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in this we investigate flows on discrete curves in @xmath0 , @xmath1 , and @xmath2 . by a discrete curve @xmath3 in @xmath4 we mean a map @xmath5 . a flow on @xmath3 is a smooth variation of @xmath3 . the description of these curves is mainly motivated by the picture of discrete curves in @xmath1 , in other words we imagine curves in @xmath0 as curves in @xmath1 , which are lifted to homogeneous coordinates ; and curves in @xmath2 , as curves in @xmath1 , which do not hit infinity . in particular this view allows us to find a novel interpretation of the wellknown one dimensional toda lattice hierarchy in terms of flows on discrete curves . the toda lattice hierarchy is a set of equations , including the toda equation : @xmath6 the toda equation is sometimes also called first flow equation of the toda lattice hierarchy , it was discovered by toda in 1967 ( @xcite ) . a good overview about the vast literature about the toda lattice can be found in ( @xcite ) . the paper is organized as follows . in section [ sec : dc : definitions ] we define discrete curves in @xmath0 and @xmath7 a discrete analog of the schwartzian derivative in terms of cross - ratios of four neighboring points will be defined . a zero curvature or lax representation for flows on that discrete curves will be given . in section [ sec : dc : arclength ] we will restrict ourselves to the case of so - called discrete ( conformal ) arc length parametrized curves . in [ subsec : dc : arclengthc2 ] we define certain flows on arc length parametrized curves which in the turn induce flows for the cross - ratios of that curve . it will be shown that for these flows the cross - ratios give solutions to equations of the volterra hierarchy . in [ subsec : dc : discreteeuclidean ] the arc length parametrized curves in @xmath0 viewed as curves in @xmath1 in homogeneous coordinates will be reduced to curves in @xmath2 , by assuming that the second coordinate does not vanish . we call this reduction euclidean reduction . it will be shown that if one starts with a flow on a discrete arc length parametrized curve in @xmath0 , such that the corresponding cross - ratios are solutions to the second flow of the volterra hierarchy , which is also called the discrete kdv flow , then the corresponding curve obtained by euclidean reduction satisfies a discrete version of the mkdv flow . this gives a discrete version of a geometrical interpretation of the miura transformation . in [ subsec : dc : discreteflows ] a nice geometrical interpretation of bcklund transformations of arc length parametrized discrete curves in @xmath0 will be given . in the turn a time discrete version of the volterra equation , which is the first flow equation of the volterra hierarchy , is derived . in section [ sec : dc : todalattice ] determinants of two and three neighboring points of a discrete curve in @xmath0 will be identified with the flaschka - manakov variables of the toda lattice . the toda flows define flow directions for curves in @xmath8 in [ sec : dc : threeflowsred ] the flows on the discrete curves in @xmath0 given by toda lattice hierarchy will be further investigated . in particular in [ subsec : dc : threeflows ] the flows corresponding to the first three flows will be given explicitly . in [ sec : dc : invariant ] we derive several geometrical features of these flows . in particular in [ subsec : dc : ellipse ] we will look at flows on curves which are compatible with the known reduction @xcite of the toda lattice hierarchy to the volterra lattice hierarchy by setting the flaschka - manakov variable @xmath9 this reduction is different from from the reduction in [ subsec : dc : arclengthc2 ] , where among others @xmath10 . nevertheless the cross ratios of such curves evolve again with equations of the volterra hierachy . hence there exist two geometrically motivated reductions of the toda lattice hierachy to the volterra hierarchy . the case @xmath11 and @xmath12 lies in the intersection of both reductions , it belongs to the trivial solutions of the toda lattice hierachy equations . the corresponding class of curves , which are invariant under this trivial flow are quadrics , as will be shown in that subsection . in the subsection [ sec : dc : elastica ] we will again look for a class of curves , which is invariant ( up to euclidian motion and tangential flow ) under a certain toda flow . this time we find that in the euclidian reduction socalled discrete planar elastic curves are invariant under the discrete mkdv flow . let @xmath13 , @xmath14 be a discrete curve in the complex projective space . we assume @xmath15 is immersed , i.e. @xmath16 and @xmath17 are pairwise disjoined . by introducing homogeneous coordinates , we can lift @xmath15 to a map : & & ^2 + k & & _ k = ( x_k + y_k ) [ def : dc : gamdef]with @xmath18 . obviously @xmath3 is not uniquely defined : for @xmath19 , @xmath20 is also a valid lift . define : g_k & = & ( _ k,_k+1 ) = x_k y_k+1-y_k x_k+1 + u_k&=&(_k-1,_k+1 ) [ def : dc : gu ] the following lemma can be straightforwardly obtained by using the above definitions ( [ def : dc : gu ] ) : @xmath21 [ lem : dc : curvesteps ] if the variables @xmath22 and @xmath23 and initial points @xmath24 and @xmath25 are given , then lemma [ lem : dc : gam1 ] is a recursive definition of a discrete curve . this may look as an odd way to define a discrete curve ; nevertheless later on the power of this definition will become more apparent . in particular the variables @xmath22 and @xmath23 will be related to the flaschka - manakov @xcite variables of the one dimensional toda lattice . note that after choosing an initial @xmath24 it is always possible ( c is immersed ) to find @xmath26 such that : g_k = 1 [ def : dc : deteins ] for all @xmath27 . we will call discrete curves @xmath28 with property ( [ def : dc : deteins ] ) conformal arc length parametrized . hence the variables @xmath23 measure the deviation from arc length parameterization . the _ cross - ratio _ of four points @xmath29 is defined by @xmath30 let us denote the cross - ratio of four neighboring points of @xmath3 by @xmath31 : @xmath32 now @xmath33 . this means that @xmath26 and @xmath34 are linearly independent . hence an arbitrary flow on ( or variation of ) @xmath3 can be written in the following way : @xmath35 the variables @xmath36 are arbitrary , for convenience we sometimes use the normalized variable : _ k = _ k ( g_k+1+g_k-1 ) . [ def : dc : betahut ] a flow on the discrete curve @xmath3 given by ( [ eq : dc : generalflow ] ) generates the following flow on the variables @xmath23 : @xmath37 [ lem : dc : gu ] differentiate @xmath23 in ( [ def : dc : gu ] ) : @xmath38 if the flow is arc length preserving , i.e. in particular @xmath39 f.a . @xmath40 then there exists an obvious trivial solution for ( [ eq : dc : dotg ] ) : choosing @xmath41 induces @xmath42 . this flow corresponds to the freedom of the initial choice of @xmath24 and has no effect on the corresponding curve @xmath15 in @xmath1 . note also that in this case ( [ eq : dc : dotg ] ) is a linear equation . so one can always add any two flows solving it . a flow on the discrete curve @xmath3 given by ( [ eq : dc : generalflow ] ) generates the following flow on the variables @xmath22 : rcl u_k & = & u_k(_k-1 + _ k+1 ) + & + & _ k-1(g_k-2+g_k-1 ) - _ k+1(g_k+g_k+1 ) + & + & u_k ( _ k+1 - _ k-1 ) [ eq : dc : flowonu ] or equivalently rcl & = & _ k-1 + _ k+1 + & & + _ k-1(g_k-2+g_k-1 ) - _ k+1(g_k+g_k+1 ) + & & + _ k+1 - _ k-1 [ lem : dc : flowonu ] u_k & = & ( _ k-1 , _ k+1)+ ( _ k-1,_k+1 ) + & = & _ k-1u_k + g_k - ( _ k-2,_k+1 ) + & & + _ k+1u_k - g_k-1 + ( _ k-1,_k+2 ) using lemma [ lem : dc : curvesteps ] one gets @xmath43 which gives the result . a flow on the discrete curve @xmath3 given by ( [ eq : dc : generalflow ] ) generates the following flow on the cross - ratios @xmath44 : & = & ( _ k -1)(_k+1(+ ) -_k ( + ) ) + & & + _ k+1 _ k+2(+ ) + & & -_k-1 _ k-1(+ ) [ eq : dc : flowonq ] using the definition of the cross - ratio ( [ eq : dc : q ] ) and lemmas [ lem : dc : flowonu ] and [ lem : dc : gu ] the assertion follows immediately . [ thm : dc : cp1flow ] a flow on the discrete curve @xmath3 given by ( [ eq : dc : generalflow ] ) generates the following flow on the non - lifted curve @xmath15 in @xmath1 whenever @xmath15 does not hit @xmath45 : @xmath46 equation ( [ eq : dc : flowoncp1 ] ) can be written in the form @xmath47 where @xmath48 denotes the harmonic mean . in general the flow of the non - lifted curve will depend on the chosen lift , since the @xmath23 depend on the choice . note however , that the @xmath49 do not contribute to the evolution of the non - lifted curve @xmath15 . for a given lift @xmath50 the non - lifted curve may be reconstructed by @xmath51 whenever @xmath52 which means that the curve does not hit @xmath45 . now insert this and the definitions of @xmath53 and @xmath54 in both sides of equation ( [ eq : dc : flowoncp1 ] ) . define : f_k = ( ^t_k + ^t_k-1)= ( x_k&y_k + x_k-1&y_k-1 ) . note that if the curve @xmath3 is conformal arc length parametrized ( [ def : dc : deteins ] ) then @xmath55 for all @xmath40 . let @xmath56 be arbitrary and @xmath23,@xmath22 be as defined in ( [ def : dc : gu ] ) . then @xmath57 with l_k = ( u_k&- + 1&0 ) v_k = ( _ k+_k & -(1 + ) + ( 1 + ) & _ k-1-_k-1 ) . [ eq : dc : vk]the compatibility equation l_k = v_k+1l_k - l_k v_k [ eq : dc : comp ] is satisfied for all @xmath56 . [ prop : dc : lv ] the compatibility equation [ eq : dc : comp ] is also called zero curvature equation . the construction of @xmath58 is obvious with lemma [ lem : dc : curvesteps ] . the construction of @xmath59 follows also quite straightforwardly from lemma [ lem : dc : curvesteps ] and the definition of the flow on @xmath3 in ( [ eq : dc : generalflow ] ) . the compatibility equation ( [ eq : dc : comp ] ) holds by construction . the flows on @xmath23 and @xmath22 were constructed by using a well defined flow on @xmath3 for which in particular @xmath60 ( which gives ( [ eq : dc : comp ] ) ) . nevertheless ( [ eq : dc : comp ] ) can also easily be checked directly . let @xmath15 be a discrete curve in @xmath1 . up mbius transformations ( or up to the choice of @xmath61 , @xmath62 , and @xmath63 ) @xmath15 is completely determined by the cross - ratios @xmath44 . if one scales all @xmath44 with a non - vanishing factor @xmath64 one gets again up to mbius transformations a new discrete curve @xmath65 . we call the family of all such curves the _ associated family _ of @xmath66 . as mentioned in the beginning , one has a choice when lifting a discrete curve from @xmath1 to @xmath0 . this choice can be fixed by prescribing the determinants of successive points . a natural coice is here to set @xmath67 which we called conformal arc length parametrization and we will discuss this choice here . however , in section [ sec : dc : todalattice ] we will see , that other normalizations are likewise meaningfull . we will now investigate flows on @xmath3 that preserve the conformal arc length . the condition for this is , that @xmath68 , which implies by equation [ eq : dc : dotg ] @xmath69 for @xmath70 and @xmath71 from equation ( [ eq : dc : generalflow ] ) . recall that the conformal arc length condition leaves us with an initial choice of @xmath26 and that this freedom corresponds to a trivial flow with @xmath72 and @xmath73 . this flow is a first example of a ( conformal ) arc length preserving flow . it does not change the curve @xmath15 in @xmath74 though , since only the @xmath75 contribute to the evolution of the non - lifted curve . if we choose @xmath76 and @xmath77 . we get for the curve @xmath78 this is what we will call the _ conformal tangential flow_. then @xmath79 and @xmath31 will solve the famous volterra model @xcite : @xmath80 if we want this equation for the whole associated family of @xmath3 we must scale time by @xmath64 : @xmath81 one obtains the next higher flow of the volterra hierarchy @xcite when one chooses @xmath82 . this implies @xmath83 * conjecture * _ let ^_k+1-^_k : = . [ def : dc : betait]given flows @xmath84 this defines the variables @xmath85 up to a constant . now observe that starting with the flow @xmath86 gives @xmath87 , where @xmath88 is an arbitrary constant . inserting these @xmath75 into the flow equation ( [ eq : dc : flowonq ] ) in the reduced case @xmath89 gives in the turn a new flow equation @xmath90 which is ( up to the constant @xmath91 ) the volterra equation . now inserting this volterra equation into equation ( [ def : dc : betait ] ) gives new @xmath75 as @xmath92 which in the turn give the following flows on the cross - ratios : & = & a_1 ( ( _ k-1)(_k+1-_k-1 ) ) + & + & a_1 ( _ k+1(_k+2+_k+1)-_k-1(_k-1+_k-2 ) ) + & + & a_2(_k+1-_k-1 ) . this is ( up the constant @xmath91 ) the next higher flow in the volterra hierarchy plus a volterra term if @xmath93 . _ we conjecture that all higher flows of the volterra hierarchy can be obtained in this way . there are strong indications that this holds also in the continuous case @xcite . a proof of this conjecture would be beyond the scope of this article , we postpone this to a later publication . to make contact with the classical results we will now derive the @xmath94-lax representation of the volterra model for our tangential flow : define the gauge matrix @xmath95 and set @xmath96 this gauge of @xmath58 implies the following change for @xmath59 : @xmath97 & = & \quadmatrix{1+\q_{k-1}}{-1}{\q_{k-1}}{\q_k } + \quadmatrix { \q_{-1}-\frac12}{0}{0}{\q_{-1}-\frac12}\thep \end{array}\ ] ] the second matrix summand may be omitted since it is constant . if we now transpose the system , reverse the direction of the @xmath27-labeling and introduce the spectral parameter @xmath64 as mentioned above we end with the two matrices : @xmath98 v^{\mathrm{v}}(\lambda ) & = & \quadmatrix{1+\lambda\q_{k+1}}{\lambda\q_{k+1}}{-1}{\lambda\q_k } \end{array}\ ] ] with the compatibility condition @xmath99 . this is up to the change @xmath100 and a gauge transformation with @xmath101 the known form of the volterra lax - pair @xcite . let @xmath15 be a discrete curve in @xmath2 and set @xmath102 . if @xmath103 holds we call @xmath15 arc length parametrized . remember , that a flow on @xmath15 can be described via lemma [ thm : dc : cp1flow ] for an arc length parameterized curve @xmath15 the _ curvature _ @xmath104 is defined as follows : @xmath105 @xmath104 can be computed in the following way : @xmath106 in the arc length parametrized case we can write @xmath107 since @xmath108 and the same for the scalar product with @xmath109 . let us compute how the discrete curvature evolves : write @xmath110 and since then @xmath111 we get @xmath112 on the other hand using equations ( [ eq : dc : dots ] ) and ( [ eq : dc : curvature2 ] ) one can calculate @xmath113 to be @xmath114 in the case @xmath115 this implies for the evolution of the discrete curvature @xmath104 @xmath116 and as the flow on the discrete curve we get the well known tangential flow @xcite : @xmath117 now let us rewrite @xmath31 to get an interpretation for the choice of @xmath71 that gives the second volterra flow @xmath118 : @xmath119 & = & ( ( 1+\frac{2i-\kappa_k}{2i+\kappa_k})(1+\frac{2i+\kappa_{k+1}}{2i-\kappa_{k+1}}))^{-1 } = -\frac1{16}(2i+\kappa_k)(2i-\kappa_{k+1})\\[0.4 cm ] & = & \frac1{16}(2i(\kappa_{k+1 } -\kappa_k ) + ( \kappa_k \kappa_{k+1 } ) + 4)\thep \end{array}\ ] ] with this on hand we can calculate @xmath120 which leaves us with @xmath121 & & \frac{1}{64 } \left ( ( \frac{\kappa_{k+1}^2}{4}+1)(\kappa_{k+2 } + \kappa_k ) - ( \frac{\kappa_{k-1}^2}{4 } + 1 ) ( \kappa_{k } + \kappa_{k-2 } ) \right ) \end{array}\ ] ] for the evolution of the discrete curvature @xmath104 . this is up to a tangential flow part which can be removed by adjusting the constant term in the choice of @xmath71a discretization of the mkdv equation : @xmath122 therefore we will call the flow that comes from the second volterra flow discrete _ mkdv flow _ : @xmath123 the discrete tangential flow and the discrete mkdv flow both preserve the discrete arc length parameterization . we calculate @xmath124 for a general flow : @xmath125 so the condition for a flow of the form @xmath126 to preserve the discrete arc length is @xmath127 for the tangential flow this clearly holds . in the case of the mkdv flow it is an easy exercise to show equation ( [ eq : dc : euklarckond ] ) . in section [ sec : dc : elastica ] we will see that the discrete mkdv flow is connected to so called discrete elastic curves . as in the previous section let @xmath3 be the lift of a immersed discrete curve in @xmath1 into @xmath0 satisfying the normalization ( [ eq : dc : arclengthconstraint ] ) . given an initial @xmath128 and a complex parameter @xmath129 there is an unique map @xmath130 satisfying normalization ( [ eq : dc : arclengthconstraint ] ) and @xmath131 we will call @xmath132 a bcklund transform of @xmath3 . solving equation ( [ eq : dc : crevol ] ) for @xmath133 gives that @xmath133 is a mbius transform of @xmath134 . if @xmath132 is a bcklund transform of @xmath3 with parameter @xmath129 then @xmath135 with @xmath136 . due to the properties of the cross - ratio ( a useful table of the identities can be found in @xcite ) we have @xmath137 multiplying the first two and the second two equations proves the second statement . if we set @xmath138 we see that @xmath139 and therefore @xmath140 which proves the first statement . if @xmath15 is a periodic curve with period @xmath141 , we can ask for @xmath142 to be periodic too . since the map sending @xmath61 to @xmath143 is a mbius transformation it has at least one but in general two fix - points . these special choices of initial points give two bcklund transforms that can be viewed as past and future in a discrete time evolution . we will now show , that this bcklund transformation can serve as a discretization of the tangential flow since the evolution on the @xmath31 s are a discrete version of the volterra model . the discretization of the volterra model first appeared in tsujimoto , e. al . 1993.we will refer to the version stated in @xcite . there it is given in the form @xmath144 with @xmath145 being the discretization constant . let @xmath146 be a bcklund transform of @xmath31 with parameter @xmath129 . the map sending @xmath44 to @xmath147 is the discrete time volterra model ( [ eq : dc : ddvolterra ] ) with @xmath148 , @xmath149 , @xmath150 and @xmath151 . with the settings from the theorem we have @xmath152 and on the other hand @xmath153 and @xmath154 this proves the theorem . the continued bcklund transformations give rise to maps @xmath155 that can be viewed as discrete conformal maps especially in the case when @xmath129 is real negative ( which is quite far from the tangential flow , that is approximated with @xmath156 ) @xcite . on the other hand in case or real @xmath129 the transformation is not restricted to the plane : four points with real cross - ratio always lie on a circle . thus the map that sends @xmath157 to @xmath158 is well defined in any dimension . maps from @xmath159 to @xmath160 with cross - ratio -1 for all elementary quadrilaterals see chapter @xcite . ] serve as discretization of isothermic surfaces and have been investigated in @xcite . let @xmath161 be arbitrary . define p_k & : = & u_k -[def : dc : p ] + e^ & : = & g_k [ def : dc : q ] clearly the above definitions are not unique . @xmath162 and @xmath163 will be identified with the flaschka - manakov variables of the toda lattice hierarchy @xcite . @xmath64 will be the corresponding spectral parameter . with the above definitions at hand we are now able to state the following correspondence with the toda lattice hierarchy . denote @xmath164 define _ k : = v_k^11 + _ k : = - . [ def : dc : alphabet ] by ( [ eq : dc : generalflow ] ) and with definition ( [ def : dc : alphabet ] ) , @xmath49 and @xmath75 define a certain flow on discrete curves in @xmath0 . let @xmath59 be a lax representation matrix corresponding to the n - th toda flow in the notations as in @xcite . then the lax matrices @xmath59 of the discrete curve flow in ( [ eq : dc : vk ] ) together with the definitions ( [ def : dc : alphabet ] ) are identical to the above @xmath165 hence the compatibility equation ( [ eq : dc : comp ] ) for a flow on discrete curves in @xmath0 corresponding to the definitions ( [ def : dc : alphabet ] ) , is the compatibility equation of the n - th toda flow . [ theor : dc : major ] setting -g^-1_k-1u_k^-1(g_k-1+g_k)_k & & v_k^12 + g^-1_k-1u_k-1 ^ -1(g_k-2+g_k-1)_k-1 & & v_k^21 results in the constraint @xmath166 but this constraint is just the 22-component of the compatibility equation ( [ eq : dc : comp ] ) for general @xmath59 and the toda @xmath58 , hence it is satisfied by all @xmath59 of the toda hierarchy . hence the variables @xmath75 are well defined . likewise the second constraint obtained by setting _ k + _ k & & v_k^11 + _ k-1 - _ k-1 & & v_k^22 together with the 22-component gives the 21 component of ( [ eq : dc : comp ] ) . hence the variables @xmath49 are well defined . the 11- and 12-component are giving the toda field equations . the above matrices can be regauged into the matrices @xmath167 and @xmath168 with @xmath169 @xmath170 and @xmath171 are then the 2 by 2 matrix representation of the usual flaschka - manakov matrices @xcite . one has @xmath172 on the other hand by the definition of the matrices @xmath59 in proposition [ def : dc : gu ] one has = ( q_k - q_k+1)= v_k+1 . [ eq : dc : trace]hence q_k = tr v_k+1 . [ eq : dc : qdotandtrace ] [ rem : dc : qs ] the flow directions for a discrete curve @xmath3 is given by a specific choice of the variables @xmath49 , @xmath75 ( compare with ( [ eq : dc : generalflow ] ) ) . in the following we will choose @xmath49 , @xmath75 in such a way that the corresponding evolution for the determinants @xmath23 , and @xmath22 ( [ def : dc : gu ] ) is the evolution of the canonical variables of the toda lattice hierachy . in this section we will look at the flow directions given by the `` first '' three flows of the toda lattice hierachy . using ( [ eq : dc : generalflow ] ) define the following flow directions for @xmath3 : _ k^tl1 & = & - ( p_k + ) = - + _ k^tl1 & = & - = - . [ def : dc : albettl1]the induced flow on the determinants @xmath23 and @xmath22 ( [ def : dc : gu ] ) is with definitions ( [ def : dc : p ] ) , ( [ def : dc : q ] ) given by the first toda lattice flow . [ prop : dc : firstflow ] in accordance with theorem [ theor : dc : major ] we obtain the following compatibility matrices : l_k&= & ( g_k(p_k + ) & - + 1&0 ) + v_k&= & ( -(p_k+ ) & g_k-1 ^ -1 + - g_k-1 ^ -1 & ( p_k-1 + ) ) these matrices are wellknown @xcite and give the toda lattice equations , which are called the first flow of the toda lattice hierachy : & = & tr v_k+1 = ( p_k - p_k+1 ) + p_k&= & g_k^-2-g_k-1 ^ -2 . due to remark [ rem : dc : qs ] the variables @xmath173 evolve with @xmath174 hence @xmath175 which is the wellknown toda lattice equation . analogously the flows for the next two higher flows @xcite can be determined : define _ k^tl2 & = & - ( p_k^2 - ^2 ) -(g_k^-2 - 2g_k-1 ^ -2+g_k-2 ^ -2 ) + _ k^tl2 & = & - . [ def : dc : albettl2]the induced flow on @xmath23 and @xmath162 ( [ def : dc : gu ] ) is with definitions ( [ def : dc : p ] ) , ( [ def : dc : q ] ) given by the second flow of the toda lattice hierachy @xcite : & = & -(p^2_k+1-p^2_k+g_k+1 ^ -2-g_k-1 ^ 2 ) [ eq : dc : evgtl2 ] + p_k&= & g_k^-2(p_k+1+p_k)-g_k-1 ^ -2(p_k+p_k-1 ) @xmath176 define _ k^tl3 & = & - ( p_k^3+^3 - 2(p_k+)g_k^-1g_k-1 ^ -1 ) + & & -g_k^-2(2 p_k+p_k+1 ) + g_k-1 ^ -2(p_k-1 + 2p_k ) + _ k^tl3 & = & - ( g_k-1 ^ -2+g_k^-2+p_k^2-p_k + ^2 ) . [ def : dc : albettl3]the induced flow on @xmath23 and @xmath162 ( [ def : dc : gu ] ) is with definitions ( [ def : dc : p ] ) , ( [ def : dc : q ] ) given by the third flow of the toda lattice hierachy @xcite : & = & -[(p_k+1 ^ 3+p_k+2g_k+1 ^ -2 + 2p_k+1g_k+1 ^ -2 + p_k+1g_k^-2 ) + & & -(p_k^3+p_k-1g_k-1 ^ -2+p_kg_k^-2 + 2p_kg_k-1 ^ -2 ) ] + p_k&=&g_k^-2 ( p_k+1 ^ 2 + p_k^2 + p_k+1p_k + g_k+1 ^ -2+g_k^-2)- + & & g_k-1 ^ -2(p_k^2+p_k-1 ^ 2+p_k p_k-1+g_k-1 ^ -2+g_k-2 ^ -2 ) @xmath177 in the following we look at the reduction @xmath178 and determine the geometric shape of curves , which are invariant under the trivial toda and the discrete kdv flow . in the previous section one reduced the discrete curves in @xmath0 to socalled arclength parametrized curves , by setting @xmath179 for all @xmath180 it was conjectured that for that reduction there exist certain flow directions ( [ def : dc : betait ] ) which let the cross - ratios evolve according to flows in the volterra hierachy . on the other hand it is a fact @xcite that the toda lattice hierachy reduces to the volterra hierachy for flows with an even enumeration number if @xmath178 . indeed , looking at the above equations one sees easily that if @xmath178 then the evolution of the @xmath163 in equation [ eq : dc : evgtl2 ] is given by the volterra equation . in the same manner the third flow admits no reduction but the fourth flow would again give an equation for the @xmath163 which can be identified with a discrete version of the kdv equation @xcite . we observe that for the case @xmath178 the cross - ratio is given by @xmath181 hence as a direct consequence if @xmath182 , @xmath178 then the flow of the cross - ratios @xmath44 , as given by the flow of the 2nth toda lattice hierachy via theorem [ theor : dc : major ] satisfies the equations of the nth volterra hierachy . as can be seen by looking at the above toda lattice hierachy equations , the reduction @xmath183 is generally not compatible with the equations . it is compatible if @xmath184 ( and especially @xmath178 , which can allways be achieved by a change of @xmath64 ) , which gives the trivial evolution @xmath185 for all flows of the toda hierachy . hence the two reductions @xmath186 and @xmath178 with their corresponding volterra hierachy flows , seem to be in some kind of duality , where only the trivial reduction @xmath187 @xmath188 @xmath189 @xmath190 @xmath183 seems to lie in the intersection of the two pictures . we were interested in what kind of curves belong to this most trivial solution of the toda flows . the following proposition shows that if the variables @xmath191 and @xmath192 are constants , then @xmath3 lies on a quadric , or in other words in this case @xmath3 defines a * discrete quadric * in @xmath8 \a ) let @xmath24 , @xmath25 be fixed initial conditions for a discrete curve @xmath193 with m _ 0,_0 > & = & 1 + m _ 1,_1 > & = & 1 + m _ 0,_1 > & = & , + [ def : dc : initial1]where @xmath194 is the naive complexification of the real scalar product to @xmath0 and @xmath195 is a symmetric 2 by 2 matrix ; @xmath196 and @xmath54 is a free parameter . then @xmath195 is uniquely defined by conditions [ def : dc : initial1 ] and @xmath197 . b)define a discrete curve with the above initial conditions recursively by : @xmath198 then m _ k,_k+1 > & = & + m _ > & = & 1 k . [ prop : dc : quadric ] \a ) after a lengthy calculation m can be derived from the conditions ( [ def : dc : initial1 ] ) as m= ( y_0 ^ 2+y_1 ^ 2 - 2y_0 y_1 & ( x_0 y_1 + y_0 x_1 ) -(x_0 y_0 + x_1 y_1 ) + ( x_0 y_1 + y_0 x_1 ) -(x_0 y_0 + x_1 y_1 ) & x_0 ^ 2+x_1 ^ 2 - 2x_0 x_1 ) \b ) @xmath199 , hence by induction @xmath200 for all @xmath40 . from this it follows that @xmath201 and hence by induction @xmath202 for all @xmath40 . looking at the variables @xmath49 , @xmath75 corresponding to the first three toda flows ( e.g.([def : dc : albettl1 ] ) ) the case @xmath203 gives the following evolution on the curves : _ k = const(_k+1-_k-1 ) [ eq : dc : tangent]where the constant @xmath204 varies corresponding to the considered toda flow . a flow direction as in [ eq : dc : tangent ] along a discrete quadric @xmath3 ( as defined in proposition [ prop : dc : quadric ] ) is tangent to the ( smooth ) quadric on which the points of @xmath3 lie . we have to show that if @xmath205 then @xmath206 now |_0 m _ k(t),_k(t)>&= & m _ k,_k > + m _ k,_k > + & = & 2 ( m _ k+1,_k > - m _ k-1,_k > ) + & = & 2 ( - ) using lemma [ lem : dc : gam1 ] the proof for the second assertion works analogously . by the above , a class of curves ( namely quadrics ) was sorted out by looking at trivial solutions to the toda lattice equations . the movement of these curves under the corresponding toda flow ( tangent to the quadric ) was very geometrical and simple . in the next section a similar construction will be done . here it will be shown in the euclidian reduction ( [ subsec : dc : discreteeuclidean ] ) , that discrete curves which evolve under the discrete mkdv flow simply by a translation , define socalled discrete elastic curves . let @xmath15 be a discrete arc length parametrized curve in @xmath2 as discussed in section [ subsec : dc : discreteeuclidean ] . a discrete regular arc length parametrized curve @xmath207 @xmath208 is called _ planar elastic curve _ if it is an critical point to the functional @xmath209 the admissible variations preserve the arc length , @xmath210 and the tangents at the end points . it can be shown @xcite that the curvature of a discrete elastic curve obeys the following equation : @xmath211 for some real constant @xmath212 . we now want to know , which class of curves is invariant under the discrete mkdv flow ( up to euclidean motion and some tangential flow ) . it will turn out that that discrete elastic curves are a special case in that class . since discrete arc length parametrized curves are determined by their curvature ( [ eq : dc : curvature ] ) up to euclidean motion , it is sufficient to impose the constraint that the curvature must not change up to the changes made by the tangential flow ( [ eq : dc : kappatangential ] ) . in other words : we ask for solutions @xmath104 for which ( [ eq : dc : kappatangential ] ) is a multiple of ( [ eq : dc : discretemkdv ] ) : @xmath213 one can `` integrate '' this equation twice and get the following lemma : the curvature of a discrete curve , that evolves up to some tangential flow by euclidean motion under the mkdv flow satisfies @xmath214 for some constants @xmath212 and @xmath215 and a function @xmath15 with @xmath216 . nothing left to show . in the case @xmath217 and @xmath218 this gives the equation for planar elastic curves ( [ eq : dc : elasticcurvature ] ) . figure [ fig : dc : foursym ] shows three closed discrete generalized elastic curves . in this paper we introduced a novel view onto the one dimensional toda lattice hierarchy in terms of flows on discrete curves . in particular this view allowed us to give a geometric meaning to the trivial toda flow and the mkdv flow which are special flows within the toda lattice hierarchy . this was achieved by classifying that class of curves , which is invariant under the corresponding flow . it would be interesting to investigate discrete curves that belong to other toda flows in this sense . in fact we view our exposition rather as a starting point for a more thorough investigation of this subject . if one finds a meaningful symplectic structure on the space of curves @xmath219 , then by the novel view onto the toda lattice hierarchy this may lead to a symplectic structure for the `` vertex operators '' @xmath220 . we are currently working on that issue . the theory of discrete curves is in close connection to the theory of discrete surfaces . in addition it was already known to darboux @xcite that certain invariants on surfaces admitting a conjugate net parametrization satisfy the two dimensional toda lattice equation . it is an interesting question to see whether our approach can be transferred to discrete surfaces and the two dimensional toda lattice hierarchy @xcite . it would be interesting to see , whether our approach could also be applied to generalized toda systems @xcite . we havent thought about that yet . we like to thank ulrich pinkall for deep and interesting discussions and many inspiring ideas . we like to thank yuri suris and leon takhtajan for helpful hints . a. doliva and santini . geometry of discrete curves and lattices and integrable difference equations . in a. bobenko and r. seiler , editors , _ discrete integrable geometry and physics _ , chapter part i 6 . oxford university press , 1999 . u. hertrich - jeromin , t. hoffmann , and u. pinkall . a discrete version of the darboux transformation for isothermic surfaces . in a. bobenko and r. seiler , editors , _ discrete integrable geometry and physics _ , pages 5981 . oxford university press , 1999 .

in this we investigate flows on discrete curves in @xmath0 , @xmath1 , and @xmath2 . a novel interpretation of the one dimensional toda lattice hierarchy and reductions thereof as flows on discrete curves will be given . [ sec : dc ]

in 1937 ettore majorana realized that the dirac equation could be modified to support a new class of particles called majorana fermions ( mfs ) , with the intriguing property that these particles are their own anti - particles.@xcite mathematically , if @xmath5 is the annihilation operator for a majorana particle then @xmath6 . these exotic particles are non - abelian anyons , which means that particle exchanges are not merely accompanied by a @xmath7 for bosons or a @xmath8 for fermions that multiplies the wave function . additionally , the exchange statistics of mfs does not follow the regular anyons observed as quasiparticles in 2d systems , where the exchange operation yields a berry phase @xmath9 multiplying the wave function.@xcite thus we end up for mfs with an exotic non - abelian exchange statistics . until now no elementary particle on nature was found as a mf . there is one possibility that neutrinos might be mfs . on going experiments are attempting to verify this hypothesis.@xcite despite its origin in high energy physics , mfs came recently in the news as a quasi - particle excitation in the low - energy field of solid state physics.@xcite thus in the last few years the pursuit for devices hosting mfs has received much attention from the scientific community , in particular working with quantum computing . such a quest is due to the possibility of bounding two far apart mfs in order to define a nonlocal qubit completely immune to the decoherence effect , which is crucial for the accomplishment of a robust topological quantum computer.@xcite to this end , experimental realizations should reveal first signatures of mfs that ensure the existence of them and hence , their application as essential blocks for quantum computing . nowadays , the most promising setups for this goal lies on the superconductor based systems.@xcite for instance , it was recently measured as a mf signature a zero - bias peak in the conductance between a normal metal and the end of a semiconductor nanowire ( insb ) that is attached to a s - wave superconductor.@xcite this superconductor induces superconductivity in the insb nanowire via the proximity effect . in the presence of a magnetic field parallel to the wire it was found a peak sticked at the midgap of the nontrivial topological superconductor . this peak is washed out for zero magnetic fields or when the magnetic field is parallel to the spin - orbit field of the wire . additionally , this peak tends to disappear when temperature increases . all these features are in agreement with theoretical works that settle the ingredients necessary to have a majorana bound state ( mbs ) in a hybrid nanowire - superconductor device.@xcite however , an alternative explanations for these measurements were later proposed.@xcite additionally , mf are expected to appear in a variety of solid state systems , namely , topological superconductors@xcite and fractional quantum hall systems@xcite . moreover , mfs are theoretically predicted to appear in a half - quantum vortex of a p - wave superconductors@xcite or at the ends of supercondutor vortices in doped topological insulators.@xcite in solid state physics the main way to probe mbss is via conductance . a few experiments use tunneling spectroscopy to probe mfs as a zero - bias anomaly . there are some theoretical proposals that deal with transport through a single level quantum dot attached to a left and to a right lead and to a mbs in the end of a quantum wire.@xcite the main transport feature found for this system is a conductance peak pinned at zero - bias with an amplitude of one - half the ballistic conductance @xmath10,@xcite valid when the left and right leads couple symmetrically to the dot . we point out that in ref . [ , e. vernek _ et al . _ have found that such a value arises from the leaking of the mbs into the quantum dot . additionally , the transport in this system was investigated in the large bias regime , revealing a non - conserving current between left and right leads.@xcite in the present paper we apply the keldysh nonequilibrium green s function technique@xcite to extend these previous works to the whole bias voltage window , ranging from the zero - bias limit up to the large bias regime . so , instead of focusing only on the zero - bias anomaly , we explore the whole @xmath0 curve in the presence of a single mbs . for comparison we also show the results to the case of a regular fermionic ( rf ) zero - mode coupled to the dot . both cases ( mbs and rf ) differ appreciably along the bias voltage window , not only in the zero - bias regime . we observe , for instance , the formation of an additional plateau in the i - v curve when the dot is coupled to a mbs . additionally , it is found a slope at the characteristic @xmath0 curve equal to one - half the quantum of conductance @xmath11 when the bias voltage tends to zero , in accordance to ref . [ . we pay particular attention to the coupling asymmetry between _ left lead - quantum dot _ and _ right lead - quantum dot_. these couplings are characterized by the tunneling rates @xmath12 and @xmath13 , respectively . we investigate the cases @xmath14 and @xmath15 . cao et al.@xcite found that in the large bias regime the current is not conserved with @xmath16 or @xmath17 depending on the asymmetry factor @xmath18 . interestingly , the nonconserving feature also affects the zero - bias conductance . the zero - bias limit departs from @xmath4 when the leads couple asymmetrically ( @xmath19 ) to the dot . we have found in the zero - bias limit @xmath20 and @xmath21 or the opposite , depending on the degree of asymmetry @xmath22 . neither @xmath23 nor @xmath24 coincide with the conductance obtained via the landauer - bttiker equation in linear response regime , except for symmetric couplings ( @xmath25 ) . this indicates that a full nonequilibrium quantum transport formulation is more suitable to describe the system with majorana bound state . thermal effects are also investigated . we observe that when the temperature is large enough both mbs and rf cases become indistinguishable for any bias voltage . the paper is organized as follows . in sec . ii we present a detailed derivation of the nonlinear transport equations obtained via keldysh technique . in sec . iii we show the main results found and in sec . iv we conclude . to describe the system presented in fig . ( [ fig1 ] ) we use the hamiltonian originally proposed by liu and baranger,@xcite @xmath26 where the first term gives the free - electron energy of the reservoirs , @xmath27 , the second term is the single level quantum dot hamiltonian , @xmath28 and the third term gives the tunnel coupling between the quantum dot and the leads , @xmath29 $ ] , with @xmath30 ( left lead ) or @xmath31 ( right lead ) . the fourth term accounts for the majorana modes , and the last two terms can be understood as follows : ( i ) @xmath32 , @xmath33 and @xmath34 we have a regular fermionic ( rf ) zero - mode attached to the quantum dot and ( ii ) for @xmath35 we obtain a mbs coupled to the quantum dot . in case ( i ) the hamiltonian becomes @xmath36 while for ( ii ) we have @xmath37 where @xmath38 and @xmath39 . in the following nonequilibrium calculation we consider this last hamiltonian . in order to compare our findings with the ones obtained previously for the large bias limit , we adopt @xmath40 , where @xmath41 gives the tunnel coupling between the dot and the nearby mbs in ref . we highlight that the present spinless hamiltonian for mfs assumes a strong magnetic field applied on the whole setup of fig . ( [ fig1 ] ) , thus resulting a large zeeman splitting where the higher levels are not energetic favorable within the operational temperatures of the system . in this case , one spin component becomes completely inert and the spin degrees of freedom can be safely ignored . as a result , the coulomb interaction between opposite spins in the quantum dot is avoided and the model becomes exactly solvable . to our best knowledge , this work is the first to obtain such a solution by using green s functions in the keldysh framework . the current in the lead @xmath42 can be calculated from the definition @xmath43 , where @xmath44 is the modulus of the electron charge . @xmath45 is the total number operator for lead @xmath46 and @xmath47 is a thermodynamics average . the time derivative of @xmath48 is calculated via heisenberg equation , @xmath49 $ ] ( we adopt @xmath50 ) , which results in@xcite @xmath51,\ ] ] where @xmath52 . after a straightforward calculation the current expression can be cast into the following form @xmath53f_{\a}+g_{d}^{<}(\w)\}.\label{i}\ ] ] here @xmath54 , with @xmath55 being the density of states of the reservoir @xmath46 , and the green s functions @xmath56 , @xmath57 and @xmath58 are the retarded , advanced and lesser green s functions of the quantum dot . these green s functions can be obtained via analytic continuation of the contour - ordered green s functions @xmath59 , where @xmath60 orders the operators along the keldysh contour . since the equation of motion for @xmath61 is structurally equivalent to the chronological time - ordered green s function @xmath62,@xcite in what follows we calculate @xmath63 via equation of motion technique . taking the time derivative with respect to @xmath64 we obtain @xmath65g_{d}(t , t ' ) & = & \delta(t - t')+\sum_{k,\alpha}v_{\alpha}^{*}g_{c_{k\a}}(t , t')\nonumber \\ & & \phantom{xxxxxx}-\lambda g_{\eta_{1}}(t , t'),\label{gd}\end{aligned}\ ] ] where the additional green s functions were defined as @xmath66 and @xmath67 . calculating the time - derivative of these new green s function with respect to @xmath64 we find @xmath68g_{c_{k\a}}(t , t')=v_{\a}g_{d}(t , t'),\label{gk}\end{aligned}\ ] ] and @xmath69 observe that two new green s functions arise at this last equation , namely , @xmath70 and @xmath71 . performing once again the time - derivative with respect to @xmath64 of these two green s functions we arrive at @xmath72 and @xmath73g_{d^{\dag}}(t , t')=-\sum_{k,\alpha}v_{\alpha}g_{c_{k\alpha}^{\dagger}}(t , t')+\lambda g_{\eta_{1}}(t , t').\nonumber \\ \label{gddagger}\end{aligned}\ ] ] one more green s function appears at this last results , @xmath74 , whose equation of motion can be easily calculated , @xmath75g_{c_{k\alpha}^{\dagger}}(t , t')=-v_{\alpha}^{*}g_{d^{\dag}}(t , t').\label{gkdagger}\end{aligned}\ ] ] equations ( [ gd ] ) , ( [ gk ] ) , ( [ geta1 ] ) , ( [ geta2 ] ) , ( [ gddagger ] ) and ( [ gkdagger ] ) constitute a complete set of six differential equations . in order to reduce to only four equations we write eqs . ( [ gk ] ) and ( [ gkdagger ] ) in their integral forms@xcite @xmath76 and use them into eqs . ( [ gd ] ) and ( [ gddagger ] ) . this gives us @xmath65g_{d}(t , t ' ) & = & \delta(t - t')+\int dt_{1}\s(t , t_{1})g_{d}(t_{1},t')\nonumber \\ & & \phantom{xxxxxx}-\lambda g_{\eta_{1}}(t , t'),\label{gdint}\end{aligned}\ ] ] and @xmath73g_{d^{\dag}}(t , t ' ) & = & \int dt_{1}\s'(t , t_{1})g_{d^{\dag}}(t_{1},t')\nonumber \\ & & \phantom{xxxxxx}+\lambda g_{\eta_{1}}(t , t'),\nonumber \\ \label{gddaggerint}\end{aligned}\ ] ] where @xmath77 and @xmath78 . equations ( [ geta1 ] ) , ( [ geta2 ] ) , ( [ gdint ] ) and ( [ gddaggerint ] ) constitute our new set of four - integrodifferential equations , which can be written in a matrix form as @xmath79 \left [ \begin{array}{c } g_{d}(t , t ' ) \\ g_{\eta_{1}}(t , t ' ) \\ g_{\eta_{2}}(t , t ' ) \\ g_{d^{\dagger}}(t , t ' ) \\ \end{array } \right ] & = & \delta(t - t')\left [ \begin{array}{c } 1 \\ 0 \\ 0 \\ 0 \\ \end{array } \right ] + \int dt_{1}\left [ \begin{array}{cccc } \s(t , t_{1 } ) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \s'(t , t_{1 } ) \\ \end{array } \right ] \left [ \begin{array}{c } g_{d}(t_1,t ' ) \\ g_{\eta_{1}}(t_1,t ' ) \\ g_{\eta_{2}}(t_1,t ' ) \\ g_{d^{\dagger}}(t_1,t ' ) \\ \end{array } \right]+\nonumber\\ & & \phantom{xxxxxxxxxxxx } \left [ \begin{array}{cccc } 0 & -\lambda & 0 & 0 \\ -\lambda & 0 & i\varepsilon_{m } & \lambda \\ 0 & -i\varepsilon_{m } & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ \end{array } \right ] \left [ \begin{array}{c } g_{d}(t , t ' ) \\ g_{\eta_{1}}(t , t ' ) \\ g_{\eta_{2}}(t , t ' ) \\ g_{d^{\dagger}}(t , t ' ) \\ \end{array } \right],\end{aligned}\ ] ] or in a more compact way as @xmath80 where the matrix @xmath81 is defined according to @xmath79 \mathbf{g}(t , t')=\d(t - t ' ) \mathbf{i},\nonumber\\\end{aligned}\ ] ] with @xmath82 being the @xmath83 identity matrix , and @xmath84+\nonumber\\ & & \d(t - t')\left [ \begin{array}{cccc } 0 & -\lambda & 0 & 0 \\ -\lambda & 0 & i\varepsilon_{m } & \lambda \\ 0 & -i\varepsilon_{m } & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ \end{array } \right].\end{aligned}\ ] ] the vectors @xmath85 and @xmath86 are defined as @xmath87\phantom{xxx}\mathrm{and}\phantom{xxx}\vec{u}=\left [ \begin{array}{c } 1 \\ 0 \\ 0 \\ 0 \\ \end{array } \right].\ ] ] iterating eq . ( [ gvec ] ) we can show that @xmath88 with the dyson equation @xmath89 writing a similar equation in the keldysh contour,@xcite @xmath90 and applying the langreth s analytical continuation rules,@xcite we obtain in the frequency domain @xmath91 to the retarded green s function and @xmath92 to the lesser green s function both already in the fourier domain . the retarded and lesser components of the self - energy can be expressed as @xmath93,\ ] ] and @xmath94 has only two nonzero elements , @xmath95,\\ { { \s}}^<_{44}(\w ) & = & i[\g_l(-\w)f_l(-\w)+\g_r(-\w)f_r(-\w)],\end{aligned}\ ] ] with eqs . ( [ gr ] ) and ( [ g < ] ) we can calculate the transport properties described below . for differing @xmath41 and symmetric case @xmath96 . both mbs and rf cases are shown , as black and blue lines , respectively . the @xmath97 gives the same results for both cases , which corresponds to a transport through a single level quantum dot . for finite @xmath41 the two cases present distinct @xmath0 profiles . the rf case shows a flat @xmath0 characteristics around zero bias and then it increases when the double - dot conduction channels cross the reservoir chemical potential . contrasting , the mbs regime yields a typical slope around zero bias which turns into the @xmath4 as predicted in the literature . in panels ( b)-(c ) we show @xmath98 for both cases . while in the mbs the conductance is pinned at 0.5 @xmath99 it is zero in the rf regime . parameters : @xmath96 , @xmath100 , @xmath101 , @xmath102 , @xmath103 , @xmath97 , @xmath104 and @xmath105 , @xmath106.,height=188 ] in fig . [ fig2](a ) we compare the characteristic @xmath0 curve in the cases of a mbs and a rf attached to the quantum dot . we adopt @xmath107 as our energy scale , so the bias voltage , the energy levels , and the coupling @xmath41 will be expressed in units of @xmath107 , while the currents in units of @xmath108 , with @xmath109 being the planck s constant . for @xmath110 both results coincide and the system behaves as a single level quantum dot . for @xmath111 distinct features arise in each case . in particular , in the linear response regime , the current presents a finite slope as the bias increases for the mbs case while it is flat for the rf situation . as the bias voltage increases above the linear response regime , we observe the formation of a plateau in the current for the mbs case and then it increases further , saturating at large enough bias voltages . in contrast , for @xmath33 ( rf ) we have a single step current profile , without the formation of an intermediate plateau . for larger biases the current coincides for both cases ( rf and mbs ) . and ( e)-(f ) current difference @xmath112 against the bias voltage in energy units of @xmath107 . we consider @xmath113 ( left panels ) and @xmath114 ( right panels ) . both @xmath115 and @xmath116 present similar features against bias voltage but distinct values . in particular the slope around zero bias and the plateaus differ from each other . in order to confirm our nonequilibrium calculation we compare the high bias plateau with the ones predicted by cao _ et al.@xcite _ via the master equation technique . the different @xmath112 increases with bias and then it saturates at the value predicted in the aforementioned reference . the zero bias value of the differential conductance @xmath98 contrasts to the one obtained for the symmetric case @xmath96 . here @xmath23 and @xmath24 are not at 0.5 and they differ from each other , with @xmath117 for @xmath118 and the opposite for @xmath119 . parameters : @xmath100 , @xmath101 , @xmath102 , @xmath103 , @xmath120.,height=302 ] in fig . [ fig2](b ) we show the differential conductance ( @xmath98 ) for the currents presented in fig . [ fig2](a ) in the presence of a mbs . for @xmath110 the conductance @xmath98 is the standard lorentzian with broadening given by @xmath121 . in contrast , for @xmath122 the conductance reveals a three peaks structure , in which one of them has an amplitude of 0.5 pinned at zero - bias , in accordance to the work of liu and baranger.@xcite for the rf , though , we find @xmath98 similar to the characteristic t - shaped quantum dot geometry,@xcite where the conductance is zero for bias voltage close to zero . it is valid to note that the currents presented in fig . ( [ fig2 ] ) for both mbs and rf cases can also be obtained from the standard landauer - bttiker expression@xcite @xmath123t(\w),\label{isymmetric}\ ] ] where @xmath124(-2)\text{{im}}[g_{dd}^{r}(\w)]$ ] , which gives the following conductance in the linear response limit @xmath125[-\frac{\partial f}{\partial\w}].\label{glinear}\ ] ] this symmetric expression is only true for charge conserving systems where @xmath126 . this is always the case when @xmath33 ( rf ) . however , for @xmath127 ( mbs ) this is valid in the symmetric coupling regime ( @xmath96 ) only . when @xmath128 the left and right currents depart from each other , and consequently the result obtained from eq . ( [ isymmetric ] ) differs from both @xmath115 and @xmath116 obtained via eq . ( [ i ] ) . in order to explore the coupling asymmetries ( @xmath129 ) in the transport , we plot separately in fig . ( [ fig3 ] ) both @xmath115 and @xmath116 for @xmath130 ( mbs ) , and their corresponding @xmath98 profiles for two asymmetry factors @xmath113 ( left panels ) and @xmath114 ( right panels ) . it is clear from the plot that the system does not conserve current ( @xmath131 ) . for larger enough bias voltages the currents @xmath115 and @xmath116 attain different plateaus , which are confirmed by the analytical results , recently derived by cao _ _ via born - markov master equation technique , namely,@xcite @xmath132,\label{cao1}\\ i_{r } & = & \frac{\g_{l}\g_{r}}{\g}[1+\frac{4(y-1)\l'^{2}}{\g^{2}+4(\e_{d}^{2}+\e_{m}^{2}+2\l'^{2})}].\label{cao2}\end{aligned}\ ] ] these large bias limiting values are plotted in fig . ( [ fig3 ] ) as dotted lines . looking at the zero - bias limit , one may note that the slopes of @xmath115 and @xmath116 vs. @xmath133 deviate from each other with @xmath134 for @xmath113 and @xmath135 for @xmath114 . the differential conductance @xmath23 and @xmath24 clearly show the difference of the slopes at zero bias , with @xmath136 and @xmath137 for @xmath113 and @xmath138 and @xmath139 for @xmath114 . this contrasts with the symmetric case , where both conductances are at 0.5 , as predicted by liu and baranger.@xcite and @xmath24 at zero - bias against @xmath22 in the presence of a mbs . @xmath23 is larger than @xmath24 for small @xmath22 , they attain the same value at @xmath96 and then @xmath24 turns greater than @xmath23 as @xmath22 becomes higher than one . for comparison we show @xmath98 obtained via linear response theory . parameters : @xmath100 , @xmath101 , @xmath102 , @xmath103 , @xmath120.,height=188 ] , @xmath104 and @xmath140 . both mbs and rf cases are shown . for small temperatures both cases differ , however as @xmath141 increases the two regimes tend to the same results . in particular , the characteristic signature @xmath142 for a mbs is washed out as the temperature enhances . parameters : @xmath96 , @xmath100 , @xmath102 , @xmath103 , @xmath120.,height=188 ] in fig . 3(e)-(f ) we plot the difference @xmath112 against bias voltage . it is clear that in the nonequilibrium regime the current is not conserved with @xmath143 for @xmath118 and the opposite for @xmath119 . as the bias voltage enlarges and all the conduction channels ( three channels in the presence of a mbs ) become inside the conduction window , the difference @xmath112 attains the plateau predicted by eqs . ( [ cao1])-([cao2 ] ) . in fig . ( [ fig4 ] ) we show how @xmath98 evolves with @xmath22 at the zero - bias limit . both @xmath23 ( black ) and @xmath24 ( blue ) are shown . as a matter of comparison we also plot @xmath98 obtained via the standard linear response expression , eq . ( [ glinear ] ) . while all results coincide for the symmetric case ( @xmath96 ) , they all differ for @xmath128 . finally , fig . ( [ fig5 ] ) shows @xmath144 vs. @xmath133 curves and the corresponding differential conductance for different temperatures in the symmetric case ( @xmath96 ) . both the mbs and rf cases are presented . as the temperature increases the curves for both regimes tend to become smoother , as expected due to the smearing out of the fermi function around the chemical potential of the electronic reservoirs . in particular , opposite behavior between mbs and rf are seen at the slope of the @xmath0 curve around zero bias . while in the mbs the slope is suppressed for increasing @xmath141 , it is amplified in the rf case for @xmath145 . this behavior can be clearly seen in the differential conductance @xmath98 at zero bias . remarkably , both mbs and rf cases coincide for large enough temperature and the @xmath0 presents a linear profile . we have studied nonequilibrium quantum transport in a quantum dot attached to two leads and to a localized majorana bound state . our approach , based on the keldysh nonequilibrium green s function , allows us to study transport through the whole bias voltage range , starting at the zero - bias limit and moving up to the large bias regime . previous works investigate separately only the zero - bias or the large bias limit . to the best of our knowledge this is the first work that covers the entire bias window . our findings include the characteristic slope of @xmath4 in the @xmath0 profile at the zero - bias limit when the two leads couple symmetrically to the quantum dot , in accordance to the prediction of ref . however , in the asymmetric case ( @xmath128 ) we find a deviation from this slope , with @xmath20 and @xmath21 or the opposite , depending on the degree of asymmetry . we also compare both @xmath23 and @xmath24 with the conductance obtained via eq . ( [ glinear ] ) . they all agree only for symmetric coupling ( @xmath96 ) . this indicates that a full nonequilibrium quantum transport formulation is required to a better description of the system . our results were also compared to those expected when a quantum dot is coupled to a rf zero - mode , instead of a mbs . the two cases ( rf and mbs ) differ appreciably in the entire bias - voltage range , not only at the zero bias regime . additionally , we observe the formation of a plateau in the @xmath0 profile for intermediate bias voltages when the dot is coupled to a mbs . this plateau is not seen in the rf case . we also note that when the reservoirs temperature is large enough the two cases coincide , thus becoming indistinguishable via transport measurements if the dot is attached to a rf level or to a mbs . this work was supported by the brazilian agencies cnpq , capes , fapemig , fapespa , vale / fapespa , eletrobras / eletronorte and prope / unesp .

we investigate theoretically nonequilibrium quantum transport in a quantum dot attached to a majorana bound state . our approach is based on the keldysh green s function formalism , which allows us to investigate the electric current continuously from the zero - bias limit up to the large bias regime . in particular , our findings fully agree with previous results in the literature that calculate transport using linear response theory ( zero - bias ) or the master equation ( high bias ) . our @xmath0 curves reveal a characteristic slope given by @xmath1 in linear response regime , where @xmath2 is the ballistic conductance @xmath3 as predicted in phys . rev . b 84 , 201308(r ) ( 2011 ) . deviations from this behavior is also discussed when the dot couples asymmetrically to both left and right leads . the differential conductance obtained from the left or the right currents can be larger or smaller than @xmath4 depending on the strength of the coupling asymmetry . in particular , the standard conductance derived from the landauer - bttiker equation in linear response regime does not agree with the full nonequilibrium calculation , when the two leads couple asymmetrically to the quantum dot . we also compare the current through the quantum dot coupled to a regular fermionic ( rf ) zero - mode or to a majorana bound state ( mbs ) . the results differ considerably for the entire bias voltage range analyzed . additionally , we observe the formation of a plateau in the characteristic @xmath0 curve for intermediate bias voltages when the dot is coupled to a mbs . thermal effects are also considered . we note that when the temperature of the reservoirs is large enough both rf and mbs cases coincide for all bias voltages .

we have recorded the flight of a fly during take off and landing using digital high speed photography . it is shown that the dynamics of flexible wings are different for these procedures . during this observation fly flew freely in a big box and it was not tethered .

in this fluid dynamics video , we demonstrated take off and landing of a fly . the deformation of wings is in focus in this video .

since early fifties @xcite the physics of manganites challenge our current understanding of transition - metal oxides , and define both theoretical and experimental research problem that involves charge , spin , lattice and orbital degrees of freedom . recently in a modern systematic experimental studies a very rich phase diagram ( see , for example , ref . ) depending on the doping concentration , temperature and pressure was obtained in the doped manganese oxides with perovskite structure @xmath3 ( where @xmath4 is trivalent rare - earth and @xmath5 is divalent alkaline ion , respectively ) . at different doping concentration a full variety of magnetically ordered states such as antiferromagnetic ( afm ) insulator , ferromagnetic ( fm ) metal and charge ordered ( co ) insulator were observed . many efforts have been made by theoreticians to understand it based on various models and approaches . historically , double exchange ( de ) model@xcite was the basic one . in this model @xmath6-electrons are localized , whereas the @xmath7-electrons are mobile and use @xmath8 @xmath9-orbitals as a bridge between @xmath10 ions . the hopping of itinerant electrons together with a very strong on - site hund s coupling drives core spins to align parallel . qualitatively de model gave appropriate interpretation of the phase diagram at the doping range @xmath11 where fm metallic behavior was observed . one of the subtle aspects of the perovskite manganites is the charge ordered state observed in almost all such compounds at half - doping.@xcite a direct evidence of the co state in half - doped manganites has been provided by the electron diffraction for @xmath12.@xcite similar observations have also been reported for @xmath13 @xcite and in @xmath14.@xcite co state is characterized by an alternating @xmath15 and @xmath16 ions arrangement in @xmath17 plane with the charge stacking in @xmath18-direction . in co state these systems show an insulating behavior with a very peculiar form of afm spin ordering . the observed magnetic structure is a ce - type and consists of quasi one - dimensional ferromagnetic zig - zag chains coupled antiferromagnetically . in addition , these systems show @xmath19/@xmath20 orbital ordering . another noteworthy observations were done by studying @xmath21 crystals with controlled one - electron bandwidth . as already mentioned above at half - doping @xmath22 has a co ce - type insulating state . however , by substitution @xmath23 with @xmath24 leading to the increase of the carrier bandwidth , one induces the collapse of the co insulating state , and the a - type metallic state with @xmath25 orbital ordering is realized in @xmath13.@xcite the coexistence of the a - type spin ordered and ce - type spin / charge ordered states has been detected in the bilayer @xmath26 @xcite and three - dimensional @xmath27.@xcite these results indicate the competition between the metallic a - type @xmath25 orbital ordering and the insulating ce - type @xmath28 orbital ordering at half - doping and demonstrate the importance of the magnetic , charge and orbital order coupling in these compounds . in recent publications it was shown that the de anisotropy resulted from the orbital degeneracy with the peculiar @xmath7 transfer amplitudes is important and to be a key point in explaining the various types of afm ordering.@xcite until now most of the theoretical studies of co state were done in the framework of the one - orbital model ignoring the double degeneracy of @xmath7 orbitals.@xcite the detailed mean - filed analysis of phase diagram of one - orbital de model in the presence of both on - site and inter - site coulomb terms has been given in ref .. it has been shown , that in the vicinity of half - doping the double - exchange gain of energy is considerably suppressed by the inter - site coulomb interaction that favors charge - ordered state . recently , the co state within the two - orbital model has been investigated by the projection perturbation techniques combined with the coherent state formalism and by monte carlo simulations in refs . and , respectively . in ref . the origin of co has been attributed to the effective particle - hole interaction and co state with c - type of the spin ordering at @xmath29 has been obtained . the authors of ref . have shown that the experimentally observed charge , spin , and orbital ordering could be stabilized due to jahn - teller phonons . in the present paper we investigate the role of the orbital degeneracy in the co state based on the two - orbital de model including the intersite coulomb interaction . we adopt the mean - field ( mf ) approximation to derive the ground state phase diagram in the two - orbital model , and compare it to that of the corresponding one - orbital model . we argue that the orbital degeneracy together with the peculiar @xmath7 transfer amplitude has a drastic effect on the phase diagram and is important in obtaining the realistic magnetic / charge / orbital ordering observed in half - doped manganites . the paper is organized as follows : in the next section the model hamiltonian is presented and the mean - field scheme is formulated . the ground state phase diagrams of the one- and two - orbital models are derived and compared in sec.iii . sec.iv summarizes our main results . in the appendix the canonical transformation diagonalizing the mf hamiltonian and the resulted band structure of various magnetic phases are presented . we start with the two orbital ferromagnetic kondo lattice model supplemented by the intersite coulomb repulsion @xmath30- j_{\rm h}\sum_{i}{\bf s}_i{\bf \sigma}_i \nonumber\\ & + & j\sum_{\langle ij\rangle}{\bf s}_{i}{\bf s}_{j } + v\sum_{\langle ij\rangle}n_{i}n_{j}-\mu\sum_{i}n_{i}. \label{1}\end{aligned}\ ] ] the first term of eq.([1 ] ) describes an electron hopping between the two @xmath31 orbitals of the nearest neighbor ( nn ) mn - ions . the orbitals @xmath32 and @xmath33 correspond to @xmath34 and 2 , respectively . due to the shape of the @xmath7 orbitals , their hybridization is different in the three cubic directions that leads to direction dependent hopping with the anisotropic transfer matrix elements @xmath35 given by @xmath36 the second term in eq.([1 ] ) describes the hund s coupling between the spins of localized @xmath6- electrons @xmath37 and the itinerant @xmath31 electrons with spin @xmath38 . the superexchange ( se ) interaction of localized spins between the nn sites is given by @xmath0 , @xmath1 represents the inter - site coulomb repulsion of @xmath31 electrons , @xmath39 is the particle number operator and @xmath40 is the chemical potential . the effect of the on - site coulomb interaction that is not included in our model hamiltonian will be discussed later . we study the hamiltonian ( [ 1 ] ) within the mf approximation , which is set up by introducing the order parameter for static charge - density wave of the form @xmath41 , with @xmath42 being the electron density and @xmath43 . further , we treat localized spin subsystem classically and assume a strong hund s coupling @xmath44 . in this limit one may take the local spin quantization axis parallel to @xmath6-spins and in the rotated bases retain only `` spin - up '' components of the mobile electrons . then the transfer integral between the nn sites is modified through relative angle of the @xmath6-spins at the @xmath45 and @xmath46 sites as @xmath47 , where @xmath48 is the relative angle of the @xmath6-spins . we consider the following magnetic phases that competes : i ) ferromagnetic configuration ( f - type spin ordering ) with @xmath49 ( @xmath50 and @xmath51 are the angels between the neighboring spin in @xmath52-plane and @xmath18-direction , respectively , ii ) layer - type antiferromagnetic configuration ( a - type spin ordering ) the local spins are parallel in the planes and antiferromagnetically aligned between the neighboring planes , that corresponds to @xmath53 and @xmath54 . iii ) chain - type antiferromagnetic configuration ( c - type spin ordering ) the local spins are parallel in the straight chains and antiferromagnetically coupled between the chains @xmath55 and @xmath56 . iv ) neel - type antiferromagnetic configuration ( g - type spin ordering ) with all spins being antiparallel @xmath57 . v ) ce - type spin ordering with zig - zag ferromagnetic chains coupled antiferromagnetically . as a result we come to the following mf hamiltonian : @xmath58- \delta\sum _ { i}e^{i{\bf q}{\bf r}_{i}}n_{i } \nonumber\\ & -&\mu\sum_{i}n_{i } + 2(d-3)js^2 n~ , \label{3}\end{aligned}\ ] ] where @xmath59 , @xmath60 for 3-dimensional cubic lattice , and @xmath61 is dimensionality of the magnetic order ( @xmath62 and @xmath63 , respectively for g- , c- , a- , and f- type spin ordering ) . in eq.([3 ] ) the zero of the energy is chosen in such a manner that the se energy vanishes in the fm state . to obtain phase diagram we need to compare the free energies of all possible magnetic configurations . in order to incorporate the role of orbital degeneracy , first we consider the one orbital model ignoring the double degeneracy of @xmath31 orbitals . retaining only the one orbital per mn - ion and assuming the isotropic transfer amplitude , the electronic part of the mf hamiltonian in @xmath64-space is written as : @xmath65 with @xmath66 the above hamiltonian ( [ 4 ] ) is easily diagonalized by the following canonical transformation : @xmath67 with @xmath68^{\frac{1}{2}},\ ; % \\ \\ % { \displaystyle v_{{\bf k}}=\frac{1}{\sqrt{2}}\left [ 1+\frac{\tilde { t}_{\bf k}}{\varepsilon_{\bf k } } \right]^{\frac{1}{2 } } , \nonumber\\ % { \displaystyle \varepsilon_{{\bf k}}&=&\sqrt{\tilde{t}_{\bf k}^2+\delta^2 } \ ; . % } % \end{array } % \right . \label{7}\end{aligned}\ ] ] in terms of the @xmath69-operators , the one particle hamiltonian reads @xmath70 at half - filling the chemical potential lies inside the gap ( @xmath71 ) and recalling that @xmath59 we receive a self - consistent equation for the order parameter @xmath72 in fig.[f1 ] the overall behavior of the order parameter @xmath73 as a function of @xmath74 is presented for various magnetic configuration . since the wave vector summation in the right hand side of eq.([9 ] ) diverges in the limit @xmath75 there exist a nontrivial solution even at @xmath76 and hence a transition from homogeneous to co state is continuous . we also note that @xmath73 diminishes exponentially with increasing the bandwidth ( see fig.[f1 ] ) indicating that the transition between the homogeneous and the co state is a result of the competition between the kinetic and the electrostatic energy . by comparing the free energies of different magnetic configuration we obtain the phase diagram as shown in fig.[f2 ] . at small @xmath1 , with increasing @xmath0 the system , starting from the f - co phase , first enters to the c - co phase and then to the g - co state . since the gain in the magnetic energy when the system moves from a- to c - phase is larger then the gain in the kinetic energy in c to a transition the a - co phase is absent in this part of phase diagram . with increasing of @xmath1 at @xmath77 the co gap in c - co phase overcomes that one in a - co phase that results in opening of small window of a - co phase in the phase digram . we also note that with increasing of @xmath1 the se coupling needed to stabilize the afm configuration decreases since the bandwidth effect is overshadowed when the gap becomes larger . the different magnetic structures in fig.[f2 ] are separated by the first - order boundaries . there is a jump in the charge order parameter across the phase boundaries , since the value of order parameter @xmath73 depends on the effective bandwidth and hence on the underlying magnetic structure , as can be seen from fig . 2 . to describe the effect of orbital degeneracy , we consider the mf hamiltonian ( [ 3 ] ) with anisotropic hopping amplitude . in the momentum space the electronic part of the hamiltonian reads as : @xmath78 d_{{\bf k}\alpha } ^{\dagger}d_{{\bf k}\beta}- \delta\sum _ { { \bf k},\alpha}d_{{\bf k},\alpha}^{\dagger } d_{{\bf k}+{\bf q } , \alpha } \label{10}\end{aligned}\ ] ] with @xmath79 and @xmath80 . the diagonalization of this hamiltonian ( see appendix ) leads to the four band model . in the case of f- and a - type spin ordering and at the filling corresponding to one electron per two mn - ions ( half - doped case ) the gap is not opened at the fermi surface , and the chemical potential moves down with the lower two bands . we solve the gap equation self - consistently with one for the chemical potential . as it seen in fig.[f3 ] , the transition to the charge ordered state is not continuous and there exists a critical value @xmath81 above which the ordered state is favorable ( @xmath82 and @xmath83 for f- and a - type spin ordering , respectively ) . as for the @xmath84-type spin ordering , there is no difference between the one and two orbital models in this sense . the existence of the additional orbital only introduce the localized level which is empty at the filling we consider . the above hamiltonian ( 10 ) describes the f- , a- , c- , and g - type spin ordering on an equal footing . however the presence of additional orbital degree of freedom , with the peculiar anisotropic transfer amplitudes @xmath85 [ see eq.([2 ] ) ] results in the anisotropic de interaction @xcite and may lead to the stabilization of ce spin ordering . let us consider one zig - zag with two ferromagnetic bonds alternated in @xmath86 and @xmath87 directions ( fig.[f4 ] ) . the corner and middle sites are denoted by @xmath88 and @xmath89 , respectively , and the unit cell is given by four nonequivalent atoms . + for further discussions it is convenient to adopt the following bases of the @xmath31 orbitals at nonequivalent sites : @xcite @xmath90 , @xmath91 , and @xmath92 @xmath93 on @xmath94 , @xmath95 and @xmath96 , @xmath97 sites , respectively [ see , fig.[f4 ] ] . in the new bases the transfer matrix elements are given by @xmath98 between @xmath94@xmath95 ( @xmath96@xmath97 ) and @xmath95@xmath96 ( @xmath97@xmath94 ) nn sites , respectively . as a result the zig - zag chain is modeled as a dimerized one with the alternating hopping amplitude and can be described by the following hamiltonian @xmath99 \right.\nonumber\\ & -&(\mu+\delta ) \{a_{i\alpha } ^{\dagger}a_{i\alpha}+ { \bar a}_{i\alpha } ^{\dagger}{\bar a}_{i\alpha}\ } -(\mu-\delta ) \left.\{b_{i\alpha } ^{\dagger}b_{i\alpha}+ { \bar b}_{i\alpha } ^{\dagger}{\bar b}_{i\alpha}\}\right\ } \label{13}\end{aligned}\ ] ] 2 where @xmath45 runs along the zig - zag and denotes the number of unit cell . diagonalization of the above hamiltonian ( see appendix ) leads to the complicated band structure consisting of bonding and antibonding bands , @xmath100 , and nonbonding states @xmath101 . due to the topology of the zig - zag structure , only the directional @xmath102 orbitals at @xmath103 sites give the input in the low energy bonding state [ see appendix for a details ] . while for the @xmath104 sites both two orthogonal orbitals do contribute . therefore , the orbital degeneracy is removed on the middle site sublattice and the carriers on this sublattice will occupy the directional orbitals , leading to the polarized orbital state with @xmath28 orbital ordering . we also emphasize , that at half - doping the bonding band is full and the system is a band insulator even in the absence of the charge ordering . the onset of the charge ordering renormalizes the gap to higher value . the behavior of the charge order parameter is depicted in fig.3 . the transition to charge ordered state takes place at @xmath105 , that is lower then that one for a- and f- type spin ordering . a smaller value of intersite coulomb repulsion is needed to introduce the charge ordered state in the state with the lower dimension of ferromagnetic component . the ground state phase diagram of the two - orbital model is given in fig.5 . the phases with different magnetic structures are separated by the first - order boundaries ( soled lines in fig . the transition from the charge disordered to the charge ordered version of the given magnetic structure is continuous , there is no jump in the charge order parameter during the transition which does not change the symmetry of underlying magnetic structure ( the corresponding second - order boundaries are denoted by the dashed lines in fig.5 ) . at small value of @xmath1 with increasing of se coupling the system goes through four magnetic phases f , a , ce , and g consequently . due to the additional orbital degree of freedom , unlike to the one orbital model , there exist a finite phase space of a - type spin ordering in the small @xmath1 region @xmath106 . however , for @xmath107 , when the charge ordering is introduced in the a phase , and hence the bandwidth effect is suppressed , ce - co phase wins and the a - co phase is never realized in the phase diagram of the present model . in summary , we have investigated the ground state phase diagram of the one- and two - orbital extended double exchange models within the mean filed approximation at half - doping . in the case of the one - orbital model the mf theory predicts the continuous phase transition to the charge ordered state due to the intersite coulomb interaction @xmath1 . in the two - orbital model the character of the transition is changed by the introducing the nonzero critical value of @xmath1 . while the transition to charge ordered state with given magnetic structure is continuous , there is jump in of charge order parameter across the phase boundaries between the states with different symmetry of magnetic structure . depending on the intersite coulomb interaction @xmath1 and and superexchange coupling @xmath0 different types of spin ordering ( f- , a- , g - type ) accompanied by the charge ordering may take place in the ground state of the one - orbital model . the presence of orbital degeneracy with the peculiar anisotropic @xmath7 transfer amplitude introduces the new magnetic state , ce - type spin ordering , in the phase diagram of the two - orbital model . the c - type spin ordering is never achieved within this model due to its instability against the effective `` dimerization '' and formation of the zig - zag ferromagnetic order . the alternation of the ferromagnetic bonds in @xmath108 directions leads to the alternation of the hopping amplitude . as a result the bare band is splited into bonding and antibonding states and the `` dimerization '' gap opens on the fermi surface at half - doping . the ce - type spin ordered state is accompanied with @xmath28 orbital ordering , originated from the topology of the zig - zag structure . the ce - type spin / charge state also wins the co state with a - type afm ordering and there exists the phase boundary between the ce spin / charge ordered state and charge disordered ( cd ) a - type state [ see fig.5 ] . the experimentally detected competition between a - cd and ce - co states in the half - doped manganites could indicate that the parameters of the system are close to this phase boundary . therefore the small change of the bandwidth may drive the system from one to other state . that is also suggested by the huge oxygen isotope effect observed in la - pr compounds . @xcite a substitution of @xmath109 by the isotope @xmath110 narrows the carrier bandwidth due to the polaronic effect and as a result the electrostatic energy might overcome the kinetic energy and the charge ordered insulating state might be established . we would like to point out that there exist other physical factors and ingredients not included in present treatment that may stabilize the particular state and modify the phase diagram ( quantum nature of the spins , coupling to the jan - teller phonons , and on site coulomb interaction ) . as it was recently discussed in ref . the interorbital on - site coulomb interaction @xmath111 could be responsible for the experimentally observed charge ordered state . in the ce structure the orbital degeneracy is removed on the sublattice composed by the middle sites and the low energy state corresponds to the directional orbitals ( see appendix of the present paper ) . if charge carriers occupy the corner sites there will be a positive contribution from the @xmath111 term to the system energy . while , if only middle site sublattice is occupied this positive contribution disappears since the onsite coulomb term has the zero matrix element in this case . therefore , @xmath111 term in ce - type spin ordered state acts as a source of the charge ordering , but in a rather deferent way then the intersite coulomb repulsion . since the middle sites are stacked in @xmath18-direction @xmath111 induces experimentally observed @xmath112 charge ordering instead of the wigner crystal type co favored by the intersite coulomb interaction considered here . nevertheless , we believe that the main effect of the orbital degeneracy on the interplay of different type of magnetic ordering in charge - ordered state is at least qualitatively captured by the present treatment indicating the crucial importance of the electronic state degeneracy on the phase diagram . financial support by the the intas program , grants no 97 - 0963 and no 97 - 11066 , are acknowledged . diagonalization of the hamiltonian ( [ 10 ] ) can be done by two subsequent canonical transformation . first , we diagonalize the free part of the hamiltonian by introducing the new fermionic operators @xmath113 with @xmath114 one finds the effective hamiltonian of the form @xmath115 . \label{a3}\end{aligned}\ ] ] further we perform the transformation similar to ( [ 6 ] ) and introduce the new sets of fermionic operators as @xmath116 where @xmath117^{\frac{1}{2}}\!\!\!,\;\;\ ; { \bar v}_{{\bf k}i}=(-1)^{i } \frac{\text{sgn}(\varepsilon^{12}_{\bf k})}{\sqrt{2}}\left [ 1+\frac{\varepsilon _ { { \bf k}i}}{e_{{\bf k}i } } \right]^{\frac{1}{2}}\nonumber \\ e_{{\bf k}i}&=&\sqrt{\varepsilon _ { { \bf k}i}^2+\delta^2 } , \hspace*{2 cm } i=1,2.\end{aligned}\ ] ] as a result we get the following four band hamiltonian @xmath118\right.\nonumber\\ & + & \left.e_{{\bf k}2}[a^{\dagger}_{{\bf k}3}a_{{\bf k}3 } -a^{\dagger}_{{\bf k}4}a_{{\bf k}4}]\right\}-\mu \sum\limits_{{\bf k},i=1}^{4}a^{\dagger}_{{\bf k}i}a_{{\bf k}i}. \label{a4}\end{aligned}\ ] ] @xmath119 \right.\nonumber\\ & -&(\mu+\delta ) \{a_{{\bf k}\alpha } ^{\dagger}a_{{\bf k}\alpha}+ { \bar a}_{{\bf k}\alpha } ^{\dagger}{\bar a}_{{\bf k}\alpha}\}-(\mu-\delta ) \left.\{b_{{\bf k}\alpha } ^{\dagger}b_{{\bf k}\alpha}- { \bar b}_{{\bf k}\alpha } ^{\dagger}{\bar b}_{{\bf k}\alpha}\ } \right\ } \label{a5}\end{aligned}\ ] ] 2 where @xmath120 and the lattice constant is set to be unity ( @xmath121 see fig.[f4 ] ) . the above hamiltonian can be simplified by transforming to new fermion operators @xmath122 , @xmath123 the operators @xmath124 , are obtained from @xmath125 by the same transformation as in eq.([a6 ] ) . one finds the effective hamiltonian of the form @xmath126 , where @xmath127 \ ; . \label{a8}\end{aligned}\ ] ] further , we consider only the first part given by ( [ a7 ] ) , generalization of the diagonalization procedure on @xmath128 is straightforward . with the help of the explicit expression of the hopping amplitude matrix we first transform operators @xmath129 to @xmath130 by the fermionic @xmath131 transformation with @xmath132 where @xmath133 . in terms of the new operators the effective hamiltonian is written as @xmath134 where @xmath135 . the transformed hamiltonian is already diagonal in the subspace given by the effective orbital index @xmath136 . however it mixes the fermionic fields @xmath137 and @xmath138 at @xmath139 . the hybridization part of the hamiltonian ( [ a10 ] ) can be diagonalized following the same rout as in eqs . ( [ a1],[a2 ] ) . as a result we come to the following diagonal form : @xmath140\right.\nonumber\\ & + & \left.\delta[\beta^ { \dagger}_{{\bf k}3}\beta_{{\bf k}3 } -\beta^{\dagger}_{{\bf k}4}\beta_{{\bf k}4}]\right\}-\mu \sum\limits_{{\bf k},i=1}^{4}\beta^{\dagger}_{{\bf k}i}\beta_{{\bf k}i } , \nonumber\\ { \bar h}&=&h\left[\beta\rightarrow { \bar \beta},{\cal e } \rightarrow { \bar { \cal e}}\right ] \label{a11}\end{aligned}\ ] ] where @xmath141 and @xmath142 , @xmath143 , and @xmath144 are given by the linear combination of @xmath145 and @xmath146 . let us now briefly comment the obtained band structure . it is given by the bonding and antibonding bands @xmath147 ( [ a12 ] ) and nonbonding bands @xmath148 in between . the nondirectional orbital from the middle sites ( @xmath149 and @xmath150 orbitals on @xmath94 and @xmath96 type sites , respectively ) is completely decoupled from the other states giving rise to the nonbolding band with energy @xmath151 corresponding to @xmath152 in eq.([a11 ] ) . the other nonbonding band with energy @xmath153 corresponding to the states @xmath154 is given by the linear combination of the degenerate orbitals on the corner sites . the state orthogonal to this nonboning state hybridizes with the directional orbital on the middle sites ( @xmath19 and @xmath20 orbitals on @xmath94 and @xmath96 type sites , respectively ) leading to the bonding and antibonding bands . as follows from the above discussion , the particular geometry of the zig - zag structure not only leads to the opening of `` dimerization''-like gap in the spectrum , but also removes the orbital degeneracy on the middle sites . the energy of the directional orbital is lowered due to the hybridization and hence delocalization , while its orthogonal orbital remains local . e. o. wollan , w. c. koehler , phys . b. * 100 * , 545 , ( 1955 ) . p. schiffer , a. p. ramirez , w. bao , and s .- w . cheong , phys . lett . * 75 * , 3336 , ( 1995 ) ; s .- w . cheong and h.y.hwang , in : colossal magnetoresistance oxides , ed . y.tokura , gordon & breach , monographs in cond.matt . science , ( 1998 ) ; a. p. ramirez , j. phys . matter * 9 * , 8171 , ( 1997 ) . c. zener , phys . rev . * 82 * , 403 , ( 1951 ) ; p. w. anderson , h. hasegawa , phys . rev . * 100 * , 675 , ( 1955 ) ; p. g. de gennes , phys . rev . * 118 * , 141 , ( 1960 ) . p. g. radaelli , d. e. cox , m. marezio , s - w . cheong , p. e. schiffer , and a. p. ramirez , phys . lett * 75 * , 4488 , ( 1995 ) . chen and s .- w . cheong , phys . 76 * , 4042 , ( 1996 ) . y.tomioka , a. asamitsu , y. moritomo , h. kuwahara , and y. tokura , phys . lett . * 74 * , 5108 , ( 1995 ) . h. kuwahara , y. tomioka , a. asamitsu , y. morimoto , and y.tokura , science * 270 * , 961 , ( 1995 ) . h. kawano , r. kajimoto , h. yoshizawa , y. tomioka , y. kuwahara , and y. tokura , phys . rev 78 * , 4253 , ( 1997 ) . m. kubota , h. yoshizawa , y. morimoto , h. fujioka , k. hirota , and y. endoh , cond - mat/9811192 . l. p. gorkov and v. z. kresin , jetp lett . * 67 * , 985 , ( 1998 ) . j. van der brink and d. i. khomskii , phys . lett . * 82 * , 1016 , ( 1999 ) . l. sheng and c. s. ting , phys . b * 57 * , 5265 , ( 1998 ) . j. d. lee and b. j. min , phys . rev . b * 55 * , r14713 , ( 1997 ) . s. yunoki , j. hu , a. l. malvezzi , a. moreo , n. furukawa , and e. dagotto , phys . rev . lett . * 80 * , 845 , ( 1998 ) . s. k. mishra , r. pandit , and s. satpathy , phys . rev . b * 56 * , 2316 , ( 1997 ) , j. phys . cond . matter * 11 * , 8561 , ( 1999 ) . s. d. shen and z. d. wang , cond - mat/9906126 . s. yunoki , t. hotta and e. dagotto , to appear in phys . ( cond - mat/9909254 ) . i.soloviev and k. terakura , phys . lett . * 83 * , 2825 , ( 1999 ) . n. a. babushkina , l. m. belova , o. yu . gorbenko , a. r. kaul , a. a. bosak , v. i. ozhogin , aand k. i. kugel , nature ( london ) * 391 * , 159 , ( 1998 ) . a. m. balagurov , v. yu . pomjakushin , d. v. sheptyakov , v. l. aksenov , n. a. babushkina , l. m. belova , a. n. taldenkov , a. v. inyushkin , p. fischer , m. gutmann , l. keller , o. yu . gorbenko , and a. r. kaul , phys . b * 60 * , 383 , ( 1999 ) . j. van der brink , g.khaliulin and d.i.khomskii , phys . lett . * 83 * , 5118 , ( 1999 ) .

