsion of @xmath84 . if @xmath116 vanishes identically , then we say that the manifold @xmath81 is a normal almost paracontact metric manifold @xcite . the normality condition implies that the almost paracomplex structure @xmath57 defined on @xmath119 by @xmath120 is integrable . here @xmath97 is tangent to @xmath81 , @xmath121 is the coordinate on @xmath122 and @xmath123 is a @xmath124 function on @xmath119 . now we recall the following proposition which characterized the normality of almost paracontact metric @xmath125-manifolds : @xcite for an almost paracontact metric @xmath125-manifold @xmath126 , the following three conditions are mutually equivalent * @xmath126 is normal , * there exist smooth functions @xmath127 on @xmath126 such that @xmath128 * there exist smooth functions @xmath127 on @xmath126 such that @xmath129 where @xmath22 is the levi - civita connection of the pseudo - riemannian metric @xmath94 . the functions @xmath127 appearing in eqs . ( [ nablaxphiy ] ) and ( [ nablaxxi ] ) are given by @xmath130 a normal almost paracontact metric @xmath125-manifold is called paracosymplectic if @xmath131 and para - sasakian if @xmath132 @xcite . one can look structure preserving mapping between almost para - hermitian and almost paracontact manifolds as analogous of the well - known holomorphic mappings in complex geometry @xcite . + let @xmath133 , @xmath1 be almost paracontact metric manifolds and @xmath64 be an almost para - hermitian manifold . then a smooth map * @xmath134 is @xmath135-paraholomorphic map if @xmath136 . for such a map @xmath137 . * @xmath138 is @xmath139-paraholomorphic map if @xmath140 . here * @xmath142 is @xmath53-paraholomorphic map if @xmath143 . in particular , @xmath144 and @xmath145 . when @xmath15 interwines the structures upto a minus sign , we say about @xmath135-anti paraholomorphic , @xmath139-anti paraholomorphic and @xmath53-anti paraholomorphic mappings . now , we prove the following result . let @xmath11 be a smooth @xmath53-paraholomorphic map between para - sasakian manifolds @xmath146 , @xmath1 . then @xmath147 where @xmath148 . since @xmath15 has values in @xmath149 so that @xmath150 and @xmath151 have values in @xmath149 . thus , we have @xmath152 in the last equality , we have used . on the other hand , we obtain @xmath153 from eqs . and , we have @xmath154 let @xmath155 be a local orthonormal frame for @xmath156 . taking the trace in and using the fact that @xmath157 is symmetric , we have . this completes the proof . following the proof of the above proposition , we can give the following remarks : for a para - sasakian manifold @xmath158 and a para - hermitian manifold @xmath64 . if * @xmath134 be a @xmath135-paraholomorphic map then we have @xmath159 where @xmath160 . + * @xmath138 be a @xmath139-paraholomorphic map then we have @xmath161 where @xmath162 . [ t1 ] let @xmath11 be a @xmath135-paraholomorphic map between para - sasakian manifold @xmath163 and para - k@xmath164hler manifold @xmath64 . then @xmath11 is harmonic . let @xmath165 be a local orthonormal adapted basis on @xmath166 , then from eqs . and , we have @xmath167 ( since for a @xmath135-paraholomorphic map @xmath137 ) . it follows by the use of equation that @xmath168 as @xmath55 is a para - k@xmath169hler manifold . therefore , @xmath170 and @xmath11 is harmonic . this completes the proof of the theorem . for @xmath1 , let @xmath171 be real distributions , respectively , on para - sasakian manifolds @xmath172 of rank @xmath173 then it admits globally defined @xmath86-form @xmath174 such that @xmath175 . clearly , @xmath176 , where @xmath177 is the real distribution of rank one defined by @xmath178 @xcite . now , we prove : [ hk1 ] for any @xmath139-paraholomorphic map @xmath11 between almost para - hermitian manifold @xmath64 and para - sasakian manifold @xmath179 , the tension field @xmath180 . before going to proof of this theorem , we first prove the following proposition : for an almost para - hermitian manifold @xmath64 , we have @xmath181\bigg\}\end{aligned}\ ] ] where @xmath182 is a local orthonormal frame on @xmath183 . it is straightforward to calculate @xmath184-j(\nabla_{e'_{i}}e'_{i})+j(\nabla_{je'_{i}}je'_{i})\big\ } \ ] ] and the result follows from and . this completes the proof . _ proof of theorem [ hk1 ] . _ since @xmath185 , @xmath186 therefore for any local orthonormal frame @xmath182 on @xmath183 , we obtain by using eqs . , , and that @xmath187 employing eq . , the above equation reduces to @xmath188\bigg),\xi_{1}\bigg).\end{aligned}\ ] ] reusing eq . in , we get @xmath189 which shows that @xmath180 . this completes the proof of the theorem . by the consequence of the above theorem we can state the following result as a corollary of the theorem [ hk1 ] . let @xmath64 and @xmath163 be para - k@xmath164hler and para - sasakian manifolds respectively . then for any @xmath139-paraholomorphic map @xmath138 , the tension field @xmath180 . in this section we define the notion of @xmath54-parapluriharmonic map which is similar to the notion of @xmath84-pluriharmonic map between almost contact metric manifold and riemannian manifold , for @xmath84-pluriharmonic map see : @xcite . a smooth map @xmath11 between almost paracontact metric manifold @xmath179 and pseudo - riemannian manifold @xmath190 is said to be @xmath54-parapluriharmonic if @xmath191 where the second fundamental form @xmath192 of @xmath11 is defined by . in particular , @xmath193 for any tangent vector @xmath97 . we recall that @xmath137 for a @xmath135-paraholomorphic map and @xmath200 is para - k@xmath201hler , and that from eq . for any vectors @xmath202 tangent to @xmath203 , we have @xmath204 using equation for a given map , we obtain @xmath205 replacing @xmath98 by @xmath206 and employing eqs . and , the above equation reduces to @xmath207 by the virtue of the fact that @xmath157 is symmetric , we obtain from above equation that @xmath208 the above expresion implies that @xmath11 is @xmath54-parapluriharmonic and thus harmonic from the proposition [ pro1 ] . this completes the proof of the theorem . let @xmath198 be a normal almost paracontact metric 3-manifold with @xmath210constant , @xmath199 be a para - k@xmath164hler manifold and @xmath211 be a smooth @xmath135-paraholomorphic map . then @xmath212 is paracosymplectic manifold . since @xmath11 is a @xmath53-paraholomorphic map then for all @xmath214 there exists a function @xmath123 on @xmath215 such that @xmath216 for any @xmath217 , we have from eqs . , and that @xmath218 from above equation and the fact that @xmath157 is symmetric , we obtain that @xmath219 replacing @xmath98 by @xmath220 in above expression and using eqs . and , we find @xmath221 this implies that @xmath222 if and only if @xmath213 . this completes the proof of the theorem . let @xmath223 be @xmath125-dimensional manifolds with standard cartesian coordinates . define the almost paracontact structures @xmath224 respectively on @xmath225 by @xmath226 where @xmath227 , @xmath228 , @xmath229 , @xmath230 , @xmath231 and @xmath232 . by direct calculations , one verifies that the nijenhuis torsion of @xmath233 for @xmath1 vanishes , which implies that the structures are normal . let the pseudo - riemannian metrics @xmath234 are prescribed respectively on @xmath225 by @xmath235=\begin{bmatrix}-x&0&0\\0&x^4+x&x^2\\0&x^2&1\end{bmatrix},\ \left[g_2\left(e_s',e_t'\right)\right]=\begin{bmatrix}v^4+v&0&v^2\\0&-v&0\\v^2&0&1\end{bmatrix},\end{aligned}\ ] ] for all @xmath236 . for the levi - civita connections @xmath237 with respect to metrics @xmath238 respectively , we obtain @xmath239 @xmath240 from above expressions and equation , we find @xmath241 , @xmath242 . hence the @xmath203 and @xmath243 are para - saakian manifolds with invariant distributions @xmath244 and @xmath245 respectively . let @xmath246 be a mapping defined by @xmath247 . then @xmath143 , _ i_.e . , @xmath11 is a @xmath53-paraholomorphic map between para - sasakian manifolds . for any @xmath248 and @xmath249 , it is not hard to see that @xmath250 , @xmath251 and @xmath252 thus theorem [ main ] is verified . : harmonic maps in general relativity and quantum field theory , in : gauduchon , p. , editor , harmonic mappings , twistors , and @xmath253-models ( luminy , 1986 ) . advanced series in math . physics * 4 * , 270 - 305 , world scientific publishing , singapore ( 1988 ) .

the purpose of this paper is to study the harmonicity of maps to or from para - sasakian manifolds . we derive the condition for the tension field of paraholomorphic map between almost para - hermitian manifold and para - sasakian manifold . the necessary and sufficient condition for a paraholomorphic map between para - sasakian manifolds to be parapluriharmonic are shown and a non - trivial example is presented for its illustrations . dst / inspire/03/2014/001552 . ]

1508.02972

evidence in the past year from experiments at the cern large hadron collider ( lhc ) @xcite and the fermilab tevatron @xcite encourages a strong sense of anticipation that the long - sought neutral higgs boson is on the verge of discovery with mass in the vicinity of 125 gev . as more data are analyzed , and the lhc energy is increased from 7 to 8 tev , experimental investigations will naturally turn toward determination of the properties of the observed mass enhancement particularly , the branching fractions into pairs of gauge bosons , standard model fermions , and possibly other states . the original formulation of electroweak symmetry breaking focused on couplings of the higgs boson to massive gauge bosons @xcite . tree - level yukawa couplings between fermions and higgs bosons came later in the current version of the `` standard model '' ( sm ) in which the higgs boson serves as the agent for generation of fermion masses as well as gauge boson masses . proposals have also been made of higgs doublets @xcite or triplets @xcite in which the higgs boson is explicitly `` fermiophobic , '' namely , without tree - level couplings to fermions . in this paper , we emphasize a measurement that offers the possibility to test a broad class of models where higgs boson couplings to fermions , if they exist , are small enough that they do not affect the branching fractions to gauge bosons . we focus on @xmath2 associated production where @xmath1 decays into a pair of photons , @xmath6 , and @xmath7 , @xmath8 decays into a pair of jets , @xmath9 . we investigate whether the peak observed near 125 gev in the diphoton @xmath10 invariant mass spectrum @xcite in the 7 tev lhc data provides support for a suppressed fermion coupling hypothesis . we show that this process offers excellent prospects for distinguishing a fermiophobic higgs boson from a standard model higgs boson . the phenomenology of a fermiophobic higgs boson is very different from the sm case . since the coupling to top quarks @xmath11 is suppressed , a fermiophobic higgs boson is not produced through the dominant sm production channel , the gluon - gluon fusion process @xmath12 , where the interaction occurs through a top - quark loop . rather , production of a fermiophobic higgs boson occurs in association with an electroweak gauge boson @xmath13 where @xmath14 , @xmath8 , or through vector boson fusion ( vbf ) , @xmath15 . between these two modes , the relative cross section favors vbf , but @xmath2 associated production offers the opportunity to observe a final state in which there are two potentially prominent resonances in coincidence , the higgs boson peak in @xmath16 along with the @xmath0 peak in the dijet mass distribution @xmath17 . the favorable branching fraction for @xmath17 guides our choice of this decay channel rather than the leptonic decays @xmath18 or @xmath19 . the lhc atlas and cms collaborations consider the fermiophobic possibility in two recent papers @xcite . in the @xmath20 channel , cms requires the transverse energy of the two jets to be larger than 30 and 20 gev , with large pseudorapidity separation between the jets ( @xmath21 ) and dijet invariant mass larger than 350 gev . these cuts are designed for the vbf production process . in the @xmath2 channel , they concentrate on the leptonic decay modes of the vector bosons . while the background is smaller , the signal is suppressed by the small branching fraction to leptons . atlas uses the inclusive diphoton channel @xmath22 . in the diphoton mass region near 125 gev , atlas sees some evidence for an increase in the signal to background ratio at large values of the transverse momentum of the diphoton pair . this increase is qualitatively consistent with the expectation of a harder higgs boson @xmath23 spectrum from vbf and associated production , compared to the sm gluon fusion mechanism . on the other hand , the ratio of the higgs signal to qcd background in the @xmath24 channel also improves with @xmath23 of the higgs boson in the sm @xcite , so the @xmath23 spectrum alone is not a good discriminator . the fermiophobic possibility must be reconciled also with a tevatron collider enhancement in the @xmath25 mass spectrum @xcite in the general vicinity of 125 gev , implying a possible coupling of the higgs boson to fermions . however , these results have yet to be corroborated by lhc data and could be interpreted in a model in which effective yukawa couplings are radiatively induced @xcite . the emphasis in this paper is placed on the investigation of the fermiophobic option in associated production , with @xmath0 decay to a pair of jets . we compute the expected signal rates from associated production and vbf , and the backgrounds from @xmath26 in perturbative quantum chromodynamics . adopting event selections similar to those used by the lhc collaborations , we show that the current @xmath27 fb@xmath3 might contain @xmath281.9 standard deviation ( @xmath4 ) evidence for a fermiophobic higgs boson in the @xmath29 channel . we argue that clear evidence ( @xmath5 ) of a fermiophobic higgs boson could be obtained by study of the @xmath13 channel at 8 tev with @xmath30 fb@xmath3 of integrated luminosity . we urge concentrated experimental effort on higgs plus vector boson associated production . fermiophobic higgs bosons are produced predominantly via @xmath2 ( @xmath7 , @xmath8 ) associated production or vector boson fusion ( vbf ) . associated production will produce hard jets if @xmath17 [ fig.[fig : feyn]a ] , with the invariant mass of the dijet system @xmath31 showing a resonance structure in the electroweak gauge boson mass region ( @xmath328091 gev ) . vector boson fusion is characterized by two hard forward jets [ fig . [ fig : feyn]b ] , and it contributes a long tail to the dijet invariant mass distribution , with few events in the @xmath33 mass region . in contrast , additional jets from production of a sm higgs boson are mostly from soft initial state radiation off the gluon - gluon fusion initial state . we exploit these different event topologies to distinguish a fermiophobic higgs boson from a standard model higgs boson . associated production with @xmath1 decay to diphotons and @xmath34 decay to dijets , and ( b ) vbf production of @xmath35dijets . ] the contribution to diphoton production from a fermiophobic higgs is surprisingly large . while the cross section for fermiophobic higgs production is suppressed compared to the sm by an order of magnitude , the branching fraction for @xmath36 is correspondingly increased . the net result is that the production cross section for @xmath37 for a fermiophobic higgs boson is predicted to be nearly identical to that of a sm higgs boson @xcite . in order to compare directly with data , we begin with a higgs to diphoton signal analysis by the atlas collaboration @xcite . atlas sees an excess of events when compared to either the fermiophobic or sm higgs models of a factor of @xmath38 @xcite . since we wish to distinguish a fermiophobic higgs signal from a sm higgs signal , we focus on predicting the _ fraction _ of the atlas @xmath36 data sample that should contain a dijet invariant mass peak @xmath31 near the @xmath39 and @xmath8 masses . hence , we normalize the total number of events in our signal predictions by this experimental factor of 2 . three fermiophobic higgs signal processes should contribute to the atlas diphoton mass peak : @xmath40 , @xmath41 , and vbf . in order to determine the proportion of each signal process we calculate the acceptance of each process at next - to - leading - order in qcd . we generate weighted signal events using mcfm @xcite , where we substitute photons for @xmath42 quarks in the final state , and use hdecay @xcite to correct for the branching fraction for @xmath36 . we impose atlas inspired @xcite acceptance cuts on the two photons in the final state : * photon candidates are ordered in transverse energy @xmath43 , and the leading ( subleading ) candidate is required to have @xmath44 gev ( 25 gev ) ; * both photons must satisfy pseudorapidity cuts of @xmath45 or @xmath46 ; * both photons must be isolated with at most 5 gev of energy deposited in a cone of @xmath47 around the candidate , where @xmath48 is the azimuthal angle , after the photon energy is removed . we determine the number of events that should appear in each production channel after atlas photon acceptance cuts by applying a photon reconstruction and identification efficiency . this efficiency is 65% for @xmath49 gev and 95% for @xmath50 gev . we do a linear extrapolation of photon efficiencies for other values of photon @xmath43 , and assume that it is 100% for a photon with @xmath51 gev . we use the atlas isolation cut acceptance of 87% for a 120 gev higgs boson @xcite . as a cross check , we calculate the diphoton acceptance for the gluon - gluon fusion channel using the same method and find a cut acceptance of 34.9% , in very good agreement with the 35% given by atlas @xcite . the numbers of events predicted in 4.9 fb@xmath3 at 7 tev from the @xmath40 , @xmath41 , and vbf channels before and after atlas acceptance cuts ( scaled by the factor of 2 above ) are shown in table [ tab : after_diphotoncuta ] . vector boson fusion supplies the largest fraction of the higgs diphoton events . however , since the distinguishing feature is a @xmath39 or @xmath8 dijet mass peak in the @xmath2 final state of interest to us , our additional cuts are optimized to select the @xmath40 and @xmath41 processes . .numbers of signal and background events after cuts expected in @xmath27 fb@xmath3 of data at 7 tev . atlas @xmath52 cuts in the second line include photon acceptances , efficiencies , and isolation . [ tab : after_diphotoncuta ] [ cols="<,>,>,>,>",options="header " , ] of data . shaded bands show the statistical uncertainty of the background . the second bin covers the vector boson mass region @xmath53 . ] in this paper , we investigate the possibility of using present @xcite and future diphoton data from the lhc to distinguish a fermiophobic higgs boson from a sm higgs boson . unlike the sm higgs , nearly 40% of fermiophobic higgs bosons are produced in association with a @xmath39 or @xmath8 vector boson . since the largest branching fraction for these vector boson decays is into jets , we focus on @xmath17 and devise cut - based analyses that attempt to provide a clean signal and large significance . we show that a @xmath4 significance excess could be found in the existing 4.9 fb@xmath3 of data at 7 tev . with the anticipated 10 fb@xmath3 to be acquired soon at 8 tev , we find that @xmath5 evidence should be possible . we expect that these significances could be improved by including additional angular correlations in the dijet system and between the diphoton and dijet systems , perhaps in a neural - net based approach , and we encourage the experimental collaborations to expand upon the analysis presented here . g. aad _ et al . _ ( atlas collaboration ) , phys . b * 710 * , 49 ( 2012 ) . s. chatrchyan _ et al . _ ( cms collaboration ) , phys . b * 710 * , 26 ( 2012 ) . tevnph ( tevatron new phenomina and higgs working group ) and cdf and d0 collaborations , arxiv:1203.3774 [ hep - ex ] . f. englert and r. brout , phys . * 13 * , 321 ( 1964 ) ; p. w. higgs , phys . * 12 * , 132 ( 1964 ) ; p. w. higgs , phys . * 13 * , 508 ( 1964 ) ; g. s. guralnik , c. r. hagen , and t. w. b. kibble , phys . * 13 * , 585 ( 1964 ) ; p. w. higgs , phys . * 145 * , 1156 ( 1966 ) ; t. w. b. kibble , phys . rev . * 155 * , 1554 ( 1967 ) . a. barroso , l. brucher , and r. santos , phys . d * 60 * , 035005 ( 1999 ) . j. f. gunion , r. vega , and j. wudka , phys . d * 42 * , 1673 ( 1990 ) ; a. g. akeroyd , m. a. diaz , m. a. rivera , and d. romero , phys . d * 83 * , 095003 ( 2011 ) . cms collaboration , cms pas hig-12 - 002 . atlas collaboration , g. aad , b. abbott , j. abdallah , s. abdel khalek , a. a. abdelalim , o. abdinov and b. abi _ et al . _ , arxiv:1205.0701 [ hep - ex ] . c. balazs , e. l. berger , p. m. nadolsky and c. -p . yuan , phys . d * 76 * , 013009 ( 2007 ) [ arxiv:0704.0001 [ hep - ph ] ] . e. gabrielli and b. mele , phys . d * 82 * , 113014 ( 2010 ) [ erratum - ibid . d * 83 * , 079901 ( 2011 ) ] [ arxiv:1005.2498 [ hep - ph ] ] ; e. gabrielli , b. mele , and m. raidal , arxiv:1202.1796 [ hep - ph ] . atlas collaboration , atlas - conf-2012 - 019 . j. m. campbell and r. k. ellis , nucl . * 205 - 206 * , 10 ( 2010 ) . a. djouadi , j. kalinowski , and m. spira , comput . phys . commun . * 108 * , 56 ( 1998 ) . j. alwall , m. herquet , f. maltoni , o. mattelaer , and t. stelzer , j. high energy phys . * 1106 * , 128 ( 2011 ) . t. sjostrand , s. mrenna , and p. z. skands , j. high energy phys . * 0605 * , 026 ( 2006 ) . j. conway _ et al . _ , http://www.physics.ucdavis.edu/ @xmath28conway / research / software / pgs / pgs.html . m. l. mangano , m. moretti , f. piccinini , and m. treccani , j. high energy phys . * 0701 * , 013 ( 2007 ) .

production in association with an electroweak vector boson @xmath0 is a distinctive mode of production for a higgs boson @xmath1 without tree - level couplings to fermions , known as a fermiophobic higgs boson . we focus on @xmath2 associated production with @xmath1 decay into a pair of photons , and @xmath0 into a pair of jets , with the goal of distinguishing a fermiophobic higgs boson from the standard model higgs boson . performing a simulation of the signal and pertinent qcd backgrounds , and using the same event selection cuts employed by the lhc atlas collaboration , we argue that existing lhc data at 7 tev with 4.9 fb@xmath3 of integrated luminosity may distinguish a fermiophobic higgs boson from a standard model higgs boson near 125 gev at about 1.9 standard deviation signal significance ( @xmath4 ) per experiment . at 8 tev we show that associated production could yield @xmath5 significance per experiment with 10 fb@xmath3 of data .

1203.6645

in the standard model ( sm ) , the electroweak symmetry breaking results in a physical neutral cp - even higgs boson . there also exist various models extended by increasing higgs fields , such as the multi - higgs doublet model where there are extra two neutral and two charged physical higgs bosons for each additional doublet . if cp is a good symmetry in the higgs sector , one of the extra neutral bosons is cp - even and the other is cp - odd . if cp is not invariant , all of the higgs states which have same quantum numbers except for the cp property can be mixed in their mass eigenstates . then , physical higgs bosons do not have definite cp parity . one of the notable advantages of a photon linear collider ( plc ) is to provide information about the cp property of the higgs boson by use of linear polarizations of colliding photons . defining @xmath1 two - photon states as @xmath2 and @xmath3 where @xmath4 indicate their helicities with @xmath5 units , the cp transformation leads to the other states : @xmath6 , @xmath7 . then , the cp eigenstates are @xmath8 and @xmath9 ; the former is a @xmath1 component of parallel polarized photons and couples to a cp - even higgs boson ( @xmath10 ) , and the latter is of perpendicularly polarized photons and couples to a cp - odd one ( @xmath11 ) . however , because colliding beams at plc are generated by the compton back - scattering between high energy electrons and laser photons , the energy spectrum of @xmath12 distributes broadly and the degrees of polarizations depend strongly on @xmath13 , where @xmath12 and @xmath14 are the center - of - mass energy of @xmath15 collisions and @xmath16 collisions at parent lc . in the case of the @xmath17 gev lc , linear polarizations can be used effectively for relatively light higgs bosons whose masses are less than about a few hundred gev . for heavier higgs bosons , it is necessary that the electron energy is raised or we use other methods , conventionally . we propose one method where we take advantage of interference between higgs - production and background amplitudes with circular polarized beams . a broad peak of the @xmath12 spectrum in the range where the degrees of circular polarizations become large is helpful to the method , because we have an interest in the energy dependence of the interference effects . as an example , we consider the process @xmath18 which receives contribution from the higgs - exchanged @xmath19-channel diagram and the top - quark - exchanged @xmath20-channel ones . the helicity - dependent cross sections are expressed as @xmath21,\end{aligned}\ ] ] where @xmath22 and @xmath23 are the helicity amplitudes of higgs resonance ( @xmath24 = @xmath10 or @xmath11 ) and top continuum processes , @xmath25 denote the helicities of colliding photons in @xmath5 units , @xmath26 and @xmath27 the final @xmath28 and @xmath29 helicities in the center - of - mass frame in @xmath30 ( we also write @xmath31 ( @xmath32 ) as @xmath33 ( @xmath34 ) . ) . there is another observable sensitive to cp parity . when the decay of a top quark is taken into account , the cross sections for the processes @xmath38 can be written by @xmath39~ re \left [ { \cal d}_{l}^{\lambda } \overline{\cal d}_{l}^{\overline{\lambda } } { \cal d}_{r}^{\lambda * } \overline{\cal d}_{r}^{\overline{\lambda } * } \right ] \nonumber \\ & - & 2~im \left [ { \cal m}^{\lambda_1 \lambda_2 ll } { \cal m}^{\lambda_1 \lambda_2 rr * } \right]~ i m \left [ { \cal d}_{l}^{\lambda } \overline{\cal d}_{l}^{\overline{\lambda } } { \cal d}_{r}^{\lambda * } \overline{\cal d}_{r}^{\overline{\lambda } * } \right ] \biggr\}. \label{sigma - bw}\end{aligned}\ ] ] here , the decay amplitudes for the processes @xmath40 and @xmath41 are defined as @xmath42 and @xmath43 , the explicit forms of which are in the appendix of ref . we notice the azimuthal angles of @xmath44 and @xmath45 in the @xmath28 and @xmath29 rest frame . describing them as @xmath24 and @xmath46 , they appear in the third and fourth terms in eq . [ sigma - bw ] : @xmath47 \propto \cos(\phi- \overline{\phi } ) , \nonumber \\ i m \left [ { \cal d}_{l}^{\lambda } \overline{\cal d}_{l}^{\overline{\lambda } } { \cal d}_{r}^{\lambda * } \overline{\cal d}_{r}^{\overline{\lambda } * } \right ] \propto \sin(\phi- \overline{\phi}).\end{aligned}\ ] ] therefore , we obtain @xmath48.\end{aligned}\ ] ] when colliding photons are polarized to be @xmath32 , @xmath49 \simeq i m \left [ { \cal m}^{++ ll}_{\phi } { \cal m}^{++ rr*}_{cont } \right]$ ] is satisfied considering @xmath50 . since the higgs - production amplitudes @xmath51 and @xmath52 have opposite signs in the mssm - type models , @xmath53 tells us the cp parity of the higgs boson . a numerical calculation is shown in fig . [ fig1](c ) . though the quantity turns out to be rather small because of cancellation between the contributions from longitudinally and transversely polarized @xmath54 s , we can recover the sensitivity to the cp parity by taking account of @xmath54-decay distributions . when the cp symmetry is not conserved in the higgs potential , the helicity amplitudes for the higgs - production are denoted by @xmath55 \left[\,\sigma\beta_t a_t -i b_t\right ] \delta_{\lambda_1 \lambda_2}\delta_{\sigma \overline\sigma}\ , . \label{mephi}\end{aligned}\ ] ] where @xmath56 and @xmath57 are proportional to the cp - even components of vertices , @xmath58 and @xmath59 , @xmath60 and @xmath61 to the cp - odd components . since \{@xmath62 } or / and \{@xmath63 } have non - zero values simultaneously , they induce complicated interference . moreover , the amplitudes include six parameters for vertices ; @xmath62 are complex , whereas @xmath63 are real . therefore , we need at least six observables to determine the cp property completely and are urged to use linear polarizations as well . all observables obtained from various polarizations ( including the mixture of linear and circular ones ) are exhibited in ref . we have discussed measurement of the cp property of the higgs boson at plc . it has been found that we can extract information about the cp property of the higgs boson from the observation of interference effects between higgs - production and background amplitudes . if the higgs boson have definite cp parity , this method can be powerful in high @xmath64 region where linear polarizations of colliding photons become useless conventionally . e. asakawa , j. kamoshita , a. sugamoto , and i. watanabe , _ eur . j. _ * c14 * , 335 ( 2000 ) . e. asakawa , phd thesis ( in preparation ) e. asakawa , s.y . choi , k. hagiwara , and j.s . lee , _ phys . rev . _ * d62 * , 115005 ( 2000 ) .

we study measurement of the cp property of the higgs boson at a photon linear collider . one method where we take advantage of interference between higgs - production and background amplitudes is proposed . a broad peak of the photon energy spectrum is helpful in observing the energy dependence of the interference effects . numerical results for the process @xmath0 are shown as an example .

hep-ph0101234

two conceptual pictures of galaxy clustering have been examined in the literature , the continuous hierarchical clustering model and the power - law cluster model ( peebles 1980 , 61 ) . in the hierarchical clustering model , which has emerged as the accepted model over the past two decades , galaxy clustering is characterized by power - law correlation functions : the @xmath3-point correlation function @xmath4 scales with configuration size as @xmath5 , where @xmath6 and the two - point correlation function goes as @xmath7 . the hierarchical model is motivated by the observed power - law behavior @xmath8 of galaxy correlations ( groth & peebles 1977 ; fry & peebles 1978 ) , with a theoretical basis in a self - similar , scale - invariant solution to the equations of motion ( davis & peebles 1977 ) . the alternative power - law cluster model has an even longer history ( neyman & scott 1952 ; peebles 1974 , 1980 ; mcclelland & silk 1977 ; scherrer & bertschinger 1991 ; sheth & jain 1997 ; valageas 1998 ; yano & gouda 1999 ) . in this model , galaxies are placed in spherical clumps that are assumed to follow a power - law density profile @xmath9 , with the centers of the clumps distributed randomly . the resulting two - point correlation function is also a power law with a logarithmic slope @xmath10 . while it is possible to reproduce the observed two - point function by an appropriate choice of the power index @xmath11 , peebles and groth ( 1975 ) pointed out that this model produces a three - point function that is too steep to be consistent with observations in the zwicky and lick catalogs . in an earlier paper ( ma & fry 2000a ) , we have shown that in the nonlinear regime , the three - point correlation function @xmath12 of the cosmological mass density field does not exactly follow the prediction @xmath13 of the hierarchical clustering model . these conclusions are drawn from study of high resolution numerical simulations of a cold dark matter ( cdm ) model with cosmological constant and of a model with scale - free initial conditions @xmath14 with @xmath2 . in experiments replacing simulation dark matter halos with power - law density profiles , @xmath15 , we have demonstrated that the behavior of the correlation functions in the nonlinear regime are determined by the halo profiles , but that it is not possible to match both the two- and three - point correlations with a single slope @xmath16 . these results differ from the predictions of both of these two conceptual models . in this paper , we expand our previous study of the nonlinear two- and three - point correlation functions by investigating a new prescription that takes into account the non - power - law profiles of halos , the distribution of halo masses , and the spatial correlations of halo centers . each of these ingredients has been well studied in the literature . we find that this halo model provides a good description of the two- and three - point correlation functions in both the @xmath2 and cdm simulations over the entire range of scales from the weak clustering , perturbative regime on large length scales , to the strongly nonlinear regime on small length scales . our result is approximately hierarchical over an intermediate range of scales , thus uniting the two pictures . an independent recent study by seljak ( 2000 ) , which appeared during completion of this work , has also examined the two - point power spectrum in a similar construction and has found that this type of approach can reproduce the power spectrum in the cdm model . the analytic model proposed here can be used to compute the two- and three - point correlation functions and their fourier transforms , the power spectrum and bispectrum , over any range of scale where the input halo properties are valid . in a subsequent paper ( ma & fry 2000c ) , we study the predictions of this analytic halo model for the asymptotic nonlinear behavior of the @xmath3-point correlation functions and the pairwise velocities and examine the conditions required for stable clustering . the outline of this paper is as follows . in 2 we describe the three input ingredients of the model : halo density profiles , halo mass functions , and halo - halo correlations . in 3 we assemble these ingredients and construct analytic expressions for the two - point correlation function @xmath17 and the power spectrum @xmath18 . in 4 we do the same for the three - point correlation function @xmath19 and its fourier transform , the bispectrum @xmath20 . in 5 we test the validity of this new model by comparing its predictions with results from numerical simulations of an @xmath2 scale free model and a low - density cdm model with a cosmological constant ( @xmath21cdm ) . we also present results of the synthetic halo replacement technique used to enhance the numerical resolution . in 6 we discuss further the physical meanings and implications of the model . in particular , we elaborate on two important implications of this model : deviations from the common assumptions of stable clustering and hierarchical clustering . section 7 is a summary . it has been suggested recently that the mass density profiles of cold dark matter halos have a roughly universal shape , generally independent of cosmological parameters ( navarro , frenk , & white 1996 , 1997 ) @xmath22 where @xmath23 is a dimensionless density amplitude , @xmath24 is a characteristic radius , and @xmath25 is the mean background density . we consider two functional forms for the density profiles @xmath26 both forms have asymptotic behaviors @xmath27 at small @xmath28 and @xmath29 at large @xmath28 , but they differ in the transition region . the first form @xmath30 with @xmath31 is found to provide a good fit to simulation halos by navarro et al . ( 1996 , 1997 ) , whereas the second form @xmath32 with a steeper inner slope @xmath33 is favored by moore et al . ( 1998 , 1999 ) . some independent simulations have produced halos that are well fit by the shallower @xmath31 inner slope ( e.g. , hernquist 1990 ; dubinski & carlberg 1991 ; huss , jain , & steinmetz 1999 ) , and others the steeper @xmath34 slope ( e.g. , fukushige and makino 1997 ) . jing & suto ( 2000 ) have recently reported a mass - dependent inner slope , with @xmath35 for galactic - mass halos and @xmath36 for cluster - mass halos . many of these authors find that the outer profile scales as @xmath37 , but steeper outer profiles have also been suggested ( hernquist 1990 ; dubinski & carlberg 1991 ) . given these uncertainties , we will consider in this paper both types of profiles in . the parameters @xmath24 and @xmath23 in equation ( [ shape ] ) are generally functions of the halo mass @xmath38 . a concentration parameter , @xmath39 can be used to quantify the central density of a halo ( navarro et al . 1997 ) , where @xmath40 is the radius within which the average density is 200 times the mean density of the universe . using @xmath41 , we can relate @xmath24 and @xmath23 to @xmath38 and @xmath42 , where the scale radius @xmath24 is @xmath43 and the density amplitude @xmath44 is @xmath45}\ , , \qquad p=1\ , , \nonumber\\ \bar\delta_{ii } & = & { 100\,c^3\over \ln(1+c^{3/2})}\ , , \qquad p={3\over 2 } \ , . \label{delbar}\end{aligned}\ ] ] typical values of @xmath42 are in the range of a few to ten for type i and perhaps a factor of three smaller for type ii . there is a weak dependence on mass , such that less massive halos have a larger central density ( e.g. , cole & lacey 1996 ; tormen , bouchet , & white 1997 ; navarro et al . 1996 , 1997 ; jing & suto 2000 ) . this is understood in general terms as reflecting the mean density at the redshift @xmath46 when the halo initially collapsed , @xmath47 . for @xmath48 this is @xmath49 , or @xmath50 in a scale - free model . the number density of halos with mass @xmath38 within a logarithmic interval is often approximated by the prescription of press & schechter ( 1974 ) , @xmath51 where @xmath52 is a parameter characterizing the linear overdensity at the onset of gravitational collapse , and @xmath53 is the linear rms mass fluctuations in spheres of radius @xmath54 @xmath55 where @xmath56 is the fourier transform of a real - space tophat window function . the mass @xmath38 is related to @xmath54 by @xmath57 . for scale free models with a power law initial power spectrum @xmath58 , this is @xmath59 . the parameter @xmath60 characterizes the mass scale at the onset of nonlinearity , @xmath61 , and is related to the nonlinear wavenumber @xmath62 ( defined as ] @xmath63 ) by @xmath64 where @xmath65^{3/(n+3 ) } \ , , \nonumber \\ b^{(3+n)/3 } & = & \sin\left[(n+2){\pi \over 2}\right ] \ , \gamma(n+2 ) \ , { 9 \ , ( 2^{-n } ) ( 3+n ) \over ( -n)(1-n)(3-n ) } \end{aligned}\ ] ] ( defined for @xmath66 ) . various modifications to the press - schechter mass function have been suggested ( e.g. , sheth & tormen 1999 ; lee & shandarin 1999 ; jenkins et al . 2000 ) to improve the accuracy of the original formula . dark matter halos do not cluster in the same way as the mass density field . on large scales , a bias parameter @xmath67 is typically used to quantify this difference . let @xmath68 be the two - point correlation function of halos with masses @xmath38 and @xmath69 , @xmath70 be the linear correlation function for the mass density field , and @xmath71 and @xmath72 be the corresponding power spectra . on large length scales , we assume a linear bias and write @xmath73 based on the peak and the press - schechter formalism , mo & white ( 1996 ) developed a model for the linear bias @xmath74 , which is later modified by jing ( 1998 ) to be @xmath75 the original formula for @xmath74 by mo & white includes only the first factor above ; the second factor , dependent on the primordial spectral index @xmath76 , is obtained empirically for an improved fit to simulation results at the lower mass end ( jing 1998 ) . in this bias model , @xmath74 is below unity for @xmath77 ( where @xmath61 ) and reaches @xmath78 for @xmath79 . small dark matter halos are therefore anti - biased relative to the mass density . for @xmath80 , @xmath74 increases monotonically with the halo mass and reaches @xmath81 at @xmath82 . nonlinear effects on the bias have been studied ( kravtsov & klypin 1999 and references therein ) , but they are unimportant in our model because the halo - halo correlation terms contribute significantly only on large length scales in the linear regime ( see 3 and 4 ) . similarly , we use higher order bias parameters to relate the higher - order correlation functions for halos and mass density . in this paper we examine the three - point correlation function @xmath19 and its fourier transform , the bispectrum @xmath20 ( see 4 for a more detailed discussion ) . on large length scales where the amplitude of @xmath83 is small , perturbation theory can be used to relate the lowest order contribution to the bispectrum of the mass density to the linear power spectrum @xmath72 ( fry 1984 ) : @xmath84 using this perturbative result and the results of mo , jing , & white ( 1997 ) , we can write the halo bispectrum as @xmath85\ , p_\lin(k_1)\,p_\lin(k_2 ) \nonumber\\ & + & \left [ b(m ) b(m ' ) b(m'')\,f_{23}+ b(m ) b_2(m ' ) b(m'')\right]\ , p_\lin(k_2)\,p_\lin(k_3 ) \nonumber \\ & + & \left [ b(m ) b(m ' ) b(m'')\,f_{31}+ b_2(m ) b(m ' ) b(m'')\right]\ , p_\lin(k_3)\,p_\lin(k_1 ) \,,\nonumber\\ & & \label{bbias}\end{aligned}\ ] ] where @xmath74 is given by , and the quadratic bias parameter @xmath86 is @xmath87 for the special equilateral case of @xmath88 , simplifies to @xmath89 \,p_\lin^2(k ) \ , . \label{bbias2}\end{aligned}\ ] ] in practice , the terms involving @xmath86 in equations ( [ bbias ] ) and ( [ bbias2 ] ) make only a small net contribution . for simplicity , we will therefore not include this term in the subsequent derivations and calculations . we now construct our analytic halo model for the two - point correlation function @xmath17 and the power spectrum @xmath18 . the two - point correlation function of the cosmological mass density field @xmath90 is @xmath91 the fourier transform of @xmath17 is the mass power spectrum @xmath92 , which is related to the density field in @xmath93-space by @xmath94 , where @xmath95 is the dirac delta - function . the two - point correlation function measures the excess probability above the poisson distribution of finding a pair of objects with separation @xmath96 ( peebles 1980 ) . the objects can be taken to be dark matter particles , most of which cluster gravitationally in the form of dark matter halos . one should therefore be able to express @xmath97 for the density field in terms of properties of dark matter halos . in this picture , we can write the contributions to @xmath97 as two separate terms , one from particle pairs in the same halo , and the other from pairs that reside in two different halos . in realistic cosmological models , dark matter halos exhibit a spectrum of masses that can be characterized by a distribution function @xmath98 , and the halo centers are spatially correlated . taking these factors into consideration , we can write the two - point correlation function for @xmath83 in terms of the halo density profile @xmath99 , halo mass function @xmath98 , and halo - halo correlation function @xmath100 discussed in 2 . we write @xmath101 where the subscripts `` @xmath102 '' and `` @xmath103 '' denote contributions from particle pairs in `` 1-halo '' and `` 2-halos '' , respectively , and @xmath104 \left[\int dm \ , { dn\over dm}\,\deltabar\ , u(r''/r_s)\,b(m ) \right ] \nonumber\\ & & \times\,\ , \xi_\lin(|\r'-\r''+\r| ) \ , . \label{xi2}\end{aligned}\ ] ] these expressions arise from averaging over displacements @xmath105 , @xmath106 of halo centers from the particle positions @xmath107 , @xmath108 , where @xmath109 . in the last expression above , we have used the bias model of to relate the halo - halo correlation function @xmath110 to the linear correlation function @xmath111 of the mass density field . as we will show in 5 , the dominant contribution to the two - point correlation function in the nonlinear regime on small length scales is from the first , 1-halo term @xmath112 for particle pairs that reside in the same halos . this makes intuitive sense , because closely spaced particle pairs are most likely to be found in the same halo . this term is determined by the convolution of the dimensionless density profile with itself , @xmath113 for many forms of @xmath99 , the angular integration in this equation is analytic , and @xmath114 can be reduced to a simple one - dimensional integral over @xmath115 . for some special cases , @xmath114 can even be reduced to an analytic expression . we leave the detailed results for @xmath114 to the appendix . in @xmath93-space , the convolutions in for @xmath17 become simple products . using @xmath116 to denote the fourier transform of @xmath99 , where @xmath117 , we can readily transform into expressions for the mass power spectrum : @xmath118 where the 1-halo and 2-halo terms are @xmath119 ^ 2 \nonumber\\ \noalign{\smallskip } p_{2h}(k ) & = & \int dm\,{dn\over dm } \,r_s^3\,\deltabar\,\ut(kr_s ) \int dm'\,{dn\over dm ' } \,r'^3_s\,\deltabar'\,\ut(kr'_s)\,p_\halo(k ) \label{pk } \\ & = & \left [ \int dm\,{dn\over dm } \,r_s^3\,\deltabar\,\ut(kr_s)\,b(m ) \right]^2\,p_\lin(k)\,.\nonumber\end{aligned}\ ] ] to arrive at the last expression above , we have again used the bias model of . for computational efficiency , we find that the algebraic expressions @xmath120/3 \ } \over ( 1 + q^{1.1})^{(2/1.1 ) } } \ , , \qquad p=1 \nonumber\\ \ut_{ii}(q ) & = & { 4\pi \ { \ln(e+1/q ) + 0.25\ln[\ln(e+1/q)]\ } \over 1 + 0.8\ , q^{1.5 } } \ , , \qquad p={3\over 2 } \label{uq}\end{aligned}\ ] ] provide excellent fits for the profiles of navarro et al . ( 1997 ) and moore et al . ( 1999 ) , with less than 4% rms error for form i and less than 1% rms error for form ii . the functional form is chosen to reproduce the asymptotic behaviors : @xmath121 at small @xmath122 ( with no radial cutoff ) , and @xmath123 ( type i ) and @xmath124 ( type ii ) at large @xmath122 . the two - point @xmath17 and @xmath18 can now be computed analytically from equations ( [ xi2 ] ) and ( [ pk ] ) . the inputs are or ( [ uq ] ) for the halo density profile @xmath99 or @xmath116 , equations ( [ rs ] ) and ( [ delbar ] ) for @xmath24 and @xmath23 , for the halo mass function @xmath98 , and for the halo - halo correlation function . since the halo density profile appears to have a nearly universal form regardless of background cosmology , @xmath17 and @xmath18 depend on cosmological parameters mainly through @xmath125 of and the halo concentration @xmath126 or central density @xmath127 . ( see ma & fry 2000c for a more detailed discussion of @xmath126 . ) here we construct our analytic halo model for the three - point correlation function @xmath128 and the bispectrum @xmath129 . the joint probability of finding three objects in volume elements @xmath130 , and @xmath131 is given by @xmath132 \,\bar{n}^3 dv_1\,dv_2\,dv_3\,,\ ] ] where @xmath17 and @xmath19 are the two- and three - point correlation functions , respectively , @xmath133 is the mean number density of objects , and @xmath134 and @xmath135 are the lengths of the sides of the triangle defined by the three objects ( peebles 1980 ) . the fourier transform of the three - point correlation function @xmath19 is the bispectrum @xmath20 , which is related to the density field in @xmath93-space by @xmath136 . the bispectrum depends on any three parameters that define a triangle in @xmath93-space . a particular simple configuration to study is the equilateral triangle ( @xmath88 ) , and in this case the bispectrum @xmath137 depends only on a single wavenumber . similar to the two - point halo model of 3 , we can write the contributions to the three - point correlation function @xmath128 of the mass density as three separate terms , each term representing particle triplets that reside in a single halo , two distinct halos , or three distinct halos . taking into account the halo mass distribution and halo - halo correlations discussed in 2 , we obtain @xmath138 where the separate 1-halo , 2-halo , and 3-halo terms are @xmath139 the dominant contribution to the three - point correlation function in the nonlinear regime is from the first term @xmath140 , which comes from particle triplets that reside in the same halo . this term is determined by the convolution @xmath141 of three factors of the density profile @xmath99 , and is analogous to the convolution @xmath114 in for the one - halo term @xmath112 in the two - point correlation function . the bispectrum of the mass density field @xmath83 in @xmath93-space can be obtained by fourier transforming the equations above . we find @xmath142 where @xmath143 \ , [ r_s^3\,\deltabar\,\ut(k_2 r_s ) ] \ , [ r_s^3\,\deltabar\ , \ut(k_3 r_s ) ] \nonumber \\ \noalign{\smallskip } b_{2h}(k_1,k_2,k_3 ) & = & \int dm\,{dn\over dm}\ , [ r_s^3\,\deltabar\,\ut(k_1 r_s)]\ , [ r_s^3\,\deltabar\,\ut(k_2 r_s ) ] \nonumber\\ & & \times \int dm'\,{dn\over dm'}\,r'^3_s\,\deltabar'\,\ut(k_3 r'_s ) \ , p_\halo(k_3 ; m , m ' ) + \hbox{sym.(1,2,3 ) } \\ \noalign{\smallskip } b_{3h}(k_1,k_2,k_3 ) & = & \int dm \,{dn\over dm}\,r_s^3 \,\deltabar\,\ut(k_1r_s ) \int dm ' \,{dn\over dm'}\,r'^3_s \,\deltabar'\,\ut(k_2r'_s ) \nonumber \\ & & \times \int dm''\,{dn\over dm''}\,r''^3_s \,\deltabar''\,\ut(k_3r''_s ) \ , b_\halo(k_1,k_2,k_3;m , m',m '' ) \ , . \nonumber\end{aligned}\ ] ] the halo - halo power spectrum @xmath144 and bispectrum @xmath145 are related to the linear mass power spectrum @xmath146 by equations ( [ pbias ] ) and ( [ bbias ] ) . the expressions for the mass bispectrum above simplify considerably for the equilateral triangle configuration , and @xmath147 where @xmath148 ^ 3 \nonumber\\ b^{\rm eq}_{2h}(k ) & = & 3\ , \left [ \int dm\,{dn\over dm}\ , [ r_s^3\,\deltabar\,\ut(kr_s)]^2\,b(m)\right ] \left[\int dm\,{dn\over dm}\,r_s^3\,\deltabar\,\ut(kr_s)\,b(m ) \right ] \ , p_{\rm lin}(k)\ , \label{bk } \\ b^{\rm eq}_{3h}(k ) & = & \left [ \int dm\,{dn\over dm}\ , r_s^3\,\deltabar\,\ut(kr_s)\,b(m ) \right]^3\ , { 12\over 7}\,p^2_{\rm lin}(k)\,.\nonumber\end{aligned}\ ] ] here we have written out explicitly the bias factors @xmath74 using equations ( [ pbias ] ) and ( [ bbias2 ] ) , and we have neglected terms with @xmath86 as discussed in 2.3 . in this section we compare the predictions of our analytical model described in 2 , 3 , and 4 with results from cosmological @xmath3-body simulations . we examine two cosmological models : an @xmath2 scale - free model and a low - density @xmath21cdm model . these are the same simulations studied in ma & fry ( 2000a ) . the @xmath2 simulation has @xmath149 particles and a plummer force softening length of @xmath150 , where @xmath151 is the box length . the @xmath21cdm model is spatially flat with matter density @xmath152 and cosmological constant @xmath153 . this run has @xmath154 particles and is performed in a @xmath155 comoving box with a comoving force softening length of @xmath156 for hubble parameter @xmath157 . the baryon fraction is set to zero for simplicity . the primordial power spectrum has a spectral index of @xmath158 , and the density fluctuations are drawn from a random gaussian distribution . the gravitational forces are computed with a particle - particle particle - mesh ( p@xmath159 m ) code ( ferrell & bertschinger 1994 ) . we compute the density field @xmath83 on a grid from particle positions using the second - order triangular - shaped cloud ( tsc ) interpolation scheme . a fast fourier transform is then used to obtain @xmath83 in @xmath93-space . the @xmath93-space tsc window function is deconvolved to correct for smearing in real space due to the interpolation , and shot noise terms are subtracted to correct for discreteness effects . we then compute the second and third moments of the density amplitudes in fourier space . we show results for the power spectrum as the dimensionless variance @xmath160 . a useful dimensionless three - point statistic is the hierarchical three - point amplitude @xmath161 the three - point amplitude @xmath1 has the convenient feature that for the lowest nonvanishing result in perturbation theory , @xmath1 is independent of time and the overall amplitude of @xmath0 ; for scale - free models with a power - law @xmath0 , @xmath1 is independent of overall scale as well . to lowest order , it follows from that the equilateral bispectrum has a particularly simple form , @xmath162 , and we have @xmath163 , independent of the power spectrum . to investigate the numerical effects of limited resolution in the simulations , we have experimented with the distribution of matter in halos identified in the simulations . in these experiments , we keep the locations and masses of the halos unchanged but redistribute the subset of particles which lies within the virial radius @xmath40 ( the radius within which the mean overdensity is 200 ) of each halo according to a prescribed density profile . we then recompute the two- and three - point statistics @xmath164 and @xmath1 from the redistributed particle positions as well as the original non - halo particles , which remain at their original positions . by using density profiles obtained empirically from higher - resolution simulations of individual halos , this recipe allows us to model accurately the inner regions of the halos on scales below the numerical softening length scale while at the same time preserving all the large - scale information available in the large parent simulation . this technique should also be useful for other studies that are sensitive to the inner halo density profiles , for example the ray - tracing method in gravitational lensing . ma & fry ( 2000a ) have used this replacement technique to experiment with synthetic halos that follow a pure power - law profile @xmath165 . it is found that @xmath166 and @xmath167 at high-@xmath93 indeed obey @xmath168 and @xmath169 as predicted by the simple power - law model of peebles ( 1974 ) . the scaling works even in the presence of the full distribution of matter outside the halo cores . here we extend this replacement technique to more realistic halo profiles of equation ( [ u ] ) . figures 1 and 2 illustrate the effects on the matter power spectrum and bispectrum when the original halos in large cosmological simulations are replaced by synthetic halos with the density profile @xmath170 of equation ( [ u ] ) . for the @xmath2 scale - free model , the concentration parameter is taken to be @xmath171 , which is consistent with navarro et al . ( 1997 ) and has the expected scaling with mass , @xmath172 , in a scale - free model . for @xmath21cdm models , we use @xmath173 as suggested by figure 3 of moore et al . we note , however , that @xmath126 from various recent simulations has shown a large scatter , and its functional form depends on the exact form of the density profile used . for the @xmath21cdm model and form @xmath170 , for example , a flatter and smaller @xmath174 appears to be preferred by jing & suto ( 2000 ) and navarro et al . the results of tormen et al . ( 1997 ) and cole & lacey ( 1996 ) are also only marginally consistent with each other . a more detailed investigation of the different forms of @xmath126 can be found in ma & fry ( 2000c ) . in figures 1 and 2 , the agreement at low values of @xmath93 between the original and synthetic halos is excellent , confirming that the correlation functions on larger length scales are insensitive to the spatial distribution of particles in the halo cores . the only significant difference between the simulation and synthetic halos appears at small length scales , where the coarser resolution of the simulation blurs out the structure of the inner halo and results in an inner profile flatter than in equation ( [ u ] ) . this effect is manifested in the bending over of the dashed curves for @xmath18 in figures 1 and 2 at high @xmath93 , and is corrected for when the synthetic halos are used . we now proceed to compare the predictions of the analytic model of 2 4 with the numerical results from cosmological simulations . figures 3 and 4 show the @xmath93-space density variance @xmath175 ( upper panel ) and the three - point amplitude @xmath176 for equilateral triangles for the @xmath2 scale - free model and the @xmath21cdm model . the solid black curves are the model predictions computed from equations ( [ pk ] ) and ( [ bk ] ) . the contribution from the single - halo and multiple - halo terms are shown separately as dashed curves . for the density profile , we use the same @xmath170 and concentration parameters as in figures 1 and 2 . for the mass function , we use the press - schechter formula but reduce its overall amplitude by 25% , which we find necessary in order to match the halo mass functions for our numerical simulations . this overestimation of halo numbers with @xmath177 by press - schechter is a well known result reported in many other studies ( see jenkins et al . 2000 and references therein ) . the mass limits for the integrals in equations ( [ pk ] ) and ( [ bk ] ) do not significantly affect the model predictions for the total @xmath164 or @xmath1 . raising the lower mass limit does reduce the contribution from lower mass halos and hence lower the high-@xmath93 amplitudes of the multiple halo terms @xmath178 , @xmath179 , and @xmath180 , but these terms make negligible contributions to the total @xmath164 and @xmath1 . as discussed in 3 and 4 , the nonlinear parts of both the two- and three - point statistics are determined by the dominant 1-halo term because the closely spaced particle pairs and triplets mostly reside in the same halos . the multiple - halo terms are therefore significant only on larger length scales comparable to the separation between halos . their inclusion , however , is necessary for the transition into the linear regime . for the @xmath2 model in figure 3 , we plot the results against the scaled @xmath181 , where @xmath62 characterizes the length scale that is becoming nonlinear and is defined by @xmath182 . three time outputs are shown , where the expansion factor ( 1 initially ) and @xmath62 ( in units of @xmath183 ) are : @xmath184 , and @xmath185 ( from left to right ) . for the two - point @xmath166 , the agreement between the model prediction and the simulations is excellent . the three simulation outputs also overlap well , indicating that self - similarity is obeyed , as reported in jain & bertschinger ( 1998 ) . for the three - point @xmath186 , however , self - similar scaling does not hold as rigorously ( ma & fry 2000a ) . it is interesting to note that the analytic prediction agrees most closely with the earliest output @xmath187 ( green curve ) . this provides further evidence to the suggestion of ma & fry ( 2000a ) that the later outputs of the @xmath2 simulation may be affected by the finite volume of the simulation box . for the @xmath21cdm model in figure 4 , the analytic model again provides a good match to the @xmath3-body results within the fluctuations among the simulations . we illustrate the numerical effects due to box sizes by showing results from two runs with volume ( 100 mpc)@xmath159 and ( 640 mpc)@xmath159 . the model predictions extend well beyond the resolution of the simulations . the real - space two - point correlation function for the @xmath2 and @xmath21cdm models is shown in figures 5 and 6 . for the halo model predictions , we have chosen to show only the results for the 1-halo term @xmath112 because this term dominates the interesting nonlinear portion of @xmath97 . the agreement between the halo model ( dashed curves ) and the simulations ( symbols ) is again excellent . for the 2-halo terms @xmath188 , the computation can be done more easily in @xmath93-space as shown in figures 3 and 4 , so we do not include them here . for comparison , we plot in figures 36 the results from the commonly used fitting formulas for the nonlinear power spectrum ( hamilton et al . 1991 ; jain et al . 1995 ; peacock & dodds 1996 ; ma 1998 ; ma et al . while the formulas provide a good approximation to @xmath166 up to @xmath189 for the @xmath2 model and @xmath190 mpc@xmath191 for the @xmath21cdm model , the figures show that significant deviations occur at higher @xmath93 , and the fitting formula and our current model predict different high-@xmath93 slopes for @xmath166 . since the high-@xmath93 behavior of the fitting formulas has been constructed to obey the stable clustering prediction , this discrepancy has an important implication for the validity of stable clustering , which we discuss briefly in the next section and at length in ma & fry ( 2000c ) . we have constructed a physical model for the correlation functions of the mass density field in which the correlations are derived from properties of dark matter halos . we have described in detail the input , construction , and results of this model in 2 5 . we now examine more closely its physical meanings and implications in three separate regimes . on scales larger than the size of the largest halo , the contributions from separate halos dominate , and ( by design ) the model reproduces the results of perturbation theory . on intermediate scales , @xmath192 , because of the exponential cutoff in the mass function @xmath98 at the high mass end , the contribution to the volume integrals in is dominated by the large-@xmath96 regime where the halo profiles are roughly @xmath37 . the correlation functions therefore behave approximately as predicted by the power - law model with @xmath193 , i.e. , @xmath194 and @xmath195 constant . this is why @xmath1 exhibits an approximately flat plateau at intermediate @xmath93 in the bottom panels of figures 3 and 4 . on the smallest and most nonlinear scales , the correlation functions probe the innermost regions of the halos . intriguingly , the halo model predicts on these scales a behavior that is different from either the frequently - assumed stable clustering result of @xmath196 with @xmath197 ( davis & peebles 1977 ) , or the power - law profile result of @xmath10 . the implication of departure from stable clustering is significant because all the fitting formulas for the nonlinear @xmath18 in the literature ( see 5.2 ) have been constructed to approach the stable clustering limit at high @xmath93 . a more detailed study on the criteria for stable clustering in this model is given in a separate paper ( ma & fry 2000c ) . the origin of the deviation from stable clustering in the model at high-@xmath93 can be understood as follows . for the two - point function , as @xmath93 becomes large , the one - halo integral @xmath198 in converges before the exponential cutoff , and is dominated by contributions near the mass scale for which @xmath199 . the behavior now depends on the mass distribution function . the various mass functions discussed in 2.2 have the same general behavior of @xmath200 , where @xmath201 . the press - schechter form assumes @xmath202 ( see eq . [ [ ps ] ] ) , while others ( e.g. , sheth & tormen 1999 ; jenkins et al . 2000 ) suggest a flatter slope of @xmath203 for the lower mass halos . since the scale radius @xmath24 depends on mass as @xmath204 , and @xmath205 ( up to logarithmic factors ) , we find from that the power spectrum at high @xmath93 goes as @xmath206 changing variables to @xmath207 , we see that @xmath208 where the first term in @xmath209 is the prediction of stable clustering . the departure arises from the factor @xmath210 in the mass function , and would vanish only if @xmath211 or @xmath212 . this is the origin of the difference in @xmath166 at high @xmath93 between the model prediction ( solid curves ) and the fitting formula ( dotted curves ) shown in figures 3 and 4 . for the three - point function , the one - halo integral @xmath213 in converges ( barely , for @xmath214 and @xmath215 ) , giving @xmath216 this again disagrees with the prediction of stable clustering that @xmath1 is constant , but it appears to be consistent with numerical simulations as shown in figures 3 and 4 . for yet higher order correlations , details of the halo profile begin to matter . for @xmath31 , the pattern of equations ( [ g2halo ] ) and ( [ g3halo ] ) persists to all orders , but for @xmath214 they apply only for the two- and three - point functions ; for four - point and higher functions the nonlinear scale @xmath217 and @xmath218 for @xmath219 . thus there seems to be some potentially interesting behavior that is tested only in the four - point function and higher . we have presented an analytic model for the two- and three - point correlation functions @xmath17 and @xmath19 of the cosmological mass density field and their fourier transforms , the mass power spectrum @xmath18 and the bispectrum @xmath20 . in this model , the clustering statistics of the density field are derived from a superposition of dark matter halos with a given set of input halo properties . these input ingredients include realistic halo density profiles of , halo mass distribution of , and halo - halo spatial correlations of equations ( [ pbias ] ) and ( [ bbias2 ] ) . the main results of the model are given by equations ( [ xi2 ] ) and ( [ pk ] ) for the two - point statistics @xmath97 and @xmath0 , and by equations ( [ zeta_123 ] ) and ( [ bk ] ) for the three - point statistics @xmath128 and @xmath129 . this model provides a rapid way to compute the correlation functions over all length scales where the model inputs are valid ; it also gives a physical interpretation of the clustering process of matter in the universe . we have tested the validity of this model by comparing its predictions with results from cosmological simulations of an @xmath2 scale - free model and a @xmath21cdm model . as figures 3 6 illustrate , the model describes well the simulation results spanning the entire range of behavior from the perturbative regime on large scales to the strongly nonlinear regime on small scales . to probe the critical high-@xmath93 range in the deeply nonlinear regime , we have used a halo replacement technique to increase the resolution of the large parent simulations . as figures 1 and 2 illustrate , this method of replacing the original halos that suffer from numerically softened cores with synthetic halos of analytic profiles is a reasonable way to improve the resolution of numerical simulations . by using density profiles obtained empirically from higher - resolution simulations of individual halos , this recipe allows us to model accurately the inner regions of the halos on scales below the numerical softening length scale , while at the same time preserving all the large - scale information available in the large parent simulation . this technique should also be useful for other studies that depend on the inner halo density profiles , for example , the ray - tracing method in gravitational lensing . given that dark matter halos in simulations ( and presumably in nature ) are not perfectly spherical , cleanly delineated objects , it is intriguing that the model constructed in this paper works as well as it does at matching the simulation results . nevertheless , this analytic model provides a good qualitative and quantitative description over the entire range of scales covered by the simulation , and it can be used to make predictions beyond these scales . this is the first model prescription that successfully reproduces both two- and three - point mass correlations . we believe that it will prove to be a generally useful framework . we have enjoyed stimulating discussions with john peacock and david weinberg . we thank edmund bertschinger for valuable comments and for providing the @xmath2 scale - free simulation . computing time for this work is provided by the national scalable cluster project and the intel eniac2000 project at the university of pennsylvania . m. acknowledges support of an alfred p. sloan foundation fellowship , a cottrell scholars award from the research corporation , a penn research foundation award , and nsf grant ast 9973461 . in this appendix we display analytic forms for the convolution of the dimensionless profile shape @xmath220 discussed in 3 . these analytic expressions are useful for computing the nonlinear two - point correlation function @xmath97 of the mass density field , which is dominated by the 1-halo term @xmath112 in and is related to @xmath114 by @xmath221 for the type - i profile @xmath222 of , the angular integration in equation ( [ lamb2 ] ) is analytic , and @xmath114 is reduced to a simple integral @xmath223 \ , . \label{lamb1a}\ ] ] for the special case @xmath31 , this integral can be further reduced to the analytical form @xmath224\ , , \qquad p=1 \ , . \label{lamb1b}\ ] ] for @xmath225 of , we are able to simplify @xmath114 to @xmath226 where the function @xmath227 represents the angular part of the integration in equation ( [ lamb2 ] ) and @xmath228 the integral in @xmath229 can be reduced to analytic forms for special values of @xmath230 . here we display the six cases @xmath231 , @xmath232 , 1 , @xmath233 , 2 , and @xmath234 : @xmath235 + \ln \left [ 1 - ( x+y ) + ( x+y)^2 \over 1 + 2(x+y ) + ( x+y)^2 \right ] \right\ } \nonumber \\ & & \qquad -{1\over 6 } \left\ { \hbox{replace $ ( x+y)$ above with $ |x - y|$ } \right\ } \\ \noalign{\smallskip } f_{1/2 } & = & { 1 \over 10 } \left\ { -2 \sqrt{10 + 2\sqrt5 } \ , \tan^{-1 } \ ! \left ( { 1 + \sqrt5 - 4 \sqrt{x+y } \over \sqrt{10 - 2\sqrt5)}}\right ) \right . \nonumber \\ & & \qquad -2 \sqrt{10 - 2 \sqrt5 } \ , \tan^{-1 } \ ! \left ( { -1 + \sqrt5 + 4 \sqrt{x+y } \over \sqrt{10 + 2\sqrt5)}}\right ) \nonumber \\ & & + 4 \ln\left ( 1 + \sqrt{x+y } \right ) - ( 1 + \sqrt5 ) \ln\left [ 1 + { 1\over 2 } ( -1 + \sqrt5 ) \sqrt{x+y } + x+y \right ] \nonumber \\ & & \left . - ( 1 - \sqrt5 ) \ln \left [ 1 - { 1\over 2 } ( 1 + \sqrt5 ) \sqrt{x+y } + x+y \right ] \right\ } \nonumber \\ & & \qquad -{1\over 10 } \left\ { \hbox{replace $ ( x+y)$ above with $ |x - y|$ } \right\ } \\ \noalign{\smallskip } f_1 & = & \tan^{-1}(x+y ) - \tan^{-1}(|x - y| ) \\ \noalign{\smallskip } f_{3/2 } & = & { 1\over 3 } \left\ { 2\sqrt{3 } \tan^{-1 } \left[{-1 + 2\sqrt{x+y } \over \sqrt{3 } } \right ] + \ln \left [ 1 + 2 \sqrt{x+y } + x+y \over 1 - \sqrt{x+y } + x+y \right ] \right\ } \nonumber\\ & & \qquad -{1\over 3 } \left\ { \hbox{replace $ ( x+y)$ above with $ |x - y|$ } \right\ } \\ \noalign{\smallskip } f_2 & = & \ln \left [ { x+y \over 1+x+y } \right ] - \ln \left [ { |x - y| \over 1+|x - y| } \right ] \\ \noalign{\smallskip } f_{5/2 } & = & { 2 \over \sqrt{|x - y| } } - { 2 \over \sqrt{x+y } } + \ln\left [ { ( 1 + 2\sqrt{x+y } + x+y ) \ , |x - y| \over ( 1 + 2\sqrt{|x - y| } + |x - y| ) \ , ( x+y)}\right ] \end{aligned}\ ] ]

we present an analytic model for the fully nonlinear power spectrum @xmath0 and bispectrum @xmath1 of the cosmological mass density field . the model is based on physical properties of dark matter halos , with the three main model inputs being analytic halo density profiles , halo mass functions , and halo - halo spatial correlations , each of which has been well studied in the literature . we demonstrate that this new model can reproduce the power spectrum and bispectrum computed from cosmological simulations of both an @xmath2 scale - free model and a low - density cold dark matter model . to enhance the dynamic range of these large simulations , we use the synthetic halo replacement technique of ma & fry ( 2000 ) , where the original halos with numerically softened cores are replaced by synthetic halos of realistic density profiles . at high wavenumbers , our model predicts a slope for the nonlinear power spectrum different from the often - used fitting formulas in the literature based on the stable clustering assumption . our model also predicts a three - point amplitude @xmath1 that is scale dependent , in contrast to the popular hierarchical clustering assumption . this model provides a rapid way to compute the mass power spectrum and bispectrum over all length scales where the input halo properties are valid . it also provides a physical interpretation of the clustering properties of matter in the universe .

astro-ph0003343

the interplay between spin density wave ( sdw ) ordering and superconductivity clearly plays a central role in the physics of a variety of quasi two - dimensional correlated electron materials . this is evident from recent studies of the phase diagram of the ferro - pnictides @xcite and the ` 115 ' family of heavy - fermion compounds @xcite . in the cuprates , it has been argued that @xmath4-wave superconductivity is induced by sdw fluctuations in a metal @xcite , and this has been the starting point for numerous studies of the complex phase diagram @xcite . in all these materials , there is a regime of co - existence between sdw ordering and superconductivity , and this opens the way to a study of the ` competition ' between these orders @xcite : this competition can be tuned by an applied magnetic field , as has been studied in a number of revealing experiments @xcite on the lsco and ybco series of superconductors . this paper will discuss a question that arises naturally in the study of such competing orders @xcite . we consider , first , the ` parent ' quantum critical point as that associated with the onset of sdw order , @xmath5 , in a metal . to access this point we have to suppress superconductivity in some manner , say by the application of a magnetic field . this parent critical point will occur at a value @xmath6 of some tuning parameter @xmath7 , which could be the carrier concentration or the applied pressure . we define @xmath7 so that @xmath8 is the sdw phase with @xmath9 ; see fig . [ totalshift ] . the value of @xmath6 is clearly material specific , and will depend upon numerous microscopic details . then , we turn our attention to the onset of sdw order within the superconductor ( sc ) ; we characterize the latter by a gap amplitude @xmath1 , and denote the critical value of @xmath7 by @xmath10 . the essence of the picture of competing orders is that the onset of superconductivity should shrink the region of sdw order , and hence @xmath11 . we will be interested here in particular in the magnitude of the shift @xmath12 . we will see that the shift is dominated by low energy physics , and so has a universal character . this shift @xmath12 played a central role in the phase diagrams presented in refs . , and applied to the cuprates . recent work has shown that similar phase diagrams also apply to the pnictides @xcite and the 115 compounds @xcite . in the pnictides , a `` backbending '' of the onset of sdw order upon entering the sc phase , consistent with the idea of @xmath13 . are for the sdw and the nematic phase transitions . the critical points in the metal are at @xmath14 , and under superconductivity , these shift to @xmath15 , towards the ordered phases . , title="fig:",width=3 ] are for the sdw and the nematic phase transitions . the critical points in the metal are at @xmath14 , and under superconductivity , these shift to @xmath15 , towards the ordered phases . , title="fig:",width=3 ] let us begin by computing the shift @xmath12 in mean - field landau theory . the simplest free energy of the sdw and sc order parameters has the form @xcite : @xmath16 here @xmath17 is the phenomenological parameter which controls the competition between the order parameters . examining the onset of a phase with @xmath18 in the superconductor with @xmath19 , we conclude immediately from eq . ( [ landau ] ) that @xmath20 such a shift was a key feature of the theory in ref . . the primary focus of the previous work was in the lower field region , where the superconductivity is well - formed , and @xmath1 is large . here it is appropriate to treat the superconductivity in a mean - field manner , and ignore pairing fluctuations , while treating spin fluctuations more carefully . the present paper turns the focus to higher fields , where eventually superconductivity is lost . here , clearly , landau theory can not be expected to apply to the superconducting order . moreover , we expect the fermi surface of the electrons to be revealed , and a more careful treatment of the electronic degrees of freedom is called for . one of the primary results of our paper will be that the landau theory result in eq . ( [ landaushift ] ) breaks down for small @xmath1 , and in particular in the limit @xmath21 . this is a consequence of the crucial importance of fermi surface physics in determining the position of the sdw transition at @xmath22 . instead , we will show from the physics of the `` hot spots '' on the fermi surface that the shift is larger , with @xmath23 for the competing order picture to hold , we require that @xmath24 . somewhat surprisingly , we will find that our results for @xmath25 are not transparently positive definite . different regions of the fermi surface contribute opposing signs , so that determining the final sign of @xmath25 becomes a delicate computation . in particular @xmath25 will depend upon the vicinity of ` hot spots ' on the fermi surface , which are special points connected by the sdw ordering wavevector . we will find that the immediate vicinity of the hot spots contributes a positive sign to @xmath25 , while farther regions contribute a negative sign . thus the primary competition between sdw and superconductivity happens at the hot spots , while other regions of the fermi surface which survive the onset of sdw order continue to yield an attraction between sdw and superconductivity . for the case where the two hotspots connected by the sdw ordering wavevector are equivalent under a lattice symmetry operation ( _ i.e. _ they have the same pairing gap and the same magnitude of the fermi velocity ) , we will find that distinct contributions to @xmath25 exactly compensate each other , so that @xmath26 . however , in the case that the two spots are not crystallographically equivalent ( which is the generic situation in both the cuprates and the pnictides ) , we will show that @xmath24 . a positive @xmath25 is indicated in fig . [ totalshift ] . we had considered the shift in sdw ordering due to superconductivity in a previous work @xcite . however , in that work , the metallic and superconducting states were not fermi liquids and bcs states respectively , but rather fractionalized states known as ` algebraic charge liquids ' @xcite . in this case , we found that the competition between sdw and superconductivity was robust , and always yielded a shrinking in the size of the sdw region . we will not consider such exotic states here , but work entirely within the framework of fermi liquid theory , in which the onset of superconductivity leads to a traditional bcs superconductor . in this context the interplay between sdw and sc in a fermi liquid is conveniently encapsulated in the ` spin - fermion ' model @xcite . we will find the same qualitative shift in the sdw critical point as found earlier @xcite , and the estimate in eq . ( [ sdwshift ] ) . we will also generalize our methods to analyze the shifts in the quantum critical points of other orderings between the metallic and superconducting phases . specifically , we will consider charge density wave ( cdw ) order and ising - nematic order , @xmath27 . we will find that the cdw shift initially appears to be formally similar to eq . ( [ sdwshift ] ) , but the coefficient @xmath25 is found to be exactly zero ; terms higher order in @xmath1 do indicate competition with superconductivity , but the cdw critical point shift is much smaller than the sdw s for ising - nematic order , @xmath27 , we will also find that the fermi surface result is similar to that in landau theory , as in eq . ( [ landaushift ] ) . this smaller shift is also illustrated in fig . [ totalshift ] . the weaker effect upon @xmath27 is due to its reduced sensitivity to the gap opened by superconductivity on the fermi surface . we will present a detailed discussion of the implication of these results for the pnictide and cuprate phase diagrams to section [ conclusion ] . however , let us highlight here an important inference that will follow from our computations . we will argue that the experimental phase diagrams imply that ising - nematic ordering and sdw ordering are independent instabilities of the fermi surface for the cuprates . in contrast , for the pnictides , our conclusion will be that the sdw ordering is the primary fermi surface instability , and the ising - nematic ordering is a secondary response to the square of the sdw order parameter . this paper is structured as follows . we will begin in section [ spin - fermion ] by introducing our starting point , the spin - fermion model with the sdw fluctuation . within the spin - fermion theory , we show that the sdw fluctuation induces the @xmath4 wave pairing for the cuprate and @xmath28 for the pnictides instead usual @xmath29 wave pairings in section [ instabilities ] . assuming the @xmath4 or @xmath28 wave pairings in each case , we extend the spin - fermion theory into the theory with pairing and other possible orders in section [ theory ] . in section [ qcpshift ] , we show the quantum critical point shifts toward the ordered phase , which explicitly shows the competition between superconductivity and the sdw phase . section [ conclusion ] presents our conclusions . we will study the system with sdw quantum phase transition in two dimensional system . the main ingredients of the spin - fermion model are fermi surfaces and the sdw order parameter . let us first consider generic microscopic hamiltonians for the cuprates and the pnictides . @xmath30 here , we consider `` minimal fermi surfaces '' for both materials , where @xmath31 correspond to the one band electron for the cuprates around the @xmath32 point , and for the pnictides , @xmath33 correspond to the hole and electron bands centered at @xmath34 and @xmath35 . all terms containing the sdw operators are in @xmath36 . in fig . [ hotspots ] , we illustrate typical hole - doping cuprates large fermi surface and pnictides two band fermi surfaces . to see general features , we consider incommensurate ordering wave - vectors for cuprates@xcite , @xmath37 for the pnictides , we represent the ordering wavevector as @xmath38 explicitly in the figure , but there is another hot spot with the ordering vector , @xmath39 . from now on , we set the lattice constant as a fundamental unit as usual . as it is well - known in the literature,@xcite the incommensurate sdw fluctuation is decribed by two complex vector wave functions . @xmath40\end{aligned}\ ] ] in the figure , two distinct hot spots are represented by filled and empty circles in the cuprate fermi surface . the filled one is farther from the nodal point than the empty one . note that the incommensurate sdw fluctuation links a filled circle with a empty circle . if we consider one special case , the commensurate sdw fluctuation , two kinds of hot spots become identical , and we only need one real @xmath41 field to describe the sdw fluctuation as usual . , and the pnictides have @xmath42 . in the upper panel , the filled and empty circles have different distances from the node , which means the gap magnitudes are different . two ordering wave vectors are represented by the dashed and thin lines in the upper panel and arrowed lines are in the bottom . for the pnictides , the electron band is distorted because there is no symmetry that guarantees identicalness of the hole and the electron band , which can also induces different gap functions.@xcite in both panels , every hot spot is numbered and the bar notation is used for the negative . , title="fig:",width=2 ] , and the pnictides have @xmath42 . in the upper panel , the filled and empty circles have different distances from the node , which means the gap magnitudes are different . two ordering wave vectors are represented by the dashed and thin lines in the upper panel and arrowed lines are in the bottom . for the pnictides , the electron band is distorted because there is no symmetry that guarantees identicalness of the hole and the electron band , which can also induces different gap functions.@xcite in both panels , every hot spot is numbered and the bar notation is used for the negative . , title="fig:",width=2 ] because the hot spots mainly contribute to the transition in the low energy theory , we write the electron annihilation operator as combination of the hot spot continuum fields as follows . for the cuprate , @xmath43 the @xmath44 fields represent the nearer and farther fields from the nodal points , so the sdw fluctuation mixes the @xmath45 and @xmath46 fields . the commensurate limit means @xmath45 and @xmath46 fields become identical . for the pnictides case , two bands can be described by hot spot fields as follows . @xmath47 as in the cuprates , the sdw fluctuation mixes @xmath45 and @xmath46 particles in this notation . we note that the electron band does not have the shape of circle , which makes the sdw possible , and there is no symmetry that guarantees the identity of the two bands . for convenience , we focus on the commensurate case in the remaining of this section , and the next two sections for describing spin - fermion models . but , later in the sec . [ qcpshift ] , we will come back to the general incommensurate cases and consider the critical point shifts in general . the spin - fermion model with commensurate sdw simply becomes @xmath48 where @xmath49 is non - zero constant when @xmath50 are connected by @xmath51 . the first and second lines describe the dynamics of the spin and fermion sectors . the third line is the `` yukawa '' coupling term . note that we explicitly include the second derivative kinetic term for the fermions in the theory , which becomes irrelevant if we only focus on the sdw phase transition . such term is not necessary for describing the sdw phase transition with non - collinear fermi velocities only , but the existence plays an important role in extending the theory to the one with pairing and nematic orders . note that the final form of the spin - fermion model is exactly the same in both the cuprates and the pnictides even though the microscopic band structure and the ordering vectors are completely different . this means the physics for the sdw transition is universal and we can focus on one case and apply the result to the other case . as usual , the effective action for the sdw phase transition is given by integrating out the fermions and expand with order parameters . @xmath52 |\varphi_a(k,\omega_n)|^2 \nonumber \\ & & + \frac{u}{4 ! } \int d^2 x d \tau ( \varphi_a^2(x,\tau))^2 \label{hertz}\end{aligned}\ ] ] this hertz - type theory is well - known and it describes the sdw fluctuation with dynamical critical exponent @xmath53 at least in the zeroth order . in this paper , we only focus on the critical point shifts rather than critical properties of the transition itself within the spin - fermion model , we can address pairing problems naturally . if we consider the sdw fluctuation as a pairing boson , then we need to investigate plausibility of the pairing instabilities by the sdw . the basic idea is following . if we assume there is infinitesimal pairing , then the pairing becomes enhanced or suppressed by the integrating out higher energy - momentum contributions depending on the possibility of the pairing channel . in the fig . [ vertex ] such a pairing vertex is illustrated . note that the fermions with opposite momentums are paired , so the participating fermions in the pairing is not the same as ones in the sdw in general . if we consider the @xmath29 wave channel in the cuprates and the @xmath54 channel for the pnictides , the pairing and its vertex correction are @xmath55 note that the relative sign between hot spots does not change , which is the main characteristic of the @xmath29 wave pairing . as we can see , the vertex becomes irrelevant in the low energy limit in the rg sense . therefore , the sdw fluctuation can not mediate usual @xmath29 wave pairing for the cuprates and @xmath54 for the pnictides . however , in the @xmath4 wave channel of the cuprates @xcite and the @xmath28 channel @xcite of the pnictides , the relative sign between the hotspots changes , given the pairing symmetries . such relative sign changes allow the pairing channel s enhancement . the pairing and its vertex correction are @xmath56 clearly , the alternative sign change induces enhancement of the @xmath4 , @xmath28 wave pairings in the low energy limit . therefore , the @xmath4 and @xmath28 pairing is natural under the sdw fluctuations rather than usual @xmath29 and @xmath54 pairings in conventional theory . in the next sections , we assume the existence of the pairings in each case and incorporate them into the spin - fermion model in a manner consistent with symmetry . moreover , by symmetry consideration , we also introduce other possible order parameters such as a nematic order and charge density wave and extend our theory to incorporate them . notice that the vertex correction in the eq . ( [ pairing ] ) is logarithmically divergent if we have finite correlation length . such behavior is a well - known signature of the conventional bcs theory . however , in the quantum critical region , the pairing boson is softened and the quantum fluctuation becomes important . the nature of this quantum critical pairing has been discussed in refs . . to extend the spin - fermion model to the one with pairing terms and nematic order parameter , let us consider microscopic symmetries and thier transformations because the square lattice symmetry should be respected in the low energy theory . hereafter , we analyze the symmetry in terms of the cuprate problem unless otherwise stated . it is easy to extend it to the pnictides case . due to the @xmath57 wave property , the pairing term s rotation and reflection needs additional factors . in appendix [ app : symmetry ] , we show the explicit transformation properties of fields and bilinear terms to avoid notation ambiguity . nambu spinors for particles are defined in a usual way : @xmath58 in table [ table1 ] , we summarize the transformation rules of the spinor fields . 2 .symmetry transformations of the spinor fields under square lattice symmetry operations . @xmath59 : translation by one lattice spacing along the @xmath60 direction ; @xmath61 : 90@xmath62 rotation about a lattice site ( @xmath63 ) ; @xmath64 : reflection about the @xmath65 axis ( @xmath66 ) ; @xmath67 : time - reversal , defined as a symmetry ( similar to parity ) of the imaginary time path integral and the conjugate fields are transformed to @xmath68 . note that such a @xmath67 operation is not anti - linear . also , the notation , @xmath69 is used for convenience . [ cols="^,^,^,^,^",options="header " , ] in this section , we set the notation for the symmetry transformation of the square lattice . mean field hamiltonian with the @xmath70 needs the pairing term as @xmath71 . @xmath72 the first line describes usual hopping terms on the square lattice , which gives the fermi surface . we exclude special ` nesting ' type fermi surfaces and assume there are points linked by the spin density wave ordering vector . we start with lattice field transformations . @xmath73 the rotation and reflection transformation attaches the factor @xmath74 , which makes the @xmath4 wave pairing term invariant . after writing the lattice fields with continuum field , we can obtain the transformation in table . [ spottable ] . it is worthwhile to mention that at low energy or long - wavelength scale the fields at hot spots can be treated as independent fields . time reversal symmetry is obtained with low energy fields instead of the lattice fields , so we do not consider it here . see the caption of the table . [ table1 ] . @xmath58 after introducing the nambu spinors , bilinear spinors transformations can be done easily . physical quantities are described with bilinear terms such as density and pairing interactions . below several important bilinear terms are listed up to constants . as we saw in the above , the critical point shift for the cdw is determined by the function , @xmath76 \nonumber \\ & & = \frac{1}{4 \pi^2 } \int d \theta d r \biggr [ \frac{1}{|\cos(\theta)|+|\sin(\theta)| } \nonumber \\ & & - \frac{r}{\sqrt{r^2 \cos^2(\theta)+1}+\sqrt{r^2 \sin^2(\theta)+1}}\nonumber \\ & & \times\biggl(1 + \frac{1}{\sqrt{r^2 \cos^2(\theta)+1}\sqrt{r^2 \sin^2(\theta)+1 } } \biggr ) \biggr ] \nonumber \\ & & = \lim_{\lambda \rightarrow \infty } \int d \theta f_{\lambda } ( \theta ) .\end{aligned}\ ] ] case formally . the blue , green , dark - yellow correspond to the cutoff @xmath80 , @xmath81 , @xmath82 . as we can see , the integrand is positive in any cases , but the numbers are getting smaller with larger cutoffs . , width=3 ] r. m. fernandes , d. k. pratt , w. tian , j. zarestky , a. kreyssig , s. nandi , m. g. kim , a. thaler , n. ni , s. l. budko , p. c. canfield , r. j. mcqueeney , j. schmalian , and a. i. goldman , phys . b * 81 * , 140501(r ) ( 2010 ) . b. lake , h. m. rnnow , n. b. christensen , g. aeppli , k. lefmann , d. f. mcmorrow , p. vorderwisch , p. smeibidl , n. mangkorntong , t. sasagawa , m. nohara , h. takagi , and t. e. mason , nature * 415 * , 299 ( 2002 ) . j. chang , ch . niedermayer , r. gilardi , n. b. christensen , h. m. rnnow , d. f. mcmorrow , m. ay , j. stahn , o. sobolev , a. hiess , s. pailhes , c. baines , n. momono , m. oda , m. ido , and j. mesot , phys . b * 78 * , 104525 ( 2008 ) . j. chang , n. b. christensen , ch . niedermayer , k. lefmann , h. m. rnnow , d. f. mcmorrow , a. schneidewind , p. link , a. hiess , m. boehm , r. mottl , s. pailhes , n. momono , m. oda , m. ido , and j. mesot , phys . lett . * 102 * , 177006 ( 2009 ) . e. demler , s. sachdev and y. zhang , phys . . lett . * 87 * , 067202 ( 2001 ) ; y. zhang , e. demler , and s. sachdev , phys . b. * 66 * , 094501 ( 2002 ) . r. k. kaul , a. kolezhuk , m. levin , s. sachdev , and t. senthil , phys . b * 75 * , 235122 ( 2007 ) . m. vojta , y. zhang , and s. sachdev , phys . lett . * 85 * , 4940 ( 2000 ) ; erratum * 100 * , 089904(e ) ( 2008 ) . m. vojta , y. zhang , and s. sachdev , int . b * 14 * , 3719 ( 2000 ) . m. d. prato , a. pelissetto , and e. vicari , phys . b. * 74 * , 144507 ( 2006 ) . a. pelissetto , s. sachdev , and e. vicari , phys . rev * 101 * , 027005 ( 2008 ) . e. a. kim , m. j. lawler , p. oreto , s. sachdev , e. fradkin , and s. a. kivelson , phys . b * 77 * , 184514 ( 2008 ) . y. huh and s. sachdev , phys . rev . b * 78 * , 064512 ( 2008 ) .

we compare the position of an ordering transition in a metal to that in a superconductor . for the spin density wave ( sdw ) transition , we find that the quantum critical point shifts by order @xmath0 , where @xmath1 is pairing amplitude , so that the region of sdw order is smaller in the superconductor than in the metal . this shift is larger than the @xmath2 shift predicted by theories of competing orders which ignore fermi surface effects . for ising - nematic order , the shift from fermi surface effects remains of order @xmath3 . we discuss implications of these results for the phase diagrams of the cuprates and the pnictides . we conclude that recent observations imply that the ising - nematic order is tied to the square of the sdw order in the pnictides , but not in the cuprates . = 10000

1005.3312

m _ spitzer_/irac image of ngc 5907 . spectra from @xmath9 10 to 37 @xmath0mwere taken at the nucleus and at distances of 5 , 10 , and 15 kpc from the nucleus along the galaxy s major axis.,width=377 ] the physical conditions and excitation mechanisms of atomic and molecular gas in the outer disks of nearby spiral galaxies are only beginning to be explored . most of this gas is thought to be neutral atomic hydrogen ( ) , or cold ( t @xmath10 50 k ) molecular hydrogen ( @xmath2 ) . the presence of cold @xmath2 is usually inferred only by indirect means via observations of carbon monoxide ( co ) , and quantified by assuming an ( uncertain ) empirical conversion factor between the two molecules . direct detection of @xmath2 is preferable . however , since @xmath2 has no allowed dipole radiative transitions , it has to be heated above @xmath9 100 k to radiate significantly via quadrupole pure - rotational transitions in the mid - infrared ( mid - ir ) or through ro - vibrational transitions from even warmer gas emerging in the near - infrared . furthermore , the other direct observational window the detection through the absorption of uv radiation in the electronic lyman werner bands is challenging , and only under rare conditions has it been possible to detect the presence of cold @xmath2 in the galaxy through fuv absorption ( e.g. , * ? ? ? * ; * ? ? ? * ) . the _ infrared space observatory ( iso ) _ provided the first opportunity to directly observe warm extragalactic molecular hydrogen in nearby galaxies , unhampered by the atmosphere ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? more recently the _ spitzer space telescope _ @xcite has provided a wealth of new data on rotational @xmath2 emission lines in dozens of nearby galaxies , ranging from normal galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) to ultraluminous infrared galaxies ( ulirgs ; * ? ? ? unusually strong intergroup @xmath2 emission associated with a large - scale ( @xmath9 30 kpc ) x - ray emitting shock has recently been found associated with the compact stephan s quintet galaxy group @xcite and the _ taffy galaxy _ bridge ( b. w. peterson et al . 2010 , in preparation ) . similarly large @xmath2 line fluxes have been found in 17 galaxies in a sample of 55 low - luminosity radio galaxies @xcite . in stephan s quintet and in the low - luminosity radio galaxies , very weak thermal continua are detected , suggesting shock excitation of @xmath2 , rather than excitation via photodissociation regions ( pdrs ) associated with star formation ( e.g. , * ? ? ? other sources of @xmath2 heating , for example cosmic ray heating , have also been suggested to explain the strong @xmath2 emission in the orion bar @xcite . it is therefore of interest to examine the strength of @xmath2 emission in the outer regions of nearby galaxies , where star formation and cosmic ray heating are much reduced , and where the dominant gas component is usually assumed to be neutral atomic hydrogen rather than molecular gas . by outer regions in this paper we mean the radii 10 kpc and beyond from the nucleus . _ iso _ observations of the nearby edge - on galaxy ngc 891 directly detected the abundant warm @xmath2 out to a distance of 11 kpc from the galaxy center @xcite , with warm @xmath2 mass surface densities of @xmath9 3000 m@xmath6 pc@xmath7 . this suggested a dominant contribution of molecular hydrogen to the mass - density of the disk , and perhaps that molecular hydrogen could contribute a significant part of the `` missing mass '' in this galaxy . intrigued by these early _ iso _ results , we pursued _ spitzer _ infrared spectrograph ( irs ; * ? ? ? * ) observations of two local , nearly edge - on galaxies , ngc 4565 and ngc 5907 , to explore the possibility of massive reservoirs of warm molecular gas far from the nuclei . these early - mission irs high resolution spectra cover infrared wavelengths from 10 @xmath0 m to 37 @xmath0 m , and target the 00 s(0 ) and 00 s(1 ) @xmath2 lines , which are known to contain the strongest emission from the mass in warm molecular gas in nearby galaxies ( e.g. , * ? ? ? the spectral range also covered several other mid - ir lines which assisted us in exploring the importance of star formation as an excitation mechanism in these regions . although the current observations are not extremely sensitive , they provide interesting constraints on the nature of @xmath2 emission in the outer disks of galaxies . to assist in our analysis , we also utilized _ spitzer _ infrared array camera ( irac ) 8 @xmath0 m images of ngc 4565 ( figure [ fig1 ] ) and ngc 5907 ( figure [ fig2 ] ) taken in _ program pid 3 ( p.i . giovanni fazio ; m. l. n. ashby 2009 , private communication ) . finally , we utilized archival _ spitzer _ multiband imaging photometer ( mips ) images of ngc 4565 and ngc 5907 at 24 , 70 , and 160 @xmath0 m ( figure [ fig3 ] shows these maps for ngc 4565 ) . ngc 4565 is a nearby ( we adopted a distance of 10 mpc for our observations ) , sb - type nearly edge - on ( inclination 88 ; @xcite @xcite ) large ( d@xmath11=158 ) disk galaxy with a nucleus classified as sy1.9 @xcite . a sharp dust lane delineates the disk plane of the galaxy , and there is significant obscuration caused by dust within the galactic plane . @xcite found that this galaxy has a nuclear molecular disk as well as a molecular gas ring at a distance of @xmath9 12 ( 36 kpc ) from the nucleus , and weaker extended molecular gas emission . the molecular gas ring has an associated dust ring , which is seen in the _ spitzer _ 8 @xmath0 m image shown in figure [ fig1 ] . the distribution is asymmetric along the disk plane , with substantially more emission coming from the northwestern side , and there is a strong , continuous warp in the emission starting at @xmath9 7 on both sides of the nucleus @xcite . @xcite showed that at a radius of @xmath9 10 kpc the interstellar medium ( ism ) transitions from being dominated by molecular gas to being dominated by atomic gas . ngc 5907 is a similarly large ( d@xmath11=126 ) , nearby ( adopted distance 11 mpc ) , almost edge - on ( inclination 87 ; @xcite @xcite ) , disk galaxy . co observations show a fast - rotating nuclear molecular disk with bar - like non - circular motions beyond the nucleus @xcite . the distribution shows a warp at both the southeastern and northwestern sides of the nucleus , starting at @xmath9 5 radius @xcite . cccccc + position & [ neii]12.81@xmath0 m & [ neiii]15.55@xmath0 m & [ siii]18.71@xmath0 m & [ siii]33.48@xmath0 m & [ siii]34.81@xmath0 m + + se 5 kpc ex & 187@xmath126 & 44@xmath125 & 65@xmath126 & 335@xmath1213 & 397@xmath1215 + se 5 kpc pt & 283@xmath129 & 73@xmath129 & 118@xmath1210 & 531@xmath1221 & 650@xmath1215 + se 10 kpc ex & 95@xmath125 & 28@xmath1212 & 48@xmath124 & 284@xmath126 & 234@xmath1213 + se 10 kpc pt & 144@xmath127 & 47@xmath1220 & 88@xmath128 & 386@xmath129 & 383@xmath1213 + se 15 kpc ex & 30@xmath127 & @xmath10 14 & 13@xmath123 & & + se 15 kpc pt & 45@xmath1211 & @xmath10 23 & 24@xmath125 & & + nw 5 kpc ex & 207@xmath126 & 40@xmath125 & 87@xmath123 & 411@xmath128 & 487@xmath127 + nw 5 kpc pt & 314@xmath129 & 67@xmath129 & 158@xmath126 & 652@xmath1213 & 799@xmath1211 + nw 10 kpc ex & 69@xmath123 & 41@xmath125 & 36@xmath123 & 199@xmath126 & 144@xmath125 + nw 10 kpc pt & 104@xmath125 & 68@xmath127 & 65@xmath125 & 316@xmath1210 & 236@xmath128 + nw 15 kpc ex & 28@xmath128 & 28@xmath128 & @xmath10 10 & 40@xmath125 & 69@xmath128 + nw 15 kpc pt & 43@xmath1212 & 47@xmath1213 & @xmath10 19 & 64@xmath128 & 113@xmath1213 + cccccccccc pt & 55@xmath129 & 322@xmath126 & 47@xmath123 & 354@xmath128 & 238@xmath1213 & 52@xmath127 & 431@xmath1211 & 726@xmath1213 & 867@xmath1221 + ex & 39@xmath126 & 213@xmath124 & 30@xmath122 & 212@xmath125 & 131@xmath127 & 37@xmath125 & 302@xmath128 & 457@xmath128 & 529@xmath1213 + lccc nucleus ex & 189@xmath125 & 170@xmath127 & 29@xmath126 + nucleus pt & 281@xmath128 & 298@xmath1214 & 43@xmath1210 + se 5 kpc ex & 176@xmath123 & 121@xmath124 & @xmath10 12 + se 5 kpc pt & 261@xmath125 & 213@xmath127 & @xmath10 17 + se 10 kpc ex & 88@xmath121 & 52 @xmath122 & @xmath10 17 + se 10 kpc pt & 130@xmath122 & 92@xmath124 & @xmath10 25 + se 15 kpc ex & & @xmath10 14 & @xmath10 16 + se 15 kpc pt & & @xmath10 25 & @xmath10 23 + nw 5 kpc ex & 221@xmath124 & 83@xmath123 & @xmath10 13 + nw 5 kpc pt & 327@xmath126 & 145@xmath125 & @xmath10 19 + nw 10 kpc ex & 89@xmath125 & 36@xmath123 & @xmath10 19 + nw 10 kpc pt & 132@xmath127 & 64@xmath125 & @xmath10 27 + nw 15 kpc ex & 36@xmath122 & @xmath10 13 & @xmath10 13 + nw 15 kpc pt & 53@xmath123 & @xmath10 22 & @xmath10 18 + we observed ngc 4565 and ngc 5907 with both high resolution modules of _ spitzer s _ irs instrument on 2005 january 10 and 2005 june 6 , respectively ( pid 3319 ; and ) . the short - high ( sh ) module covers wavelengths from 9.9 to 19.6 @xmath0 m and has a slit size of 47 @xmath13 113 , while the long - high ( lh ) module brackets the 18.7 to 37.2 @xmath0 m wavelength range with a slit size of 111 @xmath13 223 . we took one cycle of `` staring mode '' observations with a 120 s ramp time with the sh module and a 240 s ramp time with the lh module . the effective integration times were approximately doubled to @xmath9 240 s ( sh ) and @xmath9 480 s ( lh ) because each cycle takes two spectra , moving the target to positions 1/3 and 2/3 slit lengths away from the end of the slit along the slit long axis . we observed three positions along the galaxy major axes on both sides of the nuclei at distances of 5 , 10 , and 15 kpc from the nucleus . we also observed the nucleus of ngc 4565 . projections of the sh and lh slits on the 8 @xmath0 m irac galaxy images are shown in figures [ fig1 ] and [ fig2 ] . by overlaying our observed positions on visible light and maps we confirmed that the gaseous and stellar warps start beyond the outermost observed locations in these two galaxies . only in the northwest 15 kpc pointing in ngc 5907 could a very small amount of @xmath2 have been missed if it strictly follows the distribution . however , even in that position the majority of emission comes from along the major axis of the galaxy . cccccc se 5 kpc ex & 187@xmath125 & 33@xmath122 & 70@xmath124 & 386@xmath1221 & 553@xmath129 + se 5 kpc pt & 284@xmath127 & 55@xmath124 & 127@xmath127 & 613@xmath1233 & 906@xmath1214 + se 10 kpc ex & 48@xmath122 & 21@xmath123 & 30@xmath122 & 197@xmath1214 & 201@xmath127 + se 10 kpc pt & 72@xmath123 & 35@xmath125 & 54@xmath124 & 312@xmath1222 & 329@xmath1211 + se 15 kpc ex & @xmath10 12 & @xmath10 13 & @xmath10 13 & @xmath10 21 & @xmath10 31 + se 15 kpc pt & @xmath10 18 & @xmath10 22 & @xmath10 24 & @xmath10 34 & @xmath10 51 + nw 5 kpc ex & 168@xmath128 & 31@xmath125 & 50@xmath122 & 379@xmath1212 & 481@xmath124 + nw 5 kpc pt & 254@xmath1212 & 52@xmath128 & 90@xmath124 & 602@xmath1219 & 789@xmath126 + nw 10 kpc ex & 28@xmath122 & 8@xmath120.4 & 13@xmath122 & 115@xmath1210 & 146@xmath124 + nw 10 kpc pt & 43@xmath123 & 14@xmath120.6 & 24@xmath123 & 182@xmath1216 & 239@xmath127 + nw 15 kpc ex & @xmath10 14 & @xmath10 7 & @xmath10 18 & @xmath10 22 & @xmath10 32 + nw 15 kpc pt & @xmath10 21 & @xmath10 12 & @xmath10 33 & @xmath10 35 & @xmath10 53 + the background brightnesses ( due to ecliptic emission ) were @xmath9 29 mjy sr@xmath8 for ngc 4565 , and 1718 mjy sr@xmath8 for ngc 5907 . no separate background spectra were taken , since the recommended observing strategy during the first cycle of _ spitzer _ observations was still evolving and no clear recommendations existed at the time . this considerably complicated the removal of bad pixels from the spectra , discussed below . we did not use the `` peak - up '' option since our targets are extended , but the intrinsic irs pointing accuracy of @xmath9 1@xmath14@xmath14 was sufficient for our purposes . we used spectra that were processed through the standard _ spitzer _ irs pipeline ( version s13.2.0 ) . we first edited the basic calibrated data frames to remove bad or `` rogue '' pixels , using a custom - made software script that allows interactive removal of isolated bad pixels from these data . we then ran the spectra through the s17 version of the custom spectral extraction software spice provided by the _ spitzer _ science center , using the whole slit width extractions and initially the standard point source calibration for flux calibration . corrections were later made to line fluxes for both point and extended source calibration using the slit - loss factors provided in spice , and these two extreme limits bracket the true ( unknown ) distribution of gas in the slits . a comparison of the @xmath2 28 @xmath0 m line fluxes for the two nod positions in both galaxies shows differences of @xmath1010% , suggesting the gas distribution is relatively smooth on the size scale of the nods ( five and nine arcseconds , respectively , for sh and lh slits ) . extended emission is also suggested by structure along the slit in individual images . we consider the distribution of the emission - line gas to be close to flat and extended , and apply the corresponding calibration , but also quote , for reference , the very unlikely values derived by applying the point source calibration , when interpreting the properties of the observed galaxies . for the lh spectra of ngc 4565 , we encountered a low - level `` fringing '' effect not seen in the spectra of ngc 5907 . this effect would normally have been removed had we obtained dedicated `` off '' observations ( not obtained during our cycle-1 observations ) , and appears to be the result of incomplete `` jail bar '' removal in the pipeline ( an effect in which parts of the detector array show a patterning , which in this case was brighter than usual ) . to remove this effect , we decided to use the observations taken at 15 kpc southeast ( se ) of the nucleus of ngc 4565 as a reference , and subtracted this lh spectrum from all the others . this led to a significant improvement in the spectra . nonetheless , such a procedure runs the risk that there may have been faint emission at that reference position which would be removed from all other lh points ( this primarily affects the @xmath2 00 s(0 ) line which lies in the lh module . however , unlike the 15 kpc northwest ( nw ) point , which clearly shows 28 @xmath0 m s(0 ) emission even before we performed the subtraction , the se point appears devoid of emission as can be seen in figure [ fig4 ] . we feel confident , therefore , that the use of the se 15 kpc spectrum as a reference has not adversely affected our conclusions . although faint emission might have been present in this reference spectrum , based on measurements of the raw spectrum of ngc 4565 at the 15 kpc se position , we believe that an rms upper limit to such emission is 1.2 @xmath13 10@xmath15 w m@xmath7 hz@xmath8 at the position of the 00 s(0 ) line , corresponding to less than 10@xmath16 of the faintest @xmath2 emission detected at the 15 kpc nw point . in other words , the subtraction of the reference spectrum from the observations introduces a systematic error ( not to be confused with a random error ) which is estimated to be less than 10% of the faintest emission detected , a result that does not affect the conclusions of this paper . none of the sh spectra were affected by this instrumental effect . finally , we combined ( by averaging ) the spectra obtained at the two nod positions at each separate radial distance . the final spectra are shown in figures [ fig5 ] and [ fig6 ] . the 00 s(0 ) and 00 s(1 ) transitions of @xmath2 were detected at the 5 and 10 kpc distances from the nucleus in both galaxies . @xmath2 was also detected at the northwest 15 kpc location in ngc 4565 . on this side of the galaxy there is also substantially more emission @xcite . the 00 s(2 ) transition of @xmath2 was also detected in the seyfert nucleus of ngc 4565 . a variety of forbidden lines ( [ ] 12.81 @xmath0 m , [ ] 15.55 @xmath0 m , [ ] 18.71/33.48 @xmath0 m , and [ ] 34.82 @xmath0 m ) were detected in most locations of both galaxies . the continuum emission from the nucleus of ngc 4565 appears relatively flat , although it shows the broad pah emission feature around 17 @xmath0 m . because the irs apertures cover several hundred parsecs , most of this pah emission is likely emitted by the disk of this galaxy . in addition to the forbidden lines detected elsewhere in this galaxy , [ ] 10.51 @xmath0 m , [ ] 14.32/24.31 @xmath0 m , and [ ] 25.89 @xmath0 m were detected in the nucleus . the indicator lines of active galactic nuclei ( agns ) , [ ] 14.32 @xmath0 m and [ ] 24.31 @xmath0 m ( e.g. , * ? ? ? * ) , were both detected at a signal - to - noise ratio of @xmath910 . we extracted line fluxes by fitting gaussians to the lines using the smart software package @xcite . we fitted the broad aromatic features ( indicated in figures [ fig5 ] and [ fig6 ] ) with lorentzian profiles ( cf . the lorentzian method used by * ? ? ? * ) . the extracted fluxes are given in tables [ table1a ] , [ table1b ] , [ table1c ] , [ table2a ] , and [ table2b ] , where `` pt '' indicates point source calibrated spectra and `` ex '' indicates fluxes corresponding to a flat , infinitely extended distribution , as explained in detail in section 2 . the uncertainties were estimated by taking into account the quality of the profile fit . generally high signal - to - noise ratios ( @xmath17 10 ) were obtained for most of the lines except in the outer regions of the disks . upper limits were estimated as 4 @xmath18 , where @xmath19 is the rms noise in the region of the expected line and @xmath20@xmath21 is the width of an unresolved line at the corresponding wavelength ( which corresponds essentially to the width of the bandpass for the high resolution modules , @xmath21/600 ) . to assist in diagnosing the gas excitation conditions , we also estimated the flux densities in the irac 8 @xmath0 m images of ngc 4565 and ngc 5907 , under the areas covered by the irs slits in our observations . we used the same irac 8 @xmath0 m filter width as @xcite to convert the flux densities from jy into fluxes in w m@xmath7 , but we have not attempted to subtract the stellar emission from the irac image as it is generally only a few per cent of the total emission at 8 @xmath0 m . cccc se 5 kpc ex & 180@xmath127 & 92@xmath125 & @xmath10 16 + se 5 kpc pt & 267@xmath1210 & 162@xmath128 & @xmath10 24 + se 10 kpc ex & 51@xmath122 & 18@xmath122 & @xmath10 12 + se 10 kpc pt & 75@xmath123 & 32@xmath123 & @xmath10 18 + se 15 kpc ex & @xmath10 17 & @xmath10 9 & @xmath10 9 + se 15 kpc pt & @xmath10 25 & @xmath10 16 & @xmath10 14 + nw 5 kpc ex & 188@xmath123 & 104@xmath123 & @xmath10 13 + nw 5 kpc pt & 278@xmath124 & 182@xmath126 & @xmath10 20 + nw 10 kpc ex & 51@xmath123 & 15@xmath122 & @xmath10 15 + nw 10 kpc pt & 76@xmath124 & 27@xmath123 & @xmath10 22 + nw 15 kpc ex & @xmath10 18 & @xmath10 13 & @xmath10 16 + nw 15 kpc pt & @xmath10 26 & @xmath10 23 & @xmath10 24 + to investigate the gas excitation conditions we compared the strengths of various emission lines by forming line ratios . when comparing line ratios formed from lines taken with two different modules ( lh and sh ) , we scaled the fluxes by a factor of 4.66 , the ratio of the areas of the two module apertures . we also applied the extended source calibration correction to the line fluxes before taking the ratio to be consistent with this approach . we show the line ratios in figures [ fig7 ] and [ fig8 ] . in ngc 4565 the [ ] 33.48 @xmath0m/ [ ] 18.71 @xmath0 m ratios are close to 1 , typical for extranuclear regions seen in the sings sample of nearby galaxies @xcite , except at 10 kpc se where the ratio drops below 0.4 . this would imply a drop in the electron density by factors of a few hundreds @xcite . this ratio is slightly higher , between 1 and 2 , in ngc 5907 , covering very well the region in which most of the extranuclear areas studied by @xcite fall . the [ ] 34.81 @xmath0m/ [ ] 33.48 @xmath0 m ratio in ngc 4565 has a surprisingly low value of just above 1 at the nucleus , which is at the lower end of values seen in the nuclei of agn galaxies in the sample of @xcite . lcccc nuc & 164 ( 122 ) & 2.6 ( 2.0 ) & 4.1 ( 59 ) & 6.6 ( 95 ) + se 5 & 149 ( 112 ) & 2.5 ( 1.9 ) & 4.8 ( 74 ) & 7.7 ( 119 ) + se 10 & 146 ( 110 ) & 2.4 ( 1.8 ) & 2.4 ( 40 ) & 3.8 ( 64 ) + nw 5 & 135 ( 106 ) & 2.3 ( 1.7 ) & 7.5 ( 113 ) & 12 . ( 182 ) + nw 10 & 132 ( 104 ) & 2.3 ( 1.7 ) & 3.3 ( 51 ) & 5.2 ( 82 ) + nw 15 & @xmath10136 ( 103 ) & 2.3 ( 1.6 ) & 1.1 ( 21 ) & 1.8 ( 34 ) + lcccc se 5 & 139 ( 107 ) & 2.4 ( 1.8 ) & 5.8 ( 90 ) & 9.3 ( 145 ) + se 10 & 129 ( 101 ) & 2.2 ( 1.6 ) & 2.0 ( 32 ) & 3.3 ( 51 ) + nw 5 & 142 ( 109 ) & 2.4 ( 1.8 ) & 5.7 ( 89 ) & 9.2 ( 144 ) + nw 10 & 124 ( 100 ) & 2.1 ( 1.6 ) & 2.3 ( 33 ) & 3.6 ( 53 ) + the [ ] 15.55 @xmath0m/ [ ] 12.81 @xmath0 m ratio behaves as expected in both galaxies . it is higher in low - metallicity regions ( towards larger radii in both galaxies ) , and lower towards the center in regions that are expected to have a higher metallicity . it achieves a high value in the nucleus of ngc 4565 , consistent with what was seen by @xcite in the sings sample , presumably due to the higher excitation conditions near an agn . the @xmath2 s(0 ) to @xmath2 s(1 ) ratio hovers around 0.5 in both galaxies and is seen to increase towards the outer disk in both galaxies on both sides of the disk . this may primarily be an effect of the temperature , and it will be discussed in more detail in section [ excisection ] . we also show the @xmath2 s(0 ) to [ ] 33.48 @xmath0 m ratio . [ ] 33.48 @xmath0 m is mostly excited by star formation , and thus this ratio can be used as a rough indicator of the significance of star formation induced excitation of the @xmath2 molecule . we see that the ratio stays fairly constant at around 0.5 in both galaxies , but goes up at the 15 kpc nw point in ngc 4565 . this is consistent with the ionization level of molecules dropping in the outermost disk , as discussed in section [ excitdiscussion ] below . figure [ fig9 ] shows the ratios of the fluxes in the 11.3 and 7.7 @xmath0 m pah features versus the distance from the nucleus on the nw side of ngc 4565 . the irac 8 @xmath0 m fluxes measured in the irac image within the sh aperture and at the same spatial locations as the spectra were used as a proxy for the 7.7 @xmath0 m pah flux . the ratio is increasing towards 15 kpc nw , which most likely implies that the ism is becoming less ionized towards the outer disk , as the 7.7 @xmath0 m pah feature consists of more ionized dust material than the 11.3 @xmath0 m pah feature ( e.g. , * ? ? ? * ) . we constructed excitation diagrams ( figures [ fig10 ] and [ fig11 ] ) from the @xmath2data in order to place constraints on the molecular gas properties . these diagrams plot the column density ( n@xmath22 ) of @xmath2 in the upper level of each transition , normalized by its statistical weight , versus the upper level energy e@xmath22 ( e.g. , * ? ? ? * ) , which we derived from the measured fluxes assuming local thermodynamic equilibrium for each position observed . both extended and point source flux distributions are shown for the detected lines . the extended source results include a wavelength - dependent slit - loss correction which makes the extended flux calibration differ from the point source flux calibration typically by a factor of 1.5 , and has a further geometrical correction of 4.66 for the different areas of the sh and lh slits . the grey area between the two limiting cases ( point and extended ) is the parameter space that most likely encompasses the actual case . however , we stress that the point source assumption is very unrealistic , as discussed in section [ observ ] and given the small slit aperture and the thickness of the disks . as we will see , this assumption also leads to unlikely low gas temperatures close to 100 k , and therefore high implied @xmath2 gas surface densities . for this reason we prefer to consider the extended source limit to be much closer to the actual situation , but the point source provides a useful ( although unrealistic ) boundary . the uncertainty in the @xmath2 properties is governed largely by this uncertainty in the slit loss corrections rather than the formal errors , which are quite small because the lines were all detected with quite high signal to noise ratios ( snr ; they vary from 10 to 50 in most cases ) . the solid lines indicate the best fits to the s(0 ) , s(1 ) , and s(2 ) data points assuming @xmath2 in thermal equilibrium with a single - temperature component ( we will discuss the consequences of relaxing this assumption below ) . the fits also assume a thermal equilibrium ratio for the ortho - to - para species ( o / p ) , as is reasonable if the density of the @xmath2 is above the critical density ( which , for the low j transitions , is typically @xmath9 100 mol @xmath23 ) , a condition probably satisfied in most cases . for temperatures less than @xmath9 300 k , this leads to o / p ratios @xmath10 3 . for example , at t = 115120 k , o / p = 2 for thermal equilibrium . however , it is far from clear that thermal equilibrium is appropriate in all cases . for example , if the excitation mechanism were a shock , then the passage of the shock could leave the @xmath2 molecules in a state where they do not have enough time to equilibrate . this is another source of uncertainty . for example , if o / p was 3 instead of 2 ( a case where the gas has not had time to come into thermal equilibrium ) , then this would change the calculated temperature from t = 120 k to 113 k with a corresponding increase in the total @xmath2 mass surface density . this uncertainty in the o / p ratio is comparable with the uncertainty in the clumpiness in the @xmath2 distribution which leads to the broad range of possible temperatures as shown in figures [ fig10 ] and [ fig11 ] . the single - temperature fits to these data are shown in each panel of the figures . outside the nucleus only an upper limit is available for the s(2 ) line . therefore , fitting more than one thermal component is not statistically justifiable the fits are the formal solutions . we note that the assumption that the source of @xmath2 is a point source always yields very low @xmath2 temperatures , bordering on becoming physically unreasonable . thus we believe that the warmer temperatures implied by the extended source calibration are more physically reasonable for the case of these edge - on galaxies . one exception is the nucleus of ngc 4565 , where a point - source assumption may be reasonable , as it contains a seyfert nucleus . based on these assumptions , tables [ excipars ] and [ excipars2 ] summarize the derived @xmath2 physical parameters for ngc 4565 and ngc 5907 , respectively : temperature ( k ) , equilibrium ortho / para ratio , column density of @xmath2 ( mol cm@xmath7 ) , and the mass surface density of @xmath2 ( m@xmath6 pc@xmath7 ) . the temperatures and mass surface densities for ngc 4565 and ngc 5907 are also shown in figures [ fig12 ] and [ fig13 ] . the gas is colder in the outer disk where the mass surface density is also lower , creating the apparent impression of a correlation between temperature and mass surface density . the derived ( extended source ) mass surface densities are more than 100 times smaller than those found in ngc 891 by @xcite . see section [ darkmatter ] for more discussion about the implication of the implied warm @xmath2 mass surface densities . if one adopts the extended source assumption , and excludes the nucleus of ngc 4565 which is significantly warmer , there is no obvious change in the fitted temperature with radius within the uncertainty from @xmath1 = 5 kpc to @xmath1 = 10 kpc on both sides of this galaxy ( t = 146149 k on the southeastern side and t = 132134 k on the northwestern side ) . even at @xmath1 = 15 kpc on the northwestern side , the upper limit to the s(1 ) flux provides a temperature limit which is at least consistent with a flat temperature distribution . the situation is different in ngc 5907 , where the outermost 10 kpc points seem more than 10 k cooler than those measured at 5 kpc . for this galaxy the radial temperature profile is also symmetric , unlike that in ngc 4565 . in the previous discussion we have made an assumption that a single - temperature model is reasonable . this is clearly not the case for the nucleus of ngc 4565 , where the 00 s(2 ) line was detected . the first panel of figure [ fig10 ] shows that the single - temperature fits do not pass through the s(2 ) point . indeed , in general , extragalactic sources almost always show a range of allowable temperatures and often a multiple - component fit is required . one consequence of fitting a multiple - temperature model is that the warmer component softens the slope of the fit in the excitation diagram . this means that the lower temperature component becomes even cooler , once a warm component is subtracted . to illustrate this we have fitted a two - component model to the nucleus of ngc 4565 and derived the following temperatures and column densities . instead of a single ( in this case point - like ) nuclear source with t = 122@xmath124 k and a column density n@xmath25 of 5.9 @xmath13 10@xmath26 mol cm@xmath7 , we obtain t(1 ) = 115@xmath123 k , n(1)@xmath25 = 7.4 @xmath13 10@xmath26 mol cm@xmath7 , and t(2 ) = 450550 k , n(2)@xmath25 = 34 @xmath13 10@xmath27 mol cm@xmath7 . the warmer component is less constrained because the error bar on the s(2 ) line is larger than that of the s(0 ) and s(1 ) lines . note that the effect in this case of relaxing the single - temperature model is to increase the cold component column density by 25% . the warmer component adds a negligible amount to the final column density . it is very likely that the nucleus of ngc 4565 is different from the disk because it contains a seyfert component which may contribute additional heating to the @xmath2emission . this is reflected in the generally higher single - temperature fits shown in figure [ fig10 ] . it may seem odd that the two - component fit gives a temperature for the cold component in the nuclear pointing that is colder than elsewhere in the disk . however , the thermodynamics of the @xmath2 molecule is likely to be very complex , involving a multi - phase medium with unknown heating and cooling conditions . it is also possible that the density distribution of the clouds near the nucleus is very different from elsewhere in the disk , and there may be more very dense cold clouds near the nucleus . with the spatial resolution afforded by the irs , we can not resolve this question . furthermore , magnetohydrodynamic shocks may be driven into a clumpy medium near the nucleus , which will lead to a range of temperatures . it should also be noted that it is impossible to estimate what the separate contributions of the disk and the nucleus are to the observed line fluxes in the nuclear pointings . thus , depending on the strength of a second or third component , the temperature of the coolest component is always lowered relative to a single - temperature fit - case . because we used only single - temperature fits , it is possible that we underestimated the total @xmath2 column density if warmer components were present , because a cooler @xmath2 temperature implies a larger total @xmath2 column density . since we have , in general , no information about a warmer component , we can not do more than fit a single - temperature component and accept a degree of uncertainty in the final @xmath2 column densities and masses . the line fluxes measured in the seyfert nucleus of ngc 4565 are listed in table [ table1b ] . the continuum appears relatively flat , although it shows a signature of the broad pah emission feature around 17 @xmath0 m . since the apertures are relatively large ( covering several hundreds of pc ) , a lot of this pah emission is likely to come from the disk of ngc 4565 . the 11.3 @xmath0 m and 12.9 @xmath0 m pah features are also strong , but weaker than in the spectra taken at 5 kpc from the nucleus . the detected emission lines come from @xmath2 , o , ne , s , and si . the agn indicator lines of [ ] 14.32 @xmath0 m and [ ] 24.31 @xmath0 m ( e.g. , * ? ? ? * ) are both detected at s / n @xmath9 10 . the @xmath2 s(0)/@xmath2 s(1 ) ratio reaches its minimum at the nuclear position ( see figure [ fig7 ] ) , implying the highest gas temperatures , as can also be seen in figure [ fig10 ] . the [ ] 15.55 @xmath0m/ [ ] 12.81 @xmath0 m ratio reaches a peak in the nucleus . the value of @xmath9 1 for this ratio indicates a moderate nuclear starburst @xcite . it is also consistent with the classification of ngc 4565 as a sy1.9 galaxy @xcite . in both galaxies , ngc 4565 and ngc 5907 , we see that the emission line intensities and the derived mass surface densities of @xmath2 emission ( as well as the intensity of the forbidden lines ) decrease with increasing radius , while the temperature decreases only slightly . also , the 20-cm radio continuum , for ngc 4565 shown in figure [ fig14 ] @xcite , decreases strongly towards the 15 kpc radius ( which in ngc 4565 is actually outside the detected radio continuum emission on the nw side ) . we calculated the ratio of the @xmath2 luminosity surface density over the total infrared ( tir ) emission luminosity surface density ( figure [ fig15 ] ) . tir was calculated as in equation ( 9 ) of @xcite over the lh slit area , measuring surface brightness values in the 8 , 24 , 70 , and 160 @xmath0 m spitzer irac and mips maps that were all smoothed to the resolution of the 160 @xmath0 m map , at the positions of the observed irs slits . this ratio is relatively constant with the radius at about 0.2%0.4% . this value is somewhat higher than the 0.05%0.1% typically seen in the sings sample , but since we could not match the resolution and aperture of the broad - band images , from which tir was estimated , to the single slit observations taken in the staring mode , such a bias is expected . when plotting the @xmath2 emission power over the irac 8 @xmath0 m power ( figure [ fig16 ] ) we see that the points in ngc 4565 and ngc 5907 lie generally above the star formation region points in @xcite . specifically , we see an increase in the ratio towards the outer 15 kpc nw point in ngc 4565 . we also see no change in the @xmath2 s(0)/11.3 @xmath0 m pah ratio ( figure [ fig17 ] ) on the northwestern side of the disk of ngc 4565 , but we see an increase in the 11.3 @xmath0m/7.7 @xmath0 m pah feature ratio ( figure [ fig9 ] ) from 10 kpc to 15 kpc . one explanation is that the pahs become more neutral in the lower uv excitation environment of the outer disk at 15 kpc nw in ngc 4565 . the possible change in pah excitation from ionized to neutral changes the relative strengths of the 11.3 @xmath0 m with respect to the 7.7 @xmath0 m pahs because the 11.3 @xmath0 m pah feature becomes more dominant as the pahs become more neutral . this might naturally explain why the 15 kpc nw point in figure [ fig15 ] stands out . it is not due to the @xmath2 emission becoming relatively stronger at 15 kpc , but due to the 7.7 @xmath0 m pah feature becoming weaker . we also measured the 24 @xmath0 m flux densities in the areas covered by the irs slits in ngc 4565 , and noticed that the 24 @xmath0 m flux density decreases with radius . since the 24 @xmath0 m emission is a relatively good proxy of the star formation intensity ( e.g. , * ? ? ? * ) , this implies that the uv flux intensity is decreasing with radius , therefore producing a more neutral ism at larger radii , consistent with our results derived from the pah flux ratio . @xcite have shown that there is an apparently strong coupling between the surface densities of neutral hydrogen and dark matter in spiral galaxies , with a significant and pronounced peak in @xmath28/@xmath29 @xmath9 9 . to see whether this holds in ngc 5907 , we used the multicomponent dynamical model of ngc 5907 by @xcite . for ease of calculation we approximated the dark matter distribution by a singular isothermal sphere . at a radial distance of 10 kpc from the nucleus we calculate a @xmath30 = 188 m@xmath6 pc@xmath7 . @xmath31 @xmath9 17.4 m@xmath6 pc@xmath7 @xcite and the ratio of the two is 10.8 . including the warm @xmath2 gas only reduces the dark matter to gas ratio to 9 . thus ngc 5907 appears to follow the relationship found by @xcite . at a radial distance of 15 kpc from the nucleus the warm @xmath2 is undetected . assuming , as suggested by the data , an extended , smooth emission distribution , @xmath32 @xmath33 3 m@xmath6 pc@xmath7 . using our approximation to the barnaby thronson model , @xmath30/@xmath32 @xmath34 42 . at 15 kpc , @xmath35 = 10.3 m@xmath6 pc@xmath7 and @xmath30/@xmath35 = 12.2 . it is therefore clear that the mass of the ism in ngc 5907 , including the mass of the warm molecular gas , is too small by more than an order of magnitude to account for the requisite dark matter . unfortunately there is no model of the mass distribution and dynamics of ngc 4565 similar to that of @xcite for ngc 5907 . the neutral atomic hydrogen properties were studied by @xcite , and the cold molecular gas properties by @xcite while visible light surface photometry of ngc 4565 was performed by @xcite . we assumed that the stars and the ism are confined to a thin disk and the dark halo can be described , again , by a singular isothermal sphere . the rotation curve is given by @xcite . @xcite showed that the optical disk is truncated at a radius of 24.9 kpc , comparable to where @xcite sees a warp . using a @xmath36-band luminosity of the old disk of 1.4@xmath1310@xmath37 l@xmath6 and a median value of 7.5 ( corrected for hubble constant h@xmath38 = 75 km s@xmath8 mpc@xmath8 ) for the mass to luminosity ratio of sab sb galaxies from @xcite , the mass of the luminous stellar disk is 10.5@xmath1310@xmath37 m@xmath6 . this may be an overestimate as some fraction of the mass quoted by @xcite is dark . the total mass of the neutral atomic hydrogen is 5.96@xmath1310@xmath39 m@xmath6 @xcite and that of the cold molecular hydrogen 2.4@xmath1310@xmath39 m@xmath6 @xcite . we assumed that the ism and the stars are confined to a thin disk and all components are truncated at 25 kpc . @xcite suggests that the velocity at the truncation radius is @xmath40 where @xmath41 is 25 kpc for ngc 4565 and @xmath42 is the total mass . beyond @xmath41=25 kpc the velocity is assumed to decline in a keplerian fashion . at a radial distance of 35 kpc , the observed circular velocity of the galaxy is about 214 km s@xmath8 @xcite . from the mass and velocity components of our model we find that the halo contributes a velocity of 150 km s@xmath8 to the system . we reflect these values back to a radius of 15 kpc at which we observed the most distant emission from warm @xmath2 , and we recalculate the mass surface densities . assuming a singular isothermal sphere for the dark matter , the mass surface density of dark matter is 86 m@xmath6 pc@xmath7 , the ratio of the mass surface densities of dark matter and warm @xmath2 ( with the much likelier smoothly distributed extended source emission calibration ) is @xmath34 86/1.8 = 48 , and that of dark matter to all of the ism components ( neutral atomic hydrogen , warm molecular hydrogen , and cold molecular hydrogen ) is @xmath34 15 . from our analysis it is clear that the mass surface densities of the warm molecular gas can not produce the observed rotation velocities at large radii in ngc 4565 and ngc 5907 . the `` missing mass '' in these two galaxies can not be accounted for by warm @xmath2 gas . the molecular gas at the 15 kpc nw point in the outer disk of ngc 4565 , as probed by the @xmath2 rotational lines , has roughly the same temperature and the same ratio of the @xmath2 to far - ir power as in the inner disk . however , the star formation rate at the 15 kpc nw point , traced by the mid - ir emission , has substantially decreased , compared to the inner disk . in other words , although the intensity of the exciting radiation field has been reduced ( as seen also in the change in the ratio of the ionized to neutral pah molecules and in a reduction in the tir intensity when comparing the 10 kpc nw and 15 kpc nw points ) , the molecular gas is heated to a similar temperature throughout the disk . therefore , something other than star formation may be heating the gas at the 15 kpc nw point . cosmic ray ( cr ) heating of the @xmath2 does not appear viable at the 15 kpc nw point because we see a dramatic decrease in the strength of the synchrotron radio continuum emission , which is a tracer of cosmic rays accelerated in the magnetic field of the galaxy , between the 10 and 15 kpc nw points , as shown in figure [ fig14 ] . the 15 kpc nw point lies outside the detectable signal in the radio continuum maps of @xcite . the upper limit of the 20-cm radio continuum ( 150 @xmath0jy / beam ; 3@xmath19 ) at the 15 kpc nw point suggests a difference of a factor of @xmath17 64 in the 20-cm radio continuum flux density between the 10 and 15 kpc points . this change is not reflected in the decrease in the @xmath2 line luminosity which is only a factor of 7 . however , despite the lack of detected radio continuum at the 15 kpc nw point , we can not completely rule out cr excitation . if we assume an equipartition of energy between crs in the disk and the magnetic energy density , the upper limit to the radio continuum flux density corresponds to an upper limit for the equipartition magnetic field strength of @xmath44 @xmath33 1 @xmath0 g , following the assumptions discussed in @xcite . this corresponds to a magnetic energy density ( and a comparable cr energy density ) of @xmath9 9.6 @xmath13 10@xmath45 ergs @xmath23 . for a canonical synchrotron lifetime in the mid - plane of 10@xmath46 yrs , the crs could potentially provide @xmath47 @xmath33 4.7 @xmath13 10@xmath48 w / kpc@xmath49 of power if such a population of crs existed below the detection limit of the radio continuum observations . interestingly , this is only a factor of two lower than the @xmath2line luminosity in the s(0 ) line at 15 kpc nw ( 7 @xmath13 10@xmath48 w / kpc@xmath49 ) , and therefore cr heating , although unlikely ( it would require very rapid deposition timescales of much less than 10@xmath46 yrs and high heating efficiency ) , can not be completely ruled out . we note that the equivalent @xmath50 and @xmath51 for the 10 kpc nw point in ngc 4565 are 3.3 @xmath0 g and 5.0 @xmath13 10@xmath52 ergs @xmath23 , respectively = 10 kpc and 5 kpc at @xmath1 = 15 kpc . ] , and the ratio of @xmath51/@xmath53(@xmath2 ) @xmath9 0.5 at that point . this suggests that within the radio continuum emitting disk of ngc 4565 , trapped cosmic rays can , in principle , provide energy to heat the @xmath2 in the disk ( again high heating efficiency would be needed ) . in those same regions , star formation , through pdr heating , provides a more likely channel for heating the @xmath2 gas ( see * ? ? ? the feasibility of cosmic ray heating at @xmath1 = 15 kpc can also be estimated using an independent ionization argument @xcite . if we assume that the cosmic rays heat the gas through partial ionization of the @xmath2 to h@xmath54@xmath55 , then the rate of ionization through cosmic ray heating must balance the rate of cooling of the @xmath2 per molecule . for a column density @xmath56 @xmath9 1.1 @xmath13 10@xmath57 mol cm@xmath7 and the observed luminosity in the s(0 ) line , we estimate the @xmath2 cooling per molecule to be 6.7 @xmath13 10@xmath58 w mol@xmath8 . @xcite estimate the cr heating rate per molecule to be 8 @xmath13 10@xmath59 @xmath60@xmath61 w , where @xmath60@xmath61 is the @xmath2ionization rate . under these assumptions , for cr heating to balance the @xmath2 cooling would require an ionization rate of @xmath9 10@xmath62 s@xmath8 . this value is comparable to that measured in the galactic center ( see * ? ? ? * ) , but is unlikely to be realized in the outer disk of ngc 4565 . this again suggests that cr excitation is an unlikely source of heating for the s(0 ) line at @xmath1 = 15 kpc unless conditions there are very unusual . the two remaining options for @xmath2 excitation in the outer disk are heating within extended pdr regions , or shock heating . we have already shown that the ratio of the @xmath2 power to the far - ir emission power is consistent with pdr heating ( in comparison with the models of * ? ? ? * ) , but it is not clear if this process works in the outer disk . indeed , it is very likely ( and observations with the _ herschel space observatory _ will help to resolve this issue ) that a large component of the tir flux we see in the outer disk of ngc 4565 comes from cirrus clouds heated by the general radiation field of the outer disk , and not from young stars . thus one is left with the puzzling result that the @xmath2 excitation remains constant to within a factor of two in the outer disk which does not contain a high concentration of young stars . widely distributed pdr regions around a smoothly distributed set of faint young stars may be responsible for the excitation , but this can not be demonstrated with our observations . another possible way of heating the @xmath2 is by shocks , perhaps through a recent passage of a disturbance through the outer disk of ngc 4565 . a recent model of how @xmath2 can be excited in a powerful shock propagating through a multi - phase medium in stephan s quintet has been presented by @xcite . however , this model was tuned to the specific problem of how to generate large amounts of power in the @xmath2lines in a 1000 km sec@xmath8 shock moving through a clumpy medium . to explain the emission in the outer regions of ngc 4565 via shocks would require considerably less energy input , but there is no obvious source of energy to drive the shocks . curiously , recent _ spitzer _ observations of a high - latitude cirrus cloud within the galaxy @xcite have revealed unusually strong @xmath2emission from regions that are clearly not associated with pdrs . shock heating is one possible explanation . these observations suggest that the excitation of @xmath2 in galaxy disks is not yet well understood . we have examined the excitation of gas , dust , and pahs and the physical conditions of the ism in two nearby normal edge - on disk galaxies , ngc 4565 and ngc 5907 , out to 15 kpc from the nucleus of each galaxy . our most important conclusions can be summarized as follows . 1 . we have detected the rotational 17 @xmath0 m s(0 ) and 28 @xmath0 m s(1 ) @xmath2 line transitions at 5 and 10 kpc , and most interestingly , the s(0 ) @xmath2 line at 15 kpc nw from the nucleus of ngc 4565 . we have discovered that in these two edge - on galaxies , ngc 4565 and ngc 5907 , the warm molecular gas temperature ( although uncertain ) and the ratio of the @xmath2 line luminosity surface density to the total infrared luminosity surface density are rather flat with radius . however , the active star formation rate , as measured by , e.g. , the 24 @xmath0 m emission , falls rapidly with radius . this result is potentially inconsistent with excitation of the @xmath2 emission by photodissociation regions in the outer disks of these galaxies . alternatives to the @xmath2 excitation in the outer disk are cosmic ray heating and shocks . based on the midplane radio continuum emission intensities , excitation by cosmic rays and photodissociation regions are both viable in the inner disk . however , in the outer disk the non - detection of radio continuum implies that cosmic rays are less important there . therefore , extended photodissociation regions or shocks can excite the emission at the outermost disk , as seen in ngc 4565 at the 15 kpc nw point . we see an increase of the 11.3 @xmath0m/7.7 @xmath0 m pah feature strength ratio ( where we used the irac 8 @xmath0 m band to be a proxy of the 7.7 @xmath0 m emission ) at the 15 kpc nw position in ngc 4565 . we also see that the summed @xmath2 line intensity over the 8 @xmath0memission intensity ratio increases at the same position . our interpretation is that the @xmath2 s(0 ) 28 @xmath0 m emission at the 15 kpc nw position may still be excited by ( weaker ) emission from photodissociation regions , coming from a more neutral medium at this large distance from the nucleus , as the strength of the 7.7 @xmath0 m pah feature , which traces more highly ionized dust , decreases with respect to the strength of the 11.3 @xmath0 m pah feature , which traces more neutral dust . the observations strongly suggest that the warm molecular gas is smoothly distributed . assuming such an extended distribution , the detected mass surface densities of warm molecular hydrogen are very low at large radii in both galaxies . it is very unlikely that this component of the ism contributes at any significant level to the `` missing mass '' in the outer regions of these two edge - on disk galaxies . the seyfert 1.9 nucleus in ngc 4565 revealed [ ] 14.32 @xmath0 m , [ ] 24.31 @xmath0 m , [ ] 10.51 @xmath0 m , and [ ] 25.89 @xmath0mlines , as well as the 12.28 @xmath0 m @xmath2 s(2 ) line . the higher excitation forbidden lines are expected to be seen in seyfert nuclei . we are grateful to tom jarrett at ipac for helping us to correct the rogue pixels with his custom - made iraf script . we thank the anonymous referee for very helpful and detailed comments . we acknowledge stimulating discussions with eric murphy on the mid - ir and far - ir properties of nearby galaxies . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . smart was developed by the irs team at cornell university and is available through the _ spitzer _ science center at caltech . the irs was a collaborative venture between cornell university and ball aerospace corporation funded by nasa through the jet propulsion laboratory and ames research center . this work is based on observations made with the _ spitzer space telescope _ , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . support for this work was provided by nasa through an award issued by jpl / caltech .

we have observed warm molecular hydrogen in two nearby edge - on disk galaxies , ngc 4565 and ngc 5907 , using the _ spitzer _ high - resolution infrared spectrograph . the 00 s(0 ) 28.2 @xmath0 m and 00 s(1 ) 17.0 @xmath0 m pure rotational lines were detected out to 10 kpc from the center of each galaxy on both sides of the major axis , and in ngc 4565 the s(0 ) line was detected at @xmath1 = 15 kpc on one side . this location is beyond the transition zone where diffuse neutral atomic hydrogen starts to dominate over cold molecular gas , and marks a transition from a disk dominated by high surface - brightness far - ir emission to that of a more quiescent disk . it also lies beyond a steep drop in the radio continuum emission from cosmic rays in the disk . despite indications that star formation activity decreases with radius , the @xmath2 excitation temperature and the ratio of the @xmath2 line and the far - ir luminosity surface densities , @xmath3(l@xmath4)/@xmath3(l@xmath5 ) , change very little as a function of radius , even into the diffuse outer region of the disk of ngc 4565 . this suggests that the source of excitation of the @xmath2 operates over a large range of radii , and is broadly independent of the strength and relative location of uv emission from young stars . although excitation in photodissociation regions is the most common explanation for the widespread @xmath2 emission , cosmic ray heating or shocks can not be ruled out . at @xmath1 = 15 kpc in ngc 4565 , outside the main uv and radio continuum - dominated disk , we derived a higher than normal @xmath2 to 7.7 @xmath0 m pah emission ratio , but this is likely due to a transition from mainly ionized pah molecules in the inner disk to mainly neutral pah molecules in the outer disk . the inferred mass surface densities of warm molecular hydrogen in both edge - on galaxies differ substantially , being 4(60 ) m@xmath6 pc@xmath7 and 3(50 ) m@xmath6 pc@xmath7 at @xmath1 = 10 kpc for ngc 4565 and ngc 5907 , respectively . the higher values represent very unlikely point - source upper limits . the point source case is not supported by the observed emission distribution in the spectral slits . these mass surface densities can not support the observed rotation velocities in excess of 200 km s@xmath8 . therefore , warm molecular hydrogen can not account for dark matter in these disk galaxies , contrary to what was implied by a previous _ iso _ study of the nearby edge - on galaxy ngc 891 .