phase diagram of half - doped perovskite manganites is studied within the extended double - exchange model . to demonstrate the role of orbital degrees of freedom both one- and two - orbital models are examined . a rich phase diagram is obtained in the mean - filed theory at zero temperature as a function of @xmath0 ( antiferromagnetic ( afm ) superexchange interaction ) and @xmath1 ( intersite coulomb repulsion ) . for the one - orbital model a charge - ordered ( co ) state appears at any value of @xmath1 with different types of magnetic order which changes with increasing @xmath0 from ferromagnetic ( f ) to afm ones of the types a , c and g . the orbital degeneracy results in appearance of a new ce - type spin order that is favorable due to opening of the `` dimerization '' gap at the fermi surface . in addition , the co state appears only for @xmath2 for f and ce states while c - type afm state disappears and a - type afm state is observed only at small values of @xmath1 as a charge disordered one . the relevance of our results to the experimental data are discussed . 2

according to common belief , @xcite in order to obtain the differential cross - section @xmath0 for the scattering on hermitian scalar spherically symmetric potential @xmath1 , it is sufficient to have the on - shell scattering amplitudes @xmath2 , appeared as a coefficients near outgoing or incoming spherical waves in the first order of asymptotic expansion for the scattering wave functions @xmath3 , when @xmath4 , for @xmath5 , @xmath6 , @xmath7 , @xmath8 : @xmath9 is the total ( elastic ) cross - section . but what about the term @xmath10 in eq . ( [ 1 _ ] ) ? according to common belief @xcite it does not change both the definitions ( [ 3 _ ] ) . on the other hand the recent investigations @xcite of ( anti- ) neutrino processes at a short distances from the sources reveal a possible significance of such corrections for the event rate . since the macroscopic parameter @xmath11 has very specific meaning in the field theoretical consideration of ref.@xcite , it seems natural and convenient to elucidate this question at first for nonrelativistic quantum mechanical scattering . in the next sections a closed formula and simple recurrent relation for coefficients of asymptotic expansion of wave functions @xmath3 in all orders of @xmath12 are obtained in terms of the on - shell scattering amplitudes @xmath2 only . this asymptotic expansion together with some additional hints gives an exhaustive answer to above question , which unlike a common belief reveals the necessity of corresponding modification of the first definition ( [ 3 _ ] ) for differential cross - section . at the same time for the second definition ( [ 3 _ ] ) of total cross - section and for unitarity condition with the optical theorem all that asymptotic integer power corrections exactly disappear . in order to understand the nature of asymptotic expansion , we have to remember some properties @xcite of wave functions and amplitodes ( [ 2 _ ] ) . the functions @xmath3 , as a solution of schrdinger equation for the energy @xmath13 satisfy integral lippman - schwinger equation : @xmath14 where the differential vector operators in the spherical basis @xmath15 have the following properties : @xmath16 , \label{7}\end{aligned}\ ] ] and the well known representation at point @xmath17 for the spherical wave coming from point @xmath18 , as a free schrdinger 3-dimensional green function,@xcite is used : @xmath19 which for @xmath20 satisfies the inhomogeneous equation : @xmath21 the index of power is defined in the sense of analytical continuation with a small real negative addition @xcite as : @xmath22 , which is not written here and below but is everywhere assumed . the following lemma is in order . for @xmath23 , @xmath24 , @xmath25 , @xmath26 , with operator @xmath27 ( or @xmath28 ) defined by eqs . ( [ 3])([7 ] ) and positively defined operator @xmath29 , so that @xmath30 is positively defined : @xmath31 } { s!(\mp 2ikr)^s}\right\ } e^{\mp ik({{\vec{\rm n}}}\cdot{{\vec{\rm x}}})}. \label{18 } \end{aligned}\ ] ] the expression ( [ 18_0 ] ) is formal operator rewrite for @xmath32 of the usual multipole expansion of the green function@xcite ( [ 8 ] ) via the corresponding expansion of the plane wave,@xcite given also by formulas ( 8.533 ) , ( 8.534 ) of ref.@xcite , with the formally introduced instead @xmath33 , but not really appeared self - adjoint operator @xmath34 : @xmath35 here the spherical functions @xmath36 and legendre polynomials @xmath37 for @xmath38 or @xmath39 , as eigenfunctions of self - adjoint operator ( [ 6 ] ) , ( [ 7 ] ) on the unit sphere , satisfy well known orthogonality , parity , completeness and other conditions @xcite ( [ 21])([a_5 ] ) with delta - function @xmath40 on the unit sphere . the solutions @xmath41 , @xmath42 of free radial schrdinger equation : @xmath43\frac{\psi_{l\,0}(kr)}{r}= l(l+1)\frac{\psi_{l\,0}(kr)}{r } , \label{26}\ ] ] are defined by macdonald @xmath44 and bessel @xmath45 functions,@xcite ( [ a_1])([24_0 ] ) , that for integer @xmath33 i.e. half integer @xmath46 are reduced to elementary functions : @xmath47 the function @xmath44 ( [ a_1 ] ) is a whole function@xcite of @xmath48 , that is the reason , why the above introduced in ( [ 18_0 ] ) well defined operator @xmath49 does not appear explicitly . the expansion ( [ 18 ] ) is the known asymptotic series for expression ( [ 18_0 ] ) at the large @xmath11 as an infinite asymptotic version @xcite of the sum ( [ 24 ] ) for the case of arbitrary non - integer @xmath33 , @xmath50 , supplemented by observation @xcite for the product : @xmath51 , \label{28}\ ] ] which thus may be factored out from the sum over @xmath33 as an operator product in the right hand side of eq . ( [ 18 ] ) , converting @xcite the expression ( [ 19 ] ) into the expansion ( [ 18 ] ) . _ remark . _ the operator @xmath52 in eq . ( [ 18 ] ) by the same success may be replaced by operator in square brackets of the left hand side of eq . ( [ 26 ] ) or by the same brackets with interchanging @xmath53 . let the potential @xmath54 have finite first absolute moment and decrease for @xmath55 faster than any power of @xmath56 , then the integral @xmath57 in eq . ( [ 5 _ ] ) for sufficiently large @xmath11 admits asymptotic expansion defined by the on - shell scattering amplitudes @xmath2 only . this expansion has asymptotic sense @xcite even though the potential @xmath54 in eq . ( [ 4 _ ] ) has a finite support : @xmath58f^\pm(k{{\vec{\rm n}}};{{\vec{\rm k } } } ) , \;\mbox { or : } \label{30 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! h^\pm_s(k{{\vec{\rm n}}};{{\vec{\rm k}}})=\frac{{\cal l}_{{{\vec{\rm n}}}}-s(s-1)}{s } h^\pm_{s-1}(k{{\vec{\rm n}}};{{\vec{\rm k } } } ) , \quad { { \vec{\rm k}}}=k{{\vec{\omega } } } , \label{30_0}\end{aligned}\ ] ] and is equivalent to infinite reordering of its asymptotic multipole expansion : @xcite @xmath59 as usual on - shell scattering amplitude @xcite . supposing at first the @xmath54 has a finite support at @xmath60 , one can directly substitute for @xmath61 the expression ( [ 18_0 ] ) into representation ( [ 5 _ ] ) of @xmath57 with the following result after interchanging the order of differentiation and integration for fourier transform ( [ 32_2 ] ) , well justified also @xcite for the asymptotic series ( [ 18 ] ) : @xmath62 } { s!(\mp 2ikr)^s}\right\}f^\pm(k{{\vec{\rm n}}};{{\vec{\rm k } } } ) . \label{33}\end{aligned}\ ] ] this is exact asymptotic expansion ( [ 29 ] ) with the coefficients @xmath63 , defined by eq . ( [ 30 ] ) . however that is not the case for the potential @xmath54 with infinite support . estimating it for @xmath64 as @xmath65 with arbitrary finite @xmath66 , the two pieces of correction that should be added , are easily estimated as : @xmath67 , \label{34_1 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \mbox{with the finite norm \cite{ng , ar } : $ ||\psi||=\underset{{{\vec{\rm x}}}}{\sup}|\psi({{\vec{\rm x}}})|$ of functions } \;\psi^\pm_{{\vec{\rm k}}}({{\vec{\rm x } } } ) . \label{34_2}\end{aligned}\ ] ] due to the arbitrariness of @xmath66 for that corrections the asymptotic expansion conserves its form ( [ 29 ] ) , ( [ 33 ] ) but acquires an additional asymptotic meaning @xcite in comparison with expansion ( [ 18 ] ) . indeed , according to the partial wave decomposition ( [ 32_1 ] ) of scattering amplitude @xmath68 , the expression ( [ 33_0 ] ) is a formal operator rewrite of the asymptotic multipole expansion @xcite ( [ 31 ] ) of @xmath57 . unlike its exact value in eq . ( [ 5 _ ] ) the expansion ( [ 31 ] ) due to eqs . ( [ 26 ] ) , ( [ 24 ] ) is a solution of free schrdinger equation like eq . ( [ 9 ] ) with @xmath69 . for @xmath70 , when @xmath71 , both schrdinger equations ( [ 4 _ ] ) and ( [ 9 ] ) coincide for @xmath61 , and the asymptotical relations ( [ 31 ] ) , ( [ 33_0 ] ) become exact expressions due to the convergence in the usual sense @xcite for @xmath61 of expansion ( [ 31 ] ) as well as expansions ( [ 19 ] ) , ( [ 20 ] ) . at the same time the expansions ( [ 29 ] ) , ( [ 33 ] ) conserve their asymptotic meaning acquired according to lemma 1 . the potential @xmath54 implied above , for the case with infinite support has only finite effective radius @xcite and provides for partial waves slower behavior @xcite at @xmath72 , like @xmath73 , @xmath74 for yukawa - type potential . that is enough for convergence of partial wave decompositions ( [ 32 ] ) , ( [ 32_1 ] ) , but can not provide the convergence of multipole expansion ( [ 31 ] ) , which itself acquires now the asymptotical meaning . its infinite reordering : ( [ 31 ] ) @xmath75 ( [ 29 ] ) , ( [ 32 ] ) given here simply `` displaces '' this asymptotical meaning from the summation over angular momentum @xmath76 to the always asymptotical summation over the integer powers of @xmath12 , coefficients of which are well defined now by derivatives ( [ 30 ] ) of scattering amplitude with respect to @xmath77 or by convergent partial wave decompositions ( [ 32 ] ) . so , all coefficients are observable . _ from the estimations ( [ 34_0])([34_2 ] ) it is also clear @xcite that already the standard asymptotic ( [ 1 _ ] ) implies for the potential at least @xmath78 . more generally this estimations mean that for @xmath79 with @xmath55 the asymptotic expansion ( [ 33 ] ) is applied till @xmath80 . thus , whole further consideration is possible only for @xmath54 defined in above conditions of theorem 1 . in order to make careful analysis of cross - sections the following lemma 2 is useful . the function @xmath81}$ ] as a distribution on the space of infinitely smooth functions @xmath82 on the unit sphere @xmath83 parametrized by ( [ 4 ] ) has the following exact operator representation for @xmath39 . let s @xmath84}\,{\cal h}({{\vec{\rm n}}})\equiv \int\limits^1_{-1}\!dc\,e^{ikr(1-c)}\,\overline{\cal h}(c)= \label { s_0 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! = \int\limits^1_{-1}\!dc\left[\delta(1-c)-e^{2ikr}\delta(1+c)\right ] \left(-ikr+\partial_c\right)^{-1}\overline{\cal h}(c ) . \label { s_1}\end{aligned}\ ] ] using @xmath85 the result is obtained by integration by parts on @xmath77 infinite number of times , supposing the operator in ( [ s_1 ] ) as a formal series of powers of differential operator @xmath86 . the well known standard relation @xcite of the first order on @xmath56 here corresponds to @xmath87 . now let s consider the elementary flux of nondiagonal current @xmath88 through a small element of spherical surface @xmath89 , for @xmath5 , @xmath90 , @xmath7 , and @xmath91 , @xmath92 according to ( [ 3])([6 ] ) . the total flux through any surface is zero because the current is conserved @xcite due to eq . ( [ 4 _ ] ) : @xmath93 , \qquad \left(\vec{\nabla}_{{{\vec{\rm r}}}}\cdot{\vec{\rm j}}_{{{\vec{\rm q}}},{{\vec{\rm k}}}}({{\vec{\rm r}}})\right)=0 , \label { s_2 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! r^2 d\omega({{\vec{\rm n}}})\left({{\vec{\rm n}}}\cdot{\vec{\rm j}}_{{{\vec{\rm q}}},{{\vec{\rm k}}}}({{\vec{\rm r}}})\right)=r^2 d\omega({{\vec{\rm n } } } ) \frac 1{2i}\left[\left(\psi^+_{{\vec{\rm q}}}({{\vec{\rm r}}})\right)^*\overset{\leftrightarrow}{\partial}_r \psi^+_{{\vec{\rm k}}}({{\vec{\rm r}}})\right]\longmapsto \label { s_3 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \longmapsto r^2 d\omega({{\vec{\rm n}}})\,\frac{k}2\ , e^{ikr({{\vec{\rm n}}}\cdot({{\vec{\omega}}}-{{\vec{\rm v}}}))}({{\vec{\rm n}}}\cdot({{\vec{\omega}}}+{{\vec{\rm v } } } ) ) + \frac { d\omega({{\vec{\rm n}}})}{2i}\cdot \label { s_4 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \cdot\left[\overset{*}{f}{}^+(k{{\vec{\rm n}}};k{{\vec{\rm v } } } ) \left(\chi_{\overset{\leftarrow}{\lambda}{\!}_{{{\vec{\rm n}}}}}(ikr ) \overset{\leftrightarrow}{\partial}_r \chi_{\overset{\rightarrow}{\lambda}{\!}_{{{\vec{\rm n}}}}}(-ikr)\right)f^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})\right]- \frac { d\omega({{\vec{\rm n}}})}{2i}\cdot \label { s_5 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \cdot\left\{\left(e^{ikr[1-({{\vec{\rm n}}}\cdot{{\vec{\rm v}}})]}\ , e^z\left[z({{\vec{\rm n}}}\cdot{{\vec{\rm v}}})+1-z\frac d{dz}\right ] \chi_{\overset{\rightarrow}{\lambda}{\!}_{{{\vec{\rm n}}}}}(z)f^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})\right)_{z\,=\,0-ikr } \!\!\!- \right . \label { s_6 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \left . -\biggl({{\vec{\rm v}}}\rightleftharpoons{{\vec{\omega}}}\biggr)^*\right\ } , \label { s_7}\end{aligned}\ ] ] where for the wave function @xmath94 the expressions ( [ 5 _ ] ) and ( [ 33_0 ] ) were used for sufficiently large @xmath11 . the arrows point out the directions of action for the operators @xmath95 and so on from lemma 1 , that in fact are directions for the operators @xmath96 ( [ 6 ] ) , ( [ 7 ] ) and so on . integration of separate terms over solid angle @xmath97 here gives the following interesting results . for the flux of incoming plane waves ( [ s_4 ] ) , since @xmath98 , one has @xcite @xmath99 since angular momentum ( [ 7 ] ) is self - adjoint operator on the unit sphere , and since wronskian@xcite @xmath100 , @xmath101 , then ignoring arrows of @xmath49 one gets : @xmath102= \label { s_9}\\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! = k \int\ ! d\omega({{\vec{\rm n}}})\overset{*}{f}{}^+(k{{\vec{\rm n}}};k{{\vec{\rm v}}})f^+(k{{\vec{\rm n}}};k{{\vec{\omega } } } ) . \label { s_10}\end{aligned}\ ] ] this is the full scattered flux from ( [ s_5 ] ) as the right hand side of unitarity condition , @xcite but now with taking into account all possible power asymptotic corrections . it is clear that the same result follows from partial wave decomposition ( [ 32_1 ] ) with the help of eq . ( [ a_2 ] ) ( clf . ( [ s_24 ] , ( [ s_25 ] ) below ) . the lines ( [ s_6 ] ) , ( [ s_7 ] ) reflect the interference between incoming and scattered fluxes , which , according to lemma 2 for the first exponential of ( [ s_6 ] ) , takes place only in forward and backward directions . note that any averaging over @xmath11 due to rapidly oscillating exponent @xmath103 eliminates @xcite the second term with backward contribution in eq . ( [ s_1 ] ) . with this elimination and definition ( [ s_00 ] ) the line ( [ s_6 ] ) for @xmath104 recasts : @xmath105 \chi_{\overset{\rightarrow}{\lambda}{\!}_{{{\vec{\rm n}}}}}(z)f^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})\right)_{z\,=\,0-ikr}. \label { s_11 } \end{aligned}\ ] ] by moving the operator from denominator into the exponential for @xmath106 : @xmath107 after simple commutations , one obtains for ( [ s_11 ] ) @xmath108 with @xmath109 where it is possible . for the arbitrary term of partial wave decomposition ( [ 32_1 ] ) the scattering amplitude here is effectively replaced by legendre polynomial : @xmath110 , that immediately replaces @xmath111 , thus permiting all the remaining @xmath112- integration and @xmath86- differentiations in the closed form by using the relations ( [ a_3 ] ) , ( [ a_6 ] ) , and the wronskian @xmath113 , wherefrom for : @xmath114 it follows : @xmath115 thus , the contribution of the lines ( [ s_6 ] ) , ( [ s_7 ] ) into the full flux in accordance with the left hand side of unitarity condition @xcite becomes equal to : @xmath116= -4\pi\,im\,f^+(k{{\vec{\rm v}}};k{{\vec{\omega } } } ) , \label { s_19}\ ] ] with taking into account now all possible power asymptotic corrections . it is clear that the case @xmath117 in eqs . ( [ s_10 ] ) , ( [ s_19 ] ) corresponding to optical theorem @xcite is not also changed by these corrections . moreover , because the operator of angular momentum square ( [ 7 ] ) is reduced for this case to @xmath118 , the answer ( [ s_18 ] ) may be checked directly from ( [ s_11 ] ) on operator level up to a few first orders on @xmath12 . the fate of differential cross - section becomes now more complicated . defined @xcite as the ratio of scattered elementary flux ( [ s_5 ] ) over the incoming flux ( [ s_4 ] ) for @xmath117 it still contains the power corrections defined by ( [ 33_0 ] ) , ( [ 33 ] ) of theorem 1 : @xmath119= \label { s_20 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! = |f^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})|^2-\frac 1{kr}\,im\left[\overset{*}{f}{}^+(k{{\vec{\rm n}}};k{{\vec{\omega } } } ) \overset{\rightarrow}{\cal l}{}_{{\vec{\rm n}}}f^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})\right]+\frac 1{4(kr)^2}\cdot \label { s_21 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \cdot\left\{\left|\overset{\rightarrow}{\cal l}{}_{{\vec{\rm n}}}f^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})\right|^2 - re\left[\overset{*}{f}{}^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})\overset{\rightarrow}{\cal l}{}^2_{{\vec{\rm n}}}f^+(k{{\vec{\rm n}}};k{{\vec{\omega}}})\right]\right\}+o\left(\frac 1{r^3}\right ) . \label { s_22 } \end{aligned}\ ] ] in terms of partial wave decomposition ( [ 32_1 ] ) with corresponding phase shifts @xmath120 , for @xmath121 , @xmath122 , @xmath123 , @xmath124 , it reads : @xmath125 , \label { s_24 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \mbox { where : } \ ; \left[\frac{(\chi_l(ikr)\overset{\leftrightarrow}{\partial}_r \chi_j(-ikr))}{2ik}\right]= 1-\frac{\delta_{jl}}{2ik}\int\limits^\infty_r\!\frac{dr}{r^2}\chi_l(ikr)\chi_j(-ikr))= \label { s_25 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! = 1-\frac{\delta_{jl}}{2ikr}-\frac{\delta^2_{jl}}{8(kr)^2}+ o\left(\frac 1{r^3}\right ) . \qquad \mbox { ( see also ( \ref{a_7 } ) ) . } \label { s_25_0}\end{aligned}\ ] ] in fact the power corrections in this two - fold series appear only for @xmath126 . they contain only real or imaginary parts of the products @xmath127 and automatically disappear for @xmath128 in total cross - section . since the born approximation for amplitudes @xmath129 or @xmath130 is real @xcite for real potential , it is not enough to obtain first order correction @xmath131 , corresponding to eqs . ( [ s_21 ] ) , ( [ s_25_0 ] ) to be non zero . for the case of mutual scattering of identical particles @xcite with spin @xmath132 one faces symmetrical or antisymmetrical scattering wave functions , amplitudes and corresponding cross - sections , generalizations of which are straightforward . since one has instead of ( [ 5 _ ] ) , ( [ 33_0 ] ) @xmath133 . \label { s_28 } \end{aligned}\ ] ] of course the partial wave decompositions like ( [ 32_1 ] ) , ( [ s_24 ] ) contain now only even @xmath76 for @xmath134 and only odd @xmath76 for @xmath135 . the differential cross - sections for scattering of nonpolarized identical particles are defined by the usual way @xcite as : @xmath136 with the well known @xcite probabilities @xmath137 . using the operator expansion of free green function for helmholtz equation a full asymptotic expansion on inverse integer power of distance is obtained for the wave function of potential scattering . it is shown how the power corrections affect the definition of differential cross - section but are exactly cancelled in total cross - section , unitarity relation and optical theorem , the validity of above is thus essentially expanded . it is worth to note that all obtained corrections are defined by observable on - shell amplitude or partial phase shifts . on the other hand the real observation of this dependence involves reevaluation of the phase shifts extracted previously from differential cross - section without such finite distance corrections . in spite of the asymptotic expansion by its nature has no sense as infinite sum , the obtained asymptotic expansions for the wave function and cross - section have sense for any finite order of @xmath12 if potential @xmath1 has a finite support or decrease for @xmath138 faster than any power of @xmath139 . otherwise the maximal order of their validity is governed directly by the potential according to remark to theorem 1 . so , following the authors of ref.@xcite we again can conclude : `` therefore the physics under consideration seems to comply naturally with the mathematical requirements '' . [ [ section ] ] the following relations for spherical functions and legendre polynomials are necessary @xcite for @xmath140 , with @xmath83 , @xmath141 parametrized by eq . ( [ 4 ] ) and lemma 1 , and @xmath77 may be equal to any of values @xmath142 : @xmath143 bessel and macdonald functions@xcite in definitions ( [ 26 ] ) , ( [ 24 ] ) are defined by the relations with @xmath144 , as : @xmath145 , \label{24_1 } \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \mbox{with : } \ ; \int\limits^\infty_0\ ! dr\ , \psi_{j\,0}(kr)\psi_{j\,0}(qr)=\frac{\pi}2\delta(q - k ) . \label{24_0 } \end{aligned}\ ] ] by making use of ( [ 24 ] ) and ( [ a_5 ] ) for integer @xmath33 and @xmath146 one finds : @xmath147 for integral ( [ s_25 ] ) the eq . ( [ 24 ] ) also gives : @xmath148 authors thank v. naumov , d. naumov , and n. ilyin for useful discussions . d. v. naumov , v. a. naumov , d. s. shkirmanov , _ neutrino flavor transitions in quantum field theory and generalized grimus - stockinger theorem , in rep . on 41-st itep int . winter school of phys . _ , ( moscow , russia , 2013 ) . ( http://www.itep.ru/ws/2013/lec/yss/feb15/feb15/5-prezentaciya_itep.pdf )

the possibly new very fine quantum mechanical effect is discussed in this letter . it is shown that in potential scattering the differential cross - section in fact nontrivialy depends on the scattering center distance , but this dependence is eliminated by integration on the solid angle for total cross - section , unitarity relation and optical theorem , the validity of above is thus essentially expanded .

due to its importance as a base material for the fabrication of optoelectronic devices in the blue and ultraviolet spectral region , gan has been extensively studied in recent years . the comprehensive study of material and device properties@xcite has recently been complemented by an increasing number of studies concerning surface structures@xcite and associated growth mechanisms , in particular for growth by plasma - assisted molecular - beam epitaxy ( pambe ) on the ga - polar ( 0001 ) surface.@xcite one of the fundamental results of these studies is that gan growth by pambe ought to be carried out under ga - rich conditions in order to obtain a smooth surface morphology and optimized material properties . at low growth temperatures , however , this leads to ga accumulation and droplet formation , which is detrimental to the gan epilayer quality.@xcite as a consequence , it was thought that optimum gan growth conditions must be as close as possible to ga / n stoichiometry . it has been recently observed that such ga accumulation can be prevented when growing gan at high temperatures and small ga excess fluxes.@xcite in particular , it has been shown that , at high growth temperatures , a wide range of ga fluxes exists , for which a finite amount of excess ga is present on the gan surface whose quantity is independent of the value of the ga flux.@xcite such conditions may provide a `` growth window '' for gan pambe , i.e. a region , where the growth mechanisms and the surface morphology are independent of fluctuations of ga flux and growth temperature.@xcite however , the quantitative description of a gan `` growth diagram '' , which describes the ga surface coverage during growth as a function of ga flux and growth temperature , has not yet been achieved : the results on the temperature dependence of the critical excess ga flux at the onset of ga droplet formation are contradictory , yielding activation energies of 2.8ev ( ref . ) and 4.8ev ( ref . ) , respectively . this suggests that the underlying mechanisms of ga accumulation are not yet understood . to address this discrepancy , we have performed ga adsorption measurements on ( 0001 ) gan . we discuss the results in the framework of a lattice - gas growth model for adsorption , which is based on _ ab initio _ calculated parameters . this model explains the origin of the apparently contradicting parameters derived from previous experimental studies@xcite and gives a consistent description . the adsorption experiments were performed in a meca2000 molecular - beam epitaxy chamber equipped with a standard effusion cell for ga evaporation . the chamber also contains a rf plasma cell to provide active nitrogen for gan growth . the pseudo - substrates used were 2@xmath1 m thick ( 0001 ) ( ga - polarity ) gan layers grown by mocvd on sapphire . the substrate temperature @xmath2 was measured by a thermocouple in mechanical contact to the backside of the molybdenum sample holder and shielded from direct heating . prior to all experiments , a 100 nm thick gan layer was grown under ga - rich conditions on the pseudosubstrates to remove the influence of a possible surface contamination layer . ga fluxes @xmath3 have been calibrated to ga effusion cell temperatures by reflection high - energy electron diffraction ( rheed ) intensity oscillations during n - rich gan growth at a substrate temperature of @xmath4@xmath5c . in these conditions , the growth rate is actually proportional to the impinging ga flux . it is sound to assume that the ga adatom sticking coefficient is unity at such a low substrate temperature , which permits an absolute calibration of the ga flux _ in terms of the gan surface site density_. the ga surface coverage was assessed by analyzing the specularly - reflected rheed intensity by a method described in refs . and . this method uses the oscillatory transients in specular rheed intensity , which are observed during ga adsorption and desorption on / from ( 0001 ) gan surfaces . it has been shown that the duration of these transients can be qualitatively related to the amount of adsorbing or desorbing ga.@xcite in general , the relation between intensity and ga coverage is unknown . although tempting , the interpretation of these electron reflectivity transients in terms of rheed oscillations is not obvious . furthermore , it must been noted that the shape of the transients ( albeit not their duration ) depends on the diffraction conditions , notably the incidence angle . of course , the modeling of electron reflection would allow to directly relate the rheed intensity to the ga coverage ( and the surface structure ) but this is beyond the scope of this work . however , the total duration of the transients occurring during ga desorption can be used to qualitatively estimate the amount of ga adsorbed on the surface . in ref . , an indirect method has been used to draw limited _ quantitative _ information from the desorption transients ; below we will demonstrate a fully quantitative calibration relating the ga desorption transient time to absolute ga surface coverages . this allows us to circumvent the problem that the rheed intensity can not in general be easily related to adatom coverage . the experimental procedure is thus as follows : to assess the ga quantity present after ga adsorption for different impinging ga fluxes , the ga flux has been interrupted after a fixed adsorption time . the subsequent variation of the specular rheed intensity due to ga evaporation under vacuum has been recorded . specular rheed intensity during ga desorption from a ( 0001 ) gan surface . beforehand , ga adsorption has been carried out at ga fluxes @xmath3 as indicated . for @xmath6ml / s , the desorption transients correspond to equilibrium adsorption , i.e. they do not change as a function of the previous adsorption time . for @xmath7ml / s , this does not hold and the desorption transients depend on the adsorption time ( here 1min ) . the substrate temperature is @xmath8@xmath5c.,width=302 ] figure [ fig : or ] shows the variation of the specular rheed intensity after 1min of ga adsorption ( substrate temperature @xmath8@xmath5c , ga fluxes @xmath3 as indicated ) and subsequent interruption of the ga flux ( at @xmath9 ) . we observe oscillatory transients during ga desorption from the ( 0001 ) gan surface . to quantify the desorption process , we define a desorption time @xmath10 as the time interval between the shuttering of the ga flux and the last inflection point in rheed intensity ( fig . [ fig : or ] ) . the desorption time depends on the amount of ga present on the gan surface after adsorption.@xcite the fundamental finding is that for ga fluxes below @xmath11 monolayers ( ml)/s , the desorption transients and thus the desorption time @xmath10 are independent of the previous adsorption time , i.e. of the amount of nominally impinged ga . this is consistent with the results in ref . and is visualized in fig . [ fig : sat ] , which shows @xmath10 as a function of the amount of nominally impinged ga @xmath12 ( defined as the product of the ga flux and the adsorption time ) . the derivative of this curve gives the ( coverage and flux dependent ) ga sticking coefficient . data are shown in fig . [ fig : sat ] for a substrate temperature of @xmath8@xmath5c and two different ga fluxes : for @xmath13ml / s , the desorption time ( and hence the amount of adsorbed ga ) monotonously increases until it saturates after @xmath14ml . at larger @xmath12 , the adsorption has reached equilibrium and the coverage remains constant . ga desorption time as a function of the amount of nominally impinged ga , @xmath12 , for two different ga fluxes , as indicated . @xmath8@xmath5c.,width=321 ] ga desorption time as a function of impinging ga flux @xmath3 after equilibrium has been attained for regions 1 and 2 , and after 1min of ga adsorption in region 3 . the substrate temperature is @xmath8@xmath5c.,width=321 ] for a higher ga flux of @xmath15ml / s , we observe the same behavior for @xmath16ml , but thereafter , we find no saturation . instead , a continuous increase of the desorption time is observed , corresponding to ga accumulation on the gan surface . a more detailed analysis finds that the desorption transients for @xmath17ml / s correspond to finite equilibrium ga surface coverages . for higher ga fluxes , no finite equilibrium coverages exist and ga will infinitely grow and finally form macroscopic droplets on the surface . these results of fig . [ fig : or ] are summarized in fig . [ fig : isotherme ] , which shows the ga desorption time related to the amount of adsorbed ga as a function of the impinging ga flux at a constant substrate temperature @xmath8@xmath5c . it can be regarded as a ga adsorption isotherm . we can discriminate three different regions:(1 ) an s - shaped increase of the ga coverage for @xmath18ml / s , ( 2 ) a constant ga coverage up to @xmath19ml / s , independent of the ga flux , and ( 3 ) ga accumulation and no finite equilibrium ga coverages for higher @xmath3 . it is worth noting that the transitions between the three regimes are discontinuous within the experimental precision of 1@xmath5c of the ga effusion cell ( @xmath20ml / s for fluxes around 0.5ml / s ) . in particular , no intermediate equilibrium coverages have been found between regimes 1 and 2 . therefore , the transition fluxes between the different regimes are well defined . ga adsorption phase diagram indicating the ga surface coverage as a function of impinging ga flux @xmath3 and substrate temperature @xmath2 . the definition of regions 13 follows fig . [ fig : isotherme ] . the inset shows an arrhenius plot of the data.,width=321 ] [ cols="<,<,^,^,^,^,^",options="header " , ] to fully assess the thermodynamics of the adsorption process , the variation of the adsorption isotherm has to be known as a function of substrate temperature . in the following we will restrict ourselves to the study of the variation of the transition fluxes between the different regimes . the result is plotted in fig . [ fig : ddp ] . we see that the transition fluxes vary exponentially with substrate temperature . the inset shows an arrhenius plot of the data , yielding activation energies of @xmath21ev and @xmath22ev for the @xmath23 and @xmath24 transition , respectively . these values , the prefactors , and the corresponding values for the growth phase diagram in refs . ( @xmath23 corresponds to @xmath25 and @xmath24 corresponds to @xmath26 ) and ( discussing the @xmath24 transition only ) are summarized in tab . [ tab : fitresults ] and will be discussed in sec . the adsorption isotherm in fig . [ fig : isotherme ] is given in terms of the ga desorption time , which is only _ qualitatively _ related to the amount of adsorbed ga . for a fully _ quantitative _ treatment of ga adsorption , one must know the dependence of the ga desorption rate @xmath27 on the ga surface coverage @xmath28 . this would allow the modeling of the ga re - evaporation , as the desorption rate is given by @xmath29 with the initial condition @xmath30 , which denotes the amount of ga adsorbed in equilibrium conditions , i.e. , before ga desorption sets in . after the time interval @xmath10 , the ga coverage becomes zero . the knowledge of @xmath10 would thus allow us to compute @xmath31 if @xmath32 was known ( which it is not ) . this requirement can be circumvented by considering that in equilibrium , the impinging ga flux @xmath3 must exactly balance the evaporation rate , hence @xmath33)$ ] . integrating eq . ( [ eq : rateeq ] ) , taking the first derivative with respect to @xmath10 , and using the above substitution leads to @xmath34 this expression allows the computation of @xmath31 from @xmath10 as a function of @xmath3 ( which is known from the experimental data ) and can be evaluated numerically . note that @xmath31 only depends on the derivative of @xmath35 , which means that @xmath10 does not necessarily have to denote the very end of ga adsorption but can be taken after any time interval , as long as it is well - defined in the rheed signal . any further ga desorption after the end of the chosen time interval will lead to a constant offset @xmath10 , which does not contribute to @xmath31 in eq . ( [ eq : modeladsorptionisotherm ] ) . calibrated ga adsorption isotherm at @xmath8@xmath5c using eq . ( [ eq : modeladsorptionisotherm ] ) . the data are derived from fig . [ fig : isotherme ] . as the method can only be applied to equilibrium coverages , only regions 1 and 2 are represented.,width=321 ] using eq . ( [ eq : modeladsorptionisotherm ] ) , we can now calibrate the ga adsorption isotherm in fig . [ fig : isotherme ] . since the relation @xmath36 implies steady - state conditions , only the data in regions 1 and 2 can be treated . applying eq . ( [ eq : modeladsorptionisotherm ] ) to the data in region 3 does not lead to the amount of ga adsorbed after a finite time but to incorrect values because under these conditions @xmath37 . since experimental data are available only for @xmath38ml / s , the isotherm has been extrapolated by an exponential for smaller fluxes . the contribution of the interval @xmath39ml / s to the overall integral in eq . ( [ eq : modeladsorptionisotherm ] ) is 0.04ml . this is only a minor correction , suggesting that the specific form of the extrapolation function is not important . the result of the calibration is plotted in fig . [ fig : calib ] . we observe that in region 1 , the ga coverage increases from almost zero to a value of 0.98ml , close to 1ml . the coverage then increases abruptly to a value of 2.5ml in region 2 . typical systematical errors can be estimated to be of the order of @xmath40ml . what is the detailed structure of such adsorbed ga films ? it must be kept in mind that the ga fluxes have been calibrated by gan rheed oscillations and are hence given in terms of the gan surface site density . a gan coverage of 1ml thus indicates the adsorption of a ga adatom on each gan site . this suggests that ga adsorbs in region 1 as a coherent ( pseudomorphic ) adlayer . at the @xmath23 transition , the ga coverage increases by about 1.5ml , i.e. by more than a pseudomorphic adlayer . this compares favorably to the laterally - contracted bilayer model , which has been calculated to be the most stable structure of ga - rich ( 0001 ) gan surfaces.@xcite it consists of two adsorbed ga adlayers on top of a ga - terminated gan surface . the first layer is found to be pseudomorphic to the gan surface but the second one has an in - plane lattice parameter of 2.75 , about 13.8% smaller than that of gan ( 3.189 ) . the second layer thus contains 1.3ml of ga in terms of the gan surface site density , in good agreement with the experimental value of 1.5ml . * ( a ) * rheed pattern ( azimuth @xmath41 ) of a ( 0001 ) gan surface with a ga bilayer present at @xmath42@xmath5c . the white arrows indicate additional streaks due to the ga film . * ( b ) * rheed intensity profile in the @xmath43 direction evidencing the additional rheed streaks due to the ga bilayer.,width=302 ] the formation of a laterally - contracted bilayer structure in regime 2 is further corroborated by the observation of supplementary streaks in the rheed pattern after ga adsorption in regime 2 and rapid quenching of the sample down to substrate temperatures below about @xmath44@xmath5c [ see fig . [ fig : rheed](a ) ] . at higher temperatures , the supplementary streaks are too weak to be detected unambiguously . the lattice parameter corresponding to these streaks is found to be @xmath45 by a fit using pseudo - voigt functions and assuming the bulk lattice parameter for the gan layer . this value compares very favorably to that obtained by the _ ab inito _ calculations ( 2.75).@xcite combined with the adsorption results , this strongly suggests that adsorption in region 2 leads to the formation of a laterally - contracted ga bilayer . the shape of the oscillation transients in fig . [ fig : or ] suggests that , in region 3 , ga droplets are formed on top of this ga bilayer . this ga de - wetting transition may indicate that the attractive interaction energy of a ga adatom with the surface is maximum in the second layer and lower in the third layer.@xcite this conclusion is consistent with the first principles results which will be discussed in the next section . ga thus grows in a stranski - krastanow mode on ( 0001 ) gan . finally , the different adsorption regimes can be summarized as follows : ( 1 ) ga coverage @xmath46ml , i.e. successive formation of a coherent ga monolayer , ( 2 ) a ga coverage of @xmath47ml , forming a laterally - contracted ga bilayer , and ( 3 ) ga accumulation and droplet formation on top of a ga bilayer . in the following , we will derive an ab initio based growth model which describes the temperature dependence of the transition fluxes between the different regimes . comparison between the experimental growth phase diagram in ref . ( open symbols ) and the diagram derived from the the ga adsorption data in this work ( solid lines ) . the different regimes 1 , 2 , and 3 of the adsorption phase diagram are indicated , which correspond to regimes b , c , and d in ref . , respectively.,width=302 ] an intuitive connection between adsorption and growth phase diagrams can be made by assuming that the _ excess _ ga in ga - rich growth conditions behaves as if it would be adsorbed on a gan surface . then , adding the gan growth rate ( 0.28ml / s in the experiments of ref . ) to the adsorption phase transition fluxes should reproduce the growth phase diagram of ref . . this comparison is shown in fig . [ fig : comp ] . it demonstrates good overall agreement , suggesting that the assumption is valid . however , it must be noted that the critical fluxes derived from adsorption measurements fall below those in the growth phase diagram , although differences are of the order of the experimental precision . yet , a closer look at the activation barriers and prefactors ( tab . [ tab : fitresults ] ) , as derived from the experimental phase diagrams , poses a number of questions . first , why are the activation energies and prefactors in the adsorption and growth phase diagram so different , even though absolute values of transition fluxes are close ? second , why are the values for the same transition ( @xmath24 for growth ) in refs . and so different ? finally , while the barrier of 2.8ev for the @xmath24 transition in the growth phase diagram in ref . is close to the cohesive energy of bulk ga and has thus been directly interpreted as a ga desorption barrier , what is the physical origin of the 5.1ev activation energy and the meaning of a prefactor of 10@xmath48hz ? the latter value is fundamentally different from prefactors typically observed / calculated for diffusion or desorption processes ( which are of the order of @xmath49hz ) . to address the above questions , we have analyzed the data in terms of a simple growth model and in combination with first principles total energy calculations . in order to simplify the discussion of the growth model , we divide the problem in two parts : first , we derive how the density of ga adatoms @xmath50 on the surface@xcite depends on parameters such as temperature @xmath51 , ga flux @xmath3 , and growth rate @xmath52 . second , we calculate the critical adatom density , at which nucleation occurs and the system undergoes a phase transition . the adatom coverage is given by : @xmath53 here , the desorption time @xmath54 is given by @xmath55 with @xmath56 the desorption barrier and @xmath57 the boltzmann constant . @xmath58 is the surface diffusion constant for ga adatoms and @xmath59 gives the lateral position on the surface . for step flow growth ( as realized in ga - rich growth conditions@xcite ) , incorporation occurs essentially at the step edges . since these are moving , the incorporation rate @xmath60 is in general inhomogeneous and time dependent . for ideal step flow it will be zero on the terraces and @xmath61 only at the step edges . the incorporation rate and the growth rate @xmath52 are directly related by : @xmath62 here the integration is performed over the total surface area @xmath63 . since in our experimental setup stationary conditions are realized , the explicit time dependence in the growth rate disappears ( i.e. @xmath64 ) . furthermore , in stationary conditions eq . ( [ eq : drhodta ] ) also simplifies : @xmath65 , i.e. the left hand side becomes zero . a further simplification in eq . ( [ eq : drhodta ] ) can be made by taking into account that the phase transitions we are interested in occur exclusively under ga - rich conditions . under these conditions the surface steps on the surface are ga - terminated@xcite and the incorporation rate @xmath66 at such a step will be small . this is in contrast to the conventional step flow picture where the sticking probability of an atom at the step edge is assumed to be close to one . the difference to the conventional model is due to the fact that we have two species with very different concentrations . for nitrogen ( which is the minority species for very ga - rich conditions ) the sticking coefficient at the ga - terminated step edges will be close to one and steps act as sinks to the nitrogen concentration which becomes highly inhomegenous . for ga atoms , however , the sticking probablity is low . thus , the effect of steps on ga will be small and the ga - adatom density is virtually homogeneous , i.e. @xmath67 . based on the above discussion , eq . ( [ eq : drhodta ] ) can be written as : @xmath68 here , @xmath69 denotes the equilibrium adatom density . we note that this equation holds for both growth ( @xmath70 ) and adsorption ( @xmath71 ) . the solution is easily found to be @xmath72 nucleation occurs if the ( stable ) nuclei are in thermodynamic equilibrium with the lattice gas ( which is described by the adatom density ) on the surface . at low densities ( @xmath73 ) , interactions in the lattice gas itself can be neglected and we obtain @xmath74 here , @xmath75 is the number of adatoms in the lattice gas , @xmath76 is the total number of surface sites which can be occupied by the adatoms , and @xmath77 is the energy the adatom gains if it is attached to a subcritical nucleus making the latter stable . based on the above model , we can directly obtain the critical ga flux @xmath78 at which the phase transitions occur . combining eqs . ( [ eq : rho ] ) and ( [ eq : rhocrit ] ) gives : @xmath79 this equation can be rewritten using eq . ( [ eq : des ] ) as @xmath80 and applies both to the adsorption and growth phase diagram . the activation energy is thus expected to represent the total binding energy of a ga atom in a critical ga cluster . a comparison with the experimental results ( see tab . [ tab : fitresults ] ) shows that energy and prefactor are _ not _ constant [ as expected from eq . ( [ eq : critflux2 ] ) ] but vary largely with the growth conditions . as a general trend , one finds that the activation energy and the prefactor decrease with increasing growth rate ( n flux ) . it is also interesting to note that only in the case of high growth rate ( @xmath81ml / s ) the prefactor ( @xmath82hz ) is close to the typical attempt frequencies observed / expected for desorption , i.e. in the @xmath83hz range . for conditions where growth is slow ( @xmath84ml / s ) or absent ( adsorption ) , prefactors are found which are many orders of magnitude larger . in order to identify the origin of these apparent discrepancies , we have explicitly calculated the desorption and the formation of small ga clusters on the ga bilayer surface employing density functional theory . in the following we will focus on the ga - bilayer structure ( which corresponds to the @xmath24 transition ) . based on the almost identical energies of the @xmath85 and @xmath24 transitions in the adsorption phase diagram we expect the mechanisms / energetics to be rather similar . schematic top view of an adatom on the ga bilayer structure . the white balls mark the positions of the ga atoms in the second layer . the gray balls mark the positions of the ga atoms in the contracted ga epilayer and the black ball the ga adatom in the t4 position . the dashed line shows the @xmath86 surface unit cell of the ideal bulk truncated gan ( 0001 ) surface . the solid line shows the surface unit cell of the ga bilayer structure.,width=302 ] schematic top view of a 2 atom island ( dimer ) on the ga bilayer structure . the solid line shows the @xmath87 surface unit cell which has been used to describe this structure.,width=302 ] specifically , we use soft troullier - martins pseudopotentials@xcite and the perdew - burke - ernzerhof generalized - gradient approximation ( pbe - gga ) to describe exchange / correlation.@xcite the ga @xmath88 semicore states were described in the frozen core approximation ( nlcc).@xcite details about the method can be found elsewhere.@xcite the calculations have been performed with a plane wave basis set ( energy cutoff : 50ry ) . the brillouin zone has been sampled by a @xmath89 monkhorst - pack mesh for the @xmath90 unit cell and @xmath91 for the @xmath92 unit cell.@xcite the ga bilayer with one up to four adatoms has been modeled by a slab consisting of 2 double layers and @xmath93 ( to describe the adsorption of a single adatom , see fig . [ fig : schema1 ] ) and larger @xmath90 ( to describe adatom islands , see fig . [ fig : schema2 ] ) unit cells . increasing the slab thickness to 4 double layers changes the surface energy by less than 0.5mev per surface unit cell . adatoms are added only on the upper surface of the slab . the lower surface has been passivated by pseudo - hydrogen to remove the electrically active surface states . the adatom(s ) and the first two surface layers have been fully relaxed . detailed convergence checks can be found in refs . . based on these studies , we have calculated the desorption energy of an adatom and the binding energy of adatoms in small islands . the desorption energy ( the energy needed to remove the adatom from the surface ) is defined by @xmath94 where @xmath95 is the total energy of the surface including the adatom , @xmath96 that of the free surface and @xmath97 that of the ( spin - polarized@xcite ) ga atom . using this expression we find an adatom binding energy of 2.52ev for the @xmath93 structure and 2.41ev for the @xmath98 unit cell . in the equilibrium configuration , the adatom sits on a three - fold coordinated hollow site . ga bond length between adatom and surface layer is 2.68 , i.e. very close to the nearest neighbor distance in @xmath99-ga of 2.71.@xcite the binding energy of an adatom in an island consisting of @xmath100 adatoms is given by : @xmath101 \quad .\ ] ] for an island consisting of 2 adatoms we find @xmath102ev . for larger islands consisting of 3 adatoms @xmath103ev and @xmath104ev for a 4 atom island . the numbers are the island formation energies as calculated in a @xmath105 cell . it is interesting to note here that all islands are unstable against the formation of ga - droplets : the formation energy of a ga atom in an island ( @xmath106ev ) is smaller than the cohesive energy of bulk ga of 2.8ev.@xcite therefore the islands act as nucleation centers for ga droplet formation . natural logarithm of the experimental prefactors @xmath107 as a function of the activation energy @xmath108 for various data in this work and refs . and ( full circles ; see tab . [ tab : fitresults ] ) . the solid line represents a linear fit.,width=302 ] we can now compare these energies with those obtained from the analysis of the experimental data ( tab . [ tab : fitresults ] ) . as can be seen , the activation energy is close to our calculated desorption energy only for the high growth case ( 1.1ml / s ) . for lower growth rates or adsorption , activation energies are found which are way too large . a closer analysis of the prefactors and the activation energies shows a clear relation ( see fig . [ fig : lnnuea ] ) : @xmath109 the supercript `` exp '' indicates experimental data . for the values given in tab . [ tab : fitresults ] we get @xmath110ev@xmath111 and @xmath112 . the observed relation between prefactor and activation barrier can be explained in terms of a temperature dependent activation energy . since the experimentally accessible temperature range is rather small ( @xmath113k ) , we assume a linear dependence : @xmath114 with @xmath115 the temperature independent contribution , @xmath116 the temperature offset , and @xmath99 the linear temperature coefficient . equation ( [ eq : critflux2 ] ) then becomes @xmath117 a comparison between eqs . ( [ eq : fit ] ) and ( [ eq : critfluxlinea ] ) gives the following relations : @xmath118 this leads to @xmath119 using these relations and @xmath120ev ( as found from our first principles calculations ) , we obtain @xmath121@xmath5c ( which is close to the experimentally accessed temperature range , so the linear approximation for the temperature dependence of @xmath122 is well justified ) and @xmath123hz ( which is close to typical attempt frequencies ) . using the above parameters and eq . ( [ eq : expea ] ) , the linear temperature coefficient @xmath124 can be computed for all transitions . the result is shown in tab . [ tab : fitresults ] . based on these values we can renormalize the prefactors following @xmath125 . the resulting numbers calculated with @xmath123hz are listed in tab . [ tab : fitresults ] . the good agreement with the experimental prefactors @xmath126 shows that the model consistently describes all previous experimental studies . the large variation in frequencies and activation energies can be explained by assuming that the n flux affects the temperature dependence : it is largest if no n flux is present and monotonously decreases with increasing n flux ( growth rate ) . it is interesting to note that , although the temperature dependence has a huge effect on the experimentally measured apparent activation barrier @xmath127 ( by almost a factor of two ) and prefactor @xmath126 ( by up to 12 orders of magnitude ) , the actual change in the effective desorption energy @xmath114 is small : in the case of adsorption ( @xmath128 ) , the activation barrier changes from 2.84ev to 2.73ev within the experimentally measured temperature window ( 700@xmath5c to 750@xmath5c ) . in the case of gan growth , the variation is even smaller . this can be intuitively understood by considering that the experimentally measured activation energy represents a linear projection to zero temperature even when its real temperature dependence deviates strongly from linear behavior outside our experimental temperature window . based on a combination of experimental rheed studies and first - principle growth models we have ( i ) quantitatively determined the ga coverage on the gan ( 0001 ) surface during adsorption as a function of ga flux and substrate temperature and ( ii ) derived a model , which consistently describes the adsorption of ga on gan surfaces as well as the accumulation of ga during ga - rich gan growth . this model resolves the discrepancy in previous measurements of the activation energy characterizing the critical ga flux for the onset of ga droplet fromation during gan growth.@xcite the model also explains the origin of the experimentally - observed unphysically high prefactors in terms of a temperature dependent desorption barrier . at the moment , we can only speculate about possible mechanisms which reduce the activation barrier at higher temperatures . a possible scenario emerges from our first principles calculations where we find that the number of atoms in the compressed ga layer of the bilayer structure and thus the lateral lattice constant of the top ga layer significantly changes with temperature . this change in the surface geometry is expected to have an important effect on the island formation energy and will be discussed in a forthcoming paper.@xcite the authors would like to thank o. briot ( university of montpellier , france ) for providing the gan templates and f. rieutord ( cea grenoble , drfmc / si3 m ) for valuable discussions . e. bellet - amalric ( cea grenoble , drfmc / sp2 m ) is acknowledged for the analysis of fig . [ fig : rheed ] . l. l. and j. n. like to thank the eu tmr program ipam and j. n. the deutsche forschungsgemeinschaft sfb 296 . a. r. smith , v. ramachandran , r. m. feenstra , d. w. greve , m .- s . shin and m. skowronski , j. neugebauer , and j. e. northrup , j. vac . b * 16 * , 1641 ( 1998 ) ; a. r. smith , r. m. feenstra , d. w. greve , m. s. shin , m. skowronski , j. neugebauer , and j. e. northrup , _ ibid . _ * 16 * , 2242 ( 1998 ) . xue , q .- k . xue , r. z. bakhtizin , y. hasegawa , i. s. t. tsong , t. sakurai , and t. ohno , phys . b * 59 * , 12604 ( 1999 ) ; q .- k . xue , q .- z . xue , r. z. bakhtizin , y. hasegawa , i. s. t. tsong , t. sakurai , and t. ohno , phys . lett . * 82 * , 3074 ( 1999 ) . here , @xmath50 denotes the adatom density on top of the last completed ga layer ( a monolayer for the @xmath23 transition and a bilayer for the @xmath24 transition ) . this is contrasted by the ga coverage @xmath28 , which gives the _ total _ amount of ga present on the surface ( including completed ga mono- or bilayers ) .

we study the adsorption behavior of ga on ( 0001 ) gan surfaces combining experimental specular reflection high - energy electron diffraction with theoretical investigations in the framework of a kinetic model for adsorption and _ ab initio _ calculations of energy parameters . based on the experimental results we find that , for substrate temperatures and ga fluxes typically used in molecular - beam epitaxy of gan , _ finite _ equilibrium ga surface coverages can be obtained . the measurement of a ga / gan adsorption isotherm allows the quantification of the equilibrium ga surface coverage as a function of the impinging ga flux . in particular , we show that a large range of ga fluxes exists , where @xmath0 monolayers ( in terms of the gan surface site density ) of ga are adsorbed on the gan surface . we further demonstrate that the structure of this adsorbed ga film is in good agreement with the laterally - contracted ga bilayer model predicted to be most stable for strongly ga - rich surfaces [ j. e. northrup _ et al . _ , phys . rev . b * 61 * , 9932 ( 2000 ) ] . for lower ga fluxes , a discontinuous transition to ga monolayer equilibrium coverage is found , followed by a continuous decrease towards zero coverage ; for higher ga fluxes , ga droplet formation is found , similar to what has been observed during ga - rich gan growth . the boundary fluxes limiting the region of 2.5 monolayers equilibrium ga adsorption have been measured as a function of the gan substrate temperature giving rise to a ga / gan adsorption phase diagram . the temperature dependence is discussed within an _ ab initio _ based growth model for adsorption taking into account the nucleation of ga clusters . this model consistently explains recent contradictory results of the activation energy describing the critical ga flux for the onset of ga droplet formation during ga - rich gan growth [ b. heying _ et al . _ , j. appl . phys . * 88 * , 1855 ( 2000 ) ; c. adelmann _ et al . _ , j. appl . phys . * 91 * , 9638 ( 2002 ) . ] .

it is well known that the manifestation of the dynamical aspects of quantum chaos is possible only within its characteristic timescales , the heisenberg time in the regular case and the logarithmic timescale in the chaotic case , where typical phenomena with a semiclassical description are possible such as relaxation , exponential sensitivity , etc @xcite . the logarithmic timescale determines the time interval where the wave packet motion is as random as the classical trajectory by spreading over the whole phase space @xcite . it should be noted that some authors consider the logarithmic timescale as a satisfactory resolution of the apparent contradiction between the correspondence principle and the quantum to classical transition in chaotic phenomena @xcite . concerning the chaotic dynamics in quantum systems , the kolmogorov sinai entropy ( ks entropy ) @xcite has proven to be one of the most used indicators . the main reason is that the behavior of chaotic systems of continuous spectrum can be modeled from discretized models such that the ks entropies of the continuous systems and of the discrete ones coincide for a certain time range . taking into account the graininess of the quantum phase space due the indetermination principle , non commutative quantum extensions of the ks entropy can be found @xcite . thus , the issue of graininess is intimately related to the quantum chaos timescales @xcite . to complete this picture , it should be mentioned that a relevant property in dynamical systems is the ergodicity , i.e. when the subsets of phase space have a correlation decay such that any two subsets are statistically independent in time average " for large times . this property , that is assumed as an hypothesis in thermodynamics and in ensembles theory @xcite , is at the basis of the statistical mechanics by allowing the approach to the equilibrium by means of densities that are uniformly distributed in phase space . in this sense , in previous works @xcite quantum extensions of the ergodic property were studied , from which characterizations of the chaotic behavior of the casati prosen model @xcite and of the phase transitions of the kicked rotator were obtained @xcite . the main goal of this paper is to exploit the graininess of the quantum phase space and the properties of the ks entropy in an ergodic dynamics in order to get an estimation of the logarithmic timescale in the semiclassical limit . the paper is organized as follows . in section 2 we give the preliminaries and an estimation of the ks entropy in an ergodic dynamics is presented . in section 3 we show that this estimation can be extended for the classical analogue of a quantum system in the semiclassical limit . from this estimation and a time rescaled ks entropy of the classical analogue , we obtain the logarithmic timescale . section 4 is devoted to a discussion of the results and its physical relevance . finally , in section 5 we draw some conclusions , and future research directions are outlined . the definitions , concepts and theorems given in this section are extracted from the ref . @xcite . we recall the definition of the ks entropy within the standard framework of measure theory . consider a dynamical system given by @xmath0 , where @xmath1 is the phase space , @xmath2 is a @xmath3-algebra , @xmath4 $ ] is a normalized measure and @xmath5 is a semigroup of measure preserving transformations . for instance , @xmath6 could be the classical liouville transformation or the corresponding classical transformation associated to the quantum schrdinger transformation . @xmath7 is usually @xmath8 for continuous dynamical systems and @xmath9 for discrete ones . let us divide the phase space @xmath1 in a partition @xmath10 of @xmath11 small cells @xmath12 of measure @xmath13 . the entropy of @xmath10 is defined as @xmath14 now , given two partitions @xmath15 and @xmath16 we can obtain the partition @xmath17 which is @xmath18 , i.e. @xmath17 is a refinement of @xmath15 and @xmath16 . in particular , from @xmath10 we can obtain the partition @xmath19 being @xmath20 the inverse of @xmath21 ( i.e. @xmath22 ) and @xmath23 . from this , the ks entropy @xmath24 of the dynamical system is defined as @xmath25 where the supreme is taken over all measurable initial partitions @xmath10 of @xmath1 . in addition , the brudno theorem expresses that the ks entropy is the average unpredictability of information of all possible trajectories in the phase space . furthermore , pesin theorem relates the ks entropy with the exponential instability of motion given by the lyapunov exponents . thus , from the pesin theorem it follows that @xmath26 is a sufficient condition for chaotic motion @xcite . by taking @xmath0 as the classical analogue of a quantum system and considering the timescale @xmath27 within the quantum and classical descriptions coincide , the definition can be expressed as @xmath28 now since @xmath29 one can recast as @xmath30 finally , from this equation one can express @xmath24 as @xmath31 the main role of the time rescaled ks entropy @xmath32 is that allows to introduce the timescale @xmath27 as a parameter . this concept will be an important ingredient for obtaining the logarithmic timescale . in dynamical systems theory , the correlation decay of ergodic systems is one of the most important properties for the validity of the statistical description of the dynamics because different regions of phase space become statistical independent in time average " when they are enough separated in time . more precisely , if one has a dynamical system @xmath33 that is ergodic then the correlations between two arbitrary sets @xmath34 that are sufficiently separated in time satisfy @xmath35 where @xmath36 is the correlation between @xmath37 and @xmath38 with @xmath37 the set @xmath39 at time @xmath40 . this equation expresses the so called _ ergodicity property _ which guarantees the equality between the time average and the space average of any function along the trajectories of the dynamical system . ergodicity property is satisfied by all the chaotic systems , like chaotic billiards , chaotic maps , including systems described by ensemble theory . furthermore , the calculation of the ks entropy is intimately related with the ergodicity property , as we shall see . in order to calculate the kolmogorov sinai entropy @xmath24 the concept of generator is of fundamental importance . a numerable partition @xmath41 of @xmath1 is called a _ generator _ of @xmath1 for an invertible measure preserving @xmath42 if @xmath43 this equation expresses that the entire @xmath3algebra @xmath2 can be generated by means of numerable intersections of the form + @xmath44 where @xmath45 for all @xmath46 . it can be proved that if @xmath47 is a generator and @xmath48 then @xmath49 which reduces the problem of taking the supreme in the formula of the @xmath24 to find a generator @xmath47 . in practice , still having found a generator , the calculation of @xmath50 turns out a difficult task due to the large number of subsets of the partition @xmath51 as soon as @xmath52 increases . however , a good estimation of the @xmath24 can be made by means of the existence of finite generators . this is the content of the following theorem . [ estimation ks](estimation of the ks entropy by means of finite generators ) if @xmath0 and @xmath53 is an ergodic invertible measure preserving transformation with @xmath54 then @xmath42 has a finite generator @xmath47 @xmath55 such that @xmath56 with the help of the theorem [ estimation ks ] and taking into account the graininess of the quantum phase space , one can obtain a semiclassical version of the theorem [ estimation ks ] from which the logarithmic timescale can be deduced straightforwardly . we begin by describing the natural graininess of the quantum phase space . [ fig1 ] the region @xmath57 that the classical analogue occupies has a volume that is approximately the sum of the volumes of the rigid boxes @xmath58 contained in @xmath57 . the region @xmath2 corresponding to the rigid boxes that intersect the frontier of @xmath57 can be neglected in the limit @xmath59.,width=302 ] in quantum mechanics , the uncertainty principle leads to a granulated phase space composed by rigid boxes of minimal size @xmath60 where @xmath61 is the dimension of the phase space . this is the so called _ graininess _ of the quantum phase space . in a typical chaotic dynamics the motion in phase space @xmath1 is bounded , occupying the system a finite region @xmath57 of volume @xmath62 . in turn , in the semiclassical limit @xmath63 the value of @xmath62 can be approximated by the sum of all the rigid boxes @xmath58 that are contained in @xmath57 . let us call @xmath64 to these boxes . in such situation the region @xmath2 corresponding to the rigid boxes that intersect the frontier of @xmath57 can be neglected . an illustration for @xmath65 is shown in fig . , since there is no subset in grained phase space having a volume smaller than @xmath58 it follows that @xmath66 is the unique generator of @xmath57 , that we will call _ quantum generator_. moreover , one has that any @xmath3-algebra in quantum phase space can be only composed by unions of the rigid boxes @xmath64 . in order to obtain a semiclassical version of theorem [ estimation ks ] , we consider a quantum system having a classical analogue @xmath33 provided with a finely grained phase space @xmath1 in the semiclassical limit @xmath59 , as is shown in fig . 1 . also , the partition @xmath47 of is the quantum generator of the region @xmath57 occupied by the classical analogue . let us assume that the dynamics in phase space is ergodic . then , we can arrive to our main contribution of this work , established by means of the following result . [ estimation logarithmic](estimation of the time rescaled ks entropy of the classical analogue ) assume one has a quantum system having a classical analogue @xmath0 that occupies a region @xmath57 of a discretized quantum phase space of dimension @xmath61 . if @xmath67 is an ergodic and invertible measure preserving transformation with @xmath68 the time rescaled ks entropy of the classical analogue , then in the semiclassical limit @xmath59 one has @xmath69 where @xmath70 is the quasiclassical parameter @xmath71 and @xmath72 is the quantum generator . it is clear that the partition @xmath72 of eq . is the only quantum generator in the quantum phase space and since @xmath62 is @xmath52 times the volume @xmath73 of each rigid box contained in @xmath57 then one obtains @xmath74 . then , the result follows by applying the theorem [ estimation ks ] to the classical analogue in the semiclassical limit . from the equation one has @xmath75 from which follows that @xmath76 now , assuming a chaotic motion of the classical analogue by means of the condition @xmath26 , then one can make the approximation @xmath77 . replacing this in the last inequality one has @xmath78 which is precisely the logarithmic timescale . here we provide a discussion about the physical relevance of the results obtained at the light of the quantum chaos theory . several previous work based on the quantum dynamics of observable values @xcite , quantization by means of symmetric and ordered expansions @xcite , and the wave packet spreadings along hyperbolic trajectories @xcite among others , show that a unified scenario for a characterization of the quantum chaos timescales is still absent . furthermore , the mathematical structure used in most of these approaches makes difficult to visualize intuitively the quantum and classical elements that are present , or even in some cases the results are restricted to special initial conditions @xcite . nevertheless , we can mention the aspects of our contribution in agreement with some standard approaches used . below we quote some results of the literature and discuss them from the point of view of the present work . * _ the timescale @xmath79 may be one of the universal and fundamentals characteristic of quantum chaos accessible to experimental observation @xcite . in fact , the existence of @xmath79 @xmath80 where @xmath81 are constants has been observed and discussed in detail for some typical models of quantum chaos @xcite . _ + the relation results as a mathematical consequence of theorem [ estimation logarithmic ] for any phase space of arbitrary dimension @xmath61 . in fact , from one obtains @xmath82 , and @xmath83 the volume occupied by the system along the dynamics . * _ every classical structure with a phase space area smaller than planck s constant @xmath84 has no quantum correspondence . only the classical structures extending in phase space over scales larger than planck s constant are susceptible to emerge out of quantum mechanical waves @xcite . _ + from theorem [ estimation logarithmic ] one can see that the classical structure of ks entropy estimation ( theorem [ estimation ks ] ) emerges in terms of the quasiclassical parameter @xmath71 in the semiclassical limit , as is expressed in . * _ if strong chaos occurs in the classical limit , then for a rather short time @xmath79 the wave packet spreads over the phase volume : @xmath85 where @xmath86 is the characteristic lyapunov exponent . therefore , for the time scale @xmath79 , one has : @xmath87 , where @xmath88 is of the order of the number of quanta of characteristic wave packet width @xcite . _ + in fact , from with @xmath65 it follows that for the case of the wave packet one has : @xmath89 , @xmath90 ( pesin theorem ) , and @xmath91 . in this way , the number of quanta of the characteristic wave packet width is equal to the number @xmath52 of the members of the quantum generator given by . [ fig2 ] showing the necessary elements for obtaining the logarithmic timescale.,width=529 ] a panoramic outlook of the content of theorem [ estimation logarithmic ] is shown in fig . we have presented , in the semiclassical limit , an estimation of the logarithmic timescale for a quantum system having a classical analogue provided with an ergodic dynamics in its phase space . the three ingredients that we used were : 1 ) the fine granularity of the quantum phase space in the semiclassical limit , 2 ) the existence of an estimation of the ks entropy in terms of finite generators of the region that the system occupies in phase space , and 3 ) a time rescaling of the ks entropy that allows to introduce the characteristic timescale as a parameter . in summary , our contribution is three fold . on the one hand , the logarithmic timescale arises , in the semiclassical limit , as a formal result of the ergodic theory applied to a discretized quantum phase space of the classical analogue in an ergodic dynamics , thus providing a theoretical bridge between the ergodic theory and the graininess of the quantum phase space . on the other hand , the theorem [ estimation logarithmic ] makes more visible and rigorous the simultaneous interaction of the effects of the quantum dynamics and the classical instability in phase space . in fact , the semiclassical parameter @xmath71 is expressed as the number of members of the quantum generator of the region that the classical analogue occupies in phase space . finally , one can consider the theorem [ estimation logarithmic ] as a mathematical proof of the existence of the logarithmic timescale when the dynamics of the classical analogue is chaotic , i.e. a positive ks entropy . however , since the quasiclassical parameter @xmath71 and the ks entropy are system specific , in each example the parameters of the logarithmic timescale must be determined by the experimental observation . one final remark . it is pertinent to point out that , in addition to theorem [ estimation ks ] , the techniques employed ( i.e. the existence of a single quantum generator of the quantum phase space and the time rescaling property of the ks entropy ) can be used to extend semiclassically others results of the ergodic theory . this work was partially supported by conicet and universidad nacional de la plata , argentina . gomez , i. & castagnino , m. [ 2014 ] `` on the classical limit of quantum mechanics , fundamental graininess and chaos : compatibility of chaos with the correspondence principle , '' _ chaos , sol . * 68 * , 98 - 113 . ikeda , k. [ 1993 ] `` quantum and chaos : how incompatible ? , '' _ proceeding of the 5th yukawa international seminar . _ * 116*. landsman , n. [ 2007 ] `` between classical and quantum , '' _ philosophy part a _ eds . butterfield , j. & earman , j. ( elsevier , amsterdam ) , pp .

an estimation of the logarithmic timescale in quantum systems having an ergodic dynamics in the semiclassical limit of quasiclassical large parameters , is presented . the estimation is based on the existence of finite generators for ergodic measure preserving transformations having a finite kolmogorov sinai ( ks ) entropy and on the time rescaling property of the ks entropy . the results are in agreement with the obtained in the literature but with a simpler mathematics and within the context of the ergodic theory .

some of the most intriguing properties of 4d @xmath3 gauge theories are those related to the dependence on the @xmath1 parameter associated with a topological term in the ( euclidean ) lagrangian @xmath13 where @xmath14 is the topological charge density , @xmath15 the dependence on @xmath1 vanishes perturbatively , therefore it is intrinsically nonperturbative @xcite . the recent renewed activity in the study of the topological properties of gauge field theories , and of @xmath1 dependence in particular , has been triggered by two different motivations . from the purely theoretical point of view @xmath1-related topics naturally appear in such disparate conceptual frameworks as the semiclassical methods @xcite , the expansion in the number of colors @xcite , the holographic approach @xcite and the lattice discretization ( see , e.g. , @xcite for a review of the main results ) . from the phenomenological point of view , the nontrivial @xmath1 dependence is related to the breaking of the axial @xmath16 symmetry and related issues of the hadronic phenomenology @xcite , such as the @xmath17 mass . moreover , it is related to the axion physics ( see , e.g. , @xcite for a recent review ) , put forward to provide a solution of the strong cp problem @xcite , i.e. to explain the fact that the experimental value of @xmath1 is compatible with zero , with a very small bound @xmath18 from neutron electric dipole measurements @xcite . axions are also natural dark matter candidates @xcite and , given the absence of susy signals from accelerator experiments , this is becoming one of the most theoretically appealing possibility . the ground - state energy density of 4d @xmath3 gauge theories is an even function of @xmath1 . it is expected to be analytic at @xmath5 , thus it can be expanded in the form @xmath19 where @xmath7 is the topological susceptibility , and the dimensionless coefficients @xmath8 parametrize the non - quadratic part of the @xmath1 dependence . they are related to the cumulants of the topological charge distribution at @xmath5 ; in particular @xmath8 quantify the deviations from a simple gaussian distribution . standard large-@xmath0 arguments predict the large-@xmath0 behavior @xcite @xmath20 since @xmath7 and @xmath8 can not be computed analytically , these large-@xmath0 scaling relations can be only tested numerically . earlier studies have mainly focused on the investigation of the large-@xmath0 scaling of the topological susceptibility , reporting a good agreement with the corresponding large-@xmath0 expectations ( see , e.g. , @xcite ) . instead , the numerical determination of the higher - order coefficients of the @xmath1 expansion turns out to be a difficult numerical challenge . most efforts have been dedicated to the @xmath21 case @xcite , reaching a precision corresponding to a relative error below 10% only recently @xcite . some higher-@xmath0 results were reported in ref . @xcite , presenting a first attempt to investigate the large-@xmath0 scaling of @xmath22 ; the numerical precision that could be reached was however quite limited , with a signal for @xmath23 at two standard deviation from zero and only an upper bound for @xmath24 . in order to further support the evidence of the large-@xmath0 scaling scenario beyond the quadratic term of the expansion of the ground - state energy dentity ( [ thdep ] ) , we investigate the scaling of the higher - order terms , in particular those associated with @xmath22 and @xmath25 . from the computational point of view , the most convenient method to perform such an investigation exploits monte carlo simulations of @xmath3 gauge theories in the presence of an imaginary @xmath1 angle , which are not plagued by the sign problem . their analysis allows us to obtain accurate estimates of the coefficients of the expansion around @xmath5 . analogous methods based on computations at imaginary values of @xmath1 have been already employed in some numerical studies of the @xmath21 gauge theory @xcite and @xmath4 models @xcite . 2d @xmath4 models share with 4d @xmath3 gauge theories many physically interesting properties , like asymptotic freedom , dynamical mass generation , confinement , instantons and @xmath1 dependence ; moreover their large-@xmath0 expansion can be studied by analytical methods @xcite . as a consequence they are an attractive laboratory where to test theoretical ideas that might turn out to be applicable to qcd . an expansion of the form eq . applies also to the @xmath1 dependence of 2d @xmath4 models . similarly to 4d @xmath3 gauge theories , large-@xmath0 scaling arguments predict the large-@xmath0 behavior @xmath26 and @xmath27 . these large-@xmath0 scaling behaviors are confirmed by explicit analytical computations , see @xcite . in this paper , following the approach introduced in @xcite , we present a systematic and easily automated way of computing the leading large-@xmath0 terms of @xmath8 . the paper is organized as follows . in section [ sec : sun ] we present the results obtained for the case of the 4d @xmath3 gauge theories : first we discuss the numerical setup used and the reasons for some specific algorithmic choices adopted , then we present the physical results obtained . in section [ sec : cpn ] the case of the 2d @xmath4 models is discussed and a determination of the leading order large-@xmath0 expansion for the coefficients @xmath8 is presented . finally , in section [ sec : concl ] , we draw our conclusions . in appendices some technical details are examined regarding a comparison between smoothing algorithms in @xmath24 ( app . [ sec : cgf ] ) and an attempt to reduce the autocorrelation time using a parallel tempering algorithm ( app . [ sec : pt ] ) . tables of numerical data are reported in app . [ sec : data ] . the traditional procedure that has been used in past to compute the coefficients entering eq . consists in relating them to the fluctuations of the topological charge @xmath28 at @xmath5 . the first few coefficients of the expansion can indeed be written as ( see e.g. @xcite ) @xmath29_{\theta=0}}{360 \langle q^2\rangle_{\theta=0 } } , \label{b4}\end{aligned}\ ] ] etc . , where @xmath30 is the 4d volume and all the averages are computed using the action with @xmath5 . while this method is obviously correct from the theoretical point of view , it is numerically inefficient for the determination of the @xmath8 coefficients . indeed fluctuation observables are not self - averaging @xcite and , in order to keep a constant signal to noise ratio , one has to dramatically increase the statistics of the simulations when increasing the volume ( see e.g. @xcite for a numerical example ) . as a consequence it is extremely difficult to keep finite size effects under control and to extract the infinite - volume limit . to avoid this problem , one can introduce a source term in the action , which allows us to better investigate the response of the system . this can be achieved by performing numerical simulations at imaginary values of the @xmath1 angle , @xmath31 , in order to maintain the positivity of the path integral measure and avoid a sign problem , and study for example the behaviour of @xmath32 as a function of @xmath33 . it is indeed easy to verify that @xcite @xmath34 also higher cumulants of the topological charge distribution ( for which relations analogous to eq . exist ) can be used for this purpose , however the numerical precision quickly degrades for higher cumulants . nevertheless , since the computation of these higher cumulants does not require any additional cpu time , the optimal strategy seems to be to perform a common fit to a few of the lowest cumulants of the topological charge @xcite ( of course by taking into account the correlation between them ) . after this general introduction to motivate the computational strategy adopted , we describe the details of the discretization setup . for the @xmath21 case we use results already reported in the literature , while new simulations are performed for the @xmath23 and @xmath24 cases . the lattice action used in the sampling of the gauge configurations is @xmath35 = s_w[u ] - \theta_{l } q_l[u ] \ , , \ ] ] where @xmath36 $ ] is the standard wilson plaquette action @xcite and @xmath37 . for the topological charge density we adopt the discretization @xcite : @xmath38 where @xmath39 is the plaquette , @xmath40 coincides with the usual levi - civita tensor for positive entries and it is extended to negative ones by @xmath41 and complete antisymmetry . the discretization eq . of the topological charge density makes the total action in eq . linear in each gauge link , thus enabling the adoption of standard efficient update algorithms , like heat - bath and overrelaxation , a fact of paramount importance , since we have to deal with the strong critical slowing down of the topological modes @xcite . a practical complication is due to the fact that the discretization of the topological charge density induces a finite renormalization of @xmath14 @xcite and thus of @xmath1 . denoting this renormalization constant by @xmath42 , we thus have @xmath43 where @xmath44 is the numerical value that is used in the actual simulation . two different strategies can be used to cope with this complication : in one case @xmath42 is computed separately and eq . can then be directly used ( see @xcite for more details ) . another possibility consists in rewriting eq . , and the analogous equations for the higher cumulants , directly in term of @xmath44 , in such a way that by performing a common fit to the cumulants it is possible to evaluate both @xmath42 and the parameters appearing in eq . ( see @xcite for more details ) . in our numerical work we adopt the second strategy , that turn out to be slightly more efficient from the numerical point of view . all results that we present are obtained by analyzing the @xmath33 dependence of the first four cumulants of the topological charge distribution . in order to avoid the appearance of further renormalization factors , the topological charge is measured on smoothed configurations . the smoothing procedure adopted uses the standard cooling technique @xcite , which is the computationally cheapest procedure ( especially for large values of @xmath0 ) . cooling is implemented _ la _ cabbibbo - marinari , using the @xmath45 diagonal @xmath46 subgroups of @xmath3 , and we follow @xcite in defining the measured topological charge @xmath47 by @xmath48 where @xmath49 is the integer closest to @xmath50 and the coefficient @xmath51 is the value that minimize @xmath52\right)^2\right\rangle \ .\ ] ] this procedure is introduced in order to avoid the necessity for prolongated cooling , and in fact we observe no significant differences in the results obtained by using a number of cooling steps between @xmath53 and @xmath54 , while more than @xmath55 cooling steps would be needed to reach a plateaux using just @xmath56 instead of the @xmath47 defined by eq . . at finite lattice spacing , the two definitions ( rounded vs. non - rounded ) can lead to different results corresponding to different lattice artefacts , however it has been shown that the same continuum limit is reached in the two cases @xcite . the results that we present in the following are obtained using @xmath57 cooling steps and the definition of @xmath47 in eq . . we also mention that the results of this cooling procedure are compatible with those of other approaches proposed in the literature , see @xcite and app . [ sec : cgf ] . seven @xmath44 values are typically used in the simulations , going from @xmath58 to @xmath59 with steps @xmath60 ; when expressed in term of the renormalized parameter @xmath61 this range of @xmath44 values corresponds ( for the couplings used in this work ) to @xmath62 . we verify that this range of values is large enough to give a clear signal , but not so large to introduce systematic errors . the results of all tests performed using a smaller interval of @xmath44 values give perfectly compatible results . for the update we use a combination of standard heat - bath @xcite and overrelaxation @xcite algorithms , implemented _ la _ cabibbo - marinari @xcite using all the @xmath45 diagonal @xmath46 subgroups of @xmath3 . the topological charge is evaluated every 10 update steps , one update step being composed of a heath - bath and five overrelaxation updates for all the links of the lattice , updated in a mixed checkerboard and lexicographic order . the total statistic acquired for each coupling value is typically of @xmath63 measures . in order to apply the analytic continuation method in an actual computation , it is necessary to truncate the expansion in eq . ( or , which is the same , in eq . ) in order to fit the numerical data . we actually perform a global fit to the first four cumulants which , when rewritten in terms of @xmath44 , read @xmath64 an example of such global fit , with a truncation including up to @xmath65 terms in the ground state energy density ( i.e. setting @xmath66 ) , is reported in fig . [ fig : glob ] for the case of the @xmath23 gauge theory . to quantify the systematic error associated with this procedure we consider two different truncations : in one case all the terms of eq . up to @xmath67 are retained ( i.e. up to @xmath25 ) , while in the other case a truncation up to @xmath68 ( i.e. up to @xmath22 ) is used . both truncations nicely fit the numerical data and the estimates of the coefficient @xmath25 turn out to be compatible with zero in all the cases . this is not surprising , since even for @xmath21 only upper bounds on @xmath69 exist ( see , e.g. , @xcite ) and its value is expected to approach zero very quickly as the number of colors is increased , see eq . . we verify that the values of @xmath70 and @xmath22 obtained by using the two different truncations are perfectly compatible with each other , indicating that no sizable systematic error is introduced by the truncation procedure , see the example in fig . [ fig : sys ] . for this reason we decide to use the @xmath68 truncation to estimate @xmath42 , @xmath7 and @xmath22 , while the @xmath67 truncation is obviously needed to obtain an upper bound for @xmath69 . possible further systematic errors are checked by varying the fitted range of @xmath44 and verifying the stability of the fit parameters . terms : data refer to the @xmath71 lattice at coupling @xmath72 for the @xmath23 gauge theory . continuous lines are the result of a combined fit of the first four cumulants . ] in @xmath24 using different truncations of eq . . ] ( top , in lattice units ) and @xmath22 ( bottom ) on the lattice size for @xmath24 at coupling @xmath73 . ] for @xmath24 gauge theory . the results obtained in this work are compared with the determination of @xcite ( data have been slightly shifted horizontally to improve the readability ) . ] l|l|l|l ' '' '' @xmath0 & @xmath74 & @xmath75 & @xmath76 + 3 & 0.0289(13 ) & @xmath770.0216(15 ) & @xmath780.0001(3 ) + 4 & 0.0248(8 ) & @xmath770.0155(20 ) & @xmath770.0003(3 ) + 6 & 0.0230(8 ) & @xmath770.0045(15 ) & @xmath770.0001(7 ) + with the number of colors . the dashed line is the result of a best fit with a linear functional dependence . ] hypercubic lattices of size @xmath79 are used in all cases : they are expected to be large enough to provide the infinite - volume limit within the typical errors of our simulations ( see e.g. @xcite ) . this is explicitly verified in some test cases : for example the @xmath24 simulations at coupling @xmath73 were replicated on lattices of size @xmath80 and for the coupling @xmath81 on lattices with @xmath82 ; in all cases no statistically significant volume dependence is observed , see fig . [ fig : su6fs ] . the possibility of using such large lattices in the determination of @xmath22 and higher cumulants is a consequence of the numerical setup adopted , with simulations performed at imaginary @xmath1 values . before starting to discuss our main subject , namely the determination of @xmath22 and its large @xmath0 behavior , we show that our data reproduce the well known large @xmath0 scaling of @xmath83 . for @xmath21 we use results already available in the literature ( those reported in tab . 1 of @xcite ) and for the scale setting in the @xmath23 and @xmath24 cases we used the determination of the string tension reported in @xcite . for @xmath23 we observe no improvement with respect to the old results of @xcite , since the final error on @xmath83 is dominated by the error on the string tension . this is also the case for the final continuum result in the @xmath24 case , indeed we obtained @xmath84 to be compared with the value @xmath85 reported in ref . @xcite , however the continuum extrapolation of the new results is much more solid , as shown in fig . [ fig : chisu6 ] . the continuum values of @xmath83 for @xmath86 and @xmath87 are reported in tab . [ tab : contvalues ] , their scaling with @xmath0 is shown in fig . [ fig : chilargen ] and the result of a linear fit in @xmath88 gives @xmath89 which slightly improves the previous result of ref . @xcite . assuming the standard value @xmath90 mev , we obtain @xmath91 mev . as noted before , the dominant source of error in @xmath83 is the error on the string tension . as a consequence , to improve this result it would be enough to improve the precision of the @xmath92 determination or to use different observables to set the scale . since our main interest in this work is the analysis of the higher order cumulants @xmath8 , which are dimensionless , we have not pursued this investigation any further . in fig . [ fig : b2alln ] the results obtained for @xmath22 with @xmath93 are shown as a function of the ( square of the ) lattice spacing . the values of @xmath94 for the @xmath21 data have been computed using @xmath95 from @xcite to plot the @xmath22 data from @xcite . for @xmath23 , data are precise enough to perform a linear fit in @xmath94 and check for the systematics of the continuum extrapolation by varying the fit range ; the final result obtained is reported in tab . [ tab : contvalues ] ( for @xmath21 we use the value obtained in @xcite , where a similar analysis was performed ) . for the case of @xmath24 we could not reach lattice spacings as small as the ones used for @xmath21 and @xmath23 due to the dramatic increase of the autocorrelation times of the topological charge . to boost the sampling we tried using parallel tempering switches between different @xmath44 simulations but this did not result in a significant improvement ( see app . [ sec : pt ] for more details ) . as a consequence , the analysis of the @xmath24 results can not be as statistically accurate as those for @xmath21 and @xmath23 . in spite of this , a clear trend can be seen in the @xmath24 data shown in fig . [ fig : b2alln ] : @xmath22 flattens for @xmath96 , which is the region in which also @xmath23 data show no significant dependence on the lattice spacing , and we use the conservative estimate @xmath97 , which is displayed in fig . [ fig : b2alln ] by the horizontal blue dashed lines . for both @xmath23 and @xmath24 we increased significantly the precision of the @xmath22 determination with respect to results available in the literature : the previous estimates were indeed @xmath98 and @xmath99 from @xcite , to be compared with the numbers reported in tab . [ tab : contvalues ] . values on the lattice spacing for the case of three , four and six colors . see the main text for the details of the fitting procedure . ] with the number of colors . lines are result of a best fit performed using the linear dependence expected from large @xmath0 arguments ( dashed line fitting all data , full line fitting only those for @xmath100 ) , and adding also a quadratic contribution ( dotted - dashed line ) . ] the estimates of @xmath22 versus the number of colors are shown in fig . [ fig : b2largen ] . they decrease with increasing @xmath0 , strongly supporting a vanishing large-@xmath0 limit . fitting the data to the ansatz @xmath101 we obtain @xmath102 , fully supporting the @xmath88 scaling predicted by the large-@xmath0 scaling arguments . we now analyze the data assuming the @xmath88 scaling . some fits are shown in fig . [ fig : b2largen ] . the leading form @xmath103 of the expected @xmath0 dependence is used with two different fit ranges : in one case all the data are fitted , which gives @xmath104 , while in the other case only data with @xmath105 are used , obtaining @xmath12 . these results are in perfect agreement with those of the fit performed using also the nlo correction , i.e. to @xmath106 , that gives @xmath107 and @xmath108 , further indicating the absence of significant nlo correction . as our final estimate we report @xmath109 the previous estimate for this quantity in the literature was @xmath110 from @xcite and it should be stressed that not only the error of the final result gets reduced in the present study , but also the whole analysis is now much more solid , since the old result relied heavily on the @xmath21 result . some estimates of the @xmath67 coefficient @xmath25 of the ground - state energy density are shown in fig . [ fig : b4alln ] . various fits , in particular linear fits to take into account the leading scaling corrections , lead to the estimates reported in table [ tab : contvalues ] . as previously anticipated , they are still compatible with zero . assuming the large-@xmath0 scaling @xmath111 for @xmath112 , we obtain the bound latexmath:[\[\label{barb4 } our results for the large-@xmath0 coefficients @xmath114 may be compared with the analytical calculations by holographic approaches @xcite . in particular , a compatible ( negative sign ) result for @xmath115 is reported in ref . @xcite . finally in fig . [ fig : zalln ] we present our determinations of the renormalization factor @xmath42 for @xmath86 and @xmath87 and for the various lattice spacings used ( again @xmath21 data come from @xcite ) . it can be noted that all the data approximately collapse on a common curve , i.e. @xmath42 at fixed lattice spacing has a well defined large-@xmath0 limit . this behaviour could have been guessed by noting that the perturbative computation of @xmath42 performed in @xcite is in fact ( up to subleading corrections ) an expansion in the t hooft coupling @xmath116 . for @xmath9 . ] values on the lattice spacing for the case of three , four and six colors . ] the 2d @xmath4 ( euclidean ) lagrangian in the presence of a @xmath1 term is : @xmath117 where @xmath118 is an @xmath0-component complex vector satisfying @xmath119 , @xmath120 and @xmath121 . in order to analyze the large-@xmath0 behavior of the models one must introduce the lagrange multiplier fields @xmath122 and @xmath51 and perform a gaussian integration , thus obtaining the effective action @xmath123 - \frac{n}{2\,f } \int { \ensuremath{\mathrm{d}}}^2 x \,[i\,\alpha ] \\ & -i \frac{\theta}{4\pi } \int { \ensuremath{\mathrm{d}}}^2x \,\ , \epsilon_{\mu \nu } f_{\mu \nu}\ , \end{aligned}\ ] ] where now @xmath124 and @xmath125 . the multiplier fields become dynamical and in particular @xmath122 develops a massless pole , thus behaving as a bona fide ( abelian ) gauge field . the functional evaluation of @xmath126 in the large-@xmath0 limit can now be performed starting from the computation of the effective potential @xmath127 as a function of the constant vacuum expectation values @xmath128 and @xmath129 . in ref @xcite it has been shown that @xmath130 \ , \end{aligned}\ ] ] where @xmath131 and @xmath132 is the standard gamma function . it is now apparent that the natural expansion parameter for the large-@xmath0 evaluation of @xmath133 is @xmath134 @xcite . to the purpose of evaluating @xmath2 one must then solve the saddle point equations @xmath135 the first equation may be employed in order to find the function @xmath136 , independent of @xmath1 , and to generate the large @xmath0 effective lagrangian for the gauge degrees of freedom @xmath137 $ ] . the dependence on @xmath138 of the large @xmath0 vacuum energy can now be found immediately from the relationship @xmath139\ , \ ] ] where @xmath140 is the solution of the equation @xmath141 one must appreciate that solving the last equation implies a continuation from real to complex values of @xmath142 , that can be easily performed in the perturbative regime by observing that @xmath143 admits an asymptotic expansion in the even powers of @xmath142 . therefore it is possible to find a solution for purely imaginary @xmath142 in the form of a power series in the odd powers of @xmath138 . the first few terms of the expansion of @xmath140 are @xmath144 where @xmath145 is a square mass scale . beside the leading large-@xmath0 behavior of the topological susceptibility @xcite @xmath146 we obtain the rescaled coefficients @xmath147 of the @xmath1 expansion of the ground - state energy density : @xmath148 etc ... these results for @xmath8 extend those reported in ref . @xcite ( in particular they correct the value of @xmath149 ) . an analysis of several higher order coefficients shows that they are all negative and grow very rapidly , as one might have expected as a consequence of the nonanalytic dependence of the effective lagrangian on @xmath142 already observed in ref . @xcite . in turn this phenomenon can be related to the fact that the full - fledged dependence on @xmath1 of the vacuum energy for any finite value of @xmath0 must exhibit a @xmath150 periodicity which disappears in the large @xmath0 limit , thus implying a noncommutativity of the expansions and a vanishing radius of convergence in the variable @xmath151 . we finally mention that the large-@xmath0 behavior ( [ chiln ] ) of the topological susceptibility has been confirmed by numerical results of lattice @xmath4 models @xcite . instead , numerical results for the @xmath1-expansion coefficients @xmath8 have never been obtained yet . we study the large-@xmath0 scaling behavior of the @xmath1 dependence of 4d @xmath3 gauge theories and 2d @xmath4 models , where @xmath1 is the parameter associated with the lagrangian topological term . in particular , we focus on the first few coefficients @xmath8 of the expansion ( [ thdep ] ) of their ground - state energy @xmath2 beyond the quadratic approximation , which parametrize the deviations from a simple gaussian distribution of the topological charge at @xmath5 . we present a numerical analysis of monte carlo simulations of 4d @xmath3 lattice gauge theories for @xmath9 in the presence of an imaginary @xmath1 term . this method , based on the analytic continuation of the @xmath1 dependence from imaginary to real @xmath1 values , allows us to significantly improve earlier determinations of the first few coefficients @xmath8 . the results provide a robust evidence of the large-@xmath0 behavior predicted by standard large-@xmath0 scaling arguments , i.e. , @xmath10 . in particular , we obtain @xmath11 with @xmath12 . the results for the next coefficient @xmath25 of the @xmath1 expansion ( [ thdep ] ) show that it is very small , in agreement with the large-@xmath0 prediction that @xmath152 . assuming the large-@xmath0 scaling @xmath153 , we obtain the bound @xmath154 . an important issue concerns the consistency between the @xmath155 dependence in the large-@xmath0 limit and the @xmath156 periodicity related to the topological phase - like nature of @xmath1 . indeed , the large-@xmath0 scaling behavior is apparently incompatible with the periodicity condition @xmath157 , which is a consequence of the quantization of the topological charge , as indicated by semiclassical arguments based on its geometrical meaning for continuous field configurations . indeed a regular function of @xmath158 can not be invariant for @xmath159 , unless it is constant . a plausible way out @xcite is that the ground - state energy @xmath2 tends to a multibranched function in the large-@xmath0 limit , such as @xmath160 where @xmath161 is a generic function . @xmath2 is then periodic in @xmath1 , but not regular everywhere . as a consequence , the physical relevance of the large-@xmath0 scaling of the @xmath1 dependence should be only restricted to the power - law expansion ( [ thdep ] ) around @xmath5 , and of analogous expansions of other observables , thus to the @xmath0 dependence of their coefficients . our results significantly strengthen the evidence of the large-@xmath0 scaling scenario of the @xmath1 dependence , extending it beyond the @xmath162 expansion . we note that the large-@xmath0 scaling of the @xmath1 expansion is not guaranteed . indeed there are some notable cases in which this does not apply . for example this occurs in the high - temperature regime of 4d @xmath3 gauge theories : for high temperatures the dilute instanton - gas approximation is expected to provide reliable results and one gets ( see e.g. @xcite ) the result @xmath163 for any @xmath0 value . while the diga approximation is a priori expected to be valid only at asymptotically high temperatures , the switch from the large @xmath0 behavior to the instanton gas behavior occurs at the deconfinement transition temperature @xmath164 @xcite . the analytic continuation method that we used to compute the @xmath1 dependence can be also exploited in finite - temperature simulation , where it is typically even more efficient term increases the critical temperature @xcite . ] . as an example of its application in finite - temperature runs , fig . [ fig : b2 t ] presents an updating of the results presented in @xcite regarding the change of @xmath1 dependence across the deconfinement transition . while the results for @xmath165 were precise enough also in the original publication , the region below deconfinement is much more difficult ( see the discussion in @xcite ) . by combining the result for @xmath21 obtained in @xcite and the present ones for @xmath24 , in the left side of fig . [ fig : b2 t ] we can now display the continuum extrapolated zero temperature value of @xmath22 for @xmath24 and much more precise results for the finite temperature values of @xmath22 . these results confirm the results of @xcite to an higher accuracy : in the low - temperature phase the large-@xmath0 scaling holds true ( @xmath7 and @xmath8 being almost temperature independent ) , an abrupt change of the @xmath1 dependence of the free energy happens at deconfinement and the @xmath8 values do not depend on @xmath0 in the high - temperature phase . finally , this paper also reports a study of the large-@xmath0 @xmath1 dependence of the 2d @xmath4 models , whose leading behavior can be computed analytically . the results confirm the predicted large-@xmath0 scaling behavior @xmath166 for the coefficients of the expansion of the ground - state energy around @xmath5 . across the deconfinement transition for @xmath21 and @xmath24 ( @xmath167 is the reduced temperature defined by @xmath168 ) . the horizontal bands denote the zero temperature values . updated version of the figure originally presented in @xcite . ] we acknowledge useful discussions with francesco bigazzi and haris panagopoulos . numerical simulations have been performed on the galileo machine at cineca ( under infn project npqcd ) , on the csn4 cluster of the scientific computing center at infn - pisa and on grid resources provided by infn . it was shown in @xcite that cooling and the gradient flow with wilson action give identical results for the topological charge when the number of cooling steps @xmath169 is related to the dimensionless flow time @xmath170 by the relation @xmath171 . this relation was explicitly verified by simulation in @xmath21 gauge theory and it was later extended to improved gauge actions @xcite . during the early stages of this work we numerically verified on a subsample of configurations that , as theoretically expected , the same relation holds true also in the @xmath24 case . an example of the comparison between the two methods is reported in fig . [ cgf_fig ] , which displays some generic features : in @xmath24 the topological charge is much more stable than in @xmath21 , to reach a plateau of @xmath56 around 100 cooling steps are needed , for very prolongated smoothing both cooling and gradient flow evolutions tunnel to the topologically trivial configuration and the tunneling typically happens first for the gradient flow . configurations . ] parallel tempering @xcite , also know as replica exchange monte carlo , is the most widely used variant of the simulated tempering algorithm @xcite and was originally introduced to speed up simulations of spin glasses . in this appendix we report the results of some tests performed to investigate the effectiveness of parallel tempering to reduce the autocorrelation of the topological charge in @xmath24 . parallel tempering is typically used in systems with complicated energy landscapes to reduce the autocorrelation times . the original idea is to perform standard simulations at various temperatures ( with higher temperatures decorrelating faster than the lower ones ) and once in a while try to exchange the configurations at different temperatures with a metropolis - like step , that guarantees the detailed balance and hence the stochastic exactness of the algorithms . in this way the quickly decorrelating runs `` feed '' the slow ones and autocorrelations are drastically reduced . for the case of gauge theories the first natural choice would be to use parallel tempering between runs at different @xmath172 values , with the runs at large values of @xmath172 playing the role of the slowly decorrelating ones . although from a theoretical point of view this should work , one is faced with an efficiency problem : in order for the exchanges to be accepted with reasonable probability the @xmath172 values have to be close to each other , in fact closer and closer as the volume is increased , thus making the algorithm not convenient apart from extreme cases . see e.g. ref . @xcite for applications to the 2d @xmath4 models . this is the reason why alternative procedures have been proposed to work with different @xmath172 values , that are closer in spirit to the idea of multi - level simulations , see e.g. @xcite . an exchange was proposed every 4 measures , while in @xmath173 it was proposed every 40 measures . ] since we are using simulations at nonvanishing values of the @xmath1 angle , an alternative possibility is to perform the switch step of the parallel tempering between runs at different @xmath44 values @xcite . in this case there are no `` fast '' and `` slow '' runs , but since the mean values of the topological charge are different for different @xmath44 values , the switch step characteristic of the parallel tempering is expected to effectively increase the tunneling rate of the topological charge . as a testbed for the parallel tempering in @xmath44 we used @xmath24 with coupling @xmath174 and @xmath44 values from @xmath175 to @xmath176 with @xmath60 . using the standard algorithm described in sec . [ sec : numset ] the autocorrelation time of the square of the topological charge is around @xmath55 measures ( with 1 measure every 10 updates ) and we tried two different exchange frequencies in the parallel tempering : in the run denoted by @xmath177 an exchange was proposed every 4 measures , while in @xmath173 it was proposed every 40 measures ; in both the cases the proposed switch was accepted with a probability of about @xmath178 . l|l|l|l|l|l|l|l|l @xmath172 & @xmath179 & @xmath180 & @xmath181 & @xmath42 & @xmath182 & @xmath83 & @xmath22 & @xmath25 + ' '' '' 10.720 & 12 & 0.2959(14 ) & 0.80(5 ) & 0.09828(26 ) & @xmath183 & 0.02995(58 ) & -0.01628(62 ) & -0.00065(26 ) + ' '' '' 10.816 & 12 & 0.2642(7 ) & 1.6(1 ) & 0.11231(49 ) & @xmath184 & 0.02905(34 ) & -0.01658(79 ) & 0.00022(26 ) + ' '' '' 10.912 & 12 & 0.2368(6 ) & 2.5(5 ) & 0.12586(66 ) & @xmath185 & 0.02853(36 ) & -0.01617(67 ) & -0.00010(14 ) + ' '' '' 11.008 & 14 & 0.2160(8 ) & 6.0(5 ) & 0.13792(88 ) & @xmath186 & 0.02776(47 ) & -0.01526(84 ) & 0.00002(19 ) + ' '' '' 11.104 & 16 & 0.1981(5 ) & 14(1 ) & 0.1518(14 ) & @xmath187 & 0.02681(48 ) & -0.0167(11 ) & -0.00020(23 ) + l|l|l|l|l|l|l|l|l @xmath172 & @xmath179 & @xmath180 & @xmath181 & @xmath42 & @xmath182 & @xmath83 & @xmath22 & @xmath25 + ' '' '' 24.500 & 12 & 0.3420(19 ) * & 1.8(2 ) & 0.08338(25 ) & @xmath188 & 0.02828(63 ) & -0.01030(92 ) & 0.00008(60 ) + ' '' '' 24.624 & 10 & 0.3239(8 ) & 4.2(3 ) & 0.09386(54 ) & @xmath189 & 0.02412(27 ) & -0.0096(10 ) & 0.00011(31 ) + ' '' '' 24.768 & 12 & 0.2973(5 ) & 11(1 ) & 0.10278(73 ) & @xmath190 & 0.02375(23 ) & -0.0056(13 ) & 0.00009(42 ) + ' '' '' 24.845 & 12 & 0.2801(13 ) * & 22(3 ) & 0.10832(78 ) & @xmath191 & 0.02509(50 ) & -0.0049(11 ) & -0.00006(32 ) + ' '' '' 25.056 & 12 & 0.2534(6 ) & 80(10 ) & 0.11822(85 ) & @xmath192 & 0.02370(28 ) & -0.0047(10 ) & -0.00004(23 ) + the autocorrelation times of @xmath193 for the different values of @xmath44 and the various run are shown in fig . [ fig : pt ] . as was to be expected given the range of @xmath44 used in the parallel tempering , small @xmath44 runs decorrelate faster than the ones with large @xmath44 , and in all the cases an important decrease of @xmath181 is observed , that is more significant for the case of @xmath177 , in which exchanges were proposed at higher rate than in @xmath173 . in the best case the autocorrelation time was reduced by around an order of magnitude with respect to the standard runs . with respect to the single run at @xmath58 this reduction of @xmath181 is however not sufficient to compensate for the cpu time required to perform the update of the 11 replicas used in the parallel tempering , since simulations at nonvanishing @xmath44 values are about 2.5 more time consuming than simulation at @xmath58 . on the other hand , the idea of the method of analytic continuation in @xmath1 for computing the @xmath8 coefficients is exactly to use several @xmath44 values anyway , so that one can still hope to have an efficiency gain . this is however not the case : the simulations performed at different @xmath44 values are obviously correlated in the parallel tempering and , taking this correlation into account , no gain is apparently obtained by using the parallel tempering in the computation e.g. of @xmath22 . a possible explanation of this result ( i.e. strong reduction of the autocorrelation for the single @xmath44 run and strong correlation between different @xmath44 runs ) is the following . while on average the lattice operator @xmath56 is obviously related to the operator @xmath47 , the specific form of their uv fluctuations can be different and are larger , in particular , for @xmath56 . as a consequence , the metropolis test for the exchange of configurations , which is solely based on @xmath56 , could be easier , but then not accompained by a fast decorrelation of the global topological content @xmath47 after the exchange , which would proceed with a decorrelation time likely comparable with the @xmath181 of the standard simulation . if this interpretation is correct , then the observed reduction of the autocorrelation times at fixed @xmath44 is just a consequence of the reshuffling of the configurations induced by the exchanges , which are very frequent due to the largest uv fluctuations of @xmath56 . the update of the global information contained in the time histories at different @xmath44 values , which is the one used in the global fit , suffers instead from the usual autocorrelation problems . one possibility , in order to improve the performance of the parallel tempering algorithm , could be to adopt an improved discretization of @xmath56 , e.g. a smeared definition of the topological charge density , such as those considered in refs . @xcite ; this would require to abandon the heatbath and overrelaxation algorithms in favour of an hybrid monte carlo approach @xcite . however , it is not clear a priori whether that would result in an improvement of the global decorrelation properties , i.e. in a final net gain , or rather in a deterioration of the autocorrelation time for the single trajectory at fixed @xmath44 , because of the rarer configuration reshuffling . in tab . 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we study the large-@xmath0 scaling behavior of the @xmath1 dependence of the ground - state energy density @xmath2 of four - dimensional ( 4d ) @xmath3 gauge theories and two - dimensional ( 2d ) @xmath4 models , where @xmath1 is the parameter associated with the lagrangian topological term . we consider its @xmath1 expansion around @xmath5 , @xmath6 where @xmath7 is the topological susceptibility and @xmath8 are dimensionless coefficients . we focus on the first few coefficients @xmath8 , which parametrize the deviation from a simple gaussian distribution of the topological charge at @xmath5 . we present a numerical analysis of monte carlo simulations of 4d @xmath3 lattice gauge theories for @xmath9 in the presence of an imaginary @xmath1 term . the results provide a robust evidence of the large-@xmath0 behavior predicted by standard large-@xmath0 scaling arguments , i.e. @xmath10 . in particular , we obtain @xmath11 with @xmath12 . we also show that the large-@xmath0 scaling scenario applies to 2d @xmath4 models as well , by an analytical computation of the leading large-@xmath0 @xmath1 dependence around @xmath5 .

in our recent work , the co@xmath0 molecule was identified for the first time in the spectra of brown dwarfs observed with the infrared astronomical satellite _ akari _ * hereafter referred to as paper i ) . we tried to interpret the observed behavior of the co@xmath0 band with the use of the model photosphere of brown dwarfs , referred to as the unified cloudy model ( ucm ; * ? ? ? * ; * ? ? ? * ) . in modeling the photospheres of brown dwarfs , one problem is how to consider the chemical composition , since no direct abundance analysis is known for brown dwarfs . we thought it reasonable to assume a typical composition for the disk stellar population such as the sun . however , the solar composition itself experienced drastic changes in the past decades , and the true chemical composition of the sun is still by no means well established . nevertheless , we thought it appropriate to use the latest version of the solar abundances as possible proxies for the chemical abundances in the brown dwarfs . in our earlier version of ucms @xcite , we assumed the solar abundances largely based on the classical lte analysis using one dimensional ( 1d ) hydrostatic model photospheres and , in particular , c & o abundances were log@xmath5 and log@xmath6 on the scale of log@xmath9 ( e.g. * ? ? ? * ; * ? ? ? * ) . at about the same time as we were computing our first version of ucms , a new result for the solar c & o abundances based on three dimensional ( 3d ) time - dependent hydrodynamical model of the solar photosphere was published @xcite . since the classical 1d model may be too simplified for the real solar photosphere , this new approach seemed to be a useful contribution to the solar abundance analysis . the current version of ucms @xcite is thus based on this new result ( log@xmath7 and log@xmath8 ) by @xcite as noted elsewhere @xcite . the new c & o abundances are about 0.2dex smaller as compared to the classical values referred to above . we applied our current version of ucms to the brown dwarfs observed with _ akari _ in paper i , and we could explain about half of our sample of spectra almost perfectly . for the other half of our targets , we could explain the overall seds by this version of ucms , but we could not explain their strong co@xmath0 bands . one explanation is that this may be due to an unknown process related to co@xmath0 , since anomalously strong co band depths have also been explained by a special process now known as vertical mixing . however , we happened to try our old version of ucms based on the classical c & o abundances and found that the co@xmath0 band appeared to be much stronger in the spectra based on the old models than on the present models . at the first glance , this is rather surprising because c & o abundances in the old models are only about 0.2dex larger than those in the present models . however , we realized immediately that the co@xmath0 abundance is extremely sensitive to both c & o abundances because the co@xmath0 abundance depends on the cube of c & o abundances ( @xmath10 ) . we recall that the strong dependence of the co@xmath0 abundance on metallicity , [ fe / h],= log@xmath11 - log@xmath12 . ] was previously known by a detailed thermochemical analysis of the c , n , and o bearing gaseous molecules @xcite . the above result demonstrates that at least two different series of model photospheres are needed for the analysis of the co@xmath0 band observed with _ akari_. for this purpose , we reconsider our old version of ucms based on the classical c & o abundances to represent a case of rather high c & o abundances . our current version of ucms based on the new c & o abundances will serve as representing a case of the reduced c & o abundances compared to the old version of the ucms . an important implication of this result is that the metallicity ( c & o abundances ) in brown dwarfs should have a variety of values . the interpretation and analysis of the spectra of cool dwarfs already have a rather long history ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and the effect of metallicity has been discussed by some authors . for example , @xcite have measured the strengths of the major h@xmath0o and ch@xmath13 bands in the 1.02.5@xmath1 m region in a large sample of t dwarfs , and found that the resultant spectral indices plotted against spectral type revealed considerable scatter . several reasons for this result including the effects of dust , gravity , and metallicity have been considered , but it appeared difficult to separate the effect of metallicity from the remaining paremeters . @xcite have shown that the effect of metallicity on the seds of t dwarfs should be significant , but noted that other parameters such as gravity can affect the seds similarly . this result again showed the difficulty in determining metallicity uniquely from seds . also , the so - called `` blue '' l dwarfs classified as l subdwarfs have been interpreted to have low metallicity with [ fe / h ] from @xmath14 to @xmath15 ( e.g. * ? ? ? * ) , but those l dwarfs with unusually blue near - infrared colors can also be explained by a patchy cloud model @xcite . the brief survey outlined above reveals that the problem of metallicity in brown dwarfs is still unresolved . in this paper , we will show clear evidence of metallicity variations in brown dwarfs for the first time . in fact , the most important significance of the discovery of co@xmath0 with _ akari _ is that it demonstrated the variations of the c & o abundances by at least 50% in brown dwarfs and that it provided a means by which to estimate the c & o abundances in very cool dwarfs . in paper i , we have analyzed the _ spectra and discussed the basic physical parameters of our objects in detail . there we have applied the conventional method based on a direct comparison of the observed and predicted spectra . in this paper , we examine the results of paper i by a more detailed numerical method in section [ sec : fitcmp ] , and we confirm that the physical parameters determined in paper i mostly agree with those based on the reduced - chi - square minimization method within the estimated errors . a problem , however , is that an adequate application of such a rigorous numerical method requires the input data of sufficient accuracy . unfortunately , the input data our present models of brown dwarfs are not precise enough for this purpose , as discussed in section 4.5 of paper i. therefore the numerical method does not necessarily provide the best answer and the traditional fitting method `` by eye '' can still be useful for some cases . for these reasons , we use the physical parameters determined in paper i and adopt the same approach as paper i , eye - fitting , throughout this paper . cccl series & log @xmath2 @xmath16 & log @xmath3 @xmath16 & note on chemical composition + ucm - a & 8.60 & 8.92 & 1d solar abundances ( e.g. * ? ? ? * ) @xmath17 + ucm - c & 8.39 & 8.69 & 3d solar abundances ( e.g. * ? ? ? * ) @xmath18 + notes . + a ) the logarithmic abundance on the scale of log @xmath19 = 12.0 . b ) actually , we have applied a slightly updated version as summarized in table 1 of @xcite . c ) a full listing of the abundances is given in http://www.mtk.ioa.s.u-tokyo.ac.jp/@xmath20ttsuji/export/ucm/tables/table1.dat . our two series of ucms are referred to as ucm - a and ucm - c , and they only differ in c & o abundances as summarized in table [ tbl : tbl1 ] . the ucm - a series is based on the classical c & o abundances , which we refer to as 1d solar abundances for simplicity , and the ucm - c series on the new abundances , which we refer to as 3d solar abundances . we first examine the effects of c & o abundances on the thermal structure of the photosphere . for this purpose , the models of the ucm - c series are taken from our databasettsuji / export / ucmlm/ and /ucm/. ] . since our code to compute ucms has been modified to some extent over the last 10 years , we recompute all the models of the ucm - a series used in the present paper . therefore the models of the ucm - a and ucm - c series are now computed by exactly the same code , except for the rosseland and planck mean opacities , which of course differ according to the chemical composition adopted . we show a simple comparison of the photospheric structures of the ucm - a and ucm - c series for the cases of @xmath21 = 900 , 1200 , and 1500k in figure [ fig : fig1 ] . other parameters such as @xmath22 and log @xmath23 are chosen to be those actually found for our objects ( see table [ tbl : fit ] ) . inspection of figure [ fig : fig1 ] reveals that the models of the ucm - a series shown by the dashed lines are generally warmer by up to about 100k as compared to the models of the ucm - c series shown by the solid lines . since the major opacity sources such as co and h@xmath0o are more abundant in the ucm - a than in the ucm - c series , the blanketing effect due to molecular bands should be more effective and hence the models of the ucm - a series are warmer than those of the ucm - c series . next , we examine the effects of c & o abundances on the co@xmath0 and other molecular abundances . we present the abundances of h@xmath0o , co , co@xmath0 , and ch@xmath13 for the case of @xmath21 = 1500k , @xmath24 = 1700k , and @xmath25 for the models of the ucm - c ( applied to 2mass j152322@xmath263014 in section [ sec : spc ] ) and ucm - a ( applied to sdss j083008@xmath264828 ) series in figure [ fig : fig2 ] as the solid and dashed lines , respectively . the increased c & o abundances result in the increases of co , co@xmath0 , and h@xmath0o abundances as expected . the increase of the co@xmath0 abundance in the ucm - a series is quite significant for the reason noted before . on the contrary , the ch@xmath13 abundance shows a decrease in the ucm - a series and this unexpected result may be because the direct effect of the increased carbon abundance on the ch@xmath13 abundance is superseded by the dissociation of ch@xmath13 due to the elevated temperatures in the model of the ucm - a series ( figure [ fig : fig1 ] ) . as another example , we show the case of @xmath21 = 1200k , @xmath24 = 1900k , and @xmath25 in figure [ fig : fig3 ] . the results are again shown for the ucm - c and ucm - a series with the solid and dashed lines , respectively . in this case , the effects of the abundance changes are more pronounced , especially for co@xmath0 . we will see in section [ sec : spc ] that the case of the ucm - a series is approximately realized in 2mass j055919@xmath271404 in which the co@xmath0 band appears to be very strong . we compare the observed spectra of six brown dwarfs with the predicted ones based on the models of the ucm - a and ucm - c series in figure [ fig : six ] ( a)(f ) . the effect of c & o abundances on the spectra of brown dwarfs can be seen most clearly in a comparison of 2mass j152322@xmath263014 and sdss j083008@xmath264828 shown in figure [ fig : six](c ) and figure [ fig : six](d ) , respectively . these two brown dwarfs were found to have nearly the same physical parameters ( @xmath28k , @xmath29k , and log@xmath23 = 4.5 ) but the spectra looked to be quite different ( paper i ) . in particular , the co@xmath0 band at 4.2@xmath1 m appeared to be much stronger in sdss j083008@xmath264828 than in 2mass j152322@xmath263014 . in 2mass j152322@xmath263014 , the observed spectrum could be accounted for by the model of the ucm - c series ( paper i ) , as confirmed by curve 1 in figure [ fig : six](c ) : especially , the regions of the h@xmath0o 2.7@xmath1 m and the co@xmath0 4.2@xmath1 m bands as well as the @xmath30-branch of ch@xmath13 band appeared to be well explained by the model of the ucm - c series . on the other hand , the predicted spectrum based on the model of the ucm - a series shown by curve 2 can not explain those features . in sdss j083008@xmath264828 shown in figure [ fig : six](d ) , the observed spectrum could not be accounted for by the model of the ucm - c series ( paper i ) , as confirmed by curve 1 . on the other hand , the strong co@xmath0 band at 4.2@xmath1 m can now be explained reasonably well by the model of the ucm - a series , as shown by curve 2 in figure [ fig : six](d ) . thus the large depression due to the co@xmath0 band turns out to be due to the high c & o abundances in a lte model . it is to be noted that this result is due to the increase of both c & o abundances . in fact , a change of the c abundance alone , for example , produced only minor change on the co@xmath0 band strength ( paper i ) . the large depression over the 2.7@xmath1 m region mostly due to h@xmath0o can also be better explained with the model of the ucm - a series . on the contrary , the @xmath30-branch of the ch@xmath13 band appears to be weaker with the model of the ucm - a than with that of the ucm - c series , consistent with the decrease of the ch@xmath13 in the ucm - a compared to the ucm - c series ( figure [ fig : fig2 ] ) . also , the observed ch@xmath13 @xmath30-branch indeed agrees better with the predicted one based on the model of ucm - a . thus , we conclude that the ucm - a series should be applied to sdss j083008@xmath264828 rather than the ucm - c series . clcccccc no . & object & ucm series@xmath31 & @xmath21(k ) & @xmath22(k ) & @xmath32 & @xmath33 & @xmath34(k)@xmath35 + 1 & & ucm - c & 1800 & 1800 & 5.5 & 0.804 & @xmath36 + 2 & & ucm - c & 1700 & 1700 & 4.5 & 0.716 & @xmath37 + 3 & & ucm - c & 1500 & 1700 & 4.5 & 0.684 & @xmath38 + 4 & & ucm - a & 1500 & 1700 & 4.5 & 0.610 & @xmath39 + 5 & & ucm - a & 1200 & 1900 & 4.5 & 1.122 & @xmath40 + 6 & & ucm - a & 900 & @xmath41 & 4.5 & 0.676 & @xmath42 + notes . + a ) the ucm series applied and indicates approximate c & o abundances ( see table [ tbl : tbl1 ] ) . b ) radius @xmath43 relative to the jupiter s radius @xmath44 . c ) @xmath45(empirical values by @xcite ) @xmath27 @xmath21(column 4 in this table ) . in figure [ fig : six](a ) , we compare the observed spectrum of sdss j053952@xmath270059 with the predicted ones based on the models of the ucm - a and ucm - c series . we already know that this spectrum is well explained by a model of ucm - c series in paper i ( curve 1 ) . on the other hand , the observed spectrum can not be explained by a model of the ucm - a series of the same parameters ( curve 2 ) . we obtain more or less similar result for sdss j144600@xmath260024 , namely the observed spectrum of this object can be well explained by the model of the ucm - c ( curve 1 in figure [ fig : six](b ) ) , but not with that of the ucm - a series ( curve 2 ) . in figure [ fig : six](e ) , we examine the case of 2mass j055919@xmath271404 in which the observed co@xmath0 band is very strong . we could not explain the co@xmath0 and co bands in this object with our ucm - c series ( paper i ) , as confirmed by curve 1 . but we can now explain the strong co@xmath0 band approximately with our model of the ucm - a series , as shown by curve 2 . this result is fairly consistent with the very large increase of the co@xmath0 abundance for this model as noted in figure [ fig : fig3 ] . the fit of curve 2 based on the ucm - a series can in principle be improved further by a fine tuning of c & o abundances . but we defer such a detailed abundance analysis to future works and we only note here that the strength of the co@xmath0 band is adjustable by changing c & o abundances . lastly , we examine the case of 2mass j041519@xmath270935 in figure [ fig : six](f ) . this is a case in which the fitting parameters had to be changed from the result of paper i : we previously found that @xmath46k based on the model of the ucm - c series . however , the h@xmath0o band at 2.7@xmath1 m appears to be too strong if we apply the same parameters to the model of the ucm - a series . as shown in figure [ fig : j0415](a ) , the observed spectrum can be explained by the model of a higher effective temperature of @xmath47k with the model of the ucm - a series . also , we find that the case of log@xmath48 provides the best fit as shown in figure [ fig : j0415](b ) . thus , we conclude that ( @xmath49 , @xmath24 , log@xmath23 ) = ( 900 , @xmath50 , 4.5 ) for 2mass j041519@xmath270935 for the models of the ucm - a series . we show the best possible predicted spectrum based on the model of the ucm - c series with @xmath46k and that based on the model of the ucm - a series with @xmath47k by curve 1 and curve 2 , respectively , in figure [ fig : six](f ) . we see that the model of the ucm - a series appears to match better with the observed spectrum than the model of the ucm - c series . finally , we summarize the basic parameters of the six brown dwarfs in table [ tbl : fit ] . the major change from table 4 of paper i is to have introduced abundance classes in column 3 indicated by the ucm series applied : ucm - a and ucm - c means that c & o abundances should be close to those of the 1d and 3d solar abundances , respectively . the physical parameters are re - examined with the models of the ucm - a series in the same manner as in paper i. a major change in physical parameter is @xmath21 for 2mass j041519@xmath270935 , which is changed from 800k to 900k as shown in figure [ fig : j0415](a ) . as for other five objects , the physical parameters for the models of the ucm - a series remain the same as for the ucm - c series . we confirm that the overall seds based on the models of the ucm - a and ucm - c series of the same physical parameters agree well ( see figure [ fig : six ] ) even though some local features due to the co@xmath0 and h@xmath0o bands differ somewhat . thus , it is natural that the physical parameters mainly derived from the fits of the overall seds remain the same for ucm - a and ucm - c series . the @xmath51 values for sdss j083008@xmath264828 , 2mass j055919@xmath271404 , and 2mass j041519@xmath270935 are changed slightly , reflecting the changes of models from the ucm - c to the ucm - a series . fitting of the model spectra to the observed spectra is carried out by `` eyes '' throughout in this paper as well as paper i. the reader might wonder whether the eye - fitting can find reliable `` best '' models for various objects . in this subsection we assess our eye - fitting by comparing with the numerical fitting results . we evaluate the goodness of fit by the reduced - chi - square ( hereafter @xmath52 ) defined as , @xmath53 where @xmath54 and @xmath55 are fluxes of the observed and model spectra at @xmath56-th wavelength grid , respectively . the uncertainty of the observed flux is indicated as @xmath57 , and @xmath58 is the degree of freedom . @xmath59 is the scaling factor that minimizes @xmath52 and is given by @xmath60 these definitions are in principle equivalent to `` goodness - of - fit '' statistics @xmath61 by @xcite for the equal weight case . from our experience in paper i we know that the current ucm can not fit the observations beyond 4@xmath1 m at least in some objects . therefore we limit the wavelength range for calculating @xmath52 to 2.644.15@xmath1 m ( cf . model spectra are available from 2.64@xmath1 m and co@xmath0 band starts from 4.17@xmath1 m ) . rccccc no . & @xmath21 & @xmath32 & @xmath22 & @xmath59 & @xmath52 + & ( k ) & & ( k ) & ( @xmath62 ) + + & * 1800 * & * 5.5 * & * 1800 * & * 6.32 * & * 1.127 * + 2 & 1900 & 5.5 & 1800 & 5.82 & 1.206 + 3 & 1900 & 5.0 & 1800 & 6.37 & 1.288 + + 1 & 2000 & 4.5 & 1700 & 1.53 & 0.496 + 2 & 1900 & 4.5 & 1700 & 1.61 & 0.513 + 3 & 1800 & 4.5 & 1700 & 1.69 & 0.553 + * 10 * & * 1700 * & * 4.5 * & * 1700 * & * 1.79 * & * 0.695 * + + 1 & 1600 & 5.5 & 1700 & 1.89 & 0.675 + 2 & 1600 & 5.0 & 1700 & 1.91 & 0.733 + 3 & 1700 & 5.5 & 1800 & 1.69 & 0.740 + * 4 * & * 1500 * & * 4.5 * & * 1700 * & * 2.45 * & * 0.793 * + + 1 & 1600 & 4.5 & 1800 & 3.78 & 0.679 + 2 & 1700 & 4.5 & 1800 & 3.52 & 0.711 + 3 & 1800 & 5.0 & 1900 & 3.44 & 0.746 + * 9 * & * 1500 * & * 4.5 * & * 1700 * & * 4.54 * & * 0.841 * + + 1 & 1200 & 4.5 & @xmath41 & 21.8 & 0.389 + * 2 * & * 1200 * & * 4.5 * & * 1900 * & * 20.3 * & * 0.418 * + 3 & 1100 & 4.5 & 1700 & 25.3 & 0.482 + + & * 800 * & * 4.5 * & * @xmath41 * & * 29.1 * & * 0.170 * + 2 & 900 & 4.5 & @xmath41 & 20.6 & 0.173 + 3 & 900 & 5.0 & @xmath41 & 17.2 & 0.195 + notes . + models of the ucm - c series are adopted throughout as in paper i. the eye - fitting results quoted from paper i are indicated in bold - face . in table [ tbl : fitcmp ] we list the three best models based on the @xmath52 value and the model quoted in paper i ( by eye - fitting ) for each object in our sample . the eye - fitting results are indicated in bold - face . the models selected by the eye - fitting achieve the minimum @xmath52 for sdss j053951@xmath270059 ( l5 ) and 2mass j041519@xmath270935 ( t8 ; for this particular object we search for the best model among those of @xmath63 for the reason outlined in section 4.3.6 of paper i ) and the second minimum @xmath52 for 2mass j055919@xmath271404 ( t4.5 ) . the difference of @xmath52 between the first and second model for the last case is tiny , and the model parameters are within the uncertainty we stated in paper i ( @xmath64 k for @xmath21 and @xmath22 , and @xmath65 dex for @xmath32 ) . for two late - l objects , 2mass j152322@xmath263014 ( l8 ) and sdss j083008@xmath264828 ( l9 ) , the differences in the model parameters are mostly within the uncertainty of eye - fitting , although the models used in paper i are not included in the numerical best three for these dwarfs . altough @xmath32 of 2mass j152322@xmath263014 differs by 1.0 dex , the eye - selected model is in the 4th position in the list , and we consider that it is still in the accepted range . a significant difference between the two methods is found in the l5 dwarf sdss j144600@xmath260024 . the numerical fitting suggests @xmath66 k as the best , which is 300 k higher than the one selected by the eye - fitting . the second and third are of @xmath21 = 1900 and 1800 k. @xmath32 and @xmath22 are the same in all fours models . in fact we regarded such high @xmath21 values to be unrealistic for an l5 dwarf , and did not consider them in the eye - fitting . the empirical @xmath21 derived by @xcite of this object is even low as 1592 k. a key feature is the ch@xmath13 3.3@xmath1 m band , which appears only in the @xmath67 k model . the observed spectrum of this source is rather noisy and the detection of this band is marginal . if the tiny dip seen near 3.3@xmath1 m in the observed spectrum is actually the ch@xmath13 band , the eye - fitting results , even if it is not perfect , are justified . the nature of mid - l to early - t type dwarfs are still under debate and their effective temperatures might actually spread to higher values . incomplete atmosphere modeling is another possible reason . as we discuss in paper i , current atmosphere models for brown dwarfs are still exploratory and the ucm is one of such models . there are many physical and chemical processes not yet understood in brown dwarfs . these problems shall be attacked and eventually incorporated into future model atmospheres , but it is beyond the scope of the current paper . in addition , some of our _ akari _ spectra have relatively low s / n . under such circumstances , numerical fitting may not always return a unique and physically reasonable solution . on the other hand the eye - fitting would give weight to some key features and consider balance over the wavelength range . our goal in this paper is to highlight the effects of chemical abundance in the brown dwarf atmosphere . the comparisons described above well demonstrate that the eye - fitting is , even if it is not perfect , useful to find reasonable models for our purpose . therefore , we apply the model parameters based on our eye - fitting for the six objects including j144600@xmath260024 in the analysis throughout this paper . it is noted that the differences between the models appear much more prominently over the shorter wavelength range especially in @xmath68-band , even if the spectra in the _ akari _ wavelength range are similar to each other . consideration of near - infrared data such as 2mass photometry or ground - based spectroscopy will enable constraining the model parameters better ( sorahana et al . in preparation ) . such improvements in spectral range will also help us to evaluate the goodness of the fit in the wavelengths beyond 4@xmath1 m . the very strong co@xmath0 feature observed with _ akari _ in some brown dwarfs has remained a puzzle ( paper i ) , but we find that this is simply due to the effect of c & o abundances . generally , a small change in the chemical composition does not have a large effect on the predicted spectra at low resolution nor on the thermal structure of the photosphere in hotter stars . in fact , this is the reason why one dimensional spectral classification ( e.g. harvard system ) is possible for such stars . but in the case of cool stars , a small change of the chemical composition is amplified in molecular abundances . a drastic example is the spectral branching of cool giant stars into m , s , and c types according to whether the c / o ratio is smaller or larger than unity . in cool dwarfs , the change of c & o abundances also produces significant effect on the strengths of molecular bands as well as on the photospheric structures because of the large molecular opacities . the results of section [ sec : spc ] reveal that half of the brown dwarf spectra observed with _ akari _ ( i.e. sdss j053952@xmath270059 , sdss j144600@xmath260024 , and 2mass j152322@xmath263014 ) can be fitted by the predicted spectra based on the models of the ucm - c series ( paper i ) . although the fits are by no means perfect , the fits with the predicted spectra based on the ucm - c series are better than those based on the umc - a series . for this reason , c & o abundances in these three brown dwarfs should be closer to the recent 3d solar abundances rather than to the classical 1d solar abundances . on the other hand , the remaining half ( i.e. sdss j083008@xmath264828 , 2mass j055919@xmath271404 , and 2mass j041519@xmath270935 ) of our sample can be reasonably accounted for by the models of the ucm - a series . therefore , c & o abundances in these three objects should be closer to the classical 1d solar abundances rather than to the recent 3d solar abundances . since [ fe / h ] of the main sequence stars in the galactic disk covers the range from @xmath69 to + 0.2 ( e.g. * ? ? ? * ) , the same metallicity distribution may apply to brown dwarfs . it is certainly only by chance that the brown dwarfs we have observed are divided into two groups by c & o abundances . our sample is too small to investigate the metallicity distribution in brown dwarfs , and we hope that this problem can be pursued further with a larger sample . the problem of the solar c & o abundances is still under intensive discussion ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? although our problem here is not the solar composition , it is of some interest to know which of the proposed solar composition results more realistic for the sun . if the recent 3d result is more realistic for the sun , three of our sample may have about solar composition and the remaining three may be about 0.2dex more metal rich . this means that the proportion of metal rich objects with the highest [ fe / h ] of about + 0.2 is quite high in our present sample of brown dwarfs . we have shown that c & o abundances have significant effects on the 2.55.0@xmath1 m spectra of brown dwarfs , and we now examine their effect on the 0.92.5@xmath1 m spectra . as examples , we compare the predicted 0.92.5@xmath1 m spectra of the models of ucm - a and ucm - c series for the case of @xmath29k , @xmath28k , and log@xmath23 = 4.5 , together with those for the 2.55.0@xmath1 m spectra in figure [ fig : nir ] . the major difference between the ucm - a and ucm - c series is that h@xmath0o bands at 1.1 , 1.4 , 1.9 , and 2.7@xmath1 m are all stronger in the ucm - a ( curve 2 in figure [ fig : nir ] ) than in the ucm - c series ( curve 1 ) , and this is due to a direct effect of the increased oxygen abundance ( see figure [ fig : fig2 ] ) . thus the h@xmath0o band strengths depend sensitively on oxygen abundance , and we may hope to determine oxygen abundance from the h@xmath0o bands . however , we must remember that the h@xmath0o band strengths also depend on other parameters such as @xmath24 , @xmath49 , and log@xmath23 , and we should encounter the same difficulty due to a degeneracy of the parameters as noted before by other authors ( e.g. * ? ? ? * ; * ? ? ? this fact reconfirms the unique role of co@xmath0 as a metallicity ( c & o abundances ) indicator in brown dwarfs . thanks to the _ akari _ spectra , we are for the first time able to demonstrate that the metallicity , more specifically c & o abundances , are important parameters to understand brown dwarf atmospheres . until now , we have assumed that it was sufficient to use one sequence of model photospheres based on a representative chemical composition in analyzing low resolution spectra of cool dwarfs . we must now admit that such an assumption is inappropriate , and we should consider abundance effects more carefully , especially of c & o , in our future analysis of cool dwarfs . also , we can not use any solar composition for cool dwarfs unless this substitution can be justified by a direct analysis of the spectra of cool dwarfs . it is true that a detailed abundance analysis of brown dwarfs is difficult especially with low resolution spectra , but well defined molecular bands , even at low resolution , can be potential abundance indicators . we know already that co@xmath70 is a fine indicator of c & o abundances . unfortunately , however , co@xmath0 is accessible only from space telescopes and , moreover , spectroscopic observations in the near infrared are mostly neglected by the recent space infrared missions . from the view point of the study on cool dwarfs ( and other cool stars ) , the importance of observing the near infrared spectra ( especially between 2.5 and 5.0@xmath1 m ) from space can not be emphasized too much . although the spectra of brown dwarfs appear to be complicated , we are now convinced that the spectra of brown dwarfs can basically be understood on the basis of the lte model photospheres , but only if the chemical composition is properly considered . this is a reasonable result for such high density photospheres as of brown dwarfs in which frequent collisions easily maintain thermal equilibrium . thus the chemical composition is the most important ingredient in the interpretation and analysis of even low resolution spectra . now , with better observed data for brown dwarfs including those from space , analysis of the spectra and abundance determination can be done iteratively for brown dwarfs as for ordinary stars . finally , we must remember that a major difficulty in the analysis of the spectra of brown dwarfs is that we have no model of comparable accuracy as for ordinary stars yet . for this reason , even the accurate numerical method such as outlined in section [ sec : fitcmp ] can not be infallible . in fact , we have no model reproducing all the observable features correctly , and the model found by the numerical method as well as by the eye - fitting method may prove incorrect even if they are relatively satisfactory among the models currently available . within this limitation , we hope that our main results on the differential effects of c & o abundances are relatively free of present brown dwarf model uncertainties . we thank an anonymous referee for critical reading of the text and for invaluable suggestions regarding the method of analysis of the spectra of brown dwarfs . we are grateful to dr . poshak gandhi for his careful checking of the manuscript and many suggestions to improve the text . this research is based on observations with _ akari _ , a jaxa project with the participation of esa . we acknowledge jsps / kakenhi(c ) no.22540260 ( pi : i. yamamura ) . allende prieto , c. , lambert , d. l. , & asplund , m. 2002 , , 573 , l137 anders , e. , & grevesse , n. 1989 , geochimi . acta , 53 , 197 asplund , m. , grevesse , n. , sauval , a. j. , & scott , p. 2009 , , 47 , 481 ayres , t. r. , plymate , c. , & keller , c. u. 2006 , , 165 , 618 burgasser , a. j. , burrows , a. , & kirkpatrick , j. d. 2006a , , 639 , 1095 burgasser , a. j. , geballe , t. r. , leggett , s. k. , kirkpatrick , j. d. , & golimowski , d. a. 2006b , , 637 , 1067 burgasser , a. j. , witte , s. , helling , c. , sanderson , r. e. , bochanski , j. j. , & hauschildt , p. h. 2009 , , 697 , 148 caffau , e. , ludwig , h .- g . , steffen , m. , ayres , t. r. , bonifacio , p. , cayrel , r. , freytag , b. , & plez , b. 2008 , , 488 , 1031 cushing , m. c. , et al . 2008 , , 678 , 1372 edvardsson , b. , andersen , j. , gustafsson , b. , lambert , d. l. , nissen , p. e. , & tomkin , j. 1993 , , 275 , 101 folkes , s. l. , pinfield , d. j. , kendall , t. r. , & jones , h. r. a. 2007 , , 378 , 901 grevesse , n. , lambert , d. l. , sauval , a. j. , van dishoeck , e. f. , farmer , c. b. , & norton , r. h. 1991 , , 242 , 488 griffith , c. a. , & yelle , r. v. 1999 , , 519 , l85 kirkpatrick , j. d. 2005 , , 43 , 195 leggett , s. k. , et al . 2009 , , 695 , 1517 leggett , s. k. , marley , m. s. , freedman , r. , saumon , d. , lie , m. c. , geballe , t. r. , golimowski , d. a. , & stephens , d. c. 2007a , , 667 , 537 leggett , s. k. , saumon , d. , marley , m. s. , geballe , t. r. , & fan , x. 2007b , , 655 , 1079 lodders , k. , & fegley , b. 2002 , icarus , 155 , 393 marley , m. s. , saumon , d. , & goldblatt , c. 2010 , , 723 , l117 noll , k. s. , geballe , t. r. , & marley , m. s. 1997 , , 489 , l87 oppenheimer , b. r. , kulkarni , s. r. , matthews , k. , & van kerkwijk , m. h. 1998 , , 502 , 932 saumon , d. , geballe , t. r. , leggett , s. k. , marley , m. s. , freedman , r. s. , lodders , k. , fegley , b. , jr . , & sengupta , s. k. 2000 , , 541 , 374 stephens , d. c. , et al . 2009 , , 702 , 154 tsuji , t. 2002 , , 575 , 264 tsuji , t. 2005 , , 621 , 1033 tsuji , t. , nakajima , t. , & yanagisawa , k. 2004 , , 607 , 511 vrba , f. j. et al . 2004 , , 127 , 2948 yamamura , i. , tsuji , t. , & tanab , t. 2010 , , 722 , 682 ( paper i )

recent observations with the infrared astronomical satellite _ akari _ have shown that the co@xmath0 bands at 4.2@xmath1 m in three brown dwarfs are much stronger than expected from the unified cloudy model ( ucm ) based on recent solar c & o abundances . this result has been a puzzle , but we now find that this is simply an abundance effect : we show that these strong co@xmath0 bands can be explained with the ucms based on the classical c & o abundances ( log@xmath2 and log@xmath3 ) , which are about 0.2dex larger compared to the recent values . since three other brown dwarfs could be well interpreted with the recent solar c & o abundances , we require at least two model sequences based on the different chemical compositions to interpret all the _ akari _ spectra . the reason for this is that the co@xmath0 band is especially sensitive to c & o abundances , since the co@xmath0 abundance depends approximately on @xmath4 the cube of c & o abundances . for this reason , even low resolution spectra of very cool dwarfs , especially of co@xmath0 , can not be understood unless a model with proper abundances is applied . for the same reason , co@xmath0 is an excellent indicator of c & o abundances , and we can now estimate c & o abundances of brown dwarfs : three out of six brown dwarfs observed with _ _ should have high c & o abundances similar to the classical solar values ( e.g. log@xmath5 and log@xmath6 ) , but the other three may have low c & o abundances similar to the recent solar values ( e.g. log@xmath7 and log@xmath8 ) . this result implies that three out of six brown dwarfs are highly metal rich relative to the sun if the recent solar c & o abundances are correct .