1007.4194

more than two decades after pioneering works @xcite , the phenomenology related to the deconfined phase of qcd , _ i.e. _ the quark - gluon plasma ( qgp ) is still a fascinating topic both experimentally and theoretically . on the experimental side , the qcd matter was or is studied in heavy - ion collisions ( rhic , sps , fair , lhc ) . these experiments seem to show that the qgp behaves like a perfect fluid . on the theoretical side , the study of qcd at finite temperature deserves also much interest because it is a challenging problem in itself and because of the many connections with experiments . the aim of this work is to study the thermodynamic features of qgp by resorting to a @xmath0-matrix approach . the power of this approach is that the bound states and scattering states of the system can be studied in a whole picture . such an approach has already proved to give relevant results in the study of hadronic matter above the critical temperature of deconfinement ( @xmath1 ) @xcite but has not yet been applied to compute the equation of state ( eos ) . this observable will be performed here thanks to the dashen , ma and bernstein s formulation of statistical mechanics in terms of the @xmath4-matrix ( or @xmath5-matrix ) @xcite . such a formulation is particularly well suited for systems whose microscopic constituents behave according to relativistic quantum mechanics . the qgp is indeed identified to a quantum gas of gluons and quarks , which are seen as the effective degrees of freedom propagating in the plasma . this assumption is actually common to all the so - called quasiparticle approaches @xcite , with the crucial difference that the use of a @xmath5-matrix formulation allows us to investigate the behavior of the qgp in a temperature range where it is strongly interacting . this strong interaction means here that bound states are expected to still survive above @xmath1 . although the above formulation can be applied to the full qgp , this paper is dedicated to the description of the gluon plasma . dealing with only one particle species simplifies drastically the problem while the main feature of the description , _ i.e. _ the explicit inclusion of interactions in a quasiparticle approach , is kept . moreover , the pure gauge thermodynamic features ( in particular , the eos ) are well - known in lattice qcd ; this will allow an accurate comparison between our phenomenological approach and the lattice qcd calculations . a particularity of this paper is the generalization of the formalism to any gauge groups , with a particular attention for su(@xmath2 ) and the large-@xmath2 limit , and for g@xmath6 . this group has originally attracted attention because , the center of g@xmath6 being trivial , models relating deconfinement to the breaking of a center of symmetry are no longer valid as for su(@xmath2 ) . however , it still exhibits a first - order phase transition as su(@xmath2 ) does @xcite . hence , g@xmath6 appears quite attractive from a theoretical point of view . the paper is organized as follows . ii is dedicated to the presentation of the general quasiparticle approach based on the @xmath5-matrix formalism proposed in @xcite . in sec . iii , the model is particularized to a yang - mills plasma with the inclusion of 2-body interactions and , in sec . iv , useful analytic comments concerning the thermodynamic observables in the su(@xmath2 ) and g@xmath6 cases are discussed . the model parameters are fixed in sec . v and the existence of the bound states inside the gluon plasma is discussed in sec . vi . in sec . vii , the computation of the eos is presented . finally , sec . viii is devoted to the conclusions and perspectives . the results of @xcite can be summarized as follows : the grand potential @xmath7 , expressed as an energy density , of an interacting particle gas is given by ( in units where @xmath8 ) . @xmath9.\ ] ] in the above equation , the first term , @xmath10 , is the grand potential of the free relativistic particles , _ i.e. _ the remaining part of the grand potential if the interactions are turned off . the second term accounts for interactions in the plasma and is a sum running on all the species , the number of particles included , and the quantum numbers necessary to fix a channel . the set of all these channels is generically denoted @xmath11 . the vectors @xmath12 and @xmath13 contain the chemical potentials and the particle number of each species taking part in a given scattering channel . the contributions above and below the threshold . ] @xmath14 are separated . below the threshold , one has @xmath15 the grand potential coming from bound states , seen as free additional species in the plasma and appearing as poles of the @xmath4-matrix . above the threshold , one has the scattering contribution , where the trace is taken in the center of mass frame of the channel @xmath11 and where @xmath16 is the @xmath4-matrix , depending in particular on the total energy @xmath17 . the symmetrizer @xmath18 enforces the pauli principle when a channel involving identical particles is considered , and the subscript @xmath19 means that only the connected scattering diagrams are taken into account . notice that @xmath20 is the modified bessel function of the second kind , that @xmath21 is linked to the temperature @xmath0 thanks to @xmath22 , and that the notation @xmath23 is used . by definition , @xmath24 , where @xmath25 is the off - shell @xmath0-matrix and where @xmath26 is the free hamiltonian of the system . a convenient way to compute @xmath25 is to solve the lippmann - schwinger equation for the off - shell @xmath0-matrix , schematically given by @xmath27 with @xmath28 the free propagator and @xmath29 the interaction potential . once the @xmath5-matrix is known , two problems can be simultaneously addressed : the existence of bound states in the plasma and its eos . the @xmath0-matrix formalism has the advantage of treating bound and scattering states on the same footing , and is particularly suited for the present situation where we expect bound states to become less and less bound when the temperature increases , eventually crossing over and melting into the continuum . this dissociation mechanism has been shown to provide considerable threshold enhancement effects in heavy quark anti - quark correlation functions @xcite . the plasma eos is obtained from ( [ pot0 ] ) . then , the pressure is simply given by @xmath30 and the other thermodynamic observables can derived from @xmath31 . for example , the trace anomaly ( @xmath32 ) and the entropy density ( @xmath33 ) read @xmath34 _ { { \cal v } \text{,}\ , \beta\vec \mu } , \quad s=-\beta^2\left [ \partial_\beta p\right]_{{\cal v } \text{,}\ , \vec \mu}\ ] ] where @xmath35 is the volume of the system . for later convenience , the thermodynamic quantities will be normalized to the stefan - boltzmann pressure , which is defined as @xmath36 @xmath37 being the masses of the particles propagating in the medium . the explicit computation of @xmath7 obviously requires the knowledge of the on - shell @xmath0-matrix that can be derived in particular from ( [ ls ] ) . a key ingredient of the present approach is thus the potential @xmath29 , encoding the interactions between the particles in the plasma . in the following , @xmath29 is chosen as pairwise : for a @xmath38-body channel , @xmath39 with @xmath40 the potential between two particle species @xmath41 and @xmath42 . having in mind the building of an effective framework describing the deconfined phase of a non - abelian gauge theory , each particle composing the plasma should be in a given representation of the considered gauge ( or color ) group . it is therefore reasonable to assume that the potential @xmath29 between two particles in the representations @xmath43 and @xmath44 of the considered gauge group has the color - dependence of a ( screened ) one - gluon - exchange potential , that is , in momentum space , @xmath45 where @xmath46 denotes the generator of the considered gauge algebra in the representation @xmath47 , and where the real function @xmath48 only depends on the temperature and two momenta ( no dependence on the mass or other attributes of the particle ) . we keep the name gluon for the gauge particle even if the gauge group can formally be arbitrary . in the above definition , it has to be remembered that @xmath49 and that @xmath50 , @xmath51 being the adjoint representation of the gauge group under study and @xmath52 being the value of the quadratic casimir in the representation @xmath47 . note that in the case of su(@xmath2 ) , @xmath53 is the t hooft coupling ( fixed in the large-@xmath2 limit ) . introducing quadratic casimirs , one can rewrite ( [ v00 ] ) as @xmath54 with @xmath55 the pair representation and @xmath56 again , the real function @xmath57 only depends on the temperature and on two momenta an explicit form for @xmath58 will be given later . the validity of the form ( [ v0 ] ) for @xmath40 has partially been checked in pure gauge su(3 ) lattice calculations , showing that the static potential between two sources , in different representations and bound in a color singlet , follows the casimir scaling expected from a process of one - gluon - exchange type @xcite . the peculiar scaling ( [ v0 ] ) also leads to a relevant large-@xmath2 behavior of the eos when the gauge group su(@xmath2 ) is chosen , as it will be shown in sec . [ sunt ] . among the various possible representations , the case where a singlet ( denoted @xmath59 ) appears in the tensor product @xmath60 is particularly relevant : since @xmath61 and the other quadratic casimirs are positive , the singlet is the most attractive channel in any two - body scattering process , so the most favorable one for the formation of bound states . such two - particle bound states should presumably be the lowest - lying ones and , being color - singlet , would give rise to low - lying glueballs or mesons for instance . the scattering term in ( [ pot0 ] ) , given by @xmath62 can be considerably simplified by using the born approximation , _ i.e. _ by noticing that if the interactions are weak enough , @xmath63 . such conditions are generally expected to be valid at high enough temperatures , where the typical interaction energy is small with respect to the typical thermal energy of the particles . note also that , in some cases , this approximation can be relevant when the factor @xmath64 is negligible , irrespective of the temperature : such cases will be encountered when the gauge group is su(@xmath2 ) ( see sec . [ sunt ] ) . to the first order in @xmath29 , ( [ pot_s ] ) becomes @xmath65 let us write explicitly @xmath66 , where @xmath38 is the total number of particles involved in a given scattering channel , and where @xmath67 are the remaining quantum numbers . a useful remark to be done at this stage is that the pairwise structure of @xmath29 causes @xmath68 to be always vanishing excepted in two - body channels . here , at the born approximation , @xmath38 is always equal to 2 . then , @xmath69 , with @xmath70 and @xmath71 . note that the color channel @xmath55 and the particles species @xmath41 , @xmath42 are part of @xmath67 . after having extracted from the trace the color and @xmath72 dependences ( @xmath73 if the charge conjugation is not relevant ) , one is led to @xmath74 with @xmath75 is the pair representation dimension , @xmath76 the remaining trace on the momentum space and @xmath77 the potential with the angular symmetry of the considered channel . among the various summations to be performed in @xmath78 , two are of particular interest : the one over the different interacting species , that can be denoted @xmath79 , and the one over the representations appearing in @xmath60 , that is @xmath80 . because of @xmath81 , ( [ pot_s3 ] ) is thus proportional to a factor @xmath82 for a given pair @xmath41 , @xmath42 in a given @xmath72 channel . when the combinations of species does not have to respect a symmetry principle , this last sum runs on all the representations appearing in @xmath60 ; one can then show that @xmath83 indeed , it is known in group theory that the second order dynkin indices @xmath84 in a tensor product obey a sum rule that can be rewritten using our notations as @xmath85 @xcite . using @xmath86 @xcite , one straightforwardly shows that ( [ group ] ) holds . note that ( [ group ] ) and ( [ pot_s3 ] ) are thus _ a priori _ nonzero when a symmetry principle has to be respected : the summation can not then be performed on all possible color representations . let us now particularize the general formalism presented in the previous section to a genuine yang - mills plasma , _ i.e. _ with no matter fields . the bosonic degrees of freedom propagating in the plasma are then quasigluons , that are transverse spin-1 bosons in the adjoint representation of the gauge group . the baryonic potential can be set equal to 0 and according to standard formulas in statistical mechanics , one has @xmath87 where the quasigluons are _ a priori _ supposed to have a mass @xmath88 , and where @xmath89 is the grand potential per degree of freedom associated to a bosonic species with mass @xmath90 . equation ( [ psb ] ) leads to @xmath91 let us recall that in the following , the term gluon indifferently denotes the gauge field of yang - mills theory and the quasigluons . the sum @xmath92 appearing in ( [ pot0 ] ) now explicitly reads @xmath93 , where @xmath94 is the number of gluons involved in the interaction process . as soon as @xmath95 , the determination of the allowed color channels and of the correct symmetrized gluon states generally becomes a painful task , to which the problem of finding the @xmath0-matrix in many - body scattering must be added . intuitively , one can nevertheless expect the dominant scattering processes to be two - gluon ones , and thus only consider @xmath96 2 in a first approach . after simplification , the grand potential ( [ pot0 ] ) eventually reads @xmath97 \bigg\rbrace , \nonumber\end{aligned}\ ] ] where the symbol prime " is the derivative respective to the energy and @xmath98 is the mass of the two - gluon bound state with color @xmath55 and quantum numbers @xmath99 , if it exists . the index @xmath100 in the @xmath72 channel is dropped since the charge conjugation is always positive for a two - gluon state @xcite . in the remaining trace , it is understood that the @xmath0-matrix has been computed in a given two - body channel with color @xmath55 and quantum numbers @xmath99 , and that the dirac @xmath101 reads @xmath102 , with the dispersion relation @xmath103 . note also that , in connection with nuclear many - body approaches , ( [ omega2 ] ) can be rewritten in terms of a weighted thermal average of scattering phase shifts by means of unitarity of the @xmath16-matrix . the computation of the two - gluon @xmath0-matrix is explained in detail in the following section . jacob and wick s helicity formalism @xcite can be applied to describe a two - gluon state , where the gluons are seen as transverse spin-1 particles . let us generically define @xmath104 the quantum state of a particle with momentum @xmath105 , spin @xmath33 , and helicity @xmath53 . if the particle is transverse , only @xmath106 is allowed , while all the projections from @xmath107 to @xmath108 are allowed if the particle has a usual spin degree of freedom . then it can be deduced from @xcite that the quantum state @xmath109,\ ] ] with @xmath110 and @xmath111^{\frac{1}{2}}\int^{2\pi}_0d\phi\int^\pi_0d\theta\ , \sin\theta\ { \cal d}^{j*}_{m,\lambda_1-\lambda_2}(\phi,\theta,-\phi)\ , r(\phi,\theta,-\phi)\nonumber\\ & & \qquad\qquad\qquad\qquad\qquad\times \ , a^{\dagger}_{\lambda_1}(\vec p\,)a^{\dagger}_{\lambda_2}(-\vec p\,)\left|0\right\rangle,\end{aligned}\ ] ] is a two - particle helicity state in the rest frame of the system which is also an eigenstate of the total spin @xmath112 and of the parity , _ i.e. _ @xmath113 , @xmath114 , and @xmath115 with @xmath116 and @xmath117 the intrinsic parity and spin of particle @xmath41 . moreover , @xmath118 . in the above definition , @xmath119 is the rotation operator of euler angles @xmath120 and @xmath121 are the wigner @xmath122-matrices . the coordinates @xmath123 are the polar angles of @xmath105 . when both particles have a spin degree of freedom , the helicity basis , spanned by the helicity states ( [ hstate ] ) , is equivalent to a standard @xmath124 basis up to an orthogonal transformation @xcite . when at least one of the particles is transverse , both basis are no longer equivalent , but the helicity states can still be expressed as particular linear combinations of @xmath124 states @xcite . this will be convenient in view of future computations . when the two particles are identical ( @xmath125 , @xmath126 ) , it is relevant to study the action of the permutation operator @xmath127 . one finds @xcite @xmath128\left|j^p , m;\lambda_1,\lambda_2,\eta\right\rangle=\left|j^p , m;\lambda_1,\lambda_2,\eta\right\rangle+(-1)^{j}\left|j^p , m;\lambda_2,\lambda_1,\eta\right\rangle,\ ] ] where the operator @xmath129={\cal s}$ ] is nothing else than a projector on the symmetric ( @xmath33 integer ) or antisymmetric ( @xmath33 half - integer ) part of the helicity state . it is readily seen in ( [ symdef ] ) that symmetrizing the state will eventually lead to selection rules for @xmath130 ( this is particularly clear if one sets @xmath131 ) . when extra degrees of freedom are added , it is also of interest to use the antisymmetrizer @xmath132={\cal a}$ ] as done in table [ tabs ] . a general discussion about the two - gluon helicity states can be found in @xcite , to which we refer the interested reader . for the present work , it is sufficient to recall that four families of helicity states can be found , separated in helicity singlets @xmath133 and doublets @xmath134 following the pioneering work @xcite . the corresponding quantum numbers are given in table [ tabs ] , as well as the average value of the squared orbital angular momentum , @xmath135 , computed with these states . .symmetrized and antisymmetrized two - gluon helicity states , following the notation of @xcite , with the corresponding quantum numbers and averaged squared orbital angular momentum . [ cols="^,^,^,^",options="header " , ] [ tab4 ] the lattice data that are used as a starting point to build our interaction potential are those of @xcite , _ i.e. _ the static free energy between a quark - antiquark pair bound in a color singlet for @xmath136 . for numerical convenience , it is preferable to deal with a fitted form of these , rather than with interpolations of the available points . to fit the data of @xcite , the analytic form proposed by satz in @xcite is used : @xmath137-\frac{4}{3}\frac{\alpha}{r}\big[{\rm e}^{-\mu(t ) r}+\mu(t ) r\big].\ ] ] the way of obtaining this formula is the following . first , it is known that the static quark - antiquark energy at zero temperature is accurately fitted by a so - called funnel shape @xmath138 see _ e.g. _ @xcite . when @xmath139 , one can imagine that this potential is progressively screened by thermal fluctuations . an effective theory for studying the screening of a given potential is the debye - hckel theory , in which the thermal fluctuations are all contained in a screening function @xmath140 , that modifies the zero - temperature potential and eventually leads to the form ( [ f1lat ] ) . the explicit form of @xmath140 is unknown _ a priori _ and has to be fitted on the lattice data . as it can be seen in fig . [ fig1a ] , the form @xmath141 with @xmath142 provides an accurate fit of the lattice data in the range 1 - 3 @xmath1 . a more complete fit should be such that @xmath143 , but our model is not intended to be able to cross " the phase transition in @xmath1 . the simple form ( [ mufit ] ) is already satisfactory . the corresponding internal energy @xmath144 is plotted in fig . [ fig2a ] . of a quark - antiquark pair bound in a color singlet , computed in su(3 ) quenched lattice qcd and plotted for different temperatures ( symbols ) . data are taken from @xcite and expressed in units of @xmath145 , with @xmath146 the quark - antiquark separation . the fitted form ( [ f1lat])-([afit ] ) is compared to the lattice data ( solid lines ) . , width=321 ] of a quark - antiquark pair bound in a color singlet , computed from the fitted form ( [ f1lat])-([afit ] ) and plotted for different temperatures ( solid lines ) . , width=321 ] to compute ( [ omega2 ] ) , it is necessary to use a given representation . in order to use the calculation of the @xmath0-matrix proposed in @xcite , the scattering part of ( [ pot_s3 ] ) must be computed in the momentum space representation ( the two first term are simply free gas contributions and can be easily computed ) . let us focus on @xmath147 . \end{aligned}\ ] ] using the following definitions concerning the trace of an operator @xmath148 in momentum space @xmath149 and the partial wave expansion @xmath150 where @xmath151 is the legendre polynomial of order @xmath152 , ( [ oms ] ) reads @xmath153 \right . \\ & & + \left . \displaystyle\frac{1}{16 \pi^2 } \displaystyle\int_{2m_g}^\infty d\epsilon \ , \epsilon^4 \ , \left(\frac{\epsilon^2}{4 } - m_g^2\right ) k_2(\beta\epsilon ) \left[\left({\rm re } { \cal t}_{j^p}(\epsilon ; q_\epsilon , q_\epsilon)\right ) ' { \rm i m } { \cal t}_{j^p}(\epsilon ; q_\epsilon , q_\epsilon ) \right ] \right).\end{aligned}\ ] ] 99 j.c . collins and m.j . perry , phys . lett . * 34 * , 1353 ( 1975 ) . shuryak , phys . rep . * 61 * , 71 ( 1980 ) . d. cabrera and r. rapp , phys . d * 76 * , 114506 ( 2007 ) . r. dashen , s .- k . ma , and h. j. bernstein , phys . rev . * 187 * , 345 ( 1989 ) . gorenstein , s.n . yang phys . f * 52 * , 5206 ( 1995 ) ; f buisseret and g lacroix , eur . phys . j. c * 70 * , 1051 ( 2010 ) . m. pepe and u .- j . wiese , nucl . b * 768 * , 21 ( 2007 ) . s. gupta , k. huebner , and o. kaczmarek , phys . d * 77 * , 034503 ( 2008 ) . j. fuchs and c. schweigert , _ symmetries , lie algebras and representations _ , cambridge university press ( 1997 ) . n. boulanger , f. buisseret , v. mathieu , and c. semay , eur . j. a * 38 * , 317 ( 2008 ) . m. jacob and g. c. wick , annals phys . * 7 * , 404 ( 1959 ) . v. mathieu , n. kochelev , and v. vento , int . e * 18 * , 1 ( 2009 ) , and references therein . m. i. haftel and f. tabakin , nucl . a * 158 * , 1 ( 1970 ) . f. buisseret and g. lacroix , phys . b * 705 * , 405 ( 2011 ) . b. lucini , m. teper , and u. wenger , jhep * 02 * , 033 ( 2005 ) . lu . liptak and s. olejnik , d * 78 * , 074501 ( 2008 ) ; 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the strongly coupled phase of yang - mills plasma with arbitrary gauge group is studied in a @xmath0-matrix approach . the existence of lowest - lying glueballs , interpreted as bound states of two transverse gluons ( quasi - particles in a many - body set up ) , is analyzed in a non - perturbative scattering formalism with the input of lattice - qcd static potentials . glueballs are actually found to be bound up to 1.3 @xmath1 . starting from the @xmath0-matrix , the plasma equation of state is computed by resorting to dashen , ma and bernstein s formulation of statistical mechanics and favorably compared to quenched lattice data . special emphasis is put on su(@xmath2 ) gauge groups , for which analytical results can be obtained in the large-@xmath2 limit , and predictions for a @xmath3 gauge group are also given within this work .

1210.1716

accurate measurements of the cosmic microwave background ( cmb ) anisotropies provide an excellent mean to probe the physics of the early universe , in particular the hypothesis of inflation . recently , scientists working with the boomerang @xcite and maxima-1 @xcite cmb experiments announced the clear detection of the first acoustic peak at an angular scale @xmath2 , which confirms the most important prediction of inflation : the universe seems to be spatially flat @xcite . in the framework of inflation cmb anisotropies follow from the basic principles of general relativity and quantum field theory . to predict the multipole moments of these cmb anisotropies two ingredients are necessary : the initial spectra of scalar and tensor perturbations and the `` transfer functions '' , which describe the evolution of the spectra since the end of inflation . the transfer functions depend on cosmological parameters such as the hubble constant ( @xmath3 ) , the total energy density ( @xmath4 ) , the density of baryons ( @xmath5 ) , the density of cold dark matter ( @xmath6 ) and the cosmological constant ( @xmath7 ) . for the analysis of cmb maps it is a reasonable first step to test the most simple and physical model of the early universe : slow - roll inflation with a single scalar field . slow - roll inflation predicts a logarithmic dependence of the power spectra on the wave number @xmath8 @xcite . however , in most studies of the cmb anisotropy the spectral shape of power - law inflation @xcite , corresponding to an exponential potential for the inflaton field , has been considered . this case is unphysical , since power - law inflation does never stop . two of us @xcite have shown , using analytical techniques , that the predictions of power - law and slow - roll inflation can differ significantly . here , we confirm these results and calculate the cmb anisotropies with a full boltzmann code developed by one of us ( a. r. ) . the numerical accuracy of this code has been tested by comparison to analytical results ( low @xmath9 ) and to cmbfast v3.2 @xcite . in general both codes agree within @xmath10 . we use the new cmb data to test slow - roll inflation , assuming the two most popular versions of cold dark matter ( cdm ) models ( our `` priors '' ) : the standard cdm model ( scdm : @xmath11 , @xmath12 ) and the cosmic concordance model ( @xmath13cdm : @xmath14 , @xmath15 , @xmath16 ) , which is motivated by the results of the high-@xmath17 supernovae searches @xcite . in particular , we take @xmath18 in agreement with the most important prediction of inflation , @xmath19 , which is consistent with supernovae type ia measurements [ @xmath20 at @xmath21 c.l . @xcite ] , and @xmath22 , as inferred from the observed abundance of d and primordial nucleosynthesis [ @xmath23 @xcite ] . in this letter we recall the basic predictions of slow - roll inflation ( sec . 2 ) and correct errors and misconceptions that have been recently made in the literature on this issue ( sec . 3 ) . in section 4 we compare for the first time the predictions of slow - roll inflation with the recent data of boomerang and maxima-1 ( without any elaborated statistical technique ; we remind that only @xmath24 of the boomerang data have been analyzed so far ) . the power spectra from power - law inflation , for which the scale factor behaves as @xmath25 with @xmath26 , change with a fixed power of the wavenumber @xmath8 . for the bardeen potential and for gravitational waves the power spectra in the matter - dominated era are respectively given by @xcite @xmath27 where @xmath28 is a pivot scale and where @xmath29 . the factor @xmath30 is predicted from inflation , its expression is given in @xcite . here , @xmath30 is _ a priori _ free and must be tuned such that the angular spectrum is cobe - normalized . the choice of @xmath31 fixes @xmath32 and @xmath33 and we always have @xmath34 . the predictions of power - law inflation are the same for any value of the pivot scale , since @xmath28 can be included into the definition of @xmath30 . let us now turn to slow - roll inflation which is certainly physically more relevant , since it covers a wide class of inflationary models . slow - roll is essentially controlled by two parameters : @xmath35 and @xmath36 , where @xmath37 is the hubble rate . these two parameters can be related to the shape of the inflaton potential @xcite . all derivatives of @xmath38 and @xmath39 have to be negligible , e.g. @xmath40 , only then the slow - roll approximation is valid . slow - roll inflation corresponds to a regime where @xmath38 and @xmath41 are constant and small in comparison with unity . the power spectra of the bardeen potential and gravitational waves can be written as @xcite @xmath42 , \\ \label{specsrgw } k^3p_{\rm h } & = & \frac{400a}{9 } \biggl[1 - 2\epsilon \biggl(c+1+\ln \frac{k}{k_0}\biggr)\biggr],\end{aligned}\ ] ] where @xmath43 , @xmath44 being the euler constant . slow - roll inflation predicts the value of @xmath30 , which is given in @xcite and has not necessarily the same numerical value as for power - law inflation . one important difference to power - law inflation is that the choice of the pivot scale @xmath28 now matters . it has been shown in @xcite that the slow - roll error in the scalar multipoles is minimized at the multipole index @xmath45 if @xmath46 , where @xmath47 is the comoving distance to the last scattering surface and @xmath48 $ ] with @xmath49 . for @xmath50 this gives @xmath51 , where @xmath52 for scdm and @xmath53 for @xmath54cdm . usually the choice @xmath55 is made , which corresponds to @xmath56 . in this letter we also consider the case @xmath57 , which roughly corresponds to the location of the first acoustic peak . finally , from eqs . ( [ specsrd ] ) ( [ specsrgw ] ) the spectral indices are inferred @xmath58 an important consequence of these formulas is that the relation @xmath59 does not hold for slow - roll inflation , except in the particular case @xmath60 . we found five misconceptions in the literature , which do have an important impact on the extraction of cosmological parameters from the measured cmb multipole moments : a- from eqs . ( [ specpls ] ) and ( [ specsrd ] ) ( [ specsrgw ] ) we see that the shapes of the spectra are not the same in power - law inflation and in slow - roll inflation ( even if @xmath60 ) . unfortunately , the unphysical power - law shape ( [ specpls ] ) is assumed frequently , although the relevance of deviations from the power - law shape has been discussed earlier [ see e.g. @xcite ] . this difference in the shape affects the estimates of cosmological parameters in @xcite and @xcite , since this misconception is built in into the most commonly used numerical codes : cmbfast and camb @xcite . in @xcite it has been demonstrated that the difference is important and increases with @xmath61 . for instance , with the usual choice @xmath62 , the error is @xmath63 at @xmath64 for @xmath65 , see fig . [ compsr - pl ] . it has been suggested @xcite to move the pivot scale to @xmath66 , which decreases the difference from the spectral shapes . for the case considered before , the difference reduces to @xmath67 with @xmath66 , as can be seen in fig . [ compsr - pl ] . for cases @xmath68 the error from the wrong shape increases [ for the primordial spectra this has been studied by @xcite ] . thus , for the accurate estimation of the cosmological parameters , one must not mistake power - law inflation for slow - roll inflation . we suggest to place the scale for which the slow - roll parameters @xmath38 and @xmath39 are determined in the region of the acoustic peaks , rather than in the cobe region , which decreases the error from the slow - roll approximation and one can get rid of the limitations from cosmic variance for the normalization . b- in various publications @xcite and codes ( cmbfast and camb ) @xmath69 and @xmath70 are allowed in the data analysis , while working with power - law spectra ( the prediction of power - law inflation ) . this is meaningless in the context of inflationary perturbations . for the case @xmath71 the scalar amplitude is divergent and the linear approximation breaks down [ see eq . ( [ specpls ] ) ] . if , nevertheless , the power law shape is assumed , @xmath72 should be fulfilled . on the contrary , in slow - roll inflation , as can be checked on eqs . ( [ specindsr ] ) , one can have @xmath73 or @xmath74 , only @xmath75 is compulsory . c- a third misconception is that gravitational waves are not taken into account properly . this is an important issue since a non - vanishing contribution of gravitational waves modifies the normalization and changes the height of the first acoustic peak . in @xcite ( see the footnote [ 13 ] ) , it was assumed that if @xmath74 , there are no gravitational waves at all , a supposition in complete contradiction with the predictions of slow - roll inflation . also , in that article , the relation @xmath76 was used . it is valid for power - law inflation only . in @xcite gravitational waves have been neglected , which restricts the analysis for their choice of @xmath71 to the case @xmath77 , such that tensors contribute less than about @xmath78 of the power . d- by default in the cmbfast and camb codes the contribution of gravitational waves is calculated according to the relation @xmath79 . @xcite argued , based on this relation , that power - law models with large tilt can not explain the observed anisotropies . however , this relation is only valid for power - law inflation and the scdm model . in particular this is no longer true when @xmath80 ( @xmath54cdm model ) . the reason is the so - called `` late integrated sachs - wolfe effect '' , which has been well known for a long time @xcite . the normalization must be performed utilizing the power spectra themselves and not the quadrupoles in order not to include an effect of the transfer function . in fig . [ wrongts ] , we display the @xmath54cdm multipole moments ( for @xmath65 ) in the case where the wrong normalization is used together with the case where the normalization is correctly calculated with the help of eqs . ( [ specsrd ] ) ( [ specsrgw ] ) . the error is @xmath81 at @xmath64 . this weakens the mentioned argument of @xcite and in fact questions any analysis that uses the cmbfast default scalar - tensor ratio together with a non - vanishing cosmological constant . e- finally , cmbfast and the pre - july 2000 versions of camb calculate the low-@xmath9 multipoles in the tensorial sector inaccurately . in the case of power - law inflation and the scdm model they can be well approximated by @xmath82 with , @xmath83 where @xmath84 is a spherical bessel function . for @xmath85 , this gives in agreement with @xcite : @xmath86 . the code developed by one of us ( a.r . ) reproduces this value with a precision better than @xmath87 , whereas cmbfast gives @xmath88 , i.e. an error of @xmath89 . above @xmath90 both codes agree reasonably well . this problem has been fixed in the july 2000 version of camb . we now consider the most simple and physical model for inflation ( i.e. slow - roll inflation optimized with @xmath91 ) for the scdm and @xmath54cdm scenarios and compare its predictions with the observational data of cobe / dmr @xcite , boomerang and maxima-1 . we demonstrate that a large region of the parameter space @xmath1 [ or equivalently @xmath92 , forbidden in the case of power - law inflation but allowed in the case of slow - roll inflation , contains models which fit the data as good as the models usually considered in the data analysis . the data are very often presented in terms of band - power @xmath93 , with @xmath94 . for any value of @xmath95 and @xmath39 , @xmath96 can be approximatively expressed in terms of the band - power for @xmath38 , @xmath97 @xmath98.\ ] ] a corresponding formula for power - law inflation has been presented in @xcite [ see remark before eq . ( 33 ) ] . for the scdm and @xmath54cdm scenarios considered here ( see the introduction ) , we have respectively for the first peak : @xmath99 , @xmath100 , and for the second peak @xmath101 , @xmath102 . in the previous equation , we have assumed @xmath103 which is valid only if @xmath104 in order for the tensorial modes to be negligible . the quantity @xmath105 is defined by @xmath106 and appears because the spectrum is normalized to the multipole @xmath107 . at the leading order , it can be expressed as @xmath108 and at the next - to - leading order , it is given by @xmath109.\ ] ] eq . ( [ bandpower ] ) permits us to roughly understand how the spectrum is modified when the slow - roll parameters are changed . for fixed @xmath110 , i.e. for a fixed scalar spectral index @xmath32 , increasing @xmath38 , i.e. increasing the value of @xmath111 , lowers @xmath96 . increasing @xmath112 ( i.e. decreasing @xmath32 ) while @xmath38 ( i.e. @xmath33 ) remains constant has the same effect . in figs . [ scdm ] ( scdm scenario ) and [ lambdacdm ] ( @xmath54cdm scenario ) , we display the theoretical predictions of slow - roll inflation for some values of the slow - roll parameters . without performing a @xmath113-analysis , our main conclusion is that there exist models that reasonably fit available cmb data , which were not included in the estimates of cosmological parameters before , in particular in the data analysis of the recent cmb maps @xcite . this includes models with @xmath114 and non - negligible gravitational waves contribution . for instance , the model @xmath115 , @xmath116 ( i.e. @xmath73 and @xmath117 ) in the @xmath54cdm scenario , see fig . [ lambdacdm ] goes through all the maxima-1 data points ( at 1@xmath118 ) but one . in this particular case , gravitational waves represent @xmath119 of the power at @xmath120 , i.e. @xmath121 . this provides a good example which violates common ( unjustified ) believes about inflation . let us stress that for both figures we did not optimize the fits by exploring the @xmath78 resp . @xmath122 uncertainty in the calibration of the boomerang and maxima-1 results , nor did we optimize the fits by varying @xmath3 and @xmath123 nor any other parameters . it is impossible to extract the values of cosmological parameters from the cmb anisotropy data without assumptions on the initial spectra . for this purpose , slow - roll inflation is the best model presently known and is consistent with presently available data . unfortunately , very often , only power - law inflation is considered . the difference between both models is in general significant , which implies that only a limited part of the space of parameters has been correctly studied so far . data analysis has been based on unjustified prejudices that @xmath32 may be greater than one in power - law inflation , that the relation @xmath0 must hold in general and that , gravitational waves are negligible in general . we want to stress that a subdominant effect ( as the contribution of gravitational waves in many inflation models ) is not necessarily negligible . although very important on the conceptual side , the previous misconceptions were not crucial for the cobe / dmr experiment . for the next generation of measurements , which aim to extract cosmological parameters with a precision of a few percent , distinguishing power - law inflation from slow - roll inflation becomes mandatory . we think that a correct analysis of the cmb data should start from the spectra given in eqs . ( [ specsrd])([specsrgw ] ) and should be performed in the whole space of parameters ( @xmath124 . this should result in the determination of the best @xmath38 and @xmath39 . the present letter hopefully motivates more detailed tests of the most simple inflationary scenario : slow - roll inflation . _ note added in web version : _ after the proofs of our paper have been sent , version 4.0 of cmbfast was posted on the web . in this version the inaccuracy in the predictions of the tensor multipole moments has been fixed . such that the scalar and tensor spectral indices agree in both cases ( @xmath65 , @xmath125 ) , which means for the power - law model @xmath126 . for @xmath62 , the usual pivot , the difference between the power - law and slow - roll spectra is large , which improves for a pivot @xmath127 . the contribution of gravitational waves is displayed for power - law and slow - roll inflation ( @xmath62 ) . ]

we emphasize that the estimation of cosmological parameters from cosmic microwave background ( cmb ) anisotropy data , such as the recent high resolution maps from boomerang and maxima-1 , requires assumptions about the primordial spectra . the latter are predicted from inflation . the physically best - motivated scenario is that of slow - roll inflation . however , very often , the unphysical power - law inflation scenario is ( implicitly ) assumed in the cmb data analysis . we show that the predicted multipole moments differ significantly in both cases . we identify several misconceptions present in the literature ( and in the way inflationary relations are often combined in popular numerical codes ) . for example , we do not believe that , generically , inflation predicts the relation @xmath0 for the spectral indices of scalar and tensor perturbations or that gravitational waves are negligible . we calculate the cmb multipole moments for various values of the slow - roll parameters and demonstrate that an important part of the space of parameters @xmath1 has been overlooked in the cmb data analysis so far .

astro-ph0006392

investigation of the mass function of globular clusters is of great importance for a variety of problems in astrophysics covering star formation processes , the dynamical evolution of stellar systems and the nature of dark matter in the galaxy . large progress has been made in recent years both by ground based observation and , more recently , thanks to observations by hst . nevertheless most of issues concerning the shape of the initial mass function ( imf ) , its dependence on cluster parameters , the actual relevance of dynamical processes in its evolution and the relation between the imf and the present - day mass function ( pdmf ) are still matters of debate . the first investigation addressing the dependence of the slope of the mass function on cluster structural parameters and metallicity was carried out by mcclure et al . ( 1986 ) who found the slope of the pdmf for a sample of six galactic clusters to be correlated with their metallicity , the low - metallicity clusters having steeper mass functions . in subsequent work capaccioli , ortolani & piotto ( 1991 ) , piotto ( 1991 ) and capaccioli , piotto & stiavelli ( 1993 ) have considered a larger sample of clusters and have questioned the conclusion of mcclure et al . and showed the correlation between the slope of the pdmf and the position of the cluster in the galaxy to be stronger than that with the metallicity . finally djorgovski , piotto & capaccioli ( 1993 ) have addressed this problem again by multivariate statistical methods and have concluded that both the position in the galaxy ( galactocentric distance and height above the disk ) and the metallicity play some role in determining the slope of the pdmf but the former is more important than the latter . the observed correlation is in the sense of clusters more distant from the galactic center or from the disk having steeper mass functions . the data used in the above works are from ground based observations and the slopes are measured for a limited range of star masses ( @xmath2 ) . recent investigations of the luminosity function of some galactic globular clusters by hst have been able to extend the available data to fainter magnitudes ( paresce , demarchi & romaniello 1995 , de marchi & paresce 1995ab , elson et al . 1995 , piotto , cool & king 1996,1997 , santiago , elson & gilmore 1996 ) . hst data for for , , 6 , m15 and m30 are now available . these clusters span a wide range of values of metallicity , their structural parameters suggest they have undergone a very different dynamical evolution and the issue concerning the origin of the shape of the pdmf has been addressed again in the light of this new data . de marchi & paresce ( 1995b ) compare the mf of 6 , m15 and showing that all these clusters have a flat mf for low - masses ; they point out that the mf is flat at the low - mass end for both 6 and m15 and that these mfs are very similar though these clusters are likely to have had a different dynamical history . as for , this is shown to have a different mf from m15 and 6 . noting that the metallicity of is very different from that of 6 and m15 de marchi & paresce make the hypothesis that the differences between the mfs of these clusters might be due to a different initial mass function ( imf ) depending on the metallicity , thus giving new support to the conclusion of mcclure et al . ( 1986 ) , with the subsequent dynamical evolution playing no relevant role . however in a recent work , santiago et al . ( 1996 ) show that the mf of , whose metallicity is similar to that of , is steeper than the mf of , and cast some doubt on the scenario supported by de marchi and paresce . santiago et al . point out that if one assumes a universal imf , the comparison of the mf of with those of 6 , m15 , would indicate that the latter clusters have experienced significant dynamical evolution with strong depletion of low - mass stars . finally piotto et al . ( 1996,1997 ) argue that the reason why de marchi and paresce get a similar mf for 6 and m15 is that they compare only the low - mass end and show that , by comparing the lf including the data for the bright end , 6 appears to be markedly deficient in faint stars . as the metallicities of 6 and m15 are very similar , this result lends strong support to the hypothesis that the mf of 6 is flattened by dynamical processes . king ( 1996 ) notes that this hypothesis is further enforced by the comparison of the orbits of 6 and m15 , as obtained by dauphole et al . ( 1996 ) ; according to this work 6 would be more affected by the tidal interaction with the galaxy as it would cross the disk more frequently and would have a smaller perigalactic distance than m15 . additional observations covering a larger range of cluster parameters are necessary , as well as theoretical investigations addressing both the problems connected with the stellar evolution models ( see alexander et al . 1997 , dantona & mazzitelli 1996 for two recent works in this direction ) allowing a better determination of the mass - luminosity relationship for low - mass low - metallicity stars ( see e.g. elson et al . 1995 for a clear presentation of the problems due to the uncertainties on @xmath3 relation ) and those connected with the dynamical evolution , thus clarifying the efficiency of evolutionary processes in modifying the imf . as for this latter aspect the situation is far from being clear : simple semi - analytical models by stiavelli et al . ( 1991 ) , stiavelli , piotto & capaccioli ( 1992 ) and capaccioli et al . ( 1993 ) suggest that disk shocking could play a relevant role in establishing the observed correlation between the slope of the pdmf and the position in the galaxy and some indications on the role of evaporation due to two - body relaxation come from many multi - mass fokker - planck investigations of the dynamical evolution of clusters ( see e.g. chernoff & weinberg 1990 , weinberg 1994 ) but no firm conclusion has been reached on the relevance of various dynamical processes in giving rise to the observed correlation between the slope of the mass function and position in the galaxy and the interplay between different dynamical processes has not been fully explored . in this paper we show the results of a set of @xmath0- body simulations including the effects of the presence of the tidal field of the galaxy , stellar evolution , disk shocking , two - body relaxation and spanning a range of different initial conditions for the mass and concentration of the cluster , the slope of the imf and the distance from the galactic center . the main goal of our theoretical investigation is that of assessing the importance of various evolutionary processes in altering the mass function of a globular cluster , trying to establish to what extent these processes can be responsible for the differences observed between the mfs in galactic globular clusters . having included the effects of stellar evolution in our investigation we will be able to address some other issues concerning the evolution of the stellar content of globular clusters . in particular we will focus our attention on the fraction of white dwarfs expected to be present in a cluster during the different stages of its dynamical evolution . these are of increasing interest for a variety of reasons . it had long been realised that white dwarfs should exist in some abundance in globular clusters , and they have been included routinely in dynamical models . otherwise , however , little attention was paid to them while they remained unobservable . that situation has changed dramatically in recent years ( e.g. richer et al . 1995 ) , and it is now claimed that white dwarfs formed up to 9 gyr ago have been detected ( richer et al . furthermore recent suggestions have been made that white dwarfs could account for as much as 50 percent of the total mass of some clusters ( heggie & hut 1996 ) or up to 85 percent in some locations in other clusters ( gebhardt et al . this sudden burst of interest in white dwarfs will stimulate renewed attention in their influence on cluster dynamics , and has motivated the work reported in section 3.3 . in sect.2 we describe the method used for our investigation . in sect.3 the results of simulations including the effects of stellar evolution and two - body relaxation only are described , and some analytical expressions are derived for the main quantities of interest for this work . in sect.4 the results of runs including also the effects of disk shocking are described and , where possible , we generalize the analytical expressions obtained in sect.3 . the dependence of the results on the initial number of particles used in the simulations are discussed in sect.5 . summary and conclusions are in sect.6 . all the @xmath0-body simulations made for our investigation have been carried out using a version of the code nbody4 ( aarseth 1985 ) including mass loss of single stars due to stellar evolution , the effects of disk shocking and of the tidal field of the galaxy . the code is a direct summation code adopting an hermite integration scheme and a binary hierarchy of time - steps ; a grape-4 board containing 48 harp chips ( makino , kokubo & taiji 1993 ) connected to a dec alphastation 3000/700 has been used for the evaluation of the forces and force derivatives required by the hermite scheme . most of the simulations carried out in this work start with a total number of particles @xmath4 and it took a cpu time ranging from about 3 to 15 hours for each of them to be completed depending on the initial concentration of the system and its galactocentric distance ( for further information on the code and the performance of the board see e.g. aarseth 1994 , aarseth & heggie 1997a ) . in our simulations we have assumed the cluster to be on a circular orbit and to move in a keplerian potential determined by a point mass @xmath5 equal to the total mass of the galaxy inside the adopted galactocentric distance @xmath6 ; the value of the circular speed has been taken equal to @xmath7 km / s . if we denote the angular velocity of the cluster around the galaxy by @xmath8 and take the coordinates of a star relative to the center of the cluster with the @xmath9-axis pointing in the direction away from the galactic center and the y - axis in the direction of the cluster motion we can write the equations of motion for a star in the cluster as ( see e.g. chandrasekhar 1942 ) @xmath10 where @xmath11 represent the acceleration from other stars in the cluster and the terms involving @xmath8 on the left - side of the equations are due to the coriolis , centrifugal and tidal field accelerations . king s models ( king 1966 ) with different concentrations have been used to produce initial conditions and the standard @xmath0- body units ( heggie & mathieu 1986 ) ( total mass @xmath12 , @xmath13 and initial energy equal to -1/4 ) have been adopted . the value of @xmath8 in @xmath0-body units is given by its initial relationship with the tidal radius ( see aarseth & heggie 1997a ) r_t^*=(3^*2)^-1/3 , where the @xmath14 denotes quantities in @xmath0-body units . masses of stars have been assigned according to a power - law mass function @xmath15 between @xmath16 and @xmath17 and initially there is no equipartition of energies of stars with different masses . disk shocking has been included according to the model described in chernoff , kochanek & shapiro ( 1986 ) . following spitzer ( 1958 ) , chernoff et al . describe the motion of a single star in the cluster as that of an harmonic oscillator with a frequency @xmath18 perturbed by the force due to the disk . assuming the orbit to be perpendicular to the plane of the disk and choosing the direction of motion along the @xmath19-axis , the equation of motion of a star in a frame at rest with respect to the center of the cluster is + ^2 y = g(t)y where @xmath20 is the distance between the star and the center of the cluster , @xmath19 the height of the center of the cluster over the disk and @xmath21 is the differential acceleration due to the disk . modelling the disk as a one - component isothermal self - gravitating gas and assuming an exponential decay for the disk surface density , the acceleration due to the disk is given by k_y = k_0 ( y / y_0)^-r_g / h . where @xmath6 is the galactocentric distance and @xmath22 and @xmath23 are the disk scale height and characteristic scale length respectively . starting from the above equations chernoff et al . show that the change of velocity in the @xmath19 direction suffered by a star due to the disk shocking can be written as v_y = y i_c ( ) [ dvds ] where @xmath24 is i_c=2 ^2 ( /2)k_0 z_0 ^ -r_g / h and @xmath25 is the frequency of disk crossing . when @xmath26 , the above expression provides the same result which would be obtained in the impulse approximation ( see e.g. spitzer 1987 ) while for @xmath27 the crossing is adiabatic and the change in star s velocity tends exponentially to zero . notice that , in the pure impulse approximation , ( as adopted , for example , by capaccioli et al . ( 1993 ) ) , the effect of a disk shock is minimised by choosing an orbit which crosses the disk perpendicularly , as we have done , because then @xmath28 is maximised ( for a given circular speed ) . the adiabatic correction , however , suppresses low - frequency disk shocking , and the net result is that the effect of disk shocking is almost independent of the inclination of the orbit over a wide range ( chernoff , kochanek & shapiro 1986 ) . in our investigation we have adopted the two - component disk model obtained by chernoff et al . from fitting bahcall s ( 1984 ) determination of acceleration in the solar neighbourhood k_y(r_g=8 , y)=_i k_0i(y / y_0i ) with @xmath29 @xmath30 and the scale length has been taken equal to @xmath31 kpc ( see e.g. bahcall , schmidt & soneira 1982 ) . in the simulations including the effects of disk shocking , each half - orbital period of the cluster around the galaxy the @xmath19-component of the velocities of all the stars in the system have been changed according to eq.([dvds ] ) ( where @xmath24 has been replaced by the sum , @xmath32 , with @xmath33 being the function @xmath24 evaluated for the two disk components ) with the frequency @xmath18 taken to be equal to the ratio of the current value of tangential velocity of the star to its radial distance from the cluster center . stellar evolution has been modelled by assuming that the mass lost by each star immediately escapes from the cluster ( this is likely to be a reasonable approximation since the escape velocity from a typical globular cluster is of the order of 10 km s@xmath34 ) and thus the mass of each star has been decreased at the appropriate time by an amount depending on the initial value of the mass itself . the fraction of mass lost by stellar evolution and the time when mass loss has to take place in the simulation have been calculated adopting the same model used in chernoff & weinberg ( 1990 ) ( see also fukushige & heggie 1995 ) : stars whose initial mass is larger than @xmath35 end their life as neutron stars with a mass equal to @xmath36 , stars with @xmath37 are assumed to leave no remnant and stars less massive than @xmath38 produce white dwarfs with a mass equal to @xmath39 ; the times the mass must be removed are determined by a linear interpolation of the main sequence times calculated by iben & renzini ( 1983 ) and reported in table 1b of chernoff & weinberg ( 1990 ) . we have assumed that there is no kick velocity from the supernova explosion for stars ending their evolution as neutron stars . though consistent with the assumptions of chernoff & weinberg ( 1990 ) this assumption is very likely to be incorrect ( drukier 1996 ) . nevertheless , a more realistic treatment would make almost no difference to our results , as the mass fraction in neutron stars rarely exceeds 1 percent for our assumed imf ( fig.1d ) . the last issue to be addressed concerns the scaling from time in @xmath0-body units to astrophysical units ; as we want to include the effects of disk shocking and stellar evolution , an appropriate time scaling is necessary not only for a correct application of @xmath0-body results to real clusters but also for a proper determination of the times when the mass of stars must be removed due to stellar evolution and the change in the velocities of stars due to disk shocking have to be made . an extensive investigation of this issue has been carried out by aarseth & heggie ( 1997a ) ( see also fukushige & heggie 1995 ) who have shown that , depending on the time scale of the physical process ( stellar evolution or relaxation ) determining the evolution of the cluster , the scaling must ensure that the ratio of `` real '' time to @xmath0-body time must be equal to the ratio of `` real '' to @xmath0-body crossing or relaxation time . in fact , as discussed in aarseth & heggie ( 1997a ) , if the cluster lifetime is shorter than the time scale in which relaxation effects become important ( e.g. because of strong mass loss due to stellar evolution causing the cluster to disrupt ) the proper factor to scale @xmath0-body time to real time is given by the ratio of the crossing time of @xmath0-body system to that of the real system . on the other hand if the cluster survives for a time long enough to be affected by relaxation the proper conversion factor is provided by the ratio of relaxation times of @xmath0-body and real system thus ensuring that the number of relaxation times elapsed is the same in the @xmath0-body system and in the real cluster . possibly a variable time scaling , taking into account the transition from a phase dominated by stellar evolution effects to one dominated by relaxation , might be the best choice ( see aarseth & heggie 1997a for a detailed discussion of this point ) . since all the systems considered in our work do not disrupt quickly due to the effects of stellar evolution but survive for a lapse of time during which relaxation effects become important , we have scaled time by the ratio of relaxation times ( half - mass relaxation times have been used ) of @xmath0-body to real system . a set of simulations not including disk shocking have been run to establish to what extent differential escape due to two - body relaxation ( and mass loss due to stellar evolution ) can alter the mass function of a globular cluster . since the evaporation rate due to two - body relaxation is larger for low - mass stars than it is for the high mass ones ( see e.g. spitzer 1987 , giersz & heggie 1997 for some recent @xmath0-body simulations where this is clearly shown ) a flattening of the imf as dynamical evolution of a cluster proceeds is expected . this process could also be responsible for the observed correlation between the slope of the mass function and the galactocentric distance ; in fact , as the evaporation rate is inversely proportional to the relaxation time , the largest changes in the imf , for a fixed value of the cluster mass , are to take place closer to the galactic center where the size of clusters are smaller and the relaxation time shorter . table 1 summarizes the initial conditions and the main results for the set of runs done . the parameter @xmath40 is defined as f_cw where @xmath41 is the initial mass of the cluster , @xmath0 the total initial number of stars , @xmath6 the distance from the galactic center and @xmath42 the circular velocity around the galaxy . this parameter , introduced by chernoff & weinberg ( 1990 ) , is proportional to the relaxation time and thus clusters having the same value of @xmath40 form a family of models evolving in the same way provided that relaxation dominates ( obviously this is true for clusters all with the same initial concentration and imf ) . both the initial concentration and the slope of the imf have been varied in order to investigate the dependence of the results on these quantities . the slope of the mass function , at @xmath43 gyr , reported in table 1 is that measured for main sequence stars with @xmath44 . it is evident that evaporation through the tidal boundary due to two - body relaxation gives rise to a significant trend between the slope of the mf and the distance from the galactic center . as the results of runs starting with @xmath45 show , the trend established does not depend significantly on the initial concentration . on the other hand , the slope of the imf seems to have a more important effect on the formation of this correlation , and more in general on the extent the imf evolves , as shown by the results of the runs with @xmath46 . in this case the differences between the initial and the final value of @xmath47 are much smaller than for runs starting with a flatter mf and the trend between @xmath47 and the galactocentric distance is much weaker . as is evident from the data in the table , there is a clear relationship between the amount of mass which has escaped from the cluster and the variation of its mass function ( we will go through this point in greater detail later in the paper ) and the smaller changes in @xmath47 for these runs simply reflect the smaller mass loss rate for these systems . we now focus our attention on the runs with @xmath48 . fig.1 shows the time evolution of some global quantities for this set of runs . panel ( c ) , where the time evolution of the total number of particles in the first and the fifth bin ( corresponding to a range of masses @xmath49 and @xmath50 respectively ) of the mass function for the run at @xmath51kpc and for that at @xmath52 kpc has been plotted , clearly shows the preferential escape of low - mass stars which is responsible for the flattening of the imf . as shown in panel ( f ) the evolution of @xmath47 is quicker for clusters at smaller galactocentric distances and this gives rise to the correlation between @xmath47 and @xmath6 . finally it is interesting to note from panel ( e ) that the fraction of the total mass of a system in white dwarfs increases as the total mass of the cluster decreases ( see sect.3.2 for further discussion on all the issues concerning white dwarfs ) . in fig.2 the mf at @xmath53 gyr for main sequence stars for the run at @xmath54 kpc is plotted to show the progressive flattening of the mf . the data available and discussed above refer to a very limited set of initial conditions ; nevertheless , besides being important for the indications they have provided , the results of these runs can be used to derive some general analytical expressions for the main quantities of interest for our work which will allow the results to be extended to a larger set of initial conditions . from the data of these runs it is possible to derive expressions providing the total mass of a cluster , the slope of the mf and the fraction of white dwarfs at any time @xmath1 , for any value of the initial mass and any value of the galactocentric distance within reasonable limits we will see below . let s start by investigating the behaviour of the total mass of the cluster . the total mass of a cluster decreases as a result of the mass loss due to stellar evolution and two - body relaxation . for a given initial value of @xmath47 ( and fixed values of the lower , @xmath55 , and upper , @xmath56 , limits of the mf ) the fractional mass loss due to stellar evolution is the same for all clusters no matter what their initial mass is or what their galactocentric distance is . for the stellar evolution model adopted in this work this can be explicitly written as a function of the stellar mass at turn - off , @xmath57 . having defined the following functions @xmath58 we can write the fractional mass loss due to stellar evolution as @xmath59 to calculate the total mass loss due to stellar evolution at any time @xmath1 it is necessary to know the turn - off mass at that time ; this is derived by interpolating the data provided in table 1b of chernoff & weinberg ( 1990 ) . in fig.3a the time evolution of the total mass is presented . we plot separately that due to mass loss associated with stellar evolution and that due to other processes , i.e. mainly two - body relaxation and the effect of the tide . the latter shows clearly its dependence on the initial conditions of the mass loss due to the latter process and its linear dependence on time . we point out that while the amount of mass loss due to stellar evolution obtained by eq . ( [ mlst ] ) does not consider the possibility that some stars might escape from the cluster before losing their mass , the curves obtained from @xmath0-body data include only mass loss from stars inside the tidal boundary of the cluster ; for the imf chosen in our simulations the great majority of escaping stars have masses smaller than those evolving before 20 gyr and thus the theoretical estimate and the data from simulations are almost coincident . in fig.3b we show that , scaling the time by the parameter @xmath40 , all the curves representing the time evolution of the total mass ( considering only the mass loss by two - body relaxation ) at different galactocentric distances coincide . thus we can write = 1-m_st.ev . m_i - f_cwt [ mlo]where @xmath60 is a constant that can be determined by a linear fit of the curves shown in fig.3b and is found to be @xmath61 ( calculated as the mean value of the slopes of the curves shown in figure 3b where time is measured in myr ) . eq.([mlo ] ) allows us to calculate the mass of a cluster at any time @xmath1 for a cluster with initial mass @xmath41 and located at a distance from the galactic center @xmath6 ; only a slight difference in the value of @xmath60 has been found for systems starting with @xmath45 , which in this case is @xmath62 . fig.4 shows a 3-d plot of @xmath63 as a function of @xmath41 and @xmath6 and a contour plot of this function . the more massive clusters and those located at larger distances from the galactic center are those which preserve a larger fraction of their initial mass as two - body relaxation is less and less efficient and the only mass loss is due to stellar evolution . though it is not the purpose of this paper to consider the lifetimes of the globular clusters , it interesting in passing to compare our result with some others which have been used in discussions of the evolution of the galactic globular cluster system . it is shown in aarseth & heggie ( 1997a ) that the lifetime obtained by the methods we use are in satisfactory agreement with those determined by chernoff & weinberg ( 1990 ) using the fokker - planck method . aguilar et al . ( 1988 ) used two different expressions for the lifetime ( due to escape by two - body relaxation ) of a tidally bound cluster , but only for clusters with stars of equal mass . we find that the lifetimes for a typical model given by their formulae exceed our by a factor of about three or four . chernoff , kochanek & shapiro ( 1986 ) also neglected stellar evolution but did include several additional mechanisms such as disk shocking . even so their lifetimes exceed ours ( which in this section do _ not _ include disk shocking ) by a factor of two in typical cases . typical lifetimes to evaporation ( by two - body effects alone , but with stars of equal mass ) given by gnedin & ostriker ( 1997 ) exceed ours by about 40 percent . in order to obtain an analytical expression for the slope of the mass function it is necessary to estimate the evaporation rate for stars with different masses . if the mass function is @xmath64 we can write for any values of the mass of stars @xmath65 and @xmath66 = - ( dn / dm)_1-(dn / dm)_2 m_1-m_2.we estimate @xmath67 ( dndm)_1,2=n_1,2(t)m_1,2where @xmath68 is the total number of stars with mass between @xmath69 and @xmath70 at time @xmath1 . it is found from our results that the time evolution of total number of particles with a given mass is approximately linear and the slope is inversely proportional to the parameter @xmath40 n_1=n_1(0)(1-_1tf_cw ) it follows that the slope of the mass function at time @xmath1 can be written as ( t)=(0)-m_1-m_2 [ nt ] we have chosen @xmath71 and @xmath72 ( this being the range where our values of @xmath73 are calculated ) . @xmath74 has been estimated by a linear fit of the curve @xmath75 for the run with @xmath52 kpc ( for @xmath76 where it has a definite linear behaviour ) . by fitting the values of @xmath77 from our runs at different @xmath6 with the function ( [ nt ] ) we have determined @xmath78 . thus we get the following estimate for the slope of the mass function at time @xmath1 for a cluster with initial mass @xmath41 orbiting at a distance @xmath6 from the galactic center @xmath79 \label{nemp } \end{aligned}\ ] ] where we have taken @xmath80 and @xmath81 is the initial mean stellar mass for the imf adopted in our work . by eq.([mlo ] ) it is also possible to calculate @xmath73 at any given time @xmath1 , for a cluster located at @xmath6 and whose mass at time @xmath1 is @xmath82 . fig.5 shows the curves of @xmath83 as a function of the galactocentric distance for clusters having the same initial mass . eq.([nemp ] ) is valid as long as the number of particles is a linear function of time ; it is possible to see from fig . 1a that towards the end of the cluster lifetime , the behaviour of @xmath84 deviates from the linear scaling with time and slows significantly . the linear regime ends approximately at @xmath85 gyr ( or equivalently when @xmath86 ) ; for values of @xmath1 larger than this , eq.([nemp ] ) will provide a slope of the mf smaller ( i.e. a flatter mf ) than the real one . thus , in the plane @xmath87 , stellar mass loss and evaporation due to two - body relaxation cause points , initially on the line @xmath88 , to spread into the strip shown in fig.5 . while the curves on this diagram are qualitatively consistent with the correlation between @xmath73 and @xmath6 observed for galactic globular clusters ( see below ) , much caution would be needed in the use of these results in any _ detailed _ interpretation of observations . in the first place the observational data themselves are subject to considerable uncertainty from a variety of sources , including the distinction between global and local luminosity functions , the questionable reliability of ground - based data , and our poor knowledge of the mass - luminosity relationship for low - mass stars . next , there are limitations in our modelling , including the use of circular orbits and a single initial mass function , and measurement of the mass function over a single range in mass . also , we have found that the time scale on which the mf evolves is heavily determined by the initial total mass , which is unknown for observed clusters . one assumption of our models we have been able to check is the influence of primordial binaries , by comparison with models discussed by aarseth & heggie ( 1997b in preparation ) . inclusion of 10 percent hard binaries by number appears to decrease @xmath47 by an amount of order 0.05 ( i.e. the mass function is flattened ) ; this is small compared to the evolutionary changes and fluctuations in the data ( cf . fig.6 ) , and may not be significant . despite the caveats it is still interesting to compare our results with a representative sample of mass function slopes from the literature . for example capaccioli et al . ( 1993 ) have summarised ground - based data for 17 clusters and they find a dependence of @xmath73 on the current value of @xmath6 which is quite similar ( qualitatively and quantitatively ) to that of fig.5 for relatively low initial masses ( @xmath89 ) . to go further it would be necessary to give attention to most caveats stated above . for example , fig.13 ( below ) shows that disk shocking increases the evolution of @xmath73 beyond that indicated in fig.5 , while our values of @xmath73 are measured for stars of lower mass than is the case with the observational data . furthermore , it might be more realistic to compare our simulations with observational data for the dependence of @xmath73 on the estimated _ perigalactic _ distance of the clusters . incidentally it is often implied that the observed correlation is a signature of disk shocking . we have , however , shown that relaxation by itself can significantly affect the shape of the imf in a manner qualitatively consistent with observation . we turn now our attention to the differences between the global / initial mass function and the local mass function measured at various distances from the cluster center . since in some cases observational data are taken for different clusters at different distances from the cluster center it is important to understand how large the difference between a local mf and the global one can be and where , inside the cluster , this difference is larger , as well as to establish whether and where the pdmf keeps memory of the imf . we have calculated the slope of the mf in three shells : the innermost shell extends from the center to @xmath90 , where @xmath91 is the radius containing the innermost @xmath92 of the total number of stars ; the intermediate shell extends from @xmath90 to @xmath93 and the outermost shell extends from @xmath93 to @xmath94 ( both 3-d and 2-d ( projected distance ) shells have been considered ) . the limits of the three shells compared with the half - mass radius of the cluster , @xmath95 , are approximately @xmath96 . we focus our discussion on the results for the run at @xmath54 kpc and for that at @xmath52 kpc . the former undergoes a strong mass loss and evolution of the global value of @xmath73 while , in the latter , mass loss due to two - body relaxation is negligible and @xmath73 does not change significantly . if only mass segregation takes place and no star escapes from the cluster due to relaxation , the mass function of the innermost shell becomes flatter during the evolution while the mf of the outermost one becomes steeper . what happens when there is some mass loss depends on how strong this is and on how efficient it is in counteracting the trend given by mass segregation . in fact , while the mf of the inner shell will always tend to become flatter , the shape of the mf of the outermost shell depends on the relative efficiency of two processes acting in opposite directions : mass segregation tends to steepen the mf , evaporation tends to flatten it down . 6a - d show the time evolution of the mf in 2-d and 3-d shells for the runs at @xmath54 kpc and @xmath52 kpc . the above qualitative scenario is evident from figs . 6a - b , relative to the run at @xmath54 kpc : the mf of the innermost shell immediately flattens , that of the intermediate shell initially preserves its initial slope but eventually flattens , and the mf of the outermost shell after an initial stage during which tends to undergo a slight steepening eventually flattens when mass loss effects dominate mass segregation . at @xmath97 kpc the only relevant effects are those due to mass segregation and , as shown in figs . 6c - d , the slope of the mf is flatter than the initial one in the inner shells and steeper than the initial mf in the halo . we note that in this case no significant mass loss occurs and the mf near the half - mass radius resembles quite well the imf ( richer et al . 1991 , by an analysis of multi - mass king models , arrived at the same conclusion ) , but we emphasize that the results for the run at @xmath54 kpc , show this is not always the case , since , if a significant mass loss takes place , the pdmf at @xmath95 has no relationship with the imf after a few gyr . to provide a quantitative estimate of the difference between the initial / global mf and the local mf we have calculated the following quantities = _ shell-_glob , and = _ shell-_i . a negative ( positive ) value of these quantities means that the local mf is flatter ( steeper ) than the global / initial mf . the time evolution of the above quantities for the 3-d shells ( no qualitative difference is present in their behaviour for the 2-d shells ) is shown in figs 7a - d . at @xmath54 kpc a strong mass loss takes place which eventually makes the mf of all the shells flatter than the imf . on the other hand when the local mfs are compared with the global mf , the effects of mass segregation are evident as the outermost shell mf and the innermost shell mf are respectively steeper and flatter than the global one . the intermediate shell mf quite well resembles the global one during the entire evolution . at @xmath52 kpc the mass loss due to two - body relaxation is negligible and the global mf barely changes during the entire simulation . in this case mass segregation is the only process at work and the results obtained are in agreement with what is expected . from the above results we can draw the following conclusions : 1 ) the mf in the outer shells of a cluster is always steeper than the global one but it can be much flatter than the imf if strong mass loss occurred , thus changing it significantly ; this means that observational data taken in the outer regions of a cluster may be of no help in getting any information about the imf if the cluster has undergone a significant dynamical evolution ; 2 ) the mf near the half - mass radius is the most similar to the global one and thus it can be significantly different from the mf in more external regions of the cluster , always in the sense that the external mf is steeper ; much caution should be used in comparing observational data taken at different distances for different clusters and modelling of mass segregation effects ( see e.g. king , sosin & cool 1995 ) when any comparison is to be done is necessary.though it is not the purpose of this paper to investigate mass segregation as such , it is of interest to see how well its effect on the local mass function slope can be modelled with a multi - mass anisotropic king model . for the sake of illustration we took a model with initial mass @xmath98 and imf index @xmath99 at galactocentric radius @xmath100 kpc , and examined its structure at 20 gyr . this is a post - collapse model which has lost a little over half its mass and about 20 percent of its remaining mass is in the form of white dwarfs ( see section 3.3 ) . the stars were binned uniformly in @xmath101 in the range @xmath102 and in lagrangian radii corresponding to 10 uniformly spaced bins by total number . the white dwarfs were added to the bin of stars with average mass about @xmath103 . next a king model was selected to fit the central parameters of all species and , by variation of the remaining free parameters , the density profile of the stars of average mass about @xmath104 . finally the spatial profile of @xmath47 in the king model , computed as in the @xmath0-body models , was compared with the @xmath0-body result . it was found that the value of @xmath47 in the king model was slightly but sistematically smaller ( by an average of about 0.4 ) than in the @xmath0-body model , except in the inner 10 percent , where the models were forced to agree . undoubtedly a post - collapse model is a strenuous test of model fitting , and such a significant discrepancy would not be expected in less evolved models . any difference in the mfs of two clusters observed at different distances from the center does not necessarily imply their global mf are different and in any case their difference is likely not to signify the real difference between the global mfs . the code we have used for our investigation includes the effects of stellar evolution according to the same model used in chernoff & weinberg ( 1990 ) . besides studying the effects of this process on the dynamical evolution of a cluster it is thus possible to address the issue of the presence of degenerate remnants in the cluster , white dwarfs and neutron stars . as for the neutron stars these are found to be a very small fraction of the total mass after 15 gyr and no systematic investigation has been possible due to the small numbers involved . it has been possible to do much more for the white dwarfs . in table 1 the ratio of the total mass in white dwarfs to the total mass at @xmath43 gyr for all the runs done is shown . the first point coming out from the data is that the smaller the final value of the total mass , the larger the fraction of the mass in white dwarfs . the fraction of mass in white dwarfs results from two opposite processes : the production of white dwarfs from stars with initial mass less than 4.7 @xmath105 evolving from the main sequence and the loss by evaporation through the tidal boundary of the white dwarfs which have already formed or , possibly , of their main sequence progenitors . for a power - law imf with lower limit , @xmath55 , and upper limit @xmath56 , the ratio of the total mass in white dwarfs , @xmath106 , produced at a time @xmath1 , when the turn - off mass is @xmath57 , to the initial mass @xmath41 is given by @xmath107 & \mbox{if $ \tilde m/\mo<4.7$}. \end{array } \right . \label{wdt}\ ] ] the above expression would provide the total mass of white dwarfs if no loss of stars took place . actually a certain fraction of the white dwarfs produced escapes and this expression provides only an upper limit for @xmath106 . the real value of @xmath108 can be written as the product of the eq.([wdt ] ) by a function of time taking into account the reduction of the expected mass in white dwarfs due to the loss of stars through the tidal boundary ( m_wdm_i)_real=(m_wdm_i)_prodf(t ) [ wdr ] the function @xmath109 has to be determined by the results of @xmath0-body simulations . in fig . 8a we show the time evolution of @xmath108 for the runs starting with @xmath110 , @xmath111 and @xmath112 kpc . also shown in the figure is @xmath113 which , as expected , always lies above the curves corresponding to the data from @xmath0-body simulations . the function @xmath109 obtained from the ratio of @xmath108 from @xmath0-body data to @xmath114 ( eq.[wdt ] ) is shown for the same runs in fig . 8b . in fig . 8c the same function , this time for all the runs ( both @xmath45 and @xmath110 ) , versus the time scaled by the parameter @xmath40 is shown . as expected some spread is still present among curves corresponding to different initial conditions ; in fact even though the effect of two - body relaxation in causing stars to evaporate from a cluster is the same for the same value of the scaled time , the properties of the population of white dwarfs ( their mass distribution essentially ) is different and a difference in the fraction of escaped white dwarfs is thus expected . nevertheless this difference is not very large and we can approximately write @xmath109 as a unique function of the scaled time , @xmath115 . in order to obtain an analytic expression for this function , we have done a polynomial fit of the data obtained from the two runs at @xmath54 kpc starting with @xmath110 and @xmath45 ( fig.8d ) f(t , r_g , m_i)=f(t / f_cw)=1 - 0.794tf_cw+7.12 10 ^ -5(tf_cw)^2 -3.82 10 ^ -9(tf_cw)^3 [ polfit ] which provides a good approximation for @xmath116 gyr ( or equivalently for @xmath117 ) . by the above equations we can predict the fraction of the total ( initial or at any time @xmath1 ) mass in white dwarfs at any time @xmath1 for clusters at a galactocentric distance @xmath6 ( kpc ) with an initial mass @xmath118 or equivalently for clusters having a family parameter @xmath119 ( 0.28 being the mean value of the mass for the imf we are considering and for which the analytic expression of @xmath120 is valid ) . in fig . 9 we show the plot of @xmath121 versus @xmath40 : @xmath121 is a decreasing function of @xmath40 and this means that the fraction of the total mass in white dwarfs is larger for clusters undergoing a strong dynamical evolution and losing a large fraction of their initial mass ; the smaller the ratio of the final to the initial total mass is , the larger the fraction of the final mass in white dwarfs is . fig.10 shows the plot of @xmath121 versus the logarithm of the total mass of the cluster at @xmath43 gyr for different values of @xmath6 . not very much is currently known about the fraction of white dwarfs in globular cluster from observations ; the only estimates are supplied by authors using observed light ( and velocity dispersion ) profiles to model the structure and the stellar content of individual globular clusters usually by multi - mass king models ( see e.g. meylan 1987 and references therein ) . following this method , the fraction of white dwarfs is estimated assuming that the current value of the slope of the mf is equal to that of the imf and , consistently , that the only mass loss that occurred during the cluster lifetime is due to stellar evolution ( hereafter we will refer to this estimate as the observational estimate ) . it is clear that if mass loss due to relaxation takes place and , as a consequence of this the imf is different from the present one , this procedure will not provide a correct estimate of the content of white dwarfs . assuming that the slope of the imf is equal to @xmath122 , we can use the results of our simulations to provide a quantitative estimate of the error incurred . fig.11a shows the contour plot of @xmath123 in the plane @xmath124 providing the value of @xmath123 for any cluster once the slope of its mass function and the time this value of the slope is reached are known ( assuming the initial value of @xmath73 is equal to 2.5 ) ; dashed lines show the evolution of @xmath73 for clusters with different values of @xmath40 obtained from eq . ( [ nemp ] ) . fig.11b shows an analogous plot but in this case the values of @xmath123 are the observational estimates . finally fig.11c shows the contour plot of the ratio of the two above estimates of @xmath123 ( the estimate from @xmath0-body data to the observational one ) and it provides the correction to be made to the observational estimate ; the latter overestimates the content of white dwarfs but the error made for the fraction of white dwarfs at @xmath43 gyr is never too large , as the correction to be made is never smaller than about 0.65 at @xmath43 gyr . figs . 12a - c show plots analogous to those of figs . 11a - c but in the plane @xmath125-@xmath6 . in this section we will show the results of a set of simulations including the effects of disk shocking . the main goal is that of investigating to what extent this process can alter the results described in the previous section in which only stellar evolution and relaxation were considered . we begin by making some preliminary qualitative considerations about the expected changes due to disk shocking in the three quantities ( total mass , white dwarf content and mf slope ) we have focussed our attention on until now . as for the total mass , it is obvious that disk shocking will cause an additional mass loss and the total mass of a cluster at any time @xmath1 will be ( depending on the strength of disk shocking ) less than or equal to the corresponding value for a cluster starting with the same initial conditions but not undergoing disk shocking . it is not so obvious what to expect for the content of white dwarfs ; no simple qualitative argument can predict if the same correlation between the fraction of total mass in white dwarfs and the ratio of total to initial mass , shown to hold in the previous section , still holds in this case . in fact , it is not possible to know , _ a priori _ , if the mass lost by disk shocking will contain the same fraction of white dwarfs it would contain if this additional mass loss were due to two - body relaxation . analogous information is required to predict the effects of disk shocking on the evolution of the mf . as for this latter issue one difference between the mass loss due to disk shocking and that due to two - body relaxation is that , contrary to what happens for mass loss due to two - body relaxation , disk shocking does not produce any differential escape of stars with different mass unless the masses are segregated by some other mechanism ; the change in the velocity of a star due to disk shocking , for a cluster at a given galactocentric distance , depends only on the distance from the cluster center and not on the mass of the star . this means that only with the joint effect of mass segregation , driving low mass stars into the halo and high - mass stars into the core , can the mass loss due to disk shocking alter the slope of the mass function . the extent to which disk shocking can alter the mass function thus depends not only on the mass loss but also on mass segregation . initial conditions and the main results for the runs including disk shocking are shown in table 2 . besides running simulations with the same initial conditions adopted for runs without disk shocking , we have investigated a set of initial conditions having the same galactocentric distance but different values of the initial mass ( and thus different values of @xmath40 ) and a set of initial conditions all having the same value of @xmath40 but different galactocentric distances . the former set have been investigated in order to determine the differences in the effects of a sequence of interactions with a disk having the same strength ( fixed @xmath6 ) on clusters having different values of @xmath40 , while the latter have been done to study the effects of disk shocking with a varying strength on clusters all having the same initial value of @xmath40 . the results of these runs , besides providing useful indications on the effects of disk shocking on the stellar content of globular clusters , should make possible the derivation of more general expressions for the total mass , the slope of the mf and the fraction of the total mass in white dwarfs ( eqs.([mlo],[nemp],[wdr ] ) ) . in fig.13 we have plotted the slope of the mf after 15 gyr as a function of the galactocentric distance , showing to what extent disk shocking has changed the final result expected for runs starting with the same initial conditions but not undergoing disk shockings . besides the same quantities already shown in table 1 for the runs without disk shocking , the fraction of the initial mass lost by disk shocking @xmath126 and the additional change in the slope of the mf due to disk shocking have been included in table 2 . both of these quantities have been calculated as the difference between the values obtained from the simulations with disk shocking and the results expected for the same initial conditions without the effects of disk shockings , the latter being calculated by eqs.([mlo],[nemp ] ) . in figs . 14a - b the evolution of the total mass and of the slope of the mf for the runs not including disk shocking is compared with that obtained from runs with the same initial conditions and including the effects of disk shocking . in agreement with what is expected , the evolution of both the total mass and the mf is faster for systems undergoing disk shocking ( except for the system at @xmath52 kpc where the effects of disk shocking are negligible and the differences are due to statistical fluctuations ) . these plots provide generic information on the effects of disk shocking , but the most important information to answer the questions raised above concerning the fraction of white dwarfs contained in the mass lost by disk shocking and the effect of this process on the evolution of the slope of the mass function comes from the plots shown in figs . 15 - 16 . in these figures the fraction of the total mass at @xmath127 gyr in white dwarfs and the difference between the initial slope of the mf and that at @xmath127 gyr are plotted against the ratio of the total mass at @xmath43 gyr to the initial one ; the values predicted from the analytical expression derived in the previous section for the runs without disk shocking are also shown . both the change in the slope of the mf and the content of white dwarfs depend only on the fraction of mass lost during the evolution no matter whether relaxation is the only process causing the evaporation of stars or disk shocking is also responsible for a part of the escaping stars . this means that the content of the fraction of mass lost by disk shocking is similar to that lost by two - body relaxation . we note that , even though the values of the final mass we have plotted in fig.16 are determined both by mass loss due stellar evolution and relaxation / disk shocking / tidal stripping , obviously the relevant quantity in determining the variation in the slope of the imf ( for main sequence stars ) is only the escape of stars due to relaxation / disk shocking / tidal stripping . since the data plotted refer to systems all with the same imf and thus all losing by stellar evolution the same fraction of their initial mass , the difference between the real final mass and the final mass calculated considering only the escape of stars by relaxation / disk shocking is equal to a constant ( @xmath128 for the imf adopted in our work ; in fact @xmath129 tends to 0 for @xmath130 that is when the only mass loss is due to stellar evolution ) . should data from systems losing a different fraction of their initial mass by stellar evolution be considered for this plot , this correction should be properly taken into account . the same remark applies to fig.15 ; in this case also the dependence on the imf of the total mass of white dwarfs produced should be considered . if , analogously to what was done for the runs without disk shocking , we could get an analytical expression for the time evolution of the mass , we might then use the relationship between this quantity and the fraction of white dwarfs and the slope of the mf to obtain these quantities at any time @xmath1 and for a quite general set of initial conditions . 17 shows the time evolution of the total mass for four different runs including disk shocking , two for initial conditions with the same value of @xmath40 and two for initial conditions having the same galactocentric distances . the mass lost by stellar evolution has been added so that the plotted lines show only the mass lost by disk shocking and two - body relaxation . similarly to what happens when disk shocking is not included , the mass is a linear function of time . this and the results coming from figs . 15 - 16 suggest that it is possible for any initial condition to define an _ equivalent family parameter _ , @xmath131 having the following meaning : the evolution of a cluster with a family parameter @xmath40 whose evolution is driven by both two - body relaxation and disk shocking is equivalent to that of cluster whose evolution is driven by two - body relaxation only but whose family parameter is @xmath131 , where @xmath132 always . of course @xmath131 depends on @xmath40 and on @xmath6 and the main goal is now that of finding out this dependence by which we will be able to calculate analytically any quantity we are interested in once the initial conditions are specified , exactly as we have done in the previous section . in the previous section we have shown that the mass loss is a linear function of time and that the mass loss rate is inversely proportional to the family parameter @xmath40 ; in order to get an analytic expression for @xmath131 we have calculated from the results of our @xmath0-body simulations an empirical analytical expression for the mass loss rate as a function of @xmath40 and @xmath6 and derived from this @xmath131 . analogously to what was done for the runs without disk shocking , we can write = 1-m_st.ev . m_i - t [ mlods ] ; as shown in the previous section , when two - body relaxation is the only process considered , @xmath133 depends only on @xmath40 ( @xmath134 , see eq.([mlo ] ) ) ; if the effects of disk shocking also are taken into account @xmath133 depends both on @xmath40 and on @xmath6 . from our data we find = 0.6931 - 1.46 r_g-1.134f_cw+0.2916f_cw r_g [ slp ] . the above expression is completely empirical and it has been derived from data spanning a limited range of values of @xmath6 ( @xmath135 ) and @xmath40(@xmath136 ) but it nevertheless provides useful qualitative and quantitative information on the evolution of clusters including the effects of disk shocking . the plot of @xmath133 obtained from @xmath0-body data versus the value calculated from eq.([slp ] ) , fig . 18 , shows that eq.([slp ] ) approximates well the dependence of @xmath133 on @xmath40 and @xmath6 . as explained above , once @xmath133 is known , we can easily calculate @xmath131 ; fig.19a shows the ratio of the total mass at @xmath127gyr to the initial mass versus @xmath40 for all the runs done , both with and without disk shocking , with the solid line showing the values predicted from eq.([mlo ] ) . as expected the real value of @xmath40 does not provide a good indication of the mass loss for the runs with disk shocking ; if @xmath40 is replaced by @xmath131 ( of course no difference exists between these two quantities for runs without disk shocking ) all the data are located , with a good approximation , along the curve predicted by eq.([mlo ] ) ( fig.19b ) . in fig . 20 families of models having given values of @xmath40 and @xmath131 are shown ; the qualitative behaviour agrees with what one would expect : two clusters located at the same galactocentric distance , one undergoing disk shocking and one not , evolve in the same way if the initial mass of the former is larger than that of the latter , while if they have the same initial mass , they evolve in the same way if the former is located at a galactocentric distance larger than the latter . analogous information is contained in fig . 21 where curves of equal - mass at @xmath43 gyr are shown for clusters evolving with and without disk shocking in the plane @xmath137 . we can now calculate analytically @xmath47 and the content of white dwarfs at any time @xmath1 and for any initial condition ( within the limits mentioned in this and in the previous section and for the particular choice of imf adopted in our work ) taking into account also the effects of disk shocking . 22 shows the changes due to disk shocking in the curves of @xmath138 versus @xmath6 for different values of the initial mass . as for the formation of a correlation between the slope of the mass function and the galactocentric distance , as could be seen already from fig . 13 , disk shocking has the effect of increasing the trend established by two - body relaxation . by the simulations described in the previous sections , starting with @xmath4 particles , we are trying to model the evolution of stellar systems typically having @xmath139 stars ; the time scaling adopted to convert time in @xmath0-body units to astrophysical units should allow a proper use of the @xmath0-body data to investigate the evolution of real clusters . nevertheless , as shown also in aarseth & heggie ( 1997a ) , a slight dependence of the results on @xmath0 still exists . in order to give a quantitative estimate of the extent of the @xmath0-dependence we have done seven runs starting with the same initial conditions @xmath140 but with a different initial number of particles ( four runs with @xmath4 , two runs with @xmath141 and one run with @xmath142 so as to have quantities all with the same statistical significance for all the values of @xmath0 investigated ) . in table 3 we summarize the relevant information on these runs . in agreement with the results of aarseth & heggie ( 1997a ) , we find that the larger the number of particles is , the faster the evolution of the system is . part of the explanation of this is likely to reside in the differences in the structure of clusters with different @xmath0 at the end of the initial phase dominated by mass loss due to stellar evolution . in fact , as explained in larger detail in aarseth & heggie ( 1997a ) , scaling time from @xmath0-body units to astrophysical units by the ratio of @xmath0-body to real relaxation time implies that the smaller the number of stars is , the smaller the ratio of time scale of mass loss due stellar evolution to the crossing time scale is . this is quite important since it is well known ( see e.g. hills 1980 , lada , marculis & dearborn 1984 , fukushige & heggie 1995 , aarseth & heggie 1997a ) that the rate at which mass loss occurs plays an important role in determining the evolution of a stellar system , spanning from an expansion of the entire cluster proportional to @xmath143 without the escape of any star in the case of slow mass loss to the complete disruption of the cluster in the case of rapid mass loss exceeding 1/2 of the entire initial mass of the cluster ; this difference is likely to produce a difference in the final structure of clusters losing mass by stellar evolution in the impulsive or adiabatic regime . the subsequent differences in the mass loss rate are likely to be due , at least partially , to this difference even though further investigation on this point is needed . some simulations with @xmath4 and @xmath141 not including stellar evolution have been carried out to test to what extent the above effect was actually able to explain the observed differences . the trend for system with larger values of @xmath0 to evolve faster has been found to be still present , even though to a smaller extent . this means that part of the differences observed are unlikely to be due to the chosen scaling and their origin must be due to a real difference in the evolution of systems with different @xmath0 not simply scaling with the relaxation time . we could not find a convincing explanation for this result but possibly the dependence on @xmath0 of the depth of the potential well and the consequent dependence of the fraction of stars ejected from the core reaching the outer regions ( see e.g. giersz & heggie 1994 ) might explain the observed differences . 23 , in which we have plotted the slope of @xmath144 , i.e. @xmath60 , ( excluding the mass loss due to stellar evolution ) versus @xmath0 , clearly shows the increase in the mass loss rate as @xmath0 increases . a correction of approximately @xmath145 seems to be necessary to the conversion from time in @xmath0-body units to time in astrophysical units adopted in the set of runs with @xmath4 . as shown in figs . 15 and 16 , however , the runs for all values of n yield the same result for the way in which both the variation in the slope of the mass function and the content of white dwarfs at @xmath43 gyr depend on the fraction of initial mass left after 15 gyr . while further investigation on the @xmath0 dependence of the results is still necessary , particularly for what concerns the origin of this dependence , in the light of this last result it is clear that the correction to be made just implies that our results from simulations with @xmath4 at @xmath43 gyr would actually be more relevant for real clusters at @xmath146 gyr and that results from simulations at @xmath147 gyr would be those relevant for real clusters at @xmath43 gyr . in this work we have carried out a large set of @xmath0-body simulations to investigate the dynamical evolution of globular clusters , focusing our attention on the effects of dynamical evolution on the stellar content of the systems ; in particular we have investigated the evolution of the mf and of the fraction of the total mass in white dwarfs . the code used includes the effects of stellar evolution , two - body relaxation and disk shocking and takes the presence of the tidal field of the galaxy into account . a set of different initial conditions for the structure of the cluster and for the galactocentric distance has been considered . the dependence of the slope of the mf and of the fraction of white dwarfs on the initial conditions of the clusters has been explored and in particular we have determined to what extent the observed correlation between the slope of the mf and the galactocentric distance can result from the effects of dynamical evolution ; we have tried to provide a quantitative estimate of the role played by relaxation and disk shocking . the dependence of the slope of the mass function on the location inside the cluster has been also investigated , our attention being focused on the differences between the local mf , the imf and the global pdmf . the main conclusions we can draw are : 1 . in agreement with what is expected , as a result of mass loss through the tidal boundary , both due to two - body relaxation and to disk shocking , the global mass function becomes flatter . for given initial parameters , mass loss is stronger for clusters closer to the galactic center , and , consequently , a trend between the slope of the mf and the galactocentric distance forms as evolution goes on . this trend is stronger for low - mass clusters , as these have shorter relaxation times and thus evolve more quickly than massive clusters . both mass loss by two - body relaxation and disk shocking are important in causing the mf to flatten . by the results of @xmath0-body simulations we have derived an analytical expression for the slope of the mass function at any time @xmath1 and for any initial value of the mass and of the galactocentric distance both with and without the effects of disk shocking . + the difference between the initial and the final ( at @xmath43 gyr ) slope of the mf has been shown to depend approximately only on the fraction of the initial mass lost and this dependence is the same no matter whether disk shocking is included or not . the mf near the half - mass radius is the one which , during the entire evolution , is least affected by mass segregation and quite well resembles the present - day global mass function . the extent to which the mf observed near the half - mass radius can provide us with useful information on the imf thus depends on the difference between the imf and the pdmf . possibly , when the the pdmf is different from the imf , observations of the mf at radii larger than the half - mass radius can supply indications on the imf , but it is important to note that , in cases of strong mass loss and evolution of the imf , we have shown that , even the mf in the outer regions of the clusters eventually becomes flatter than the imf whose memory is thus completely erased from the pdmf . the mf in the inner regions is always significantly flatter than the pdmf as a result of mass segregation . 3 . the ratio of the total mass of white dwarfs retained in the cluster to the total mass of the cluster , @xmath123 , increases during the evolution : as the fraction of the initial mass left in the cluster decreases , the fraction of this in white dwarfs increases . the total mass of white dwarfs is determined by the interplay between the rate of production determined by the time scales of stellar evolution and the rate of escape through the tidal boundary of the white dwarfs , or possibly of their progenitors , determined by the relaxation time and disk shocking time scale . we have obtained an analytical expression for the fraction of white dwarfs present in a cluster as a function of the initial conditions and of time as the product of the production rate , that can be easily derived analytically once the imf has been given , by the escape rate , which is derived instead from a fit to the @xmath0-body data . analogously to what was shown for the difference between the initial and the final slope of the mf , the dependence of @xmath123 on @xmath148 is the same no matter whether the effects of disk shocking are included or not . + we have made a comparison between our estimate of the fraction of white dwarfs and the one which would be obtained by extrapolating the present day main sequence mass function ( a procedure often used in literature ) and we have shown that the former is , in most cases , smaller than the latter , with the ratio of the two estimates ranging , in most relevant cases , from about 0.65 to 1 depending on the initial conditions . all the simulations we have carried out for our work started with an initial number of particles @xmath4 . the dependence of our results on @xmath0 has been checked by an additional set of simulations starting with larger number of particles ( @xmath141 , @xmath142 ) . in agreement with the results by aarseth & heggie ( 1997a ) we have shown that the scaling of time from @xmath0-body units to astrophysical units adopted for @xmath4 requires a correction of about @xmath145 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iau symp . 174 , eds . p.hut and j.makino , p.29 + king , i.r . , sosin , c. , cool , a.m . , 1995 , apjl , 452 , l33 + lada c.j . , margulis m. , dearborn d.s . , 1984 , apj , 285 , 141 + makino , j. , kokubo , e. , taiji , m. , 1993 , pasj , 45 , 349 + mcclure , r.d.,et al . , 1986 , apj , 307,l49 + meylan , g. , 1987 , a&a , 184 , 144 + paresce , f. , de marchi , g. , romaniello , m. , 1995 , apj , 440 , 216 + piotto , g. , 1991,in formation and evolution of star clusters , ed . k.janes , aspcs vol.13 , p.200 + piotto , g. , cool , a.m. , king i.r . , 1996 , in dynamical evolution of star clusters , iau symp . 174 , eds . p.hut and j.makino , p.71 + piotto , g. , cool , a.m. , king i.r . , 1997 , aj , in press + richer , h.b . , fahlman , g.g . , buonanno , r. , fusi pecci , f. , searle , l. , thompson , i.b . , 1991 , apj , 381,147 + richer h.b . , fahlman g.g . , ibata r.a . , stetson p.b . , bell r.a . , bolte m. , bond h.e . , harris w.e . , hesser j.e . , mandushev g. , pryor c. , vandenberg d.a . , 1995 , apj , 451 , l17 + richer h.b . , fahlman g.g . , ibata r.a.,pryor c.,bell r.a . , bolte m. , bond h.e . , harris w.e . , hesser j.e . , holland s. , ivanans n. , mandushev g. , stetson p.b . , wood m.a . , 1997 , apj , submitted + santiago , b.,elson , r. , gilmore , g. , 1996 , mnras , 281 , 1363 + spitzer , l. , 1958 , apj , 127 , 17 + spitzer , l. , 1987 , dynamical evolution of globular clusters , princeton university press + stiavelli , m. , piotto , g.,capaccioli , m. , ortolani , s. , 1991,in formation and evolution of star clusters , ed . k.janes , aspcs vol.13 , 449 + stiavelli , m. , piotto , g.,capaccioli , m. , 1992 , in morphological and physical classification of galaxies , ed . g.longo , m.capaccioli and g. busarello , p.445 + weinberg , m. , 1994 , aj , 108 , 1414 + + @xmath149&@xmath150 ( kpc)&@xmath151&@xmath152 & @xmath153&@xmath154&@xmath155&@xmath156&@xmath157 + @xmath158 & 4&@xmath159&7 & 2.5 & 0.205 & 0.279 & 1.46 & 1.04 + @xmath158 & 5&@xmath160 & 7 & 2.5 & 0.307 & 0.199 & 1.81 & 0.69 + @xmath158 & 8&@xmath161 & 7 & 2.5 & 0.524 & 0.149 & 2.21 & 0.29 + @xmath158 & 16&@xmath162 & 7 & 2.5 & 0.657 & 0.128 & 2.41 & 0.09 + @xmath158 & 4&@xmath159 & 5 & 2.5 & 0.195 & 0.273 & 1.38 & 1.12 + @xmath158 & 5 & @xmath163 & 5 & 2.5 & 0.344 & 0.196 & 1.96 & 0.54 + @xmath158 & 8 & @xmath164 & 5 & 2.5 & 0.542 & 0.150 & 2.31 & 0.19 + @xmath158 & 16 & @xmath165 & 5 & 2.5 & 0.670 & 0.130 & 2.39 & 0.11 + @xmath158 & 4 & @xmath166 & 7 & 3.5 & 0.633 & 0.013 & 3.27 & 0.23 + @xmath158 & 5 & @xmath167 & 7 & 3.5 & 0.702 & 0.012 & 3.39 & 0.11 + @xmath158 & 8&@xmath168 & 7 & 3.5 & 0.808 & 0.014 & 3.42 & 0.08 + @xmath158 & 16&@xmath169 & 7 & 3.5 & 0.890 & 0.019 & 3.46 & 0.04 + + + + @xmath149&@xmath6 ( kpc)&@xmath151&@xmath170&@xmath155&@xmath171&@xmath157 & @xmath172 & @xmath126 + @xmath158 & 4&@xmath173 & 0.080 & 0.340 & 0.37 & 2.13&1.05 & 0.113 + @xmath158 & 5&@xmath160 & 0.254 & 0.242 & 1.50 & 1.00&0.39 & 0.064 + @xmath158 & 8 & @xmath174&0.489 & 0.149 & 2.18 & 0.32 & 0.04 & 0.015 + @xmath158 & 16&@xmath162 & 0.670 & 0.127 & 2.38 & 0.12 & 0.002 & -0.011 + @xmath175 & 2.1&@xmath160 & 0.166 & 0.292 & 1.12 & 1.38 & 0.77 & 0.156 + @xmath175 & 3.3 & @xmath174&0.418 & 0.168 & 2.10 & 0.40 & 0.11 & 0.083 + @xmath175 & 6.7&@xmath162 & 0.610 & 0.123 & 2.35 & 0.15 & 0.03 & 0.050 + @xmath176 & 5 & @xmath177 & 0.082 & 0.357 & 0.39 & 2.11 & 0.99 & 0.107 + @xmath178 & 5 & @xmath179 & 0.337 & 0.179 & 1.91 & 0.59 & 0.16 & 0.068 + @xmath180 & 5 & @xmath181&0.391 & 0.176 & 2.02 & 0.48 & 0.14 & 0.073 + @xmath182&5&@xmath183&0.471 & 0.148 & 2.13 & 0.37 & 0.12 & 0.057 + @xmath184&5 & @xmath185&0.563 & 0.142 & 2.30 & 0.20 & 0.03 & 0.043 + @xmath186&5&@xmath187&0.658 & 0.127 & 2.40 & 0.10&0.02 & 0.047 + @xmath188 & 14.0 & @xmath122&0.345 & 0.195 & 1.95 & 0.55 & -0.07 & -0.029 + @xmath189 & 9.2&@xmath160&0.323 & 0.210 & 1.85 & 0.65 & 0.03 & -0.006 + @xmath190 & 7.45 & @xmath122&0.266 & 0.222 & 1.64 & 0.86&0.24 & 0.051 + @xmath191 & 3.95 & @xmath122 & 0.233 & 0.255 & 1.53 & 0.97 & 0.35 & 0.084 + @xmath192 & 3.2&@xmath160&0.191 & 0.257 & 1.53 & 0.97&0.35 & 0.127 + @xmath186 & 1.1&@xmath160 & 0.173 & 0.289 & 1.25 & 1.25&0.63 & 0.143 + + + @xmath193&@xmath0&@xmath170&@xmath155&@xmath156&@xmath194(gyr ) + i&4096&0.195 & 0.273 & 1.38 & 6.0 + ii&4096&0.242&0.268 & 1.50 & 7.5 + iii&4096&0.195&0.281&1.44 & 5.5 + iv&4096&0.202&0.286 & 1.32 & 7.1 + i&8192&0.127 & 0.365 & 0.86 & 6.3 + ii&8192&0.156&0.325&1.25 & 6.9 + i&16384&0.075 & 0.422 & 0.54 & 7.6 + + + + + + figure 1 time evolution of the main properties of systems starting with @xmath195 , @xmath110 , @xmath196 and having galactocentric distances equal to @xmath54 kpc ( solid line ) , @xmath197 kpc ( dotted line ) , @xmath198 kpc ( short - dashed line ) , @xmath52 kpc ( long - dashed line ) . ( a ) evolution of the total number of stars;(b ) evolution of the ratio of total mass to initial mass ; ( c ) evolution of the total number of stars having a mass in the range @xmath199 ( upper curves ) and @xmath200 ( lower curves ) for the systems located at @xmath54 kpc ( solid lines ) and at @xmath52 kpc ( long - dashed lines ) ; ( d ) evolution of the ratio of the total mass of neutron stars to the total mass at time @xmath1 ; ( e ) evolution of the ratio of total mass of white dwarfs to the total mass at time @xmath1 ; ( f ) evolution of the slope of the mass function calculated for main sequence stars in the range @xmath201 . + figure 2 time evolution of the mass function for main sequence stars for the system with @xmath195 , @xmath110 , @xmath202 and @xmath54 kpc . the four curves shown correspond ( from the upper to the lower one ) to @xmath203 gyr , @xmath204 gyr , @xmath205 gyr , @xmath43 gyr . + figure 3 ( a ) time evolution of the ratio of total mass to initial mass for systems with @xmath195 , @xmath110 , @xmath196 and @xmath112 kpc ( symbols as in figure 1 ) . straight lines take into account only the mass loss by relaxation , curved lines only the mass loss by stellar evolution . the latter curve does not depend significantly on the galactocentric distance of the system and the four curves are almost indistinguishable ; ( b ) evolution of the ratio of total mass to initial mass for systems with @xmath195 , @xmath110 , @xmath196 and @xmath112 kpc ( symbols as in figure 1 ) including only the mass loss by relaxation versus time scaled by the value of @xmath40 for each system . + figure 4 ( a ) fraction of the initial mass remaining in a cluster after @xmath206 gyr , @xmath207 , as a function of @xmath208 and @xmath6 for systems with @xmath195 and @xmath110 ; ( b ) contours of constant values of @xmath207 , as indicated on the right end of the curves shown in the plot , as a function of the galactocentric distance and of the logarithm of the initial mass . + figure 5 slope of the mass function at @xmath127 gyr measured for main sequence stars with masses in the range @xmath209~m_{\odot}$ ] from the analytic expression derived in the text ( see eq.([nemp ] ) ) as a function of galactocentric distance for the following values of the initial mass of the cluster ( from the upper to the lower curve ) : @xmath210 . initial conditions are @xmath195 , @xmath110 . dots represent values obtained from @xmath0-body simulations for @xmath211 . + figure 6 time evolution of the slope of the mass function of main sequence stars with @xmath201 calculated in three different 3-d and 2-d shells ( see text for their exact limits ) : data for the innermost shell are plotted by a solid line and circles , for the intermediate shell by short - dashed line and triangles , for the outermost shell by long - dashed line and crosses . + initial conditions are @xmath195,@xmath110 , @xmath202 and galactocentric distance as indicated below . ( a ) 3-d shells , @xmath51 kpc ; ( b ) 2-d shells , @xmath51 kpc ; ( c ) 3-d shells , @xmath52 kpc ; ( d ) 2-d shells , @xmath52 kpc . + figure 7 ( a ) evolution of the difference between the slope of the mass function in 3-d shells and the slope of the initial mass function ( all measured for main sequence stars with @xmath201 ) for the system at @xmath54 kpc . initial conditions of the systems : @xmath212 . ( b ) evolution of the difference between the slope of the mass function in 3-d shells and the slope of the global mass function at time @xmath1 ( all measured for main sequence stars with @xmath201 ) for the system at @xmath54 kpc . + ( c)-(d ) same as ( a ) and ( b ) but for the system at @xmath52 kpc . symbols as in figure 6 . + figure 8 ( a ) time evolution of the ratio of total mass in white dwarfs at time @xmath1 to the total initial mass for systems with @xmath195 , @xmath110 , @xmath213 and @xmath54 kpc ( solid line ) , 5 kpc ( dotted line ) , 8 kpc ( short - dashed line ) , 16 kpc ( long - dashed line ) . the dot - dashed line is the fraction of total mass in white dwarfs expected if all the white dwarfs produced were retained in the cluster and none escaped due to relaxation ( see eq.([wdt ] ) in the text ) , @xmath214 . + ( b ) ratio of the total mass of white dwarfs retained in a system at time @xmath1 to the total mass of white dwarfs actually produced for systems with @xmath195 , @xmath110 , @xmath196 and @xmath54 kpc ( solid line ) , 5 kpc ( dotted line ) , 8 kpc ( short - dashed line ) , 16 kpc ( long - dashed line ) versus time . + ( c ) same as ( b ) but with time scaled by the parameter @xmath40 . data from runs with @xmath45 are also shown in this plot . ( symbols distinguishing different @xmath6 as in ( b ) ) . + ( d ) ratio of total mass in white dwarfs at time @xmath1 to the total mass of white dwarfs produced at time @xmath1 versus @xmath115 ( average values of the two runs starting @xmath45 and @xmath110 both with @xmath195 , @xmath54 kpc , @xmath215 ) . the line superimposed is the result of a polynomial fit ( see eq.([polfit ] ) in the text ) . + figure 9 fraction of the total mass at @xmath43 gyr in white dwarfs as a function of the parameter @xmath40 calculated by eq.([wdr ] ) ( see text ) . + figure 10 fraction of the total mass at @xmath43 gyr in white dwarfs as a function of @xmath216 at different distances from the galactic center ( calculated by eq.([wdr ] ) in the text ) . + figure 11 in all figures dashed lines show the time evolution of the slope of the mass function for clusters with ( from the lower to the upper curve ) @xmath217 calculated by eq.([nemp ] ) in the text . + ( a ) contours of equal values of @xmath218 as a function of time , @xmath1 , and slope of the mass function at time @xmath1 calculated according to eq.([wdr ] ) in the text taking @xmath195 and @xmath110 . + ( b ) contours of equal values of @xmath218 as a function of time , @xmath1 , and slope of the mass function at time @xmath1 calculated by extrapolating the current properties of a cluster back in time ( `` observational '' procedure ; see text for further details ) . ( c)contours of equal values of the ratio of the theoretical estimate of @xmath218 to the `` observational '' one as a function of time , @xmath1 , and slope of the mass function at time @xmath1 . + figure 12 as in figure 11 but as a function of the logarithm of the mass of the cluster at @xmath43 gyr and of its galactocentric distance . + figure 13 same as figure 5 with arrows showing the additional change in the slope of the mass function due to the effects of disk shocking ( data from @xmath0-body simulations ) . + figure 14 ( a ) comparison of the time evolution of @xmath148 for systems with the effects of disk shocking and without them . all the systems start with @xmath195 and @xmath110 , @xmath202 ; galactocentric distances are @xmath112 kpc ( solid , dotted , short - dashed and long - dashed line respectively).for each pair the lower curve is the one relative to the run with disk shocking , with the exception of the runs at @xmath52 kpc for which the lower one refers to the run without disk shocking . + ( b ) same as ( a ) for the evolution of the slope of the mass function . + figure 15 fraction of the total mass at @xmath43 gyr in white dwarfs as a function of the fraction of the initial mass left in the cluster at @xmath43 gyr . solid line is calculated by the analytical expression ( eq.([wdr ] ) ) derived in the text . dots are data from @xmath0-body simulations with @xmath195 ( full dots from runs without disk shocking , see table 1 and table 3 ; circles from runs with disk shocking , see table 2 ; triangles from runs starting with @xmath141 and the cross refers to the run with @xmath142 see table 3 ) . + figure 16 difference between the initial value of the slope of the mass function ( @xmath195 ) and its value at @xmath43 gyr as a function of the fraction of the initial mass left in the cluster at @xmath43 gyr . solid line is calculated by the analytical expression ( eq.([nemp ] ) ) derived in the text . dots are data from @xmath0-body simulations ( symbols as in figure 15 ) . + figure 17 time evolution of the ratio of mass at time @xmath1 to initial mass not considering mass loss due to stellar evolution for four runs with disk shocking . straight lines show the best linear fit of the curves obtained from @xmath0-body data . from the lower to the upper one , the curves refer to the following initial conditions @xmath219 , @xmath220 , @xmath221 , @xmath222 . + figure 18 slope of @xmath148 ( not taking into account mass loss by stellar evolution ) for runs with disk shocking from @xmath0-body data versus the analytical expression derived in the text ( see eq.([slp ] ) ) . the line is given by @xmath223 . + figure 19 ( a ) @xmath224 versus the parameter @xmath40 for runs with ( circles ) and without ( full dots ) disk shocking . solid line is calculated by the analytical expression derived in the text ( eq.([mlo ] ) ) . ( b ) same as ( a ) but with the family parameter @xmath40 replaced by the equivalent family parameter @xmath131 defined in the text . + figure 20 contours of equal values ( given at the right side of each pair of curves ) of the family parameter @xmath40 ( solid lines ) and of the equivalent family parameter , @xmath131 , defined in the text . + figure 21 contours of equal values of the mass of a cluster at @xmath43 gyr with ( dashed lines ) and without ( solid lines ) the effects of disk shocking ; initial conditions @xmath110 and @xmath195 . the number beside each curve gives @xmath125 . + figure 22 slope of the mass function at @xmath127 gyr for main sequence stars with masses in the range @xmath209~m_{\odot}$ ] without ( solid lines ) and with ( dashed lines ) the effects of disk shocking calculated by the analytic expression derived in the text ( eq.([nemp ] ) with @xmath40 replaced by @xmath131 for the case with disk shocking ) as a function of galactocentric distance for the following values of the initial mass of the cluster ( from the upper to the lower curve ) : @xmath210 . initial conditions are @xmath195 , @xmath110 . + figure 23 slope of @xmath144 for systems starting with @xmath225 and @xmath202 versus the initial number of stars in the simulation @xmath0 . disk shocking was not included in these runs . four runs have been done for @xmath4 and two runs for @xmath141 .

in this paper we show the results of a large set of @xmath0-body simulations modelling the evolution of globular clusters driven by relaxation , stellar evolution , disk shocking and including the effects of the tidal field of the galaxy . we investigate the evolution of multi - mass models with a power - law initial mass function ( imf ) starting with different initial masses , concentrations , slopes of the imf and located at different galactocentric distances . we show to what extent the effects of the various evolutionary processes alter the shape of the imf and to what extent these changes depend on the position of the cluster in the galaxy . both the changes in the global mass function and in the local one ( measured at different distances from the cluster center ) are investigated showing whether and where the local mass function keeps memory of the imf and where it provides a good indication of the current global mass function . the evolution of the population of white dwarfs is also followed in detail and we supply an estimate of the fraction of the current value of the total mass expected to be in white dwarfs depending on the main initial conditions for the cluster ( mass and position in the galaxy ) . simple analytical expression by which it is possible to calculate the main quantities of interest ( total mass , fraction of white dwarfs , slope of the mass function ) at any time @xmath1 for a larger number of different initial conditions than those investigated numerically have been derived . 6ngc 6397 globular clusters : general stellar dynamics

astro-ph9705073

bl lac objects ( hereinafter bl lacs or bll ) are active galactic nuclei ( agn ) characterized by luminous , rapidly variable uv to nir non thermal continuum emission and polarization , strong compact flat spectrum radio emission and superluminal motion . similar properties are observed also in flat spectrum radio quasars ( fsrq ) and the two types of active nuclei are often grouped together in the blazar class . from the spectroscopical point of view bl lacs are characterized by quasi featureless optical spectra . in fact their spectra are often dominated by the non thermal continuum that arises from the nucleus . to this emission it is superimposed a thermal contribution due to the stellar component of the host galaxy . like in other agn , emission lines could be generated by fluorescence in clouds surrounding the central black hole . moreover , as it happens for high z quasars in some cases absorption lines due to intervening gas in the halo of foreground galaxies can be observed in the spectra of bl lacs and one can derive a lower limit to the redshift of the object . the detectability of spectral features depends on the brightness of the nuclear source : in fact during low brightness states , intrinsic absorption features can be more easily revealed , while during high states one can better discover intervening absorption systems . because of the strong contribution from the continuum the equivalent width ( ew ) of all these spectral features is often very small and their detection represents a challenging task . in the past decade a number of projects were carried out to derive the redshift of bl lac objects . most of these works were based on optical spectra collected with 4 m class telescopes , and are therefore limited by relatively low signal - to - noise ratio ( s / n ) , low spectral resolution and limited wavelength range ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? recently , however , some observations with 8 m class telescopes were carried out @xcite . despite these efforts , a significant fraction of known bl lacs ( e. g. 50 % in @xcite catalogue ) have still unknown redshift . in order to improve the knowledge of the redshift of bl lacs we carried out a project to obtain optical spectra of sources with still unknown or uncertain redshift using the european southern observatory ( eso ) 8-m very large telescopes ( vlt ) . this allows one to improve significantly the s / n of the spectra and therefore the capability to detect faint spectral features . a first report on this work , giving the redshift of 12 objects , has been presented by ( * ? ? ? * paper i ) , and here we refer on the results for the full sample of 42 observed sources . the outline of this paper is the following . in section [ sec : sample ] we give some characterization of the 42 observed objects . the observation and analysis procedures are described in section [ sec : obsred ] . in sections [ sec : results ] and [ sec : notes ] we report the results of our spectroscopic study . finally in section [ sec : discussion ] a summary and conclusions of this study are given . throughout this paper we adopted the following cosmological parameters : h@xmath1= 70 km s@xmath2 mpc@xmath2 , @xmath3=0.7 , @xmath4=0.3 . the sample of bl lac objects ( and candidates ) observed with the vlt telescopes was selected from two extended lists of bl lacs : the @xcite collection of bl lacs and the sedentary survey ( * ? ? ? * ; * ? ? ? * in the following addressed as ss ) . the @xcite list contains all objects identified as bl lacs belonging to the complete samples existing at the time of its compilation , selected in the radio , optical and x - ray bands ( e.g. : 1 jansky survey 1-jy , @xcite , palomar - green survey pg , @xcite , extended medium sensitivity survey emss , @xcite , slew survey , @xcite , white - giommi - angelini catalogue wga @xcite ) . it includes also sources from the @xcite and @xcite catalogues ( in the latter case we checked that the source was still included in the 2001 version ) , for a total of 233 objects . the criteria used to define a bl lac object in @xcite depend on the sample of origin . in most cases , the ew of the lines is required to be @xmath55 , but also uv excess , optical polarization and variability , radio - to - optical spectral index are used as selecting criteria . the ss was obtained cross - correlating the national radio astronomy observatory ( nrao ) very large array ( vla ) sky survey ( nvss ) data @xcite with the rosat all sky survey bright source catalogue ( rbsc ) list of sources @xcite . ss selected a complete sample of 150 high energy peaked bl lacs ( hbl , see * ? ? ? * for definition ) down to a 3.5 mjy radio flux limit . bl lac classification in the ss is based on the position of the sources on the @xmath6 plane . the @xcite and ss datasets lead to a combined list containing 348 objects . the distribution of the v magnitude for these objects is reported in fig . [ fig : distv ] . the bulk of them have v between 15 and 20 , and the the fraction of objects with unknown redshift increases with the apparent magnitude and reaches @xmath0 50% at the faintest magnitudes . note , however , that also at v @xmath0 15 - 17 about 20% of the sources have not known redshift . the total number of objects with unknown redshifts is 105 . from the combined list we selected sources with @xmath7 + 15@xmath8 , for observability from the vlt site . moreover to grant a sufficiently high s / n level of the optical spectra we required v@xmath922 . thus we gathered a list of 59 objects . during three observational campaigns , performed in service mode , we completed this optical spectroscopy program , obtaining data for @xmath070% of the sample ( 42 sources ) . our sample is similar to the parent sample of 348 objects in terms of mean apparent magnitude and subdivision in low ( lbl ) and high energy peaked bl lacs . optical spectra were collected in service mode with the focal reducer and low dispersion spectrograph ( fors1 , * ? ? ? * ) on the vlt . the observations were obtained from april 2003 to march 2004 with ut1 ( antu ) and from april to october 2004 with ut2 ( kueyen ) . we used the 300v+i grism combined with a 2 slit , yielding a dispersion of 110 / mm ( corresponding to 2.64 / pixel ) and a spectral resolution of 1520 covering the 3800@xmath108000 range . the seeing during observations was in the range 0.5@xmath102.5 , with an average of @xmath01 . relevant informations on the sample objects are given in table [ tab : results ] . data reduction was performed using iraf@xcite following standard procedures for spectral analysis . this includes bias subtraction , flat fielding and cleaning for bad pixels . for each target we obtained three spectra in order to get a good correction of cosmic rays and to check the reality of weak features . the individual frames were then combined into a single average image . wavelength calibration was performed using the spectra of a helium / neon / argon lamp obtained during the same observing night , reaching an accuracy of @xmath0 3 (rms ) . from these images we extracted one - dimensional spectra adopting an optimal extraction algorithm @xcite to improve the s / n . although this program did not require optimal photometric conditions , most of the observations were obtained with clear sky . this enables us to perform a spectrophotometric calibration of the acquired data using standard stars @xcite observed in the same nights . from the database of sky conditions at paranal we estimate that a photometric accuracy of 10% was reached during our observing nights . the spectra were also corrected for galactic extinction , using the law by @xcite and assuming values of e@xmath11 from @xcite . in fig . [ fig : spec ] we give the optical spectrum of each source . in order to show more clearly the continuum shape and the faint features we report both the flux calibrated and the normalized spectrum for each object . the main emission and absorption features are identified . those due to the galactic interstellar gas are indicated as `` ism '' and `` dib '' ( diffuse interstellar bands , see section [ sec : resb ] ) , while telluric absorptions are marked as @xmath12 . in a first approximation , the optical continuum of a bl lac object is due to the superposition of two components : the non - thermal emission of the active nucleus , doppler - enhanced because of the alignment of the jet with the line of sight , and the emission of the host galaxy . depending on the relative strength of the nucleus with respect to the galaxy light , the spectral signature of the latter can be either easily detected or diluted beyond the point of recognition . taking into account the robust evidence that the host galaxies are giant ellipticals ( e.g. * ? ? ? * ) , to describe the continuum and derive the optical spectral index of the non - thermal component , we fitted a power law ( @xmath13 , the spectral indices are given in table [ tab : results ] ) plus the spectrum of a typical elliptical galaxy as described by the @xcite template . while in most cases the contribution of the host galaxy was negligible , in 6 sources it was not , and the luminosity of the host can thus be derived . for these six sources ( three of them were presented in paper i ) we give the best fit decomposition in fig.[fig : gfit2 ] and report the parameters in table [ tab : gfit ] . the derived absolute magnitudes of the host galaxies are consistent with the distribution of m@xmath14 of bl lac hosts given by @xcite . the detection and the measurement of very weak spectral features is difficult to assess because it depends on the choice of the parameters used to define the spectral line and the continuum . in order to apply an objective method for any given spectrum we evaluate the minimum measurable equivalent width ( ew@xmath15 ) defined as twice the rms of the distribution of all ew values measured dividing the normalized spectrum into 30 wide bins ( details for this automatic routine are given in paper i ) . we checked that the s / n ratio dependence inside the considered spectral range varies at most by 20 % , remaining @xmath910% over a large wavelength range . this reflects into a similar variation of ew@xmath15 . the procedure for calculating ew@xmath15 was applied to all featureless or quasi - featureless spectra to find faint spectral lines . all features above the ew@xmath15 threshold , ranging from @xmath0 1 to 0.1 in our data , were considered as line candidates and were carefully visually inspected and measured . the results are summarized in tab . [ tab : results ] . based on the detected lines and the shape of the continuum we confirm the bl lac classification for 36 objects , while 6 sources were reclassified . depending on the observed spectral properties the objects can be assembled in three groups . twelve objects belonging to this group were reported in paper i. six more are presented here ( table [ tab : results ] ) . three have redshift derived from emission lines ( 0723008 , z=0.128 ; 2131021 , z=1.284 ; 2223114 , z=0.997 ) and three from absorption lines ( 1212 + 078 , z=0.137 ; 1248296 , z=0.382 ; 2214313 , z=0.460 ) . details on each source are given in section [ sec : notes ] . despite their classification as bl lac objects in one or more input catalogues , six sources have spectra incompatible with this identification . five of them were reclassified either as quasars ( 0420 + 022 , 1320 + 084 ) or stars ( 1210 + 121 , 1222 + 102 , 1319 + 019 ) , while object 0841 + 129 remains of uncertain nature . in spite of the high s / n 18 objects exhibit spectra lacking any intrinsic feature . in several spectra we clearly see absorption features from the interstellar medium ( ism ) of our galaxy . in particular , we are able to detect caii @xmath163934,3968 , nai @xmath175892 atomic lines , and a number of dibs @xmath16 4428,4726,4882,5772 , generated by complex molecules in the ism ( e.g. * ? ? ? * and references therein ) . in fig . [ fig : ism ] we report the average spectrum of the interstellar absorptions . in three cases absorption lines from intervening gas are detected , leading to lower limits on the redshift of the objects ( 0841 + 129 , z@xmath182.48 ; 2133449 , z@xmath180.52 ; 2233148 , z@xmath180.49 ) . for these 18 sources we have estimated a redshift lower limit based on the ew@xmath15 of their spectra and the apparent magnitudes of the nuclei . we report * these * in table [ tab : results ] . the procedure to obtain these limits is described in section [ sec : ew2z ] . in this section we describe the procedure to obtain redshift lower limits for bl lacs with lineless spectra ( see table [ tab : results ] ) from the ew@xmath15 of the spectrum and the observed magnitude of the object . under the assumption that the host galaxy luminosity is confined in a narrow range @xcite it is in fact possible to constrain the position of the source on the nucleus - to - host flux ratio ( @xmath19 ) _ vs _ redshift plane . we assume that the observed spectrum of a bl lac object is given by the contribution of two components : 1- a non - thermal emission from the nucleus that can be described by a power law ( @xmath20 , where @xmath21 is the normalization constant ) ; 2 - a thermal component due to the host galaxy . depending on the relative contribution of the two components the optical spectrum will be dominated by the non - thermal ( featureless ) emission or by the spectral signature of the host galaxy . the observed equivalent width ( ew@xmath22 ) of a given spectral absorption line is diluted depending on the ratio of the two components . detection of this spectral feature requires a spectrum with a sufficiently high s / n . this is illustrated in fig [ fig : specsim ] , where a simulated spectrum ( @xmath19=5 , z=0.5 ) is reproduced with two different s / n ratios . the s / n=300 spectrum grants a secure detection of the caii features , while with s / n=30 the lines are undetected . in order to estimate the redshift of an object from the ew@xmath15 we need to know the relation between ew@xmath22 and the nucleus - to - host flux ratio @xmath19 . for a spectral absorption line of intrinsic equivalent width ew@xmath1 the observed equivalent width is given by the relation ( see also * ? ? ? * ) : @xmath23 the nucleus - to host ratio @xmath19 can be represented by @xmath24 where @xmath25 is the giant elliptical spectral template by ( * ? ? ? * see also section [ sec : gfit ] ) , and a(@xmath26 ) is a correction term that takes into account the loss of light inside the observed aperture . in this work the aperture is a 2@xmath276 slit that captures @xmath28 90% of the nuclear light , but not the whole surrounding galaxy that is more extended than the aperture ( in particular for low z targets ) . in order to estimate this effect we evaluated the amount of light lost from the galaxy through the aperture in use from simulated images of bl lacs ( point source plus the host galaxy ) . the main parameters involved are the shape and the size of the host galaxy . according to the most extensive imaging studies of bll @xcite we assumed that the host galaxy is a giant elliptical of effective radius r@xmath29= 10 kpc . the fraction of starlight lost then depends on the redshift of the object and is particularly significant at z@xmath9 0.2 , producing the bending of the curves in fig . [ fig : apeffect ] . since we want to refer the observed equivalent width to the nucleus - to - host ratio @xmath30 at a fixed wavelength @xmath31 , equation ( [ eq:1 ] ) can be rewritten as : @xmath32 where @xmath33 is the nucleus - to - host ratio normalized to that at @xmath31 ( @xmath34 ; see fig . [ fig : delta ] ) . on the other hand the quantity @xmath35 depends also on the observed magnitudes of the object , since @xmath36\ ] ] where m@xmath37 is the nucleus absolute magnitude and m@xmath38 is the host absolute magnitude , and @xmath39 where m@xmath37 is the nucleus apparent magnitude , d@xmath40(@xmath26 ) is the luminosity distance and k@xmath37(@xmath26 ) is the nucleus k - correction , computed following @xcite . the absolute magnitude of the host is @xmath41 where m@xmath42=22.9 is the average r band magnitude of bl lac hosts at z=0 and e(@xmath26 ) is the evolution correction , as given by @xcite . an example of the procedure described above is given in fig . [ fig : ew2z ] , where the relationships between log(@xmath35 ) and the redshift for a given value of ew@xmath15 and m@xmath37 are shown . the intersection of the two curves yields a lower limit to the redshift of the target . when it goes beyond the observed spectral range , we set the redshift limit to the value corresponding to the considered feature reaching the upper limit of the observed wavelength range ( z@xmath01 in the case of caii @xmath173934 line ) . the uncertainty of this procedure depends mainly on the spread of the distribution of the host galaxy luminosity . this issue is discussed in @xcite and in @xcite , where it is shown that the 64 bl lacs hosts of known redshift resolved with hst are well represented by an elliptical of m@xmath14=-22.9 , with 68% of them in the interval -23.4 -22.4 . this procedure can be used for any absorption line belonging to the host galaxy and for which an estimate of the un - diluted ew is available . in this work we considered the caii absorption line at @xmath17=3934 (ew@xmath1=16 ) , we assumed a power law spectral index @xmath43=0.7 @xcite , and we referred to the effective wavelength of the r band ( @xmath31=6750 ) to compute @xmath35 ( which implies @xmath44=4.3 ) . in order to test this procedure we considered eight bl lacs for which the caii line of the host galaxy has been measured . five of these objects derive from the observations discussed here and in paper i , three others are from observations obtained at the eso 3.6 @xcite . these spectra are reported in fig . [ fig : zestspec ] and the relevant parameters are given in table [ tab : resultscomp ] . the comparison between the redshifts estimated by our procedure with the spectroscopic ones indicates a reasonable good agreement ( see fig . [ fig : zestcomp ] ) . [ [ section ] ] 0047 + 023 + + + + + + + + this compact and flat spectrum radio source was classified as a bl lac by @xcite on the basis of uv color and featureless spectra . further featureless optical spectra obtained by @xcite confirmed the bl lac . even in our s / n@xmath0 80 spectrum no spectral features were found . based on the minimum detectable ew the source is most likely at z @xmath18 0.82 . [ [ section-1 ] ] 0048097 + + + + + + + + previous optical observations of this well known bl lac object belonging to the 1-jy sample , reported a featureless spectrum @xcite . @xcite , however , suggested the presence of an emission line at 6092 ( possibly identified with [ oii ] @xmath173727 at z=0.634 or [ oii ] @xmath175007 at z=0.216 ) . @xcite proposed z@xmath180.5 , based on the non detection of the host galaxy in the optical images of the source . our s / n=250 optical spectrum does not confirm the presence of the emission line at 6092 , and apart of some telluric lines and a number of galactic absorptions it is found featureless . from our ew@xmath15 estimate , we infer that this source is at z@xmath180.3 . [ [ section-2 ] ] 0420 + 022 + + + + + + + + @xcite classified this source as a bl lac candidate on the basis of a featureless ( although noisy ) optical spectrum . @xcite through an unpublished optical spectrum propose a redshift z=2.277 and classify the source as a radio loud qso . in our optical spectrum we are able to clearly detect emission lines ly@xmath451419 ovi ] @xmath171034 , civ @xmath171549 and ciii ] @xmath171909 , at z = 2.278 . a recent spectrum obtained by @xcite also confirm our findings and the classification of the object as a qso . [ [ section-3 ] ] 0422 + 004 + + + + + + + + this object is a well known radio selected bl lac , included in the @xcite catalogue . @xcite detected the host galaxy with ground - based imaging , proposing z@xmath00.20.3 . the optical spectrum taken by @xcite is featureless . our spectrum ( s / n=230 ) does not show evidence for intrinsic spectral features from the host , suggesting a very high n / h ratio . interstellar absorptions from nai @xmath175892 and dibs at 5772 and 4726 are well detected . based on ew@xmath15 , we estimate z@xmath180.31 . [ [ section-4 ] ] 0627199 + + + + + + + + @xcite obtained a lineless spectrum for this radio selected bl lac object . our vlt spectrum , of moderate s / n ( 50 ) , shows no spectral features . from ew@xmath15we set z@xmath180.63 . [ [ section-5 ] ] 0723008 + + + + + + + + @xcite classified this source as a narrow emission line radio galaxy based on an optical spectrum , giving z=0.127 . @xcite report an optical polarization of 1.5 % , classifying the source as a low polarization qso . @xcite report this source as a bl lac object . @xcite gives broad band indices @xmath46=0.7 and @xmath47=1.0 , which are compatible with a bl lac or a fsrq classification . our optical spectrum is clearly dominated by a non thermal emission with spectral index @xmath43=0.7 . superposed to this , strong narrow emission lines and absorption lines from the underlying host galaxy at z=0.127 are detected , confirming the redshift . from the values of the spectral indices and the measured ew for the spectral lines , we suggest that this object is of intermediate nature between a bl lac and a quasar . [ [ section-6 ] ] 0841 + 129 + + + + + + + + this source , first identified by c. hazard ( see * ? ? ? * and references therein ) , is a damped lyman @xmath43 absorption ( dla ) qso at z@xmath182.48 as derived from the two very strong dlas at @xmath04100 and @xmath04225 ( see for example * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . the classification as a bl lac object was motivated by the absence of prominent emission lines @xcite . our spectrum , in addition to several absorption lines , exhibit three possible broad emission structures at@xmath04310 , @xmath04850 and @xmath05370 . these could be interpreted as nv @xmath171240 , siiv @xmath171397 and civ @xmath171549 at z@xmath02.47 . this is consistent with z@xmath02.5 , deduced from the observed position of the onset of the absorption of the ly@xmath48 forest @xcite . an alternative explanation is , however , that these structures are pseudo - emissions resulting from the depression of the continuum caused by the envelope of many unresolved narrow absorption features . higher resolution spectra of the object in the spectral range 4200 to 5800 are needed to distinguish between the two possibilities . [ [ section-7 ] ] 1210 + 121 + + + + + + + + @xcite proposed that this object was the optical counterpart of a radio source in the molongo catalogue ( mc2 * ? ? ? * ) ; the separation was however 16 . @xcite reported large optical variability and polarization , apparently reinforcing the identification . @xcite found a featureless optical spectrum . our vlt spectrum clearly shows that the source is a type b star in our galaxy . [ [ section-8 ] ] 1212 + 078 + + + + + + + + our vlt spectrum clearly shows the presence of a strong thermal component due to the host . we detected caii @xmath163934 , 3968 , g band @xmath174305 , mgi @xmath175175 and h@xmath456563 in emission at z=0.137 , confirming the redshift estimated by @xcite . the contribution of the non - thermal component is visible in the bluest part of the spectrum . the best fit decomposition of the spectrum gives @xmath43=1.17 for the non - thermal component and m@xmath14=-22.0 for the host . though this is somewhat fainter than expected for a bl lac host galaxy , we can not exclude that given the low redshift and the consequent large apparent size of the host , part of the light did not enter in the slit . [ [ section-9 ] ] 1222 + 102 + + + + + + + + this is a blue stellar object in the direction of the virgo - coma cluster . its apparent position in the sky is very close to the center of the galaxy ngc 4380 , still well inside the galaxy boundaries . the projected separation to the nucleus at the redshift of the galaxy is @xmath010 kpc . the object is considered a candidate bl lac in the @xcite list , selected because of its uv excess . @xcite reports the observation of a featureless spectrum . @xcite estimates this object a possible candidate of expulsion from a galactic nucleus . the sharp absorption lines detected in our spectrum clearly indicate a stellar origin . the measured colors lead to a temperature of @xmath010000 k. if the object were a main sequence or a supergiant star , the corresponding distance will put it outside the galaxy , but not at the distance of ngc 4380 . we are therefore led to consider a white dwarf , which would be at 100200 pc . the absence of h lines indicates a dq or dxp white dwarf @xcite . some of the lines are referable to hei and ci transitions . the object clearly deserves further study ; in particular polarization measurements would be interesting . [ [ section-10 ] ] 1248296 + + + + + + + + @xcite obtained a low s / n spectrum of this source , and proposed a bl lac at z=0.487 based on the possible detection of the host galaxy features . in our vlt spectrum caii , g band , h@xmath49 are clearly detected at z=0.382 , confirming the findings of @xcite , while in the blue part the contribution of a non - thermal component is clearly visible . the best fit decomposition gives @xmath43=0.92 for the non - thermal component visible below 5000 , and m@xmath14=-22.7 for the host , in good agreement with result from the direct detection of the host in hst imaging @xcite . [ [ section-11 ] ] 1319 + 019 + + + + + + + + this object was initially selected as a bl lac candidate on the basis of the university of michigan objective prism survey @xcite designed to find agn and it is included as bl lac in the @xcite catalogue . no radio counterpart for this source has been found in literature . later @xcite proposed its classification as a bl lac , based on a low s / n optical spectrum that was found featureless . in our much better quality spectrum we clearly see many absorption features that characterize the object as a galactic star of spectral type @xmath0a . our findings are also in agreement with the spectral classification of the 2df qso redshift survey ( 2qz , see * ? ? ? [ [ section-12 ] ] 1320 + 084 + + + + + + + + this source is part of the bl lac sample extracted from the einstein slew survey and a radio counterpart was reported by @xcite . our vlt data show the source has a qso like spectrum at z=1.5 , in contrast with a featureless spectrum observed by @xcite . several intervening absorption lines , in particular mgii at z=1.347 were also detected . [ [ section-13 ] ] 1349439 + + + + + + + + the spectrum of this x - ray selected bl lac @xcite , shows a number of absorption lines from the interstellar medium : caii @xmath163934,3968 , the 5772 dib , nai @xmath175892 . no intrinsic features were detected , and the deduced redshift lower limit is z@xmath180.39 . as already pointed out by @xcite , the value z=0.05 sometime reported for this object is consequence of a confusion with the nearby seyfert 1 galaxy q 1349 - 439 . [ [ section-14 ] ] 1442032 + + + + + + + + this x - ray source , the radio counterpart of which was found in the nvss survey , was first classified as a bl lac in the rbsc - nvss sample by @xcite , and then confirmed by the ss . there are no published optical spectra for this source . our optical spectrum is featureless , with the exception of the nai @xmath175892 absorption feature from our galaxy ism . the ew@xmath15 value for this objects leads to z@xmath180.51 . [ [ section-15 ] ] 1500 - 154 + + + + + + + + this x - ray selected bl lac is part of the rsbc - nvss sample @xcite and enters in ss . no previous optical spectroscopy has been found in the literature . our spectrum is completely featureless , leading to z@xmath18 0.38 from the obtained ew@xmath15 . [ [ section-16 ] ] 1553 + 113 + + + + + + + + this source is an optically selected bl lac from the palomar - green survey . the redshift estimate z=0.360 given in the @xcite catalogue was disproved by later spectroscopy @xcite . while no intrinsic features were detected in our s / n=250 vlt spectrum , a number of absorption lines due to our galaxy ism were revealed : caii @xmath163934,3968 , nai @xmath175892 and dibs at 4428,4726,4882,5772 . the ew@xmath15 estimate for this object gives a limit z@xmath180.09 . [ [ section-17 ] ] 1722 + 119 + + + + + + + + @xcite reported a tentative redshift z=0.018 for this x - ray selected , highly polarized bl lac . this estimate was not confirmed by more recent observations @xcite . our vlt spectrum ( s / n=350 ) shows only absorption features due to our galaxy ism : caii @xmath163934,3968 , nai @xmath175892 and dibs at 4428 , 4726 , 4882 , 5772 , with no evidence of intrinsic features . from the minimum ew@xmath15 we derive z@xmath180.17 . [ [ section-18 ] ] 2012017 + + + + + + + + consistently with previous observations of this radio selected bl lac @xcite , also our s / n=130 vlt spectrum is featureless . the optical spectral index is @xmath43=0.49 , in marginal agreement with @xmath43=0.33@xmath500.12 reported by @xcite . from ew@xmath15 we derive z@xmath180.94 . [ [ section-19 ] ] 2128254 + + + + + + + + the spectrum of this x - ray selected bl lac candidate is reported as featureless by ss . we confirm this result and set a lower limit of z@xmath180.86 for the redshift . [ [ section-20 ] ] 2131021 + + + + + + + + @xcite and @xcite proposed a redshift of 1.285 for this source , based on the detection of ciii ] @xmath171909 , mgii @xmath172798 and [ oii ] @xmath173727 , opposed to the z=0.557 suggested by @xcite . while [ oii ] falls outside our spectral range , we confirm the presence of ciii ] and mgii emission lines at z=1.283 , also detecting the fainter cii ] @xmath172326 feature at the same redshift . [ [ section-21 ] ] 2133449 + + + + + + + + this source was discovered because of its optical variability by @xcite . optical spectroscopy by @xcite and @xcite led to completely featureless spectra . our vlt observations clearly show the presence of an intervening absorption feature at 4250 , a tentative identification of which is intervening mgii at z=0.52 ( see * ? ? ? the lower limit on z derived from ew@xmath15 is z@xmath180.98 . [ [ section-22 ] ] 2136428 + + + + + + + + the spectrum obtained by @xcite , who discovered this source studying its optical variability , is completely featureless . our vlt observations shows several absorption features due to the ism of our galaxy : dib at 4428 , 4726 , 4882 and 5772 , caii @xmath163934 , 3968 and nai@xmath17 5892 atomic lines . the feature at 5942 could be caii @xmath173968 at z=0.497 . since at this redshift the caii @xmath173934 should fall at 5890 , where it will be strongly contaminated by the interstellar nai absorption , the redshift estimate is only tentative . the lower limit deduced by the minimum measurable ew is z@xmath180.24 . [ [ section-23 ] ] 2214313 + + + + + + + + our vlt spectrum of this object clearly shows the typical spectral signature of the host galaxy ( caii @xmath163934,3968 and g band @xmath174305 ) at z=0.46 . the best fit decomposition gives @xmath43=0.9 for the non - thermal component and m@xmath14=22.3 for the host . previous optical spectroscopy performed by @xcite with the eso 3.6 m telescope failed to detect any spectral feature . [ [ section-24 ] ] 2223114 + + + + + + + + optical observations of this radio source obtained by @xcite did not show any intrinsic spectral feature . in our spectrum , that extends further in the red , we detect a single narrow emission line at @xmath177367 ( ew = 5 ) . this is a real feature since it clearly appears on each of the 3 individual spectra ( see section 3 ) . a possible identification of this line is [ oii]@xmath17 3727 at z=0.977 , while mgii @xmath172798 gives z=1.633 . we discarded this second classification because the line fwhm ( 1200 km s@xmath2 ) , is typical for a narrow line such [ oii ] , while for mgii a larger value would be expected . moreover , with a mgii identification both civ @xmath171549 and ciii ] @xmath171909 broad lines would be expected inside the observed spectral range , but no other features are detected . [ [ section-25 ] ] 2233148 + + + + + + + + the redshift z=0.325 reported by @xcite is due to confusion with the source hb89 2233 + 134 in @xcite . @xcite report an intervening system at z=0.609 , but without giving an identification of the corresponding absorption feature . we detect several absorption features on the spectrum . in particular we propose to identify the features at 4165 and 4183 as mgii at z=0.492 , while , using the ew@xmath15 estimate from the spectrum , z@xmath180.65 is found . [ [ section-26 ] ] 2254204 + + + + + + + + previous optical spectroscopy @xcite of this bl lac object from the 1jy sample showed completely featureless spectra . with vlt we are able to detect faint interstellar absorptions of caii @xmath163934 , 3968 and nai@xmath17 5892 , but no intrinsic or intervening spectral lines are found . the inferred redshift limit is z@xmath180.47 . [ [ section-27 ] ] 2307375 + + + + + + + + this source was first classified as a bl lac in the rsbc - nvss sample @xcite . the classification was then confirmed by the ss . no previous optical spectroscopy has been published . our vlt spectrum is featureless , allowing us to set only a lower limit to the redshift of z@xmath181 . [ [ section-28 ] ] 2342153 + + + + + + + + this source is part of the emss sample of bl lac objects . our vlt data , as well as previous optical spectroscopy with the 6.5 m telescope of multi mirror telescope observatory @xcite showed a featureless spectrum . from ew@xmath15 we derive z@xmath181 . [ [ section-29 ] ] 2354021 + + + + + + + + this object was discussed in paper i. here we report only the spectrum , in fig . [ fig : spec ] . [ [ section-30 ] ] 2354175 + + + + + + + + this x - ray source from rosat all sky survey , is classified as bl lac candidate in the rbsc - nvss sample @xcite and in the ss . no previous spectroscopy was published in literature . our s / n=150 vlt spectrum is featureless , allowing only to set a lower limit of z@xmath180.85 to the redshift . out of 42 objects observed we confirm the bl lac classification for 36 sources and for 18 of them we are able to measure / confirm the redshift . this information allows us to derive the luminosity of the objects . the distribution in the v band luminosity - distance plane is indeed fully consistent with what observed for bl lacs of known redshift in the combination of the @xcite and the ss sample ( see fig . [ fig : distz ] ) . we note that the sources in this combined list are affected by the typical selection effect of incomplete , flux limited samples : the envelope of the objects follows in fact the expected behavior for sources with constant v magnitude , centered around v=18 , with a spread of @xmath0 3 mag . the objects discussed here ( filled circles and lower limits ) follow the same distribution , with absolute magnitudes ranging between -21.5 and -27.5 , slightly increasing with the redshift . in 18 cases the optical spectra remain lineless in spite of the high s / n of the obtained optical spectra . this indicates that if they are hosted by galaxies of standard luminosity they have likely very luminous or extremely beamed nuclei ( see also * ? ? ? * ) . in the latter case one may expect to see the most extreme cases of relativistic beaming , making these sources ideal targets for milli - arcsec resolution radio observations . alternatively , if the host galaxies were under - luminous , these objects could be rare examples of dwarf galaxies hosting an agn ( see * ? ? ? the high s / n of most of the optical spectra obtained at vlt represents a frontier for the determination of the redshift of bl lacs with current instrumentation and further improvement of the issue will not be easy to get . in particular the four brightest objects ( r@xmath916 : 0048099 , 1553 + 113 , 1722 + 119 , 2136 - 428 ) we observed at the vlt have values of ew@xmath15 smaller than 0.25 . these objects belong to an interesting sub - population of bl lac objects with extreme nuclear ( and/or host ) properties for which it is actually not possible to derive the intrinsic physical parameters . one possibility is to image the source when it is particularly faint in order to improve the detection of the host galaxy and to derive an imaging redshift @xcite . deep spectroscopic observations in the near - ir may also prove to be effective in the determination of the redshift considering this region of the spectrum is poorly known . , spectral lines are marked by the line identification , intervening mgii absorption systems are reported as `` int . mgii '' , unidentified intervening systems are indicated with * , absorption features from atomic species in the interstellar medium of our galaxy are labeled by ism , diffuse interstellar bands by dib . ] , referred to @xmath31=6750 (effective wavelength for r band magnitude ) , as a function of the wavelength . the assumed spectral index for the nuclear component is @xmath43=0.7 . for caii @xmath173934 absorption feature , @xmath44 equals 4.3 . ] values for 1rxs j150343.0@xmath10154107 . the thick solid line represents the n / h vs z limit obtained from the ew@xmath15 value . dotted curves correspond to a 0.1 uncertainty on ew@xmath15 . the dashed line gives the n / h vs. z relation for a bl lac with a host galaxy with m@xmath14=-22.9 and nuclear apparent magnitude r=17.7 . dotted lines correspond to the uncertainty due to the range of variation of host galaxy magnitude ( 0.5 mag ) and observational photometric errors ( 0.1 mag ) . the intersection between the two solid lines gives the lower limit on the redshift . the analytic form of the curves is described by eqs . [ eq : ewnh ] , [ eq : magnh ] ; further details can be found in @xcite ] lllllllllcl pks 0047 + 023 & 0047 + 023 & 00 49 43.2 & + 02 37 04.8 & 05 aug 03 & 1800 & 80 & 19.0 & 0.61 & 0.36 & @xmath180.82 + pks 0048@xmath1009 & 0048@xmath10097 & 00 50 41.3 & @xmath1009 29 05.2 & 17 sep 03 & 1800 & 250 & 16.0 & 0.95 & 0.22 & @xmath180.30 + pks 0420 + 022 & 0420 + 022 & 04 22 52.2 & + 02 19 26.9 & 19 nov 03 & 2325 & 90 & 18.9 & * & 0.41 & 2.278 + pks 0422 + 00 & 0422 + 004 & 04 24 46.8 & + 00 36 06.3 & 27 nov 03 & 2325 & 230 & 16.2 & 0.88 & 0.25 & @xmath180.31 + pks 0627@xmath10199 & 0627@xmath10199 & 06 29 23.8 & @xmath1019 59 19.7 & 16 dec 03 & 2325 & 50 & 19.3 & 0.56 & 0.92 & @xmath180.63 + pks 0723@xmath1000 & 0723@xmath10008 & 07 25 50.6 & @xmath1000 54 56.5 & 25 dec 03 & 2325 & 250 & 16.0 & * & 0.23 & 0.127 + h 0841 + 1256 & 0841 + 129 & 08 44 24.1 & + 12 45 48.0 & 30 dec 03 & 2325 & 100 & 18.0 & * & 0.38 & @xmath182.48 + hb89 1210 + 121 & 1210 + 121 & 12 12 33.9 & + 11 50 56.9 & 24 jan 04 & 2325 & 180 & 17.8 & * & 0.28 & * + 1es 1212 + 078 & 1212 + 078 & 12 15 10.9 & + 07 32 03.8 & 25 jan 04 & 2325 & 100 & 17.3 & 1.17 & 0.39 & 0.137 + 1222 + 102 & 1222 + 102 & 12 25 23.1 & + 09 59 35.0 & 26 jan 04 & 2325 & 160 & 17.7 & 2.67 & 0.30 & * + 1es 1248@xmath10296 & 1248@xmath10296 & 12 51 34.9 & @xmath1029 58 42.9 & 24 jan 04 & 2325 & 50 & 19.5 & 0.92 & 0.57 & 0.382 + um566 & 1319 + 019 & 13 19 55.1 & + 01 52 58.3 & 30 apr 03 & 1800 & 100 & 18.2 & * & 0.36 & * + 1es 1320 + 084n & 1320 + 084 & 13 22 54.9 & + 08 10 10.0 & 30 apr 03 & 2325 & 50 & 19.5 & * & 0.54 & 1.500 + pks 1349@xmath10439 & 1349@xmath10439 & 13 52 56.5 & @xmath1044 12 40.4 & 30 apr 03 & 1800 & 240 & 16.9 & 0.82 & 0.32 & @xmath180.39 + 1rxs j144505.9@xmath10032613 & 1442@xmath10032 & 14 45 05.8 & @xmath1003 26 12.8 & 28 aug 04 & 2325 & 100 & 17.7 & 1.21 & 0.35 & @xmath180.51 + 1rxs j150343.0@xmath10154107 & 1500@xmath10154 & 15 03 42.9 & @xmath1015 41 07.0 & 28 aug 04 & 2325 & 40 & 17.8 & 1.52 & 0.78 & @xmath180.38 + hb89 1553 + 113 & 1553 + 113 & 15 55 43.0 & + 11 11 24.4 & 01 aug 03 & 1800 & 250 & 14.0 & 0.84 & 0.25 & @xmath180.09 + h 1722 + 119 & 1722 + 119 & 17 25 04.4 & + 11 52 15.2 & 06 apr 03 & 1800 & 350 & 14.7 & 1.30 & 0.18 & @xmath180.17 + pks 2012@xmath10017 & 2012@xmath10017 & 20 15 15.2 & @xmath1001 37 33.0 & 31 jul 03 & 1800 & 130 & 19.3 & 0.49 & 0.34 & @xmath180.94 + 1rxs j213151.7@xmath10251602 & 2128@xmath10254 & 21 31 51.6 & @xmath1025 16 00.8 & 10 jul 04 & 2325 & 70 & 19.0 & 1.28 & 0.32 & @xmath180.86 + pks 2131@xmath10021 & 2131@xmath10021 & 21 34 10.3 & @xmath1001 53 17.0 & 18 jul 04 & 2325 & 80 & 19.2 & 0.29 & 0.43 & 1.284 + mh 2133@xmath10449 & 2133@xmath10449 & 21 36 18.4 & @xmath1044 43 49.0 & 12 jul 04 & 2325 & 60 & 19.5 & 1.02 & 0.37 & @xmath180.98 + mh 2136@xmath10428 & 2136@xmath10428 & 21 39 24.1 & @xmath1042 35 21.3 & 03 jul 03 & 1800 & 490 & 15.6 & 0.84 & 0.24 & @xmath180.24 + rx j22174@xmath103106 & 2214@xmath10313 & 22 17 28.4 & @xmath1031 06 19.0 & 10 jul 04 & 2325 & 50 & 19.7 & 0.90 & 0.68 & 0.460 + pks 2223@xmath10114 & 2223@xmath10114 & 22 25 43.6 & @xmath1011 13 40.0 & 02 sep 04 & 2325 & 20 & 21.5 & 0.31 & 1.04 & 0.997 + pks 2233@xmath10148 & 2233@xmath10148 & 22 36 34.0 & @xmath1014 33 21.0 & 02 sep 04 & 2325 & 170 & 18.5 & 0.15 & 0.30 & @xmath180.65 + pks 2254@xmath10204 & 2254@xmath10204 & 22 56 41.2 & @xmath1020 11 40.3 & 31 jul 03 & 1800 & 220 & 17.1 & 0.86 & 0.25 & @xmath180.47 + 1rxs j231027.0@xmath10371926 & 2307@xmath10375 & 23 10 26.9 & @xmath1037 19 26.0 & 10 jul 04 & 2325 & 80 & 19.6 & 1.15 & 0.34 & @xmath181.03 + ms 2342.7@xmath101531 & 2342@xmath10153 & 23 45 22.4 & @xmath1015 15 06.7 & 26 jul 03 & 2325 & 20 & 21.4 & 1.02 & 1.72 & @xmath181.03 + 1rxs j235730.1@xmath10171801 & 2354@xmath10175 & 23 57 29.7 & @xmath1017 18 05.3 & 12 jul 04 & 2325 & 150 & 18.2 & 1.44 & 0.17 & @xmath180.85 + llllllll 0224 + 018 & 0.456 & 1.50 & 19.9 & @xmath1023.2 & 2.2 & @xmath1023.1 & ( 1 ) + 0316@xmath10121 & 0.443 & 1.44 & 20.2 & @xmath1022.8 & 6.4 & & ( 1 ) + 0557@xmath10385 & 0.302 & 1.61 & 18.3 & @xmath1023.4 & 5.5 & & ( 1 ) + 1212 + 078 & 0.137 & 1.17 & 17.4 & @xmath1022.0 & 0.4 & @xmath1023.0 & + 1248@xmath10296 & 0.382 & 0.92 & 19.7 & @xmath1022.7 & 0.8 & @xmath1023.7 & + 2214@xmath10313 & 0.460 & 0.90 & 20.8 & @xmath1022.3 & 2.1 & & + llllllllr 0420 + 022 & qso & 2.278 & & & & & & + & & & ly@xmath51 & 4020 & 2.278 & e & 9400 & -73.0 + & & & siii & 4285 & 2.279 & e & 3900 & -4.0 + & & & cii & 4381 & 2.281 & e & 5500 & -3.0 + & & & siiv & 4587 & 2.283 & e & 6300 & -24.0 + & & & civ & 5077 & 2.278 & e & 4900 & -50.0 + & & & ciii ] & 6250 & 2.274 & e & 4500 & -45.0 + 0723@xmath10008 & qso / bll & 0.127 & & & & & & + & & & [ nev ] & 3858 & 0.126 & e & 1100 & -0.4 + & & & [ oii ] & 4200 & 0.127 & e & 1200 & -2.1 + & & & [ neiii ] & 4359 & 0.127 & e & 1300 & -0.7 + & & & caii & 4433 & 0.127 & g & & 0.6 + & & & b band & 4477 & 0.127 & g & & 0.1 + & & & g band & 4847 & 0.127 & g & & 0.1 + & & & h@xmath52 & 4897 & 0.128 & e & 3700 & -1.0 + & & & h@xmath49 & 5477 & 0.127 & e & 1100 & -1.0 + & & & [ oiii ] & 5587 & 0.127 & e & 900 & -1.9 + & & & [ oiii ] & 5642 & 0.127 & e & 900 & -5.5 + & & & mg i & 5830 & 0.127 & g & & 0.4 + & & & nai & 6645 & 0.127 & g & & 0.1 + & & & [ oii ] & 7098 & 0.127 & e & 1000 & -1.2 + & & & h@xmath48 & 7402 & 0.128 & e & 1900 & -8.7 + & & & sii & 7561 & 0.126 & e & 600 & -1.1 + & & & & & & & & + 1212 + 078 & bll & 0.137 & & & & & & + & & & caii & 4473 & 0.137 & g & & 5.7 + & & & caii & 4510 & 0.137 & g & & 4.8 + & & & g band & 4890 & 0.136 & g & & 5.3 + & & & h@xmath49&5529&0.137 & g & & 3.5 + & & & mg i & 5883 & 0.137 & g & & 16.2 + & & & na i & 6696 & 0.137 & g & & 3.9 + & & & h@xmath48&7481&0.139 & e & 700 & -2.0 + 1248@xmath10296 & bll & 0.382 & & & & & & + & & & caii & 5436 & 0.382 & g & 2100 & 6.1 + & & & caii & 5482 & & g & 1700 & 4.2 + & & & g band & 5942 & & g & 2700 & 5.6 + & & & h@xmath49&6719&0.382 & g & 1600 & 1.5 + 1320 + 084 & qso & 1.500 & & & & & & + & & & civ & 3873 & 1.500 & e & 2200 & 1.9 + & & & heii & 4095 & 1.500 & e & 1900 & -9.0 + & & & ? & 4234 & & a & & -80.2 + & & & niii ] & 4376 & 1.500 & e & 2300 & -5.8 + & & & ciii ] & 4770 & 1.500 & e & 2200 & -40.9 + & & & ? & 6071 & & a & & 0.6 + & & & ? & 6130 & & a & & 1.4 + & & & mgii & 6563 & 1.347 & a & & 1.8 + & & & mgii & 6578 & 1.347 & a & & 1.6 + & & & mgii & 7001 & 1.502 & e & 6700 & -60.0 + 2131@xmath10021 & bll & 1.283 & & & & & & + & & & ciii ] & 4357 & 1.283 & e & 2000 & -4.4 + & & & cii ] & 5312 & 1.284 & e & 700 & -1.4 + & & & mgii & 6383 & 1.281 & e & 3000 & -3.8 + 2133@xmath10449 & bll & @xmath180.52 & & & & & & + & & & mgii & 4250 & 0.519 & a & 2500 & 1.5 + & & & & & & & & + 2214@xmath10313 & bll & 0.460 & & & & & & + & & & caii & 5746 & 0.461 & g & 3000 & 3.7 + & & & caii & 5792 & 0.460 & g & 4300 & 3.3 + & & & gband & 6280 & 0.459 & g & 3200 & 2.6 + & & & & & & & & + 2223@xmath10114 & bll & 0.977 & & & & & & + & & & [ oii ] & 7367 & & e & 1200 & -5.0 + & & & & & & & & + 2233@xmath10148 & bll & @xmath180.49 & & & & & & + & & & mgii & 4165 & 0.490 & a & & 0.7 + & & & mgii & 4183 & 0.493 & a & & 0.9 + & & & ? & 4514 & & a & & 1.7 + & & & ? & 4598 & & a & & 0.5 + & & & & & & & & + lllllll 0224 + 018 & 0.457 & 19.0 * & 1.7 & 0.42 & 0.12 & vlt + 0316@xmath10261 & 0.443 & 18.1 * & 0.6 & 0.47 & 0.15 & vlt + 0557@xmath10385 & 0.302 & 16.9 * & 0.9 & 0.22 & 0.08 & vlt + 1212 + 078 & 0.137 & 18.5 * & 5.7 & 0.14 & 0.05 & vlt + 1248@xmath10296 & 0.382 & 19.8 * & 6.1 & 0.28 & 0.08 & vlt + 1440 + 122 & 0.162 & 17.2 & 3.5 & 0.09 & 0.04 & eso3.6 + 1914@xmath10194 & 0.137 & 15.8 & 0.5 & 0.17 & 0.09 & eso3.6 + 2005@xmath10489 & 0.071 & 14.1 & 0.4 & 0.07 & 0.05 & eso3.6 + 2214@xmath10313 & 0.460 & 19.9 * & 3.7 & 0.41 & 0.10 & vlt +

we report on eso very large telescope optical spectroscopy of 42 bl lacertae objects of unknown redshift . nuclear emission lines were observed in 12 objects , while for another six we detected absorption features due to their host galaxy . the new high s / n spectra therefore allow us to measure the redshift of 18 sources . five of the observed objects were reclassified either as stars or quasars , and one is of uncertain nature . for the remaining 18 the optical spectra appear without intrinsic features in spite of our ability to measure rather faint ( ew @xmath00.1 ) spectral lines . for the latter sources a lower limit to the redshift was set exploiting the very fact that the absorption lines of the host galaxy are undetected on the observed spectra .

astro-ph0601506

the problem of simultaneous localization and mapping ( slam ) has a rich history over the past two decades , which is too broad to cover here , see e.g. @xcite . the extended kalman filter ( ekf ) based slam ( the ekf - slam ) has played an important historical role , and is still used , notably for its ability to close loops thanks to the maintenance of correlations between remote landmarks . the fact that the ekf - slam is inconsistent ( that is , it returns a covariance matrix that is too optimistic , see e.g. , @xcite , leading to inaccurate estimates ) was early noticed @xcite and has since been explained in various papers @xcite . in the present paper we consider the inconsistency issues that stem from the fact that , as only relative measurements are available , the origin and orientation of the earth - fixed frame can never be correctly estimated , but the ekf - slam tends to think " it can estimate them as its output covariance matrix reflects an information gain in those directions of the state space . this lack of observability , and the poor ability of the ekf to handle it , is notably regarded as the root cause of inconsistency in @xcite ( see also references therein ) . in the present paper we advocate the use of the invariant ( i)-ekf to prevent covariance reduction in directions of the state space where no information is available . the invariant extended kalman filter ( iekf ) is a novel methodology introduced in @xcite that consists in slightly modifying the ekf equations to have them respect the geometrical structure of the problem . reserved to systems defined on lie groups , it has been mainly driven by applications to localization and guidance , where it appears as a slight modification of the multiplicative ekf ( mekf ) , widely known and used in the world of aeronautics . it has been proved to possess theoretical local convergence properties the ekf lacks in @xcite , to be an improvement over the ekf in practice ( see e.g. , @xcite and more recently @xcite where the ekf is outperformed ) , and has been successfully implemented in industrial applications to navigation ( see the patent @xcite ) . in the present paper , we slightly generalize the iekf framework , to make it capable to handle very general observations ( such as range and bearing or bearing only observations ) , and we show how the derived iekf - slam , a simple variant of the ekf - slam , allows remedying the inconsistency of ekf - slam stemming from the non - observability of the orientation and origin of the global frame . the issue of ekf - slam inconsistency has been the object of many papers , see @xcite to cite a few , where empirical evidence ( through monte - carlo simulations ) and theoretical explanations in various particular situations have been accumulated . in particular , the insights of @xcite have been that the orientation uncertainty is a key feature in the inconsistency . the article @xcite , in line with @xcite , also underlines the importance of the linearization process , as linearizing about the true trajectory solves the inconsistency issues , but is impossible to implement in practice as the true state is unknown . it derives a relationship that should hold between various jacobians appearing in the ekf equations when they are evaluated at the current state estimate to ensure consistency . a little later , the works of g.p . huang , a.i . mourikis , and s. i. roumeliotis @xcite have provided a sound theoretical analysis of the ekf - slam inconsistency as caused by the ekf inability to correctly reflect the three unobservable degrees of freedom ( as an overall rotation and translation of the global reference frame leave all the measurements unchanged ) . indeed , the filter tends to erroneously acquire information along the directions spanned by those unobservable transformations . to remedy this problem , the above mentioned authors have proposed various solutions , the most advanced being the observability constrained ( oc)-ekf . the idea is to pick a linearization point that is such that the unobservable subspace seen " by the ekf system model is of appropriate dimension , while minimizing the expected errors of the linearization points . our approach , that relies on the iekf , provides an interesting alternative to the oc - ekf , based on a quite different route . indeed , the rationale is to apply the ekf methodology , but using alternative estimation errors to the standard linear difference between the estimate and the true state . any non - linear error that reflects a discrepancy between the true state and the estimate , necessarily defines a local frame around any point , and the idea underlying the iekf amounts to write the kalman jacobians and covariances in this frame . we notice and prove here that an alternative nonlinear error defines a local frame where the unobservable subspace is _ everywhere _ spanned by the same vectors . using this local frame at the current estimate to express kalman s covariance matrix will be shown to ensure the unobservable subspace seen " by the ekf system model is _ automatically _ of appropriate dimension . we thus obtain an ekf variant which automatically comes with consistency properties . moreover , we relate unobservability to the inverse of the covariance matrix ( called information matrix ) rather than on the covariance matrix itself , and we derive guarantees of information decrease over unobservable directions . contrarily to the oc - ekf , and as in the standard ekf , we use here the latest , and thus best , state estimate as the linearization point to compute the filter jacobians . in a nutshell , whereas the key fact for the analysis of @xcite is that the choice of the linearization point affects the observability properties of the linearized state error system of the ekf , the key fact for our analysis is that the choice of the error variable has similar consequences . theoretical results and simulations underline the relevance of the proposed approach . robot - centric formulations such as @xcite , and later @xcite are promising attempts to tackle unobservability , but they unfortunately lack convenience as the position of all the landmarks must be revised during the propagation step , so that the landmarks estimated position becomes in turn sensitive to the motion sensor s noise . they do not provably solve the observability issues considered in the present paper , and it can be noted the oc - ekf has demonstrated better experimental performance than the robocentric mapping filter , in @xcite . in particular , the very recent papers @xcite propose to write the equations of the slam in the robot s frame under a constant velocity assumption . using an output injection technique , those equations become linear , allowing to prove global asymptotic convergence of any linear observer for the corresponding deterministic linear model . this is fundamentally a deterministic approach and property , and as the matrices appearing in the obtained linear model are functions of the observations , the behavior of the filter is not easy to anticipate in a noisy context : the observation noise thus corrupts the very propagation step of the filter . some recent papers also propose to improve consistency through local map joining , see @xcite and references therein . although appealing , this approach is rather oriented towards large - scale maps , and requires the existence of local submaps . but when using submap joining algorithm , inconsistency in even one of the submaps , leads to an inconsistent global map " @xcite . this approach may thus prove complementary , if the iekf slam proposed in the present paper is used to build consistent submaps . note that , the iekf slam can also be readily combined with other measurements such as the gps , whereas the submap approach is tailored for pure slam . from a methodology viewpoint , it is worth noting our approach does not bring to bear estimation errors written in a robot frame , as @xcite . although based on symmetries as well , the estimation errors we use are slightly more complicated . finally , nonlinear optimization techniques have become popular for slam recently , see e.g. , @xcite as one of the first papers . links between our approach , and those novel methods are discussed in the paper s conclusion . the paper is organized as follows . in section [ sec:1 ] , the standard ekf equations and ekf - slam algorithm are reviewed . in section [ sec:2 ] we recall the problem that neither the origin nor the orientation of the global frame are observable , but the ekf - slam systematically tends to think " it observes them , which leads to inconsistency . in section [ sec:22 ] we introduce the iekf - slam algorithm . in section [ sec:3 ] we show how the linearized model of the iekf always correctly captures the considered unobservable directions . in section [ sect::tools ] we derive a property of the covariance matrix output by the filter that can be interpreted in terms of fisher information . in section [ sec:4 ] simulations support the theoretical results and illustrate the benefits of the proposed algorithm . finally , the iekf theory of @xcite is briefly recapped in the appendix , and the iekf slam shown to be an application of this theory indeed . the equations of the iekf slam in 3d are then also derived applying the general theory . consider a general dynamical system in discrete time with state @xmath0 associated to a sequence of observations @xmath1 . the equations are as follows : @xmath2 @xmath3 where @xmath4 is the function encoding the evolution of the system , @xmath5 is the process noise , @xmath6 an input , @xmath7 the observation function and @xmath8 the measurement noise . the ekf propagates the estimate @xmath9 obtained after the observation @xmath10 , through the deterministic part of : @xmath11 the update of @xmath12 using the new observation @xmath13 is based on the first - order approximation of the non - linear system , around the estimate @xmath14 , with respect to the estimation errors @xmath15 defined as : @xmath16 using the jacobians @xmath17 , @xmath18 , and @xmath19 , the combination of equations , and yields the following first - order expansion of the error system @xmath20 where the second order terms , that is , terms of order @xmath21 have been removed according to the standard way the ekf handles non - additive noises in the model ( see e.g. , @xcite p. 386 ) . using the linear kalman equations with @xmath22 the gain @xmath23 is computed , and letting @xmath24 , an estimate @xmath25 of the error @xmath26 accounting for the observation @xmath13 is computed , along with its covariance matrix @xmath27 . the state is updated accordingly : @xmath28 the detailed equations are recalled in algorithm [ algo::ekf ] . the assumption underlying the ekf is that through first - order approximations _ of the state error _ evolution , the linear kalman equations allow computing a gaussian approximation of the error @xmath29 after each measurement , yielding an approximation of the sought density @xmath30 . however , the linearizations involved induce inevitable approximations that may lead the filter to inconsistencies and sometimes even divergence . define @xmath31 and @xmath32 through and . define @xmath33 as @xmath34 and @xmath35 as @xmath36 . * propagation * @xmath37 @xmath38 * update * @xmath39 @xmath40 , @xmath41 @xmath42p_{n|n-1 } $ ] for simplicity s sake let us focus on the standard steered " bicycle ( or unicycle ) model @xcite . the state is defined as : @xmath43 where @xmath44 denotes the heading , @xmath45 the 2d position of the robot / vehicle , @xmath46 the position of unknown landmark @xmath47 ( landmarks or synonymously features , constitute the map ) . the equations of the model are : @xmath48 where @xmath49 denotes the odometry - based estimate of the heading variation of the vehicle , @xmath50 the odometry - based indication of relative shift , @xmath51 and @xmath52 their associated noises , and @xmath53 is the matrix encoding a rotation of angle @xmath54 : @xmath55note that a forward euler discretization of the continuous time well - known unicycle equations leads to @xmath50 having its second entry null . more sophisticated integration methods or models including side slip may yet lead to non - zero values of both entries of @xmath56 so we opt for a more general model with @xmath50 . the covariance matrix of the noises will be denoted by @xmath57 with @xmath58 . a general landmark observation in the robot s frame reads : @xmath59+v_n^1 \\ \vdots \\ \tilde h \left [ r(\theta_n)^t \left ( p^k - x_n \right ) \right]+v_n^k \end{pmatrix}\ ] ] where @xmath60 ( or @xmath61 for monocular visual slam ) is the observation of the features at time step @xmath62 , and @xmath8 the observation noise , and @xmath63 is any function . only a subset of the features is actually observed at time @xmath62 . however , to simplify the exposure of the filters equations , we systematically assume in the sequel that all features are observed . we let the output noise covariance matrix be @xmath64 note that , the observation model encompasses the usual range and bearing observations used in the slam problem by letting @xmath65 . if we choose instead the one dimensional observation @xmath66 we recover the 2d monocular slam measurement . note also we do not provide any specific form for the noise in the output : this is because the properties we are about to prove are related to the observability and thus only depend on the deterministic part of the system , so they are in fact totally insensitive to the way the noise enters the system . we merely apply here the methodology of ekf to the slam problem described in section [ sect::slam_problem ] . the first - order expansions , applied to equations , yield : @xmath67 with @xmath68 , @xmath69 and @xmath70 denotes the jacobian of @xmath63 computed at @xmath71 \in { { \mathbb r}}^2 $ ] . the obtained ekf - slam algorithm is recaped in algorithm [ algo::ekf_slam_linear ] . define @xmath31 and @xmath32 as in . define @xmath33 , @xmath35 as in and . * propagation * for all @xmath72 @xmath38 * update * @xmath73 \\ \vdots \\ \tilde h \left [ r(\hat \theta_{n|n-1})^t \left ( \hat p^k_{n|n-1}- \hat x_{n|n-1 } \right ) \right ] \end{pmatrix}$ ] @xmath40 , @xmath41 @xmath42p_{n|n-1 } $ ] in this section we come back to the general framework , . the standard issue of observability @xcite is fundamentally a deterministic notion so the noise is systematically turned off . [ def::non_obs ] we say a transformation @xmath74 of the system - is unobservable if for any initial conditions @xmath75 and @xmath76 the induced solutions of the dynamics with noise turned off , i.e. , @xmath77 yield the same output at each time step @xmath78 , that is : @xmath79 it concretely means that ( with all noises turned off ) if the transformation is applied to the initial state then none of the observations @xmath13 are going to be affected . as a consequence , there is no way to know this transformation has been applied . in line with @xcite we will focus here on the observability properties of the linearized system . to that end we define the notion of non - observable ( or unobservable ) shift which is an infinitesimal counterpart to definition [ def::non_obs ] , and is strongly related to the infinitesimal observability @xcite : [ def::non_obs_first_order ] let @xmath80 denote a solution of with noise turned off . a vector @xmath81 is said to be an unobservable shift of - around @xmath82 if : @xmath83 where @xmath32 is the linearization of @xmath7 at @xmath84 and where @xmath85 is the solution at @xmath62 of the linearized system @xmath86 initialized at @xmath87 , with @xmath88 denoting the jacobian matrix of @xmath89 computed at @xmath90 . in other words ( see e.g. @xcite ) , for all @xmath91 , @xmath85 lies in the kernel of the observability matrix between steps @xmath92 and @xmath62 associated to the linearized error - state system model , i.e. , @xmath93=0 $ ] . the interpretation is as follows : consider another initial state shifted from @xmath82 to @xmath94 . saying that @xmath87 is unobservable means no difference on the sequence of observations up to the first order could be detected between both trajectories . formally , this condition reads : @xmath95 , i.e. , @xmath96 . an estimation method conveying its own estimation uncertainty as the ekf , albeit based on linearizations , should be able to detect such directions and to reflect that accurate estimates along such directions are beyond reach . in the present paper we consider unobservability corresponding to the impossibility to observe the position and orientation of the global frame @xcite . the corresponding shifts have already been derived in the literature . [ prop::first_order_rotations_prelim]@xcite let @xmath97 be an estimate of the state . only one feature is considered , the generalization of the proposition to several features is trivial . the first - order perturbation of the estimate corresponding to an infinitesimal rotation of angle @xmath98 of the global frame consists of the shift @xmath99 with @xmath100 . in the same way , the first - order perturbation of the estimate corresponding to an infinitesimal translation of the global frame of vector @xmath101 consists of the shift @xmath102 . when rotating the global frame the heading becomes : @xmath103 the position of the robot becomes : @xmath104 the position of the feature becomes : @xmath105 stacking these results we obtain the first - order variation of the full state vector ( regarding the rotation only , the effect of infinitesimal translation being trivial to derive ) : @xmath106 [ prop::first_order_rotations]@xcite the shifts of proposition [ prop::first_order_rotations ] that correspond to infinitesimal rotations , are unobservable shifts of - in the sense of definition [ def::non_obs_first_order ] . the intuitive explanation is clear @xcite : `` if the robot and landmark positions are shifted equally along those vectors , it will not be possible to distinguish the shifted position from the original one through the measurements . '' this section recalls using the notations of the present paper , a result of @xcite . it shows the infinitesimal rotations defined in proposition [ prop::first_order_rotations ] are not , in general , unobservable shifts of the system linearized about the trajectory estimated by the ekf . indeed , applying definition [ def::non_obs_first_order ] to in the case of a single feature ( the generalization being straightforward ) with @xmath107 and @xmath108 yields the condition for an infinitesimal rotation of the initial state to be unobservable for the linearized system . this condition writes @xmath109 and boils down to have for any @xmath110 ( see @xcite ) : @xmath111=0\ ] ] where @xmath112 is the jacobian of @xmath63 computed at @xmath113 . for example , if @xmath63 is invertible the condition boils down to @xmath114=0.\label{never_verified}\end{aligned}\ ] ] we see the quantities involved are the updates of the state . as they depend on the noise , there is a null probability for the condition to be respected , and it is always violated in practice . but the point of the present paper is to show that the problem is related to the ( arbitrary in a non - linear context ) choice to represent the estimation error as the linear difference @xmath115 , not to an inconsistency issue inherent to ekf - like methods applied to slam . by devising an ekf - slam based on another estimation error variable , which in some sense amounts to change coordinates , the false observability problem can be corrected . the qualitative reason why this is sufficient is related to the basic cause of false observability : a given fixed shift may or may not be observable depending on the linearization point @xmath116 , as proved by proposition [ prop::first_order_rotations ] . it turns out that the latter property is not inherently related to the slam problem : it is in fact a mere consequence of the errors definition . defining those errors otherwise can dramatically modify the condition . this is the object of the remainder of this article . building upon the theory of the invariant ( i)ekf on matrix lie groups , as described and studied in @xcite , we introduce in this section a novel iekf for slam . in appendix [ primer]-[gen : iekf ] the general theory of the iekf is recalled and slightly extended to account for the very general form of output , and the algorithm derived herein is shown to be a direct application of the theory . to spare the reader a study of the lie group based theory , we attempt to explain in simple terms the iekf methodology on the particular slam example throughout the present section . consider the model equations with state @xmath84 given by . exactly as the ekf , the iekf propagates the estimated state obtained after the observation @xmath10 of through the deterministic part of i.e. , @xmath117 , @xmath118 , @xmath119 for all @xmath72 . to update the predicted state @xmath120 using the observation @xmath13 we use a first order taylor expansion of the error system . but , _ instead of considering the usual state error @xmath121 _ , we rather use the ( linearized ) estimation error defined as follows @xmath122and @xmath123 is analogously defined . for close - by @xmath124 , this represents an error variable in the usual sense indeed , as @xmath125 if and only if @xmath126 . as in the standard ekf methodology , let us see how this _ alternative _ estimation error is propagated through a first - order approximation of the error system . using the propagation equations of the filter , and , we find @xmath127 where terms of order @xmath128 , @xmath129 , and @xmath130 have been neglected as in the standard the ekf handles non - additive noises @xcite . to derive we have used the equalities @xmath131 , @xmath132 : @xmath133 note that , the odometer outputs @xmath134 have miraculously vanished . this is in fact a characteristics - and a key feature - of the iekf approach . let us now compute the first - order approximation of the observation error , using the alternative state error . define @xmath32 as the matrix , depending on @xmath120 only , such that for all @xmath135 defined by , the innovation term @xmath136 \\ \vdots \\ \tilde h \left [ r(\theta_n)^t \left ( p^k - x_n \right ) \right ] \end{pmatrix}-\begin{pmatrix } \tilde h \left [ r(\hat \theta_{n|n-1 } ) ^t \left(\hat p^1_{n|n-1}-\hat x_{n|n-1 } \right ) \right ] \\ \vdots \\ \tilde h \left [ r(\hat \theta_{n|n-1 } ) ^t \left ( \hat p^k_{n|n-1}-\hat x_{n|n-1 } \right ) \right ] \end{pmatrix}\]]is equal to @xmath137 . using that @xmath138 $ ] , and @xmath139 from , we see that @xmath32 is defined as in below . thus the linearized ( first - order ) system model with respect to alternative error writes @xmath140 with @xmath141 , and @xmath142 where @xmath143 is the jacobian of @xmath63 computed at @xmath144 . as in the standard ekf methodology , the matrices @xmath145 allow to compute the kalman gain @xmath23 and covariance @xmath146 . letting @xmath147 be the standardly defined innovation ( see algorithm [ algo::iekf_slam ] just after update " ) , @xmath148 is an estimate of the linearized error @xmath149 accounting for the observation @xmath13 , and @xmath27 is supposed to encode the dispersion @xmath150 . the final step of the standard ekf methodology is to update the estimated state @xmath120 thanks to the estimated linearized error @xmath148 . there is a small catch , though : @xmath151 being not anymore defined as a mere difference @xmath121 , simply adding @xmath123 to @xmath152 would not be appropriate . the most natural counterpart to in our setting , would be to choose for @xmath152 the values of @xmath153 making the right member of equal to the just computed @xmath123 . however , the iekf theory recalled in appendix [ gen : iekf ] , suggests an update that amounts to the latter to the first order , but whose non - linear structure ensures better properties @xcite . thus , the state is updated as follows @xmath154 , with @xmath155 defined by @xmath156where @xmath157 . algorithm [ algo::iekf_slam ] recaps the various steps of the iekf slam . define @xmath31 and @xmath32 as in . define @xmath33 , @xmath35 as in and . * propagation * for all @xmath72 @xmath38 * update * @xmath73 \\ \vdots \\ \tilde h \left [ r(\hat \theta_{n|n-1})^t \left ( \hat p^k_{n|n-1}- \hat x_{n|n-1 } \right ) \right ] \end{pmatrix}$ ] @xmath40 , @xmath41 @xmath42p_{n|n-1 } $ ] in this section we show the infinitesimal rotations and translations of the global frame are unobservable shifts in the sense of definition [ def::non_obs_first_order ] regardless of the linearization points used to compute the matrices @xmath158 and @xmath32 of eq . , a feature in sharp contrast with the usual restricting condition on the linearization points . in other words we show that infinitesimal rotations and translations of the global frame are always unobservable shifts of the system model _ linearized _ with respect to error regardless of the linearization point , a feature in sharp contrast with previous results ( see section [ sect::eks_slam8inconsistency ] and references therein ) . we can consider only one feature ( @xmath159 ) without loss of generality . the expression of the linearized system model has become much simpler , as the linearized error has the remarkable property to remain constant during the propagation step in the absence of noise , since @xmath160 in - . first , let us derive the impact of first - order variations stemming from rotations and translations of the global frame on the error as defined by , that is , an error of the following form @xmath161 [ prop::first_order_rotations_non_linear ] let @xmath162 be an estimate of the state . the first - order perturbation of the _ linearized _ estimation error defined by around 0 , corresponding to an _ infinitesimal _ rotation of angle @xmath98 of the global frame , reads @xmath163 in the same way , an _ infinitesimal _ translation of the global frame with vector @xmath101 implies a first - order perturbation of the error system of the form @xmath164 according to proposition [ prop::first_order_rotations_prelim ] , an infinitesimal rotation by an angle @xmath165 of the true state corresponds to the transformation @xmath166 . @xmath167 and @xmath168 . regarding @xmath151 of eq it corresponds to the variation @xmath169this direction of the state space is seen " by the _ linearized _ error system as the vector @xmath170 . similarly , a translation of vector @xmath171 of the global frame yields the transformation @xmath172 . the effect on the linearized error @xmath151 of is obviously the perturbation @xmath173 neglecting terms of order @xmath174 . we can now prove the first major result of the present article : the infinitesimal transformations stemming from rotations and translations of the gobal frame are unobservable shifts for the iekf linearized model . [ slam : thm : obs ] consider the slam problem defined by equations and , and the iekf - slam algorithm [ algo::iekf_slam ] . let @xmath87 denote a linear combination of infinitesimal rotations and translations @xmath175 of the whole system defined as follows @xmath176 then @xmath87 is an unobservable shift of the linearized system model - of the iekf slam in the sense of definition [ def::non_obs_first_order ] , and this whatever the sequence of true states and estimates @xmath177 : the very structure of the iekf is consistent with the considered unobservability . note that definition [ def::non_obs_first_order ] involves a propagated perturbation @xmath85 , but as here @xmath158 is @xmath178 : we have @xmath179 . thus , the only point to check is : @xmath180 i.e. , @xmath181 . this is straightforward replacing @xmath87 with alternatively @xmath182 and @xmath183 . we obtained the consistency property we were pursuing : the linearized model correctly captures the unobservability of global rotations and translations . as a byproduct , the unobservable seen by the filter is automatically of appropriate dimension . the standard ekf is tuned to reduce the state estimation error @xmath184 defined through the original state variables @xmath124 of the problem . albeit perfectly suited to the linear case , the latter state error has in fact absolutely no fundamental reason to rule the linearization process in a non - linear setting . the basic difference when analyzing the ekf and the iekf is that * in the standard ekf , there is a trivial correspondence between a small variation of the true state and a small variation of the estimation error . but the global rotations of the frame make the error vary in a non - trivial way as recalled in section [ sec:2 ] . * in the iekf approach , the effect of a small rotation of the state on the variation of the estimation error becomes trivial as ensured by proposition [ prop::first_order_rotations_non_linear ] . but the error is non - trivially related to the state , as its definition explicitly depends on the linearization point @xmath116 . many consistency issues of the ekf stem from the fact that the updated covariance matrix @xmath27 is computed before the update , namely at the predicted state @xmath120 , and thus does not account for the updated state s value @xmath152 , albeit supposed to reflect the covariance of the updated error . this is why the oc - ekf typically seeks to avoid linearizing at the latest , albeit best , state estimate , in order to find a close - by state such that the covariance matrix resulting from linearization preserves the observability subspace dimension . the iekf approach is wholly different : the updated covariance @xmath27 is computed at the latest estimate @xmath120 , which is akin to the standard ekf methodology . but it is then indirectly adapted to the updated state , since it is _ interpreted _ as the covariance of the error @xmath123 . and contrarily to the standard case , the definition of this error depends on @xmath152 . more intuitively , we can say the confidence ellipsoids encoded in @xmath27 are attached to a basis that undergoes a transformation when moved over from @xmath120 to @xmath152 , this transformation being tied to the unobservable directions . this prevents spurious reduction of the covariance over unobservable shifts , which are not identical at @xmath120 and @xmath152 . finally , note the alternative error is all but artificial : it naturally stems from the lie group structure of the problem . this is logical as the considered unobservability actually pertains to an _ invariance _ of the model - , that is the slam problem , to global translations and rotations . thus it comes as no surprise the _ invariant _ approach , that brings to bear invariant state errors that encode the very symmetries of the problem , prove fruitful ( see the appendix for more details ) . our approach can be related to the previous work @xcite . indeed , according to the latter article , failing to capture the right dimension of the observability subspace in the linearized model leads to `` spurious information gain along directions of the state space where no information is actually available '' and results in `` unjustified reduction of the covariance estimates , a primary cause of filter inconsistency '' . theorem [ slam : thm : obs ] proves that infinitesimal rotations and translations of the global frame , which are unobservable in the slam problem , are always `` seen '' by the iekf linearized model as unobservable directions indeed , so this filter does not suffer from `` false observability '' issues . this is our major theoretical result . that said , the results of the latter section concern the system with noise turned off , and pertain to an automatic control approach to the notion of observability as in @xcite . the present section is rather concerned with the estimation theoretic consequences of theorem [ slam : thm : obs ] . we prove indeed , that the iekf s output covariance matrix correctly reflects an absence of `` information gain '' along the unobservable directions , as mentioned above , but where the information is now to be understood in the sense of fisher information . as a by - product , this allows relating our results to a slightly different approach to slam consistency , that rather focuses on the fisher information matrix than on the observability matrix , see in particular @xcite . the exposure of the present section is based on the seminal article @xcite . see also @xcite for related ideas applied to slam . consider the system with output . define the collection of state vectors and observations up to time @xmath62 : @xmath185the joint probability distribution of the @xmath186 vector @xmath187 and of the @xmath188 vector @xmath189 is @xmath190the bayesian fisher information matrix ( bifm ) is defined as the following @xmath191 matrix based upon the dyad of the gradient of the log - likelihood : @xmath192[\nabla_{\tilde x_n } \log p(\tilde y_n,\tilde x_n]^t)\]]and note that , for the slam problem it boils down to the matrix of @xcite . this matrix is of interest to us as it yields a lower bound on the accuracy achievable by any estimator used to attack the filtering problem - . indeed let @xmath193 be defined as the _ inverse _ of the @xmath194 right - lower block of @xmath195^{-1}$ ] . this matrix provides a lower bound on the mean square error of estimating @xmath84 from past and present measurements @xmath189 and prior @xmath196 . indeed , for any unbiased estimator @xmath197 : @xmath198[t(\tilde y_n)]^t)\succeq j_n^{-1}\ ] ] where @xmath199 means @xmath200 is positive semi - definite . @xmath201 is called the bayesian or posterior cramr - rao lower bound for the filtering problem @xcite . most interestingly , in the case where @xmath4 and @xmath7 are linear , the prior distribution is gaussian , and the noises are additive and gaussian , we have @xmath202where @xmath27 is the covariance matrix output by the kalman filter . thus , in the linear gaussian case , @xmath203 reflects the statistical information available at time @xmath62 on the state @xmath84 . by extension in the slam literature @xmath203 is often simply referred to as the information matrix , in non - linear contexts also , e.g. when using extended information filters @xcite . in the last section , we have recalled that in the linear gaussian case , the inverse of the covariance matrix output by the kalman filter is the fisher information available to the filter ( this is also stated in @xcite p. 304 ) . in the light of those results , it is natural to expect from any ekf variant , that the inverse of the output covariance matrix @xmath203 reflect an absence of information gain along unobservable directions indeed . if the filter fails to do so , the output covariance matrix will be too optimistic , that is , inconsistent , and wrong covariances yield wrong gains @xcite . the following theorem shows the linearized system model of the iekf allows ensuring the desired property of the covariance matrix . it is our second major result . [ slam : big : thm ] consider the slam problem defined by equations - and the iekf - slam algorithm [ algo::iekf_slam ] . let @xmath87 denote a linear combination of infinitesimal rotations and translations @xmath175 of the whole system , as defined in theorem [ slam : thm : obs ] . @xmath87 is thus an unobservable shift . if the matrix @xmath27 output by the iekf remains invertible , we have at all times : @xmath204 as @xmath205 in , the unobservable shifts remain fixed i.e. @xmath206 . at the propagation step we have : @xmath207 as @xmath208 is positive semi - definite . and at the update step ( see the kalman information filter form in @xcite ) we have : @xmath209 \delta x_0 = \delta x_0^t p_{n|n-1}^{-1 } \delta x_0\ ] ] as @xmath210 as shown in the proof of theorem [ slam : thm : obs ] . thus @xmath211 is non - increasing over time @xmath62 . note that , the proof evidences that if @xmath27 is not invertible , the results of the theorem still hold , writing the iekf in information form . our result essentially means the _ linearized model _ of the iekf has a structure which guarantees that the covariance matrix at all times reflects an absence of spurious " ( bayesian fisher ) information gain over directions that correspond to the unobservable rotations and translations of the global frame . in this section we verify in simulation the claimed properties on the one hand , and on the other we illustrate the striking consistency improvement achieved by the iekf slam . to that end , we propose to consider a similar numerical experiment as in the sound work @xcite dedicated to the inconsistency of ekf and the benefits of the oc - ekf . the iekf is compared here to the standard ekf , the ukf , the oc - ekf , and the ideal ekf , which is the - impossible to implement - variant of the ekf where the state is linearized about the _ true _ trajectory . . we see the indicator stays around 1 for iekf slam and oc - ekf slam over the whole time interval , as expected from a consistent estimation method . the ideal " ekf , where the system is linearized on the true value of the state , yields similar results . to the opposite , we see the ekf is inconsistent , and the ukf also . ] uncertainty envelope computed by the filter . filters whith theoretical properties regarding non - observable directions ( iekf , oc - ekf and ideal ekf ) remedy this problem . ] . bottom plot is a zoom of the first time steps . the information over an infinitesimal perturbation corresponding to a rotation of the whole system is decreasing for the iekf slam , which is a consistent behavior as this perturbation is unobservable . ideal ekf and ocekf give similar results , but ekf and ukf do not . the plot also confirms ekf and ukf slam tend to acquire spurious information over this unobservable direction.,title="fig : " ] . bottom plot is a zoom of the first time steps . the information over an infinitesimal perturbation corresponding to a rotation of the whole system is decreasing for the iekf slam , which is a consistent behavior as this perturbation is unobservable . ideal ekf and ocekf give similar results , but ekf and ukf do not . the plot also confirms ekf and ukf slam tend to acquire spurious information over this unobservable direction.,title="fig : " ] the simulation setting we chose is ( deliberately ) similar to the one used in @xcite ( section 6.2 ) . the vehicle ( or robot ) drives a 15m - diameter loop ten times in the 2d plane , finding on its path 20 unknown features as displayed on figure [ fig::map ] . the velocity and angular velocity are constant ( 1 m / s and 9 deg / s respectively ) . the relative position of the features in the reference frame of the vehicle is observed once every second ( where @xmath63 of is the identity ) . the standard deviation @xmath212 of the velocity measurement on each wheel of the vehicle is @xmath213 of the velocity . this yields the standard deviation @xmath214 of the resulting linear velocity and @xmath215 of the rotational velocity : @xmath216 and @xmath217 , where @xmath218 is the distance between the drive wheels ( see @xcite ) . the features are visible if they lie within a sensing range of 5 m , in which case they are observed with an isotropic noise of standard deviation 10 cm . the initial uncertainty over position and heading is zero - which will prove a condition not sufficient to prevent failure of the ekf . each time a landmark is seen for the first time , its position is initialized in the earth frame using the current estimated pose of the robot , the associated uncertainty is set to a very high value compared to the size of the map , then a kalman update is performed to correlate the position of the new feature withe the other variables . each second , all visible landmarks ( i.e. those in a range of 5 m ) are processed simultaneously in a stacked observation vector . five algorithms are compared : 1 . the classical ekf , described in algorithm [ algo::ekf_slam_linear ] . 2 . the proposed iekf slam algorithm described in algorithm [ algo::iekf_slam ] . the ideal ekf as defined in @xcite , i.e. , a classical ekf where the riccati equation is computed at the true trajectory of the system instead of the estimated trajectory . although not usable in practice , the latter is a good reference to compare with , as it is supposed to be an ekf with consistent behavior . the oc - ekf described in @xcite , which is so far the the only method that guarantees the non - observable subspace has appropriate dimension . 5 . the unscented kalman filter ( ukf ) , known to better deal with the non - linearities than the ekf . before going further , the next subsection introduces the nees indicator used in the simulations to measure the consistency of these methods . classical criteria used to evaluate the performance of an estimation method , like root mean squared ( rms ) error do not inform about consistency as they do not take into account the uncertainty returned by the filter . this point is addressed by the normalized estimation error squared ( nees ) , which computes the average squared value of the error , normalized by the covariance matrix of the ekf . for a sample @xmath219 of error values having dimension @xmath220 , each of them with a covariance matrix @xmath221 of size @xmath222 , the nees is defined by : @xmath223 if each @xmath224 is a zero mean gaussian with covariance matrix @xmath221 , then for large @xmath225 we have nees @xmath226 . the case nees @xmath227 reveals an inconsistency issue : the actual uncertainty is higher than the computed uncertainty . this situation typically occurs when the filter is optimistic as it believes to have gained information over a non - observable direction . the nees indicator will be used , along with the usual rms , to illustrate our solution to slam inconsistency in the sequel . figure [ fig::nees ] displays the nees indicator of the vehicle pose estimate ( heading and position ) over time , computed for 50 monte - carlo runs of the experiment described in section [ sect::exp_setting ] . as expected , the profile of the nees for classical ekf , ideal ekf and oc - ekf is the same as in the previous paper @xcite which inspired this experimental section . note that we used here a normalized version of the nees , making its swing value equal to 1 . we see also that the result is similar for oc - ekf slam and ideal ekf slam : the nees varies between 1 and 1.7 , in contrast to the ekf slam and ukf slam which exhibit large inconsistencies over the robot pose we see here that the iekf remedies inconsistency , with a nees value that remains close to 1 . note that , it performs here even better than oc - ekf and ideal ekf ( whose results are very close to each other ) , in terms of consistency . the basic difference between iekf and these filters lies in using or not the current estimate as a linearization point . uncertainty directions being very dependent from the estimate , what figure [ fig::nees ] suggests is that they may not be correctly captured if computed on a different point . the other aspect of the evaluation of an ekf - like method is performance : regardless of the relevance of the covariance matrix returned by the filter ( i.e. consistency ) , pure performance can be evaluated through rms of the heading and position error , whose values over time are displayed in figure [ fig::heading_rms ] . they confirm an expectable result : solving consistency issues improves the accuracy of the estimate as a byproduct , as wrong covariances yield wrong gains @xcite . selecting a _ single run _ , we can also illustrate the inconsistency issue in terms of covariance and information . figure [ fig::ekf_enveloppe ] displays the heading error for ekf , ukf , iekf , oc - ekf and ideal ekf slam , and the @xmath228 envelope returned by each filter . this illustrates both the false observability issue and the resulting inconsistency of ekf and ukf : the heading uncertainty is reduced over time while the estimation error goes outside the @xmath228 envelope . to the opposite , the behavior of iekf , oc - ekf and ideal ekf is sound . figure [ fig::ekf_enveloppe2 ] shows the map and the landmarks @xmath228 uncertainty ellipsoids : similarly , both ekf and ukf fail capturing the true landmarks positions within the @xmath228 ellipsoids whereas the three over filters succeed to do so . finally , figure [ fig::info_rot ] displays the evolution of the information over a shift corresponding to infinitesimal rotations as defined in theorem [ slam : big : thm ] , that is the evolution over time of the quantity @xmath229 . the theorem is successfully illustrated : the latter quantity is always decreasing for the iekf , ideal ekf , oc - ekf but not for the ekf and the ( slightly better to this respect ) ukf . this work evidences that the ekf algorithm for slam is not inherently inconsistent - at least regarding inconsistency related to unobservable transformations of the global frame - but the choice of the right coordinates for the linearization process is pivotal . we showed that applying the recent theory of the iekf - an ekf ( slight ) variant - leads to provable properties regarding observability and consistency . extensive monte - carlo simulations have illustrated the consistency of the new method and the striking improvement over ekf , ukf , oc - ekf , and more remarkably over the ideal ekf also , which is the - impossible to implement - variant of the ekf where the system is linearized about the true trajectory . note that , the iekf approach may prove relevant beyond slam to some other problems in robotics as well , such as autonomous navigation ( see @xcite ) , and in combination with controllers , notably for motion planning purposes , see @xcite . in @xcite the iekf has proved to possess _ global _ asymptotic convergence properties on a simple localization problem of a wheeled robot , which is a strong property . the iekf has also been patented for navigation with inertial sensors @xcite . nowadays nonlinear optimization based slam algorithms are becoming popular as compared with ekf slam , see e.g. @xcite for one of the first papers on the subject . we yet anticipate a simple ekf slam with consistency properties will prove useful to the research community , the ekf slam having been abandoned in part due to its inconsistency . the general ekf has proved useful in numerous industrial applications , especially in the field of guidance and navigation . it has the benefits of being 1-recursive , avoiding to store the whole trajectory and 2-suited to on - line real - time applications . moreover the aerospace and defense industry has developed a corpus of experience for its industrial implementation and validation . and the iekf is a variant that , being in every respect similar to ekf , retains all its advantages , but which possesses additional guaranteed properties . note also that , all the improvements of the ekf for slam such as e.g. , the slam of @xcite and sparse extended information filters @xcite , can virtually be turned into their invariant counterpart . the high dimensional optimization formulation of the slam problem being prone to local minima , having an accurate initial value ( i.e. a small initial estimation error ) is very critical @xcite . the iekf slam algorithm proposed in the present paper may thus be advantageously used to initialize those methods in challenging situations . besides , we anticipate our approach based on symmetries could help improve ( at least first order ) optimization techniques for slam . to understand why , assume by simplicity the sensors to be noise free . then , moving a candidate trajectory along unobservable directions will not change the cost function , and an efficient optimization algorithm should account for this . and when a gradient descent algorithm is used , only a first - order expansion of the cost function is considered . our lie group approach will allow defining steepest descent directions in a alternative geometric way , that will `` stick '' to the unobservable directions , and the corresponding update will move along the ( lie group ) state space in a non - linear yet relevant way . this issue is left for future work , but a thorough understanding of the interest of the invariant approach for the ekf , is a first step in this direction . the authors would like to thank cyril joly for his advice . in this section we provide more details on the iekf theory on matrix lie groups , and show how the underlying lie group structure of the slam problem has been used indeed to build the iekf slam algorithm [ algo::iekf_slam ] . we also provide the iekf equations for 3d slam . for more information on the iekf see @xcite and references therein . a matrix lie group @xmath230 is a subset of square invertible @xmath231 matrices @xmath232 verifying the following properties : @xmath233 where @xmath234 is the identity matrix of @xmath235 . if @xmath236 is a curve over @xmath230 with @xmath237 , then its derivative at @xmath238 necessarily lies in a subset @xmath239 of @xmath240 . @xmath239 is a vector space and it is called the lie algebra of @xmath230 . it has same dimension @xmath241 as @xmath230 . thanks to a linear invertible map denoted by @xmath242 , one can advantageously identify @xmath239 to @xmath243 . besides , the vector space @xmath239 can be mapped to the matrix lie group @xmath230 through the classical matrix exponential @xmath244 . thus , @xmath245 can be mapped to @xmath230 through the lie exponential map defined by @xmath246 for @xmath247 . this map is invertible for small @xmath151 , and we have @xmath248 . the well - known baker - campbell - hausdorff ( bch ) formula gives a series expansion for the product @xmath249 . in particular it ensures @xmath250 , where @xmath251 is of the order @xmath252 . for any @xmath253 , the adjoint matrix @xmath254 is defined by @xmath255 for all @xmath256 . we now give explicit formulas for two groups of particular interest for the slam problem . this famous group in robotics can be defined using homogeneous matrices , i.e. , @xmath258 . @xmath259 , then @xmath260 , where @xmath261 and @xmath262 . we have @xmath263 . the lie exponential writes @xmath264 where @xmath157 . we have @xmath265 . we now introduce a simple extension of @xmath257 , inspiring from preliminary remarks in @xcite . for @xmath267 and @xmath268 , consider the map @xmath269 defined by @xmath270 \vdots & \\[-0.5ex ] 0_{1,2 } & \end{array } \right)\ ] ] and let @xmath271 be defined by @xmath272and denote it by @xmath273 . note that , we recover @xmath257 for @xmath159 , i.e. , @xmath274 . letting @xmath261 and @xmath275 yields @xmath276 \vdots & \\[-0.5ex ] 0_{1,2 } & \end{array } \right)$ ] and @xmath277 . it turns out , by extension of the @xmath257 results , that there exists a closed form for the lie exponential @xmath278 that writes @xmath279 with @xmath157 . the @xmath280 is also easily derived by extension of @xmath257 , but to save space , we only display it once : @xmath281 is defined as the matrix @xmath282 of eq . this section is a summary of the iekf methodology of @xcite . let @xmath230 be a matrix lie group . consider a general dynamical system @xmath283 on the group , associated to a sequence of observations @xmath1 , with equations as follows : @xmath284 @xmath285 where @xmath286 is an input matrix which encodes the displacement according to the evolution model , @xmath287 is a vector encoding the model noise , @xmath288 is the observation function and @xmath289 the measurement noise . the iekf propagates an estimate obtained after the previous observation @xmath10 through the deterministic part of : @xmath290 to update @xmath291 using the new observation @xmath13 , one has to consider an estimation error that is _ well - defined _ on the group . in this paper we will use the following right - invariant errors @xmath292 which are equal to @xmath234 when @xmath293 . the terminology stems from the fact they are invariant to right multiplications , that is , transformations of the form @xmath294 with @xmath253 . note that , one could alternatively consider left - invariant errors but it turns out to be less fruitful for slam . the iekf update is based upon a first - order expansion of the non - linear system associated to the errors around @xmath234 . first , compute the full error s evolution @xmath295note that the term @xmath296 has disappeared ! this is a key property for the successes of the invariant filtering approach @xcite . to linearize this equation we define @xmath297 around @xmath234 through @xmath298as in the standard non - additive noise ekf methodology @xcite all terms of order @xmath299 , are assumed small and are neglected . using the bch formula , and neglecting the latter terms , we get @xmath300 using the local invertibility of @xmath301 around @xmath302 , we get the following linearized error evolution in @xmath243 : @xmath303 where @xmath304 and @xmath305 . to linearize the output error , we now slightly adapt the iekf theory @xcite to account for the general form of output . note that , @xmath306 . as @xmath149 is assumed small , and as @xmath307 , a first - order taylor expansion in @xmath247 arbitrary , allows definit @xmath32 as follows @xmath308 as in the standard theory , the kalman gain matrix @xmath23 allows computing an estimate of the linearized error after the observation @xmath13 through @xmath309 , where @xmath310 . recall the state estimation errors defined by - are of the form @xmath311 , that is , @xmath312 . thus an estimate of @xmath313 after observation @xmath13 which is consistent with - , is obtained through the following lie group counterpart of the linear update @xmath314 the equations of the filter are detailed in algorithm [ algo::iekf ] . choose initial @xmath315 and @xmath316 [ algo::iekf ] define @xmath32 as in and let @xmath317 and @xmath318 . define @xmath33 as @xmath34 and @xmath35 as @xmath36 . * propagation * @xmath319 @xmath320 * update * @xmath321 @xmath322 , @xmath41 @xmath42p_{n|n-1 } $ ] the lie group that underlies the slam problem , is @xmath273 introduced in appendix [ sect::tuto_se23 ] . let us apply the general theory of the iekf to this group . to define the lie group counterpart @xmath313 of the state @xmath84 defined by , we let @xmath323 . the model equations write @xmath324 with @xmath325 . at the propagation step , the iekf propagates the estimate through the corresponding deterministic equations @xmath326with @xmath327 . a simple matrix multiplication shows that @xmath328 , where @xmath329 \\ p^1_n-[r \left ( \theta_n -{\hat \theta}_{n|n-1}\right ) \hat p^1_{n|n-1 } ] \\ \vdots \\ p^k_n- [ r \left ( \theta_n -{\hat \theta}_{n|n-1}\right ) \hat p^k_{n|n-1 } ] \end{pmatrix}\]]and where @xmath330 is defined analogously . the linearized error is defined by , that is , @xmath331 . as terms of order @xmath332 are to be neglected in the linearized equations , it suffices to compute a first - order approximation of @xmath333 . first note that @xmath149 defined by is a linear approximation to @xmath334 , that is , @xmath335 . this readily implies @xmath336recalling @xmath337 we see is a first approximation of @xmath333 as defined in indeed . the way the linearized error propagates has already been computed and consists of , which is the same as the first equation of . it exactly matches what can be expected from the general theory , that is , equation , recalling that @xmath282 of is the map @xmath338 of the group @xmath273 indeed . extending the group @xmath266 to the 3d case , and applying the general iekf theory of section [ gen : iekf ] , we derive in the present section an iekf for 3d slam . due to space limitations an as it is not the primary object of the present paper we pursue extreme brevity of exposure . see also @xcite . note that , although the 3d slam equations make use of rotation matrices , they are in fact totally intrinsic : when using quaternions ( recommended ) or euler angles ( not recommended ) they write the same as the group @xmath342 we introduce does in fact not depend on a specific representation of rotations . the equations of the robot in 3d and in continuous time write : @xmath343 where @xmath344 is a rotation matrix that represents the robot s orientation at time @xmath345 , @xmath346 denotes the angular velocity of the robot measured by a gyrometer or by odometry ( in combination with a unicycle model for a terrestrial vehicle ) , @xmath347 the velocity in the robot s frame , and @xmath348 is the position of landmark @xmath47 , and where @xmath349 for @xmath350 denotes the skew symmetric matrix of @xmath351 such that for any @xmath352 we have @xmath353 . finally @xmath354 and @xmath355 denote ( resp . ) the noise on angular and linear velocities . although the theory of iekf could very well be applied directly to this continuous time dynamics as in @xcite , we apply it here to a discretized model , to be consistent with the rest of the article . although exact discretization of the noisy model on the group is beyond reach @xcite , letting @xmath356 be the time step , the following first - order integration scheme is widely used : @xmath357\omega_n , \quad x_n = x_{n-1}+r_{n-1 } ( v_n+ w_n^v ) , \\ p_n^j & = p_{n-1}^j,\quad 1\leq j\leq k \end{aligned}\ ] ] where the increments @xmath358 are obtained solving the noise - free initial conditions during the @xmath62-th time step with initial condition @xmath359 , and where the following discrete noise @xmath360 is obtained by integration of the corresponding white noises . note that , this scheme is accurate to first - order terms in @xmath356 . a general landmark observation in the car s frame reads : @xmath361+v_n^1 \\ \vdots \\ \tilde h \left [ r_n^t \left ( p^k - x_n \right ) \right]+v_n^k \end{pmatrix}\ ] ] where @xmath362 ( or @xmath363 for monocular visual slam ) is the observation of the features at time step @xmath62 , and @xmath8 the observation noise . we let the output noise covariance matrix be @xmath364 ( not to be confused with the rotation @xmath35 ) . the lie group that underlies the problem is the group @xmath365 that we introduce as follows . for @xmath366 and @xmath367 let @xmath368 \vdots & \\[-0.5ex ] 0_{1,3 } & \end{array } \right)\ ] ] and let @xmath369 be defined as @xmath370and denote it by @xmath365 . we then have @xmath371 \vdots & \\[-0.5ex ] 0_{1,3 } & \end{array } \right)$ ] and @xmath372 . for @xmath373 , by extension of the @xmath374 results , we have the closed form : @xmath375 where @xmath376 . as easily seen by analogy with @xmath374@xmath377 \vdots & \\[-0.5ex ] ( p^k)_\times r & \end{array } \right)\end{aligned}\ ] ] let the state be @xmath378 , and let @xmath379 be @xmath380 , and let @xmath116 and @xmath381 be their estimated counterparts . it is easily seen that up to terms that will disappear in the linearization process anyway , the model for the state is mapped through @xmath382 defined at , to a model of the form . using the matrix logarithm , define @xmath384 as the solution of @xmath385 = r\hat r^{t}$ ] . neglecting terms of order @xmath386 , we have @xmath387 . a first order identification as in appendix [ lnz : sec ] thus yields @xmath388 as a vector that satisfies the definition @xmath389 up to terms of order @xmath390 ) . as in the standard ekf theory , the iekf propagates an estimate obtained after the previous observation @xmath10 through the deterministic part of , or equivalently in matrix form . thus , the propagation equation is given by where @xmath391 and @xmath392 as a direct application of the theory . let @xmath391 , @xmath32 as in , @xmath396 using define @xmath33 by . @xmath397 is the observation noise cov . propagation * for all @xmath72 @xmath38 * update * @xmath398 \\ \vdots \\ \tilde h \left [ r_{n|n-1}^t \left ( \hat p^k_{n|n-1}- \hat x_{n|n-1 } \right ) \right ] \end{pmatrix}$ ] @xmath399 , @xmath41 @xmath42p_{n|n-1 } $ ] t. bailey , j. nieto , j. guivant , m. stevens , and e. nebot . consistency of the ekf - slam algorithm . in _ intelligent robots and systems , 2006 ieee / rsj international conference on _ , pages 35623568 . ieee , 2006 . s. bonnabel , p. martin , and e. salaun . invariant extended kalman filter : theory and application to a velocity - aided attitude estimation problem . in _ ieee conference on decision and control _ , pp . 1297 - 1304 , 2009 . g. p. huang , a. mourikis , st . an observability - constrained sliding window filter for slam . in _ intelligent robots and systems ( iros ) , 2011 ieee / rsj international conference on _ , pages 6572 . ieee , 2011 . g. p. huang , a. i. mourikis , and s. i roumeliotis . analysis and improvement of the consistency of extended kalman filter based slam . in _ robotics and automation , 2008 . icra 2008 . ieee international conference on _ , pages 473479 . ieee , 2008 . s. j. julier and j. k. uhlmann . a counter example to the theory of simultaneous localization and map building . in _ robotics and automation , 2001 . proceedings 2001 icra . ieee international conference on _ , volume 4 , pages 42384243 . ieee , 2001 . k. w. lee , w. s. wijesoma , and j. i. guzman . on the observability and observability analysis of slam . in _ intelligent robots and systems ( iros ) , 2006 ieee / rsj international conference on _ , pages 35693574 . ieee , 2006 . a. martinelli , n. tomatis , and r. siegwart . some results on slam and the closing the loop problem . in _ intelligent robots and systems ( iros ) , 2012 ieee / rsj international conference on _ , pages 334339 . ieee , 2005 . s. thrun , y. liu , d. koller , a. y. ng,2 . ghahramani and h. durrant - whyte . simultaneous localization and mapping with sparse extended information filters . in _ the international journal of robotics research _ , 23(7 - 8 ) : 693 - 716 , 2004 . l. zhao , s. huang , and g. dissanayake . linear slam : a linear solution to the feature - based and pose graph slam based on submap joining . in _ intelligent robots and systems ( iros ) , 2013 ieee / rsj international conference on . _ ieee , 2013 .

in this paper we address the inconsistency of the ekf - based slam algorithm that stems from non - observability of the origin and orientation of the global reference frame . we prove on the non - linear two - dimensional problem with point landmarks observed that this type of inconsistency is remedied using the invariant ekf , a recently introduced variant of the ekf meant to account for the symmetries of the state space . extensive monte - carlo runs illustrate the theoretical results .

1510.06263

ultra - cold atomic gases offer possibilities for realizations of complex mathematical models used in different fields of physics with an unprecedented level of the experimental control @xcite . for example , condensed matter phenomena like the superfluid - mott insulator transition and the bose - glass phase or the anderson localization effects can be experimentally investigated @xcite . fermionic gases , in particular fermi superfluids , have received a lot of attention , especially after the observation of the transition between the superfluid bardeen - cooper - schrieffer ( bcs ) pairs and the bose - einstein condensate ( bec ) of diatomic molecules @xcite . the behavior of a small _ object _ immersed in degenerate quantum gases has been investigated by several authors @xcite . for example , weak interactions between a single impurity atom and particles of a large bec can be described by the perturbation theory . for stronger interactions an effective mass of an impurity atom diverges indicating the breakdown of the perturbation approach and the self - localization of the impurity _ object _ in a close analogy to the small polaron problem , i.e. localization of an electron in a surrounding cloud of lattice distortions @xcite . in ultra - cold fermionic gases an example of polaron effects with a small number of spin - up fermions immersed in a large cloud of spin - down fermi particles has been studied theoretically @xcite and recently realized experimentally @xcite . employing a feshbach resonance , that allows tuning the interaction strength between atoms , experimentalists have been able to investigate a transition from the nearly non - interacting case , through the polaron regime to the limit where pairs of unlike fermions form tightly bound molecules . in the present publication we consider a small number of bose particles immersed in a large , homogeneous , superfluid and balanced mixture of spin - up and spin - down fermions and analyze the self - localization phenomenon . another limit , investigated already in the literature , concerns bose - fermi mixtures with a number of bosons comparable to ( or even larger than ) a number of fermions and effects of the phase separation @xcite . the latter corresponds to instability of a homogeneous solution when boson - fermion interaction reaches a critical strength . in the case of small boson numbers , the boson - boson interactions can be neglected and the uniform density solution is unstable as soon as the boson - fermion coupling constant becomes non - zero . however , this does not mean the self - localization of bose particles . we show that the self - localization takes place for stronger interactions when the boson - fermion coupling constant is greater than a non - zero critical value . the possibility of solitonic behavior in bose - fermi mixtures with fermions both in the normal and superfluid states has been investigated in the literature @xcite . for a large number of bosons , if the attractive boson - fermion interaction is sufficiently strong , the boson - boson repulsion may be outweighed and the whole bose and fermi clouds reveal solitonic behavior . we consider bose - fermi mixtures in the opposite limit of small boson numbers . in that regime different kind of solitons exists . indeed , in the 1d case description of the system may be reduced to a simple model where bosons and a single pair of fermions at the bottom of the fermi sea are described by a vector soliton solution . the paper is organized as follows . in sec . [ model ] we introduce the model used in the description of bose - fermi mixtures . the results for the case of three - dimensional ( 3d ) and 1d spaces are collected in sec . [ results ] and we conclude in sec . [ conclusions ] . let us consider a small number @xmath0 of bosonic atoms in the bose - einstein condensate state immersed in a homogeneous , dilute and balanced mixture of fermions in two different internal spin states in a 3d volume . interactions of ultra - cold atoms can be described via contact potentials @xmath1 with strengths given in terms of @xmath2-wave scattering lengths @xmath3 as @xmath4 , where @xmath5 stands for a reduce mass of a pair of interacting atoms . in our model we consider attractive interactions between fermions in different spin states , i.e. negative coupling constant @xmath6 . interactions between bosons and fermions are determined by the spin - independent parameter @xmath7 . we neglect mutual interactions of bosonic atoms in the assumption that either their density remains sufficiently small or the coupling constant is negligible . the system is described by the following hamiltonian [ h ] & = & ^3 r , & & where @xmath8 . @xmath9 and @xmath10 refer , respectively , to the field operators of bosonic and fermionic atoms where @xmath11 indicates a spin state . @xmath12 stands for the chemical potential of the fermi sub - system and @xmath13 and @xmath14 are masses of bosons and fermions , respectively . we look for a thermal equilibrium state assuming that the bose and fermi sub - systems are separable . for instance in the limit of zero temperature it is given by a product ground state [ produkt ] |= |_f |_b . we also postulate that the fermi sub - system can be described by the bcs mean - field approximation @xcite with the paring field @xmath15 and the hartree - fock potential @xmath16 affected by a potential proportional to the density of bosons @xmath17 . assuming a spherical symmetry of particle densities , the description of the system reduces to the bogoliubov - de gennes equations for fermions [ bg ] ( h_0+w + g_bfn_b||^2)u_nlm+v_nlm&=&e_nl u_nlm^ * u_nlm-(h_0+w + g_bfn_b||^2 ) v_nlm&=&e_nlv_nlm , & & where @xmath18 and @xmath19 stand for angular momentum quantum numbers and w & = & -|g_ff|_nlm , & & + & = & |g_ff| _ nlm ( 1 - 2f_nl ) u_nlm ( ) v^*_nlm ( ) , [ delta ] with the fermi - dirac distribution f_nl= , which have to be solved together with the gross - pitaevskii equation for bosons [ bosony ] ( r)= _ b(r ) , where [ veff ] v(r)=-w(r)=g_bf_f(r ) . the effective potential @xmath20 for bosons comes from contact interactions between bosons and fermions . @xmath21 is density of fermions and @xmath22 is the chemical potential for bosons . we consider the temperature much lower than the critical temperature for bose - einstein condensation therefore we can neglect thermal excitations of bosons . the coupled equations ( [ bg ] ) and ( [ bosony ] ) are solved numerically in a self - consisted manner . in the calculations we adopt [ units ] e_0&=&2e_f= , l_0&= & , units for energy and length , respectively , where @xmath23 is the fermi wave - number of a uniform ideal fermi gas of density @xmath24 . in these units the coupling constants are the following g_ff&=&4k_fa_ff , g_bf&=&2k_fa_bf(1 + ) , and we deal with six independent parameters in the system : number of bosons @xmath0 , chemical potential of fermi sub - system @xmath12 , ratio of the masses @xmath25 , scattering lengths @xmath26 and @xmath27 and radius @xmath28 of the 3d volume we consider . in the 3d case the coupling constant @xmath6 in @xmath29 [ eq . ( [ delta ] ) ] has to be regularized in order to avoid ultraviolet divergences . that is , @xmath30 where @xmath31 the logarithmic term in ( [ regular ] ) results from the sum over bogoliubov modes corresponding to the energy above a numerical cutoff @xmath32 performed in the spirit of the local density approximation , see @xcite for details . without interactions between bosons and fermions the ground state of the system corresponds to uniform particle densities . for the non - zero coupling constant @xmath7 , the uniform solution become unstable and , depending on the sign of @xmath7 , the bosonic and fermionic clouds tend to separate from each other or try to stick together . for sufficiently strong interactions , the effect of the self - localization may be expected ( see the similar problem in the case of an impurity atom immersed in a large bose - einstein condensate considered in ref . @xcite ) . indeed , for @xmath33 bosons repel fermions and create a potential well in their vicinity where they may localize if the well is sufficiently large . for attractive interactions the density of fermions increases in the vicinity of bose particles . due to the fact that @xmath34 , the bosons experience the density deformation in a form of a potential well and they may localize . we begin with the 3d model and focus on the repulsive boson - fermion interactions . analysis of both zero - temperature limit and thermal effects are performed . then we consider the 1d case where the self - localization phenomenon may be related to the presence of a vector soliton solution . na atoms in a superfluid mixture of @xmath35k atoms . panel ( a ) shows the pairing function @xmath36 , panel ( b ) fermion density @xmath37 and panel ( c ) density of bosons @xmath38 . solid black lines correspond to boson - fermion interaction strength @xmath39 and dotted - dashed blue lines to @xmath40 . in panel ( c ) the dotted - dashed blue line is hardly visible because for @xmath40 bosons are delocalized and their density very small . number of bosons @xmath41 and fermions @xmath42 ( chemical potential @xmath43 ) and fermion - fermion coupling constant @xmath44 . ] of two fermion pairs at the bottom of the fermi sea with angular momentum @xmath45 . panel ( a ) corresponds to the ground state ( @xmath46 ) of the radial degree of freedom and panel ( b ) to the first excited state ( @xmath47 ) . solid black lines correspond to boson - fermion interaction strength @xmath39 and dotted - dashed blue lines to @xmath40 . all parameters are the same as in fig . [ one ] . ] [ panel ( a ) ] and the standard deviation @xmath48 [ panel ( b ) ] versus boson - fermion coupling constant @xmath7 . black full symbols correspond to a mixture of @xmath49na and @xmath35k atoms while red open symbols to a mixture of @xmath50li and @xmath35k atoms . note the abrupt transitions to localized states when critical values of @xmath7 are reached . all the other parameters are the same as in figs . [ one]-[two ] . ] na atoms in a superfluid mixture of @xmath35k atoms for non - zero temperature . panel ( a ) shows the pairing function @xmath36 , panel ( b ) fermion density @xmath37 and panel ( c ) density of bosons @xmath38 . solid black lines correspond to @xmath51 and @xmath43 , red dashed lines to @xmath52 and @xmath53 , blue dotted - dashed lines to @xmath54 and @xmath55 . boson - fermion interaction strength @xmath39 , fermion - fermion coupling constant @xmath44 and number of bosons @xmath41 and fermions @xmath56 . in panel ( a ) the dotted - dashed blue line is not visible because for @xmath57 the pairing function is equal zero . in panels ( c ) and ( d ) the solid black and dashed red lines are hardly distinguishable . ] figure [ one ] shows the densities of bosons and fermions and the pairing function corresponding to the ground state of the system for @xmath40 and @xmath39 . without boson - fermion interactions the quantities are flat and uniform ( except a small region close to the edge of the 3d volume due to assumed open boundary conditions ) . however , when the considerable interactions are turned on it becomes energetically favorable to separate bosons and fermions , the @xmath58 is depleted around the center and bosons form a bound state localized in small area around @xmath59 . it is clear , that the localization effect is the result of boson - fermion interactions . it relies on a local deformation of the density of fermions and is not affected by the boundary conditions . the response of the fermi sub - system to bosons , that tend to localize , can be investigated by monitoring deformation of the bogoliubov quasi - particle modes . the density of fermions is the sum of the bogoliubov modes @xmath60 . the modes with zero angular momentum contribute only to the density around @xmath59 . consequently , the modification of these modes is primarily responsible for preparation of the potential well in which bosons localize . in fig . [ two ] we illustrate the deformation of two modes with @xmath45 corresponding to fermions at the bottom of the fermi sea but we should keep in mind that all modes with @xmath45 become affected by the interactions with bosons . the deformation of modes for fermions at the fermi level is reflected by a change of a shape of the pairing field visible in fig . [ one ] , because those modes contribute mainly to @xmath61 . the interaction of fermions and the _ impurity _ bose particles influences the pairing function @xmath29 only locally , see fig . it implies that the superfluidity is not destroyed even when the interaction is so strong that the localization of the _ impurity _ object takes place . the data in figs . [ one]-[two ] are related to @xmath41 @xmath49na atoms and the mixture of @xmath56 @xmath35k atoms ( chemical potential @xmath43 ) in two different hyperfine states . we set the scattering lengths @xmath44 and @xmath39 with the assumption that they can be realized by the use of the feshbach resonances ( e.g. magnetic resonance for fermions and optical resonance between bosons and fermions @xcite ) . in fig . [ three ] we show the average radius of the bose cloud @xmath62 and the standard deviation @xmath48 as a function of the coupling constant @xmath7 . the self - localization means that both @xmath62 and @xmath63 are much smaller than the radius of the 3d volume . one can see that there is a critical non - zero value of @xmath7 leading to the self - localization . this critical @xmath7 is different from the critical value for the instability of the homogeneous solution ( i.e. phase separation condition ) . the latter , for the case without boson - boson interactions , corresponds to @xmath33 . if we replace the sodium atoms by @xmath50li atoms , it turns out that the critical value of @xmath7 for the self - localization increases . this is , because compressing the cloud of light lithium particles costs more energy than in the case of heavier sodium atoms . a small non - zero temperature mostly affects superfluidity and has little effect on the self - localization phenomenon . indeed , in fig . [ four ] we see that even for @xmath52 when the pairing function is very small the densities of bosons and fermions hardly change . increasing temperature to @xmath64 ( which is still much smaller than critical temperature for bose - einstein condensation of @xmath41 bosonic atoms localized in a volume of the radius @xmath65 , i.e. @xmath66 ) we observe effects of thermal fluctuations in the fermion density and a modification of the density of bosons but the self - localization persists . thus , bosons self - localize both for the normal and superfluid phase of the fermi sub - system . we have considered the repulsive boson - fermion interaction . for the attractive interaction we do not observe the self - localization regardless on the phase of the fermi sub - system . for @xmath34 the particle densities may collapse to dirac - delta distributions . for sufficiently small @xmath67 a metastable state may appear . however , it turns out that the existence of such a metastable state is not the result of self - localization in the system . indeed , it is an effect of a compromise between the requirement of minimal kinetic energies and restrictions related to the boundary conditions . in the following we consider a 1d model where there is no problem with the collapse of the densities and show that bose particles can localize in the fermi sub - system for attractive boson - fermion interactions too . ) in a superfluid mixture of fermions in 1d space . panel ( a ) shows the pairing function @xmath68 , panel ( b ) fermion density @xmath69 and panel ( c ) boson density @xmath70 . solid black lines correspond to boson - fermion interaction strength @xmath71 and dotted - dashed blue lines to @xmath40 . number of fermions @xmath72 ( chemical potential @xmath43 ) and fermion - fermion coupling constant @xmath73 . ratio of masses of bose and fermi particles @xmath25 fulfills eq . ( [ msol ] ) . the configuration space extends from @xmath74 to @xmath75 . in panel ( c ) the dotted - dashed blue line is hardly visible , because the boson is delocalized and its density very small for @xmath40 . red dashed line in panel ( c ) indicates the solitonic solution eq . ( [ soliton ] ) . ] corresponding to fermion pairs located close to the bottom of the fermi sea . panel ( a ) is related to the pair of fermions at the bottom of the fermi sea , panel ( b ) and ( c ) to the next pairs . solid black lines correspond to the numerical solutions . red dashed line in panel ( a ) indicates solitonic solution eq . ( [ soliton ] ) . all the others parameters are the same as in fig . [ five ] . ] , versus boson - fermion coupling constant @xmath7 . black full symbols correspond to the numerical values and red open symbols to the solutions eq . ( [ soliton ] ) . the configuration space extends from @xmath76 to @xmath77 . all the others parameters are the same as in fig . [ five ] . ] if in @xmath78 and @xmath79 directions we apply harmonic potentials of frequency @xmath80 and @xmath81 is much greater than the chemical potentials , the 3d system becomes effectively one - dimensional . then , the description reduces to the 1d version of eqs . ( [ bg])-([veff ] ) with the following coupling constants [ in the units ( [ units ] ) ] [ 1dcoup ] g_ff^1d&=&a_ff , g_bf^1d&=&a_bf , which have been obtained assuming that the @xmath78 and @xmath79 degrees of freedom of each atom are in the ground states of the harmonic potentials . in the 1d case there is no ultraviolet divergence and the pairing function does not require regularization . nevertheless numerical simulations converge much faster if the bogoliubov modes , above a numerical cut - off energy @xmath82 , are included in the spirit of the local density approximation . that is , the coupling constant in @xmath29 is substituted by = - . for repulsive boson - fermion interactions , we observe the self - localization of bosons with the behaviour of the particle densities similar as in the 3d case . therefore we focus on attractive interactions only . figure [ five ] shows the results for @xmath71 , obtained with periodic boundary conditions for fermions and open boundary conditions for bosons . for the attractive interactions bosons and fermions try to stick together which leads to an increase of the fermion density in the vicinity of the boson concentration and the creation of a potential well for localization of bose particles . analyzing the bogoliubov modes @xmath83 ( see fig . [ six ] ) we find out that the probability density @xmath84 of a pair of fermions at the bottom of the fermi sea becomes strongly localized . the bogoliubov mode @xmath85 of the next fermion pair forms also a bound state . since @xmath85 is an antisymmetric function it is nearly zero in the area around @xmath86 . probability densities of other fermions are deformed and almost all of them drop to zero in the region where @xmath87 is localized . this may be interpreted as a realization of the pauli exclusion rule . in the bcs limit only particles close to the fermi level contribute to the pairing function @xmath29 and there is practically no contributions from fermions located deeply in the fermi sea . therefore there is also no contribution from the pair of fermions at the bottom of the fermi sea . that is why @xmath68 , contrary to the fermion density , reveals a minimum at @xmath86 , see fig . [ five ] . the analysis of the bogoliubov modes suggests a simple model of self - localization in the case of attractive boson - fermion interactions . suppose , that in the vicinity of the localized bosons we may neglect the pairing field and the density of all fermions except a fermion pair at the bottom of the fermi sea . then , we obtain the following set of equations ( -e_0)v_0&=&v_0 , [ v01 ] + & & _ b&=&. [ eqsol ] for = + , [ msol ] there exists analytical solution of eqs . ( [ v01])-([eqsol ] ) , [ soliton ] ( z)=v_0(z)=(z ) , with & = & |g_bf| , e_0&=&+ , _ such a solution resembles vector solitons . they appear in non - linear optics when interactions of several field components are described by a set of coupled non - linear schrdinger equations @xcite . note that for the self - localization of an impurity atom in a large bec considered in ref . @xcite , the 1d system is described by a parametric soliton with the state of the impurity atom given by the hyperbolic secant squared function . a comparison of the analytical solutions ( [ soliton ] ) with numerical results of the full set of equations is shown in figs . [ five]-[six ] . the agreement is very good and increases with the strength of boson - fermion interactions . indeed , for the strong interaction , due to the pauli exclusion rule , there is negligible probability density to find other fermions than the localized pair in the vicinity of @xmath86 . as a consequence , the localized bosons interact almost exclusively with the localized fermion pair and the set of eqs . ( [ v01])-([eqsol ] ) becomes exact . figure [ six]b shows that the bogoliubov mode @xmath85 forms an antisymmetric bound state . in the vicinity of @xmath86 ( where the fermion density is dominated by @xmath88 and the pairing function drops to zero ) this mode should fulfill equation similar to eq . ( [ v01 ] ) , that is ( -e_1)v_1=v_1 . [ v1 ] if @xmath89 and @xmath90 are given by eq . ( [ soliton ] ) the antisymmetric solution of eq . ( [ v1 ] ) forms a marginal bound state v_1(z)&~ & ( z ) , + e_1&=&. in the full description of the system , the state govern by the equation ( 19 ) may become either truely bound or unbound . in the considered system , it turns out that the state is pushed towards a true bound state as visible in fig . 6b . when the boson - fermion coupling constant @xmath7 is decreased , we observe the increasing discrepancy between analytical and numerical solutions , see fig . [ seven ] . the width of the boson probability density obtained numerically is significantly greater than the corresponding analytical value . this is due to the fact , that in the effective potential experienced by the bosons a considerable contribution comes from other fermions , and not only from the pair at the bottom of the fermi sea . the density of such fermions , contrary to the localized fermion pair , possesses a minimum at @xmath86 and thus effectively makes the potential well for bosons weaker . consequently , bosons occupy a much larger space than can be expected on the basis of solutions eq . ( [ soliton ] ) . we have considered a small number of bosons immersed in a superfluid mixture of fermions in two different spin states . with negligible boson - boson interactions , homogeneous densities of the particles become unstable as soon as the boson - fermion coupling constant is non - zero . it corresponds to the phase separation transition . we show that in 3d space for sufficiently strong repulsive boson - fermion interactions another transition takes place , i.e. the self - localization of bose particles . that is , the repulsion between particles creates a local potential well for bosons where , if the well is sufficiently large , they can localize . the self - localization is present both for the superfluid and the normal state of fermions . it modifies properties of the fermi sub - system locally without destroying the superfluidity . low non - zero temperature affects the pairing function but has little effect on the self - localization phenomenon . we do not observe the self - localization for attractive boson - fermion interactions in the 3d case . in this context the self - localization requires sufficiently strong boson - fermion interactions . however , for strong attractive interactions no metastable state of the system exists and the densities of the atoms collapse to dirac - delta distributions indicating a breakdown of the description with the contact interaction potentials . in the 1d case there is no collapse for attractive boson - fermion interactions . the self - localization of bosons is accompanied by localization of a pair of fermions at the bottom of the fermi sea . this phenomenon can be described by a simple model where the self - localization is related to the existence of a vector soliton solution . to realize experimentally the self - localization of bosons in a fermi system , ultra - cold clouds of bosons and fermions have to be prepared in a laboratory . for a sufficiently large boson - fermion coupling constant , that can be achieved by means of a feshbach resonance , the self - localization takes place . signatures of the self - localization can be visible in expansion of the atomic clouds after trapping potentials are turned off . that is , if during the time of flight the boson - fermion interactions are kept negligibly weak , the initially strongly localized boson cloud will show much faster expansion than the fermi cloud due to release of a large kinetic energy . the simplest experiment would employ a fermi sub - system in a normal phase . in order to observe the self - localization in a superfluid fermi mixture a manipulation of a fermion - fermion coupling constant is also needed and two feshbach resonances must be employed , e.g. one resonance controlled by magnetic field and the other by optical means . this work is supported by the polish government within research projects 2009 - 2012 ( kt ) and 2008 - 2011 ( ks ) . m. inguscio , w. ketterle , and c. salomon ( editors ) , _ ultra - cold fermi gases _ , proceedings of the international school of physics `` enrico fermi , '' course clxiv , varenna 2006 , ( ios press , amsterdam ) 2007 .

we consider self - localization of a small number of bose particles immersed in a large homogeneous superfluid mixture of fermions in three and one dimensional spaces . bosons distort the density of surrounding fermions and create a potential well where they can form a bound state analogous to a small polaron state . in the three dimensional volume we observe the self - localization for repulsive interactions between bosons and fermions . in the one dimensional case bosons self - localize as well as for attractive interactions forming , together with a pair of fermions at the bottom of the fermi sea , a vector soliton . we analyze also thermal effects and show that small non - zero temperature affects the pairing function of the fermi - subsystem and has little influence on the self - localization phenomena .