calculating cyclic homology of the crossed product algebra is an attractive problem studied extensively in cyclic homology theory . when @xmath9 is a discrete group or a compact lie group and @xmath1 is an algebra or a @xmath10 manifold acted by @xmath9 , the cyclic homology of the crossed product algebra @xmath11 is considered by b.l . feigin and b.l . tsygan @xcite , j.l . brylinski @xcite , v. nistor @xcite , and e. getzler and j.d.s . jones @xcite . when @xmath1 is an @xmath12-module algebra , where @xmath12 is a hopf algebra with an invertible antipode , r. akbarpour and m. khalkhali @xcite investigated the cyclic homology of the crossed product algebra @xmath13 . their results generalize the work of getzler and jones in @xcite . in recent decades there have appeared many kinds of products of different types of algebras in the research of hopf algebras , for instance , the crossed product , or called ( classical ) smash product , of a hopf algebra and its module algebra , takeuchi s smash product @xcite of a left comodule algebra and a left module algebra where the action and the coaction are taken over one hopf algebra , the tensor product of two algebras in the natural sense or in a braided tensor category , the ( generalized ) drinfeld double , double crossproduct of hopf algebras , etc . these concepts are closely related with the factorization of an algebra into two subalgebras . the algebra factorization is described by s. majid @xcite , the generalized factorization problem is stated by s. caenepeel et al . @xcite . a ` generalized braiding ' , which is quasitriangular and normal , is associated closely with the algebra factorization . when it is a bijection , we call the product algebra a _ strong smash product algebra_. in this paper , we generalize both works of getzler and jones @xcite , and of akbarpour and khalkhali @xcite to strong smash product algebras . indeed , the crossed product algebras discussed in @xcite and @xcite are special examples of the strong smash product algebras . we organize this paper as follows . in section [ 111 ] , we give the explicit definition of the strong smash product algebra @xmath0 . in section [ 222 ] , we construct a cylindrical module @xmath14 . using diagrammatical presentations we prove that @xmath7 the cyclic module related to the diagonal of @xmath6 is isomorphic to the cyclic module of @xmath0 . in section [ 33 ] , we recall some notations and apply the generalized eilenberg - zilber theorem for cylindrical modules due to getzler and jones @xcite . in section [ 44 ] , we construct a spectral sequence converging to the cyclic homology of @xmath0 . in section [ 55 ] , we apply our theorems to majid s double crossproduct of hopf algebras after showing that they pertain to the class of strong smash product algebras . as any drinfeld s quantum double has a double crossproduct structure ( see @xcite ) , the notion of strong smash product algebras does cover a wild range of the recent interesting examples , for instance , the two - parameter or multiparameter quantum groups , and the pointed hopf algebras arising from nichols algebras of diagonal type ( see @xcite and references therein ) . besides these , another concrete example for the computation of the cyclic homology of the pareigis hopf algebra @xmath15 is given to illuminate our results . we assume that @xmath16 is a field containing @xmath17 in the whole paper unless otherwise stated . every algebra in this paper is assumed to be a unital associative @xmath16-algebra . majid defined in his book @xcite an algebra factorization . a unital and associative algebra @xmath18 _ factorizes _ through its subalgebras @xmath1 and @xmath2 , if the product map defines a linear isomorphism @xmath19 . the necessary and sufficient conditions for the existence of an algebra factorization is the existence of a linear map @xmath3 from @xmath4 to @xmath20 , which is quasitriangular and normal . in @xcite , the algebra which can be factorized is called a smash product and denoted by @xmath21 . in addition , if @xmath3 is also an isomorphism of vector spaces , we call @xmath22 a _ strong _ smash product algebra . the explicit definitions are as follows : let @xmath1 and @xmath2 be two algebras , and @xmath23 be a linear map . @xmath3 is called _ quasitriangular _ if it obeys @xmath24 where @xmath25 is the product map , @xmath26 and @xmath27 . @xmath3 is called _ normal _ if it obeys @xmath28 the _ smash product algebra _ of @xmath1 and @xmath2 with a quasitriangular and normal @xmath3 , denoted by @xmath0 , is defined to be @xmath20 as a vector space equipped with product @xmath29 the smash product algebra @xmath0 defined above is a unital associative algebra with the unit @xmath30 . the product of @xmath21 appeared first in @xcite , where a sufficient condition is given for the product to be associative . the smash product algebra @xmath0 is said to be _ strong _ , if @xmath3 is invertible . [ 1.2 ] @xmath0 is a strong smash product algebra if and only if @xmath31 is a strong smash product algebra . indeed , @xmath3 is quasitriangular ( resp . normal ) if and only if @xmath32 is quasitriangular ( resp . normal ) . since @xmath3 is invertible with @xmath33 , we have @xmath34 the normalization conditions are clear . it proves very convenient to do computations using diagrammatical presentations . present the multiplication of an algebra by @xmath35!{\xcaph-@(1.5)}[l(0.5)][d(0.75)]!{\xcapv[0.5]@(0)}}$ ] and @xmath3 from @xmath4 to @xmath20 by @xmath35!{\xoverv}[d(0.5)][l(0.3 ) ] { a}[r(1.5)]b[uu][r(0.3)]a[l(1.5)]b}$ ] . thus @xmath32 can be presented by @xmath35!{\xunderv}[d(0.5)][l(0.3 ) ] { b}[r(1.5)]a[uu][r(0.3)]b[l(1.5)]a}$ ] . the quasitriangular conditions can be diagrammatically expressed as follows : @xmath36{mr.eps}\ar@{}[r]|-{\cong}&\includegraphics[scale=0.5]{rrm.eps}}\ ] ] @xmath36{1mr.eps}\ar@{}[r]|-{\cong}&\includegraphics[scale=0.5]{m1rr.eps}}\ ] ] the concept of smash product algebra @xmath0 recovers the crossed product algebra ( or called classical smash product algebra ) @xmath13 and takeuchi s smash product algebra @xmath37 ( defined in @xcite ) where @xmath12 is a hopf algebra , @xmath1 is an @xmath12-module algebra and @xmath2 is an @xmath12-comodule algebra . the two subalgebras play different roles in @xmath13 and @xmath37 . one algebra produces action on the other . however , in the strong smash product algebra @xmath0 , the status of @xmath1 and @xmath2 is equal . they act on each other . the strong smash product algebra @xmath0 is a more natural concept , as in physics the general principle is that every action has a ` reaction ' . many smash product algebras are strong smash product algebras . the tensor product of two algebras in a braided tensor category is a strong smash product algebra . here @xmath3 is deduced directly from the braiding in that category , so @xmath3 is invertible . [ eg ch1.5 ] let @xmath12 be a hopf algebra with an invertible antipode @xmath38 . @xmath1 is a left @xmath12-module algebra and @xmath2 is a left @xmath12-comodule algebra . takeuchi s smash product @xmath39 is an algebra with the multiplication @xmath40}.a')\#b_{[0]}b'$ ] and the unit @xmath30 , where @xmath41}\ot b_{[0]}$ ] is the left @xmath12-comodule structure map for @xmath42 and @xmath43 . when @xmath44 , @xmath45 is the crossed product algebra . define @xmath46 by @xmath47}.a\ot b_{[0]}.\ ] ] one can check that @xmath3 is quasitriangular and normal through the definition of the module algebra and the comodule algebra . @xmath3 has the inverse defined by @xmath48}\ot s^{-1}(b_{[-1]}).a,\ ] ] for all @xmath49 and @xmath50 . hence , the crossed product algebras discussed in @xcite and @xcite are special examples of our strong smash product algebras . * 2.1 * from getzler and jones point of view , all the operators of a cyclic module can be generated by only two operators , i.e. , the last face map and the extra degeneracy map . hence we can give an equivalent definition for cyclic modules . in this subsection , @xmath16 can be a commutative ring . a _ cyclic module _ is a sequence of @xmath16-modules @xmath51 which is endowed for each @xmath52 with two @xmath16-linear maps @xmath53 and @xmath54 , such that @xmath55 is invertible , and by setting @xmath56 for any @xmath57 , the following relations hold @xmath58 @xmath59 s are called _ face _ maps and @xmath60 s are called _ degeneracy _ maps for @xmath61 , @xmath62 is called the _ cyclic operator_. @xmath63 is called the _ extra degeneracy map _ and @xmath53 is called the _ last face _ map for @xmath64 . therefore , a cyclic module can be regarded as an underlying simplicial module @xmath65 , whose face maps , degeneracy maps and cyclic operators are generated by the last face map @xmath66 and the extra degeneracy map @xmath63 for each @xmath64 in the way expressed in and satisfying and . if the condition is replaced by @xmath67 then that sequence of @xmath16-modules is called a _ paracyclic module_. in fact , the equalities in are consequences of the cyclicity of the invertible operator @xmath62 , that is , from , one can get . for all @xmath52 , set the following operators @xmath68 [ 2.2 ] we have the equalities @xmath69 getzler and jones first introduced in @xcite the concepts of the bi - paracyclic module and the cylindrical module . we recall their definitions here . a _ bi - paracyclic module _ is a sequence of @xmath16-modules @xmath70 such that @xmath71 and @xmath72 are two paracyclic modules and the operators @xmath73 commute with the operators @xmath74 . moreover , if in addition , @xmath75 for all @xmath76 , then this bi - paracyclic module is called a _ cylindrical module_. another interesting concept named parachain complex was also given by getzler and jones @xcite . the mixed complex defined by kassel @xcite is a special case of parachain complexes . here we need only the mixed complex . the _ mixed complex _ is , by definition , a graded @xmath16-module @xmath77 endowed with two graded commutative differentials , one decreasing the degree and the other increasing the degree . that is , @xmath78 with @xmath79 and @xmath80 satisfies @xmath81 . a morphism of mixed complexes @xmath78 to @xmath82 is a sequence of morphisms @xmath83 for @xmath84 such that @xmath85 commutes with @xmath86 . for a cyclic module @xmath87 associated with the operators @xmath88 and @xmath89 defined in , @xmath90 is a mixed complex . it is usually simpler to consider the complex with one differential than to consider the mixed complex with two differentials . actually , a mixed complex can be converted into a complex . let @xmath91 be a non - negative graded @xmath16-module . denote by @xmath92 $ ] the graded @xmath16-modules of formal power series in a variable @xmath93 with coefficients in @xmath91 . set the degree of @xmath93 be @xmath94 . if @xmath91 is endowed with a degree @xmath95 endomorphism @xmath88 and a degree @xmath96 endomorphism @xmath89 , then @xmath97 is a mixed complex if and only if @xmath98,\mathrm{b}+u\mathrm{b})$ ] is a complex with the differential @xmath86 . here set @xmath99=\sum_{i\geq 0}v_{n+2i}u^i$ ] . + * 2.2 * now we return to our strong smash product algebra . the cyclic module @xmath8 of an algebra @xmath0 is defined as usual ( see @xcite etc ) . that is , @xmath100 for all @xmath101 with @xmath102 where @xmath103 . for @xmath1 and @xmath2 the subalgebras of @xmath104 , we introduce a cylindrical module denoted by @xmath6 which generalizes the cylindrical module constructed in the paper @xcite by getzler and jones where @xmath2 is a group algebra and @xmath1 is a @xmath2-module algebra , also generalizes the cylindrical module constructed in the paper @xcite by akbarpour and khalkhali where @xmath2 is a hopf algebra with an invertible antipode and @xmath1 is a @xmath2-module algebra . for @xmath105 , set @xmath106 endowed with the following operators which are mainly defined on @xmath2 s side : @xmath107 and the following operators which are mainly defined on @xmath1 s side : @xmath108 where @xmath109 is the flip map defined by @xmath110 @xmath111 is a composition of @xmath3 s defined by @xmath112 and @xmath113 is a composition of @xmath32 s defined by @xmath114 for @xmath115 and @xmath116 . define the last face maps by @xmath117 and @xmath118 . we can simply write @xmath119 and @xmath120 . graphically , present the flip map between @xmath20 and @xmath4 by @xmath35!{\xunderv@(0)}[d(0.5)]a[r]b[uu]a[l]b}$ ] , its inverse is @xmath35!{\xunderv@(0)}[d(0.5)]b[r]a[uu]b[l]a}$ ] . the identity is denoted by @xmath121!{\xcapv@(0)}}$ ] . then @xmath122 and @xmath123 can be presented by @xmath124{t.eps } } , \quad \bar{t}_{p , q}= \raisebox{-2.5pc } { \includegraphics[scale=0.5]{tbar.eps}}.\ ] ] the elements in @xmath1 are drawn with thick lines and the elements in @xmath2 are drawn with thin lines in order to show differences . since @xmath125 , we have @xmath126{rr-1.eps}}\cong \raisebox{-1.5pc}{\includegraphics[scale=0.5]{r-1r.eps}}\cong \raisebox{-1.5pc}{\includegraphics[scale=0.5]{44.eps}}\cong \raisebox{-1.5pc}{\includegraphics[scale=0.5]{4.eps}}.\ ] ] although @xmath3 does not satisfy the braid relations , the flip maps always satisfy them and are involutions . when the three crosses in one side of the braid relations consist of two flip maps and one @xmath3 or @xmath32 , we still have the `` braid '' relations . @xmath127 , @xmath128 , @xmath129 , where @xmath130 denotes @xmath131 , i.e. , the flip map of two elements . the graphical notations are @xmath132{b1.eps } } { \cong } \raisebox{-2.5pc}{\includegraphics[scale=0.5]{b11.eps}},\qquad \raisebox{-2.5pc}{\includegraphics[scale=0.5]{b2.eps } } { \cong } \raisebox{-2.5pc}{\includegraphics[scale=0.5]{b22.eps } } , \qquad \raisebox{-2.5pc}{\includegraphics[scale=0.5]{b3.eps } } { \cong } \raisebox{-2.5pc}{\includegraphics[scale=0.5]{b33.eps}}.\ ] ] for @xmath32 , we have the same relations . @xmath133 is a cylindrical module . we check the commutativity of the barred operators and unbarred operators first . we would like to use the graphical proof . @xmath134{t_tbar.eps}}\ar[r]^- { \txt{act ( i)}}&}\ ] ] @xmath36{t_tbar01.eps } & & \ar[rr]^-{\txt{act ( ii ) on\\ its crosses in\\the upper right corner}}_-{\txt { and the\\ lower left corner}}&&}\ ] ] @xmath36{t_tbar02.eps}&\ar[rr]^-{\txt{the flip maps\\ are involutions\\ and obey\\ braid relations}}&&}\ ] ] @xmath36{t_tbar03.eps}\ar[r]^-{\txt{(i ) } } & { \includegraphics[scale=0.5]{t_tbar04.eps}}}\ ] ] @xmath135 ^ -{\txt{(i ) } } & \raisebox{-5pc } { \includegraphics[scale=0.5]{tbar_t.eps}}=\bar{t}_{p , q}{t}_{p , q}}.\ ] ] \(ii ) for @xmath136 , @xmath137{djt.eps}}&\ar[rr]^- { \txt{flip map is\\ quasi-\\triangular}}&&}\ ] ] @xmath36{djt01.eps } & \ar[rr]^- { \txt{$r$ is quasi-\\triangular}}&&}\ ] ] @xmath138 { djt02.eps}}={t}_{p , q}\bar{d}_j^{p , q}}\ ] ] similar proof holds for @xmath139 , for @xmath140 . the flip map and @xmath141 are quasitriangular and normal , @xmath142 and @xmath143 , so the other commutative equalities can be proved easily . for the cylindrical condition , we use inductions on @xmath144 and @xmath145 . for @xmath146 , using the fourth picture in the process of turning @xmath147 to @xmath148 , we get @xmath149|-{\displaystyle(t_{1,1}\bar{t}_{1,1})^2=\qquad } & \includegraphics[scale=0.5,height=5cm]{t1tbar1_2_01.eps } \ar[r]^-{\txt{(i)}}&\includegraphics[scale=0.5,height=5cm]{t1tbar1_2_02.eps}\ar[r]^-{\txt{(i)}}&}\ ] ] @xmath150{t1tbar1_2_03.eps } \ar[r]^-{\txt{(i)}}&\includegraphics[scale=0.5,height=5cm]{t1tbar1_2_04.eps}\ar[r]^-{\txt{(i ) } } & \includegraphics[scale=0.5,height=5cm]{t1tbar1_2_05.eps } \ar@{}[r]|-{{\displaystyle=~ id.}}&}\ ] ] suppose that @xmath151 for @xmath152 and @xmath153 , we need to prove @xmath154 . we have @xmath155{tn+1.eps}}\quad \text{and}\quad \bar{t}_{p , q}^{\,q+1}=\raisebox{-5pc}{\includegraphics[scale=0.5,height=5cm]{tbarn+1.eps}}.\ ] ] we can use bands in graphs to stand for parallel lines , that is , lines without any intersections or crosses between themselves . if we can draw the elements @xmath156 of @xmath2 together by a grey band , and draw the elements @xmath157 of @xmath1 together by a black band , then using the movements for the case @xmath158 , we will get the proposition . we just give the equivalent moves for turning the lines @xmath159 and @xmath160 in the graph of @xmath161 to parallel lines , others can be done by similar moves . the only intersections between @xmath159 and @xmath160 occur while doing the @xmath144-th and @xmath162-th powers of @xmath122 . so we concentrate on that part of graph . @xmath163{tn+1001.eps}\ar[r]^-{\txt{(ii ) } } & \includegraphics[scale=0.5,height=4cm , width=5cm]{tn+1002.eps}}\ ] ] @xmath164 ^ -{\txt{flip maps'\\ braid\\ relations } } & & \includegraphics[scale=0.5,height=4cm , width=5cm]{tn+1003.eps}}.\ ] ] let @xmath7 be the _ diagonal _ of the cylindrical module @xmath6 , i.e. , @xmath165 it is a cyclic module with face maps @xmath166 , degeneracy maps @xmath167 and the cyclic operator @xmath168 . [ 2.6 ] @xmath7 is isomorphic to @xmath8 as cyclic modules . define morphisms @xmath169 by @xmath170 and @xmath171 by @xmath172 @xmath173{phi.eps}}\ ] ] @xmath174{psi.eps}}\ ] ] note that @xmath175 for @xmath176 . so @xmath177 and @xmath178 are inverses to each other . we need to prove that @xmath177 and @xmath178 are morphisms of cyclic modules . we only show that @xmath177 commutates with the cyclic operator and the face maps . it is similar for @xmath178 . again using the fourth picture in the process of turning @xmath147 to @xmath148 , we get @xmath149|-{\displaystyle \phi t_{n , n}\bar{t}_{n , n}= } & \quad\includegraphics[scale=0.5,height=5cm]{t_tbar_phi_1.eps } \ar[r]^-{\txt{(i)}}&\includegraphics[scale=0.5,height=5cm]{t_tbar_phi_2.eps } } \ ] ] @xmath179 ^ -{\txt{(i ) } } & \quad\includegraphics[scale=0.5,height=5cm]{t_tbar_phi_3.eps } \ar[r]^-{\txt{(i)}}_-{\txt{(ii)}}&\includegraphics[scale=0.5,height=5cm]{t_tbar_phi_4.eps } } \ ] ] @xmath179 ^ -{\txt{(ii ) } } & \quad\includegraphics[scale=0.5,height=5cm]{t_tbar_phi_5.eps } \ar[r]^-{\cong}&\includegraphics[scale=0.5,height=5cm]{phit.eps}\ar@{}[r]|-{\displaystyle = t\phi . } & } \ ] ] since @xmath3 and @xmath32 are quasitriangular , for @xmath180 , @xmath149|-{\displaystyle d_i\phi = } & \quad\includegraphics[scale=0.5,height=3.5cm , width=5.5cm]{phid.eps } \ar@{}[r]|-{\cong}&\includegraphics[scale=0.5,height=3.5cm , width=5.5cm]{ddbarphi.eps}\quad\ar@{}[r]|-{\displaystyle = \phi\bar{d}^{n , n}_id^{n , n}_i . } & } \ ] ] let @xmath181 be a cylindrical module . we can set as in the degree @xmath95 endomorphism @xmath88 ( resp.@xmath182 ) , the degree @xmath96 endomorphism @xmath89 ( resp.@xmath183 ) and the degree @xmath184 endomorphism @xmath185 ( resp . @xmath186 ) associated with @xmath187 ( resp . @xmath188 ) . the total parachain complex is a mixed complex . explicitly , let @xmath189 , @xmath190 and @xmath191 . since @xmath192 is a cylindrical module , @xmath193 , @xmath194 and @xmath195 . then by lemma [ 2.2 ] , @xmath196 @xmath197 is a mixed complex . the generalized eilenberg - zilber theorem for paracyclic modules was proved by getzler and jones @xcite using topological method , later it was reproved by khalkhali and rangipour @xcite using an algebraic method . the theorem tells us that , for a cylindrical module there exists a quasi - isomorphism from its total mixed complex to its diagonal mixed complex . due to proposition [ 2.6 ] , we have : let @xmath0 be a strong smash product algebra , @xmath14 a cylindrical module defined in and . then there exists a quasi - isomorphism of mixed complexes @xmath198 and @xmath8 . it was discovered by getzler and jones @xcite that the hochschild homology , the cyclic homology , the negative cyclic homology and the periodic cyclic homology can be unified to be cyclic homologies of a mixed complex with coefficients . specifically , let @xmath199 be a mixed complex and @xmath200 be a graded @xmath201$]-module , denote @xmath202\ot_{k[u]}w $ ] by @xmath203 . note that this tensor product is a graded tensor product . let @xmath204 be the mixed complex associated to its cyclic module structure . @xmath205 is a complex with the differential @xmath206 } id_w$ ] . call the homology of the complex @xmath205 _ the cyclic homology of the mixed complex of @xmath207 with coefficients in @xmath200 _ and denote it by @xmath208 . then for @xmath209 $ ] ( resp . @xmath210,k[u , u^{-1}]/uk[u]$ ] and @xmath201/uk[u]$ ] ) @xmath211 ( resp . @xmath212 and @xmath213 ) . if @xmath214 is the usual cyclic module associated with an algebra @xmath1 , then we simply denote @xmath215 by @xmath216 . the first author would like to thank professor getzler for pointing out the flatness condition concealed here , which turns out useful in the consequent arguments . [ 3.02 ] let @xmath16 be a field , @xmath217 a @xmath16-vector space and @xmath93 a variable . then @xmath218 $ ] is a flat @xmath201$]-module . since @xmath201 $ ] is a principal ideal domain , a @xmath201$]-module is flat if and only if it is torsion - free . clearly , @xmath218 $ ] is a torsion - free @xmath201$]-module . let @xmath219 be a ring , @xmath220 a left @xmath219-module and @xmath221 bounded below complexes of flat right @xmath222-modules . if @xmath223 and @xmath224 are quasi - isomorphic , then @xmath225 we know from @xcite that , for bounded below complexes of flat right @xmath222-modules @xmath223 and @xmath224 , @xmath226 for each @xmath52 , where @xmath227 is the hypertor . and we have spectral sequences converging to them , that is , @xmath228 @xmath229 for all @xmath230 , as @xmath231 . it yields that @xmath232 by using the mapping lemma for @xmath233 ( see @xcite ) . the above two lemmas still hold in the graded module category . [ 3.4 ] if there exists @xmath234 is a quasi - isomorphic of mixed complexes , then for any graded @xmath201$]-module @xmath200 , we have an isomorphism of cyclic homology groups @xmath235 using the generalized eilenberg - zilber theorem for paracyclic modules , we have [ 3.2 ] let @xmath0 be a strong smash product algebra , @xmath6 be the cylindrical module defined in and . then @xmath236 the following corollary will be used in the next section . [ coro 3.5 ] let @xmath237 be a complex of @xmath16-modules and @xmath200 a graded @xmath201$]-module . then for each @xmath52 , @xmath238\ot_{k[u]}w)={\mathrm{h}}_n(\mathfrak{c})[[u]]\ot_{k[u ] } w.\ ] ] since @xmath239 $ ] is a complex of flat @xmath201$]-modules , @xmath238\ot_{k[u]}w)=\mathbb{t}\mathrm{or}^{k[u]}_n(\mathfrak{c}[[u]],w).\ ] ] note that the differential of the complex @xmath239 $ ] does not depend on @xmath93 . we have a spectral sequence converging to the hypertor whose @xmath240-term is @xmath241}({\mathrm{h}}_q(\mathfrak{c})[[u]],w).\ ] ] because @xmath242 $ ] is also a flat @xmath201$]-module , the spectral sequence collapses . we get @xmath238\ot_{k[u]}w)=\mathbb{t}\mathrm{or}^{k[u]}_n(\mathfrak{c}[[u]],w)={\mathrm{h}}_n(\mathfrak{c})[[u]]\ot_{k[u]}w.\ ] ] this completes the proof . we can also construct a spectral sequence to calculate the cyclic homology of a strong smash product algebra @xmath104 . this is the same as calculating the cyclic homology of @xmath243 . the first column of the cylindrical module @xmath6 plays an important role . denote by @xmath244 this paracyclic module @xmath245 . [ 4.1]for each @xmath246 , @xmath247 is an @xmath1-bimodule via the left @xmath1-module action @xmath248 and the right @xmath1-module action @xmath249 where @xmath250 is defined in section [ 222 ] , and @xmath251 . the right action is trivial . by proposition [ 1.2 ] , @xmath32 is also quasitriangular and normal , then the left @xmath1-module action is well - defined . and both actions are compatible . for each @xmath252 , we can define a _ hochschild complex _ @xmath253 , whose homology is _ the hochschild homology of the algebra _ @xmath1 with coefficients in @xmath254 ( see @xcite ) . the hochschild complex is defined explicitly as follows : for any @xmath255 , @xmath256 the differential @xmath257 is @xmath258 denote this hochschild homology by @xmath259 . @xmath260 is a cylindrical module with the same operators defined for @xmath6 in and . indeed , for each @xmath261 , @xmath262 note that @xmath263 of @xmath6 is exactly the differential @xmath264 defined in . define @xmath265 as the _ co - invariant space _ of @xmath244 under the left and right actions of @xmath1 constructed in lemma [ 4.1 ] , i.e. , @xmath266 and we define the following operators on @xmath265 : @xmath267 use the following notations as in @xcite , for @xmath3,@xmath268 and for @xmath32 , @xmath269 where @xmath270 and @xmath271 . then one can check that these operators in are well defined on the co - invariant space . for example , @xmath272 @xmath273 is a cyclic module with operators defined in . we only check that @xmath274 . the other identities are similar to check . in the coinvariant subspace , we have @xmath275 in fact , the above proposition is a special case of the following theorem . [ h01 ] for any @xmath276 , @xmath277 is a cyclic module with @xmath278 induced from operators of @xmath6 defined in . especially , we have @xmath279 we need to check that , @xmath280 inducing on @xmath281 turns out to be identity . for any @xmath282 , @xmath283 , or equivalently , @xmath284 , @xmath285 since the barred operators commutate with the unbarred operators , all unbarred operators @xmath278 are well - defined on @xmath286 preserving the relations and . [ 4.3 ] the homology group of the complex @xmath287\stackrel{d}\longrightarrow c_{q-1}(a , c_{\bullet}({}_a^{~\natural}b))[[u]]\ra\cdots\ ] ] is @xmath288 $ ] , for each @xmath145 . by corollary [ 3.2 ] , in order to calculate the cyclic homology of the strong smash algebra @xmath0 with coefficients in @xmath200 , we can compute the cyclic homology of @xmath243 with coefficients in @xmath200 , that is , the homology of the complex @xmath289 we define a filtration on @xmath290 by rows . set @xmath291}w , \text { for } p\geq 0;\ ] ] and @xmath292 the spectral sequence @xmath293 of this filtration with @xmath294 starts from @xmath295}w,\ ] ] equipped with @xmath296 . recall that @xmath297=\sum_{p , l\geq 0}(b^{\ot ( p+2l+1)}\ot a^{\ot ( q+1)})u^l$ ] , @xmath298 so from lemma [ 4.3 ] and corollary [ coro 3.5 ] , we get : the @xmath299-term of the spectral sequence is @xmath300 equipped with @xmath301 that is induced by @xmath302 . [ lim1 ] the @xmath240-term of the spectral sequence is identified with the cyclic homology of the cyclic module @xmath303 with coefficients in @xmath200 . it converges to the cyclic homology of the strong smash product algebra @xmath0 with coefficients in @xmath200 . that is , @xmath304 in parallel , one can also consider the bottom row of the cylindrical module @xmath6 . we just state the process and indicate the differences here . we skip proofs which are similar as in previous discussions . denote the paracyclic module @xmath305 by @xmath306 . [ 4.7]for each @xmath246 , @xmath307 is a @xmath2-bimodule via the left @xmath2-module action @xmath308 and the right @xmath2-module action @xmath309 where @xmath310 is defined in section [ 222 ] , and @xmath311 . for each @xmath312 , a _ hochschild complex _ @xmath313 can be defined , its homology is _ the hochschild homology of the algebra _ @xmath2 with coefficients in @xmath314 . the hochschild complex is defined explicitly as follows : for any @xmath252 , @xmath315 the differential @xmath316 is @xmath317 denote this hochschild homology by @xmath318 . a difference occurs here , as the positions of @xmath1 s and @xmath2 s are changed . @xmath319 is a cylindrical module , which is isomorphic to @xmath6 . we give the isomorphisms between @xmath319 and @xmath6 , then the bi - paracyclic operators on @xmath319 are constructed from the operators on @xmath6 through the isomorphisms . in this way , we get the corollary . we give the isomorphisms and the operators on @xmath319 explicitly . for each @xmath320 , @xmath321 @xmath322 and @xmath323 @xmath324 we can see that @xmath325 hence , the operators @xmath326 on @xmath327 are defined as follows : @xmath328 define @xmath329 as the _ co - invariant space _ of @xmath306 under the left and right actions given in lemma [ 4.7 ] , i.e. , @xmath330 and we define the following operators on @xmath329 : @xmath331 indeed , @xmath332 is induced by @xmath333 , as @xmath334 and @xmath335 when @xmath144 is @xmath184 . one can check that these operators are well defined on the co - invariant space . [ h02 ] for any @xmath336 , @xmath337 is a cyclic module with @xmath338 induced from operators of @xmath319 defined in . especially , we have @xmath339is a cyclic module with operators defined in . we define a filtration on @xmath290 by columns . set @xmath340}w,\text { for } q\geq 0;\ ] ] and @xmath341 for @xmath342 . the spectral sequence @xmath343 of this filtration with @xmath344 starts from @xmath345}w,\end{aligned}\ ] ] equipped with @xmath346 . the @xmath299-term of the spectral sequence is @xmath347 equipped with @xmath348 that is induced by @xmath349 . [ lim2 ] the @xmath240-term of the spectral sequence is identified with the cyclic homology of the cyclic module @xmath350 with coefficients in @xmath200 . it converges to the cyclic homology of the strong smash product algebra @xmath0 with coefficients in @xmath200 . that is , @xmath351 by proposition 1.1.13 of @xcite , we can use the derived functor @xmath352 to express the hochschild homology of an algebra @xmath1 with coefficients in @xmath220 which is an @xmath1-bimodule , that is , @xmath353 where @xmath354 . for a separable algebra that is projective over its enveloping algebra , its homology with coefficients in any module is zero . hence , the spectral sequence collapses at @xmath240 , that is , @xmath355 for all @xmath230 unless @xmath356 . so we have if the algebra @xmath1 @xmath357resp . @xmath2@xmath358 is separable , then there is a natural isomorphism of cyclic homology groups @xmath359 from the above results , one can observe that our theorems take advantage of good homological property of either of two subalgebras . even in the case of the crossed product algebra @xmath13 , where @xmath12 is a hopf algebra with invertible antipode and @xmath1 is an @xmath12-module algebra , the `` nice '' homological property of @xmath1 sometimes will play a key role in computing the cyclic homology of the crossed product , by comparison with the homological property of @xmath12 being weak . we will illustrate this point by examples in the next section . * 5.1 * in this subsection , we apply our theorems to majid s double crossproduct of hopf algebras which is inspired by bismash product of groups defined by takeuchi @xcite . bismash product of groups is a generalization of semiproduct of groups . in order to define this product , he provided the notion of a matched pair of groups . given a matched pair of groups @xmath360 , the bismash product of @xmath9 and @xmath361 denoted by @xmath362 is still a group . the theory is developed by majid @xcite . he defined a matched pair of hopf algebras and constructed a product hopf algebra which he called a double crossproduct of hopf algebras . using this new definition he provided another way to construct drinfeld s quantum double . we start by recalling the definition due to majid @xcite . a pair @xmath363 of hopf algebras is said to be _ matched _ if @xmath2 is a left @xmath12-module coalgebra via @xmath364 , and @xmath12 is a right @xmath2-module coalgebra via @xmath365 , @xmath366 such that the following equalities hold for @xmath367 . @xmath368 the _ double crossproduct _ @xmath369 is a hopf algebra equipped with @xmath370 [ 5.1 ] let @xmath363 be a matched pair of hopf algebras . if @xmath12 and @xmath2 have invertible antipodes , then the double crossproduct of @xmath2 and @xmath12 denoted by @xmath369 is a strong smash product algebra . in particular , the group algebra of the bismash product of a matched pair of groups is a strong smash product algebra . since @xmath372 is a hopf algebra , then @xmath373 is also a hopf algebra . denote the convolution map on @xmath374 by @xmath375 . define the operator @xmath376 by @xmath377 we should check that @xmath378 actually , we only need to check the first equality . indeed , if it holds , then @xmath379 from and , we have @xmath380 using the same method , we can prove . \1 ) @xmath3 is quasitriangular : @xmath385 , @xmath386 the third equality holds due to the @xmath12-module coalgebra structure of @xmath2 , the forth equality holds because of . similarly , one can prove that @xmath387 . \3 ) @xmath3 is invertible : for @xmath391 , set @xmath392 @xmath393 we claim that @xmath394 is the inverse of @xmath3 . @xmath395 the third and the forth equalities above are due to the @xmath2-module coalgebra structure of @xmath12 and the @xmath12-module coalgebra structure of @xmath2 . the fifth equality is due to . the sixth and the last equalities hold because of and . @xmath396 the fifth equality above holds because of and . for a finite group @xmath9 and an arbitrary @xmath9-bimodule @xmath220 , since @xmath399 $ ] is semisimple , @xmath400,m)$ ] is @xmath184 for all @xmath52 except for @xmath401 . then by the above corollary , theorems [ h01 ] and [ h02 ] , we have [ gk ] given a matched pair of finite groups @xmath360 , then @xmath402;w)\cong { \mathrm{hc}}_n(c^{g}_{\bullet}({}_{g}^{~\natural}k);w),\\ { \mathrm{hc}}_{n}(k[g\bowtie k];w)\cong { \mathrm{hc}}_n(c^{k}_{\bullet}({g}_{k}^{\natural});w).\end{gathered}\ ] ] if @xmath12 is a finite dimensional hopf algebra , then the antipode of @xmath12 is always invertible ( see , corollary 5.6.1 of @xcite ) . using the adjoint action of @xmath12 on itself , majid in example 4.6 of @xcite constructed a matched pair @xmath403 and deduced the drinfeld s quantum double @xmath404 . by corollaries [ bh ] and [ gk ] , we have actually , any drinfeld s quantum double turns out to be of majid s double crossproduct structure ( see @xcite ) , while the recently appeared attractive objects , such as the two - parameter or the multiparameter ( restricted ) quantum ( affine ) groups , the pointed hopf algebras arising from nichols algebras of diagonal type ( cf . @xcite and references therein ) , are of drinfeld s double structures ( under certain conditions for the root of unity cases ) . thereby , our machinery established for the strong smash product algebras is indeed suitable to a large class of many interesting hopf algebras . * 5.2 * the following first example comes from the rank @xmath96 case ( modified ) of the smash product algebra @xmath409 introduced in @xcite ( p.525 , subsection * 3.5 * ) , which was used to define intrinsically and construct a quantum weyl algebra @xmath410 . although our example here is still a crossed product algebra , proposition 5.3 in @xcite , under the assumption of the hopf algebra @xmath12 being semisimple ( so automatically finite dimensional ) , does not work for our example . let @xmath411 be an @xmath412-th primitive root of unity . define @xmath413 to be @xmath414/(x^n{-}1)$ ] which is isomorphic to the group algebra @xmath415 $ ] . define @xmath416 to be the associative @xmath16-algebra generated by @xmath417 , @xmath418 , subject to relation @xmath419 . @xmath416 is a hopf algebra with the coproduct , counit , and antipode defined as follows : @xmath420 @xmath421 the antipode of @xmath416 is invertible , as @xmath422 . one can calculate that @xmath423 and @xmath424 . let @xmath425 for @xmath426 . then @xmath427 . as we stated in example [ eg ch1.5 ] , the crossed product @xmath436 is a strong smash product algebra , since @xmath416 is a hopf algebra with invertible antipode . for example , @xmath437 , @xmath438 and @xmath439 for @xmath431 . let @xmath441 be the quotient algebra of @xmath442 by the ideal @xmath443 . as @xmath443 is a hopf ideal ( owing to @xmath444 and @xmath445 ) , @xmath441 is a hopf algebra . in particular , when @xmath446 , @xmath447 is nothing but the pareigis hopf algebra @xmath448 ( see @xcite or the next subsection for definition ) . furthermore , consider the quotient hopf algebra @xmath449 of @xmath441 by the hopf ideal @xmath450 . @xmath451 is just the taft algebra . cyclic homology of the taft algebra as a special truncated quiver algebra is computed by taillefer @xcite . in @xcite , pareigis defined a noncommutative and noncocommutative hopf algebra @xmath452 , which links closely the category of complexes and the category of comodules over @xmath452 . that is , the category of complexes is equivalent as a tensor category to the category of comodules over @xmath452 . explicitly , @xmath452 is defined to be the quotient algebra of the free algebra @xmath453 by the two sided ideal that is generated by @xmath454 then @xmath452 turns out to be a hopf algebra with the following coproduct , counit and antipode , @xmath455 @xmath452 can be regarded as the crossed product algebra of @xmath456/s^2 $ ] and @xmath457 $ ] , where @xmath456/s^2 $ ] is a module algebra over @xmath457 $ ] with the conjugate action @xmath458 . denote by @xmath459 the algebra of dual number @xmath456/s^2 $ ] , and by @xmath460 the laurent polynomial ring @xmath457 $ ] . @xmath461 @xmath452 is a strong smash product algebra @xmath462 with the invertible @xmath463 defined to be @xmath464 let @xmath209/uk[u]$ ] . we would like to calculate the hochschild homology of @xmath452 first . consider the cyclic module @xmath465 . its face maps , degeneracy maps , and cyclic operators are induced by the corresponding operators defined in for the cylindrical module @xmath466 . using the following resolution of @xmath459 by projective @xmath467-modules ( see e.g. , @xcite ) @xmath468 ^ -{\nu}&d^{e}\ar[r]^{\mu}&d^{e}\ar[r]^{\nu}&d^{e } \ar[r]^{\mu}&d^ { e}\ar[r]^{\mathrm{m}}&d\ar[r ] & 0},\ ] ] where @xmath469 , @xmath470 , @xmath471 is the product of @xmath459 , we get @xmath472 where @xmath473 @xmath474 in order to specify the operators of the cyclic module @xmath475 , we should represent the elements of @xmath475 by elements of @xmath466 . according to the comparison theorem , there is a unique chain map lifting @xmath476 from the resolution @xmath477 to the bar resolution of @xmath459 up to chain homotopy equivalence . this required chain map @xmath478 is defined as follows , @xmath479^{\nu}&d^{e}\ar[r]^{\mu}\ar[d]^{\zeta_3 } & d^{e}\ar[r]^{\nu}\ar[d]^{\zeta_2}&d^{e } \ar[r]^{\mu}\ar[d]^{\zeta_1}&d^ { e}\ar[r]^{\mathrm{m}}\ar[d]^{\zeta_0 } & d\ar[r]\ar[d]^{id } & 0\\ \cdots\ar[r]^{\mathrm{b}'}&d^{\ot 5}\ar[r]^{\mathrm{b}'}&d^{\ot 4}\ar[r]^{\mathrm{b}'}&d^{\ot 3}\ar[r]^{\mathrm{b}'}&d^{\ot 2}\ar[r]^{\mathrm{b}'}&d\ar[r ] & 0}\ ] ] where @xmath480 and @xmath481 hence , @xmath482 the cyclic operator @xmath483 on @xmath484 is defined via @xmath485 we can describe the cyclic modules @xmath486 simply . let @xmath487 be the cyclic module of the algebra @xmath460 . since the face maps , degeneracy maps , and the cyclic operators of @xmath487 do not change the total degree of @xmath62 , @xmath487 can be decomposed into the direct sum of two sub - cyclic modules @xmath488 and @xmath489 with @xmath490 and @xmath491 . let @xmath492 be the cyclic module with @xmath493 and the operators @xmath494 where @xmath495 is an even integer . the cyclic homology of @xmath460 is well - known ( see e.g. , p.337 in @xcite ) @xmath512 thanks to the short exact sequences @xmath513 ( see e.g. , @xcite theorem 4.1.13 ) , where @xmath514 and @xmath515 , we get the cyclic homology of @xmath452 . the laurent polynomial ring @xmath460 is isomorphic to the group algebra @xmath517 $ ] . if making use of the results of @xcite , one can construct another spectral sequence @xmath518 with @xmath519 for @xmath520 , converging to the cyclic homology of @xmath452 . in this way , it remains to determine @xmath521 to achieve @xmath522 . since this spectral sequence collapses at @xmath522 , one then does more . * acknowledgements . * the authors are supported in part by the nnsf ( grants : 10971065 , 10728102 ) , the pcsirt and the rfdp from the moe , the national and shanghai leading academic discipline projects ( project number : b407 ) . the first author would like to express her gratitude to professor ezra getzler for pointing out the flatness condition . she is indebted to her advisor professor marc rosso for his kind help . the authors also would like to thank professor joachim cuntz for his useful comments and encouragement . y. pei , n. hu and m. rosso , _ multiparameter quantum groups and quantum shuffles , ( i ) _ , in quantum affine algebras , extended affine lie algebras , and their applications , contemp . , * 506 * , amer . math . soc . , 2010 , pp . 145171 . arxiv:0811.0129 .

for any strong smash product algebra @xmath0 of two algebras @xmath1 and @xmath2 with a bijective morphism @xmath3 mapping from @xmath4 to @xmath5 , we construct a cylindrical module @xmath6 whose diagonal cyclic module @xmath7 is graphically proven to be isomorphic to @xmath8 the cyclic module of the algebra . a spectral sequence is established to converge to the cyclic homology of @xmath0 . examples are provided to show how our results work . particularly , the cyclic homology of the pareigis hopf algebra is obtained in the way . * keywords * : cyclic homology , strong smash product algebra . + * msc(2000 ) * : 19d55 , 16s40 .

let @xmath1 be a galois extension of number fields and @xmath2 the ring of integers of @xmath3 . it is a classical problem to investigate relationships between @xmath4 as a galois module and class numbers of intermediate extensions . one shape that such relationships can take is that of unit index formulae , such as ( * ? ? ? * proposition 4.1 ) for elementary abelian galois groups , or @xcite for dihedral groups . the introductions to these papers contain an overview of some of the history of the problem and further references . in this work , we generalise these index formulae to a large class of galois groups and to various different galois modules , namely to units of rings of integers of number fields , to higher @xmath0groups thereof , and to mordell weil groups of elliptic curves over number fields . in fact , we develop an algebraic machine that produces such formulae in a great variety of contexts . in the introduction , we will begin by stating some concrete applications , and will progres to ever more general results . [ thm : ellcurvesg20 ] let @xmath5 be the semidirect product of @xmath6 and @xmath7 with faithful action . let @xmath1 be a galois extension of number fields with galois group @xmath5 , let @xmath8 be the unique intermediate extension of degree 4 , @xmath9 distinct intermediate extensions of degree 5 , and let @xmath10 be an elliptic curve . let the map @xmath11 be induced by inclusion of each summand . the remaining notation will be recalled in [ sec : ellcurves ] . assume for simplicity that @xmath12 has finite tate shafarevich groups over all intermediate extensions of @xmath1 ( we actually prove an unconditional result , see remark [ rem : uncond ] ) . then @xmath13 ^ 2}{|\ker f_e|^2 } , \end{aligned}\ ] ] where @xmath14 , and where the sum in the index is taken inside @xmath15 . the index on the right hand side carries information about the @xmath16$]-module structure of @xmath15 . this information goes far beyond the ranks in the sense that there exist @xmath16$]-modules whose complexifications are isomorphic , but which can be distinguished by the index . thus , the theorem says that tamagawa numbers and tate shafarevich groups control to some extent the fine integral galois module structure of the mordell - weil group . in fact , already the approach in @xcite easily generalises to the aforementioned galois modules , with the only necessary extra ingredient being a compatibility of standard conjectures on special values of zeta and @xmath17functions with artin formalism . however , it uses the classification of all integral representations of dihedral groups , and therefore does not directly extend to other galois groups . the techniques in @xcite on the other hand generalise to a large class of groups . but they rely on a direct computation with units and are not immediately applicable to other galois modules . in particular , theorem [ thm : ellcurvesg20 ] is not accessible by either approach . the main achievement of the present work is a unification and generalisation of the approaches of @xcite and @xcite . proposition [ prop : fixedphi ] is of central importance in this endevour . a byproduct of this unification will be a more conceptual proof of the already known index formulae and a better understanding of the algebraic concepts involved . in the next three theorems , the combination of a group @xmath5 and a set @xmath18 of subgroups of @xmath5 is one of the following : * @xmath19 a semidirect product of non - trivial cyclic groups with @xmath20 a prime and with @xmath21 acting faithfully on @xmath22 , and with @xmath18 consisting of @xmath23 distinct subgroups of @xmath5 of order @xmath23 and the normal subgroup of order @xmath20 ; or * an elementary abelian @xmath20group , with @xmath18 consisting of all index @xmath20 subgroups ; or * a heisenberg group of order @xmath24 , where @xmath20 is an odd prime , and with @xmath18 consisting of the unique normal subgroup @xmath25 of order @xmath26 and of @xmath20 non - conjugate cyclic subgroups of order @xmath20 that are not contained in @xmath25 ; or * any other group @xmath5 and set of subgroups @xmath18 such that the map @xmath27 defined by ( [ eq : phi ] ) is an injection of @xmath5-modules with finite cokernel . below , @xmath28 denotes the grothendieck group of the category of finitely generated @xmath29$]modules . for a @xmath5-module @xmath30 , @xmath31 and @xmath32 denote the @xmath5-modules @xmath33 and @xmath34 , respectively . when no confusion can arise , we will treat @xmath29$]modules synonymously with their image in @xmath28 . [ thm : u ] given @xmath5 and @xmath18 from the above list , there exists an explicitly computable group homomorphism @xmath35 and an explicit @xmath36valued map @xmath37 on conjugacy classes of subgroups of @xmath5 , such that for any galois extension @xmath1 of number fields with galois group @xmath5 and for any finite @xmath5stable set @xmath38 of places of @xmath3 containing the archimedean ones , we have @xmath39}{|\ker f_{\co}|}\nonumber,\end{aligned}\ ] ] where @xmath40 denotes the places of @xmath0 lying below those in @xmath38 , and where the map @xmath41 is induced by inclusions @xmath42 , @xmath43 . [ thm : k ] given @xmath5 and @xmath18 from the above list , there exists an explicitly computable group homomorphism @xmath35 ( the same as in theorem [ thm : u ] ) such that for any galois extension of number fields @xmath1 with galois group @xmath5 , and for @xmath44 , @xmath45}{|\ker f_k|}\nonumber,\end{aligned}\ ] ] where @xmath46 denotes equality up to an integer power of 2 . here , we have slightly abused notation by writing @xmath47 $ ] when we mean @xmath48,\ ] ] and @xmath49 instead of @xmath50 , where @xmath51 is induced by inclusions ( cf . [ sec : kgroups ] ) @xmath52 most of the literature on the structure of mordell weil groups of elliptic curves centres of course around questions about the rank . here , we address the question : assuming that we know everything about ranks , what can we say about the finer integral structure of these galois modules ? the cleanest statements are obtained if one assumes that the relevant tate shafarevich groups are finite , but we also derive an unconditional analogue . [ thm : e ] given @xmath5 and @xmath18 from the above list , there exists an explicitly computable group homomorphism @xmath35 ( the same as in theorems [ thm : u ] and [ thm : k ] ) such that for any galois extension of number fields @xmath1 with galois group @xmath5 and for any elliptic curve @xmath10 with @xmath53 finite for all @xmath54 , we have @xmath55 ^ 2}{|\ker f_e|^2}\nonumber,\end{aligned}\ ] ] where the map @xmath56 is induced by inclusions @xmath57 , @xmath43 . artin s induction theorem implies that given any @xmath29$]module @xmath58 , @xmath59 is determined by the dimensions @xmath60 for all @xmath54 . therefore , theorem [ thm : ellcurvesg20 ] is a special case of theorem [ thm : e ] , while theorem [ thm : u ] is a direct generalisation of ( * ? ? ? * proposition 4.1 ) and ( * ? ? ? * theorem 1.1 ) . [ rem : uncond ] combining equations ( [ eq : ellcurves ] ) and ( [ eq : final ] ) yields an unconditional version of ( [ eq : maine ] ) . unlike in the case of units , this is , to our knowledge , the first such index formula for elliptic curves . it seems rather striking , that the function @xmath61 is the same in all three theorems . thus , it is not only independent of the realisation of @xmath5 as a galois group ( and in particular of the base field ) , but even independent of the galois module in question . in [ sec : examples ] , we give an explicit example of how to compute @xmath61 for a concrete group , thereby deducing theorem [ thm : ellcurvesg20 ] from theorem [ thm : e ] . the function @xmath62 in theorem [ thm : u ] is trivial on cyclic subgroups . in particular , the corresponding product vanishes when @xmath38 is just the set of archimedean places . the above theorems are special cases of the representation theoretic machine we develop . in its maximal generality , our result may be stated as follows ( the necessary concepts , particularly that of brauer relations and of regulator constants , will be recalled in the next section ) : [ thm : r ] let @xmath5 be a finite group and let @xmath63\longrightarrow \bigoplus_j \z[g / h_j']=p_2\ ] ] be an injection of @xmath16$]permutation modules with finite cokernel . for a finitely generated @xmath16$]module @xmath30 , write @xmath64 , @xmath65 , and @xmath66 for the induced map @xmath67 . finally , denote by @xmath68 the regulator constant of @xmath30 with respect to the brauer relation @xmath69 . then , the quantity @xmath70 only depends on the isomorphism class of the @xmath29$]-module @xmath33 . the significance of this result is that regulator constants of galois modules can be linked to quotients of their regulators , which in turn are linked to other number theoretic invariants through special values of @xmath17functions . on the other hand , by making judicious choices of @xmath27 , one can turn the cokernel in theorem [ thm : r ] into an index , such as e.g. the one in equations ( u ) , ( k ) and ( e ) , or the index of the image of the @xmath5-module @xmath30 in @xmath71 under the norm maps , or other natural invariants of galois modules . the function @xmath61 and the map @xmath72 then change , depending on the particular index that one chooses to investigate , but the left hand side of the equations ( u ) , ( k ) and ( e ) does not . the main object of study will be regulator constants , as defined in @xcite . we begin by recalling in [ sec : numthry ] how certain quotients of classical regulators of number fields , of borel regulators , and of regulators of elliptic curves can be translated into regulator constants . in [ sec : regconsts ] and [ sec : indices ] , we will set up a framework that allows one to translate regulator constants into indices in a purely representation theoretic setting . the main new input is proposition [ prop : fixedphi ] , which is the generalisation of ( * ? ? ? * end of 4 ) to arbitrary finite groups . with that in place , we can essentially follow the strategy of @xcite . this procedure works under a certain condition on the galois group ( see beginning of [ sec : indices ] ) . it is an interesting purely group theoretic problem to determine all groups that satisfy this condition , which we will not address here . theorems [ thm : u ] , [ thm : k ] and [ thm : e ] are obtained by simply substituting the regulator constant index relationship , as expressed by ( [ eq : final ] ) , in any of the number theoretic situations discussed in the next section . the quotients of number theoretic data that appear in all these theorems come from brauer relations ( see [ sec : numthry ] ) . brauer proved that any such quotient of numbers of roots of unity in number fields is a power of two . as a completely independent result , we show in proposition [ prop : ktorsion ] that the same is true for @xmath73 , which is the analogue of numbers of roots of unity that appears in lichtenbaum s conjecture . a significant part of this research was done while both authors took part in the 2011 conference on galois module structures in luminy . we thank the organisers of the conference for bringing us together and the cirm for hosting the conference . we would also like to thank haiyan zhou for pointing out two inaccuracies in an earlier draft . we begin by recalling the definitions of brauer relations and of regulator constants and their relevance for number theory . let @xmath5 be a finite group . we say that the formal linear combination @xmath74 of subgroups of @xmath5 is a _ brauer relation _ in @xmath5 if the virtual permutation representation @xmath75^{\oplus n_h}$ ] is zero . [ not : tors ] for any abelian group @xmath76 , we write @xmath77 for @xmath78 . for any homomorphism of abelian groups @xmath79 , write @xmath80 for the induced homomorphism @xmath81 and @xmath82 for the restriction @xmath83 . let @xmath5 be a finite group and @xmath30 a finitely generated @xmath16$]module . let @xmath84 be a bilinear @xmath5invariant pairing that is non degenerate on @xmath85 . let @xmath86 be a brauer relation in @xmath5 . define the regulator constant of @xmath30 with respect to @xmath87 by @xmath88 where each determinant is evaluated on any @xmath89-basis of @xmath90 . this definition is independent of the choice of pairing ( * ? ? ? * theorem 2.17 ) and in particular lies in @xmath36 . regulator constants naturally arise in number theory . the heart of the paper will be an investigation of the regulator constants themselves and their relationship with indices of fixed submodules in a given module . the results can then be applied to any of the equations ( [ eq : units ] ) , ( [ eq : kgroups ] ) , or ( [ eq : ellcurves ] ) to derive immediate number theoretic consequences . for the rest of the section , fix a galois extension @xmath1 of number fields with galois group @xmath5 and a brauer relation @xmath91 in @xmath5 . we embed @xmath3 and all other extensions of @xmath0 that we will consider inside a fixed algebraic closure of @xmath0 . let @xmath38 be a finite @xmath5stable set of places of @xmath3 containing all the archimedean ones . then artin formalism for artin @xmath17functions , combined with the analytic class number formula , implies that @xmath92 where @xmath93 denotes @xmath38class numbers , @xmath94 denotes @xmath38regulators and @xmath95 denotes the numbers of roots of unity . one can show ( * ? ? ? * proposition 2.15 ) that @xmath96)}\cdot \prod_{h\leq g } \reg_s(f^{h})^{2n_h},\ ] ] where @xmath40 is the set of primes of @xmath0 lying below those in @xmath38 , and @xmath97 denotes the decomposition group of a prime of @xmath3 above @xmath98 . it follows that @xmath99 ) } = \cc_\theta(\co_{s , f}^\times).\end{aligned}\ ] ] the factor @xmath100)$ ] can be made very explicit for any concrete @xmath5 . for example , regulator constants of permutation modules corresponding to cyclic subgroups are always trivial ( * ? ? ? * lemma 2.46 ) , and in particular the denominator vanishes if @xmath38 is the set of archimedean places of @xmath3 . otherwise , it is an easily computable function of the number of primes in @xmath38 with given splitting behaviour . it is always a rational number , which has non trivial @xmath20adic order only for those primes @xmath20 that divide @xmath101 . see ( * ? ? ? * theorem 1.2 ) for an even stronger restriction . let @xmath102 , @xmath103 be the real places of @xmath3 and @xmath104 , @xmath105 be representatives of the complex places . let @xmath44 and set @xmath106 if @xmath23 is odd and @xmath107 if @xmath23 is even . borel has constructed a map @xmath108 with kernel precisely equal to the torsion subgroup , and showed @xcite that the image is a full rank lattice . the @xmath23th borel regulator @xmath109 is then defined as the covolume of this image . if @xmath1 and @xmath5 are as before , and @xmath110 is a subgroup of @xmath5 , then for any odd prime @xmath20 , we have @xmath111 ( see e.g. ( * ? ? ? * proposition 2.9 ) ) . we will therefore treat @xmath112 as a subgroup of @xmath113 . we have suppressed all the details concerning the normalisation of @xmath114 , for which we refer to @xcite , but note that these details will be irrelevant for us . what is important , is that if for a number field @xmath25 we define a pairing @xmath115 on @xmath116 by @xmath117 then @xmath118 is @xmath5invariant , and for @xmath119 we have @xmath120\langle\cdot,\cdot\rangle_n$ ] . moreover , by definition , @xmath121 where @xmath122 range over a basis of @xmath123 . the lichtenbaum conjecture on leading coefficients of dedekind zeta functions at @xmath124 together with aforementioned artin formalism predicts that @xmath125 where @xmath46 denotes equality up to an integer power of 2 . in fact , the lichtenbaum conjecture is known to be compatible with artin formalism @xcite , so equation ( [ eq : kartin ] ) is true unconditionally . in view of the above discussion , we also get unconditionally @xmath126 the following result is completely independent of the rest of the paper : [ prop : ktorsion ] let @xmath5 be a finite group and let @xmath127 be a brauer relation in @xmath5 . let @xmath1 be a galois extension of number fields with galois group @xmath5 . then @xmath128 is a power of two . it suffices to show that the rational number in the proposition has trivial @xmath20-part for all odd primes @xmath20 . recall ( e.g. @xcite ) that for a number field @xmath25 and an odd prime number @xmath20 , the @xmath20-part of @xmath129 is isomorphic to @xmath130 , the fixed submodule of the @xmath23th tate twist of the group of @xmath20power roots of unity . putting @xmath131 , we now make two observations . first , for every subgroup @xmath110 of @xmath5 , the @xmath20-part of @xmath132 is isomorphic to @xmath133 . secondly , since @xmath20 is odd , the automorphism group of @xmath134 is cyclic . so if @xmath135 denotes the kernel of the map @xmath136 , then @xmath135 is normal in @xmath5 , and @xmath137 is cyclic , and in particular has no non - trivial brauer relations . since for @xmath54 , @xmath138 , the result follows immediately from ( * ? ? ? * theorem 2.36 ( q ) ) . the following immediate consequence is noteworthy , since it greatly generalises several previous works on tame kernels ( see e.g. @xcite ) : let @xmath1 be a finite galois extension of totally real number fields with galois group @xmath5 , let @xmath91 be a brauer relation in @xmath5 . then @xmath139 is a power of 2 for any @xmath140 . since @xmath3 is totally real , @xmath141 is torsion for any subfield @xmath25 of @xmath3 . the assertion therefore follows from equation ( [ eq : kartin ] ) and proposition [ prop : ktorsion ] . let @xmath10 be an elliptic curve . a consequence of the birch and swinnerton dyer conjecture and of artin formalism for twisted @xmath17functions is @xmath142 where @xmath143 denotes the product of suitably normalised tamagawa numbers over all finite places of @xmath144 . see e.g. the introduction to @xcite for the details on the normalisation of the tamagawa numbers , which will be immaterial for us . formula ( [ eq : ellcurvescond ] ) can in fact be shown to be true only under the assumption that the relevant tate shafarevich groups are finite : if we write @xmath87 as @xmath145 , then the products of weil restrictions of scalars @xmath146 and @xmath147 are isogenous abelian varieties . the claim therefore follows from the compatibility of the birch and swinnerton dyer conjecture with weil restriction of scalars @xcite and with isogenies ( * ? ? ? * chapter i , theorem 7.3 ) . one can also get an entirely unconditional statement by incorporating the ( conjecturally trivial ) divisible parts of the tate shafarevich groups as follows . let @xmath148\hookrightarrow \bigoplus_j\z[g / h_j']\ ] ] be an inclusion of @xmath16$]modules with finite cokernel ( cf . [ sec : regconsts ] ) . let @xmath76 and @xmath149 denote the abelian varieties @xmath146 and @xmath147 , respectively . let @xmath150 and @xmath151 be the dual abelian varieties . then @xmath27 induces an isogeny @xmath152 and the dual isogeny @xmath153 . these in turn give maps on the divisible parts of tate shafarevich groups : @xmath154 and @xmath155 . denote their ( necessarily finite ) kernels by @xmath156 and @xmath157 , respectively . [ prop : bsd ] we have @xmath158}{|e(f^h)_{\tors}|^2}\right)^{n_h}= \frac{|\kappa^t|}{|\kappa|},\ ] ] where @xmath159 denotes the quotients by the divisible parts . see ( * ? ? ? * theorem 4.3 ) or @xcite substituting the nron tate height pairing on @xmath15 in the definition of regulator constants yields @xmath160 and so we have the unconditional statement @xmath161|\kappa|}\right)^{n_h}=\cc_\theta(e(f)).\end{aligned}\ ] ] we retain notation [ not : tors ] . in light of the previous section , our aim is to express regulator constants of an arbitrary @xmath16$]module in terms of the index of a submodule generated by fixed points under various subgroups of @xmath5 . this will be done in the next section . here , we prove the necessary preliminary results . throughout this section , @xmath5 is a finite group , @xmath162 is a brauer relation in @xmath5 , and @xmath30 is a finitely generated @xmath16$]module . we begin by giving an alternative definition of regulator constants . write @xmath163 $ ] , @xmath164 $ ] . since @xmath87 is a brauer relation , there exists an injection of @xmath16$]modules @xmath165 with finite cokernel . dualising this and fixing isomorphisms between the permutation modules and their duals , we also get @xmath166 note that if @xmath27 is given by the matrix @xmath167 with respect to some fixed bases on @xmath168 and @xmath169 , then @xmath170 is given by @xmath171 with respect to the dual bases . applying the contravariant functor @xmath172 , which we abbreviate to @xmath173 , we obtain the maps @xmath174 [ lem : regconsts ] we have @xmath175 in particular , the right hand side is independent of the @xmath16$]module homomorphism @xmath27 . consider the commutative diagram @xmath176 & ( p_2,m)_{\tors } \ar[d]^{(\phi , m)_{\tors}}\ar[r ] & ( p_2,m ) \ar[d]^{(\phi , m)}\ar[r ] & ( p_2,m)/\tors \ar[d]^{\overline{(\phi , m)}}\ar[r ] & 0\\ 0 \ar[r ] & ( p_1,m)_{\tors } \ar[r ] & ( p_1,m ) \ar[r ] & ( p_1,m)/\tors \ar[r ] & 0 , } \ ] ] and similarly for @xmath177 . by ( * theorem 3.2 ) , @xmath178 ( note that in @xcite , @xmath30 is assumed to be torsion free , but the proof extends verbatim to the general case ) . since @xmath179 is trivial , the snake lemma , applied to the above diagram , implies that @xmath180 and similarly for @xmath177 . since both torsion subgroups are finite , @xmath181 whence the result follows . [ prop : fixedphi ] for any fixed @xmath27 , the value of @xmath182 only depends on the isomorphism class of @xmath183 , and not on the integral structure of @xmath30 . as in the previous proof , we have @xmath184 and similarly for @xmath177 , and @xmath185 so it is enough to show that @xmath186 only depends on @xmath187 . for the rest of the proof , we drop the overline and assume without loss of generality that @xmath30 is torsion free . we have @xmath188 so it remains to prove that if @xmath189 , then @xmath190 only depends on @xmath183 for torsion free @xmath30 . now , @xmath191 is a full rank lattice in the vector space @xmath192 , and @xmath190 , being the expansion factor of the lattice under @xmath27 , does not depend on the choice of lattice . [ cor : cphi ] for any fixed @xmath27 , there exists a function @xmath193 that is only a function of the collection of numbers @xmath194 as @xmath110 ranges over the subgroups of @xmath5 . it is uniquely determined by its values on the irreducible rational representations of @xmath5 . the right hand side of the equation is clearly multiplicative in direct sums of representations . the result follows immediately from proposition [ prop : fixedphi ] and artin s induction theorem . for any fixed @xmath27 , we have @xmath195 this proves theorem [ thm : r ] . the name of the game will be to choose suitable injections @xmath27 , for which @xmath196 can be interpreted in terms of natural invariants of the galois module @xmath30 . from now on , we closely follow @xcite . suppose that we have a subset @xmath18 of subgroups of @xmath5 such that @xmath197 is a brauer relation and such that the map @xmath198\oplus\z^{\oplus|\cu| } & \rightarrow & \z\oplus \bigoplus_{h\in\cu } \z[g / h]\\ ( \sigma,0 ) & \mapsto & ( 1,(\sigma h)_{h\in\cu}),\nonumber\\ ( 0,(n_h)_{h\in\cu } ) & \mapsto & ( 0,(n_hn_h)_{h\in\cu}),\;\;\;n_h=\sum_{g\in g / h}g\nonumber\end{aligned}\ ] ] is an injection of @xmath5modules . see example [ ex : groups ] for some families of groups that have such brauer relations . it is important to note that , although we usually treat brauer relations as elements of the burnside ring of @xmath5 , so that subgroups of @xmath5 are only regarded as representatives of conjugacy classes , the problem of finding a map @xmath27 as above that is injective may depend on the `` right '' choice of conjugacy class representatives . the map @xmath27 commutes with taking coinvariants , so we get a commutative diagram , where the horizontal maps are the augmentation maps : @xmath199\oplus \z^{\oplus|\cu| } \ar[d]^\phi\ar@{->>}[r ] & \z^{|\cu|+1}\ar[d]^{\phi_g}\\ \z\oplus\bigoplus_{h\in \cu}\z[g / h ] \ar@{->>}[r ] & \z^{\oplus|\cu|+1}. } \ ] ] finally , applying the contravariant functor @xmath200 , we get the commutative diagram with exact rows @xmath176 & ( m^g)^{\oplus|\cu|+1 } \ar[d]^{(\phi_g , m)}\ar[r ] & m^g\oplus\bigoplus_{h\in\cu } m^{h } \ar[d]^{(\phi , m)}\ar[r ] & \bigoplus_{h\in\cu } m^{h}/m^g \ar[d]^f\ar[r ] & 0\\ 0 \ar[r ] & ( m^g)^{\oplus|\cu|+1 } \ar[r ] & m\oplus ( m^g)^{\oplus|\cu| } \ar[r ] & m / m^g \ar[r ] & 0 , } \ ] ] where the map @xmath72 is induced by inclusions @xmath201 , so that @xmath202.\ ] ] thus , the snake lemma , together with the results of the previous section , yield @xmath203 ^ 2}{|\ker f|^2 } & = & \frac{|\coker(\phi , m)|^2}{|\ker(\phi , m)|^2}\cdot\frac{|\ker(\phi_g , m)|^2}{|\coker(\phi_g , m)|^2}\nonumber\\ & = & \frac{c_{\phi}(m)}{\cc_{\theta}(m)}\cdot \frac{|m_{\tors}|^2\cdot|m^g_{\tors}|^{2|\cu|-2}}{\prod_{h\in\cu}|m^{h}_{\tors}|^2}\cdot \frac{\cc_{0}(m)}{c_{\phi_g}(m)}\cdot\frac{|m^g_{\tors}|^{2|\cu|+2}}{|m^g_{\tors}|^{2|\cu|+2}}\nonumber\\ & = & \frac{c_{\phi}(m)}{c_{\phi_g}(m)\cc_{\theta}(m)}\cdot \frac{|m_{\tors}|^2\cdot|m^g_{\tors}|^{2|\cu|-2}}{\prod_{h\in\cu}|m^{h}_{\tors}|^2}.\end{aligned}\ ] ] in an arithmetic context , the torsion quotient will generically vanish . for example , if @xmath0 is a number fields , @xmath5 is a fixed finite group , and @xmath10 is an elliptic curve , then for a generic galois extension @xmath1 with galois group @xmath5 , @xmath204 ( so here , we consider @xmath205 ) . more precisely , @xmath206=e(k)[p^\infty]$ ] for all @xmath1 of bounded degree whenever @xmath20 is sufficiently large ( the implicit bound only depending on @xmath10 and on the degree of @xmath1 ) . if , on the other hand , we set @xmath207 , then the torsion quotient is a power of 2 . for @xmath208 this is due to brauer @xcite , and for @xmath44 this is proposition [ prop : ktorsion ] . for many concrete groups @xmath5 and relations @xmath87 , the torsion quotient can be shown to always vanish . note also , that once we substitute the above formula for the regulator constant into ( [ eq : units ] ) , ( [ eq : kgroups ] ) , or ( [ eq : ellcurves ] ) , the torsion quotient cancels , so in any case , it is not present in the final index formulae . combining equation ( [ eq : final ] ) with ( [ eq : units ] ) , ( [ eq : kgroups ] ) , and ( [ eq : ellcurves ] ) proves theorems [ thm : u ] , [ thm : k ] , and [ thm : e ] , respectively , for all groups that have a brauer relation of the form ( [ eq : phi ] ) . see the next section for several examples of such groups . the function @xmath209)$ ] of theorems [ thm : u ] , [ thm : k ] and [ thm : e ] is given on an irreducible representation @xmath58 by @xmath210 for any lattice @xmath30 in @xmath58 , while @xmath211)^{-1}$ ] . [ ex : groups ] examples of groups that admit a brauer relation of the form ( [ eq : phi ] ) include the following : * elementary abelian @xmath20groups , with @xmath18 being the set of all subgroups of index @xmath20 . this recovers the main results of @xcite , and generalises these results to other galois modules ; * semidirect products @xmath212 with @xmath20 an odd prime and with @xmath21 acting faithfully on @xmath22 , and with @xmath18 consisting of @xmath22 and of @xmath23 distinct subgroups of order @xmath23 . this includes the case of ( * ? ? ? * theorem 1.1 ) and provides a vast generalisation of that result . * heisenberg groups of order @xmath24 , where @xmath20 is an odd prime , and where @xmath18 consists of the unique normal subgroup @xmath25 of order @xmath26 and of @xmath20 non - conjugate cyclic groups of order @xmath20 that are not contained in @xmath25 . let @xmath213 with @xmath7 acting faithfully on @xmath6 . as mentioned in the previous example , @xmath214 is a brauer relation , and letting @xmath18 consist of @xmath6 and of 4 distinct cyclic groups of order 4 , @xmath215\oplus \z^5\rightarrow \z\oplus \bigoplus_{h\in \cu}\z[g / h]\ ] ] defined by ( [ eq : phi ] ) is an injection of @xmath5modules with finite cokernel . given a @xmath16$]module @xmath30 , the map @xmath66 corresponding to @xmath27 is given by @xmath216 while @xmath177 is easily seen to be @xmath217 * the two 1dimensional representations @xmath218 , @xmath219 that are lifted from @xmath220 , * the direct sum @xmath221 of the remaining two 1dimensional complex representations lifted from @xmath222 , * the 4dimensional induction @xmath223 of a non trivial one dimensional character from @xmath6 to @xmath5 . in each of these , we need to choose a @xmath5invariant lattice . clearly , @xmath224 and @xmath219 contain , up to isomorphism , only one @xmath16$]module each , which we will denote by @xmath225 , @xmath226 respectively . next , let @xmath227 be the non trivial torsion free @xmath228$]module of @xmath89rank 1 . define @xmath229 to be the lift from @xmath222 of @xmath230 . this is a @xmath5invariant full rank sublattice of @xmath221 . as for @xmath223 , it will be simpler to work with @xmath231 , which can be realised as the induction from @xmath6 of @xmath232\ominus \q[c_5/c_5]$ ] . let @xmath233 be the @xmath234$]module @xmath235\big/\langle\sum_{g\in c_5}g\rangle_\z$ ] and define @xmath236 to be the induction of @xmath237 to @xmath5 , so that @xmath238 . * let @xmath243 . clearly , the left hand side of equation ( [ eq : final ] ) is trivial , so @xmath244)=1/\cc_\theta(m)=125 $ ] . for all the remaining lattices , we will have @xmath245 , so the cokernels will be trivial on corestrictions . * let @xmath246 . then , @xmath247 and @xmath248 , so @xmath66 is surjective , and @xmath249 . * the same reasoning applies to @xmath250 , but since @xmath229 has rank 2 , we have @xmath251 . * let @xmath252 . we have @xmath253 , while for any subgroup @xmath110 of order 4 , @xmath254 has @xmath89rank 4 . an explicit computation , either by hand or using a computer algebra package , yields @xmath255 . for an arbitrary rational representation @xmath58 of @xmath5 , the multiplicities @xmath260 of the irreducible rational representations in the direct sum decomposition of @xmath58 are determined by @xmath60 as @xmath110 ranges over subgroups of @xmath5 : @xmath261 solving for the multiplicities , we deduce , for an arbitrary rational representation @xmath58 , @xmath262 ) & = & 5^{3\rk v^g}\cdot 5^{-(\rk v^{d_{10 } } - \rk v^g)}\cdot25^{-(\rk v^{c_5 } - \rk v^{d_{10}})/2}\cdot 5^{-3(\rk v^{c_4 } - \rk v^g)}\\ & = & 5^{7\rk v^g - \rk v^{c_5 } - 3\rk v^{c_4}}.\end{aligned}\ ] ] * by artin s induction theorem , the function @xmath61 is also determined uniquely by its values on @xmath263 $ ] as @xmath264 ranges over the cyclic subgroups of @xmath5 , and these may be easier to calculate in specific cases , especially when it is difficult to classify all irreducible rational representations . indeed , given any irreducible rational representation @xmath58 of @xmath5 , some integer multiple @xmath265 can be written as a @xmath89-linear combination of such permutation representations . but since @xmath209)$ ] is a positive real number , the value of @xmath266)$ ] uniquely determines @xmath209)$ ] . we have already implicitly used this to calculate @xmath259)$ ] . * instead of computing @xmath267 and @xmath268 , one may compute one of the cokernels together with @xmath269 . this is particularly helpful when combined with the previous observation , since regulator constants of permutation representations are very easy to compute : using either ( * ? ? ? * proposition 2.45 ) or ( * ? ? ? * example 2.19 ) , one finds that @xmath270 ) = |g|^{1-|\cu| } \prod_{h\in \cu } \prod_{g\in c\backslash g / h}|h^g\cap c|.\ ] ] moreover , for the corresponding cokernels on the corestrictions , the regulator constant is computed with respect to the trivial brauer relation and is therefore 1 , so that @xmath271 for all integral representations @xmath30 of @xmath5 .

we prove very general index formulae for integral galois modules , specifically for units in rings of integers of number fields , for higher @xmath0groups of rings of integers , and for mordell - weil groups of elliptic curves over number fields . these formulae link the respective galois module structure to other arithmetic invariants , such as class numbers , or tamagawa numbers and tate shafarevich groups . this is a generalisation of known results on units to other galois modules and to many more galois groups , and at the same time a unification of the approaches hitherto developed in the case of units .

we call chemical evolution of galaxies the study of how the chemical elements formed in stars and were distributed in galaxies . during the big bang only the light elements ( h , d , he , li ) were synthesized , while stars have been responsible for the formation and distribution of all the elements from carbon to uranium and beyond . some light elements , such as li , be and b are formed during the spallation process , which is the interaction between the cosmic rays and c , n , o atoms present in the interstellar medium ( ism ) . therefore , stars produce chemical elements in their interiors by means of nuclear reactions of fusion and then restore these elements into the ism when they die . the fusion reactions occur up to @xmath0 which is the element with the maximum binding energy per nucleon . for elements heavier than @xmath0 the nuclear fusion is therefore inhibited and the nuclear fission is favored . the stars are born , live and die and they can die in a quiescent fashion like white dwarfs or violently as supernovae ( sne ) . the supernovae can be of various types : core - collapse sne , namely the explosion of single massive stars ( @xmath1 , type ii , ib , ic ) , and type ia sne , namely the explosion of a white dwarf occurring after accreting material from a companion in a binary system . in this lecture i will first describe the ingredients necessary to build galactic chemical evolution models . such models can be analytical or numerical and they aim at following the evolution in time and space of the abundances of the chemical elements in the ism . then , i will focuse on some highlights in the galactic chemical evolution and on the comparison models - observations . in doing so , i will show how chemical evolution models can constrain stellar nucleosynthesis and galaxy formation timescales . the basic ingredients necessary to build a chemical evolution model are : the `` yield per stellar generation '' of a single chemical element , can be defined as ( tinsley 1980 ) : @xmath16 where @xmath10 is the mass of the newly produced element @xmath11 ejected by a star of mass @xmath17 , and @xmath18 is the returned fraction . the yield @xmath19 is therefore the mass fraction of the element @xmath11 newly produced by a generation of stars relative to the fraction of mass in remnants ( white dwarfs , neutron stars and black holes ) and never dying low mass stars ( @xmath20 ) . we define `` returned fraction '' the fraction of mass ejected into the ism by an entire stellar generation , namely : @xmath21 the term fraction originates from the fact that both @xmath19 and @xmath18 are divided by the normalization condition of the imf , namely : @xmath22 in order to define @xmath19 and @xmath18 we have made a very specific assumption : the instantaneous recycling approximation ( ira ) , which states that _ all stars more massive than 1@xmath13 die instantaneously , while all stars less massive live forever_. this assumption allows us to solve the chemical evolution equations analytically , as we will see in the following , but it is a very poor approximation for chemical elements produced on long timescales such as c , n and fe . on the other hand , for oxygen , which is almost entirely produced by short lived core - collapse sne , ira is an acceptable approximation . in figure 1 we show @xmath23 and @xmath18 , computed for different initial metallicities of the stars and different imfs . as one can see , the dependence of these quantities on z is negligible whereas that on the imf is strong . in that figure it does not appear the kroupa(2001 ) universal imf , suggesting that the imf in stellar clusters is an universal one . kroupa ( 2001 ) imf is a two - slope imf with a slope for stars more massive than 0.5@xmath13 very similar to that of the salpeter ( 1955 ) imf , still widely used in model for external galaxies . the simple model of chemical evolution assumes that the system is evolving as a closed - box , without inflows or outflows , the imf is constant in time , the chemical composition of the gas is primordial and the mixing between the chemical products ejected by stars and the ism is instantaneous , plus ira . the initial gas mass is therefore , @xmath24 , @xmath25 is the fractionary mass of gas , the metallicity is @xmath26 is zero at the time t=0 . the basic equations can be written as : @xmath27 which describes the evolution of gas and where the integral is the rate at which dying stars restore both the enriched and unenriched material into the ism at the time @xmath28 . the equation for metals is : @xmath29 \psi(t-\tau_m ) \varphi(m)dm},\ ] ] where the first term in the square brackets represents the mass of pristine metals which are restored into the ism without suffering any nuclear processing , whereas the second term contains the newly formed and ejected metals ( maeder 1992 ) . when ira is assumed , the sfr can be taken out of the integrals and the equation for metals be solved analytically . its solution is : @xmath30 the metallicity yield per stellar generation @xmath31 which appears in the above equation is known as _ effective yield _ , simply defined as the yield @xmath32 that would be deduced if the system were assumed to be described by the simple model . therefore , the effective yield is : @xmath33 clearly , the true yield @xmath31 will be always lower than the effective one in both cases of winds and infall of primordial gas . the only way to increase the effective yield is to assume an imf more weighted towards massive stars than the canonical imfs . we define primary element an element produced directly from h and he a typical primary element is carbon or oxygen which originate from the 3-@xmath34 reactions we define secondary element an element produced starting from metals already present in the star at birth ( e.g. nitrogen produced in the cno cycle ) . we recall that the solution of the chemical evolution equations for a secondary element , with abundance @xmath35 , implies : @xmath36 where @xmath37 is the yield per stellar generation of the secondary element . this means that , for a secondary element the simple closed - box model predicts that its abundance increases proportionally to the metallicity squared , namely : @xmath38 in the case of a model with outflow but no inflow occurring at a rate : @xmath39 where @xmath40 is a free parameter larger than or equal to zero , the solution for the metallicity of the system is : @xmath41.\ ] ] it is clear that in the case of @xmath42 the solution is the same as that of the simple model . in the case of a model without outflow but inflow of primordial gas ( @xmath43 ) , occurring at a rate : @xmath44 with @xmath45 a positive constant different from zero and from 1 , the solution is : @xmath46.\ ] ] again , if @xmath47 , the solution is the same as that of the simple model . the case @xmath48 is a particular one , called `` extreme infall '' and it has a different solution : @xmath49 where @xmath50 . for a more extensive discussion of these and other analytical solutions see matteucci ( 2012 ) . when the stellar lifetimes are correctly taken into account the chemical evolution equations should be solved numerically . this allows us to follow in detail the temporal evolution of the abundances of single elements . a complete chemical evolution model in the presence of both galactic wind , gas infall and radial flows can be described by a number of equations equal to the number of chemical species : in particular , if @xmath51 is the mass of the gas in the form of any chemical element @xmath11 , we can write the following set of integer - differential equations which can be solved only numerically , if ira is relaxed : @xmath52 dm}+\\ \nonumber ( 1-a_b)\int_{m_{bm}}^ { m_{bm}}{\psi(t-\tau_{m})q_{mi}(t-\tau_{m)}\varphi(m)dm}+\\ \nonumber \int_{m_{bm}}^{m_u}{\psi(t-\tau_m)q_{mi}(t-\tau_m ) \varphi(m)dm } + x_{ia}(t ) a(t)\\ \nonumber - x_{i}(t ) w(t ) + x_{i}(t)i(t),\end{aligned}\ ] ] where @xmath51 can be substituted by @xmath53 , namely the surface gas density of the element @xmath11 . in several models of chemical evolution it is customarily to use normalized variables which should be substituted to @xmath54 or to @xmath55 , such as for example : @xmath56 with @xmath57 , and @xmath58 ( @xmath59 ) being the total surface mass density ( mass ) at the present time @xmath60 . the surface densities are more indicated for computing the chemical evolution of galactic disks , while for spheroids one can use the masses . the quantity @xmath61 represents the abundance by mass of the element @xmath11 and by definition the summation over all the mass abundances of the elements present in the gas mixture is equal to unity . these equations include the products from type ia sne ( third term on the right ) and the products of stars ending their lives as white dwarfs and core - collapse sne ( for a more extensive description of the equations see matteucci , 2012 ) . in this section we will illustrate some examples where predictions from chemical evolution models are compared to observations . we will start with the milky way , which is the galaxy for which we possess the majority of information . a good model for the chemical evolution of the milky way should reproduce several constraints , including the g - dwarf metallicity distribution , the [ x / fe ] vs. [ fe / h]= @xmath62 relations ( where x is a generic element from carbon up to the heaviest ones ) , abundance and gas gradients along the galactic disk , among others . in figure 2 we show the predictions from romano et al . ( 2010 ) concerning several chemical elements obtained by using different sets of stellar yields and compared to observation in stars . as one can see , the agreement is good for some chemical species whereas for others the agreement is still very poor . the reason for that probably resides in the uncertainties still existing in the theoretical stellar yields . the shapes of the [ x / fe ] vs. [ fe / h ] relation can be successfully interpreted as due to the _ time - delay model _ , namely the fact that elements such as c , n and fe are mainly produced by long living stars , while others such as @xmath34-elements ( o , mg , si , ca ) are produced by short living stars . in particular , fe is mainly produced by type ia sne and only a small fraction of it originates in core - collapse ( cc ) sne . type i a sne explode on longer timescales than core - collapse sne and therefore ratios such as [ @xmath34/fe ] , where @xmath34 indicates @xmath34-elements which are mainly produced in cc sne , can be used as cosmic clocks . the higher than solar value of the [ @xmath34/fe ] ratios for low [ fe / h ] values , is then due to the cc sne which restore the @xmath34-elements on short timescales . when type ia sne , originating from co white dwarfs which have longer lifetimes ( from 30 myr up to a hubble time ) , start restoring the bulk of fe , then the [ @xmath34/fe ] ratios start decreasing . by means of the time - delay model we can interpret any abundace ratio . the model underlying the predictions of figure 2 is an updated version of the two - infall model by chiappini et al . ( 1997 ) : this model assumes that the milky way formed by means of two main gas accretion episodes , one during which the halo and thick disk formed and another during which the thin disk formed on much longer timescales . in figure 3 we show an illustration of the so - called li - problem . in particular , we report the abundance of @xmath63 ( a(li ) = 12 + log(li / h ) versus [ fe / h ] ) . the data sources are indicated inside the figure . the upper envelope of the data should represent the evolution of the abundance of @xmath63 in the ism since the stars in the upper envelope should exhibit the li abundance present in the gas when they formed . the stars below should instead have consumed the original li which is very fragile and tends to react with protons to form @xmath64 when the temperature is @xmath65 k. according to this interpretation of the diagram , the younger stars with higher [ fe / h ] should reflect the fact that the li abundance has increased with galactic lifetime owing to li stellar production by stars . stars which can create , preserve and eject li into the ism are : low and intermediate mass stars during the asymptotic giant branch ( agb ) phase , the cc sne , cosmic rays and perhaps novae ( see izzo et al . 2015 for the possible detection of li in a nova ) . the li problem arises from the fact that both wmap and planck experiments have derived a primordial @xmath63 abundance higher than the abundance found in the upper envelope of the galactic halo stars ( those with [ fe / h]@xmath66 -1.0 in figure 3 ) . this could mean that the oldest galactic stars have consumed their primordial li and roughly by the same amount for stars in the range -3.0 -1.0 in [ fe / h ] ( the so - called spite plateau ) . the lines in figure 3 represent the predictions of a chemical evolution model ( the two - infall model ) including sne , novae , cosmic rays and agb stars as li producers and starting from the primordial li abundance deduced either by wmap or by the value of the spite plateau . there is not yet a solution for the li - problem . for solar vicinity stars and meteorites ( symbols ; see legend ) compared to the predictions of chemical evolution models ( lines and colored areas ) . the chemical evolution model is from romano et al . ( 2010).the black line represents all the stellar li producers , as described in the text . the red line is the same model but without novae . note that the dotted theoretical curve starts with the primordial li value from wmap which lies well above the li abundances of halo stars.figure from izzo et al . ( 2015 ) . ] in figure 4 we show the model predictions and data for the ufd galaxy hercules , a small ( with stellar mass of @xmath67 ) satellite of the milky way . the data refer to the [ ca / fe ] ratios measured in the stars of this galaxy and include also the data for ca in the milky way and dwarf spheroidal stars . the data show that the milky way and hercules evolved in a different way , since the [ ca / fe ] ratios are lower at the same [ fe / h ] in the ufd galaxy , relative to the milky way . the models , computed with an efficiency of star formation one thousand times lower than assumed for the milky way can well reproduce the trend . the model for hercules includes also a strong galactic wind proportional to the sfr . because of the already mentioned time - delay model , the fe delayed production coupled with a regime of slow star formation predict that the [ ca / fe ] ratios are lower . at the same [ fe / h ] , since when the type ia sne start to be important in producing fe , the ca and the other metals produced by cc sne have not yet attained the same abundances as in the miilky way which evolves with a much faster star formation . ratios as functions of [ fe / h ] . the red stars represent abundances in stars of hercules , while the blue points represent dwarf spheroidals and black points the milky way . the red lines are models assuming a star formation efficiency ( @xmath9 ) varying in the range 0.002 - 0.008 @xmath68 . figure from koch et al . ( 2012 ) . ] therefore , in order to interpret the abundance patterns in galaxies we should consider the history of star formation , since it affects strongly the shape of the [ x / fe ] vs. [ fe / h ] relations . in this way , we can reconstruct the history of stars formation in galaxies just by looking at their abundances . this approach is known as _ astroarchaeological approach_. from figure 4 it apprears that the dwarf spheroidals have a higher star formation efficiency than the ufds but a lower one than the milky way . from this figure it is difficult to conclude that stars of the ufds could have been the building blocks of the galactic halo , but more data on ufds are necessary before drawing firm conclusions on this point . i acknowledge financial support from the prin2010 - 2011 project `` the chemical and dynamical evolution of the milky way and local group galaxies '' , prot.2010ly5n2 t .

in this lecture i will introduce the concept of galactic chemical evolution , namely the study of how and where the chemical elements formed and how they were distributed in the stars and gas in galaxies . the main ingredients to build models of galactic chemical evolution will be described . they include : initial conditions , star formation history , stellar nucleosynthesis and gas flows in and out of galaxies . then some simple analytical models and their solutions will be discussed together with the main criticisms associated to them . the yield per stellar generation will be defined and the hypothesis of instantaneous recycling approximation will be critically discussed . detailed numerical models of chemical evolution of galaxies of different morphological type , able to follow the time evolution of the abundances of single elements , will be discussed and their predictions will be compared to observational data . the comparisons will include stellar abundances as well as interstellar medium ones , measured in galaxies . i will show how , from these comparisons , one can derive important constraints on stellar nucleosynthesis and galaxy formation mechanisms . most of the concepts described in this lecture can be found in the monograph by matteucci ( 2012 ) .

one of the most important and intriging issues of modern physics is the so called cosmological constant problem " @xcite , ( ccp ) , most easily seen by studying the apparently uncontrolled behaviour of the zero point energies , which would lead to a corresponding equally uncontrolled vacuum energy or cosmological constant term . even staying at the classical level , the observed very small cosmological term in the present universe is still very puzzling . furthermore , the cosmological constant problem has evolved from the old cosmological constant problem " , where physicist were concerned with explaining why the observed vacuum energy density of the universe is exactly zero , to different type of ccp since the evidence for the accelerating universe became evident , for reviews see @xcite . we have therefore since the discovery of the accelerated universe a new cosmological constant problem " @xcite , the problem is now not to explain zero , but to explain a very small vacuum energy density . this new situation posed by the discovery of a very small vacuum energy density of the universe means that getting a zero vacuum energy density for the present universe is definitely not the full solution of the problem , although it may be a step towards its solution . one point of view to the ccp that has been popular has been to provide a bound based on the anthropic principle " @xcite . in this approach , a too large cosmological constant will not provide the necessary conditions required for the existence of life , the anthropic principle provides then an upper bound on the cosmological constant . one problem with this approach is for example that it relies on our knowledge of life as we know it and ignores the possibility that other life forms could be possible , for which other ( unknown ) bounds would be relevant , therefore the reasoning appears by its very nature subjective , since of course if the observed cosmological constant will be different , our universe will be different and this could include different kind of life that may be could have adjusted itself to a higher cosmological constant of the universe . but even accepting the validity of anthropic considerations , we still do not understand why the observed vacuum energy density must be positive instead of possibly a very small negative quantity . accepting the anthropic explanation means may be also giving up on discovering important physics related to the ccp and this may be the biggest objection . nevertheless , the idea of associating somehow restrictions on the origin of the universe with the cosmological constant problem seems interesting . we will take on this point of view , but leave out the not understood concept of life out from our considerations . instead , we will require , in a very specific framework , the non - singular origin of the universe . the advantage of this point of view is that it is formulated in terms of ideas of physics alone , without reference to biology , which unlike physics , has not reached the level of an exact science . another interesting consequence is that we can learn that of a non - singularly created universe may not have a too big cosmological constant , an effect that points to a certain type of gravitational suppression of uv divergences in quantum field theory . in this respect , one should point out that even in the context of the inflationary scenario @xcite which solves many cosmological problems , one still encounters the initial singularity problem which remains unsolved , showing that the universe necessarily had a singular beginning for generic inflationary cosmologies @xcite . here we will adopt the very attractive emergent universe " scenario , where those conclusions concerning singularities can be avoided @xcite . the way to escape the singularity in these models is to violate the geometrical assumptions of these theorems , which assume i ) that the universe has open space sections ii ) the hubble expansion is always greater than zero in the past . in @xcite the open space section condition is violated since closed robertson walker universes with @xmath7 are considered and the hubble expansion can become zero , so that both i ) and ii ) are avoided . in @xcite even models based on standard general relativity , ordinary matter and minimally coupled scalar fields were considered and can provide indeed a non - singular ( geodesically complete ) inflationary universe , with a past eternal einstein static universe that eventually evolves into an inflationary universe . those most simple models suffer however from instabilities , associated with the instability of the einstein static universe . the instability is possible to cure by going away from gr , considering non perturbative corrections to the einstein s field equations in the context of the loop quantum gravity @xcite , a brane world cosmology with a time like extra dimension @xcite considering the starobinski model for radiative corrections ( which can not be derived from an effective action ) @xcite or exotic matter @xcite . in addition to this , the consideration of a jordan brans dicke model also can provide a stable initial state for the emerging universe scenario @xcite . in this review we study a different theoretical framework where such emerging universe scenario is realized in a natural way , where instabilities are avoided and a succesfull inflationary phase with a graceful exit can be achieved . the model we will use was studied first in @xcite , however , in the context of this model , a few scenarios are possible . for example in the first paper on this model @xcite a special choice of state to describe the present state of our universe was made . then in @xcite a different candidate for the vacuum that represents our present universe was made . the way in which we best represents the present state of the universe is crucial , since as it should be obvious , the discussion of the ccp depends on what vacuum we take . in @xcite we expressed the stability and existence conditions for the non - singular universe in terms of the energy of the vacuum of our candidate for the present universe . in @xcite a few typos in @xcite were corrected and also the discussion of some notions discussed was improved in @xcite and more deeper studies will be done in this review . indeed in this review , all those topics will be further clarified , in particular the vacuum structure of this model will be extended . a very important new feature that will be presented in this review is the existence of a kinetic vacuum " , that produces a vacuum energy state which is degenerate with the vacuum choice made in @xcite , this degeneracy is analyzed and the dynamical role of this kinetic vacuum in the evolution of the universe and the ccp is analyzed . we work in the context of a theory built along the lines of the two measures theory ( tmt ) . basic idea is developed in @xcite , @xcite-@xcite @xcite , @xcite-@xcite , @xcite-@xcite , @xcite and more specifically in the context of the scale invariant realization of such theories @xcite , @xcite-@xcite , @xcite-@xcite , @xcite . these theories can provide a new approach to the cosmological constant problem and can be generalized to obtain also a theory with a dynamical spacetime @xcite , furthermore , string and brane theories , as well as brane world scenarios can be constructed using two measure theories ideas @xcite-@xcite . we should also point out that the hodge dual construction of @xcite for supergravity constitutes in fact an example of a tmt . the construction by comelli @xcite where no square root of the determinant of the metric is used and instead a total divergence appears is also a very much related approach . the two measure theories have many points of similarity with `` lagrange multiplier gravity ( lmg ) '' @xcite . in lmg there is a lagrange multiplier field which enforces the condition that a certain function is zero . for a comparison of one of these lagrange multiplier gravity models with observations see @xcite . in the two measure theory this is equivalent to the constraint which requires some lagrangian to be constant . the two measure model presented here , as opposed to the lmg models of @xcite provide us with an arbitrary constant of integration . the introduction of constraints can cause dirac fields to contribute to dark energy @xcite or scalar fields to behave like dust like in @xcite and this dust behaviour can be caused by the stabilization of a tachyonic field due to the constraint , accompanied by a floating dark energy component @xcite . tmt models naturally avoid the 5th force problem @xcite . we will consider a slight generalization of the tmt case , where , we consider also the possible effects of zero point energy densities , thus softly breaking " the basic structure of tmt for this purpose . we will show how the stated goals of a stable emergent universe can be achieved in the framework of the model and also how the stability of the emerging universe imposes interesting constraints on the energy density of the ground state of the theory as defined in this paper : it must be positive but not very large , thus the vacuum energy and therefore the term that softly breaks the tmt structure appears to be naturally controlled . an important ingredient of the model considered here is its softly broken conformal invariance , meaning that we allow conformal breaking terms only though potentials of the dilaton , which nevertheless preserve global scale invariance . in another models for emergent universe we have studied @xcite , that rule of softly broken conformal invariance was taken into account . it is also a perfectly consistent , but different approach . the review will be organized as follows : first we review the principles of the tmt and in particular the model studied in @xcite , which has global scale invariance and how this can be the basis for the emerging universe . such model gives rise , in the effective einstein frame , to an effective potential for a dilaton field ( needed to implement an interesting model with global scale invariance ) which has a flat region . following this , we look at the generalization of this model @xcite by adding a curvature square or simply @xmath8 term " and show that the resulting model contains now two flat regions . the existence of two flat regions for the potential is shown to be consequence of the s.s.b . of the scale symmetry . we then consider the incorporation in the model of the zero point fluctuations , parametrized by a cosmological constant in the einstein frame . in this resulting model , there are two possible types of emerging universe solutions , for one of those , the initial einstein universe can be stabilized due to the non linearities of the model , provided the vacuum energy density of the ground state is positive but not very large . this is a very satisfactory results , since it means that the stability of the emerging universe prevents the vacuum energy in the present universe from being very large!. the transition from the emergent universe to the ground state goes through an intermediate inflationary phase , therefore reproducing the basic standard cosmological model as well . we end with a discussion section and present the point of view that the creation of the universe can be considered as a threshold event " for zero present vacuum energy density , which naturally gives a positive but small vacuum energy density . the general structure of general coordinate invariant theories is taken usually as @xmath9 where @xmath10 . the introduction of @xmath11 is required since @xmath12 by itself is not a scalar but the product @xmath13 is a scalar . inserting @xmath11 , which has the transformation properties of a density , produces a scalar action @xmath14 , as defined by eq.([1 ] ) , provided @xmath15 is a scalar . in principle nothing prevents us from considering other densities instead of @xmath11 . one construction of such alternative measure of integration " , is obtained as follows : given 4-scalars @xmath16 ( a = 1,2,3,4 ) , one can construct the density @xmath17 and consider in addition to the action @xmath14 , as defined by eq.([1 ] ) , @xmath18 , defined as @xmath19 @xmath20 is again some scalar , which may contain the curvature ( i.e. the gravitational contribution ) and a matter contribution , as it can be the case for @xmath14 , as defined by eq.([1 ] ) . for an approach that uses four - vectors instead of four - scalars see @xcite . in the action @xmath18 defined by eq.([3 ] ) the measure carries degrees of freedom independent of that of the metric and that of the matter fields . the most natural and successful formulation of the theory is achieved when the connection is also treated as an independent degree of freedom . this is what is usually referred to as the first order formalism . one can consider both contributions , and allowing therefore both geometrical objects to enter the theory and take as our action @xmath21 here @xmath15 and @xmath20 are @xmath16 independent . we will study now the dynamics of a scalar field @xmath3 interacting with gravity as given by the following action , where except for the potential terms @xmath22 and @xmath23 we have conformal invariance , the potential terms @xmath22 and @xmath23 break down this to global scale invariance . @xmath24 @xmath25 @xmath26 @xmath27 @xmath28 the suffix @xmath29 in @xmath30 is to emphasize that here the curvature appears only linearly . here , except for the potential terms @xmath22 and @xmath23 we have conformal invariance , the potential terms @xmath22 and @xmath23 break down this to global scale invariance . since the breaking of local conformal invariance is only through potential terms , we call this a soft breaking " . in the variational principle @xmath31 , the measure fields scalars @xmath16 and the matter " - scalar field @xmath3 are all to be treated as independent variables although the variational principle may result in equations that allow us to solve some of these variables in terms of others . for the case the potential terms @xmath32 we have local conformal invariance @xmath33 and @xmath16 is transformed according to @xmath34 @xmath35 where @xmath36 is the jacobian of the transformation of the @xmath16 fields . this will be a symmetry in the case @xmath32 if @xmath37 notice that @xmath38 can be a local function of space time , this can be arranged by performing for the @xmath16 fields one of the ( infinite ) possible diffeomorphims in the internal @xmath16 space . we can still retain a global scale invariance in model for very special exponential form for the @xmath22 and @xmath23 potentials . indeed , if we perform the global scale transformation ( @xmath39 = constant ) @xmath40 then ( 9 ) is invariant provided @xmath41 and @xmath42 are of the form @xcite @xmath43 and @xmath16 is transformed according to @xmath44 which means @xmath45 such that @xmath46 and @xmath47 we will now work out the equations of motion after introducing @xmath41 and @xmath42 and see how the integration of the equations of motion allows the spontaneous breaking of the scale invariance . let us begin by considering the equations which are obtained from the variation of the fields that appear in the measure , i.e. the @xmath16 fields . we obtain then @xmath48 where @xmath49 . since it is easy to check that @xmath50 , it follows that det @xmath51 if @xmath52 . therefore if @xmath52 we obtain that @xmath53 , or that @xmath54 where m is constant . notice that this equation breaks spontaneously the global scale invariance of the theory , since the left hand side has a non trivial transformation under the scale transformations , while the right hand side is equal to @xmath55 , a constant that after we integrate the equations is fixed , can not be changed and therefore for any @xmath56 we have obtained indeed , spontaneous breaking of scale invariance . we will see what is the connection now . as we will see , the connection appears in the original frame as a non riemannian object . however , we will see that by a simple conformal tranformation of the metric we can recover the riemannian structure . the interpretation of the equations in the frame gives then an interesting physical picture , as we will see . let us begin by studying the equations obtained from the variation of the connections @xmath57 . we obtain then @xmath58 if we define @xmath59 as @xmath60 where @xmath61 is the christoffel symbol , we obtain for @xmath59 the equation @xmath62 where @xmath63 . the general solution of eq.([e28 ] ) is @xmath64[e30 ] where @xmath65 is an arbitrary function due to the @xmath65 - symmetry of the curvature @xcite @xmath66 , @xmath67 z being any scalar ( which means @xmath68 ) . if we choose the gauge @xmath69 , we obtain @xmath70 considering now the variation with respect to @xmath71 , we obtain @xmath72 solving for @xmath73 from eq.([e31 ] ) and introducing in eq.[e25 ] , we obtain @xmath74 a constraint that allows us to solve for @xmath75 , @xmath76 to get the physical content of the theory , it is best consider variables that have well defined dynamical interpretation . the original metric does not has a non zero canonical momenta . the fundamental variable of the theory in the first order formalism is the connection and its canonical momenta is a function of @xmath77 , given by , @xmath78 and @xmath75 given by eq.([e33 ] ) . interestingly enough , working with @xmath77 is the same as going to the einstein conformal frame " . in terms of @xmath77 the non riemannian contribution @xmath79 dissappears from the equations . this is because the connection can be written as the christoffel symbol of the metric @xmath77 . in terms of @xmath77 the equations of motion for the metric can be written then in the einstein form ( we define @xmath80 usual ricci tensor in terms of the bar metric @xmath81 and @xmath82 ) @xmath83 where @xmath84 and @xmath85 in terms of the metric @xmath86 , the equation of motion of the scalar field @xmath3 takes the standard general - relativity form @xmath87 notice that if @xmath88 and @xmath89 also , provided @xmath90 is finite and @xmath91 there . this means the zero cosmological constant state is achieved without any sort of fine tuning . that is , independently of whether we add to @xmath23 a constant piece , or whether we change the value of @xmath55 , as long as there is still a point where @xmath92 , then still @xmath93 and @xmath94 ( still provided @xmath90 is finite and @xmath91 there ) . this is the basic feature that characterizes the tmt and allows it to solve the old " cosmological constant problem , at least at the classical level . in what follows we will study the effective potential ( [ e37 ] ) for the special case of global scale invariance , which as we will see displays additional very special features which makes it attractive in the context of cosmology . notice that in terms of the variables @xmath3 , @xmath77 , the scale " transformation becomes only a shift in the scalar field @xmath3 , since @xmath77 is invariant ( since @xmath95 and @xmath96 ) @xmath97 if @xmath98 and @xmath99 as required by scale invariance eqs . ( [ e18 ] , [ e20 ] , [ e21 ] , [ e22 ] , [ e23 ] ) , we obtain from the expression ( [ e37 ] ) @xmath100 since we can always perform the transformation @xmath101 we can choose by convention @xmath102 . we then see that as @xmath103 const . providing an infinite flat region as depicted in fig also a minimum is achieved at zero cosmological constant for the case @xmath104 at the point @xmath105 in conclusion , the scale invariance of the original theory is responsible for the non appearance ( in the physics ) of a certain scale , that associated to m. however , masses do appear , since the coupling to two different measures of @xmath15 and @xmath20 allow us to introduce two independent couplings @xmath106 and @xmath107 , a situation which is unlike the standard formulation of globally scale invariant theories , where usually no stable vacuum state exists . the constant of integration @xmath55 plays a very important role indeed : any non vanishing value for this constant implements , already at the classical level s.s.b . of scale invariance . as we have seen , it is possible to obtain a model that through a spontaneous breaking of scale invariace can give us a flat region . we want to obtain now two flat regions in our effective potential . a simple generalization of the action @xmath30 will fix this . the basic new feature we add is the presence is higher curvature terms in the action @xcite-@xcite , which have been shown to be very relevant in cosmology . in particular he first inflationary model from a model with higher terms in the curvature was proposed in @xcite . what one needs to do is simply consider the addition of a scale invariant term of the form @xmath108 the total action being then @xmath109 . in the first order formalism @xmath110 is not only globally scale invariant but also locally scale invariant , that is conformally invariant ( recall that in the first order formalism the connection is an independent degree of freedom and it does not transform under a conformal transformation of the metric ) . let us see what the equations of motion tell us , now with the addition of @xmath111 to the action . first of all , since the addition has been only to the part of the action that couples to @xmath112 , the equations of motion derived from the variation of the measure fields remains unchanged . that is eq.([e25 ] ) remains valid . the variation of the action with respect to @xmath113 gives now @xmath114 it is interesting to notice that if we contract this equation with @xmath113 , the @xmath115 terms do not contribute . this means that the same value for the scalar curvature @xmath116 is obtained as in section 2 , if we express our result in terms of @xmath3 , its derivatives and @xmath113 . solving the scalar curvature from this and inserting in the other @xmath115 - independent equation @xmath117 we get still the same solution for the ratio of the measures which was found in the case where the @xmath115 terms were absent , i.e. @xmath118 . in the presence of the @xmath119 term in the action , eq . ( [ e26 ] ) gets modified so that instead of @xmath120 , @xmath121 = @xmath122 appears . this in turn implies that eq.([e27 ] ) keeps its form but where @xmath123 is replaced by @xmath124 , where once again , @xmath125 . following then the same steps as in the model without the curvature square terms , we can then verify that the connection is the christoffel symbol of the metric @xmath77 given by @xmath126 @xmath127 defines now the einstein frame " . equations ( [ e46 ] ) can now be expressed in the einstein form " @xmath128 where @xmath129 where @xmath130 here it is satisfied that @xmath131 , equation that expressed in terms of @xmath132 becomes @xmath133 . this allows us to solve for @xmath116 and we get , @xmath134 notice that if we express @xmath116 in terms of @xmath3 , its derivatives and @xmath113 , the result is the same as in the model without the curvature squared term , this is not true anymore once we express @xmath116 in terms of @xmath3 , its derivatives and @xmath135 . in any case , once we insert ( [ e51 ] ) into ( [ e50 ] ) , we see that the effective potential ( [ e50 ] ) will depend on the derivatives of the scalar field now . it acts as a normal scalar field potential under the conditions of slow rolling or low gradients and in the case the scalar field is near the region @xmath136 . notice that since @xmath137 , then if @xmath138 , then , as in the simpler model without the curvature squared terms , we obtain that @xmath139 at that point without fine tuning ( here by @xmath140 we mean the derivative of @xmath141 with respect to the scalar field @xmath3 , as usual ) . in the case of the scale invariant case , where @xmath23 and @xmath22 are given by equation ( [ e19 ] ) , it is interesting to study the shape of @xmath142 as a function of @xmath3 in the case of a constant @xmath3 , in which case @xmath143 can be regarded as a real scalar field potential . then from ( [ e51 ] ) we get @xmath144 , which inserted in ( [ e50 ] ) gives , @xmath145 the limiting values of @xmath143 are : first , for asymptotically large positive values , ie . as @xmath146 , we have @xmath147 . second , for asymptotically large but negative values of the scalar field , that is as @xmath148 , we have : @xmath149 . in these two asymptotic regions ( @xmath150 and @xmath148 ) an examination of the scalar field equation reveals that a constant scalar field configuration is a solution of the equations , as is of course expected from the flatness of the effective potential in these regions . notice that in all the above discussion it is fundamental that @xmath151 . if @xmath152 the potential becomes just a flat one , @xmath153 everywhere ( not only at high values of @xmath154 ) . all the non trivial features necessary for a graceful exit , the other flat region associated to the planck scale and the minimum at zero if @xmath155 are all lost . as we discussed in the model without a curvature squared term , @xmath151 implies the we are considering a situation with s.s.b . of scale invariance . these kind of models with potentials giving rise to two flat potentials have been applied to produce models for bags and confinement in a very natural way @xcite . one could question the use of the einstein frame metric @xmath127 in contrast to the original metric @xmath156 . in this respect , it is interesting to see the role of both the original metric and that of the einstein frame metric in a canonical approach to the first order formalism . here we see that the original metric does not have a canonically conjugated momentum ( this turns out to be zero ) , in contrast , the canonically conjugated momentum to the connection turns out to be a function exclusively of @xmath77 , this einstein metric is therefore a genuine dynamical canonical variable , as opposed to the original metric . there is also a lagrangian formulation of the theory which uses @xmath77 , as we will see in the next section , what we can call the action in the einstein frame . in this frame we can quantize the theory for example and consider contributions without reference to the original frame , thus possibly considering breaking the tmt structure of the theory through quantum effects , but such breaking will be done softly " through the introduction of a cosmological term only . surprisingly , the remaining structure of the theory , reminiscent from the original tmt structure will be enough to control the strength of this additional cosmological term once we demand that the universe originated from a non - singular and stable emergent state . the effective energy - momentum tensor can be represented in a form like that of a perfect fluid @xmath157 here @xmath158 . this defines a pressure functional and an energy density functional . the system of equations obtained after solving for @xmath75 , working in the einstein frame with the metric @xmath159 can be obtained from a k - essence " type effective action , as it is standard in treatments of theories with non linear kinetic terms or k - essence models@xcite-@xcite . the action from which the classical equations follow is , @xmath160 \label{k - eff}\ ] ] @xmath161 @xmath162 where it is understood that , @xmath163 we have two possible formulations concerning @xmath116 : notice first that @xmath164 and @xmath116 are different objects , the @xmath164 is the riemannian curvature scalar in the einstein frame , while @xmath116 is a different object . this @xmath116 will be treated in two different ways : \1 . first order formalism for @xmath116 . here @xmath116 is a lagrangian variable , determined as follows , @xmath116 that appear in the expression above for @xmath165 can be obtained from the variation of the pressure functional action above with respect to @xmath116 , this gives exactly the expression for @xmath116 that has been solved already in terms of @xmath166 , etc , see eq . ( [ e51 ] ) . second order formalism for @xmath116 . @xmath116 that appear in the action above is exactly the expression for @xmath116 that has been solved already in terms of @xmath166 , etc . the second order formalism can be obtained from the first order formalism by solving algebraically r from the eq . ( [ e51 ] ) obtained by variation of @xmath116 , and inserting back into the action . one may also use the method outlined in @xcite to find the effective action in the einstein frame , in @xcite the problem of a curvature squared theory with standard measure was studied . the methods outlined there can be also applied in the modified measure case @xcite , thus providing another derivation of the effective action explained above . the problem that we have to solve to find the effective lagrangian is basically finding that lagrangian tat will produce the effective energy momentum tensor in the einstein frame by the variation of the @xmath77 metric @xmath167 versus the scalar field @xmath3 . we consider unit where @xmath168 , @xmath169 , @xmath170 and @xmath171 . left panel : @xmath172 , @xmath173 , @xmath174 . right panel : @xmath175 , @xmath173 , @xmath174 . [ fig - paper1],title="fig:",width=264 ] versus the scalar field @xmath3 . we consider unit where @xmath168 , @xmath169 , @xmath170 and @xmath171 . left panel : @xmath172 , @xmath173 , @xmath174 . right panel : @xmath175 , @xmath173 , @xmath174 . [ fig - paper1],title="fig:",width=264 ] in contrast to the simplified models studied in literature@xcite , it is impossible here to represent @xmath176 in a factorizable form like @xmath177 . the scalar field effective lagrangian can be taken as a starting point for many considerations . in particular , the quantization of the model can proceed from ( [ k - eff ] ) and additional terms could be generated by radiative corrections . we will focus only on a possible cosmological term in the einstein frame added ( due to zero point fluctuations ) to ( [ k - eff ] ) , which leads then to the new action @xmath178 \label{act.lambda}\ ] ] this addition to the effective action leaves the equations of motion of the scalar field unaffected , but the gravitational equations aquire a cosmological constant . adding the @xmath179 term can be regarded as a redefinition of @xmath180 @xmath181as we will see the stability of the emerging universe imposes interesting constraints on @xmath179 versus the scalar field @xmath3 . we consider unit where @xmath168 , @xmath169 , @xmath170 and @xmath182 . left panel : @xmath175 , @xmath173 , @xmath174 . right panel : @xmath183 , @xmath184 , @xmath174.[fig - paper2],title="fig:",width=264 ] versus the scalar field @xmath3 . we consider unit where @xmath168 , @xmath169 , @xmath170 and @xmath182 . left panel : @xmath175 , @xmath173 , @xmath174 . right panel : @xmath183 , @xmath184 , @xmath174.[fig - paper2],title="fig:",width=264 ] after introducing the @xmath179 term , we get from the variation of @xmath116 the same value of @xmath116 , unaffected by the new @xmath179 term , but as one can easily see then @xmath116 does not have the interpretation of a curvature scalar in the original frame since it is unaffected by the new source of energy density ( the @xmath179 term ) , this is why the @xmath179 term theory does not have a formulation in the original frame , but is a perfectly legitimate generalization of the theory , probably obtained by considering zero point fluctuations , notice that quantum theory is possible only in the einstein frame . notice that even in the original frame the bar metric ( not the original metric ) appears automatically in the canonically conjugate momenta to the connection , so we can expect from this that the bar metric and not the original metric be the relevant one for the quantum theory . in figure [ fig - paper1 ] and [ fig - paper2 ] we have plotted the effective potential as a function of the scalar field , for @xmath185 and @xmath182 respectively . we consider unit where @xmath186 , @xmath169 , @xmath170 and different values for @xmath187 . we now want to consider the detailed analysis of the emerging universe solutions and in the next section their stability in the tmt scale invariant theory . we start considering the cosmological solutions of the form @xmath188 in this case , we obtain for the energy density and the pressure , the following expressions . we will consider a scenario where the scalar field @xmath3 is moving in the extreme right region @xmath189 , in this case the expressions for the energy density @xmath190 and pressure @xmath165 are given by , @xmath191 and @xmath192 it is interesting to notice that all terms proportional to @xmath193 behave like radiation " , since @xmath194 is satisfied . here the constants @xmath195 and @xmath196 are given by , @xmath197 it will be convenient to decompose " the constant @xmath179 into two pieces , @xmath198 since as @xmath199 , @xmath200 . therefore @xmath201 has the interesting interpretation of the vacuum energy density in the @xmath202 vacuum . as we will see , it is remarkable that the stability and existence of non - singular emergent universe implies that @xmath203 , and it is bounded from above as well . the equation that determines such static universe @xmath204 , @xmath205 , @xmath206 gives rise to a restriction for @xmath207 that have to satisfy the following equation in order to guarantee that the universe be static , because @xmath206 is proportional to @xmath208 , we must require that @xmath209 , which leads to @xmath210 this equation leads to two roots , the first being @xmath211 the second root is : @xmath212 it is also interesting to see that if the discriminant is positive , the first solution has automatically positive energy density , if we only consider cases where @xmath213 , which is required if we want the emerging solution to be able to turn into an inflationary solution eventually . one can see that the condition @xmath214 for the first solution reduces to the inequality @xmath215 , where @xmath216 , since we must have @xmath217 , otherwise we get a negative kinetic term during the inflationary period , and as we will see in the next section , we must have that @xmath218 from the stability of the solution , and as long as @xmath219 , it is always true that this inequality is satisfied . before going into the subject of the small perturbations and stability of these solutions , we would like to notice the entropy like " conservation laws that may be useful in a non perturbative analysis of the theory . in fact in the @xmath4 region , we have the exact symmetry @xmath220 . and considering that the effective matter action here is @xmath221 , we have the conserved quantity @xmath222 it is very interesting to notice that @xmath223 where @xmath224 assumes the entropy density " form @xmath225 provided we identify the temperature " t with @xmath226 . we will now consider the perturbation equations . considering small deviations of @xmath226 the from the static emerging solution value @xmath207 and also considering the perturbations of the scale factor @xmath227 , we obtain , from eq . ( [ eq.density ] ) @xmath228 at the same time @xmath229 can be obtained from the perturbation of the friedmann equation @xmath230 and since we are perturbing a solution which is static , i.e. , has @xmath231 , we obtain then @xmath232 we also have the second order friedmann equation @xmath233 for the static emerging solution , we have @xmath234 , @xmath235 , so @xmath236 where we have chosen to express our result in terms of @xmath237 , defined by @xmath238 , which for the emerging solution has the value @xmath239 . using this in [ pert.fried.eq . ] , we obtain @xmath240 and equating the values of @xmath229 as given by [ eq.density-pert . ] and [ pert.fried.eq.3 ] we obtain a linear relation between @xmath241 and @xmath242 , which is , @xmath243 where @xmath244 we now consider the perturbation of the eq . ( [ fried.eq.2 ] ) . in the right hand side of this equation we consider that @xmath245 , with @xmath246 where , @xmath247 and therefore , the perturbation of the eq . ( [ fried.eq.2 ] ) leads to , @xmath248 to evaluate this , we use [ omega - eq . ] , [ v - eq . ] and the expressions that relate the variations in @xmath227 and @xmath226 ( [ delta - delta ] ) . defining the small " variable @xmath249 as @xmath250 we obtain , @xmath251 where , @xmath252,\ ] ] notice that the sum of the last two terms in the expression for @xmath253 , that is @xmath254 vanish since @xmath239 , for the same reason , we have that @xmath255 , which brings us to the simplified expression @xmath256,\ ] ] for the stability of the static solution , we need that @xmath257 , where @xmath258 is defined either by e. ( [ e2 ] ) ( @xmath259 ) or by e. ( [ e3 ] ) ( @xmath260 ) . if we take e. ( [ e3 ] ) ( @xmath260 ) and use this in the above expression for @xmath253 , we obtain , @xmath261,\ ] ] to avoid negative kinetic terms during the slow roll phase that takes place following the emergent phase , we must consider @xmath217 , so , we see that the second solution is unstable and will not be considered further . now in the case of the first solution , e. ( [ e2 ] ) ( @xmath259 ) , then @xmath253 becomes @xmath262,\ ] ] so the condition of stability becomes @xmath263 , or @xmath264 , squaring both sides and since @xmath217 , we get @xmath265 , which means @xmath218 , and therefore @xmath266 , multiplying by @xmath267 , we obtain , @xmath268 , replacing the values of @xmath269 , given by [ abc ] we obtain the condition @xmath270 now there is the condition that the discriminant be positive @xmath271 @xmath272,\ ] ] since @xmath273 > 0 $ ] , @xmath218 , meaning that @xmath274 , we see that we obtain a positive upper bound for the energy density of the vacuum as @xmath199 , which must be positive , but not very big . the emerging phase owes its existence to a strictly constant vacuum energy ( which here is represented by the value of @xmath275 ) at very large values of the field @xmath3 . in fact , while for @xmath276 the effective potential of the scalar field is perfectly flat , for any @xmath277 the effective potential acquires a non trivial shape . this causes the transition from the emergent phase to a slow roll inflationary phase which will be the subject of this section . following @xcite , we consider now then the relevant equations for the model in the slow roll regime , i.e. for @xmath226 small and when the scalar field @xmath3 is large , but finite and we consider the first corrections to the flatness to the effective potential . dropping higher powers of @xmath226 in the contributions for the kinetic energy and in the scalar curvature @xmath116 , we obtain @xmath278 here , as usual @xmath279 . in the slow roll approximation , we can drop the second derivative term of @xmath3 and the second power of @xmath226 in the equation for @xmath280 and we get @xmath281 where @xmath282 . the relevant expression for @xmath283 will be that given by ( [ effpotslow ] ) , i.e. , where all higher derivatives are ignored in the potential , consistent with the slow roll approximation . we now display the relevant expressions for the region of very large , but not infinite @xmath3 , these are : @xmath284 and @xmath285 the relevant constants that will affect our results are , @xmath196 , as given by ( [ c ] ) and @xmath286 and @xmath287 given by @xmath288 and @xmath289 respectively . ( [ effpotatlargephi ] ) we can calculates the key landmarks of the inflationary history : first , the value of the scalar field where inflation ends , @xmath290 and a value for the scalar field @xmath291 bigger than this ( @xmath292 ) and which happens earlier , which represents the horizon crossing point " . we must demand then that a typical number of e - foldings , like @xmath293 , takes place between @xmath291 , until the end of inflation at @xmath294 . to determine the end of inflation , we consider the quantity @xmath295 and consider the point in the evolution of the universe where @xmath296 , only when @xmath297 , we have an accelerating universe , so the point @xmath296 represents indeed the end of inflation . calculating the derivative with respect to cosmic time of the hubble expansion using ( [ friedmannslow ] ) and ( [ slowroll ] ) , we obtain that the condition @xmath296 gives @xmath298 working to leading order , setting @xmath299 , @xmath300 and @xmath301 , this gives as a solution , @xmath302 notice that if @xmath55 and @xmath303 have different signs and if @xmath304 , @xmath305 for the allowed range of parameters the stable emerging solution , so @xmath306 represents the absolute value of @xmath307 . we now consider @xmath291 and the requirement that this precedes @xmath290 by @xmath308 e - foldings , @xmath309 where in the last step we have used the slow roll equation of motion for the scalar field ( [ slowroll ] ) to solve for @xmath226 . solving @xmath280 in terms of @xmath283 using ( [ friedmannslow ] ) , working to leading order , setting @xmath299 and integrating , we obtain the relation between @xmath291 and @xmath290 , @xmath310 as we mentioned before @xmath305 for the allowed range of parameters the stable emerging solution , so that @xmath292 as it should be for everything to make sense . introducing eq . ( [ sol.endofinf.eq . ] ) into eq . ( [ sol.crossing . ] ) , we obtain , @xmath311 we finally calculate the power of the primordial scalar perturbations . if the scalar field @xmath3 had a canonically normalized kinetic term , the spectrum of the primordial perturbations will be given by the equation @xmath312 however , as we can see from ( [ energydensityslow ] ) , the kinetic term is not canonically normalized because of the factor @xmath313 in that equation . in this point we will study the scalar and tensor perturbations for our model where the kinetic term is not canonically normalized . the general expression for the perturbed metric about the friedmann - robertson - walker is @xmath314 dx^i dx^j,\end{aligned}\ ] ] where @xmath315 , @xmath316 , @xmath317 and @xmath318 are the scalar type metric perturbations and @xmath319 characterizes the transverse - traceless tensor perturbation . the power spectrum of the curvature perturbation in the slow - roll approximation for a not - canonically kinetic term becomes ref.@xcite(see also refs.@xcite ) @xmath320 where it was defined speed of sound " , @xmath321 , as @xmath322 with @xmath323 an function of the scalar field and of the kinetic term @xmath324 . here @xmath325 denote the derivative with respect @xmath326 . in our case @xmath327 , with @xmath328 . thus , from eq.([pec ] ) we get @xmath329 the scalar spectral index @xmath330 , is defined by @xmath331 where @xmath332 and @xmath333 , respectively . on the other hand , the generation of tensor perturbations during inflation would produce gravitational wave . the amplitude of tensor perturbations was evaluated in ref.@xcite , where @xmath334 and the tensor spectral index @xmath335 , becomes @xmath336 and they satisfy a generalized consistency relation @xmath337 versus @xmath330 for three values of @xmath338 . for @xmath169 solid line , @xmath339 dash line and @xmath340 dots line , respectively . here , we have fixed the values @xmath341 , @xmath342 , @xmath343 , @xmath174 , @xmath344 and @xmath345 , respectively . the seven - year wmap data places stronger limits on the tensor - scalar ratio ( shown in red ) than five - year data ( blue ) @xcite.[fig2],width=480 ] therefore , the scalar field ( to leading order ) that should appear in eq . ( [ primordialpert . ] ) should be @xmath346 and instead of eq . ( [ eq10 ] ) , we must use @xmath347 the power spectrum of the perturbations goes , up to a factor of order one , which we will denote @xmath348 as @xmath349 , so we have , @xmath350 this quantity should be evaluated at @xmath351 given by ( [ sol.crossing.final ] ) . solving for @xmath226 from the slow roll equation ( [ slowroll ] ) , evaluating the derivative of the effective potential using ( [ effpotatlargephi ] ) and solving for @xmath352 from ( [ friedmannslow ] ) , we obtain , to leading order , @xmath353 using then ( [ sol.crossing.final ] ) for @xmath354 , we obtain our final result , @xmath355 it is very interesting first of all that @xmath307 dependence has dropped out and with it all dependence on @xmath55 . in fact this can be regarded as a non trivial consistency check of our estimates , since apart from its sign , the value @xmath55 should not affect the results . this is due to the fact that from a different value of @xmath55 ( although with the same sign ) , we can recover the original potential by performing a shift of the scalar field @xmath3 . in fig.[fig2 ] we show the dependence of the tensor - scalar ratio @xmath356 on the spectral index @xmath330 . from left to right @xmath169 ( solid line ) , @xmath339 ( dash line ) and @xmath340 ( dots line ) , respectively . from ref.@xcite , two - dimensional marginalized constraints ( 68@xmath357 and 95@xmath357 confidence levels ) on inflationary parameters @xmath356 and @xmath330 , the spectral index of fluctuations , defined at @xmath358 = 0.002 mpc@xmath359 . the seven - year wmap data places stronger limits on @xmath356 ( shown in red ) than five - year data ( blue)@xcite , @xcite . in order to write down values that relate @xmath330 and @xmath356 , we used eqs.([ns ] ) and ( [ ration ] ) . also we have used the values @xmath341 , @xmath342 , @xmath343 , @xmath174 , @xmath344 and @xmath345 , respectively . from eqs.([n ] ) , ( [ ns ] ) and ( [ ration ] ) , we observed numerically that for @xmath360 , the curve @xmath361 ( see fig.[fig2 ] ) for wmap 7-years enters the 95@xmath357 confidence region where the ratio @xmath362 , which corresponds to the number of e - folds , @xmath363 . for @xmath339 , @xmath364 corresponds to @xmath365 and for @xmath340 , @xmath366 corresponds to @xmath367 . from 68@xmath357 confidence region for @xmath360 , @xmath368 , which corresponds to @xmath369 . for @xmath370 , @xmath371 corresponds to @xmath372 and for @xmath340 , @xmath373 corresponds to @xmath374 . we noted that the parameter @xmath338 , which lies in the range @xmath375 , the model is well supported by the data as could be seen from fig.[fig2 ] . for the discussion of the vacuum structure of the theory , we start studying @xmath143 for the case of a constant field @xmath3 , given by , @xmath376 this is necessary , but not enough , since as we will see , the consideration of constant fields @xmath3 alone can lead to misleading conclusions , in some cases , the dependence of @xmath143 on the kinetic term can be crucial to see if and how we can achieve the crossing of an apparent barrier . for a constant field @xmath3 the limiting values of @xmath143 are ( now that we added the constant @xmath377 ) : first , for asymptotically large positive values , ie . as @xmath146 , we have @xmath378 . second , for asymptotically large but negative values of the scalar field , that is as @xmath148 , we have : @xmath379 . in these two asymptotic regions ( @xmath150 and @xmath148 ) an examination of the scalar field equation reveals that a constant scalar field configuration is a solution of the equations , as is of course expected from the flatness of the effective potential in these regions . notice that in all the above discussion it is fundamental that @xmath151 . if @xmath152 the potential becomes just a flat one , @xmath380 everywhere ( not only at high values of @xmath154 ) . finally , there is a minimum at @xmath381 if @xmath155 . in summary , and if @xmath382 , @xmath217 , we have that there is a hierarchy of vacua , @xmath383 where @xmath384 . notice that we assume above that @xmath385 and @xmath386 , but @xmath387 and @xmath388 would be indistinguishable from that situation , that is , the important requirement is @xmath389 . we could have a scenario where we start the non - singular emergent universe at @xmath390 where @xmath391 , which then slow rolls , then inflates @xcite and finally gets trapped in the local minimum with energy density @xmath392 , that was the picture favored in @xcite , while here we want to argue that the most attractive and relevant description for the final state of our universe is realized after inflation in the flat region @xmath5 , since in this region the vacuum energy density is positive and bounded from above , so its a good candidate for our present state of the universe . it remains to be seen however whether a smooth transition all the way from @xmath393 to @xmath6 is possible . before we discuss the transition to the @xmath6 , it is necessary to discuss another vacuum state , which we may call the kinetic vacuum state " which is in fact degenerate with this one . the kinetic vacuum state " that , with time dependence and say for no space dependence and @xmath394 given by @xmath395 which can be solved for @xmath226 in the real domain for @xmath274 . for this case @xmath116 ( which is not a riemannian curvature ) , as given by [ e51 ] diverges , the riemannian scalar derived from the einstein frame metric is perfectly regular . in this case then @xmath396 that is , for this value of @xmath394 , regardless of the value of the scalar field , the value of @xmath283 becomes degenerate with its value for constant and arbitrarily negative @xmath3 , which is our candidate vacuum for the present state of the universe . notice that this value for @xmath394 is also the one obtained by extremizing the pressure functional in the region of very large scalar field values , so in this limit it is obvious that such configuration satisfy the euler lagrange equations , but indeed it is a general feature , the equations of motion for the kinetic vacuum are satisfied , regardless of what value we take for the scalar field . in order to discuss the possibility of transition to @xmath5 . in our case , since we are interested in a local minimum between @xmath4 or @xmath397 , we can take @xmath55 of either sign . taking for definitness @xmath385 , @xmath382 , @xmath217,@xmath304 , we see that there will be a point , given by [ effppluslambda ] , defined by @xmath398 where @xmath283 as will spike to @xmath399 , go then down to @xmath400 and then asymptotically its positive asymptotic value at @xmath6 . this has the appearance of a potential barrier . however , this is deceptive , such barrier exists for constant @xmath3 , but can be avoided by considering a transition from any @xmath3 , but with the appropriate value of @xmath394 that defines the kinetic vacuum . a detailed dynamical analysis will be presented now concerning these issues , the field equations become in the cosmological case : @xmath401 by using the definitions of @xmath41 and @xmath42 we can express @xmath190 and @xmath165 as follow : @xmath402}\,\frac{\dot{\phi}^2}{2 } + v_{eff } \,,\ ] ] @xmath403}\,\frac{\dot{\phi}^2}{2 } - v_{eff } \,,\ ] ] @xmath404 ^ 2}\bigg [ u + \epsilon \left(\frac{-\kappa(v+m ) + \frac{\kappa}{2}\dot{\phi}^2\chi}{(1 + \kappa ^{2 } \epsilon\,\dot{\phi}^2)}\right)^2\bigg ] + \lambda\ ] ] we can note that independent of @xmath226 there are a singularity in the potential ( also in @xmath190 and @xmath165 ) when @xmath405 , where @xmath406 . the only way to avoid this situation is consider @xmath407 before @xmath3 arrive to @xmath408 . we can note that , in this case , the effective potential becomes flat ( i.e. independent of @xmath3 ) and everything is finite . it is interesting to note that for the case where this model admit an static and stable universe solution in the region @xmath409 the kinetic vacuum state solution is an attractor in the region @xmath410 , see discussion below . this situation was already found in the limit @xmath411 in @xcite where the stability of the static solution was studied . in particular in the limit @xmath4 the set of equations ( [ ds1 ] , [ ds2 ] ) could be written as an autonomous system of two dimensions respect to @xmath352 and @xmath412 as follow , see @xcite : @xmath413 - h^2 , \label{dinamic1}\\ \nonumber \\ \dot{y } & = & - \frac{3a\,(1 + \kappa^2\,\epsilon \,y)\,y}{\frac{a}{2 } + a\,\kappa^2\,\epsilon \,y + 2b\,y}\,h \,\ , , \label{dinamic2}\end{aligned}\ ] ] as was mentioned in @xcite this system has five critical points where one of these points correspond to the es universe discussed previously , but there are also the critical point @xmath414 this critical point is , precisely , the kinetic vacuum state which avoid the singularity problem of @xmath283 discussed above . after we linearize the equations ( [ dinamic1 ] , [ dinamic2 ] ) near this critical point we obtain that the eigenvalues of the linearized equations are negative @xmath415 and @xmath416 , then , this critical point is an attractor . ( [ fds1 ] ) top left panel show part of the _ direction field _ of the system and four numerical solutions where we can note that the kinetic vacuum state is an attractor solution . it is interesting to note that this solution is in fact an attractor not only in the limit @xmath4 , but also in others regions . in order to study this point in more details let us write the set of equations ( [ ds1 ] , [ ds2 ] ) as an autonomous system of three dimensions as follow : @xmath417 where we have defined @xmath418 . we are consider @xmath419 because we are interested in the cases where the field moves from @xmath400 to positive values , following the emergent universe scheme . we can can note that , in general , the solutions @xmath420 , @xmath421 is stable . this solution correspond to a flat effective potential and @xmath3 rolling with constant @xmath226 . in this case , we can past over the point @xmath422 , see numerical solutions fig.([fns1 ] ) . also , we can observed that solutions near the kinetic vacuum solution can pass over this point , because this solution is an attractor . the general behaviour could be see in the fig . ( [ fds1 ] ) where it is plot the _ direction field _ for the effective two dimensional autonomous system in variables @xmath352 and @xmath423 which we obtain when evaluate the system of eqs.([sdc1 ] , [ sdc2],[sdc3 ] ) at different values of @xmath3 . the first plot correspond to the limit @xmath424 the second is for @xmath425 , and the third correspond to @xmath426 . in order to study in a more systematic way the nature of the kinetic vacuum state we linearize the eqs.([sdc1 ] , [ sdc2],[sdc3 ] ) near the critical point @xmath427 leaving @xmath3 arbitrary . we obtain the following two dimensional effective autonomous system with variables @xmath428 and @xmath429 : @xmath430 } \,,\label{pert1}\\ \nonumber \\ \delta \dot{y}= \bigg(-3h_0 + \frac{2\alpha\,m\,\epsilon\,\kappa^2(m+v)\sqrt{y_0}}{u + \kappa\,\epsilon(m+v)^2}\bigg)\delta y\,.\label{pert2}\end{aligned}\ ] ] the eigenvalues of equations ( [ pert1 ] ) , ( [ pert2 ] ) are : @xmath431 the equilibrium point is stable ( attractor ) if the eigenvalues are negative . then , depending on the values of the parameters of the models , this is the case for a large set of values of @xmath3 , not only for the case @xmath432 discussed previously in @xcite . in particular for the numerical values used in @xcite , which are consistent with the stability of the es universe , the critical point is stable for @xmath433 , see fig . ( [ fds1 ] ) . it is interesting to note that when @xmath6 , then @xmath434 for example , if we consider the numerical values used in @xcite this is a positive number . this means that at some value of @xmath435 , when the scalar field moves to @xmath400 the stable ( attractor ) equilibrium point becomes unstable ( focus ) , see fig . ( [ fds1 ] ) . in order to study numerical solutions we chose the following values for the free parameters of the model , in units where @xmath168 ; @xmath436 , @xmath437 , @xmath438 , @xmath169 , @xmath170 and @xmath341 . these values satisfy the requirements of stability of the es solution in the limit @xmath439 , see ref.@xcite . under this assumptions we obtain that : @xmath440 numerical solution to the eqs.([sdc1 ] , [ sdc2 ] , [ sdc3 ] ) are show in fig.([fns1 ] ) , where we can note that the point @xmath422 is passed on during the evolution of the scalar field . this situation is achieved by the kinetic vacuum solution , but also by others solutions which decays to the kinetic vacuum solution before arrive to the point @xmath422 , see fig . ( [ fns1 ] ) . the figure ( [ fns1 ] ) left panel shown a projection of the axis @xmath352 and @xmath3 and the evolution of six numerical solutions . the right panel shown a projection of the axis @xmath423 and @xmath3 and the evolution of six numerical solutions . and some numerical solutions.[fds1],title="fig:",width=264 ] and some numerical solutions.[fds1],title="fig:",width=264 ] and some numerical solutions.[fds1],title="fig:",width=264 ] , [ sdc2 ] , [ sdc3 ] ) . [ fns1],title="fig:",width=264 ] , [ sdc2 ] , [ sdc3 ] ) . [ fns1],title="fig:",width=264 ] we have considered a non - singular origin for the universe starting from an einstein static universe , the so called emergent universe " scenario , in the framework of a theory which uses two volume elements @xmath0 and @xmath1 , where @xmath2 is a metric independent density , used as an additional measure of integration . also curvature , curvature square terms and for scale invariance a dilaton field @xmath3 are considered in the action . the first order formalism was applied . the integration of the equations of motion associated with the new measure gives rise to the spontaneous symmetry breaking ( s.s.b ) of scale invariance ( s.i . ) . after s.s.b . of s.i . , using the the einstein frame metric , it is found that a non trivial potential for the dilaton is generated . one could question the use of the einstein frame metric @xmath127 in contrast to the original metric @xmath156 . in this respect , it is interesting to see the role of both the original metric and that of the einstein frame metric in a canonical approach to the first order formalism . here we see that the original metric does not have a canonically conjugated momentum ( this turns out to be zero ) , in contrast , the canonically conjugated momentum to the connection turns out to be a function exclusively of @xmath127 , this einstein metric is therefore a genuine dynamical canonical variable , as opposed to the original metric . there is also a lagrangian formulation of the theory which uses @xmath127 , what we can call the action in the einstein frame . in this frame we can quantize the theory for example and consider contributions without reference to the original frame , thus possibly considering breaking the tmt structure of the theory , but such breaking will be done softly " through the introduction of a cosmological term only . surprisingly , the remaining structure of the theory , reminiscent from the original tmt structure will be enough to control the strength of this additional cosmological term once we demand that the universe originated from a non - singular and stable emergent state . in the einstein frame we argue that the cosmological term parametrizes the zero point fluctuations . the resulting effective potential for the dilaton contains two flat regions , for @xmath4 relevant for the non - singular origin of the universe , followed by an inflationary phase and then transition to @xmath6 , which in this paper we take as describing our present universe . an intermediate local minimum is obtained if @xmath441 , the region as @xmath393 has a higher energy density than this local minimum and of course of the region @xmath6 , if @xmath217 and @xmath442 . @xmath217 is also required for satisfactory slow roll in the inflationary region @xmath4 ( after the emergent phase ) . the dynamics of the scalar field becomes non linear and these non linearities are instrumental in the stability of some of the emergent universe solutions , which exists for a parameter range of values of the vacuum energy in @xmath6 , which must be positive but not very big , avoiding the extreme fine tuning required to keep the vacuum energy density of the present universe small . a sort of solution of the cosmological constant problem , where an a priori arbitrary cosmological term is restricted by the consideration of the non - singular and stable emergent origin for the universe . notice then that the creation of the universe can be considered as a threshold event " for zero present vacuum energy density , that is a threshold event for @xmath443 and we can learn what we can expect in this case by comparing with well known threshold events . for example in particle physics , the process @xmath444 , has a cross section of the form ( ignoring the mass of the electron and considering the center of mass frame , @xmath318 being the center of mass energy of each of the colliding @xmath445 or @xmath446 ) , @xmath447\sqrt{\frac{e^{2}-m_{\mu}^2 } { e^{2}}}\ ] ] for @xmath448 and exactly zero for @xmath449 . we see that exactly at threshold this cross section is zero , but at this exact point it has a cusp , the derivative is infinite and the function jumps as we slightly increase @xmath318 . by analogy , assuming that the vacuum energy can be tuned somehow ( like the center of mass energy @xmath318 of each of the colliding particles in the case of the annihilation process above ) , we can expect zero probability for exactly zero vacuum energy density @xmath450 , but that soon after we build up any positive @xmath451 we will then able to create the universe , naturally then , there will be a creation process resulting in a universe with a small but positive @xmath451 which represents the total energy density for the region describing the present universe , @xmath6 or by the kinetic vacuum ( which is degenerate with that state ) . one may ask the question : how is it possible to discuss the creation of the universe " in the context of the emergent universe " ? . after all , the emergent universe basic philosophy is that the universe had a past of infinite duration . however , that most simple notion of an emergent universe with a past of infinite duration has been recently challenged by mithani and vilenkin @xcite , @xcite at least in the context of a special model . they have shown that an emergent universe , although completly stable classically , could be unstable under a tunnelling process to collapse . on the other hand , an emergent universe can indeed be created from nothing by a tunnelling process as well . an emerging universe could last for a long time provided it is classically stable , that is where the constraints on the cosmological constant for the late universe discussed here come in . if it is not stable , the emergent universe will not provide us with an appropriate intermediate state " connecting the creation of the universe with the present universe . the existence of this stable intermediate state provides in our picture the reason for the universe to prefer a very small vacuum energy density at late times , since universes that are created , but do not make use of the intermediate classically stable emergent universe will almost immediately recollapse , so they will not be selected " . finally , it could be that we arrive to the emergent solution not by quantum creation from nothing , by the evolution from something else , for example by the production of a bubble in a pre - existing state @xcite , from here we go on to inflation . notice that the bound gives a small vacuum energy density , without reference to the threshold mechanism mentioned before . for this notice that upper the bound on the present vacuum energy density of the universe contains a @xmath452 suppression . if we think of the @xmath453 term as generated through radiative corrections , @xmath115 is indeed formally infinite , in dimensional regularization goes as @xmath454 , @xcite-@xcite so it can have either sign ( depending on how we approach @xmath455 ) . in any case , a very large @xmath115 means a very strict bound on the present vacuum energy density of the universe . we would like to thank sergio del campo and ramon herrera , our coauthors in ref . @xcite , which is central for our review , for our crucial collaboration with them and for discussions on all the subjects in this review , in particular we have benefited from our discussions concerning the aspects related to inflation , slow roll , etc . in the context of the model studied here . we also want to thank zvi bern , alexander kaganovich and alexander vilenkin for very important additional discussions . ucla , tufts university and the university of barcelona are thanked for their wonderful hospitality . p. l. has been partially supported by fondecyt grant n@xmath456 11090410 , mecesup ubb0704 and universidad del bo - 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we consider a non - singular origin for the universe starting from an einstein static universe , the so called emergent universe " scenario , in the framework of a theory which uses two volume elements @xmath0 and @xmath1 , where @xmath2 is a metric independent density , used as an additional measure of integration . also curvature , curvature square terms and for scale invariance a dilaton field @xmath3 are considered in the action . the first order formalism is applied . the integration of the equations of motion associated with the new measure gives rise to the spontaneous symmetry breaking ( s.s.b ) of scale invariance ( s.i . ) . after s.s.b . of s.i . , it is found that a non trivial potential for the dilaton is generated . in the einstein frame we also add a cosmological term that parametrizes the zero point fluctuations . the resulting effective potential for the dilaton contains two flat regions , for @xmath4 relevant for the non - singular origin of the universe , followed by an inflationary phase and @xmath5 , describing our present universe . the dynamics of the scalar field becomes non linear and these non linearities produce a non trivial vacuum structure for the theory and are responsible for the stability of some of the emergent universe solutions , which exists for a parameter range of values of the vacuum energy in @xmath6 , which must be positive but not very big , avoiding the extreme fine tuning required to keep the vacuum energy density of the present universe small . the non trivial vacuum structure is crucial to ensure the smooth transition from the emerging phase , to an inflationary phase and finally to the slowly accelerated universe now . zero vacuum energy density for the present universe defines the threshold for the creation of the universe .

consider a randomly accelerated particle which moves on the half line @xmath9 with an absorbing boundary at @xmath1 . the motion is governed by @xmath10 where @xmath11 is uncorrelated white noise with zero mean . the probability @xmath12 that a particle with initial position @xmath4 and initial velocity @xmath5 has not yet been absorbed after a time @xmath13 has been studied by mckean @xcite and others @xcite . for a particle initially at @xmath1 with velocity @xmath14 , the probability decays as @xmath15 following cornell , swift , and bray @xcite , we consider a more general boundary condition . on arriving at the boundary the particle is absorbed with probability @xmath6 and reflected with probability @xmath7 . the velocities of the particle just after and before reflection are related by @xmath16 . here @xmath8 is the coefficient of restitution , and @xmath17 and @xmath18 correspond to elastic and inelastic collisions , respectively . as physical motivation , we note that temporal or spatial irregularities at the boundaries of a system or a statistical capture process may give rise to partial absorption . the properties of a particle which is subject to a random force and collides inelastically are of interest in connection with driven granular media . for the partially absorbing , inelastic boundary condition the probability that the particle has not yet been absorbed decays as @xmath19 burkhardt @xcite and de smedt _ et al . _ @xcite showed that the persistence exponent @xmath20 is non - universal , depending on @xmath7 and @xmath8 according to the relation @xmath21 for @xmath22 , @xmath23 , so eqs . ( [ qabs ] ) and ( [ qpr ] ) are consistent . solving eq . ( [ expopr ] ) for @xmath24 graphically , one finds a unique solution @xmath25 in the physical region @xmath26 , @xmath27 , representing a decay slower than in eq . ( [ qabs ] ) . the decay in eq . ( [ qpr ] ) is so slow that the mean absorption time @xmath28=\int_0^\infty dt\thinspace q(0,v;t)\label{meantime}\ ] ] is infinite , even for @xmath22 . on the other hand , the mean absorption time @xmath29 , averaged over a long but finite time @xmath30 instead of an infinite time , is a well - defined quantity , which varies as @xmath31 and diverges in the limit @xmath32 . if the particle moves on the finite line @xmath0 and is absorbed the first time it reaches either @xmath1 or @xmath2 , the probability that it has not yet been absorbed after a time @xmath13 decays as @xmath33 for long times @xcite , much more rapidly than the power law ( [ qabs ] ) for a single absorbing boundary . the mean absorption time @xmath3 for a particle with initial postion @xmath4 and initial velocity @xmath5 is well defined . masoliver and porr @xcite calculated this quantity exactly , by solving the inhomogeneous fokker - planck equation @xmath34 with the boundary conditions @xmath35 corresponding to reflection symmetry and the immediate absorption of a particle with initial conditions @xmath1 , @xmath36 . their result , which is rederived in the appendix , is given by @xmath37^{-1/6}\thinspace\left[_2f_1(1,-{2\over 3};{5\over 6};1-y)-\thinspace_2f_1(1,-{2\over 3};{5\over 6};y)\right]\label{mp}\end{aligned}\ ] ] for @xmath14 , where @xmath38 is a standard hypergeometric function @xcite . the subscript @xmath39 of @xmath40 is a reminder that the particle is absorbed the first time it reaches either boundary . for use below we note the asymptotic forms @xmath41 the second of these forms is intuitively obvious , corresponding to ballistic propagation from @xmath1 to @xmath2 in the time @xmath42 . in this paper we study the mean absorption time of a randomly - accelerated particle moving on the finite line @xmath0 for the more general boundary condition described above : absorption at the boundary with probability @xmath6 , reflection with probability @xmath7 and coefficient of restitution @xmath8 . for this boundary the mean absorption time also satisfies the fokker - planck equation ( [ fp ] ) with reflection symmetry ( [ refsym ] ) , but the absorbing boundary condition ( [ absorbingbc ] ) is replaced by @xmath43 the absorbing boundary condition ( [ absorbingbc ] ) considered by masoliver and porr is unusual in that @xmath44 is specified for @xmath36 but not @xmath14 . that the fokker - planck equation with this boundary condition and with reflection symmetry ( [ refsym ] ) is a well - posed boundary value problem was known @xcite long before the exact solution of masoliver and porr we are unaware of a similar proof for the more general boundary condition ( [ partialabsorbingbc ] ) . in this paper we show how to solve the more general case with an extension of the approach of masoliver and porr . in the appendix we show that the fokker - planck equation ( [ fp ] ) with reflection symmetry ( [ refsym ] ) has the exact green s function solution @xmath45 for @xmath14 , where @xmath46 is the same function as in eq . ( [ mp ] ) , @xmath47\;,\label{gf2}\end{aligned}\ ] ] @xmath48 and @xmath49 is a standard confluent hypergeometric function @xcite . to calculate @xmath50 from eq . ( [ gf1 ] ) , one must first determine the unknown function @xmath51 on the right - hand side . setting @xmath2 in eqs . ( [ gf1])-([gf3 ] ) and substituting @xmath52 and @xmath53 , which follow from eqs . ( [ refsym ] ) and ( [ partialabsorbingbc ] ) , leads to the integral equation @xmath54 where @xmath55\;,\label{ieq2}\ ] ] and @xmath56 is a generalized hypergeometric function @xcite . our analytical and numerical predictions for the mean absorption time are derived directly from eqs . ( [ ieq1 ] ) and ( [ ieq2 ] ) . the same symmetric kernel @xmath57 plays a central role in refs . @xcite , in determining the equilibrium distribution function @xmath58 of a randomly accelerated particle moving between inelastic walls at @xmath1 and @xmath2 with which it collides inelastically . as discussed in @xcite , the quantity @xmath59 generalizes mckean s result for the velocity distribution at first return @xcite from the half - line to the finite line . the probability that a randomly accelerated particle , which leaves @xmath1 with velocity @xmath14 , arrives with speed between @xmath60 and @xmath61 the next time it reaches either @xmath1 or @xmath2 is given by @xmath62 , where the first and second terms of eq . ( [ ieq2 ] ) correspond to arrival at @xmath1 and @xmath2 , and where @xmath63 integral equation ( [ ieq1 ] ) follows immediately from this interpretation . the first , second , @xmath64 terms in the iterative solution of the integral equation represent the mean time to reach the boundary for the first time , the mean time between the first and second boundary collisions , , weighted with a factor @xmath7 for each reflection . in eq . ( [ ieq1 ] ) the asymptotic form of @xmath44 for small and large @xmath5 is determined by the first and second terms , respectively , of the kernel ( [ ieq2 ] ) . for @xmath65 the particle is not always absorbed the first time it reaches the boundary , which implies @xmath66 . for small @xmath5 we look for a solution with the asymptotic form @xmath67 with exponent @xmath68 smaller than the value @xmath69 for @xmath22 in eq . ( [ asymp ] ) . substituting eq . ( [ asy1 ] ) in eq . ( [ ieq1 ] ) , we find that the asymptotic form is consistent with the integral equation if @xmath70 solving eq . ( [ asy2 ] ) for @xmath71 graphically , one finds a unique solution satisfying @xmath72 in the physical region @xmath26 , @xmath27 . comparing eqs . ( [ expopr ] ) and ( [ asy2 ] ) , one sees that @xmath73 . thus , the exponents of @xmath5 in the results ( [ expopr ] ) , ( [ tstar ] ) for the half line and ( [ asy1 ] ) , ( [ asy2 ] ) for the finite line are the same . this suggests that the small @xmath5 behavior ( [ asy1 ] ) , ( [ asy2 ] ) is due to repeated collisions with the same boundary . for large @xmath5 , @xmath74 , corresponding to ballistic propagation from @xmath1 to @xmath2 in the time @xmath42 . combining this relation with eqs . ( [ refsym ] ) and ( [ partialabsorbingbc ] ) yields @xmath75 equation ( [ asy3 ] ) also follows from integral equation ( [ ieq1 ] ) on using the asymptotic forms ( [ asymp ] ) for large @xmath5 and @xmath76 for large @xmath77 . for @xmath78 , the mean absorption time is expected to decrease to zero in the large @xmath5 limit . iteration of eq . ( [ asy3 ] ) and/or substitution of the ansatz @xmath79 in eq . ( [ asy3 ] ) leads to the asymptotic forms @xmath80 for @xmath81 , with @xmath78 . the large @xmath5 behavior for @xmath82 , @xmath18 , corresponding to inelastic reflection with zero absorption probability , requires special attention and is considered in section v - c . to interpolate between the asymptotic forms of @xmath44 for small and large @xmath5 , given in the preceding section , we solved integral equation ( [ ieq1 ] ) numerically . changing the integration variable from @xmath60 to @xmath83 , we solved the resulting equation , @xmath84 by iteration . to evaluate the integral in eq . ( [ ieq3 ] ) numerically , we changed the integration variable from @xmath60 to @xmath85 where @xmath86 is a positive constant , and then used simpson s rule , exact for polynomials of degree 3 , with equally spaced mesh points in @xmath77 and @xmath87 . to ensure good accuracy for small @xmath77 ( small @xmath60 ) , we subtracted off the leading asymptotic form of the integrand , fit to a power law , and then integrated it analytically . in the region where both @xmath5 and @xmath60 are small , where the first term in the kernel ( [ ieq2 ] ) diverges , the integrand was fit to @xmath88 times a power law in @xmath77 and then integrated analytically . the parameters @xmath89 and @xmath86 and the mesh size were chosen to give good convergence with about 100 mesh points in each of the intervals @xmath90 and @xmath91 . starting with @xmath92 as the first approximation to @xmath44 , one typically needs about 10 iterations for convergence . in our simulations the motion of the particle is governed by the difference equations @xmath93 here @xmath94 and @xmath95 are the position and velocity at time @xmath96 , and @xmath97 . the quantities @xmath98 and @xmath99 are independent gaussian random numbers such that @xmath100 as discussed in @xcite , this algorithm generates trajectories which are consistent with the exact probability distribution @xmath101 of a randomly accelerated particle in free space , i.e. in the absence of boundaries . in free space there is no time step error in the algorithm . the time step @xmath102 is arbitrary . however , close to the boundaries trajectories are not generated with the correct probability , because the free space distribution @xmath101 includes trajectories which wander outside the interval @xmath0 and return during the time @xmath13 . as in @xcite , we make the time step smaller near the boundaries to exclude these spurious trajectories . using the algorithm ( [ xstep])-([gauss ] ) with large time steps away from the boundaries , instead of a conventional algorithm with a constant time step , enables us to simulate the particle for much longer times . for a particle in free space with position and velocity @xmath94 , @xmath95 at time @xmath96 , the coordinate @xmath103 at a time @xmath102 is distributed according to a gaussian function @xcite , with a maximum at @xmath104 and root - mean square width @xmath105 . choosing @xmath102 so that @xmath106 where the constant @xmath107 is about 5 or larger , ensures that the gaussian distribution lies well within the interval @xmath0 . the largest @xmath102 consistent with inequality ( [ step1 ] ) is given by @xmath108 , where @xmath109 to avoid solving eq . ( [ ddef ] ) , cubic in @xmath110 , at each step of the algorithm , one may use the approximation @xmath111^{-1}\;,\end{array}\right.\quad\begin{array}{l}v>0\;,\\v<0\ ; , \end{array}\label{dapprox}\ ] ] which is asymptotically exact for @xmath112 and @xmath113 and accurate to better than @xmath114 for @xmath115 . the most efficient time step to use in the algorithm is the largest @xmath102 consistent with _ both _ inequalities ( [ step1 ] ) and ( [ step2 ] ) . using a smaller @xmath102 slows the simulation without improving the accuracy . since the inequality ( [ step2 ] ) follows from ( [ step1 ] ) on making the reflection symmetric substitution @xmath116 , @xmath117 , the optimal time step is @xmath118+\delta\;,\label{beststep}\ ] ] where min denotes the smaller of the two quantities inside the square brackets . as in @xcite a small minimum time step @xmath119 has been included in eq . ( [ beststep ] ) . without it , the step size decreases to zero as the particle approaches the boundary , and , as in zeno s paradox , the particle never gets there . after inclusion of @xmath119 , the particle not only reaches the boundary but jumps slightly pass it . we kept the overshoot small by choosing a sufficiently large value for the parameter @xmath107 in eqs . ( [ step1])-([dapprox ] ) . in the case @xmath82 , @xmath120 of highly inelastic collisions with absorption probability zero , the speed of the particle becomes extremely small after many boundary collisions , and great care is required to simulate the behavior reliably . according to eqs . ( [ vstep ] ) and ( [ beststep ] ) , the root - mean - square velocity change at each time step has the minimum average value @xmath121 after each boundary collision we set @xmath122 equal to 1/500 of the velocity just after the collision . this @xmath122 is the smallest velocity the algorithm can reliably handle . the corresponding value of @xmath119 , given by eq . ( [ deltav ] ) , is then used until the next boundary collision . the results for the mean time @xmath44 reported in section v are based on @xmath123 independent particle trajectories for each value of the initial velocity @xmath5 . in this subsection we present our various results for @xmath78 , @xmath17 . in this regime the particle is absorbed at the boundary with probability @xmath6 and reflected _ elastically _ with probability @xmath7 . this is the partial survival " model , studied on the half line in refs . @xcite . for @xmath78 , @xmath17 the mean absorption time has the asymptotic forms ( [ asy1 ] ) , ( [ asy2 ] ) , and ( [ asy4 ] ) for small and large @xmath5 , respectively . as the reflection probability @xmath7 increases from 0 to 1 , @xmath124 decreases from the masoliver - porra result @xmath125 in eq . ( [ asymp ] ) for absorption on first arrival at the boundary , to the value @xmath126 , corresponding to no absorption at all , or @xmath127 . in fig . [ pfig ] our results for the mean absorption time @xmath44 for partially absorbing , elastic boundary conditions as a function of the initial velocity @xmath5 are compared for @xmath17 and @xmath128 . the exact asymptotic forms ( [ asy1 ] ) and ( [ asy2 ] ) for small @xmath5 and ( [ asy4 ] ) for large @xmath5 , the numerical solution of eq . ( [ ieq1 ] ) , and the simulation results are clearly in excellent agreement . for fixed @xmath5 the mean absorption time in fig . [ pfig ] increases with increasing reflection probability @xmath7 , as expected . for fixed @xmath7 the mean absorption time does not vary monotonically with @xmath5 but has an absolute maximum at an intermediate velocity . for much larger initial velocities the particle is absorbed more rapidly since it bounces back and forth between the boundaries , colliding at a rapid rate . for much smaller initial velocities the particle is absorbed more rapidly because it collides repeatedly with the boundary where it starts . here we present our results for @xmath78 , @xmath18 . in this regime the particle is absorbed at the boundary with probability @xmath6 and reflected _ inelastically _ with probability @xmath7 and coefficient of restitution @xmath8 . results for the half - line geometry with this boundary condition are given in ref . @xcite . in fig . [ requalpfig ] our results for the mean absorption time @xmath44 as a function of the initial velocity @xmath5 are compared for @xmath129 , @xmath130 , and @xmath131 . again the exact asymptotic forms ( [ asy1 ] ) and ( [ asy2 ] ) for small @xmath5 and ( [ asy4 ] ) for large @xmath5 , the numerical solution of eq . ( [ ieq1 ] ) , and the simulation results are in excellent agreement . note that the large @xmath5 asymptotic forms ( [ asy4 ] ) for @xmath132 and @xmath133 are different and that the data test both . corrections to the large @xmath5 asymptotic form in the crossover regime @xmath134 lead to conspicuously slower convergence for @xmath135 . in this subsection we consider boundaries with @xmath82 , @xmath18 . at the boundary the particle is reflected inelastically with probability 1 and absorbed with probability zero . this is the case of interest in connection with driven granular matter . _ @xcite predicted that for @xmath120 , where @xmath136 the particle undergoes inelastic collapse , " making an infinite number of boundary collisions in a finite time , coming to rest , and remaining there . the prediction that the inelastic collisions localize the particle at the boundary for @xmath120 was questioned by florencio _ _ @xcite and anton @xcite on the basis of simulations . the equilibrium distribution function @xmath58 of a particle moving on the finite line between two inelastic boundaries was studied by burkhardt , franklin , and gawronski @xcite for @xmath137 and burkhardt and kotsev @xcite for @xmath120 with exact analytical and numerical calculations and simulations , similar to this paper . according to @xcite , @xmath58 varies smoothly and analytically with @xmath8 throughout the interval @xmath26 and does not collapse onto the boundaries . for @xmath120 the equilibrium boundary collision rate is infinite , but the collisions do not localize the particle at the boundary ( see footnote @xcite ) . in the case @xmath82 , @xmath18 iterating integral equation ( [ ieq1 ] ) adds the mean time to reach the boundary for the first time , the mean time between the first and second boundary collisions , etc . , taking the change in speed in each collision into account . thus the solution @xmath44 represents the mean time @xmath44 for an infinite number of boundary collisions . for @xmath82 , @xmath120 , @xmath44 has the asymptotic forms ( [ asy1 ] ) , ( [ asy2 ] ) for small @xmath5 . as the coefficient of restitution @xmath8 increases from 0 to 1 , @xmath138 decreases from the masoliver - porra result @xmath125 in eq . ( [ asymp ] ) to the value @xmath139 , confirming the result of cornell _ et al . _ for @xmath140 in eq . ( [ rc ] ) . the negative , unphysical value of @xmath71 for @xmath137 signals the breakdown of the solution with finite @xmath44 . for @xmath137 the mean time for an infinite number of collisions is infinite . in contrast with the results for @xmath78 given in eq . ( [ asy4 ] ) , in the case @xmath82 , @xmath18 , the mean time @xmath44 does not vanish in the limit @xmath81 . this may be understood by reverse iteration of the large @xmath5 recurrence relation ( [ asy3 ] ) , which yields @xmath141 this result holds for any finite velocity @xmath5 large enough so that the time to travel between the two boundaries is accurately given by the ballistic time @xmath42 . the first @xmath142 terms on the right - hand side give the time a particle with initial velocity @xmath143 takes to reach velocity @xmath5 and sum to @xmath144 in the limit @xmath145 . according to eq . ( [ tinfty ] ) , @xmath146 is finite and non - vanishing for @xmath120 . the most general asymptotic form for large @xmath5 consistent with these two properties and with the recurrence relation ( [ asy3 ] ) is @xmath147 where @xmath148 is a periodic function of @xmath149 with period @xmath150 . the periodic term in eq . ( [ asy5 ] ) came as a surprise to us . the large @xmath5 recurrence relation ( [ asy3 ] ) implies @xmath151 for @xmath82 , which suggests , but does not guarantee , that @xmath44 is a monotonically increasing function of @xmath5 . the asymptotic form ( [ asy5 ] ) for @xmath120 with the periodic term is entirely consistent with @xmath151 and with eq . ( [ tinfty ] ) . the periodicity already appears in the following crude approximation : for @xmath152 define @xmath153 , in accordance with eq . ( [ asy1 ] ) , and then use the large @xmath5 recurrence relation , eq . ( [ asy3 ] ) with @xmath82 , to obtain @xmath44 for @xmath154 . to determine @xmath44 with simulations , we measured the mean time @xmath155 for @xmath156 boundary collisions , plotted it versus @xmath157 , and then extrapolated to @xmath158 . this is shown for @xmath159 and @xmath160 and @xmath161 in fig . [ extrapfig ] . in fig . [ rfig ] our results for the mean time @xmath44 for an infinite number of boundary collisions are shown as a function of @xmath5 for @xmath162 and @xmath159 , @xmath163 , and @xmath164 . for fixed @xmath5 the mean time increases with increasing @xmath8 , as expected . solving integral equation ( [ ieq1 ] ) reliably becomes increasingly difficult as @xmath8 decreases , and in fig . [ rfig ] we only show the numerical solution for @xmath159 . the simulation data are in good agreement with this numerical solution , with the asymptotic form ( [ asy1 ] ) , ( [ asy2 ] ) for small @xmath5 , and with the onset of periodic behavior in accordance with eq . ( [ asy5 ] ) . the dashed lines for large @xmath5 were calculated from the dashed lines for small @xmath5 using recurrence relation ( [ asy3 ] ) . masoliver and porr calculated the mean absorption time @xmath3 of a randomly accelerated particle on the finite line @xmath0 , assuming that it is absorbed the first time it reaches either @xmath1 or @xmath2 . we have considered a more general boundary condition , parametrized by the reflection probability @xmath7 and the coefficient of restitution @xmath8 . we derived an integral equation which determines @xmath3 and used it to obtain the exact asymptotic form of @xmath44 for large and small @xmath5 and numerical results for intermediate @xmath5 . the asymptotic forms and numerical results are in excellent agreement with our computer simulations of the randomly accelerated particle . the case @xmath82 , @xmath18 corresponds to a single particle moving between walls with which it collides inelastically . this simple and fundamental system is of interest in connection with properties of driven granular media . we find that the mean time for an infinite number of boundary collisions is finite for @xmath120 and infinite for @xmath137 , as predicted by cornell _ the prediction that the infinite sequence of collisions leads to inelastic collapse , with localization of the particle at the boundary , has been questioned on the basis of simulations @xcite and an analysis @xcite of the equilibrium distribution function @xmath58 using the same approach as in this paper . the results for the mean time reported here are largely independent of the inelastic collapse controversy . both the integral equation and the simulation procedure sum the mean time , collision by collision , without directly addressing the question of localization at the boundary . we thank jerrold franklin for many stimulating discussions . twb also gratefully acknowledges discussions and correspondence with lucian anton and alan bray . with the substitution @xmath165 , eq . ( [ fp ] ) takes the form @xmath166 this differential equation is the same as eq . ( 3 ) of ref . @xcite , except for the inhomogeneous term @xmath39 on the right - hand side . we follow ref . @xcite very closely in solving it . first we introduce the laplace transform @xmath167 which according to eq . ( [ fp2 ] ) satisfies @xmath168 this the same as eq . ( a2 ) in @xcite , except for the extra term @xmath169 on the right - hand side . as in @xcite , we solve eq . ( [ ltfp2 ] ) in terms of airy functions and invert the laplace transform , with the help of the faltung theorem . this yields @xmath170 which is the same as eq . ( 6 ) of @xcite , except for an additional term ( the first term ) on the right - hand side . to express the unknown function @xmath171 on the right - hand side of eq . ( [ agf1 ] ) in terms of the other unknown @xmath172 , we first take the limit @xmath173 in eq . ( [ agf1 ] ) , which yields @xmath174\label{agf2}\;.\end{aligned}\ ] ] from reflection symmetry @xmath175 , and @xmath176 . substituting eq . ( [ agf2 ] ) in the first of these relations and making use of the second , we obtain @xmath177\left(\textstyle{1\over 2}u^2+\tilde{t}(0,u)\right)\;,\label{agf3}\ ] ] where we have used @xmath178 equation ( [ agf3 ] ) may be regarded as an integral equation for @xmath171 . the solution , derived in @xcite using the approach of @xcite , is @xmath179\left(\textstyle{1\over 2}u^2+\tilde{t}(0,u)\right)\ ; , \label{agf4}\ ] ] where the function @xmath180 is given in eq . ( [ gf3 ] ) . substituting eq . ( [ agf4 ] ) in ( [ agf1 ] ) yields @xmath181\nonumber\\&\ & + \int_0^\infty du\thinspace u\thinspace g(x , v , u)\tilde{t}(0,u)\;,\label{agf5}\end{aligned}\ ] ] where @xmath182 is given in eqs . ( [ gf2 ] ) . replacing @xmath183 by @xmath50 in eq . ( [ agf5 ] ) and evaluating the integral @xmath184 $ ] , we obtain the green s function solution for @xmath50 in eqs . ( [ gf1])-([gf3 ] ) and reproduce the masoliver - porr result for @xmath40 in eq . 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( [ ieq2 ] ) generalizes the result @xmath185 for a particle moving on the half line @xmath9 with a single boundary at @xmath1 , derived by mckean @xcite and independently by cornell et al . d. j. bicout and t. w. burkhardt , j. phys . a * 33 * , 6835 ( 2000 ) . j. florencio , f. c. s barreto , and o. f. de alcantara bonfim , phys . lett . * 84 * , 196 ( 2000 ) . l. anton , phys . e * 65 * , 047102 ( 2002 ) . this result , which may seem counter - intuitive , has the following mathematical origin : the equilibrium distribution @xmath58 obtained in @xcite varies asymptotically as @xmath186 for @xmath187 , where the exponent @xmath188 increases monotonically from @xmath189 to @xmath190 as @xmath8 decreases from 1 to 0 , with @xmath191 . thus , the equilibrium collision rate @xmath192 diverges at the lower limit for @xmath120 . this result only holds for the continuous time dynamics ( [ eqmo ] ) . for discrete dynamics with a nonvanishing time step , the number of collisions in a finite time is necessarily finite . d. porter and d. s. g. sterling _ integral equations _ , ( cambridge university press , cambridge , 1990 ) . and , from top to bottom , @xmath193 , @xmath194 , @xmath195 , and @xmath189 . the solid line for @xmath22 is the exact result of masoliver and porr . the other solid lines are the numerical solutions of the integral equation ( [ ieq1 ] ) . the points are the results of our simulations . the dashed lines show the exact asymptotic forms ( [ asy1 ] ) , ( [ asy2 ] ) for small @xmath5 and ( [ asy4 ] ) for large @xmath5 . the proportionality constant in eq . ( [ asy1 ] ) was chosen to fit the simulation data.,title="fig:",scaledwidth=70.0% ] + , @xmath130 , @xmath131 , @xmath196 . the bottom curve is the exact result ( [ mp ] ) of masoliver and porr . the other solid lines are the numerical solutions of the integral equation ( [ ieq1 ] ) . the points are the results of our simulations . the dashed lines show the exact asymptotic forms ( [ asy1 ] ) , ( [ asy2 ] ) for small @xmath5 and ( [ asy4 ] ) for large @xmath5 . the proportionality constant in eq . ( [ asy1 ] ) was chosen to fit the simulation data . note the slow convergence to the asymptotic form for large @xmath5 for ( 0.501,0.5 ) in the crossover region @xmath134.,title="fig:",scaledwidth=70.0% ] + for @xmath156 boundary collisions versus @xmath157 for @xmath197 and , from top to bottom , @xmath1980.001 and 0.0001 . the mean time @xmath44 for an infinite number of collisions is estimated by extrapolating to the vertical axis . this yields @xmath199 and @xmath200.,title="fig:",scaledwidth=70.0% ] + and , from top to bottom , @xmath159 , @xmath163 , @xmath164 . the upper curve is the numerical solution of the integral equation ( [ ieq1 ] ) for @xmath159 . the points are the results of our simulations , obtained by extrapolation , as in fig . [ extrapfig ] . the bottom curve ( no simulations ) is the exact masoliver - porr result ( [ mp ] ) . the dashed lines on the left show the exact asymptotic form ( [ asy1 ] ) , ( [ asy2 ] ) for small @xmath5 . the proportionality constant in eq . ( [ asy1 ] ) was chosen to fit the simulation data . for large @xmath5 , approximately periodic behavior in @xmath149 , as in eq . ( [ asy5 ] ) , is expected . the dashed lines for large @xmath5 in fig . [ rfig ] were calculated from the dashed lines for small @xmath5 using recurrence relation ( [ asy3]).,title="fig:",scaledwidth=70.0% ] +

consider a particle which is randomly accelerated by gaussian white noise on the line segment @xmath0 and is absorbed as soon as it reaches @xmath1 or @xmath2 . the mean absorption time @xmath3 , where @xmath4 and @xmath5 denote the initial position and velocity , was calculated exactly by masoliver and porr in 1995 . we consider a more general boundary condition . on arriving at either boundary , the particle is absorbed with probability @xmath6 and reflected with probability @xmath7 . the reflections are inelastic , with coefficient of restitution @xmath8 . with exact analytical and numerical methods and simulations , we study the mean absorption time as a function of @xmath7 and @xmath8 . pacs 02.50.ey , 05.40.-a , 45.70.-n