1006.2324

classical random walks on form the basis for many successful physics - inspired algorithms . the evolution of probability distributions according to simple update rules for probability spreading allows us to sample from thermal distributions ( via the metropolis algorithm @xcite ) or to look for ground - states of physical systems(with simulated annealing @xcite ) . the effectiveness of random - walk based algorithms can be characterized by its mixing time ( how fast it approaches the stationary distribution ) , or by a hitting time ( how fast it reaches a particular vertex ) . for example , the fast mixing of a random walk algorithm for sampling from the thermal distribution of the ising model @xcite forms the basis of a fully polynomial randomized approximation scheme for the permanent of a matrix @xcite . thinking about how to utilize the probabilistic nature of quantum mechanics , instead of analyzing the diffusion of probabilities , we can ask what happens if we let the amplitudes in a system whose interactions respect some graph structure evolve according to the schrdinger equation . the result of this way of thought are _ quantum walks _ @xcite , a useful tool in quantum computation . they bring new dynamics ( different wavepacket spreading @xcite ) and algorithmic applications ( e.g. in searching for graph properties @xcite , graph traversal @xcite , game evaluation @xcite ) as well as theoretical results ( universality for computation @xcite ) . we can define quantum walks in discrete time with an additional coin register , or in continuous time , with hamiltonians which are adjacency matrices of graphs . in this paper , we choose the latter approach . the mixing of quantum walks has been previously investigated for several types of graphs e.g. on a chain @xcite , a 2d lattice @xcite , hypercubes @xcite and circulant graphs @xcite . in this paper we focus on continuous quantum walks on _ necklaces _ cyclic graphs composed from many ( @xmath0 ) copies of a subgraph of size @xmath1 ( pearls ) , as depicted in fig.[figurenecklaces ] . our goal is to provide a simplified approach for finding their eigenvectors and eigenvalues , as well as for analyzing the mixing times for such walks . the motivation for analyzing this type of graph comes from hamiltonian complexity @xcite . quantum computation in the usual circuit model @xcite can be translated into a quantum walk in two ways . first , following childs @xcite , evolving a wavepacket on a graph with many wires ( representing basis states ) , connected according to the desired quantum circuit . second , we can use feynman s idea @xcite to view a computation as a `` pointer '' particle doing a quantum walk ( hopping ) in a `` clock '' register , while the computation gets done in a `` data '' register @xcite or particles holding the working data hopping along a graph @xcite . in both cases , we need to look at transmission / reflection properties of the graphs , and their long - term dynamics . specifically , we would like to know ( and ensure ) that a computation is done when we want it to be , not having the wavepacket localized ( or spread ) in undesired parts of the graph . this is why we focus on the mixing properties of quantum walks that are underlying quantum computational models based on quantum walks , looking at their spectra in detail . note that proofs of computational complexity for qma - hard problems ( e.g. @xcite ) also involves investigating the ( low - lying ) spectrum of a quantum walk . the simplest graph involved in the feynman - like models is a line or a cycle , and the dynamics for this quantum walk are well understood @xcite . we look at continuous - time quantum walks on _ necklace _ graphs , which appear in the analysis of quantum computational models @xcite that generalize the feynman approach . necklace graphs could also be viewed as implementing dynamics for quantum walks on imperfect cycles . utilizing the cyclic structure of the necklaces , we propose a bloch - type ansatz for the eigenfunctions , allowing us to obtain several results . first , in section [ sec : ansatz ] we reduce the problem of finding the eigenvectors and eigenvalues of the quantum walk on a necklace of @xmath0 pearls of size @xmath1 to diagonalizing a @xmath2 matrix @xmath0 times ( compared to full @xmath3 diagonalization . second , in section [ sec : mixing ] , we analyze average - time mixing for quantum walks on necklaces and find a general method for showing convergence to the limiting distribution . finally , in section [ sec : examples ] , we work out examples of quantum walks on particular necklaces , giving analytic ( and numerical ) results for the eigenvectors , eigenvalues and the scaling of the mixing time , concluding with open questions in section [ sec : conclusions ] . consider a quantum system with a hamiltonian @xmath4 given by the adjacency matrix of a necklace - like structure . the simplest necklace is a cycle with @xmath0 vertices . a general necklace is a collection of @xmath0 pearls ( small identical graphs with @xmath1 nodes ) , connected into a cycle as in fig . [ figurenecklaces ] . we label points in the @xmath5-th pearl @xmath6 , with @xmath7 . the endpoints of the @xmath5-th pearl ( connected to the previous and following pearls ) are @xmath8 and @xmath9 let @xmath10 be the adjacency matrix of a pearl . the hamiltonian for the whole necklace is a sum of intra - pearl terms and the connections between them : @xmath11 our goal is to find the eigenvalues and eigenvectors of @xmath4 . because of the underlying cyclic structure of a general necklace graph with @xmath0 pearls , we can assume that its eigenvectors will have a structure related to a plane wave on a cycle with @xmath0 nodes . let us then look at the @xmath0-node cycle first . there the hamiltonian has no @xmath12 s in it , allowing us to find the ( plane - wave ) eigenvectors of @xmath13 : @xmath14 corresponding to eigenvalues @xmath15 parametrized by momenta @xmath16 : @xmath17 for @xmath18 . and @xmath19.,width=249 ] consider now a general necklace with @xmath0 pearls . we expect the eigenvectors of the necklace hamiltonian to have a form resembling , also depending on the momenta @xmath16 . let us thus look for the eigenvectors of @xmath4 in the form @xmath20 where each @xmath21 is a normalized vector with support only on the @xmath5-th pearl ( the vertices @xmath22 ) . using and , we obtain @xmath23 where the last two terms correspond to the amplitudes on the endpoints of the @xmath5-th pearl coming from the endpoints of the neighboring pearls . notice that because of our parametrization , the hamiltonian is now block - diagonalized , acting in the same way on each pearl ( see figure [ figurenecklacepb ] ) . when @xmath24 is an eigenvector of @xmath4 , we also have @xmath25 using and , finding the eigenvalues of @xmath4 thus reduces to diagonalizing the @xmath2 matrix @xmath26 where @xmath10 is the adjacency matrix of a pearl , and @xmath27\end{aligned}\ ] ] has only two non - zero elements in the corners if a pearl has two distinct roots @xmath28 and @xmath29 . there is a special case when a pearl is connected to the rest of the necklace through a single root vertex @xmath28 . there , the matrix @xmath30 has a single nonzero element and reads @xmath31.\end{aligned}\ ] ] diagonalizing gives us @xmath1-dimensional vectors @xmath32 . for each @xmath33 , there will be @xmath1 of these , and we will label them @xmath34 with @xmath35 . the corresponding eigenvalues @xmath36 of @xmath37 are also the eigenvalues of the full hamiltonian @xmath4 . therefore , to find all the @xmath38 eigenvalues @xmath36 of the necklace hamiltonian with @xmath0 pearls , we need to diagonalize the @xmath2 matrix @xmath37 for each @xmath33 . to get the eigenvectors of @xmath4 from the eigenvectors of @xmath37 , we plug the coefficients @xmath39 of the vectors we just found into and . in conclusion , the ansatz simplifies the general problem of diagonalizing the @xmath3 matrix @xmath4 to diagonalizing an @xmath2 matrix @xmath0 times . this is useful especially when @xmath1 is small and @xmath0 is large . our focus in what follows will be on mixing of continuous quantum walks on many - pearled ( large-@xmath0 ) necklaces . time evolution according to the schrdinger equation with a hamiltonian that is an adjacency matrix of a graph produces a continuous time quantum walk . let the eigenvectors of the system be @xmath40 and the corresponding eigenvalues @xmath15 . when starting from an initial state @xmath41 , the probability of finding the `` walker '' at vertex @xmath42 at time @xmath43 ( measuring position @xmath44 ) is @xmath45 the evolution is unitary , so it does not mix towards a time - independent stationary distribution like a classical markov process . on the other hand , we can think about mixing for a quantum walk in a time - averaged sense , investigating a time - averaged probability distribution . it holds information about the probability of finding the system at a particular vertex at time @xmath43 , chosen uniformly at random between @xmath46 and @xmath47 ( a chosen limiting time ) : @xmath48 this time - averaged probability has a well - defined @xmath49 limit , which gives us the _ limiting probability distribution _ , expressible using the eigenvectors of @xmath4 as : @xmath50 where the final sum goes over pairs of equal eigenvalues . note that for some quantum walks this limiting distribution can be dependent on the initial state ( e.g. when we start in some eigenstate ) , so we will keep the superscript @xmath51 around . to determine how fast the time - averaged probability converges towards the limiting distribution , we need to bound the total distribution distance @xmath52 . using , integrating an exponential and realizing that the terms summed over pairs of equal eigenvalues subtract out , we arrive at @xmath53 where the sum now goes over pairs of eigenvalues that are not equal . we can put an upper bound on this expression by a technique similar to @xcite . first , we use @xmath54 and move the absolute value inside the sums , to obtain @xmath55 the cauchy - schwartz inequality @xmath56 allows us to perform the sum over @xmath44 , resulting in @xmath57 after another use of the cauchy - schwartz inequality on the terms involving @xmath41 , realizing the expression is symmetric under exchange of @xmath58 and @xmath59 , we finally obtain @xmath60 which corresponds to lemma 4.3 of @xcite . it involves a sum of the inverse of eigenvalue differences . these terms can be large , but as @xmath47 grows , the @xmath61 factor can bring the total variation difference to zero . it is our task now to investigate how fast this happens . we seek @xmath62 ( the _ mixing time _ ) , for which @xmath63 would hold for all @xmath64 , given any precision parameter @xmath65 . for our first example , we now follow @xcite and compute the limiting distribution for the case of a walk on a cycle . later , we will show that the time - averaged probability converges to it for times @xmath66 , using a more general mixing result proved in section [ sec : method ] . the eigenvalues and eigenvectors for the continuous - time quantum walk on a cycle are given by and . we obtain the limiting distribution from by summing over the few nonzero terms . the sum over the equal eigenvalues splits into a sum over @xmath67 and @xmath68 ( degenerate eigenvalue pairs ) . when the initial state @xmath41 is concentrated at a vertex @xmath69 , in the case of even @xmath0 , the limiting distribution for the quantum walk on a cycle is @xmath70 where @xmath71 for all pairs @xmath72 , with an exception for the two points @xmath73 and @xmath74 , where its value is @xmath75 . for a cycle with an odd length @xmath0 , we get @xmath76 with @xmath77 defined in the same way as for even @xmath0 , equal to zero for all pairs @xmath72 except for @xmath73 , where @xmath78 . the slight differences from a uniform distribution arise because not all of the eigenvalues are doubly degenerate . proving that the time - averaged distribution converges towards the limiting distribution for @xmath79 takes more work . we want to show that the total distribution distance @xmath52 goes to zero as @xmath79 . when computing @xmath80 , the terms with @xmath81 produce the limiting distribution @xmath82 and are thus subtracted out . however , the terms left over ( which were killed by the @xmath83 limit when computing @xmath84 ) need to be carefully accounted for and govern the convergence . in , we have a bound on the total distribution distance by a sum over pairs of inequal eigenvalues . we will upper bound this sum in section [ sec : method ] , using a general approach of lower bounding the terms @xmath85 in . this result is then applicable to several other walks on necklaces . the rate of convergence of the time - averaged distribution towards the limiting distribution is governed by a sum of @xmath86 over non - equal eigenvalues as in . we will now show a method for upper bounding it that will work in several cases . first , let us choose two particular sectors of eigenvalues , fixing @xmath87 and @xmath88 . it is often possible to bound the eigenvalue differences for this sector as @xmath89 for some constant @xmath90 , where where @xmath91 are the momenta , and the indices @xmath92 run from @xmath46 to @xmath93 , observing @xmath94 . we will rewrite using the substitution @xmath95 and @xmath96 . @xmath97 where @xmath98 while @xmath99 , and @xmath100 while @xmath101 . first , because of symmetry , we observe that it is enough to sum only over @xmath102 and multiply the resulting sum by 4 . second , it can only increase our upper bound if we count all @xmath103 , instead of having to take care with counting starting at @xmath104 . the symmetry of the term involving @xmath105 then again allows us to sum only up to @xmath106 and multiply the result by 4 . therefore , we obtain @xmath107 recalling now that @xmath108 , we can deal with each sum as @xmath109 thus finally expressing the sum in ( note that we worked only in a single @xmath110 sector ) as @xmath111 where the extra factor @xmath112 comes from the term @xmath113 , when we expect the initial state to have roughly equal overlap with all momentum states . according to and working this out for all sectors @xmath110 , this suffices to show an upper bound on the mixing time ( in the time - averaged sense ) for this type of quantum walk , which grows with the system size a @xmath114 . we will now show that for particular examples of walks on necklaces , the eigenvalues obey , and so that we can use the above approach for proving their convergence . the first example that we can deal with using this method is the cycle itself , where is an equality . therefore , we have just shown that the time - averaged distribution ( when starting from a single point ) converges to the limiting distribution for a cycle of length @xmath0 with mixing time @xmath115 ( up to logarithmic factors ) . we now look at a specific type of necklaces , which appear in the quantum - walk based model of computation @xcite . these `` combs '' are constructed from a ring of length @xmath116 by attaching an extra node ( tooth ) to the ring at every @xmath117-th vertex as in fig . [ figurecombkd ] . the `` pearl '' in this comb - like graph has size @xmath118 , and there are @xmath0 of them , so the total number of vertices in this graph is @xmath119 . we will now analyze the spectra and mixing properties on the @xmath120-comb necklaces , showing their similarity to ( and differences from ) a simple cycle . -comb graph is a ring of length @xmath121 , with an extra vertex connected at every @xmath117-th node.,width=240 ] this is the simplest of the graphs , with a pearl that has only two nodes ( the base and the tooth ) : @xmath123.\end{aligned}\ ] ] because @xmath10 has a single root , the matrix @xmath30 needed to construct @xmath37 reads @xmath124.\end{aligned}\ ] ] therefore , the matrix @xmath37 is @xmath125.\end{aligned}\ ] ] its eigenvalues are @xmath126 and the corresponding eigenvectors are @xmath127\end{aligned}\ ] ] where the upper vector component corresponds to the base ( and the lower component to the tooth ) of the comb . according to , this gives us the eigenvectors of @xmath4 as @xmath128_{(j)}\end{aligned}\ ] ] with @xmath129 for @xmath18 . the eigenvalues @xmath15 of the hamiltonian are symmetrically distributed around zero , and each of them is also doubled if @xmath0 is even . note that two of the eigenvectors @xmath130 are zero on every other base and tooth , and correspond to eigenvalues @xmath131 . the limiting distribution is analyzed in appendix [ app : comb1 ] . we find that for large @xmath0 , the limiting distribution ( when starting from a base vertex ) is @xmath132 on base vertices and @xmath133 on teeth , with corrections for the initial vertex and the vertex across from it . we now prove convergence to the time - averaged limiting distribution , showing that the ( time - averaged ) mixing on this densest comb is no different than the one we saw for a cycle . we will upper bound the sum in by the method in section [ sec : method ] , dividing the eigenvalues into 4 regions , @xmath134 , corresponding to choices of @xmath87 and @xmath88 . in the @xmath135 and @xmath136 regions , we have @xmath137 so the inverse of such terms does not govern the scaling in . the important region combinations must then be @xmath138 and @xmath139 , where a few lines of algebra give us @xmath140 as the eigenvalues are well bounded away from zero . armed with this inequality , and the fact that the overlap of a single - starting - vertex initial state with the eigenvectors scales as @xmath141 , we can now use the result of section [ sec : method ] . this gives us an upper bound on the mixing time for the @xmath122-comb necklace , scaling with @xmath0 as @xmath142 , i.e. the same as for a cycle with no teeth , up to logarithmic factors . the next example is the @xmath143 comb . it has a tooth ( extra vertex ) at every second node of the basic loop , so its pearl @xmath10 has three vertices . we label the base of the tooth as vertex 1 , and the top of the tooth as vertex 2 , giving : @xmath144.\end{aligned}\ ] ] following the procedure of section [ sec : ansatz ] , we need to find the eigenvalues and eigenvectors of the matrices @xmath145 constructed as in : @xmath146 + \left [ \begin{array}{ccc } 0 & 0 & e^{-ip_k } \\ 0 & 0 & 0 \\ e^{ip_k } & 0 & 0 \end{array } \right ] = \left [ \begin{array}{ccc } 0 & 1 & e^{-\frac{ip_k}{2 } } 2\cos \frac{p_k}{2 } \\ 1 & 0 & 0 \\ e^{\frac{ip_k}{2 } } 2\frac{\cos p_k}{2 } & 0 & 0 \end{array } \right ] .\end{aligned}\ ] ] after some algebra , we find that its three eigenvalues are @xmath147 with the corresponding eigenvectors @xmath148 , \qquad { | y^{\,(k,\pm ) } \rangle } = \frac{1}{\sqrt{2 ( 3 + 2\cos p_k ) } } \left [ \begin{array}{c } \pm \sqrt{3 + 2\cos p_k } \\ 1 \\ 2 e^{-\frac{ip_k}{2 } } \cos\frac{p_k}{2 } \\ \end{array } \right ] . \label{k2eigenvectors}\end{aligned}\ ] ] to construct the eigenvectors of the hamiltonian @xmath4 , we use in equation . let us now look for a lower bound on the gap between eigenvalues . when we choose two eigenvalues from different sectors ( @xmath46 , @xmath149 or @xmath150 ) , the differences between them are always larger than @xmath151 . the only interesting cases are thus the @xmath138 and @xmath139 choices of eigenvalue pairs . there we find @xmath152 this lower bound on the nonzero eigenvalue gaps allows us to use the results of section [ sec : method ] and prove the upper bound @xmath153 on the mixing time for the @xmath143-comb . this is once again the same upper bound we found for the cycle and the @xmath122-comb in section [ sec : k1 ] . in the last example we want to show that comb - like necklaces with @xmath117 vertices between teeth ( see figure [ figurecombkd ] ) mix similarly to cycles . we dealt with the most non - cycle - like examples in the previous sections , and now we will numerically look at combs with general spacing @xmath117 . the results for the smallest nonzero eigenvalue differences for @xmath154 combs are shown in figure [ figurecombeigs ] . in a log - log plot of the smallest eigenvalue difference vs. the number of pearls @xmath0 ( for various values of @xmath117 ) , we see the characteristic @xmath155 scaling . thus , the numerics imply that the scaling of the mixing time gets no worse than @xmath156 . however , it is likely that the eigenvalue differences also obey the cos - like scaling as we have seen for @xmath157 . if we could show this , we would again prove that the mixing time scales with @xmath0 as @xmath158 . -comb graphs ( see figure [ figurecombkd ] ) and a cycle . a straight - line fit through the datapoins indicates a @xmath155 scaling with the growing number of pearls.,width=288 ] the goal of this paper was to utilize the cyclical repetitive structure of necklace - like graphs , providing a general method for analyzing the eigenvectors and eigenvalues of continuous - time quantum walks on such graphs . using a bloch - theorem - like ansatz , we block - diagonalized the hamiltonian , decreasing the effective size of the problem from @xmath38 to @xmath1 , where @xmath1 is the size of a pearl and @xmath0 is the number of pearls in the necklace . next , we wanted to investigate the mixing times ( for approaching a limiting distribution in a time - averaged sense ) for these quantum walks . in section [ sec : method ] we have shown that proving a @xmath159-like lower bound on ( non - equal ) eigenvalue differences results in a mixing time @xmath115 for a graph with @xmath0 pearls , which is the same as for a cycle with @xmath0 nodes . note though , that the prefactor in the mixing time can depend on the size of the pearl @xmath1 . finally , in section [ sec : examples ] we exhibited the bound ( and thus the cycle - like mixing time ) for two necklace - like graphs . these graphs appear in the models of quantum computation @xcite that extend the feynman - computer with the so - called railroad switches , and finding the polynomial - time ( in @xmath0 ) scaling of the mixing - time is required for showing their effectiveness . dn acknowledges support from the slovak research and development agency under the contract no . lpp-0430 - 09 , from the projects meta - qute itms 26240120022 , vega qwaen and european project q - essence . we thank zuzana gavorov , daniel reitzner , and vladimr buek for fruitful discussions . 99 n. metropolis , a.w . rosenbluth , n.n . rosenbluth , a.h . teller , e. teller : _ equation of state calculations for fast computing machines , _ j. chem . * 21 , * 1087 ( 1953 ) w.k . hastings : _ monte carlo sampling methods using markov chains and their applications , _ biometrika * 57 , * 97 ( 1970 ) s. kirkpatrick , c.d . gelatt jr . , m.p . optimization by simulated annealing , _ science * 220 , * 671 ( 1983 ) j. ern : _ thermodynamical approach to the travelling salesman problem , _ j. opt . theory appl . * 45 , * 41 ( 1985 ) m. jerrum and a. sinclair , _ polynomial - time approximation algorithms for the ising model _ , siam journal on computing , vol . 22 , pp . 10871116 , 1993 . m. jerrum , a. sinclair , and e. vigoda , _ a polynomial - time approximation algorithm for the permanent of a matrix non - negative entries _ , journal of the acm , vol . 51 , issue 4 , pp . 671697 , 2004 . d. aharonov , a. ambainis , j. kempe , u. vazirani : _ quantum walks on graphs , _ in stoc 01 : proceedings of the thirty - third annual acm symposium on theory of computing , 5059 ( new york , ny , usa , acm , 2001 ) e. farhi , s. gutmann , _ quantum computation and decision trees _ , phys . rev . a 58 ( 2 ) , 915 ( 1998 ) a. ambainis , e. bach , a. nayak , a. vishwanath , and j. watrous : _ one - dimensional quantum walks , _ in stoc 01 : proceedings of the thirty - third annual acm symposium on theory of computing , 3749 ( new york , ny , usa , acm , 2001 ) m. santha , _ quantum walk based search algorithms _ , proc . of 5th theory and applications of models of computation ( tamc08 ) , lectures notes on computer science , vol . 4978 , pp . 3146 , 2008 . a. m. childs , r. cleve , e. deotto , e. farhi , s. gutmann , d. a. spielman , _ exponential algorithmic speedup by quantum walk _ , proceedings of the 35th acm symposium on theory of computing , pp . 59 - 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381 t. osborne , hamiltonian complexity , arxiv:1106.5875 ( 2011 ) m. a. nielsen , i. l. chuang , quantum information and computation , cambridge university press , cambridge , uk , 2000 . r. feynman , quantum mechanical computers , opt . news , vol . 11 , pp . 1146 ( 1985 ) , j. kempe , a. kitaev , and o. regev . the complexity of the local hamiltonian problem . , 35(5):10701097 , 2006 . d. nagaj and s. mozes , _ new construction for a qma complete three - local hamiltonian _ , journal of mathematical physics , 48:2104 , 2007 . d. de falco , d. tamascelli , _ grover s algorithm on a feynman computer _ , j. phys . 37 , 909 - 930 ( 2004 ) d. nagaj , _ fast universal quantum computation with railroad - switch local hamiltonians _ , journal of mathematical physics , 51 ( 6 ) , 062201 ( 2010 ) a. mizel , d. a. lidar and m. mitchell , simple proof of equivalence between adiabatic quantum computation and the circuit model , phys . rev . 99 , 070502 ( 2008 ) . d. aharonov , d. gottesman , s. irani , j. kempe , the power of quantum systems on a line , proc . 48th ieee focs , 373 - 383 ( 2007 ) d. nagaj , _ universal 2-local hamiltonian quantum computing _ a 85 , 032330 ( 2012 ) l. eldar , o. regev , quantum sat for a qutrit - cinquit pair is qma@xmath160-complete , icalp 2008 , l. aceto et al . ( eds ) , part i , lncs 5125 , pp . 881 - 892 , springer - verlag berlin , heidelberg ( 2008 ) , l. eldar , p. love , d. nagaj , o. regev , in preparation d. nagaj and p. wocjan , hamiltonian quantum cellular automata in 1d , phys . a 78 , 032311 ( 2008 ) we now analyze the limiting distribution for the continuous - time quantum walk on a @xmath122-comb in fig . [ k1lim ] , starting from a single ( base ) vertex @xmath69 . first , we do our work analytically , and end with a few high-@xmath0 numerical approximations . the conclusion is that the distribution is flat except for a few points very close to the two special positions @xmath161 and @xmath74 . first , we look at the time - averaged limiting distribution for going from base @xmath162 to base @xmath163 ( which thanks to the identity turns out to be the same as for going from tooth @xmath164 to tooth @xmath165 ) , obtaining @xmath166 where the last term involving @xmath167 occurs only for even @xmath0 . using the identities @xmath168 and denoting @xmath169 @xmath170 with @xmath77 defined in , we rewrite to finally obtain @xmath171 -comb when starting from the base of the @xmath174th pearl for odd @xmath175 ( left ) and even @xmath176 ( right).,title="fig:",width=268 ] -comb when starting from the base of the @xmath174th pearl for odd @xmath175 ( left ) and even @xmath176 ( right).,title="fig:",width=268 ] finally , let us look at the high @xmath0 approximation ( see also fig.[k1lim ] ) . in @xmath177 we replaced the sum by an integral and obtained @xmath178 . at points @xmath69 for which @xmath179 , the expression @xmath180 equals @xmath177 exactly , while it rapidly falls off to zero with growing distance of @xmath69 from @xmath44 ( or @xmath181 ) . the limiting distribution when starting from a base point for even number of pearls is thus well approximated by a flat distribution with @xmath182 on bases and @xmath183 on teeth , with the exception of the starting pearl and the pearl exactly opposite to it receiving @xmath184 and @xmath185 , respectively . it is very similar for odd @xmath0 , except that we do not have the special case of the opposite pearl . for an example of the limiting distribution with odd and even @xmath0 , see fig . [ k1lim ] .

we analyze continuous - time quantum walks on necklace graphs cyclical graphs consisting of many copies of a smaller graph ( pearl ) . using a bloch - type ansatz for the eigenfunctions , we block - diagonalize the hamiltonian , reducing the effective size of the problem to the size of a single pearl . we then present a general approach for showing that the mixing time scales ( with growing size of the necklace ) similarly to that of a simple walk on a cycle . finally , we present results for mixing on several necklace graphs .

1111.4433

the connection between long duration gamma - ray bursts ( lgrbs ) and hydrogen poor type ic supernovae has become well established based on the detection of spectroscopic signatures of these supernovae accompanying a handful of relatively local grbs ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the grb - sne sample increases when combined with a larger set of events which exhibit photometric signatures in their lightcurves , consistent with sne ic ( see e.g. , * ? ? ? * ) . although , these light curve humps " are not uniquely diagnostic of the supernova type , and are open to alternative interpretations , the emerging scenario is that at the majority of long grbs are associated with type ic supernovae ( e.g. * ? ? ? however , the picture painted by observations of such grb - sne pairs has remained unsatisfactory in some respects . on average , these local events differ substantially from the majority of the grb population in terms of energy release , with isotropic energy releases ( @xmath7 ) a factor of @xmath8 lower than the bulk population ( e.g. * ? ? ? of the bursts with the strongest evidence for sne , only grb 030329 , with @xmath9 erg appears close to a being a classical cosmological long - grb . several local grb / sne pairs exhibit @xmath10-ray emission of extremely long duration @xcite , while other very long events at larger redshift show little evidence for sne @xcite . indeed , these local , low luminosity bursts have been suggested to arise from a very different physical mechanism than the classical bursts , such as relativistic shock break - out from the supernova itself ( e.g. * ? ? ? such emission is difficult to locate in more luminous grbs due to a combination of distance and glare from the burst itself , although evidence for possible shock break - out components has been found in some grbs @xcite . however , the several order of magnitude difference in energy release between the local , low - luminosity and cosmological , high luminosity grbs could also be indicative of rather different physical mechanisms at play . given this , the nature of the connection between the most energetic grbs and their supernovae remains in urgent need of further study . here we report observations of the brightest ( highest fluence ) grb detected in the past @xmath220 years , grb 130427a . the isotropic energy release of @xmath11 erg , places it in the most luminous 5% of grbs observed to date by _ swift _ , and a factor of 100 brighter than grb 030329 @xcite which was the most luminous grb with a well studied supernova . at a redshift of @xmath12 @xcite the burst is close enough that any supernova is open to spectroscopic study , and indeed the presence of a supernova , sn 2013cq , has been established @xcite . here we use the resolution of the _ hubble space telescope _ to resolve and dramatically reduce the galaxy contribution , and its uv capability to track the afterglow , hence enabling a view of the supernova as free as possible from the host , afterglow and atmospheric hinderance . grb 130427a was discovered by _ swift _ at 07:47:57 ut on 27 april 2013 @xcite . it was also detected as an exceptionally bright grb by _ konus - wind _ and _ fermi _ with gbm @xcite and lat @xcite , and its prompt fluence of @xmath13 ergs @xmath14 in the 10 - 1000 kev band @xcite makes it the most fluent grb observed by _ swift _ , fermi or batse . it showed a bright x - ray and optical afterglow , peaking at r=7.4 before the _ swift _ grb trigger @xcite . early spectroscopy of the afterglow yielded a redshift of @xmath12 @xcite , which was confirmed from later , more detailed spectroscopic observations @xcite . a full description of the afterglow is given in @xcite . deep photometric and spectroscopic observations over the first 10 days post burst revealed a re - brightening , consistent with the presence of a type ic supernova , sn 2013cq @xcite . lllllll 20-may-2013 & 56432.10521 & 22.78 & f336w & 1266 & 23.28 @xmath15 0.02 + 20-may-2013 & 56432.12503 & 22.80 & f160w & 1048 & 21.60 @xmath15 0.01 + 20-may-2013 & 56432.37207 & 23.05 & f606w & 180 & 21.76 @xmath15 0.01 + 20-may-2013 & 56432.37603 & 23.05 & g800l & 1880 & - + 12-apr-2014 & 56759.14866 & 349.82 & f336w & 2508 & 26.17 @xmath15 0.09 + 18-apr-2014 & 56765.18459 & 355.86 & f606w & 180 & 25.67 @xmath15 0.11 + 18-apr-2014 & 56765.18459 & 355.86 & g800l & 1880 & - + [ hst ] we observed the location of grb 130427a with _ hst _ on 20 may 2013 , 23 days after the initial burst detection . a second epoch was obtained in april 2014 , almost a year after the initial burst . a log of these observations is shown in table [ hst ] . for more detailed study of the host galaxy we also utilize a longer ( 2228 s ) wfc3/f606w observation obtained on 15 may 2014 . the imaging data were reduced in the standard fashion ; with on - the - fly processed data retrieved from the archive and subsequently re - drizzled using astrodrizzle , for uvis observations we separately corrected for pixel based charge transfer inefficiency ( cte ) @xcite . photometry was performed in small apertures to minimize any contribution from underlying galaxy light , and maximise signal to noise , it was subsequently corrected using standard aperture corrections . we also use direct image subtraction to isolate the afterglow / sne light at early epochs , this effectively removes the host contribution . these magnitudes may still contain some transient light , but since the second epoch magnitudes are a factor of @xmath2 15 - 30 lower than observed at early times , this suggests that these epochs can be used for effective subtraction.the resulting photometry is shown in table [ hst ] , while our _ hst _ images are shown in figure [ hst ] . we also obtained grism spectroscopy centered at @xmath16 with the g800l grism on acs , with a position angle chosen to minimize the contribution from the underlying host galaxy ( see figure [ hst ] ) . again we utilized the on - the - fly calibrated images , corrected for cte and bias striping . we detected sources on a single f606w image , and extracted these via axe , subsequently drizzling each of the four exposures to create master spectra , which was flux calibrated using published sensitivity curves . we extracted the light from the grb counterpart in a relatively small aperture ( @xmath17 ) . in principle a given pixel in the grism image may be exposed to light of multiple different wavelengths from different spatial locations on the chip . for the second epoch of observations we force an extraction of the same width at the position of the transient ( as determined by a map between the first and second epoch of direct imaging ) . we then subtract this from the initial spectrum to obtain a host free spectrum . since we have utilised a tight aperture around the sne we subsequently scale this subtracted spectrum to the host subtracted magnitudes of the afterglow / sne . the afterglow is offset ( @xmath18 ) from the centroid of the host galaxy light in f606w , and so the latter is of little concern in our small ( 0.1 ) apertures . however , regions underlying grbs are frequently amongst the most luminous parts of their hosts @xcite , so some contamination may be expected . our late time subtraction removes this from both our broad band photometry and grism spectroscopy . for the high resolution observations reported here this contamination is small ( at most 6% in the uv and 3% in the optical ) for our photometry . however , the contribution is somewhat larger in the grism observations . these observations disperse light not only from the region directly underlying the grb , but also from other locations in the host ( which represent contributions at different wavelengths , overlapping the transient light ) . in particular , the proximate , bright star forming region contaminates the sne considerably ( @xmath1920% at the red end of our spectrum beyond 9000 ) . this region is also likely to be the dominant host contaminant in ground based spectroscopy ( since the host galaxy is resolved ) , and has quite different colours from the global host , implying potentially significant systematic errors when the broadband sdss colours of the host are used to attempt a subtraction ( e.g. xu et al . 2013 , melandri et al . 2014 ) . here we can directly remove this contribution via the subtraction of the deep second epoch . sne ib / c are generally weak uv emitters due to the strong metal line blanketing shortward of @xmath20 , so to first order our f336w observations should be free of supernova contribution . this is consistent with the uv colours of grb060218/sn2006aj ( from * ? ? ? * ) and of xrf100316d / sn2010bh ( from * ? ? ? * ) , which would predict a factor @xmath21 decrease in flux between f606w and f336w . there are few uv spectra of sn ic , however if we graft the stis uv observations of sn 2002ap onto the optical spectra of sn 1998bw following @xcite then we can obtain a first order approximation of the likely uv spectral shape ( see figure [ sn ] ) . these colours and spectra suggest it is reasonable to assume that the f336w light is dominated by the afterglow component . to confirm this we utilize the uv to ir lightcurve from @xcite . this exhibits a spectral slope of @xmath22 with @xmath23 and predicts u(ab)=23.41 @xmath15 0.10 at the time of the first epoch of _ hst _ f336w observations . the corresponding _ hst _ uv magnitude is f336w(ab)=23.28 @xmath15 0.02 . corrected for foreground absorption this is consistent with the afterglow contributing 90@xmath1510% of the measured f336w flux , confirming our assumptions above . this afterglow model predicts f160w(ab ) = 21.84 , somewhat fainter than the measured magnitude and suggesting that the supernova makes up @xmath220% of the light in this band , again in keeping with expectations of the few ir spectra of broad - lined sne ic obtained to - date ( e.g. * ? ? ? we estimate the supernova contribution by using the model above , with initial error bars accounting for the uncertainty in the f336w afterglow light , discussed above , and the intrinsic value of @xmath24 , which we adopt to be 0.92 @xmath25 . for this range of models we then subtract the afterglow spectrum from the measured grism data , and neglect any host galaxy contamination . the extremum of this model is set by the assumption that both f336w and f160w are entirely dominated by afterglow , and by subtracting the resulting power - law index . our grism spectrum , both before and after subtraction of the afterglow and host light , is shown in figure [ sn ] ( top and bottom ) . broad features , consistent with those seen in other high velocity sn ic associated with grbs are clearly visible in the spectrum , even before subtraction of the afterglow component . the absence of broad emission at h@xmath26 or he absorption rules out type ii or ib events respectively . in the lower panel of figure [ sn ] we plot rest - frame wavelength versus luminosity comparisons of the afterglow subtracted and de - reddened spectra of sn 2013cq with various grb / sne pairs . the closest match for the overall spectral shape and luminosity is that of sn 1998bw @xcite . the similarity in appearance of these sne is primarily due to the overall spectral shape , with a drop in luminosity of a factor of @xmath2 3 over the 5000 - 7000 range , substantially more than seen in other grb / sne pairs . the broad colours of these sne are similar , and if the kink at @xmath27 is interpreted as the siii ( 6355 ) blend then it would be indicative of a photospheric velocity similar to sn 1998bw at the same epoch ( @xmath28 km s@xmath4 ) , although we note that this feature is apparently stronger in sn 1998bw , where there is marked upturn in flux redward of it . this may suggest a somewhat higher velocity for sn 2013cq . taken at face value this would suggest that sn 2013cq is broadly similar in peak luminosity , @xmath29ni production and kinetic energy to sn 1998bw . while the similarity to sn 1998bw is very marked over the rest frame spectral range from 5100 - 7000 it is much poorer around the central peak of an sn ic at @xmath30 . in sn 2013cq this feature appears to be broader , and blueshifted relative to sn 1998bw , as observed by @xcite . our comparison suggests that this peak may fit better with the slightly higher velocity sn 2003dh @xcite , the only one of the comparison sne to arise from a luminous cosmological grb . in this case , the sn 2003dh model shown in figure 2 remains a factor of 50% less luminous than sn 2013cq , and is systematically redder ( i.e. a smaller decrement between the peak and @xmath31 ) . however , the positions of the broadened lines do then appear generally similar . alternatively , this could be suggestive of a much higher velocity supernova such as sn 2010bh @xcite as favoured by @xcite , who infer @xmath32 km s@xmath4 from the feii ( 5169 ) at an epoch of 12.5 rest - frame days . this does provide a good match to the location and width of the feature ; however , the overall spectral shape and luminosity of sn 2010bh are also very different from sn 2013cq . at 17 days sn 2013cq appears to be much bluer , and a factor @xmath193 times more luminous than sn 2010bh , suggesting it is not as good an analog as sn 1998bw . to obtain a match in both luminosity and general spectral shape would require that the reddening in the direction of sn 2010bh had been significantly underestimated . our plotted spectrum assumes @xmath33 and @xmath34 = 0.14 @xcite . to obtain a match would require de - reddening the spectra with an _ additional _ @xmath35 , well beyond the values allowed from the na i d doublet observed in moderate resolution x - shooter spectroscopy @xcite and even the largest extinctions allowed by the afterglow spectrum @xcite . hence , we disfavour the suggestion that the underlying sne is similar to sn 2010bh , due to the significant disparity in the luminosity . instead we favour a sne similar in @xmath29ni yield and kinetic energy to sn 1998bw and sn 2003dh , in which possible velocity structure within the ejecta gives rise to a range of features within the spectra that do not provide a unique match to any previous grb / sne pair . indeed , the recent study of @xcite also identifies a source which is much brighter than sn 2010bh , but spectrally intermediate between sn 1998bw and sn 2010bh . despite these differences , it is clear that the features observed in the spectrum of sn 2013cq are broadly compatible to the range observed in sn / grb pairs , although a full model description of the spectrum is beyond the scope of this paper . to obtain approximate bounds on the luminosity of the supernova we integrate our afterglow and host subtracted spectra ( allowing for the errors in the afterglow parameters ) through through a rest - frame v - band filter to obtain a luminosity relative to sn 1998bw . this suggests that sn 2013cq has a luminosity factor @xmath36 of @xmath37 . given the suggestions that sn 2013cq is more rapidly evolving than sn 1998bw @xcite , these observations ( at the v - band peak of sn 1998bw ) may underestimate by true peak luminosity , which may be even closer to that of sn 1998bw . grb 130427a is unusual as a luminous grb at a redshift at which the supernovae are open to detailed study . indeed , it the the highest luminosity burst for which there is spectroscopic evidence for a supernova . in this regard the similarity of the supernova to that seen in a burst ( grb 980425 ) that was six orders of magnitude less energetic is striking . as we show in figure [ eiso ] there is no correlation between the grb energetics and the inferred peak magnitudes of their sne . these similarities in sne peak luminosity , and in their spectra suggest similarities in the ejected @xmath29ni masses and kinetic energies . recent modelling @xcite has used these diagnostics to suggest that most broad - lined sn ic associated with low luminosity grbs in turn arise from a relatively small range of zams progenitor masses , perhaps between @xmath38 m@xmath6 . the detection of a similar sn in a highly luminous grb extends this across a broader range of energy . our _ hst _ observations clearly resolve the afterglow from the host galaxy , showing the grb to lie at a spatial offset of 0.83 , ( 4.0 kpc at @xmath0 ) from the centre of its host . this is a relatively large offset for a grb from its host galaxy ( @xmath39% of those in @xcite ) although by no means exceptional ( see figure 3 ) . in the f606w observations the galaxy exhibits a bar - like structure with a face - on disk visible beyond this , a weak spiral structure is also apparent , making the host one of few to be classified as a spiral @xcite . the f336w imaging shows weak star formation close to the centre of the galaxy , but the most striking feature is a strong star forming region at a similar radial offset to the grb , but offset from the grb position by 0.3(1.5 kpc ) . this region the strongest region of star formation in the host galaxy . while the magnitude of star formation underlying the grb position remains uncertain because of contamination with the late time afterglow , the offset region is at least a factor of @xmath40 brighter . it also shows readily detectable oiii emission in the late time grism spectrum . the weak spiral arm in the direction of the grb also appears distorted , and so it may be that star formation has been triggered via a tidal interaction . in the larger field around grb 130427a we note several galaxies of similar magnitude ( see figure [ hst ] ) . in our grism spectroscopy these galaxies do not exhibit strong spectral lines , but they may well imply that the host of grb 130427a lies within an association . the magnitude of the galaxy in a large aperture at late times after the afterglow / sne has significantly faded is f336w(ab)=22.84 @xmath15 0.07 at a rest - frame wavelength of approximately 2500 . this corresponds to a uv star formation rate of @xmath41 m@xmath6 yr@xmath4 ( error statistical only ) , broadly in agreement with that inferred from the sdss observed sed of the galaxy of @xmath42 m@xmath6 yr@xmath4 @xcite , and suggesting a relatively ( but not extremely ) low specific star formation rate by the standards of grb hosts @xcite . the compact star forming region close to the grb contains @xmath210% of this star formation , making a highly luminous star forming region comparable to that seen in the host of grb 980425/sn 1998bw @xcite or in a handful of other local galaxies @xcite . at late times uv emission is visible close to the grb position , indicating that lower level star formation is likely arising proximate to the grb . this may still contain some afterglow light , but this region is at least a factor @xmath43 fainter than the brightest star forming region in the host . some previously extremely energetic bursts ( e.g. grb 080319b , @xcite ) have shown extremely faint and small host galaxies , while the hosts of grbs 980425 , 030329 and 060218 are also sub - luminous and lmc - like . however , as show in figure 3 the overall population of grb - sn host luminosities is shows no discernible correlation with the energy of the grb ( see also @xcite ) , and the host of grb 130427a is in keeping with these expectations . it is intriguing to note that the grb host galaxy with the closest properties to that of grb 130427a / sn 2013cq is in fact the host of sn 1998bw . it is also a rare example of a spiral galaxy , in which the grb occurs at a moderately large offset from the nucleus , and from the strongest region of star formation within the host galaxy . this is shown graphically in figure [ 98bwcomp ] . given the similarities in supernova and environment grb 130427a would seem like a close analog of grb 980425a if it were not for the factor of @xmath44 difference in their @xmath10-ray energy releases . we have presented _ hst _ imaging and spectroscopic observations of the extremely bright grb 130427a , which show it was associated with a luminous broad line sn ic ( sn 2013cq ) . the red spectra offer good agreement with those of sn 1998bw , while the bluer spectra appear well matched in position if not shape with sn 2010bh . the host galaxy appears to be a disk galaxy of moderate luminosity and star formation rate , whose overall characteristics are consistent with those of the grb host population at large . the similar properties of the sne and hosts over six orders of magnitude in grb isotropic equivalent energy would appear to suggest that the energy of the grb is not a strong function of environment or the mass of the progenitor star , and that stars of similar mass and composition are responsible for the entire luminosity function of grbs . more complex effects within the star ( e.g. rotation ) or geometric effects are therefore needed to explain much of the diversity in the grb luminosity function . for _ swift _ grbs ( from * ? ? ? * ) , amended with the @xmath7 values for grb / sn pairs when not from _ swift_. bottom : the r - band luminosities of candidate grb / sn ( scaled to sn 1998bw , which has @xmath45 ) , against the isotropic luminosities of the grbs . points in red are those with strong spectroscopic evidence for associated supernovae ( category a in @xcite ) , blue points indicate cases where spectral features were seen at lower signal to noise ( category b in @xcite ) , and black points are those with weaker evidence for associated sne . the two errors bars marked in the case of sn 2013cq represent the error associated with a simple afterglow subtraction and the extrema of possibilities ( see text for details ) . the middle two panels show the offset from the grb host centre and the grb host b - band absolute magnitude as a function of isotropic energy release . the colour coding is the same as for the lower plot , while grey triangles indicate values from the literature for grbs without claimed sne associations ( data from @xcite ) . these data show that the properties of grb sne and host galaxies are largely unaffected by the energies of the burst , and hence that progenitors in similar environments and with similar initial masses can likely create the entire grb luminosity function . , width=302 ] we thank matt mountain and the stsci staff for rapidly scheduling our observations . ajl thanks the leverhulme trust . ajl , nrt and kw are supported by stfc . the dark cosmology centre is funded by the dnrf . based on observations made with the nasa / esa hubble space telescope , obtained at the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas 5 - 26555 . these observations are associated with program # 13230 , 13110 & 13117 .

we present _ hubble space telescope _ ( _ hst _ ) observations of the exceptionally bright and luminous _ swift _ gamma - ray burst , grb 130427a . at @xmath0 this burst affords an excellent opportunity to study the supernova and host galaxy associated with an intrinsically extremely luminous burst ( @xmath1 erg ) : more luminous than any previous grb with a spectroscopically associated supernova . we use the combination of the image quality , uv capability and and invariant psf of _ hst _ to provide the best possible separation of the afterglow , host and supernova contributions to the observed light @xmath217 rest - frame days after the burst utilising a host subtraction spectrum obtained 1 year later . acs grism observations show that the associated supernova , sn 2013cq , has an overall spectral shape and luminosity similar to sn 1998bw ( with a photospheric velocity , @xmath3 km s@xmath4 ) . the positions of the bluer features are better matched by the higher velocity sn 2010bh ( @xmath5 km s@xmath4 ) , but this sn is significantly fainter , and fails to reproduce the overall spectral shape , perhaps indicative of velocity structure in the ejecta . we find that the burst originated @xmath24 kpc from the nucleus of a moderately star forming ( 1 m@xmath6 yr@xmath4 ) , possibly interacting disc galaxy . the absolute magnitude , physical size and morphology of this galaxy , as well as the location of the grb within it are also strikingly similar to those of grb980425/sn 1998bw . the similarity of supernovae and environment from both the most luminous and least luminous grbs suggests broadly similar progenitor stars can create grbs across six orders of magnitude in isotropic energy .

1307.5338

* this article reviews the recently developed g - ratio imaging framework . * confounds in the methodology are detailed . * recent progress and applications are reviewed . the g - ratio is an explicit quantitative measure of the relative myelin thickness of a myelinated axon , given by the ratio of the inner to the outer diameter of the myelin sheath . both axon diameter and myelin thickness contribute to neuronal conduction velocity , and given the spatial constraints of the nervous system and cellular energetics , an optimal g - ratio of roughly 0.6 - 0.8 arises @xcite . spatial constraints are more stringent in the central nervous system ( cns ) , leading to higher g - ratios than in peripheral nerve @xcite . study of the g - ratio _ in vivo _ is interesting in the context of healthy development , aging , and disease progression and treatment . in demyelinating diseases such as multiple sclerosis ( ms ) , g - ratio changes and axon loss occur , and the g - ratio changes can then partially recover during the remyelination phase @xcite . the possibility that the g - ratio is dependent on gender during development , driven by testosterone differences , has recently been proposed @xcite and investigated @xcite . possible clinical ramifications of a non - optimal g - ratio include `` disconnection '' syndromes such as schizophrenia @xcite , in which g - ratio differences have been reported @xcite . the g - ratio is expected to vary slightly in healthy neuronal tissue . the relationship between axon size and myelin sheath thickness is close to , but not exactly , linear @xcite , with the nonlinearity more pronounced for larger axon size @xcite , where the g - ratio is higher . during development , axon growth outpaces myelination , resulting in a decreasing g - ratio as myelination catches up @xcite . there is relatively little literature on the spatial variation of the g - ratio in healthy tissue . values in the range 0.72 - 0.81 have been reported in the cns of small animals ( mouse , rat , guinea pig , rabbit ) @xcite . other primary pathology and disorders may lead to an abnormal g - ratio . these include leukodystropies and axonal changes , such as axonal swelling in ischemia . there are many outstanding questions in demyelinating disease that could be best answered by imaging the g - ratio _ in vivo_. for example , in ms , disease progression is still the topic of active research . most histopathological data are from patients at the latest stages of the disease . potential treatment includes agents for both immunosuppression and remyelination . however , if most demyelinated axons die quickly , and the rest remyelinate effectively on their own early in the disease , remyelination agents will be of little clinical value . detailed longitudinal study of the extent of remyelination can therefore aid in choosing avenues for therapy . while techniques exist for measurement of the g - ratio _ ex - vivo _ , measurement of the g - ratio _ in vivo _ is an area of active research . currently , there are quantitative mri markers that are sensitive to the myelin volume fraction ( mvf ) and the intra - axonal volume fraction or axon volume fraction ( avf ) . in recent work @xcite , it has been shown that measuring these two quantities is sufficient to compute one g - ratio for a voxel , or an _ aggregate _ g - ratio . the g - ratio is a function of the ratio of the mvf to the avf . the challenge then becomes how to estimate the mvf and the avf precisely and accurately with mri . the fiber density or fiber volume fraction ( fvf ) is the sum of the mvf and the avf , and the g - ratio imaging framework @xcite aims to decouple the fiber density from the g - ratio , such that a more complete picture of the microstructural detail can be achieved . this , coupled with other microstructural measures such as axon diameter @xcite , comprises the field of _ in vivo histology _ of white matter . we wish to describe microstructure in detail on a scale much finer than an imaging voxel , aggregated over the voxel . as previously defined , the g - ratio is the ratio of the inner to the outer diameter of the myelin sheath of a myelinated axon ( see fig . [ gcartoon ] ) . it has been shown in recent work @xcite that the g - ratio can be expressed as a function of the myelin volume fraction and the axon volume fraction , and hence can be estimated without explicit measurement of these diameters : @xmath0 this formulation applies to any imaging modality ( e.g. , electron microscopy ( em ) , where the mvf and avf can be measured after segmentation of the image - see fig.[gcartoon ] ) , but it is of particular interest to be able to estimate the g - ratio _ in vivo_. mri provides us with several different contrast mechanisms for estimation of these volume fractions , and given mvf@xmath1 and avf@xmath1 , we can estimate @xmath2 . we will hereafter refer to this mri - based g - ratio metric as @xmath3 for simplicity , but note that it is derived from mri images with certain contrasts sensitive but not equal to the mvf and avf . estimation of these quantities is discussed in the next sections . original ( top left ) and segmented ( top right ) electron micrograph showing axons of white matter , the intra - axonal space ( blue ) , and the myelin ( red ) the myelin appears black because of osmium preparation . the fiber g - ratio is the ratio of the inner to the outer radius of the myelin sheath surrounding an axon . the aggregate g - ratio can be expressed as a function of the myelin volume fraction ( mvf ) and the axon volume fraction ( avf ) . the myelin macromolecules , myelin water , and intra- and extra - axonal water compartments all have distinct properties , which can be exploited to generate mri images from which the respective compartment volume fractions can be estimated.,scaledwidth=50.0% ] diffusion mri is particularly well suited to aid in the estimation of the axon volume fraction . it is sensitive to the displacement distribution of water molecules moving randomly with thermal energy , and this displacement distribution is affected by the cellular structure present in the tissue . as the molecules impinge on the cellular membranes , organelles , and cytoskeleton , the displacement distribution takes on a unique shape depending on the environment . intra - axonal diffusion is said to be _ restricted _ , resembling free gaussian diffusion at short diffusion times , but departing markedly from gaussianity at longer times , where the displacement distribution is limited by the pore shape . there will be a sharp drop in the probability of displacement beyond the cell radius . extra - axonal diffusion is said to be _ hindered _ , resembling free gaussian diffusion , but with a smaller variance due to impingement of motion . many diffusion models exist for explicit estimation of the relative cellular compartment sizes . these include neurite orientation density and dispersion imaging ( noddi ) @xcite , the composite hindered and restricted model of diffusion ( charmed ) @xcite , diffusion basis spectrum imaging ( dbsi ) @xcite , restriction spectrum imaging ( rsi ) @xcite , white matter tract integrity ( wmti ) from diffusion kurtosis imaging ( dki ) @xcite , temporal diffusion spectroscopy @xcite , double pulsed field gradient ( dpfg ) mri @xcite , the spherical mean technique @xcite , the distribution of anisotropic microstructural environments in diffusion - compartment imaging ( diamond ) + @xcite , and many others @xcite . it is also possible to perform noddi with relaxed constraints ( noddida ) @xcite , and to do this calculation analytically ( lemonade ) @xcite . another approach , termed the apparent fiber density ( afd ) , uses high diffusion weighting to virtually eliminate the hindered diffusion signal , leaving only intra - axonal water @xcite . it has been used to estimate the the relative axon volume fraction of different fiber populations in a voxel . a modification , termed the tensor fiber density ( tfd ) , can be performed with lower diffusion weighting @xcite . the simplest diffusion mri models do not differentiate between the tissue compartments . for instance , the diffusion tensor @xcite models the entire displacement distribution as an anisotropic gaussian function . the parameters defining this function will change if the intra - axonal volume fraction changes , but to what extent is it practical to extract meaningful quantitative compartment volume fractions from the tensor ? recently , a framework called noddi - dti has been developed , in which the proximity of dti - based parameters to the computed noddi parameters is assessed @xcite , given certain assumptions . the fa and the mean diffusivity ( md ) are highly correlated in straight , parallel fiber bundles , and will change with changing avf , leading to estimates of the relative intra - axonal volume . however , this formulation is probably an oversimplification of the microstructural situation , and more detailed modeling is a better choice to ensure specificity to white matter fibers . the original full - brain g - ratio demonstration @xcite employed the noddi model of diffusion . it was chosen because of its suitability in the presence of complex subvoxel fiber geometry , including fiber divergence , which may occur to a significant scale in almost all imaging voxels @xcite , and its suitability on clinical scanners with relatively low gradient strength . having a fast implementation of the model fitting with numerical stability is important for large studies , hence , the convex - optimized amico implementation is beneficial @xcite . while diffusion mri is a modality of choice for imaging microstructure , it can only measure the displacement distribution of water molecules that are visible in a diffusion mri experiment . this limits us to water that is visible at an echo time ( te ) on the order of 50 - 100 ms , and therefore excludes water that is trapped between the myelin bilayers , which has a t@xmath4 on the order of 10 ms . hence , the estimates provided by these models are of the intra - axonal volume fraction _ of the diffusion visible volume_. myelin does not figure in the models . given , e.g. , the noddi model outputs , a complementary myelin imaging technique must be used to estimate the absolute axon volume fraction . the avf is given by @xmath5 with @xmath6 and @xmath7 the isotropic and restricted volume fractions from the noddi model , and the mvf obtained from a myelin mapping technique , examples of which are discussed below . diffusion contrast may not be our only window onto the axon volume fraction . recent work has shown that it is possible to disambiguate the myelin , intra - axonal , and extra - axonal water compartments using complex gradient echo ( gre ) images @xcite . the myelin water is separable from the combined intra- and extra - axonal water using multicomponent t@xmath8 reconstruction , providing a myelin marker ( see below ) . however , incorporation of the phase of the gre images potentially allows us to separate all three compartments based on frequency shifts . challenges include the fact that the frequency shift is dependent on the orientation of the axon to the main magnetic field b@xmath9 . when the axon is oriented perpendicular to b@xmath9 , the myelin water will experience a positive frequency shift , the intra - axonal water a negative frequency shift , and the extra - axonal water will not experience a frequency shift . note that the avf as defined by these diffusion mri models is specific to white matter . while it makes sense to define an axon volume fraction in grey matter , the models in general can not distinguish between axons and dendrites . the noddi model s @xmath7 parameter , for example , is `` neurite density '' , i.e. , all cellular processes that can be assumed to have infinitely restricted diffusion in their transverse plane . hence , the g - ratio from mri data is undefined in grey matter . the fiber volume fraction is the sum of the avf and the mvf . can diffusion mri , or any other mri contrast mechanism , measure the total fiber volume fraction itself ? clearly , gre images have potential , as discussed above . is diffusion imaging sensitive to the fvf , as opposed to the avf ? while myelin water is virtually invisible in diffusion mri , diffusion mri is not insensitive to myelin . first , the ratio of the intra- to extra - axonal diffusion mri visible water in a voxel will change as the myelin volume fraction in that voxel changes . hence , for example , the noddi parameter @xmath7 changes with demyelination , even if all axons remain intact . second , diffusion acquisitions are heavily t@xmath4 weighted , and t@xmath4 is myelin - sensitive . the total diffusion weighted signal thus decreases as myelin content increases . however , to robustly quantify myelin volume fraction , it is necessary to add a second contrast mechanism , even if it is additional t@xmath4 weighted images , to the scanning protocol . this is discussed below . despite the nomenclature , as noted above , even the apparent fiber density and tensor fiber density are in fact relative axon densities . they would provide a relative fvf only if the g - ratio is constant . in a recent study of the g - ratio @xcite , the tfd was equated with the fvf , not the avf , for input to the g - ratio formula . the g - ratio is a function of the ratio of the mvf to the avf ( eq . [ geq ] ) , or alternately , the ratio of the mvf to the fvf : @xmath10 this means that conclusions reached about the variation of the g - ratio found by equating the tfd with the fvf will be robust in this case . absolute g - ratios in this case were calibrated to have a mean of 0.7 in healthy white matter . in early work on the fiber g - ratio , it was shown that assuming a simple white matter model of straight , parallel cylinders , the fraction anisotropy ( fa ) of the diffusion tensor is proportional to the total fiber volume fraction , with a quadratic relationship @xcite . the model has been shown to give reasonable values in human corpus callosum @xcite , however , it suffers from several problems . first , it applies only to straight , parallel fiber bundles , such as the corpus callosum and parts of the spinal tracts . however , the regions of the brain where this model can be expected to hold at all are very limited , as there are crossing or splaying fibers in up to 95% of diffusion mri voxels in parenchyma @xcite , and curvature is almost ubiquitous at standard imaging resolution . even the axons of callosal fibers are not straight and parallel , with splay up to 18@xmath11 @xcite . second , the model assumes a relatively uniform , if random , packing of axons on the scale of the mri voxel . due to the nonlinear nature of the fa , it will depend strongly on the packing geometry . if two voxels , one with densely packed axons and one with sparsely packed axons , are combined into one , the fa for that voxel will not be the average of the two original voxels , whereas the fiber density will be . third , fa is in practice acquisition and b - value dependent . we note that if diffusion mri were capable of estimating the absolute avf or fvf as well as the ratio of the intra - axonal to extra - fiber water , the g - ratio could immediately be estimated from these two quantities , without further myelin imaging . this has yet to be done robustly , and it is therefore preferable to use a more robust independent myelin marker . there are many different contrasts and computed parameters that are sensitive to myelin @xcite . the possible sources of signal from the myelin compartment are the ultra - short t@xmath4 protons in the macromolecules of the myelin sheath itself ( t@xmath12 ) and the short t@xmath4 water protons present between the phospholipid bilayers ( t@xmath13 , see fig . [ gcartoon ] ) . most mri contrast mechanisms are sensitive to myelin content , but few are specific . the myelin phospholipid bilayers create local larmor frequency variations for water protons in their vicinity due to diamagnetic susceptibility effects . this results in myelin content modulated transverse relaxation times t@xmath8 @xcite and t@xmath4 , and longitudinal relaxation time t@xmath14 . the local larmor field shift ( fl ) and the susceptibility itself ( @xmath15 ) can be computed as well @xcite . ultra - short te ( ute ) imaging can be used to image the protons tightly bound to macromolecules @xcite . an alternate approach to isolating the myelin compartment is magnetization transfer ( mt ) imaging , where the ultra - short t@xmath4 macromolecular proton pool size can be estimated by transfer of magnetization to the observable water pool . mt based parameters sensitive to macromolecular protons include the magnetization transfer ratio ( mtr ) @xcite , the mt saturation index ( mt@xmath16 ) @xcite , the macromolecular pool size ( f ) from quantitative magnetization transfer @xcite , single - point two - pool modeling @xcite , and inhomogeneous mt @xcite . alternately , the myelin water can be imaged with quantitative multicomponent t@xmath4 @xcite or t@xmath8 @xcite relaxation , which yields the myelin water fraction ( mwf ) surrogate for myelin density . variants include gradient and spin echo ( grase ) mwf imaging @xcite , linear combination myelin imaging ( e.g. , @xcite ) , t@xmath4 prepared mwf imaging @xcite , multi - component driven equilibrium single point estimation of t@xmath4 ( mcdespot ) @xcite and direct visualization of the short transverse relaxation time component via an inversion recovery preparation to reduce long t@xmath14 signal ( vista ) @xcite . other alternate approaches exploiting myelin - modulated relaxation times include combined contrast imaging ( t@xmath17/t@xmath18 ) @xcite or independent component analysis @xcite . proton density is also sensitive to macromolecular content , and the proton - density based macromolecular tissue volume ( mtv ) @xcite has been used as a quantitative myelin marker . while these mri measures have been shown to correlate highly with myelin content @xcite , they have not been incorporated in a specific tissue model in a manner similar to the diffusion signal , and hence some calibration is needed . this is still a topic of research . caveats of improper calibration of the mvf are discussed in section [ cal ] . in this and the previous sections , we have discussed imaging techniques for both diffusion - visible microstructure and myelin . any multi - modal modal imaging protocol with contrasts such as these , sensitive to the axon and myelin volume fractions , is sensitive to the g - ratio ( e.g. , @xcite ) . the purpose of the explicit g - ratio formulation is to create a measure that is _ specific _ to the g - ratio . it is interesting to ask whether we could use a technique such as deep learning to estimate the g - ratio , skipping explicit modeling completely . in the following sections , we illustrate several important points about g - ratio imaging using experimental data acquired at our site the following describes the acquisition protocol . we acquired data from healthy volunteers and from multiple sclerosis patients . these data were acquired on a siemens 3 t trio mri scanner with a 32 channel head coil . a t@xmath17 structural mprage volume with 1 mm isotropic voxel size was acquired for all subjects . for diffusion imaging , the voxel size was 2 mm isotropic . for most experiments , the noddi diffusion protocol consisted of 7 b=0 s / mm@xmath19 , 30 b=700 s / mm@xmath19 , and 64 b=2000 s / mm@xmath19 images , 3x slice acceleration , 2x grappa acceleration , all acquired twice with ap - pa phase encode reversal . for the other experiments , as detailed below when they are introduced , the slice acceleration and phase encode reversal were not employed . for a dataset optimized for diffusion tensor reconstruction , a dataset with 99 diffusion encoding directions at b=1000 s / mm@xmath19 and 9 b=0 s / mm@xmath19 images was acquired . for magnetization transfer images , we also used 2 mm iso - tropic voxels to match the diffusion imaging voxel size . for mtr , one 3d non - selective pd - weighted rf - spoiled gradient echo ( spgr ) scan was acquired with tr=30 ms and excitation flip angle @xmath20 , and one mt - weighted scan was acquired with the same parameters and an mt pulse with 2.2 khz frequency offset and 540@xmath11 mt pulse flip angle . for mt@xmath16 computation , these same mt - on and mt - off scans were used , with one additional t@xmath14-weighted scan with tr=11 ms and excitation flip angle @xmath21 . for qmt computation , 10-point logarithmic sampling of the z - spectrum from 0.433 - 17.235 khz frequency offset was acquired , with two mt pulse flip angles for each point , 426@xmath11 and 142@xmath11 , and excitation flip angle @xmath22 . the qmt acquisition was accelerated with 2x grappa acceleration . additional scans for correction of the maps included b@xmath14 field mapping using the double angle technique @xcite , with 60@xmath11 and 120@xmath11 flip angles , b@xmath9 field mapping using the two - point phase difference technique , with te@xmath14/te@xmath4 = 4.0/8.48 ms , and t@xmath14 mapping using the variable flip angle technique @xcite , with flip angles 3@xmath11 and 20@xmath11 . additional t@xmath4-flair and pd@xmath23 images were acquired for the ms subjects to aid in lesion segmentation . in this section , we discuss pitfalls and outstanding issues in g - ratio imaging . experimental results are included in these sections to illustrate these problems . the diffusion mri post - processing techniques described in section [ avf ] give a range of outputs . some are physical quantities ( such as the diffusion displacement distribution ; kurtosis ) , while some are parameters of detailed biological models ( such as the intra - axonal volume fraction ) . models are valuable , but the user has to be aware of the assumptions made . the parameter space in existing models ranges from three free parameters in noddi to six @xcite , twenty three @xcite , and thirty one @xcite in other models . recent analysis hypothesizes that the lower number of free parameters in , e.g. , noddi and charmed , may be matched to the level of complexity possible on current clinical systems @xcite , while high gradient strength , high b - values , and more b - shells may be necessary for more complex models @xcite , and would make them more optimal . this is a general problem with multi - exponential models when diffusion weighting is weak @xcite . on standard mr systems , relaxing the constraints on fixed parameters has been shown to lead to degeneracy of solutions @xcite . regularization approaches such as the spherical mean technique ( smt ) @xcite can make the problem less ill - posed . one of the fixed parameters in the noddi model is the parallel diffusivity in the intra- and extra - axonal space , both set to the same fixed value . other models explicitly model these as unequal ; for instance , wmti assumes that the intra - axonal diffusivity is less than or equal to the extra - axonal diffusivity . the actual values are unknown , however simulations have shown that the assumption of equal parallel diffusivities leads to a 34 - 53% overestimation of the intra - axonal compartment size if the diffusivities are in fact unequal @xcite , with the intra - axonal diffusivity either greater than or less than the extra - axonal diffusivity . independent of whether the intra- and extra - axonal parallel diffusivities are equal , another source of this bias is the tortuosity model @xcite employed by many models , including noddi , diamond , and the smt . this model computes the perpendicular extra - axonal diffusivity as a function of the diffusion - visible intra - axonal volume fraction of the non - csf tissue ( v@xmath24 in the noddi model ) . this tortuosity estimate is bound to be inaccurate because the tortuosity is expected to vary as the absolute fiber volume fraction of the non - csf tissue ( @xmath25 ) , not the diffusion - visible fiber volume fraction . these two quantities are very different , as the myelin and axon volume fractions are almost equal in healthy tissue @xcite . the mvf could be explicitly included in the equation , and would be expected to result in a myelin volume dependent reduction in @xmath7 : @xmath26 however , in healthy tissue , where the fvf should scale roughly as the v@xmath24 parameter , this tortuosity model does not appear to hold when applied to experimental data with independent estimates of the parallel and perpendicular extra - axonal diffusivities @xcite . another fixed parameter in the noddi model is the t@xmath4 relaxation time of all tissue , assumed to be the same , even in csf . this leads to an overestimation of v@xmath27 , which can be corrected @xcite given t@xmath4 estimates from , e.g. , a t@xmath4 mapping technique such as mcdespot @xcite . diffusion mri is exquisitely sensitive to fiber geometry . the fractional anisotropy , as mentioned above , may be more sensitive to geometry than to any microstructural feature @xcite . hence , microstructural models must be careful to take geometry ( crossing , splaying , curving , microscopic packing configuration ) into account . a typical diffusion imaging voxel is roughly 8 mm@xmath28 , while the axons probed by microstructural models are on the order of one micron . the noddi model that has been used in several g - ratio imaging studies to date assumes there is a single fiber population with potential splay or curvature , but does not explicitly model crossings . furthermore , the tortuosity model employed is probably not correct for varying packing density on the sub - voxel scale @xcite ; it has been shown to depend on the packing arrangement and break down for tight axon packing @xcite . this probably explains the discrepancy between model and experiment mentioned above , because the geometry of axonal packing can vary considerably for a given average volume fraction . to what extent does the fiber dispersion model of noddi handle crossing fiber bundles ? we have employed the diffusion mri simulator dsim @xcite to investigate this question @xcite . we simulated realistic axonal packing @xcite in voxels with straight , parallel fibers and with two equal size bundles of straight fibers crossing at 90@xmath11 ( see fig.[sim ] ) . fiber volume fractions were set equal for both configurations and were varied from 0.3 to 0.7 . g - ratios were varied from 0.7 to 0.9 . the diffusion weighted signal was generated , and the noddi model parameters computed using the noddi matlab toolbox @xcite . the fvf was computed from the noddi parameters using the known mvf . the computed fvf was @xmath29 lower in the crossing fiber case for the noddi - based fvf . this demonstrates that the noddi model , while not explicitly designed for crossing fibers , gives acceptable results in this case , and can be used for full - brain g - ratio estimation at standard voxel size with significant subvoxel fiber crossing , with only a small decrease in the estimated fvf due to partial volume averaging of fiber orientations . simulated fibers in straight , parallel configuration ( left ) vs. crossing ( right ) , with equal fiber volume fraction and similar distributions of axon diameter and position . the noddi model underestimates the fvf by @xmath29 in the crossing fiber case , whereas the dti model underestimates the fvf by @xmath30.,scaledwidth=50.0% ] * _ experiments : comparison of dti and noddi for fvf estimation _ * [ dtivsnoddi ] noddi works optimally with diffusion mri measurements made on at least two shells in q - space , i.e. , two different nonzero b - values , although recent work has proposed solutions for single shell data , at least where certain assumptions can be made about the tissue , or where high b - values are used @xcite . in contrast , the diffusion tensor can be robustly fitted and the fiber volume fraction inferred @xcite ( see section [ fvf ] ) using a much more sparsely sampled , single shell dataset . many research programs have large databases of single - shell diffusion data , often with limited angular sampling of q - space as well . it is therefore of interest to explore to what extent such data can be used in investigation of the g - ratio . in the simulations described above , we also computed the diffusion tensor using in - house software@xcite . the fvf was computed from the fa using the quadratic relationship determined from previous simulations @xcite . as expected , the fa is not a predictor of fvf in the presence of crossing fibers : the computed fvf was @xmath30 lower in the crossing fiber case compared to the parallel fiber case for the dti - based fvf . to compare noddi and dti _ in vivo _ , diffusion and qmt data were acquired as described in section [ acq ] for one healthy volunteer , without slice acceleration or phase encode reversal . the qmt data were processed with in - house software and the noddi parameters as described above . the avf , mvf , and g - ratio were computed voxelwise from the diffusion and qmt data as described in section [ calculation ] . additionally , the diffusion tensor was calculated using the b=1000 s / mm@xmath19 diffusion shell . the fvf was then calculated from the fractional anisotropy of the diffusion tensor using the quadratic relationship @xcite . the corpus callosum was skeletonized on the fa image and a voxel - wise correlation between the fvf computed from dti and from noddi and qmt was performed for these voxels . the coefficient of proportionality between f and mvf was determined from previous em histological analysis @xcite . [ correlationscatter ] shows the fvf computed using both noddi and dti in the skeleton of the healthy human corpus callosum . the correlation between fvf measured using the two techniques was r=0.79 , with @xmath31 . this indicates a slight discrepancy between the fvf using noddi compared to dti , and a reasonably high correlation between techniques on the skeleton . correlation between dti- and noddi - derived fiber volume fraction on the skeleton of the corpus callosum.,scaledwidth=45.0% ] possible explanations for the higher estimates using noddi appear in section [ params ] , although without ground truth is is difficult to say which approach is more accurate . additionally , because the fa does not explicitly model compartments , it is subject to partial volume effects . while partial volume averaging with csf will decrease the fa , the fa - based quadratic fvf model appears to break down in this case . this effect could possibly be reduced by applying the free water elimination technique @xcite to obtain the correct fvf for the non - csf compartment and then scaling to reflect the partial volume averaging with csf afterward . to conclude , while the fa is generally a poor indicator of fvf , it may be a reasonable surrogate in certain special cases when data are limited . it is interesting to consider how useful imaging a cross - section of a white matter fascicle may be , regardless of the model used . if the g - ratio can be assumed to be constant along an axon , measurement of a cross - section is useful . however , in many pathological situations , such as wallerian degeneration , it is of interest to study the entire length of the axon . most existing diffusion models assume that extra - axonal diffusion is gaussian , hindered by the structures present , but not restricted . however , observation of tightly packed axons in microscopy ( e.g. , fig.[gcartoon ] ) indicates that the intra- and extra - axonal spaces may not be as distinguishable as the models assume . it is unclear to what extent the extra - axonal diffusion is non - gaussian . if axons are packed tightly together , is extra - axonal diffusion non - gaussian ? it is not clear whether the water mobility through the tight passageways between fibers is distinguishable from the restricted diffusion within spaces surrounded by contiguous myelin . if signal from the extra - axonal space is erroneously attributed to the intra - axonal space , the model output will be incorrect . some models make no attempt to distinguish between intra- and extra - cellular restricted diffusion , meaning the pore size estimates may reflect a mixture of the two @xcite . time - dependent ( i.e. , non - gaussian ) diffusion has recently been observed in the extra - axonal space @xcite using long diffusion times . this may be due to axon varicosity , axonal beading , or variation in axonal packing @xcite . diffusion modeling is an active field , and advances in the near future will hopefully improve precision and accuracy of avf estimates using diffusion mri . histological validation may aid in understanding the strengths and limitations of these estimates . at present , the limitations of these models propagate to the g - ratio , as do the limitations of mvf estimates , which are discussed below . how do we make a quantitative estimate of the mvf from myelin sensitive mri markers ? linear correlations have been shown between the individual myelin sensitive metrics ( such as f @xcite , mtr @xcite , r@xmath14 @xcite , mwf @xcite , and mtv @xcite ) with the mvf from histology . given the linear correlations that have been established , a logical first approximation is to assume a linear relationship between the chosen myelin - sensitive metric and the mvf . then , using the macromolecular pool size f as an example , the relationship is @xmath32 with @xmath33 and @xmath34 constants . while a non - zero value for @xmath34 has been indicated by some studies @xcite , this may be an artefact due to the inherent bias in linear regression . the assumption of a linear relationship hinges on the assumption that non - myelin macromolecular content scales linearly with myelin content , which can break down in disease , or even in healthy tissue . if the myelin and non - myelin macromolecular content do scale linearly , as is assumed here , a constant non - zero intercept is unlikely , meaning a theoretical prior that @xmath35 is reasonable . there is evidence that even if a simple scaling relationship exists between f and mvf , it is dependent on acquisition and post - processing details . for instance , a recent study calibrated f at two different sites , and found a different scaling factor for each @xcite . these scaling factors in turn differ from those obtained from other investigations @xcite . hence , careful calibration for each study must be performed . several studies have calibrated scaling factors based on a given expected g - ratio in healthy white matter @xcite , however , the g - ratio in healthy white matter is not precisely known . none of the myelin - sensitive mri markers is 100% specific to myelin , and most are sensitive to myelin in a slightly different way . magnetization transfer contrast is specific to macromolecules , and more specific to lipids than to proteins @xcite . macromolecules in the axon membrane itself , in neurofilaments within the axons , and in glial cell bodies , will contribute to the mt signal , with myelin constituting only 50% of the macromolecular content in healthy white matter @xcite . additionally , mt - based metrics such as the magnetization transfer ratio will have residual contrast from other mechanisms . we expect the mtr contrast to vary linearly with macromolecular content , but also with t@xmath14 @xcite . t@xmath14 has the reverse sensitivity to myelin than does the mt effect @xcite , meaning that these effects work against each other , reducing the dynamic range and power of mtr as a marker of myelin . furthermore , t@xmath14 is sensitive to iron and calcium content , intercompartmental exchange , and diffusion , and hence sensitive to axon size @xcite and axon count @xcite . this means the relationship between mtr and mvf may not be monotonic , and is certainly nonlinear . this nonlinearity is evident in published plots of mtr vs. f , e.g. , that shown by levesque et al . @xcite , and the lack of dynamic range of mtr is also evident @xcite . the mt@xmath16 technique aims to remove the t@xmath14 dependence in mtr . both mtr and mt@xmath16 depend on the offset frequency used in the acquisition . ihmt shows promise as a more myelin - specific mt marker due to its sensitivity to specific molecules in myelin that broaden the z - spectrum asymmetrically , although it has recently been shown that asymmetric broadening is not essential to generate a non - zero ihmt signal @xcite , and the technique suffers from low signal . qmt is the most comprehensive of the mt - based myelin markers , although its use is impeded by long acquisition times , and its parameters appear to be sensitive to the specific model and fitting algorithm . finally , mt based estimates of mylein volume will be insensitive to variation in the distance between the lipid bilayers . proton - density based techniques @xcite will , like mt , be sensitive to all macromolecules , with a different weighting on these macromolecules compared to the lipid- + dominated mt signal . relaxation - based myelin markers are also not 100% specific to myelin . the confounds with using t@xmath14 directly were mentioned above , and the dependence on iron and calcium concentration , intercompartmental exchange and diffusion will also affect t@xmath4 . t@xmath8 is also sensitive to iron concentration , as well as fiber orientation @xcite . isolating the short t@xmath4 or short t@xmath8 compartment enhances specificity to myelin , but mwf estimates vary nonlinearly with myelin content as the sheath thins and exchange and diffusion properties are modulated @xcite . variants may have biases , for example , the mc- + despot technique has been shown to overestimate the mwf @xcite . combining t@xmath14 and t@xmath4 in various ways @xcite may increase specificity , although they rely on myelin being the dominant source of contrast . in the ute technique , it is as yet unclear how to map the signal directly to myelin content . in addition to these confounds , most of these myelin markers have recently been shown to have orientation dependence . these include t@xmath8 , chi @xcite , mtr @xcite , and t@xmath4 of the macromolecular pool . while these myelin imaging techniques are certainly powerful tools in the study of healthy and diseased brain , can they be used reliably in the g - ratio imaging framework ? as an illustration of the effects of miscalibration of myelin markers , consider the following scenario @xcite . we investigate three mt - based myelin markers : mtr , mt@xmath16 , and macromolecular pool size f. we assume a simple linear scaling between our mri marker and the mvf . as described in previous work @xcite , we calibrate f using the same acquisition protocol in the macaque , coupled with _ ex - vivo _ electron microscopy of the same tissue , and then calibrate mtr and mt@xmath16 to match the mean f - based mvf in white matter . we then compute the g - ratio , using the noddi model of diffusion and the mvf derived from the myelin markers ( eq.s [ avffrnoddi],[geq ] ) . mtr , mt@xmath16 , qmt , and noddi data were acquired for one healthy volunteer and one ms patient , as described in section [ acq ] . for the ms patient , the mt@xmath16 images were computed from the qmt mt - off and mt - on ( mt pulse offset 2.732 khz , flip angle 142@xmath11 ) images and one additional t@xmath17 image with te=3.3 ms , tr=15 ms , and excitation flip angle @xmath36 . binary segmentation of white and grey matter was performed using beast @xcite , using the mprage image only . the macromolecular pool size f was computed using in - house software . the diffusion images were preprocessed using fsl @xcite , and the noddi parameters computed using the noddi matlab toolbox @xcite . mt@xmath16 was computed according to helms et al . lesion segmentation for the ms subject was performed with in - house software . a combined mri / histology dataset @xcite was used to scale each myelin marker ( mtr , mt@xmath16 , and f ) to give the mvf , with the assumption of a linear relationship ( eq . [ mvflin ] ) with intercept b=0 . correlations between the three myelin markers were computed in brain parenchyma . percent differences were computed between healthy white matter and healthy grey matter for each of the three myelin markers . the avf was computed using eq . [ avffrnoddi ] , and g - ratios were computed in the ms and healthy brains using eq . average g - ratios were computed in healthy white matter , normal appearing white matter ( nawm ) , and ms lesions . a theoretical computation was also performed , varying the mapping of an arbitrary myelin metric to mvf using eq . [ mvflin ] . we separately varied the slope ( c ) and the intercept ( b ) for a range of fiber volume fraction values and mapped the computed g - ratio as a function of fvf . when varying the slope , the intercept was fixed at the origin . mtr plotted versus f ( top ) and mt@xmath16 plotted versus f ( bottom ) in parenchyma . the mtr vs. f plot shows a marked nonlinearity ( r=0.59 ) , as is expected , while mt@xmath16 increases the linearity of the relationship ( r=0.77 ) and the dynamic range.,scaledwidth=50.0% ] the correlation of mtr with f was r=0.59 ( p@xmath370.001 ) , and of mt@xmath16 with f was r=0.77 ( p@xmath370.001 ) . [ mtr_mtsat_corrwf ] shows plots of mtr versus f ( top ) and mt@xmath16 versus f ( bottom ) in parenchyma . of note , the plot of mtr versus f appears to have a nonlinear shape , similar to that seen in the literature @xcite . when t@xmath14 effects are reduced using mt@xmath16 , the linearity and dynamic range increase . in healthy brain , the percent difference between white and grey matter was 15.02% for mtr , 40.08% for mt@xmath16 , and 45.86% for f. the narrower dynamic range of the mvf derived from mtr can also be seen in fig . [ mvfs ] , where grey matter has markedly higher values . if this simple scaling to obtain the mvf is used in the g - ratio formula , the g - ratio in healthy white matter is relatively constant . however , when lesions exist , the contrast using the different mvf markers is very different . in the ms patient , the mean g - ratio in normal appearing white matter ( nawm ) was 0.76 for all three mvf markers . in ms lesions , the mean g - ratio was 0.65 , 0.80 , and 0.80 , for mtr , mt@xmath16 , and f , respectively . [ gratios ] shows the spatial distribution of g - ratios in the ms patient for the three mvf markers . [ mvfsim ] shows the theoretical effect of having an improper slope ( top ) or intercept ( bottom ) in the relationship between an arbitrary myelin marker and the mvf , in the case where the ( theoretical ) relationship is in fact linear . the plots show that the computed g - ratio becomes fiber density dependent , in addition to being incorrect . the mtr is a commonly used myelin marker , however , due to t@xmath14 sensitivity , it lacks dynamic range . mt@xmath16 correlates more highly with f , obtained from an explicit qmt model designed to isolate the macromolecular tissue content . it is important to note , however , that this correlation may be driven to some extent by the different b@xmath14 sensitivities of the techniques ( see section [ preproc ] ) , and the mtr was not corrected for b@xmath14 induced variability @xcite . independently of this demonstration of the potential of mt@xmath16 for myelin mapping , researchers have found that mt@xmath16 may be more sensitive to tissue damage than mtr in multiple sclerosis , with more correlation with disability metrics @xcite . mt@xmath16 has recently been used by other groups in g - ratio imaging of healthy adults @xcite . plots of the mvf derived from ( from left to right ) mtr , mt@xmath16 , and f , in healthy brain.,scaledwidth=50.0% ] plots of the g - ratio computed using ( from left to right ) mtr , mt@xmath16 , and f , in the ms patient . the arrow indicates a lesion in which the apparent g - ratio is lower than in nawm for mtr , but higher than in nawm for mt@xmath16 and f.,scaledwidth=50.0% ] if the mvf is miscalibrated in this g - ratio imaging formulation , there will be a residual dependence on fiber volume fraction in our formulation . this reduces the power of the g - ratio metric , which ideally is completely decoupled from the fiber density . independent of specificity of the myelin marker , if the myelin calibration is inaccurate , this residual dependence on fiber volume fraction occurs . it is clear that the g - ratio metric we will compute is g - ratio _ weighted _ , and the better the calibration , the more weighted to the g - ratio it will be . until quantitative myelin mapping is _ accurate _ , the g - ratio metric will not be specific to the g - ratio . effect of having an improper slope ( top ) or intercept ( bottom ) in the relationship between an arbitrary myelin marker and the mvf , in the case where the ( theoretical ) relationship is in fact linear . the plots show that the computed g - ratio becomes fiber density dependent , in addition to incorrect . , title="fig:",scaledwidth=50.0% ] effect of having an improper slope ( top ) or intercept ( bottom ) in the relationship between an arbitrary myelin marker and the mvf , in the case where the ( theoretical ) relationship is in fact linear . the plots show that the computed g - ratio becomes fiber density dependent , in addition to incorrect . , title="fig:",scaledwidth=50.0% ] one possible solution for mvf calibration is to calibrate the g - ratio to a known value in certain regions of interest @xcite , as mentioned above . however , care must be taken that this step is not adjusting for differences in the diffusion part of the pipeline ( e.g. , different implementations of the diffusion model @xcite ) , and therefore still leaving a fiber density dependence . additionally , the correct value in these regions of interest must be known . calibration based on expected mvf would remove this sensitivity , but is subject to error due to partial volume averaging of white matter with other tissue . if the relationship between the myelin - sensitive metric and the mvf is not a simple scaling , such calibration will fail . particular care needs to be taken when studying disease . if the assumed relationship between the myelin marker and the mvf is incorrect , the computed g - ratio will be incorrect . is it possible to compute a g - ratio that is correct to within a scaling factor , and not sensitive to the fiber density ? this would require that the avf or fvf be estimated independent of the mvf . simple models such as the diffusion tensor , apparent fiber density @xcite , and tensor fiber density @xcite , are indicators of fiber or axon density , but detailed modeling is most likely superior . consideration of contrasts other than diffusion mri , such as gradient - echo based approaches @xcite , might also help with this problem . the g - ratio is a function of the ratio of the mvf to the avf , and a technique that measures this ratio directly would be optimal . however , gre based estimates would be of the myelin and axon _ water _ fraction , and hence would still need to be calibrated for the volumetric occupancy of water in these tissues . in summary , both specificity and accuracy are important for both avf and mvf estimation . this may require more sophisticated models in both contexts . for example , we have thus far ignored cell membranes . the axon membrane should technically be included in the avf , and its volume is up to 4% of the avf @xcite , but it would most likely be included in the mvf using mt - based mvf estimation . the methods described above include several pre - processing steps , including distortion and field inhomogeneity correction , that deserve further discussion . the mt - based contrasts are acquired with spin - warp acquisition trains , and the diffusion - based contrasts are acquired with single - shot epi . when any acquisition details are changed , the distortions in the images change , and co - registration of voxels for voxelwise quantitative computations becomes more difficult . the blip - up blip - down phase encode strategy ( section [ acq ] ) allows for precise correction of susceptibility - induced distortion in the diffusion images @xcite . lack of correction for this distortion leads to visible bands of artefactually high g - ratio near tissue - csf interfaces ( see , e.g , @xcite ) . this was illustrated by mohammadi et al . @xcite ( see fig . [ siawoosh_misreg ] ) . uncorrected diffusion mri data leads to g - ratios in the vicinity of unity at the edge of the genu of the corpus callosum , caused by voxels where the avf is artefactually high ( containing little or no csf ) , and the mvf low ( because the correctly localized voxels actually contain csf ) . the white matter - csf boundary is a region of obvious misregistration , but much of the frontal lobe suffers from susceptibility induced distortion , and would therefore have incorrect g - ratios . misregistration artefact due to susceptibility - induced distortion in diffusion weighted images . at left is an mt@xmath16 image with white matter outlined in red , for one slice ( a ) and a cropped region at the genu ( b ) . in the center ( c , d ) is an original epi diffusion scan with no diffusion weighting and contrast inverted ( ib0 ) . the misregistration with the mt@xmath16-defined white matter boundary is marked . at right ( e , f ) is the g - ratio computed with these contrasts . uncorrected diffusion mri data leads to g - ratios in the vicinity of unity at the edge of the genu , caused by voxels where the avf is artefactually high ( containing little or no csf ) , and the mvf low ( because the correctly localized voxels actually contain csf ) . reproduced from @xcite . , scaledwidth=50.0% ] multi - modal imaging protocols are a powerful tool for investigation of microstructure . we have thus far discussed combining multiple images with partially orthogonal contrasts in order to estimate the g - ratio . however , problems such as the above misregistration issue arise . can a single acquisition train provide multiple contrasts ? one such approach was described recently for simultaneous mapping of myelin content and diffusion parameters @xcite . it consists of an inversion - recovery preparation before a diffusion weighted sequence , allowing for fitting of a model that incorporates both t@xmath14 ( a myelin marker @xcite ) and axonal attributes . this approach is conceptually extensible to other myelin - sensitive preparations or modifications of a diffusion weighted sequence , such as quantitative t@xmath4 estimation @xcite . can the g - ratio be estimated using a single contrast mechanism ? this could also offer inherent co - registration , as well as possibly increase the acquisition speed . we have noted that if diffusion mri is used as an fvf marker , then diffusion is essentially a single - contrast g - ratio imaging technique . however , we reiterate that it is preferable to use a more robust myelin marker . as discussed in section [ avf ] , analysis of complex gre images may lead us to a technique for estimating the mvf and avf volume fractions from one set of images . our image processing pipeline for qmt analysis includes b@xmath9 and b@xmath14 correction @xcite . b@xmath14 correction for mtr is the subject of continuing research @xcite . as for mt@xmath16 , the acquisition strategy leads to a relatively b@xmath14 insensitive map . however , a semi - empirical b@xmath14 correction is also made @xcite to correct for higher order effects . using such correction , spatial uniformity of the mt@xmath16 map is improved . correction for field inhomogeneity should be considered regardless of the myelin mapping technique employed . b@xmath9 correction is a particular concern in multicomponent quantitative t@xmath8 modeling @xcite . the g - ratio imaging paradigm extracts a single g - ratio metric per voxel . at typical imaging resolution possible for the constituent mr images , a voxel contains hundreds of thousands of axons . the g - ratio is not constant in tissue , but takes on a distribution of values ( see fig . [ macaque_g_dist ] , which shows the g - ratio distribution in the macaque corpus callosum , measured using electron microscopy ) . the range of myelination includes some unmyelinated axons within healthy white matter . the g - ratio distribution may broaden and become bi - modal in disease . even within a single axon with intact myelin , the g - ratio may vary due to organelle swelling . fiber bundles that cross within one voxel may have different g - ratio distributions . in development , some fibers within one fiber bundle will fully develop , while others will be pruned , resulting in an interim bimodal g - ratio distribution within the fascicle . the current mri - based g - ratio framework will not be able to distinguish these cases , as it reports only an intermediate g - ratio value . it is robust to crossing fibers , in that it will report the same intermediate g - ratio value whether the separate bundles cross or lie parallel to each other . the broad g - ratio distribution is in part a resolution problem , but the g - ratio is expected to be heterogeneous on a scale smaller than we can hope to resolve with mri . g - ratio distributions from electron microscopy of the cynamolgous macaque corpus callosum , samples 1 - 8 from genu to splenium . reproduced from @xcite . , scaledwidth=50.0% ] the aggregate g - ratio we compute in the case of a distribution of values is not precisely fiber- or axon - area weighted , but is close to axon area weighted within a reasonable range of values @xcite . larger axons will have a greater weight in the aggregate g - ratio metric we measure . simply put , the aggregate g - ratio is the g - ratio one would measure if all axons had the same g - ratio . in the case of an ambiguous g - ratio distribution , what techniques can we use to infer what situation is occurring ? in multiple sclerosis , for example , two possible scenarios probably occur frequently . one is patchy demyelination , on a scale much smaller than a voxel and smaller than the diffusion distance , and the other is more globally distributed thin myelin . these two scenarios could give rise to equal avf , mvf , and aggregate g - ratio measurements . one possible way to differentiate these cases could be to look more closely at parameters available to us from diffusion models . it has been shown that the extra - axonal perpendicular diffusivity is relatively unchanged by patchy demyelination in a demyelinating mouse model @xcite , because diffusing molecules encounter normal hindrance to motion on most of their trajectory , whereas the axon water fraction is sensitive to this patchy demyelination . hence , the discrepancy between these two measures can be taken as a measure of patchy demyelination . alternatively , one can scan subjects longitudinally and infer disease progression . from the ambiguous timepoint described above , the axons in the patches that are demyelinated may die , leaving a decreased avf and mvf , and a return to a near - healthy g - ratio . in the case of globally thin myelin , the remyelination may continue , leaving a near - healthy avf , mvf , and g - ratio . note that the g - ratio metric still does not distinguish these pathologically distinct cases . there are two unknowns - the fiber density and the g - ratio ( or , alternately , the mvf and the avf ) , and one must consider both to have a full picture of the tissue . looking at the time courses , one can hypothesize what the g - ratio distribution was at the first timepoint . it would be technically challenging to measure the g - ratio distribution _ in vivo_. even with an estimate of a distribution of diffusion properties , and an estimate of the distribution of myelin - sensitive metric , the g - ratio distribution is ill defined . however , several recent acquisition strategies may help us get closer to this aim . one approach is to take advantage of the distinguishable diffusion signal between different fiber orientations . in the ir - prepared diffusion acquisition described above , the model specifies multiple fiber populations with distinct orientations , each with its own t@xmath14 value . this means the diffusion properties , including the restricted pool fraction ( a marker of intra - axonal signal from the charmed model ) , are paired with a corresponding t@xmath14 for each fiber orientation . hence , a g - ratio metric could be computed for each fiber orientation . this could be of benefit in , e.g. , microstructure informed white matter fiber tractography ( e.g. , @xcite ) of fiber populatons with distinct g - ratios . `` jumping '' from one fiber population to another is very common in tractography @xcite , and constraining tractography to pathways with consistent microstructural features could help reduce false positives in regions of closely intermingling tract systems . it may be possible , conceptually , to estimate the g - ratio distribution via a 2d spectroscopic approach . while extremely acquisition intensive , 2d spectroscopy of t@xmath4 and the diffusion coefficient has been demonstrated recently @xcite . the acquisition involves making all diffusion measurements at different echo times . if a distribution of a myelin volume sensitive metric ( here , t@xmath4 ) can be estimated simultaneously with a distribution of a diffusion - based metric sensitive to the axon volume , it may be possible to infer the distribution of g - ratios . making a multi - modal imaging protocol short enough for the study of patient populations and use in the clinic is a considerable challenge . our investigation of mt@xmath16 as a replacement for qmt was done in the interest of reducing acquisition time . there exist other short mt - based approaches , such as single - point two - pool modeling @xcite and inhomogeneous mt @xcite . another approach could be to use compressed sensing @xcite for mt - based acquisitions . gre - based myelin water fraction approaches @xcite may also offer a faster approach for estimating the myelin water fraction , and possibly , as mentioned above , a way to eliminate the diffusion imaging part of the protocol . diffusion imaging has benefited from many acceleration approaches in recent years , including parallel imaging , which can also be used in the myelin mapping protocols , slice multiplexing @xcite , and hardware advances such as the connectom gradient system . this has been an incomplete but useful list of pitfalls . now , we will consider the promise of imaging the aggregate g - ratio weighted metric , despite its pitfalls . g - ratio imaging is being explored in many different contexts , described below . the promise of g - ratio imaging is its potential to provide us with valuable _ in vivo _ estimates of relative myelination . in the last few years , studies showing the potential of this framework have begun to emerge . [ healthyg ] shows an image of the g - ratio in healthy white matter using our qmt and noddi g - ratio protocol ( see section [ acq ] ) . the map is relatively flat , with a mean g - ratio of 0.75 . other groups have explored these and other mvf and avf sensitive contrasts for g - ratio mapping in healthy white matter . these include a study of the effects of age and gender in a population of subjects aged 20 to 76 using qmt and noddi @xcite , studies of healthy adults using mt@xmath16 and the tfd @xcite and mtv and dti @xcite , and a study of healthy subjects using the vista myelin water imaging technique and noddi @xcite . the g - ratio in healthy while matter , imaged using qmt and noddi.,scaledwidth=35.0% ] a variation of the g - ratio with age appears to be detectable with this methodology @xcite . a variation with gender has not been seen , and if it exists in adolescence @xcite , a study designed for sufficient statistical power at a precise age will be required to detect it . in addition to exploring the effect of age and gender , spatial variability of the g - ratio has been investigated . an elevated g - ratio at the splenium of the corpus callosum has been seen @xcite . the splenium has been reported to contain `` super - axons '' of relatively large diameter , and these would be expected , due to the nonlinearity of the g - ratio @xcite , to have relatively thinner myelin sheaths . electron microscopy in the macaque @xcite ( see fig . [ splenium ] ) confirms this ; the `` super - axons '' dominate the aggregate g - ratio measure , which was seen to be elevated in the splenium using both em measurements and mri of the same tissue @xcite . existence of `` super - axons '' in the splenium of the corpus callosum . top : drawing based on histology by aboitiz _ ( reproduced from @xcite ) , showing large diameter at the splenium . bottom : em of the g - ratio in the cynamolgous macaque showing large diameter axons at the splenium . these will dominate the aggregate g - ratio measure , which was elevated in the splenium using both em measurements and mri of the same tissue shown here @xcite . , scaledwidth=50.0% ] g - ratio imaging has also been performed in the healthy human spinal cord @xcite , where there are considerable technical challenges , such as motion , susceptibility , and the need for significantly higher resolution than we have described for cerebral applications . et al . _ acquired g - ratio data at 0.8 mm x 0.8 mm inplane voxel size . this study used the charmed model of diffusion , more accessible on scanners with high gradient strength , on a connectom skyra scanner . it used the mtv myelin marker . of interest , the g - ratio was not found to vary significantly across white matter tracts in the spinal cord , while the diffusion metrics ( restricted fraction , diffusivity of the hindered compartment , and axon diameter ) and the mtv metric did vary across tracts . this is expected , as heterogeniety in packing and axon diameter is expected to be greater than heterogeneity of the g - ratio , and the g - ratio is also robust to partial voluming effects . multiple groups have studied the g - ratio _ in vivo _ in the developing brain @xcite . axon growth outpaces myelination during development , and therefore a decreasing g - ratio is expected as myelination reaches maturity , as was seen in these studies . imaging the g - ratio _ in vivo _ in multiple sclerosis has been explored by several groups @xcite and is of interest for several reasons . it can possibly help assess disease evolution , and can help monitor response to treatment . it has the potential to aid in the development of new therapies for remyelination . it can also help us understand which therapies might be more fruitful avenues of research . therapy for ms includes immunotherapy and remyelination therapy , however , most of our histological knowledge of ms comes from samples from older subjects at more advanced stages of the disease . if remyelination happens effectively , at least for some axons , at the earlier stages of the disease , and the myelin loss measured using myelin markers is due to fiber density loss only , then immunotherapy would be a more useful therapy , at least once most surviving axons have returned to near - normal g - ratios . despite the promise of imaging the g - ratio _ in vivo _ with ms , it is important to remember the effect of miscalibration of the myelin metric when interpreting g - ratio estimates in ms . there is evidence that fiber density drops precipitously in some lesions @xcite . we have seen ( fig . [ gratios ] ) that in this case , mtr for example does not drop enough , making ms lesions appear to have a lowered g - ratio instead of a higher g - ratio as expected . inspecting the bottom ( red ) curve in fig . [ mvfsim ] , we see that even if there is a linear relationship between the myelin marker of choice and the mvf , miscalibration leads to an apparent g - ratio metric that is elevated in regions of lower fiber density , and significantly lower in regions of healthy fiber density . this occurs when in fact all of the fibers have the same g - ratio , and could easily be interpreted as hypomyelination in an ms subject or population . to further complicate the situation , one must consider how the myelin sensitive metrics behave in the presence of astrocyte scarring , glial cell processes , and inflammatory cell swelling . the mapping of many myelin markers to mvf may change as the ratio of myelin to other visible macromolecular structures changes . it is also not clear how well the estimates of avf and mvf behave at very low fiber density . another point of concern in ms imaging is that all myelin will affect the mr signal , even if it is not part of an intact fiber . research indicates that there is acute demyelination followed by a period of clearance of myelin debris , followed by effective remyelination . during clearance , remyelination can occur , but this myelin is of poor quality @xcite . on the scale of an mri voxel , there can be myelin debris , poor remyelination , and higher quality remyelination . the extent to which myelin debris affects the myelin volume estimates may depend on the myelin mapping technique chosen . further studies of ms are ongoing , including pediatric populations , optic neuritis , and studies investigating whether gado - linium enhancing lesions have a distinct g - ratio . g - ratio imaging has potential to aid in the understanding and treatment of multiple other diseases . white matter abnormalities may underlie many developmental disorders . these include pelizaeus merzbacher disease and sturge - weber syndrome . the g - ratio can change due to axonal changes that occur with intact myelin ( for example , axonal swelling due to infarction ) could result in an increased apparent g - ratio . g - ratio differences have been seen in schizophrenia using electron microscopy @xcite , and researchers hope to be able to study such changes _ in vivo _ in schizophrenia and other psychiatric disorders . another potential application of g - ratio imaging is bridging the gap between microstructure and large - scale functional measures such as conduction delays . the theoretically optimal g - ratio for signal conduction should predict conduction delays . the true promise of g - ratio imaging will come with validation . _ ex - vivo _ validation has been performed , investigating the g - ratio explicitly @xcite and the individual metrics used to compute it , both myelin and axon volume weighted @xcite or one of these two components of the g - ratio formulation @xcite . these studies compare _ in - vivo _ or _ ex - vivo _ mri metrics to electron microscopy , optical microscopy , myelin staining , immunohistochemistry , and coherent anti - stokes raman spectroscopy ( cars ) . while no microscopy technique is perfect , microscopy provides a reasonable validation for imaging techniques , taking into account the possibility for tissue shrinkage and distortion , limitations in contrast and resolution , and segmentation techniques @xcite . interpretation of findings of demyelinating models should take into account the particularities of the demyelinating challenge . et al . _ have shown that the extra - axonal diffusivity perpendicular to axons correlates with the g - ratio in a cuprizone demyelinating model in mice @xcite . this is probably driven by a fiber volume fraction decrease , because little axon loss would be expected in this model , meaning the g - ratio and fiber volume fraction correlate highly . similarly , west _ et al . _ have shown a correlation between the discrepancy between f and mwf and the g - ratio in a knockout model in mice @xcite . this is probably a correlation with absolute myelin thickness , via exchange effects , as opposed to the g - ratio per se . we have discussed the considerable promise of g - ratio imaging with mri . computing the g - ratio metric is a useful way to interpret myelin volume - weighted and axon / fiber volume - weighted data . we have furthermore discussed the pitfalls of g - ratio imaging , including mr artefacts , lack of specificity , lack of spatial resolution , and acquisition time . with the confounds described in the text in mind , such as accuracy of myelin mapping , g - ratio imaging clearly gives us an _ aggregate _ g - ratio + _ weighted _ metric . this imaging framework provides information on two quantities : the fiber density and the g - ratio , and attempts to decouple these two quantities to the best of the ability of our current imaging technology . the promise of g - ratio imaging includes the multitude of pathological conditions in which _ in vivo _ g - ratio estimates can aid in understanding disease , developing therapies , and monitoring disease progression . it also includes the study of normal variability , development , aging , and functional dynamics . the authors would like to thank tomas paus , robert dougherty , mathieu boudreau , eva alonso - ortiz , blanche perraud , j.f . cabana , christine tardif , jessica dubois , ofer pasternak , atef badji , robert brown , masaaki hori , and david rudko for their insights and contributions to this work . this work was supported by grants from campus alberta innovates , the canadian institutes for health research ( gbp , fdn-143290 , and jca , fdn-143263 ) , the natural science and engineering research council of canada ( ns , 2016 - 06774 , and jca , 435897 - 2013 ) , the montreal heart institute foundation , the fonds de recherche du qubec - sant ( jca , 28826 ) , the quebec bioimaging network ( ns , 8436 - 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015 - 0451 - 7 . url http://dx.doi.org/10.1007/s11182-015-0451-7 . vasily l. yarnykh . pulsed z - spectroscopic imaging of cross - relaxation parameters in tissues for human mri : theory and clinical applications . _ , 470 ( 5):0 929939 , may 2002 . doi : 10.1002/mrm.10120 . url http://dx.doi.org/10.1002/mrm.10120 . vasily l. yarnykh . fast macromolecular proton fraction mapping from a single off - resonance magnetization transfer measurement . _ magn reson med _ , 680 ( 1):0 166178 , july 2012 . issn 1522 - 2594 . url http://view.ncbi.nlm.nih.gov/pubmed/22190042 . aldo zaimi , tanguy duval , alicja gasecka , daniel ct , nikola stikov , and julien cohen - adad . : open source software for axon and myelin segmentation and morphometric analysis . _ frontiers in neuroinformatics _ , 10 , august 2016 . issn 1662 - 5196 . doi : 10.3389/fninf.2016.00037 . url http://dx.doi.org/10.3389/fninf.2016.00037 . hui zhang , penny l. hubbard , geoff j. m. parker , and daniel c. alexander . axon diameter mapping in the presence of orientation dispersion with diffusion mri . _ neuroimage _ , 560 ( 3):0 13011315 , june 2011 . issn 10538119 . doi : 10.1016/j.neuroimage.2011.01.084 . url http://dx.doi.org/10.1016/j.neuroimage.2011.01.084 . hui zhang , torben schneider , claudia a. wheeler - 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the fiber g - ratio is the ratio of the inner to the outer diameter of the myelin sheath of a myelinated axon . it has a limited dynamic range in healthy white matter , as it is optimized for speed of signal conduction , cellular energetics , and spatial constraints . _ in vivo _ imaging of the g - ratio in health and disease would greatly increase our knowledge of the nervous system and our ability to diagnose , monitor , and treat disease . mri based g - ratio imaging was first conceived in 2011 , and expanded to be feasible in full brain with preliminary results in 2013 . this manuscript reviews the growing g - ratio imaging literature and speculates on future applications . it details the methodology for imaging the g - ratio with mri , and describes the known pitfalls and challenges in doing so . g - ratio , mri , myelin imaging , diffusion mri , white matter , microstructure

1701.02760

one decade has gone by since the earlier groundbreaking experiments performed by andrei geim and konstantin novoselov @xcite ( nobel laureates in physics in 2010 ) to isolate single layer membranes of graphite , graphene . soon after , theoretical @xcite and experimental @xcite groups highlighted the properties of charge carriers in this material which resemble much to ultrarelativistic electrons , thus establishing a bridge between solid state and particle physics ( see , for instance , refs . graphene has given rise to the new era of dirac materials with potential applications in nanotechnology , but also offering an opportunity to test the core of fundamental physics in a condensed matter environment . mechanical , thermal and electronic properties of this two - dimensional crystal locate it among the best candidates to replace silicon in nanotechnological devices , basically due to its hardness , yet flexibility , high electron mobility and thermal conductivity @xcite . crystal structure of graphene consists in a honeycomb array of tightly packed carbon atoms , thus allowing an accurate tight - binding description . at low energies , such a description becomes in the continuous limit the lagrangian of massless quantum electrodynamics in ( 2 + 1)-dimensions , qed@xmath7 , for the charge carriers restricted to move along the membrane @xcite , but in which the `` photon '' is allowed to move throughout space in such a way that the static coulomb interaction is still described by a potential that varies as the inverse of the distance on the plane of motion of electrons . in this form , low - energy dynamics of graphene is in accordance with the spirit of brane - world scenarios of fundamental interactions ( see , for instance , ref . @xcite ) where the gauge field ( photon ) is allowed to move throughout the bulk ( full space ) , but matter fields are restricted to a brane ( the graphene layer ) . expectedly , quantum field theoretical methods have been developed to describe phenomena in graphene which have been theorized in the high energy physics realm , but that would appear enhanced in this material due to the ratio of the speed of light in vacuum and the fermi velocity of its charge carriers , @xmath8 . theoretical objects like the effective action in external electromagnetic fields have been calculated by several authors in connection with the schwinger mechanism for pair production and the issue of minimal conductivity @xcite , ideas that have been generalized to the multilayer case @xcite . other `` relativistic '' effects discussed in literature include the klein paradox @xcite , casimir effect @xcite and the dynamical formation of a mass gap from excitonic condensates @xcite . graphene properties have been handled also from the perspective of non - conmutative quantum mechanics @xcite . a remarkable feature of graphene is the visual transparency of the membranes . its opacity has been measured @xcite to be roughly 2.3% with almost negligible reflectance . this observation has opened the possibility of using single layers of this crystal in combination with bio - materials to produce clean hydrogen by photocatalysis @xcite with visible light . the problem of light absorption in graphene can be addressed from quantum field theoretical methods @xcite . several authors have considered the dirac picture for its charge carriers in terms of the degrees of freedom of qed@xmath7 under different assumptions . parity violating effects were considered in @xcite , whereas the influence of a strong magnetic field was considered in @xcite in connection with the faraday effect . measurements of magneto - optical properties of epitaxial graphene have been reported in ref . @xcite , in particular the polarization rotation and light absorption . results seem to be in accordance with the `` relativistic '' behavior of charge carriers for a range of values of the external magnetic field intensity between 0.5 - 7 t @xcite . for the discussion of these results , the structure of the vacuum polarization tensor is the cornerstone . this operator has been calculated by several authors in the presence of a strong magnetic field perpendicularly aligned with the graphene membrane @xcite . in this work , we continue the discussion but in our considerations , the external magnetic field is weak in intensity as compared to the effective mass @xmath9 , where @xmath10 and @xmath11 are , respectively , the fermi momentum and fermi velocity of charge carriers . the article is organized as follows : we start modeling the low - energy behavior of graphene from massless qed@xmath7 subjected to an external magnetic field perpendicular to the membrane , namely , we consider the full space , but restrict the dynamics of charge carriers in graphene to an infinite plane where the third spatial component is set to zero . expanding the quasiparticle propagator in the weak field regime , we calculate the vacuum polarization tensor to the leading order in the external field intensity in sect . 2 . in sect . 3 , we introduce the polarization operator in the modified maxwell s equation to describe the propagation of electromagnetic waves in space . from the matching conditions , we calculate the transmission coefficient and from there , the intensity of transmitted light and angle of polarization rotation in terms of the longitudinal and transverse conductivities , which we derive from ohm s law . our results correspond to the weak field faraday effect . we discuss our findings and conclude in sect . some details of the calculation of the polarization tensor are presented in an appendix . tight - binding approach to the description of monolayer graphene corresponds in the continuum to a massless version of quantum electrodynamics in ( 2 + 1 ) dimensions , but with a static coulomb interaction which varies as the inverse of the distance , just as in ordinary space @xcite . we adopt the conventions of refs . @xcite and consider an infinite graphene membrane immersed in a ( 3 + 1)-dimensional space oriented along the plane @xmath12 . the action for this model is expressed as @xmath13 with @xmath14 and @xmath15 . in our considerations , greek indices take the values 0,1,2,3 , and latin indices 0,1,2 , labeling the coordinates of the graphene layer . moreover , the re - scaled dirac matrices are such that @xmath16 , @xmath17 and for later convenience , we also consider the matrix @xmath18 , where @xmath11 is the fermi velocity of quasiparticles in the crystal . in the natural units of the system ( namely , when @xmath19 ) , the form of the action has been dubbed as reduced qed and has been proposed in the context of brane - world scenarios @xcite . measuring the response of graphene to external electromagnetic fields amounts to calculate the effective action , which in turn is expressed through the vacuum polarization tensor @xmath20 . because in this case the dynamics of fermions is restricted to a plane according to fig . [ fig1 ] we can express @xmath21\;,\label{fullvp}\ ] ] where the trace is over full space and then we set @xmath22 here , @xmath23 represents the quasiparticle propagator ( electric charge @xmath24 ) and the double fermion line in the diagram specifies that the propagator is corrected by some classical external field . we consider the situation in which a uniform magnetic field is aligned perpendicularly to the graphene membrane . we think of this field as being weak in intensity , as compared to the the natural scale @xmath9 , where @xmath10 is the quasiparticles fermi momentum such that @xmath25 behaves as an effective dirac mass for the charge carriers . this situation can be formally achieved by considering the quasiparticles with a finite mass gap @xmath25 and then expand the corresponding schwinger propagator in the proper time representation @xcite , @xmath26\,,\label{pts}\end{aligned}\ ] ] in powers of @xmath27 , retaining terms up to order @xmath28 and then letting @xmath29 ) in the proper time representation does not contribute in the vacuum polarization tensor , and thus we neglect it from start . ] . we adopt a prescription where we split the transverse and parallel components with respect to the magnetic field direction of an arbitrary vector @xmath30 defined on the graphene membrane according to @xmath31 such that @xmath32 . any reference to the third spatial component has been taken into account in the @xmath33 integration in eq . ( [ fullvp ] ) and does not appear in what follows . therefore , @xmath34 and @xmath35 . furthermore , we take @xmath36 , such that @xmath37 . thus , in the weak field limit , the structure of the quasiparticle propagator becomes @xcite @xmath38\ , . \label{expansions}\end{aligned}\ ] ] here , the matrices @xmath39 and @xmath40 do not appear rescaled because the operators @xmath41 , with @xmath42 the identity matrix , correspond to the ( pseudo)spin projection operators @xcite . with the above expansion ( [ expansions ] ) , it is straightforward to verify that the structure of the vacuum polarization is @xmath43 \eta_b^\nu\;,\label{fullpi}\ ] ] where we have defined @xmath44 . the first term in the square bracket represents the polarization tensor in vacuum , whereas the second term stands for the quadratic order contribution to the polarization tensor . the linear correction in @xmath45 , @xmath46 , is absent due to the parity preserving property of the model . in other words , contributions to the polarization arising from a chern - simons term are not considered in this work . the magnetic field independent vacuum polarization tensor @xmath47 has been calculated by many authors @xcite . it is of the form @xmath48 with @xmath49 and @xmath50 as usual . moreover , @xmath51 is the magnitude of the momentum vector with components @xmath52 , and the polarization scalar @xmath53 this vacuum contribution is transverse , as demanded by gauge invariance . on the other hand , the quadratic correction has two contributions , @xmath54 \nonumber\\ & & + 2 \int \frac{d^3k}{(2\pi)^3}{\rm tr}[\tilde{\gamma}^a s_2(k ) \tilde{\gamma}^b s_0(k+p)]\;,\label{pi1120}\end{aligned}\ ] ] with a suggestive notation that the @xmath55 contributions comes from each of the quasiparticle propagators being dressed at the first order in the external field , whereas @xmath56 has one propagator without field , whereas the second one is dressed at order @xmath1 . the factor of 2 is a symmetry factor . evaluation of these integrals is cumbersome , but straightforward . our procedure was the following , we have started by inserting the expansion in eq . ( [ expansions ] ) into each of the contributions to the polarization tensor in eq . ( [ pi1120 ] ) . then , with the aid of the identity @xmath57^{p+1}}\;,\ ] ] followed by the shift of variables @xmath58 , after taking the traces over full space and performing the remaining contractions , we obtain @xmath59\ ; , \nonumber\\ \pi^{a b}_{(2)-20}&=&\frac{4i\tilde{\alpha}}{\pi^3}\bigg[\left(g_\parallel^{ab}-g_\perp^{ab}\right)\left(i^{03}_{115}(\tilde{p } ) + \tilde{p}_\perp^2 i^{23}_{105}(\tilde{p})\right)\nonumber\\ & & + g_\parallel^{ab}\left ( i^{03}_{115}(\tilde{p})+p_\parallel^2i^{23}_{015}(\tilde{p})\right ) \nonumber\\ & & -\left(\tilde{p}_\parallel^a \tilde{p}^b + \tilde{p}_\parallel^b \tilde{p}^a-\tilde{p}_\parallel^2g^{ab}\right)\left(i^{14}_{015}(\tilde{p})+\tilde{p}_\perp^2 i^{23}_{005}(\tilde{p})\right)\nonumber\\ & & + \left(\tilde{p}_\perp^a \tilde{p}^b + \tilde{p}_\perp^b \tilde{p}^a-\tilde{p}_\perp^2g^{ab}\right ) \left(i^{14}_{105}(\tilde{p})+\tilde{p}_\parallel^2 i^{23}_{005}(\tilde{p})\right)\nonumber\\ & & + \left(\tilde{p}_\perp^a \tilde{p}^b_\parallel + \tilde{p}_\parallel^a \tilde{p}^b_\perp\right ) \left(i^{23}_{015}(\tilde{p})-2i^{23}_{105}(\tilde{p } ) \right)\bigg]\;,\end{aligned}\ ] ] where the master integral @xmath60^{\rm r}}\;,\nonumber\\ & = & ( -1)^{\rm m+n - r}\frac{i\pi}{(\tilde{p}^2)^{{\rm r - m - n}-3/2 } } b\left({\rm n}+1,{\rm r - n}-1\right)\nonumber\\ & & \hspace{-34mm}\times b\left({\rm m}+\frac{1}{2},{\rm r - m - n}-\frac{3}{2 } \right ) b\left(\rm{f - r - m - n}+\frac{5}{2},{\rm g - r - m - n}+\frac{5}{2 } \right)\ ; , \label{master}\end{aligned}\ ] ] is written in terms of beta functions @xmath61 and whose explicit evaluation is presented in the appendix . making use of the master integral , the quadratic correction in the external field to the polarization tensor can be written as @xmath62 \eta_b^\nu\ , , \label{pi2}\ ] ] with the transverse tensors @xmath63 and the polarization scalars @xmath64 thus , the final expression for @xmath65 becomes @xmath66\eta_b^\nu \;.\nonumber\\ \label{final}\end{aligned}\ ] ] the above result , eq . ( [ final ] ) , comprises the main result of this section and is the basis for our discussion below . before proceeding , a few comments are at hand : * @xmath67 is a transverse tensor order by order in @xmath45 . this fact justifies that our procedure to include the influence of the external magnetic field by means of expansion of the proper time representation of the quasiparticle propagator preserves gauge invariance . * our procedure is an alternative to the traditional approach in which the vacuum polarization tensor is expressed as a double proper time integral @xcite . in fact , for the particular case of qed in ( 2 + 1)-dimensions considered in ref . @xcite , the weak field expansion of the polarization scalars , eqs . ( 48)-(50 ) of that reference , match our findings in the massless limit , when we set @xmath19 . we shall use the expressions for @xmath20 developed in this section to discuss the problem of light absorption in graphene . from the action of our model , eq . ( [ action ] ) , we can describe the propagation of electromagnetic waves throughout space according to the modified maxwell s equations @xmath68 which fulfill the conditions @xmath69 following refs . @xcite , we interpret the delta function in eq . ( [ maxwell ] ) as a current along the graphene plane . thus , from ohm s law , @xmath70 where the indices @xmath71 take the values 1 and 2 , emphazising that they refer to the spatial coordinates of the graphene membrane . assuming a varying electric field with frequency @xmath3 expressed in a temporal gauge @xmath72 , namely , @xmath73 and noticing , from the generalized maxwell s equations ( [ maxwell ] ) that @xmath74 , we can identify the transverse conductivity as @xmath75 for the problem of light absorption , let us consider a plane wave of frequency @xmath3 , which travels along the @xmath76-direction from below the graphene layer with a linear polarization along the @xmath77 direction . these assumptions allow us to write @xcite @xmath78 where @xmath79 is the levi - civita symbol and @xmath4 , @xmath5 represent the longitudinal and transverse conductivities . moreover , considering that the wave insides on the graphene plane , the reflected and transmitted waves can be described as @xmath80 where @xmath81 are the unit vectors along the directions @xmath82 and @xmath83 on the membrane . thus , from the general form of the vacuum polarization tensor , eq . ( [ inverse ] ) , the boundary conditions ( [ bc ] ) simplify to @xmath84 where @xmath85\;.\ ] ] thus , the transmission coefficients can be straightforwardly obtained @xcite @xmath86 with @xmath87 , accounting for the degrees of freedom of charge carriers . therefore , the intensity of transmitted light is @xmath88 in terms of the conductivity tensor @xmath89 , @xmath90 and the angle of polarization rotation can be expressed as @xmath91 substituting the explicit form of the polarization scalars , we finally arrive at the main results of this article , namely , @xmath92 and angle of polarization rotation @xmath93 as a function of the incoming electromagnetic wave frequency @xmath3 ( in arbitrary units ) for different values of the external magnetic field , also in arbitrary units , but preserving the weakness of the intensity of our approximation . solid red curve corresponds to the case of @xmath94 in this set of arbitrary units , dot - dashed black curve , @xmath95 , short - dashed blue curve , @xmath96 and long - dashed purple curve , @xmath97.,title="fig:",scaledwidth=40.0% ] and angle of polarization rotation @xmath93 as a function of the incoming electromagnetic wave frequency @xmath3 ( in arbitrary units ) for different values of the external magnetic field , also in arbitrary units , but preserving the weakness of the intensity of our approximation . solid red curve corresponds to the case of @xmath94 in this set of arbitrary units , dot - dashed black curve , @xmath95 , short - dashed blue curve , @xmath96 and long - dashed purple curve , @xmath97.,title="fig:",scaledwidth=40.0% ] these quantities are plotted in the left and right panel , respectively , of fig . [ fig2 ] as a function of the frequency of incident light @xmath3 for several values of the external magnetic field . comparing with the measured universal absorption rate @xmath98 @xcite , we conclude that in the weak field limit , the intensity of transmitted light and angle of polarization rotation get corrected by factors @xmath6 , in consistency with the experimental and theoretical findings for these quantities in absence of external fields as well as in and the presence of a strong magnetic field @xcite . in this work , we have calculated the vacuum polarization tensor in a low energy effective model of graphene based on massless qed@xmath7 . we have considered a uniform magnetic field aligned perpendicularly to the graphene membrane and expanded the charge carrier propagator in the weak field regime , as compared to the effective mass @xmath9 of the quasiparticles . we have considered the explicit limit @xmath29 . the passarino veltman - type of integrals involved in the calculation of the polarization operator were obtained after a lengthy , but straightforward procedure from a single master integral that yields a transverse @xmath20 , eq . ( [ final ] ) , in every order of expansion on the intensity of the external field . one piece of this object is inherited from the form of the polarization tensor in vacuum and receives a leading correction of order @xmath1 , whereas the second piece is transverse in the coordinates on the graphene membrane and vanishes in the absence of the field . direct calculation not always renders a manifestly transverse polarization operator @xcite , for instance , in ordinary qed . spurious terms might arise as a consequence of a regularization procedure . nevertheless , careful treatment of the regulators ensure gauge invariance is preserved for arbitrary magnetic field strength . qed@xmath7 being superrenormalizable , lacks of uv - regularization issues . nevertheless , we have presented an alternative calculation to the standard representation of the polarization tensor as a double proper time integral @xcite , which manifestly preserves gauge invariance . as an application of the vacuum polarization tensor , we have estimated the light absorption in graphene and the angle of rotation of polarization of light passing through a membrane of this material . we observe a deviation of the form @xmath6 as compared to the vacuum result for graphene opacity . the same behavior is observed for the angle of polarization rotation . our findings are in agreement with previously reported theoretical calculations @xcite as well as the experimental light absorption of @xmath99 per graphene membrane @xcite . further applications of the polarization tensor presented here and the effective action derived from it are under scrutiny and will be presented elsewhere . we acknowledge valuable discussions from cristin villavicencio , ngel snchez and mara elena tejeda . ar and sho acknowledge conacyt ( mxico ) for financial support for sabbatical and short visit at puc , respectively and cic - umsnh under grant no . 4.22 as well as the hospitality of puc , where the main part of this work was carried out . ml acknowledges final support from fondecyt ( chile ) grants nos . 1130056 and 1120770 . dv acknowledges support from conicyt ( chile ) . in this appendix , we compute the master integral in eq . ( [ master ] ) . for this purpose , we write @xmath100 with @xmath101^{\rm r}}\;.\ ] ] after wick rotating to euclidean space , writing @xmath102 and with the aid of the identity @xmath103 we immediately obtain @xmath104^{{\rm r - m - n}-3/2 } } \;.\end{aligned}\ ] ] then , the remaining integral over @xmath82 in eq . ( [ master2 ] ) can be performed from the definition of the beta function @xmath105 which finally lead us to the result ( [ master ] ) . 00 novoselov k s et al 2005 nature 438 197 . gusynin v p and sharapov s g 2005 phys . 95 146801 . zhang y et al 2005 nature 438 20 . gusynin v p , sharapov s g and carbotte j p 2007 , _ int . b _ * 21 * 4611 . geim a k and novoselov k s 2007 , _ nature mat . _ * 6 * , 183 . savage n 2012 , _ nature _ * 483 * , s30 . gorbar e v , gusynin v p and miransky v a 2001 , _ phys . d _ * 64 * 105028 ; teber s 2012 , _ phys . d _ * 86 * 025005 ; kotikov a v and teber s 2013 , _ phys . d _ * 87 * 087701 ; kotikov a v and teber s 2014 , _ phys . * 89 * 065038 ; teber s 2014 , _ phys . * 89 * 067702 . beneventano c g , giaconni p , santangelo e m and soldati r 2007 , _ j. phys . a _ * 40 * , f35 . katsnelson m i , volovik g e and zubkov m a 2013 , _ ann . phys . _ * 331 * , 160 . katsnelson m i , geim a k and novoselov k s 2006 , _ nature phys . _ * 2 * , 620 dobson j f , white a and rubio a 2006 , _ phys . lett . _ * 96 * , 073201 ; gmez - santos g 2009 , _ phys . b _ * 80 * , 245424 ; sernelius b e 2011 , _ eur . phys . lett . _ * 95 * , 57003 ; sarabadani j , naji a , asgari r and podgornik r 2011 , _ phys . b _ * 84 * , 155407 . khsevshenko d v 2009 , _ j. phys cond . * 21 * , 075303 ; sabio j , sols f y guinea f 2010 , _ phys . b _ * 82 * 121413 ; gonzalez j 2010 , _ phys . b _ * 82 * , 155404 ; gamayun o v , gorbar e v and gusynin v i 2010 , _ phys . rev . b _ * 81 * 075429 ; wang j - r and lui g z 2011 , _ j. phys * 23 * , 155602 ; wang j - r and lui g z 2011 , _ j. phys . * 23 * , 345601 . falomir h , gamboa j , loewe m , and nieto m 2012 , _ j. physics a _ * 45 * , 135308 . nair r r , blake p , grigorenko a n , novoselov k s , booth t j , stauber t , peres n m r and a. k. geim a k 2008 , _ science _ * 320 * , 1308 . wang p et al 2014 , acs - nano 8 , 7995 . fialkovsky i and vassilevich d v 2012 , _ int . * 27 * 1260007 ; fialkovsky i and vassilevich d v 2012 , int . * 14 * 88 - 99 fialkovsky i and vassilevich d v 2009 , _ j. phys . a _ * 42 * 422001 fialkovsky i and vassilevich d v 2012 , _ eur . j. b _ * 85 * 384 . grassee i , levallois j , walter a l , ostler m , bostwick a , rotenberg e , seyller t , van der marel d and kuzmenko a b 2011 , _ nat . phys . _ * 7 * , 48 . gorbar e v , gusynin v p , miransky v a and shovkovy i a 2002 , _ phys . b _ * 66 * , 045108 ; gusynin v p and sharapov s g 2006 , _ phys . rev . b _ * 73 * , 245411 ; gusynin v p , sharapov s g , carbotte j p 2007 , _ j. phys . condens . matter _ * 19 * , 026222 ; gusynin v p , sharapov s g , j.p . carbotte j p 2009 , _ new j. phys . _ * 11 * , 095013 ; pyatkovskiy p k 2009 , _ j. phys . : condens * 21 * , 025506 ; pyatkovskiy p k and gusynin v p 2011 , _ phys . rev . b _ * 83 * , 075422 ; schwinger j s 1951 , _ phys . rev . _ * 82 * 664 . chyi t - k , hwang c - w , kao w - f , lin g - l , ng k - w and tseng j - j 2000 , _ phys . d _ * 62 * , 105014 . appelquist t w , bowick m j , karabali d , wijewardhana l c r 1986 , _ phys . d _ * 33 * 3704 . dittrich w and reuter m 1985 , _ `` effective lagrangians in quantum electrodynamics '' _ , springer - verlag , berlin ; dittrich w and gies h 2000 , _ `` probing the quantum vacuum '' _ , springer - verlag , berlin - heidelberg . schubert c and varlamov v 2013 , _ math . meth . appl . sci . _ * 34 * , 1638 . shpagin a v 1996 , _ dynamical mass generation in ( 2 + 1 ) dimensional electrodynamics in an external magnetic field _ , preprint hep - ph/9611412 ( unpublished ) . chao j , yu l and huang m 2014 , _ phys . d _ * 90 * , 045033 .

we carry out an explicit calculation of the vacuum polarization tensor for an effective low - energy model of monolayer graphene in the presence of a weak magnetic field of intensity @xmath0 perpendicularly aligned to the membrane . by expanding the quasiparticle propagator in the schwinger proper time representation up to order @xmath1 , where @xmath2 is the unit charge , we find an explicitly transverse tensor , consistent with gauge invariance . furthermore , assuming that graphene is radiated with monochromatic light of frequency @xmath3 along the external field direction , from the modified maxwell s equations we derive the intensity of transmitted light and the angle of polarization rotation in terms of the longitudinal ( @xmath4 ) and transverse ( @xmath5 ) conductivities . corrections to these quantities , both calculated and measured , are of order @xmath6 . our findings generalize and complement previously known results reported in literature regarding the light absorption problem in graphene from the experimental and theoretical points of view , with and without external magnetic fields .

1410.5501

entanglement generation between two atomic qubits has attracted considerable attention during the last two decades due to its importance in various quantum information processes @xcite . those ideal processes , such as quantum teleportation , quantum cryptography , and quantum computation algorithms are strongly related to the capability of generating bipartite entanglement @xcite . however , in real quantum systems there are uncontrollable interactions with the surrounding environment which usually lead to a decoherence resulting in the destruction of the entanglement . recently , several effects of different kinds of noisy environments , specifically bosonic environment @xcite , and fermionic environment @xcite on the entanglement dynamics have been extensively studied . especial effort was applied to find decoherence free entangled states @xcite . for instance , b. kraus and j. i. cirac @xcite show that two atoms can get entangled by interacting with a common source of squeezed light and the steady state is maximally entangled even though the modes are subjected to cavity losses . s. b. zheng and g. c. guo @xcite proposed a scheme to generate two - atom epr states in such a way that the cavity is only virtually excited . s. schneider and g. j. milburn @xcite show how the steady state of a dissipative many - body system , driven far from equilibrium , may exhibit nonzero quantum entanglement . molmer and sorensen have proposed a scheme for the generation of multiparticle entangled states in ion traps without the control of the ion motion . although the effect of the environment on the atomic entanglement is usually destructive , in some specific situations two quantum systems can get entangled in the process of their decaying to a common thermal bath @xcite . a similar effect was discussed in @xcite where a method of generation of entangled light from a noisy field has been proposed . it was also shown @xcite that the interaction between two spins and an itinerant electron environment leads to entanglement of the initially unentangled spins . in this article we study how an effective atomic environment modifies the atomic entanglement generated in the course of resonant interaction of a single mode of the cavity field with a couple of two - level atoms ( the so - called dicke or tavis - cummings @xcite model ) . evolution of entanglement in the two - atom dicke model was previously studied in the case of an ideal cavity in @xcite and in the presence of a dissipative environment in @xcite . our study is motivated by the following physical situation : consider a cluster of two - level atoms ( resonant with a mode of a cavity field ) placed in a strong electric field ( see e.g. @xcite ) . physically it could be a cluster of polar moleculae . the electric field generates a noticeable stark shift so that most of the atoms are detuned far from the resonance , except a very small portion of them , whose dipole moments are approximately orthogonal to the field . because the atom-(quantum ) field interaction times are much shorter than the typical times of atomic diffusion , we can consider that the orientation of the dipole moment is frozenand that the physical mechanism of changing the atomic dipole orientation is a collision with the cavity walls , since collisions between the atoms in an atomic cluster are practically improbable . in the process of interaction with the cavity field the resonant atoms become entangled . we will study the simplest situation where there are only two resonant atoms . nevertheless , the effect of non - resonant atoms on the dynamics of resonant ones is not trivial . the dispersive interaction of the field mode with non - resonant atoms leads to a modification of the field s phase which , in turn , affects the evolution of resonant atoms . thus , the non - resonant atoms play the role of an effective dispersive environment whose whole effect could be expected to reduce to a phase dumping @xcite , and thus to the entanglement decaying . nevertheless , as it will be shown , the influence of such effective environment is not always destructive but also leads to a constructive interference , which reflects in , appearance of a system of collapses and revivals of the atomic concurrence even in the presence of just a single photon in a cavity . the article is organized as follows : in section [ limit ] we analytically show , for some specific initial conditions ( non - excited atoms and the field in a fock state ) , that the entanglement of formation in a bipartite system of two - level atoms interacting with a quantized mode reaches its maximum value when only one excitation is involved and it decays as @xmath0 when @xmath1 , being @xmath2 the number of photons in the initial fock field state . in section [ dickeaat ] we derive the effective hamiltonian of noninteracting two - level ( resonant ) atoms and a cluster of @xmath3 atoms ( far from resonance ) interacting simultaneously with a quantized mode and we find the evolution operator when only one excitation is considered . in section [ eoff ] we study the effect of the dispersive atomic environment on the entanglement dynamics generated by one excitation for two different initial conditions . in section [ conclusions ] we summarize our results . by entanglement of two subsystems we mean the quantum mechanics feature whose state can not be written as a mixed sum of products of the states of each the the subsystems . in this case the entangled subsystems are no longer independent even if they are spatially far separated . a measure , @xmath4 , of the degree of entanglement for a pure @xmath5 state of a bipartite system can be given by means of the entropy of von neumann , of any of the two subsystems . for a mixed state @xmath6 the entanglement of formation @xmath7 between two bidimensional systems is defined as the infimum of the average entanglement over all possible pure - state ensemble decompositions of @xmath6 @xcite . wootters found an analytic solution to this minimization procedure in terms of the eigenvalues of the @xmath8 or @xmath9 non - hermitian operators , where the tilde denotes the spin flip of the quantum state . the solution for the @xmath10 concurrence associated with the entanglement of formation of a mixed state of a bipartite of bidimensional subsystems is given by @xmath11 , where the @xmath12 s are the square roots of eigenvalues of the @xmath13 operator and the eigenvalues of the @xmath14 operator , decreasingly ordered . throughout this article we consider this @xmath10 as a measure of the entanglement degree between the two resonance atoms @xmath15 and @xmath16 . let us consider two identical two - level atoms resonantly interacting with a single - mode cavity field . the interaction hamiltonian has the form @xmath17 where @xmath18 and @xmath19 , with @xmath20 and @xmath21 being the excited and ground eigenstates of @xmath22 of the @xmath23th atom ( @xmath24 ) , @xmath15 and @xmath25 are , respectively , the creation and annihilation operators for the cavity mode , @xmath26 is the atom - cavity coupling strength . considering initially a fock field state and both atoms in their ground states , the reduced atomic density operator , at time @xmath27 , is @xmath28 ^ 2}{(2n-1)^{2}}|0\rangle_a|0\rangle_{b\hspace{0.03in}a}\langle 0|_b\langle 0| + \frac{ns_n^2(t)}{2n-1}|\psi^+\rangle _ { ab\hspace{0.03in}ab}\langle\psi^{+}| + \frac{n ( n-1)[1-c_n(t)]^2 } { ( 2n-1)^{2}}|1\rangle _ { a}|1\rangle_{b\hspace{0.03in}a}\langle 1|_{b}\langle 1| , \label{rabb}\ ] ] where we have defined the functions : @xmath29 and the symmetric state : @xmath30 so , the @xmath31 concurrence of the ( [ rabb ] ) density operator is given by @xmath32 when it is positive and is zero otherwise . both terms on the right side hand of eq . ( [ concurrence ] ) are zero for @xmath33 whereas the second term is also zero for @xmath34 and , for other values of @xmath2 , the second term always reduce the concurrence . therefore , as a function of the number @xmath2 of excitations , the concurrence ( [ concurrence ] ) acquires its maximum value for @xmath34 at any time instant and it is given by @xmath35 . on the other hand , for @xmath1 the concurrence ( [ concurrence ] ) behaves as @xmath36 the behavior of @xmath37 in the limit @xmath1 was found numerically by tessier _ figure [ fig0 ] shows the concurrence as a function of the @xmath2 initial fock state , and the @xmath38 adimensional time . black means value @xmath39 , maximum entanglement , white means value zero , whereas greys mean partial values of entanglement . it can be seen that maximun value @xmath39 is only reached for the initial condition @xmath40 . [ t ] ) of the concurrence for initially unexcited @xmath15 and @xmath16 atoms and the field in a @xmath2 fock state . black means value @xmath39 , maximum entanglement , white means value zero , and greys mean partial values of entanglement.,title="fig:",scaledwidth=40.0% ] in the next section we study how the concurrence is affected by the presence of an effective atomic environment when only one excitation is involved . we consider a collection of @xmath41 non - identical two - level atoms interacting with a single mode of a quantized field in an ideal cavity . two atoms , labelled by subindexes @xmath15 and @xmath16 , are resonant with the mode whereas the other @xmath3 atoms interact dispersively with the mode . the hamiltonian which drives the unitary dynamics of the whole system under the rotating wave approximation has the form @xmath42 where @xmath43 and @xmath15 are the usual one mode field operators , and @xmath44 are the @xmath45 components of the pauli operators corresponding to the @xmath46th two - level atom ( @xmath47 ) . atomic operators obey the standard @xmath48 commutation relations , @xmath49 = 2s_{zi}\delta_{ij}$ ] and @xmath50 = \pm s_{\pm i}\delta_{ij}$ ] . since the total number of excitations , represented by the operator @xmath51 , is an integral of motion , the above hamiltonian can be rewritten as follows : @xmath52 with @xmath53 where @xmath54 , @xmath55 are the detunings between the transition of the @xmath56th atom and the mode frequency . now , we assume that all @xmath3 atoms are far from the resonance , so that @xmath57 , @xmath58 . the effective hamiltonian , approximately describing the interaction process , can be obtained from the interaction hamiltonian ( [ hint ] ) by using the method of lie rotations @xcite , namely by applying to the hamiltonian ( [ h ] ) the following unitary transformation @xmath59 where @xmath60 . neglecting terms of order higher than @xmath61 , we obtain the following effective hamiltonian : @xmath62 where we have defined the operator @xmath63 . the last term in ( [ heff1 ] ) represents an effective dipolar interaction between the non - resonant atoms , and its contribution to the system dynamics strongly depends on the internal resonance condition between atomic frequencies . let us consider randomly distributed frequencies , such that they satisfy the condition @xmath64 , @xmath65 . then , the terms @xmath66 in the last sum of the effective hamiltonian ( [ heff1 ] ) rapidly oscillate and can be neglected . finally , the effective hamiltonian , up to a constant energy shift , becomes : @xmath67 in the given approximation the total number of excitations storedin the non - resonant atoms is a constant of motion , which reflects a dispersive character of interaction . the first term in the above equation represents just transition frequency shifts of the non - resonant atoms , and commutes with the rest of the terms ( so , it can be taken out of the hamiltonian ) . the second term is the dynamic stark shift and its contribution to the resonant dynamics , described by the last term , strongly depends on the state of the non - resonant atoms , which can be considered as a kind of atomic environment . since the maximum entanglement in the system of two resonant atoms is reached when the total number of excitation is one , @xmath68 , we consider exclusively this situation . so , under the constraint that there is only one photon , the corresponding evolution operator can be found and , in the standard tensor product basis , it is given by @xmath69 , \label{eo}\ ] ] where we have defined the operators : @xmath70 and the rabi frequencies @xmath71 depend on the field and on the environment variables as follows : @xmath72 ^{1/2},\quad \frac{\hat{y}}{2 } = \sum_{j=1}^{a}g_{j}\epsilon _ { j}s_{zj}. \nonumber\ ] ] so , the dynamics depends on the distribution of the different rabi frequencies which appear as a contribution of @xmath3 non - resonance distinguishable two - level atoms and it also depends on the initial state of the whole system . now , let us suppose that the environment atoms are prepared in a coherent superposition of excited and ground states : @xmath73 where the last sum is taken over all possible binary vectors @xmath74 , @xmath75 . we capture the key features of the concurrence evolution by considering two particular cases of initial conditions . first , we suppose that the resonance atoms are initially in the ground state and that the field is in the one photon fock state : @xmath76 applying the evolution operator ( [ eo ] ) to the ( [ ini1 ] ) state and tracing up over the field and the off - resonance atomic environment , we obtain for the resonant atoms the following reduced density operator : @xmath77 with @xmath78 where @xmath79 is a certain arrangement of the @xmath80 vector and @xmath81 . we have neglected order corrections higher than or equal to @xmath82 on the amplitudes . from now on the coupling constants for field - environment atoms are taken to be equal , @xmath83 for all @xmath23 . and @xmath16 atoms and the field in the one - photon fock state , with @xmath84 , @xmath85 , and @xmath86.,scaledwidth=40.0% ] the concurrence corresponding to the density matrix ( [ dens1 ] ) takes the form @xmath87 if the resonant atoms are initially prepared in the symmetric one excitation state and the field is in the vacuum fock state , then the initial state of the whole system is the following tensor product : @xmath88 ) the concurrence takes the form @xmath89 it is worth noting that the concurrences ( [ c1],[c2 ] ) are composed of many rabi frequencies , which leads to a structure similar to the collapses and revivals in the jaynes - cummings model @xcite . nevertheless , in the present case the set of different frequencies is due to the presence of the dispersive atomic environment in contrast to the standard jcm where different rabi frequencies appear as contributions of different fock field states ( recall that a well - defined collapse - revival structure requires a significant number of excitations @xcite ) . let us consider randomly distributed numbers @xmath90 , @xmath91 with the mean @xmath92 and the standard deviation @xmath93 . then , the numbers @xmath94 , @xmath61 have the following mean values and standard deviations : @xmath95 first , we will find the distribution of the ( [ q ] ) quantities . we assume that @xmath96 and @xmath97 . then , there are @xmath98 peaks corresponding to different values of the number of positive components of @xmath80 , @xmath99 ; for a given value of @xmath56 there are @xmath100 $ ] values of @xmath101 and @xmath102 which are normally distributed in accordance with the central limit theorem . for the @xmath56th peak , the mean value and the standard deviation are given by @xmath103 note that the first and the last peaks are infinitely narrow . the rabi frequency distribution has @xmath98 peaks ( now the summation in ( [ c1 ] , [ c2 ] ) is from @xmath104 to @xmath3 ) , and the frequency corresponding to the @xmath56th peak can be approximated as follows : @xmath105 . \nonumber\ ] ] thus , the expressions ( [ c1 ] ) and ( [ c2 ] ) for the concurrence can be approximated as follows : @xmath106 . \label{cc}\ ] ] the @xmath107 separation between the @xmath56th and the @xmath108th peaks , and the @xmath109 width of the @xmath56th peak are @xmath110 , \nonumber \\ \delta_{k } & = & \sqrt{2}g[1+(k-\frac{a}{2})+\frac{\tilde{g } ^{2}}{4g^{2}}(k-\frac{a}{2})^{2 } ] \sigma_{\epsilon^{2},k}. \nonumber\end{aligned}\ ] ] then , considering the approximation of narrow peaks , @xmath111 , i.e. @xmath112 , the sum in ( [ cc ] ) can be represented as a sum of gaussians , that is @xmath113 with the width @xmath114 which grows with time . the ( [ c22 ] ) sum reveals the collapse - revival structure of the concurrences ( [ c1 ] ) and ( [ c2 ] ) . the first collapse happens when @xmath115 and it is followed by revival at time : @xmath116 in figure [ fig1 ] we show the exact evolution of the concurrence for initially unexcited atoms and the field in the one - photon fock state in the presence of the environment atoms . one can observe that the entanglement also reaches its maximum value . we can estimate from ( [ tcol ] ) and ( [ trev ] ) the time scale required to observe the environment induced collapse - revival structure . taking the typical values of the interaction constant from @xcite : @xmath117 and @xmath118 , we obtain @xmath119 which is of order of the passage time of the atom through the cavity ( cold atoms , @xmath120 ) and less than the photon lifetime @xmath121 . the collapse time is @xmath122 times less than @xmath123 . in summary , we have studied the dynamics of the concurrence of two atoms resonantly interacting with a cavity mode in the presence of many off - resonance atoms . we have shown that , for random distribution of atomic detunings and initially symmetric excitation of non - resonant atoms , the coherent influence of the environment can be separated from the dephasing . the coherence influence of the environment reflects in the appearance of the collapse - revival structure of the concurrence , with an average value of one half . this behavior is induced only by the presence of the dispersive environment . it is worth noting that the entanglement of formation in revival periods has as extreme value @xmath39 , which means that the state of the bipartite system becomes pure at those times . 99 p. w. shor , phys . a * 52 * , 2493 ( 1995 ) ; d. p. divicenzo , science * 270 * , 255 ( 1995 ) ; l. k. grover , phys . rev . lett . * 79 * , 325 ( 1997 ) ; j. i. cirac , p. zoller , nature * 404 * , 579 ( 2000 ) . m. a. nielsen and i. l. chuang , _ quantum computation and quantum information _ ( cambridge university press , cambridge , 2000 ) ; g. alber _ et al . _ , _ quantum information _ ( springer , berlin , 2001 ) ; d. bouwmeester , a. ekert , and a. zeilinger , _ the physics of quantum information _ ( spring - 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we study entanglement dynamics of a couple of two - level atoms resonantly interacting with a cavity mode and embedded in a dispersive atomic environment . we show that in the absence of the environment the entanglement reaches its maximum value when only one exitation is involved . then , we find that the atomic environment modifies that entanglement dynamics and induces a typical collapse - revival structure even for an initial one photon fock state of the field .

quant-ph0509201

by comparing observations of galaxies in clusters at @xmath7 with those in clusters at @xmath5 , we infer that some environmental effects in clusters have influences on the evolution of the galaxies . @xcite found that clusters at @xmath7 have a high fraction of blue galaxies in comparison with nearby clusters , and subsequent works have confirmed this trend ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? recent observations with the _ hubble space telescope _ ( hst ) revealed details of the blue galaxies . @xcite and @xcite found that most of the blue galaxies are normal spirals with active star formation . on the other hand , observations have shown that the fraction of so galaxies decreases rapidly with redshift in contrast with the normal spirals @xcite . these suggest that the blue normal spirals observed in high redshift clusters evolve into the non - blue so galaxies observed in nearby clusters . in fact , observations show that in distant clusters there are galaxies in post - starburst phase.@xcite . these galaxies may be the ones for which star formation activity is dying down . several mechanisms are proposed that can lead to the color and morphological transformations between galaxy classes in clusters , such as galaxy mergers @xcite , tides by the cluster potential @xcite , and tidal interactions between galaxies @xcite . one of the strongest candidates is the ram - pressure stripping proposed by @xcite . if a cluster galaxy moves through hot intracluster medium ( icm ) , it feels ram - pressure from the icm , which is given by @xmath8 , where @xmath9 is the icm density and @xmath10 is the relative velocity between the galaxy and the icm . if the ram - pressure becomes large enough , the interstellar medium ( ism ) of the galaxy can not be hold by the gravity of the galaxy and is swept away . numerical simulations demonstrate that in a cluster environment , the stripping is likely to occur @xcite . in particular , high - resolution three dimensional numerical simulations show that the ram - pressure stripping is so effective that it removes 100% of the atomic hydrogen content of luminous galaxies within @xmath11 yr @xcite . on the other hand , @xcite investigated the influence of ram - pressure on the star formation activity of a disk galaxy . they found that just before the atomic hydrogen content is stripped , the star formation rate increases at most a factor of 2 , but rapidly decreases on a timescale of @xmath11 yr after the stripping . after the star formation activity , which mainly occurred in the disk , ceases , the galaxy looks like a s0 galaxy in both color and morphology @xcite . hi deficient galaxies and galaxies with no strong emission - lines seen in cluster cores support the theoretical predictions ( e.g. * ? ? ? * ; * ? ? ? although ram - pressure stripping alone does not explain the detailed morphological features of s0 galaxies , such as their large bulge to disk ratios or their conspicuous thick disks @xcite , it may be a principal mechanism of the transformation of spirals with active star formation into s0 galaxies with inactive star formation . however , most of the previous studies dealt with the ram - pressure stripping of a model galaxy in a given cluster with arbitrary initial conditions . moreover , they did not take the evolution of cluster structure into account ; as will be seen in [ sec : model ] , the structure of high - redshift clusters is different from that of nearby clusters even for the same mass . since it affects the icm density , the velocity of the galaxies , and the efficiency of ram - pressure stripping , it must be considered when we compare the theoretical models with observations of high - redshift clusters . in this paper , we investigate ram - pressure stripping in clusters at various redshifts , which grow according to a hierarchical clustering scenario ; the initial position and velocity of galaxies are given by a spherical collapse model of cluster formation . moreover , since a cluster breaks into smaller progenitors as time goes backwards , galaxies in the cluster might have been affected by ram - pressure stripping when they were in the progenitors before the present - day cluster formed . thus , we also consider the ram - pressure stripping of galaxies in these progenitors . since ram - pressure is proportional to the density of icm , the icm distribution of a cluster may be related to the evolution of the cluster galaxies feeling ram - pressure . x - ray observations of nearby clusters show that their icm distributions are generally different from their dark matter distributions . in particular , for low temperature clusters , the distributions of icm are significantly flatter than those of dark matter and the icm fraction in their central regions is small @xcite . a possible explanation of the icm distributions is that the icm has been heated non - gravitationally . in fact , @xcite indicated that the entropy of the icm in the central regions of low - temperature or less massive clusters is higher than can be explained by gravitational collapse alone , although it is not understood what heats the icm ( e.g. supernovae or agn ) and where the icm is heated , that is , outside or inside clusters . heating other than the gravity of a cluster makes the icm distribution flatter and different from the dark matter distribution . thus , we expect that the position where a galaxy suffers from ram - pressure stripping depends on whether the icm of the cluster ( or the gas accreted by the cluster later on ) has been heated non - gravitationally or not . in particular , we expect that the position where ram - pressure stripping occurs is more sensitive to the non - gravitational heating in the past . this is because a cluster breaks into progenitors or less massive clusters as time goes backwards , and because the heat required to explain the observations should have more influence on the icm distributions of the less massive clusters @xcite . therefore , ram - pressure stripping in the progenitors may tell us when the icm of present - day clusters was heated non - gravitationally ; it will be a clue to the heating sources . our paper is organized as follows . in [ sec : model ] we describe our models of the dark matter distribution and the icm distribution within clusters , the ram - pressure stripping of a radially infalling galaxy , and the evolution of cluster progenitors . in [ sec : result ] we give the results of our calculations . we compare them with observations in [ sec : disc ] . conclusions are given in [ sec : conc ] . the virial radius of a cluster with virial mass @xmath12 is defined as @xmath13 where @xmath14 is the critical density of the universe and @xmath15 is the ratio of the average density of the cluster to the critical density at redshift @xmath16 . the former is given by @xmath17 where @xmath18 is the critical density at @xmath19 , and @xmath20 is the cosmological density parameter . the latter is given by @xmath21 for the einstein - de sitter universe and @xmath22 for the flat universe with non - zero cosmological constant @xcite . in equation ( [ eq : dc_lam ] ) , the parameter @xmath23 is given by @xmath24 . we assume that a cluster is spherically symmetric and the density distribution of gravitational matter is @xmath25 where @xmath26 and @xmath27 are constants , and @xmath28 is the distance from the cluster center . the normalization , @xmath26 , is given by @xmath29 we choose @xmath30 , because the slope is consistent with observations @xcite . moreover , the results of numerical simulations show that the mass distribution in the outer region of clusters is approximately given by equation ( [ eq : rho_m ] ) with @xmath31 @xcite . we consider two icm mass distributions . one follows equation ( [ eq : rho_m ] ) except for the normalization and the core structure ; @xmath32^{-\alpha/2 } } { [ 1+(r_{\rm vir}/r_{\rm c})^2]^{-\alpha/2}}\:.\ ] ] the icm mass within the virial radius of a cluster is @xmath33 the normalization @xmath34 is determined by the relation @xmath35 . where @xmath36 is the gas or baryon fraction of the universe . this distribution corresponds to the case where the icm is in pressure equilibrium with the gravity of the cluster and is not heated by anything other than the gravity . we introduce the core structure to avoid the divergence of gas density at @xmath37 and use @xmath38 . we call this distribution the non - heated icm distribution. we use @xmath39 , where the present value of the hubble constant is written as @xmath40 . the value of @xmath41 is the observed icm mass fraction of high - temperature clusters @xcite , for which the effect of non - gravitational heating is expected to be small . however , as mentioned in [ sec : intro ] , observations show that icm is additionally heated non - gravitationally at least for nearby clusters . thus , we also model the distribution of the heated icm using the observed parameters of nearby clusters as follows . in this paper , we assume that the icm had been heated before accreted by clusters . however , the distribution will qualitatively be the same even if the icm is heated after accreted by clusters ( see * ? ? ? following @xcite , we define the adiabat @xmath42 , where @xmath43 is the gas pressure , @xmath44 is its density , and @xmath45 is a constant . if icm had already been heated before accreted by a cluster , the entropy prevents the gas from collapsing into the cluster with dark matter . in this case , the icm fraction of the cluster is given by @xmath46\:,\ ] ] where @xmath47 @xcite . the virial temperature of a cluster is given by @xmath48 where @xmath49 is the boltzmann constant , @xmath50 is the mean molecular weight , @xmath51 is the hydrogen mass , and @xmath52 is the gravitational constant . when the virial temperature of a cluster is much larger than that of the gas accreted by the cluster , a shock forms near the virial radius of the cluster @xcite . the temperature of the postshock gas ( @xmath53 ) is related to that of the preshock gas ( @xmath54 ) and is approximately given by @xmath55 @xcite . since the gas temperature does not change very much for @xmath56 ( @xcite ; see also @xcite ) , the icm temperature of the cluster is given by @xmath57 . since we assume that the density profile of gravitational matter is given by equation ( [ eq : rho_m ] ) with @xmath30 , the density profile of icm is given by @xmath58^{-3\beta/2 } } { [ 1+(r_{\rm vir}/r_{\rm c})^2]^{-3\beta/2}}\:,\ ] ] where @xmath59 ( see * ? ? ? * ) . observations show that @xmath60 kev although it depends on the distribution of the gravitational matter in a cluster @xcite . we choose @xmath61 kev , hereafter . the normalization @xmath34 is determined by the relation @xmath62 . when @xmath63 , a shock does not form but the gas accreted by a cluster adiabatically falls into the cluster . the icm profile for @xmath56 is obtained by solving the equation of hydrostatic equilibrium , @xmath64 where @xmath65 is the mass of the cluster within radius @xmath28 . generally , equation ( [ eq : stat ] ) does not have analytical solutions for the matter distribution we adopted ( equation [ [ eq : rho_m ] ] with @xmath30 ) . thus , we use the solution for the isothermal distribution ( @xmath66=\rho_{\rm mv , iso } [ r / r_{\rm vir}]^{-2}$ ] ) as an approximation ; assuming @xmath67 , it is given by @xmath68^{3/2}\:,\ ] ] where @xmath69 @xcite . the parameter @xmath34 is determined by the relation @xmath62 . in [ sec : result ] , we use the profile ( [ eq : icm_h ] ) for @xmath70 and the profile ( [ eq : rho_ad ] ) for @xmath71 . we refer to this icm distribution as ` the heated icm distribution ' . from observations , we use the value of @xmath72 , which is assumed to be independent of cluster mass @xcite . we consider a radially infalling disk galaxy accreted by a cluster with dark matter . the initial velocity of the model galaxy , @xmath73 , is given at @xmath74 , and it is @xmath75 where @xmath76 is the turnaround radius of the cluster . assuming that @xmath77 on the basis of the virial theorem , the initial velocity is @xmath78 the virial radius @xmath79 is given by equation ( [ eq : r_vir ] ) . the velocity of the model galaxy is obtained by solving the equation of motion ; @xmath80 as the velocity of the galaxy increases , the ram - pressure from icm also increases . the condition of ram - pressure stripping is @xmath81 where @xmath82 is the gravitational surface mass density , @xmath83 is the surface density of the hi gas , @xmath84 is the rotation velocity , and @xmath85 is the characteristic radius of the galaxy @xcite . @xcite have numerically confirmed that this analytic relation provides a good approximation . we define the cluster radius at which the condition ( [ eq : strip ] ) is satisfied for the first time as the stripping radius , @xmath86 . since we assume that the icm is nearly in pressure equilibrium for @xmath56 , the relative velocity @xmath10 is equivalent to the velocity of the galaxy relative to the cluster for @xmath56 . since the mass distribution of a cluster within the virial radius does not change rapidly , the time that a galaxy takes from @xmath79 to @xmath86 is approximately given by @xmath87 where @xmath88 thus , when the ism of the galaxy that was located at @xmath89 at @xmath90 is stripped at @xmath91 at @xmath92 , the virial radius of the cluster becomes larger than @xmath93 . we also investigate ram - pressure stripping in progenitors of a cluster . in this subsection , we construct a model of the growth of clusters . the conditional probability that a particle which resides in a object ( ` halo ' ) of mass @xmath94 at time @xmath95 is contained in a smaller halo of mass @xmath96 at time @xmath97 ( @xmath98 ) is @xmath99d m_1 \;,\ ] ] where @xmath100 is the critical density threshold for a spherical perturbation to collapse by the time @xmath101 , and @xmath102 $ ] is the rms density fluctuation smoothed over a region of mass @xmath103 for @xmath104 and 2 @xcite . in this paper , we use an approximative formula of @xmath100 @xcite and an fitting formula of @xmath105 for the cdm fluctuation spectrum @xcite . we define the typical mass of halos at @xmath106 that become part of a larger halo of mass @xmath107 at later time @xmath108 as @xmath109 where @xmath110 is the lower cutoff mass . we choose @xmath111 , which corresponds to the mass of dwarf galaxies . in [ sec : prog ] , we investigate a cluster progenitor whose virial mass is given by @xmath112 in that subsection , we will often represent @xmath113 with @xmath114 or @xmath12 unless it is misunderstood , and we will often abbreviate a cluster progenitor to a ` cluster ' . the parameters for a model galaxy are @xmath115 , @xmath116 kpc , and @xmath117 . although the values are those for our galaxy @xcite , the condition of ram - pressure stripping ( relation [ [ eq : strip ] ] ) is not much different even in the case of smaller galaxies . taking m33 for instance , the relation ( [ eq : strip ] ) turns to be @xmath118 , because the values are @xmath119 , @xmath120 kpc , and @xmath121 @xcite . as cosmological models , we consider a standard cold dark mater model ( scdm model ) and a cold dark matter model with non - zero cosmological constant ( @xmath122cdm model ) . the cosmological parameters are @xmath123 , @xmath124 , @xmath125 , and @xmath126 for the scdm model , and @xmath127 , @xmath128 , @xmath129 , and @xmath130 for the @xmath122cdm model . as will be seen , the results do not much depend on the cosmological models . we first discuss the results of the non - heated icm models . figure [ fig : m_rst ] shows the relations between the stripping radius , @xmath86 , and virial mass , @xmath12 , of clusters at several redshifts . for a given redshift , @xmath86 is an increasing function of @xmath12 . one of the reasons is the mass - dependence of @xmath79 ( figure [ fig : m_rvir ] ) . that is , massive clusters are large . to see the effect of ram - pressure stripping , we illustrate @xmath131 in figure [ fig : m_rstvir ] . for a given redshift , @xmath131 is larger for more massive clusters . since not all galaxies have the pure radial orbits in real clusters , we suppose that in the real clusters with larger @xmath132 , more galaxies are affected by ram - pressure stripping . in that sense , ram - pressure stripping is more effective in more massive clusters . the mass dependence can be explicitly shown as follows . for a given redshift , the virial density of clusters , @xmath133 , does not depend on mass ( equations [ [ eq : rho_crit]]-[[eq : dc_lam ] ] ) . thus , the virial radius of a cluster is given by @xmath134 . therefore the velocity of a galaxy infalling into the cluster has the relation , @xmath135 . since the typical icm density of a cluster is proportional to the virial density , the ram - pressure of the galaxy follows the relation , @xmath136 . the virial radius , @xmath79 , of a cluster with a given mass is smaller at higher redshift ( figure [ fig : m_rvir ] ) , because the virial density of the cluster @xmath137 is increases as a function of redshift ( equations [ [ eq : r_vir]]-[[eq : dc_lam ] ] ) . on the other hand , @xmath86 of a cluster with a given mass is larger at higher redshift in the non - heated icm models ( figure [ fig : m_rst ] ) . this is because the typical icm density of clusters increases as the virial density increases . since ram - pressure is proportional to icm density , it is more effective at higher redshift ( figure [ fig : m_rstvir ] ) . this can be clarified easily in the scdm and non - heated icm model as follows . the icm density of clusters follows @xmath138 . for a fixed mass , the virial radius is represented by @xmath139 . thus , the velocity of a galaxy is given by @xmath140 , and the ram - pressure is given by @xmath141 . this explains large @xmath131 of high - redshift clusters ( figure [ fig : m_rstvir ] ) . next , we show the results of the heated icm models . in comparison with the non - heated icm models , @xmath86 and @xmath131 for less massive clusters at a given redshift in the heated icm models are small ( figures [ fig : m_rst ] and [ fig : m_rstvir ] ) . this is because of the small icm density of the model clusters . as the mass of clusters decreases , @xmath142 in equation ( [ eq : icm_h ] ) decreases , then the icm distribution changes from equation ( [ eq : icm_h ] ) to ( [ eq : rho_ad ] ) , then the icm fraction changes from @xmath143 to @xmath144 ( equation [ [ eq : f_icm ] ] ) . in figures [ fig : m_rst ] and [ fig : m_rstvir ] , the small jump at @xmath145 and @xmath19 in the @xmath122cdm model is due to the shift from equation ( [ eq : icm_h ] ) to ( [ eq : rho_ad ] ) . figure [ fig : m_rstvir ] shows that in the heated icm model , ram - pressure stripping is ineffective for galaxy clusters ( or groups ) with @xmath146 at @xmath5 and for those with @xmath147 at @xmath3 . however , for rich clusters with @xmath148 observed at @xmath149 , the ram - pressure stripping should be more effective than the clusters with the same mass at @xmath5 regardless of the non - gravitational heating . using the model constructed in [ sec : cluster ] , we investigate the history of ram - pressure stripping of galaxies in clusters . figure [ fig : mass ] shows the evolutions of @xmath150 for @xmath151 and @xmath152 , that is , the evolutions of clusters with typical mass . the redshifts corresponding to @xmath153 are @xmath154 and 0.5 . we show the evolutions of @xmath79 and @xmath86 in figures [ fig : r_vir ] and [ fig : r_st ] , respectively . at @xmath155 , the stripping radius @xmath86 in the model of the heated icm is not much different from that in the model of the non - heated icm . in fact , the non - gravitational heating does not affect ram - pressure stripping for clusters with mass of @xmath156 at @xmath5 ( figure [ fig : m_rstvir ] ) . for clusters in that mass range , even if the gas had been heated before accreted by clusters , the clusters have gathered a large amount of gas and their virial temperatures have become large enough until @xmath155 . in our model of the heated icm , these respectively mean that @xmath157 ( equation [ [ eq : f_icm ] ] ) and the icm distribution is given by equation ( [ eq : icm_h ] ) . at @xmath155 , the values of @xmath9 are not much different between the heated icm distribution and the non - heated icm distribution at @xmath158 , although the slope of the former distribution is smaller . as @xmath16 increases , @xmath86 in the scdm model decreases faster than that in the @xmath122cdm model ( figure [ fig : r_st ] ) . this is because @xmath12 and @xmath79 decrease faster in the former model ( figures [ fig : mass ] and [ fig : r_vir ] ) . moreover , @xmath86 decreases faster in the model of the heated icm than that in the model of the non - heated icm . in order to see the effect of icm heating rather than that of the decrease of cluster size , we show the evolutions of @xmath131 in figure [ fig : r_stvir ] . as can be seen , @xmath131 decreases rapidly at high redshifts in the models of the heated icm , while it does not change significantly in the models of the non - heated icm . the changes of the slope in the former models are due to the shift in the icm distribution from equation ( [ eq : icm_h ] ) to ( [ eq : rho_ad ] ) and the shift in the icm fraction from @xmath157 to @xmath144 ( equation [ [ eq : f_icm ] ] ) . taking the heated icm model of @xmath159 , @xmath154 , and scdm as an example ( figure [ fig : r_st]a and [ fig : r_stvir]a ) , the shift in the icm distribution occurs at @xmath160 and the shift in @xmath161 occurs at @xmath162 . the rapid decrease of @xmath131 in the model of the heated icm is chiefly attributed to the decrease of @xmath163 . on the other hand , the almost constant @xmath131 in the models of the non - heated icm is explained by the fact that although the mass of cluster progenitors and the velocity of galaxies in them decrease with @xmath16 ( figure [ fig : mass ] ) , the average mass density of progenitors , @xmath137 , and thus the average icm density of progenitors , @xmath164 , increase ( equations [ [ eq : rho_crit]][[eq : dc_lam ] ] ) . we take the model of @xmath165 , @xmath154 , and scdm as an example again . figures [ fig : mass ] and [ fig : r_vir ] respectively show that @xmath166 and @xmath167 . thus , the velocity of the model galaxy has the redshift dependence such as @xmath168 . on the other hand , @xmath169 . therefore , the ram - pressure is almost independent of redshift or @xmath170 . in summary , figure [ fig : r_stvir ] shows that if the icm ( or the gas accreted by a cluster later on ) is heated non - gravitationally at @xmath2 , ram - pressure stripping does not occur in cluster progenitors at @xmath171 . on the other hand , if the icm had not been heated non - gravitationally until @xmath5 , ram - pressure stripping occurs even at @xmath6 . the difference can be explained by the small mass of cluster progenitors . at @xmath3 , the masses of progenitors are @xmath172 ( figure [ fig : mass ] ) . figure [ fig : m_rstvir ] shows that the non - gravitational heating significantly reduces the effect of ram - pressure at this mass range at @xmath3 . in the previous sections , we have modeled the ram - pressure stripping in clusters and their progenitors . in this section , we compare the results with several observations . since the direct observation of ram - pressure stripping of galaxies , such as the observation of hi distribution in galaxies , is difficult except for nearby clusters at present , we discuss the morphology and color of galaxies . in [ sec : m_r ] , we show that the ram - pressure stripping in clusters should be more effective in more massive clusters for a given redshift ( figure [ fig : m_rstvir ] ) . @xcite investigate observational data of nearby clusters ( @xmath173 ) and found that the spiral fraction decreases and the s0 fraction increases with the x - ray temperature and luminosity ( see also * ? ? ? * ; * ? ? ? the clusters with high x - ray temperature and luminosity are generally massive ones ( e.g. * ? ? ? thus , the relations confirmed by @xcite are consistent with our prediction , if the ram - pressure stripping converts spiral galaxies into s0 galaxies . recently , clusters at @xmath174 are observed in detail @xcite . @xcite observed a cluster cl @xmath175 at @xmath176 and found that the fraction of s0 galaxies is large . the ratio of s0 galaxies to elliptical galaxies in the cluster is @xmath177 , which is larger than that in intermediate redshift clusters ( @xmath178 ) . on the contrary , the fraction of spiral galaxies is only @xmath179% . @xcite estimated that the virial mass of the cluster is @xmath180 , which seems to be exceptionally massive at that redshift . our model predicts that at @xmath181 , most of the galaxies infalling into clusters of the mass should be subject to ram - pressure stripping ( figure [ fig : m_rstvir ] ) . thus , the high s0 fraction may be due to the transformation of the field spiral galaxies by ram - pressure stripping . if this is the case , the transformation by stripping must be rapid . this is because the infall rate of field galaxies , most of which are blue spiral galaxies , increases with @xmath16 @xcite ; the ram - pressure stripping must convert rather part of the blue spiral galaxies into s0 galaxies in a short time . on the other hand , most of the galaxies in a poor cluster cl @xmath182 at @xmath183 are normal spiral galaxies @xcite . the population suggests that the ram - pressure stripping is not effective in the cluster . since this cluster consists of two small components of @xmath184 and @xmath185 , it is qualitatively consistent with our prediction ( figure [ fig : m_rstvir ] ) . more observations of galaxy groups with @xmath186 at @xmath174 may give us information about the non - gravitational heating of icm for @xmath181(see figure [ fig : m_rstvir ] ) . however , for the quantitative comparison between the theory and the observations , we may need to use a so - called semi - analytic model of cluster formation or a numerical simulation including the effects of the galaxy infall rate , the variation of galaxy orbits , and the ram - pressure stripping . @xcite found that uv - excess red galaxies are abundant in a rich cluster near the quasar b2 @xmath187 at @xmath188 ; the fraction of such galaxies is relatively small at @xmath189 @xcite . @xcite estimated the abell richness of the cluster and found that it is class one . by comparing the abell catalogue @xcite with x - ray catalogues ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , it is shown that the icm temperature of the class one clusters is @xmath190 kev . if @xmath191 , the temperature corresponds to @xmath192 in our models . thus , our model predicts that most of the galaxies infalling into the cluster should be subject to ram - pressure stripping ( figure [ fig : m_rstvir ] ) . since the color of the uv - excess red galaxies can be explained by the superposition of very weak star formation activity on old stellar population , the galaxies may be the ones suffering from ram - pressure stripping and the very weak star formation may be ember although alternative interpretations may be possible . if this is true , the abundance of the galaxies may also indicate that ram - pressure stripping happens very often at @xmath174 , although more sophisticated models are required for quantitative arguments as is the case of cl @xmath175 and cl @xmath182 . in the future , it will be useful to investigate the morphology of the uv - excess red galaxies in high - redshift clusters in order to know whether the galaxies are the ones subject to ram - pressure stripping ; if the galaxies have red disks , they are probably the ones @xcite . it is to be noted that in a rich cluster ms 1054 - 03 at @xmath193 , most of the spiral galaxies are red @xcite . at intermediate redshift @xmath194 , observations show that the fraction of blue galaxies in rich clusters , @xmath41 , is larger than that at @xmath5 @xcite . on the other hand , our model predicts that the fraction of galaxies affected by ram - pressure stripping increases with @xmath16 . it seems that these contradict to each other , because the ram - pressure stripping suppresses star formation in galaxies @xcite . however , as is mentioned above , the infall rate of field spiral galaxies is higher at the intermediate redshift in comparison with @xmath5 . thus , the increase of @xmath41 that occurs in galaxy clusters at the redshift may suggest that the effect of the increase of the infall rate overwhelms the effect of the ram - pressure stripping in snapshots of galaxy population , and that the increase of @xmath41 is not inconsistent with our prediction ; we predict that the for a given mass , ram - pressure stripping at the intermediate redshift is not as effective as that at @xmath174 ( figure [ fig : m_rstvir ] ) . moreover , the average mass of the clusters observed at the intermediate redshift is expect to be smaller than that at higher redshift ( @xmath174 ) , because the clusters at the intermediate redshift are nearer to us and easier to be observed . on the other hand , observations also suggest that the star formation of the infalling galaxies are ultimately truncated in clusters at the intermediate redshift , although there is a contradiction over the way the star formation is suppressed @xcite . considering that most of the clusters observed at the intermediate redshift are fairly rich , ram - pressure stripping should be fairly effective in the clusters , although it is not as effective as that at @xmath174 ( figure [ fig : m_rstvir ] ) . thus , our model appears to be consistent with the truncation of star formation . moreover , it is qualitatively consistent with the observed transformation of spiral galaxies into s0 galaxies in rich clusters at the intermediate redshift ( see [ sec : intro ] ) . except for extremely massive clusters ( @xmath195 ) , figure [ fig : m_rst ] shows that @xmath196 mpc regardless of redshift @xmath16 and the icm heating . for nearby clusters at @xmath5 , observations show that the fraction of s0 galaxies increases at @xmath197 mpc @xcite , which appears to be larger than the prediction by our ram - pressure model if the transformation into s0 galaxies is due to ram - pressure stripping . moreover , for clusters observed at @xmath198 , the fraction of blue galaxies also decreases at @xmath197 mpc @xcite . if ram - pressure stripping is the main mechanism of the transformation of blue spiral galaxies into s0 galaxies , the observations suggest that some galaxies at @xmath199 mpc have already been affected by the stripping when they were in the progenitors of the clusters before accreted by the main cluster progenitors . in fact , figure [ fig : r_stvir ] shows that the ram - pressure stripping in progenitors is effective at least for @xmath200 if the icm is not heated non - gravitationally and it is effective for @xmath201 even if the icm is heated non - gravitationally . moreover , even if galaxies were inside the stripping radius of the main progenitor at some earlier time , some of them may have been scattered to large apocenter orbits during the merger process of cluster progenitors @xcite . analyzing observational data of galaxies in rich clusters , @xcite investigated the star formation history of all the galaxies in the central regions of the clusters . they found that star the formation rate per galaxy mass declines more rapidly than in the field environment at @xmath202 ; it suggests the truncation of star formation in most of the galaxies . this may imply that ram - pressure stripping has been effective in the clusters or in their progenitors at least for @xmath202 , because the star formation rate of galaxies should decline after the ism , from which stars are born , is stripped . since figure [ fig : r_stvir ] shows that the ram - pressure stripping has been effective for @xmath202 for rich clusters regardless of the non - gravitational heating , the results of @xcite are consistent with our predictions . unfortunately , because of large uncertainty , their results can not constrain the star formation rate of the galaxies in the cluster progenitors for @xmath181 . thus , we can not discuss the effect of the non - gravitational heating ( figure [ fig : r_stvir ] ) . we have studied ram - pressure stripping of galaxies in clusters and their progenitors . in particular , we pay attention to its dependence on redshift and the mass of clusters . as a model galaxy , we consider a radially infalling disk galaxy ; the initial position and velocity are given by a spherical collapse model of structure formation . since x - ray observations show that the icm of nearby clusters is heated non - gravitationally , we also investigate the effect of the heating on the ram - pressure stripping . our main findings are the following : \1 . for a given redshift , ram - pressure stripping of galaxies is more effective in more massive clusters . this is because the velocity of the radially infalling galaxy increases with the virial mass of the cluster . if ram - pressure stripping transforms spiral galaxies into s0 galaxies , our model is consistent with the observed relation between galaxy populations and cluster luminosities ( or temperatures ) . \2 . for a given mass of clusters , ram - pressure stripping of galaxies in the clusters is more effective at higher redshift . this is because the density of the intracluster medium increases with the redshift . in particular , at @xmath181 , most of the galaxies radially infalling into the centers of rich clusters are affected by ram - pressure stripping . the relatively high fraction of s0 galaxies and the abundance of uv - excess galaxies in rich clusters at @xmath174 may be due to the ram - pressure stripping . the non - gravitational heating reduces the effect of ram - pressure stripping for clusters with @xmath203 at @xmath5 and for those with @xmath204 at @xmath174 . however , for clusters with @xmath205 , it does not have an influence on the effect of ram - pressure stripping . if the icm ( or the gas accreted by a cluster later on ) is heated non - gravitationally at @xmath2 , ram - pressure stripping does not occur in cluster progenitors at @xmath171 , because the heat makes the icm fraction of the cluster progenitors small . on the other hand , if the icm had not been heated non - gravitationally until @xmath5 , ram - pressure stripping occurs even at @xmath6 . i am grateful to t. yamada , m. nagashima , i. tanaka , t. kodama , t. tsuchiya , and d. a. dale for useful discussions and comments . comments from an anonymous referee led to significant improvements in the quality of this paper .

we have investigated the ram - pressure stripping of disk galaxies in clusters at various redshifts and in cluster progenitors ; the clusters grow up on a hierarchical clustering scenario . we consider a radially infalling galaxy whose initial position and velocity are given by a spherical collapse model of structure formation . moreover , since observations show that the intracluster medium ( icm ) of nearby clusters is non - gravitationally heated , we study the effect of the non - gravitational heating on the ram - pressure stripping . for a given redshift , we find that ram - pressure stripping has more influence on galaxies in more massive clusters . on the other hand , for a given mass , it has more influence on galaxies in the clusters at higher redshifts . in particular , we predict that in rich clusters at @xmath0 , most of the galaxies are affected by the ram - pressure stripping . while the non - gravitational heating significantly reduces the influence of ram - pressure stripping on galaxies in clusters with mass smaller than @xmath1 , it does not affect the influence in richer clusters . if the icm is heated non - gravitationally at @xmath2 , ram - pressure stripping does not occur at @xmath3 in the progenitors of clusters observed at @xmath4 , because the heat makes the icm fraction of the cluster progenitors small . on the other hand , if the icm is heated non - gravitationally at @xmath5 for the first time , the ram - pressure stripping occurs even at @xmath6 . we compare the results with the observations of galaxies in clusters at various redshifts .

astro-ph0012252

this talk based on refs . @xcite@xcite ( where also an extensive list of references can be found ) follows closely an earlier report of ref . we shall discuss the scaling law , called geometrical scaling ( gs ) , which has been introduced in the context of dis @xcite . it has been also shown that gs is exhibited by the @xmath4 spectra at the lhc @xcite@xcite and that an onset of gs can be seen in heavy ion collisions at rhic energies @xcite . at low bjorken @xmath5 gluonic cloud in the proton is characterized by an intermediate energy scale @xmath6 , called saturation scale @xcite . @xmath6 is defined as the border line between dense and dilute gluonic systems ( for review see _ e.g. _ refs . @xcite ) . in the present paper we study the consequences of the very existence of @xmath6 ; the details of saturation phenomenon are here not of primary importance . here we shall focus of four different pieces of data which exhibit both emergence and violation of geometrical scaling . in sect . [ method ] we briefly describe the method used to assess the existence of gs . secondly , in sect . [ dis ] we describe our recent analysis @xcite of combined hera data @xcite where it has been shown that gs in dis works surprisingly well up to relatively large @xmath7 ( see also @xcite ) . next , in sect . [ pplhc ] , on the example of the cms @xmath1 spectra in central rapidity @xcite , we show that gs is also present in hadronic collisions . for particles produced at non - zero rapidities , one ( larger ) bjorken @xmath8 may be outside of the domain of gs , _ i.e. _ @xmath9 , and violation of gs should appear . in sect . [ ppna61 ] we present analysis of the pp data from na61/shine experiment at cern @xcite and show that gs is indeed violated once rapidity is increased . finally in sect . [ gsids ] we analyze identified particles spectra where the particle mass provides another energy scale which may lead to the violation of gs , or at least to some sort of its modification @xcite . we conclude in sect . [ concl ] . geometrical scaling hypothesis means that some observable @xmath10 depending in principle on two independent kinematical variables , like @xmath11 and @xmath12 , depends in fact only on a given combination of them , denoted in the following as @xmath13 : @xmath14 here function @xmath15 in eq . ( [ gsdef ] ) is a dimensionless universal function of scaling variable @xmath13 : @xmath16 and @xmath17 is the saturation scale . here @xmath18 and @xmath19 are free parameters which , however , are not of importance in the present analysis , and exponent @xmath20 is a dynamical quantity of the order of @xmath21 . throughout this paper we shall test the hypothesis whether different pieces of data can be described by formula ( [ gsdef ] ) with _ constant _ @xmath20 , and what is the kinematical range where gs is working satisfactorily . as a consequence of eq . ( [ gsdef ] ) observables @xmath22 for different @xmath23 s should fall on a universal curve , if evaluated in terms of scaling variable @xmath13 . this means that ratios @xmath24 should be equal to unity independently of @xmath13 . here for some @xmath25 we pick up all @xmath26 which have at least two overlapping points in @xmath12 . for @xmath27 points of the same @xmath28 but different @xmath11 s correspond in general to different @xmath13 s . therefore one has to interpolate @xmath29 to @xmath30 such that @xmath31 . this procedure is described in detail in refs . @xcite . by adjusting @xmath20 one can make @xmath32 for all @xmath33 in a given interval . in order to find an optimal value @xmath34 which minimizes deviations of ratios ( [ rxdef ] ) from unity we form the chi - square measure@xmath35 where the sum over @xmath36 extends over all points of given @xmath23 that have overlap with @xmath37 , and @xmath38 is a number of such points . in the case of dis the relevant scaling observable is @xmath39 cross section and variable @xmath11 is simply bjorken @xmath11 . in fig . [ xlamlog ] we present 3-d plot of @xmath40 which has been found by minimizing ( [ chix1 ] ) . obtained by minimization of eq . ( [ chix1]).,width=302 ] qualitatively , gs is given by the independence of @xmath41 on bjorken @xmath11 and by the requirement that the respective value of @xmath42 is small ( for more detailed discussion see refs . one can see from fig . [ xlamlog ] that the stability corner of @xmath41 extends up to @xmath43 , which is well above the original expectations . in ref . @xcite we have shown that : @xmath44 spectra khachatryan:2010xs at 7 tev to 0.9 ( blue circles ) and 2.36 tev ( red triangles ) plotted as functions of @xmath45 ( left ) and scaling variable @xmath46 ( right ) for @xmath47 . , title="fig : " ] spectra khachatryan:2010xs at 7 tev to 0.9 ( blue circles ) and 2.36 tev ( red triangles ) plotted as functions of @xmath45 ( left ) and scaling variable @xmath46 ( right ) for @xmath47 . , title="fig : " ] in hadronic collisions at c.m . energy @xmath48 particles are produced in the scattering process of two patrons ( mainly gluons ) carrying bjorken @xmath11 s @xmath49 for central rapidities @xmath50 . in this case charged particles multiplicity spectra exhibit gs @xcite @xmath51 where @xmath15 is a universal dimensionless function of the scaling variable ( [ taudef ] ) . therefore the method of ratios can be applied to the multiplicity distributions at different energies ( @xmath52 taking over the role of @xmath53 in eq . ( [ rxdef ] ) ) as an inverse of ( [ rxdef ] ) ] . for @xmath54 we take the highest lhc energy of 7 tev . hence one can form two ratios @xmath55 with @xmath56 and @xmath57 tev . these ratios are plotted in fig . [ ratios1 ] for the cms single non - diffractive spectra for @xmath58 and for @xmath59 , which minimizes ( [ chix1 ] ) in this case . we see that original ratios plotted in terms of @xmath4 range from 1.5 to 7 , whereas plotted in terms of @xmath60 they are well concentrated around unity . the optimal exponent @xmath20 is , however , smaller than in the case of dis . why this so , remains to be understood . for @xmath61 two bjorken @xmath11 s can be quite different : @xmath62 . therefore by increasing @xmath63 one can eventually reach @xmath64 and gs violation should be seen . for that purpose we shall use pp data from na61/shine experiment at cern @xcite at different rapidities @xmath65 and at five scattering energies @xmath66 , and @xmath67 gev . as functions of @xmath60 for the lowest rapidity @xmath68 : a ) for @xmath69 when @xmath70 and b ) for @xmath71 which corresponds to gs.,title="fig:",width=226 ] as functions of @xmath60 for the lowest rapidity @xmath68 : a ) for @xmath69 when @xmath70 and b ) for @xmath71 which corresponds to gs.,title="fig:",width=226 ] as functions of @xmath60 for @xmath71 and for different rapidities a ) @xmath72 and b ) @xmath73 . with increase of rapidity , gradual closure of the gs window can be seen.,title="fig:",width=226 ] as functions of @xmath60 for @xmath71 and for different rapidities a ) @xmath72 and b ) @xmath73 . with increase of rapidity , gradual closure of the gs window can be seen.,title="fig:",width=226 ] in fig . [ y01 ] we plot ratios @xmath74 ( [ rxdef ] ) for @xmath75 spectra in central rapidity for @xmath69 and 0.27 . for @xmath68 the gs region extends down to the smallest energy because @xmath76 is as large as 0.08 . however , the quality of gs is the worst for the lowest energy @xmath77 . by increasing @xmath63 some points fall outside the gs window because @xmath78 , and finally for @xmath79 no gs is present in na61/shine data . this is illustrated nicely in fig . in ref . @xcite we have proposed that in the case of identified particles another scaling variable should be used in which @xmath80 is replaced by @xmath81 ( @xmath82 scaling ) , _ i.e. _ @xmath83 this choice is purely phenomenological for the following reasons . firstly , the gluon cloud is in principle not sensitive to the mass of the particle it finally is fragmenting to , so in principle one should take @xmath80 as an argument of the saturation scale . in this case the proper scaling variable would be @xmath84 however this choice ( @xmath85@xmath80 scaling ) does not really differ numerically from the one given by eq . ( [ taumtdef ] ) . to this end let us see how scaling properties of gs are affected by going from scaling variable @xmath86 ( [ taudef ] ) to @xmath87 ( [ taumtdef ] ) and what would be the difference in scaling properties if we had chosen @xmath80 as an argument in the saturation scale leading to scaling variable @xmath88 ( [ taumtpdef ] ) . this is illustrated in fig . [ ratiosmandpt ] where we show analysis @xcite of recent alice data on identified particles @xcite . in fig . [ ratiosmandpt].a - c full symbols refer to the @xmath80 scaling ( [ taudef ] ) and open symbols to @xmath85 scaling or @xmath85@xmath80 scaling . one can see very small difference between open symbols indicating that scaling variables @xmath89 ( [ taumtdef ] ) and @xmath90 ( [ taumtpdef ] ) exhibit gs of the same quality . on the contrary @xmath80 scaling in variable @xmath91 ( [ taudef ] ) is visibly worse . finally in fig . [ ratiosmandpt].d , on the example of protons , we compare @xmath85 scaling ( open symbols ) and @xmath92 scaling ( full symbols ) in variable @xmath93 for @xmath71 . one can see that no gs has been achieved in the latter case . qualitatively the same behavior can be observed for other values of @xmath20 . , @xmath94 and @xmath95 for @xmath71 . full symbols correspond to ratios @xmath96 plotted in terms of the scaling variable @xmath91 , open symbols to @xmath94 and @xmath97 , note negligible differences between the latter two forms of scaling variable . panel a ) corresponds to pions , b ) to kaons and c ) to protons . in panel d ) we show comparison of geometrical scaling for protons in scaling variables @xmath94 and @xmath98 , no gs can be achieved in the latter case.,title="fig : " ] , @xmath94 and @xmath95 for @xmath71 . full symbols correspond to ratios @xmath96 plotted in terms of the scaling variable @xmath91 , open symbols to @xmath94 and @xmath97 , note negligible differences between the latter two forms of scaling variable . panel a ) corresponds to pions , b ) to kaons and c ) to protons . in panel d ) we show comparison of geometrical scaling for protons in scaling variables @xmath94 and @xmath98 , no gs can be achieved in the latter case.,title="fig : " ] + , @xmath94 and @xmath95 for @xmath71 . full symbols correspond to ratios @xmath96 plotted in terms of the scaling variable @xmath91 , open symbols to @xmath94 and @xmath97 , note negligible differences between the latter two forms of scaling variable . panel a ) corresponds to pions , b ) to kaons and c ) to protons . in panel d ) we show comparison of geometrical scaling for protons in scaling variables @xmath94 and @xmath98 , no gs can be achieved in the latter case.,title="fig : " ] , @xmath94 and @xmath95 for @xmath71 . full symbols correspond to ratios @xmath96 plotted in terms of the scaling variable @xmath91 , open symbols to @xmath94 and @xmath97 , note negligible differences between the latter two forms of scaling variable . panel a ) corresponds to pions , b ) to kaons and c ) to protons . in panel d ) we show comparison of geometrical scaling for protons in scaling variables @xmath94 and @xmath98 , no gs can be achieved in the latter case.,title="fig : " ] in ref . @xcite we have shown that gs in dis works well up to rather large bjorken @xmath11 s with exponent @xmath99 . in pp collisions at the lhc energies in central rapidity gs is seen in the charged particle multiplicity spectra , however , @xmath59 in this case @xcite . by changing rapidity one can force one of the bjorken @xmath11 s of colliding patrons to exceed @xmath76 and gs violation is expected . such behavior is indeed observed in the na61/shine pp data @xcite . finally we have shown that for identified particles scaling variable @xmath13 of eq . ( [ taudef ] ) should be replaced by @xmath94 defined in eq . ( [ taumtdef ] ) and the scaling exponent @xmath100 @xcite . many thanks are due to the organizers of this successful series of conferences . this work was supported by the polish ncn grant 2011/01/b / st2/00492 . 99 l. mclerran and m. praszalowicz , acta phys . pol . b * 41 * ( 2010 ) 1917 and acta phys . pol . b * 42 * ( 2011 ) 99 . m. praszalowicz , phys . * 106 * ( 2011 ) 142002 . m. praszalowicz , acta phys . pol . b * 42 * ( 2011 ) 1557 and arxiv:1205.4538 [ hep - ph ] . m. praszalowicz and t. stebel , jhep * 1303 * ( 2013 ) 090 and * 1304 * ( 2013 ) 169 . m. praszalowicz , phys . d * 87 * ( 2013 ) 071502(r ) . m. praszalowicz , arxiv:1308.5911 [ hep - ph ] . m. praszalowicz , acta phys . pol . b proceeding supplement * 6 * ( 2013 ) 815 . a. m. stasto , k. j. golec - biernat and j. kwiecinski , phys . * 86 * ( 2001 ) 596 . l. v. gribov , e. m. levin and m. g. ryskin , phys . rept . * 100 * ( 1983 ) 1 ; + a. h. mueller and j - w . qiu , nucl . phys . * 268 * ( 1986 ) 427 ; a. h. mueller , nucl . * b558 * ( 1999 ) 285 . k. j. golec - biernat and m. wsthoff , phys . d * 59 * ( 1998 ) 014017 and phys . d * 60 * ( 1999 ) 114023 . a. h. mueller , _ parton saturation : an overview _ , arxiv : hep - ph/0111244 . l. mclerran , acta phys . pol . b * 41 * ( 2010 ) 2799 . c. adloff _ et al . _ [ h1 collaboration ] , eur . j. c * 21 * ( 2001 ) 33 ; s. chekanov _ et al . _ [ zeus collaboration ] , eur . j. c * 21 * ( 2001 ) 443 . f. caola , s. forte and j. rojo , nucl . a * 854 * ( 2011 ) 32 . n. abgrall _ et al . _ [ na61/shine collaboration ] , _ report from the na61/shine experiment at the cern sps _ cern - spsc-2012 - 029 , spsc - sr-107 ; + a. aduszkiewicz , ph.d . thesis in prepartation , university of warsaw , 2013 ; + sz . pulawski , talk at 9th polish workshop on relativistic heavy - ion collisions , krakw , november 2012 and private communication . k. aamodt _ et al . _ [ alice collaboration ] , eur . j. c * 71 * ( 2011 ) 1655 [ arxiv:1101.4110 [ hep - ex ] ] ; a. ortiz velasquez [ alice collaboration ] , nucl . a * 904 - 905 * ( 2013 ) 763c [ arxiv:1210.6995 [ hep - ex ] ] ( alice preliminary ) .

we argue that geometrical scaling ( gs ) proposed originally in the context of deep inelastic scattering ( dis ) at hera works also in pp collisions at the lhc energies and in na61/shine experiment . we show that in dis gs is working up to relatively large bjorken @xmath0 . as a consequence negative pion multiplicity @xmath1 distributions at na61/shine energies exhibit gs in mid rapidity region . for @xmath2 clear sign of scaling violations can be seen when one of the colliding partons has bjorken @xmath3 . finally , we argue that in the case of identified particles gs scaling is still present but the scaling variable is a function of transverse mass rather than @xmath1 . + + pacs number(s ) : 13.85.ni,12.38.lg

1309.0933

the continuous progress in the fabrication of small electronic circuits has kept active the theoretical study of quantum electronic transport . for example , transport properties in point contacts , atomic chains , carbon nanotubes , and quantum wires are currently under experimental investigations . although the theoretical study of electronic transport in mesoscopic systems where the phase coherence of electrons is preserved along the whole device has been of great interest for several decades , these recent experimental advances have further renewed the motivation of theoreticians for studying the electronic transport through one - dimensional structures . in a disordered mesoscopic wire the random position of impurities gives a stochastic character to the electronic transport . therefore , it is of particular relevance the analysis of the statistical properties of transport quantities like the conductance . on the theoretical side , there is a full description of the statistical properties of the conductance in quasi - one - dimensional disordered systems within an independent - electron picture at zero temperature @xmath1 and `` zero bias '' ( actually , infinitesimally small voltage ) @xmath2 . @xcite this degree of detail in the theoretical description is , however , not available for disordered wires at finite temperatures and bias voltage , even in the simple case of one - dimensional ( 1d ) wires . this is an unfortunate fact since experiments are usually performed within a wide range of @xmath1 and @xmath2 . for example , the effect of finite bias voltage has been found to modify the behavior of the conductance fluctuations . @xcite several efforts have been made in order to incorporate the information of a finite @xmath1 into the statistical description of quantum electronic transport in 1d - disordered systems . in ref . , at zero voltage , a model of zero - width resonances represented by @xmath5-functions was introduced in order to calculate the dependence on the temperature of the averages of the conductance @xmath6 and @xmath7 . assuming full coherent transport , in ref . the effect of the thermal smearing on the mean resistance of a 1d wire was studied . at finite temperature and bias voltage , the distribution of the conductance @xmath3 was calculated in ref . by using a statistical model of resonant tunneling transmission . the methodology there employed was , however , restricted to disordered wires of length @xmath8 , where @xmath9 is the mean free path , and very small values of @xmath1 and @xmath2 ( although sizable ) : typically @xmath10 and @xmath11 were considered of the order or smaller than the mean spacing between energy levels @xmath12 . within this regime of temperatures and voltages it can be assumed that only one resonance the closest to the fermi energy contributes to the transport . even with these limitations , that work shows that the conductance distribution do display novel features as a consequence of finite @xmath1 and @xmath2 values . particularly , it was shown that the distribution of conductances narrows as the value of @xmath1 and/or @xmath2 is increased . we would like to remark that the assumption of phase coherent transport adopted in previous works , as well as in the present one , is restricted by the phase relaxation @xmath13 at finite temperature and voltage . for example , inelastic scattering and thermally activated hopping processes,@xcite which might be important at finite values of @xmath1 , @xmath2 are not considered in our analysis . in ref . , however , it is discussed the possibility of satisfying @xmath14 even at finite temperature and bias voltage for short enough wires , which opens the possibility of observing the effect of finite @xmath15 on the statistical properties of the conductance in realistic systems . it is also interesting to note that the range of voltage @xmath2 where the averaged charge current @xmath16 is a linear function of @xmath2 or departs slightly from this behavior , i.e. , the so - called _ linear response _ regime , can be rather wide as we will show in section [ results ] . the main goal of the present paper is to compute the distribution of the conductance @xmath4 at values of @xmath10 and @xmath11 typically larger than the mean level spacing , where several resonances contribute to the conductance . we show that , within this regime of temperatures and voltages , the distribution of the conductance can be well described in terms of the convolutions of the zero temperature / voltage conductance distribution @xmath17 . the number of distributions to be convolved is determined by the number of resonances in an energy window where electron transport can take place . this energy window is defined by the value of the temperature and voltage . the method introduced in the present work applies to any degree of disorder i.e. _ to any value _ of the ratio @xmath18 , provided several resonances contribute to the transport , allowing for the investigation of short wires where coherent transport is more likely to be observed . for completeness in the sense that all regimes of temperature and voltage will be covered in this paper , we also study the regime of small @xmath1 and @xmath2 following ref . this paper is organized as follows . in sec . [ method ] we present the methodology to analytically calculate the conductance distribution @xmath3 . we divide this section into two subsections . the first one , [ resonant_model ] , is devoted to review the case of small temperatures and voltage regimes where one can assume that only the closest resonance to the fermi energy contribute to the conductance . @xcite in the second one , subsection [ convolution_method ] , we introduce a method based on the convolution of the known distribution of conductances at zero temperature and bias voltage in order to study the regime of high temperatures and bias voltages . in section [ results ] we present the model used in our simulations and show some general features of our 1d - disordered system obtained numerically , which support the assumptions introduced in our theoretical proposal of section [ method ] . in section [ results ] we also compare the results of the distribution @xmath3 from the numerical simulations to the predictions of the theoretical approach presented in the previous section , for the different regimes of @xmath1 and @xmath2 . finally , in section [ summary ] we give a summary and conclusions of our study of the distribution of conductances at finite temperature and bias voltage . we consider the usual setup where the disordered conductor is placed between left and right reservoirs at the same finite temperature @xmath1 and chemical potentials @xmath19 and @xmath20 , respectively . the conductance @xmath21 can be written as ( in units of the conductance quantum @xmath22 ) @xmath23 where @xmath24 + 1\right\}^{-1}$ ] is the fermi function , being @xmath25 the boltzmann constant , while @xmath26 is the dimensionless conductance for @xmath27 at @xmath28 . it is clear from the above integral expression for @xmath4 that the energy window where electronic transport takes place will depend on the values of @xmath1 and @xmath2 . this can be emphasized recasting the difference in the fermi functions in the above expression , eq.([gtv ] ) , as @xcite @xmath29 where the symbol @xmath30 denotes the convolution in energy , @xmath31 is the unit step function , and @xmath32 is the thermal smearing function . therefore , the difference of the fermi functions can be expressed as a convolution in energy of two functions : one depends only on the applied voltage @xmath2 and the other one , only on temperature @xmath1 . this implies that thermal and voltages effects are statistically independent , and this will be useful to calculate the distribution of @xmath4 . in this subsection we briefly review the resonant model introduced in ref . . this model is appropriate for systems with non overlapping resonances , which is satisfied for disordered systems with @xmath33 . assuming a lorentzian line shape for the resonances , the dimensionless conductance @xmath26 at @xmath34 can be written as @xcite @xmath35 where @xmath36 is the total width , @xmath37 $ ] are the left and right partial widths , and @xmath38 are the localization lengths . on resonance @xmath39 , the terms of the sum eq . ( [ g_model ] ) , @xmath40^{-2 } $ ] , depend on the location @xmath41 of the state , and are maximum for @xmath42 ( center of wire ) . substituting eq . ( [ g_model ] ) into eq . ( [ gtv ] ) , it is found that the conductance @xmath4 is given by @xmath43,\ ] ] where @xmath44 . we now assume that @xmath45 . under this condition only the resonance closest to the fermi level ( @xmath46 ) contributes to the conductance @xmath4 and therefore eq . ( [ g_model ] ) can be simplified to @xcite @xmath47 . \nonumber\\\end{aligned}\ ] ] in order to calculate the distribution of the conductance @xmath3 a statistical model for the resonances is introduced : the resonances @xmath48 and their positions @xmath49 are considered uniformly distributed in a interval @xmath12 and @xmath50 , respectively , while the distribution of the inverse of the localization lengths @xmath51 follows a gaussian distribution with mean @xmath52 and @xmath53 . this distribution for @xmath54 is good for systems with @xmath55 . in section [ results ] we will show some examples for the distribution @xmath3 obtained within this resonant tunneling model . we now go to the main goal of this work and propose a description of the conductance distribution , which is valid for arbitrary length of the wire and for voltages and temperatures satisfying @xmath56 , but small enough to satisfy that the behavior of the average current does not depart from linear response . firstly , we discretize the integral in eq . ( [ gtv ] ) as @xmath57 , \end{aligned}\ ] ] where we have assumed the same width @xmath58 for all the elements in the sum , eq . ( [ gdiscretet ] ) . let us first consider the simple case of finite bias voltage and zero temperature . in this case , the thermal broadening function in eq . ( [ dff ] ) is a delta function and therefore @xmath26 in eq . ( [ gtv ] ) is only multiplied by a rectangular function of width @xmath59 . ( [ gdiscretet ] ) is reduced to @xmath60 where the number of terms in the sum @xmath61 satisfies @xmath62= @xmath59 . this approximation is exact in the limit @xmath63 . however , being our aim the evaluation of @xmath3 , we approximate ( [ gdiscrete ] ) by a finite number @xmath61 of _ statistically independent _ contributions . we now assume that this number corresponds to the mean number of levels in an energy window @xmath10 , i.e. , @xmath64 . this is a natural assumption since for a given disorder realization @xmath26 is a spectral function with peaks at the energy levels of the wire ( resonances ) . in a non - interacting electron picture , the height of the resonance peaks , as well as the energy levels change for different disorder realizations , but the average separation between levels is a well defined quantity and the spectral weights centered at the different energy levels are uncorrelated . in addition , we assume that @xmath26 is a random stationary function of the energy , at least in the energy window where transport takes place . then the statistical properties of @xmath26 do not change in such energy window and therefore the distributions @xmath65 are actually independent of the energy @xmath66 , i.e. @xmath67 . under the above assumptions the distribution @xmath68 can be computed from the convolution of @xmath61 distributions @xmath17 , i.e. , the @xmath61th autoconvolution of the distribution at zero temperature and bias voltage : @xmath69 where we have defined @xmath70 , and the upper indices enumerate the number of distributions @xmath17 that enters into the convolution . as we mentioned in the introduction , at zero temperature and infinitesimal small bias voltage the statistical properties of the transport quantities , in particular the distribution @xmath17 for 1d and quasi-1d disordered systems is well known . in a framework of random - matrix theory , a diffusion equation known as dorokhov - mello - pereyra - kumar ( dmpk ) equation has been successful in describing the evolution of the conductance distribution @xmath17 as a function of the system length @xmath50 in quasi - one dimensional systems @xcite . for strictly 1d wires the dmpk equation is reduced to the melnikov equation whose solution @xmath17 can be written as @xcite @xmath71 where @xmath72 and @xmath73 is the length of the system @xmath50 in units of the mean free path @xmath9 . in the limit of @xmath74 and @xmath75 closed analytical expressions for @xmath17 can be obtained . we remark that the only parameter that enters in the distribution @xmath17 is the so - called disorder parameter @xmath73 . the simple relationship between the random variables @xmath76 and @xmath77 , eq . ( [ gdiscrete ] ) , encloses important conclusions since this means that the cumulant generating function for @xmath0 and @xmath78 are also simple related : @xmath79 with @xmath80 being @xmath81 , while for the average and the first three central moments @xmath82 and @xmath83 it is satisfied : @xmath84 @xmath85 by definition . then , eq.([cumu ] ) shows that the mean value of the conductance @xmath86 is independent of the bias voltage @xmath2 . ( color online ) sketch of the behavior of the function ( [ finit ] ) and its approximation by a rectangular function , in black dashed and red solid lines . the behavior of the integral of the function ( [ finit ] ) is also indicated in dashed - dotted lines . ] let us now consider a more realistic situation where temperature and bias voltage are both finite . at finite temperature , the difference of fermi functions in eq . ( [ gtv ] ) reads @xmath87 . \nonumber\\\end{aligned}\ ] ] in fig . [ fig_sketch ] it is shown the behavior of eq . ( [ finit ] ) . in order to keep our statistical analysis of the conductance as simple as possible , we approximate the bell - shaped function ( [ finit ] ) by a rectangular function of height of @xmath88 and width @xmath89 . the area of the rectangle is , therefore , @xmath10 as the area of the original bell - shaped function eq . ( [ finit ] ) . this simplification allows us to proceed as in subsection [ subt0v ] : from the width @xmath89 we define an `` effective number of resonances '' , @xmath90 , given by @xmath91 we can verify that for @xmath92 , @xmath93 , as expected . thus to calculate @xmath3 we can follow exactly the same procedure of the previous subsection [ subt0v ] , we just replace @xmath61 by @xmath90 . also , at finite @xmath1 and @xmath2 it is possible to define a simple relation between cumulants and moments of the distribution @xmath3 and @xmath17 ; again , substituting @xmath61 by @xmath90 in eq . ( [ cumu2 ] ) . in this section we verify our theoretical study of the statistical properties of the conductance , in particular the conductance distribution , by comparing to numerical simulations of a 1d - disordered wire . the results presented here give numerical evidence that supports the main hypothesis of our theoretical model and benchmark the quality of the approximation of the distribution function at finite bias and temperature on the basis of convolutions of the distribution in eq . ( [ pofg ] ) against exact numerical results . in our simulations we model a 1d - disordered wire of length @xmath50 using the standard tight binding hamiltonian of spinless electrons with a single atomic orbital per lattice site and nearest neighbors hopping parameter @xmath94 : @xmath95 with @xmath96 , being @xmath97 the lattice constant , which we set as the unit of length and @xmath98 , the number of sites of the 1d lattice . the on site energies @xmath99 are chosen randomly from an uniform distribution of width @xmath100 ( anderson model ) . all the energy scales are taken dimensionless in units of the hopping parameter @xmath94 . the disordered conductor is connected to the left and to the right to 1d clean semi - infinite leads , which we assume to be at chemical potentials @xmath19 and @xmath20 , respectively , and temperature @xmath1 . the hamiltonian describing the contacts between the reservoirs and the disordered wire reads : @xmath101 being @xmath102 the contact sites of the left and right semi - infinite leads , respectively . for each disorder realization , the current is numerically calculated from the expression ( [ gtv ] ) , with @xmath103 being @xmath104 , which corresponds to reservoirs modeled by semi - infinite tight - binding chains with hopping @xmath94 , while @xmath105 is the retarded green s function for the disordered chain connected to the reservoirs . @xcite typically , we generate numerically @xmath106 to @xmath107 realizations of disorder . for several values of the disorder parameter @xmath73 , we have verified that for very small bias ( @xmath108 ) and @xmath28 we are able to reproduce the distribution @xmath17 given by eq . ( [ pofg ] ) . the mean free path @xmath9 is extracted from the numerical simulations through the relation @xmath109 . in all numerical simulations we present below we have fixed @xmath110 and the disorder strength to @xmath111 which sets @xmath112 . the theoretical distributions @xmath3 , as described by eq . ( [ pofgconvolutions ] ) of the previous section , are obtained by collecting the data for @xmath4 from an ensemble of conductances generated numerically using a metropolis monte carlo algorithm . before presenting our results for the probability distribution at finite temperature and bias voltage , we would like to establish the range of the bias voltage @xmath2 where the averaged current @xmath113 varies linearly with @xmath2 . to this end we have included in eq . ( [ ham ] ) the potential drop due to the presence of the bias by modifying the local energies as @xmath114 . in fig . [ fig_j ] we show the disorder averaged current @xmath115 as a function of the bias voltage , at @xmath28 . a linear behavior of @xmath116 with the bias @xmath2 is observed up to @xmath117 for different lengths of the system . therefore if we want to restrict our study to the linear response regime , we can not take arbitrary large values of @xmath1 and @xmath2 . in the present model , this implies energy values for @xmath10 and @xmath118 . let us note , that this range of energy is actually rather wide : it is larger than 10@xmath119 of the total band width of the clean system ( equal to @xmath120 ) . since we focus in voltages within the linear response regime , in what follows we consider the wire model ( eq . ( [ ham ] ) ) without including the effect of the potential drop in the local energies @xmath99 . @xmath115 as a function of @xmath2 for @xmath111 , @xmath110 and different lengths @xmath121 ( solid ) , @xmath122 ( dotted ) , @xmath123 ( dashed ) , and @xmath124 ( dashed - dotted ) . inset : @xmath125 vs @xmath50 for @xmath28 and @xmath126 ( black crosses ) , @xmath127 ( blue squares ) and @xmath128 ( red diamonds ) . note that the different symbols are overlapped . ] in the inset of fig . [ fig_j ] we also show the average of the conductance @xmath125 for different values of @xmath1 , @xmath2 , as a function of the system length @xmath50 . we observe that @xmath125 is independent of the values of the temperature and voltage and decreases exponentially with @xmath50 as it is expected . this behavior is in agreement with previous results @xcite as well as with the prediction based on eq . ( [ cumu ] ) . another important point to verify is the reliability of our assumption that the mean energy spacing between levels of eq.([ham ] ) sets the energy scale where the spectral weights of @xmath26 are statistically independent . to this end , we have investigated the behavior of the auto - correlation function @xmath129 , where the over line means energy average . in fig . [ fig_levels ] we have plotted @xmath130 ( normalized by dividing by @xmath131 ) for three different wire lengths : @xmath132 and @xmath133 . in all the cases @xmath130 decays between @xmath134 of its value at @xmath135 for @xmath136 . in order words , @xmath130 is a vanishing function in an energy scale @xmath137 smaller than the mean level spacing . we have also verified numerically that in the linear response regime @xmath138 is in fact independent of @xmath139 , as we assumed in the previous section . ( color online ) mean number of levels @xmath140 of the hamiltonian ( [ ham ] ) in an energy interval equal to @xmath141 , as a function of the system length @xmath50 . inset : the autocorrelation function @xmath130 is plotted for @xmath142 , @xmath143 and @xmath144 ( red open circles , black solid circles and blue square symbols , respectively ) . see the text for details . ] on the other hand , it is instructive to show the evolution of the conductance distribution @xmath3 with the number of autoconvolutions of @xmath17 , as described in the previous section . we have chosen the simple case of @xmath28 with and finite @xmath145 with @xmath146 . in fig . [ fig_illustrative ] , the histogram in dashed line corresponds to the distribution obtained from our numerical simulations , while the histograms in solid line correspond to @xmath147 obtained by @xmath148 autoconvolutions of @xmath17 , for @xmath149 , and 10 . we can observe that as @xmath148 increases from 4 to 6 the theoretical @xmath147 evolves to the numerical distribution . when @xmath150 , which corresponds to the mean number of levels @xmath61 for @xmath123 , see fig . [ fig_levels ] , a very good agreement is found with the numerical distribution . the fact that the optimum number of autoconvolutions of @xmath17 coincides at @xmath28 with the mean number of levels within ev , supports the validity of our approach of subsection [ subt0v ] . ( color online ) histograms obtained from eq.([pofgconvolutions ] ) performing different number of convolutions @xmath148 , for a finite voltage @xmath145 and @xmath123 . a ) @xmath151 , b ) @xmath152 , c ) @xmath153 and d ) @xmath154 . for comparison the distribution @xmath147 ( histogram in red dashed line ) obtained numerically is also plotted . for @xmath155 convolutions the distribution @xmath147 is quite well reproduced . the size bins of the numerical and theoretical histograms are slightly different to distinguish better between both histograms . ] in this subsection we show some numerical results for the distribution @xmath147 when the values of the temperature and bias voltage are small enough that only one resonance contributes to the conductance @xmath4 ( subsection [ resonant_model ] ) . we also indicate the limitations of this method by showing an example of the conductance distribution at regimes of @xmath1 and @xmath2 beyond the scope of the resonant model . ( color online ) histograms obtained with the resonant model of subsection [ resonant_model ] ( solid black lines ) for @xmath123 . results in dashed red lines correspond to the numerical simulations on the disordered tight - binding chain . plots in the main panel correspond to @xmath156 while the inset corresponds to @xmath157 . the distribution @xmath17 is also shown in the main panel ( blue dotted line ) . ] the theoretical distribution @xmath3 is calculated by generating numerically an ensemble of conductances @xmath0 accordingly to eq . ( [ g_t_inter ] ) , where the random variables @xmath48 and @xmath49 are obtained numerically from a uniform distribution , while @xmath158 is extracted from a gaussian distribution , as it is described in [ resonant_model ] . in figure [ reso_fig ] we show the distribution of @xmath159 ( we have chosen the variable @xmath159 since the details of the distribution can be better seen in the logarithm scale ) from the resonant tunneling model ( histogram in solid line ) for the case of @xmath160 and @xmath161 . a good agreement with the numerical simulation ( histogram in dashed line ) is seen . note that , although this situation corresponds to a small bias voltage , the exact distribution ( solid lines ) clearly departs from the behavior of the `` zero '' bias distribution @xmath17 ( histogram in dotted lines ) . therefore the resonant model reasonably describes the numerical behavior of @xmath147 , in particular the decrease of the width of the distribution as @xmath162 increases ( with @xmath163 ) . however , if we increase the value of @xmath1 , keeping @xmath164 , in our previous numerical example , the quality of the approximation provided by this model deteriorates , as it is illustrated in the inset of the figure [ reso_fig ] . this happens in general when the values @xmath1 and/or @xmath2 are increased in such a way that more than one resonance might contribute to the conductance . in the next subsection we will see that the convolution method describe correctly the regime of @xmath1 and @xmath2 when several resonances are involved in the transport problem . let us now go to the analysis of the probability distribution function at finite @xmath2 with @xmath28 using the methodology introduced in subsection [ subt0v ] . in figure [ fig_vtzero ] we show the distributions @xmath68 for different lengths @xmath50 of the disordered wire with @xmath165 . we recall that @xmath112 in all cases . for each length @xmath50 , the distribution @xmath147 obtained from eq.([pofgconvolutions ] ) ( histogram in solid line ) and the numerical distribution ( histogram in dashed line ) are both displayed for their comparison . a very good agreement can be seen . in order to provide evidence of the strong effect of the finite bias voltage on the conductance distribution , @xmath17 is also shown in dotted lines in the same figure . ( color online ) distributions @xmath147 for @xmath165 and @xmath28 ( red dashed line histogram ) obtained numerically for different lengths a ) @xmath166 , b ) @xmath167 , c ) @xmath122 , and d ) @xmath123 . the solid black line histograms correspond to the distribution of @xmath0 obtained as a convolution of @xmath61 terms a ) @xmath168 , b)@xmath169 , c)@xmath170 , d)@xmath171 . the blue dotted curves show @xmath17 for zero bias and temperature . the size of the bins of the numerical and theoretical histograms are slightly different to distinguish better between both histograms . ] we have also verified the relation between the second and third moments , eq . ( [ cumu2 ] ) . for a bias voltage @xmath165 , in table [ table1 ] we show the values of @xmath172 extracted from the numerical simulation and those for @xmath173 ( @xmath174 ) calculated from then integral expression of @xmath17 , eq . ( [ pofg ] ) . again , a good agreement between numerics and theory is obtained . .moments @xmath175 ( @xmath176 ) for bias voltage @xmath165 , @xmath111 for different lengths @xmath50 obtained numerically and @xmath177 from the distribution @xmath17 , eq.([pofg ] ) . [ cols="^,^,^,^,^,^ " , ] [ table2 ] finally , it worth mentioning that we have verified that the approach based in convolutions of @xmath17 also provides a good approximation to the exact @xmath3 at finite @xmath1 in cases with larger @xmath10 ( e.g. @xmath178 ) . in an independent electron picture and assuming full phase - coherent electronic transport , we have studied the distribution of the conductance @xmath3 in 1d - disordered systems at finite temperature and bias voltage . ( color online ) numerical distributions @xmath147 ( dashed red line histograms ) of a system with length @xmath123 , @xmath179 , and different temperatures : a)@xmath180 , b)@xmath181 , c)@xmath182 , d)@xmath183 . the theoretical distribution from @xmath90 autoconvolutions of @xmath17 ( solid line histograms ) are compared with the corresponding numerical results : a)@xmath184 , b)@xmath185 , c)@xmath186 , d)@xmath187 . a good agreement is observed . ] we have observed a strong effect of finite @xmath1 and @xmath2 at the level of the conductance distribution . in general @xmath3 is narrower compared to the conductance distribution at @xmath188 . the average of the conductance is , however , independent of @xmath1 and @xmath2 ( see fig . [ fig_j ] ) . this implies that higher moments have to be analyzed to see the effect of the temperature and voltage . when temperatures and voltages are small ( less than the mean - level spacing ) , only one resonance is relevant to the conductance . in this regime , the distribution of the conductance is well described by a simplified resonant - tunneling model , eq . ( [ g_t_inter ] ) , as we have verified numerically . as the temperature and bias voltage is increased several resonances contribute to @xmath4 . in this regime , the conductance distribution @xmath3 can be obtained from the convolutions of the known distribution of conductance at zero temperature and bias voltage @xmath17 . the number of autoconvolutions of @xmath17 is determined by the width of the energy window where transport can take place . in the case of zero temperature and finite bias voltage , the width of the energy window is trivially @xmath10 and the number of convolutions is given by the mean number of levels @xmath189 . in the case of finite temperature and a finite bias voltage , we have simplified the problem by introducing an effective number of resonances that allow us to reduce the problem of finite @xmath1 and @xmath2 to the simpler case of zero temperature , and finite @xmath2 . the results of our theoretical method have been compared to numerical simulations of a 1d - disordered system at different regimes of temperature , voltage , and different values of the length of the system . a good agreement has been found in all cases . we point out that for small @xmath1 and @xmath2 the line shape of the resonances was relevant in the calculation of @xmath4 ( see section [ resonant_model ] ) . when the values of @xmath1 and @xmath2 are such that several resonances contribute to the conductance , the line shape is seen to be irrelevant . to conclude , we remark that with the resonant model and the convolution method of sections [ resonant_model ] and [ convolution_method ] , respectively , we are able to analyze the conductance distribution at all regimes of temperature and bias voltage , under the assumptions described in the paper . f. f is grateful to the hospitality of bifi in zaragoza and cab in bariloche where part of this work has been done , and to auip for a travel grant . m.j.s . and l.a . acknowledge financial support from pict 0311609 and ( m.j.s . ) from fundacin antorchas and pict 0313829 from anpcyt , argentina . l. a. and v. a. g. also acknowledge financial support from the ministerio de educacin y ciencia , spain , through the ramn y cajal program . this work was supported by grant `` grupo de investigacin de excelencia dga '' and ( l.a . ) bfm2003 - 08532-c02 - 01 from mceyc of spain . l.a and m.j.s . are staff members and f.f is fellow of conicet , argentina .

we calculate the distribution of the conductance @xmath0 in a one - dimensional disordered wire at finite temperature @xmath1 and bias voltage @xmath2 in a independent - electron picture and assuming full coherent transport . at high enough temperature and bias voltage , where several resonances of the system contribute to the conductance , the distribution @xmath3 can be represented with good accuracy by autoconvolutions of the distribution of the conductance at zero temperature and zero bias voltage . the number of convolutions depends on @xmath1 and @xmath2 . in the regime of very low @xmath1 and @xmath2 , where only one resonance is relevant to @xmath4 , the conductance distribution is analyzed by a resonant tunneling conductance model . strong effects of finite @xmath1 and @xmath2 on the conductance distribution are observed and well described by our theoretical analysis , as we verify by performing a number of numerical simulations of a one - dimensional disordered wire at different temperatures , voltages , and lengths of the wire . analytical estimates for the first moments of @xmath3 at high temperature and bias voltage are also provided .

cond-mat0606715

describing the laws of physics in terms of underlying symmetries has always been a powerful tool . lie algebras and lie superalgebras are central in particle physics , and the space - time symmetries can be obtained by an inn - wigner contraction of certain lie ( super)algebras . @xmath0lie algebras @xcite , a possible extension of lie ( super)algebras , have been considered some times ago as the natural structure underlying fractional supersymmetry ( fsusy ) @xcite ( one possible extension of supersymmetry ) . in this contribution we show how one can construct many examples of finite dimensional @xmath0lie algebras from lie ( super)algebras and finite - dimensional fsusy extensions of the poincar algebra are obtained by inn - wigner contraction of certain @xmath0lie algebras . the natural mathematical structure , generalizing the concept of lie superalgebras and relevant for the algebraic description of fractional supersymmetry was introduced in @xcite and called an @xmath0lie algebra . we do not want to go into the detailed definition of this structure here and will only recall the basic points , useful for our purpose . more details can be found in @xcite . let @xmath5 be a positive integer and @xmath6 . we consider now a complex vector space @xmath7 which has an automorphism @xmath8 satisfying @xmath9 . we set @xmath10 , @xmath11 and @xmath12 ( @xmath13 is the eigenspace corresponding to the eigenvalue @xmath14 of @xmath8 ) . hence , @xmath15 we say that @xmath7 is an @xmath0lie algebra if : 1 . @xmath16 , the zero graded part of @xmath7 , is a lie algebra . @xmath17 @xmath18 , the @xmath19 graded part of @xmath7 , is a representation of @xmath16 . 3 . there are symmetric multilinear @xmath20equivariant maps @xmath21 where @xmath22 denotes the @xmath0fold symmetric product of @xmath23 . in other words , we assume that some of the elements of the lie algebra @xmath16 can be expressed as @xmath0th order symmetric products of `` more fundamental generators '' . 4 . the generators of @xmath7 are assumed to satisfy jacobi identities ( @xmath24 , @xmath25 , @xmath11 ) : @xmath26,b_3\right ] + \left[\left[b_2,b_3\right],b_1\right ] + \left[\left[b_3,b_1\right],b_2\right ] = 0 , \nonumber \\ \left[\left[b_1,b_2\right],a_3\right ] + \left[\left[b_2,a_3\right],b_1\right ] + \left[\left[a_3,b_1\right],b_2\right ] = 0,\nonumber \\ \left[b,\left\{a_1,\dots , a_f\right\}\right ] = \left\{\left[b , a_1 \right],\dots , a_f\right\ } + \cdots + \left\{a_1,\dots,\left[b , a_f\right ] \right\ } , \nonumber \\ \sum\limits_{i=1}^{f+1 } \left [ a_i,\left\{a_1,\dots , a_{i-1 } , a_{i+1},\dots , a_{f+1}\right\ } \right ] = 0 . \label{rausch : eq : jac}\end{aligned}\ ] ] the first three identities are consequences of the previously defined properties but the fourth is an extra constraint . more details ( unitarity , representations , _ etc . _ ) can be found in @xcite . let us first note that no relation between different graded sectors is postulated . secondly , the sub - space @xmath27 @xmath28 is itself an @xmath0lie algebra . from now on , @xmath0lie algebras of the types @xmath29 will be considered . most of the examples of @xmath0lie algebras are infinite dimensional ( see _ e.g. _ @xcite ) . however in @xcite an inductive theorem to construct finite - dimensional @xmath0lie algebras was proven : + * theorem 1 * _ let @xmath30 be a lie algebra and @xmath31 a representation of @xmath30 such that _ \(i ) @xmath32 is an @xmath0lie algebra of order @xmath33;. in this case the notion of graded @xmath34lie algebra has to be introduced @xcite . @xmath35 , is a graded @xmath34lie algebra if ( i ) @xmath36 a lie algebra and @xmath31 is a representation of @xmath36 isomorphic to the adjoint representation , ( ii ) there is a @xmath37 equivariant map @xmath38 such that @xmath39 + \left[f_2 , \mu(f_1 ) \right ] = 0 , f_1,f_2 \in \ { g}_1 $ ] . ] \(ii ) @xmath40 admits a @xmath41equivariant symmetric form @xmath42 of order @xmath43 . then @xmath44 admits an @xmath0lie algebra structure of order @xmath45 , which we call the @xmath0lie algebra induced from @xmath46 and @xmath42 . + by hypothesis , there exist @xmath47equivariant maps @xmath48 and @xmath49 . now , consider @xmath50 defined by @xmath51 where @xmath52 and @xmath53 is the group of permutations on @xmath54 elements . by construction , this is a @xmath47equivariant map from @xmath55 , thus the three first jacobi identities are satisfied . the last jacobi identity , is more difficult to check and is a consequence of the corresponding identity for the @xmath0lie algebra @xmath46 and a factorisation property ( see @xcite for more details ) . an interesting consequence of the theorem of the previous section is that it enables us to construct an @xmath0lie algebras associated to _ any _ lie ( super)algebras . consider the graded @xmath34lie algebra @xmath56 where @xmath57 is a lie algebra , @xmath58 is the adjoint representation of @xmath57 and @xmath59 is the identity . let @xmath60 be a basis of @xmath57 , and @xmath61 the corresponding basis of @xmath58 . the graded @xmath34lie algebra structure on @xmath7 is then : @xmath62 = f_{ab}^{\ \ \ c } j_c , \qquad \left[j_a , a_b \right ] = f_{ab}^{\ \ \ c } a_c , \qquad \mu(a_a)= j_a,\end{aligned}\ ] ] where @xmath63 are the structure constants of @xmath57 , the second ingredient to construct an @xmath0lie algebra is to define a symmetric invariant form on @xmath64 . but on @xmath64 , the adjoint representation of @xmath65 , the invariant symmetric forms are well known and correspond to the casimir operators @xcite . then , considering a casimir operator of order @xmath66 of @xmath67 , we can induce the structure of an @xmath0lie algebra of order @xmath68 on @xmath69 . one can give explicit formulae for the bracket of these @xmath0lie algebras as follows . let @xmath70 be a casimir operator of order @xmath66 ( for @xmath71 , the killing form @xmath72 is a primitive casimir of order two ) . then , the @xmath0bracket of the @xmath0lie algebra is @xmath73 for the killing form this gives @xmath74 if @xmath75 , the @xmath0lie algebra of order three induced from the killing form is the @xmath0lie algebra of @xcite . the construction of @xmath0lie algebras associated to lie superalgebras is more involved . we just give here a simple example ( for more details see @xcite ) : the @xmath0lie algebra of order @xmath76 @xmath77 induced from the ( i ) lie superalgebra @xmath78 and ( ii ) the quadratic form @xmath79 , where @xmath8 is the invariant symplectic form on @xmath80 and @xmath81 the invariant symplectic form on @xmath82 . let @xmath83 be a basis of @xmath84 and @xmath85 be a basis of @xmath86 . let @xmath87 be a basis of @xmath88 . then the four brackets of @xmath7 take the following form @xmath89 it is interesting to notice that this @xmath0lie algebra admits a simple matrix representation @xcite : @xmath90 and @xmath91 . it is well known that supersymmetric extensions of the poincar algebra can be obtained by inn - wigner contraction of certain lie superalgebras . in fact , one can also obtain some fsusy extensions of the poincar algebra by inn - wigner contraction of certain @xmath0lie algebras as we now show with one example @xcite . let @xmath92 be the real @xmath0lie algebra of order three induced from the real graded @xmath34lie algebra @xmath93 and the killing form on @xmath94 ( see eq . [ eq:3-lie ] ) . using vector indices of @xmath95 coming from the inclusion @xmath96 , the bosonic part of @xmath97 is generated by @xmath98 , with @xmath99 and the graded part by @xmath100 . letting @xmath101 after the inn - wigner contraction , @xmath102{\lambda } } q_{\mu \nu } , & j_{4 \mu } \to \frac{1}{\sqrt[3]{\lambda } } q_{\mu } , \end{array}\end{aligned}\ ] ] one sees that @xmath103 and @xmath104 generate the @xmath105 poincar algebra and that @xmath106 are the fractional supercharges in respectively the adjoint and vector representations of @xmath107 . this @xmath0lie algebra of order three is therefore a non - trivial extension of the poincar algebra where translations are cubes of more fundamental generators . the subspace generated by @xmath108 is also an @xmath0lie algebra of order three extending the poincar algebra in which the trilinear symmetric brackets have the simple form : @xmath109 where @xmath110 is the minkowski metric . in this paper a sketch of the construction of @xmath0lie algebras associated to lie ( super)algebras were given . more complete results , such as a criteria for simplicity , representation theory , matrix realizations _ etc . _ , was given in @xcite . 9 rausch de traubenberg m and slupinski m. j. 2000 j. math . phys 41 4556 - 4571 [ hep - th/9904126 ] . rausch de traubenberg m and slupinski m. j. 2002 _ proceedings of institute of mathematics of nas of ukraine _ , p 548 - 554 , vol . 43 , editors a.g . nikitin , v.m . boyko and r.o . popovych , kyiv , institute of mathematics [ arxiv : hep - th/0110020 ] . rausch de traubenberg m and slupinski m. j. 2002 _ finite - dimensional lie algebras of order @xmath5 _ , arxiv : hep - th/0205113 , to appear in j. math . durand s 1993 mod . lett a 8 23232334 [ hep - th/9305130 ] . rausch de traubenberg m and slupinski m. j. 1997 mod . a 12 3051 - 3066 [ hep - th/9609203 ] . rausch de traubenberg m 1998 hep - th/9802141 ( habilitation thesis , in french ) . chevalley c and eilenberg s 1948 trans . 63 85 - 124 . ahmedov h , yildiz a and ucan y 2001 j. phys . a 34 6413 - 6424 [ math.rt/0012058 ] .

@xmath0lie algebras are natural generalisations of lie algebras ( @xmath1 ) and lie superalgebras ( @xmath2 ) . we give finite dimensional examples of @xmath0lie algebras obtained by an inductive process from lie algebras and lie superalgebras . matrix realizations of the @xmath0lie algebras constructed in this way from @xmath3 are given . we obtain a non - trivial extension of the poincar algebra by an inn - wigner contraction of a certain @xmath0lie algebras with @xmath4 .

hep-th0209144

37 com is the primary star of a wide triple system ( tokovinin 2008 ) , but the synchronisation effect plays no role for its fast rotation and activity . its significant photometric and caii h&k emission variabilities were presented by strassmeier et al . ( 1997 ; 1999 ) and de medeiros et al . ( 1999 ) and interpreted as signatures of magnetic activity . observational data for 37 com were obtained with two twin fiber - fed echelle spectropolarimeters narval ( 2 m tbl at pic du midi observatory , france ) and espadons ( 3.6 m cfht ) . we have collected 11 stokes v spectra for 37 com in the period january 2010 july 2010 . the least squares deconvolution ( lsd ) multi - line technique was applied and the surface - averaged longitudinal magnetic field b@xmath1 was computed using the first - order moment method ( donati el al . 1997 ; wade et al . the zeeman doppler imaging ( zdi ) tomographic technique was employed for mapping the large - scale magnetic field of the star ( donati et al . with radial velocity ( rv ) , s - index , h@xmath0 and caii irt ( 854.2 nm ) . * center : * normalized stokes v profiles observed profiles ( black ) ; synthetic fit ( red ) ; zero level ( dashes lines ) . the error bars are on the left of each profile . * right : * the magnetic map of 37 com . the magnetic field strength is in gauss . the vertical ticks on top of the radial map show the phases when there are observations.,title="fig:",width=151,height=226 ] with radial velocity ( rv ) , s - index , h@xmath0 and caii irt ( 854.2 nm ) . * center : * normalized stokes v profiles observed profiles ( black ) ; synthetic fit ( red ) ; zero level ( dashes lines ) . the error bars are on the left of each profile . * right : * the magnetic map of 37 com . the magnetic field strength is in gauss . the vertical ticks on top of the radial map show the phases when there are observations.,title="fig:",width=113,height=226 ] with radial velocity ( rv ) , s - index , h@xmath0 and caii irt ( 854.2 nm ) . * center : * normalized stokes v profiles observed profiles ( black ) ; synthetic fit ( red ) ; zero level ( dashes lines ) . the error bars are on the left of each profile . * right : * the magnetic map of 37 com . the magnetic field strength is in gauss . the vertical ticks on top of the radial map show the phases when there are observations.,title="fig:",width=226,height=226 ] there are significant variations of b@xmath1 in the interval from -2.5 g to 6.5 g with at least one sign reversal during the observational period ( fig . [ fig : zdi ] left ) . also , radial velocity , s - index and line activity indicators h@xmath0 and caii irt ( 854.2 nm ) show significant variations , and clear correlations with each other as well as the longitudinal field . the zdi mapping ( fig . [ fig : zdi ] center and right ) reveals that the large - scale magnetic field has a dominant poloidal component , which contains about 88% of the reconstructed magnetic energy . the star has a differential rotation with the following parameters : @xmath2 rad / d ( the rotation rate at the equator ) and @xmath3 rad / d ( the difference in the rotation rate between the polar region and the equator ) ( petit et al . 2002 ) . 37 com shows simpler surface magnetic structure than the fast rotators v390 aur ( konstantinova - antova et al . 2012 ) and hd 232862 ( aurire et al . in prep . ) and shows more complex structure than the slow rotators ek eri ( aurire et al . 2011 ) and @xmath4 ceti ( tsvetkova et al . 2013 ) , which are suspected of being descendants of ap - stars . the location of 37 com on the hertzsprung - russell diagram was determined on the basis of state - of - the - art stellar evolution models ( charbonnel & lagarde 2010 ) and the mass is found to be 5.25 @xmath5 , in a good agreement with the literature . synthetic spectra in the region containing @xmath6cn and @xmath7cn molecular lines were calculated and compared to our spectra in order to infer the @xmath6c/@xmath7c ratio . the best fit was achieved for @xmath6c/@xmath7c @xmath8 . from these results , it appears that 37 com is in the core helium - burning phase .

we present the first magnetic map of the late - type giant 37 com . the least squares deconvolution ( lsd ) method and zeeman doppler imaging ( zdi ) inversion technique were applied . the chromospheric activity indicators h@xmath0 , s - index , caii irt and the radial velocity were also measured . the evolutionary status of the star has been studied on the basis of state - of - the - art stellar evolutionary models and chemical abundance analysis . 37 com appears to be in the core helium - burning phase .