cs@xmath0cucl@xmath1 is a quasi-2d heisenberg antiferromagnet with @xmath9=1/2 cu@xmath10 spins arranged in a triangular lattice with spatially - anisotropic couplings.@xcite the weak interlayer couplings stabilize magnetic order at temperatures below 0.62 k into an incommensurate spin spiral . the ordering wavevector is largely renormalized from the classical large-@xmath9 value and this is attributed to the presence of strong quantum fluctuations enhanced by the low spin , geometric frustrations and low dimensionality.@xcite the purpose of the thermodynamic measurements reported here is to probe the phase diagrams in applied magnetic field and see how the ground state spin order evolves from the low - field region , dominated by strong quantum fluctuations , up to the saturated ferromagnetic phase , where quantum fluctuations are entirely suppressed by the field . intermediate fields are particularly interesting as the combination of ( still ) strong quantum fluctuations , potentially degenerate states due to frustration and an effective `` cancelling '' of small anisotropies by the applied field may stabilize non - trivial forms of magnetic order . the hamiltonian of cs@xmath0cucl@xmath1 has been determined from measurements of the magnon dispersion in the saturated ferromagnetic phase.@xcite the exchanges form a triangular lattice with spatially - anisotropic couplings as shown in fig . 1(b ) with exchanges @xmath11 mev ( 4.34 k ) along @xmath3 , @xmath12 along the zig - zag bonds in the @xmath13 plane , and weak interlayer couplings @xmath14 along @xmath2 . in addition there is also a small dzyaloshinskii - moriya interaction @xmath15 , which creates an easy - plane anisotropy in the ( @xmath13 ) plane ( for details see ref . ) . neutron diffraction measurements have shown rather different behavior depending on the field direction with respect to the easy - plane . for perpendicular fields ( along @xmath2-axis ) incommensurate cone order with spins precessing around the field axis is stable up to ferromagnetic saturation , however for fields applied along the @xmath4-axis ( in - plane ) the incommensurate order is suppressed by rather low fields , 2.1 t compared to the saturation field of 8.0 t along this axis.@xcite the purpose of the present magnetization and specific heat measurements is to explore in detail the phase diagram in this region of intermediate to high fields . from anomalies in the thermodynamic quantities we observe that for in - plane field several phases occur in - between the low - field spiral and the saturated ferromagnetic states . from the magnetization curve we extract the work required to fully saturate the spins and from this we derive the total ground state energy in magnetic field and the component due to zero - point quantum fluctuations . dc - magnetization of a high - quality single crystal of cs@xmath0cucl@xmath1 grown from solution was measured at temperatures down to 0.05k and high fields up to 11.5 t using a high - resolution capacitive faraday magnetometer@xcite . a commercial superconducting quantum interference device magnetometer ( quantum design mpms ) was used to measure the magnetization from 2k to 300 k. the specific heat measurements were carried out at temperatures down to 0.05 k in magnetic fields up to 11.5 t using the compensated quasi - adiabatic heat pulse method @xcite . ( a ) temperature dependence of the susceptibility @xmath16 of cs@xmath0cucl@xmath1 along the three crystallographic axes . labels indicate magnetic long range order ( lro ) , short - range order ( sro ) and paramagnetic ( pm ) . ( b ) susceptibility divided by the @xmath17-factor squared compared to calculations for a 2d anisotropic triangular lattice ( see inset ) with @xmath18 and @xmath19k ( thick solid line ) , and non - interacting 1d chains with @xmath20 and @xmath21k ( thick dashed line).,height=377 ] ( a ) magnetization curves of cs@xmath0cucl@xmath1 measured at the base temperature for the field applied along the three crystallographic axes . the curves for @xmath22 and @xmath4 are shifted by 0.3 and 0.6@xmath23cu , respectively . ( b ) susceptibility @xmath24 vs field . vertical arrows indicate anomalies associated with phase transitions ( see text).,width=302 ] we first discuss the temperature - dependence of the magnetic susceptibility at low field and compare with theoretical predictions for an anisotropic triangular lattice as appropriate for cs@xmath0cucl@xmath1 . figure [ chi](a ) shows the measured susceptibility @xmath25 in a field of 0.1 t. a curie - weiss local - moment behavior is observed at high temperatures and a broad maximum , characteristic of short - range antiferromagnetic correlations , occurs around @xmath26 k , in agreement with earlier data.@xcite upon further cooling the @xmath3- and @xmath4-axes susceptibilities show a clear kink at @xmath27=0.62k , indicating the transition to long - range magnetic order . no clear anomaly at @xmath27 is observed for @xmath28 . this is because the magnetic structure has ordered moments spiralling in a plane which makes a very small angle ( @xmath29 ) with the @xmath13 plane.@xcite in this case the near out - of - plane ( @xmath2-axis ) susceptibility is much less sensitive to the onset of magnetic order compared to the in - plane susceptibility ( along @xmath3 and @xmath4 ) . fitting the high - temperature data ( @xmath30 k ) to a curie - weiss form @xmath31 with @xmath32 gives @xmath330.2 k and @xmath17-factors @xmath34=2.27 , @xmath35=2.11 and @xmath36=2.36 for the @xmath2- , @xmath3- and @xmath4-axes , respectively . the @xmath17-factors are in good agreement with the values obtained by low - temperature esr measurements @xmath17=(2.20 , 2.08 , 2.30)@xcite . when the susceptibility is scaled by the determined @xmath17-values , @xmath37 , the data along all three crystallographic directions overlap within experimental accuracy onto a common curve for temperatures above the peak , indicating that the small anisotropy term in the hamiltonian ( estimated to @xmath385% @xmath6 ) are only relevant at much lower temperatures . in the temperature range @xmath39 we compare the data with high - temperature series expansion calculations@xcite for a 2d spin-1/2 hamiltonian on an anisotropic triangular lattice ( see fig . [ chi](b ) inset ) . very good agreement is found for exchange couplings @xmath18 and @xmath6=4.46k ( 0.384 mev ) ( solid line in fig . [ chi](b ) ) . in contrast , the data departs significantly from the expected bonner - fisher curve for one - dimensional chains ( @xmath20 and @xmath21 k).@xcite figure [ m ] shows the magnetization @xmath40 and its derivative @xmath41 as a function of applied field at a base temperature of 0.05 k for the @xmath2- and @xmath3-axes and 0.07 k for the @xmath4-axis . for all three axes the magnetization increases linearly at low field but has a clear overall convex shape and saturates above a critical field @xmath42=8.44(2 ) , 8.89(2 ) and 8.00(2 ) t along the @xmath2- , @xmath3- and @xmath4-axis , respectively . when normalized by the @xmath17-values the saturation fields are the same within 2% for the three directions , the difference being the same order of magnitude as the relative strength 5% of the anisotropy terms in the hamiltonian.@xcite the saturation magnetizations @xmath43 are obtained to be only 1 - 2.5% below the full spin value of 1/2 , which might be due to experimental uncertainties in the absolute units conversion or a slight overestimate of the @xmath17-values by this amount . including such a small uncertainty in the @xmath17-values has only a small effect on the normalized susceptibility @xmath37 in fig . [ chi](b ) and does not change significantly the results of the comparison with the series expansion calculation for the anisotropic triangular lattice . ( a)reduced magnetization , ( b ) susceptibility , and ( c ) ground - state energy , vs reduced field . black ( solid ) , blue ( dashed ) , green ( dash - dotted ) and red ( dotted ) lines show experimental data for @xmath44a , semiclassical mean - field prediction , linear spin - wave theory including 1st order quantum corrections and bethe - ansatz prediction for 1d chains @xmath20 , respectively.,width=302 ] before analyzing in detail the various transitions in field identified by anomalies in the susceptibility @xmath41 ( vertical arrows in fig . [ m]b ) we briefly discuss how the ground - state energy varies with the applied field , as this gives important information about the effects of quantum fluctuations . the ground state energy is obtained by direct integration of the magnetization curve , i.e. @xmath45 where the energy ( per spin ) above the saturation field takes the classical value @xmath46 because the ferromagnetic state is an exact eigenstate of the hamiltonian with no fluctuations.@xcite here @xmath47 is the sum of all exchange interactions equal to @xmath48 for the main hamiltonian in cs@xmath0cucl@xmath1 [ see fig . [ chi]b ) inset ] . figure [ energyanal ] shows comparisons between the experimental data for @xmath49@xmath50@xmath2 ( black solid lines , similar results obtained using @xmath3- or @xmath4-axis data ) , a mean field calculation ( blue dashed lines ) , a linear spin - wave theory(lswt ) with 1st order quantum correction ( green dash - dotted lines ) and bethe - ansatz prediction for 1d chains @xmath20 ( red dotted lines ) . in magnetic field , a cone structure is predicted by the classical mean field calculation @xcite @xmath51-g\mu_b b s \sin\theta$ ] ( blue dashed lines ) where @xmath52 is the classical ordering wavevector @xmath53 $ ] , @xmath54 , the saturation field is @xmath55 $ ] and @xmath56 . here we use @xmath57 mev and @xmath58 for the main hamiltonian in cs@xmath0cucl@xmath1 . the experimentally - determined ground state energy ( black solid line ) is lower than the classical value ( blue dashed line ) due to zero - point quantum fluctuations . the energy difference in zero field is 85% of the expected classical energy @xmath59 , indicating rather strong quantum fluctuations in the ground state . the strongly non - linear ( convex ) shape of the magnetization curve compared to the classically - expected linear form @xmath60 [ see fig . [ energyanal](a ) ] is a direct indication of the importance of zero - point quantum fluctuations . including 1st order quantum correction to the classical result in a linear spin - wave approach gives@xcite @xmath61-g\mu_bb(s+1/2)\sin\theta+\langle \omega_{\bm{k } } \rangle /2 $ ] where @xmath62 is the average magnon energy in the 2d brillouin zone of the triangular lattice . this improves the agreement with the data ( green dash - dotted lines ) . particularly at high fields it captures better the divergence of the susceptibility [ see fig . [ energyanal](b ) ] at the transition to saturation . the saturation field is underestimated slightly because we have here neglected the weak inter - layer couplings @xmath63=4.5% @xmath6 and the dm interaction @xmath64=5.3%j , both of which increase the field required to ferromagnetically - align the spins . it is also illuminating to contrast the data with a model of non - interacting chains ( @xmath20 , red dotted lines ) . this would largely ( by 48% ) underestimate the observed saturation field and would predict a rather different functional form for the magnetization @xmath65 and susceptibility @xmath66 compared to the data , indicating that the 2d frustrated couplings are important . specific heat as a function of temperature in magnetic fields along @xmath3-axis . specific heat data in fields 4 , 5 and 6 t are shifted upwards by 0.5 , 1.0 and 1.5 j / mol@xmath67k@xmath68 , respectively.,width=302 ] when the magnetic field is applied along the @xmath2-axis perpendicular to the plane of the zero - field spiral the ordered spins ca nt towards the field axis and at the same time maintain a spiral rotation in the @xmath13 plane thus forming a cone . the cone angle closes continuously at the transition to saturation and as expected in this case the susceptibility @xmath69 observes a sharp peak followed by a sudden drop as the field crosses the cone to saturated ferromagnet transition , see fig . however for fields applied along the @xmath3- and @xmath4-axes several additional anomalies are present in the magnetization curve apart from the sharp drop in susceptibility upon reaching saturation , indicating several different phases stabilized at intermediate field . before discussing in detail the experimental phase diagrams we note that for all field directions the magnetization increases in field up to saturation and no intermediate - field plateaus are observed , in contrast to the isostructural material cs@xmath0cubr@xmath1 , where a narrow plateau phase occurs for in - plane field when the magnetization is near @xmath70 of saturation.@xcite such a plateau phase is expected for the fully - frustrated ( @xmath71 ) triangular antiferromagnet and originates from the formation of the gapped collinear up - up - down state in field . the absence of a plateau in cs@xmath0cucl@xmath1 is probably related to the weaker frustration ( @xmath72=0.34(3 ) ) compared to cs@xmath0cubr@xmath1(@xmath73 ) @xcite . a difference in the phase diagrams in field applied along the @xmath2-axis and in the @xmath13 plane in cs@xmath0cucl@xmath1 is expected based on the presence of small dm terms in the spin hamiltonian , which create a weak easy - plane anisotropy in the @xmath13 plane.@xcite semi - classical calculations which take this anisotropy into account predict two phases below saturation:@xcite a distorted spiral at low field separated by a spin - flop like transition from a cone at intermediate field . the data in fig . [ m](b ) however observe more complex behavior with several different intermediate - field ordered phases . also early neutron scattering measurements did not observe the characteristic incommensurate magnetic bragg peaks expected for a cone structure at @xmath742.1 t @xmath50@xmath4-axis @xcite , suggesting that the magnetic structure at intermediate field may be quite different from the classical prediction and may be stabilized by quantum fluctuations beyond the mean - field level . to map out the extent of the various phases in in - plane field we have made a detailed survey of the @xmath75 phase diagram using both temperature and field scans in magnetization and specific heat and the resulting phase diagrams are shown in fig . [ b - t ] . below we describe in detail the signature of those transitions in specific heat and magnetization data , first for field along the @xmath3-axis , then @xmath4-axis . magnetization and differential susceptibility @xmath69 in field along @xmath3 are shown in fig . @xmath69 shows a sharp peak at @xmath49=2.76 t and an additional small peak at @xmath49=8.57 t and those two anomalies indicate two new phases at base temperature below the saturation field and above the spiral phase . to probe the extent in temperature of those phases we show in fig . [ c_b ] specific heat measurements as a function of temperature at constant magnetic field . at 3 t two peaks are clearly observed indicating two successive phase transitions upon cooling from high temperatures . the lower critical temperature increases rapidly with increasing field and gradually approaches the upper transition at 5 t and the two peaks appear to merge at 6 t. magnetization normalized by applied field @xmath76 ( thick solid lines , left axis ) and its derivative @xmath77 ( thin solid lines , right axis ) as a function of temperature for @xmath22 . vertical arrows indicate anomalies.,width=321 ] raw capacitance data as a function of temperature in magnetic field of 4 and 6 t applied along @xmath3-axis . data are vertically shifted for clarity . filled and open arrows indicate anomalies . thick ( thin ) solid lines correspond to measurements with ( without ) gradient field . inset shows the temperature derivative of capacitance data in gradient field 10t / m . the curves are shifted vertically for clarity.,width=302 ] expanded plots of magnetization and susceptibility @xmath41 as a function of field along @xmath4-axis . data are identical to those from fig . 2(a ) and ( b).,width=302 ] specific heat as a function of temperature in magnetic fields along @xmath4-axis . specific heat data in fields 4 , 5 and 6 t are shifted upwards by 0.5 , 1.0 and 1.5 j / mol@xmath67k@xmath68 , respectively.,width=283 ] magnetization normalized by applied field @xmath76 ( thick solid lines , left axis ) and its derivative @xmath77 ( thin solid lines , right axis ) as a function of temperature for @xmath78 . vertical arrows indicate anomalies.,width=302 ] complementary magnetization data vs. temperature for field along @xmath3 is shown in fig . [ m(t)b ] . at 3 and 4 t two anomalies are observed also in @xmath79 and its derivative @xmath77 , at essentially the same temperatures as the peaks in specific heat , indicating that those two anomalies are associated with magnetic phase transitions . the anomalies appear as kinks in @xmath79 and steps in @xmath77 . at 5 t however the scaled magnetization @xmath79 only observes a clear anomaly at the lower of the two critical temperatures observed in specific heat . at 6 t no anomaly is visible in @xmath79 , but only the derivative @xmath77 shows a kink . the missing anomalies can however be seen in the raw capacitance data , plotted in fig . [ torque ] , which also contain information not only on the ( longitudinal ) magnetization but also the transverse spin components . at 4 t the capacitance in both zero and non - zero gradient field show two successive transitions indicated by solid and open arrows . those are in good agreement with the peaks observed in specific heat . although there is no anomaly visible in the magnetization at 6 t , the raw capacitance shows an anomaly ( see derivative of @xmath80 in inset of fig . [ torque ] ) ) at the same temperature as the peak in specific heat . the capacitance in non - zero gradient field contains information on the torque of the sample in addition to the magnetization , while that in zero gradient field does not depend on magnetization but only on the torque . the torque contribution is subtracted by measuring the capacitance in zero gradient field ( details of measurement technique are described in ref . [ 5 ] ) . the fact that there is no anomaly in magnetization implies that subtraction of torque effect cancels out the anomaly in the raw data . therefore only the torque ( transverse magnetization ) has an anomaly and the longitudinal magnetization has no anomaly at the critical temperatures for these missing anomalies . for field along @xmath4 it has been reported from neutron scattering study that the spiral phase at zero field is suppressed by magnetic field of 1.4 t and above this field ordered spins form an incommensurate elliptical structure with elongation along the field direction @xcite . the elliptical phase is suppressed at 2.1 t where the intensity of incommensurate magnetic bragg peaks vanishes and the properties of the phase above 2.1 t are still unknown . as shown in fig . [ m](b ) , the suppression of the spiral phase is clearly seen as a step in magnetization ( a sharp peak in @xmath69 ) at 1.40 t . in fig . [ anormm(b)c ] the magnetization at 0.07k and its derivative @xmath69 are expanded in order to show the four anomalies above the spiral phase . in fig . [ anormm(b)c](a ) , @xmath40 shows a small step ( a peak in @xmath81 ) at 2.05 t , corresponding to the suppression of the elliptical phase . as indicated by an open arrow in fig . [ anormm(b)c](a ) , a step in susceptibility at 2.18 t is clearly seen , indicating possibly a new phase which may exist only in a very small range of fields from 2.05 to 2.18 t . the next transition occurs at 3.67 t ( for increasing field ) shown in fig . [ anormm(b)c](b ) . @xmath40 has a step accompanied by a hysteresis , indicative of a first order transition . figure [ anormm(b)c](c ) shows another transition at 7.09 t with a clear hysteresis . [ c_c ] shows specific heat in magnetic fields along the @xmath4-axis . at 3 t only one transition is observed upon cooling , whereas at 4 and 5 t two successive transitions are observed . the lower temperature transition is very sharp , related to the first order behavior ( hysteresis ) on this transition line also observed in magnetization data @xmath40 at 3.67 t shown in fig . [ anormm(b)c](b ) . the lower temperature transition shifts to higher temperatures with increasing field and almost merges with the upper transition at 6 t . fig . [ m(t)c ] shows complementary magnetization data vs. temperature . at 3 t @xmath79 and @xmath77 show a kink and a step at 0.35k , respectively . at 4 t the position of the kink ( step in @xmath77 ) is shifted to slightly higher temperature and another step - like anomaly ( a negative peak in @xmath77 ) appears at lower temperatures 0.22k . at 5 t this lower temperature step shifts to higher temperatures and the upper temperature anomaly ( kink ) can not be seen in @xmath79 but is manifested as a kink in @xmath77 at 0.38k . again the anomaly is missing in @xmath79 , but the raw capacitance data ( not shown ) exhibits an anomaly at 0.38k in good agreement with the specific heat result . as shown in the top panel of fig . [ m(t)c ] , @xmath79 and @xmath77 have two anomalies at 6 t . note that due to the first order character of the lower temperature transition the anomalies of @xmath76 ( @xmath77 ) indicated by open arrows in fig . [ m(t)c ] are steps ( peaks ) rather than kinks ( steps ) . @xmath75 phase diagrams of cs@xmath0cucl@xmath1 for @xmath44 @xmath3- and @xmath4-axis . data points of open circles ( magnetization ) , squares ( specific heat ) and triangles ( neutrons @xcite ) connected by solid lines indicate phase boundaries . solid circles show positions of the maximum in the temperature dependence of the magnetization and indicate a cross - over from paramagnetic to short - range order(sro ) . `` e '' on the phase diagram for @xmath78-axis denotes the elliptical phase . @xcite , width=302 ] the phase diagrams for @xmath22- and @xmath4-axis constructed using the anomalies discussed above are shown in fig . [ b - t ] . the new data agree with and complement earlier low - field neutron diffraction results ( open triangles).@xcite apart from the phase transition boundaries identified above we have also marked the cross - over line between paramagnetic and antiferromagnetic short - range ordered(sro ) region , determined by the location of the peak in the temperature dependence of the magnetization such as in fig . [ chi](a ) . the peak position @xmath82 decreases with increasing field and disappears above @xmath83 , indicating suppression of antiferromagnetic correlations by magnetic field . for the field along @xmath3 and @xmath4-axis the phase diagrams are much more complicated than that for @xmath28 which shows only one cone phase up to saturation field @xcite . for @xmath22 three new phases appear above the spiral phase . two of these phases occupy small areas of the @xmath75 phase diagram . for @xmath78 four new phases are observed in addition to the spiral and elliptical phases . we note that the absence of an observable anomaly in the temperature dependence of the magnetization upon crossing the phase transitions near certain fields ( 6 t along @xmath3 and 5 t along @xmath4 ) is consistent with ehrenfest relation and is related to the fact that the transition boundary @xmath84 is near flat around those points . the relation between the shape of the phase boundary and the anomaly in @xmath85 was discussed by t. tayama , _ et . al._@xcite and is @xmath86 where @xmath87 is the discontinuity of quantity @xmath88 , @xmath89 is the specific heat and @xmath90 is the field - dependent critical temperature of second order phase transition . this shows that the discontinuity in @xmath91 vanishes when @xmath92 , i.e. when the phase boundary is flat in field . this is indeed the case for 6 t @xmath93 and at 5 t @xmath94 [ see fig . [ b - t ] ] , and here only a kink and no discontinuity is seen in @xmath91 . we have studied the magnetic phase diagrams of cs@xmath0cucl@xmath1 by measuring magnetization and specific heat at low temperatures and high magnetic fields . the low - field susceptibility in the temperature range from below the broad maximum to the curie - weiss region is well - described by high - order series expansion calculations for the partially frustrated triangular lattice with @xmath7=1/3 and @xmath6=0.385mev . the extracted ground state energy in zero field obtained directly from integrating the magnetization curve is nearly a factor of 2 lower compared to the classical mean - field result . this indicates strong zero - point quantum fluctuations in the ground state , captured in part by including quantum fluctuations to order @xmath95 in a linear spin - wave approach . the obtained @xmath75 phase diagrams for in - plane field ( @xmath96 and @xmath4-axis ) show several new intermediate - field phases . the difference between the phase diagrams for @xmath97 and @xmath4 can not be explained by a semi - classical calculation for the main hamiltonian in cs@xmath0cucl@xmath1 , of a frustrated 2d heisenberg model on an anisotropic triangular lattice with small dm terms . further neutron scattering experiments are needed to clarify the magnetic properties of these new phases . we would like to thank v. yushankhai , d. kovrizhin , d.a . tennant , m. y. veillette and j.t . chalker for fruitful discussions , z. weihong for sending the series data , p. gegenwart and t. lhmann for technical support .

we report magnetization and specific heat measurements in the 2d frustrated spin-1/2 heisenberg antiferromagnet cs@xmath0cucl@xmath1 at temperatures down to 0.05 k and high magnetic fields up to 11.5 t applied along @xmath2 , @xmath3 and @xmath4-axes . the low - field susceptibility @xmath5 shows a broad maximum around 2.8k characteristic of short - range antiferromagnetic correlations and the overall temperature dependence is well described by high temperature series expansion calculations for the partially frustrated triangular lattice with @xmath6=4.46k and @xmath7=1/3 . at much lower temperatures ( @xmath8 k ) and in in - plane field ( along @xmath3 and @xmath4 -axes ) several new intermediate - field ordered phases are observed in - between the low - field incommensurate spiral and the high - field saturated ferromagnetic state . the ground state energy extracted from the magnetization curve shows strong zero - point quantum fluctuations in the ground state at low and intermediate fields .

advances in complex condensed matter systems @xcite and in quantum information @xcite have stimulated the study of ultracold atomic systems @xcite such as bose - einstein condensates ( becs ) @xcite , mott insulators @xcite , graphene @xcite , superconductors @xcite , and others . among these systems too are fermi gases @xcite , which is a large ensemble of particles that obey fermi - dirac statistics . in this study , we will deal with a fermi gas trapped by an optical lattice , which wecker et al . @xcite showed to be useful in quantum computing @xcite by determining the phases diagram and ground state properties . other applications of fermi gases in optical lattices are in the cooling of magnetically trapped gas @xcite ; observation of quantized vortices , quenching of moment of inertia , and spin polarization in superfluid helium @xcite ; entanglement in luttinger liquids @xcite ; double - photo - ionization of molecular hydrogen @xcite ; landau - zener tunneling in double - well potential in deep bec regime @xcite ; topological phase transitions driven by real - next - nearest neighbor hopping @xcite ; fermion condensation quantum phase transition in @xmath0 @xcite ; interaction quantum quenches in one - dimensional model with spin imbalance @xcite ; realization of bardeen - cooper - schrieffer ( bcs)-bec state crossover in regime of resonant interactions @xcite ; among other applications . to engineer optical lattice systems , the hubbard model is used since only nearest neighbor interactions among lattice sites allow the interaction of the fermi gas particles in the lattice @xcite . furthermore , external field parameters can be varied over time using this model @xcite . this description of fermionic systems allows us the study of the possibility of quantum phenomena such as spin - exchange interaction and to design properties like anisotropy and sign by proper choices of optical potentials @xcite . considering the lattice as an open quantum system , i.e. connecting it to an external environment such as a heat bath or a quantum particle ( i.e. harmonic oscillator ) bath , gives a more realistic description of the system . this is because the energy of the lattice system is not constant due to its exposure to the environment and the number of particles in the lattice is not constant because particles from the bath can be trapped into the lattice and particles in the lattice can be expelled to the bath @xcite . since ultracold atomic gases can be trapped by an optical lattice , and taking into account its interaction with the environment , can information be transported from a particular lattice site to another over time such that ultracold fermions trapped in an optical lattice can be used to engineer information transport systems ? this is answerable by studying the dissipative dynamics of the fermi - hubbard model , i.e. the effect of connecting the optical lattice containing a trapped fermi gas to a fermionic bath . the characteristics of this system , in particular its ability to transport information and other quantities , can be explored through time by the particle population of the lattice , the existence of spin exchange through time , the particle occupation of each lattice site , and the entanglement in the lattice . this article studies a system of a fermi gas trapped by a three - site optical lattice interacting with a fermionic bath in order to observe phenomena such as spin exchange ( which can be observed easily through fermions ) and entanglement , which can be used to describe information dynamics in systems that can be mapped onto lattices . the discussion is organized as follows : in section [ sec : fermihubbard ] , based on discussions from jaksch and zoller @xcite and breuer and petruccione @xcite , the model is described by defining the field operators of the system and the bath . the free and interaction hamiltonians and the born - markov master equation of the system are written . the derived dissipative dynamics are then used in section [ sec : occprobfid ] to describe the transport of information through the lattice using fidelity , occupation probability , and entanglement . conclusions , recommendations , and possible future extensions are stated in section [ sec : conclusion ] . a fermi gas is an ultracold atomic system of particles obeying the pauli exclusion principle @xcite . suppose that the gas is trapped by an optical lattice with a potential @xmath1 and whose lattice sites all accommodate just one energy level , with the magnitude of that energy level being identical for all three lattice sites . since there is only one energy level , each lattice site can be occupied by up to two particles of different spins , i.e. one spin - up and one spin - down . ( see figure [ fig : schematic ] . ) mathematically , this system can be represented using the field operator @xmath2 where @xmath3 is the vector pointing to a specific lattice site and @xmath4 is the spin of the particle . the annihilation operator corresponding to the spin of a particle in the lattice site located at @xmath3 is @xmath5 to describe the occupation or non - occupation of a particular spin state , the basis @xmath6 is used . @xmath7 denotes occupation and @xmath8 , non - occupation . since a lattice site can be occupied by at most a spin - up and a spin - down particle , the basis for the site s state vectors is @xmath9 , where @xmath10 is the number of spin - up and spin - down particles respectively . therefore , the basis for the state of the particles occupying the lattice site consists of the following column matrices : @xmath11 it then follows that the annihilation operator for the spin - up and the spin - down particles in a lattice site at @xmath3 are @xmath12 respectively . since the lattice interacts with a bath , it is an open quantum system therefore , the state of the system is described using the density matrix [ rhos ] @xmath13 where @xmath14 is the probability of finding @xmath15 particles of certain spin @xmath16 in a lattice site pointed by @xmath17 at a time @xmath18 . the potential of the lattice can be split as @xmath19 . @xmath20 is the external trapping potential of the lattice , which varies slowly compared to the solved lattice potential @xmath21 . an example of trapping potentials are the magnetic trapping potentials , which enable the trapping or expulsion of particles from the optical lattice . examples of @xmath21 are those of particle in a box , harmonic oscillator , and hydrogen atom . through rabi and raman lasers , we observe that the particle spin splits the known potential such that @xmath22 . this means that the rabi laser potential @xmath23 causes the particles to scatter through the lattice with unchanged spin and the raman laser potential @xmath24 changes the particle spin via scattering . the free hamiltonian is defined as @xmath25 where @xmath26 is the reduced mass and @xmath27 is the interaction strength between two particles . if the only interaction is @xmath28-wave scattering , @xmath29 where @xmath30 is the @xmath28-wave scattering length . this free hamiltonian can be expressed in terms of the fermionic annihilation and creation operators @xmath31 and @xmath32 provided we let the following quantities be constant : the free particle energy @xmath33 the energy of interaction between two particles @xmath34 the rabi laser energy @xmath35 and the raman laser energy @xmath36 these quantities are made constant by assumption that they are uniform for all lattice sites in this study . thus , in terms of fermionic annihilation and creation operators , the free hamiltonian is @xmath37 we consider the case where the optical lattice interacts with a fermionic bath defined by a set of fermionic annihilation ( creation ) operators @xmath38 ( @xmath39 ) . therefore , the bath is mathematically described using the field operator of the form @xmath40 where @xmath41 is the desnity of the fermionic bath , @xmath42 is the volume of the bath , @xmath43 and @xmath44 are constants . the state of the bath is described by the density matrix @xmath45 . the interaction with the bath causes fermionic particles to either be trapped by or expelled from the lattice . this adds to the effective potential , causing scattering and the spin exchange . thus , this interaction can be expressed in terms of both the bath and the system field operators such that @xcite @xmath46 with this study limited to two - body interactions , terms of order @xmath47 can be neglected . therefore , in terms of annihilation and creation operators of both the system and environment particles , the interaction hamiltonian of the system is @xmath48 see appendix [ app : intham ] for the detailed outline of the derivation of the interaction hamiltonian . the interaction of the system and the bath evolves through time . therefore , using the baker - campbell - hausdorff formula , the time - evolved interaction hamiltonian is @xmath49 see appendix [ app : inthamt ] for the outline of the derivation of eq . ( [ fermifermiintht ] ) . as stated earlier , the interaction of the fermions trapped in the lattice with a bath of fermionic particles causes them to either get trapped by or expelled from the lattice system over time . our task is now to determine the time evolution equation for the fermions in the lattice . the assumptions made in deriving the equation are : ( 1 ) the system has a negligible effect on the bath and ( 2 ) the state of the system at a particular time step depends only on its state in the previous time step . these assumptions constitute the born - markov approximation . the time evolution of systems that follow the born - markov approximation is described by the equation @xcite @xmath50.\ ] ] substituting the interaction hamiltonian in eq . ( [ inthfermi ] ) and using the anticommutator relations of the fermionic operators @xcite @xmath51 together with the rotating wave approximation , we then find that the master equation of the system in this study is @xmath52 ] - [ \hat{c}_{\vec{l ' } , \uparrow}^\dagger \hat{c}_{\vec{l ' } + \vec{r ' } , \uparrow } , [ \hat{c}_{\vec{l } , \downarrow}^\dagger \hat{c}_{\vec{l } + \vec{r } , \downarrow } , \rho_s ( t ) ] ] + h.c . \right ) \nonumber \\ & + \left ( [ \hat{c}_{\vec{l ' } , \downarrow}^\dagger \hat{c}_{\vec{l ' } + \vec{r ' } , \downarrow } , [ \hat{c}_{\vec{l } + \vec{r } , \downarrow}^\dagger \hat{c}_{\vec{l } , \downarrow } , \rho_s ( t ) ] ] - [ \hat{c}_{\vec{l ' } , \uparrow}^\dagger \hat{c}_{\vec{l ' } + \vec{r ' } , \uparrow } , [ \hat{c}_{\vec{l } + \vec{r } , \uparrow}^\dagger \hat{c}_{\vec{l } , \uparrow } , \rho_s ( t ) ] ] - h.c . \right ) \}. \label{fermifermimasteqn } \end{aligned}\ ] ] see appendix [ app : mastereqn ] for the detailed outline of the derivation master equation . with the dynamics of particles in the lattice described by eq . ( [ fermifermimasteqn ] ) , the system can be simulated numerically . there are two types of lattices considered for the fermi - hubbard model . first is an open lattice where the first and the third sites do not interact directly ( see figure [ fig : schematicopen ] ) . second is a closed lattice where there is nearest neighbor interaction between the sites , including the first and third ( see figure [ fig : schematicclosed ] ) . to describe the behavior of the gas in the lattice , we can observe its probability of occupying a state @xmath53 , @xmath54 known as the fidelity . from the state of the system , we can know if information is being transferred from one lattice site through the particles and their specific spins . to observe the fidelity ( and the occupation probability and the entanglement in the system ) , we denote four different initial states of the system : 1 . pure state in which the lattice has no particles : @xmath55 2 . maximally entangled state such that the lattice is half - empty and half - full : @xmath56 3 . 70% spin - down in the first site and 30% spin - down in the third site : @xmath57 4 . 30% spin - down in the first site and 70% spin - down in the third site : @xmath58 figure [ fig : fidelitycomp1 ] summarizes the results of computations of the fidelity for cases of different initial states . the states @xmath53 with respect to which the fidelity is computed are the following : 1 . the lattice is empty : @xmath55 2 . the lattice is full : @xmath59 3 . 50% probability that a spin - down particle is found in the first site and 50% probability that a spin - down particle is found in the third site : @xmath60 4 . 50% probability that a spin - up particle is found in the first site and 50% probability that a spin - up particle is found in the third site : @xmath61 5 . 50% probability that a spin - down particle is found in the first site and 50% probability that a spin - up particle is found in the third site : @xmath62 6 . 50% probability that a spin - up particle is found in the first site and 50% probability that a spin - down particle is found in the third site : @xmath63 the choice of these states are based on the following : the first two @xmath53s tell if the whole lattice would be filled with information while the latter four @xmath53s ( particularly the application of the @xmath64 @xmath53 on the @xmath65 and @xmath66 initial state cases indicated earlier ) would indicate if a spin - exchange happened or not . we can then see in figures [ fig:3openemptyfidelity ] and [ fig:3closedemptyfidelity ] that an empty lattice does not remain empty through time for there is a nonzero probability that at least one spin - up and one spin - down particle occupies either the first or the third lattice site at some time @xmath67 . on the other hand , when the system is initially in the maximally entangled state where there is 50% probability of finding the lattice empty and 50% probability of finding the lattice full of particles as seen in figures [ fig:3openmaxentfidelity ] and [ fig:3closedmaxentfidelity ] , the 50% chance of finding the system full is constant though time . however , the probability that the lattice is found empty becomes zero at @xmath68 and that the other states are occupied . lastly , we consider the cases wherein there is only one particle in the lattice and there is a nonzero probability , ( either 30% or 70% ) that it is in the first site or in the third site . the results in figures [ fig:3open3070fidelity ] to [ fig:3closed7030fidelity ] show the fidelity such that there is 50% probability of finding a spin - down particle in the first site and 50% probability of finding a spin - up particle in the third site . through the numerical results , we can draw the conclusion that fermionic particles from the bath enter into and exit from the lattice through time . spin exchange can also happen such that the spin of the particle on either the first or third lattice flips from down to up at time @xmath68 . another quantity that describes the dynamics of the particles in the lattice through time is the occupation probability . this is the probability in which at least one particle , regardless of spin , occupies the lattice site . for the @xmath69 site , the equation for the occupation probability is @xmath70 where @xmath71 a @xmath72 diagonal matrix where any entry pertaining to non - occupation of the @xmath69 lattice site is zero and the rest is equal to one . for each lattice site , it can be stated as follows : [ occprobforms ] @xmath73 where @xmath74 . the sum of all the occupation probabilities is not normalized because the individual sites , not the lattice as a whole , are observed . as seen in figure [ fig : occprob ] , for all cases , there is a nonzero probability that at least one particle , regardless of spin , would occupy each lattice site at @xmath68 even if the lattice is initially empty . this confirms the statement in section [ subsec : fidelity ] that the lattice does not remain empty through time as observed using fidelity . for the first two initial conditions as seen in figures [ fig:3openemptyoccprob ] to [ fig:3closedmaxentoccprob ] , the occupation probability of each lattice site increases through time until it approaches 1 as @xmath75 . the occupation probability of the first site , however , is the one that approaches the value of unity fastest among the three sites . on the other hand , as seen through figures [ fig:3open3070occprob ] to [ fig:3closed7030occprob ] , the occupation probabilities of all sites stabilize at unity for open lattices . however , for closed lattices , only the occupation probability of the first site approaches 1 as @xmath75 while the occupation probability of the other sites approach @xmath76 as @xmath75 . this means that there is a chance that no particle is present in the second and third site for long times . lastly , the transport of information through the lattice can be described by measuring the entanglement of the optical lattice with entropy . in this study , the measure of entanglement is the linear entropy , @xmath77 wherein @xmath78 is the dimension of the system , i.e. the number of the lattice sites . ( in this study , @xmath79 . ) linear entropy was chosen as the entanglement measure in this study as it is measure that can be applicable to any system of any dimension ( as measures such as concurrence fail for higher dimensions ) . it is then seen in figure [ fig : entropy ] that in general , entanglement as measured by the linear entropy of the time - evolved state peaks at some time @xmath68 then decreases , stabilizing at a certain value as @xmath75 . for open lattices , we can consider as an illustrative case that shown in figure [ fig:3openmaxentent ] , where the lattice system is considered as maximally entangled at @xmath80 as there is a 50% probability of finding the system full of fermionic particles and 50% probability of finding it empty . comparing this to the other cases for the open lattice , this shows that if the initial state of the lattice has a nonzero probability of having the lattice full of particles , nonzero linear entropy can be achieved as @xmath75 . this means that the initial probability of having the lattice full can partially preserve the entanglement through time . on the other hand , for closed lattices , the linear entropy @xmath81 as @xmath75 in general however , the linear entropy stabilizes at a maximum value of @xmath82 as @xmath75 compared to the other cases with respect to the initial condition . therefore , in general , closing the lattice and starting the lattice in a state such that there is a nonzero initial probability of finding the lattice full of particles helps increase the long - time value of entanglement in the system . in this article , we examine the dissipative dynamics of a fermi gas in a three - site optical lattice exposed to a fermionic environment as illustrated in figure [ fig : schematic ] . since fermion - fermion scattering takes place due to the interaction of the lattice and environment , intrinsic properties of the particle , i.e. spin , are regarded as information passed through the lattice . using the fidelity and the occupation probability , we find that even if at @xmath83 , the lattice is empty , particles from the environment will enter into it . furthermore , we observe through the fidelity that spin exchange happens in the system such that a spin - up particle can turn into a spin - down particle upon the exposure of the system to the fermionic bath . lastly , closing the lattice and starting it in a state such that there is nonzero probability of finding the lattice full helps preserve a higher value of entanglement . the results imply that the three - site optical lattice system can be used to realize quantum technologies , such as quantum wires or a series of quantum computers by means of quantum transport of information through lattice sites , which may be parts of a quantum wire or nodes in a network of quantum computers . furthermore , the system can be used for quantum encryption via spin exchange within a quantum or classical network . this work is limited to a three - site optical lattice due to limitations in available computing resources . however , this work can be extended to a lattice with four or more sites , which we intend to do in future work . this research is funded by a grant from the national research council of the philippines ( nrcp ) as nrcp project p-022 . roland caballar would like to acknolwedge m. a. a. estrella for valuable technical and conceptual assistance during the research . vladimir villegas would like to acknowledge the financial support from the department of science and technology ( dost ) via the advanced science and technology human resource development ( asthrdp ) , j. l. duanan for valuable technical assistance and b. villegas , e. galapon , r. j. abuel , g .- flores , and d. m. antazo for valuable conceptual assistance during the research . finally , v. villegas and r. caballar would like to acknowledge t. lindberg , a. bjrler , j. bjrler , m. larsson , a. erlandsson , b. hannigan , a. netrebko , k. te kanawa , d. damrau , c. bartoli , t. bangalter , g - m . de homem - christo , e. l. b. gil , and h. e. soberano for conceptual discussions . the wannier functions @xmath84 have the following properties . 1 . an alternative notation is as follows : @xmath85 2 . it then follows that translation of the wannier functions would be @xmath86 3 . the bloch functions @xmath87 can be expressed in terms of wannier functions through the fourier transform @xmath88 wherein @xmath89 is a normalization constant . 4 . using the expression of wannier functions in terms of bloch functions , we know that the wannier functions are orthonormal , i.e. @xmath90 the interaction hamiltonian is expressed in terms of the lattice and bath field operators in eq . ( [ inthamgen ] ) . however , we can write the interaction hamiltonian in terms of the lattice and bath annihilation and creation operators in order to express the effects of the particles on the bath onto the lattice . we substitute the field operators of the optical lattice in eq . ( [ latoptr ] ) and the field operators of the fermionic bath in ( [ bathfieldop ] ) , we get the following . @xmath91 multiplying the lattice field operators with the first terms of the bath field operators would just result to orthonormality conditions . however , note that we can only apply orthonormality on the position but not on the spin . on the other hand , for terms with bath creation and annihilation operators , we use the properties of the wannier functions to evaluate the integrals and drop terms of the order @xmath92 ( since we restrict ourselves to two - body interaction ) . the overlap integral is expressed as @xcite @xmath93 since the interaction hamiltonian must be hermitian , then we know that @xmath94 is real . therefore , we can also impose that @xmath95 . therefore , we have eq . ( [ inthfermi ] ) . after the derivation of the interaction hamiltonian , the next step is to evolve it through time . it can be done using the baker - campbell - hausdorff formula . @xmath96 - \frac{1}{2 } \left [ h_0 , \left [ h_0 , h_i \right ] \right ] + ... \label{bch}\end{aligned}\ ] ] @xmath97 is the free hamiltonian given in eq . ( [ freehop ] ) and @xmath98 is the interaction hamiltonian given in eq . ( [ inthfermi ] ) . the pauli exclusion principle states that only one particle can occupy an energy level . in our case , since we are particular with the spin , two fermions - one spin - up and one spin - down - can occupy the same lattice site at a time . therefore , in evaluating the nested commutators in eq . ( [ bch ] ) , we make use of the anticommutator relations given in eq . ( [ anticomm1 ] ) and eq . ( [ anticomm2 ] ) @xcite . on the other hand , we are just concerned with two - body interactions . therefore , we can neglect terms with three or more fermionic operators . therefore , the time - evolved interaction hamiltonian for the fermion - fermion scattering case is eq . ( [ fermifermiintht ] ) . in this study , we apply the born - markov approximation , meaning that the action of the optical lattice on the fermionic bath is negligible such that they have minimal coupling and the state at time @xmath18 depends only on the immediate previous time step . therefore , the dynamics of the system is guided by the born - markov master equation given in eq . ( [ bornmarkovdef ] ) @xcite . next , due to turbulent fluctuations in the exponential term , the integral vanishes at infinity . furthermore , we will observe terms with the factor of @xmath100 , which signify resonance and are dropped by the use of the rotating wave approximation . by applying further the anticommutator relations in eq . ( [ anticomm1 ] ) and eq . ( [ anticomm2 ] ) , we observe that terms with the patterns of alternating spins i.e. @xmath101 and vice versa cancel each other out . @xmath104 ] + [ \hat{c}_{\vec{l ' } , \downarrow}^{\dagger } \hat{c}_{\vec{l ' } + \vec{r ' } , \downarrow } , [ \hat{c}_{\vec{l } , \uparrow}^{\dagger } \hat{c}_{\vec{l } + \vec{r } , \uparrow } , \rho_s ( t ) ] ] \nonumber \\ & + [ \hat{c}_{\vec{l ' } + \vec{r ' } , \downarrow}^{\dagger } \hat{c}_{\vec{l ' } , \downarrow } , [ \hat{c}_{\vec{l } + \vec{r } , \uparrow}^{\dagger } \hat{c}_{\vec{l } , \uparrow } , \rho_s ( t ) ] ] - [ \hat{c}_{\vec{l ' } + \vec{r ' } , \uparrow}^{\dagger } \hat{c}_{\vec{l ' } , \uparrow } , [ \hat{c}_{\vec{l } + \vec{r } , \downarrow}^{\dagger } \hat{c}_{\vec{l } , \downarrow } , \rho_s ( t ) ] ] \nonumber \\ & - [ \hat{c}_{\vec{l ' } , \uparrow}^{\dagger } \hat{c}_{\vec{l ' } + \vec{r ' } , \uparrow } , [ \hat{c}_{\vec{l } + \vec{r } , \uparrow}^{\dagger } \hat{c}_{\vec{l } , \uparrow } , \rho_s ( t ) ] ] + [ \hat{c}_{\vec{l ' } , \downarrow}^{\dagger } \hat{c}_{\vec{l ' } + \vec{r ' } , \downarrow } , [ \hat{c}_{\vec{l } + \vec{r } , \downarrow}^{\dagger } \hat{c}_{\vec{l } , \downarrow } , \rho_s ( t ) ] ] \nonumber \\ & + [ \hat{c}_{\vec{l ' } + \vec{r ' } , \uparrow}^{\dagger } \hat{c}_{\vec{l ' } , \uparrow } , [ \hat{c}_{\vec{l } , \uparrow}^{\dagger } \hat{c}_{\vec{l } + \vec{r } , \uparrow } , \rho_s ( t ) ] ] - [ \hat{c}_{\vec{l ' } + \vec{r ' } , \downarrow}^{\dagger } \hat{c}_{\vec{l ' } , \downarrow } , [ \hat{c}_{\vec{l } , \downarrow}^{\dagger } \hat{c}_{\vec{l } + \vec{r } , \downarrow } , \rho_s ( t ) ] ] \ } \end{aligned}\ ] ]

in this article , we investigate the dissipative dynamics of a fermi gas trapped in a three - site optical lattice exposed to a fermionic environment . the lattice sites admit at most one spin - up and one spin - down particle at a time and its interaction with the fermionic environment cause particles to either be trapped by or expelled from it . it is then shown that apart from each lattice site being populated by at least one particle , spin exchange is observed , which allows the possibility for the system to be used to encrypt information via quantum cryptography . furthermore , we also observe entanglement among the lattice sites , denoting that transport of quantum information among particles is possible in this ultracold atomic system .

rich clusters can be used to constrain cosmological models of large - scale structure formation . rich clusters are the largest virialized objects in the universe and , hence , their abundance and evolution can be simply related to the linear mass power spectrum , @xmath9 . their x - ray temperature can be used to infer the cluster mass . then , press - schechter ( 1974 ) theory can be used to relate the observed cluster abundance as a function of mass to the @xmath2 fluctuation on 8 @xmath3 mpc scales , @xmath1 . the result is a constraint on a combination of parameters , @xmath10 , where @xmath5 is a function of model parameters . in this paper , we obtain a general expression for @xmath5 which applies to a wide range of models including the standard cold dark matter ( scdm ) model , models with a mixture of cosmological constant ( @xmath11 ) and cold dark matter ( @xmath11cdm ) , and qcdm models with a mixture of cold dark matter and quintessence , a dynamical , time - evolving , spatially inhomogeneous component with negative pressure . an example of quintessence would be a scalar field ( @xmath12 ) rolling down a potential @xmath13 . it has been shown that a spectrum of @xmath11cdm and qcdm models satisfy all current observational constraints ( wang et al . our expression for @xmath5 includes dependence on the spectral index @xmath6 , the hubble constant @xmath7 ( in units of 100 km sec@xmath14 mpc@xmath14 ) , and the equation - of - state of the quintessence component @xmath8 . in section 2 , we discuss our derivation of the cluster abundance constraint , which follows earlier derivations to some degree but includes some new features when applied to quintessence . some may wish to proceed directly to the resulting constraint , given in section 3 . our result appears to differ slightly from some earlier works , but we explain in this section the reasons for those differences . our purpose in determining @xmath5 is to transform observations into a powerful constraint on models . in section 4 , we discuss the further constraint derived from studying the evolution of cluster abundance from redshift @xmath15 to @xmath16 . in section 5 , we conclude by applying the cluster abundance constraint on @xmath1 in combination with the constraint on cobe normalization of @xmath9 to pick out a spectrum of best - fitting @xmath11cdm and qcdm models . in this section , we present the derivation of the cluster abundance constraint on @xmath1 for qcdm models . we take a pedagogical approach in which we first discuss each step for a @xmath11cdm model and then point out the differences that arise in qcdm models . the final result can be found in section 3 . press - schechter theory relates the number density of clusters to their mass . observations determine directly the temperature instead of the mass of clusters . therefore , the mass - temperature relation derived from the virial theorem ( lahav et al . 1991 ; lilje 1992 ) is used to apply the press - schechter relation . first , let s consider a universe with vacuum energy density @xmath17 . for a spherical overdensity , we have @xmath18 where @xmath19 is the kinetic energy , @xmath20 is the potential energy associated with the spherical mass overdensity , and @xmath21 is the potential energy associated with @xmath11 . the kinetic energy is @xmath22 where @xmath23 is the mass of the cluster and @xmath24 is the mean square velocity of particles in the cluster . the gravitational potential is @xmath25 where @xmath26 is the gravitational constant and @xmath27 is the radius of the cluster . the potential due to vacuum energy is @xmath28 . re - expressing the background matter energy density at redshift @xmath29 as @xmath30 where @xmath31 the is the hubble parameter , @xmath32 is the ratio of the matter density to the critical density , and @xmath29 is the redshift , the virial relation ( [ virial ] ) becomes : @xmath33 where @xmath34 is the ratio of the cluster to background density , and @xmath35 and @xmath36 are the density parameters for vacuum energy and matter at redshift @xmath29 , respectively . the observed temperature of the gas is @xmath37 where @xmath38 is the mean mass of particles ; @xmath39 is the line - of - sight velocity dispersion ; @xmath40 is the ratio of the kinetic energy to the temperature and @xmath41 is the boltzmann constant . the mass of the cluster is then : @xmath42^{-1/2 } \left[1-\frac{2}{\delta_c } \frac{\omega_{\lambda}(z)}{\omega_{m}(z)}\right]^{-3/2 } h^{-1}10^{15 } m_{\odot}\ ] ] where @xmath43 where @xmath44 is a fudge factor of order unity that allows deviation from the simplistic spherical model . using this relation , the mass - temperature relation of a virialized cluster can be computed at any @xmath29 . however , if we evaluate at redshift @xmath45 , twice the turn - around time ( that is , @xmath46 where @xmath47 is the redshift at which the cluster turns around , @xmath48 ) , then @xmath49 becomes a function of @xmath4 only ( for @xmath11 and open universes ) . note that @xmath50 is the redshift at which the cluster formally collapses to @xmath51 according to an unperturbed spherical solution . in quintessence models , the principal difference is that the energy density in @xmath12 decreases with time , whereas vacuum energy remains constant . the @xmath12-component does not cluster on scales less than 100 mpc ( caldwell , dave , & steinhardt 1998 ) . consequently , the only effect of @xmath12 on the abundance of rich clusters with size less than 100 mpc is through its modification of the background evolution . we will restrict ourselves to cases where the equation - of - state @xmath8 is constant or slowly varying . in this case , the ratio @xmath52 above can be replaced by @xmath53 . for quintessence models , the ratio of cluster to background density , @xmath54 , is a function of two variables : @xmath55 where @xmath56 ; @xmath57 and @xmath58 are the radius of the cluster at @xmath59 and at virialization , respectively . ( the second equality utilizes the standard assumption that the cluster has virialized at @xmath45 . ) the factor @xmath60 has been computed by solving for the evolution of a spherical overdensity in a cosmological model with constant @xmath8 ( see appendix a ) . we find that @xmath61 is weakly model - dependent : @xmath62 for @xmath63 . the fact that this expression is weakly model - dependent means that we can also apply it to models with time - varying @xmath8 . by the virial theorem and energy conservation , we have : @xmath64 this leads to an approximate solution @xmath65 where @xmath66 and @xmath67 . the fact that @xmath68 is time - dependent if @xmath69 is because the energy within the cluster is not exactly conserved and the temperature is shifting after virialization due to the change in the @xmath12-energy , @xmath70 , within the cluster . however , since the matter density @xmath71 is much larger than @xmath70 in a collapsed cluster , this correction is negligible . the dispersion of the density field on a given comoving scale @xmath27 is @xmath72 where @xmath73 and @xmath74 @xmath75 is the power spectrum and @xmath76 is the fourier transform of the fractional density perturbation @xmath77 @xmath78 for constant or slowly varying @xmath79 , the bbks approximation to the power spectrum ( bardeen et al . 1986 ) is reliable . however , if @xmath80 or if @xmath8 is rapidly varying , we find no general fitting formula ; instead , @xmath9 must be obtained numerically . the @xmath2 mass fluctuation @xmath81 can be expressed as @xmath82 , where @xmath83 is the growth factor . the growth factor is proportional to the linear density perturbation @xmath84 and normalized to @xmath85 . we find that a good approximation to the the growth index is given by ( see appendix b ) @xmath86 where @xmath87 is the scale factor and @xmath88 the growth factor @xmath83 is obtained from the integral expression @xmath89\ ] ] we tested the expression for @xmath90 obtained from eq . ( [ alpha ] ) against the value obtained by numerically integrating the density perturbation equations ; for @xmath91 between zero and 0.8 , the accuracy is better than 1% . according to press - shechter theory , the comoving number density of virialized objects with mass @xmath92 is : @xmath93d\ , m\ ] ] where @xmath94 is the perturbaton that would be predicted for a spherical overdensity of radius @xmath27 and mass @xmath23 according to linear theory . given the observed number density @xmath95 within a certain temperature range @xmath96 , eqs . ( [ m_vir ] ) and ( [ ps ] ) can be used to determine the normalization of the mass power spectrum @xmath1 . the major uncertainties in this method are the observational error in the number density @xmath95 and the systematic error in determining the model parameters @xmath97 in eq . ( [ ps ] ) and @xmath98 in eq . ( [ m_vir ] ) . specifically @xmath99 \frac{\delta f_{\beta}}{f_{\beta}}\ ] ] where @xmath100^{-1}$ ] , @xmath101 and @xmath102 are positive and of order unity in the range of interest . by studying spherical models , we find that @xmath97 varies slowly as a function of @xmath4 , @xmath103 . we also find that @xmath98 does not depend on the cosmological model and can be determined by numerical simulation . according to eq . ( [ m_vir ] ) , the virial temperature corresponding to a given virial mass depends on the redshift at which a cluster is virialized . therefore , to get the number density of clusters of a given temperature range today , we need to find out the virialization rate and integrate from @xmath15 to @xmath104 . assuming that the merger of clusters is negligible , the press - schechter relation , eq . ( [ ps ] ) can be re - expressed as : + @xmath105 where @xmath106 . lacey & cole ( 1993,1994 ) and sasaki ( 1994 ) have estimated the corrections due to cluster merging ( see also , viana & liddle 1996 ) . the corrections are small . in our results , we average the two estimates of the merging correction to obtain our final result . the cluster abundance constraint on @xmath1 is obtained by comparing the theoretical prediction discussed in the previous section to observations . the observed x - ray cluster abundance as a function of temperature was presented by edge et al ( 1990 ) and henry & arnaud ( 1991 , hereafter ha ) . after a recent correction ( henry 1997 ) to ha , the two results agree . we have fit the theoretically predicted number density _ vs. _ temperature curve ( the temperature function ) to the ha data . our results can be fit by @xmath107 where @xmath108 where @xmath109 is the spectral index of primordial energy density perturbations and @xmath7 is the present hubble constant in units of @xmath110 . for qcdm models with equation - of - state @xmath8 ( including @xmath11cdm with @xmath111 ) , our fit to @xmath5 is @xmath112 for open models @xmath113 many groups have presented similar constraints on @xmath1 for @xmath11cdm and ocdm models . in general , all of them are in reasonable agreement with one another and with our result . we identify below the sources of the discrepancies , some real and some only apparent , when compared to white , efstatiou , & frenk ( 1993 , hereafter wef ) ; eke , cole , & frenk ( 1996 , hereafter ecf ) ; viana and liddle ( 1996 , hereafter vl ) ; pen ( 1997 , hereafter pen ) ; kitayama and suto ( 1997 , hereafter ks ) . \(1 ) as we argued , an integration ( with merger correction ) of eq . ( [ dps ] ) is necessary since the mass - temperature relation is redshift dependent . most groups only applied eqs . ( [ m_vir ] ) and ( [ ps ] ) at redshift @xmath15 to obtain their main results , which leads to an overestimate of @xmath1 by as much as @xmath114 . however , some groups ( ecf and pen ) fit the number density and temperature relations to numerical simulations to normalize their coefficient @xmath98 in eq . ( [ m_vir ] ) . in so doing , they effectively incorporated the integration correction into the coefficient @xmath98 for the cases that were numerically tested . consequently , their fitted value of @xmath98 does not represent precisely @xmath115 as defined in eq . ( [ m_vir ] ) . we shall call their corresponding coefficient @xmath116 to emphasize that the physical meaning of this coefficient has been modified to include intregration over redshift . since the contribution of the integration is not proportional to @xmath98 in all cosmological models , it is more precise and physically meaningful to do separately the redshift integration . \(2 ) the shape of the theoretical cluster temperature function , eq . ( [ dps ] ) , does not agree equally well with observations for all parameters . the fit is particularly poor for models with large @xmath117 and positive tilt ( @xmath118 ) . to handle this problem , some groups ( wef ; pen ; vl ) only fit the observed number density at one particular temperature ; this introduces some arbitrariness and leads to much larger uncertainties depending on which temperature is chosen . in our analysis , we fit the theoretical temperature function to all the data points provided by ha . \(3 ) the recent correction to the ha data results in a correction to the vl results of 20% . \(4 ) most groups ( wef ; ecf ; vl ; pen ) assume a fixed `` shape parameter '' @xmath119 . we found that no single @xmath120 is valid for all qcdm models . instead of expressing our results in terms of fixed @xmath120 , we fix @xmath121 . we include the dependence on @xmath7 ( @xmath122 ) explicitly . \(5 ) most recent analyses ( ecf ; vl ; pen ; ks ) adopted similar modeling of the mass - temperature relation . however , there is still about 10% to 20% disagreement on the value of @xmath98 in eq . ( [ m_vir ] ) due to the uncertainties of the numerical simulations . we found that the most extensive simulation results , those of ecf and pen , agree very well with each other . by normalizing our theoretical calculations to their simulations , we found @xmath123 . notice that their reported values for the coefficient are @xmath124 and @xmath125 , respectively . this discrepancy is due to rolling into @xmath116 the integration effect described under ( 1 ) . recent simulations by bryan and norman ( 1997 ) also indicate a similar result , once one corrects for their slightly higher value of @xmath97 . \(5 ) some groups ( vl ) used the differential temperature function ( the cluster abundance within unit temperature interval around a center value @xmath126 ) while others ( wef ; ecf ; penn ; ks ) used cumulative temperature function ( the cluster abundance with temperature above a critical temperature @xmath127 ) . these two approaches give similar results because the cluster abundance drops exponentially with temperature and the cumulative cluster abundance is well approximated by counting the cluster abundance around @xmath127 . we compared the results obtained by using the differential temperature function given by ha and that obtained by using the cumulative temperature function provided by ecf and found them to be in good agreement . however , the error bar of the latter is much smaller : most models were excluded by 95% confidence level by the temperature function fitting . to be conservative , we used the former to get our results and errors . ( [ error ] ) can be used to estimate the total error for @xmath1 . from the present scatter of numerical simulation results , @xmath98 has about 20% uncertainty , @xmath97 has about 10% uncertainty and another 15% uncertainty comes from the observation . therefore , the net uncertainty quoted in eq . ( [ sigma8 ] ) is about 20% corresponding to 95% confidence level . by applying the same theoretical tools , we can also study the evolution of the cluster abundance to obtain further constraints on @xmath1 and @xmath4 . the current redshift survey results were converted to number densities of clusters with their comoving-1.5 mass ( the mass within comoving radius @xmath128 ) greater than a given mass threshold @xmath129 ( carlberg et al . 1997 ; bahcall , fan , & cen 1997 ) . if the mass profile for the cluster obeys @xmath130 near @xmath131 , and the average virial overdensity is equal to @xmath132 as calculated in eq . ( [ delta_c ] ) , then the virial mass @xmath23 is related to @xmath129 by @xmath133 eq . ( [ ps ] ) can be used to estimate the number density of observed objects at a given redshift . we adopted @xmath134 as suggested by carlberg , yee , & ellingson ( 1997 ) . the log - abundance as a function of @xmath29 , @xmath135 , is roughly linear as a function of @xmath29 for @xmath136 for the models of interest . a useful parameter to characterize the evolution of cluster abundance at redshift @xmath137 is @xmath138 , defined by @xmath139 where @xmath140 is the number density of clusters with comoving-1.5 mass greater than @xmath129 observed at redshift @xmath29 . the smaller @xmath138 is , the stronger the evolution is . by applying this analysis to models , we found + ( 1 ) cluster abundance evolution strongly depends on @xmath1 : low @xmath1 leads to strong evolution . this agrees with what bahcall , fan , & cen ( 1997 ) have found . this is a general feature of gaussian - distributed random density peaks . + ( 2 ) cluster abundance is also sensitive to the equation - of - state of quintessence @xmath8 : low @xmath8 leads to strong evolution . with the same @xmath32 , the growth of density perturbations gets suppressed earlier in high @xmath8 models , therefore , they have a weaker evolution in recent epochs ( @xmath137 ) . + in figure 1 , we show @xmath141 as a function of @xmath1 for some sample models which have been chosen because they all fit current observations well ( see discussion in following section ) . we allow @xmath1 to vary from 0.5 to 1.0 with the cobe normalized @xmath1 shown as opaque circles . the current redshift survey data ( bahcall & fan 1998 ) indicate @xmath142 at the 3@xmath143 level ( with mean equal to -1.7 ) , which is consistent with all six cobe normalized models . the cluster abundance and evolution constraints , when combined with future measurements of the cosmic microwave background , may be an effective means of discriminating quintessence and @xmath11 models . the cosmic microwave background ( cmb ) anisotropy provides a constraint on the mass power spectrum on the horizon scale . for a given model , this constraint from large - scale anisotropy as measured by the cobe - dmr satellite ( smoot et al . 1992 ; bennett et al . 1996 ) can be extrapolated to obtain a limit on @xmath1 that is completely independent of the cluster abundance constraint . in figure 2 , we plot the dependence of @xmath1 on @xmath144 . for each @xmath8 , a different curve is shown . along each curve is highlighted the range of @xmath145 consistent with the cluster abundance constraint derived in this paper . hence , the best - fit models are those near the middle of the highlighted regions . these are the same models used as examples in figure 1 . near future satellite experiments , such as the nasa microwave anisotropy probe ( map ) and the esa planck mission , will greatly improve upon cobe by determining the temperature anisotropy power spectrum to extremely high precision from large to small angular scales . even a full - sky , cosmic variance limited measurement of the cmb anisotropy , though , may not be sufficient to discriminate @xmath11cdm from qcdm models . there is a degeneracy in parameter space such that , for any given @xmath11cdm models , there is a continuous family of qcdm models which predicts the same cmb power spectrum ( huey et al . 1998 ; white 1998 ) it is possible that the data points to a qcdm model , say , which lies outside this degenerate set of models , _ e.g. _ , a model with rapidly varying @xmath8 . however , if the data points to the degenerate set of models , then it is critically important to find a method of discriminating models further . not only does degeneracy mean that @xmath11 can not be distinguished from quintessence , but also that large uncertainty in @xmath4 and @xmath7 . here we wish to illustrate how cluster abundance may play an important role . an example of a degeneracy curve " is shown in figure 3 . given a value of @xmath7 for any one point along the curve , values of @xmath7 can be chosen for other points along the curve such that the models are all indistinguishable from cmb measurements . the near - future satellites are capable of limiting parameter space to a single degeneracy curve . the cmb anisotropy also narrowly constrains @xmath109 , @xmath146 , and @xmath147 . however , even when combined with the cluster abundance constraint and other cosmological constraints from the age , hubble constant , baryon fraction , lyman-@xmath148 opacity , deceleration parameter and mass power spectrum , a substantial degeneracy can remain . figure 3 includes a shaded region which exemplifies the range which these models might allow , based on current measurements ( wang et al . 1998 ; huey et al . 1998 ) . because of the uncertainty in cluster abundance at @xmath15 and other cosmic parameters , the overlap between the degeneracy curve and the shaded region allows a wide range of @xmath4 , @xmath7 and @xmath8 . cluster evolution offers a promising approach for breaking the degeneracy . figure 4 illustrates the variation of @xmath138 as a function of @xmath8 for models along the degeneracy curve and inside the shaded region of figure 3 . the variation in @xmath138 is nearly 2 , corresponding to nearly two orders of magnitude variation in abundance at redshift @xmath149 . the range of @xmath138 is between -3.5 and -5.5 in this case , but this could be shifted upward or downward by adjusting cosmic parameters . the point is that models which are degenerate in terms of cmb anisotropy are spread out in @xmath138 . if the measurements can be refined so that @xmath138 is know to better than @xmath150 , then cluster evolution may play an important role in discriminating between quintessence and vacuum density and , thereby , determining @xmath4 and @xmath7 . we have benefitted greatly from many discussions with n. bahcall and r. caldwell . we also thank p. bode , g. bryan , a. liddle , y. suto and p. viana for useful suggestions . this research was supported by the department of energy at penn , de - fg02 - 95er40893 . bahcall , n.a . , fan , x. , & cen , r. 1997,apj,485,l53 bahcall , n.a . , & fan , x. 1998,apj , in press , astro - ph/9803277 bardeen , j.m . , bond , j.r . , kaiser , n. , & szalay , a.s . 1986,apj,304,15 bennett , c.l . 1996,apj,464,l1 bryan , g.l . , & norman , m.l . 1997,astro - ph/9710107 caldwell , r.r . , dave , r. , & steinhardt , p.j . 1998,in preparation carlberg , r.g . , morris , s.m . , yee , h.k.c . , & ellingson , e. 1997,apj,479,l19 carlberg , r.g . , yee , h.k.c . , & ellingson , e. 1997,apj,478,462 edge , a.c . , stewart , g.c . , fabian , a.c . , & arnaud , k.a . 1990,mnras,245,559 eke , v.r . , cole , s. , & frenk , c.s . 1996,mnras,282,263 henry , j.p . 1997,apj,489,l1 henry , j.p . , & arnaud , k.a . 1991,apj,372,410 huey , g. , wang , l. , dave , r. , caldwell , r.r . , & steinhardt , p.j . 1998,in preparation kitayama , t. , & suto , y. 1997,apj,490,557 lacey , c. , & cole , s. 1993,mnras,262,627 lacey , c. , & cole , s. 1994,mnras,271,676 lahav , o. , lilje , p.b . , primack , j.r . , & rees , m.j . 1991,mnras,251,128 lilje , p.b . 1992,apj,386,l33 pen , u .- l . 1997,astro - ph/9610147 press , w.h . , & schechter , p. 1974,apj,187,452 sasaki , s. 1994,pasj,46,427 smoot , g. et al . 1992,apj,396,l1 viana , p.t.p . , & liddle , a.r . 1996,mnras,281,323 wang , l , caldwell , r.r . , ostriker , j.p . , & steinhardt , p.j . 1998,in preparation white , m 1998 , astro - ph/9802295 white , s.d.m . , efstathiou , g. , & frenk , c.s . we study a spherical overdensity with uniform matter density @xmath151 and radius @xmath27 in a background that satisfies the friedmann equation : @xmath152 where @xmath153 is the scale factor , @xmath154 is the background matter energy density and @xmath70 is the energy density in @xmath12 . quintessence does nt cluster at the interesting scales ; the energy density in @xmath12 remains the same both inside and outside the overdensity patch . because the curvature is not a constant inside the overdensity patch . we use the time - time component of the einstein equations ( which does not involve the curvature term ) to solve for the growth of the overdensity patch @xmath155\end{aligned}\ ] ] we have used @xmath156 to obtain the second equality . now , we define @xmath157 where @xmath158 and @xmath57 are the scale factor and the radius at turn - around time , then @xmath159 where @xmath160 is the matter energy density parameter at @xmath161 , @xmath162 is the hubble constant at turn - around time and @xmath163 . eqs . ( [ background ] ) and ( [ overdensity ] ) can be then written as @xmath164\end{aligned}\ ] ] where @xmath165 . with the boundary condition @xmath166 and @xmath167 , @xmath60 is uniquely determined by eqs . ( [ x ] ) and ( [ y ] ) , given the function form of @xmath160 . for constant @xmath8 , we have @xmath168 and @xmath60 obtained from eqs . ( [ x ] ) and ( [ y ] ) can be well fitted by @xmath169 the linear overdensity @xmath97 at @xmath46 can also be calculated by evolving eqs . ( [ x ] ) and ( [ y ] ) . at early time , the perturbation is linear @xmath170 once @xmath97 is known at some @xmath171 , then it is easily obtained at an arbitrary time @xmath172 where @xmath90 is the growth factor that can be calculated by using eq . ( [ g ] ) the q - component does not participate directly in cluster formation , but it alters the background cosmic evolution . the linear perturbation equation can be written as : @xmath173 where @xmath153 is the scale factor of the universe , dot means derivative with respect to physical time @xmath174 , @xmath175 , @xmath71 and @xmath176 are the density and overdensity of the matter respectively . the background evolution equations in a flat universe are : @xmath177 where @xmath70 is the energy density of the q - component and @xmath178 is the equation - of - state of the q - component . now , we can define a matter energy density parameter @xmath179 so that : @xmath180 from eqs . ( [ friedman1 ] ) , ( [ friedman2 ] ) and ( [ omega ] ) , we can get : @xmath181 by using eqs . ( [ friedman1 ] ) , ( [ omega ] ) and conservation of stress energy @xmath182 : @xmath183 by using eqs . ( [ linear ] ) , ( [ omega ] ) and ( [ dotaomega ] ) we get : @xmath184=\frac 3 2 \omega\ ] ] the growth index @xmath185 is defined as : @xmath186 by using eqs . ( [ aomega ] ) and ( [ lineara ] ) , we are able to get the equation for @xmath185 in terms of @xmath187 : @xmath188 + f^2=\frac 3 2 \omega\ ] ] now , we introduce variable @xmath148 , so that @xmath189 , and eq . ( [ main ] ) becomes : @xmath190 for slowly varying equation - of - state ( @xmath191 ) , we shall get : @xmath192 by following a similar derivation , we found that eq . ( [ alpha_f ] ) is valid for an open universe if we set @xmath193 . hence , @xmath148 is weakly dependent on @xmath4 . the result is @xmath194 for @xmath11cdm ( w=-1 ) and @xmath195 for ocdm .

the abundance of rich clusters is a strong constraint on the mass power spectrum . the current constraint can be expressed in the form @xmath0 where @xmath1 is the @xmath2 mass fluctuation on 8 @xmath3 mpc scales , @xmath4 is the ratio of matter density to the critical density , and @xmath5 is model - dependent . in this paper , we determine a general expression for @xmath5 that applies to any models with a mixture of cold dark matter plus cosmological constant or quintessence ( a time - evolving , spatially - inhomogeneous component with negative pressure ) including dependence on the spectral index @xmath6 , the hubble constant @xmath7 , and the equation - of - state of the quintessence component @xmath8 . the cluster constraint is combined with cobe measurements to identify a spectrum of best - fitting models . the constraint from the evolution of rich clusters is also discussed . epsf _ subject headings : _ cosmology : theory - dark matter - large - scale structure of universe - galaxies : clusters : general - x - rays : galaxies

historically , capillarity problems attracted attention because of their applications in physics , for instance in the study of wetting phenomena @xcite , energy minimizing drops and their adhesion properties @xcite , as well as because of their connections with minimal surfaces , see e.g. @xcite and references therein . in this paper we are interested in the study of the evolution of a droplet flowing on a horizontal hyperplane under curvature driven forces with a prescribed ( possibly nonconstant ) contact angle . although there are results in the literature describing the static and dynamic behaviours of droplets @xcite , not too much seems to be known concerning their mean curvature motion . various results have been obtained for mean curvature flow of hypersurfaces with dirichlet boundary conditions @xcite and zero - neumann boundary condition @xcite . it is also worthwhile to recall that , when the contact angle is constant , the evolution is related to the so - called mean curvature flow of surface clusters , also called space partitions ( networks , in the plane ) : in two dimensions local well - posedness has been shown in @xcite , and authors of @xcite derived global existence of the motion of grain boundaries close to an equilibrium configuration . see also @xcite for related results . in higher space dimensions short time existence for symmetric partitions of space into three phases with graph - type interfaces has been derived in @xcite . very recently , authors of @xcite have shown short time existence of the mean curvature flow of three surface clusters . if we describe the evolving droplet by a set @xmath0 @xmath1 the time , where @xmath2 is the upper half - space in @xmath3 the evolution problem we are interested in reads as @xmath4 where @xmath5 is the normal velocity and @xmath6 is the mean curvature of @xmath7 supplied with the contact angle condition on the contact set ( the boundary of the wetted area ) : @xmath8 where @xmath9 is the outer unit normal to @xmath10 at @xmath11 and @xmath12 $ ] is the cosine of the prescribed contact angle . we do not allow @xmath13 to be tangent to @xmath11 i.e. we suppose @xmath14 on @xmath15 for some @xmath16.$ ] following @xcite , in appendix [ sec : short_time_existence ] we show local well - posedness of - . short time existence describes the motion only up to the first singularity time . in order to continue the flow through singularities one needs a notion of weak solution . concerning the case without boundary , there are various notions of generalized solutions , such as brakke s varifold - solution @xcite , the viscosity solution ( see @xcite and references therein ) , the almgren - taylor - wang @xcite and luckhaus - sturzenhecker @xcite solution , the minimal barrier solution ( see @xcite and references therein ) ; see also @xcite for other different approaches . in the present paper we want to adapt the scheme proposed in @xcite , and later extended to the notions of _ minimizing movement _ and _ generalized minimizing movement _ ( shortly gmm ) by de giorgi @xcite ( see also @xcite ) to solve - . let us recall the definition . [ def : general_gmm ] let @xmath17 be a topological space , @xmath18 $ ] be a functional and @xmath19 we say that @xmath20 is a generalized minimizing movement associated to @xmath21 ( shortly gmm ) starting from @xmath22 and we write @xmath23 if there exist @xmath24 and a diverging sequence @xmath25 such that @xmath26 ) = u(t)\quad \text{for any $ t\ge0,$}\ ] ] and the functions @xmath27 @xmath28 @xmath29 are defined inductively as @xmath30 for @xmath31 and @xmath32 if @xmath33 consists of a unique element it is called a minimizing movement starting from @xmath34 in the sequel , we take @xmath35 @xmath36 $ ] defined by @xmath37 where @xmath38 is the initial set , @xmath39 is the distance to @xmath40 and @xmath41 is the capillary functional . if @xmath42 ( hence when the term @xmath43 is not present ) , the weak evolution ( gmm ) has been studied in @xcite and @xcite , see also @xcite for the dirichlet case . further when no ambiguity appears we use @xmath44 to denote the gmm starting from @xmath45 after setting in section [ sec : preliminaries ] the notation , and some properties of finite perimeter sets , in section [ sec : capillarity_functionals ] we study the functional @xmath46 and its level - set counterpart @xmath47 including lower semicontinuity and coercivity , which will be useful in section [ sec : comparison_principles ] . in particular , the map @xmath48 is @xmath49-lower semicontinuous if and only if @xmath50 ( lemma [ lem : lsc_of_f0 ] ) . although we can also establish the coercivity of @xmath51 ( proposition [ prop : coercivity_of_the_capillarity_functional ] ) , compactness theorems in @xmath52 can not be applied because of the unboundedness of @xmath53 however , in theorem [ teo : unconstrained minimizer ] we prove that if @xmath38 is bounded and @xmath54 then there is a minimizer in @xmath55 of @xmath51 , and any minimizer is bounded . in lemma [ lem : behav_big_lambda ] we study the behaviour of minimizers as @xmath56 in proposition [ prop : minimizer_of_f_0 ] we show existence of constrained minimizers of @xmath46 , which will be used in the proof of existence of gmms and in comparison principles . in appendix [ sec : error_est ] we need to generalize such existence and uniform boundedness results to minimizers of functionals of type @xmath57 under suitable hypotheses on @xmath58 in section [ sec : density_estimates ] we study the regularity of minimizers @xmath51 ( theorem [ teo : regularity ] ) . we point out the uniform density estimates for minimizers of @xmath51 and constrained minimizers of @xmath46 ( theorem [ teo : lower_density_est ] and proposition [ prop : dens_est_for_cap ] ) , which are the main ingredients in the existence proof of gmms ( section [ sec : existence_of_gmm ] ) , and in the proof of coincidence with distributional solutions ( section [ sec : weak_curvature ] ) . in section [ sec : comparison_principles ] we prove the following comparison principle for minimizers of @xmath51 ( theorem [ teo : e_0_and_f_0 ] ) : _ if @xmath59 are bounded , @xmath60 , @xmath61 and @xmath62 then _ * there exists a minimizer @xmath63 of @xmath64 containing any minimizer of @xmath65 * there exists a minimizer @xmath66 of @xmath67 contained in any minimizer of @xmath68 if in addition @xmath69 then any minimizer @xmath70 and @xmath71 of @xmath67 and @xmath64 respectively , satisfy @xmath72 as a corollary , we show that if @xmath73 is a bounded minimizer of @xmath46 in the collection @xmath74 of all finite perimeter sets containing @xmath75 and if @xmath76 , then for any @xmath77 , a minimizer @xmath70 of @xmath51 satisfies @xmath78 ( proposition [ prop : boundedness_of_minimizers ] ) . in section [ sec : existence_of_gmm ] we apply the scheme in definition [ def : general_gmm ] to the functional @xmath79 : as in @xcite we build a locally @xmath80-hlder continuous generalized minimizing movement @xmath81 starting from a bounded set @xmath38 ( theorem [ teo : existence_of_gmm ] ) . moreover , using the results of section [ sec : comparison_principles ] , we prove that any gmm starting from a bounded set stays bounded . in general , for two gmms one can not expect a comparison principle ( for example in the presence of fattening ) . however , the notions of _ maximal _ and _ minimal _ gmms ( definition [ def : max_min_gmm ] ) are always comparable if the initial sets are comparable ( theorem [ teo : comp_princ_for_gmm ] ) . this requires regularity of minimizers of @xmath79 and @xmath82 see sections [ sec : almgren_taylor_wang_functional ] and [ sec : density_estimates ] . finally , in section [ sec : weak_curvature ] we prove that , under a suitable conditional convergence assumption and if @xmath83 our gmm solution is , in fact , a _ distributional solution _ to - . @xmath84 stands for the characteristic function of the lebesgue measurable set @xmath85 and @xmath86 denotes its lebesgue measure . the set of @xmath49-functions having bounded total variation in an open set @xmath87 is denoted by @xmath88 and @xmath89 given @xmath90 we denote by @xmath91 the _ perimeter _ of @xmath92 in @xmath93 i.e. @xmath94 by @xmath95 the essential boundary of @xmath96 and by @xmath97 the measure - theoretical exterior normal to @xmath92 at @xmath98 since lebesgue equivalent sets in @xmath99 have the same perimeter in @xmath93 we assume that any set @xmath100 we consider coincides with the set @xmath101 of points of density one , where @xmath102 is the ball of radius @xmath103 centered at @xmath104 recall that @xmath105 for simplicity , set @xmath106 we say that @xmath107 has locally finite perimeter in @xmath3 if @xmath108 for every bounded open set @xmath109 the collection of all sets of locally finite perimeter is denoted by @xmath110 we refer to @xcite for a complete information about @xmath52-functions and sets of finite perimeter . for a fixed nonempty @xmath38 set @xmath111 which is @xmath49-closed . given @xmath112 and @xmath113 let @xmath114 stand for the truncated cylinder in @xmath115 of height @xmath116 whose basis is an open ball @xmath117 centered at the origin of radius @xmath118 also set @xmath119 by ( * ? ? ? * theorem ii ) , for every @xmath120 the additive set function @xmath121 defined on the open sets @xmath122 extends to a measure @xmath123 defined on the borel @xmath124-algebra of @xmath53 moreover , @xmath125 is strongly subadditive , i.e. @xmath126 let @xmath99 be an open set with lipschitz boundary and @xmath127 we denote the interior and exterior traces of the set @xmath92 on @xmath15 respectively by @xmath128 and @xmath129 and we recall that @xmath130 moreover , the integration by parts formula holds @xcite : @xmath131 where @xmath132 is the outer unit normal to @xmath133 if @xmath134 is an open set with lipschitz boundary , then @xmath135 the trace set of @xmath136 on @xmath15 is denoted by @xmath137 with a slight abuse of notation we set @xmath138 note that @xmath139 in general , even if @xmath140 the traces @xmath141 are in @xmath142 but not in @xmath143 for instance , if @xmath144 and @xmath145,$ ] then @xmath146 whereas @xmath147 in lemma [ lem : betacond ] we show that @xmath148 for any @xmath140 provided that @xmath99 is a half - space . from now on we fix @xmath149 we often identify @xmath150 with @xmath151 so that @xmath152 means @xmath153 and @xmath154 denotes the projection @xmath155 the following lemma shows that the @xmath156-norm of the trace of @xmath157 is controlled by @xmath158 [ lem : betacond ] for any @xmath157 and for any @xmath159 the relations @xmath160 hold . in particular , @xmath161 the last inequality of is immediate . the first inequality is enough to be shown for @xmath162 * step 1 . * if @xmath163 is locally lipschitz , then follows from the divergence theorem . indeed , suppose that @xmath164 is compact . since @xmath165 we have @xmath166 hence nonnegativity of @xmath163 implies that @xmath167 if @xmath164 is not compact , we use @xmath168 in instead of @xmath169 where @xmath170 is lipschitz , linear in @xmath171,$ ] @xmath172 in @xmath173 $ ] and @xmath174 in @xmath175 now follows from the monotone convergence theorem . in particular , when @xmath176 we have @xmath177 * step 2 . * assume that @xmath178 for some open set @xmath179 consider a sequence @xmath180 of nonnegative locally lipschitz functions converging @xmath181-almost everywhere to @xmath163 on @xmath15 such that @xmath182 and @xmath183 by fatou s lemma and step 1 we get @xmath184 * step 3 . * assume that @xmath185 where @xmath186 is a measurable set . fix @xmath187 a closed set @xmath188 such that @xmath189 and a decreasing sequence @xmath190 of open sets such that @xmath191 using step 2 for every @xmath192 we establish @xmath193 since @xmath194 there exists @xmath195 such that for any @xmath196 one has @xmath197 thus , for any @xmath198 and @xmath192 we have @xmath199 this and imply @xmath200 in addition @xmath201 from - and the inequality @xmath202 we obtain @xmath203 and arbitrariness of @xmath204 implies the assertion . * step 4 . * if @xmath205 @xmath206 where @xmath207 are disjoint measurable subsets of @xmath11 then the result follows from step 3 . finally , if @xmath159 is any nonnegative function , as in the proof of step 2 , approximation of @xmath163 with an increasing sequence of step functions and fatou s lemma conclude the proof . from lemma [ lem : betacond ] it follows that @xmath157 if and only if @xmath208 if @xmath209 then its trace belongs to @xmath143 indeed , it is well - known that @xmath210 @xmath211 in particular , @xmath212 for a.e . @xmath213 and @xmath214 using with @xmath215 for a.e . @xmath213 and @xmath216 we get @xmath217 and we obtain @xmath218 notice that for every @xmath159 one has also @xmath219 the following lemma is the analog to comparison theorem in @xcite and any closed convex set @xmath220 the inequality @xmath221 holds ; equality occurs if and only if @xmath222 . ] . [ ambrosio_lemma ] let @xmath157 and @xmath223 be a closed half - space such that @xmath224 then @xmath225 note that if @xmath226 then follows from @xcite . so we assume that @xmath227 translating if necessary we may suppose that @xmath228 let @xmath229 denote the @xmath230-dimensional subspace orthogonal to @xmath231 which is spanned by @xmath232 and @xmath233 take a unit vector @xmath234 such that @xmath235 and @xmath236 and let @xmath237 be the open halfspace of @xmath115 such that @xmath238 notice that by construction , @xmath239 therefore @xmath240\\ = & p(e , l)-p(e\cap h).\end{aligned}\ ] ] hence , we need just to show @xmath241 since @xmath242 we have @xmath243 and @xmath244 where @xmath245 is the interior of @xmath246 applying lemma [ lem : lower_bound_of_f0 ] below with @xmath247 @xmath248 @xmath249 and @xmath250 the orthogonal projection over @xmath251 ( so that @xmath252 ) , using also @xmath100 we get @xmath253 now follows from - and the inequality @xmath254 [ muhim_corollary ] let @xmath255 be a closed convex set such that @xmath256 @xmath181-a.e . on @xmath257 then @xmath258 for every @xmath259 since @xmath255 is convex , we can choose countably many @xmath260 dense in @xmath261 such that @xmath262 where @xmath263 is the closed half space whose outer unit normal is @xmath264 then an inductive application of lemma [ ambrosio_lemma ] and the lower semicontinuity of perimeter imply the assertion . let @xmath265 the capillary functional @xmath266 and its `` level set '' version @xmath267 are defined as @xmath268 and @xmath269 respectively . note that @xmath270 is convex , @xmath271 for any @xmath272 , and @xmath273 for any @xmath274 moreover , when @xmath275 by the functional @xmath46 is nonnegative , and the same holds for @xmath270 as by - one has @xmath276 the functional @xmath270 will be useful for the comparison principles ( section [ sec : comparison_principles ] ) . if @xmath277 then @xmath278 and hence , for any @xmath272 one has @xmath279where @xmath280 is the @xmath281-st component of the vector measure @xmath282 hence , the functional @xmath270 can also be represented as @xmath283 the next lemma is a localized version of ( * ? ? ? * lemma 4 ) , which is needed to prove coercivity of @xmath46 and @xmath270 and will be frequently used in the proofs ( see for example the proofs of theorem [ teo : unconstrained minimizer1 ] and theorem [ teo : lower_density_est ] ) . [ lem : lower_bound_of_f0 ] assume that @xmath50 and @xmath274 then for any open set @xmath284 with @xmath285 and @xmath286\cap \omega \cap { \partial}^*e\big)=0\ ] ] the inequality @xmath287\ ] ] holds . let us first show that if @xmath288 has locally finite perimeter in @xmath3 then @xmath289 set @xmath290 for any @xmath291 take an open set @xmath292 such that @xmath293 and @xmath294 since @xmath295 one has @xmath296 let @xmath297 denote the ball of radius @xmath298 centered at the origin . recall that for any @xmath299 the following estimate @xcite holds : @xmath300 then using @xmath301 we establish @xmath302 now letting @xmath303 we get @xmath304 and follows from letting @xmath305 we have @xmath306 where in the second equality we used . moreover , from with @xmath307 we get @xmath308 now , using lemma [ lem : betacond ] with @xmath163 replaced with @xmath309 from and we obtain @xmath310 finally , adding the identities @xmath311 and using we deduce @xmath312 this relation yields . [ prop : coercivity_of_the_capillarity_functional ] if @xmath313 @xmath181-a.e . on @xmath15 for some @xmath314,$ ] then @xmath315 moreover , if @xmath316 for some @xmath317,$ ] then @xmath318 the inequality @xmath319 follows from lemma [ lem : lower_bound_of_f0 ] with @xmath320 moreover , it is immediate to see that @xmath321 now follows from the inequalities @xmath322 for a.e . @xmath323 and @xmath324 for a.e . @xmath325 from - , and by ( * ? ? * remark 2.14 ) , possibly after extending @xmath20 to @xmath326 outside @xmath53 [ rem : posit_negat_part ] from the proof of proposition [ prop : coercivity_of_the_capillarity_functional ] it follows that if @xmath327 then holds for any @xmath159 with @xmath328 if @xmath329 is valid whenever @xmath330 [ emtyset_minimizer ] if @xmath331 on a set of infinite @xmath181-measure , then @xmath46 is unbounded from below . note also that if @xmath275 then @xmath332 is the unique minimizer of @xmath46 in @xmath333 indeed , clearly , @xmath334 if there were a minimizer @xmath335 of @xmath82 there would exist @xmath113 such that @xmath336 now since @xmath337 by @xcite we have @xmath338 a contradiction . [ lem : lsc_of_f0 ] assume that @xmath339 then the functionals @xmath46 and @xmath270 are @xmath49-lower semicontinuous if and only if @xmath340 assume that @xmath340 in this case the lower semicontinuity of @xmath46 is proven in ( * ? ? ? * lemma 2 ) . let us prove the lower semicontinuity of @xmath341 take @xmath342 such that @xmath343 in @xmath344 by we may assume that @xmath345 as @xmath346 for a.e . @xmath347 then using the nonnegativity of summands , the lower semicontinuity of @xmath46 and fatou s lemma in we establish @xmath348 now assume that @xmath349 i.e. the set @xmath350 has positive @xmath181-measure . let for some @xmath351 the set @xmath352 satisfy @xmath353 by lusin s theorem , for any @xmath354 there exists @xmath355 such that @xmath356 and @xmath357 let @xmath358 be so large that @xmath359 and choose an open set @xmath360 of finite perimeter such that @xmath361 define the sequence of sets @xmath362 clearly , @xmath363 in @xmath49 as @xmath364 then , indicating by @xmath365 the perimeter of @xmath366 in @xmath151 from the relations @xmath367 we establish @xmath368 since @xmath369 one has also @xmath370 hence @xmath46 and @xmath270 are not @xmath49-lower semicontinuous . finally , let for some @xmath371 the set @xmath372 satisfy @xmath373 again by lusin s theorem for any @xmath374 there exists @xmath375 such that @xmath376 and @xmath377 we may choose @xmath378 so large that @xmath379 let us choose an open set @xmath380 of finite perimeter such that @xmath381 now define the sequence of sets @xmath382 clearly , @xmath383 in @xmath49 as @xmath364 then from the relations @xmath384 we establish @xmath385 in particular , @xmath386 [ rem : lsc_cap_f_omega_bounded ] if @xmath99 is an arbitrary bounded open set with lipschitz boundary and @xmath50 , then the lower semicontinuity of @xmath46 is a consequence of ( * ? ? ? * theorem 3.4 ) . in this case @xmath46 is bounded from below by @xmath387 hence again fatou s lemma and yield lower semicontinuity of @xmath341 in the sequel , for a given nonempty set @xmath388 @xmath389 stands for the distance function from the boundary of @xmath390 in @xmath391 @xmath392 the function @xmath393 is called the _ signed distance function _ from @xmath390 in @xmath99 negative inside @xmath394 the distance from the empty set is assumed to be equal to @xmath395 notice that for @xmath396 @xmath397 @xmath398 provided @xmath399 moreover , we assume @xmath400 whenever @xmath401 given @xmath402 @xmath38 and @xmath28 recalling the definition of @xmath46 in , we define the _ capillary almgren - taylor - wang - type _ functional @xmath403 $ ] with contact angle @xmath163 , as @xmath404 so that @xmath405 whenever @xmath406 we always suppose that @xmath407 and in this section we assume that @xmath408 : -1\le \beta \le 1 - 2\kappa ~\text{${\mathcal{h}}^n$-a.e on}~{\partial}\omega . \end{cases}\ ] ] hence , there exists a cylinder @xmath409 containing @xmath255 whose basis is an open ball @xmath410 of radius @xmath411 and height @xmath412 define @xmath413 where @xmath414 the proof of the next result is essentially postponed to appendix [ sec : error_est ] , since the main idea does not differ too much from @xcite . [ teo : unconstrained minimizer ] suppose that holds . then the minimum problem @xmath415 has a solution @xmath70 . moreover , any minimizer is contained in @xmath416 . let @xmath417 and @xmath418,\quad { \mathcal{v}}(e):=\int_e fdx.\ ] ] then @xmath419 satisfies hypothesis [ hyp:2 ] and by remark [ rem : muhim_coef ] @xmath420 now the proof directly follows from theorem [ teo : unconstrained minimizer1 ] . if @xmath421 then has a unique solution @xmath422 moreover , for some choices of @xmath407 and @xmath423 the empty set solves . for example , let @xmath424 be the ball centered at @xmath425 such that @xmath426 if @xmath427 then as in @xcite , one can show that @xmath428 is the unique minimizer of @xmath429 [ rem : unconst_min ] let @xmath430 minimize @xmath51 in @xmath431 then @xmath430 is an unconstrained minimizer , i.e. @xmath432 indeed , let @xmath70 be any minimizer of @xmath433 clearly , @xmath434 on the other hand , by theorem [ teo : unconstrained minimizer ] @xmath435 and by minimality of @xmath430 in @xmath416 we have @xmath436 which implies . recalling remark [ emtyset_minimizer ] and definition of @xmath437 we have also the following result . [ prop : minimizer_of_f_0 ] under assumptions the constrained minimum problem @xmath438 has a solution . in addition , any minimizer @xmath73 satisfies @xmath439 where @xmath440 is given by , and @xmath73 is also a solution of @xmath441 set @xmath442,\quad { \mathcal{v}}(e):= \begin{cases } 0 & \text{if}\,\ , e\in { { \mathcal e}}(e_0),\\ + \infty & \text{if}\,\ , e\in bv(\omega,\{0,1\})\setminus { { \mathcal e}}(e_0 ) . \end{cases}\ ] ] then @xmath419 satisfies hypothesis [ hyp:2 ] and @xmath420 now existence of a minimizer @xmath73 of @xmath46 in @xmath437 and the inclusion @xmath443 follow from theorem [ teo : unconstrained minimizer1 ] . to show the last statement we observe that the inclusion @xmath77 implies @xmath444 hence the minimality of @xmath73 yields the inequality @xmath445 for any @xmath446 solutions of will be called constrained minimizers of @xmath46 in @xmath447 [ exa : decreas ] suppose that @xmath448 is a closed convex set so that @xmath449 @xmath181-a.e . on @xmath257 then for every @xmath450)$ ] the set @xmath255 is a constrained minimizer of @xmath46 in @xmath447 indeed , by corollary [ muhim_corollary ] @xmath258 for all @xmath451 therefore @xmath452 the following lemma shows the behaviour of @xmath70 as @xmath56 [ lem : behav_big_lambda ] assume and @xmath453 then any minimizer @xmath70 satisfies : * @xmath454 * @xmath455 * @xmath456 * if @xmath457 then @xmath458 as @xmath459 where @xmath460 denotes kuratowski convergence @xcite . \a ) we have @xmath461 moreover , from @xmath462 and we get @xmath463 hence @xmath464 recall from theorem [ teo : unconstrained minimizer ] that @xmath435 for all @xmath465 hence , by compactness , from every diverging sequence @xmath466 we can select a subsequence @xmath467 such that @xmath468 for some @xmath469 from we deduce that @xmath470 and thus , since @xmath471 and by assumption @xmath472 we get @xmath473 now arbitrariness of @xmath25 implies a ) . \b ) clearly , @xmath474 for all @xmath465 then by a ) and by the @xmath49-lower semicontinuity of @xmath46 ( lemma [ lem : lsc_of_f0 ] ) we establish @xmath475 and b ) follows . \c ) follows from b ) and nonnegativity of @xmath476 since @xmath477 = 0.\ ] ] d ) it suffices to show that every diverging sequence @xmath25 has a subsequence @xmath478 such that @xmath479 choose any sequence @xmath480 by compactness of closed sets in kuratowski convergence @xcite , there exists a closed set @xmath481 such that up to a not relabelled subsequence @xmath482 as @xmath483 let us show first that @xmath484 take any @xmath485 ; we may suppose that @xmath486 since @xmath487 is closed , there exists a ball @xmath488 such that @xmath489 since @xmath490 as @xmath491 we have @xmath492 for @xmath493 large enough . therefore , @xmath494 and by a ) and lower semicontinuity , @xmath495 this yields @xmath496 and thus @xmath497 now suppose that there exists @xmath498 then there exists @xmath298 such that @xmath499 since @xmath500 there exists @xmath501 such that @xmath502 choose @xmath503 so large that @xmath504 and @xmath505 where @xmath506 is defined in . by proposition [ prop : uniform_l_infty_est ] below , we have @xmath507 on the other hand , by construction , @xmath508 which leads to a contradiction . this yields @xmath509 , and d ) follows . in this section we assume that @xmath510 : ~ \|\beta\|_\infty \le 1 - 2\kappa . \end{cases}\ ] ] define @xmath511 and @xmath512 where @xmath513 is the relative isoperimetric constant for the ball . the aim of this section is to prove the following uniform density estimates for minimizers of @xmath514 needed to prove regularity of minimizers ( theorem [ teo : regularity ] ) and proposition [ prop : uniform_estimate ] . [ teo : lower_density_est ] assume that @xmath255 and @xmath163 are as in and @xmath515 is a minimizer of @xmath516 then either @xmath428 or @xmath517 @xmath518 for every @xmath519 and @xmath520 in particular , @xmath521 we postpone the proof after several auxiliary results . first we show a weaker version of theorem [ teo : lower_density_est ] ; the difference stands in that proposition [ prop : lower_density_est_weak ] holds for @xmath522 and @xmath523 depends on @xmath524 whereas theorem [ teo : lower_density_est ] is valid for @xmath525 and @xmath526 is independent of @xmath527 [ prop : lower_density_est_weak ] under the assumptions of theorem [ teo : lower_density_est ] , setting @xmath528 for any nonempty @xmath529 @xmath530 and @xmath531 the density estimates - hold . for completeness we give the full proof of the proposition using the methods of @xcite . we recall that one could also employ the density estimates for almost minimizers of the capillary functional ( see for instance ( * ? ? ? * lemma 2.8 ) ) . set @xmath532 and fix @xmath533 let @xmath534 be the ball of radius @xmath535 centered at @xmath536 we can choose @xmath537 such that @xmath538 first we show that @xmath70 satisfies @xmath539 comparing @xmath540 with @xmath541 for a.e . @xmath542 we establish @xmath543 sending @xmath544 we get @xmath545 by theorem [ teo : unconstrained minimizer ] @xmath435 and thus , since @xmath546 for any @xmath547 @xmath548 moreover , using for @xmath549 we get : @xmath550 now by the isoperimetric inequality , @xmath551 set @xmath552 then @xmath553 is absolutely continuous , @xmath554 @xmath555 for all @xmath103 and @xmath556 for a.e . @xmath557 consequently , and give @xmath558 since @xmath559 and @xmath560 from the last inequality we obtain @xmath561 integrating we get the lower volume density estimate @xmath562 let us prove the upper volume density estimate in . since @xmath563 if @xmath564 the inequality @xmath565 is trivial . so assume that @xmath566 since @xmath567 arguing as in the proof of we get @xmath568 from the isoperimetric inequality , , and also , it follows that @xmath569 repeating the same arguments as before we establish @xmath570 let us now show . from we get @xmath571 r^{n}\end{aligned}\ ] ] for a.e @xmath557 since @xmath572 is a nonnegative measure , this inequality holds for all @xmath557 this proves the upper perimeter estimate in . the lower perimeter density estimate in follows from and the relative isoperimetric inequality ( see for example @xcite ) . [ teo : regularity ] assume that @xmath255 and @xmath163 satisfy . then any nonempty minimizer @xmath70 is open in @xmath115 and @xmath573 is an @xmath574-dimensional manifold of class @xmath575 for a suitable @xmath576 , and @xmath577 for all @xmath578 moreover , if @xmath579 then * @xmath580 * @xmath581 is a set of finite perimeter in @xmath15 and @xmath582 where @xmath583 denotes the boundary of @xmath584 in @xmath133 moreover , if @xmath585 then @xmath586 * there exists a relatively closed set @xmath587 with @xmath588 such that in a neighborhood of any @xmath589 the set @xmath590 is a @xmath591-manifold with boundary , and @xmath592 since @xmath70 is a minimizer of @xmath51 in every ball @xmath593 we can apply ( * ? ? ? * theorem 5.2 ) to prove that @xmath70 is open and @xmath573 is @xmath575 with @xmath577 for all @xmath578 moreover , if @xmath579 by the remaning assertions follow from ( * ? ? ? * lemma 2.16 , theorem 1.10 ) . [ rem : uncons_dens_est ] ( compare with ( * ? ? ? * remark 1.4 ) and @xcite . ) \a ) assume that @xmath594 and @xmath103 are such that @xmath595 then @xmath596 in @xmath597 and from we get @xmath598 then proceeding as in the proof of proposition [ prop : lower_density_est_weak ] we get @xmath599 moreover , from it follows that @xmath600r^n.\ ] ] \b ) similarly , if @xmath601 and @xmath602 then @xmath599 observe that in both cases @xmath537 _ need not be _ in @xmath603 and the assumption @xmath519 is not necessary . the following proposition is the analog of ( * ? ? ? * lemma 2.1 ) and ( * ? ? ? * proposition 3.2.1 ) . [ prop : uniform_l_infty_est ] assume that @xmath255 and @xmath163 are as in and @xmath515 is a minimizer of @xmath516 then @xmath604 let @xmath605 suppose by contradiction that there exist @xmath187 @xmath407 and @xmath606 such that @xmath607 consider first the case @xmath608 by regularity of @xmath70 ( theorem [ teo : regularity ] ) we may assume that @xmath609 note that @xmath610 where @xmath611 @xmath612 since @xmath613 and @xmath614 for any @xmath615 from we establish @xmath616\rho^n.\ ] ] this and remark [ rem : uncons_dens_est ] ( a ) yield is of order @xmath617 in general , we can not apply it with @xmath618 @xmath619\rho^n,\ ] ] or equivalently , recalling the definition of @xmath620 @xmath621 which is a contradiction . a similar contradiction is obtained when @xmath622 we repeat the same procedures of the proof of proposition [ prop : lower_density_est_weak ] with improved estimates for the volume term of @xmath433 let @xmath623 @xmath624 fix @xmath625 and choose @xmath626 such that @xmath627 from it follows @xmath628 therefore , using the obvious inequality @xmath629 from we establish that @xmath630 since @xmath631 and @xmath632 similarly to from we deduce @xmath633 by the definition of @xmath634 one has @xmath635 thus , @xmath636 integrating this differential inequality we get the lower volume density estimate in . let us prove the upper volume density estimate in . due to we may suppose that @xmath637 as above one can estimate @xmath39 in @xmath638 as follows : @xmath639 since @xmath640 in @xmath641 plugging in and proceeding as above we establish @xmath642 from which the upper volume density estimates in follows . the proof of is exactly the same as the proof of perimeter density estimates in proposition [ prop : lower_density_est_weak ] . finally , is a standard consequence of a covering argument . let us prove the following @xmath643-estimate for the minimizers of @xmath514 the analog of ( * ? ? ? * lemma 1.5 ) and ( * ? ? ? * proposition 3.2.3 ) . notice carefully the exponent @xmath644 of @xmath645 in . [ prop : uniform_estimate ] assume that @xmath255 and @xmath163 satisfy and the uniform volume density estimates holds for @xmath527 then for any minimizer @xmath70 of @xmath646 the estimate @xmath647 holds , where @xmath648 and @xmath649 is the constant in besicovitch covering theorem . set @xmath650 by chebyshev inequality @xmath651 let us estimate @xmath652 since @xmath255 is bounded , by besicovitch s covering theorem there exist at most countably many balls @xmath653 @xmath654 such that any point of @xmath655 belongs to at most @xmath649 balls , @xmath656 and @xmath657 since the balls @xmath658 cover @xmath659 by the density estimates and the relative isoperimetric inequality we get @xmath660 therefore @xmath661 now follows from the estimates for @xmath662 and from @xmath599 a specific choice of @xmath663 will be made in the proof of theorem [ teo : existence_of_gmm ] . we conclude this section with a proposition about the regularity of minimizers of @xmath664 [ prop : dens_est_for_cap ] assume that @xmath255 and @xmath163 satisfy and there exist @xmath665 such that for every @xmath666 and @xmath667 the inequalities @xmath668 hold . let @xmath73 be a constrained minimizer of @xmath46 in @xmath447 then for every @xmath669 and @xmath670 @xmath671 in particular , @xmath672 let @xmath673 and @xmath667 be such that @xmath674 where @xmath675 we start with the upper volume density estimate in . we may suppose @xmath676 since the case @xmath677 is trivial . using @xmath678 as in we establish @xmath679 adding @xmath680 to both sides and proceeding as in we get @xmath681 and hence as in the proof of theorem [ teo : lower_density_est ] @xmath682 this implies the upper volume density estimate in . the lower volume density estimate is a little delicate , since in general we can not use the set @xmath683 as a competitor since it need not belong to @xmath447 if @xmath684 then @xmath666 and , hence , using @xmath685 and the lower volume density estimate for @xmath255 we establish @xmath686 if @xmath687 and @xmath688 then we may use comparison set @xmath689 and as in the proof of we obtain @xmath690 suppose @xmath691 since one can extend to @xmath692 $ ] by continuity , if @xmath693 then @xmath694 let @xmath695 and @xmath696 be such that @xmath697 then using @xmath698 the lower density estimate for @xmath255 and @xmath699 we obtain @xmath700 now the lower perimeter estimate follows from the volume density estimates and the relative isoperimetric inequality . the upper perimeter estimate is obtained from : @xmath701 finally , the relation @xmath702 is a consequence of the density estimates together with a covering argument the main result of this section is the following comparison between minimizers of @xmath433 [ teo : e_0_and_f_0 ] assume that @xmath703 @xmath704 satisfy . suppose that @xmath60 and @xmath705 then * there exists a minimizer @xmath706 of @xmath64 containing any minimizer of @xmath65 * there exists a minimizer @xmath707 of @xmath67 contained in any minimizer of @xmath708 if in addition @xmath709 then all minimizers @xmath70 and @xmath71 of @xmath67 and @xmath64 respectively satisfy @xmath710 we do not exclude the case that either @xmath70 or @xmath71 is empty . [ rem : exis_max_min ] for any @xmath524 @xmath163 satisfying , using theorem [ teo : e_0_and_f_0 ] with @xmath711 and @xmath712 we establish the existence of unique minimizers @xmath66 and @xmath713 of @xmath514 such that any other minimizer @xmath70 satisfies @xmath714 [ def : exis_max_min ] we call @xmath713 and @xmath66 the maximal and minimal minimizer of @xmath51 respectively . before proving theorem [ teo : e_0_and_f_0 ] we need the following observations . given @xmath163 satisfying , @xmath715 @xmath716 and @xmath717 @xmath718 a.e . in @xmath719 define the convex functional @xmath720)\to ( -\infty,+\infty],$ ] a sort of level - set capillary almgren - taylor - wang - type functional , as @xmath721 set @xmath722 where @xmath414 by example [ exam:1 ] the functional @xmath723,\qquad{\mathcal{v}}(e):=\int_evdx\ ] ] satisfies hypothesis [ hyp:2 ] . thus , by theorem [ teo : unconstrained minimizer1 ] the functional @xmath724 has a minimizer , and every minimizer @xmath725 satisfies @xmath726 notice that by and , @xmath727),\ ] ] which yields that @xmath728 is a minimizer of @xmath729 in @xmath730).$ ] the following remark is in the spirit of ( * ? ? ? * section 1 ) . [ rem : level_sets_of_minimizers ] from it follows that @xmath731)$ ] is a minimizer of @xmath729 in @xmath730)$ ] if and only if @xmath732 is a minimizer of @xmath729 for a.e . @xmath733.$ ] indeed , let for some @xmath731)$ ] the function @xmath732 be a minimizer of @xmath729 for a.e . @xmath733.$ ] then for any @xmath734)$ ] and for a.e . @xmath733 $ ] one has @xmath735 therefore , @xmath736 conversely , if @xmath731)$ ] is a minimizer of @xmath737 then for a.e . @xmath733 $ ] one has @xmath738 hence , from it follows that @xmath739 for a.e . @xmath733.$ ] in particular , if @xmath731)$ ] is a minimizer of @xmath737 then by @xmath740 for a.e . @xmath741,$ ] i.e. @xmath742 a.e . in @xmath743 hence , @xmath744 ) } { { \mathcal b}_{\beta}}(u , v , c)= \min\limits_{\begin{substack } u\in bv(\omega,[0,1]),\\ \text{$u=0 $ a.e . in $ \omega\setminus { \ensuremath{c_{{\mathcal{r}}_1(c , v)}^{h}}}$ } \end{substack } } { { \mathcal b}_{\beta}}(u , v , c).\ ] ] [ lem : rel_min_of atw_and_f ] let @xmath524 @xmath163 satisfy , and @xmath440 be defined as in . then @xmath70 is a minimizer of @xmath51 if and only if @xmath745 is a minimizer of @xmath746 where @xmath747 by we have @xmath748 now if @xmath70 minimizes @xmath514 we have @xmath435 ( theorem [ teo : unconstrained minimizer ] ) and thus , for any @xmath731)$ ] with @xmath742 a.e . in @xmath749 from - we deduce @xmath750 by @xmath745 is a minimizer of @xmath751 conversely , assume that @xmath745 is a minimizer of @xmath746 then by @xmath435 is a minimizer of @xmath51 in @xmath431 hence , by remark [ rem : unconst_min ] @xmath70 is a minimizer of @xmath433 [ prop : strong_max_pr ] assume that @xmath752 @xmath753 a.e . in @xmath99 and @xmath754 a.e . in @xmath755 suppose also that @xmath756 satisfy . let @xmath757)$ ] be minimizers of @xmath758 and @xmath759 respectively . then @xmath760 a.e . in @xmath53 adding the inequalities @xmath761 and @xmath762 and using @xmath763 we establish @xmath764 since @xmath765 and @xmath766 this inequality holds if and only if @xmath767 i.e. @xmath760 a.e . in @xmath53 [ prop : weak_comp_pr ] assume that @xmath752 @xmath768 a.e . in @xmath99 and @xmath754 a.e . in @xmath755 suppose also that @xmath756 satisfy . then : * there exists a minimizer @xmath769 of @xmath758 such that @xmath770 for any minimizer @xmath771 of @xmath772 * there exists a minimizer @xmath773 of @xmath759 such that @xmath774 for any minimizer @xmath775 of @xmath776 \a ) take @xmath777 since @xmath778 a.e . in @xmath93 by proposition [ prop : strong_max_pr ] any minimizer @xmath779)$ ] of @xmath780 and @xmath759 respectively , satisfies @xmath781 let @xmath782 by minimality , @xmath783 and since by remark [ rem : level_sets_of_minimizers ] @xmath784 a.e . in @xmath785 recalling we get @xmath786 by compactness , there exists @xmath787)$ ] such that , up to a ( not relabelled ) subsequence , @xmath788 in @xmath49 and a.e . in @xmath99 as @xmath789 then any minimizer @xmath771 of @xmath759 satisfies @xmath770 a.e . in @xmath53 it remains to show that @xmath769 is a minimizer of @xmath776 by we may consider only those @xmath731)$ ] with @xmath742 a.e . in @xmath790 as a competitor . in this case , the continuity of @xmath791 the minimality of @xmath792 and the lower semicontinuity of @xmath270 imply @xmath793 \b ) can be proven in a similar manner . let @xmath794 where @xmath795 and @xmath796 are defined as in . then by theorem [ teo : unconstrained minimizer ] any minimizer @xmath70 ( resp . @xmath797 ) of @xmath67 ( resp . @xmath64 ) is contained in the cylinder @xmath798 where @xmath799 set @xmath800 and @xmath801 since @xmath802 we have @xmath803 moreover , by there exists a cylinder @xmath804 such that @xmath754 in @xmath755 \a ) since @xmath768 and @xmath766 by proposition [ prop : weak_comp_pr ] b ) there exists a minimizer @xmath805 of @xmath759 such that any minimizer @xmath775 of @xmath758 satisfies @xmath806 by remark [ rem : level_sets_of_minimizers ] there exists @xmath807 such that @xmath808 is a minimizer of @xmath809 then , recalling the expression of @xmath810 by lemma [ lem : rel_min_of atw_and_f ] @xmath811 is a minimizer of @xmath708 moreover , if @xmath70 is a minimizer of @xmath812 then by lemma [ lem : rel_min_of atw_and_f ] @xmath745 is a minimizer of @xmath813 and by @xmath814 in particular , @xmath815 \b ) is analogous to a ) using proposition [ prop : weak_comp_pr ] a ) . the last assertion follows with the same arguments from lemma [ lem : rel_min_of atw_and_f ] and proposition [ prop : strong_max_pr ] , since implies that @xmath816 one useful case is when @xmath255 is a constrained minimizer of @xmath46 in @xmath817 in this case @xmath255 acts as a barrier for minimizers of @xmath433 [ prop : el_in_e0 ] assume that @xmath818 satisfy . let @xmath766 @xmath255 be a constrained minimizer of @xmath819 in @xmath437 and @xmath515 be a minimizer of @xmath820 then @xmath821 comparing @xmath70 with @xmath822 we get @xmath823 from the constrained minimality of @xmath255 we have @xmath824 i.e. @xmath825 adding these inequalities we obtain @xmath826 then the condition @xmath756 and yield that @xmath827 since @xmath828 outside @xmath829 the last inequality is possible only if @xmath830 i.e. @xmath821 proposition [ prop : el_in_e0 ] gives the following monotonicity principle . [ prop : el_monotone ] assume that @xmath831 satisfy , @xmath255 is a constrained minimizer of @xmath46 in @xmath437 such that @xmath832 and @xmath833 is a minimizer of @xmath834 for @xmath835 then @xmath836 for any @xmath837 moreover , every @xmath838 @xmath839 is also a constrained minimizer of @xmath46 in @xmath840 comparison between @xmath70 and @xmath841 gives @xmath842 similarly , for @xmath843 and @xmath844 we have @xmath845 adding the above inequalities and using we obtain @xmath846 by hypothesis @xmath472 according to proposition [ prop : el_in_e0 ] , @xmath847 thus @xmath848 in @xmath849 but since @xmath850 is possible only if @xmath851 i.e. @xmath852 to prove the final assertion take any set @xmath853 then using @xmath854 @xmath855 and @xmath856 we get @xmath857 moreover , since @xmath858 from we obtain @xmath859 i.e. @xmath860 [ prop : boundedness_of_minimizers ] suppose that @xmath255 and @xmath163 satisfy . * let @xmath861 be a constrained minimizer of @xmath46 in @xmath447 then every minimizer @xmath70 of @xmath51 satisfies @xmath862 * let @xmath861 be a bounded constrained minimizer of @xmath46 in @xmath863 then for every @xmath77 and for every minimizer @xmath70 of @xmath51 one has @xmath864 moreover , @xmath73 can be chosen such that @xmath865 \a ) by proposition [ prop : minimizer_of_f_0 ] @xmath73 is a constrained minimizer of @xmath866 in @xmath863 let @xmath867 be the maximal minimizer of @xmath868 ( definition [ def : exis_max_min ] ) . by proposition [ prop : el_in_e0 ] we have @xmath869 take any minimizer @xmath70 of @xmath433 since @xmath870 by theorem [ teo : e_0_and_f_0 ] a ) we have @xmath871 \b ) the proof of the first part is exactly the same as the proof of a ) . to prove the second part , we take any @xmath872 satisfying the hypotheses of proposition [ prop : dens_est_for_cap ] and containing @xmath527 by theorem [ prop : minimizer_of_f_0 ] there exists a constrained minimizer @xmath73 of @xmath46 in @xmath873 in particular , @xmath73 is bounded , and by proposition [ prop : dens_est_for_cap ] @xmath874 since @xmath875 we have @xmath865 consider the functional @xmath876 $ ] given by @xmath877 where @xmath878 $ ] denotes the integer part of @xmath879 for any @xmath880 we build the family of sets @xmath881 iteratively as follows : @xmath882 and @xmath883 @xmath884 is a minimizer of @xmath885 in @xmath886 notice that existence of minimizers follows from theorem [ teo : unconstrained minimizer ] . from now on , we omit the dependence on @xmath358 of @xmath887 and we use the notation @xmath888 [ teo : existence_of_gmm ] let @xmath255 and @xmath889 satisfy . then @xmath44 is nonempty , i.e. there exist a map @xmath81 and a diverging sequence @xmath890 such that @xmath891)\delta e(t)| = 0 , \qquad t\in [ 0,+\infty).\ ] ] moreover , every gmm @xmath892 starting from @xmath255 is contained in a bounded set depending only on @xmath255 and @xmath893 and belongs to @xmath894 in the sense that @xmath895 where @xmath896 and @xmath897 is defined in . if in addition @xmath898 then holds for any @xmath899 with @xmath900 finally , @xmath901)}{\mathcal{h}}^n { \mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\partial}^*e_{\lambda_j } ( [ \lambda_jt ] ) \overset{w^*}{\rightharpoonup } \nu_{e(t)}{\mathcal{h}}^n{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\partial}^*e(t ) \quad { \text{for all $ t\ge0 $ as $ \lambda_j\to+\infty.$}}\ ] ] given @xmath902 set @xmath903 then for @xmath904 the minimality of @xmath881 entails @xmath905 i.e. @xmath906 in particular , the sequence @xmath907 is nonincreasing and @xmath908 let @xmath213 and set @xmath909.$ ] then yields @xmath910 ) ) \le { { \mathcal c}_{\beta}}(e_\lambda([\lambda t]),\omega ) \le p(e_0).\ ] ] take @xmath911 @xmath912 and let @xmath407 be large enough that for some @xmath913 @xmath914 @xmath915,\quad k_0+n-1=[\lambda t_2],\ ] ] i.e. @xmath916 then @xmath917 since all @xmath918 @xmath919 satisfy uniform density estimates - ( theorem [ teo : lower_density_est ] ) , by proposition [ prop : uniform_estimate ] we have since a priori we do not know whether @xmath255 satisfies the density estimates , we can not start summing from @xmath920 @xmath921 for any @xmath922 the first sum can be estimated using : @xmath923 moreover , for any @xmath924 by @xmath925 and thus @xmath926 using and the nonnegativity of @xmath46 we get @xmath927 thus , from , and @xmath928 now take @xmath645 so large that @xmath929 so that proposition [ prop : uniform_estimate ] holds for @xmath930 from and we obtain @xmath931)\delta e_{\lambda } ( [ \lambda t_2 ] ) \big| \le & \frac{c_{n,\kappa}p(e_0)}{\kappa}\,\frac{n-2}{\lambda |t_2-t_1|^{1/2 } } + \frac{1}{\lambda } \,\frac{c_{n,\kappa}p(e_0)}{\kappa|t_2-t_1|^{1/2}}+ p(e_0 ) \ , |t_2-t_1|^{1/2}\\ \le & \theta(n,\kappa)p(e_0)\,|t_2-t_1|^{1/2}+ \frac{1}{\lambda}\,\frac{c_{n,\kappa}p(e_0)}{\kappa|t_2-t_1|^{1/2}}. \end{aligned}\ ] ] by proposition [ prop : boundedness_of_minimizers ] b ) there exists a constrained minimizer @xmath932 of @xmath46 in @xmath74 such that @xmath933 and @xmath934 by induction , we can show that @xmath935 for all @xmath936 consider now an arbitrary diverging sequence @xmath937 compactness and a diagonal process yield the existence of a subsequence ( still denoted by @xmath25 ) such that @xmath938)$ ] converges in @xmath49 to a set @xmath939 for any rational @xmath940 as @xmath483 if @xmath941 with @xmath942 letting @xmath943 in we get latexmath:[\[\label{gelder_condit } completeness of @xmath49 we can uniquely extend @xmath945 to a family @xmath946 preserving the hlder continuity in @xmath947 now we show . if @xmath948 @xmath949 in @xmath49 as @xmath483 if @xmath325 take any @xmath950 and @xmath951 such that @xmath952 by the choice of @xmath953 holds for @xmath954 and thus , using - we get @xmath955)\delta e(t)| \le & \limsup\limits_{j\to+\infty } |e_{\lambda_j}([\lambda_jt])\delta e_{\lambda_j}([\lambda_jt_{\varepsilon}])|\\ & + \limsup\limits_{j\to+\infty } |e_{\lambda_j}([\lambda_jt_{\varepsilon}])\delta e(t_{\varepsilon})| + \le & 2 \theta(n,\kappa)p(e_0 ) |t - t_{\varepsilon}|^{1/2 } < 2 \theta(n,\kappa)p(e_0)\sqrt{\varepsilon}.\end{aligned}\ ] ] therefore , letting @xmath956 we get . when @xmath472 for any @xmath957 choosing @xmath645 sufficiently large , from we obtain @xmath958 by lemma [ lem : behav_big_lambda ] a ) the last term on the right hand side converges to @xmath326 as @xmath56 hence letting @xmath959 in we get the @xmath960-hlder continuity of @xmath961 in @xmath962 now let us prove . we need to show that for any @xmath963 @xmath964 ) } \phi\cdot \nu_{e_{\lambda_j}([\lambda_jt])}\,d{\mathcal{h}}^n= \int_{{\partial}^*e(t ) } \phi\cdot\nu_{e(t)}\,d{\mathcal{h}}^n \quad\forall \phi\in c_c({\mathbb{r}}^{n+1},{\mathbb{r}}^{n+1}).\ ] ] if @xmath965 by the generalized divergence formula and by we have @xmath966 ) } \phi\cdot \nu_{e_{\lambda_j}([\lambda_jt])}\,d{\mathcal{h}}^n&= \lim\limits_{j\to+\infty } \int_{e_{\lambda_j}([\lambda_jt ] ) } { \mathop{\mathrm{div}}}\phi\,d{\mathcal{h}}^n \\ & = \int_{e(t ) } { \mathop{\mathrm{div}}}\phi\,d{\mathcal{h}}^n= \int_{{\partial}^*e(t ) } \phi\cdot\nu_{e(t)}\,d{\mathcal{h}}^n . \end{aligned}\ ] ] in general , we approximate @xmath967 uniformly with @xmath968 @xmath904 and use the previous result . finally , if @xmath969 then by construction and proposition [ prop : boundedness_of_minimizers ] b ) one has @xmath970)\subseteq e^+,$ ] where @xmath971 is a bounded minimizer of @xmath46 in @xmath972 therefore @xmath973 for all @xmath974 [ def : max_min_gmm ] let @xmath831 satisfy , and @xmath25 be a diverging sequence such that @xmath975)^*\qquad\forall t\ge0\ ] ] exist in @xmath976 where @xmath938)^*$ ] is the maximal minimizer of @xmath977 - 1)^*,\lambda_j)$ ] with @xmath978 ( definition [ def : exis_max_min ] ) . we call @xmath979 the _ maximal _ gmm associated to the sequence @xmath937 analogously , @xmath980)_*\qquad\forall t\ge0,\ ] ] obtained using the minimal minimizers @xmath938)_*$ ] of @xmath981 - 1)_*,\lambda_j)$ ] with @xmath982 is called the minimal gmm associated to the sequence @xmath937 observe that if @xmath961 is any gmm obtained by the sequence @xmath953 then according to the proof of theorem [ teo : existence_of_gmm ] ( possibly passing to nonrelabelled subsequences ) there exist the maximal gmm @xmath983 and the minimal gmm @xmath984 associated to @xmath937 now by remark [ rem : exis_max_min ] one has @xmath985 for all @xmath974 [ teo : comp_princ_for_gmm ] let @xmath986 satisfy with @xmath60 and @xmath987 if @xmath988 and @xmath989 are minimal gmms associated to a sequence @xmath953 then @xmath990 for all @xmath974 analogously , if @xmath979 and @xmath991 are maximal gmms associated to @xmath992 then @xmath993 for all @xmath974 since @xmath994 and @xmath766 by definition of @xmath995 and @xmath996 ( resp . @xmath997 and @xmath998 ) and by theorem [ teo : e_0_and_f_0 ] , we have @xmath999 ( resp . @xmath1000 ) which implies @xmath990 ( resp . @xmath993 ) for all @xmath974 from the proof of theorem [ teo : existence_of_gmm ] and propositions [ prop : el_in_e0 ] -[prop : el_monotone ] we get the following result ( compare with @xcite ) , that could be applied , for instance , to @xmath255 as in example [ exa : decreas ] . [ teo : homot_shrink ] let @xmath255 be a constrained minimizer of @xmath46 in @xmath437 such that @xmath453 then every maximal ( minimal ) gmm @xmath961 starting from @xmath255 satisfies @xmath1001 provided @xmath1002 applying propositions [ prop : el_in_e0 ] and [ prop : el_monotone ] inductively to maximal minimizers @xmath1003 of @xmath1004 we get @xmath1005 for all @xmath904 and @xmath465 hence , if @xmath1006 then @xmath1007)^*\subseteq e_\lambda([\lambda t'])^*.$ ] now the assertion of the theorem follows from . the arguments for minimal minimizers are the same . the aim of this section is to prove that under suitable assumptions gmm is in fact a distributional solution of - . let us start with the following a vector field @xmath1008 is called _ admissible _ if @xmath1009 on @xmath133 observe that if @xmath1010 is admissible , then for any @xmath1011 with @xmath291 sufficiently small , the vector field @xmath1012 is a @xmath1013-diffeomorphism that satisfies @xmath1014 @xmath1015 [ prop : first_var ] suppose that @xmath524 @xmath163 satisfy assumptions and let @xmath157 be bounded with @xmath1016 then @xmath1017 where @xmath1018 is the essential boundary of @xmath1019 on @xmath15 and @xmath1020 is the outer unit normal to @xmath1021 from ( * ? ? ? * theorem 17.5 ) @xmath1022 moreover , ( * ? ? ? * theorem 17.8 ) and the admissibility of @xmath1023 imply that @xmath1024 finally , since @xmath1019 is a set of finite perimeter in @xmath1025 again using ( * ? ? ? * theorem 17.8 ) we get @xmath1026 [ rem : w_curvature ] under assumptions and @xmath579 if @xmath70 is a minimizer of @xmath514 and if @xmath1027 is a @xmath1028-manifold with @xmath1029- rectifiable boundary , then the mean curvature @xmath1030 of @xmath1031 is equal to @xmath1032 indeed , using the tangential divergence formula for manifolds with boundary we have @xmath1033 where @xmath1034 is the outer unit conormal to @xmath1035 at @xmath1036 by minimality of @xmath529 we have @xmath1037 i.e. @xmath1038 this implies @xmath1039 and @xmath1040 notice that from the latter in particular , we get @xmath1041 accordingly for instance with theorem [ teo : regularity ] . remark [ rem : w_curvature ] motivates the following definition @xcite . [ def : dist_curv ] let @xmath1042 the function @xmath1043 is called _ distributional mean curvature of @xmath1044 _ if for every @xmath1045 the generalized tangential divergence formula holds : @xmath1046 given @xmath1047 and @xmath213 set @xmath1048 - 1)}(x ) & \text{if $ t\ge \frac1\lambda,$}\\ 0&\text{if $ t\in[0,\frac1\lambda).$ } \end{cases}\ ] ] [ rem : traces ] by theorem [ teo : regularity ] , @xmath1049))\in bv({\mathbb{r}}^n,\{0,1\}).$ ] the next result relates gmm with distributional solutions of - . [ teo : distr_sol ] let @xmath831 satisfy , @xmath472 @xmath1050 be a gmm starting from @xmath255 obtained along the diverging sequence @xmath25 . suppose that @xmath1051 ) ) \overset{w^*}{\rightharpoonup } { \mathcal{h}}^n{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}(\omega\cap{\partial}^ * e(t ) ) \quad \text { as $ j\to+\infty$ for a.e . $ t\ge0.$}\ ] ] then there exist a function @xmath1052 with @xmath1053 and a ( not relabelled ) subsequence such that @xmath1054 ) } \phi v_{\lambda_j}\,d{\mathcal{h}}^ndt = \int_0^{+\infty } \int_{\omega\cap { \partial}^*e(t ) } \phi v \,d{\mathcal{h}}^ndt,\ ] ] @xmath1055 ) } v_{\lambda_j}\,\nu_{e_{\lambda_j}([\lambda_jt])}\cdot \psi\ , d{\mathcal{h}}^ndt = \int_0^{+\infty}\int_{\omega\cap { \partial}^*e(t ) } v\ , \nu_{e(t)}\cdot \psi \,d{\mathcal{h}}^n dt\ ] ] for any @xmath1056 @xmath1057 where @xmath1058\mathfrak{b}(n)}{(\kappa/2)^{n+1}\omega_{n+1}}.$ ] moreover , @xmath1050 solves - with initial datum @xmath255 in the following sense : * for a.e . @xmath1 the set @xmath1059 has distributional mean curvature @xmath1060 and if @xmath83 for every @xmath1061 @xmath1062 * if @xmath1063 and there exists @xmath1064 such that @xmath1065)))\le h(t ) \qquad \text{for all $ j\ge1 $ and a.e . $ t\ge0,$}\ ] ] then @xmath1066 for a.e . @xmath213 and @xmath1067 for every admissible @xmath1068 the need for assumption is not surprising ; see @xcite for conditional results obtained in other contexts in a similar spirit . we postpone the proof after several auxiliary results . [ prop : dist_m_c ] assume that @xmath255 and @xmath163 satisfy . then for any @xmath407 and a.e . @xmath1069 the function @xmath1070 is the distributional mean curvature of @xmath1007).$ ] set @xmath1071).$ ] remark [ rem : traces ] and imply that @xmath1072 hence , it suffices to prove @xmath1073 for a.e . @xmath1074 and since @xmath1075 this follows from lemma [ lem : l2bound_velocity ] below . from definition [ def : dist_curv ] , proposition [ prop : dist_m_c ] and lemma [ lem : l2bound_velocity ] it follows that @xmath1076)}(t , x)\quad \text{for a.e . $ t\ge1/\lambda$ and $ { \mathcal{h}}^n$-a.e . $ x\in \omega\cap{\partial}e_{\lambda}([\lambda t]).$}\ ] ] this is a discretized version of equation . [ lem : l2bound_velocity ] under assumptions the inequality @xmath1077 ) } ( v_\lambda)^2\,d{\mathcal{h}}^ndt \le \alpha(n,\kappa)p(e_0)\ ] ] holds . the proof is analogous to the proof of ( * ? ? ? * lemma 3.6 ) . given @xmath291 and @xmath157 let @xmath1078 for @xmath1079 and @xmath1080 such that @xmath1081,$ ] where @xmath506 is given by , define @xmath1082 - 1)\big)_{r(n,\kappa ) \lambda^{-1/2}}^+:\,\,2^{\ell}<|v_\lambda(x , t)|\le 2^{\ell+1}\big\}.\ ] ] by proposition [ prop : uniform_l_infty_est ] @xmath1083)\delta e_\lambda([\lambda t]-1)\subseteq \cup_\ell k(\ell ) . $ ] take @xmath1084).$ ] then @xmath1085 - 1)=\emptyset$ ] and hence , by remark [ rem : uncons_dens_est ] the following density estimates hold : @xmath1086)\cap b_{\frac{2^{\ell-1}}{\lambda}}(x)|\ge \left(\frac{\kappa}{2}\right)^{n+1 } \omega_{n+1}\left(\frac{2^{\ell-1}}{\lambda}\right)^{n+1},\\ { \mathcal{h}}^n(b_{\frac{2^{\ell-1}}{\lambda}}(x)\cap\omega\cap { \partial}e_\lambda([\lambda t ] ) ) \le \big[(n+1)\omega_{n+1}+\omega_n\big ] \left(\frac{2^{\ell-1}}{\lambda}\right)^n . \end{aligned}\ ] ] using @xmath1087 for any @xmath1088 from we deduce @xmath1089 ) } ( v_\lambda)^2 \,d{\mathcal{h}}^n \le & 25[(n+1)\omega_{n+1}+\omega_n ] ( 2^{\ell-1})^2\left(\frac{2^{\ell-1}}{\lambda}\right)^n\\ \le & \frac{25 [ ( n+1)\omega_{n+1}+\omega_n]}{(\kappa/2)^{n+1}\omega_{n+1}}\ , \lambda\int_{b_{\frac{2^{\ell-1}}{\lambda}}(x)\cap ( e_\lambda([\lambda t])\delta e_\lambda([\lambda t]-1))}|v_\lambda|\,dx.\end{aligned}\ ] ] application of the besicovitch covering theorem to the collection of balls @xmath1090)\}$ ] gives @xmath1091 ) } ( v_\lambda)^2 \,d{\mathcal{h}}^n \le \frac{25[(n+1)\omega_{n+1}+\omega_n]\mathfrak{b}(n)}{(\kappa/2)^{n+1}\omega_{n+1}}\ , \lambda \int_{\{2^{\ell-1}\le |v_\lambda| \le 2^{\ell+2}\ } \cap ( e_\lambda([\lambda t])\delta e_\lambda([\lambda t]-1))}|v_\lambda|\,dx.\ ] ] now summing up these inequalities over @xmath1092 with @xmath1093,$ ] and using the properties of @xmath1094 and the definition of @xmath1095 we get @xmath1096 ) } ( v_\lambda)^2 \,d{\mathcal{h}}^n \le \alpha(n,\kappa)\,\lambda\int_{e_\lambda([\lambda t])\delta e_\lambda([\lambda t]-1)}|v_\lambda|\,dx.\ ] ] observe that by for any @xmath1069 one has @xmath1097)\delta e_\lambda([\lambda t]-1)}|v_\lambda|\,dx \le{{\mathcal c}_{\beta}}(e_\lambda([\lambda t]-1),\omega ) - { { \mathcal c}_{\beta}}(e_\lambda([\lambda t]),\omega).\ ] ] thus @xmath1096 ) } ( v_\lambda)^2 \,d{\mathcal{h}}^n \le \alpha(n,\kappa)\,\lambda \big({{\mathcal c}_{\beta}}(e_\lambda([\lambda t]-1),\omega ) - { { \mathcal c}_{\beta}}(e_\lambda([\lambda t]),\omega)\big).\ ] ] fixing @xmath1098 and integrating this inequality in @xmath1099 $ ] we get @xmath1100 ) } ( v_\lambda)^2 \,d{\mathcal{h}}^ndt \le & \alpha(n,\kappa)\ , \sum\limits_{k=1}^{[t\lambda]+1}\big({{\mathcal c}_{\beta}}(e_\lambda(k-1),\omega ) - { { \mathcal c}_{\beta}}(e_\lambda(k),\omega)\big)\\ \le & \alpha(n,\kappa)\ , { { \mathcal c}_{\beta}}(e_0,\omega ) \le \alpha(n,\kappa)\,p(e_0),\end{aligned}\ ] ] where we used . now letting @xmath1101 completes the proof . the following error estimate can be demonstrated along the same lines of @xcite , therefore the proof is omitted . [ prop : error_estimate ] let @xmath1102 under assumption , for every @xmath1103 the following error - estimate holds : @xmath1104 ) } -\chi _ { e_{\lambda_j}([\lambda_jt]-1)})\phi\,dx -\int_{\omega\cap{\partial}e_{\lambda_j}([\lambda_jt])}{\tilde d}_{e_{\lambda_j}([\lambda_jt]-1)}\ , \phi \,d{\mathcal{h}}^n \bigg)dt \to0.\ ] ] lemma [ lem : l2bound_velocity ] , and ( * ? ? ? * theorem 4.4.2 ) imply that there exist a ( not relabelled ) subsequence and a function @xmath1105 satisfying - . in particular , from it follows that @xmath1106 for a.e . @xmath1107 let us prove that @xmath6 is the distributional mean curvature of @xmath939 for a.e . @xmath974 fixing @xmath1108 by the divergence formula for any @xmath1109 one has @xmath1110 ) } { \mathop{\mathrm{div}}}\phi dx - \int_{\omega\cap { \partial}^*e_{\lambda_j}([\lambda_jt ] ) } \phi\cdot\nu_{e_{\lambda_j}([\lambda_jt])}\,d{\mathcal{h}}^n = \int_{{\partial}\omega\cap { \partial}^ * e_{\lambda_j}([\lambda_jt])}\phi_{n+1}d{\mathcal{h}}^n.\ ] ] hence , from and we get @xmath1111))}\phi_{n+1}d{\mathcal{h}}^n.\ ] ] the left - hand - side of is @xmath1112 therefore , @xmath1113))\overset{w^*}{\rightharpoonup } { \mathcal{h}}^n{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\mathrm{tr}}(e(t ) ) \qquad \text{as $ j\to+\infty.$}\ ] ] combining this with we get @xmath1114 ) \overset{w^*}{\rightharpoonup } { \mathcal{h}}^n{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\partial}^ * e(t ) \quad \text { as $ j\to+\infty$ for a.e . $ t\ge0.$}\ ] ] take @xmath1115 and an admissible @xmath1116 by and ( * ? ? ? * formula ( 4.2 ) ) for a.e . @xmath1 and for every @xmath1117 one has @xmath1118 ) } f(x,\nu_{e_{\lambda_j}([\lambda_jt])}(x))\,d{\mathcal{h}}^n = \int_{\omega\cap { \partial}^*e(t)}f(x,\nu_{e(t)}(x))\,d{\mathcal{h}}^n.\ ] ] in particular , taking @xmath1119 such that @xmath1120 in @xmath1121 by the dominated convergence theorem , and , for @xmath1122 we establish @xmath1123 ) } \eta(t ) f(x,\nu_{e_{\lambda_j}([\lambda_jt])})d{\mathcal{h}}^ndt\\ = & \lim\limits_{j\to+\infty } \int_0^{+\infty } \int_{\omega\cap { \partial}^*e_{\lambda_j}([\lambda_jt ] ) } v_{\lambda_j}\nu_{e_{\lambda_j}([\lambda_jt])}\cdot \psi(t , x)d{\mathcal{h}}^ndt\\ = & \int_0^{+\infty } \int_{\omega\cap { \partial}^*e(t ) } v \nu_{e(t)}\cdot \psi(t , x)d{\mathcal{h}}^ndt = \int_0^{+\infty}\eta(t ) \int_{\omega\cap { \partial}^*e(t ) } h_{e(t ) } \nu_{e(t)}\cdot x\,d{\mathcal{h}}^ndt . \end{aligned}\ ] ] since @xmath1115 is arbitrary , for a.e . @xmath1 we get @xmath1124 hence @xmath6 is the generalized mean curvature of @xmath1125 let us show . take @xmath1126 by a change of variables we have @xmath1127 ) } \phi dx - & \int_{e_{\lambda_j}([\lambda_jt]-1 ) } \phi dx \big]dt \\ = & \int_{1/\lambda_j}^{+\infty } \int_{e_{\lambda_j}([\lambda_jt ] ) } ( \phi(t , x ) - \phi(t+1/\lambda_j , x))dxdt -\frac{1}{\lambda_j}\int_{e(0 ) } \phi(x,0)\,dx.\end{aligned}\ ] ] since @xmath1128 , from we get @xmath1129 ) } \phi dx - \int_{e_{\lambda_j}([\lambda_jt]-1 ) } \phi dx \big]dt = -\int_0^{+\infty } \int_{e(t ) } \frac{{\partial}\phi}{{\partial}t}\,(t , x)\,dxdt - \int_{e_0 } \phi(x,0)dx.\ ] ] therefore , , and the definition of @xmath6 imply @xmath1130)}v_{\lambda_j } \phi \,d{\mathcal{h}}^n dt\\ = & \int_{0}^{+\infty } \int_{\omega\cap{\partial}^ * e(t ) } h_{e(t ) } \phi \,d{\mathcal{h}}^n dt.\end{aligned}\ ] ] \(ii ) take an admissible @xmath1010 and @xmath1131 from @xmath1132 ) } \big({\mathop{\mathrm{div}}}x - \nu_{e_{\lambda_j}([\lambda_jt ] ) } \cdot ( \nabla x ) \nu_{e_{\lambda_j}([\lambda_jt])}\big)\,d{\mathcal{h}}^ndt\\ - \int_0^{+\infty } \eta(t)\int_{\omega\cap { \partial}^*e_{\lambda_j}([\lambda_jt ] ) } v_{\lambda_j}\,x\cdot \nu_{e_{\lambda_j}([\lambda_jt])}\,d{\mathcal{h}}^ndt\\ = \int_0^{+\infty}\eta(t)\int_{{\partial}^ * { \mathrm{tr}}(e_{\lambda_j}([\lambda_jt ] ) ) } \beta\,x'\cdot \nu_{{\mathrm{tr}}(e_{\lambda_j}([\lambda_jt]))}'\,d{\mathcal{h}}^{n-1}. \end{gathered}\ ] ] let @xmath1133 be any subsequence of @xmath937 by the uniform bound on the perimeters and by compactness there exists a further subsequence @xmath1134 of @xmath1133 and a set @xmath1135 such that @xmath1136 ) ) \to \hat f$ ] in @xmath1137 and @xmath1138 ) ) } ' \ , { \mathcal{h}}^{n-1}{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\partial}^*{\mathrm{tr}}(e_{\lambda_{j_{l_k}}}([\lambda_{j_{l_k}}t ] ) ) \overset{w^*}{\rightharpoonup } \nu_{\hat f}'\,{\mathcal{h}}^{n-1}{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\partial}^ * \hat f\quad\text{as $ k\to+\infty$}\ ] ] for a.e . @xmath974 by for every @xmath1139 we have @xmath1140 ) ) } \phi\,d{\mathcal{h}}^n= \int _ { \hat f } \phi\,d{\mathcal{h}}^n.\ ] ] whence , @xmath1141 therefore @xmath1142 ) ) } ' \ , { \mathcal{h}}^{n-1}{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\partial}^*{\mathrm{tr}}(e_{\lambda_j}([\lambda_jt ] ) ) \overset{w^*}{\rightharpoonup } \nu_{{\mathrm{tr}}(e(t))}'\,{\mathcal{h}}^{n-1}{\mathop{\hbox{\vrule height 7pt width 0.5pt depth 0pt \vrule height 0.5pt width 6pt depth 0pt}}\nolimits}{\partial}^ * { \mathrm{tr}}e(t)\quad\text{as $ j\to+\infty.$}\ ] ] now taking limit in , using , and applying the dominated convergence theorem on the right - hand - side we get . in this section we prove an existence result for minimum problems of type @xmath1143 where @xmath1144.$ ] since @xmath46 is finite in @xmath1145 the functional @xmath1146 is well - defined in @xmath333 we study under the following hypotheses on @xmath1147 [ hyp:2 ] * @xmath419 is bounded from below in @xmath55 and there exists a cylinder @xmath1148 @xmath1149 such that @xmath1150 * @xmath1151 for any @xmath140 @xmath1152,$ ] and @xmath1153 * @xmath1154 for any @xmath157 and @xmath1155 * @xmath419 is @xmath49-lower semicontinuous in @xmath333 [ exam:1 ] besides the following functionals @xmath1144 $ ] satisfy hypothesis [ hyp:2 ] : * given @xmath1156 with @xmath1157 a.e . in @xmath1158 for some @xmath1159 @xmath1160 in particular , we may take @xmath417 with @xmath1161 and @xmath1162 so that by @xmath1146 coincides with @xmath1163 * given a bounded set @xmath1164 @xmath1165 @xmath1166 given @xmath419 satisfying hypothesis [ hyp:2 ] set @xmath1167 clearly , @xmath1168 hence @xmath1169 in view of the previous observation , once we prove the next theorem , the proof of theorem [ teo : unconstrained minimizer ] follows . [ teo : unconstrained minimizer1 ] suppose that hypothesis [ hyp:2 ] holds . suppose also @xmath1170 and there exists @xmath1171 $ ] such that @xmath1172 @xmath181-a.e on @xmath133 then the minimum problem @xmath1173 has a solution . moreover , any minimizer is contained in @xmath1174 where using the isoperimetric inequality @xcite , but we do not need this here . ] @xmath1175 and @xmath1176 is defined in section [ subsec : exist_min_atw ] . [ rem : muhim_coef ] in case of example [ exam:1 ] 1 ) with @xmath1177 for some @xmath1178 @xmath1179 hence , @xmath1180 where @xmath440 is defined in . the same is true if @xmath419 is as in . the assumption on @xmath163 and the @xmath49-lower semicontinuity of @xmath46 ( lemma [ lem : lsc_of_f0 ] ) imply the @xmath49-lower semicontinuity of @xmath1181 moreover , the coercivity of @xmath82 hypothesis [ hyp:2 ] ( a ) and imply the coercivity of @xmath1182 @xmath1183 the main problem in the proof of existence of minimizers of @xmath1146 is the lack of compactness due to the unboundedness of @xmath53 however , for every @xmath1184 inequality , the compactness theorem in @xmath1185 ( see for instance ( * ? ? ? * theorems 3.23 and 3.39 ) ) and the lower semicontinuity of @xmath1146 imply that there exists a solution @xmath1186 of @xmath1187 to prove theorem [ teo : unconstrained minimizer1 ] we mainly follow ( * ? ? ? * section 4 ) , where the existence of volume preserving minimizers of @xmath46 has been shown . we need two preliminary lemmas . as in ( * section 3 ) first we show that one can choose a minimizing sequence consisting of bounded sets . [ lem : minmizers in cylindr ] suppose that hypothesis [ hyp:2 ] holds . then @xmath1188 we need two intermediate steps . the first step concerns truncations with horizontal hyperplanes . * step 1 . * we have @xmath1189 indeed , it suffices to show that if @xmath1190 then @xmath1191 clearly , @xmath92 and @xmath1192 have the same trace on @xmath15 and thus @xmath1193\ , \chi_e \,d{\mathcal{h}}^n = \int_{{\partial}\omega } [ 1+\beta]\ , \chi _ { e\cap\overline{\omega_{{k}-\frac12 } } } \,d{\mathcal{h}}^n.\ ] ] from the comparison theorem of @xcite we have @xmath1194 by hypothesis [ hyp:2 ] ( b ) we have also @xmath1195 therefore from the definition of @xmath1146 we get even the strict inequality @xmath1196 the second step is more delicate and concerns truncations with the lateral boundary of vertical cylinders . * for any @xmath950 there exists @xmath1197 and @xmath1198 such that @xmath1199 indeed , according to step 1 and hypothesis [ hyp:2 ] ( a ) , given @xmath291 there exists @xmath1200 with @xmath1201 such that @xmath1202 since @xmath1203 for sufficiently large @xmath1204 one has @xmath1205 hence there exists @xmath1206 such that @xmath1207 the perimeter of @xmath1208 does not charge those portions of @xmath1209 on the lateral part of @xmath1210 now , let @xmath1211 since @xmath1212 we have @xmath1213 by hypothesis [ hyp:2 ] ( a ) , @xmath1214 thus employing we get @xmath1215 by lemma [ lem : lower_bound_of_f0 ] applied with @xmath1216 and @xmath1217 we have @xmath1218 consequently , from the choice of @xmath1208 and @xmath1219 we get @xmath1220 this concludes the proof of step 2 . now , observe that @xmath1221 on the other hand , since the mapping @xmath1222 is nonincreasing , step 2 implies @xmath1223 therefore follows . as in ( * lemma 3 ) the following lemma holds . [ lem : key_lemma1 ] suppose that @xmath163 satisfies and hypothesis [ hyp:2 ] holds . let @xmath1224 be a minimizer of @xmath1146 in @xmath1225 then for any @xmath1226 there exists @xmath1227 $ ] such that @xmath1228 hence @xmath1229 the idea of the proof is to cut the @xmath1224 with vertical cylinders , similarly to ( * ? ? ? * lemma 5 ) where cuts with horizontal hyperplanes are performed . for @xmath1226 by the isoperimetric - type inequality ( * ? ? ? * theorem vi ) , , the minimality of @xmath1224 and by the definition of @xmath1230 we have @xmath1231 thus , for any @xmath1232 one has @xmath1233 take @xmath1234 such that @xmath1235 and set @xmath1236 @xmath1237 * step 1 . * we claim that @xmath1238 where @xmath1239 it suffices to prove that @xmath1240 we have @xmath1241 similarly , @xmath1242 hence @xmath1243 comparing @xmath1244 with @xmath1245 we get @xmath1246 therefore from hypothesis [ hyp:2 ] ( c ) we obtain @xmath1247\ , \chi_{e^r\cap ( { \ensuremath{c_{r_3}^{{k}}}}\setminus \overline{{\ensuremath{c_{r_1}^{{k } } } } } ) } \,d{\mathcal{h}}^n . \end{aligned}\ ] ] inserting in the identity @xmath1248 we get @xmath1249 by lemma [ lem : lower_bound_of_f0 ] applied with @xmath1250 and @xmath1251 the left - hand - side of is not less than @xmath1252 hence @xmath1253 then from it follows that @xmath1254 this finishes the proof of step 1 . before going to step 2 we need some preliminaries . choose any @xmath1255 let @xmath1256 @xmath1257 given @xmath1258 @xmath1259 define @xmath1260 by @xmath1261 hence @xmath1262 therefore , for @xmath1263 it is possible to find @xmath1264 @xmath1265 and @xmath1266 such that @xmath1267 we choose @xmath1268 let @xmath1269 * step 2 . * using the definition of @xmath1270 we show that @xmath1271 indeed , according to , and the definition of @xmath1272 one has @xmath1273 by construction , @xmath1274 i.e. @xmath1275 by induction one can check that @xmath1276 where @xmath1277 note that @xmath1278 since @xmath1279 and @xmath1280 by , the choice of @xmath1270 in implies @xmath1281 moreover @xmath1282 since @xmath1283 now follows from these estimates and . * step 3 . * let @xmath1284 be such that @xmath1285 since @xmath1286 @xmath1287 is nondecreasing and @xmath1288 is nonincreasing , there exists @xmath1227 $ ] such that @xmath1289 ( possibly up to a subsequence ) . then , by step 2 , @xmath1290 which concludes the proof of the lemma . let us prove the existence of a minimizer of @xmath1181 for @xmath1226 let @xmath1227 $ ] be as in lemma [ lem : key_lemma1 ] . then from and @xmath1291 we get @xmath1292 by and the isoperimetric - type inequality @xmath1293 thus from @xmath1294 hence , @xmath1295 satisfies @xmath1296 from and the minimality of @xmath1297 we get @xmath1298 and thus , by compactness there exists @xmath1299 such that ( up to a subsequence ) @xmath1300 in @xmath49 as @xmath1301 from the @xmath49-lower semicontinuity of @xmath1146 and from we conclude that @xmath92 is a minimizer of @xmath1181 now we prove that any minimizer @xmath92 of @xmath1146 satisfies @xmath1302 arguing as in the proof of one can show that @xmath1303 * claim . * there exists @xmath1304 ( possibly depending on @xmath419 and @xmath537 ) such that @xmath1305 for any @xmath1306 such that @xmath1307 by the minimality of @xmath92 we have @xmath1308 i.e. @xmath1309 by lemma [ lem : lower_bound_of_f0 ] @xmath1310 moreover , by the isoperimetric - type inequality , @xmath1311 therefore , and imply @xmath1312 set @xmath1313 clearly , @xmath1314.$ ] moreover , @xmath1315 is absolutely continuous , nonincreasing , @xmath1316 and @xmath1317 for a.e . @xmath1318 by @xmath1319 if @xmath92 is unbounded , then @xmath1320 for any @xmath1321 and thus , for any @xmath1322 @xmath1323 we have @xmath1324 now letting @xmath1325 we obtain @xmath1326 a contradiction . consequently , there exists @xmath1304 such that @xmath1327 i.e. @xmath1305 from the claim it follows that @xmath92 is a minimizer of @xmath1146 also in @xmath1225 by lemma [ lem : key_lemma1 ] we can find @xmath1227 $ ] such that @xmath1328 then using @xmath1329 the relations - applied with @xmath92 in place of @xmath1224 imply @xmath1330 therefore , the minimality of @xmath92 yields @xmath1331 i.e. @xmath1332 since @xmath1333 the conclusion follows . in this appendix we sketch the proof of short time existence and uniqueness of smooth hypersurfaces moving with normal velocity equal to their mean curvature in @xmath99 and meeting the boundary @xmath15 at a prescribed ( not necessarily constant ) angle . the following theorem is a generalization of ( * ? ? ? * theorem 1 ) , where short time existence and uniqueness have been proven for constant @xmath1334 [ teo : short_time_existence ] let @xmath1335 @xmath1336 @xmath1337 $ ] and @xmath448 be a bounded open set such that @xmath1338 is a bounded @xmath1339-hypersurface , @xmath1340 assume that @xmath1341 is a bounded open set with @xmath1339-boundary , @xmath1342 is a parametrization of @xmath1343 such that @xmath1344 in @xmath1345 @xmath1346 on @xmath1347 and @xmath1348 \quad \text{on $ { \partial}{\mathcal{u}},$}\ ] ] where @xmath1349 is the outward unit normal to @xmath1347 @xmath1350 is the outward unit normal of @xmath1343 at @xmath1351 and @xmath1352 = \sum\limits_{j=1}^n n_j^0p_{\sigma_j}^0.$ ] then there exists @xmath1353 a unique family of bounded open sets @xmath1354\}$ ] with a parametrization @xmath1355 \times\overline{{\mathcal{u}}},{\mathbb{r}}^{n+1})$ ] of @xmath1356 solving the parabolic system @xmath1357 where @xmath1358 and @xmath1359 , coupled with the initial condition @xmath1360 the boundary conditions @xmath1361,$ } \\ e_{n+1}\cdot \nu(p(t,\cdot ) ) = \beta(p(t,\cdot ) ) & \text{on $ { \partial}{\mathcal{u}}$ for any $ t\in [ 0,t_0],$ } \end{cases}\ ] ] and the orthogonality conditions @xmath1362\cdot { \tau_0}_i = 0\,\ , \text{on $ [ 0,t_0]\times{\partial}{\mathcal{u}}$ for every $ i=1,\ldots , n-1,$ } \end{aligned}\ ] ] where @xmath1363 is the outward unit normal to @xmath1364 at @xmath1365 and @xmath1366 is a basis for the tangent space of @xmath1367 at @xmath1368 assumption on @xmath1351 is not restrictive . indeed , if @xmath1369 is a @xmath1339 parametrization of the contact set , we may extend it to a sufficiently small tubular neighborhood @xmath1370 of @xmath1371 in @xmath1372 with the properties that @xmath1373 is a @xmath1339 diffeomorphism , @xmath1374 and @xmath1375 where @xmath1376 is the projection of @xmath1377 on @xmath1378 since @xmath1379 it follows @xmath1380 which is . now we may arbitrarily extend @xmath1373 to a @xmath1339 diffeomorphism in @xmath1381 such that @xmath1382 * step 1 . * let us linearize system fixing some @xmath1386 let @xmath1387\times \overline{{\mathcal{u}}},{\mathbb{r}}^{n+1})$ ] be the nonempty convex set consisting of all functions @xmath1388\times \overline{{\mathcal{u}}},{\mathbb{r}}^{n+1})$ ] such that for @xmath1395 set @xmath1396.$ ] then is equivalent to @xmath1397 + f(t , w).\ ] ] notice that @xmath1398\times \overline{{\mathcal{u } } } ) } \|w - p_0\|_{c^{0,1}([0,t_0]\times \overline{{\mathcal{u}}})},\ ] ] where @xmath1399 now we linearize the contact angle condition . since we have @xmath1400 from remark [ rem : normal_to_surface ] it follows that @xmath1401 let @xmath1402 $ ] , where @xmath1403 and @xmath1404 = \begin{bmatrix } \sum\limits_{i=1}^n d_{p_{\sigma_i}}\tilde \nu^1 \cdot q_{\sigma_i}\\ \sum\limits_{i=1}^n d_{p_{\sigma_i}}\tilde \nu^2 \cdot q_{\sigma_i}\\ \vdots \\ \sum\limits_{i=1}^n d_{p_{\sigma_i}}\tilde \nu^{n+1 } \cdot q_{\sigma_i}\\ \end{bmatrix}= \begin{bmatrix } \sum\limits_{i=1}^n \sum\limits_{j=1}^{n+1 } d_{(p_j)_{\sigma_i}}\tilde \nu^1 \cdot(q_j)_{\sigma_i}\\ \sum\limits_{i=1}^n \sum\limits_{j=1}^{n+1 } d_{(p_j)_{\sigma_i}}\tilde \nu^2 \cdot(q_j)_{\sigma_i}\\ \vdots \\ \sum\limits_{i=1}^n \sum\limits_{j=1}^{n+1 } d_{(p_j)_{\sigma_i}}\tilde \nu^{n+1 } \cdot(q_j)_{\sigma_i}\\ \end{bmatrix}\ ] ] clearly , @xmath1405\times \overline{{\mathcal{u}}})}^2\big).$ ] moreover , @xmath1406 + h_2(t , w)\ ] ] with @xmath1407\times \overline{{\mathcal{u}}})}^2\big).$ ] finally , since @xmath1408 we have @xmath1409 + h_3(t , w),\end{aligned}\ ] ] where @xmath1410\times \overline{{\mathcal{u}}})}}^2\big ) . $ ] thus , is equivalent to @xmath1411 = ( e_{n+1 } - \beta(p^0 ) \nu(p^0 ) ) \cdot d\tilde\nu(p_\sigma^0)[p_\sigma^0 ] + h_4(t , w),\ ] ] where @xmath1412\times \overline{{\mathcal{u}}})}}^2\big).$ ] thus we have the following linear parabolic system of equations @xmath1413 subject to the boundary conditions @xmath1414 on @xmath1393\times{\partial}{\mathcal{u}},$ ] where @xmath1415 + h_4(t , w ) , \,\ , \underbrace{0 , \,\,\ldots \,\,,0}_{(n-1)-\text{times } } \right]^t\ ] ] and , under the notation @xmath1416 @xmath1417 @xmath1418 the @xmath1419-matrices @xmath1420 and @xmath1421 @xmath1422 @xmath1423 are defined as follows : @xmath1424 @xmath1425 where the first row must be intended as @xmath1426.$ ] * step 2 . * now we check the compatibility conditions @xcite . take any @xmath1427 and let @xmath1428 be in the tangent space of @xmath1371 at @xmath1429 let @xmath1430 be a solution of the quadratic equation @xmath1431 in @xmath1432 with positive imaginary part . notice that @xmath1433 and @xmath1434 in order to prove the compatibility conditions we should prove that the rows of matrix @xmath1435 are linearly independent modulo the polynomial @xmath1436 whenever @xmath1437 @xmath1438 according to the definitions of @xmath1439 and @xmath1440 one checks @xcite that the compatibility conditions are equivalent to the conditions @xmath1441 since a basis of the tangent space @xmath1442 of @xmath1443 belongs to the horizontal subspace of @xmath115 and @xmath1444 is normal to @xmath1443 at @xmath1351 we have @xmath1445 moreover , since @xmath1446 and @xmath1343 satisfies the contact angle condition , @xmath1447 and @xmath1444 are linearly independent , i.e. @xmath1448 * step 3 . * by ( * ? ? ? * theorem 4.9 ) since @xmath1449 @xmath1450 and the compatibility conditions hold , for any @xmath1451\times\overline{{\mathcal{u}}}),$ ] @xmath1452 there exists a unique solution @xmath1388\times\overline{{\mathcal{u}}})$ ] such that @xmath1453,$}\\ & ( e_{n+1 } - \beta(p^0 ) \nu(p^0 ) ) \cdot d\tilde \nu(p^0)[w_\sigma ] = ( e_{n+1 } - \beta(p^0 ) \nu(p^0 ) ) \cdot d\tilde \nu(p^0)[p_\sigma^0 ] + \tilde f(t , x ) \quad \text{on $ [ 0,t_0]\times { \partial}{\mathcal{u}}$},\\ & \left(\sum\limits_{j=1}^n n_j^0 w_{\sigma_j}\right)\cdot { \tau_0}_i = 0\qquad \text{on $ [ 0,t_0]\times { \partial}{\mathcal{u}}$ and $ i=1,\ldots , n-1.$}\end{aligned}\ ] ] * step 4 . * finally , mimicking @xcite we can prove the existence of and uniqueness of solution - in time interval @xmath1454 $ ] for some sufficiently small @xmath1455 depending on @xmath1456 and @xmath1457 let @xmath1458 @xmath1459 be bounded sets such that @xmath1460 are @xmath1339 hypersurfaces , and the smooth flows @xmath1461 starting from @xmath1462 exist in @xmath1454,$ ] @xmath1463 if @xmath756 and @xmath1464 then @xmath1465 for all @xmath1466.$ ] the first author would like to express his gratitude to the international centre for theoretical physics ( ictp ) in trieste for its hospitality and facilities . the second author is very grateful to the international centre for theoretical physics ( ictp ) and the scuola internazionale superiore di studi avanzati ( sissa ) in trieste , where this research was made . sulla propriet isoperimetrica dellipersfera , nella classe degli insiemi aventi frontiera orientata di misura finita . atti accad . lincei mem . i ( 8) , * 5 * ( 1958 ) , 33 - 44 . evolution of nonparametric surfaces with speed depending on curvature . iii . some remarks on mean curvature and anisotropic flows . degenerate diffusions ( minneapolis , mn , 1991 ) , 141 - 156 , i m a vol . 47 * , springer , new york , 1993 .