1310.8083

pilot systems play an important role in negative discharge development process . in metre - scale spark pilots appear as bipolar formations from a point in space called `` space stem '' and grow in both directions away from and towards the high - voltage electrode . in longer gaps pilots can transform into space leaders @xcite . such space leaders also appear in front of a lightning leader channel , grow in both directions and finally attach to the main channel . the stepped propagation is a common feature of both negative long laboratory sparks and lightning leaders . lightning leader steps are also closely associated with discrete intense bursts of x - ray radiation @xcite and may even be responsible for terrestrial gamma - ray flashes ( tgfs ) , as was previously suggested in @xcite and recently modeled in @xcite . similarly , as shown in @xcite , pilots are involved in x - ray burst generation in long laboratory sparks . thus , experimental study of pilot system formation and development can provide more information about lightning leader x - rays and tgfs . however , our understanding of pilots is mostly based on streak photographs obtained in the last century . the existence of bipolar structures in long laboratory discharges was first shown in 1960 s @xcite . in 1981 the les renardires group performed a fundamental study on negative discharges using various electrodes , gap distances and voltage rise times @xcite . this study provided the first systematic description of the various phases of negative discharge development . phenomena such as negative leader , space stem , pilot system , and space leader were photographed , identified and described . theoretical efforts and models to explain long atmospheric discharges were presented by gallimberti ( @xcite and citations therein ) but these do not explain how pilot systems form and develop @xcite . it is assumed that pilots appear from a `` space stem '' ahead of the leader tip in virgin air . in 2003 vernon cooray @xcite formulated the situation as follows : _ `` the pilot system consists of a bright spot called the space stem of short duration , from which streamers of both polarity develop in opposite directions''_. this is consistent with our observations . we will demonstrate below that pilots are preceded by negative streamer heads and often contain several bright spots . + in this work we first show the pilot system development in the laboratory with high spatial and temporal resolution from the very beginning till attachment to the hv electrode . it is demonstrated how a single negative streamer creates such a complex bipolar structure . then we propose a 1d model of the collective ionization front evolution that is capable to capture most important observed details , such as glowing beads , bipolarity and ionization front collisions . striking similarity between pilots and high - altitude sprites will be discussed at the end . the setup was available at the high - voltage laboratory in eindhoven university of technology from 2008 till 2014 but it is currently dismantled . a 2.4 mv haefely marx generator was used to create metre - long sparks . the voltage was set at 1 mv with 1.2/50 @xmath0s rise / fall time when not loaded . the generator was connected to a spark gap between two conical electrodes . the setup and all measuring equipment was exhaustively described in series of publications @xcite and only its optical part will be repeated here for consistency . to obtain images of the pre - breakdown phenomena between two electrodes , a ns - fast 4 picos camera @xcite was located at 4 m distance from the gap , perpendicular to developing discharge . the camera contains a charge coupled image sensor preceded by a fast switched image intensifier ( iccd ) . the image intensifier is a micro - channel plate that allows adjustment of the camera sensitivity by varying the applied voltage between 600 and 1000 v. the ccd is read out with 12-bit and 780x580 pixels resolution . lenses were either nikon 35 mm f2.8 fixed focus or sigma 70 - 300 mm f/45.6 zoom . the field of view of the camera covers the region below the hv electrode . the camera has a black and white ccd and is not calibrated . the applied color scheme is linked to the light intensity and intended to increase visual perception , but it does not represent the actual plasma temperature . the camera was placed inside an emc cabinet . appropriate shielding protected the camera and its communication cables against electromagnetic interference . more emc aspects of the setup have been discussed in @xcite . in 2009 cooray et al . proposed a mechanism of x - ray generation in long laboratory sparks @xcite . it was suggested that the x - ray bursts are caused by encounters of negative and positive streamer fronts . although the negative discharge development process was simplified in this model , the main idea of streamer encounter as the emission source has recently been experimentally supported . with positive high - voltage pulse , x - rays indeed appear at the moment when positive corona from the hv electrode merge with negative corona from the grounded electrode @xcite . many encounters between individual streamers of opposite polarity occur at this moment . the development of negative discharges is more complex . we photographed the x - ray source region with ns - fast camera , and showed that positive streamers appear in the vicinity of the negative hv electrode @xcite . the positive streamers originate from bipolar pilot systems and encounter nearby negative streamers . for more details and properties of the x - ray emission we refer to @xcite ; a statistical analysis is given in @xcite . it is assumed that the x - rays are generated by high energy electrons in bremsstrahlung process . the first attempt to measure such electrons directly has recently been published in @xcite . here it should be noted that a recent simulation of the x - ray emission questioned the streamer encounter mechanism @xcite . while it was confirmed that such encounters dramatically increase the electric field between two streamer fronts to values much higher than the cold runaway breakdown , the field persists only for several picoseconds due to rapid rise of electron density . the ionization quickly collapses the field , giving no time for the electrons to accumulate high energy . it is possible , however , that the action of electric field on electron acceleration was underestimated in that work , as discussed in subsection [ ssec : discuss_xray ] . nevertheless , in measurements the x - rays bursts and pilots coincide in space and time , so the precise role of the latter requires further investigation . for the sake of consistency , we reproduce here the pilot development as reported in @xcite , and discuss additional features . figure [ fig : collage ] shows the pilot system development process . every image was exposed for 50 ns and represents a single individual discharge at the moment of the most intense x - ray emission . the time delay between two consecutive discharges was at least 10 s. the depicted area is located below the hv electrode . the hv electrode tip is only visible in images ( b ) and ( f ) at the top in the middle . the electrical signals of the full discharge are shown in figure [ fig : plot ] , the voltage @xmath1 over the gap , the currents @xmath2 and @xmath3 through both electrodes and the x - ray signals . two labr@xmath4 scintillation x - ray detectors were placed next to each other . both simultaneously registered a 400 kev signal . all images of figure [ fig : collage ] are taken with 50 ns shutter time that fell within the x - ray time window , or between @xmath5s . the pilot systems occur at an advanced stage of the discharge development . returning to figure [ fig : collage ] , we observe the negative streamers ( 1 ) that originate from the hv electrode ( image ( a ) ) . some of them leave isolated beads ( 2 ) behind during the propagation . we will call them `` streamer beads '' or just beads ; the reader should not confuse these with bead lightning @xcite . in 2d images the beads appear at @xmath6 cm intervals . some beads become branching points of the negative streamer . eventually the channels and heads of the branched streamers form the negative streamer corona . the streamer propagation velocity is measured by two different techniques : ( i ) by measuring the streamer trace length in an image with known exposure time and ( ii ) by measuring the displacement of a streamer head comparing two consecutive short exposure images taken by two cameras . the velocities are measured for many different streamers in different discharges ; we further refer to @xcite for details on negative streamer and corona development . we found that the negative streamer heads propagate at @xmath710@xmath8 m / s , which is in agreement with previously reported in @xcite for 2 m gaps . since the projection into the camera plane always reduces distances , the actual distance and velocity is likely to be closer to the highest measured value than to the average . at the same moment , positive streamers start at the beads , either as single intense streak or as branches . the branches first move perpendicularly ( image ( c ) ) to the streamer channel driven by the streamer electric field , and then they start to propagate towards the hv electrode ( images ( b ) - ( f ) ) . this gives the branches the appearance of a stack of the greek letters @xmath9 . to substantiate the movement towards the electrode , we used two high - speed cameras triggered in sequence ; the results have earlier been published in @xcite and are reproduced here for consistency . figure [ fig : two_cameras ] shows two subsequent images made with 3 ns exposure time . the first image was placed on the red layer of an rgb picture , the second image was delayed by 10 ns with respect to the first one , and placed on the blue layer of the same picture . the arrows indicate the displacement from the red to the blue images . clearly , many streamers move towards the cathode . the positive streamer velocity is difficult to measure correctly due to the limited extension in space and apparent dependence on other factors , such as the proximity of the hv electrode . we estimated velocity of the positive streamer head at @xmath1010@xmath8 m / s , or half the speed of the negative ones . as a result , positive streamers appear brighter in the images than negative assuming equal intrinsic brightness . the entire structure - the negative corona , streamer beads with @xmath9s , is called a pilot system ( 4 ) in @xcite . pilots are encircled in images ( c ) - ( e ) by dashed ellipses . in our setup the last bead appears at @xmath11 cm distance from the tip . we can use the streamer velocity data of figure 6 in @xcite to go back to the moment that the negative streamer head was at this point ( 0.5 @xmath0s in figure [ fig : plot ] ) . then the applied voltage @xmath1 was 500 kv , or the local electric field @xmath12 kv / cm assuming a @xmath13-dependence for @xmath14 . this field is close to the so - called `` stability field '' @xmath15 @xcite . it also indicates that the streamer experiences a shortage in charge and electric field which hinders smooth propagation . the blockade may lead to beading . as a support for this suggestion we refer to @xcite where it was demonstrated that a shorter voltage rise time , i.e. larger @xmath16 , leads to smoother discharge development . the upper parts of figure [ fig : collage](c ) and ( d ) show many @xmath9s that are about to collide with negative streamers or the cathode . such collisions provide the kick - off for the few electrons that become run - away in the electric field and produce x - ray by bremsstrahlung a few nanosecond later @xcite . we observed high frequency cathode current oscillations @xcite simultaneously with the x - rays , and also attribute the oscillation to the collisions . figure [ fig : sketch ] summarizes the pilot system development . it starts with a negative streamer that leaves one or more beads behind , that may grow into @xmath9s , and that finally transforms into the bipolar structure . an attempt to combine this history with the streak photographs of @xcite is hindered by the space - time convolution inherent to streak images . to this adds the larger gaps of 2 and 7 m versus ours of 1 up to 1.5 m and the longer voltage rise time of 6 @xmath0s versus 1.2 @xmath0s . figure 6.1.5(b ) in @xcite shows at least three stationary beads at 10 times larger separation than ours that last for about 0.5 @xmath0s ; the beads seem to appear out of the blue in virgin air . our images demonstrate that a precursor streamer initiates the beads . though it is not apparent from the photographs presented here , previous observations on longer spark gaps show that the pilot system becomes a hot space leader if given sufficient time @xcite . figure 6.3.2 in @xcite shows that the instantaneous velocity of the space stem ranges from @xmath17 to @xmath18 m / s ; the higher velocity goes with the shorter voltage rise time . such space stem behavior can be explained by subsequent launching of positive streamers starting from the first bead to the last . naturally , this will appear as a moving space stem on streak photographs . two types of positive corona emanate from the beads . the first is a single positive streamer , for instance image(a ) in figure [ fig : positive_part ] ; the second , our stack of @xmath9s in image ( b ) , has many streamers . the single positive streamer is significantly brighter and wider than all other streamers . the @xmath9s are shown in detail in figure [ fig : positive_part ] image ( b ) , and also in figure [ fig : collage ] images ( c ) - ( f ) . the streamers follow the local electric field lines towards the hv electrode . there are no visible beads anymore , which indicates that they fade away quickly , in fact about 10 times quicker than those in figure 6.1.5(b ) of @xcite . the faint speckle trace is visible in images ( a ) and ( b ) and indicated by small arrows . in image ( a ) it runs from the hv electrode tip down in the middle of the picture . in image ( b ) it comes from the left upper corner and goes through the structure down . it is a camera artifact . the camera s electronic shutter is switched off during the final breakdown , but some light can still leak through it and appear in the images as a trace . it helps to identify the streamer that grows into the final spark . + figure [ fig : reconnection ] shows an example of the pilot reconnection . both types of pilot systems , as described above , are clearly visible . a reconnection between positive streamers in stp ambient air was previously shown in @xcite and possible mechanism proposed in @xcite . it is shown here that the reconnection also occurs between negative streamers , in this case interconnecting two pilot systems . the characteristic curvature of the streamer path and termination on the edge of another streamer channel indicates that they merged . numerical modeling of streamer development in air has currently reached a rather advanced stage . the streamers discharge has been modelled in full 3d space and based solely on microscopic physical mechanisms , e.g. @xcite . however , obtaining numerically in this way a fully developed branching fractal streamer pattern is computationally difficult and still under development . to study the streamer structures with limited computational resources , one can introduce macroscopic physics , i.e. , assumptions about the details of streamers which are not modelled microscopically @xcite . the `` pilots '' occur only at an advanced stage of a streamer discharge , by which we mean that the streamer corona have been fully developed and the individual streamers may have undergone possibly multiple branching . in order to understand the pilots , we are thus forced to follow the route of macroscopic ( simplified ) modeling . so that we can simplify the modeling , we make a rather crude assumption of spherical symmetry in the developed structured streamer discharge , with physical values being functions only of a single ( radial ) coordinate . the individual streamer branches are thus not considered but are treated collectively , and we represent the collective streamer effects in transverse ( angular ) direction in terms of the effective curvature , which is thus different from the curvature of individual streamer heads . the overall schematic representation of the model used is shown in figure [ fig : model_cartoon ] . we shall model the development of a discharge in quasi - electrostatic approximation , i.e. , neglecting the effects of electromagnetic waves . this can be done since the ratios of typical length and time scales , including the typical streamer velocities @xmath19 m / s ( given above in this paper ) , are much less than the speed of light . in principle , the relatively large @xmath20 m size of the electrode gap can give importance to electromagnetic effects of very fast phenomena happening at time scales @xmath211 ns , which could occur , e.g. , during the streamer collisions ; this is a topic for our future research . the quasi - electrostatic equations may be represented as : @xmath22 in these equations , @xmath23 , @xmath24 , @xmath25 are electric field , potential , and charge density , respectively , @xmath26 is the electric conductivity , @xmath27 is the electron density , @xmath28 and @xmath29 are effective ionization and attachment rates which describe propagation of streamers , which are different from the physical ( microscopic ) ionization and attachment rates ( denoted here by @xmath30 ) , and @xmath31 is the photoionization source . this model includes a row of simplifying assumptions . first , the electron mobility @xmath32 is assumed constant because it varies very little in a large range of electric fields , e.g. , from @xmath33 m@xmath34s@xmath35v@xmath35 at @xmath36 to @xmath37 m@xmath34s@xmath35v@xmath35 at @xmath38 , where @xmath39 v / m is the electric breakdown field @xcite . we may therefore justify our assumption of constant electron mobility by arguing that the discharge - related processes occurring at @xmath40 are much less important than at fields @xmath41 . second , we neglect ion conductivity due to the fact that ion mobility is at least by a factor of @xmath42 smaller @xcite . third , we neglect electron advection effects , since the velocity of electron drift is much smaller than the streamer velocity . this , in particular , leads to having no difference in propagation of positive and negative ionization fronts in our model , while the observed velocities differed by a factor of @xmath432 in the same background field . fourth , we neglected electron diffusion @xmath44 , valued at @xmath45@xmath46 m@xmath34/s in the range of electric fields of interest @xcite , because the characteristic kolmogorov - petrovskii - piskunov velocity of the ionization @xmath47 m / s @xcite is also much smaller than the observed streamer velocities . the electric breakdown occurs above @xmath48 , where the ionization prevails over attachment . however , in a 1d situation the propagation of negative streamers occurs above the negative streamer sustainment field @xmath49 , which is lower than @xmath48 . thus , we choose the functional dependence of the effective 1d rates @xmath50 in such a way that the ionization occurs at @xmath51 v / m @xcite . we approximate these rates with power functions : @xmath52 where @xmath53 ns is the typical ionization time . we note that the empirical data for the physical values of @xmath30 @xcite may also be fitted with power - law functions ( [ eq : nu_powerlaw ] ) , but we of course must use @xmath48 instead of @xmath49 in these formulas . in the case of the physical values @xmath30 , the best - fitting coefficients are @xmath54 and @xmath55 . another source of ionization growth is @xmath31 , the photoionization , which is one of the mechanisms responsible for ionization front propagation ( the other mechanisms being the neglected electron advection and electron diffusion ) . the photoionization is a non - local source @xmath56 where @xmath57 is the `` local '' ionization rate . we model it as the `` exponential profile '' model @xcite : @xmath58 where @xmath59 and @xmath60 may be understood as the characteristic `` length '' and the `` strength '' of photoionization , respectively . this model is chosen for computational efficiency , because we can find @xmath31 as the solution of helmholtz equation : @xmath61 the electrode gap is modelled as 1-dimensional interval @xmath62 $ ] , where the emitting electrode ( the cathode in the case of the negative discharge study considered here ) is at @xmath63 , while the grounded electrode is at @xmath64 m. the system is assumed to be symmetric in the transverse direction . voltage @xmath65 is applied at @xmath63 , while @xmath66 is held at @xmath67 . the discharge is started by small initial ionization at @xmath68 , @xmath63 . the curvature of ionization front @xmath69 is included through the expression for divergence of vectors @xmath70 : @xmath71 for example , a spherically - symmetric system may be modelled by taking @xmath72 . for the results presented here , we take a constant value @xmath73 for simplicity , which is equivalent to the transverse area of the discharge growing exponentially with distance . this may be justified by the streamers branching repeatedly with a fixed interval , and the transverse area being proportional to the number of streamers . in the process of the nonlinear development of ionization growth , the ionization developed into multiple persistent peaks ( seen in figure [ fig : reverse_streamers ] ) with complex dynamics ( i.e. , moving in both directions ) which may be interpreted as various luminous features of streamers . in particular , the first peak in an ionization wave may be interpreted as the streamer head , while the consequent peaks , which move in the same direction or become stationary , may be interpreted as beads . under some conditions ( specified in the next paragraph ) we observed a particular class of such peaks , which exhibited the following stages of evolution : ( 1 ) an ionization peak appears at the most advanced point of the ionization wave ; ( 2 ) the ionization is extinguished in the part of the gap between this peak and the emitting electrode ; ( 3 ) as the voltage at the emitting electrode increases further , a reverse ionization wave separates from the peak and moves towards the emitting electrode , while a direct ionization waves moves towards it , until they collide ; ( 4 ) after even further voltage increase , the ionization peak continued its movement toward the grounded electrode . in view of the observations reported above , these ionization peaks may be interpreted as the pilots which exhibit similar behavior , namely that they launch ionization waves in both directions : positive streamers towards the emitting electrode ( cathode ) and the negative streamers towards the grounded electrode ( anode ) . figure [ fig : reverse_streamers ] presents a snapshot of the process described above ( at stage 3 ) . note that we used a slower - growing shape of ionization function ( @xmath74 instead of 5.5 ) . the higher values of power coefficient @xmath75 create steeper ionization edges . the simulations with other values of @xmath75 were also performed ( e.g. , @xmath76 , @xmath77 ) and they also produce similar results ( i.e. , formation of the `` reverse streamers '' and the hv electrode current pulsations , see below ) . the chosen lower value of @xmath75 in a 1d case creates a smoother front , and thus simulates the uncertainty in the position of the individual streamer heads ( they may be distributed around some average position in @xmath78 ) ; unfortunately , the exact value of @xmath75 which is best suited for this is not known to us at the current stage of research . we also chose the value of the photoionization length @xmath59 so that the simulation produces approximately the observed streamer velocities , while the value of @xmath79 is about the same as can be obtained from experimental data @xcite . ) $ ] kv , @xmath80 m , @xmath81 , @xmath82 , @xmath74 , @xmath83 . arrows show the apparent movement of ionization enhancements . ] as we see , the variations in the ionization rate functional dependencies on @xmath84 did not change qualitatively the result of having `` islands '' of ionization and reverse ionization waves ( stages 23 described above ) . by varying other parameters we preliminary conclude that this stages only appear when ( 1 ) the voltage has a stage when it gradually increases with time and ( 2 ) the photoionization effect is rather large . although the propagation of streamers does need a large photoionization , and the `` islands '' of stage 2 appeared even at small values of @xmath79 , the reverse ionization waves ( stage 3 ) only appeared when @xmath79 exceeded a certain threshold . beside pilots and reverse streamers , another interesting outcome of the presented 1d model were the quasi - periodic current pulses at the high - voltage electrode ( cathode ) , shown in figure [ fig : current_pulses ] , which reproduce , at least qualitatively , the experimentally observed pulses shown in figure [ fig : plot ] . the current was calculated taking into account both conductivity and displacement currents : @xmath85 where the area of the cathode @xmath86 m@xmath34 is chosen so that the current is of the same order as those in figure [ fig : plot ] . , except @xmath87 m was taken in order to capture more pulses . ] in the past , the pilots and stepping mechanism were studied numerically in long negative sparks ( leaders ) by 1d modeling @xcite . however , these models treated separately the stages of the discharge which lead to the development of the pilots and stepping and therefore also take into account transition from the streamer corona to a leader discharge . in contrast to these models , we demonstrated that the leader development is not necessary for the pilot formation , and the similar stepping stages appear automatically from the nonlinear nature of the system without artificially subdividing the process . the colliding individual streamers were considered by @xcite in the context of x - ray production by high electric field , but the streamer configuration was taken as an input on the basis of previous results and only @xmath84 field was calculated . the modeling of a streamer collision , taking into account the microscopic physical processes , was also performed by @xcite . we emphasize that , in contrast to the presented model , the last two works describe collision of two pre - existing streamers and do not deal with the development of the global streamer discharge system . in most existing theoretical treatments ( see chapter 12.3 in @xcite ) the streamer channel is understood as a linear structure along which the ionization ( and therefore the electric field and current ) vary monotonously . for example , the ionization is highest at the head of the streamer channel and gradually decreases in the backward direction . however , the high - speed videos of sprites @xcite reveal luminous `` beads '' which move along each individual streamer channel . we may therefore propose a hypothesis that a streamer channel is not monotonous but has peaks of enhanced conductivity and/or field which manifest themselves as `` beads '' . the presented simulation results supports this hypothesis . namely , there are multi - peak structures in figure [ fig : reverse_streamers ] even when the peaks are moving in the same direction ( so that they are not parts of disconnected streamers ) . however , another interpretation of multi - peak structures in the simulation results is also possible . namely , since we modelled a whole group of streamers , the peaks may be just heads of separate monotonous streamer channels or groups thereof . the beads and periodic structures in ionization waves were attributed to the attachment instability by @xcite , who presented a linear - wave analysis of this phenomenon in their appendix a. the mechanisms included in the two models are slightly different ( e.g. , electron advection in @xcite vs. the effective curvature in the present work ) , so this similarity requires further investigation . pilots strikingly resemble high - altitude discharges known as sprites @xcite . such peculiar features as glowing beads , streamer branching on beads , counter - propagating streamers originated from the beads , difference in brightness of the top and bottom parts and finally reconnection , characterise both phenomena . for visual comparison we refer to figure 16 in @xcite . two known types of sprites , carrot and column , can be directly compared to two types of pilots , as shown in section [ sec : positive_part ] . similarity is clear , despite the fact that in the laboratory pilots are pointed towards the sharp electrode tip , while sprites in nature originate from a dispersed charge region and appear vertically in 2d images . the last concern of scientific community regarding different polarity of pilots and sprites has recently been eliminated . it has been known since 1999 that sprites are not uniquely associated with positive cloud - to - ground ( @xmath88cg ) lightnings , but can also be triggered by negative @xmath89cg flashes @xcite . however , the community continued to be skeptical in accepting sprite polarity asymmetry and existence of `` negative sprites '' [ private conversations at agu / egu meetings ] . five more evidences of sprites following a negative @xmath89cg discharge were published in 2016 @xcite . with simple considerations put forward in the next subsection , we can not yet explain the existence of pilots with opposite polarity . in this case , new experimental studies of positive laboratory discharges with different voltage rise times are highly desirable . as it was mentioned in the introduction , the pilots may transform into space leaders @xcite . we may also speculate how pilots determine the polarity asymmetry between the modes of propagation of negative and positive leaders . let us consider the conditions for formation of a system of forward and reverse streamers moving towards each other , such as one that appeared during stage 3 of the simulation described in section [ ssec : model_results ] , and consider the differences for two different leader polarities . in a negative leader discharge , the electrostatic field in front of the leader converges towards its tip . the negative forward streamers , being closer to the electrode , experience a higher field than the positive reverse streamers that are initiated from a position further away from the leader tip and are on their way to encounter the negative forward streamers . this is consistent with the fact that negative streamers need a higher field to support their propagation than the positive streamers . thus , both forward and reverse streamers may exist at the same time . on the other hand , in a positive leader discharge , the forward streamers are positive while the reverse streamers ( if they appeared ) would be negative . we see that the lower field , which the reverse negative streamers would experience , can not support their propagation . thus , we do not expect formation of reverse streamers in the positive - leader case , the ionization gap ( if it appeared ) would be filled only by forward positive streamers . the role of the decreasing field in the differences between positive and negative leader propagation was also discussed by @xcite , where they suggested that in the negative leader corona the forward - moving electrons attach in the lower field , which interrupts the current and leads to stepping . another implication of experiments and modelling is that the positive and negative streamers , which travel towards each other , will collide at a certain moment of the discharge . this may lead to a increased electric field in the gap between them , which subsequently leads to generation of high - energy electrons @xcite . it has been proposed as the possible mechanism of x - ray production in laboratory spark discharges @xcite . this mechanism has been modelled by @xcite who found that the number of x - ray photons produced may not be sufficient to explain observations . however , we think that the electric field in @xcite has been underestimated . the boundary conditions with fixed potential , represent a perfectly conducting boundary and lead to image charges , whose field reduces the field in the modelling domain . thus , the actual field may be higher and also occupy a bigger volume . on the other hand , a higher field between the two colliding streamers would also lead to an increased streamer velocity , which would reduce the time of the existence of the region with high field . since there are two counteracting effects , it is not clear without repeating the full simulation whether corrected boundary conditions would lead to increase of x - ray production . here , we may note that the observations confirmed the coincidence of occurrence of x - rays in laboratory sparks with colliding positive and negative streamers @xcite , even if the exact mechanism of electron acceleration is still open for discussion . in this work we first tracked a single pilot system development in the laboratory between two conical electrodes under 1 mv applied voltage . it was demonstrated for the first time that pilots do not develop from `` nowhere '' , as was thought before @xcite , but from isolated streamer beads , created in the wake of the negative streamer head . the beads , in principle , can be called a `` space stem '' for consistency with the previous studies , but it is important to highlight that they do not appear in virgin air , but behind a negative streamer . the 1d model of the ionization front evolution demonstrated that such beads and reverse ionization waves can appear with certain photoionization parameters . however , we have not yet demonstrated the differences in the discharge polarity , as the electron drift was neglected ; this is a subject of a future work . taking into account many similarities between pilot systems and sprites , not only in appearance but also in progression , we conclude that these are two manifestations of the same phenomenon . it is very desirable to investigate the pilot system development under different conditions , i.e. temperature , pressure , voltage rise time etc . in addition , the main modelling parameters as electron and photon density , their energy spectrum , the local e - field and potential with respect to ground remain hidden in all measurements . some specially designed probes would solve some ambiguities . this work was supported by the european research council under the european union s seventh framework programme ( fp7/2007 - 2013)/erc grant agreement n. 320839 and the research council of norway under contracts 208028/f50 and 223252/f50 ( coe ) . pavlo kochkin acknowledges financial support by stw - project 10757 , where stichting technische wetenschappen ( stw ) is part of the netherlands organization for scientific research nwo . van deursen a and kochkin p 2015 some emc aspects of a 2 mv marx generator with sensitive diagnostic equipment in the immedeate vicinity _ 2015 ieee international symposium on electromagnetic compatibility ( emc ) _ ( ieee ) pp 13881391

the pilot system development in metre - scale negative laboratory discharges is studied with ns - fast photography . the systems appear as bipolar structures in the vicinity of the negative high - voltage electrode . they appear as a result of a single negative streamer propagation and determine further discharge development . such systems possess features like glowing beads , bipolarity , different brightness of the top and bottom parts , and mutual reconnection . a 1d model of the ionization evolution in the spark gap is proposed . in the process of the nonlinear development of ionization growth , the model shows features similar to those observed . the visual similarities between high - altitude sprites and laboratory pilots are striking and may indicate that they are two manifestations of the same natural phenomenon . version of +

1701.03300

the be / x - ray systems represent the largest sub - class of all high mass x - ray binaries ( hmxb ) . a survey of the literature reveals that of the @xmath0240 hmxbs known in our galaxy and the magellanic clouds ( liu et al . , 2005 , 2006 ) , @xmath150% fall within this class of binary . in fact , in recent years it has emerged that there is a substantial population of hmxbs in the smc comparable in number to the galactic population . though unlike the galactic population , all except one of the smc hmxbs are be star systems . in these systems the orbit of the be star and the compact object , presumably a neutron star , is generally wide and eccentric . x - ray outbursts are normally associated with the passage of the neutron star close to the circumstellar disk ( okazaki & negueruela 2001 ) , and generally are classified as types i or ii ( stella , white & rosner , 1986 ) . the type i outbursts occur periodically at the time of the periastron passage of the neutron star , whereas type ii outbursts are much more extensive and occur when the circumstellar material expands to fill most , or all of the orbit . this paper concerns itself with type i outbursts . general reviews of such hmxb systems may be found in negueruela ( 1998 ) , corbet et al . ( 2008 ) and coe et al . ( 2000 , 2008 ) . this paper reports data acquired over 10 years using the rossi x - ray timing explorer ( rxte ) on the hmxb population of the smc . during the period of these observations there have been many opportunities to study spin changes arising from accretion torques . this extremely homogeneous population permits the first high quality tests to be carried out of the work of ghosh & lamb ( 1979 ) and joss & rappaport ( 1984 ) . the simplified source naming convention used in this work follows that established by coe et al ( 2005 ) . the smc has been the subject of extensive monitoring using the rxte proportional counter array ( pca ) over the last 10 years . the pca instrument has a full width half maximum ( fwhm ) field of view of @xmath2 and data in the energy range 3 - 10 kev were used . most of the observations were pointed at the bar region of the smc where the majority of the known x - ray pulsar systems are located . sources were identified from their pulse periods using lomb - scargle ( lomb 1976 , scargle 1982 ) power spectral analysis of the data sets . laycock et al . ( 2005 ) and galache et al . ( 2008 ) present full details of the data analysis approach that has been used to determine which pulsars were active during each observation . in their work , for each x - ray outburst , the pulse amplitude and history of the pulse periods were determined . those results are used in this work . since a database of @xmath110 years of observations of the smc exists it was therefore possible to use these data to search for evidence of spin period changes in the systems . the pca is a collimated instrument , therefore interpreting the strength and significance of the signal depends upon the position of the source within the field of view . in all the objects presented here the target was assumed to be located at the position of the known optical counterpart . only observations that had a collimator response @xmath125% and a detection significance of @xmath199% were used in this work . a total of 24 sources were chosen for this study . in each case three possible measurements of period changes were obtained : * individual active periods lasting typically 50 - 500 days were studied and used to determine the @xmath3 for a particular source data subset ( referred to in this work as the short@xmath3 ) . a simple best fit straight line to the pulse history plot was determined using a @xmath4 minimising technique . no attempt was made to fit more complicated profiles to the data , though in some cases higher order changes are suggested . an excellent example of the spin period changes seen in these systems is presented in figure [ fig1 ] which shows two outbursts from sxp59.0 . clearly both outbursts indicate an initially higher @xmath3 which levels off as the activity period progresses , but only the weighted average is used in this work . one other factor that could also modify the spin period history would be doppler - related changes . however , attempts to fit period histories with binary models have always proved difficult ( but see , for example , schurch et al , 2008 , for one possible success ) suggesting that the changes are dominated by accretion driven variability . * in addition to the short@xmath3 , where possible a longer term value was determined for each source from the whole data set covering @xmath010 years of observing - referred to here as the long@xmath3 . this typically included several periods of source activity with significant inactive gaps in between . * for many sources the orbital period is clearly apparent in the sequence of x - ray outbursts . for others , optical data from the optical gravitational lensing experiment ( ogle ) project ( udalski , kubiak and szymanski 1997 ) have been used with good success to discover the orbital modulation - see section 3.4 below for further discussion on this point . it was not always possible to determine both a short@xmath3 and long@xmath3 for every source in this work due to several possible reasons ; one being the observational coverage and another the activity history of the system . details of the recorded spin period changes are given in table [ rxte ] . full records of the behaviour of each source may be found in galache et al . ( 2008 ) . the strong link between the equilibrium spin period and the rate of change of spin period seen during outbursts is shown in figure [ fig2 ] . in this figure , the straight line represents @xmath3 = k@xmath5 - as predicted for accretion from a disk on to a neutron star ( see equation 15 in ghosh & lamb , 1979 ) . it is interesting to note that the four spin - down systems ( sxp8.80 , sxp59.0 , sxp144 & sxp1323 ) fit comfortably on this relationship alongside the much larger number of spin - up systems . to pursue the understanding of the accretion process further , a value for the x - ray luminosity , @xmath6 , is needed during each outburst . so the x - ray luminosity was calculated from the observed peak pulse amplitude in counts / pcu / s that occurred during the outburst that produced the short@xmath3 values listed in table [ rxte ] . this amplitude is converted to luminosity assuming that 1 rxte count / pcu / s = 0.4@xmath7 erg / s at the distance of 60kpc to the smc ( though the depth of the sources within the smc is unknown and _ may _ affect this distance by up to @xmath810kpc ) . the x - ray spectrum was assumed to be a power law with a photon index = 1.5 and an @xmath9 . furthermore it was assumed that there was an average pulse fraction of 33% for all the systems and hence the correct total flux is 3 times the pulse component . thus the values shown in table [ rxte ] were determined using the relationship : + + @xmath10 = 0.4 @xmath11 @xmath12 3@xmath13 erg s@xmath14 + + where @xmath13 = rxte counts pcu@xmath14 s@xmath14 note that though the values for @xmath6 obviously scale linearly with pulsed fraction , the neutron star mass determinations discussed below in section 3.3 are very insensitive to any value between 10% and 60% for the pulsed fraction . [ cols="^,^,^,^,^,^,^,^,^ " , ] figure [ fig3 ] ( upper panel ) shows a histogram of all the known pulse periods for accreting pulsar systems in the smc ( see galache et al , 2008 for details of the vast majority of the sources ) . because of the length of the each observation ( typically 10ks - 15ks ) there is undoubtedly an instrumental cut - off starting around @xmath11ks affecting the ability to detect longer periods . the sharpness of this cutoff is partially a function of the pulse profile . at the short period end , the data were regularly searched down to periods of 0.5s ( galache et al , 2008 ) and should be complete down to that number . the lower panel in figure [ fig3 ] shows the distribution of known binary periods determined from either the sequence of x - ray outbursts and/or a coherent modulation of the optical light from the system . figure [ fig4 ] shows a histogram of the peak outbursts observed from the systems studied in this work . the significantly larger outburst seen from sxp2.37 is excluded from this plot - this one outburst may well be tending towards a type ii outburst . it is interesting to note that of the 15 systems for which short term @xmath3 changes were measured , 4 ( sxp8.80 , sxp144 , sxp59.0 and sxp1323 ) show spin - downs during outburst rather than spin - ups . such reverse torques have been reported before and even transitions from spin - up to spin - down observed for example , 4u1626 - 67 ( bildsten et al , 1997 ) . no such transitions are observed in the sample presented here , but the numbers do give a clear indication of the frequency with which these reverse torques occur . in addition , it is obviously a phenomenon that is not restricted to a particular spin period regime since our three object have periods ranging from 8.9s to 1325s . five further systems show long term spin - down changes - see table [ rxte ] . however , all these other systems show spin - up during outbursts and the long - term change may be attributed to the gradually slowing down of the neutron star in the absence of repeated accretion episodes , rather than reverse torques . . the relationship between short spin period changes ( seen during an outburst of typically 1 - 2 months ) and the long term period changes seen over @xmath010 years worth of study is shown in figure [ fig5 ] . the sources broadly follow a relationship indicating that long@xmath3 = 0.2 @xmath12 short@xmath3 . so individual outbursts typically spin up ( or down ) a neutron star @xmath05 times faster than the longer , time - averaged changes . in reality , the neutron star is subjected to a series of `` kicks '' during each short outburst , followed by an intervening recovery period . this effect is illustrated clearly in figure [ fig1 ] . ghosh & lamb ( 1979 ) established the relationships for accretion onto neutron stars formulating the now well known relationship between @xmath3 and x - ray luminosity ( see equation 15 in their paper ) . this relationship was re - iterated by joss & rappaport ( 1984 ) and others , with all sets of authors comparing the theoretical predictions with available data from galactic accreting pulsar systems . the quality of the observational data used was very variable , with many objects represented simply by upper limits . in this work the consistently higher quality of the measurements allows us to compare theory and observation much more precisely . this comparison is shown in figure [ fig6 ] , where it is immediately obvious that the data provide strong support for the models over two orders of magnitude . the probability of getting this level of correlation with an uncorrelated data set is 2.5@xmath15 . from the results presented here there is a strong support for the average mass of neutron stars in these systems to lie between 1.3 and 1.9 . in fact , if the sample shown in figure [ fig6 ] is used , then the average mass of the neutron stars in the smc sample is found to be 1.62@xmath80.29m@xmath16 . the only significant deviation from the average value is that of sxp7.78 ( = smc x-3 ) which strongly suggests a much lower neutron star mass for that system . sxp59.0 has the largest error bars of all the points and lies @xmath02@xmath17 away from the m=1.9m@xmath16 ; hence is probably not significant measurement of a large neutron star mass . there are currently 56 systems with known pulse periods , but only 27 with confirmed binary periods . many of the binary periods have been determined or confirmed from ogle iii long term data which frequently show evidence of modulation believed to be at the binary period . results such as those for sxp46.6 ( mcgowan et al , 2008 ) demonstrate the strength of this link by revealing regular optical outbursts coincident with x - ray outbursts . however , for many of the smc systems the binary period has yet to be confirmed and , in particular , the longer term periods ( @xmath11 - 2 years ) are increasingly difficult to ascertain . this may be due to either missing x - ray outbursts or / and a lack of optical modulation due , perhaps , to the size and shape of the orbit . however the period histogram should reflect accurately the distribution of pulse periods found in accreting pulsar systems ( at least in the smc ) . there is obviously a strong preference to forming systems with spin periods of @xmath18100s . since the corbet diagram ( corbet et al . , 2008 ) provides strong evidence that the equilibrium spin period is driven by the orbital period , then this spin period result probably reflects the relative difficulty in long orbital period systems surviving the supernova explosion that produces the neutron star . this is supported by the binary period distribution seen in the lower panel of figure [ fig3 ] ( but with all the observational caveats mentioned above ) . it is also worth noting that there is no evidence from the corbet diagram that smc sources show any difference from their counterparts in the milky way ( corbet et al . this work presents results from @xmath010 years of x - ray monitoring of the pulse period histories of 24 hmxb systems in the smc . this homogenous group of objects provides an excellent test bed for accretion theory and , in general , the results are very supportive of current models . since current models have all been developed based upon milky way systems , this work strongly supports the conclusion that there is little , or no difference between hmxb behaviour in the smc and the galaxy . we are grateful to lee townsend for help with some of the data analysis . in addition , we are grateful to the helpful comments of the referee which enhanced the robustness of this paper . corbet , r. h. d. , coe , m. j. , mcgowan , k. e. , schurch , m. p. e. , townsend , l. j. , galache , j. l. , marshall , f. e. 2008 the magellanic system : stars , gas , and galaxies , proceedings of the international astronomical union , iau symposium , volume 256 , p. 361 - 366 .

the understanding of the accretion process on to compact objects in binary systems is an important part of modern astrophysics . theoretical work , primarily that of ghosh & lamb ( 1979 ) , has made clear predictions for the behaviour of such systems which have been generally supported by observational results of considerably varying quality from galactic accreting pulsar systems . in this work a much larger homogeneous population of such objects in the small magellanic cloud ( smc ) is used to provide more demanding tests of the accretion theory . the results are extremely supportive of the theoretical models and provide useful statistical insights into the manner in which accreting pulsars behave and evolve . [ firstpage ] stars : neutron - x - rays : binaries

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currently , one of the key quests of astrophysics is to understand and model the processes that guide the formation and evolution of galaxies . great strides have been made over the past few decades and with the advancement of technology , such as ever larger telescopes taking ever larger surveys of thousands of galaxies within an ever larger redshift range ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the advent of new techniques such as gravitational lensing ( e.g. * ? ? ? * ; * ? ? ? * ) , and galaxy surveys using integral field spectroscopy ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? recent observational evidence suggests that the star formation rate of the universe peaked at @xmath5 and that by @xmath3 half of the stellar mass of the universe today was already in place @xcite . the decreasing star formation rate , referred to as quenching , is mass dependent with the more massive galaxies being quenched earlier . also , the comparison of the most massive galaxies ( @xmath6 ) at high and low redshifts show that these quiescent galaxies have undergone a size evolution ; with the size of the galaxies increasing with decreasing redshift @xcite . this size evolution has been associated with minor mass growth , suggesting that these growths may be driven by minor merger where the size of the galaxy grows to the second power of the added mass through virial arguments , unlike major mergers where the size grows linearly to the increase in mass @xcite . additionally , recent works have pointed out that a significant part of the observed size growths in the populations of quiescent galaxies , especially at lower masses , may be due to progenitor bias , wherein the addition of large recently quenched galaxies contribute to the observed increase in the mean size of the population ( e.g. * ? ? ? * ; * ? ? ? regardless of what the process for the growth of the galaxy size , and its stellar mass may be , there is strong evidence indicating that , for the most massive galaxies , most of the additional stellar mass is added to the outskirts of the galaxies , while the central regions remain mostly unperturbed @xcite . the end result of this merging process are the most massive galaxies in the nearby universe which are found to be slowly rotating @xcite , they have cores in their surface brightness profiles @xcite , and are embedded in extended stellar envelopes @xcite . the situation appears radically different for less massive ( @xmath7 ) passive galaxies . at the present day , they are structurally different , and appear to have followed a different evolution path @xcite . they are axisymmetric @xcite , they contain disks like spiral galaxies @xcite and are dominated by rotation @xcite . these fast rotating galaxies follow the same mass - size relation , and have the same mass distribution , both in dense clusters as in the field @xcite , indicating they experienced an insignificant amount of merging during their evolution , in agreement with redshift evolution studies @xcite . due to the recent advances in the techniques of stellar population modelling and redshift surveys , a key addition to this emerging picture of galaxy evolution is provided by studies of the stellar populations of galaxies through cosmic time . the work of @xcite , using spectra from the deep2 survey @xcite , compared to local sdss @xcite results , suggests that the evolution of the red - sequence galaxy population is not consistent with a passive evolutionary model . instead , they propose that the red - sequence population should either continue to host some level of star formation ( `` frosting '' ) to present day or have newly quenched galaxies joining the red - sequence galaxies between @xmath8 and today . @xcite study quiescent high redshift galaxies via a full spectrum fitting of stacked galaxy spectra to derive the stellar ages and elemental abundances of fe , mg , c , n and ca . the work uses optical spectra of local galaxies taken from the sdss and spectra from the ages @xcite survey within a redshift range of @xmath9 . they find negligible evolution in elemental abundances at fixed stellar mass . for the most massive galaxies they measure an increase in stellar age consistent with passive evolution since @xmath10 . while at masses below @xmath11 , the data permit the addition of newly quenched galaxies . * hereafter g14 ) study a sample of 70 quiescent and star - forming galaxies at @xmath10 , above a stellar mass of @xmath12 . they derive the stellar age - mass relation of the galaxies , which they compare with the one derived in a similar manner in the local universe . they find that taken as a whole , passive evolution can not represent the evolution of galaxies in the last @xmath13 gyr . in fact , although the shape of the stellar age - mass relationship between the two redshifts is similar , the offset is inconsistent with passive evolution . this is agreement with their observed metallicity differences with redshift . they propose a mass - dependent star formation history ( sfh ) to explain the observations . here we use full - spectrum fitting to explicitly determine trends in the star formation history of a sample of 154 galaxies at @xmath14 . furthermore , we investigate the correlation between the stellar population and the physical parameters of the galaxies . we also present results on the dynamical modelling of a subset of 68 galaxies . this subsample is the same we analysed in our previous work @xcite , where we studied the initial mass function ( imf ) mass normalisation and concluded it is consistent with a @xcite slope . here , we improve upon the dynamical models by accounting for the dark matter of the galaxies via abundance matching techniques . in section 2 of the paper , we describe the observational data that we use within this study while in section 3 we discuss the selection criteria that have been implemented in the course of this analysis , along with a comparison of our galaxy sample with the parent sample . in section 4 , we present the various methods used to analyse the data followed by a discussion of the results in section 5 . in section 6 , we provide a summary of the results . in this study , we assume a flat universe with the following cosmological parameters : @xmath15 , @xmath16 , @xmath17 mpc@xmath18 . the spectral data used in this study was taken from the deep2 survey . the survey is an apparent magnitude limited , @xmath19 , spectroscopic redshift survey of 4 fields covering an area of @xmath20 deg@xmath21 across the sky characterised by low extinction . the observation targets for the survey were selected using an algorithm based on the canada - france - hawaii telescope ( cfht ) observed object size , luminosity , ( b - r ) and ( r - i ) colours @xcite and a surface brightness cut - off which eliminated wrongly classified objects @xcite . of the four fields of deep2 , only the extended groth strip ( egs ; @xcite ) has the required high resolution _ hst _ data to construct dynamical models and accurately measure galaxy siz . hence this study is restricted to this field of the deep2 . the observations for the survey were done using the deimos spectrograph on the keck-2 telescope , covering an observed wavelength range of 6,100 - 9,100 . the highest resolution grating , 1,200 line mm^-1^ , was used to achieve a spectral resolution of r @xmath22 within the @xmath23 slitlets of the instrument . the observations were taken in 3 exposures of 20 minutes each , with the fwhm generally ranging between @xmath24 and @xmath25 due to seeing . the reduction of the 2d spectrum of each galaxy and the extraction of its 1d spectra was done by the deimos _ spec2d _ pipeline . in our study we use the 1d spectra extracted via the boxcar technique from the central @xmath26 window . due to the design of the survey , the observed 1d spectra is divided into 2 halves along wavelength . here we use the blue half of the two , since the cak ( 3,934 ) and g band ( 4,304 ) stellar lines fall within this half for galaxies in the redshift range that we are studying . our photometric data was taken from the aegis data product @xcite , specifically from the _ hubble space telescope _ ( _ hst _ ) go program 10134 ( p.i . : m. davis ) . these are _ hst_/acs images taken in the f606w ( v band ) and f814w ( i band ) filters . the observations are limited to 5@xmath27 measurements of @xmath28 and @xmath29 within a circular aperture of @xmath30 . this program covered @xmath31 deg@xmath21 of the egs , overlapping entirely with the region observed by deep2 . the observations were made in a four point dither pattern , which was processed by the stsdas multidrizzle package to produce a final mosaic of the field with a pixel scale of @xmath32 . in this work , we use the f814w photometry as this translates to the vega b band at the redshifts that we are studying . in this study we use two samples extracted from the large deep2 survey catalog . the first one consists of 154 galaxies and was used to measure trends in the galaxies star formation histories via full - spectrum fitting . the second one is a subsample of 68 galaxies of the first , for which we construct dynamical models . details of the adopted selection criteria are presented below . [ sample ] our first selection criterion is based on the quality flags provided by the deep2 survey catalogue , i.e. we selected galaxies that have either `` secure '' or `` very secure '' redshift estimation based on the automated and visual spectral redshift estimation by the deep2 team . we select galaxies with redshift @xmath14 . in this study we have only used galaxies in the egs field due to the lack of availability of high resolution _ hst _ photometry in the remaining fields . by extracting @xmath33 thumbnails , from the aegis mosaics , centred at the galaxy coordinates provided by the deep2 catalogue , we visually confirm the presence of photometry of the galaxies . these thumbnails are later used to parametrize the photometry of the galaxies and further details on this is provided in section [ qualmge ] . from a total of 43,569 galaxies in the deep2 galaxy redshift survey 1,249 galaxies passed this stage of the selection process . of these , 1,240 ( @xmath34 ) galaxies have an apparent @xmath35 . the second selection criterion is based on the quality of the galaxy spectra . we calculate the mean signal - to - noise ( @xmath36 ) of a galaxy spectrum as the median flux in the galaxy spectrum divided by the standard deviation of the residuals of the spectral fit to that spectrum . in this instance , the spectral fitting was done with a set of empirical stellar spectral library , while masking gas emission features . further details on this spectral fitting is presented in section [ qualspec ] . we only select galaxies with @xmath37 . we further eliminate galaxies that do not appear to have significant absorption features , i.e. clear spectral absorptions which stood out visually above the noise . this selection criteria reduced the galaxy sample to 175 galaxies . our third selection criteria is based on the visual inspection of the thumbnails of the galaxies , taken from the f814w images by the _ we have removed 21 irregular galaxies from our final sample as these galaxies are likely to be disturbed and would confuse the trends being investigated in this study . the selection criteria based on the galaxy spectra and imaging quality , dramatically reduces our galaxy sample to 154 galaxies . in this study , we also create dynamical models for our galaxy sample , however this requires some additional selection criteria . to create the dynamical models we need to parametrize the photometry of our galaxies to high accuracy within the central regions from which the stellar kinematics are observed . we exclude galaxies that have disturbed photometry , such as dust lanes , or non - axisymmetric features , and hence this galaxy subsample is biased towards early - type galaxies with smooth light profiles . further details on this selection criterion is given in section [ qualmge ] . also , the dynamical modelling of galaxies assume a spatially constant stellar mass - to - light ratio ( @xmath38 ) , however this assumption becomes inaccurate for galaxies with multiple significant star formation events . therefore , after deriving the sfh of our galaxies , we eliminate galaxies that require more than one significant star formation episode to reproduce their spectrum . in addition to this , the atlas@xmath39 group found that young galaxies , as inferred by their strong @xmath40 absorption feature , tend to show strong gradients in the stellar @xmath38 . for this reason , these galaxies where removed from their imf studies ( * ? ? * ; * ? ? ? * hereafter c13b ) . in this study , we use full - spectrum fitting to the galaxy spectra to identify the best fitting single stellar population ( assuming solar metallicity ) of the galaxies . we then classify any galaxy with the best fitting population of 1.2 gyrs or younger as a young galaxy and eliminate them from our secondary sample . this narrows our secondary galaxy sample to 68 galaxies for which dynamical models were created . this sample is identical to that used in @xcite . given our strict selection criteria , our sample can not be representative of the general galaxy population at @xmath3 . to illustrate the effect of the selection criteria on the sample , in fig . [ sample_plot ] we compare our galaxy samples to a larger parent sample of galaxies . this parent sample contains @xmath41 galaxies selected to have `` _ secure _ '' or `` _ very secure _ '' redshift @xmath14 in the egs field of the deep2 survey . we compare the galaxy samples using data from the photometric catalogue provided by the deep2 survey , which is derived from @xcite . the catalogue contains the observed @xmath42 , @xmath43 and @xmath44 magnitudes of the galaxies calculated from observations made by canada - france - hawaii telescope using the cfh12k camera . the catalogue also contains an estimate of the gaussian radius of the galaxies which we use as a proxy for the size of the galaxies . for further details on the the catalogue , we refer the reader to @xcite . in the left panel of fig . [ sample_plot ] , we compare the @xmath45 colour of the galaxies against their absolute @xmath43 observed band magnitude . the plot depicts the @xmath43 magnitude limit of the deep2 survey and clearly separates the blue cloud and red sequence of the field . the plot demonstrates that our sample have a higher @xmath43 cut - off than the deep2 which can be explained by the s / n cut - off that we place on our galaxy spectra . this is further demonstrated in the right panel of fig . [ sample_plot ] , where we plot the @xmath43 of the galaxies against a measure of the size of the galaxies . due to the design of the survey , particularly the fact that each galaxy has a fixed exposure of @xmath46 minutes and fixed aperture size , galaxies with lower surface brightness will be of lower quality . hence , our s / n cut - off can be roughly approximated as a surface brightness cut - off . the left panel of fig . [ sample_plot ] also demonstrates that our secondary sample , indicated by the red points in the plot , is a good representation of the red sequence along with a few galaxies in the blue cloud . this is likely the consequence of the additional selection that we place on the secondary sample , whereby only galaxies with smooth light profiles and without significant star formation younger than 1.2 gyrs are selected . our full sample , indicated by the red and the blue points , appears to represent all but the bluest galaxies over a certain magnitude cut - off . this under - representation may be the result of the visual inspection of the galaxy spectra , where the presence of hydrogen emission or apparent high frequency noise make the spectrum fits look unreliable . we measure the mean @xmath36 by fitting the observed galaxy spectra with an empirical stellar spectral library . our choice for an empirical stellar library , rather than population models , to estimate the @xmath36 is done to allow for maximum freedom to the optimal template combination , especially for very young ages . this avoids potential bias in the stellar velocity dispersion , which is also extracted during this step . however our results are quite insensitive to this choice , due to the rather low @xmath36 of our spectra . the spectral fitting is done using the ppxf method and code @xcite which uses a penalised maximum likelihood fitting technique in pixel space . before fitting we logarithmically rebin the observed galaxy spectrum to 60 per pixel . as templates we use a subset of empirical stellar spectra taken from the indo - us library of coud feed stellar spectra library @xcite . the subset of 53 templates is made such that : ( i ) each spectrum is gap - free and ( ii ) the subset is a good representation of the library s atmospheric parameter range ( @xmath47 vs @xmath48 $ ] ) . we use ppxf to fit for the velocity and velocity dispersion of the galaxies using 4 additive and one multiplicative polynomial . during the fit of the galaxy spectra gas emission features are masked . the residuals of the fits are found to be random , suggesting that our spectral fitting is not biased by template mismatch . using these fits , we derive the @xmath36 of each galaxy spectra as the ratio of the median of the galaxy spectra and the standard deviations of the residuals . we remove galaxies that have @xmath49 from our final sample as we found the fits to their spectra unreliable . we derive the star formation history of our main sample of 154 galaxies via a full - spectrum fitting technique using the ppxf code @xcite . for our templates , we use the miles stellar population models @xcite . these models are based on the miles empirical stellar spectra library @xcite and have a resolution of 2.5 fwhm @xcite . to ensure realistic fits of the galaxy spectrum with the template models , we placed certain constraints on our template grid . we use stellar population models logarithmically spaced into 40 age bins within the age range of 0.087.9 gyrs to restrict the age of stellar populations to that less than the age of the universe at the redshift under study . we have also restricted our template metallicities to [ m / h ] of -0.4 , 0.0 ( solar metallicity ) and 0.22 . this constrain is based on results by @xcite where the authors studied the absorption line strengths of stacked spectra of the deep2 sample at redshifts @xmath50 and concluded that these galaxies have metallicities close to solar . since @xcite use [ fe / h ] to measure their metallicities , we should note that for our template models the [ m / h ] is equivalent to [ fe / h ] @xcite . for this analysis we adopt the salpeter imf @xcite for our stellar population models , however the derived trends in the sfh are insensitive to the imf choice , as demonstrated in @xcite . in this study , we do a full - spectrum fitting for the spectral features and continuum shape of the observed galaxy spectra . this is done using four multiplicative polynomials to account for effects of reddening and calibration uncertainties in the observed galaxy spectra . the fractional weight assigned to each template indicates the fraction of the galaxy spectrum contributed by that stellar population . we do not use any additive polynomials when fitting for the stellar populations as these polynomials can artificially change the line strengths of the model templates . during the fitting with ppxf , the code convolves the model spectra with a gaussian line - of - sight velocity distribution ( losvd ) , which is optimized to best fit the spectrum . , are presented in the top left corner of the plot . the solid red line is the best fitting relation between the two quantities , while the dashed and dotted red lines enclose regions of @xmath51 and @xmath52 respectively , where @xmath53 is the observed rms scatter around the relation . for comparison , the solid black line represents the one - to - one relation between the quantities . ] we derive the star formation history of the galaxies by fitting their spectra with the above mentioned template grid while enforcing a smoothness criterion on the distribution of the weights in a mass weighted manner . this smoothness criterion is enforced by penalising the @xmath54 for weight distributions with non - zero second partial derivatives . in this way , for infinite regularization , the 2-dimensional solution converges to a plane . regularization is a standard method to solve general ill - posed inverse problems ( e.g. * ? ? ? * ) like the recovery of the sfh from a spectrum . it is implemented in ppxf using the classic linear - regularization approach as in equation ( 19.5.10 ) of @xcite and is enforced via the _ regul _ keyword in ppxf . the value assigned to the _ regul _ keyword specifies the strength of the penalisation of the @xmath54 . a common guideline is to penalise the model till @xmath55 , where @xmath56 is the @xmath54 of the model without penalisation and @xmath57 is the number of degrees of freedom @xcite . we should note that the regularization approach is not similar to smoothing the solution after the fit . in fact regularization _ does not _ prevent sharp features in the recovered star formation or metallicity distribution , as long as these are required to fit the data . the regularization only influence the weight distribution when the solution is degenerate and many different and noisy ones can fit the data equally well . in this case the smoothest solution is preferred over the noisy ones . this is a general feature of all regularization approaches . for a few galaxies , we find the value of _ regul _ is poorly constrained by the @xmath58 criterion , due to the rather large noise in the galaxy spectra . for this reason ppxf is able to fit the spectrum using a perfectly smooth weight distribution before @xmath59 . hence , to be conservative , and to avoid our results from being driven by the regularization , we lowered the requirement on @xmath58 from the standard one mentioned above to @xmath60 . as a consistency check , we have compared the stellar masses of 151 common galaxies derived in this study and that of @xcite , where the authors use brik colour to derive the stellar mass for a sample of @xmath61 galaxies in the deep2 survey under the assumption of a chabrier imf @xcite . we calculate our galaxy stellar mass ( @xmath62 ) by multiplying the absolute luminosity of the galaxies by the @xmath63 inferred from the spectral fits @xmath64 where @xmath65 is the weight attached to the @xmath66th template by the regularised mass - weighted fit to the galaxy spectrum , @xmath67 is the mass of the template that is in stars and stellar remnants , and @xmath68 is the @xmath69-band ( vega ) luminosity of the template . please note that all m / ls derived in this study are in the b ( vega ) band . we derive this luminosity using the mge parametrization of the _ hst_/acs galaxy thumbnails , mentioned in section [ qualmge ] , using the galaxy redshift and equation ( 10 ) of c13b . the comparison between the 151 common galaxies is shown in fig . [ bundy ] , where we compare the two stellar masses using the lts_linefit method ( c13b ) which has been modified to derive the best fit with a fixed slope . the solid red line in the plot represents the best fitting unitary - slope relation to the points , with the coefficients presented in the top left corner . the @xmath53 parameter is the observed rms scatter around the relation , while the dashed and dotted red lines indicate the @xmath53 and @xmath52 deviations from the best - fit . the solid black line indicates the one - to - one relation of the quantities . we see that a relation with slope unity is a good fit between the two datasets , though there is a significant offset of 0.11 dex and some scatter . the cause of the scatter and offset is likely the effect of differences in the assumed imf between the two quantities , ie this study assumes a salpeter imf while @xcite assume a chabrier imf , and the differences between the photometry and its analysis . also , the work by @xcite found that based on the techniques used to model the @xmath63 of the galaxies , uncertainties and systematics of the order 0.10.2 dex are expected . given that the stellar masses derived in this study and that of @xcite are completely independent , use very different data and approaches , and suffer from different systematics , we adopt @xmath70 dex as our very conservative errors in the estimation of the stellar mass of an individual galaxy via stellar population modelling . , are presented in the top left corner of the plot . the solid red line is the best fitting relation between the two quantities , while the dashed and dotted red lines enclose regions of @xmath51 and @xmath52 respectively , where @xmath53 is the observed rms scatter around the relation . the solid black line is the one - to - one relation between the quantities . the independence of the quantities of systematic bias indicates that our fits do not suffer from template mismatch . ] to estimate a conservative uncertainty in our derived @xmath71 , we derive our @xmath63 under 2 model sets and fitting conditions : ( i ) a regularised fitting to the complete model grid of 120 single stellar population models and , ( ii ) a non - regularised best fit to the model grid limited to 40 models with solar metallicities . our comparison of the two in fig . [ regulvsunregul ] shows a linear relation between the two values . the observed scatter is @xmath72 dex , which corresponds to an error of @xmath73 dex or 22% in each @xmath63 , assuming their errors are comparable . for comparison we also derived the error on the derived @xmath71 via a more standard bootstrapping technique . we reshuffled the residual and repeated the determination of @xmath71 multiple times . this yielded a median error of 0.05 dex in @xmath71 for the full sample , which is smaller than the error of 0.08 dex derived above . to be conservative , and to account at least in part for systematic effects , in this study we adopt the intrinsic scatter of 0.08 dex between the two quantities as the error in the derived @xmath63 . to study the scaling relations , we measure the luminosities and size by parametrising the photometry of the galaxies . in this study , we choose the f814w photometry taken by the _ hst_/acs to do so . the need for _ hst _ photometry comes from the requirement , in the dynamical models , of an accurate description of the tracer population producing the observed kinematics . this also motivates our choice for the f814w filter , which is as close as possible to the wavelength range from which the stellar kinematics is obtained . we parametrize the photometry of the galaxies using the multiple - gaussian expansion technique @xcite . this technique fits a series of two dimensional gaussians , each defined by an amplitude , standard deviation and axial ratio , to the observed surface brightness of the galaxy image in a non - parametric manner , i.e. without the use of a predefined light profile function like the srsic profiles . the advantage of this technique is that the convolution and deprojection ( i.e. conversion of the 2d model into the intrinsic 3d luminosity distribution ) of the mge models is analytically simple . the fit is performed with the robust method and software ( see footnote 1 ) by @xcite . in this study , we use the mge parametrisation of the photometry to construct our dynamical models . these models assume axisymmetry , but galaxies can show non - axisymmetric features , such as bars . 3.2.1 ) tried to reduce the effect of bars in the models by essentially ignoring them during the fits , while forcing the mge to only describe the light distribution of the underlying galaxy disk . in this study , we also adopted the same approach for all galaxies . the luminosity of the galaxies are calculated from the analytic sum of the gaussian luminosities equation ( 10 ) of c13b , while the projected half - light radii ( ) were computed from the circularised gaussians using equation ( 11 ) , of c13b . for our galaxies , we also calculate the @xmath74 , the major axis of the isophote containing half of the light of the galaxy as these are found to be a more robust measurement of the galaxy size due to its lower dependency on galaxy inclination ( * ? ? ? we derive this value as described in section 3.3.1 of c13b . in @xcite we show that , for the subset of 68 galaxies for which we construct dynamical models , the derived using the mge parametrization are consistently offset from the one can predict from the virial equation @xmath75 , when using the coefficient of @xcite . we found that those galaxies required a 0.16 dex increase in , for @xmath76 to match the @xmath77 derived via jam models . since we use the same data , at the same redshift , for all 154 galaxies , we apply the same fractional correction to the and @xmath74 of all galaxies of this study . since the f814w filter approximates to the b band at the redshift range of our galaxies , we state all luminosity and photometric quantities in the b band . to further understand the evolution of galaxies since @xmath3 , we created dynamical models for a subsample of our galaxies . in this study we use the jeans anisotropic modelling ( jam ; see footnote 1 ) method and code of @xcite , which solves the @xcite equations under the assumptions of axisymmetry , while allowing for orbital anisotropy and rigorously accounting for seeing and aperture effects . the models require as input an accurate description of the projected light distribution of the galaxy and a parametrization for the unknown total mass . as the galaxy inclination is generally unknown , we assume a standard inclination of 60@xmath78 , the average inclination for a set of random orientations . when this inclination is not allowed by the mge model , we adopt the lowest allowed one . here an inclination of 90@xmath78 corresponds to an edge - on view . another quantity required by jam is the velocity anisotropy @xmath79 , where @xmath80 and @xmath81 are the velocity dispersion along the symmetry axis and along the cylindrical radius , respectively . in this study , we adopt @xmath82 as this is found to be a typical value for etgs in the local universe @xcite . tests by @xcite and @xcite show that the derived m / l through the jam code is weakly sensitive to the effects of inclination and spatially constant velocity anisotropy . the 68 galaxies in our modelling subsample are the same for which we created dynamical models in @xcite , and we adopt the stellar velocity dispersion from that study . the stellar kinematics were derived from the deep2 spectra within a 1 arcsec slit . in @xcite our jam models assumed the total mass is distributed like the luminous one , hence our derived dynamical m / l incorporates the effect of stellar and dark matter . here we improve our dynamical models by explicitly including dark matter using results from the abundance matching techniques . this infers a relation between the stellar and halo mass by matching simulated dark halo mass functions to observed galaxy luminosity functions , under the assumption that the most massive galaxies reside in the most massive halos ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . here we try two different relations between the stellar and halo mass . the first is taken from @xcite , where the authors fit a log - normal stellar - to - halo mass relation ( shmr ) , with a log - normal scatter , to the results of weak galaxy - galaxy lensing , galaxy clustering and galaxy number density analysis of the cosmos survey @xcite . we use equation ( 13 ) of @xcite and the parameter values for model _ sig_mod1 _ within the relevant redshift bin to derive the halo mass associated with a galaxy of given stellar mass . the second shmr we use is equation ( 2 ) of @xcite . here the authors use high resolution n - body simulations to derive the dark halo mass function at different redshifts , and match them to the observed galaxy stellar mass functions , hence allowing one to derive the halo mass for a galaxy of a given stellar mass . we further assume that the dark matter of our galaxies is distributed spherically with an nfw profile @xcite . we constrain the concentration of the halo using equation ( 12 ) of @xcite , where the authors study the evolution of the concentration of the galaxies with redshift using the bolshoi simulation . using the above equations , we calculate the nfw density profile , which is fit using the one dimensional mge fitting code of ( * ? ? ? * see footnote 1 ) , to use in jam . in practice , starting from a trial @xmath83 for a galaxy , we compute the total stellar mass by multiplication with the total luminosity . we then associate a corresponding dark halo mass using the shmr , and a nfw profile using the halo mass versus concentration relation mentioned above . for given stellar and dark matter distributions we use jam to predict the stellar @xmath84 , integrated within the adopted slit and convolved by the seeing . the model @xmath85 is matched to the observed velocity dispersion by varying the only free parameter , @xmath63 of our model , using a root finding algorithm . in this manner we account for the dark matter content of the 68 galaxies in our sample , and derive their dynamical stellar @xmath86 ( @xmath83 ) . , where we have coadded the galaxy spectra based on their stellar mass . the bin sizes of these stellar mass bins are represented here in vertical red dashed lines at stellar masses . ] in fig.[sfh ] , we have plotted the sfh of the galaxies after having collapsed them along the metallicity axis . the galaxies are sorted along the x - axis of the plot in order of increasing stellar mass as derived by stellar population modelling . in the top panel , we present the regularised sfh and we can clearly see that there is a gradual trend in the presented sfh with stellar mass . it appears that galaxies with low stellar mass tend to have a significant mass fraction of their stellar populations younger than 1 gyr , unlike the typical sfh of high stellar mass galaxies , where the significance of this young stellar population weakens . for comparison in the bottom panel we have presented the unregularised mass distribution for the full - spectrum fitting of the galaxies . in this case the weights distribution is much noisier and consists of discrete peaks , as expected . but the main trend in sfh with galaxy mass can still be recognized . this shows that the recovered trend is robust . to further illustrate this trend , we have coadded the galaxy spectra into 3 bins : @xmath87 , @xmath88 and @xmath89 . the full - spectrum fitting of these coadded spectra , along with their associated sfh , are shown in fig . [ weight_coadd ] . as the s / n is much higher for the coadded spectrum of the galaxies , we adopt the standard @xmath90 criterion for regularization ( for further information , please refer to section [ sec_sfh ] ) . also , the significantly high s / n allows us to confidently fit the coadded spectra using the entire metallicity range of the miles models , [ m / h ] = -1.7 to 0.22 . we should point out that the [ m / h ] is not equivalent to [ fe / h ] for the low metallicity models , however @xcite state that the this does nt effect the galaxy age and metallicity estimates significantly . the same trend that we could see in the individual spectra appears cleaner in the stacked ones : ( i ) the most massive galaxies require a maximally old and nearly solar population to reproduce the observations , with little room for recent star formation . ( ii ) the galaxies with intermediate masses still peak at the oldest ages , but allow for a more extended star formation history . ( iii ) the lowest mass galaxies do not peak at the oldest ages any more and additionally show clear evidence for ongoing star formation . the well known age - metallicity degeneracy of stellar population models implies that the effect on a galaxy spectrum due to a decrease in its mean age can be negated by an increase of its metallicity @xcite . this degeneracy is reduced by the use of full - spectrum fitting , which tries to reproduce a large number of spectral features @xcite , but the general trend is still present in our recovered age and metallicity distributions . an example of this can be seen in the bottom right panel of fig . [ weight_coadd ] : the ridge of equally - large weights going from an age of 2 gyr with nearly solar metallicity , to an age of 7 gyr with the lowest metallicity is likely due to the degeneracy , combined to our relatively narrow wavelength range . in table . 1 , we present the relevant stellar modelling quantities of the full galaxy sample of the study . the complete version of this table is available online with the published article . in this study , we calculate the mean age of the galaxies from the sfh as follows : @xmath91 where @xmath92 is the weight associated to the @xmath66th template during the regularised luminosity - weighted full - spectrum fit and @xmath93 is the corresponding age . to obtain luminosity - weighted quantities one can simply normalize all the ssps to the same mean flux within the fitted wavelength range , before the ppxf fit . of course the luminosity weighting refer to the fitted wavelength , which in our case lies between the @xmath94 and @xmath69 band . when the ssp spectra used in the ppxf fit have the flux as given by the models , namely corresponding to a unitary stellar mass , this equation provides mass - weighted quantities . [ cols="^,^,^,^,^,^,^,^,^,^ " , ] column ( 1 ) : deep2 galaxy identifier . column ( 2 ) : velocity dispersion as measured by full - spectrum fitting of the galaxy spectra . column ( 3 ) : error on the derived velocity dispersion of the galaxy through a bootstrapping technique . column ( 4 ) : b - band @xmath63 derived through dynamical modelling , where the dark matter halo was assigned using the shmr of @xcite . column ( 5 ) : the log of the stellar mass , in units of , of the galaxy derived using the absolute luminosity ( column ( 5 ) of table . 1 ) and the m / l of column ( 4 ) . column ( 6 ) : fraction of dark matter within 1 of the galaxy . this was derived by using the shmr of @xcite . in the work presented in @xcite , the imf normalisation of the 68 dynamically modelled galaxies was shown in fig . the plot illustrated that the imf normalisation for massive etgs at @xmath3 was salpeter - like , however a caveat of the work was the assumption that the dark matter fractions in the central regions of high redshift galaxies was not significant . in this study we include dark matter explicitly and in fig . [ imf_norm ] present an updated version of the plot , with the dark matter contribution explicitly removed . this confirms the assumption that dark matter fraction has a negligible effect on the imf normalisation of the galaxies , i.e. the _ average _ imf normalisation of massive etgs at redshift of @xmath95 is still consistent with a salpeter imf . this result is independent of the shmr used . this confirmation of our @xcite result is also in agreement with recent work by @xcite , where the authors have used a joint lensing and stellar dynamics modelling of massive galaxies out to @xmath96 , including about ten galaxies within our resdhift range . the authors found that the stellar imf normalisation was close to salpeter imf for @xmath97 . also , work by @xcite , using imf sensitive spectral features , determined that more massive galaxies have a bottom heavier imf at redshifts between 0.9 and 1.5 . the salpeter normalization of the imf is also consistent with the non - universality of the imf reported in the nearby universe ( e.g. * ? ? ? * ; * ? ? ? in fact , if the centres of massive galaxies at @xmath3 passively evolve into the massive etgs population , their imf is expected to have a salpeter normalization as observed locally @xcite . in this study , we have determined the sfh of the galaxies at redshift @xmath1 in a non - parametric manner , using full - spectrum fitting . from a parent sample of @xmath41 galaxies from the deep2 survey , we apply strict quality selection criteria to extract 154 galaxies with @xmath0 for which we derive our sfhs . for a subsample of 68 galaxies , with @xmath2 , we additionally construct dynamical models . due to these selection criteria , our galaxy samples have a higher magnitude cut - off than the deep2 survey and preferentially selects galaxies with high surface brightness . however , we show that the full galaxy sample is representative of the red sequence and all but the bluest galaxies of the blue cloud , while the secondary sample mainly consists of galaxies from the red sequence . the derived sfh for the full galaxy sample indicates that the most massive galaxies formed the bulk of their stars in the early epoch of the universe unlike low mass galaxies which are forming stars at a significant rate even at redshift of @xmath98 . this is qualitatively consistent with previous fossil record studies in the local universe , where authors have found that the star formation rate of the most massive galaxies peaked early in the age of the universe , and hence provides a robust and independent test of these results and the narrative of the formation and evolution of galaxies that these results have produced . our results based on fig . [ sfh ] and fig . [ weight_coadd ] demonstrate that the difference between the sfh of galaxies evolves gradually and is a function of stellar mass . we study the distribution of galaxy ages on the mass - size diagram . this demonstrates that the velocity - dispersion dependence in the age of the stellar populations of the central regions of the galaxies was already in place by @xmath3 . we also place an upper limit of a factor of @xmath99 to the size growth of individual galaxies since @xmath3 , in agreement with other studies . finally , using the dynamical models of the 68 galaxies in our secondary sample , which account for the dark matter in the galaxies using results from abundance matching techniques , we measure a median dark matter fraction @xmath100 per cent per cent , within a sphere of radius , for the most massive galaxies , with small variations depending on the adopted stellar - halo mass relation . comparing this to the dark matter fraction determined locally , we find that the dark matter fraction of galaxies has marginally increased in the last 8 gyrs , but is otherwise insignificant . this result confirm the study of @xcite stating that the average imf normalisation of the most massive galaxies is on average consistent with a salpeter imf . m.c . acknowledges support from a royal society university research fellowship . e. j. , kristian j. a. , lynds r. , oneil , jr . e. j. , balsano r. , rhodes j. , wfpc-1 idt , 1994 , in bulletin of the american astronomical society , vol . 26 , american astronomical society meeting abstracts , p. 1403

we study the star formation history for a sample of 154 galaxies with stellar mass @xmath0 in the redshift range @xmath1 . we do this using stellar population models combined with full - spectrum fitting of good quality spectra and high resolution photometry . for a subset of 68 galaxies ( @xmath2 ) we additionally construct dynamical models . these use an axisymmetric solution to the jeans equations , which allows for velocity anisotropy , and adopts results from abundance matching techniques to account for the dark matter content . we find that : ( i ) the trends in star formation history observed in the local universe are already in place by @xmath3 : the most massive galaxies are already passive , while lower mass ones have a more extended star formation histories , and the lowest mass galaxies are actively forming stars ; ( ii ) we place an upper limit of a factor 1.5 to the size growth of the massive galaxy population ; ( iii ) we present strong evidence for low dark matter fractions within 1 ( median of 9 per cent and 90th percentile of 21 per cent ) for galaxies with @xmath4 at these redshifts ; and ( iv ) we confirm that these galaxies have , on average , a salpeter normalisation of the stellar initial mass function . [ firstpage ] galaxies : evolution - galaxies : formation - galaxies : stellar content - galaxies : haloes - galaxies : high redshift

1508.05009

the effective potential in quantum field theory plays a crucial role in connection with the problem of the spontaneous symmetry breaking . in this field there are three classic papers @xcite . coleman and weinberg@xcite were the first ones to calculate the higher - order effective potential of a scalar field at one loop level by summing up an infinite number of feynman graphs . jackiw@xcite has used the feynman path - integral method to obtain a simple formula for the effective potential . he has succeeded in representing each loop order containing an infinite set of conventional feynman graphs by finite number of graphs using this algebraic method which can formally be extended to the arbitrary higher - loop order . in ref . @xcite the functional integral is explicitly evaluated using the steepest descent method at two - loop level . higher - loop calculations with this method are very difficult . the purpose of this paper is to show that there is a missing portion in the two - loop effective potential of the massless @xmath0 @xmath2 theory obtained by jackiw @xcite . in this paper we employ the dimensional regularization method @xcite instead of the cutoff regularization method used in ref . @xcite and for the sake of brevity we confine ourselves to the case of single component theory ( @xmath3 ) . the lagrangian for a theory of a self - interacting spinless field @xmath4 is given as ( ( x))&=&1+z2_^ -m^2+m^22 ^ 2-+4!^4 , [ lg ] where the quantities @xmath4 , @xmath5 , and @xmath6 are the renormalized field , the renormalized mass , and the renormalized coupling constant respectively , whereas @xmath7 , @xmath8 , and @xmath9 are corresponding ( infinite ) counterterm constants . we will confine ourselves to the massless theory ( @xmath10 ) . the effective potential is most suitably defined , when the effective action @xmath11)$ ] , being the generating functional of the one - particle - irreducible ( 1pi ) green s functions ( @xmath12 ) , is expressed in the following local form ( the so - called derivative expansion ) : & = & d^4x , [ loc ] where @xmath13 is the vacuum expectation value of the field operator @xmath14 in the presence of an external source . by setting @xmath13 in @xmath15 to be a constant field @xmath16 , we obtain the effective potential @xmath17 v_eff()(_cl(x))|__cl(x)=.[ve ] following the field - shift method of jackiw @xcite for the calculation of the effective potential , we first obtain the shifted lagrangian with the constant field configuration @xmath16 ( ; ( x))&=&1+z2_^ -12(m^2++2 ^ 2)^2 + & -&+6 ^ 3-+4!^4 . [ slg ] the feynman rules for this shifted lagrangian are given in fig . 1 . without introducing any new loop - expansion parameter , which is eventually set to be unity , we will use @xmath18 as a loop - counting parameter @xcite . this is the reason why we have kept all the traces of @xmath18 s in the feynman rules above in spite of our employment of the usual `` god - given '' units , @xmath19 . in addition to the above feynman rules , fig . 1 , which are used in constructing two- and higher - loop vacuum diagrams , we need another rule ( fig . 2 ) solely for a one - loop vacuum diagram which is dealt with separately in jackiw s derivation of his prescription and is essentially the same as that of coleman and weinberg @xcite from the outset . using the rules , fig . 1 and fig . 2 , and including the terms of zero - loop order , we arrive at the formal expression of the effective potential up to two - loop order : v_eff()&=&+ + + .[onn ] the last three ( bracketed- ) terms on the right - hand side in the above equation appear in fig . 3 . for the purposes of renormalization we first expand the counterterm constants in power series , beginning with order @xmath18 : m^2&=&m_1 ^ 2+^2m_2 ^ 2 + , + &=&_1+^2_2 + , + z&=&z_1+^2z_2 + . in what follows we will use the following notation for the effective potential up to the @xmath20-loop order : v_eff^[l]()=_i=0^l ^i v_eff^(i ) ( ) . the zero - loop part of the effective potential is given as v_eff^(0)()=4!^4 . [ ve0 ] the one - loop part of the effective potential is readily obtained as v_eff^(1)()&=&m_1 ^ 22 ^ 2+_14 ! ^4 -^2 ^ 48(4)^2 + ^2 ^ 4(4)^2,[ve1 ] where @xmath21 is the usual euler constant and @xmath22 is an arbitrary constant with mass dimension . the @xmath23 poles in this equation are readily cancelled out by choosing the counterterm constants @xmath24 and @xmath25 as follows : m_1 ^ 2=a_1 , _1=3^2(4)^2+b_1 , [ 11 ] where @xmath26 and @xmath27 are unspecified but finite constants at this stage . one may put @xmath26 ( and @xmath28 below ) to be zero from the beginning because the theory is massless . in our dimensional regularization scheme the pole part of @xmath24 vanishes , but this is not the case in the cutoff regularization method . besides @xmath24 and @xmath25 , there is another counterterm constant . it is @xmath29 . in jackiw s calculation , @xmath29 is set to be zero . this is matched to the standard condition for the defining the scale of the field afflicted by the infrared singularity , as remarked by coleman and weinberg @xcite . ( in fact , this singularity can not be seen in @xmath30 , the one - loop order contribution to @xmath31 . the infrared singularity appears for the first time in the two - loop order @xcite . ) now let us determine @xmath29 so as to meet the following modified condition which avoids the infrared singularity : |_^2=m^2=1 . to this end , we use the following relation @xcite |_^2=m^2= ~^(2)_(p^2)p^2|_p^2=0,^2=m^2 , [ mc ] where @xmath32 is the ( momentum - conserving ) 1pi two - point green s function in the shifted theory . the right - hand side of eq . ( [ mc ] ) is calculated as 1+6(4)^2+z_1 , from which we find z_1=-6(4)^2c_1.[c1 ] note that this wave function renormalization constant @xmath29 is free of @xmath23 singularity . but in a higher - loop order the wave function renormalization constant @xmath33 may have the @xmath23 singularity . the two - loop part of the effective potential is obtained as v_eff^(2)()&=&m_2 ^ 22 ^ 2+_24!^4 -a_1^22(4)^2 -3^3 ^ 4 8(4)^4 ^ 2 + ^3 ^ 48(4)^4 -b_1^44(4)^2 + c_1^2 ^ 44(4)^2 + & + & a_1^2 ^ 2(4)^2+b_1^4(4)^2 + & -&c_1^2 ^ 4(4)^2 + & + & ^3 ^ 4(4)^4 .[ve2 ] in the above equation , @xmath34 is a constant whose value is defined in eq . ( [ ab ] ) . notice that the so - called `` dangerous '' pole terms such as @xmath35\ln^n [ \l\fh^2/(4\p m^2)]$ ] , @xmath36 @xmath37 , in the above equation , which can not be removed by terms of counterterm constants ( @xmath38 and @xmath39 ) , have been completely cancelled out among each other . the counterterm constants @xmath40 and @xmath41 are determined as m_2 ^ 2&=&a_1(4)^2+a_2 , + _2&=&9^3(4)^4 ^ 2 -3^3(4)^4 + 6b_1(4)^2 -6c_1^2(4)^2+b_2 , [ 22 ] where @xmath28 and @xmath42 are also unspecified but finite constants . in the massive @xmath0 @xmath2 theory , the renormalization conditions ~^(2 ) ( 0)=-m^2 , ~^(4 ) ( 0)=- , are respectively translated into |_=0=m^2 , d^4 v_eff()d^4|_=0= . in our massless theory , however , we encounter the infrared singularity in the defining condition for a coupling constant . to avoid this difficulty we follow coleman and weinberg @xcite and require @xmath26 , @xmath28 , @xmath27 , and @xmath42 are determined order by order as follows : a_1&=&a_2=0 , + b_1&=&-^2(4)^2 , + b_2&=&^3(4)^4 + & & + c_1^2(4)^2.[1122 ] after disposing successfully all divergent terms in eqs . ( [ ve1 ] ) and ( [ ve2 ] ) by the counterterm constants in eqs . ( [ 11 ] ) and ( [ 22 ] ) , we eventually arrive at our new result satisfying the conditions in eq . ( [ rc ] ) : v_eff^[2]()&= & + ^2 ^ 4(4)^2+^2^3 ^ 4(4)^4 + & + & .[2p ] our result differs from that of jackiw ( see eq . ( 3.17 ) in ref . @xcite ) just by the underlined square - bracket term in eq . ( [ 2p ] ) which is the missing portion in his calculation of the two - loop effective potential . after substituting the value of @xmath43 of eq . ( [ c1 ] ) , we have the final form of the effective potential up to the two - loop order as follows : v_eff^[2]()&= & + ^2 ^ 4(4)^2+^2^3 ^ 4(4)^4.[f2p ] let us now apply a renormalization condition @xmath44 to the two - loop effective potential with most general form of @xmath45 as an assumed series solution to the renormalization group equation : v_eff^[2]()&= & + ^2 ^ 4(4)^2+^2^3 ^ 4(4)^4,[im ] where @xmath46 , @xmath47 , @xmath48 are constants . then we readily obtain |_^2=m^2&=&+ .[rrc ] the boxed term ( @xmath6-cubic term ) on the right - hand side of the above equation is an unwanted term . thus it should vanish . next let us require the parametrization invariance of the theory . the renormalization mass , @xmath22 , is indeed an arbitrary parameter , with no effect on the physics of the problem . if we pick a different mass , @xmath49 , then we define a new coupling constant =d^4v_eff^[2]()d^4|_^2=m^2=+ p_1^2+p_2^3,[rp ] where p_1&=&32(4)^2(m^2m^2 ) , + p_2&=&^2(4)^4 . ( [ rp ] ) is readily inverted iteratively as = -p_1^2+(2p_1 ^ 2-p_2)^3+o(^4 ) . we now substitute this @xmath6 into eq . ( [ i m ] ) . then the two - loop effective potential is given in terms of @xmath50 and @xmath49 as follows : v_eff^[2]()&= & + ^2 ^ 4(4)^2 + & & + ^2^3 ^ 4(4)^4+o(^4).[uw ] the parametrization invariance requires that the boxed term in eq . ( [ uw ] ) should vanish . from this and the vanishing boxed term of eq . ( [ rrc ] ) we obtain v_eff^(2)()&=&^2^3 ^ 4(4)^4.[2lp ] this shows us that even if one has an arbitrary value of @xmath47 , the parametrization invariance still holds . this is the reason why the jackiw s result ( eq . ( 3.17 ) in ref . @xcite ) is safe from the check of the parametrization invariance . in the above equation @xmath47 is fixed not by the parametrization invariance but by the correct two - loop calculation of the effective potential . in summary , jackiw used a wrong renormalization condtion , eq . ( [ wzc ] ) , in the massless @xmath0 @xmath2 theory and obtained such an incorrect value of @xmath47 as @xmath51 , but the correct value of @xmath47 is @xmath52 as given by our eq . ( [ f2p ] ) . this work was supported in part by ministry of education , project number bsri-97 - 2442 and one of the authors ( j. -m . c. ) was also supported in part by the postdoctoral fellowship of kyung hee university . in this appendix , the momenta appearing in the formulas are all ( wick - rotated ) euclidean ones and the abbreviated integration measure is defined as _ k = m^4-n , where @xmath53 is the space - time dimension in the framework of dimensional regularization @xcite and @xmath22 is an arbitrary constant with mass dimension . for the sake of completeness , we simply list one - loop and two - loop integrals needed in our calculations though they are well known . for the two - loop integrations one may refer to ref . @xcite . s_1&&_k(1+^2k^2+^2 ) = -(^2+^2)^2(4)^2 ( ^2+^24m^2)^-/2(2 - 2 ) + , + s_2&&_k1k^2+^2 = ^2(4)^2(^24m^2 ) ^-/2(2 - 1 ) , + s_3 & & _ k1(k^2+^2)^2=1(4)^2(^24m^2 ) ^-/2(2 ) , + s_4&&_k , p1(k^2+^2 ) ( p^2+^2)[(p+k)^2+^2 ] = ^2(4)^4(^24m^2)^- ^2(1+/2)(1-)(1-/2).[www ] in the above equation , @xmath21 is the usual euler constant , @xmath54 , and the numerical value of the constant @xmath34 in eq . ( [ www ] ) is a = f(1,1)=-1.1719536193,[ab ] where & & f(a , b)_0 ^ 1dx , z = ax+b(1-x)x(1-x ) . 99 s. coleman and e. weinberg , phys . d * 7 * , 1888 ( 1973 ) . r. jackiw , phys . d * 9 * , 1686 ( 1974 ) . j. iliopoulos , c. itzykson and a. martin , rev . phys . * 47 * , 165 ( 1975 ) . g. t hooft and m. veltman , nucl . phys . * b44 * , 189 ( 1972 ) . y. nambu , phys . b26 * , 626 ( 1968 ) . c. itzykson and j. -b . zuber , _ quantum field theory _ ( mcgraw - hill , new york , 1980 ) : p. 455 . r. grigjanis , r. kobes , and y. fujimoto , can . * 64 * , 537 ( 1986 ) . j. van der bij and m. veltman , nucl b231 * , 205 ( 1984 ) ; f. hoogeveen , nucl . phys . * b259 * , 19 ( 1985 ) ; c. ford , i. jack , and d. r. t. jones , nucl . phys . * b387 * , 373 ( 1992 ) ; a. i. davydychev and j. b. tausk , nucl . phys . * b397 * , 123 ( 1993 ) ; m. misiak and m. mnz , phys . lett . * b344 * , 308 ( 1995 ) .

we point out that there is a missing portion in the two - loop effective potential of the massless @xmath0 @xmath1 theory obtained by jackiw in his classic paper , phys . rev . d 9 , 1686 ( 1974 ) . ~

hep-th9804079

the star formation rate ( sfr ) of galaxies sets the rate at which galaxies grow and evolve and is the one of the most important measures for understanding the hierarchical build - up of our universe over cosmic time . large scale simulations , however , have shown that unregulated star formation leads to an overabundance of high mass galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? therefore some process ( or processes ) must be able to stop , or `` quench , '' star formation before the galaxy grows to be too big . the answer seems to lie in supermassive black holes ( smbh ) which nearly all massive galaxies harbor in their centers . smbhs grow through accretion of cold material ( active galactic nuclei ; agn ) , and the huge loss of gravitational energy of the cold material is converted into radiation that is evident across the whole electromagnetic spectrum and manifests itself as a bright point source in the nucleus of galaxies . the agn can deposit this energy into the ism of its host galaxy through jets ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or powerful outflows that either heat the gas or remove it altogether , i.e. `` feedback '' processes ( e.g * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . indirect evidence of this `` feedback '' has been observed through the simple , scaling relationships between the mass of the smbh and different properties of the host galaxy such as the stellar velocity dispersion in the bulge , the bulge mass , and the bulge luminosity ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the relative tightness of these relationships suggests a strong coevolution of the host galaxy and smbh . much debate remains however as to the exact mechanism of agn feedback and whether or not it plays a dominant role in the overall evolution of galaxies especially in light of new observations at both low and high @xmath5 that seem to deviate from the well - established relationships ( see * ? ? ? * for a detailed review ) . evidence for agn feedback though should also manifest itself in the sfr of its host galaxy , therefore much work has also focused on the so - called starburst - agn connection ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the problem lies in determining accurate estimates of the sfr in agn host galaxies . well - calibrated indicators , such as h@xmath6 emission and uv luminosity , are significantly , if not completely , contaminated by the central agn . many studies therefore turn to the infrared ( ir ) regime ( @xmath7 @xmath1 m ) where dust re - emits the stellar light from young stars . dust fills the interstellar medium ( ism ) of galaxies and plays an important part in the heating and cooling of the ism and the general physics of the galaxy . while dust contributes very little to the overall mass of a galaxy ( @xmath8 ) , the radiative output , mainly in the infrared ( ir ) regime , can , on average , constitute roughly half of the bolometric luminosity of the entire galaxy @xcite , although there is an enormous range in the fraction . dust efficiently absorbs optical and uv emission and re - radiates it in the mid- and far - infrared ( mir , fir ) depending on the temperature as well as grain size @xcite . recently formed o and b stars produce the majority of the optical and uv light in galaxies , therefore measuring the total ir light from dust provides insights into the current ( @xmath9 myr ) star formation rate ( sfr ) ( e.g. * ? ? ? * ) , although for very passive galaxies where the current sfr is much lower than it was earlier , ir emission can be an overestimate due to dust heating by an older stellar population.(e.g . * ) however , dust is also the key component in obscuring our view of agn . dust heated by the agn is thought to primarily live in a toroidal - like structure that encircles the agn and absorbs its radiative output for certain lines of sight . the dusty torus is used to explain the dichotomy of agn into seyfert 1 ( sy 1 ) and seyfert 2 ( sy 2 ) within a unified model @xcite . like o and b stars in star - forming regions , the agn outputs heavy amounts of optical and uv light , and like dust in the ism the dusty torus absorbs and re - emits this as ir radiation . spectral energy distribution ( sed ) models @xcite as well as observations @xcite suggest the torus mainly emits in the mir ( @xmath10 @xmath1 m ) with the flux density dropping rapidly in the fir ( @xmath11 @xmath1 m ) . further the sed for stellar dust re - radiation peaks in the fir @xcite , making the fir the ideal waveband to study star - formation in agn host galaxies . space - based telescopes such as the _ infrared astronomical satellite _ ( iras ; * ? ? ? * ) , _ spitzer space telescope _ @xcite , and _ infrared space observatory _ @xcite greatly expanded our knowledge of the ir universe and provided a window into the fir properties of galaxies . but , before the launch of the _ herschel space observatory _ @xcite , the fir sed was limited to @xmath12 @xmath1 m , except for studies of small samples of the brightest galaxies using ground - based instruments such as _ scuba _ ( e.g. * ? ? ? * ; * ? ? ? herschel _ with the spectral and photometric imaging receiver ( spire ; * ? ? ? * ) has pushed into the submillimeter range with observations in the 250 , 350 , and 500 @xmath1 m wavebands , probing the rayleigh - jeans tail of the modified blackbody that accurately describes the broadband fir sed of galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? these wavebands are crucial for measuring dust properties ( i.e. temperature and mass ) as @xcite and @xcite show . further , @xcite found that fir and submillimeter data are important for estimating the sfr of agn host galaxies . recent studies , such as @xcite and @xcite , focusing on the dust and star - forming properties of agn have shown the power of long wavelength _ herschel _ data to better constrain the sfr , dust mass , and dust temperature in agn host galaxies . @xcite analyzed the ir seds of low redshift ( @xmath13 ) , quasi - stellar objects ( qsos ) broadly finding most of the fir emission can be attributed to thermally - heated dust . @xcite looked at the ir seds of 24 @xmath14 m selected agn at slightly higher redshift ( @xmath15 ) around galaxy clusters finding a strong correlation between the agn and star - forming luminosity which could be due to their shared correlation with galaxy stellar mass . both studies , however , rely on agn selection using different wavebands ( optical vs. mid - infrared ) and generally probe the higher agn luminosity population . therefore , we have assembled a large ( @xmath16 ) , low redshift ( @xmath0 ) sample of agn selected using ultra - hard x - ray observations with the _ swift_/_burst alert telescope _ ( bat ) and imaged each one with _ this sample focuses on moderate luminosity seyfert galaxies ( @xmath17 ) . in @xcite , we presented the pacs data of the _ herschel_-bat agn which provided photometry at 70 and 160 @xmath1 m . in this paper , we complete the fir sed of the bat agn with the creation and analysis of the spire images . we focus on the overall luminosity distributions at the spire wavebands as well as the spire colors ( @xmath2 and @xmath3 ) to determine the likely heating sources of cold dust in agn host galaxies . we also look for correlations with a proxy for the bolometric agn luminosity to potentially reveal any indication that agn heated dust is contributing to the fir sed . this paper sets us up for a complete study of the mid - far ir sed to fully explore the star - forming properties of agn host galaxies and reveal the global starburst - agn connection in the nearby universe ( shimizu et al , in preparation ) . throughout this paper we assume a @xmath18cdm cosmology with @xmath19 km s@xmath20 mpc@xmath21 , @xmath22 , and @xmath23 . luminosity distances for each agn were calculated based on their redshift and assumed cosmology , except for those with @xmath24 where we referred to the _ _ extragalactic distance database__. swift_/bat @xcite operates in the 14195 kev energy range , continuously monitoring the sky for gamma - ray bursts . this constant monitoring has also allowed for the most complete all - sky survey in the ultra - hard x - rays . to date , bat has detected 1171 sources at @xmath25 significance corresponding to a sensitivity of @xmath26 ergs s@xmath20 @xmath27 @xcite . over 700 of those sources have been identified as a type of agn ( seyfert , blazar , qso , etc . ) we selected our sample of 313 agn from the 58 month _ swift_/bat catalog @xcite , imposing a redshift cutoff of @xmath0 . all different types of agn were chosen only excluding blazars / bl lac objects which most likely introduce complicated beaming effects . to determine their agn type , for 252 sources we used the classifications from the bat agn spectroscopic survey ( koss et al , in preparation ) which compiled and analyzed optical spectra for the _ swift_/bat 70 month catalog @xcite . seyfert classification was determined using the standard scheme from @xcite and @xcite . for the remaining 61 agn we used the classifications provided in the 70 month catalog . in total the sample contains 30 sy 1 , 30 sy 1.2 , 79 sy 1.5 , 1 sy 1.8 , 47 sy 1.9 , 121 sy 2 , 4 liners , and 1 unclassified agn . for the purpose of broad classification , in the rest of this paper we choose to classify all sy 1 - 1.5 as sy 1 s , and all sy 1.8 - 2 as st 2 s . in table [ tbl : bat_info ] we list the entire _ herschel_-bat sample along with positions and redshifts taken from the _ nasa / ipac extragalactic database _ ( ned ) . selection of agn by ultra - hard x - rays provides multiple advantages over other wavelengths . due to their high energy , ultra - hard x - rays easily pass through compton - thin gas or dust in the line of sight providing a direct view of the agn . using optical or mid - infrared selection can be problematic due to contamination by the host galaxy . also , ultra - hard x - rays are unaffected by any type of absorption by material obscuring the agn provided it is optically thin to compton scattering ( @xmath28 ) which is a concern for hard x - rays in the 2 - 10 kev energy range . numerous studies have been done on the bat sample in the past that span nearly the entire electromagnetic spectrum . @xcite and @xcite used _ spitzer_/irs spectra to study the mid - infrared properties of the bat agn . @xcite and @xcite studied the x - ray spectral properties for a subsample , while @xcite looked at the optical host galaxy properties and @xcite analyzed the optical spectra . along with these , many of the bat agn are detected at radio wavelengths with the first @xcite and nvss @xcite survey as well . one key ingredient missing though is the far - infrared ( fir ) where emission from ultraviolet - heated dust peaks . [ cols="<,<,<,^,^,<,^,^ " , ] while all the relationships show some amount of correlation with very low ( @xmath29 1% ) probabilities of occurring by chance , the strongest ones occur between wavelengths that are nearest each other . the 160 vs. 250 @xmath1 m and 250 vs. 350 @xmath1 m correlations have a correlation coefficient @xmath30 . this makes sense within the context of multiple temperature components . photometry from nearby wavelengths should be produced from closely related temperature components . the weak correlation between 70 and 500 @xmath1 m indicates the emission in these wavebands does not originate from closely related processes . 70 @xmath1 m emission comes from much hotter and smaller dust grains than 500 @xmath1 m and several processes could provide an explanation . since this is an agn sample , there could be a strong contribution from agn heated dust at 70 @xmath1 m , whereas at 500 @xmath1 m , agn related emission would likely be negligible . this is supported by our findings in @xcite where we showed that the 70 @xmath1 m luminosity is weakly correlated with agn luminosity . further , in @xcite we found that the bat agn morphologies at 70 @xmath1 m were concentrated in the nucleus potentially indicating an agn contribution . the weak correlation , however , can also be explained if non - star - forming processes also contribute to the 500 @xmath1 m emission . while in non - agn galaxies , the majority of 70 @xmath1 m emission is most likely due to small , stochastically heated dust grains around hii regions , @xmath31 @xmath1 m emission is likely produced by the heating of larger dust grains in the diffuse ism by older stars ( e.g. * ? ? ? therefore , the disconnect between the stellar populations would produce significant scatter in the correlation between 70 and 500 @xmath1 m . a third possibility is that synchrotron radiation produced by radio jets associated with agn can contribute to the fir , especially the longest wavelengths as seen in some radio - loud galaxies @xcite . this non - thermal emission would be completely unrelated to the thermal emission at 70 @xmath1 m , thereby producing a weaker correlation between the luminosities at those wavebands . in a later section we will show there are indeed some radio - loud sources in our sample where synchrotron emission dominates the spire emission , although the fraction of sources is quite low . when we break the sample down into sy 1 s and 2 s we do not find much difference between the correlation coefficients . this shows that sy 1 s and 2 s are not different in terms of their overall fir emission and the same processes are likely producing the fir emission . sy 1 s do show a slightly weaker correlation between the _ herschel _ luminosities especially the ones involving 500 @xmath1 m . this is likely due to the fact that most radio - loud agn are classified as sy 1 s so synchrotron emission is contributing strongest at 500 @xmath1 m compared to the other wavebands . correlations between each spire waveband luminosity and the bat 14195 kev luminosity . blue circles in the left column represent sy 1 s . red squares in the right column are sy 2 s . sources with gray arrows indicate @xmath32 upper limits . ] ultra - hard x - ray luminosity directly probes the current strength of the agn because it likely originates very close to the smbh . the 14195 kev luminosity then provides an unambiguous measure of the agn power especially for compton - thin sources . if we want to determine whether the agn contributes in any way to the fir luminosity , the first check would be to correlate the 14195 kev luminosity with each waveband s luminosity . @xcite ran correlation tests for the pacs wavebands finding a weak , but statistically significant correlation between the 70 and 160 @xmath1 m luminosity and the 14195 kev luminosity for sy 1 s but not for sy 2 s . using the same methods as we did to measure strengths of the correlations between each _ herschel _ luminosity , we measured the correlation between each spire and 14195 kev luminosity . the last lines of each section of table [ tab : wave_corrs ] lists the results of the correlation tests and figure [ fig : lum_spire_bat ] plots the correlations with gray arrows indicating upper limits . for the agn sample as a whole , no significant correlation exists between the spire and 14195 kev luminosity . all of the @xmath33 , after accounting for the partial correlation with distance , are below 0.1 with @xmath34 either at or above 5% . however , when we break the sample up into sy 1 s and sy 2 s and redo the correlation tests , we find a very weak correlation between the 250 and 350 @xmath1 m luminosity and ultra - hard x - ray luminosity for sy 1 s only ( @xmath35 and 0.10 ) . sy 2 s @xmath33 are consistent with no correlation with @xmath36 for all three wavebands . this continues the trend with what was found in @xcite where only sy 1 s were found to have a weak correlation between the bat luminosity and the pacs waveband luminosities . the partial correlation coefficients were @xmath37 and @xmath38 for sy 1s and @xmath39 and @xmath40 for sy 2 s at 70 and 160 @xmath1 m respectively ( see table 3 of * ? ? ? * ) * * * * . we note however that except for the 70 @xmath1 m waveband , none of the correlation coefficients are @xmath41 away from a null correlation coefficient . so even though @xmath42 , these are all quite weak correlations between the _ herschel _ wavebands and bat luminosity for sy 1 s . at 500 @xmath1 m , the correlation completely disappears . as we discuss in @xcite , this extends the trend observed in the mir where strong correlations have been measured between the 9 , 12 , and 18 @xmath1 m luminosities and the bat luminosity @xcite but moving towards longer wavelengths the correlation degrades rapidly as shown in @xcite for 90 @xmath1 m emission . clearly then , at long wavelengths ( @xmath43 @xmath1 m ) , emission from dust unrelated to the agn dominates most galaxies . however , we must still explain why sy 1 s retain a weak correlation while sy 2 s do not . @xcite discussed in detail several theories for why sy 1 s would show a different correlation between the _ herschel _ luminosities and bat luminosity . these included an intrinsically different bat luminosity distribution for sy 1 s and sy 2 s and the addition of compton - thick ( ct ) agn in the sy 2 sample . several authors have found that the sy 2 luminosity function breaks at a significantly lower luminosity than for sy 1s ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . at low bat luminosity , then , there are more sy 2 s than sy 1 s as is evident in figure [ fig : lum_spire_bat ] . @xcite showed that at low agn luminosity the correlation between sfr and agn luminosity flattens . this can be explained one of two ways : 1 . ) only at high agn luminosity is there a direct connection between star formation and agn activity . 2 . ) at high agn luminosity , the ir - related agn emission overwhelms any star - forming related ir emission even at long wavelengths . regardless of the physical reason , the flattening of the sf - agn relationship at low luminosity could explain the correlation differences seen between sy 1s and sy 2s since sy 2s are preferentially found at lower luminosity than sy 1s . @xcite tested this for the pacs wavebands and found that only using high luminosity objects did not improve the x - ray - to - ir correlation for sy 2 s . we repeated this test with the spire luminosities and limited the samples to only agn with bat luminosity greater than @xmath44 ergs s@xmath20 . for both sy 1 s and sy 2 s the correlations become insignificant , likely because of the reduction in number of sources used in the analysis . it is then inconclusive whether or not a difference in intrinsic agn luminosity is the cause of the differences in correlations between x - ray and ir luminosity for sy 1 s and sy 2 s . the other possibility is that ct sources are contaminating the sy 2 sample . this would have an effect if the high column density ( @xmath45 @xmath27 ) material obscuring the agn scatters 14 - 195 kev photons out of our line sight causing a lower measured bat luminosity . @xcite identified 44 either confirmed ct agn or likely ct agn based on x - ray hardness ratios in our sample . we removed these likely ct sources and redid the correlation tests , finding no difference from before just as @xcite found . therefore , it is unlikely that ct sources are the cause of the difference between the sy 1 and sy 2 correlations . given the inconclusiveness of the first test limiting the sample to high luminosity objects , we can only speculate about the reason for the difference in correlations . however , @xcite did find that restricting the sample to high luminosity objects increased the strength of the correlation for sy 1 s but not sy 2 s in the pacs wavebands . it is possible then that either a direct physical link between the sfr and agn luminosity that is only evident in high luminosity agn or increased contamination of the agn to the ir sed is causing the relatively stronger correlation in sy 1s but not sy 2s . what is conclusive is that the spire emission from the agn host galaxies on average is not strongly contaminated by agn - related emission given the small values for the correlation coefficients even for sy 1 s . while in the previous sections , we examined the absolute luminosities of each spire waveband and the correlations between each other and other wavebands ( pacs and bat ) , in this section we examine the spire colors ( i.e. flux ratios ) . colors in general provide measures of the shape of the sed . different objects and mechanisms produce significantly different sed shapes across the same wavelength regime , therefore colors can be used to separate distinct populations from each other especially when groups display the same absolute brightnesses . we investigate two colors , @xmath2 and @xmath3 , that probe the rayleigh jeans tail of a modified blackbody if the dominant process producing the emission is cold dust . figure [ fig : hist_colors ] plots the kde of the two colors . the top row compares the distribution of the colors ( @xmath2 on the left and @xmath3 on the right ) from the bat agn and hrs samples . while the hrs galaxies are local like the bat agn , one major difference is the stellar mass distribution . the hrs sample contains more low stellar mass galaxies while the bat agn are strictly found in galaxies with stellar mass ( @xmath46 ) values above @xmath47 m@xmath48 @xcite . as @xcite show , fir colors can be affected by the physical properties of the galaxy , especially the colors probing the cold dust such as the ones we are investigating here . therefore , we broke the hrs sample into two groups , a high mass group ( @xmath49 m@xmath48 ) and low mass one ( @xmath49 m@xmath48 ) indicated in figure [ fig : hist_colors ] by the solid and dashed green lines . stellar masses for the hrs were obtained from @xcite . we also plot the theoretical color of the modified blackbody with a dust temperature of 20 k and emissivity ( @xmath50 ) of 2.0 and 1.5 , values typical of normal , star - forming galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . the hrs high mass group and bat agn display nearly identical color distributions for both colors whereas the hrs low mass group is skewed toward lower colors . results of a k - s test show that the hrs high mass group and bat agn colors are drawn from the same parent population with a @xmath51 and 22% for @xmath2 and @xmath3 respectively . on the other hand the hrs low mass group colors are significantly different from the bat agn with @xmath52 values much less than 1% . this is consistent with what was found in @xcite , who showed that the spire colors for the hrs sample were affected by the metallicity of the galaxy with metal rich galaxies displaying larger flux ratios and a higher @xmath50 than metal poor ones . given the strong , positive relationship between metallicity and @xmath46 ( e.g. * ? ? ? * ) , this is exactly in line with what is seen in figure [ fig : hist_colors ] . the hrs high mass group and bat agn display colors closer to the ones expected for a modified blackbody with @xmath53 while the low mass hrs group are closer to @xmath54 . in figure [ fig : color - color ] we plot both colors together for the hrs and bat agn . nearly all of the hrs galaxies are concentrated along a main locus as well as many of the bat agn . we also plot the expected colors for a modified blackbody with varying temperature between 10 and 60 k and an emissivity of either 2.0 ( green line and squares ) or 1.5 ( purple line and diamonds ) . each square or diamond represents an increase of 5 k starting at 10 k in the lower left . the main locus for both samples is clearly aligned with a modified blackbody with temperatures between 1530 k. @xcite fit the fir sed of the hrs sample using a single temperature modified blackbody finding exactly this range of temperatures and an average emissivity of 1.8 . further these values are consistent with dust in the milky way , andromeda , and other nearby galaxies @xcite . in the bottom rows , we compare sy 1 s and sy 2 s . based on the results of our analysis in section [ sec : det_rate_lum_dist ] , we would expect sy 1 s and sy 2 s to show the same distribution of colors . indeed this is the case as both distributions in both colors peak at nearly the same values and have nearly the same spread . k - s tests reveal the colors for the two seyfert types are drawn from the same parent population with @xmath55 for @xmath2 and @xmath56 for @xmath3 . while the bulk of the spire colors are very similar between the hrs and bat , and the two seyfert types , one noticeable difference is a distinct bump in the color distribution around 0.75 . this bump is absent in the hrs sample and mainly is made up of sy 1s . with both flux ratios less than one , this indicates a monotonically rising sed that is in stark contrast with the rapidly declining sed characteristic of a modified blackbody . the equation for a modified blackbody is @xmath57 where @xmath58 is the standard planck blackbody function with a temperature of @xmath59 . the bump seen in figure [ fig : hist_colors ] is very evident in figure [ fig : color - color ] as a separate population in the lower left - hand corner . specifically 6 bat agn and one hrs galaxy occupy the region of color - color space where @xmath60 and @xmath61 . based on the theoretical curves , these exceptional colors can not be explained as either a different temperature or emissivity . rather an entirely different process is producing the fir emission in these galaxies and since the colors indicate essentially a rising sed , we suspected synchrotron radiation as the likely emission mechanism with its characteristic increasing power law shape with wavelength . further there seems to be a horizontal spread in the distribution of the bat agn in figure [ fig : color - color ] that is clearly not evident in the hrs . also this effect is not seen figure [ fig : hist_colors ] and the kdes because it only becomes evident when analyzing the two colors together . both samples span the same range of colors , however their distribution in color - color space is different . this is characterized by a large group of bat agn above and to the left of the main locus and @xmath62 line ( purple ) as well as a smaller group of agn below and to the right of the main locus and @xmath63 line ( green ) . the latter group can be explained simply from a decrease in temperature and increase in emissivity up to a beta value of 3.0 ( cyan line in figure [ fig : hist_colors ] ) , indicating the prevalence of large amounts of cold dust . the former group could be explained by a decrease in the emissivity closer to around values of 1.0 ( gray line ) , however this would require the dust temperature to increase to values above 60 k , not typical of regular star - forming galaxies . rather these high temperatures ( 70100 k ) are near the expected temperatures for dust heated by the agn , which show characteristic peaks in their sed between 2040 @xmath1 m @xcite . if the agn is affecting the colors of these sources more than the ones on the main locus then there should be some correlation between the offset from the main locus and an indicator of agn strength such as x - ray luminosity . to quantify the offset from the main locus , we fit the sed of all of the sources in figure [ fig : color - color ] using a modified blackbody ( eq . [ eq : mod_blackbody ] ) with a fixed emissivity of 2.0 to measure the excess or deficiency of observed 500 @xmath1 m emission compared to the model . with the emissivity fixed at 2.0 , there are only two free parameters , the dust temperature and normalization . we fit the sources within a bayesian framework using uniform priors for the logarithm of the normalization and dust temperature and a standard gaussian likelihood function . to sample the posterior probability density function , we use the ` emcee ` package @xcite that implements the affine - invariant ensemble sampler for markov chain monte carlo ( mcmc ) originally proposed by @xcite . for the model fitting , we only use 160 , 250 , and 350 @xmath1 m flux densities . we exclude the 500 @xmath1 m data point because our aim is to compare the expected 500 @xmath1 m emission from the model with the observed one and do not want the fitting influenced by the observed emission . we also exclude the 70 @xmath1 m flux density because it can be dominated by emission from hotter dust heated by young stars in dense star - forming regions or the agn itself @xcite . each sample from the mcmc chain contains values for the parameters of the modified blackbody that are likely given the posterior distribution . from all of these parameters , we calculated 40000 modeled 500 @xmath1 m emission and `` excess '' using the following equation : @xmath64 @xmath4 then represents a fractional excess ( or deficiency ) as compared to the model emission . a deficiency would be indicated by a negative value for @xmath4 . the final excess value associated with the source is then determined as the median of all of the excess values . in figure [ fig : color - color_excess ] we plot the same color - color diagram as in figure [ fig : color - color ] with each point colored by its measured @xmath4 . in general , points with low values of the @xmath3 color show high values of @xmath4 and vice versa for high values of the @xmath3 color . points along the main locus are scattered around @xmath65 . thus , @xmath4 can quantify a source s distance from the main locus and allows us to study possible causes for this excess emission at 500 @xmath1 m . we first measure the correlation between @xmath4 and radio loudness . agn historically have been classified into two groups based on how bright their radio emission is compared to another waveband , usually optical . these groups are `` radio - loud '' and `` radio - quiet '' agn with the former group showing bright radio emission and the latter faint radio emission relative to the optical or x - ray emission @xcite . while originally radio - loud and radio - quiet agn seemed to form a dichotomy , the consensus now seems to be that there is a broad distribution of radio - loudness rather than a bimodality @xcite . further , the original radio loudness parameter , @xmath66 which measured the ratio of the radio to optical luminosity , was shown to underestimate the radio loudness especially for low - luminosity seyfert galaxies @xcite . rather @xmath67 which measures the nuclear radio to x - ray luminosity ratio was confirmed to be a better radio - loudness indicator given x - rays are less affected by obscuration and contamination from the host galaxy . therefore , for the bat agn , we use @xmath68 to measure the radio - loudness with @xmath69 and @xmath70 . for @xmath71 we first cross - correlated the bat agn with the first and nvss databases which provide 1.4 ghz flux densities over all of the northern sky . first flux densities were preferred over nvss due to the much better angular resolution ( 5 `` vs. 45 '' ) . since _ swift_/bat was an all - sky survey , nearly half of the bat agn were not included in either first or nvss . for these southern sources we turned to the sydney university molonglo sky survey ( sumss ; * ? ? ? * ) which surveyed the southern sky at 843 mhz . finally , for the remaining sources missing radio data , we performed a literature search and found 5 ghz fluxes from various other studies @xcite . to convert all flux densities to 1.4 ghz , we assumed a power - law spectrum , @xmath72 , that is typical for synchrotron emission . figure [ fig : excess_vs_rx ] plots @xmath73 against @xmath68 to test our hypothesis that the excess 500 @xmath1 m emission is related to the radio loudness of the agn . in the left panel we plot all of the sources together to show the full range of @xmath73 . indeed , the six agn with the largest values of @xmath73 exhibit high values of radio loudness ( @xmath74 ) . these six agn are hb 890241 + 622 , 2masx j23272195 + 1524375 , 3c 111.0 , 3c 120 , pictor a , and pks 2331 - 240 and all are well known radio - loud agn . they correspond to the six sources in figures [ fig : color - color ] and [ fig : color - color_excess ] that lie in the lower left hand corner . further , the lone hrs galaxy seen in figure [ fig : color - color ] among the six bat agn is the radio galaxy m87 , whose jets and radio activity have been studied extensively . based on this , we prescribe color cutoffs that can easily separate radio - loud agn from radio - quiet agn and normal star - forming galaxies : @xmath60 and @xmath61 ( see dashed lines in figure [ fig : color - color_excess ] ) . while radio - loudness can explain the most extreme values of @xmath73 , it does not explain the more moderate ones . in the right panel of figure [ fig : excess_vs_rx ] , we zoom in on the agn with @xmath75 . visually there does not appear to be any strong correlation between @xmath68 and @xmath73 and the spearman rank correlation coefficient between them is -0.15 , weak and in the opposite sense of what would be expected if synchrotron emission was contaminating the 500 @xmath1 m emission . to explore even further , we analyzed the correlations between @xmath73 and two agn - related indicators , the _ swift_/bat luminosity , @xmath76 , and the 3.4 to 4.6 @xmath1 m flux ratio ( @xmath77 ) . the 3.4 ( @xmath78 ) and 4.6 ( @xmath79 ) @xmath1 m fluxes for the bat agn were obtained from the _ wide - field infrared survey explorer _ ( _ wise _ ; * ? ? ? * ) allwise catalog accessed through the _ nasa / ipac _ infrared science archive ( irsa ) . details of the compilation of _ wise _ fluxes for the bat agn will be available in an upcoming publication ( shimizu et al . in preparation ) . @xcite showed that @xmath76 can be used as a measure of the intrinsic bolometric luminosity of the agn , unaffected by host galaxy contamination or line - of - sight absorption . @xmath77 has been shown to be an effective discriminator between agn - dominated and normal star - forming galaxies that has both high reliability and completeness @xcite in magnitude units for selecting agn . in flux units this changes to @xmath80 . @xcite also show that as the fraction of emission coming from the host galaxy increases @xmath77 increases as well making it a good measure of the relative contribution of the agn to the infrared luminosity . [ fig : excess_vs_agn ] shows the relationships between both @xmath81 ( left panel ) and @xmath77 ( right panel ) with @xmath73 after removing the six radio - loud agn . both parameters display noticeable correlations with @xmath73 with @xmath81 positively correlated and @xmath77 negatively correlated . we calculated spearman rank correlation coefficients finding values of 0.49 and -0.49 respectively . pearson correlation coefficients are 0.30 and -0.50 respectively reflecting the more linear relationship between @xmath73 and @xmath77 than the one between @xmath73 and @xmath81 . all correlations have a probability of a null correlation less than 0.01% . in the right panel we also plot the @xcite cutoff for agn - dominated galaxies where values to the left of this line indicate agn - dominated colors . both panels indicate that the strength of the agn in the host galaxy is possibly having an effect on the spire colors . a stronger agn in relation to the host galaxy is causing deviations from a standard modified blackbody in the form of a small but noticeable 500 @xmath1 m offset . without longer wavelength data , however , its impossible to determine the exact cause of the 500 @xmath1 m excess so we can only speculate . submillimeter excess emission has been observed in a number of objects including dwarf and normal star - forming galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) as well as the small and large magellanic clouds @xcite and even our own milky way @xcite . various explanations have been proposed including the presence of a very cold ( @xmath82 k ) component @xcite , grain coagulation that causes the emissivity to increase for colder temperatures @xcite , fluctuations in the cosmic microwave background @xcite , and an increase in magnetic material in the ism @xcite . while all of these explanations are certainly still possible to explain the excess seen in the bat agn , they lack any direct connection to the strength of the agn . further , a key result from all of the previous work is that the submillimeter excess is more prevalent in very metal - poor galaxies ( @xmath83 ) . all of the bat agn reside in high stellar mass galaxies @xcite and given the mass - metallicity relationship @xcite should also be quite metal rich . rather , we speculate the excess is related to radio emission more closely associated with the agn itself . several studies of the radio properties of agn have revealed a millimeter excess around 100 ghz @xcite that is likely due to either an inverted or flat sed between cm and mm wavelengths . because @xcite found the excess mainly in low luminosity agn similar to sgr a * , they invoked advection dominated accretion flows ( adaf ) that produce compact nuclear jets to explain the inverter or flat seds . however the sample of @xcite was composed of x - ray bright agn including high eddington ratio ( @xmath84 , a measure of the accretion rate relative to the eddington limit ) sources where an adaf is unlikely . @xcite instead use the radio - to - x - ray luminosity ratio to argue that the high - frequency radio emission originates near the x - ray corona of the accretion disk given the ratio s similarity to that found for stellar coronal mass ejections ( e.g * ? ? ? * ) as well as the compact nature of the radio emission . magnetic activity around the accretion disk in the core of the agn would then be responsible for the excess and if magnetic activity increases with @xmath84 , this could explain the relationship seen with @xmath81 as well as @xmath77 . this strengthens the need for a more comprehensive survey of agn in the mm wavelength range as it could clearly reveal interesting physics possibly occurring near the accretion disk . we have produced the _ herschel_/spire maps for 313 agn selected from the _ swift_/bat 58 month catalog in three wavebands : 250 , 350 , and 500 @xmath1 m . combined with the pacs photometry from @xcite , the spire flux densities presented in this paper form the complete fir seds for a large , nearby , and relatively unbiased sample of agn . we used two methods for measuring the flux densities : timeline fitting for point sources and aperture photometry for extended and undetected sources . we summarize below the results of our statistical analysis and comparison to the _ herschel _ reference survey sample of normal star - forming galaxies . * sy 2s are detected at a higher rate than sy 1s , and after accounting for upper limits , sy 2 s have slightly higher spire luminosities than sy 1 s . however the effect is small and indicates that on average , the global fir properties of agn are independent of orientation . * using a partial correlation survival analysis to account for the luminosity - distance effect and upper limits , we find all of the _ herschel _ luminosities are correlated with each other suggesting the process ( or processes ) producing the emission from 70500 @xmath1 m is connected . luminosities with the smallest wavelength difference ( i.e. 160 and 250 @xmath1 m ) are much more correlated than pairs further apart ( i.e. 70 and 500 @xmath1 m ) , in agreement with different temperature components associated with different wavebands . while this could point to the agn affecting the shorter wavebands more than the longer ones and increasing the scatter , it can also be explained by an increased contribution from older stellar populations to the emission at longer wavelengths . * none of the spire luminosities are well correlated with the 14195 kev luminosity , a proxy for the bolometric agn luminosity . the agn , in general , is unlikely to be strongly affecting either the 250 , 350 , or 500 @xmath1 m emission , however sy 1s do show a very weak correlation at 250 and 350 @xmath1 m . removing ct sources does not improve the correlation for sy 2 s . it remains to be seen what the exact explanation is for the difference in correlations between sy 1s and sy 2s but possible explanations include a direct link between star - formation and agn luminosity that is evident only at high luminosity or increased contamination by the agn . * we compared the spire colors , @xmath2 and @xmath3 , with the colors of the hrs galaxies . the bat agn have statistically similar spire color distributions as the high stellar mass ( @xmath85 m@xmath48 ) hrs galaxies . this further emphasizes that on average , the fir emission of agn host galaxies is likely produced by cold dust in the ism heated by stellar radiation just as in normal star - forming galaxies without an agn . * we did find anomalous colors for 6 bat agn with @xmath60 and @xmath61 . the fir seds for these agn are dominated by synchrotron emission from a radio jet rather than thermally heated dust . * another group of agn with less anomalous colors but still removed from the main locus were analyzed by fitting the seds with a modified blackbody and calculating a 500 @xmath1 m excess . we found the 500 @xmath1 m excess is not related to radio loudness , but is well correlated with the 14195 kev luminosity and @xmath77 ( 3.4/4.6 @xmath1 m ) color from _ wise_. we speculate this is possibly related to the millimeter excess emission recently seen in agn caused by coronal emission above the accretion disk . future work will focus on combining the photometry from @xcite and this paper as well as archival data to perform detailed sed modeling to investigate the local starburst - agn connection and the agn contribution to the fir . we thank the anonymous referee whose comments and suggestions contributed and improved the quality of this paper . the herschel spacecraft was designed , built , tested , and launched under a contract to esa managed by the herschel / planck project team by an industrial consortium under the overall responsibility of the prime contractor thales alenia space ( cannes ) , and including astrium ( friedrichshafen ) responsible for the payload module and for system testing at spacecraft level , thales alenia space ( turin ) responsible for the service module , and astrium ( toulouse ) responsible for the telescope , with in excess of a hundred subcontractors . spire has been developed by a consortium of institutes led by cardiff university ( uk ) and including univ . lethbridge ( canada ) ; naoc ( china ) ; cea , lam ( france ) ; ifsi , univ . padua ( italy ) ; iac ( spain ) ; stockholm observatory ( sweden ) ; imperial college london , ral , ucl - mssl , ukatc , univ . sussex ( uk ) ; and caltech , jpl , nhsc , univ . colorado ( usa ) . this development has been supported by national funding agencies : csa ( canada ) ; naoc ( china ) ; cea , cnes , cnrs ( france ) ; asi ( italy ) ; mcinn ( spain ) ; snsb ( sweden ) ; stfc , uksa ( uk ) ; and nasa ( usa ) . this research has made use of the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this publication makes use of data products from the wide - field infrared survey explorer , which is a joint project of the university of california , los angeles , and the jet propulsion laboratory / california institute of technology , funded by the national aeronautics and space administration .

we present far - infrared ( fir ) and submillimeter photometry from the _ herschel _ _ space observatory s spectral and photometric imaging receiver _ ( spire ) for 313 nearby ( @xmath0 ) active galactic nuclei ( agn ) . we selected agn from the 58 month _ swift _ burst alert telescope ( bat ) catalog , the result of an all - sky survey in the 14195 kev energy band , allowing for a reduction in agn selection effects due to obscuration and host galaxy contamination . we find 46% ( 143/313 ) of our sample is detected at all three wavebands and combined with our pacs observations represents the most complete fir spectral energy distributions of local , moderate luminosity agn . we find no correlation between the 250 , 350 , and 500 @xmath1 m luminosities with 14195 kev luminosity , indicating the bulk of the fir emission is not related to the agn . however , seyfert 1s do show a very weak correlation with x - ray luminosity compared to seyfert 2s and we discuss possible explanations . we compare the spire colors ( @xmath2 and @xmath3 ) to a sample of normal star - forming galaxies , finding the two samples are statistically similar , especially after matching in stellar mass . but a color - color plot reveals a fraction of the _ herschel_-bat agn are displaced from the normal star - forming galaxies due to excess 500 @xmath1 m emission ( @xmath4 ) . our analysis shows @xmath4 is strongly correlated with the 14195 kev luminosity and 3.4/4.6 @xmath1 m flux ratio , evidence the excess is related to the agn . we speculate these sources are experiencing millimeter excess emission originating in the corona of the accretion disk . [ firstpage ] galaxies : active galaxies : seyfert galaxies : photometry infrared : galaxies methods : data analysis

1512.02733

quantum entanglement is at the heart of the current development of quantum information and quantum computation [ 1 ] . it is a special quantum correlation and has been recognized as an important resource in quantum information processing [ 2 - 4 ] . the experimental demonstrations of two - particle entanglement and multi - particle entanglement in the cavity quantum electrodynamics ( qed ) have been reported [ 5 , 6 ] . some applications which focus on the entanglement or the nonclassical correlations have also been realized in recent experiments [ 7 - 10 ] . however , the entanglement is not the only type of quantum correlation and there exist quantum tasks that display quantum advantage without entanglement [ 11 - 14 ] . it has been demonstrated both theoretically [ 15 - 17 ] and experimentally [ 18 ] that other nonclassical correlation , namely , quantum discord [ 19 ] can be responsible for the computational speedup for certain quantum tasks . quantum discord , introduced in [ 19 ] , is defined as the difference between the quantum mutual information and the classical correlation and is nonzero even for separate mixed states . therefore , the quantum discord may be regarded as a more general and fundamental resource in quantum information processing . recently , the dynamics of entanglement and quantum discord for some open systems has attracted much attention [ 20 - 25 ] . it has been shown that the quantum discord can be completely unaffected by certain decoherence environment during an initial time interval [ 22 ] and this phenomenon has been verified by the recent experiment [ 23 ] . the interaction between the environment and quantum system of interest can destroy quantum coherence and lead to decoherence . it is therefore of great importance to prevent or minimize the influence of environmental noise in the practical realization of quantum information processing . one of the protocols to prevent the quantum decoherence is dynamical decoupling strategies [ 26 - 28 ] by means of a train of instantaneous pulses(``bang - bang '' pulses ) . recently , experimental suppression of polarization decoherence in a ring cavity using bang - bang decoupling technique has also been reported [ 29 ] . in this letter , we propose a scheme of increasing quantum correlations for the cavity quantum electrodynamics system consisting of two noninteracting two - level atoms interacting with their own quantized field mode [ 25 ] by means of a train of instantaneous pulses . the two atoms are initially prepared in the extended werner - like states(ewl ) [ 30 ] and the cavity fields are prepared in the fock states or thermal states . we investigate how the bang - bang pulses affect the dynamics of quantum discord , entanglement , quantum mutual information and classical correlation between the two atoms . it is found that the amount of quantum discord and entanglement of the two atom can be improved by applying the bang - bang pulses , because the increased amount of quantum mutual information is greater than classical correlation by the bang - bang pulses . in this section we investigate the dynamical evolution for the cavity quantum electrodynamics system consisting of two noninteracting two - level atoms each locally interacting with its own quantized field mode with bang - bang pulses . the hamiltonian of one atom interacting with its own quantized field mode with bang - bang pulses is given by @xmath0 with @xmath1 where @xmath2 and @xmath3 denote the annihilation and creation operators for the cavity field and @xmath4 , @xmath5 , @xmath6 are the atomic operators . @xmath7 is the hamiltonian for a train of identical pulses of duration @xmath8 , i.e. , @xmath9 where @xmath10 is the time interval between two consecutive pulses and the amplitude @xmath11 of the control field is specified to be @xmath12 , which means that we consider the @xmath13-pulse only . it is not difficult to write down the time evolution operator in the absence of control pulses field directly as @xmath14 $ ] . with the help of an @xmath15 dynamical algebraic structure [ 31 ] , we can rewrite the time evolution operator as @xmath16-i\frac{\sin[\omega(k)t]}{\omega(k)}[\frac{\delta}{2}\sigma_z+\nonumber\\ \quad g(\sigma_+a+\sigma_-a^\dag)]\}\exp[-i\omega(k-\frac{1}{2})t],\qquad\end{aligned}\ ] ] where @xmath17 is a constant of motion in the hamiltonia , @xmath18 denotes detuning given by @xmath19 , and @xmath20 . when the control pulses field is present , the time evolution operator for the duration @xmath8 is given by @xmath21.\ ] ] for the case that the pulses are strong enough , i.e. the duration @xmath22 , this time evolution operator reduces to @xmath23,\ ] ] which leads to @xmath24.\ ] ] the time evolution operator of an elementary cycle between @xmath25 and @xmath26 is described by the unitary operator @xmath27 if we focus on the stroboscopic evolution at times @xmath28 , the evolution is driven by an effective average hamiltonian [ 26 ] @xmath29^n=\exp[-ih_{eff}t_{2n}].\ ] ] if @xmath10 is sufficiently short , then the effective hamiltonian is accurately represented by the following hamiltonian @xmath30 the coupling parameter @xmath31 is proportional to the detuning @xmath18 and the time interval @xmath10 between two successive pulses . obviously , the interaction between the atom and field is averaged to zero by the `` bang - bang '' pulses when @xmath32 . with the help of the @xmath15 algebraic structure as before , the evolution operator at times @xmath28 can be expressed as @xmath33-i\frac{\sin[\omega_{eff}(k)t_{2n}]}{\omega_{eff}(k)}[\frac{\delta}{2}\sigma_z-\nonumber\\ \quad ig_{eff}(\sigma_+a-\sigma_-a^\dag)]\}\exp[-i\omega(k-\frac{1}{2})t_{2n}],\qquad\end{aligned}\ ] ] where @xmath34 . the expression for the evolution operator @xmath35 in the closed subspace @xmath36 can be obtained easily from eq . ( 11 ) . in general , at a certain time @xmath37 , the evolution operator is given by @xmath38^n & \text{$0\leq\overline{t}<t$}\\ u_0(\overline{t}-t)u_{p}u_0(t)[u_c]^n & \text{$t\leq\overline{t}<2t$ } , \end{cases}\end{gathered}\ ] ] where @xmath39 $ ] , @xmath40 $ ] denotes the integer part , and the @xmath41 is the residual time after @xmath42 cycles . notice that the evolution operators @xmath43 and @xmath44 are closed in the subspace @xmath36 and the elements of matrixes for @xmath43 and @xmath44 in this subspace can be calculated as @xmath45 and @xmath46+\nonumber\\ i\frac{\delta}{2\omega(n)}\sin[\omega(n)t]\}\exp[-i\omega(n-\frac{1}{2})t]\\ \langle e|\langle n-1|u_0(t)|e\rangle|n-1\rangle=\{\cos[\omega(n)t]-\nonumber\\ i\frac{\delta}{2\omega(n)}\sin[\omega(n)t]\}\exp[-i\omega(n-\frac{1}{2})t]\\ \langle g|\langle n|u_0(t)|e\rangle|n-1\rangle=\langle e|\langle n-1|u_0(t)|g\rangle|n\rangle\nonumber\\ = -ig\sqrt{n}\frac{\sin[\omega(n ) t]}{\omega(n)}\exp[-i\omega(n-\frac{1}{2})t].\end{aligned}\ ] ] then , the explicit expression for the evolution operator @xmath47 in this subspace can be obtained from eqs . ( 12)-(17 ) . we assume that the two cavity fields are initially in the thermal state @xmath48 , where @xmath49 , and @xmath50 is the mean photon number at the inverse temperature @xmath51 . the two atoms are initially in the extended werner - like states defined by latexmath:[\ ] ] where @xmath102 is the reduced density matrices of the ith atom . for a two - qubit @xmath103 state , the quantum discord @xmath104 between two atoms can be obtained [ 33 , 34 ] . in order to quantify the entanglement dynamics of the two atoms and make a comparison with the quantum discord dynamics , we use the wootters concurrence [ 32 ] as a entanglement measure . for the state that the density matrix have @xmath103 structure [ 33 ] as eqs . ( 20)-(21 ) , the explicit expression of concurrence between two atoms is @xmath105 generally , for mixed quantum state , the quantum discord does not coincide with entanglement . however , for the two - qubit density matrix of the form @xmath106 , where @xmath107 is maximally entangled state orthogonal to @xmath108 and @xmath109 $ ] , the concurrence is equal to the quantum discord [ 35 ] : @xmath110 . the state described by the density matrix @xmath98(with @xmath111 , @xmath112 ) is the concrete situation of this density matrix with @xmath113 quantum discord @xmath114 at the points @xmath28 are plotted as a function of time @xmath28 ( a ) with @xmath115 for different @xmath18 : @xmath116(dot line ) , @xmath117(dash line ) , @xmath118(solid line ) and ( b ) with @xmath119 for different @xmath10 : @xmath120(dot line ) , @xmath121(dash line ) , @xmath97(solid line).,title="fig:",width=302,height=151 ] in fig . 1 , we plot the concurrence and quantum discord for the state described by the density matrix @xmath98 as a function of time @xmath58 and the parameter @xmath2 in the absence of control pulses field . the cavity modes are prepared initially in the vacuum states and the two atoms are prepared initially in a werner state(@xmath122 ) . it can be seen from fig . 1(a ) that the entanglement of two atoms vanishes and revives periodically with time as @xmath123 and is always zero for @xmath124 [ 33 ] , which means that the entanglement sudden death(esd ) phenomenon [ 25 ] appears for the system . the dark period of time , during which the concurrence is zero , is shorter for larger value of @xmath2 . however the quantum discord of two atoms vanishes asymptotically as shown in fig . the concurrence @xmath89(a ) and quantum discord @xmath90(b ) are plotted as a function of time @xmath58 with @xmath125 for different @xmath10 : @xmath95(dot - dash line ) , @xmath96(dash line ) , @xmath97(solid line ) and without control pulses(dot line).,title="fig:",width=302,height=151 ] quantum discord @xmath114 at the points @xmath28 are plotted as a function of time @xmath28 with @xmath126 for different @xmath10 : @xmath120(dot - dash line ) , @xmath121(dash line ) , @xmath97(solid line ) and without control pulses(dot line).,title="fig:",width=302,height=151 ] when the bang - bang control pulses field is present , the quantum and classical correlations are displayed as a function of the time in figs . the parameters @xmath127 , @xmath122 are chosen in figs . 2 , 3 and 5 , i.e. , the two atoms are initially prepared in the maximally entangled state . we can see clearly from fig . 2 that the quantum discord between the two atoms can be enhanced by the pulses because the increased amount of quantum mutual information is always larger than the classical correlation and the quantum discord recovers to its initial value at the points @xmath28 when the detuning @xmath128 . the increased amount of the quantum correlations is larger for shorter time intervals @xmath10 of the control pulses . it is worth pointing out that different choices of @xmath55 and @xmath56 do not give dynamics of quantum correlations qualitatively different from the case treated here . focusing on the evolution at times @xmath28 , the quantum discord fluctuates with period @xmath129 for the case of the detuning @xmath130 , as displayed in fig . 3(a ) and the amplitudes are independent from the detuning @xmath18 . when the cavity fields are prepared in the fock states , similar results are found for sufficiently short time intervals between two consecutive pulses . these phenomenon can be understood for both @xmath131 and @xmath132 in eq . ( 11 ) are proportional to @xmath18 and @xmath133 is not sensitive to @xmath134 when @xmath10 is small . it is quite clear from fig . 3(b ) that the amplitude of quantum discord between two atoms is smaller for pulses with shorter time intervals @xmath10 . in fact , from eqs . ( 11 ) , ( 21 ) and ( 31 ) , the quantum discord displayed in fig . 3 can be given by @xmath135 for small @xmath10 . this expression is in accord with fig . 3 . for the werner state with @xmath136 , the concurrence and quantum discord are plotted in fig . 4 as a function of time @xmath58 for different intervals of control pulses . we find that both the concurrence and quantum discord can be enhanced by applying the bang - bang control pulses . it is interesting to point out that the phenomenon of esd may disappear if the time interval @xmath10 of the control pulses is sufficiently short . when the cavity modes are prepared initially in the thermal states , the quantum discord between the two atoms can also be enhanced by the pulses with short time interval as shown in fig . the changes of quantum discord with different time intervals of the control pulses are synchronous as the same period as the fig . however , different from the case of vacuum states , the maximums of quantum discord are decreasing slightly with time . they decrease more slowly for shorter time intervals @xmath10 of the control pulses . in this paper , we propose a scheme of increasing quantum correlations for the cavity quantum electrodynamics system consisting of two noninteracting two - level atoms each locally interacting with its own quantized field mode by making use of bang - bang pulses . the two atoms are initially prepared in the ewl states and the cavity fields are prepared in the fock states or thermal states . it is found that the amount of quantum discord and entanglement of two atom can be enhanced by applying the bang - bang pulses and the increased amount is larger for shorter time intervals of the control pulses . particularly , the phenomenon of esd may disappear if the time interval @xmath10 of the control pulses is sufficiently short . in addition , the quantum correlations recover to their initial values at the points @xmath28 when the detuning @xmath128 . the values of quantum discord at times @xmath28 fluctuates with period @xmath129 for the case of the detuning @xmath130 and the amplitude is smaller for the pulses with shorter time intervals @xmath10 . it is worth noting that the pulses used in this paper are also suitable to the system with the hamiltonian of interaction has the form of @xmath137(here @xmath138 and @xmath139 are any operators of the cavity field or reservoir ) . the approach adopted here may be used to improve the implementation of tasks based on quantum correlations in quantum information processing . nielsen m. a. and chuang i. l. , _ quantum computation and quantum information _ ( cambridge university press , cambridge , 2000 ) . kempe j. , phys . a * 60 * ( 1999 ) 910 . bennett c. h. , brassard g. , crepeau c. , jozsa r. , peres a. , and wootters w. k. , phys . rev . lett . * 70*(13 ) , ( 1993 ) 1895 ; 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we propose a scheme of increasing quantum correlations for the cavity quantum electrodynamics system consisting of two noninteracting two - level atoms each locally interacting with its own quantized field mode by bang - bang pulses . we investigate the influence of the bang - bang pulses on the dynamics of quantum discord , entanglement , quantum mutual information and classical correlation between the two atoms . it is shown that the amount of quantum discord and entanglement of the two atoms can be improved by applying the bang - bang pulses .

1208.5381

spatial indexes has always been an important issue for multi dimensional data sets in relational databases ( dbs ) , in particular for those dealing with spherical coordinates , e.g. latitude / longitude for earth locations or ra / dec for celestial objects . some db servers offer built - in capabilities to create indexes on these ( coordinate ) columns which consequently speed up the execution of queries involving them . however 1 . the use of these facilities could be not easy , 2 . they typically use a syntax quite different from the astronomical one , 3 . their performance is inadequate for the astronomical use . within the mcs library project ( calderone & nicastro 2007 ; nicastro & calderone 2006 , 2007 ; ) we have implemented the dif package , a tool which performs and manages in a fully automatic way the sky pixelisation with both the htm ( kunszt et al . 2001 ) and healpix ( grski et al . 2005 ) schema . using a simple tool , any db table with sky coordinates columns can be easily indexed . this is achieved by using the facilities offered by the mysql db server ( which is the only server mcs supports at the moment ) , i.e. triggers , views and plugins . having a table with sky coordinates , the user can make it fully indexed in order to perform quick queries on rectangular and circular regions ( cone ) or to create an healpix map file . an sql query to select objects in a cone will look like this : ` select * from mycatalogue where ` ` entriesincone(20 , 30 , 5 ) ` , where ( 20,30 ) are the coordinates of the center in degrees and 5 is the radius in arcmin . the important thing to note is that the db manager needs to supply only a few parameters in the configuration phase , whereas the generic user does not need to know anything about the sky pixelisation either for ` select ` or ` insert ` or ` update ` queries . it also demonstrates that there is no need to extend standard sql for astronomical queries ( see adql ) , at least if mysql is used as db server . in terms of db table indexing , mapping a sphere with a pixel scheme means transforming a 2d into a 1d space , consequently a standard b tree index can be created on the column with the pixel ids . on a large astronomical table , depending on the `` depth '' of the pixelisation , this could lead to a gain of a 45 orders of magnitude in search efficiency . the htm and healpix schema are widely used in astronomy and are now well mature to be considered as candidates for indexing tables containing astronomical data . they are both open source and distributed as c++ libraries . htm uses triangular pixels which can recursively be subdivided into four pixels . the base pixels are 8 , 4 for each hemisphere . these `` trixels '' are not equal - area but the indexing algorithm is very efficient for selecting point sources in catalogues . healpix uses equal - area pseudo - square pixels , particularly suitable for the analysis of large - scale spatial structures . the base pixels are 12 . using a 64 bit long integer to store the index ids leads to a limit for the pixels size of about 7.7 and 0.44 milli - arcsec on a side for htm and healpix , respectively . being able to quickly retrieve the list of objects in a given sky region is crucial in several projects . for example hunting for transient sources like grbs requires fast catalogues lookup so to quickly cross match known sources with the detected objects . the ir / optical robotic telescope rem ( nicastro & calderone 2006 ) uses htm indexed catalogues to get the list of objects in @xmath0 regions . in this case accessing one billion objects catalogues like the gsc2.3 takes some 10 msec . having a fully automatic htm and healpix indexing would be crucial for the management of the dbs of future large missions like gaia . also the virtual observatory project would greatly benefit from adopting a common indexing scheme for all the various types of archive it can manage . the relevant parameters for the two pixelisations are : max res . ( @xmath1 ) : = @xmath2 $ ] = = @xmath3 ( where @xmath4 ) * htm * * healpix * + @xmath5 : @xmath6 @xmath3 ( where @xmath4 ) + i d range : @xmath7 $ ] @xmath8 $ ] + max @xmath9 : @xmath10 @xmath11 + max res . ( @xmath1 ) : @xmath12 @xmath13 ( @xmath14 ) + ' '' '' + @xmath15 ( depth ) : @xmath16 $ ] ; @xmath17 ( order @xmath18 resolution parameter ) : @xmath19 $ ] + as mentioned the maximum resolution is related to the usage of 64 bit integers and it is intrinsic to the htm and healpix c++ libraries . mcs is a set of c++ high level classes aimed at implementing an application server , that is an application providing a service over the network . mcs provides classes to interact with , manage and extend a mysql db server . the included myro package allows a per row management of db grants whereas the dif package allows the automatic management of sky pixelisation with the htm and healpix schema . see the for more information . to enable dif , when installing mcs it is enough to give to the configure script the two options ` --enable - dif --with - mysql - source = path ` where ` path ` is the path to the mysql source directory . the htm and healpix c++ libraries are included in the dif package . a db named ` dif ` will be created containing an auxiliary table ` tbl ` and a _ virtual _ table ` dif ` which is dynamically managed by the dif db engine . now let s assume one has a db ` mydb ` with a table ` mycat ` containing the two coordinates column ` racs ` and ` deccs ` representing the centi - arcsec converted j2000 equatorial coordinates ( this requires 4 bytes instead of the 8 necessary for a double value ) . to make the table manageable using both the htm and healpix pixelisation schema it is enough to give the command : where ` dif ` is the name of the script used to perform administrative tasks related to dif - handled tables , 6 is the htm depth and 8 is the healpix order whereas the 0 ( 1 ) selects the ring ( nested ) scheme . the last two parameters are the sql expressions which convert to degrees the coordinate values contained in the table fields ` racs ` and ` deccs ` . if the coordinates where already degrees , then it would have been enough to give their names , e.g. ` dif ... ra dec ` . the mysql root password is needed . in a future release we ll add the possibility to perform simple cross matching between ( dif managed ) catalogues . having an htm indexed catalogue , the query string to obtain the list of objects in a circular region centred on @xmath20 and @xmath21 with radius @xmath22 will be : + ` select * from mycat_htm where dif_htmcircle(60,30,40 ) ; ` + note the table name ` _ htm ` suffix which is needed to actually access the view handled by dif . for a rectangle with the same centre and sides @xmath23 along the @xmath24 axis and @xmath25 along the @xmath26 axis : + ` select * from mycat_htm where dif_htmrect(60,30,50,20 ) ; ` + giving only three parameters would imply a square selection . having chosen to use both htm and healpix indexing , one could request all the healpix ids of the objects in a @xmath23 square by using an htm function : + ` select healpid from mycat_htm where dif_htmrect(60,30,50 ) ; ` + to simply get the ids of the pixels falling into a circular / rectangular region one can simply ` select i d from dif.dif where ... ` , i.e. no particular dif managed table is required . to obtain the order 10 ids in ring scheme one can calculate them on the fly : + ` select dif_healplookup(0,10,racs/3.6e5,deccs/3.6e5 ) ` + ` from mycat_htm where dif_htmcircle(60,30,20 ) ; ` + giving 1 instead of 0 would give nested scheme ids . having ` ra ` and ` dec ` in degrees one would simply type ` ( 0,10,ra , dec ) ` . if one has just the healpix ids then entries on a circular region can be selected like in : + ` select * from mycat_healp where dif_healpcircle(60,30,40 ) ; ` + note the table name ` _ healp ` suffix . rectangular selections for only - healpix indexed tables will be available in the future . the current list of functions is : + ` dif_htmcircle ` , ` dif_htmrect ` , ` dif_htmrectv ` , ` dif_healpcircle ` , + ` dif_htmlookup ` , ` dif_healplookup ` , ` dif_sphedist ` . + ` dif_htmrectv ` accepts the four corners of a rectangle which can then have any orientation in the sky . ` dif_sphedist ` calculates the angular distance of two points on the sphere by using the haversines formula . a first version of idl user contributed library and demo programs aimed at producing healpix maps from the output of sql queries is available at the . calderone , g. , & nicastro , l. 2006 , in neutron stars and pulsars , mpe - report no . 291 , astro - ph/0701102 nicastro , l. , & calderone , g. 2006 , in neutron stars and pulsars , mpe - report no . 291 , astro - ph/0701099 nicastro , l. , & calderone , g. 2007 , grski k. m. , et al . 2005 , , 622 , 759 kunszt p. z. , szalay a. s. , & thakar a. r. 2001 , in mining the sky : proc . of the mpa / eso / mpe workshop , ed . a. j. banday , s. zaroubi , m. bartelmann , 631

in various astronomical projects it is crucial to have coordinates indexed tables . all sky optical and ir catalogues have up to 1 billion objects that will increase with forthcoming projects . also partial sky surveys at various wavelengths can collect information ( not just source lists ) which can be saved in coordinate ordered tables . selecting a sub - set of these entries or cross - matching them could be un - feasible if no indexing is performed . sky tessellation with various mapping functions have been proposed . it is a matter of fact that the astronomical community is accepting the htm and healpix schema as the default for object catalogues and for maps visualization and analysis , respectively . within the mcs library project , we have now made available as mysql - callable functions various htm and healpix facilities . this is made possible thanks to the capability offered by mysql 5.1 to add external plug - ins . the dif ( dynamic indexing facilities ) package distributed within the mcs library , creates and manages a combination of views , triggers , db - engine and plug - ins allowing the user to deal with database tables indexed using one or both these pixelisation schema in a completely transparent way .

0711.4964

one of the most exciting topics in nuclear physics is the study of the variation of hadron properties as the nuclear environment changes . in particular , the medium modification of the light vector - mesons is receiving a lot of attention , both theoretically and experimentally . recent experiments from the helios-3 @xcite and the ceres @xcite collaborations at the sps / cern energies have shown that there exists a large excess of the @xmath3 pairs in central s + au collisions . those experimental results may give a hint of some change of hadron properties in nuclei @xcite . forthcoming , ultra - relativistic heavy - ion experiments ( eg . at rhic ) are also expected to give significant information on the strong interaction ( qcd ) , through the detection of changes in hadronic properties ( for a review , see ref.@xcite ) . theoretically , lattice qcd simulations may eventually give the most reliable information on the density and/or temperature dependence of hadron properties in matter . however , current simulations have been performed only for finite temperature systems with zero baryon density @xcite . therefore , many authors have studied the hadron masses in matter using effective theories : the vector dominance model @xcite , qcd sum rules @xcite and the walecka model @xcite , and have reported that the mass decreases in the nuclear medium ( see also ref.@xcite ) . in the approach based on qcd sum rules , the reduction of the mass is mainly due to the four - quark condensates and one of the twist-2 condensates . however , it has been suggested that there may be considerable , intrinsic uncertainty in the standard assumptions underlying the qcd sum - rule analyses @xcite . in hadronic models , like quantum hadrodynamics ( qhd ) @xcite , the on - shell properties of the scalar ( @xmath0 ) and vector ( @xmath1 ) meson with vacuum polarization were first studied by saito , maruyama and soutome @xcite , and later by many authors @xcite . ( good physical arguments concerning the @xmath1 meson in medium were found in ref.@xcite . ) the main reason for the reduction in masses in qhd is the polarization of the dirac sea , where the _ anti - nucleons _ in matter play a crucial role . from the point of view of the quark model , however , the strong excitation of @xmath4-@xmath5 pairs in medium is difficult to understand @xcite . recently guichon , saito , rodionov and thomas @xcite have developed an entirely different model for both nuclear matter and finite nuclei , in which quarks in non - overlapping nucleon bags interact _ self - consistently _ with ( structureless ) scalar ( @xmath0 ) and vector ( @xmath1 and @xmath2 ) mesons in the mean - field approximation ( mfa ) the quark - meson coupling ( qmc ) model . ( the original idea was proposed by guichon in 1988 @xcite . several interesting applications to the properties of nuclear matter and finite nuclei are also given in a series of papers by saito and thomas @xcite . ) this model was recently used to calculate detailed properties of static , closed shell nuclei from @xmath6o to @xmath7pb , where it was shown that the model can reproduce fairly well the observed charge density distributions , neutron density distributions etc . blunden and miller @xcite have also considered a model for finite nuclei along this line . to investigate the properties of hadrons , particularly the changes in their masses in nuclear medium , one must also consider the structure of the mesons , as well as the nucleon . saito and thomas @xcite have studied variations of hadron masses and matter properties in _ infinite _ nuclear matter , in which the vector - mesons are also described by bags , but the scalar - meson mass is kept constant , and have shown the decrease of the hadron mass . now it would be most desirable to extend this picture to _ finite _ nuclei to study the changes of hadron properties in the medium , _ quantitatively_. our main aim in this paper is to give an effective lagrangian density for finite nuclei , in which the structure effects of the mesons ( @xmath0 , @xmath1 and @xmath2 ) as well as the nucleon are involved , and to study quantitative changes in the hadron ( including the hyperon ) masses by solving relativistic hartree equations for spherical nuclei derived from the lagrangian density . ( using this model , we also calculate some static properties of closed - shell nuclei . ) in the present model the change in the hadron mass can be described by a simple formula , which is expressed in terms of the number of non - strange quarks and the value of the scalar mean - field ( see also ref.@xcite ) . this is accurate over a wide range of nuclear density . we then find a new , simple scaling relation for the changes of hadron masses in the medium : ~ ~ , ~ , where @xmath8 , with the effective hadron mass , @xmath9 ( @xmath10 ) . an outline of the paper is as follows . in sec . [ sec : qmc ] , the idea of the qmc model is first reviewed . then , the model is extended to include the effect of meson structure . in sec . [ sec : numerical ] , parameters in the model are first determined to reproduce the properties of infinite nuclear matter , and the hadron masses in the medium are then discussed . a new scaling relationship among them is also derived . the static properties of several closed - shell nuclei are studied in subsection [ subsec : finite ] , where we also show the changes of the masses of the nucleon , the mesons ( @xmath0 , @xmath1 and @xmath2 ) and the hyperons ( @xmath11 , @xmath12 and @xmath13 ) in @xmath14ca and @xmath7pb . the last section gives our conclusions . let us suppose that a free nucleon ( at the origin ) consists of three light ( u and d ) quarks under a ( lorentz scalar ) confinement potential , @xmath15 . then , the dirac equation for the quark field , @xmath16 , is given by _ q(r ) = 0 , [ dirac1 ] where @xmath17 is the bare quark mass . next we consider how eq.([dirac1 ] ) is modified when the nucleon is bound in static , uniformly distributed ( iso - symmetric ) nuclear matter . in the qmc model @xcite it is assumed that each quark feels scalar , @xmath18 , and vector , @xmath19 , potentials , which are generated by the surrounding nucleons , as well as the confinement potential ( see also ref.@xcite ) . since the typical distance between two nucleons around normal nuclear density ( @xmath20 @xmath21 ) is surely larger than the typical size of the nucleon ( the radius @xmath22 is about 0.8 fm ) , the interaction ( except for the short - range part ) between the nucleons should be colour singlet ; e.g. , a meson - exchange potential . therefore , this assumption seems appropriate when the baryon density , @xmath23 , is not high . if we use the mean - field approximation for the meson fields , eq.([dirac1 ] ) may be rewritten as _ q(r ) = 0 . [ dirac2 ] the potentials generated by the medium are constants because the matter distributes uniformly . as the nucleon is static , the time - derivative operator in the dirac equation can be replaced by the quark energy , @xmath24 . by analogy with the procedure applied to the nucleon in qhd @xcite , if we introduce the effective quark mass by @xmath25 , the dirac equation , eq.([dirac2 ] ) , can be rewritten in the same form as that in free space , with the mass @xmath26 and the energy @xmath27 , instead of @xmath17 and @xmath28 . in other words , the vector interaction has _ no effect on the nucleon structure _ except for an overall phase in the quark wave function , which gives a shift in the nucleon energy . this fact _ does not _ depend on how to choose the confinement potential , @xmath15 . then , the nucleon energy ( at rest ) , @xmath29 , in the medium is @xcite e_n = m_n^(v_s^q ) + 3v_v^q , [ efmas ] where the effective nucleon mass , @xmath30 , depends on _ only the scalar potential _ in the medium . now we extend this idea to finite nuclei . the solution of the general problem of a composite , quantum particle moving in background scalar and vector fields that vary with position is extremely difficult . one has , however , a chance to solve the particular problem of interest to us , namely light quarks confined in a nucleon which is itself bound in a finite nucleus , only because the nucleon motion is relatively slow and the quarks highly relativistic @xcite . thus the born - oppenheimer approximation , in which the nucleon internal structure has time to adjust to the local fields , is naturally suited to the problem . it is relatively easy to establish that the method should be reliable at the level of a few percent @xcite . even within the born - oppenheimer approximation , the nuclear surface gives rise to external fields that may vary appreciably across the finite size of the nucleon . our approach in ref.@xcite was to start with a classical nucleon and to allow its internal structure to adjust to minimise the energy of three quarks in the ground - state of a system under constant scalar and vector fields , with values equal to those at the centre of the nucleon . in ref.@xcite , the mit bag model was used to describe the nucleon structure . blunden and miller have also examined a relativistic oscillator model as an alternative model @xcite . of course , the major problem with the mit bag ( as with many other relativistic models of nucleon structure ) is that it is difficult to boost . we therefore solve the bag equations in the instantaneous rest frame ( irf ) of the nucleon using a standard lorentz transformation to find the energy and momentum of the classical nucleon bag in the nuclear rest frame . having solved the problem using the meson fields at the centre of the @xmath31nucleon ( which is a quasi - particle with nucleon quantum numbers ) , one can use perturbation theory to correct for the variation of the scalar and vector fields across the nucleon bag . in first order perturbation theory only the spatial components of the vector potential give a non - vanishing contribution . ( note that , although in the nuclear rest frame only the time component of the vector field is non - zero , in the irf of the nucleon there are also non - vanishing spatial components . ) this extra term is a correction to the spin - orbit force . as shown in refs.@xcite , the basic result in the qmc model is that , in the scalar ( @xmath0 ) and vector ( @xmath1 ) meson fields , the nucleon behaves essentially as a point - like particle with an effective mass @xmath30 , which depends on the position through only the @xmath0 field , moving in a vector potential generated by the @xmath1 meson , as mentioned near eq.([efmas ] ) . although we discussed the qmc model using the specific model , namely the bag model , in ref.@xcite , _ the qualitative features we found are correct in any model _ in which the nucleon contains _ relativistic quarks _ and the ( middle- and long - range ) _ attractive _ and ( short - range ) _ repulsive _ n - n forces have _ lorentz - scalar _ and _ vector characters _ , respectively . let us suppose that the scalar and vector potentials in eq.([dirac2 ] ) are mediated by the @xmath0 and @xmath1 mesons , and introduce their mean - field values , which now depend on position @xmath32 , by @xmath33 and @xmath34 , respectively , where @xmath35 ( @xmath36 ) is the coupling constant of the quark-@xmath0 ( @xmath1 ) meson . furthermore , we shall add the isovector , vector meson , @xmath2 , and the coulomb field , @xmath37 , to describe finite nuclei realistically @xcite . then , the effective lagrangian density for finite nuclei , involving the quark degrees of freedom in the nucleon and the ( structureless ) meson fields , in mfa would be given by @xcite _ qmc - i&= & [ i - m_n^((r ) ) - g_(r ) _ 0 + & - & g _ b(r ) _ 0 - ( 1+^n_3 ) a(r ) _ 0 ] + & - & [ ( ( r))^2 + m_^2 ( r)^2 ] + [ ( ( r))^2 + m_^2 ( r)^2 ] + & + & [ ( b(r))^2 + m_^2 b(r)^2 ] + ( a(r))^2 , [ qmclag ] [ qmc-1 ] where @xmath38 and @xmath39 are respectively the nucleon and the @xmath2 ( the time component in the third direction of isospin ) fields . @xmath40 , @xmath41 and @xmath42 are respectively the ( constant ) masses of the @xmath0 , @xmath1 and @xmath2 mesons . @xmath43 and @xmath44 are respectively the @xmath1-n and @xmath2-n coupling constants , which are related to the corresponding quark-@xmath1 , @xmath45 , and quark-@xmath2 , @xmath46 , coupling constants as @xmath47 and @xmath48 @xcite . we call this model the qmc - i model . if we define the field - dependent @xmath0-n coupling constant , @xmath49 , by m_n^((r ) ) m_n - g_((r ) ) ( r ) , [ coup ] where @xmath50 is the free nucleon mass , it is easy to compare with qhd @xcite . @xmath49 will be discussed further below . the difference between qmc - i and qhd lies only in the coupling constant @xmath51 , which depends on the scalar field in qmc - i while it is constant in qhd . ( the relationship between qmc and qhd has been already clarified in ref.@xcite . see also ref.@xcite . ) however , this difference leads to a lot of favorable results , notably the nuclear compressibility , @xcite . detailed calculated properties of both infinite nuclear matter and finite nuclei can be found in refs.@xcite . here we consider the nucleon mass in matter further . the nucleon mass is a function of the scalar field . because the scalar field is small at low density the nucleon mass can be expanded in terms of @xmath0 as m_n^ = m_n + ( ) _ = 0 + ( ) _ = 0 ^2 + . [ nuclm ] in the qmc model the interaction hamiltonian between the nucleon and the @xmath0 field at the quark level is given by @xmath52 , and the derivative of @xmath30 with respect to @xmath0 is ( ) = -3g_^q dr _ q _ q -3g_^q s_n ( ) . [ deriv ] here we have defined the quark - scalar density in the nucleon , @xmath53 , which is itself a function of the scalar field , by eq.([deriv ] ) . because of a negative value of @xmath54 , the nucleon mass decreases in matter at low density . furthermore , we define the scalar - density ratio , @xmath55 , to be @xmath56 and the @xmath0-n coupling constant at @xmath57 to be @xmath51 ( i.e. , @xmath58 ) : c_n ( ) = s_n()/s_n(0 ) g _ = 3g_^q s_n(0 ) . [ cn ] comparing with eq.([coup ] ) , we find that ( ) = -g _ c_n ( ) = - , [ deriv2 ] and that the nucleon mass is m_n^ = m_n - g _ - g _ c_n^(0 ) ^2 + . [ nuclm2 ] in general , @xmath59 is a decreasing function because the quark in matter is more relativistic than in free space . thus , @xmath60 takes a negative value . if the nucleon were structureless @xmath59 would not depend on the scalar field , that is , @xmath59 would be constant ( @xmath61 ) . therefore , only the first two terms in the right hand side of eq.([nuclm2 ] ) remain , which is exactly the same as the equation for the effective nucleon mass in qhd . by taking the heavy - quark - mass limit in qmc we can reproduce the qhd results @xcite . if the mit bag model is adopted as the nucleon model , @xmath62 is explicitly given by @xcite s_n ( ) = , [ sss ] where @xmath63 is the kinetic energy of the quark in units of @xmath64 and @xmath65 is the eigenvalue of the quark in the nucleon in matter . we denote the bag radius of the nucleon in free space ( matter ) by @xmath22 ( @xmath66 ) . in actual numerical calculations we found that the scalar - density ratio , @xmath56 , decreases linearly ( to a very good approximation ) with @xmath67 @xcite . then , it is very useful to have a simple parametrization for @xmath59 : c_n ( ) = 1 - a_n ( g _ ) , [ paramcn ] with @xmath68 in mev ( recall @xmath69 ) and @xmath70 ( mev@xmath71 ) for @xmath17 = 5 mev and @xmath22 = 0.8 fm . this is quite accurate up to @xmath72 . as a practical matter , it is easy to solve eq.([deriv2 ] ) for @xmath49 in the case where @xmath73 is linear in @xmath68 , as in eq.([paramcn ] ) . then one finds m^_n = m_n - g _ , [ mstar ] so that the effective @xmath0-n coupling constant , @xmath74 , decreases at half the rate of @xmath56 . in the previous section we have considered the effect of nucleon structure . it is however true that the mesons are also built of quarks and anti - quarks , and that they may change their properties in matter . to incorporate the effect of meson structure in the qmc model , we suppose that the vector mesons are again described by a relativistic quark model with _ common _ scalar and vector mean - fields @xcite , like the nucleon ( see eq.([dirac2 ] ) ) . then , again the effective vector - meson mass in matter , @xmath75 , depends on only the scalar mean - field . however , for the scalar ( @xmath0 ) meson it may not be easy to describe it by a simple quark model ( like a bag ) because it couples strongly to the pseudoscalar ( @xmath76 ) channel , which requires a direct treatment of chiral symmetry in medium @xcite . since , according to the nambu jona - lasinio model @xcite or the walecka model @xcite , one might expect the @xmath0-meson mass in medium , @xmath77 , to be less than the free one , we shall here parametrize it using a quadratic function of the scalar field : ( ) = 1 - a _ ( g _ ) + b _ ( g _ ) ^2 , [ sigmas ] with @xmath68 in mev , and we introduce two parameters , @xmath78 ( in mev@xmath71 ) and @xmath79 ( in mev@xmath80 ) . ( we will determine these parameters in the next section . ) using these effective meson masses , we can find a new lagrangian density for finite nuclei , which involves the structure effects of not only the nucleons but also the mesons , in the mfa : _ qmc - ii&= & [ i - m_n^ - g_(r ) _ 0 - g _ b(r ) _ 0 - ( 1+^n_3 ) a(r ) _ 0 ] + & - & [ ( ( r))^2 + m_^2 ( r)^2 ] + [ ( ( r))^2 + m_^2 ( r)^2 ] + & + & [ ( b(r))^2 + m_^2 b(r)^2 ] + ( a(r))^2 , [ qmc-2 ] where the masses of the mesons and the nucleon depend on the scalar mean - fields . we call this model qmc - ii . at low density the vector - meson mass can be again expanded in the same way as in the nucleon case ( eq.([nuclm ] ) ) : m_v^ & = & m_v + ( ) _ = 0 + ( ) _ = 0 ^2 + , + & & m_v - 2 g_^q s_v(0 ) - g_^q s_v^(0 ) ^2 , + & & m_v - g__v / n c_v^(0 ) ^2 , [ vmm ] where @xmath81 is the quark - scalar density in the vector meson , ( ) = - g__v / n c_v ( ) , [ deriv3 ] and @xmath82 . in eqs.([vmm ] ) and ( [ deriv3 ] ) , we introduce a correction factor , @xmath83 , which is given by @xmath84 , because the coupling constant , @xmath51 , is defined specifically for the nucleon by eq.([cn ] ) . in this section we will show our numerical results using the lagrangian density of the qmc - ii model that is , including self - consistently the density dependence of the meson masses . we have studied the qmc - i model , and have already shown the calculated properties of finite nuclei in refs.@xcite . for infinite nuclear matter we take the fermi momenta for protons and neutrons to be @xmath85 ( @xmath86 or @xmath87 ) . this is defined by @xmath88 , where @xmath89 is the density of protons or neutrons , and the total baryon density , @xmath23 , is then given by @xmath90 . let the _ constant _ mean - field values for the @xmath0 , @xmath1 and @xmath2 fields be @xmath91 , @xmath92 and @xmath93 , respectively . > from the lagrangian density eq.([qmc-2 ] ) , the total energy per nucleon , @xmath94 , can be written ( without the coulomb force ) e_tot / a = _ i = p , n^k_f_i d + ^2 + _ b + _ 3 ^ 2 , [ tote ] where the value of the @xmath1 field is now determined by baryon number conservation as @xmath95 , and the @xmath2-field value by the difference in proton and neutron densities , @xmath96 , as @xmath97 @xcite . on the other hand , the scalar mean - field is given by a self - consistency condition ( scc ) : = & - & + & + & ( ) + ( ) - ( ) . [ scc ] using eqs.([deriv2 ] ) , ( [ sigmas ] ) and ( [ deriv3 ] ) , eq.([scc ] ) can be rewritten & = & + g _ ( ) ^2 + & - & ( ) . [ scc2 ] now we need a model for the structure of the hadrons involved . we use the mit bag model in static , spherical cavity approximation @xcite . as in ref.@xcite , the bag constant @xmath98 and the parameter @xmath99 ( which accounts for the sum of the c.m . and gluon fluctuation corrections @xcite ) in the familiar form of the mit bag model lagrangian are fixed to reproduce the free nucleon mass ( @xmath50 = 939 mev ) under the condition that the hadron mass be stationary under variation of the free bag radius ( @xmath22 in the case of the nucleon ) . furthermore , to fit the free vector - meson masses , @xmath100 = 783 mev and @xmath42 = 770 mev , we introduce new @xmath101-parameters for them , @xmath102 and @xmath103 . in the following we choose @xmath104 fm and the free quark mass @xmath17 = 5 mev . variations of the quark mass and @xmath22 only lead to numerically small changes in the calculated results @xcite . we then find that @xmath105 = 170.0 mev , @xmath99 = 3.295 , @xmath106 = 1.907 and @xmath107 = 1.857 . thus , @xmath59 is given by eq.([sss ] ) , and @xmath108 is given by a similar form , with the kinetic energy of quark and the bag radius for the vector meson . we find that the bag model gives @xmath109 = 0.9996 . therefore , we may discard those correction factors in practical calculations . next we must choose the two parameters in the parametrization for the @xmath0-meson mass in matter ( see eq.([sigmas ] ) ) . in this paper , we consider three parameter sets : ( a ) @xmath110 ( mev@xmath71 ) and @xmath111 ( mev@xmath80 ) , ( b ) @xmath112 ( mev@xmath71 ) and @xmath113 ( mev@xmath80 ) , ( c ) @xmath114 ( mev@xmath71 ) and @xmath111 ( mev@xmath80 ) . the parameter sets a , b and c give about 2% , 7% and 10% decreases of the @xmath0 mass at saturation density , respectively . we will revisit this issue in the next subsection . now we are in a position to determine the coupling constants . @xmath115 and @xmath116 are fixed to fit the binding energy ( @xmath117 mev ) at the saturation density ( @xmath20 @xmath21 ) for symmetric nuclear matter . furthermore , the @xmath2-meson coupling constant is used to reproduce the bulk symmetry energy , 35 mev . we take @xmath40 = 550 mev . the coupling constants and some calculated properties for matter are listed in table [ ccc ] . the last three columns show the relative changes ( from their values at zero density ) of the nucleon - bag radius ( @xmath118 ) , the lowest eigenvalue ( @xmath119 ) and the root - mean - square radius ( rms radius ) of the nucleon calculated using the quark wave function ( @xmath120 ) at saturation density . .coupling constants and calculated properties for symmetric nuclear matter at normal nuclear density ( @xmath17 = 5 mev , @xmath22 = 0.8 fm and @xmath40 = 550 mev ) . the effective nucleon mass , @xmath30 , and the nuclear compressibility , @xmath121 , are quoted in mev . the bottom row is for qhd . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] while there are still some discrepancies between the results and data , the present model provides reasonable results . in particular , as in qmc - i , it reproduces the rms charge radii for medium and heavy nuclei quite well . in figs.[hmca40 ] and [ hmpb ] we present the changes of the nucleon , @xmath0 and @xmath1 meson masses in @xmath14ca and @xmath7pb , respectively . the interior density of @xmath14ca is much higher than @xmath122 , while that in @xmath7pb is quite close to @xmath122 . accordingly , in the interior the effective hadron masses in @xmath14ca are smaller than in @xmath7pb . we can also see this in fig.[fsso ] , where the strength of the scalar field in the interior part of @xmath14ca is stronger than in @xmath7pb . using the local - density approximation and eq.([hm3 ] ) , it is possible to calculate the changes of the hyperon ( @xmath11 , @xmath12 and @xmath13 ) masses in @xmath14ca and @xmath7pb , which are respectively illustrated in figs.[hypca40 ] and [ hyppb ] . our quantitative calculations for the changes of the hyperon masses in finite nuclei may be quite important in forthcoming experiments concerning hypernuclei @xcite . we have extended the quark - meson coupling ( qmc ) model to include quark degrees of freedom within the scalar and vector mesons , as well as in the nucleons , and have investigated the density dependence of hadron masses in nuclear medium . as several authors have suggested @xcite , the hadron mass is reduced because of the scalar mean - field in medium . our results are quite consistent with the other models . in the present model the hadron mass can be related to the number of non - strange quarks and the strength of the scalar mean - field ( see eq.([hm3 ] ) ) . we have found a new , simple formula to describe the hadron masses in the medium , and this led to a new scaling relationship among them ( see eq.([scale ] ) ) . furthermore , we have calculated the changes of not only the nucleon , @xmath0 , @xmath1 and @xmath2 masses but also the hyperon ( @xmath11 , @xmath12 and @xmath13 ) masses in finite nuclei . we should note that the origins of the mass reduction in qmc and qhd are completely different @xcite . it would be very interesting to compare our results with forthcoming experiments on hypernuclei . by applying this extended qmc model to finite nuclei , we have studied the properties of some static , closed shell nuclei . our ( self - consistent ) calculations reproduce well the observed static properties of nuclei such as the charge density distributions . in the present model , there are , however , still some discrepancies in energy spectra of nuclei , in particular , the spin - orbit splittings . to overcome this defect , we have discussed one possible way , in which a constituent quark mass ( @xmath123 mev ) is adopted , in refs.@xcite . as an alternative , jin and jennings @xcite and blunden and miller @xcite have proposed variations of the bag constant and @xmath101 parameter in medium , which have been suggested by the fact that quarks are partially deconfined in matter . to help settle this problem , one should perhaps consider the change of the vacuum properties in the medium @xcite . our lagrangian density , eq.([qmc-2 ] ) , provides a lot of effective coupling terms among the meson fields because the mesons have structure ( cf . ref.@xcite ) . in particular , the lagrangian automatically offers self - coupling terms ( or non - linear terms ) with respect to the @xmath0 field . using eq.([sigmas ] ) , the lagrangian density gives the non - linear @xmath0 terms ( up to @xmath124 ) as : _ qmc - ii^nl & = & - m_^()^2 ^2 , + & & -m_^2 ^2 + g_a_m_^2 ^3 - g_^2 ( a_^2 + 2 b _ ) m_^2 ^4 . [ nls ] on the other hand , in nuclear physics , qhd with non - linear @xmath0 terms has been extensively used in mfa to describe realistic nuclei @xcite . the most popular parametrizations are called nl1 , nl2 @xcite and nl - sh @xcite , and the non - linear terms in those parametrizations are given as _ qhd^nl = - m_^2 ^2 + g_2 ^3 + g_3 ^4 , [ nls2 ] where @xmath125 and @xmath126 take respectively a positive ( negative ) [ positive ] and positive ( negative ) [ positive ] values in nl1 ( nl2 ) [ nl - sh ] . since the non - linear @xmath0 terms provide the self - energy of @xmath0 meson , it changes the @xmath0 mass in matter . comparing eq.([nls2 ] ) with eq.([nls ] ) , we can see that the effective @xmath0 mass in nl2 _ increases _ at low nuclear density while the @xmath0 mass _ decreases _ in nl1 and nl - sh in mfa . however , from the point of view of a field theory , like the nambu jona - lasinio model , an increase of the @xmath0 mass in the medium seems unlikely @xcite . ( we should note that the values of @xmath125 in those parametrizations are small compared with the corresponding one in eq.([nls ] ) . ) furthermore , from the point of view of field theory , @xmath126 in eq.([nls2 ] ) should be negative because the vacuum must be stable @xcite . therefore , we can conclude that one would expect to find @xmath127 and @xmath128 in eq.([nls2 ] ) . unfortunately , the above three parametrizations used in nuclear physics do not satisfy the condition , while our lagrangian , eq.([nls ] ) , does . it will be very interesting to explore the connection between various coupling strengths found empirically in earlier work and those found in our approach . finally , we would like to give some caveats concerning the present calculation . the basic idea of the model is that the mesons are locally coupled to the quarks . therefore , in the present model the effect of short - range correlations among the quarks , which would be associated with overlap of the hadrons , are completely neglected . at very high density these would be expected to dominate and the present model must eventually break down there ( probably beyond @xmath129 ) . furthermore , the pionic cloud of the hadron @xcite should be considered explicitly in any truly quantitative study of hadron properties in medium . we note that subtleties such as scalar - vector mixing in medium and the splitting between longitudinal and transverse masses of the vector mesons @xcite have been ignored in the present mean - field study . although the former appears to be quite small in qhd the latter will certainly be important in any attempt to actually measure the mass shift . 99 m. masera ( helios-3 collaboration ) , nucl . phys . * a590 * , 93c ( 1995 ) . p. wurm ( ceres collaboration ) , nucl . phys . * a590 * , 103c ( 1995 ) . ko and g.e . brown , nucl . phys . * a606 * , 568 ( 1996 ) ; + g. chanfray , r. rapp , and j. wambach , phys . rev 76 * , 368 ( 1996 ) . quark matter 95 , nucl . * a590 * ( 1995 ) . lattice 94 , nucl . b42 * ( proc . suppl . ) ( 1995 ) . m. asakawa , c.m . ko , p. lvai and x.j . qiu , phys . rev . * c46 * , r1159 ( 1992 ) . t. hatsuda and su h. lee , phys . rev . * c46 * , r34 ( 1993 ) ; + m. asakawa and c.m . ko , phys . rev . * c48 * , r526 ( 1993 ) ; + t. hatsuda , y. koike and su h. lee , nucl . phys . * b394 * , 221 ( 1993 ) . k. saito , t. maruyama and k. soutome , phys c40 * , 407 ( 1989 ) ; + k. soutome , t. maruyama and k. saito , nucl . phys . * a507 * , 731 ( 1990 ) . h. kurasawa and t. suzuki , prog . . phys . * 84 * , 1030 ( 1990 ) ; + j.c . caillon and j. labarsouque , phys . lett . * b311 * , 19 ( 1993 ) . c . jean , j. piekarewicz and a.g . williams , phys . rev . * c49 * , 1981 ( 1994 ) . h. shiomi and t. hatsuda , phys . lett . * b334 * , 281 ( 1994 ) ; + h. kuwabara and t. hatsuda , prog . . phys . * 96 * , 1163 ( 1995 ) . t. hatsuda , to be published in proc . of the international symposium on non - nucleonic degrees of freedon detected in nucleus , osaka , sept . 25 ( 1996 ) ( world scientific , singapore ) . d.k . griegel and thomas . d. cohen , phys . lett . * b333 * , 27 ( 1994 ) . walecka , ann . ( n.y . ) * 83 * , 491 ( 1974 ) ; + b.d . serot and j.d . walecka , advan . * 16 * , 1 ( 1986 ) . k. saito and a.w . thomas , phys . rev . * c52 * , 2789 ( 1995 ) . guichon , k. saito , e. rodionov and a.w . thomas , nucl . phys . * a601 * , 349 ( 1996 ) ; + p.a.m . guichon , k. saito and a.w . thomas , nucl - th/9602022 , to appear in australian journal of physics ( 1996 ) . guichon , phys . lett . * b200 * , 235 ( 1988 ) . k. saito , a. michels and a.w . thomas , phys . rev . * c46 * , r2149 ( 1992 ) ; + a.w . thomas , k. saito and a. michels , aust . * 46 * , 3 ( 1993 ) ; + k. saito and a.w . thomas , nucl . phys . * a574 * , 659 ( 1994 ) ; phys . lett . * b335 * , 17 ( 1994 ) ; ibid . * b363 * , 157 ( 1995 ) . k. saito and a.w . thomas , phys . lett . * b327 * , 9 ( 1994 ) . k. saito and a.w . thomas , phys . rev . * c51 * , 2757 ( 1995 ) . k. saito , k. tsushima and a.w . thomas , nucl . phys . * a609 * , 339 ( 1996 ) . blunden and g.a . miller , phys . rev . * c54 * , 359 ( 1996 ) . x. jin and b.k . jennings , phys . lett . * b374 * ( 1996 ) 13 ; phys . rev . * c54 * , 1427 ( 1996 ) ; nucl - th/9608023 . in our earlier works , the phenomenological center - of - mass correction to the bag energy was used . in this paper , however , we do not use it because the c.m . and gluon fluctuation corrections may be absorbed into the familiar form , @xmath130 . see ref.@xcite . t. hatsuda and t. kunihiro , phys . rep . * 247 * , 221 ( 1994 ) . v. bernard and ulf - g . meissner , nucl . a489 * , 647 ( 1988 ) . a. chodos , r.l . jaffe , k. johnson and c.b . thorn , phys . rev . * d10 * , 2599 ( 1974 ) . k. tsushima , k. saito and a.w . thomas , a self - consistent calculation for hypernuclei , to be published . c.j . horowitz and b.d . serot , nucl . phys . * a368 * , 503 ( 1981 ) ; + c.j . horowitz , d.p . murdoch and b.d . serot , computational nuclear physics 1 , eds . k. langanke , j.a . maruhn and s.e . koonin ( springer - verlag , berlin , 1991 ) , p.129 . b.b.p . sinha _ et al . _ , b53 * , 217 ( 1971 ) . the numerical calculation was carried out by modifying the technique described by horowitz _ et al . _ it was performed with a maximum radius of 12 ( 15 ) fm on a mesh of 0.04 fm for medium mass ( pb ) nuclei . since the numerical convergence is slow we have improved the program by mixing appropriately the meson potentials given by the @xmath131-th iteration and those by the ( @xmath132)-th iteration as is usually done in non - relativistic calculations . for the qmc - ii model , it is needed to calculate several 10 iterations for middle nuclei while about 100 iterations for pb . we find that it becomes harder to find a good initial condition for the meson fields in the case where the nuclear compressibility is smaller , for example , in the parameter set a rather than in c. probably a more powerful technique is required to make the convergence of the calculation rapid . phillips , m.c . birse and s.j . wallace , nucl - th/9610007 . furnstahl , b.d . serot and h .- b . tang , nucl . phys . * a598 * , 539 ( 1996 ) ; + h. mller and b.d . serot , nucl . phys . * a606 * , 508 ( 1996 ) . reinhard , rep . 52 * , 439 ( 1989 ) ; + g.a . lalazissis , m.m . sharma , p. ring and y.k . gambhir , nucl . phys . * a608 * , 202 ( 1996 ) . s.j . lee _ et al . _ , * 57 * , 2916 ( 1986 ) ; + p.g . _ , z. phys . * a323 * , 13 ( 1986 ) . sharma , m.a . nagarajan and p. ring , phys . lett . * b312 * , 377 ( 1993 ) . t.d . lee and g.c . wick , phys . rev . * d9 * , 2291 ( 1974 ) . thomas , advan . . phys . * 13 * , 1 ( 1984 ) ; + g.a . miller , int . * 2 * , 190 ( 1984 ) .

the quark - meson coupling model , based on a mean - field description of non - overlapping nucleon bags bound by the self - consistent exchange of @xmath0 , @xmath1 and @xmath2 mesons , is extended to investigate the change of hadron properties in finite nuclei . relativistic hartree equations for spherical nuclei have been derived from a relativistic quark model of the structure of bound nucleons and mesons . using this unified , self - consistent description of both infinite nuclear matter and finite nuclei , we investigate the properties of some closed - shell nuclei , and study the changes in the hadron masses of the non - strange vector mesons , the hyperons and the nucleon in those nuclei . we find a new , simple scaling relation for the changes of the hadron masses , which can be described in terms of the number of non - strange quarks in the hadron and the value of the scalar mean - field in a nucleus . = 0 cm = 0 cm = 0 cm = 23 cm = 16 cm adp-96 - 40/t236 variation of hadron masses in finite nuclei k. saito + physics division , tohoku college of pharmacy + sendai 981 , japan + k. tsushima and a. w. thomas + department of physics and mathematical physics + university of adelaide , south australia , 5005 , australia pacs numbers : 12.39.ba , 21.60.-n , 21.90.+f , 24.85.+p

nucl-th9612001

in our solar system , zodiacal dust grains are warm ( @xmath4150k ) and found within @xmath23au of the sun . slow but persistent collisions between asteroids complemented by material released from comets now replenish these particles . similar warm dust particles around other stars are also expected and would be manifested as excess mid - infrared emission . the implication of `` warm '' excess stars for the terrestrial planet - building process has prompted many searches including several pointed observing campaigns with _ however , a lack of consensus of what constitutes a `` warm excess '' has resulted in ambiguity and some confusion in the field . for example , spitzer surveys with mips revealed a number of stars with excess emission in the 24@xmath0 band . however , very few of these may turn out as genuine `` warm excess '' stars because the detected 24@xmath0 emission is mostly the wien tail of emission from cold ( t @xmath5 150k ) dust grains @xcite . for black - body grains , @xmath6 = t@xmath7(r@xmath7/(2r@xmath8)@xmath9 , where r@xmath10 is the distance of a grain from a star of radius r@xmath7 and temperature t@xmath7 . due to the dependence of @xmath6 on t@xmath7 and r@xmath7 , the terrestrial planetary zone ( tpz ) around high mass stars extends further out than that around low mass stars . therefore , r@xmath10 is not a good way to define the tpz while dust equilibrium temperature is equally applicable to all main - sequence stars . in our solar system , t@xmath10 is 150k near the outer boundary of the asteroid belt ( @xmath23.5au ) , and the zodiacal dust particles are sufficiently large ( @xmath230@xmath0 ) that they do radiate like blackbodies . to specify a tpz independent of the mass of the central star , we define the tpz to be the region where t@xmath11 150k . then an a0 star has 25au and an m0 star has 0.9au as the outer boundary of their tpz . because of the way it is defined , tpz applies only to the location of grains that radiate like a blackbody . according to the spitzer surveys listed above , the presence of dust in the tpz characterized by excess in the mid - ir is quite rare for stars @xmath410myrs old . for ages in the range of @xmath12myr , a posited period of the terrestrial planet formation in our solar system , only a few stars appear to possess warm dust according to our analysis ( see @xmath13 5 and table 1 ) : @xmath14 cha , a b8 member of 8 myr old @xmath14 cha cluster @xcite , @xmath14 tel & hd 172555 , a0- and a7- type members of the 12 myr old @xmath15 pic moving group @xcite , hd 3003 , an a0 member of the 30 myr old tucana / horologium moving group @xcite , and hd 113766 , an f3 binary star ( 1.2@xmath16 separation , @xcite ) , in the lower centaurus crux ( lcc ) association @xcite . in this paper , we present the a9 star ef cha , another example of this rare group of stars with warm dust at the epoch of terrestrial planet formation . hipparcos , 2mass and mid course experiment ( msx , @xcite ) sources were cross - correlated to identify main - sequence stars with excess emission at mid - ir wavelengths . out of @xmath268,000 hipparcos dwarfs with @xmath17 @xmath18 6.0 ( ) - 2.0 ( see @xcite for an explanation of this @xmath17 constraint ) in a search radius of 10@xmath16 , @xmath21000 stars within 120 pc of earth were identified with potential msx counterparts . spectral energy distributions ( sed ) were created for all @xmath21,000 msx identified hipparcos dwarfs . observed fluxes from tycho-2 @xmath19 and @xmath20 and 2mass @xmath21 , @xmath22 , and @xmath23 , were fit to a stellar atmospheric model @xcite via a @xmath24@xmath25 minimization method ( see @xcite , for detailed description of sed fitting ) . from these sed fits , about 100 hipparcos dwarfs were retained that showed apparent excess emission in the msx 8@xmath0 band ( that is , the ratio [ msx flux - photosphere flux ] / msx flux uncertainty must be @xmath26 3.0 ) . since a typical positional 3@xmath27 uncertainty of msx is @xmath26@xmath16 @xcite and msx surveyed the galactic plane , a careful background check is required to eliminate contamination sources . by over - plotting the 2mass sources on the digital sky survey ( dss ) images , we eliminated more than half of the apparent excess stars that included any dubious object ( i.e. , extended objects , extremely red objects , etc . ) within a 10@xmath16 radius from the star . among the stars that passed this visual check , ef cha was selected for follow - up observations at the gemini south telescope . independent iras detections at 12 and 25@xmath0 made ef cha one of the best candidates for further investigation . an n - band image and a spectrum of ef cha were obtained using the thermal - region camera spectrograph ( t - recs ) at the gemini south telescope in march and july of 2006 ( gs-2006a - q-10 ) , respectively . thanks to the queue observing mode at gemini observatory , the data were obtained under good seeing and photometric conditions . the standard `` beam switching '' mode was used in all observations in order to suppress sky emission and radiation from the telescope . data were obtained chopping the secondary at a frequency of 2.7 hz and noddding the telescope every @xmath230sec . chopping and nodding were set to the same direction , parallel to the slit for spectroscopy . standard data reduction procedures were carried out to reduce the image and the spectrum of ef cha at n - band . raw images were first sky - subtracted using the sky frame from each chop pair . bad pixels were replaced by the median of their neighboring pixels . aperture photometry was performed with a radius of 9 pixels ( 0.9@xmath16 ) and sky annuli of 14 to 20 pixels . the spectrum of a standard star ( hd 129078 ) was divided by a planck function with the star s effective temperature ( 4500k ) and this ratioed spectrum was then divided into the spectrum of ef cha to remove telluric and instrumental features . the wavelength calibration was performed using atmospheric transition lines from an unchopped raw frame . the 1-d spectrum was extracted by weighted averaging of 17 rows . for the n - band imaging photometry , the on - source integration time of 130 seconds produced s / n @xmath26 30 with fwhm @xmath20.54@xmath16 . for the n - band spectrum , a 886 second on - source exposure resulted in s / n @xmath26 20 . a standard star , hip 57092 , was observed close in time and position to our target and was used for flux calibration of the n - band image of ef cha . for spectroscopy , hd 129078 , a kiii 2.5 star , was observed after ef cha at a similar airmass and served as a telluric standard . while our paper was being reviewed , spitzer multiband imaging photometer for spitzer ( mips ) archival images of ef cha at 24 and 70@xmath0 were released from the gould s belt legacy program led by lori allen . ef cha was detected at mips 24@xmath0 band but not at mips 70@xmath0 band . we performed aperture photometry for ef cha at 24@xmath0 on the post - bcd image produced by spitzer science center mips pipeline . we used aperture correction of 1.167 for the 24@xmath0 image given at the ssc mips website ( http://ssc.spitzer.caltech.edu/mips/apercorr/ ) with aperture radius of 13and sky inner and outer annuli of 20 and 32 , respectively . for mips 70@xmath0 data , we estimated 3 @xmath27 upper limits to the non - detection on the mosaic image that we produced using mopex software on bcd images . table 2 lists the mid - ir measurements of ef cha from msx , iras , mips , and gemini t - recs observations . the t - recs n - band image ( fov of 28.8@xmath16 @xmath28 21.6 @xmath16 ) confirmed that no other mid - ir source appears in the vicinity of ef cha and that the mid - ir excess detected by the space observatories ( iras & msx ) originates solely from ef cha . a strong silicate emission feature in the n - band spectrum ( figure 1 ) indicates the presence of warm , small ( @xmath29 @xmath1 5@xmath0 , see figure 6 in @xcite ) dust particles . amorphous silicate grains dominate the observed emission feature . however , crystalline silicate structure , probably forsterite , appears as a small bump near 11.3@xmath0 @xcite . polycyclic aromatic hydrocarbon ( pah ) particles can also produce an emission feature at 11.3@xmath0 . however , absence of other strong pah emission features at 7.7 and 8.6@xmath0 indicates that the weak 11.3@xmath0 feature does not arise from pahs . furthermore , although pah particles do appear in some very young stellar systems , they have not been detected around stars as old as 10 myr . in contrast , crystalline silicates such as olivine , forsterite , etc . are seen in a few such stellar systems @xcite . the dust continuum excess of ef cha was fit with a single temperature blackbody curve at 240k by matching the flux density at 13@xmath0 and the mips 70@xmath0 upper limit ( figure 1 ) . the 3 @xmath27 upper limit at mips 70@xmath0 band indicates that the dust temperature should not be colder than 240k . figure 1 shows that mips 24@xmath0 flux is @xmath230mjy lower than iras 25@xmath0 flux . due to the small mips aperture size compared with iras , mips 24@xmath0 flux often comes out smaller when nearby contaminating sources are included in the large iras beam . the ground - based t - recs ( 28.8@xmath2821.6 ) image of ef cha at n - band , however , shows no contaminating source in the vicinity of ef cha . thus , the higher flux density at iras 25@xmath0 perhaps indicates the presence of a significant silicate emission feature near 18@xmath0 included in the wide passband of the iras 25@xmath0 filter ( 18.5 @xmath30 29.8@xmath0 ) . a recent spitzer irs observation of another warm excess star , bd+20 307 @xcite , shows a similar discrepancy between the iras 25@xmath0 flux and mips 24@xmath0 flux in the presence of a significant silicate emission feature at @xmath218@xmath0 ( weinberger et al . 2007 in preparation ) , consistent with our interpretation . mips 24@xmath0 flux is slightly above our 240k dust continuum fit . the wide red wing of an 18@xmath0 silicate emission feature could contribute to a slight increase in mips 24@xmath0 flux . ef cha was detected in the rosat x - ray all sky survey with @xmath31 which suggests a very young age for an a9 star ( see figure 4 in @xcite ) . on the basis of this x - ray measurement , hipparcos distance ( 106 pc ) , location in the sky ( ra = 12@xmath3207@xmath33 , dec= -79@xmath32 ) , and proper motion ( pmra = -40.2@xmath341.2 & pmde = -8.4@xmath341.3 in mas / yr ) , ef cha is believed to be a member of the `` cha - near '' moving group ( avg . ra = 12@xmath3200@xmath33 & avg . dec = -79@xmath32 , avg . pmra = -41.13@xmath341.3 & avg . pmdec = -3.32@xmath340.86 in mas / yr , @xcite ) , which is @xmath210 myr old and typically @xmath290pc from earth . large blackbody grains in thermal equilibrium at 240k , would be located @xmath24.3au from ef cha while small grains , especially those responsible for the silicate emission features in our n - band spectrum , radiate less efficiently and could be located at @xmath264.3au . recent spitzer mips observations confirmed that all aforementioned ( @xmath13 1 ) warm excess stars do not have a cold dust population , indicating few large grains at large distances @xcite . lack of cold large grains , in turn , suggests local origin of the small grains seen in these warm excess stars . without cold excess from spitzer mips 70@xmath0 data , small grains in ef cha should originate in the tpz , probably by the breakup of large grains in the tpz , rather than inward migration from an outer disk . even in the unlikely event that silicate emission comes from small grains in an outer disk that were blown away by radiation pressure as in vega @xcite , the dominant carrier of 240k continuum emission would still be large grains ( aigen li 2007 , private communication ) . the fraction of the stellar luminosity reradiated by dust , @xmath35 , is @xmath36 , which was obtained by dividing the infrared excess between 7@xmath0 and 60@xmath0 by the bolometric stellar luminosity . this @xmath35 is @xmath210,000 times larger than that of the current sun s zodiacal cloud ( @xmath37 ) but appears to be moderate for known debris disk systems at similar ages ( see figure 4 in @xcite ) . @xcite show that the ratio of dust mass to @xmath35 of a debris system is proportional to the inverse - square of dust particle semimajor axis for semimajor axes between @xmath29au and @xmath2100au . for systems with dust radius @xmath59au , this relationship overestimates the dust mass . instead , we calculate the mass of a debris ring around ef cha using @xmath38 ( eqn . 4 in @xcite ) , where @xmath39 is the density of an individual grain , @xmath40 is the dust luminosity , and @xmath41 is the average grain radius . because @xcite analyzed @xmath42 lep , a star of similar spectral type to ef cha , we adopt their model for grain size distribution . assuming r@xmath10 = 4.3au , @xmath39 = 2.5 g @xmath43 , and @xmath41 = 3@xmath0 , the dust mass is @xmath44 g ( @xmath210@xmath45 @xmath46 ) . grains with @xmath29 @xmath47 3@xmath0 will radiate approximately as blackbodies at wavelengths shorter than @xmath22@xmath48@xmath29 ( @xmath220@xmath0 ) . as may be seen from figure 1 , most of the excess ir emission at ef cha appears at wavelengths @xmath120@xmath0 . for blackbody grains at @xmath24.3au with , for example , radius @xmath29 @xmath47 3@xmath0 , the poynting robertson ( p - r ) drag time scale was used for the stellar luminosity of ef cha with a bolometric correction of -0.102 @xcite . absolute visual magnitude of ef cha is 2.35 based on the hipparcos distance of 106 pc . ] is only @xmath49 years , much less than 10 myrs . yet smaller grains with @xmath50 1.3@xmath0 would be easily blown away by radiation pressure on a much shorter time - scale . successive collisions among grains can effectively remove dust particles by grinding down large bodies into smaller grains , which then can be blown out . the characteristic collision time ( orbital period/@xmath35 ) of dust grains at 4.3au from this a9 star is @xmath51 years . both pr time and collision time were derived assuming no gas was present in the disk . while gas has not been actively searched for in ef cha , few debris disk systems at @xmath4 10 myr show presence of gas , indicating early dispersal of gas @xcite . a possibility of an optically thin gas disk surviving around a @xmath210 myr system was investigated by @xcite . however their model of gas disk is pertinent to cool dust at large distances ( @xmath4 120 au ) , but not to warm dust close to the central star as in ef cha . in addition , a recent study of ob associations shows that the lifetime of a primordial inner disk is @xmath1 3myr for herbig ae / be stars @xcite . based on the very short time scales of dust grain removal , essentially all grains responsible for significant excess emission at ef cha in the mid - ir are , therefore , likely to be second generation , not a remnant of primordial dust . [ [ debris - disk - systems - in - the - tpz - during - the - epoch - of - planet - formation ] ] debris disk systems in the tpz during the epoch of planet formation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the presence of hot dust has been recognized around other @xmath210myr stars , for example , tw hya , hd 98800 , & hen 3 - 600 in the tw hydrae association . interesting characteristics of these systems are their large @xmath35 ( @xmath52 ) and late - k and m spectral types . tw hya and hen 3 - 600 show flat ir sed up to 160@xmath0 consistent with active accretion in their disks @xcite . combined with the presence of substantial gas emission lines from tw hya , the observed infrared excess emission , at least for these two stars , appears to arise from gaseous dusty disks left over from the protostellar environment . on the other hand , a lack of gas emission ( @xcite ) and the quadruple nature of the hd 98800 system has invoked a flared debris disk as an alternative explanation for its large infrared excess emission @xcite . many young stars come in multiple systems . however , they hardly display such a high @xmath35 as that of hd 98800 . thus , the dust disk around hd 98800 might be an unusual transient pheonomeon such that it still contains a dust population composed of a mixture of promordial grains and replenished debris . some stars display a mixture of warm and cold grains where the overall infrared excess emissions is dominated by the cold dust . table 1 summarizes the currently known disk systems with warm dust regardless of spectral type and the presence of cold excess or remnant primordial dust at @xmath53myr . what separates ef cha from the stars described in the previous paragraph is that most of the infrared excess emission , if not all , arises from warm dust in the tpz , and as described in @xmath135.1 , these grains are , clearly , not a remnant of the protostellar disk . recent spitzer mips observation shows null detection of ef cha at 70@xmath0 band , leaving the presence of substantial cold dust unlikely ( see figure 1 ) . this result is consistent with the recent spitzer observations of other similar table 1 early - type warm excess systems ( @xmath14 tel , hd 3003 , hd 172555 and hd 113766 ) in which cold dust from a region analogous to the sun s kuiper belt objects is missing @xcite . in the following discussion , we characterize warm excess stars as those with warm dust in the tpz only and without cold excess ( i.e. , we exclude stars like @xmath15 pictoris ) . the fact that all currently known warm excess stars at ages between @xmath12 myr belong to nearby stellar moving groups offers an excellent opportunity to address how frequently warm excess emission appears among young stars in the solar neighborhood . @xcite list suggested members of stellar moving groups and clusters ( i.e. , @xmath14 cha cluster , tw hydra association , @xmath15 pictoris moving group , cha - near moving group , and tucana / horologium association ) at ages @xmath12 myr , within 100 pc of earth . currently spitzer mips archive data are available for all 18 members of the @xmath14 cha cluster , all 24 members of twa , all 52 members of tucana / horologium , 25 out of 27 @xmath15 pic moving group , and 9 out of 19 members of cha - near moving group . multiple systems were counted as one object unless resolved by spitzer . for example , in the @xmath15 pictoris moving group , hd 155555a , hd 155555b & hd 155555c were counted as a single object ; however , hip 10679 & hip 10680 were counted as two objects . table 1 shows that the characteristics of dust grains depend on the spectral type of the central star . all six warm debris disks considered in this paper harbor early - type central stars ( earlier than f3 ) . however , late - type stars in table 1 ( for example , pds 66 , a k1v star from lcc ) sometimes show characteristics of t tauri - like disk excess ( i.g . , @xmath54 , flat ir sed , etc . ) even at @xmath410 myr @xcite . such apparent spectral dependency perhaps arises from the relatively young ages ( @xmath210myr ) of these systems in which late - type stars still possess grains mixed with primordial dust due to a longer dust removal time scale @xcite . in the above - mentioned five nearby stellar moving groups , 38 out of 129 stars with spitzer mips measurements have spectral types earlier than g0 . therefore , we find @xmath213% ( 5/38 ) occurrence rate for the warm excess phenomenon among the stars with spectral types earlier than g0 in the nearby stellar groups at @xmath12 myr . ( beta pic is the only early - type star among the remaing 33 which has both warm and cold dust . ) for lcc , at least one ( hd 113766 ) out of 20 early - type members is a warm excess star giving 5% frequency ( see table 1 & 2 in @xcite ) . this rate can reach a maximum of 30% when we take into account five early - type lcc members that show excess emission at 24@xmath0 but have only upper - limit measurements at mips 70@xmath0 band . g0 type was chosen to separate the two apparently different populations because no g - type star except t cha appears in table 1 . furthermore , the spectral type of t cha is not well established ( g2-g8 ) and it may be a k - type star like other k / m stars in table 1 . rhee et al . ( 2007 , in preparation ) analyze all spectral types in the young nearby moving groups and conclude that the warm excess phenomenon with @xmath55 occurs for between 4 and 7% ; this uncertainty arises because some stars have only upper limits to their 70@xmath0 fluxes . we thank aigen li for helpful advice and the referee for constructive comments that improved the paper . this research was supported by nasa grant nag5 - 13067 to gemini observatory , a nasa grant to ucla , and spitzer go program # 3600 . based on observations obtained at the gemini observatory , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the particle physics and astronomy research council ( united kingdom ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , cnpq ( brazil ) and conicet ( argentina ) . this research has made use of the vizier catalogue access tool , cds , strasbourg , france and of data products from the two micron all sky survey ( the latter is a joint project of the university of massachusetts and the infrared processing and analysis center / california institute of technology , funded by the national aeronautics and space administration and the national science foundation ) . acke , b. , & van den ancker , m. e. 2004 , , 426 , 151 beichman , c. a. , et al . 2005 , , 626 , 1061 beichman , c. a. , et al . 2006 , , 639 , 1166 bryden , g. et al . 2006 , apj , 636 , 1098 chen , c. h. , & jura , m. 2001 , , 560 , l171 chen , c. h. , jura , m. , gordon , k. d. , & blaylock , m. 2005 , , 623 , 493 chen , c. h. , et al . 2006 , , 166 , 351 clarke , a. j. , oudmaijer , r. d. , & lumsden , s. l. 2005 , , 363 , 1111 cox , a. n. 2000 , allen s astrophysical quantities , 4th ed . edited by arthur n. cox . publisher : new york : aip press ; springer , 2000 . isbn : 0387987460 dent , w. r. f. , greaves , j. s. , & coulson , i. m. 2005 , , 359 , 663 dommanget , j. , & nys , o. 1994 , communications de lobservatoire royal de belgique , 115 , 1 egan , m. p. , et al . 2003 , vizier online data catalog , 5114 , 0 furland et al . 2007 , apj in press , astro - ph/07050380 . gorlova n. , et al . 2004 , apjs , 154 , 448 gorlova , n. , rieke , g. h. , muzerolle , j. , stauffer , j. r. , siegler , n. , young , e. t. , & stansberry , j. h. 2006 , , 649 , 1028 hauschildt , p. h. , allard , f. , & baron , e. 1999 , , 512 , 377 hernndez , j. , calvet , n. , hartmann , l. , briceo , c. , sicilia - aguilar , a. , & berlind , p. 2005 , , 129 , 856 hines , d. c. et al . 2006 , , 638 , 1070 kessler - silacci , j. , et al . 2006 , , 639 , 275 lisse , c. m. , beichman , c. a. , bryden , g. , & wyatt , m. c. 2007 , , 658 , 584 low , f. , smith , p. s. , werner , m. , chen , c. , krause , v. , jura , m. , & hines , d. c. 2005 , apj 631 , 1170 luhman , k. l. , & steeghs , d. 2004 , , 609 , 917 mamajek , e. m , lawson , w. a. , & feigelson , e. d. 1999 , , 516 , l77 mamajek , e. e. , lawson , w. a. , & feigelson , e. d. 2000 , , 544 , 356 mamajek , e. e. , meyer , m. r. , & liebert , j. 2002 , , 124 , 1670 megeath , s. t. , hartmann , l. , luhman , k. l. , & fazio , g. g. 2005 , , 634 , l113 pascucci , i. , et al . 2006 , , 651 , 1177 rhee , j. h. , & larkin , j. e. 2006 , , 640 , 625 rhee , j. h. , song , i. , zuckerman , b. , & mcelwain , m. 2007 , , 660 , 1556 rieke , g. h. , et al . 2005 , , 620 , 1010 schuetz , o. , meeus , g. , & sterzik , m. f. 2005 , , 431 , 175 silverstone , m. d. et al . 2006 , , 639 , 1138 smith , p. s. , hines , d. c. , low , f. j. , gehrz , r. d. , polomski , e. f. , & woodward , c. e. 2006 , , 644 , l125 song , i. , zuckerman , b. , weinberger , a. j. , becklin , e. e. 2005 , nature , 436 , 363 song , i. , zuckerman , b. , bessell , m. 2007 , submitted to apj . su , k. y. l. , et al . 2005 , , 628 , 487 takeuchi , t. , & artymowicz , p. 2001 , , 557 , 990 wyatt , m. c. , smith , r. , greaves , j. s. , beichman , c. a. , bryden , g. , & lisse , c. m. 2007 , , 658 , 569 zuckerman , b. , song , i. , bessell , m. s. , & webb , r. a. 2001 , , 562 , l87 zuckerman , b. , song , i. , & webb , r. a. 2001 , , 559 , 388 zuckerman , b. , & song , i. 2004 , , 42 , 685 zuckerman , b. , song , i. , weinberger , a. , bessell , m. 2007 apj letter in preparation . ccccccccccccccccc @xmath14 cha & b8 & 5.5 & 97 & 2.37 & 11000 & 320 & 1.5 & 8 & yes & debris & @xmath14 cha & 1 + ef cha & a9 & 7.5 & 106 & 1.92 & 7400 & 240 & 10 & 10 & yes & debris & cha - near & 2 + hd 113766 & f3v & 7.5 & 131 & ... & 7000 & 350 & 150 & 10 & yes & debris & lcc & 3,5,6 + hd 172555 & a7v & 4.8 & 29 & 1.52 & 8000 & 320 & 8.1 & 12 & yes & debris & @xmath15 pic & 3,4,6 + @xmath14 tel & a0v & 5.0 & 48 & 1.61 & 9600 & 150 & 2.1 & 12 & yes & debris & @xmath15 pic & 3,6 + hd 3003 & a0v & 5.1 & 46 & 1.59 & 9600 & 200 & 0.92 & 30 & yes & debris & tucana / horologium & 7,8 + + @xmath15 pic & a5v & 3.9 & 19 & 1.37 & 8600 & 110 & 26 & 12 & no & debris & @xmath15 pic & 4 + hd 98800 & k5ve & 9.1 & 47 & ... & 4200 & 160 & 1100 & 8 & no & primordial / debris & twa & 2,7 + tw hya & k8ve & 11.1 & 56 & 1.11 & 4000 & 150 ? & @xmath262200 & 8 & no & primordial / debris & twa & 2,7 + hen 3 - 600 & m3 & 12.1 & 42 & ... & 3200 & 250 & 1000 & 8 & no & primordial / debris & twa & 2,7 + ep cha & k5.5 & 11.2 & 97 & 1.39 & 4200 & ... & @xmath261300 & 8 & no & primordial / debris & @xmath14 cha & 10,11 + echa j0843.3 - 7905 & m3.25 & 14.0 & 97 & 0.87 & 3400 & ... & @xmath262000 & 8 & no & primordial / debris & @xmath14 cha & 10,11 + ek cha & m4 & 15.2 & 97 & 0.79 & 3300 & ... & @xmath26590 & 8 & no & primordial / debris & @xmath14 cha & 10,11 + en cha & m4.5 & 15.0 & 97 & ... & 3200 & ... & @xmath26480 & 8 & no & primordial / debris & @xmath14 cha & 10,11 + echa j0841.5 - 7853 & m4.75 & 14.4 & 97 & 0.51 & 3200 & ... & @xmath26480 & 8 & no & primordial / debris & @xmath14 cha & 10,11 + echa j0844.2 - 7833 & m5.75 & 18.4 & 97 & 0.40 & 3000 & ... & @xmath26400 & 8 & no & primordial / debris & @xmath14 cha & 10,11 + t cha & g2-g8 & 11.9 & 66 & ... & ... & ... & ... & 10 & no & primordial / debris & cha - near & 7,12 + pds 66 & k1ve & 10.3 & @xmath285 & 1.35 & 4400 & ... & @xmath262200 & 10 & no & primordial / debris & lcc & 6,9 + cccccccc 8@xmath0 & 8.28 & 167 @xmath349 & 119 & 48 & msx + n & 10.4 & 164 @xmath3414 & 84 & 80 & gemini / trecs + n & 7.7 - 12.97 & & & & trecs , @xmath56/@xmath57 100 spectrum + 12@xmath0 & 11.5 & 152 @xmath3432 & 58 & 94 & iras + 24@xmath0 & 24.0 & 80 @xmath344 & 14 & 66 & mips + 25@xmath0 & 23.7 & 110 @xmath3421 & 14 & 96 & iras + 70@xmath0 & 70.0 & @xmath5 22.4 & 1.7 & @xmath5 20.7 & mips +

most vega - like stars have far - infrared excess ( 60@xmath0 or longward in _ iras , iso , or spitzer mips _ bands ) and contain cold dust ( @xmath1150k ) analogous to the sun s kuiper - belt region . however , dust in a region more akin to our asteroid belt and thus relevant to the terrestrial planet building process is warm and produces excess emission in mid - infrared wavelengths . by cross - correlating hipparcos dwarfs with the msx catalog , we found that ef cha , a member of the recently identified , @xmath210myr old , `` cha - near '' moving group , possesses prominent mid - infrared excess . n - band spectroscopy reveals a strong emission feature characterized by a mixture of small , warm , amorphous and possibly crystalline silicate grains . survival time of warm dust grains around this a9 star is @xmath3 yrs , much less than the age of the star . thus , grains in this extra - solar terrestrial planetary zone must be of `` second generation '' and not a remnant of primodial dust and are suggestive of substantial planet formation activity . such second generation warm excess occurs around @xmath2 13% of the early - type stars in nearby young stellar associations .

0706.1265

* samaniego : * it s a real pleasure to be back at florida state , myles . i spent my first postdoctoral year in the statistics department here , and i have many fond memories . though we ve been friends for over 35 years , there are many details of your life and career that i m looking forward to hearing more about . let s start somewhere near the beginning . i know that you began your college career at carnegie mellon as an engineering major . can you tell me how you got interested in statistics ? * hollander : * i came to carnegie mellon , it wascarnegie tech when i entered in 1957 , with the aim of becoming a metallurgical engineer , but all the engineering students took more or less the same curriculum , including calculus , chemistry , english , history of western civilization . as the year progressed i found i liked math and chemistry the best so near the end of the year , i went to see the heads of metallurgy and math . the metallurgy chair was informative but laid back and said it was my decision . the math chair , david moscovitz , was much more enthusiastic . he said , `` hollander , we want you . '' well , i was only 17 , impressionable , and i liked being wanted so i became a math major . i did nt encounter a formal course in statistics until my junior year . that year , morrie degroot ( who had come to carnegie the same year i did1957he with a ph.d . from the university of chicago ) taught a course that i really enjoyed . it was based on mood s `` introduction to the theory of statistics . '' degroot wrote some encouraging comments on a couple of my exams and i began thinking i might become a statistician . then in my senior year , i took two more excellent statistics courses from ed olds . olds at that point was a senior faculty member who had actually done some work on rank correlation but was , i think , more known for his consulting with nearby industry , westinghouse , u.s . steel and others . in the afternoon he taught a statistical theory course from cramr s `` mathematical methods in statistics . '' in the evening he taught a course on quality control . i liked the juxtaposition of beautiful theory that could also be useful in an important applied context . i would say those three courses , those two teachers , sealed the deal for me . carnegie wanted me to stay on and do my ph.d . there in the math department but the lure of california , palo alto , stanford s statistics department , was too great , so i headed west . * samaniego : * let me ask a quick question about the books you mentioned . cramr is even today thought of as a very high - level book mathematically . it s surprising that it was used in an undergraduate course . * hollander : * in retrospect it is surprising but olds taught a beautiful course and it helped me later on in my studies . i still have the book in my library and i look at it from time to time . * samaniego : * i see it and it s clearly well worn . * samaniego : * you were attracted to math and science in your early years . was that your main focus in high school ? * hollander : * i was on an academic track in high school and studied mostly math and science . i attended an excellent public high school , erasmus hall , in the heart of the flatbush avenue section of brooklyn . it was a three - block walk from my apartment house . naturally , i also took other types of courses , english , social studies , history , mechanical drawing , and spanish . math was my best subject and that seemed fortunate for a kid who wanted to be an engineer . * samaniego : * how did a kid from brooklyn end up choosing to go to a private college in pittsburgh ? i suppose that once the dodgers left town , you felt free to leave , too . * hollander : * i could have stayed in brooklyn and gone to brooklyn college , thereby saving a lot of money . i could have stayed in new york state and gone to rensselaer polytechnic institute , where several of my close friends chose to go . i wanted something different , and pittsburgh , despite its reputation then as a smoggy city , due to the steel industry , appealed to me . that the dodgers were leaving brooklyn the same time i was ( 1957 was their last season in ebbets field and also my senior year of high school ) did nt affect my thinking . i did get to see them play a few times at forbes field in pittsburgh during my years at carnegie . forbes field was actually a short walk from carnegie and you could enter the ball game for free after the seventh inning . * samaniego : * tell me about your parents and their influence on your academic development . * hollander : * my mom and dad were committed to education , wanted me to go to college , and worked hard to make it happen . my dad had one year of college . he was at brooklyn polytechnic institute in the 19271928 academic year majoring in civil engineering . then the following year the depression hit and my father , as the oldest of three siblings , went to work to help support his family . he never got back to college . my dad went on to open a sequence of haberdashery stores , mostly selling pants and shirts , in the boroughs of manhattan , queens and brooklyn . my mother did not have college training but worked as a bookkeeper , mostly for a firm that managed parking lots throughout the city . they both left early in the morning and came back at dinner time . i was a latch - key kid before the term became popular . i lived on the first floor of an apartment house on linden boulevard , directly across the street from a branch of the brooklyn public library . the library was a good place to study and in my senior year i would thumb through books on engineering . civil , mechanical , electrical , aeronautical were the popular areas but metallurgy appealed to me : the chemistry labs , blast furnaces , protective masks , etc . i looked for schools that offered it and i also thought that by applying to a less popular field , i would increase my chances of being accepted , and getting a scholarship . * samaniego : * i know you had scholarship support from the ladish forging company while at carnegie mellon , and also worked for them in the summers . what was the work like ? did it play a role in your decision to go to graduate school ? * hollander : * when i switched from metallurgy to math at the end of my freshman year , i contacted the ladish forging company . they said that was fine , they would still support me , which i obviously appreciated . then in the summer of my junior and senior years i worked for them in cudahy , wisconsin . i estimated the costs of drop forgings using the costs of materials , the geometrical shapes of the parts , labor costs . i did some of that each summer and also wrote some programs in basic for the ibm 1401 . my supervisor told me on the parts i estimated for which the company was low bidder , the company lost money . i was biased low . but he said it was fine because the workers needed the work . ladish actually wanted me to work for them after graduation but i wanted to study statistics and my heart was set on stanford . ladish was nt my last position in the private sector . in the summers of 19621963 , after my first and second years of grad school , i worked for the sylvania reconnaissance laboratories in mountain view . there i did get to use some of the material i was learning at stanford , particularly markov chains and stochastic processes . in the summer of my junior year , i had an internship at the presbyterian medical center in san francisco . gerry chase and i rode the southern pacific railroad from palo alto to san francisco two or three times a week and worked on medical data . nevertheless , even though i liked these summer jobs , as my years in graduate school increased my inclination to join the private sector decreased . * samaniego : * your graduate study at stanford heavily impacted your career choices and the statistical directions you have taken . tell me about your cohort of students at stanford . * hollander : * it was a terrifically talented cohort . brad efron , howie taylor , joe eaton , carl morris , grace wahba , barry arnold , jim press , paul holland , jean donio , galen shorack , gerry chase , and many more . i should really name them all . we were all excited about the material . we wanted to learn what our professors taught and we wanted to learn how to do it ourselves . we were very cooperative and friendly among ourselves . i have many memories , howie taylor working on ( and talking about ) a probability problem at the blackboard in our office in cedar hall , carl morris and i talking about pitman efficiency at a blackboard in an empty classroom in sequoia hall and carl shedding light on what was going on , barry arnold and i discussing a mathematical statistics problem in cedar many , many such instances . brad efron was a senior student to our group who interacted with us and helped us in many ways , including discussing geometrical interpretations of theorems . we typically took the qualifying exams in the middle of our third year . to help us prepare , we would each choose a topic and write a 1012-page focused summary with solutions to problems , theorems , key ideas . i did one on nonparametrics , howie taylor did one on advanced probability , and so forth . we put the summaries together , made copies and passed them amongst ourselves . when we took our orals we were pumped , prepared , and , to the extent that one can be for such a momentous test , we were confident . also , of course , we were nervous . my exam committee was lincoln moses , rupert miller , charles stein and gerry lieberman and i see them sitting there today just as i am looking at you and i remember most of the questions to this day . * samaniego : * give me an example of a question that was asked . * hollander : * well , lincoln moses asked about nonparametric tests for dispersion and i decided to mention one of his rank tests . then gerry lieberman turned to lincoln and said in mock surprise , `` lincoln , you have a test ? '' they were close friends so gerry could tease him in this way but lincoln was nt particularly happy about my answer and then he threw a tough question at me about the asymptotic distribution of the kolmogorov smirnov statistic . charles stein asked me about decision theory and i was ready for that . i went to the blackboard and outlined the framework of a decision theory problem just like he did at the beginning of many of his lectures . * samaniego : * he did nt ask any testy inadmissibility questions , did he ? * hollander : * i had covered the blackboard and used a lot of time but he did ask about the relationship between admissibility and invariance . it had been covered in his course so i was ready for it . * samaniego : * which faculty members at stanford had the greatest influence on you , personally and professionally ? * hollander : * lincoln ellsworth moses had the greatest influence . i was lucky at the start because my first ta assignment in fall quarter , 1961 , was to be a grader in the elementary decision theory course he was teaching out of chernoff and moses . he gave the main lectures and five or six tas graded papers and met with sections to go over homework . i got to know lincoln through this activity and he also encouraged me to attend the biostatistics seminar that he and rupert miller were giving in the medical school . i would also be invited to his home in los trancos woods and got to know his wife jean and their children . i was close to him throughout and after he married mary lou coale , glee and i remained very close with them . beginning in the fall of 1963 , lincoln taught a two - quarter course on nonparametric statistics . it was a beautiful contemporary sequence and there was lots of nonparametric research in that period , particularly by erich lehmann and joe hodges at berkeley , lincoln , rupert miller , vernon johns at stanford . lincoln named me the ta for that course even though i was taking it at the same time . there i was , grading the papers of my really talented fellow students , like joe eaton and so i had to be good . i was determined to excel , to be one of the best if not the best in the class . later , motivated by this course , i wrote a thesis on nonparametrics under lincoln s direction . lincoln became my role model , the statistician i most admired and tried to emulate . he showed me how to be a professional , the joy of statistics , and the great pleasure of being a university professor . in my career i have tried to do for my students what lincoln did for me . * samaniego : * what you say about lincoln moses rings very true . from my own few interactions with him , and from things i ve heard about him over the years , he was both a fine teacher and scholar and a true gentleman . tell me about your interactions with other stanford faculty . * hollander : * i was also strongly influenced by other professors from whom i took courses . rupert miller via the biostatistics seminar , ingram olkin through the problems seminar he co - taught with shanti gupta , who was visiting in 1961 ( they started out assigning problems in cramr s book and that was a break for me as a beginning student because i had seen most of the problems at carnegie ) . ingram also taught multivariate analysis which i also took . i took charles stein s decision theory sequence and manny parzen s time series sequence . kai - lai chung taught the advanced probability sequence . they were all dedicated to their subjects , made them come alive , each had his own style , and each was at the top of his game . then on two sabbaticals at stanford , working in the medical center , i became friendly with bill and jan brown and reinforced my friendship with rupert and barbara miller . bill and jan became the godparents to our children . one special bond that existed between rupert and barbara and glee and me : jennifer ann miller and layne q hollander were delivered the same day , october 29 , 1964 , at stanford hospital , and glee and barbara shared the same hospital room for three or four days . over the years , i ve grown closer to ingram through the various international conferences on reliability that you and i have attended and to manny through his work with the nonparametrics section of asa . * samaniego : * all of the people that you ve mentioned have written very good books in probability or statistics . i m wondering , since you ve co - authored three books yourself , whether these people and the way they wrote influenced you ? * hollander : * i did put a high premium on clear writing in the three books i ve co - authored . i think the person who influenced me the most in that regard was frank proschan , who insisted on clear writing . when i took the course on stochastic processes , it was based on manny s notes ( his book was not yet out ) and it was taught by don gaver . when i took ingram s multivariate analysis course , he used his notes and ted anderson s book . i used rupert s book on multiple comparisons for research , but i did nt take that subject as a course . kai - lai used the notes that would become his beautiful book on advanced probability . certainly manny , ingram , rupert and kai - lai wrote in clear , captivating ways . * samaniego : * you met your wife glee at stanford and the two of you were married in the memorial church on the stanford campus . many of your friends feel that your bringing glee into the extended statistics community is your greatest contribution to the field ! tell me how you met glee and how you managed to persuade her to marry you . ( laughs ) * hollander : * i was sitting in my office on the second floor of ventura hall at stanford in october , 1961 . it s a spacious office and even though it had four desks , only two students would come regularly , jon kettenring and me . ( a year later pat suppes would take over that office . ) i was working on a hard problem and i paused to look out the window . i saw a young girl walking briskly , determined , in high heels , with blond hair , bouncing along with remarkable energy ( past ventura , maybe to the computer center ) . a california girl ! clearly i could never even approach a person like that . she passed out of my view and i went back to my homework , probably a waiting time problem in stochastic processes . the expected waiting time for me to approach the girl i had just seen was no doubt infinite . eight months later , in june , 1962 , my friend heinz , an engineering student from germany , and i decided to go on a double date . we decided to meet on a friday night at el rancho , a restaurant on el camino real , in palo alto . in addition to dinner , el rancho also had a dance floor and a lively band . when i arrived i realized that heinz s date was unmistakably the girl i had seen when gazing from my ventura hall office in the fall glee . the evening was going well and i was totally enthralled by glee , her brightness , her wit , her energy , her enthusiasm , her bounce . after about an hour the band played `` it s cherry pink and apple blossom white''a cha cha . i asked my date to dance but she said she did nt cha cha . i mustered the courage to ask glee . she said , `` i ll try . '' of course she was and still is a great dancer and i was on cloud nine . i thought i d made a good impression . a week later i called her on the phone and said , `` hi glee , it s myles hollander . '' she said , `` who ? '' obviously i did not impress her as much as she had impressed me . clearly i needed to go into high gear . i took her sailing on san francisco bay . i took her horseback riding in the foothills behind stanford . i took her skiing at heavenly valley . eventually my persistence triumphed . we hit it off over a period of about a year , and got married at stanford memorial church on the stanford campus in august , 1963 . we went on to have two fine sons , layne q and bart q , who , with their wives , tracy and catherine , also gave us five wonderful grandchildren taylor , connor , andrew , robert and caroline . glee earned her ph.d . at fsu in an excellent clinical psychology program and worked in private practice , and also at florida state hospital in chattahoochee . i like to say it all started with the cha cha and we re still dancing after all these years ! * samaniego : * on the statistical front , you published a major portion of your thesis in a pair of _ annals _ papers . what was the main focus of this work ? * hollander : * my thesis was devoted to rank tests for ordered alternatives in the two - way layout . lincoln moses , in his nonparametric sequence in the third year of my graduate work , had covered ordered alternatives in the one - way layout and that suggested to me some ideas for randomized blocks . i proposed a test based on a sum of overlapping signed rank statistics that is not strictly distribution - free but can be made asymptotically distribution - free . kjell doksum at berkeley was also working on closely related problems at the same time and in the end our two papers were published adjacently in the 1967 _ annals _ ( doksum , @xcite ; hollander , @xcite ) . in my thesis i also pointed out a certain multiple comparison procedure , thought by peter nemenyi ( nemenyi , @xcite ) to be distribution - free , was not , but could be made asymptotically distribution - free . i published the asymptotically distribution - free multiple comparison procedure in the 1966 _ annals _ ( hollander , @xcite ) . * samaniego : * you ve written quite a few papers on classical nonparametric testing problems . give us an idea of the range of problems you have worked on in this area . * hollander : * in my early years at fsu i wrote nonparametric papers on bivariate symmetry , regression , uncorrelated nonparametric statistics , and did a little more on ordered alternatives . i also worked with my first ph.d . student , ron randles , on a paper that was decision - theoretic rather than nonparametric . we developed @xmath0-minimax procedures for selection procedures and it was published in the 1971 _ annals _ ( randles and hollander , @xcite ) . ron took my class in nonparametrics and even though his thesis was not nonparametric in character , he did excellent work , went on to be a leader in nonparametrics and set a very high bar for my subsequent ph.d . students . thus in the beginning i was working on my own and with students . that was the way the senior leaders in the department , ralph bradley and richard savage , wanted it . work on your own , prove your mettle , and move away from your thesis topic . later on , when i began to collaborate with frank proschan and jayaram sethuraman , two great statisticians , my scope of topics vastly increased and my research got better ! whenever i received an offer or feeler from another place , i had to ponder whether i could find and establish working relationships with such superb collaborators at the next stop . i alwaysdoubted it . * samaniego : * your research over the years has been distinctly nonparametric , including , of course , interesting and important contributions to bayesian nonparametrics . you and your doctoral student , ramesh korwar , were the first to develop inference procedures for the hyperparameter of ferguson s dirichlet process , establishing the foundations for an empirical bayes treatment of nonparametric estimation . i see that it s an interest you ve sustained up to the present time . how did you get interested in this latter problem area ? * hollander : * my interest in the dirichlet process arose from tom ferguson s seminal paper ( ferguson , @xcite ) . that was the principal motivation . i had obtained a preprint before its publication . i had read some earlier papers at stanford on bayesian nonparametrics but ferguson s paper was the most tractable , the most promising . i ca nt remember the exact timing but i went to a bayesian nonparametric conference at ohio state where tom was the principal speaker . he was also aware of some of the results by ramesh korwar and me and mentioned them in his lectures . his wonderful lectures got me further fired up and i went on to do more bayesian nonparametrics with ramesh , and then later with two more of my ph.d . students , greg campbell and bob hannum , and more recently with sethu ( campbell and hollander , @xcite ; hannum , hollander and langberg , @xcite ; hannum and hollander , @xcite ; sethuraman and hollander , @xcite ) . * samaniego : * which ideas or results in your bayesian nonparametric papers seem to have had the most impact ? * hollander : * ramesh korwar and i had several interesting results in our 1973 paper in the _ annals of probability _ ( korwar and hollander , @xcite ) . we showed that when the parameter @xmath1 of the dirichlet process is nonatomic and @xmath2-additive , @xmath3 can be estimated from a sample from the process . the estimator we devised is @xmath4 , where @xmath5 is the number of distinct observations in the sample . we proved that estimator converges almost surely to @xmath6 where @xmath1 is a finite nonnull measure on a space @xmath7 that comes equipped with a @xmath8-field of subsets . we also showed in the nonatomic and @xmath8-additive case , that given @xmath5 , the @xmath5 distinct sample values are i.i.d . with distribution @xmath9 . this result has been used by others . for example , in an _ annals _ paper doksum and lo ( doksum and lo , @xcite ) considered bayes procedures when @xmath10 is chosen by a dirichlet prior and used the result to study consistency properties of posterior distributions . another result that ramesh and i had in that 1973 paper gave the joint distribution of the indicators that tell if the @xmath11th observation is distinct from the previous @xmath12 . the indicators are independent , but not identically distributed , bernoulli random variables . diaconis and freedman ( diaconis and freedman , @xcite ) used this result in their study of inconsistent bayes estimators of location . in our 1976 _ annals _ paper ( korwar and hollander , @xcite ) ramesh and i used the dirichlet process to define a sequence of empirical bayes estimators of a distribution function . one interesting consequence of that paper was a result reminiscent of the famous james stein result on the inadmissibility of multivariate @xmath13 when the dimension is @xmath14 . ramesh and i showed that if there are at least three distribution functions to be estimated , one could do better than estimating each distribution by its sample distribution . in a 1981 _ annals of probability _ paper ( hannum , hollander and langberg , @xcite ) bob hannum , naftali langberg , and i studied the distribution of a random functional @xmath15 of a dirichlet process . we related the cumulative distribution of that functional evaluated at @xmath16 , say , to the distributions of random variables @xmath17 and we obtained the characteristic function of @xmath17 . it has been surprising and gratifying to see some recent uses of this result . for example , it is used ( cifarelli and melilli , @xcite ) to study the distribution of the variance functional . the result is also used ( regazzini , guglielmi and di nunno , @xcite ) to study the probability distribution of the variance of a dirichlet measure and the probability distribution of the mean of a dirichlet measure . thus the result is getting a little play in the italian school . * samaniego : * you ve been at florida state for 42 years ! i d like to ask you about your extensive and fruitful collaborations with some of your colleagues here . tell me about your first joint paper with frank proschan . it was , i believe , one of the first papers in which tests were developed to detect particular nonparametric ( nbu ) alternatives to the exponential distribution . * hollander : * frank came here in 1971 from the boeing research labs . he was very open , very dedicated to his research . our offices were close and we became friends . one day he walked into my office and said , `` let s write a paper . '' i said , `` great . '' i was excited he asked . his main area was reliability and mine was nonparametric statistics , so we aimed to work in the intersection , namely nonparametric methods in reliability . the first paper we wrote covered our nbu ( new better than used ) test ( hollander and proschan , @xcite ) . the test is based on a @xmath18-statistic , partially reminiscent of the wilcoxon whitney statistic . we enjoyed working on it and there was a mild surprise . in calculating the probability that the statistic assumes its maximum value , the fibonacci sequence pops up . the sequence had not arisen in frank s longer research experience , nor in my shorter one . it is nice to have a mild connection with a famous pre - renaissance mathematician . i believe the paper stimulated more research in testing and estimation for the various nonparametric classes arising naturally in reliability , including more research avenues for us . * samaniego : * in a subsequent paper , you and frank discovered an interesting new context in which the total - time - on - test statistic arose . i m sure that was a pleasant surprise . * hollander : * frank and i wrote a testing paper on mean residual life ( hollander and proschan , @xcite ) that was published in _ biometrika_. we considered the decreasing mean residual life class , the new better than used in expectation class , and their duals . we defined measures of dmrlness and nbueness based on @xmath10 , plugged in the empirical for @xmath10 , and used those plug - in statistics as test statistics , standardized to make them scale - invariant . in the nbue case , we obtained the total - time - on - test statistic . up to that time it had been viewed as a test of exponentiality versus ifr or ifra alternatives . we showed its consistency class contained the larger set of nbue distributions , thus broadening its interpretation and applicability . large nonparametric classes of life distributions captured our attention for awhile . for example , we co - directed our student frank guess on a project where we defined new classes relating to a trend change in mean residual life . in our 1986 _ annals _ paper ( guess , hollander and proschan , @xcite ) we considered the case where the change point is known . later ( kochar , loader and hawkins , @xcite ) procedures were given for the situation where the change point is unknown . * samaniego : * on a personal level , what was it like to collaborate with frank proschan ? give us a feeling for his sense of humor , his work ethic , the `` reliability club '' and his overall influence on you . * hollander : * frank , as you know , had a deadpan sense of humor . he would often remind me of the comedian fred allen who was very funny but never cracked a smile , never laughed at his own jokes . when he gave a lecture frank would adroitly use transparencies , and there was always a parallel processing taking place , the material in the lecture , and humorous asides . he was dedicated to his research . he would come to the office very early , work for a few hours , go to the university pool for a swim , go home for lunch , then come back and work again . we both would come in on saturday mornings , talk about what we wanted to show , go back to our offices , try to get a result , write up the progress , then put a copy in the other person s mailbox . this went back and forth . some mornings we would come in and do this without talking face to face . the results would accumulate , and then we would have a paper . later , frank started the reliability club which met on saturday mornings to present and discuss topics on reliability . many students , several faculty and visitors would attend , and it would lead to dissertations , joint work , research grants , more papers . i had the habit of working some weekends ( including some sunday nights ; glee and i lived very close to campus then , about an 8-minute drive ) before frank arrived but frank solidified it and showed me i was not crazy doing it ( or else we were both crazy ) . without trying or fully realizing it , frank s style and work ethic became a part of mine . * samaniego : * you ve nicely integrated the parallel processing of material and humor into your own presentation strategy . * hollander : * frank , i ve always tried to be funny . it s both a strength and a weakness . i like to make people laugh but every once in a while it s not the time to be funny . over the years i ve become better at resisting the temptation to try to say something funny . but i still like to make witty remarks . i like to present to people the notion that statisticians have pizazz . * samaniego : * i ve found myself that in teaching our subject , a little bit of well - timed humor not the stand - up comedy type but the things that actually have something to do with the material we are talking about helps people stay aboard ; most people listen and enjoy it . * hollander : * at this point your advice on how to teach gets much higher marks than mine because you have just won an outstanding award at the university of california , davis . i wo nt even mention the figure here ; otherwise people will come by your house at night and break in . * samaniego : * well , myles , i ve always enjoyed your presentation style and have probably stolen more than i care to acknowledge from the talks i ve heard you give . * samaniego : * you ve written a good many papers with jayaram sethuraman . what would you consider to be the highlights of that work ? * hollander : * even before sethu and i worked together , i would go to his office for consulting . he is a brilliant statistician and he can often point you in a direction that will help , or lead you to a breakthrough when you are stuck . his entire career he has been doing that for all those wise enough to seek his assistance . my first paper with sethu is also joint with frank . it is the dt ( decreasing in transposition ) paper ( hollander , proschan and sethuraman , @xcite ) and is certainly a highlight . it is a paper on stochastic comparisons which yields many monotonicity results . among the applications in that paper were power inequalities for many rank tests . al marshall and ingram olkin later changed the dt term to ai ( arrangements increasing ) . in a later paper ( hollander and sethuraman , @xcite ) , sethu and i gave a solution to a problem posed to us by sir maurice kendall during his short visit to tallahassee in 1976 . it was `` how should one test if two groups of judges , each giving a complete ranking to a set of @xmath19 objects , agree , that is , have a common opinion ? '' we proposed a conditionally distribution - free test using the wald wolfowitz statistic . * samaniego : * tell me about your more recent work with sethu . * hollander : * sethu and i have , on and off , been working on repair models in reliability for the last 15 years . our interest was sparked by your groundbreaking paper with lyn whitaker ( whitaker and samaniego , @xcite ) in which you developed what is now called the whitaker samaniego estimator of the distribution @xmath10 of the time to first repair in imperfect repair models . with brett presnell , we considered the problem in a counting process framework ( hollander , presnell and sethuraman , @xcite ) and also developed a simultaneous confidence band for @xmath10 as well as a wilcoxon - type two - sample test in the repair context . many other important parameters , such as the expected time between repairs , depend on @xmath10 and the nature of the repair process , so the problem of estimating @xmath10 is important . five years later , with cris dorado ( dorado , hollander and sethuraman , @xcite ) we proposed a very general repair model that contains most of the models in the literature . we also introduced the notion of life supplements or boosts , so not only could the repairman move the effective age of the system to a point better than , say , minimal repair , he could also boost the residual life . recently we finished a paper on bayesian methods for repair models ( sethuraman and hollander , @xcite ) . for example , if you put a dirichlet prior on @xmath10 in , say , the imperfect repair model , and take two observations , the posterior distribution of @xmath10 is no longer dirichlet . thus there is , for these complicated repair processes which induce dependencies , a need for a broader class of priors which are conjugate . we introduced partition - based priors and showed they form a conjugate class . beyond repair models , we believe this new method for putting priors on distributions has potential in many other areas . * samaniego : * one of my favorites among your papers is a jasa paper you wrote with chen and langberg on the fixed - sample - size properties of the kaplan meier estimator . it was based on a simple but very clever idea . can you describe that work and how it came about ? * hollander : * i was interested in the kme s exact bias and its exact variance . brad efron ( efron , @xcite ) , in his fundamental article on the two - sample problem for censored data , had given bounds on the bias . proportional hazards provided a clean way to get exact results . earlier , allen ( allen , @xcite ) proved that when the cumulative hazard function of the censoring distribution is proportional to that of the survival distribution , the variables @xmath20 and the indicator function @xmath21 are independent , where @xmath22 is the time to failure , @xmath23 is the time to censorship . in his 1967 paper , efron used this result for obtaining efficiencies for his generalized wilcoxon statistic in the case when the censoring and survival distributions are exponential , and he thanked jayaram sethuraman for bringing the result to his attention . in the kme setting we ( chen , hollander and langberg , @xcite ) obtained an exact expression for moments of the kme by conditioning on @xmath24 and using allen s result . getting exact results in this setting was a natural consequence of my interest in rank order probabilities . erich lehmann really planted the seed with his famous work on the power of rank tests ( lehmann , @xcite ) where he obtained exact powers against what are now called lehmann alternatives . my natural tendency is to first try hard to get exact results , then move to asymptotics . * samaniego : * you ve done extensive joint work with some of your doctoral students . perhaps your collaboration with edsel pea is the most varied and most productive . tell me a little about that work . * hollander : * edsel is an amazingly dynamic and energetic researcher . he loves to do research and his enthusiasm is infectious . he is also very talented . we have worked on a broad range of problems . we started ( hollander and pea , @xcite ) with obtaining exact conditional randomization distributions for various tests used to compare treatments in clinical trials that use restricted treatment assignment rules , such as the biased coin design . we have also worked on confidence bands and goodness - of - fit tests in censored data settings . for example , in our 1992 jasa paper we ( hollander and pea , @xcite ) defined agoodness - of - fit test for randomly censored data that reduces to pearson s classical test when there is no censoring . we considered the simple null hypothesis and later li and doss ( li and doss , @xcite ) extended it to the composite case . thus , although not ideal , there are secondary gains in not solving the more general problem straight out . you inspire others and your paper gets cited . edsel and i have also worked on interesting reliability models . for example , in hollander and pea ( @xcite ) we used a markovian model to describe and study system reliability for systems or patients subject to varying stresses . as some parts fail , more stresses or loads may be put on the still - functioning parts . we use the failure history to incorporate the changing degrees of loads and stresses on the components . shortly after that ( hollander and pea , @xcite ) we addressed the problem about how a subsystem s performance in one environment can be used to predict its performance in another environment . another idea that may attract some interest is our class of models proposed in 2004 in the _ mathematical reliability _ volume ( pea and hollander , @xcite ) . we introduced a general class of models for recurrent events . the class includes many models that have been proposed in reliability and survival analysis . our model simultaneously incorporates effects of interventions after each event occurrence , effects of covariates , the impact of event recurrences on the unit , and the effect of unobserved random effects ( frailties ) . edsel and his colleagues and students have been studying asymptotic properties of the estimators and also applying them to various data sets . * samaniego : * tell me about your three books . * hollander : * the nonparametric books with doug , the first and second editions , were very successful ( hollander and wolfe , @xcite , @xcite ) . one important feature of these books are the real examples from diverse fields . it helped us broaden our audience beyond statisticians . doug and i also taught a short course for about nine years , mid70s to 80 s , at the george washington university continuing engineering education center . the audience at those courses consisted mainly of people in government and industry so again , in a way , we were bringing the nonparametric ideas and techniques to a different audience . wiley has sought a third edition , but doug and i have not yet committed to it . bill brown and i began writing the medical statistics book ( brown and hollander , @xcite ) in 1972 when i was on sabbatical at stanford . we also featured real examples and it was adopted at many medical schools . i also used it for many years at fsu for a basic course on statistics in the natural sciences . wiley always wanted a second edition , but bill and i never got around to it . wiley is now going to publish the original book as a paperback in its wiley classics library series . the book _ the statistical exorcist _ with frank proschan ( hollander and proschan , @xcite ) was great fun to write . the book consisted of vignettes that treated a variety of problems . we wrote in a way to explain to the readers what statistics does , rather than give a formulaic approach on how to do statistics . in fact , we did nt use any mathematical formulas or symbols . one interesting feature is the cartoons , about half of which were drawn by frank and pudge s daughter virginia and half drawn by glee . frank and i described the scenes and supplied the captions and ginny ( virginia ) and glee did the drawings . we also opened the vignettes with epigraphs , relating to statistics , from novels . some of the epigraphs are real and some were created by frank and me . in an appendix we informed the reader which ones were from our imagination . for a text , however , students found it difficult without a few formulas upon which to hang their hats , for example , when to multiply probabilities , when to add , and so on . marcell dekker also wanted a second edition and it is not beyond the realm of possibility . this semester i ve been teaching an advanced topics course . the material was an eclectic mixture of survival analysis and reliability theory where i focused on some of the parallels between the two subjects . the course title is `` nonparametric methods in reliability and survival analysis . '' whenever i look at the syllabus , it occurs to me that the material would make a good monograph . the problem is that most books on reliability are not big sellers although some are beautiful and informative . when i write , i do it not so much to make a few extra dollars , but to be read and thus a potentially large audience is the draw . * samaniego : * did any specific examples in _ the statistical exorcist _ come out of your joint research with frank ? * hollander : * some of the subject matter was motivated by the joint research . for example , we had vignettes on reliability which are unusual in an elementary book . we also had vignettes on nonparametric statistics , so the vignettes were influenced to some extent by our favorite subjects . * samaniego : * myles , _ the statistical exorcist _ is , i would say , unique in the field as an introduction to statistical thinking . the book is distinctive in a variety of ways including its general content , the humor of its cartoons and epigraphs and even the titles of some of its sections . there is one entitled , `` a tie is like kissing your sister . '' tell me about that section . * hollander : * there was a time when college football games could end in ties ; that time has long passed and now they play extra sessions to determine a winner . but the conventional wisdom of most coaches was that a tie was no good . it leaves everybody frustrated and unhappy , the players and fans on both teams . some coach coined the phrase `` a tie is like kissing your sister . '' which meant , you love your sister but you do nt get much satisfaction out of kissing her . the vignette considered an optimal strategy for near the end of the game , taking into account the chance of making an extra point ( one - point ) play , the chance of making a two - point play and the relative value of winning the game versus the relative value of tying the game . * samaniego : * with all these activities , plus your teaching and the mentoring of your graduate students , one would think that there might have been little time for other responsibilities . but , in fact , you served for nine years as the chair of your department . what were the main challenges you encountered as chair , and what achievements are you proudest of ? * hollander : * a chair obviously has many priorities : the faculty , the students , the staff , the administration . they are all important and you have to serve and contribute to the well - being of each . however , in my mind the top priority is to recruit well , get the best people possible . then everything desirable follows : a stronger curriculum , research grants , better students , and so forth . in my first term , 19781981 , my most significant hire was ian mckeague who , in 1979 , came from unc , chapel hill . he stayed 25 years , participated in grants , became an expert in survival analysis , and served a three - year term as chair . we co - directed jie yang on a topic on confidence bands for survival functions and have two papers that emanated from that work and related work on quantile functions with gang li ( hollander , mckeague and yang , @xcite ; li , hollander , mckeague and yang , @xcite ) . in my second and third terms , 19992005 , among the tenure - earning people i hired , flori bunea , from u. washington , eric chicken from purdue , dan mcgee from university of south carolina medical school , and marten wegkamp from yale , seem the most likely to contribute and hopefully stay at fsu for a long time . each filled an important gap in our curriculum , taught new courses , got involved with grants . i recruited dan as a senior biostatistician and he has been a driving force in establishing our new m.s . and ph.d . programs in biostatistics . he also succeeded me as chair . * samaniego : * tell me a bit about how you tried to broaden the department s focus and reach . what are some aspects beyond recruiting ? * hollander : * in fall 1999 i called ron randles , who was statistics chair at the university of florida ( uf ) at that time , and suggested we create an fsu - uf biannual statistics colloquium series . ron liked the idea and after getting approval from our faculties it began and continues today . the idea is that it provides the opportunity for the recent appointees of each faculty to get some outside exposure by giving a talk in the other department . thus in one semester uf comes to tallahassee and a uf person talks , and the next semester fsu goes to gainesville and an fsu person gives the colloquium talk . i also hope it leads to some joint research . some people have had discussions , but to my knowledge it hasnt happened yet . when i was chair , i was a mentor to all of our students , many of whom i recruited . i tried to teach them how to become professionals . i helped them get summer jobs and of course wrote reference letters for them . when i was younger i played intramural basketball and softball with some of them . i ve gone to some of their weddings . many students still stay in close touch with me . of course you do nt have to be the chairman to engage in these mentoring activities , but as chair one gets many opportunities to give extra advice at , for example , orientation and frequent student visits to the chair s office . here s a chair s story that goes into the highlight category . ron hobbs , an m.s . graduate of our department in 1967 , and his wife carolyn hobbs , who earned a b.s . in recreation studies from fsu in 1965 , endowed a chair in our department . it worked like this . each year for six consecutive years , ron and carolyn contributed $ 100,000 . then after six years , the state contributed $ 400,000 . then the university had one million dollars to help support the chair . one year , in early december , ron attended a meeting with me in my office and handed me an envelope with roughly $ 100,000 worth of america on line shares of stock . i thought for a moment , there s a delta jet with connecting flights to hawaii leaving in about an hour . i could promptly turn the envelope over to the university s chief fundraiser at the time , pat martin , who was also attending the meeting . or i could excuse myself , take the envelope with me ostensibly to return in a moment with the shares in a more carefully labeled envelope , and instead catch that jet.@xmath25 later that morning i noticed the sky was blue and clear as the engines roared and we took off to the west . * samaniego : * myles , that could be the beginning of the next great american novel ! * samaniego : * you served as the editor of the theory and methods section of jasa in 19941996 . i know this is an extremely labor - intensive job . you seemed to thrive on the experience . what did you enjoy most about it ? * hollander : * i had a great board of associate editors , including you , and you gave me the luxury of three reviews per paper . i liked working with the board . i also enjoyed reading the submissions one year i had 503!and the reviews . i tried to encourage authors , and with the reviews , improve the papers . even if a paper was declined , i wanted the disappointed author to feel his / her paper was treated with respect and got a fair shake . i helped to get a page increase and in some of my issues i had over 30 papers in theory and methods . i also increased the t&m acceptance rate to around 30% . i suspect it is significantly lower now . it was just a great experience . many nights and weekends i would bring a stack of folders home . if an ae was very tardy , i threatened to send in a swat team or toss him in a dark cellar until i received the reviews . one of my main goals was to make the papers readable and understandable . i insisted the authors write for the readers . i believe that was a mark of my editorship and your editorship , frank , as well . i enjoyed being jasa editor and a jasa ae before the editorship . i served on the boards of paul switzer , ray carroll and ed wegman , learned a lot from them , and was grateful for the opportunities . i ve also continued with editorial activities after my jasa term ended . in 1993 , the first volume of the _ journal of nonparametric statistics _ , founded by ibrahim ahmad , appeared and i have been a board member since then , with one break . in 1995 , mei - ling lee launched _ lifetime data analysis _ , and i have been a board member since the beginning . both of those journals publish important papers and the profession should be , and i believe is , grateful to ibrahim and mei - ling for their visions and dedicated work . * samaniego : * in 2003 , you received the noether senior scholar award for your work in nonparametric statistics . that must have been extremely satisfying . what do you see as the important open problems that current and future researchers in this area might wish to focus on ? * hollander : * the noether award is very special to me . the list of awardees consists of distinguished people with major accomplishments in nonparametrics and i am very grateful for the honor . the awardees thus far are erich lehmann , bob hogg , pranab sen , me , tom hettmansperger , manny parzen , brad efron and peter hall . stephen hawking , the great physicist , says you can not predict the great innovations in the future ; that s partially why they are termed great innovations . if , however , dennis lindley is correct about this being a bayesian century , and it seems to be going in that direction , then i would like nonparametrics to play a major role . thus i would wish for new , important innovations in bayesian nonparametrics . in my department we have at least three faculty members , anuj srivastava , victor patrangenaru and wei wu , working in image analysis , target recognition , face recognition and related areas . i would like to see nonparametric developments in these areas which are obviously important in many arenas including medical diagnoses and national security . as a field , i m glad we are pushing hard in interdisciplinary work , and it s good for our future role in science . it s valuable for the quality of research in the outside areas with which we participate and for scientific research overall . i m hopeful statisticians will contribute significantly to many of the important open questions in other fields and many already do . in academic settings , it s critical that university administrations recognize the importance of strong statistical support raising the quality of research . i want to be surprised in the future but , like hawking says , it s hard to guess at the surprises . what do you think , frank ? * samaniego : * in the 20th century , especially from say , 1940 to 1990 , the mathematical aspects of statistics were emphasized in both teaching and research . mathematical statistics was prime . the power of computation changed that considerably . then , applied problems , real applications with large and complex data sets , changed it even more . today , there are areas like data mining that are of great interest and importance but havent yet been mathematized . i wonder if it s just too early to mathematize challenging problems like these . i m guessing that some sort of theory of optimality , some sense of what s good and what s better than something else , will be part of the future development of these evolving problem areas . it s just simply too hard to do this with tools we have available now . * hollander : * it is true that you can do a lot of things now with computer - intensive methods and not worry about getting the exact results . it s a little reminiscent of when karl pearson was classifying curves . there are a lot of data - based methods , but the mathematical foundations may have to be solidified . i think now that we are pushing applied stuff , computer - intensive methods , we can get results relatively easily , for example , nonparametrics with bootstrapping and bayesian methods with mcmc . we may have to go back a little bit and shore up some of the methods , study their performance and properties as you suggest . but i think that will be considered only by theoretical statisticians . the computer - intensive surge is of course going to keep rolling , yield many new discoveries , and is great for the field . * samaniego : * you have many collateral interests , not the least of which is baseball . you once told me that you were as pleased with your published letters to the editor of _ sports illustrated _ as you were with many of your professional accomplishments . tell me about your interest in the dodgers and in sports in general . * hollander : * i was just kidding about the importance of the si letters . getting a statistical paper published is much more satisfying and represents a long - term and dedicated effort . however , the letters arose this way . my friend bob olds , a psychiatrist in st . augustine , used to live in tallahassee and write columns for the local newspaper . his future wife , ann , took a few classes from me when she was an undergraduate at fsu . bob sent a few letters to si and they were not published . he is a wonderful writer , much better than me , but just for fun i submitted two and , surprisingly , both were accepted . the first was about dodger pitcher fernando valenzuela during a period of fernandomania in la . the second was a comparison of the stanford and florida state marching bands . the latter was prompted by that bizarre play in november , 1982 , at the cal stanford big game . you may recall that the stanford band prematurely went on the field near the end of the game thinking stanford had won and they inadvertently ended up as blockers on cal s game - ending touchdown . my interest in the dodgers came about naturally during my childhood in brooklyn . during my summers in high school most of my friends were away at what was then called sleep - away camp . my parents could have afforded to send me , and i wanted to go , but i was an only child and they liked having me around . so i had summer jobs in the city and then on weekends , and on some evenings , i would walk to ebbets field , sit in the bleachers or the grandstand , and watch the bums , as they were affectionately called . this was the era of jackie robinson who displayed tremendous courage when he broke the color line in baseball . branch rickey , the dodgers general manager at the time , also deserves a lot of credit for giving robinson the opportunity . i enjoyed talking baseball to strangers at the game , seeing afro - americans and caucasians get along , and i loved the teamwork on the field . i ve lived my life with respect for people from all walks of life , from different backgrounds and cultures , and the dodgers played a role in teaching me that . during my years as chair , i tried to instill the same kind of teamwork in the department . i liked playing sports , mostly basketball , baseball and tennis . in my childhood , on the streets of brooklyn , i played city sports like punchball and stickball . i also played basketball in schoolyards and baseball at the parade ground in brooklyn . i played some tennis in high school but did nt get reasonably skilled at it until the early 70s . * samaniego : * one of the things that i ve noticed about you over the years is that you and glee like to go down to vero beach to see some spring training games . how long has that tradition been going on ? * hollander : * i would say it dates back to the 70s , almost the time we first came to tallahassee . we came to tallahassee in 1965 . we used to go to see the dodgers . it was a different era . we could actually go up to them and talk to them and chat about baseball , whereas today they re much more isolated . there are fences . i had some good conversations with players over the years . i remember once we went to vero beach and the game was rained out . it was a game against boston . fernando valenzuela was practicing with his pitching coach , ron perranoski . they were tossing the ball on a practice field so glee and i went up to them and started talking to them and they also posed for pictures . we have many pictures from those years . one with our sons layne and bart and hall - of - fame dodger pitcher sandy koufax is here on the office wall . * samaniego : * has any of your work involved sports in statistics ? * hollander : * i havent done serious sports statistics like the type that interests the sports statistics section of the asa . in the early 70s , however , woody woodward , who had been a player on fsu s baseball team , came to my office for help on the design and analysis of a study on different methods of rounding first base . i helped him and it became part of his master s thesis . later doug and i put the example in our nonparametrics book . in appreciation for the consulting , woodward sent me a baseball glove from spring training when he was a member of the cincinnati reds . i used it when i played intramural and city league softball at fsu and i still take it with me to spring training games and major league games , hoping to catch a foul ball . * samaniego : * i m visiting fsu on the occasion of a conference honoring your contributions to statistics and your department and university and commemorating your upcoming retirement . i know that you re looking forward to spending more time with family . i m sure your sons and your grandkids will soak up plenty of your freed - up time . any special plans ? * hollander : * you re right . glee and i do want to spend more time with our sons layne , and his children taylor and connor , and bart , his wife catherine , and their children andrew , robert and caroline . one set lives in plantation , florida , one in amherst , massachusetts . that will prompt some traveling . also , glee has siblings in hilton head , south carolina and spokane , washington and i have family in la , so we will get around . i also hope to go to a few statistical events . i love the international travel to conferences . you and i often attend the ones featuring reliability with the usual reliability club , ingram olkin , nozer sinpurwalla , allan sampson , nancy fluornoy , henry block , edsel pea , mark brown , phil boland , jim lynch , joe glaz , nikolaos limnios , misha nikulin , many more . glee and i own a beach house at alligator point , florida . it s about an hour drive from our home in tallahassee . we expect to be there a lot , walk on the beach , take bike rides to the western end of the point where there is a bird sanctuary , read novels , and so forth . * samaniego : * i ve got to believe that you have at least one more book in you . do you hope to do some writing once you are officially retired ? * hollander : * possibly i ll write a book . realistically , i think it s more likely i ll stay involved by writing a paper every now and then and recycling back to fsu from time to time to teach . lincoln moses said , `` there are no facts for the future . '' despite being a statistician , i ca nt predict . * samaniego : * you ve had a long and productive career as a research statistician . looking back , what would you say is your `` signature '' result ? * hollander : * i ll interpret the word `` signature '' literally and take the opportunity to say i greatly enjoyed the work we did together on your elegant concept of signatures in reliability theory during my sabbatical visit to uc davis in spring , 2006 ( hollander and samaniego , @xcite ) . for comparison of two coherent systems , each having i.i.d . components with a common distribution @xmath10 , we suggested the distribution - free measure @xmath26 where @xmath22 is the lifelength of system 1 and @xmath23 is the lifelength of system 2 . we found a neat way to calculate the measure directly in terms of the systems signatures and probabilities involving order statistics . among other things , we resolved the noncomparability issues using stochastic ordering , hazard rate ordering and likelihood ratio ordering that you ( kochar , murkerjee and samaniego , @xcite ) encountered for certain pairs of systems . in the bigger picture , my signature career quest was to promote nonparametric statistics , bring it into other areas , get more people to use it , and get students to study the subject and make contributions to the field . * samaniego : * it seems that , over the period of your career , nonparametric methods have become more and more important and pervasive . there is no question that your work has helped that direction significantly . * hollander : * thank you , frank . when i look in journals there are a lot of papers that are nonparametric in nature and the adjective nonparametric does not appear in the titles . it s just a natural way to start a problem now , letting the underlying distributions be arbitrary . * samaniego : * you ve worked with some of the legendary figures in our discipline including ralph bradley , frank wilcoxon , richard savage . these colleagues , and others , have played important roles in your professional evolution . how did the general environment at florida state help shape your career ? * hollander : * i came to florida state because of ralph , frank and richard . they were three luminaries in nonparametric statistics and i wanted to do nonparametrics research . frank and i shared an office ; he and his wife feredericka and glee and i became friends , but he died three months after i arrived . i never did research with frank , ralph or richard . but i was close to them . ralph and his wife marion and richard and his wife jo ann were always friendly to glee and me . although i did nt write with richard , up to the time he left for yale in 1973 he carefully read each one of my technical reports and often made valuable suggestions . i gave the bradley lectures at the university of georgia in 1999 , and after ralph passed away , i was asked by his family to deliver a eulogy at his memorial service in athens which i did , with pleasure . the environment at fsu was dedicated to research and i liked that . i came to a place where that was the top priority . also i came in 1965 , only six years after the department was founded by ralph , so there was the excitement of building . as it turned out , i was there when the first ph.d . graduated and thus far i have seen all of our ph.d . students graduate . * samaniego : * my recollection is that there s a famous quote attributed to you about the discipline of statistics . tell me about it . * hollander : * the saying is : `` statistics means never having to say you re certain . '' i saw the movie `` love story '' in 1971 . it was a big hit . it was based on a book of the same title by erich segal . i read the book after i saw the movie . as the title indicates , it s a love story . a wealthy harvard law student , oliver barrett , falls for a poor radcliffe girl , jennifer cavilleri , and eventually they marry . at one point , after a spat , oliver apologizes and jenny replies , `` love means never having to say you re sorry . '' in statistics we give type - i and type - ii error probabilities , confidence coefficients , confidence bands , false discovery rates , posterior probabilities and so forth , but we hedge our bets . we assess the uncertainty . with the movie fresh in my mind i transformed segal s phrase to `` statistics means never having to say you re certain . '' * samaniego : * thanks , myles . this excursion has been most enjoyable ! * hollander : * frank , we have had a long friendship that has stood the test of a continental divide between us . i look forward to its future pleasures . thank you for the conversation . it was highly enjoyable and i m grateful for the opportunity to interact in this manner and offer my musings . myles hollander and frank samaniego thank pamela mcghee , candace ooten and jennifer rivera for carefully transcribing the conversation . hollander , m. and samaniego , f. j. ( 2008 ) . the use of stochastic precedence in the comparison of engineered systems . in _ proceedings of the 2007 international conference on mathematical methods in reliability_. to appear . pea , e. and hollander , m. ( 2004 ) . models for recurrent phenomena in survival analysis and reliability . in _ mathematical reliability an expository perspective _ ( t. mazzuchi , n. singpurwalla , and r. soyer , eds . ) 105123 kluwer , norwell , massachusetts .

myles hollander was born in brooklyn , new york , on march 21 , 1941 . he graduated from carnegie mellon university in 1961 with a b.s . in mathematics . in the fall of 1961 , he entered the department of statistics , stanford university , earning his m.s . in statistics in 1962 and his ph.d . in statistics in 1965 . he joined the department of statistics , florida state university in 1965 and retired on may 31 , 2007 , after 42 years of service . he was department chair for nine years 19781981 , 19992005 . he was named professor emeritus at florida state upon retirement in 2007 . hollander served as editor of the _ journal of the american statistical association , theory and methods _ , 19941996 , and was an associate editor for that journal from 1985 until he became _ theory and methods _ editor - elect in 1993 . he also served on the editorial boards of the _ journal of nonparametric statistics _ ( 19931997 ; 20032005 ) and _ lifetime data analysis _ ( 19942007 ) . hollander has published over 100 papers on nonparametric statistics , survival analysis , reliability theory , biostatistics , probability theory , decision theory , bayesian statistics and multivariate analysis . he is grateful for the generous research support he has received throughout his career , most notably from the office of naval research , the u.s . air force office of scientific research , and the national institutes of health . myles hollander has received numerous recognitions for his contributions to the profession . he was elected fellow of the american statistical association ( 1972 ) and the institute of mathematical statistics ( 1973 ) , and became an elected member of the international statistical institute ( 1977 ) . at florida state university he was named distinguished researcher professor ( 1996 ) , he received the professorial excellence award ( 1997 ) , and in 1998 he was named the robert o. lawton distinguished professor , an award made to only one faculty member per year and the university s highest faculty honor . myles hollander was the ralph a. bradley lecturer at the university of georgia in 1999 , and in 2003 he received the gottfried e. noether senior scholar award in nonparametric statistics from the american statistical association . he was the buckingham scholar - in - residence at miami university , oxford , ohio in september , 1985 , and had sabbatical visits at stanford university ( 19721973 ; 19811982 ) , the university of washington ( 19891990 ) and the university of california at davis ( spring , 2006 ) . the following conversation took place in myles hollander s office at the department of statistics , florida state university , tallahassee , on april 19 , 2007 . .

1102.3780

thermodynamics was the great product of nineteenth century physics ; it is epitomised by the concept that there is an upper bound on the efficiency of any thermodynamic machine , known as the carnot limit . this concept survived the quantum revolution with little more than a scratch ; at present few physicists believe that a quantum machine can produce a significant amount of work at an efficiency exceeding the carnot limit . however , carnot s limit is only achievable for vanishing power output . it was recently observed that quantum mechanics imposes a _ stricter _ upper bound on the efficiency at finite power output@xcite . this upper bound coincides with that of carnot at vanishing power output , but decays monotonically as one increases the desired power output . this upper bound was found for two - terminal thermoelectric machines . in recent years , there has been a lot of theoretical @xcite and experimental @xcite interest in three - terminal thermoelectrics , see fig . [ fig : three - term ] . in particular , it is suggested that chiral three - terminal thermoelectrics@xcite could have properties of great interest for efficient power generation . all these three - terminal systems are quantum versions of traditional thermocouples @xcite , since they have one terminal in contact with a thermal reservoir and two terminals in contact with electronic reservoirs . see fig . [ fig : three - term ] . they turn heat flow from the thermal reservoir into electrical power in the electronic reservoirs , or vice versa . we refer to such three - terminal systems as _ quantum thermocouples _ , since they are too small to be treated with the usual boltzmann transport theory . there are two quantum lengthscales which enter into consideration ; the electron s wavelength and its decoherence length . in this work we will be interested in devices in which the whole thermocouple is much smaller than the decoherence length @xcite . such thermocouples would typically be larger than the electron wavelength , although they need not be . the crucial point is that electrons flow elastically ( without changing energy or thermalizing ) through the central region in fig . [ fig : three - term]a . this can also be a simple phenomenological model of the system in fig . [ fig : three - term]c , see section [ sect : voltage - probe ] . in these systems , quantum interference effects can have a crucial effect on the physics . such phase - coherent transport effects are not captured by the usual boltzmann transport theory , but they can be modelled using christen and bttiker s nonlinear scattering theory @xcite , in the cases where it is acceptable to treat electron - electron interactions at the mean - field level . such three - terminal systems are about the simplest self - contained quantum machines . there is a heat current into the system from reservoir m , @xmath0 , but no electrical current from reservoir m , @xmath3 see fig . [ fig : phenomenological ] . if reservoir l and r are at the same temperature @xmath4 , and reservoir m is hotter at @xmath5 , we can use the heat flow @xmath0 to drive an electrical current from l to r. if this electrical current flows against a potential difference , then the system turns heat into electrical power , and so is acting as a thermodynamic _ heat - engine_. alternatively , we can make the system act as a _ refrigerator _ , by applying a bias which drives a current from l to r , and `` sucks '' heat out of a reservoir m ( peltier cooling ) taking it to a lower temperature than reservoirs l and r , @xmath6 . in this work , we consider arbitrary phase - coherent three - terminal quantum systems that fall in to the category described by christen and buttiker s nonlinear scattering theory @xcite . we find upper bounds on such a system s efficiency as a heat - engine or a refrigerator at finite power output . we will show that these bounds coincide with those of two - terminal quantum systems considered in ref . [ ] , irrespective of whether the three - terminal system s time - reversal symmetry is broken ( by an external magnetic field ) or not . thus our bound applies equally to normal and _ chiral _ thermoelectrics@xcite . when the system acts as a heat - engine ( or energy - harvester@xcite ) , the input is the heat current out of the thermal reservoir ( reservoir m ) , @xmath0 , and the output is the electrical power generated by the system , @xmath7 . this power flows into a load attached between reservoirs l and r ; this load could be a motor turning electrical work into mechanical work , or some sort of work storage device . the heat - engine ( eng ) efficiency is defined as @xmath8 this never exceeds carnot s limit , @xmath9 where we recall that @xmath5 . carnot s limit is the upper bound on efficiency , but it is only achievable at zero power output . for the refrigerator the situation is reversed , the load is replaced by a power supply , and the system absorbs power , @xmath10 , from that supply . the cooling power output is the heat current out of the colder reservoir ( reservoir m ) , @xmath0 . thus the refrigerator ( fri ) efficiency or _ coefficient of performance _ ( cop ) is , @xmath11 this never exceeds carnot s limit , @xmath12 where we have @xmath6 ( which is the opposite of heat - engine ) . carnot s limit is the upper bound on efficiency , but it is only achievable at zero cooling power . bekenstein @xcite and pendry @xcite independently noted that there is an upper bound on the heat that can flow through a single transverse mode . as a result , the heat that any wave ( electron , photon , etc ) can carry away from reservoir @xmath13 at temperature @xmath14 through a cross - section carrying @xmath15 transverse modes is @xmath16 where the number of transverse modes is of order the cross - section in units of the wavelength of the particles carrying the heat . this bekenstein - pendry bound was observed experimentally in point - contacts@xcite , and recently verified to high accuracy in quantum hall edge - states @xcite . [ ] pointed out that this upper bound on heat flow , must place a similar upper bound on the power generated by a heat - engine ( since the efficiency is always finite ) . those works used the nonlinear version of landauer scattering theory @xcite to find this upper bound on the power generated , which they called the quantum bound ( qb ) , since its originates from the wavelike nature of electrons in quantum mechanics . it takes the form @xmath17 where @xmath18 . [ ] then calculated the upper bound on an heat engines efficiency for given power generation @xmath7 and showed that it is a monotonically decaying function of @xmath19 . there is no closed form algebraic expression for this upper bound at arbitrary @xmath19 , it is given by the solution of a transcendental equation . however , for @xmath20 one has @xmath21 with the next term in the expansion being of order @xmath22 . in the limit of maximum power generation , @xmath23 , the upper bound onefficiency is @xmath24 refs . [ ] calculated similar expressions for the upper bound on refrigerator efficiency as a function of cooling power . in this case , the upper bound is found to be half the bekenstein - pendry bound on heat - flow . again , the maximum efficiency equals that of carnot for cooling powers much less than the bekenstein - pendry bound , and decays monotonically as one increases the desired cooling power towards its upper limit . in the naive classical limit of vanishing wavelength compared to system size , one has @xmath25 and so the quantum bound @xmath26 and @xmath27 become irrelevant ( they go to infinity ) . so in this limit , it appears that one can achieve carnot efficiency for any power output . however quantum mechanics says that this is not the case , that for any power output that is a significant fraction of @xmath26 or @xmath27 , the upper bound on efficiency is lower than carnot efficiency . this efficiency bound was derived for two - terminal quantum systems , here we will show that exactly the same bounds apply to three - terminal quantum systems . a sketch of a system for which the voltage - probe model discussed in section [ sect : voltage - probe ] is correct . the role of reservoir m is played by the island which is large enough that any electron entering it thermalizes at temperature @xmath28 before escaping back into the dot . the electro - neutrality of the island ensures that @xmath1 in the steady - state . however , the fact the island exchanges heat ( in the form of photons or phonons ) with a thermal reservoir , means that it can still deliver heat to the three - terminal system . the island is in a steady - state at temperature @xmath28 , for which the heat flow out of the island due to electrons , @xmath0 , equals the heat flow into the island due to photons ( or phonons ) . ] here we discuss two examples of systems for which the bounds we derive here apply . the first example is the _ chiral thermoelectric _ sketched in fig . [ fig : three - term]b , as discussed in refs . [ ] . this is a three - teminal system exposed to such a strong external magnetic field that the electron flow only occurs via edge - states ( all bulk states are localized by the magnetic field ) . these edge - state are chiral , which means they circulate in a preferred direction in the scattering region ( anticlockwise in fig . [ fig : three - term]b ) , this is an intriguing situation for a heat - engine in which one wants to generate electrical power by driving a flow of electrons from reservoir l ( at lower chemical potential ) to reservoir r ( at higher chemical potential ) . the b - field alone generates an electron flow directly from l to r _ without _ a corresponding direct electron flow from r to l. thus it would seem plausible that one could take advantage of this , with a suitable choice of reservoir m and of the central scattering region to achieve higher efficiencies than in a two - terminal device ( where every flow from l to r has a corresponding flow from r to l ) . unfortunately , our general solution for a three - terminal system will show that the upper bound on efficiency at given power output is independent of the external magnetic field , so it is the same for chiral or non - chiral systems . the second example is the quantum thermocouple sketched in fig . [ fig : three - term]c . here , the third terminal ( reservoir m ) supplies heat in the form of photons ( or phonons ) . such systems have been considered using microscopic models of the photon flow @xcite , however here we instead use a phenomenological argument to replace the reservoir of photons sketched in fig . [ fig : three - term]c by the reservoir of electrons sketched in fig . [ fig : three - term]a . this is the `` voltage probe '' model @xcite , in which inelastic scattering ( such as electrons scattering from photons ) is modelled by a reservoir of electrons whose chemical potential is chosen such that that on average every electron that escapes the system into that reservoir is replaced by one coming into the system from that reservoir , so @xmath1 . [ fig : phenomenological ] shows a system for which this voltage probe model is correct . the island is large enough that any electron entering it thermalizes at temperature @xmath28 before escaping back into the dot . since the island is in a steady - state at temperature @xmath28 , the heat flow out of the island due to electrons must equal the heat flow into the island due to photons . however , one can also argue phenomenologically that the same model is a simplified description of the system sketched in fig . [ fig : three - term]c . this phenomenological model treats the exchange of a photon between the dot and reservoir m , as the replacement of an electron in the dot which has the dot s energy distribution , with an electron which has reservoir m s energy distribution . of course , this is not the most realistic model of electron - photon interactions . in particular , it assumes that each electron entering from reservoir l or r either escapes into one of those two reservoirs without any inelastic scattering from the photon - field , or it escapes after undergoing so many scatterings from the photon - field that it has _ completely _ thermalized with the photon - field . as such , this model does not capture the physics of electrons that undergo one or two inelastic scatterings from the photon - field before escaping into reservoir l or r. at this simplistic level of modelling , nothing would change if it were phonons rather than photons coming from reservoir m. while this voltage probe model has been successfully used to understand the basics of many inelastic effects in nanostructures , it should not be considered a replacement for a proper microscopic theory ( see e.g. refs . [ ] for a discussion of how the voltage probe model fails to capture aspects of inelastic scattering in ultra - clean nanostructures ) . one should be cautious about applying results for a system of the type in fig . [ fig : three - term]a to a system of the type in fig . [ fig : three - term]c , but it is none the less a reasonable first step to understanding its physics . consider a system with a scattering matrix , @xmath29 , then the transmission matrix for electrons at energy @xmath30 made of elements @xmath31 , \label{eq : def - t}\end{aligned}\ ] ] where the trace is over all transverse modes of leads @xmath13 and @xmath32 . the electrical current out of reservoir @xmath13 is then @xmath33 where lead @xmath13 has @xmath34 modes for particles at energy @xmath30 , and we define the fermi function in reservoir @xmath32 as @xmath35 \right)^{-1}. \label{eq : fermi}\end{aligned}\ ] ] the heat - current out of reservoir @xmath13 is @xmath36 the unitarity of @xmath37 places the following constraints on the transmission functions . firstly , @xmath38 secondly , @xmath39 where for compactness in what follows we define @xmath40\ . \label{eq : nmin}\end{aligned}\ ] ] it has been shown@xcite that any system obeying the above theory automatically satisfies the laws of thermodynamics , if one takes the clausius definition of entropy for the reservoirs . this means that the rate of entropy production is @xmath41 , where the sum is over all reservoirs . a system with three terminals has a three - by - three transmission matrix , meaning it has nine transmission functions . however , eq . ( [ eq : constraint1 ] ) means that only four of them are _ independent_. there are many possible choices for these four , we choose @xmath42 the remaining five transmission functions are written in terms of these functions ; [ eq:5othertransmissions ] @xmath43 given the relationships between transmission matrix elements in the previous section , we can write the electrical currents out of the reservoirs ( @xmath44 ) into the quantum system as @xmath45 \nonumber \\ & & \qquad \qquad \quad + { \cal t}_{\rm lr}({\epsilon } ) \,\big[f_{\rm l}({\epsilon } ) -f_{\rm r}({\epsilon})\big ] \big ) , \label{eq : i_l - integral } \\ i_{\rm r } & = & { e^{\operatorname{- } } } \int_{-\infty}^\infty { { { \rm d}}{\epsilon}\over h}\ \big ( { \cal t}_{\rm rm}({\epsilon } ) \,\big[f_{\rm r}({\epsilon } ) -f_{\rm m}({\epsilon})\big ] \nonumber \\ & & \qquad \qquad \quad + { \cal t}_{\rm rl}({\epsilon } ) \,\big[f_{\rm r}({\epsilon } ) -f_{\rm l}({\epsilon})\big ] \big ) , \label{eq : i_r - integral } \\ i_{\rm m } & = & -i_{\rm l}-i_{\rm r } , \label{eq : i_h - integral}\end{aligned}\ ] ] we chose to measure chemical potentials from that of reservoir m , so @xmath46 . then the heat current out of reservoir m is @xmath47 \,\big[f_{\rm r}({\epsilon } ) -f_{\rm l}({\epsilon})\big ] \nonumber \\ & & \qquad \qquad \quad + \,{\cal t}_{\rm lm}({\epsilon } ) \,\big[f_{\rm m}({\epsilon } ) -f_{\rm l}({\epsilon})\big ] \nonumber \\ & & \qquad \qquad \quad + \ , { \cal t}_{\rm rm}({\epsilon } ) \,\big[f_{\rm m}({\epsilon } ) -f_{\rm r}({\epsilon})\big ] \big ) . \label{eq : j_m - integral}\end{aligned}\ ] ] the power generated is @xmath48 our objective is to find the transmission functions , @xmath49 , @xmath50 , @xmath51 , and @xmath52 , that maximize the heat - engine efficiency for given power generation , @xmath7 . this is equivalent to finding the transmission functions that minimize heat flow out of reservoir m , @xmath0 , for given @xmath7 . to find these optimal transmission functions we must start with completely arbitrary @xmath30 dependences of the transmission functions . as in refs . [ ] , we do this by considering each transmission function as consisting of an infinite number of slices , each of vanishing width @xmath53 . we define @xmath54 as the height of the @xmath55th slice of @xmath56 , which is the slice with energy @xmath57 . we then want to optimize the biases of reservoirs l and r ( @xmath58 and @xmath59 ) and each @xmath54 ; this requires finding the value of each of this infinite number of parameters that minimize @xmath0 under the constraints that @xmath1 and that @xmath7 is fixed at the value of interest . the central ingredient in this optimization are the rate of change of @xmath7 , @xmath60 and @xmath0 with @xmath54 . here , @xmath61 , \label{eq : dpgen / dtau}\end{aligned}\ ] ] where @xmath62 means the derivative is taken for fixed @xmath58 , @xmath59 and fixed @xmath63 for all @xmath64 . doing the same for @xmath60 and @xmath0 , we get for @xmath65 , @xmath66 for a heat - engine , we consider the case where @xmath67 and @xmath5 , while @xmath68 . the fermi functions in this case are sketched in fig . [ fig : fermi - functions]a . we observe that @xmath69 & \hbox { is } & \hbox{positive for all } { \epsilon}\ , \label{eq : sign - of - f - difference2 } \\ \big[f_{\rm m}({\epsilon})-f_i({\epsilon})\big ] & \hbox { is } & \left\ { \begin{array}{l } \hbox { positive for $ { \epsilon}>{\epsilon}_{0i}$ } \,,\\ \hbox { negative for $ { \epsilon}<{\epsilon}_{0i}$ } \ , , \end{array}\right . \label{eq : sign - of - f - difference1}\end{aligned}\ ] ] where we define @xmath70 we will take @xmath71 such that @xmath1 , and @xmath72 to proceed with the derivation it is more convenient to assume we are interested in minimizing the heat - flow @xmath0 for given @xmath7 and given @xmath60 . only at the end will we take @xmath1 , to arrive at the situation of interest . we start with the assumption that the four transmission functions , @xmath73 , @xmath74 , @xmath75 and @xmath76 , each have a complete arbitrary energy dependence , and can be optimized independently . only in section [ sect : constraints ] do we take into account the relations between these transmission functions imposed by combining eq . ( [ eq : constraint2 ] ) with eq . ( [ eq:5othertransmissions ] ) . by the end of this section we will have shown that the maximal efficiency for given power generated is for transmission functions of the form shown in fig . [ fig : boxcars]a . let us define @xmath77 where @xmath78 indicates that the derivative is for fixed @xmath79 and fixed transmission functions . then , for an inifinitesimal change of @xmath54 , @xmath58 and @xmath59 we have @xmath80 we are interested in fixed @xmath7 and @xmath60 , so we want @xmath81 . this means eqs . ( [ eq : deltai_h],[eq : deltapgen ] ) form a pair of simultaneous equations , which we solve to get @xmath82 \ , \delta \tau^{(\gamma)}_{ij } , \nonumber \\ \delta v_{\rm r } \!\ ! & = & \!\ ! \left[-{\partial_{\rm l } i_{\rm m } \over { \cal a } } { { { \rm d}}p_{\rm gen } \over { { \rm d}}\tau^{(\gamma)}_{ij } } \bigg|_{v,\tau } \!\!+ { \partial_{\rm l } p_{\rm gen } \over { \cal a } } { { { \rm d}}i_{\rm m } \over { { \rm d}}\tau^{(\gamma)}_{ij } } \bigg|_{v,\tau } \right ] \delta \tau^{(\gamma)}_{ij } , \nonumber\end{aligned}\ ] ] where we define @xmath83 we substitute these results for @xmath84 and @xmath85 into eq . ( [ eq : deltaj_h ] ) and use eqs . ( [ eq : ih - to - pgen],[eq : jh - to - pgen ] ) to cast everything in terms of @xmath86 . then for @xmath65 , @xmath87 \ { { { \rm d}}p_{\rm gen } \over { { \rm d}}\tau^{(\gamma)}_{ij } } \bigg|_{v,\tau } \ , \qquad \label{eq : dj_h - result}\end{aligned}\ ] ] where we define @xmath88 , with @xmath89 , as @xmath90 thus , using eq . ( [ eq : dpgen / dtau ] ) , we conclude that @xmath0 shrinks upon increasing @xmath54 ( for fixed @xmath7 and fixed @xmath60 ) if @xmath91 \ , \big[f_j({\epsilon}_\gamma ) -f_i({\epsilon}_\gamma)\big ] & < & 0 \ , \label{eq : nearly - at - boxcars}\end{aligned}\ ] ] and otherwise @xmath0 grows upon increasing @xmath54 . the sign of the difference of fermi functions is given by eqs . ( [ eq : sign - of - f - difference2],[eq : sign - of - f - difference1 ] ) . hence , @xmath0 is reduced for fixed @xmath7 and fixed @xmath60 by * increasing @xmath50 up to @xmath92 for @xmath30 between @xmath93 and @xmath94 , while reducing @xmath50 to zero for all other @xmath30 . * increasing @xmath49 up to @xmath95 for @xmath30 between @xmath96 and @xmath97 , while reducing @xmath49 to zero for all other @xmath30 . * increasing @xmath52 up to @xmath98 for @xmath99 , while reducing @xmath49 to zero for @xmath100 . * increasing @xmath51 up to @xmath98 for @xmath101 , while reducing @xmath49 to zero for @xmath102 . here , it is eq . ( [ eq : constraint2 ] ) that stops us reducing these functions below zero , or increasing @xmath56 beyond @xmath103 . while it is hard to guess the form of @xmath97 and @xmath94 from their definition in eq . ( [ eq : eps1 ] ) . by inspecting eqs . ( [ eq : i_l - integral],[eq : i_r - integral ] ) one sees that a heat - engine should have @xmath104 and @xmath105 to ensure that both terms contributing to @xmath7 in eq . ( [ eq : pgen - integral ] ) are positive . while refrigerator are not discussed until section [ sect : fri ] , we will show there that the situation is basically the same for refrigerators . however , it is clear that refrigerators must absorb electrical power ( negative @xmath7 ) , so they will have @xmath106 and @xmath107 . section [ sect : fri ] will also show that @xmath94 is positive , while @xmath97 is negative . thus , we will consider two situations , @xmath108 as sketched in fig . [ fig : boxcars]a and [ fig : boxcars]b , respectively . the problem with the above solution is that it does not satisfy the constraints imposed by combining eq . ( [ eq : constraint2 ] ) with eq . ( [ eq:5othertransmissions ] ) . specifically , it does not satisfy the constraints on @xmath109 and @xmath110 that require [ eq : important - constraints ] @xmath111 respectively . the proposed solution violates the lower bound in eq . ( [ eq : important - constraints - a ] ) for all @xmath112 . similarly , it violates the lower bound in eq . ( [ eq : important - constraints - b ] ) for all @xmath113 . in addition , if @xmath114 , as in fig . [ fig : boxcars]b , then the proposed solution violates the upper bound in eq . ( [ eq : important - constraints - b ] ) for all @xmath115 . if @xmath116 , the solution also violates the upper bound in eq . ( [ eq : important - constraints - a ] ) for all @xmath117 . we will fix this by explicitly adding these bounds in the next section . here , we consider carrying out the optimization given by the list ( a - d ) in the previous section within the limits given by the constraints in eq . ( [ eq : constraint2 ] ) . as we are considering a heat - engine , we know that the @xmath96 , @xmath97 , @xmath93 and @xmath94 are ordered as in eq . ( [ eq : eng - eps - ordering ] ) , see fig . [ fig : boxcars]a . the optimization for @xmath30 in the window between @xmath97 and @xmath94 is trivial , since there the transmission functions in the above list ( a - d ) does not violate the constraints in eq . ( [ eq : constraint2 ] ) . this leaves us with the less trivial part of the optimization under the constraints , for @xmath112 and @xmath101 . for @xmath112 the independent optimization of the transmission functions , required increasing @xmath75 while decreasing @xmath74 and @xmath76 but doing this comes into conflict with the constraint that @xmath118 due to eq . ( [ eq : important - constraints - a ] ) . thus we do the unconstrained optimization up to the point allowed by the constraint , after which @xmath119 we then ask if @xmath0 decreases when we increase slice @xmath55 of @xmath75 and @xmath74 by infinitesimal amounts @xmath120 and @xmath121 respectively , given that one must also change slice @xmath55 of @xmath76 by @xmath122 not to violate the above constraint . with this observation , we find that for @xmath0 to decrease we need @xmath123 \nonumber \\ & & + \delta\tau^{(\gamma)}_{\rm lm}\ ( { \epsilon}_\gamma-{\epsilon}_{\rm 1l})\,\left [ f_{\rm m}({\epsilon}_\gamma)- f_{\rm r}({\epsilon}_\gamma ) \right ] \ < \ 0.\end{aligned}\ ] ] since all brackets in the above expression are positive for @xmath124 , we see that to minimize @xmath0 we should minimize both @xmath75 and @xmath74 . thus we conclude that for @xmath112 , it is optimal that all transmission functions are zero . the situation where @xmath113 can be treated in the same manner as above , upon interchanging the labels `` l '' and `` r '' . thus , the optimal situation is when all transmission functions are zero for @xmath113 . the transmission functions which optimize heat - engine efficiency for a given power output as [ eq : final - ts ] @xmath125 where @xmath103 is defined in eq . ( [ eq : nmin ] ) . these functions are sketched in fig . [ fig : boxcars]c . ( [ eq : final - t_rl ] ) means the optimal system has no direct flow of electrons between reservoirs l and r. given eqs . ( [ eq : t_ml],[eq : t_mr ] ) , this means that @xmath126 hence , the optimal three terminal situation is one that can be thought of as a pair of two - terminal problems much like those already considered in refs . [ ] . to be more explicit , eqs . ( [ eq : final - ts ] ) tell us that the optimal transmission is one that can be split into a problem of optimizing transmission between m and r through @xmath127 transverse modes ( with @xmath128 at all @xmath30 ) and another problem of optimizing transmission between m and l through @xmath129 transverse modes ( with @xmath130 at all @xmath30 ) . these two optimization problems could be treated independently were it not for the fact they are coupled by the constraint that the electrical currents in the two problems @xmath131 and @xmath132 must sum to zero to get eq . ( [ eq : im=0 ] ) . given the observation in the previous section that the optimal transmission for a three - terminal system is one which can be split into a pair of two - terminal transmission problems , we can draw two conclusions . firstly , the optimal transmission for a three - terminal system does not require any time - reversal symmetry breaking of the type generated by an external magnetic field . thus , the optimal transmission can be achieved in a system without an external magnetic field . we wish to be clear that this proof does not mean that magnetic fields may not be helpful in specific situations ; for example , a magnetic field may be helpful in tuning the transmission of a given system to be closer to the optimal one . however , it does mean that there is no requirement to have a magnetic field ; other parameters ( which do not break time - reversal symmetry ) can be tuned to bring the system s transmission to the optimal one . this is the first main conclusion of this work . secondly , it is not hard to show that a three - terminal system can not exceed the bounds found in refs . [ ] for a pair of two - terminal systems with the same number of transverse modes . to be more specific , it can not exceed the bound for a pair of two - terminal systems where one of the two - terminal systems has @xmath95 transverse modes and the other has @xmath92 transverse modes , see eq . ( [ eq : nmin ] ) . to prove this bound , it is sufficient to remark that the optimization of the three - terminal system in eq . ( [ eq : final - ts ] ) is exactly that of the optimization of a pair of two - terminal systems , with an additional constraint that the electrical currents in the two problems ( @xmath131 and @xmath132 ) sum to zero . this constraint couples the two problems and makes them much harder to resolve . however , if we simply drop the constraint on @xmath133 and @xmath132 and perform the optimization , we can be certain that we are over - estimating the efficiency at given power output . once we drop this constraint the two optimization problems become completely decoupled from each other . thus , we can optimize the transmission between m and r using the method in refs . [ ] , and independently optimize the transmission between m and l using the same method . as a result , an over - estimate of the three - terminal efficiency at given power output is bounded by the maximum two - terminal efficiency of a pair of two - terminal systems , with this bound being the one found in refs . [ ] . this is the second main conclusion of this work . having found an upper bound on the efficiency at given power output by using a process that over - estimates the efficiency , we can be sure that no three terminal system can be _ more _ efficient than a pair of optimal two - terminal systems . this makes it natural to ask if any three terminal system can be _ as _ efficient as this pair of optimal two - terminal systems . to answer this question , we present an example of a three - terminal system which is as efficient as the pair of optimal two - terminal systems . this will be our proof that the upper bound on the efficiency of a three - terminal system coincides with the upper bound on the efficiency of a pair of two - terminal systems . to proceed we take a three - terminal system with @xmath134 . given eq . ( [ eq : nmin ] ) , this could be a system with @xmath135 , or it could be a system with @xmath136 less than both @xmath137 and @xmath138 , in this case , one can take a pair of optimal two - terminal solutions from refs . [ ] , in the cases where @xmath139 . they have [ eq : eps-2term ] @xmath140 with @xmath141 where we have written the results of refs . [ ] in terms of the notation of this article , with the derivatives defined in eq . ( [ eq : define - dl - and - dr ] ) . here , @xmath142 is the part of the heat carried out of reservoir m by electron flow between reservoir m and reservoir @xmath13 , and @xmath143 is the part of the total power generated by that electron flow , so @xmath144 conservation of electrical current gives @xmath145 . as the only dependence on @xmath79 within @xmath60 , @xmath0 and @xmath7 are in @xmath146 , @xmath147 and @xmath143 , respectively , we have @xmath148 with some thought about the symmetries between l and r , we see that the derivatives have the following symmetries between l and r , [ eq : symmetries - of - partials ] @xmath149 we recall that eqs . ( [ eq : eps-2term]-[eq : symmetries - of - partials ] ) are all for an optimal pair of _ two - terminal _ systems . we now take the information in eqs . ( [ eq : eps-2term]-[eq : symmetries - of - partials ] ) , and verify that they _ also _ give an optimal solution of the three - terminal problem . for this we note that the definition of @xmath150 and @xmath151 are the same in the two- and three - terminal problems , however the definition of @xmath152 and @xmath153 are different , with that for three - terminals being eq . ( [ eq : eps1 ] ) and that for two - terminals being eq . ( [ eq : eps1-two - term ] ) . however , if we now take the symmetry relations in eq . ( [ eq : symmetries - of - partials ] ) , we see that eq . ( [ eq : eps1 ] ) reduces to eq . ( [ eq : eps1-two - term ] ) . thus , the solution of the optimization problem for a pair of two - terminal systems in eqs . ( [ eq : eps-2term]-[eq : symmetries - of - partials ] ) , is _ also _ a solution of the optimization problem for the three - terminal problem . all currents are the same in the three - terminal system as in the pair of two - terminal systems , so the efficiency and power output are also the same . finally , we note that this solution has @xmath154 , so it satisfies @xmath1 as in eq . ( [ eq : im=0 ] ) . hence , we have shown that an optimal three - terminal system can be as good as a pair of optimal two - terminal systems . this is the third main conclusion of this work ( after the two in the previous section ) . combining this conclusion with the others , we find that the upper bound on efficiency at given power output is the same for a three - terminal system as for a pair of two terminal systems . this also means that the upper bound on the efficiency for a three - terminal machine with @xmath155 is that of an _ equivalent _ three - terminal machine with @xmath156 . here , we define an `` equivalent '' system as one with the same @xmath157 ( see section [ sect : optimization - procedure ] ) . indeed , it seems likely that the maximal efficiency is achieved for @xmath156 , since this is probably the only case where the optimal efficiency with the constraint that @xmath1 is as large as that without this constraint . we wish to make it clear that while we claim the optimal three - terminal system has no advantage over a pair of optimal two - terminal systems , this does not tell us in which geometry it is easier to engineer a system which is close to that optimal value . in section [ sect : eng ] , we found that the optimal transmission functions are of the boxcar shape , eq . ( [ eq : final - ts ] ) , sketched in fig . [ fig : boxcars]c . then in section [ sect : achieving ] , we showed that when @xmath156 the optimal solution is the same as that for a pair of two - terminal systems , which were already treated in refs . [ ] . one can also consider cases where @xmath155 , where the optimal solution will have @xmath93 , @xmath94 , @xmath96 and @xmath97 different from that in refs . however , section [ sect : achieving ] already showed that this efficiency can not be larger than that of an _ equivalent _ optimal three - terminal system with @xmath156 . here we use `` equivalent '' to mean a machine with the same @xmath158 . in the case where @xmath159 , this is the same as saying that for given @xmath160 an optimal machine with @xmath161 can not be better than an optimal machine with @xmath135 . while for @xmath162 , all systems have @xmath156 . this greatly reduces practical interest in optimizing a system with @xmath155 , since optimizing implies a significant amount of control over the system , in which case it is better to engineer the system to have @xmath156 , and optimize that . the optimization procedure for @xmath155 is heavy , as well of being of little practical interest . thus , we do not carry it out here , we simply list the principle steps . * write explicit results for the currents and power in terms of four parameters @xmath97 , @xmath94 , @xmath58 and @xmath59 ( noting that @xmath96 and @xmath93 are given by @xmath58 and @xmath59 in eq . ( [ eq : eps0 ] ) ) . use these to calculate the derivatives that appear on the right hand side of eq . ( [ eq : eps1 ] ) , getting them as explicit functions of @xmath97 , @xmath94 , @xmath58 and @xmath163 . this step is straight - forward , and is carried out in appendix [ sect : currents - powers - derivatives ] . * substitute these derivatives into the right hand side of eq . ( [ eq : eps1 ] ) for @xmath164 and @xmath165 , this gives a pair of transcendental equations for the four parameters @xmath97 , @xmath94 , @xmath58 and @xmath59 . since we are interested in @xmath154 , with @xmath133 and @xmath132 being algebraic functions calculated in step ( i ) above ( see appendix [ sect : currents - powers - derivatives ] ) , this gives a third transcendental equation for these four parameters . * solve the three simultaneous transcendental equations numerically . as we have four unknown parameters and only three equations , we will get three parameters in terms of the fourth . we propose getting @xmath97 , @xmath94 , and @xmath58 as functions of @xmath59 . this involves solving the set of three simultaneous equations once for each value of @xmath59 . this is the heavy part of the calculation , which we do not do here . * once we have @xmath97 , @xmath94 , and @xmath58 as a function of @xmath59 , we can get all electrical and heat currents as a function of @xmath59 alone . since step ( iii ) was performed numerically , we are forced to do this step numerically as well . the electrical currents give us the power generated , @xmath7 , as a function of the voltage @xmath59 , which we must invert ( again numerically ) to get the voltage as a function of the power generated , @xmath166 . we then take the result for @xmath0 as a function of @xmath59 , and substitute in @xmath166 . this will give us @xmath167 , the optimal ( minimum ) heat flow out of reservoir m for a given power generated . then the maximal heat - engine efficiency @xmath168 . in refs . [ ] an upper bound on refrigerator efficiency for _ given cooling power _ was calculated directly for two - terminal devices . the result looked extremely similar to those works result for the upper bound on heat - engine efficiency for _ given power output_. it has since become clear to us how to get the result for refrigerators from the result for heat - engines . the trick it to make the physically plausible assumption that the upper bound on the cooling power of a refrigerator , @xmath0 , is a monotonic function of the electrical power it absorbs , @xmath10 . then the curve of maximum efficiency versus cooling power , @xmath0 , is the same as the curve of maximum efficiency versus absorbed power @xmath10 ( upon transforming the horizontal axis from @xmath10 to @xmath0 using the maximal efficiency curve ) . this is a great simplification of the problem , as it turns out that finding the refrigerator with maximal efficiency at given at given absorbed power , is a rather straightforward extension of the above calculation of the optimal heat - engine at given power output . here we take this point of view , we find the three - terminal refrigerator with maximal efficiency for given absorbed power , by a few straightforward modifications of the heat - engine calculation . a system absorbing power @xmath10 is the same as a system generating negative power @xmath169 . the crucial modification for maximizing refrigerator efficiency , @xmath170 , is that we must to _ maximize _ @xmath0 for given negative @xmath7 , when in the case of the heat engine we were _ minimizing _ @xmath0 for given positive @xmath7 . inspecting the calculation in section [ sect : eng ] , we see that everything follows through for a refrigerator with @xmath67 , @xmath6 , and @xmath171 . except that now we maximize @xmath0 , and that now the fermi functions in this case are those sketched in fig . [ fig : fermi - functions]b , obeying @xmath69 & \hbox { is } & \hbox{negative for all } { \epsilon}\ , \label{eq : fri - sign - of - f - difference2 } \\ \big[f_{\rm m}({\epsilon})-f_i({\epsilon})\big ] & \hbox { is } & \left\ { \begin{array}{l } \hbox { negative for $ { \epsilon}>{\epsilon}_{0i}$ } \,,\\ \hbox { positive for $ { \epsilon}<{\epsilon}_{0i}$ } \ , , \label{eq : fri - sign - of - f - difference1}\end{aligned}\ ] ] where eq . ( [ eq : eps0 ] ) is more conveniently written as @xmath172 by a careful comparison with section [ sect : eng ] , we note that all relevant differences of fermi functions in the refrigerator case have the opposite sign from in the heat - engine case . thus if a given change of transmission reduces @xmath0 for the heat - engine , then that same change will increase @xmath0 for the refrigerator . thus we conclude that the procedure that optimizes a heat - engine ( minimizing @xmath0 for given @xmath7 and @xmath60 ) also optimizes a refrigerator ( maximizing @xmath0 for given @xmath7 and @xmath60 ) . the independent optimization of @xmath73 , @xmath74 , @xmath75 and @xmath76 follows exactly as in section [ sect : independent ] . as with the heat - engine , it is difficult to guess the values of @xmath94 and @xmath97 from their definition in eq . ( [ eq : eps1 ] ) . however , for maximal refrigeration we want both terms in @xmath7 in eq . ( [ eq : pgen - integral ] ) to be negative ( so the absorbed power @xmath173 ) . by inspection of eqs . ( [ eq : i_l - integral],[eq : i_r - integral ] ) we see that this requires @xmath174 and @xmath175 . further , we can see that @xmath97 must be negative . to do this we inspect the terms in eqs . ( [ eq : j_m - integral],[eq : pgen - integral ] ) which depend on @xmath97 , and we see that making @xmath97 positive will increase @xmath10 , while reducing the cooling power @xmath0 , which is clearly not a way to maximize the efficiency , @xmath170 . a similar argument convinces us that @xmath94 must be positive . thus , we are interested in the case summarized in eq . ( [ eq : fri - eps - ordering ] ) . as we have @xmath176 , the result of independently optimizing the transmission functions is that shown in fig . [ fig : boxcars]b . for @xmath30 between @xmath97 and @xmath94 , no constraint are violated by that result ; so the optimal solution remains that all transmission functions are zero in this window . the optimization for @xmath177 and @xmath178 follows the same logic as in section [ sect : optimization+constraint - large - eps ] , except that now we want to maximize @xmath0 and the differences of fermi functions have the opposite signs . we find that the system is optimized by having all transmission functions equal to zero for @xmath179 and for @xmath178 . for @xmath30 in the window @xmath115 , the independent optimization ( maximizing @xmath73 and @xmath75 , while minimizing all other transmissions ) violates both the lower bound in eq . ( [ eq : important - constraints - a ] ) and the upper bound in eq . ( [ eq : important - constraints - b ] ) . thus this case need to be treated with care . we start by increasing @xmath73 and @xmath75 while reducing @xmath74 and @xmath76 , until we reach the limit of the bounds in eqs . ( [ eq : important - constraints - a ] ) and ( [ eq : important - constraints - b ] ) ; this occurs at @xmath181 we then ask if @xmath0 increases when we increase slice @xmath55 of @xmath75 and @xmath76 by infinitesimal amounts @xmath120 and @xmath182 respectively , given that the above constraint means that one must also change slice @xmath55 of @xmath74 by @xmath183 , and change slice @xmath55 of @xmath73 by @xmath184 . with this observation , we find that for @xmath0 to increase we need @xmath185 \nonumber \\ & & + \delta\tau^{(\gamma)}_{\rm lr}\ ( { \epsilon}_{\rm 1r}-{\epsilon}_{\rm 1l})\,\left [ f_{\rm r}({\epsilon}_\gamma)- f_{\rm m}({\epsilon}_\gamma ) \right ] \ > \ 0.\end{aligned}\ ] ] since all brackets in the above expression are negative for @xmath186 , we see that to maximize @xmath0 we should minimize both @xmath75 and @xmath76 . thus the optimum for @xmath30 between @xmath94 and @xmath93 is that @xmath73 is maximal ( @xmath187 ) while the other transmission functions are zero . the same logic can be applied to the energies @xmath30 between @xmath96 and @xmath97 , and we conclude that the optimal there is that @xmath74 is maximal ( @xmath188 ) while the other transmission functions are zero . to summarize , the transmission functions which maximize refrigerator cooling power @xmath0 for given absorbed power @xmath10 are [ eq : fri - final - ts ] @xmath189 where @xmath103 is defined in eq . ( [ eq : nmin ] ) . these transmission functions are sketched in fig . [ fig : boxcars]d . given these results and eqs . ( [ eq : t_ml],[eq : t_mr ] ) we also have @xmath190 every statement made in sections [ sect : over - estimate ] and [ sect : achieving ] about heat - engines has its analogue for refrigerators . in particular , we have proven that direct transmission between left and right is detrimental to the efficiency of the refrigerator . once this left - right transmission is suppressed , the three terminal problem for refrigerator can be thought of as a pair of two - terminal problems of the form in refs . the role of chirality is then irrelevant in the refrigerator , by which we mean that the optimal transmission can be achieved with or without the time - reversal symmetry breaking that an external magnetic field induces . we can use exactly the same logic as applied to the heat - engine in section [ sect : over - estimate ] to say that a three - terminal refrigerator can not exceed the upper bound on efficiency for given cooling power given in refs . [ ] , for a pair of two - terminal thermoelectric refrigerators ( one with @xmath95 transverse modes and the other with @xmath92 transverse modes ) . as in section [ sect : achieving ] , this two - terminal bound can be achieved in a three - terminal refrigerator with @xmath191 . ref . [ ] showed that the efficiency at given power output immediately gives the entropy production at that power output . the rate of entropy production of a heat - engine at power output , @xmath7 , is @xmath192 where @xmath193 is given in eq . ( [ eq : carnot - eng ] ) . while for a refriegrator at cooling power @xmath194 , it is @xmath195 where @xmath196 is given in eq . ( [ eq : carnot - fri ] ) . it is straight - forward to prove that these formulas apply equally to the three - terminal systems that we consider here . hence , an upper bound on efficiency at given power output immediately gives a lower bound on the rate of entropy production at that power output . this means that the proofs in this work also tell us that the lower bound on entropy production by a three - terminal system at given power output is the same as the lower bound on two - terminal systems discussed in ref . we have used scattering theory to find the upper bound on the efficiency of a three - terminal thermoelectric quantum machine at given power output . we find that this bound can be achieved at any external magnetic fields , so the bound is the same for chiral thermoelectrics as for those with no external field . this bound is the same as already found for two - terminal thermoelectric systems in refs . the upper bound on efficiency equals the carnot efficiency when the power output is vanishingly small , but it decays monotonically for increasing power output , as shown in fig . 2 of ref . [ ] . it would be worth seeing if one can derive the similar bound for the system in fig . [ fig : three - term]c with a microscopic model of the photon ( or phonon ) exchange , rather than the phenomenological model used here . most real quantum systems also loose heat to the environment ( through photon or phonon exchange ) , this can be modelled as a fourth terminal which exchanges heat but not charge with the system . a similar four - terminal geometry was discussed in ref . [ ] , which showed that such system operates in a non - thermal state and so exhibits non - local laws of thermodynamics . intriguingly , such a system can have a higher efficiency under given conditions than an equivalent classical machine operating in a thermal state , even if that classical machine is carnot efficient . it would be interesting to see how this bound behaves in such a situation , although we doubt that the pedestrian ( brute - force ) optimization used in this work will be extendable to more than three - terminals . in what follows , it is useful to define two functions , @xmath198 the first of these integrals can be evaluated by defining @xmath199 , so @xmath200 . \label{eq : g_j - result}\end{aligned}\ ] ] with a shift of integration variable , we find that @xmath201 where second line is based on substituting @xmath202 into the standard integral expression for the dilogarithm function @xmath203 . eqs . ( [ eq : i_l - integral]-[eq : i_h - integral ] ) with eqs . ( [ eq : final - ts ] ) give @xmath204 with @xmath145 . remember that @xmath205 is a derivative with respect to @xmath59 for fixed @xmath197 and @xmath88 , and the only @xmath59 dependence is in @xmath206 , we use eq . ( [ eq : useful - derivative1 ] ) to get @xmath207 with @xmath208 and @xmath209 . similarly , the only @xmath58 dependence is in @xmath210 , hence @xmath211 , \qquad\end{aligned}\ ] ] with @xmath212 and @xmath213 . then @xmath214 and @xmath215 . the heat - current out of reservoir m can be split into the two contributions defined above eq . ( [ eq : j - parts ] ) , for which @xmath216 using eq . ( [ eq : useful - derivative2 ] ) , we get @xmath217 , \qquad\end{aligned}\ ] ] and @xmath218 . \qquad\end{aligned}\ ] ] for any function @xmath219 @xmath220 \nonumber\end{aligned}\ ] ] where we defined @xmath221 for @xmath222 , and use the fact that @xmath79 only appears in these limits on the integral . thus , for @xmath223 in eq . ( [ eq : g_j+f_j ] ) we have @xmath224 similarly for @xmath225 in eq . ( [ eq : g_j+f_j ] ) , we have @xmath226 \nonumber \\ & = & { e^{\operatorname{- } } } \left ( g_j({\epsilon})+ { { \epsilon}\over h } \ f_j({\epsilon } ) \right ) . \label{eq : useful - derivative2}\end{aligned}\ ] ] finally , we mention the limits of the dilogarithm functions that appear in @xmath225 . the series expansion of the dilogarithm at small @xmath227 is @xmath228 . one can then extract the behaviour at @xmath229 for large @xmath230 using the equality @xmath231 . inserting the above small @xmath227 expansion into this , gives @xmath232 o. entin - wohlman , y. imry , a. aharony , `` three - terminal thermoelectric transport through a molecular junction '' , phys . b * 82 * , 115314 ( 2010 ) . r. snchez , m. bttiker `` optimal energy quanta to current conversion '' , phys . b * 83 * , 085428 ( 2011 ) b. sothmann , r. snchez , a.n . jordan , m. bttiker , `` rectification of thermal fluctuations in a chaotic cavity heat engine '' phys . b * 85 * , 205301 ( 2012 ) . o. entin - wohlman , a. aharony , `` three - terminal thermoelectric transport through a molecule placed on an aharonov - bohm ring '' , phys . b * 85 * , 085401 ( 2012 ) . jian - hua jiang , ora entin - wohlman , yoseph imry , `` thermoelectric three - terminal hopping transport through one - dimensional nanosystems '' , phys . b * 85 * , 075412 ( 2012 ) . m. horvat , t. prosen , g. benenti , g. casati , `` railway switch transport model '' , phys . e * 86 * , 052102 ( 2012 ) . k. brandner , k. saito , u. seifert , `` strong bounds on onsager coefficients and efficiency for three terminal thermoelectric transport in a magnetic field '' , phys . * 110 * , 070603 ( 2013 ) . v. balachandran , g. benenti , g. casati , `` efficiency of three - terminal thermoelectric transport under broken - time reversal symmetry '' , phys . b * 87 * , 165419 ( 2013 ) . h . jiang , o. entin - wohlman , y. imry , `` three - terminal semiconductor junction thermoelectric devices : improving performance '' , new j. phys . * 15 * , 075021 ( 2013 ) . o. entin - wohlman , a. aharony , y. imry , `` mesoscopic aharonov - bohm interferometers : decoherence and thermoelectric transport '' , _ in memory of akira tonomura : physicist and electron microscopist _ , eds . kazuo fujikawa and yoshimasa a. ono ( world scientific , singapore , 2013 ) . r. snchez , b. sothmann , a.n . jordan , m. bttiker , `` correlations of heat and charge currents in quantum - dot thermoelectric engines '' , new j. phys . * 15 * , 125001 ( 2013 ) . j .- h . jiang , `` enhancing efficiency and power of quantum - dots resonant tunneling thermoelectrics in three - terminal geometry by cooperative effects '' , j. appl . phys . * 116 * , 194303 ( 2014 ) . f. mazza , r. bosisio , g. benenti , v. giovannetti , r. fazio , and f. taddei , `` thermoelectric efficiency of three - terminal quantum thermal machines '' , new j. phys . * 16 * , 085001 ( 2014 ) . f. mazza , s. valentini , r. bosisio , g. benenti , v. giovannetti , r. fazio , and f. taddei , `` separation of heat and charge currents for boosted thermoelectric conversion '' , phys . b * 91 * , 245435 ( 2015 ) . hofer , and b. sothmann , `` quantum heat engines based on electronic mach - zehnder interferometers '' , phys . b * 91 * , 195406 ( 2015 ) . r. snchez , b. sothmann , and a.n . jordan , `` chiral thermoelectrics with quantum hall edge states '' , phys . rev . lett . * 114 * , 146801 ( 2015 ) r. snchez , b. sothmann , and a.n . jordan , `` effect of incoherent scattering on three - terminal quantum hall thermoelectrics '' , to appear in physica e , eprint - arxiv:1507.00162 j .- h . jiang , b. kumar agarwalla , d. segal , `` efficiency statistics and bounds of time - reversal symmetry broken systems '' , phys . lett . * 115 * , 040601 ( 2015 ) . b. roche , p. roulleau , t. jullien , y. jompol , i. farrer , d.a . ritchie and d.c . glattli , nature comm . * 6 * , 6738 ( 2015 ) . f. hartmann , p. pfeffer , s. hfling , m. kamp and l. worschech , phys . lett . * 114 * , 146805 ( 2015 ) h. thierschmann , r. snchez , b. sothmann , f. arnold , c. heyn , w. hansen , h. buhmann , l. w. molenkamp , nature nanotech . , in press ( 2015 ) ; 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we consider the nonlinear scattering theory for three - terminal thermoelectric devices , used for power generation or refrigeration . such a system is a quantum phase - coherent version of a thermocouple , and the theory applies to systems in which interactions can be treated at a mean - field level . we consider an arbitrary three - terminal system in any external magnetic field , including systems with broken time - reversal symmetry , such as chiral thermoelectrics , as well as systems in which the magnetic field plays no role . we show that the upper bound on efficiency at given power output is of quantum origin and is stricter than carnot s bound . the bound is exactly the same as previously found for two - terminal devices , and can be achieved by three - terminal systems with or without broken time - reversal symmetry . thus the bound appears to be universal for two - terminal and three - terminal ( chiral and non - chiral ) thermoelectrics . -25 mm ( a ) the three - terminal machine ( heat - engine or refrigerator ) that we consider , the exchange of electrons with reservoir m carries a heat current , @xmath0 , but not an electrical current , @xmath1 . ( b ) a chiral thermoelectric device reproduced from ref . [ ] . ( c ) a system in which photons deliver the heat , this can be phenomenologically modelled by ( a ) , see section [ sect : voltage - probe ] . , title="fig : " ] @xmath2

1603.09216

since the discovery of gamma - ray burst ( grb ) afterglows there has been growing evidence linking grbs to massive stars : the host galaxies of grbs are star - forming galaxies and the position of grbs appear to trace the blue light of young stars @xcite ; some of the host galaxies appear to be dusty with star - formation rates comparable to ultra - luminous infrared galaxies @xcite . on smaller spatial scales , there is growing evidence tying grbs to regions of high ambient density @xcite and the so - called dark grbs arise in or behind regions of high extinction @xcite . however , the most direct evidence linking grbs to massive stars comes from observations of underlying supernovae ( sne ) and x - ray lines . the presence of x - ray lines would require a significant amount of matter on stellar scales ( e.g. @xcite ) , as may be expected in models involving the death of massive stars . however , to date , these detections ( e.g. @xcite ) have not been made with high significance . if grbs do arise from the death of massive stars , then it is reasonable to expect associated sne . the grb - sn link was observationally motivated by two discoveries : the association of grb 980425 with the peculiar type ic sn 1998bw @xcite and an excess of red light superposed on the rapidly decaying afterglow of grb 980326 @xcite . however , these two discoveries were not conclusive . the sn association would require grb 980425 to be extra - ordinarily under - energetic as compared to all other cosmologically located grbs and the case for grb 980326 is weakened by the lack of a redshift for the grb or the host galaxy . nonetheless , the two discoveries motivated searches for similar underlying sn components . as summarized in section [ sec : conclusions ] , suggestions of similar red `` bumps '' in the light curves of various other grb afterglows have been made ( to varying degrees of confidence ) . however , there is little dispute that the well - studied red bump in the afterglow of grb 011121 is most easily explained by an underlying supernova @xcite . furthermore , from radio and ir observations of the afterglow @xcite , there is excellent evidence that the circumburst medium was inhomogeneous with ambient density @xmath2 , as expected from a massive star progenitor @xcite ; here , @xmath3 is the distance from the progenitor . these developments are in accordance with the expectation of the `` collapsar '' model @xcite . in this model , the core of a rotating massive star collapses to a black hole which then accretes matter and drives a relativistic jet . internal shocks within this jet first cause bursts of @xmath4-rays and then subsequently result in afterglow emission as the jet shocks the ambient medium . it is important to appreciate that the sn light is primarily powered by radioactive decay of the freshly synthesized @xmath5ni whereas the burst of @xmath4-rays are powered by the activity of the central engine . in the current generation of collapsar models , there is sufficient flexibility to allow for a large dispersion of @xmath5ni and the energy of the engine . thus , the next phase of understanding the grb - sn connection will benefit from ( and require ) observational measures of these parameters . motivated thus , we have an ongoing program of searches for sne in grb afterglows with the _ hubble space telescope _ ( hst ) . here , we present a systematic search for a sn underlying grb 010921 . in [ sec : observations ] we present our observations and the details of photometry in [ sec : subphot ] . we fit afterglow models and constrain the brightness of an underlying sn in [ sec : discussion ] . we then present an overview of previous such efforts and conclude in [ sec : conclusions ] . grb 010921 was detected by the high energy transient explorer ( hete-2 ) satellite at 2001 september 21.219 ut @xcite and the position was refined by the interplanetary network error - box @xcite . using the 5-m hale telescope and the very large array we discovered the afterglow of this event as well as the redshift of the host galaxy @xcite . the low redshift of this event , @xmath6 , made it a prime candidate for a search for an underlying sn . accordingly , as a part of our large _ hubble space telescope _ ( hst ) cycle 9 program ( go-8867 , p. i. : kulkarni ) , we triggered a series of observations with the wide field planetary camera 2 ( wfpc2 ) aboard hst . owing to the lateness in identifying the afterglow candidate , the first observation was on day 35 , slightly after the expected peak of the sn . at each of epochs 13 we obtained @xmath7 s exposures in each of five filters ( f450w , f555w , f702w , f814w and f850lp ) with a single diagonal dither by 2.5 pixels to recover the under - sampled point - spread function ( psf ) . the fourth epoch was optimized for photometry of the host galaxy and , accordingly , we increased the exposure time to @xmath8 s. we used `` on - the - fly '' pre - processing to produce debiased , flattened images . the images were then drizzled @xcite onto an image with pixels smaller than the original by a factor of 0.7 using a pixfrac of 0.8 . after rotation to a common orientation the images were registered to the first epoch images using the centroids of common objects in the field . the typical r.m.s . registration errors were less than 0.15 drizzled pixels . the host galaxy of grb 010921 has an integrated magnitude of @xmath9 mag or about 5@xmath10jy @xcite . consequently great care has to be taken to properly photometer the fading afterglow . below , we review various photometric techniques . * total magnitudes : * the simplest technique is to perform aperture photometry ( e.g. @xcite ) . the afterglow flux is obtained by subtracting the host flux estimated from a very late time measurement . a major concern is that the host flux is dependent upon the choice of aperture ( both center and size ) . thus , if different images have different seeing then it is possible to obtain an artificial bump in the light curve . * host subtraction : * the above concern can be alleviated by subtracting a late - time image from the earlier images . the afterglow may then be easily photometered in the host - subtracted images . this method has been used with considerable success by those observing sne ia ( e.g. @xcite ) . * @xmath11 subtraction : * in this technique , each image is subtracted from every other image and the afterglow residual photometered . the flux at each epoch can be fit through least - squares , assuming the flux at the final epoch is zero ( novicki and tonry , personal communication ) . this method makes use of the fact that the host galaxy has not been observed only once at late times , but at each epoch and thus better s / n can be obtained from the over - constrained system . we employed the @xmath11 subtraction technique to photometer the grb 010921 afterglow in our hst images . the images were subtracted using a modified version of isis @xcite and photometered using the analytic psf - fitting routine within vista ( j. tonry , personal communication ) . we used the synphot package within iraf to calculate the response of the instrument and filter combination to a source with constant flux of 1 mjy ; the resulting values are ab magnitudes @xcite , expressed as fluxes . corrections were made for charge - transfer ( in)efficiency ( cte ) using the prescription of @xcite and aperture - corrected to infinity . we have also re - analyzed and photometered ground - based images @xcite of the afterglow , applying @xmath11 subtraction . since this technique assumes that the flux of the afterglow in the final epoch is zero , which may not be correct for these images , we subtracted the appropriate fourth - epoch hst observation ( which we have assumed contains no afterglow ) from the final ground - based images , measured the flux of the afterglow and added this value to the fluxes derived from the @xmath11 subtraction . the results of the photometry are host - subtracted fluxes for the afterglow in each of the images , under the assumption that the afterglow flux in the final hst image ( 2001 dec 21 ) is zero ( or negligible ) . these values are presented in tables [ tab : hst ] and [ tab : ground ] . the values in table [ tab : ground ] supersede the corresponding measurements presented in @xcite and @xcite . we plot the afterglow light curves in figure [ fig : lc ] . the light - curves are monotonically decreasing ( i.e. do not level off ) , and hence we deduce that our assumption of negligible flux in the final hst image is justified . temporal breaks in optical light - curves have been seen in many afterglows and are usually attributed to a `` jet '' geometry ( see @xcite ) . we adopt a standard optical afterglow model , consisting of a broken power - law temporal decay with power - law indices @xmath12 and @xmath13 , and a power - law spectral index , @xmath14 @xcite . each of these indices are functions of the electron energy distribution index , @xmath15 , dependent upon the location of the cooling break relative to the optical bands , and so we consider two cases : the cooling break is redward of the optical ( hereafter , case r ) ; and the cooling break is blueward of the optical ( case b ) . we consider , in addition to a constant circumburst medium , an inhomogeneous circumburst medium , @xmath16 ( see @xcite and @xcite ) . we apply the parametric extinction curves of @xcite and @xcite using the interpolation calculated by @xcite . these extinction curves are characterized by two values , the magnitude of the extinction in the rest - frame of the host galaxy , @xmath17 , and the slope of the uv extinction curve , @xmath18 ( see @xcite ) . following @xcite , we adopt @xmath19 , corresponding to an lmc - like extinction curve . adopting other extinction curves ( e.g. mw , smc ) yields similar , but more - constraining results ( i.e. any underlying sn must be even fainter than the upper limit we derive below ) ; see @xcite . the main purpose of this analysis is to determine whether the light curves contain an sn component . to this end , we use the observations of sn 1998bw for an sn template since it is one of the well observed bright ib / c sne which may be related to a grb @xcite . specifically , we used the @xmath20 photometry of @xcite and derived the flux distribution of sn 1998bw , using the zero - points and filter curves of @xcite . the resulting low resolution spectrum ( consisting of 5 points at the effective wavelength of each broadband filter ) , is redshifted to @xmath6 @xcite , assuming a flat lambda cosmology with @xmath21 and @xmath22 km s@xmath23 mpc@xmath23 . the redshifted spectrum , which represents what sn 1998bw would look like at cosmological distances , is integrated with the appropriate filter curve to derive the apparent brightness at this redshift . sn 1998bw at @xmath6 would peak in the rest - frame @xmath24-band at approximately 4 @xmath25jy . it is evident from figure [ fig : lc ] that the afterglow is much fainter than this , and , further , that there is no clear bump in the afterglow light curve . we therefore allow the sn component to be scaled by @xmath26 magnitudes in our model . the sn is placed behind the same foreground ( i.e. milky way ) and host galaxy extinction as the afterglow ( which can be inferred by demanding that the temporal and intrinsic spectral slopes , which both depend on the electron distribution index , @xmath15 , be consistent ; see e.g. @xcite ) . to calculate the sn detection limit of our observations , we fit the model by minimizing @xmath27 . the afterglow was not detected in any of the f450w images , and so we exclude them from our analysis . subtracting the host f450w image from our ground - based @xmath28 image left a large residual at the position of the host galaxy ( not of the ot ) . this poor subtraction is likely due to the filter mis - match , and so we do not include this point in our analysis . our analyses are summarized in table [ tab : fit ] . in short , we find no evidence for an underlying sn . in order to calculate the formal limits , we re - fit the data for a range of values of the sn brightness and computed the probability distribution from the resultant @xmath29 . as can be seen from table [ tab : sn ] , the least constraining limit comes from the case where the afterglow evolves in a wind - stratified medium with the cooling break redward of the optical band , and even in this case , a sn brighter than @xmath30 mag is excluded at 99.7% confidence , and a sn as bright as sn 1998bw ( @xmath31 mag ) is ruled out at greater than 99.999% confidence . the peak brightness and the time scales for sne ib / c are generally correlated such that fainter sne may peak earlier @xcite . it may be important to take this into account for our analysis , since the observations most sensitive to the presence of an underlying sn are all after the peak . to do this , we shifted the @xmath20 photometry of the ( intrinsically-)fainter type ic sn 1994i @xcite to @xmath0 , and derived the transformation between the redshifted sn 1998bw and 1994i light curves using a similar method as @xcite . this method is analogous to the `` stretch '' method for sne ia @xcite . if we use this transformation in our model to transform the redshifted sn 1998bw light curve to the light curve of a sn fainter than sn 1998bw by @xmath26 magnitudes , then our least - constraining limit on an underlying sn becomes @xmath32 mag fainter than sn 1998bw ( at 99.7% confidence ) . the agreement with the above limit indicates that the uncertainty in our knowledge of the the light - curve shape and luminosity scaling light - curve is not important for this analysis . leaving aside the sn issue , our fits provide a jet - break time of approximately 35 days . from the fregate 8 400 kev fluence of @xmath33 erg @xmath34 , we calculate the @xmath35-corrected isotropic - equivalent energy release @xcite in the @xmath4-ray band , @xmath36 erg . applying the geometric correction from our measurement of the jet break ( using the formulation and normalization of @xcite ) , we obtain a jet opening angle of @xmath37 . thus the true energy release is @xmath38 erg consistent with the clustering of energy releases around @xmath39 erg @xcite . here we report the search for an underlying sn in the afterglow of grb 010921 . thanks to the superb photometric stability of hst and the @xmath11 subtraction technique , we have been able to trace the light curve of the afterglow of grb 010921 over two months . the resulting photometry is unbiased by aperture effects that are so prevalent in simple aperture and psf - fitting photometry . we report two results . first , we find a jet break time of 35 days , using only optical data . second , we find no evidence for an sn . a sn , if present , must be fainter than sn 1998bw by @xmath40 mag at 99.7% confidence . to our knowledge , to date , this is the most stringent limit for an underlying sn associated with a cosmologically located grb . as noted in [ sec : introduction ] , the collapsar model as currently understood has little power in predicting the dispersion in the amount of @xmath5ni synthesized as compared to the energy in relativistic ejecta . underlying sne are directly powered by the former whereas the grb is powered by the latter . observations are needed to start mapping the distribution in these critical explosion parameters . progress can be expected with such observational inputs accompanied by further refinements in the model . motivated thus , we summarize in table [ tab : previous ] the status of sn searches for all table [ tab : previous ] all known grbs with redshift less than 1.2 . the most secure case for an sn is that for grb 011121 @xcite . grb 980326 shows a strong red excess at about a month but unfortunately a redshift is lacking . grb 970228 shows a less clear excess but benefits from a known redshift . stated conservatively , a sn as bright as that of sn 1998bw can be ruled out in grb 000911 . in all cases , save that of grbs 980326 and 011121 , the presence of a host with a magnitude comparable to the brightness of the peak of the sn , makes it difficult to identify an sn component . as noted in [ sec : subphot ] , `` bumps '' can arise from host contamination . combining hst and ground based measurements ( as is the case for grb 970228 ) is prone to considerable errors ( [ sec : subphot ] ) . in summary , there is good evidence for an sn comparable in brightness to sn 1998bw in grb 011121 @xcite . for grb 010921 , using the hst observations reported here , we constrain any putative underlying sn to be 1.34 mag fainter than sn 1998bw . in the collapsar framework , this absence could be most readily attributed to the well known dispersion of the peak luminosity of type ib / c sne . an alternative possibility is that there may be more than one type of progenitor for long duration grbs . along these lines we note that @xcite claim that some afterglows ( e.g. grb 990123 ) are incompatible with a @xmath16 inhomogeneous circumburst distribution whereas other afterglows ( e.g. grbs 970228 and 970508 ) are better explained by invoking an inhomogeneous circumburst medium . progress requires both searches for underlying sne as well as characterizing the circumburst medium via modeling of the early - time afterglow ( e.g. grb 011121 , see @xcite ) . finally , we note that the afterglow of grb 010921 ( and any coincident sn ) was extincted by @xmath41 mag of dust in the foreground , and @xmath42 mag of dust in the host galaxy ( table [ tab : fit ] ) . thus , in the future , using acs aboard hst it should be possible to extend sn searches to at least 3 mag fainter than sn 1998bw , at which point it will be possible to detect more typical sne ib / c coincident with grbs . we thank pete challis for helpful discussions about wfpc2 reduction , and megan novicki and john tonry for an advance copy of their @xmath11 subtraction paper . srk and sgd thank nsf for supporting our ground - based grb observing program . bps and pap thank the arc for supporting australian grb research . support for proposal number hst - go-08867.01-a was provided by nasa through a grant from space telescope science institute , which is operated by the association of universities for research in astronomy , incorporated , under nasa contract nas5 - 26555 . kh is grateful for support under grant hst - go-09180.07-a . , s. g. _ et al . _ 2001b , in gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 218 + . , v. v. 2001 , in gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 136 . ccccc oct 26.731 & f450w & -0.031 @xmath43 0.022 + nov 06.956 & f450w & 0.001 @xmath43 0.028 + nov 24.990 & f450w & 0.067 @xmath43 0.029 + oct 26.791 & f555w & 0.157 @xmath43 0.015 + nov 07.015 & f555w & 0.087 @xmath43 0.017 + nov 25.121 & f555w & 0.063 @xmath43 0.018 + oct 26.859 & f702w & 0.231 @xmath43 0.013 + nov 07.149 & f702w & 0.096 @xmath43 0.015 + nov 25.203 & f702w & 0.045 @xmath43 0.015 + oct 26.932 & f814w & 0.433 @xmath43 0.024 + nov 08.359 & f814w & 0.209 @xmath43 0.024 + nov 25.621 & f814w & -0.003 @xmath43 0.025 + oct 26.992 & f850lp & 0.471 @xmath43 0.092 + nov 08.418 & f850lp & 0.207 @xmath43 0.088 + nov 25.687 & f850lp & 0.030 @xmath43 0.096 + ccccc oct 19.178 & @xmath28 & 0.671 @xmath43 0.097 & p200 + sep 22.144 & @xmath44 & 46.104 @xmath43 0.722 & p200 + sep 22.148 & @xmath44 & 44.995 @xmath43 0.661 & p200 + sep 27.354 & @xmath44 & 2.13 @xmath43 1.223 & p200 + oct 17.145 & @xmath44 & 0.086 @xmath43 0.379 & p200 + oct 18.088 & @xmath44 & 0.189 @xmath43 0.382 & p200 + oct 19.109 & @xmath44 & 0.256 @xmath43 0.285 & p200 + oct 17.165 & @xmath45 & 0.560 @xmath43 0.197 & p200 + oct 18.110 & @xmath45 & 0.523 @xmath43 0.191 & p200 + oct 19.130 & @xmath45 & 0.649 @xmath43 0.153 & p200 + oct 19.149 & @xmath46 & 1.293 @xmath43 4.273 & p200 + sep 22.3038 & @xmath47 & 11.319 @xmath43 0.981 & nofs1.0 + oct 19.253 & @xmath47 & 0.623 @xmath43 0.675 & p60 + sep 22.2976 & @xmath48 & 24.727 @xmath43 1.078 & nofs1.0 + oct 19.206 & @xmath48 & 0.229 @xmath43 0.720 & p60 + sep 22.2930 & @xmath49 & 39.116 @xmath43 5.072 & nofs1.0 + sep 22.3210 & @xmath49 & 36.135 @xmath43 4.486 & nofs1.0 + oct 19.272 & @xmath49 & 0.916 @xmath43 4.284 & p60 + nov 17.151 & @xmath49 & 0.470 @xmath43 4.238 & nofs1.0 + sep 22.2893 & @xmath24 & 84.688 @xmath43 5.778 & nofs1.0 + sep 22.795 & @xmath24 & 40.277 @xmath43 3.950 & ts + sep 22.825 & @xmath24 & 47.281 @xmath43 7.230 & ts + sep 22.878 & @xmath24 & 50.926 @xmath43 3.671 & ts + sep 22.954 & @xmath24 & 41.321 @xmath43 3.636 & ts + nov 17.093 & @xmath24 & 1.229 @xmath43 1.057 & nofs1.0 + lrrrr ism / wind , b & 2.67 @xmath43 0.06 & 33.0 @xmath43 6.5 & 0.95 @xmath43 0.08 & 19.9 + ism , r & 3.03 @xmath43 0.04 & 37.5 @xmath43 4.9 & 1.16 @xmath43 0.07 & 19.2 + wind , r & 2.33 @xmath43 0.10 & 30.3 @xmath43 9.5 & 1.35 @xmath43 0.08 & 23.1 + lrcrrll 970228 & 0.695 & @xmath49 & 25.5 & 25.2 & both & plausible but aperture , color effects with hst . ( 1 ) + 970508 & 0.835 & @xmath24 & 23.6 & 24.0 & ground & aperture effects . ( 2 ) + 980326 & ? ? ? & @xmath49 & 25 & @xmath51 27 & ground & plausible . ( 3 ) + 980613 & 1.096 & @xmath49 & & 24.0 & ground & faint afterglow , no search . ( 4 ) + 980703 & 0.966 & @xmath49 & 24 & 22.6 & ground & consistent with no sn . ( 5 ) + 990705 & 0.840 & @xmath49 & & 22.8 & ground & no search . + 990712 & 0.433 & @xmath48 & 23.8 & 21.2 & ground & aperture effects ? ( 6 ) + 991208 & 0.706 & @xmath49 & 23.9 & 24.4 & ground & bad afterglow fit . ( 7 ) + 991216 & 1.020 & @xmath49 & & 24.85 & ground & no search , consistent with no sn . ( 8) + 000418 & 1.119 & @xmath49 & & 23.8 & ground & consistent with no sn . ( 9 ) + 000911 & 1.058 & @xmath24 & 24.7 & 24.4 & ground & 2@xmath50 detection , sn @xmath52 0.9 @xmath43 0.3 @xmath53 sn1998bw . ( 10 ) + 011121 & 0.365 & @xmath49 & 23 & 26 & hst & secure .

grb 010921 was the first hete-2 grb to be localized via its afterglow emission . the low - redshift of the host galaxy , @xmath0 , prompted us to undertake intensive multi - color observations with the _ hubble space telescope _ with the goal of searching for an underlying supernova component . we do not detect any coincident supernova to a limit 1.34 mag fainter than sn 1998bw at 99.7% confidence , making this one of the most sensitive searches for an underlying sn . analysis of the afterglow data allow us to infer that the grb was situated behind a net extinction ( milky way and the host galaxy ) of @xmath1 mag in the observer frame . thus , had it not been for such heavy extinction our data would have allowed us to probe for an underlying sn with brightness approaching those of more typical type ib / c supernovae .

astro-ph0207187

there is a considerable theoretical and practical interest in the dynamics of systems of interacting particles in confined geometries @xcite . single - file diffusion ( sfd ) refers to a one - dimensional ( 1d ) process where the motion of particles in a narrow channel ( e.g. , _ quasi_-1d systems ) is limited such that particles are not able to cross each other . as a consequence , the system diffuses as a whole resulting in anomalous diffusion . the mechanism of sfd was first proposed by hodgkin and keynes @xcite in order to study the passage of molecules through narrow pores . since the order of the particles is conserved over time , this results in unusual dynamics of the system @xcite , different from what is predicted from diffusion governed by fick s law . the main characteristic of the sfd phenomena is that , in the long - time limit , the msd ( mean - square displacement , defined as @xmath3^{2 } \rangle_{\delta t}$ ] ) scales with time as @xmath4 this relation was first obtained analytically in the pioneering work of harris @xcite . recent advances in nanotechnology have stimulated a growing interest in sfd , in particular , in the study of transport in nanopores @xcite . ion channels of biological membranes and carbon nanotubes @xcite are examples of such nanopores . the macroscopic flux of particles through such nanopores is of great importance for many practical applications , e.g. , particle transport across membranes is a crucial intermediate step in almost all biological and chemical engineering processes . sfd was observed in experiments on diffusion of molecules in zeolite molecular sieves @xcite . zeolites with unconnected parallel channels may serve as a good realization of the theoretically investigated one - dimensional systems . sfd is also related to growth phenomena @xcite . the theoretical background of sfd was developed in early studies on transport phenomena in 1d channels @xcite . it is also interesting to learn how the size of the system will influence the diffusive properties of the system . sfd in finite size systems has been the focus of increasing attention since there are few exact theoretical results to date @xcite , which showed the existence of different regimes of diffusion . colloidal systems , complex plasmas and vortex matter in type - ii superconductors are examples of systems where sfd may occur . the use of colloidal particles is technically interesting since it allows real time and spatial direct observation of their position , which is a great advantage as compared to atoms or molecules , as shown recently in , e.g. , the experimental study of defect induced melting @xcite . one typically uses micro - meter size colloidal particles in narrow channels , as shown in @xcite . the paramagnetic colloidal spheres of 3.6 @xmath5 were confined in circular trenches fabricated by photolithography and their trajectories were followed over long periods of time . several other studies have focused on the diffusive properties of complex plasmas . a complex plasma consists of micrometer - sized ( `` dust '' ) particles immersed in a gaseous plasma background . dust particles typically acquire a negative charge of several thousand elementary charges , and thus they interact with each other through their strong electrostatic repulsion @xcite . systems of particles moving in space of reduced dimensionality or submitted to an external confinement potential exhibit different behavior from their free - of - border counterparts @xcite . the combined effect of interaction between the particles and the confinement potential plays a crucial role in their physical and chemical properties @xcite . @xcite , it was found that sfd depends on the inter - particle interaction and can even be suppressed if the interaction is sufficiently strong , resulting in a slower subdiffusive behavior , where @xmath1 @xmath2 , with @xmath6 . in this paper , we will investigate the effects of confinement potential on the diffusive properties of a q1d system of interacting particles . in the limiting case of very narrow ( wide ) channels , particle diffusion can be referred to sfd ( 2d regime ) characterized by a subdiffusive ( normal diffusive ) long - time regime where the mean - squared displacement ( msd ) @xmath1 @xmath7 ( @xmath8 ) . recall that the msd of a tagged hard - sphere particle in a one dimensional infinite system is characterized by two limiting diffusion behaviors : for time scales shorter than a certain crossover time @xmath9 , where @xmath10 is the diffusion coefficient and @xmath11 is the particle concentration , @xmath1 @xmath8 which is referred to as the normal diffusion regime @xcite . for times larger than @xmath12 , the system exhibits a subdiffusive behavior , with the msd @xmath1 @xmath7 , which characterizes the single - file diffusion regime . between these two regimes , there is a transient regime exhibiting a non - trivial functional form . however , in case of a _ finite _ system of diffusing particles ( e.g. , a circular chain or a straight chain in the presence of periodic boundary conditions ) , the sfd regime ( i.e. , with @xmath1 @xmath7 ) does not hold for @xmath13 , unlike in an infinite system . instead , for sufficiently long times , the sfd regime turns to the regime of _ collective _ diffusion , i.e. , when the whole system diffuses as a single `` particle '' with a renormalized mass . this diffusive behavior has been revealed in experiments @xcite and theoretical studies @xcite . this collective diffusion regime is similar to the initial short - time diffusion regime and it is characterized by either @xmath1 @xmath8 , for overdamped particles ( see , e.g. , @xcite ) or by @xmath1 @xmath14 ( followed by the msd @xmath8 ) , for underdamped systems @xcite . correspondingly , the time interval where the sfd regime is observed becomes _ finite _ in finite size systems . it depends on the lenght of the chain of diffusing particles : the longer the chain the longer the sfd time interval . therefore , in order to observe a clear power - law behavior ( i.e. , @xmath1 @xmath2 ) one should consider sufficiently large systems . here we focus on this intermediate diffusion regime and we show that it can be characterized by @xmath1 @xmath2 , where @xmath15 , depending on the width ( or the strenght of the confinement potential ) of the channel . we analyze the msd for two different channel geometries : ( i ) a linear channel , and ( ii ) a circular channel . these two systems correspond to different experimental realizations of diffusion of charged particles in narrow channels @xcite . the latter one ( i.e. , a circular channel ) has obvious advantages : ( i ) it allows a long - time observation of diffusion using a relatively short circuit , and ( ii ) it provides constant average particle density and absence of density gradients ( which occur in , e.g. , a linear channel due to the entry / exit of particles in / from the channel ) . thus circular narrow channels were used in diffusion experiments with colloids @xcite and metallic charged particles ( balls ) @xcite . furthermore , using different systems allows us to demonstrate that the results obtained in our study are generic and do not depend on the specific experimental set - up . this paper is organized as follows . in sec . ii , we introduce the model and numerical approach . in sec . iii diffusion in a system of interacting particles , confined to a straight hard - wall or parabolic channel , is studied as a function of the channel width or confinement strength . in sec . iv , we discuss the possibility of experimental observation of the studied crossover from the sfd to 2d diffusive regime . for that purpose , we analyze diffusion in a realistic experimental set - up , i.e. , diffusion of massive metallic balls embedded in a circular channel with parabolic confinement whose strength can be controlled by an applied electric field . the long - time limit is analyzed in sec . v using a discrete site model . finally , the conclusions are presented in sec . our model system consists of @xmath16 identical charged particles interacting through a repulsive pair potential @xmath17 . in this study , we use a screened coulomb potential ( yukawa potential ) , @xmath18 . in the transverse direction , the motion of the particles is restricted either by a hard - wall or by a parabolic confinement potential . thus the total potential energy of the system can be written as : @xmath19 the first term in the right - hand side ( r.h.s . ) of eq . ( [ eq1 ] ) represents the confinement potential , where @xmath20 is given by : @xmath21 for the hard - wall confinement , @xmath22 for parabolic one - dimensional potential ( in the @xmath23-direction ) , and by @xmath24 for parabolic circular confinement . here @xmath25 is the width of the channel ( for the hard - wall potential ) , @xmath26 is the mass of the particles , @xmath27 is the strenght of the parabolic 1d confining potential , @xmath28 is the coordinate of the minimum of the potential energy and @xmath29 is the displacement of the @xmath30th particle from @xmath28 ( for the parabolic circular potential ) . note that in case of a circular channel , @xmath31 , where @xmath32 is the radius of the channel . the second term in the r.h.s . of eq . ( [ eq1 ] ) represents the interaction potential between the particles . for the screened couloumb potential , @xmath33 where @xmath34 is the charge of each particle , @xmath35 is the dieletric constant of the medium , @xmath36 is the distance between @xmath30th and @xmath37th particles , and @xmath38 is the debye screening length . substituting ( [ eq3 ] ) into eq . ( [ eq1 ] ) , we obtain the potential energy of the system @xmath39 : @xmath40 in order to reveal important parameters which characterize the system , we rewrite the energy @xmath39 in a dimensionless ( @xmath41 ) form by making use of the following variable transformations : @xmath42 , @xmath43 , where @xmath44 is the mean inter - particle distance . the energy of the system then becomes @xmath45 where @xmath46 is the screening parameter of the interaction potential . in our simulations in sec . iii , we use a typical value of @xmath47 for colloidal systems and @xmath48 m . the hard - wall confinement potential is written as @xmath49 where @xmath50 is scaled by the inter - particle distance @xmath44 . we also introduce a dimensionless parameter @xmath51 which is a measure of the strenght of the parabolic 1d confinement potential . for colloidal particles moving in a nonmagnetic liquid , their motion is overdamped and thus the stochastic langevin equations of motion can be reduced to those for brownian particles @xcite : @xmath52.\end{aligned}\ ] ] note , however , that in sec . iv we will deal with massive metallic balls and therefore we will keep the inertial term in the langevin equations of motion . in eq . ( [ eq7 ] ) , @xmath53 , @xmath54 and @xmath55 are the position , the self - diffusion coefficient ( measured in m@xmath56/s ) and the mass ( in kg ) of the @xmath30th particle , respectively , @xmath57 is the time ( in seconds ) , @xmath58 is the boltzmann constant , and @xmath59 is the absolute temperature of the system . finally , @xmath60 is a randomly fluctuating force , which obeys the following conditions : @xmath61 and @xmath62 , where @xmath63 is the friction coefficient . ( [ eq7 ] ) can be written in dimensionless form as follows : @xmath64,\end{aligned}\ ] ] where we use the following transformation @xmath65 , @xmath66 , and introduced a coupling parameter @xmath67 , which is the ratio of the average potential energy to the average kinetic energy , @xmath68 , such that @xmath69 . the time @xmath70 is expressed in seconds and distances are expressed in units of the interparticle distance @xmath44 . in what follows , we will abandon the prime ( @xmath71 ) notation . we have used a first order finite difference method ( euler method ) to integrate eq . ( [ eq8 ] ) numerically . in the case of a straight channel , periodic boundary conditions ( pbc ) were applied in the @xmath72-direction while in the @xmath23-direction the system is confined either by a hard - wall or by a parabolic potential . also , we use a timestep @xmath73 and the coupling parameter is set to @xmath74 . for a circular channel , we use polar coordinates @xmath75 and model a 2d narrow channel of radius @xmath32 with parabolic potential - energy profile across the channel , i.e. , in the @xmath76-direction . in order to characterize the diffusion of the system , we calculate the msd as follows : @xmath77^{2 } \big\rangle_{\delta t},\ ] ] where @xmath16 is the total number of particles and @xmath78 represents a time average over the time interval @xmath79 . note that in the general case ( e.g. , for small circular channels with the number of particles @xmath80 see sec . iv ) the calculated msd was averaged over time _ and _ over the number of ensembles @xcite . however , we found that for large @xmath16 ( i.e. , several hundred ) the calculated msd for various ensemble realizations coincide ( with a maximum deviation within the thickness of the line representing the msd ) . to keep the inter - particle distance approximately equal to unity , we defined the total number of particles @xmath16 for a 1d and q1d system as @xmath81 where @xmath82 is the size of the simulation box ( in dimensionless units ) in the @xmath72-direction . in our simulations for a straight channel geometry , we typically used @xmath83 particles . we study the system for two different types of confinement potential : ( i ) a parabolic 1d potential in the @xmath23-direction , which can be tuned by the confinement strength @xmath84 and ( ii ) a hard - wall potential , where particles are confined by two parallel walls separated by a distance @xmath25 . the results of calculations of the msd as a function of time for different values of the confinement strength @xmath84 [ eq . ( [ ki ] ) ] and the width of the channel @xmath25 are presented in fig.[msdparabolic](a)-(c ) and fig.[msdhardwall](a)-(c ) , respectively . as a function of time for different values of @xmath84 . different diffusion regimes can be distinguished : normal diffusion regime ( @xmath85 ) and intermediate subdiffusive regime ( itr , @xmath86 ) . note that for the case of @xmath84 = 1.5 , there is a normal diffusion regime ( i.e. @xmath85 ) after the itr . the dashed and solid lines in ( a)-(c ) are a guide to the eye . panel ( d ) shows the dependence of the slope ( @xmath0 ) of the msd curves ( in the itr , characterized by an apparent power - law ; @xmath1 @xmath2 ) on the confinement strength @xmath84.,width=264 ] as a function of time for different values of @xmath25 . different diffusion regimes can be distinguished : normal diffusion regime ( @xmath85 ) and intermediate subdiffusive regime ( itr , @xmath86 ) . note that for the case of @xmath25 = 0.60 , there is a normal diffusion regime ( i.e. @xmath85 ) after the itr . the dashed and solid lines in ( a)-(c ) are a guide to the eye . panel ( d ) shows the dependence of the slope ( @xmath0 ) of the msd curves ( in the itr , characterized by an apparent power - law ; @xmath1 @xmath2 ) on the confinement parameter @xmath25.,width=264 ] initially , in both cases ( i.e. , a parabolic and a hard - wall confinement potential ) , the system exhibits a short - time normal diffusion behavior , where @xmath1 @xmath8 . this is the typical initial `` free - particle '' diffusion regime . after this initial regime , there is an intermediate subdiffusive regime ( itr ) . as discussed in ref . @xcite , the itr shows an apparent power - law behavior @xcite , where @xmath15 , and it was also found previously in different diffusion models @xcite . in the itr , we found a sfd regime for either a channel with strong parabolic confinement [ @xmath87 ( fig . [ msdparabolic](a ) ) ] or a narrow hard - wall channel [ @xmath88 ( fig . [ msdhardwall](a ) ) ] . this is due to the fact that for large ( small ) values of @xmath84 ( @xmath25 ) , the confinement prevents particles from passing each other . the results for @xmath0 in the itr are shown as a function of @xmath84 and @xmath25 in fig . [ msdparabolic](d ) and fig . [ msdhardwall](d ) , respectively . as can be seen in fig . [ msdparabolic](d ) [ fig . [ msdhardwall](d ) ] , @xmath0 increases with decreasing @xmath84 [ with increasing @xmath25 ] and thus the sfd condition turns out to be broken . the values of @xmath0 presented in these figures correspond to the minimum of the effective time dependent exponent @xmath90 . following ref . @xcite , @xmath90 is calculated using the `` double logarithmic time derivative '' @xmath91 and the results are shown in fig . [ alphatime ] . as a function of time , calculated from eq . ( [ alphatime ] ) for different values of the confinement parameters @xmath84 and @xmath25 , respectively.,width=302 ] the different diffusive regimes , i.e. normal diffusion regime ( @xmath85 ) and sfd ( @xmath92 ) , were also found recently in finite - size systems @xcite although the transition from sfd to normal diffusion was not analyzed . the @xmath0-dependence on both the confinement parameters ( i.e. , @xmath93 and @xmath94 ) presents a different qualitative behavior , namely , the sfd regime is reached after a smoother crossover in the parabolic confinement case as compared to the hardwall case . a similar smoother crossover is also found in the case of a circular channel with parabolic confinement in the radial direction . a more detailed discussion on these two different types of the behavior of @xmath0 will be provided in sec . [ circularchannel ] . for small values of the parabolic confinement ( e.g. , @xmath96 ) , the msd curves present three different diffusive regimes : ( i ) a short - time normal diffusion regime , where msd @xmath1 @xmath8 ; ( ii ) a subdiffusive regime with @xmath1 @xmath2 , where @xmath15 and ( iii ) a `` long - time '' diffusion regime , which is characterized by @xmath1 @xmath8 . note that the `` long - time '' term used here is not to be confused with the long - time used for _ infinite _ systems , as discussed in the introduction . however , for large values of the parabolic confinement ( e.g. , @xmath87 ) , we observe only two distinct diffusive regimes , namely : ( i ) a short - time normal diffusion regime ( @xmath1 @xmath8 ) and ( ii ) a sfd regime ( i.e. , @xmath1 @xmath7 ) . one question that arises naturally is whether this normal diffusion regime ( i.e. , @xmath1 @xmath8 for `` long - times '' ) is an effect of the _ colletive _ motion of the system ( center - of - mass motion ) or an effect of the single - particle jumping process , since the confinement potential @xmath96 allows particles bypass . in order to answer this question , we calculate the number of crossing events @xmath95 as a function of time and results are shown in fig . [ ncross](a ) . we found that for small values of the confinement potential ( e.g. , @xmath96 ) the number of crossing events grows linearly in time , i.e. , @xmath97 , where @xmath98 is the rate of crossing events . on the other hand , a strong confinement potential ( e.g. , @xmath87 ) prevents particles from bypassing , and thus @xmath99 during the whole simulation time . therefore , the `` long - time '' normal diffusive behavior ( i.e. , @xmath1 @xmath8 for `` long - times '' ) found in our simulations for the case where the sf ( single - file ) condition is broken ( e.g. , @xmath96 ) is _ not _ due to a collective ( center - of - mass ) diffusion . instead , this normal diffusive behaviour is due to a single - particle jumping process , which happens with a constant rate @xmath100 for the case of small values of the confinement ( @xmath96 ) and @xmath101 ( for @xmath87 ) . the same analysis was done for the case of the hard - wall confinement potential , and the results are found to be the same as for the parabolic confinement . nevertheless , we point out that the collective diffusion does indeed exist , but our results from simulations do not allow us to observe this collective ( center - of - mass ) diffusion regime because of the large size of our chain of particles ( @xmath102 ) . simulations with @xmath103 , and excluding the possibility of mutual bypass ( strong confinement potential ) , allowed us to observe that the @xmath1 @xmath8 regime is recovered in the `` long - time '' limit . in sec . v , we will further discuss the long - time limit using a model of discrete sites . as we demonstrated above , the transition from pure 1d diffusion ( sfd ) characterized by @xmath92 to a quasi-1d behavior ( with @xmath104 ) could be either more `` smooth '' ( as in fig . 1(d ) , for a parabolic confinement ) or more `` abrupt '' ( as in fig . 2(d ) , for a hard - wall confinement ) . one can intuitively expect that this difference in behavior can manifest itself also in the crossing events rate @xmath98 , i.e. , that @xmath98 as a function of @xmath84 ( or @xmath105 ) should display a clear signature of either `` smooth '' or `` abrupt '' behavior . however , the link between the two quantities , i.e. , the exponent , @xmath106 , and the crossing events rate , @xmath107 is not that straightforward . to understand this , let us refer to the long - time limit ( which will be addressed in detail within the discrete - site model in sec . as we show , in the long - time limit the exponent @xmath0 is defined by one of the two conditions : @xmath101 ( then @xmath92 ) or @xmath100 ( then @xmath108 ) and it does _ not _ depend on the specific value of @xmath98 provided it is nonzero . therefore , in the long - time limit the transition between 1d to 2d behavior _ is not sensitive _ to the particular behavior of the function @xmath107 . although for `` intermediate '' times ( considered in this section ) the condition @xmath101 _ or _ @xmath109 is not critical , nevertheless , very small change in the crossing events rate @xmath107 strongly influences the behavior of the exponent @xmath106 . this is illustrated in figs . 4(b , c ) . in fig . 4(b ) , the function @xmath110 gradually decreases from 1.45 to 0 for @xmath84 varying in a _ broad _ interval from 1.5 to 3 ( note that the segment of @xmath110 for @xmath111 is nonzero which can be seen in the inset of fig . 4(b ) showing the derivative @xmath112 ) . correspondingly , the transition from @xmath92 to @xmath113 in that interval of @xmath84 is `` smooth '' ( see fig . 1(d ) ) . on the other hand , the function @xmath114 shown in fig . 4(c ) mainly changes ( note the change of the slope @xmath115 shown in the inset of fig . 4(c ) ) in a _ narrow _ interval @xmath116 . respectively , the transition for the function @xmath117 occurs in the narrow interval @xmath116 and thus is ( more ) `` abrupt '' . as a function of time for @xmath118 particles , for @xmath96 ( black open circles ) and @xmath87 ( green open diamonds ) . the solid red line is a linear fit to @xmath95 . panels ( b ) and ( c ) show the rate of the crossing events @xmath119 as a function of the confinement potential parameters ( @xmath84 and @xmath25 ) . the insets in the panels ( b ) and ( c ) show the derivatives , @xmath112 and @xmath115 , correspondingly . , width=302 ] for the ideal 1d case , particles are located on a straight line . increasing the width @xmath25 of the confining channel will lead to a zig - zag transition @xcite . this zig - zag configuration can be seen as a distorted triangular configuration in this transition zone . further increase of @xmath25 brings the system into the 2d regime , where the normal diffusion behavior is recovered ( see fig . [ hardsnap ] ) . for the parabolic 1d confinement , we can see [ fig . [ distpy](a ) ] that the distribution of particles @xmath120 along the channel is symmetric along the axis @xmath121 . also , for large values of @xmath84 ( e.g. , @xmath87 ) particles are confined in the @xmath23-direction and thus can move only in the @xmath72-direction , forming a single - chain structure . as the confinement decreases ( @xmath122 ) , the distribution of particles @xmath120 broadens resulting in the crossover from the sfd regime ( @xmath87 ) to the 2d normal diffusion regime ( @xmath123 ) . note that for small values of @xmath84 ( e.g. , @xmath123 ) , the system forms a two - chain structure ( represented by two small peaks of @xmath120 in fig . [ distpy](a ) ) , thus allowing particles to pass each other . md simulation steps ) confined by the channel of width ( a ) @xmath88 , ( b ) @xmath124 and ( c ) @xmath125.,width=302 ] along the @xmath23-direction are shown for ( a ) different values of @xmath84 ( parabolic 1d confinement ) and ( b ) four different values of the width @xmath25 of the channel ( hard - wall confinement).,width=302 ] in the previous section , we analyzed the transition ( crossover ) from the sfd regime to 2d diffusion in narrow channels of increasing width . the analysis was performed for a straight channel with either hard - wall or parabolic confinement potential . however , in terms of possible experimental verification of the studied effect , one faces an obvious limitation of this model : although easy in simulation , it is hard to experimentally fulfill the periodic boundary conditions at the ends of an _ open _ channel . therefore , in order to avoid this difficulty , in sfd experiments @xcite circular channels were used . in this section , we investigate the transition ( crossover ) from sfd to 2d - diffusion in a system of interacting particles diffusing in a channel of _ circular _ shape . in particular , we will study the influence of the strength of the confinement ( i.e. , the depth of the potential profile across the channel ) on the diffusive behavior . without loss of generality , we will adhere to the specific conditions and parameters of the experimental set - up used in ref . an additional advantage of this model is that the motion of the system of charged metallic balls @xcite is _ not _ overdamped , and we will solve the full langevin equations of motion to study the diffusive behavior of the system . we consider @xmath16 particles , interacting through a yukawa potential [ eq . ( [ eq3 ] ) ] , which are embedded in a ring channel of radius @xmath32 . we define a parabolic confinement potential across the channel in the form ( [ vc ] ) where parameter @xmath126 is chosen as follows : @xmath127}{2r_{ch } \sin\left(\frac{\phi_i-\phi_j}{2}\right)},\ ] ] when all the particles are equidistantly distributed along the bottom of the circular channel . it should be noted that in this case , @xmath128 is approximately equal to @xmath129 due to the weak yukawa interaction , which slightly shifts the particles away from the bottom of the channel . such a choice of @xmath128 is related to the fact that we study the influence of the confinement on the diffusion and , therefore , the potential energy of the particles must be of the order of the inter - particle interaction energy . parameter @xmath130 characterizes the distance where the external potential reaches the value @xmath131 , and @xmath129 is the energy of the ground state of the system of @xmath16 particles as defined by eq . ( [ eq4 ] ) . parameter @xmath132 plays the role of a control parameter . by changing @xmath132 we can manipulate the strength of the confinement and , therefore , control the fulfillment of the single - file condition . increase in @xmath132 corresponds to a decrease in the depth of the confinement ( [ vc ] ) which leads to the expansion of the area of radial localization of particles . therefore , an increase of @xmath132 results in a similar effect ( i.e. , spatial delocalization of particles ) as an increase of temperature , i.e. , parameter @xmath132 can be considered as an `` effective temperature '' . note that such a choice of the parameter that controls the confinement strength is rather realistic . in the experiment of ref . @xcite with metallic balls , the parabolic confinement was created by an external electric field , and the depth of the potential was controlled by tuning the strength of the field . to study diffusion of charged metallic balls , we solve the langevin equation of motion in the general form ( i.e. , with the inertial term @xmath133 ) , @xmath134 where @xmath135 kg @xcite is the mass of a particle , @xmath63 is the friction coefficient ( inverse to the mobility ) . here all the parameters of the system were chosen following the experiment @xcite , and @xmath136 , @xmath137 ( which is a typical experimental value , see , e.g. , also @xcite ) . correspondingly , mass is measured in kg , length in m , and time in seconds . also , following ref . @xcite , we took a channel of radius @xmath138 mm ( in the experiment @xcite , the external radius of the channel was 10 mm , and the channel width 2 mm ; note that in our model we do not define the channel width : the motion of a particle in the transverse direction is only restricted by the parabolic confinement potential ) . we also took experimentally relevant number of diffusing particles , @xmath16 , varying from @xmath139 to @xmath140 ( in the experiment @xcite , the ring channel contained @xmath139 or @xmath141 diffusing balls ) . [ figsnapshots ] shows the results of calculations of the trajectories of @xmath80 particles diffusing in a ring of radius @xmath142 mm for the first 10@xmath143 md steps for various values of the parameter @xmath132 . as can be seen from the presented snapshots , the radial localization of particles weakens with increasing @xmath132 . at a certain value of @xmath132 this leads to the breakdown of the single - file behavior ( figs . [ figsnapshots](c)-(f ) ) . particles diffusing in a ring of radius @xmath142 mm for @xmath144 consequent time steps for different values of @xmath132 . @xmath132=1 ( a ) , 2 ( b ) , 3 ( c ) , 5 ( d ) , 7 ( e ) , 9 ( f ) . , width=321 ] it is convenient to introduce the distribution of the probability density of particles in the channel @xmath145 along the radial direction @xmath76 . in order to calculate the function @xmath146 we divided the circular channel in a number of coaxial thin rings . the ratio of the number of observations of particles in a sector of radius @xmath147 to the total number of observations during the simulation is defined as the probability density @xmath148 . in fig . [ figprob ] , the probability density @xmath146 is presented for different values of @xmath132 . with increasing @xmath132 , the distribution of the probability density @xmath146 broaden and the maximum of the function @xmath146 shifts away from the center of the channel ( see fig . [ figprob ] ) . the latter is explained by the softening of the localization of particles with increasing @xmath132 , which tend to occupy an area with a larger radius due to the repulsive inter - particle interaction . simultaneously , the distribution of the probability density @xmath146 acquires an additional bump indicating the nucleation of a two - channel particle distribution @xcite . the observed broadening and deformation of the function @xmath146 is indicative of a gradual increase of the probability of mutual bypass of particles ( i.e. , the violation of the sf ( single - file ) condition , also called the `` overtake probability '' @xcite ) with increasing @xmath132 . in a circular channel of radius @xmath142 mm along the radial direction @xmath76 . the different curves correspond to various @xmath132 . increasing @xmath132 the width of the distribution @xmath146 increases due to a weakening of the confinement . , width=321 ] created by a particle ( red ( grey ) circle ) and the qualitative distribution of the probability density of particles in circular channel @xmath146 ( green ( light grey ) line ) along the radial direction @xmath76 . the function @xmath149 determines an approximate radial distance between particles when the potential barrier @xmath150 becomes `` permeable '' for given temperature @xmath59 . the function @xmath151 characterizes a width of the distribution @xmath146 at this temperature @xmath59 . , width=283 ] let us now discuss a qualitative criterion for the breakdown of sfd , i.e. , when the _ majority _ of particles leave the sfd mode . for this purpose , let us consider a particle in the potential created by its close neighbor ( which is justified in case of short - range yukawa interparticle interaction and low density of particles in a channel ) shown in fig . different lines show the interparticle potential @xmath152 as a function of angle @xmath153 for different radii @xmath76 . for small values of @xmath132 , the center of the distribution @xmath146 ( see fig . 7 ) almost coincides with the center of the channel ( i.e. , with the minimum of the confinement potential profile ) and the distribution @xmath146 is narrow . therefore , mutual passage of particles is impossible , i.e. , the sf condition is fulfilled . the asymmetric broadening of the function @xmath146 with increasing @xmath132 results in an increasing probability of mutual bypass of particles which have to overcome a barrier @xmath150 ( see fig . [ fig7 ] ) . this becomes possible when @xmath154 . in other words , the thermal energy @xmath155 determines some minimal width @xmath149 between adjacent particles when the breakdown of the sf condition becomes possible . it is clear that `` massive '' violation of the sf condition ( i.e. , when the majority of particles bypass each other ) occurs when the halfwidth @xmath151 of the distribution of the probability density @xmath146 obeys the condition : @xmath156 the function @xmath151 is defined by the ratio of the thermal energy @xmath155 to the external potential @xmath157 and is of the same order as @xmath158 : @xmath159 therefore the criterion ( [ nsf01 ] ) can be presented in the form : @xmath160 this qualitative analysis of the breakdown of the sfd regime clarifies the role of the width and the shape of the distribution of the probability density influenced by the asymmetry of the circular channel . the msd @xmath161 is calculated as a function of time @xmath57 as : @xmath162 ^ 2 \right\rangle_t,\ ] ] where @xmath163 is the total number of particles of an ensemble and @xmath164 is the total number of ensembles . in our calculations , the number of ensembles was chosen 100 for a system consisting of 20 particles . the time dependence of the msd for different values of @xmath132 is shown in fig . [ figsfdring](a)(c ) . initially the system exhibits normal diffusion , where @xmath165 @xmath8 . this regime is followed by an intermediate subdiffusive regime , where the @xmath165 @xmath2 ( @xmath15 ) . for longer times , the system recovers `` long - time '' normal diffusion ( see discussions in sec . iiib ) , with @xmath165 @xmath8 . as in the case of straight channel geometry , this second crossover ( i.e. , from intermediate subdiffusion to `` long - time '' normal diffusion ) can also be due to two other reasons : ( i ) due to a collective ( center - of - mass ) diffusion or ( ii ) due to a single - particle jumping process . however , for the simulations in the case of a circular geometry , the number of particles is relatively small ( taking the fact that this is a finite - size system ) , and therefore , the crossover from sublinear to linear regime is due to a collective ( center - of - mass ) diffusion . we further address this issue in sec . v , where we consider a discrete site model and we exclude the center - of - mass motion . [ figsfdring](d ) shows @xmath0 as a function of @xmath132 . the function @xmath166 experiences a monotonic gradual crossover from the @xmath92 to a @xmath167-regime . note that the observed deviation from the normal diffusion behavior for large @xmath132 ( fig . [ figsfdring ] ) is related to the presence of , though weak but nonzero , external confinement in the radial direction . this change of the diffusive behavior is explained by a weakening of the average radial localization of particles with increase of @xmath132 ( fig . [ figprob ] ) and , as a consequence , by an increase of the probability of mutual bypass of particles . as a function of time for different values of the `` effective '' temperature @xmath132 = ( a ) 1 , ( b ) 2 , and ( c ) 3 . here ( d ) the diffusion exponent @xmath0 as a function of @xmath132 . increase of the `` effective '' temperature @xmath132 leads to the gradual transformation of the single - file regime of diffusion into the diffusion regime of free particles.,width=264 ] the observed crossover between the 1d single - file and 2d diffusive regimes , i.e. , @xmath166-dependence , shows a significant different qualitative behavior as compared to the case of a hard - wall confinement potential considered in sec . [ straight ] , where a rather sharp transition between the two regimes was found [ fig . [ msdhardwall](d ) ] . the different behavior is due to the different confinement profiles and can be understood from the analysis of the distribution of the probability density of particles for these two cases . in the case of a hard - wall channel , the uncompensated ( i.e. , by the confinement ) interparticle repulsion leads to a higher particle density near the boundaries rather than near the center of the channel ( see fig . [ hardsnap ] and fig . [ distpy](b ) ) . as a consequence , the breakdown of the sf condition with increasing width of the channel happens simultaneously for _ many _ particles in the vicinity of the boundary resulting in a sharp transition ( see fig . [ msdhardwall](d ) ) . on the contrary , in the case of parabolic confinement , the density distribution function has a maximum sharp or broad , depending on the confinement strength near the center of the channel ( see figs . [ distpy](a ) and [ figprob ] ) . with increasing the `` width '' of the channel ( i.e. , weakening its strength ) , only a small _ fraction _ of particles undergoes the breakdown of the sf condition . this fraction gradually increases with decreasing strength of the confinement , therefore resulting in a smooth crossover between the two diffusion regimes . the calculated msd for different geometries and confinement potentials allowed us to explain the evolution of the subdiffusive regime with varying width of the channel ( or potential strength in case of a parabolic potential ) . however , the obtained results are only valid for the intermediate regime and therefore they only describe the `` onset '' of the long - time behavior . the problem of accessing the long - time behavior in a finite chain is related to the fact that sooner or later ( i.e. , depending on the chain length ) the interacting system will evolve into a collective , or `` single - particle '' , diffusion mode which is characterized by @xmath85 . thus the question is whether the observed behavior holds for the long - time limit , i.e. , is the transition from @xmath169 to @xmath170 behavior smooth ? to answer this question , we considered a simple model , i.e. , a linear discrete chain of fixed sites filled with either particles or `` holes '' ( i.e. , sites not occupied by particles ) ( for details , see ref . @xcite ; this model was also recently used in ref . the particles can move along the chain only due to the exchange with adjacent vacancies ( i.e. , with holes ) . within this model , the long - time diffusion behavior was described _ analytically _ for an infinite linear chain as well as for a finite cyclic chain @xcite . in particular , this model predicts that : ( i ) if the chain is infinite then the long - time power law of the diffusion curve @xmath0 is @xmath171 ( i.e. , msd @xmath1 @xmath7 ) ; ( ii ) if the chain is finite then the subdiffusive regime with @xmath172 is followed by either @xmath173 regime ( if the cyclic boundary condition is realized ) , or by @xmath174 regime , i.e. , the regime of saturation ( if no cyclic boundary condition is imposed @xcite ) . the latter regime is reached for times longer than the `` diffusion time '' of a `` hole '' along the whole chain @xmath175 . let us now apply this model to a finite - size chain of particles . for this purpose , we assume that adjacent particles are able to exchange their positions with some probability @xmath176 at every time step . for example , probability @xmath177 means that a couple of any adjacent particles certainly exchange their positions once for every @xmath178 time steps . the results of our calculations of the msd performed using this model are presented in fig . [ figsfdsc](a ) . we used the following parameters : the chain length is @xmath179 sites and @xmath180 hole . averaging was done over @xmath181 ensembles . the calculation was performed for the following values of the probability : @xmath182 , and @xmath183 . ( a ) and corrected msd @xmath1@xmath184 ( b ) as a function of time for different values of the probability @xmath176 of bypassing . averaging was done over @xmath185 ensembles.,width=283 ] we see in fig . [ figsfdsc](a ) clearly the above - mentioned two diffusion regimes , i.e. , with the msd @xmath1 @xmath7 and @xmath8 . the characteristic time @xmath175 shifts towards lower values with increasing @xmath176 . however this analysis ( fig . [ figsfdsc](a ) ) does not allow to distinguish the contributions to the long - time behavior ( @xmath8 ) due to : ( i ) the breakdown of single - file condition ( i.e. , diffusion due to particle exchanges ) , and ( ii ) the `` collective '' diffusion ( chain `` rotation '' ) . to overcome this difficulty , we exclude the `` collective '' diffusion of the system and introduce a modified msd @xmath1@xmath184 ( which is so - called `` roughness '' of the system of particles , as discussed in ref . @xcite ) as follows : @xmath186 where @xmath187 is the average over time ; @xmath188 is the average of an ensemble of particles at a given time , or `` collective '' coordinate . it should be noted that @xmath189 . if the system does not experience `` collective '' diffusion then @xmath190 and the modified msd coincides with the conventional one : @xmath191 the diffusion curves calculated by using the modified msd are presented in fig . [ figsfdsc](b ) . for @xmath192 , the diffusion curve ( shown by black open squares ) after the subdiffusive regime reaches saturation ( i.e. , @xmath1@xmath184 = const ) . the observed behavior is similar to that of a finite linear chain with fixed ends ( see ref . @xcite ) . for @xmath193 , all the diffusion curves in the long - time limit are characterized by @xmath85 , _ independent _ of the value of the probability @xmath176 , as seen in fig . [ figsfdsc](b ) . in other words , the long - time diffusion does _ not _ depends on the probability of mutual exchanges of particles and has the same long - time behavior for _ any _ probability @xmath193 . here we would like to emphasize again that the long - time behavior of the diffusion curves is free from the `` collective '' diffusion effect and is only determined by particle jump diffusion . increasing a number of sites in the model corresponds , in fact , approaching to the model of infinite chain . we have found that the increasing a number of sites leads to growth of the @xmath1@xmath184 limit of saturation , on the one hand , and to a shift of @xmath175 to larger @xmath57 , on the other hand . hence , extrapolating our results to the case of infinite chain , we can conclude that in this case as well as in the case of finite - size chain , the breakdown of single - file condition leads to an abrupt transition from subdiffusive to the normal diffusion regime . the difference in the diffusive curves is just the time @xmath194 from subdiffusive regime to the normal regime : for low @xmath176 it ( @xmath194 ) is long enough while for high @xmath176 it ( @xmath194 ) is short . it is easy to see that @xmath195 . thus , we can conclude that in the long - time limit the transition from @xmath196 to @xmath197 behavior is _ note that our calculations performed using the modified msd @xmath1@xmath184 reproduce the results of ref . @xcite for a closed `` box '' . this is explained by the fact that in the closed `` box '' geometry the center of mass ( or collective ) diffusion is zero , and it is natural that the roughness ( see ref . @xcite ) and the particles diffusion coincide . we have studied a monodisperse system of interacting particles subject to three types of confinement potentials : ( i ) a 1d hardwall potential , ( ii ) a 1d parabolic confinement potential which both characterize a _ quasi_-1d system , and ( iii ) a circular confining potential , which models a finite size system . in order to study the diffusive properties of the system , we have calculated the mean - squared displacement ( msd ) numerically through molecular dynamics ( md ) simulations . for the case where particles diffuse in a straight line in a q1d channel , different diffusion regimes were found for different values of the parameters of the confining potential ( @xmath84 or @xmath25 ) . we have found that the normal diffusion is suppressed if the channel width @xmath25 is between @xmath198 and @xmath199 ( or by @xmath200 , for the case of parabolic 1d confinement ) , leading the system to a sfd regime for intermediate time scales . for values of @xmath201 , particles will be able to cross each other and the sfd regime will be no longer present . the case of a circular channel corresponds to , e.g. , the set - up used in experiments with sub - millimetric metallic massive balls diffusing in a ring with a parabolic potential profile created by an external electric field . the strength of the potential ( which determines the effective `` width '' of the channel ) can be tuned by the field strength . contrary to the case of hard - wall confinement , where the transition ( regarding the calculation of the scaling exponent ( @xmath0 ) of the msd @xmath1@xmath2 ) is sharp , a smooth crossover between the 1d single - file and the 2d diffusive regimes was observed . this behavior is explained by different profiles for the distribution of the particle density for the hard - wall and parabolic confinement profiles . in the former case , the particle density reaches its maximum near the boundaries of the channel resulting in a massive breakdown of the sf condition and thus in a sharp transition between the different diffusive regimes . in the latter case , on the contrary , the density distribution function has a maximum near the center which broadens with decreasing strength of the confinement . this results in a smooth crossover between the two diffusion regimes , i.e. , sfd and 2d regime . the analysis of the crossing events , i.e. , the rate of the crossing events @xmath98 as a function of the confinement parameter @xmath84 or @xmath105 , supports these results : the function @xmath107 displays a clear signature of either `` smooth '' or `` abrupt '' behavior . we also addressed the case of a finite discrete chain of diffusing particles . it was shown that in this case the breakdown of the single - file condition ( i.e. , when the probability @xmath176 of particles bypassing each other is non - zero ) leads to an abrupt transition from a subdiffusive regime to the normal diffusion regime . this work was supported by cnpq , funcap ( pronex grant ) , the `` odysseus '' program of the flemish government , the flemish science foundation ( fwo - vl ) , the bilateral program between flanders and brazil , and the collaborative program cnpq - fwo - vl . as recently shown when studying single - trajectory averages @xcite , time average ( ta ) may differ from ensemble average ( ea ) msd not only for nonergodic processes ( for example , for anomalous diffusion described by continuous time random walks ( ctrws ) models @xcite ) , but also for some ergodic processes in small complex systems . this apparent power - law behavior is characterized by msd @xmath1@xmath2 , and it is an intermediate phenomena due to the interplay between the crossover from the msd @xmath1@xmath7 regime to @xmath1@xmath8 regime . for details , see ref . @xcite .

diffusive properties of a monodisperse system of interacting particles confined to a _ quasi_-one - dimensional ( q1d ) channel are studied using molecular dynamics ( md ) simulations . we calculate numerically the mean - squared displacement ( msd ) and investigate the influence of the width of the channel ( or the strength of the confinement potential ) on diffusion in finite - size channels of different shapes ( i.e. , straight and circular ) . the transition from single - file diffusion ( sfd ) to the two - dimensional diffusion regime is investigated . this transition ( regarding the calculation of the scaling exponent ( @xmath0 ) of the msd @xmath1 @xmath2 ) as a function of the width of the channel , is shown to change depending on the channel s confinement profile . in particular the transition can be either smooth ( i.e. , for a parabolic confinement potential ) or rather sharp / stepwise ( i.e. , for a hard - wall potential ) , as distinct from infinite channels where this transition is abrupt . this result can be explained by qualitatively different distributions of the particle density for the different confinement potentials .

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