we study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle . using the minimizing movements method we show the existence of a weak evolution , and its compatibility with a distributional solution . we also prove various comparison results .

the central role played by rare decays on our understanding of elementary particle physics , is well known , where `` rare '' stands here for flavor changing neutral currents ( fcnc ) , which are either small or practically vanishing in the sm . some highlights : are : + 1 . in @xmath2 physics : the first appearance of charm in loops from which @xmath3 gev was predicted @xcite . + 2 . in @xmath4 physics : the importance of @xmath5 in the sm and for extracting limits on beyond the sm ( bsm ) scenarios @xcite . the top quark fcnc provide an excellent tool to investigate various extensions of the sm @xcite . the experimental upper limit of the decay @xmath6 @xcite , places severe limits on extensions of the sm . in view of the above and prompted by the recent discussion of a giga-@xmath0 option at tesla @xcite in which the center - of - mass energy will be lowered to @xmath7 , producing more than @xmath8 @xmath0 bosons ( _ i.e. _ @xmath9 times the number produced at lep ) , one should investigate rare decays of @xmath0s . now since the important subject of rare leptonic @xmath10 decays , for which the sm branching ratio is @xmath11 , was covered by illana @xcite , we concentrate here on purely hadronic fcnc @xmath12 decays . in fact we only discuss @xmath13 which in most models , including the sm , has the largest branching ratio among hadronic fcnc @xmath0 decays . note however that experimentally , the latter is practically inseparable from the @xmath14 mode . let us also note that when referring to the @xmath15 mode , we actually mean @xmath16 . we note here that in the sm @xcite @xmath17 . in the following sections we will discuss two variants of 2 higgs doublet models ( 2hdm ) and two of supersymmetry ( susy ) . of the latter the first one will be : susy with squark mixing , while in the second one fcnc will result from susy with r parity violation ( denoted by rpv , or @xmath18 ) . as we will see , @xmath19 can be either smaller , the same or above the sm with a maximal value of @xmath20 . experimentally , the attention devoted to fcnc in hadronic @xmath0 decays at lep and sld has been close to nil . the best upper limit is @xcite @xmath21 . this is a preliminary delphi limit ( which will probably remain as such forever ... ) based on about @xmath22 hadronic decays . experimentalists who are privy to lep and sld data should be encouraged to look in their data and improve the above limit . due to space limitations , the following discussion of various bsm models and their predictions for @xmath23 , will be sketchy . many more details and a more complete set of references can be found in @xcite . in fact , almost each reference should start with : `` see _ e.g._@xmath24 '' and end with : `` @xmath24 and references therein . '' we start with a generic calculation of the diagrams which modify ( at one loop ) the @xmath25 vertex , due to charged or neutral scalar , as depicted in fig . [ fig1 ] . in our case @xmath26 , @xmath27 and @xmath28 . the indices @xmath29 and @xmath30 indicate which fermions and scalars we are considering , respectively . one - loop diagrams that contribute to the flavor changing transition @xmath31 , due to scalar - fermion exchanges.,title="fig:",height=340 ] + the feynman rules are : + @xmath32 + @xmath33 + @xmath34 , + where @xmath352 $ ] . + there are 4 one - loop amplitudes , each corresponding to one of the 4 one - loop diagrams . each amplitude is proportional to @xmath36 times + @xmath37 v(p_s).$ ] @xmath38 are momentum dependent form factors , calculable from the diagrams . there are 4 per diagram , thus we have 16 form factors . @xmath39 for diagram ( 1 ) is : + @xmath40 , + and similarly for the other 15 form factors . @xmath41 is one of the usual one - loop scalar functions @xcite at @xmath42 + finally : @xmath43 , \end{aligned}\ ] ] where @xmath44 is the total sum of @xmath39s from the 4 diagrams , and similarly for @xmath45 , @xmath46 and @xmath47 . the stage is now ready for identifying , for each model , the relevant scalars @xmath48 , fermions @xmath49 and the couplings @xmath50 ( with the appropriate indices ) , as expressed in the feynman rules above . then , the route for obtaining @xmath51 using the generic equation in the previous section is clear . in 2hdm with flavor diagonal couplings of the neutral higgs to down - quarks , the fcnc @xmath52 go through the one - loop diagrams in fig . the scalars are the charged higgs bosons , @xmath53 and the fermions are @xmath54 . the couplings are : @xmath55 is as in the sm ( therefore only @xmath56 survives ) , @xmath57 is derived from the kinetic energy part of the lagrangian @xmath58 and @xmath59 is obtained from the yukawa part which , in common notation is @xcite : @xmath60.\end{aligned}\ ] ] a choice of @xmath61 and @xmath62 , which are @xmath63 matrices in flavor space , leads to a specific 2hdm . we now study two variants of 2hdm . in this model , called 2hdmii , @xmath64 , @xmath52 was considered before @xcite . using realistic values in the @xmath65 plane , we obtain : @xmath66 , two orders of magnitude below the sm . in this variant @xcite , named t2hdm , the top is rewarded for its `` fatness '' by having its mass proportional to the large @xmath67 , while all other masses are proportional to @xmath68 . it therefore makes sense to consider here only @xmath69 . using t2hdm parameters consistent with data we find : @xmath70 . fcnc in susy can emanate from squark mixing in : @xmath71 with the usual notation for the squark fields @xcite and where @xmath29 are generation indices . furthermore , @xmath72 + where @xmath73 ... are @xmath74 matrices . under certain assumptions @xcite and taking only @xmath75 or @xmath76 mixing into account : @xmath77 the above @xmath78s represent squark mixing from non - diagonal bilinears in @xmath79 . @xmath80 is a common squarks mass scale , obeying : @xmath81 . also , @xmath82s will stand for squark mixing from non - diagonal trilinears in @xmath79 @xcite . for them we adopt the ansatz of @xcite , leading to @xmath83 , where @xmath84 is a common trilinear soft breaking parameter for both up and down squarks . @xmath85 become @xmath86 matrices in the weak bases @xmath87 they are diagonalized to obtain the mass eigenstates @xmath88 with the help of @xmath89 which rotates @xmath90 to @xmath91 . we can now describe two cases of squark mixing : @xmath75 and @xmath76 mixing . the scalars here are @xmath93 , since @xmath75 admixture states run in the loops . the gluon is the only fermion in the loops , thus @xmath94 . the @xmath95 couplings are @xmath96 , since @xmath97 . in other words , one diagram ( out of the four generic diagrams ) vanishes . the @xmath98 and @xmath99 couplings are functions of elements of the rotation matrix @xmath100 mentioned above @xcite . since the two @xmath101 @xcite , we neglect them . for the other four @xmath78s we assume a common value , _ i.e. _ @xmath102 , and vary @xmath103 . the parameters needed for masses , mixing and @xmath104 are : @xmath105 and @xmath106 . we vary them subject to @xmath107 gev and have plots of practically everything as a function of everything @xcite . we find @xmath108 , where the highest value is attained for @xmath109 and one @xmath110 @xmath111 the ew scale , while @xmath112 , @xmath113 @xmath111 few tev . such splitting requires `` heavy '' susy mass scale with soft breaking parameters , which is consistent with the non - observability of susy particles so far . in this scenario the scalars are @xmath115 , similarly to the previous case , except for @xmath116 . obviously , @xmath76 admixture states run in the loops . the loop fermions are the two charginos @xmath117 , and all four generic diagrams contribute to @xmath52 . the feynman rules @xcite involve elements of the rotation matrix @xmath118 mentioned above and the chargino mixing matrices . at the end of the day , running with the parameters over all values consistent with the data , and with @xmath119 gev and @xmath120 gev we obtain : @xmath70 , which we could have anticipated since br(through @xmath76 mixing ) : br(through @xmath75 mixing ) @xmath121 . since there is no sacred principle which guarantees r - parity conservation , we assume in this part of the talk that @xmath122 is violated . then , @xmath18 terms in the susy superpotential @xmath123 lead to fcnc . @xmath124 terms ( pure @xmath125 ) in @xmath123 are irrelevant at the 1-loop level . in addition we assume , for the pure @xmath126 terms , that @xmath127 , and also neglect the bilinear term in the @xmath18 part of @xmath123 . then : @xmath128 in addition , if @xmath18 is ok then the @xmath122 conserving soft susy breaking is extended . we need only the bilinear : @xmath129 , where @xmath130 , @xmath131 are the scalar components of the hatted @xmath132 and @xmath131 , respectively . we therefore have two types of fcnc : + * type a : * trilinear - trilinear : @xmath133 . + * type b : * trilinear - bilinear : @xmath134 . we further sub - divide type a contributions to 6 groups according to the scalars and fermions running in the loops . for instance , in type a1 the scalars are @xmath135 and the fermions are @xmath54 . the @xmath95 couplings are identical to their sm values , @xmath136 ( for all @xmath29 and @xmath137 ) , @xmath138 and @xmath139 . unfortunately , going over all type a groups , taking into account the available limits on @xmath140s and on the other relevant parameters , we obtain for the trilinear - trilinear case : @xmath66 . our results are in agreement with the special cases in @xcite . in this case , a higgs exchanged in the loop mixes with a slepton , through @xmath141 , assuming that only @xmath142 . we choose to work in the `` no vev '' basis @xmath143 in which : @xmath144 @xmath145 @xmath146 where @xmath147 , @xmath148 , @xmath149 are su(2 ) cp - even , cp - odd @xmath150-sneutrinos , @xmath151 , respectively . in the basis @xmath152 we wrote the mass matrix in the charged scalar sector , in the basis @xmath153 we wrote the mass matrix in the cp - even neutral scalar sector , and in the basis @xmath154 we wrote the mass matrix in the cp - odd neutral scalar sector . the new charged scalar and cp - even and cp - odd neutral scalar mass - eigenstates are obtained by diagonalizing the above - mentioned matrices . they are : @xmath155 , and @xmath156 . in the limit @xmath157 : @xmath158 become the usual ones . rotating with the diagonalizing @xmath159 ( for charged , even - cp , odd - cp ) matrices , one goes from the @xmath90s to the @xmath91s . all depends on the four parameters @xmath160 ( the masses in the limit @xmath161 ) , @xmath162 and @xmath163 . let us sub - divide type b into two types according to the scalar and fermion in the loop : + here @xmath164 with @xmath165 . the @xmath95 couplings are equal to their values in the sm . the @xmath98 couplings include elements of the rotation matrix ( for the charged fields ) @xmath166 and @xmath167 , and @xmath168 + in this case @xmath169 and @xmath170 with @xmath171 , and @xmath172 . this is the only case for which our generic form is insufficient . this fact results from the appearance of two new diagrams proportional to a scalar - vector - vector coupling ( @xmath173 in our case ) . the other eight diagrams are special cases of the generic ones in fig . [ fig1 ] . inserting parameters consistent with the data we found that for type b : @xmath174 . let us briefly comment about the feasibility of observing ( or limiting ) a signal of @xmath175 , at a linear collider producing @xmath8 z - bosons . such a signal leads to one @xmath98-jet and one @xmath176-jet , where @xmath176 stands for quarks lighter than @xmath98 . the main background is from @xmath177 . using what , we think , are realistic efficiencies we find that a new physics signal @xmath178 , with a branching ratio of order @xmath179 , can reach beyond the 3-sigma level @xcite . we can also get a clue about how low one can go in the value ( or limit ) of @xmath180 with @xmath8 z - bosons , from the fact that the delphi preliminary result reached @xcite @xmath181 with @xmath182 @xmath0-bosons . scaling this limit , especially with the expected advances in @xmath98-tagging and identification of non-@xmath98 jets methods , an @xmath183 branching ratios should be easily attained at a giga-@xmath0 factory . one needs realistic simulations as feasibility studies for this important rare @xmath0 decay mode . our results are best summarized in table 1 which shows the best values for @xmath23 in extensions of the sm discussed in this talk . the sm result is given for comparison . note that we have not included interference with the sm as each of the values `` stands alone '' . in some cases such interference may increase the branching ratio to @xmath184 . we conclude that giga-@xmath0 experiments will have the opportunity to place significant limits , or hopefully discover the scenario beyond the sm , by searching for hadronic ( and leptonic @xcite ) neutral current flavor changing transitions . + : + i would like to thank my collaborators , especially shaouly bar - shalom , for teaching me so much . i would also like to express my appreciation to the organizers of the meeting for a job well done . thanks also to members of the theory group in desy ( hamburg ) who gave me the peace of mind i needed to prepare my talk . this research was supported in part by the us - israel binational science foundation , by the israel science foundation and by the fund for promotion of research at the technion . j. fuster , f. martinez - vidal and p. tortosa , preprint delphi 99 - 81 conf 268 , june 1999 . the preprint can be downloaded from : http://documents.cern.ch/cgi-bin/setlink?base=preprint&categ=cern&id + = open-99 - 393 . sld plans to improve this limit ; see : s. walston s talk at dpf2002 .

motivated by the well known impact of rare decays of hadrons and leptons on the evolution of the standard model and on limits for new physics , as well as by the proposal for giga-@xmath0 option at tesla , we investigate the rare decay @xmath1 in various extensions of the standard model .

it is a fundamental fact that on an almost complex manifold with hermitian metric ( almost hermitian manifold ) , the action of the almost complex structure on the tangent space at each point of the manifold is isometry . there is another kind of metric , called a norden metric or a @xmath0-metric on an almost complex manifold , such that the action of the almost complex structure is anti - isometry with respect to the metric . such a manifold is called an almost complex manifold with norden metric @xcite or with @xmath0-metric @xcite . see also ref . for generalized @xmath0-manifolds . it is known @xcite that these manifolds are classified into eight classes . the purpose of the present paper is to exhibit , by construction , almost complex structures with norden metric on lie groups as 6-manifolds , which are of a certain class , called quasi - khler manifold with norden metric . it is proved that the constructed 6-manifold is isotropic khlerian @xcite if and only if it is scalar flat or it has zero holomorphic sectional curvatures . let @xmath1 be a @xmath2-dimensional almost complex manifold with norden metric , i. e. @xmath3 is an almost complex structure and @xmath4 is a metric on @xmath5 such that @xmath6 for all differentiable vector fields @xmath7 , @xmath8 on @xmath5 , i. e. @xmath9 . the associated metric @xmath10 of @xmath4 on @xmath5 given by @xmath11 for all @xmath12 is a norden metric , too . both metrics are necessarily of signature @xmath13 . the manifold @xmath14 is an almost complex manifold with norden metric , too . further , @xmath7 , @xmath8 , @xmath15 , @xmath16 ( @xmath17 , @xmath18 , @xmath19 , @xmath20 , respectively ) will stand for arbitrary differentiable vector fields on @xmath5 ( vectors in @xmath21 , @xmath22 , respectively ) . the levi - civita connection of @xmath4 is denoted by @xmath23 . the tensor field @xmath24 of type @xmath25 on @xmath5 is defined by @xmath26 it has the following symmetries @xmath27 further , let @xmath28 ( @xmath29 ) be an arbitrary basis of @xmath21 at a point @xmath30 of @xmath5 . the components of the inverse matrix of @xmath4 are denoted by @xmath31 with respect to the basis @xmath28 . the lie form @xmath32 associated with @xmath24 is defined by @xmath33 a classification of the considered manifolds with respect to @xmath24 is given in ref . . eight classes of almost complex manifolds with norden metric are characterized there according to the properties of @xmath24 . the three basic classes are given as follows @xmath34 \phantom{\mathcal{w}_1 : f(x , y , z)=\frac{1}{4n } } \left . + g(x , j y)\theta(j z ) + g(x , j z)\theta(j y)\right\};\\[4pt ] { \mathcal{w}}_2 : \mathop{{\mathfrak{s } } } \limits_{x , y , z } f(x , y , j z)=0,\quad \theta=0;\\[8pt ] { \mathcal{w}}_3 : \mathop{{\mathfrak{s } } } \limits_{x , y , z } f(x , y , z)=0 , \end{array}\ ] ] where @xmath35 is the cyclic sum by three arguments . the special class @xmath36 of the khler manifolds with norden metric belonging to any other class is determined by the condition @xmath37 . let @xmath38 be the curvature tensor field of @xmath23 defined by @xmath39}z.\ ] ] the corresponding tensor field of type @xmath40 is determined as follows @xmath41 the ricci tensor @xmath42 and the scalar curvature @xmath43 are defined as usual by @xmath44 let @xmath45 be a non - degenerate 2-plane ( i. e. @xmath46 ) spanned by vectors @xmath47 . then , it is known , the sectional curvature of @xmath48 is defined by the following equation @xmath49 the basic sectional curvatures in @xmath21 with an almost complex structure and a norden metric @xmath4 are _ holomorphic sectional curvatures _ if @xmath50 ; _ totally real sectional curvatures _ if @xmath51 with respect to @xmath4 . the square norm @xmath52 of @xmath53 is defined in ref . by @xmath54 having in mind the definition of the tensor @xmath24 and the properties , we obtain the following equation for the square norm of @xmath53 @xmath55 where @xmath56 . [ ik ] an almost complex manifold with norden metric satisfying the condition @xmath57 is called an _ isotropic khler manifold with norden metric_. it is clear , if a manifold belongs to the class @xmath36 , then it is isotropic khlerian but the inverse statement is not always true . the only class of the three basic classes , where the almost complex structure is not integrable , is the class @xmath58 the class of the _ quasi - khler manifolds with norden metric_. let us remark that the definitional condition from implies the vanishing of the lie form @xmath32 for the class @xmath58 . let @xmath59 be a @xmath2-dimensional vector space and consider the structure of the lie algebra defined by the brackets @xmath60=c_{ij}^ke_k , $ ] where @xmath61 is a basis of @xmath59 and @xmath62 . let @xmath63 be the associated connected lie group and @xmath64 be a global basis of left invariant vector fields induced by the basis of @xmath59 . then the jacobi identity has the form @xmath65,x_k\bigr]=0.\ ] ] next we define an almost complex structure by the conditions @xmath66 let us consider the left invariant metric defined by the following way @xmath67 g(x_j , x_k)=0,\qquad j\neq k \in \{1,2,\dots,2n\}. \\ \end{array}\ ] ] the introduced metric is a norden metric because of . in this way , the induced @xmath2-dimensional manifold @xmath68 is an almost complex manifold with norden metric , in short _ almost norden manifold_. the condition the norden metric @xmath4 be a killing metric of the lie group @xmath63 with the corresponding lie algebra @xmath69 is @xmath70 , where @xmath71 and @xmath72 $ ] . it is equivalent to the condition the metric @xmath4 to be an invariant metric , i. e. @xmath73,z\right)+g\left([x , z],y\right)=0.\ ] ] [ th : kil_g ] if @xmath68 is an almost norden manifold with a killing metric @xmath4 , then it is : a @xmath58-manifold ; a locally symmetric manifold . \(i ) let @xmath23 be the levi - civita connection of @xmath4 . then the condition implies consecutively @xmath74,\quad i , j\in\{1,2,\dots,2n\},\label{invlc}\\ f(x_i , x_j , x_k)=\frac{1}{2}\bigl\{g\bigl ( [ x_i , jx_j],x_k\bigr ) - g\bigl([x_i , x_j],jx_k \bigr ) \bigr\}.\label{f - inv}\end{aligned}\ ] ] according to the last equation implies @xmath75 i. e. the manifold belongs to the class @xmath58 . \(ii ) the following form of the curvature tensor is given in ref . @xmath76,x_k\bigr],x_l]\bigr).\ ] ] using the condition for a killing metric , we obtain @xmath77,[x_k , x_l]\bigr).\ ] ] according to the constancy of the component @xmath78 and and , we get the covariant derivative of the tensor @xmath38 of type @xmath40 as follows @xmath79 \phantom{\qquad\qquad\qquad } = \frac{1}{8 } \bigl\ { g\bigl(\bigl[[x_i , x_j],x_k\bigr]-\bigl[[x_i , x_k],x_j\bigr],[x_l , x_m]\bigr]\bigr)\\[4pt ] \phantom{\qquad\qquad\qquad } + g\bigl(\bigl[[x_i , x_l],x_m\bigr]-\bigl[[x_i , x_m],x_l\bigr],[x_j , x_k]\bigr]\bigr ) \bigr\}. \end{array}\ ] ] we apply the the jacobi identity to the double commutators . then the equation gets the form @xmath80\bigr],[x_l , x_m]\bigr)\\[4pt ] \phantom{\left ( \nabla_{x_i } r \right)(x_j , x_k , x_l , x_m)=\frac{1}{8}\bigl\ { } + g\bigl(\bigl[x_i,[x_l , x_m]\bigr],[x_j , x_k]\bigr)\bigr ) \bigr\}. \end{array}\ ] ] since @xmath4 is a killing metric , then applying to we obtain the identity @xmath81 , i. e. the manifold is locally symmetric . let @xmath68 be a 6-dimensional almost norden manifold with killing metric @xmath4 . having in mind theorem [ th : kil_g ] we assert that @xmath68 is a @xmath58-manifold . let the commutators have the following decomposition @xmath82 } [ x_i , x_j]=\gamma_{ij}^k x_k,\quad \gamma_{ij}^k \in { \mathbb{r}},\qquad i , j , k\in\{1,2,\dots,6\}.\ ] ] further we consider the special case when the following two conditions are satisfied @xmath83,[x_k , x_l]\bigr)=0 , \qquad g\bigl([x_i , jx_i],[x_i , jx_i]\bigr)=0 \ ] ] for all different indices @xmath84 in @xmath85 . in other words , the commutators of the different basis vectors are mutually orthogonal and moreover the commutators of the holomorphic sectional bases are isotropic vectors with respect to the norden metric @xmath4 . according to the condition for a killing metric @xmath4 , the jacobi identity and the condition , the equations take the form given in table [ table2- ] . [ table2- ] the lie groups @xmath63 thus obtained are of a family which is characterized by three real parameters @xmath86 @xmath87 . therefore , for the manifold @xmath68 constructed above , we establish the truthfulness of the following [ th : ex ] let @xmath68 be a 6-dimensional almost norden manifold , where @xmath63 is a connected lie group with corresponding lie algebra @xmath69 determined by the global basis of left invariant vector fields @xmath88 ; @xmath3 is an almost complex structure defined by and @xmath4 is an invariant norden metric determined by and . then @xmath68 is a quasi - khler manifold with norden metric if and only if @xmath63 belongs to the 3-parametric family of lie groups determined by table [ table2- ] . let us remark , the killing form @xmath89 , @xmath90 , on the constructed lie algebra @xmath69 has the following form @xmath91 obviously , it is degenerate . let @xmath68 be the 6-dimensional @xmath58-manifold introduced in the previous section . @xmath92 \phantom{\lambda_1 } & = 2f_{256}=2f_{265}=-f_{322}=-f_{355}=2f_{413}=2f_{431 } \\[4pt ] \phantom{\lambda_1 } & = 2f_{446}=2f_{464}=-2f_{526}=-2f_{562}=2f_{535}=2f_{553}\\[4pt ] \phantom{\lambda_1 } & = -f_{611}=-f_{644}=-2f_{113}=-2f_{131}=-2f_{146}=-2f_{164}\\[4pt ] \phantom{\lambda_1 } & = -2f_{226}=-2f_{262}=2f_{235}=2f_{253}=f_{311}=f_{344}=2f_{416}\\[4pt ] \phantom{\lambda_1 } & = 2f_{461}=-2f_{434}=-2f_{443}=-2f_{523}=-2f_{532}=-2f_{556}\\[4pt ] \phantom{\lambda_1 } & = -2f_{565}=f_{622}=f_{655},\\[4pt ] \end{array}\ ] ] @xmath93 \phantom{\lambda_2 } & = -2f_{332}=-2f_{356}=-2f_{365}=-2f_{412}=-2f_{421}=-2f_{445}\\[4pt ] \phantom{\lambda_2 } & = -2f_{454}=f_{511}=f_{544}=-2f_{626}=-2f_{662}=2f_{635}=2f_{653}\\[4pt ] \phantom{\lambda_2}&=2f_{112}=2f_{121}=2f_{145}=2f_{154}=-f_{211}=-f_{244}=-2f_{326}\\[4pt ] \phantom{\lambda_2 } & = -2f_{362}=2f_{335}=2f_{353}=-2f_{415}=-2f_{451}=2f_{424}\\[4pt ] \phantom{\lambda_2 } & = 2f_{442}=-f_{533}=-f_{566}=2f_{623}=2f_{632}=2f_{656}=2f_{665},\\[4pt ] \end{array}\ ] ] @xmath94 \phantom{\lambda_3 } & = 2f_{331}=2f_{346}=2f_{364}=-f_{422}=-f_{455}=2f_{512}=2f_{521 } \\[4pt ] \phantom{\lambda_3 } & = 2f_{545}=2f_{554}=2f_{616}=2f_{661}=-2f_{634}=-2f_{643}=f_{122}\\[4pt ] \phantom{\lambda_3 } & = f_{155}=-2f_{212}=-2f_{221}=-2f_{245}=-2f_{254}=2f_{316}\\[4pt ] \phantom{\lambda_3 } & = 2f_{361}=-2f_{334}=-2f_{343}=f_{433}=f_{466}=-2f_{515}=-2f_{551}\\[4pt ] \phantom{\lambda_3 } & = 2f_{524}=2f_{542}=-2f_{613}=-2f_{631}=-2f_{646}=-2f_{664}.\\[4pt ] \end{array}\ ] ] let @xmath38 be the curvature tensor of type ( 0,4 ) determined by and on @xmath68 . we denote its components by @xmath96 ; @xmath97 . using , , and table [ table2- ] we get the nonzero components of @xmath38 as follows @xmath98 -r_{1331}=r_{4664}=\frac{1}{4}\left(\lambda_1 ^ 2+\lambda_3 ^ 2\right),\quad & -r_{1661}=r_{3443}=\frac{1}{4}\left(\lambda_1 ^ 2-\lambda_3 ^ 2\right),\quad\\[4pt ] -r_{2332}=r_{5665}=\frac{1}{4}\left(\lambda_1 ^ 2+\lambda_2 ^ 2\right),\quad & -r_{2662}=r_{3553}=\frac{1}{4}\left(\lambda_1 ^ 2-\lambda_2 ^ 2\right),\quad\\[4pt ] \end{array}\ ] ] @xmath99 r_{1251}=r_{3253}=-r_{4254}=-r_{6256}=\frac{1}{4}\lambda_2 ^ 2,\\[4pt ] r_{2142}=r_{3143}=-r_{5145}=-r_{6146}=\frac{1}{4}\lambda_3 ^ 2,\\[4pt ] \end{array}\ ] ] @xmath100 \phantom{r_{1561}}=r_{4264}=r_{5265}=-r_{1351}=-r_{2352}=r_{4354}\\[4pt ] \phantom{r_{1561}}=r_{6356}=r_{1231}=-r_{4234}=-r_{5235}=-r_{6236}=\frac{1}{4}\lambda_1\lambda_2,\\[4pt ] \end{array}\ ] ] @xmath101 \phantom{-r_{1341}}=-r_{5135}=-r_{6136}==r_{1461}=r_{2462}=r_{3463}\\[4pt ] \phantom{-r_{1341}}=-r_{5465}=-r_{2162}=-r_{3163}=r_{4164}=r_{5165}=\frac{1}{4}\lambda_1\lambda_3,\\[4pt ] \end{array}\ ] ] @xmath102 \phantom{r_{1561}}=r_{5245}=r_{6246}==-r_{2152}=-r_{3153}=r_{4154}\\[4pt ] \phantom{r_{1561}}=r_{6156}=r_{1451}=r_{2452}=r_{3453}=-r_{6456}=\frac{1}{4}\lambda_2\lambda_3.\\[4pt ] \end{array}\ ] ] having in mind and the components of @xmath38 , we obtain the components @xmath103 @xmath104 of the ricci tensor @xmath42 and the the value of the scalar curvature @xmath43 as follows @xmath105 \rho_{22}=\rho_{55}=-\rho_{25}=-\lambda_2 ^ 2,\qquad \rho_{13}=- \rho_{16}=-\rho_{34}=\rho_{46}=\lambda_1\lambda_3,\\[4pt ] \rho_{33}=\rho_{66}=-\rho_{36}=-\lambda_1 ^ 2,\qquad \rho_{23}=-\rho_{26}=-\rho_{35}=\rho_{56}=\lambda_1\lambda_2,\\[4pt ] \end{array } \end{array}\ ] ] @xmath106 then , using , and the components of @xmath38 , we obtain the corresponding sectional curvatures @xmath118 \end{array } \ ; \\[4pt ] \begin{array}{ll } -k(\alpha_{12})=k(\alpha_{45})=\frac{1}{4}\left(\lambda_2 ^ 2+\lambda_3 ^ 2\right),\;\;\;\ ; & k(\alpha_{15})=-k(\alpha_{24})=\frac{1}{4}\left(\lambda_2 ^ 2-\lambda_3 ^ 2\right),\;\\[4pt ] -k(\alpha_{13})=k(\alpha_{46})=\frac{1}{4}\left(\lambda_1 ^ 2+\lambda_3 ^ 2\right),\ ; & k(\alpha_{16})=-k(\alpha_{34})=\frac{1}{4}\left(\lambda_1 ^ 2-\lambda_3 ^ 2\right),\;\\[4pt ] -k(\alpha_{23})=k(\alpha_{56})=\frac{1}{4}\left(\lambda_1 ^ 2+\lambda_2 ^ 2\right),\ ; & k(\alpha_{26})=-k(\alpha_{35})=\frac{1}{4}\left(\lambda_1 ^ 2-\lambda_2 ^ 2\right).\\[4pt ] \end{array } \end{array}\ ] ] therefore we have the following

a 3-parametric family of 6-dimensional quasi - khler manifolds with norden metric is constructed on a lie group . this family is characterized geometrically . the condition for such a 6-manifold to be isotropic khler is given .

the analytic structure of the bloch functions for 1d crystals with inversion symmetry was investigated in ref . . among the major conclusions of this paper , is the fact that the entire band structure can be characterized by a single , though multi - valued , analytic function @xmath11[@xmath14 , with branch points that occur at complex @xmath15 . a similar conclusion holds also for the bloch functions . the positions of the branch points determine the exponential decay of the wannier functions . for insulators , they also determine the exponential decay of the density matrix and other correlation functions . it was recently shown that the order of the branch points determines the additional inverse power law decay of these functions.@xcite we can say that , although the branch points occur at complex @xmath15 , their existence and location have very important implications on the properties and dynamics of the physical states . the complex @xmath15 wavefunctions are also important when describing surface and defect states , metal - insulator junctions and electrical transport across finite crystals and linear molecular chains.@xcite the methods developed in ref . could not be extended to higher dimensions , where the results are much more limited . the first major step here was made by des cloizeaux , who studied the analytic structure near real @xmath16s , for crystals with a center of inversion.@xcite his conclusion was that the bloch functions and energies of an isolated simple band are analytic ( and periodic ) in a complex strip around the real @xmath16s . the restriction to crystals with center of symmetry was later removed.@xcite the analytic structure has been reconsidered in a study by avron and simon,@xcite who gave answers to some important , tough qualitative questions . for example , one of their conclusion was that all isolated singularities of the band energy are algebraic branch points . the topology of the riemann surface of the bloch functions for finite gap potentials in two dimensions has been investigated in a series of studies by novikov et al.@xcite the analytic structure has been also investigated by purely numerical methods . since it is impossible to explore numerically the entire complex plane , numerical methods can not provide the global structure . even so , they can provide valuable information . for example , in the complex band calculations for si,@xcite or linear molecular chains,@xcite one can clearly see how the bands are connected , even though these studies explored only the real axis of the complex energy plane . in this paper we consider linear molecular chains , described by a hamiltonian of the form @xmath17 with @xmath1 periodic with respect to one of the cartesian coordinates of @xmath18 , let us say @xmath2 : @xmath19 the wavefunctions of @xmath4 are bloch waves , @xmath5 , with the fundamental property @xmath20 we derive the generic analytic structure ( the riemann surface ) of @xmath5 and of the corresponding energy @xmath8 as functions of complex @xmath9 . we shall see that they are different branches of two analytic functions , @xmath21 and @xmath11 , with an essential singularity at @xmath13 and additional branch points which , generically , are of order 3 , respectively 1 . we show where this branch points come from , how they move when the potential is changed and , in some cases , how to estimate their location . our strategy is as follows . according to bloch theorem , we can restrict @xmath2 to @xmath22 $ ] and study the following class of hamiltonians , which will be called bloch hamiltonians : @xmath23,\ ] ] with @xmath9 referring to the following boundary conditions : @xmath24 which define the domain of @xmath25 . for @xmath22 $ ] , the eigenvectors of @xmath25 and their corresponding energies coincide with @xmath5 and @xmath8 . let @xmath26 denote the resolvent set of @xmath25 , which is composed of those points in the complex energy plane for which @xmath27 is bounded . now , it was long known that the green s function , @xmath27 , evaluated at some arbitrary point @xmath28 , is analytic of @xmath9 , for any @xmath9 in the complex plane.@xcite in section ii , we derive the local analytic structure of the eigenvectors and eigenvalues as functions of @xmath9 , starting from this observation alone . to obtain the global analytic structure , we start with a simple @xmath1 , with known global analytic structure , and then study how this structure changes when @xmath1 is modified . when this formalism is applied to 1d periodic systems , all the conclusions ( and a few additional ones ) of ref . follow with no extra effort . this is done in section iii . section iv presents the results for linear chains . we have two applications : a compact expression for the green s function , which is developed in section v , and the computation of the asymptotic behavior of the density matrix for insulating linear molecular chains , which is done in section vi . we conclude with remarks on how to calculate the analytic structure for real systems . we also have two appendices with mathematical details . we consider here , at a general and abstract level , an analytic family , @xmath29 , of closed , possibly non selfadjoint operators . the analyticity is considered in the sense of kato,@xcite which means that , for any @xmath28 , the green s function can be expanded as @xmath30 with the power series converging in the topology induced by the operator norm , for any @xmath31 in a finite vicinity of @xmath9 . we have already mentioned that the bloch hamiltonians form an analytic family . in this section , we discuss the analytic structure of the eigenvalues and the associated eigenvectors of @xmath25 , as functions of @xmath9 , based solely on the analyticity of the green s function . for this , we need to find ways of expressing the eigenvalues and eigenvectors using only the green s function . the major challenge will be posed by the degeneracies . suppose @xmath32 has an isolated , non - degenerate eigenvalue @xmath33 , for @xmath9 near @xmath34 . then there exists a closed contour @xmath35 separating @xmath36 from the rest of the spectrum . for @xmath9 in a sufficiently small vicinity of @xmath34 , @xmath37 remains the only eigenvalue inside @xmath35 and we can express @xmath33 as @xmath38 as shown in appendix a , eqs . ( [ kato ] ) and ( [ formula1 ] ) automatically imply that @xmath37 is analytic at @xmath34 . since @xmath34 was chosen arbitrarily , we can conclude that the non - degenerate eigenvalues are analytic functions of @xmath9 , as long as they stay isolated from the rest of the spectrum . , @xmath37 and @xmath39 , that become equal for @xmath40 . the figure also shows the contours of integration , @xmath35 , @xmath41 and @xmath42 used in the text.,title="fig:",width=226 ] + suppose now that @xmath25 has two isolated , non - degenerate eigenvalues , @xmath37 and @xmath43 , which become equal at some @xmath44 ( see fig . 1 ) : @xmath45 we are interested in the analytic structure of these eigenvalues and the associated eigenvectors , for @xmath9 in a vicinity of @xmath44 . ( [ formula1 ] ) is no longer useful , since there is no such @xmath9-independent contour @xmath35 that isolates one eigenvalue from the rest of the spectrum , for all @xmath9 in a vicinity of @xmath44 . the key is to work with both eigenvalues , since we can still find a @xmath9-independent contour @xmath35 , separating @xmath37 and @xmath39 from the rest of the spectrum ( see fig . 1 ) , as long as @xmath9 stays in a sufficiently small vicinity of @xmath44 . we define , @xmath46 the main observation is that we can use the green s function to express @xmath47 : @xmath48 then , as shown in appendix a , it follows from eqs . ( [ kato ] ) and ( [ formula2 ] ) that @xmath47 are analytic functions near and at @xmath44 . the functions introduced in eq . ( [ thef ] ) provide the following representation : @xmath49 \\ e^{\prime}_\lambda & = & \frac{1}{2}\left[f_1(\lambda)- \sqrt{2f_2(\lambda)-f_1(\lambda)^2}\right].\nonumber\end{aligned}\ ] ] thus , we managed to express the eigenvalues in terms of the green s function alone . the analytic function , @xmath50 must have a zero at @xmath44 , so its generic behavior near @xmath44 is @xmath51 with @xmath52 an integer larger or equal to 1 and @xmath53 analytic and non - zero at @xmath44 . the cases when @xmath54 are very special and will not be considered here . only the following two possibilities are relevant to us : @xmath55 where @xmath53 is analytic and has no zeros in a vicinity of @xmath44 . for a type i degeneracy , as @xmath9 loops around @xmath44 , @xmath37 becomes @xmath56 and vice versa . the two eigenvalues are different branches of a double - valued analytic function , with a branch point of order 1 at @xmath44 : @xmath57 for a type ii degeneracy , both @xmath37 and @xmath58 are analytic near @xmath44 . we consieder now the spectral projectors , @xmath59 and @xmath60 , associated with @xmath37 and @xmath58 , respectively . for @xmath61 , they have the following representation : @xmath62 where @xmath41 is defined by @xmath63 , with @xmath64 small enough ( thus @xmath9 dependent ) so @xmath58 lies outside @xmath41 ( see fig . 1 ) . @xmath60 has a completely equivalent representation . we list the following properties : @xmath65 for a type i degeneracy , as @xmath9 loops around @xmath44 , @xmath59 becomes @xmath66 and vice versa . thus , @xmath59 and @xmath66 are different branches of a double - valued analytic function , with branch point of order 1 at @xmath44 . in other words , they are given by the same function ( identified from now on with @xmath59 ) , which is evaluated on different riemann sheets . these sheets have @xmath44 as a common point . thus , if @xmath59 does not diverge at @xmath44 , then @xmath67 but this will contradict , for example , @xmath68 . we must conclude that @xmath59 diverges at @xmath44 . to find out the form of the singularity , we observe that , if @xmath69 denotes the difference @xmath70 , then @xmath71 has a well defined limit at @xmath44 , which can be seen from the following representation : @xmath72 since @xmath73 , we can conclude that the singularity of @xmath59 is of the form @xmath74 . in addition , we mention that the total spectral projector , @xmath75 is analytic near and at @xmath44 , as it can be seen from the representation @xmath76 and that the green function for @xmath40 and @xmath2 near @xmath77 has the following structure : @xmath78 with @xmath79 analytic near @xmath77 . this expression is interesting because the green s function of self - adjoint operators , viewed as functions of @xmath2 , always have simple poles . thus , @xmath80 can not be a self - adjoint operator . in other words , type i degeneracies can not occur for those values of @xmath9 for which @xmath25 is self - adjoint . we now turn our attention to the eigenvectors . since , for @xmath61 , @xmath59 is rank one , it can be written as @xmath81 with @xmath82 ( @xmath83 ) the eigenvector to the left ( right ) , normalized as @xmath84 as explained in ref . , passing from the projector to the eigenvectors is not a trivial matter , since these vectors are defined up to a phase factor . the question is , can we choose or define these phases so that no additional singularities are introduced ? we can give a positive answer when there is an anti - unitary transformation , @xmath85 , such that : @xmath86 as we shall later see , such @xmath85 exists , for example , when @xmath87 . now fix an arbitrary @xmath88 and observe that @xmath89 one immediate problem with the above expression is that the denominator can be zero . this can happen only for isolated values of @xmath9 for , otherwise , the denominator will be identically zero . let @xmath34 be such value , assumed different from the branch point ( we can always choose a @xmath88 satisfying this condition ) . we know that the only singularity of @xmath59 is at @xmath44 , so what we have is two functions that are equal on a domain surrounding @xmath34 but excluding @xmath34 , and one of them , @xmath59 , is analytic at @xmath34 . then it is a fact that both functions are analytic at @xmath34 . this means the numerator in the right hand side of eq . ( [ proj ] ) must also cancel at @xmath34 , with the same power as the denominator and the problem disappears . then , it is natural to think that we can define the left and right eigenvectors of @xmath25 as : @xmath90 and @xmath91 using the properties of @xmath85 , we can equivalently write : @xmath92 ) behaves like @xmath93 near @xmath34 , with @xmath94 some unknown power . as we already seen , the numerator of eq . ( [ proj ] ) must have exactly the same behavior . now let us look at eq . ( [ def ] ) : the denominator of @xmath82 behaves like @xmath95 , but the numerator can behave like @xmath96 with @xmath97 arbitrary , as long as the numerator of @xmath83 behaves like @xmath98 . however , we show in the following the @xmath97 is exactly @xmath99 . if @xmath85 can not be defined , then this is no longer true . you may think that , in such cases , we should modify the denominator of @xmath82 to @xmath100 . the problem is that @xmath34 is not unique so this may fix the problem at @xmath34 but not at other similar points . now let @xmath101 be the expansion of @xmath59 near @xmath34 . if the numerator of eq . ( [ proj ] ) cancel at @xmath34 , an expansion in powers of @xmath9 will show that this cancelation is equivalent to : @xmath102 with @xmath52 an integer larger or equal to 1 . this also means @xmath103 let us assume , for the beginning , that @xmath104 , in which case @xmath105 . the numerator of eq . ( [ proj ] ) becomes @xmath106 and the denominator @xmath107 although @xmath105 , we are not automatically guaranteed that @xmath108 . however , we already argued that numerator and denominator must cancel with the same power , so this must be so . we can repeat the same arguments for arbitrary @xmath52 and the conclusion will be the same : @xmath109 with @xmath53 and @xmath110 analytic and non - zero near @xmath34 . thus , we can take the square root in eq . ( [ def ] ) without introducing a branch point . we can also see that the denominator and numerator in eq . ( [ def ] ) cancel at @xmath34 with the same power , so there is no pole at @xmath34 . the conclusion is that @xmath82 is analytic at @xmath34 . there will be , inherently , a branch point at @xmath44 , where @xmath82 and @xmath111 behave as @xmath112 i.e. @xmath82 and @xmath111 have a branch point of order 3 at @xmath44 . if the operator @xmath85 with the above mentioned properties exists , this is their only singularity near @xmath44 . for a type ii degeneracy , @xmath59 and @xmath66 are analytic near @xmath44 . the eigenvectors can be introduce in the same way as above and , if @xmath85 exists , they are analytic functions of @xmath9 . _ analytic deformations_. we analyze now what happens when an analytic potential @xmath113 is added : @xmath114 by analytic potential we mean that @xmath115 is an analytic family in the sense of kato , in both @xmath9 and @xmath41 ( see appendix b ) . for any fixed @xmath41 , the isolated , non - degenerate eigenvalues of @xmath116 remain analytic of @xmath9 . the interesting question is what happens with the degeneracies . suppose that , at some fixed @xmath117 , there are two isolated , non - degenerate eigenvalues , @xmath118 and @xmath119 , which become equal at @xmath44 . for @xmath9 in a small vicinity of @xmath44 and @xmath41 in a small vicinity of @xmath117 , we can define the functions @xmath120 and @xmath121 as before , which are now analytic functions in both arguments , @xmath122 , near @xmath123 . if at @xmath117 , @xmath44 is a type i degeneracy , the only effect of a variation in @xmath41 is a shift of @xmath44 . indeed , since @xmath124 the analytic implicit function theorem assures us that there is a unique @xmath125 such that @xmath126 moreover , the zero is simple , i.e. @xmath125 remains a type i degeneracy . for @xmath41 near @xmath117 , @xmath127 where the dots indicate higher order terms in @xmath128 . from eqs . ( [ energy ] ) and ( [ vect ] ) we readily find : @xmath129.\ ] ] if @xmath116 and @xmath130 have complex conjugate eigenvalues , then : @xmath131 in this case , if @xmath44 is located on the real axis , so it is @xmath125 . if not , then @xmath132 which contradicts the uniqueness of @xmath125 . generically , a type ii degeneracy splits into a pair of type i degeneracies when @xmath41 is varied . indeed , since @xmath133 the generic structure of @xmath121 near @xmath123 is : @xmath134 thus , for @xmath135 , @xmath121 will have , generically , two simple zeros ( type i degeneracies ) at : @xmath136+\ldots.\ ] ] the coefficients @xmath137 , @xmath138 and @xmath139 can be derived from a perturbation expansion of eqs . ( [ thef ] ) and ( [ theg ] ) , leading to @xmath140,\end{aligned}\ ] ] plus higher orders in @xmath128 . if a type ii degeneracy does not split , it remains a type ii degeneracy for all values of @xmath41 . this is a consequence of the fact that , if two analytic functions are equal on an interval , they are equal on their entire domain . is a spiral , which have been cut along the dotted lines in individual sheets , indexed by @xmath141 . the solid dots indicate the type ii degeneracies and the arrows indicate how they pair . b ) the riemann surface of @xmath11 at @xmath142 . the empty dots represent the branch points and the arrows indicate how the riemann sheets are connected . in both panels , the thick line shows the trajectory on the riemann surface , when @xmath9 moves on the unit circle.,title="fig:",width=325 ] + we now summarize the findings of this section . the isolated , non - degenerate eigenvalues of @xmath25 are analytic of @xmath9 . double degeneracies can be of different kinds . type i degeneracies are equivalent to branch points . near such degeneracies , @xmath37 behaves as a square root . type i degeneracies are robust to analytic perturbations : as long as they stay isolated , variations of the periodic potential can not destroy or create but only shift them . at type ii degeneracies , the eigenvalues are analytic . type ii degeneracies are unstable to analytic perturbations : generically , they split into two type i degeneracies . the other types of degeneracies were considered rare and not discussed here . the analytic structure of the spectral projectors can be automatically deduced from the analytic structure of @xmath11 . if there exists an anti - unitary transformation , @xmath85 , such that @xmath143 , then the phase of the eigenvectors can be chosen in a canonical way and their analytic structure follows automatically from the analytic structure of @xmath11 . if such @xmath85 does not exists , the eigenvectors will still have singularities of the type @xmath144 at the branch points of @xmath37 but the present analysis does not rule the existence of additional singularities . a similar analysis can be developed for higher degeneracies . for a triple degeneracy , for example , we will have to consider the functions @xmath47 , with @xmath145 . however , we regard higher degeneracies as non - generic , i.e. more rare than simple degeneracies and will not be considered in this paper . ( @xmath141 ) as functions of real @xmath146 ( @xmath147 ) . b ) the generic band structure when the periodic potential is turned on.,title="fig:",width=226 ] + we apply here the abstract formalism to an already well studied problem : the analytic structure of bloch functions in 1d , i.e the wave functions of the following hamiltonian : @xmath148 according to bloch theorem , finding the wave functions and their corresponding energies is equivalent to studying the following analytic family of hamiltonians:@xcite @xmath149,\ ] ] defined in the hilbert space of square integrable functions over the interval @xmath150 $ ] . the index @xmath9 refers to the boundary conditions @xmath151 which define the domain of @xmath25 . the energy spectrum of @xmath4 consists of all eigenvalues of @xmath25 , when @xmath9 sweeps continuously the unit circle . if @xmath152 is the normalized eigenvector of @xmath25 corresponding to the eigenvalue @xmath8 , then @xmath152 coincides on @xmath150 $ ] with the bloch wave of the same energy . if we extend these functions to the entire real axis by using @xmath153,\ ] ] they will automatically satisfy the standard normalization , @xmath154 here are a few elementary properties of @xmath25 . @xmath25 is an analytic family in the sense of kato , for all @xmath155 . @xmath25 is self - adjoint if and only if @xmath9 is on the unit circle . in general , @xmath156 is the adjoint of @xmath25 . if @xmath157 is the complex conjugation , @xmath158 . thus , @xmath25 and @xmath159 have complex conjugated eigenvalues and @xmath25 and @xmath160 have identical eigenvalues . this tells us that the analytic structure is invariant to @xmath161 and @xmath162 . because of these symmetries , it is sufficient to consider only the domain @xmath163 . for @xmath9 not necessarily on the unit circle , the spectral projector on @xmath164 is given by @xmath165 we consider now the following class of hamiltonians @xmath166 and we adiabatically switch @xmath41 from zero to one . as it is shown in appendix b , potentials @xmath167 with square integrable singularities@xcite are analytic . thus , the theory developed in the previous section can be applied to a large class of potentials . encircles the origin at a small radius.,title="fig:",width=264 ] + the eigenvalues and the associated eigenvectors of @xmath168 are given by : @xmath169 they are the different branches of the following multi - valued analytic functions : @xmath170 the riemann surface of @xmath171 is shown in fig . there are only type ii degeneracies and they occur at @xmath172 : @xmath173 a plot of these eigenvalues for @xmath9 on the unit circle is given in fig now we turn on the periodic potential . typically , the energy spectrum of @xmath4 consists of an infinite set of bands separated by gaps , denoted here by @xmath174 ( see fig . when @xmath167 is modified , some of the gaps may close and other may open . if we assume that all the gaps open when the periodic potential is turned on , as in ref . , then all type ii degeneracies split into pairs of type i degeneracies , @xmath125 and @xmath175 , located on the real axis . @xmath125 and @xmath175 are branch points for @xmath11 , which connect different sheets of the original riemann surface ( see fig . since they are constrained on the real axis , the trajectory of different branch points can not intersect ( they stay on the same riemann sheet ) , i.e. the branch points remain isolated as @xmath41 is increased . thus , as the previous section showed , they move analytically as we increase the coupling constant and we can conclude that the analytic structure can not change , qualitatively , as we increase @xmath41 . for @xmath176 , we move from one sheet to another as @xmath9 moves continuously on the unit circle , as fig . the situation is different for @xmath177 ( see fig . 2b ) : when @xmath9 completes one loop on the unit circle , we end up on the same point of the riemann surface as where we started . we can then re - cut the riemann surface , so that we stay on the same sheet when @xmath9 moves on the unit circle ( see fig . 4 ) . and @xmath178 as functions of @xmath179 , for a typical kramers function @xmath180 . the solutions to eq . ( [ deg2 ] ) are given by the intersection points a , b , c , .,title="fig:",width=325 ] + thus , we rediscovered one of the main conclusions of ref . . the eigenvalues @xmath8 are different branches of a multi - valued analytic function @xmath11 with a riemann surface shown in fig . 4 : there are branch points or order 1 at @xmath181 , @xmath182 , @xmath183 , and an essential singularity at 0 . for @xmath41 small , eq . ( [ split ] ) leads to @xmath184,\ ] ] where @xmath174 is the @xmath185-th energy gap and @xmath186 is the energy in the middle of the gap . if @xmath187 , we can construct @xmath85 as @xmath188 , where @xmath157 is the complex conjugation and @xmath189 is the inversion relative to @xmath190 . thus , for systems with inversion symmetry , we can also conclude at once that the only branch points of the bloch functions are @xmath181 , @xmath182 , , which are of order 3 ( see eqs . ( [ vect ] ) ) . the present analysis actually adds something new to the results of ref . , where the author studied two particular phase choices of the bloch functions , namely , those imposed by @xmath191 ( @xmath192 ) being real at the points of inversion symmetry , @xmath190 and @xmath193 . the author warns that other choices can introduce additional singularities and thus reduce the exponential localization of the corresponding wannier functions . a frequently used method of generating wannier functions localized near an arbitrary @xmath194 is to impose @xmath195 . since such a phase choice corresponds to choosing @xmath196 in eq . ( [ def ] ) , we can automatically conclude that it does not introduce additional singularities and that the exponential localization of the corresponding wannier functions is maximal . and @xmath197 as functions of real @xmath146 ( @xmath147 ) for case a. the dashed lines shows the bands after the non - separable potential was turned on . b ) the riemann sheets of @xmath198 and @xmath197 and the type ii degeneracies ( solid circles ) , with arrows indicating how they pair ( case a ) . c ) the riemann surface at @xmath142 . the empty circles represent the branch points and the arrow indicate how they connect different points of the riemann surface.,title="fig:",width=302 ] + we specialize our discussion to 3d and consider hamiltonians of the form : @xmath199 using the bloch theorem , we can find the spectrum and the wave functions of @xmath4 by studying the following family of analytic hamiltonians : @xmath200,\ ] ] with the boundary conditions @xmath201 the general facts about @xmath25 listed in the previous section are still valid . again , the analytic structure is invariant to @xmath162 and @xmath202 so we can and shall restrict the domain of @xmath9 to the unit disk , @xmath163 . ( case a ) . the thick line shows a trajectory on the riemann surface when @xmath9 completes a loop around the origin.,title="fig:",width=264 ] + we apply the analytic deformation strategy , as we did for the 1d case . we start from a hamiltonian with known global analytic structure . for this we consider a separable potential , @xmath203 and then adiabatically introduce the non - separable part of the hamiltonian , @xmath204 we assume , for simplicity , that @xmath205 has only discrete , non - degenerate spectrum . for this , we will have to constrain @xmath206 and @xmath207 in a finite region , which can be arbitrarily large . to be specific , we assume @xmath208 $ ] and impose periodic boundary conditions . this is actually the most widely used approach in numerical calculations involving linear chains . we also assume that @xmath209 has square integrable singularities , which warranties that it is an analytic potential ( see appendix b ) . and @xmath197 as a function of real @xmath146 ( @xmath147 ) for case a. the dashed lines shows the bands after the non - separable potential was turned on . b ) the riemann sheets of @xmath198 and @xmath197 and the type ii degeneracies , with arrows indicating how they pair ( case c ) . c ) the riemann surface at @xmath142 . the empty circles represent the branch points and the arrow indicate how they connect different points of the riemann surface.,title="fig:",width=302 ] + let @xmath210 , @xmath211 and @xmath212 , @xmath8 denote the eigenvectors and the corresponding eigenvalues of @xmath213 and of @xmath214 respectively . then the eigenvectors and the corresponding eigenvalues of @xmath215 are given by : @xmath216 the global analytic structure of @xmath217 is known : for @xmath218 fixed , they are different branches of a multi - valued analytic function @xmath219 , with a riemann surface as in fig . 4 . we now look for type ii degeneracies : @xmath220 which can occur only for @xmath9 on the unit circle or on the real axis but away from the branch cuts . indeed , if @xmath221 denotes the kramers function for the strictly one dimensional hamiltonian @xmath222 , @xcite then eq . ( [ deg1 ] ) is equivalent to @xmath223 in ref . , it was shown that the equation @xmath224 has solutions only for @xmath179 on the real axis . using exactly the same arguments , one can show that all the solutions of eq . ( [ deg2 ] ) are on the real axis ( see fig . 5 ) . given that @xmath225 with @xmath226 real , it follows that @xmath9 must lie either on the unit circle or on the real axis ( away from the branch cuts ) . for example , the solutions a and c in fig . 5 have @xmath9 on the unit circle , while the solution b has @xmath9 on the real axis , inside the unit circle . we will refer to this three situations as cases a , b and c. since the analytic structure is symmetric to @xmath227 , the type ii degeneracies on the unit circle always come in pair , symmetric to the real axis . ( case c ) . the thick lines shows a trajectory on the riemann surface when @xmath9 completes a loop around the origin.,title="fig:",width=264 ] + we consider first the case a , which corresponds to the case when two bands intersect as in fig . 6b shows the riemann sheets corresponding to these two bands . there are two type ii degeneracies , marked with solid circles , on the unit circle and symmetric to the real axis . we assume for the beginning that these are the only type ii degeneracies that split when the non - separable potential is turned on . when the type ii degeneracies split , avoided crossings occur and the bands split ( see the dashed lines in fig . we denote the upper / lower band by @xmath228 . when the non - separable part of the potential is turned on , the already existing branch points shift along the real axis . for small @xmath41 , the shifts can be calculated from eq . ( [ shift ] ) . in addition , two type i degeneracies appear ( and another two outside the unit disk ) , introducing branch points that connect the original riemann sheets . the connected sheets are shown in fig . 6c , where we can also see that , when @xmath9 makes a complete loop on the unit circle , we end up at the same point as where we started . this means we can re - cut the two , now connected , sheets so that we stay on the same sheet when @xmath9 moves on the unit circle . these new sheets are shown in fig . 7 and correspond now to the upper / lower bands @xmath228 . we consider now the case c , which corresponds to a situation when two bands intersect as in fig . when the type ii degeneracies split , the bands split in @xmath229 and a gap appears . this is the only qualitative difference between cases a and c. the riemann surfaces , before and after the non - separable potential was turned on , are shown in figs . 8b and 8c . again , we can re - cut the riemann surface so one sheet corresponds to one band . these new riemann sheets are shown in fig . the case b goes completely analogous . the qualitative differences are that the branch points split from the real axis and we do nt have to re - cut the riemann surface . we analyze now a more involved possibility , namely when we have more type ii degeneracies on the same riemann sheet : @xmath230 such a situation appears when , for example , we have bands crossing as in fig . the riemann sheets for these bands and the type ii degeneracies are shown in fig . when the perturbation is turned on , the type ii degeneracies split in pairs of type i degeneracies , introducing branch points connecting the original sheets as shown in fig . again , when @xmath9 completes one loop on the unit circle , we end up at the same point of the riemann surface as where we started . we can then re - cut the riemann surface such that we stay on the same sheet when @xmath9 moves on the unit circle ( see fig . the only new element is a riemann sheet ( corresponding to the middle band ) with 6 branch points . intersects with @xmath197 and @xmath231 . the dashed lines shows the bands at @xmath142 . b ) the riemann sheets for these bands and the type ii degeneracies , with arrows indicating how they pair . c ) the riemann sheets at @xmath142.,title="fig:",width=302 ] + the last situation we consider is the emergence of a complex band . suppose that @xmath4 has a symmetry with an irreducible representation of dimension 2 . suppose that this symmetry is also present for the bloch hamiltonian at @xmath232 . in this case , the separable hamiltonian will have bands that touch like in fig . such situations are no longer accidental . in this case , the function @xmath233 introduced in eq . ( [ g ] ) behaves as @xmath234 with @xmath53 non - zero at @xmath232 . following our previous notation , this will be a type iv degeneracy ( see fig . when the non - separable potential is turned on , the degeneracy at @xmath232 can not be lifted because of the symmetry . this means @xmath233 must continue to have a zero at @xmath232 . generically , the order of this zero is reduced to 2 and two other zero s split , symmetric relative to the unit circle . in other words , the type iv degeneracy splits into a pair of type i degeneracies plus a type ii degeneracy . @xmath11 remains analytic at @xmath13 , but now the two bands are entangled , in the sense that we need to loop twice on the unit circle to return back to the same point of the riemann surface ( see fig . 12c ) and we can no longer cut the riemann surface so that we stay on the same sheet when @xmath9 moves on the unit circle . a complex band can involve an arbitrary number of bands . for example , the new bands shown with dashed lines in fig . 12a can entangle with other bands at @xmath235 , through the same mechanism , and so on . rather than cutting the riemann surface in individual sheets , we think it is much more convenient to think of a complex band as living on a surface made of all the individual sheets that are entangled through the mechanism described in fig . 12 . we note that the complex band can split in simple bands as soon as the symmetry is broken . .,title="fig:",width=264 ] + we can continue with further examples but we can already draw our main conclusions . the eigenvalues of @xmath116 are different branches of a multi - valued analytic function @xmath11 . the riemann surface of @xmath11 can be cut in sub - surfaces , such that each sub - surface describes one band . for a simple band , this subsurface consists of the entire unit disk , with cuts obtained by connecting a finite number of branch points to the essential singularity at @xmath13 . for a complex band , the sub - surface consists of a finite number of unit disks that are connected as in fig . on this sub - surface we can have an arbitrary number of branch points , that connect this sub - surface to the rest of the riemann surface . for both simple and complex bands , the branch points are symmetric relative to the real axis and , generically , they are of order 1 ( accidental " higher degeneracies can lead to branch points of higher order ) . as we analytically deform the hamiltonian , the un - split type ii degeneracies stay on the unit circle or real axis and the positions of the branch points shift smoothly . in contradistinction to the 1d case , the branch points can move from one sheet to another and their trajectories can intersect . when two of them intersect , they either become branch points of order 2 or recombine into a type ii degeneracy . higher order branch points are not stable , in the sense that small perturbations split them into two or more branch points of order 1 . the riemann surface of the spectral projector @xmath59 is the same as for @xmath11 . when the inversion symmetry is present , the analytic structure of the bloch functions can also be completely determined from the analytic structure of @xmath11 : the riemann sheets of @xmath191 are the same as for @xmath11 , but the branch points are generically of order 3 ( see eq . ( [ vect ] ) ) . with the analytic structure at hand , it is a simple exercise to find a compact expression for the green s function @xmath236 , which is a generalization of the well known sturm - liouville formula in 1d . indeed , using the eigenfunction expansion , @xmath237 where the sum goes over all unit disks of the riemann surface . changing the variable from @xmath9 to @xmath238 if necessary , the above expression can also be written as : @xmath239 where @xmath240 if @xmath241 , @xmath242 if @xmath243 , and similarly for @xmath244 . this step is necessary because we will deform the contour of integration inside the unit circle . we could deform the contour outside the unit circle , but since we chose to exclude this part of the domain we do nt have this liberty anymore , and we need to re - arange the arguments before deforming the contour . the integrand ( including the summation over @xmath185 ) is analytic at the branch points . also , for @xmath245 , @xmath246 and @xmath247 , so there is no singularity at @xmath13 . then , apart from poles , which occur whenever @xmath248 , the integrand is analytic . using the residue theorem , we conclude @xmath249 where the sum goes over all @xmath250 on the riemann surface such that @xmath251 . this expression is valid for systems with and without inversion symmetry , since it is the projector , not the individual bloch functions , that enters into the above equations . ( [ gf ] ) is closely related to the surface adapted expression of the bulk green s function derived in ref . . and @xmath198 , touch tangentially at @xmath252 ( @xmath232 ) . the dashed lines show the bands after the non - separable potential was turned on . b ) the riemann sheets for these bands and the type iv degeneracy ( solid circle ) . c ) the riemann sheets after the non - separable potential was turned on.,title="fig:",width=302 ] + as a simple application , we derive the asymptotic behavior of the density matrix @xmath253 for large @xmath254 , when there is an insulating gap between the occupied and un - occupied states . we start from @xmath255 where @xmath256 is a contour in the complex energy plane surrounding the energies of the occupied states . using eq . ( [ gf ] ) , we readily obtain @xmath257 where @xmath41 is the pre - image of the contour @xmath256 on the riemann surface of @xmath11 . we now restrict @xmath258 and @xmath259 to the first unit cell and calculate the asymptotic form of @xmath260 for large @xmath261 . using the fundamental property of the bloch functions , we have @xmath262 we deform the contour @xmath41 on the riemann surface such that the distance from its points to the unit circle is maximum . in this way , we enforced the fastest decay , with respect to @xmath261 , of the integrand . this optimal contour , will surround ( infinitely tide ) the branch cuts enclosed by the original contour . the asymptotic behavior comes from the vicinity of the branch points @xmath44 and @xmath263 ( they always come in pair ) that are the closest to the unit circle . using the behavior of the bloch functions near the branch points , we find @xmath264 where the integral is taken along the branch cut of @xmath44 . this integral is equal to @xmath265 , with @xmath266 the beta function . we conclude : @xmath267.\ ] ] again , this expression holds for systems with and without inversion symmetry , since it is the projector , not the individual bloch functions , that enters in the above equations . first , we want to point out that the formalism presented here can be also applied to cubic crystals , to derive the analytic structure of the bloch functions with respect to @xmath146 , while keeping @xmath268 and @xmath269 fixed . preliminary results and several applications have been already reported in ref . . we come now to the question of how to locate the branch points for a real system . in a straightforward approach , one will have to locate those @xmath9 inside the unit disk where @xmath25 displays degeneracies . although such a program can be , at least in principle , carried out numerically , there are few chances of success without clues of where these points are located . this is because , in more than one dimension , these degeneracies occur , in general , at complex energies . one possible solution is to follow the lines presented in this paper : locate the type ii degeneracies for a separable potential @xmath270 , chosen as close to the real potential @xmath271 as possible , and follow the trajectory of the branch points as the potential is adiabatically changed @xmath272 , from @xmath176 to 1 . we plan to complete such a program in the near future . the analytic structure of the band energies and bloch functions of 3d crystals , viewed as functions of several variables @xmath268 , @xmath269 and @xmath146 is a much more complex problem , with qualitatively new aspects . it will be interesting to see if this problem can be tackled by the same analytic deformation technique . the main part of this work was completed while the author was visiting department of physics at uc santa barbara . this work was part of the nearsightedness " project , initiated and supervised by prof . walter kohn and was supported by grants no . nsf - dmr03 - 13980 , nsf - dmr04 - 27188 and doe - de - fg02 - 04er46130 . the complex bands were investigated while the author was a fellow of the princeton center for complex materials . we prove here that if @xmath273 is an analytic family in the sense of kato,@xcite then @xmath47 defined in eq . ( [ thef ] ) are analytic functions . if @xmath274 , by definition,@xcite the limit @xmath275 exist in the topology induced by the operator norm , for any @xmath276 and @xmath277 . we denote this limit by @xmath278 . consider now an arbitrary @xmath279 , and a contour @xmath280 in @xmath281 surrounding @xmath282 eigenvalues of @xmath283 . note that the bloch hamiltonians considered in this paper have compact resolvent so their spectrum is always discrete . for @xmath9 in a small neighborhood of @xmath34 , @xmath35 remains in @xmath284 and we can define @xmath285 and @xmath286 . @xmath287 is an analytic family of rank @xmath282 operators for @xmath9 in a small neighborhood of @xmath34 . indeed , if @xmath288 then @xmath289 as @xmath290 , since it can be bounded by @xmath291 this means the limit @xmath292 exists and is equal to @xmath293 . since @xmath293 is the difference of rank @xmath282 operators , it is at most rank @xmath294 . in particular , @xmath295 . then @xmath296 as @xmath290 , since @xmath297\right|\nonumber \\ \leq 4n\left \|\frac{\hat{f}_m(\lambda)-\hat{f}_m(\lambda^\prime ) } { \lambda-\lambda^\prime}-\hat{f}^\prime_m(\lambda)\right\|,\end{aligned}\ ] ] and eq . ( [ inter1 ] ) follows from eq . ( [ inter2 ] ) . thus , the limit @xmath298 exists and is equal to @xmath299 . we discuss here the analytic perturbations for linear molecular chains . as we did in the main text , we constrain @xmath206 and @xmath207 in finite intervals . the bloch functions are determined by the following hamiltonian : @xmath300 \ \text { and } \ z\in[0,b],\ ] ] where @xmath301 is the laplace operator with periodic boundaries in @xmath206 and @xmath207 and the usual bloch conditions in @xmath2 . we show that if @xmath302^{1/2}<\infty,\ ] ] then @xmath116 is an analytic family for all @xmath303 . for this we need the following technical result . _ proposition . _ suppose @xmath113 satisfies eq . ( [ condition ] ) . then , for @xmath137 positive and sufficiently large , there exists a positive @xmath304 such that @xmath305 and : @xmath306 for any @xmath307 in the domain of @xmath168 . now , pick an arbitrary @xmath117 , let @xmath308 and denote @xmath309 , where @xmath310 denotes the operator norm . since @xmath311 for any @xmath307 in the domain of @xmath168 , taking @xmath137 sufficiently large so that @xmath312 , we obtain : @xmath313}{1-\epsilon_a |\gamma_0|}<\infty.\ ] ] if @xmath314 denotes the right hand side of the above equation , then @xmath315 is bounded for @xmath316 and has the following norm convergent expansion : @xmath317^n.\end{aligned}\ ] ] thus , @xmath315 is analytic at the arbitrarily chosen @xmath117 . we now give the proof of the proposition . for @xmath307 in the domain of @xmath168 , let @xmath318 . if @xmath319 , with @xmath137 assumed sufficiently large so that @xmath320 exists , we have @xmath321 and schwartz inequality gives ( @xmath322 ) @xmath323^{1/2 } \|g\|_{l^2}.\ ] ] if we denote @xmath324^{1/2},\ ] ] with the aid of eq . ( [ basic ] ) , we obtain : @xmath325 i.e. eq . ( [ goal ] ) , if we identify @xmath326 . we remark that @xmath327 defined in eq . ( [ alpha ] ) is optimal , in the sense that there are @xmath328s when we do have equality in eq . ( [ final ] ) . it remains to show that @xmath329 . if @xmath330 , with @xmath331 the laplace operator over the entire @xmath18 , then we have the following representation : @xmath332 where the sum goes over all points of the lattice @xmath35 defined by @xmath333 . since @xmath334 is real and positive , we can readily see that @xmath335 . moreover , @xmath336 and we have the following representation : @xmath337 where @xmath338 . note that the diagonal part of @xmath339 is independent of @xmath340 . @xmath341 can be explicitly calculated , leading to : @xmath342^{1/2},\ ] ] with equality for @xmath9 real and positive . the right hand side is finite for @xmath137 sufficiently large and goes to zero as @xmath343 . w. kohn , phys . * 115 * , 809 ( 1959 ) . l. he and d. vanderbilt , phys . 86 * , 5341 ( 2001 ) . v. heine , phys . a * 138 * , 1689 ( 1965 ) . d.n . beratan and j.j . hopfield , j. am . soc . * 106 * , 1584 ( 1984 ) h.j . choi and j. ihm , phys . b * 59 * , 2267 ( 1999 ) . p. mavropoulos , n papanikolaou and p.h . dederichs , phys . ref . lett * 85 * , 1088 ( 2000 ) . des cloizeaux , phys . rev . * 135 * , 685 ( 1964 ) . des cloizeaux , phys . 135 * , 698 ( 1964 ) . g. nenciu , comm . 91 * , 81 ( 1983 ) . avron and b. simon , ann . of physics * 110 * , 85 ( 1978 ) . i. krichever and s.p . novikov , inverse problems * 15 * , r117 ( 1999 ) . y. chang and j.n . schulman , phys . b * 25 * , 3975 ( 1982 ) . f. picaud , a. smogunov , a. dal corso and e. tosatti , j. phys . : condens . matter * 15 * , 3731 ( 2003 ) . tomfohr and o.f . sankey , phys . b * 65 * , 245105 ( 2002 ) . t. kato , perturbation theory for linear operators , springer , berlin ( 1966 ) . g. nenciu , rev . of mod . * 63 * , 91 ( 1991 ) . m. reed and b. simon , methods of modern mathematical physics vol . iv : analysis of operators , academic press , new york ( 1978 ) . by a square integrable singularity we mean @xmath344 , with the integral taken over a finite vicinity of the singularity . kramers , physica * 2 * , 483 ( 1935 ) . allen , phys . b * 19 * , 917 ( 1979 ) ; * 20 * , 1454 ( 1979 ) . e. prodan and w. kohn , pnas * 102 * , 11635 ( 2005 ) .

this paper deals with hamiltonians of the form @xmath0 , with @xmath1 periodic along the @xmath2 direction , @xmath3 . the wavefunctions of @xmath4 are the well known bloch functions @xmath5 , with the fundamental property @xmath6 and @xmath7 . we give the generic analytic structure ( i.e. the riemann surface ) of @xmath5 and their corresponding energy , @xmath8 , as functions of @xmath9 . we show that @xmath8 and @xmath10 are different branches of two multi - valued analytic functions , @xmath11 and @xmath12 , with an essential singularity at @xmath13 and additional branch points , which are generically of order 1 and 3 , respectively . we show where these branch points come from , how they move when we change the potential and how to estimate their location . based on these results , we give two applications : a compact expression of the green s function and a discussion of the asymptotic behavior of the density matrix for insulating molecular chains .

the idea of quantization has proved its importance to bridge the commutative and noncommutative versions of certain algebraic structures and promote better understanding various aspects of the latter versions . one popular structure studied for the last three decades is the quantized coordinate ring @xmath4 of @xmath1 matrices over a field @xmath2 , where @xmath5 is a nonzero element of @xmath2 ; it is usually called the _ algebra of @xmath1 quantum matrices_. here @xmath6 is the @xmath2-algebra generated by the entries ( indeterminates ) of an @xmath1 matrix @xmath7 subject to the following ( quasi)commutation relations due to manin @xcite : for @xmath8 and @xmath9 , @xmath10 this paper is devoted to quadratic identities for minors of quantum matrices ( usually called quantum minors or quantized minors or @xmath5-minors ) . for representative cases , aspects and applications of such identities , see , e.g. , @xcite ( where the list is incomplete ) . we present a novel , and rather transparent , combinatorial method which enables us to completely characterize and efficiently verify homogeneous quadratic identities of universal character that are valid for quantum minors . the identities of our interest can be written as @xmath11_q\ , [ i'_i|j'_i]_q \colon i=1,\ldots , n)=0,\ ] ] where @xmath12 , @xmath13 , and @xmath14_q$ ] denotes the quantum minor whose rows and columns are indexed by @xmath15 $ ] and @xmath16 $ ] , respectively . ( hereinafter , for a positive integer @xmath17 , we write @xmath18 $ ] for @xmath19 . ) the homogeneity means that each of the sets @xmath20 is invariant of @xmath21 , and the term `` universal '' means that ( [ eq : q_ident ] ) should be valid independently of @xmath22 and a @xmath5-matrix ( a matrix whose entries obey manin s relations and , possibly , additional ones ) . note that any cortege @xmath23 may be repeated in ( [ eq : q_ident ] ) many times . our approach is based on two sources . the first one is the _ flow - matching method _ elaborated in @xcite to characterize quadratic identities for usual minors ( viz . for @xmath24 ) . in that case the identities are viewed simpler than ( [ eq : q_ident ] ) , namely , as @xmath25\ , [ i'_i|j'_i ] \colon i=1,\ldots , n)=0.\ ] ] ( in fact , @xcite deals with natural analogs of ( [ eq : com_ident ] ) over commutative semirings , e.g. the tropical semiring @xmath26 . ) in the method of @xcite , each cortege @xmath27 is associated with a certain set @xmath28 of _ feasible matchings _ on the set @xmath29 ( where @xmath30 denotes the symmetric difference @xmath31 , and @xmath32 the disjoint union of sets @xmath33 ) . the main theorem in @xcite asserts that ( [ eq : com_ident ] ) is valid ( universally ) if and only if the families @xmath34 and @xmath35 of corteges @xmath36 with signs @xmath37 and @xmath38 , respectively , are _ balanced _ , in the sense that the total families of feasible matchings for corteges occurring in @xmath34 and in @xmath35 are equal . the main result of this paper gives necessary and sufficient conditions for the quantum version ( in theorems [ tm : nec_q_bal ] and [ tm : suff_q_bal ] ) . it says that ( [ eq : q_ident ] ) is valid ( universally ) if and only if the families of corteges @xmath34 and @xmath35 along with the function @xmath39 are @xmath5-_balanced _ , which now means the existence of a bijection between the feasible matchings for @xmath34 and @xmath35 that is agreeable with @xmath39 in a certain sense . the proof of necessity ( theorem [ tm : nec_q_bal ] ) considers non-@xmath5-balanced @xmath40 and explicitly constructs a certain graph determining a @xmath5-matrix for which ( [ eq : q_ident ] ) is violated when @xmath2 is a field of characteristic 0 and @xmath5 is transcendental over @xmath41 . the second source of our approach is the path method due to casteels @xcite . he associated with an @xmath1 cauchon diagram @xmath42 of @xcite a directed planar graph @xmath43 with @xmath44 distinguished vertices @xmath45 in which the remaining vertices correspond to white cells @xmath46 in the diagram @xmath42 and are labeled as @xmath47 . an example is illustrated in the picture . the labels @xmath47 , regarded as indeterminates , are assumed to ( quasi)commute as @xmath48 ( which is viewed `` simpler '' than ( [ eq : xijkl ] ) ) . these labels determine weights of edges and , further , weights of paths of @xmath49 . the latter give rise to the _ path matrix _ @xmath50 of size @xmath1 , of which @xmath46-th entry is the sum of weights of paths starting at @xmath51 and ending at @xmath52 . the path matrix @xmath53 has three important properties . ( i ) it is a @xmath5-matrix , and therefore , @xmath54 gives a homomorphism of @xmath6 to the corresponding algebra @xmath55 generated by the @xmath56 . ( ii ) @xmath50 admits an analog of lindstrm s lemma @xcite : for any @xmath15 $ ] and @xmath16 $ ] with @xmath57 , the minor @xmath14_q$ ] of @xmath50 can be expressed as the sum of weights of systems of _ disjoint paths _ from @xmath58 to @xmath59 in @xmath49 . ( iii ) from cauchon s algorithm @xcite interpreted in graph terms in @xcite it follows that : if the diagram @xmath42 is maximal ( i.e. , has no black cells ) , then @xmath50 becomes a _ generic @xmath5-matrix _ , see corollary 3.2.5 in @xcite . in this paper we consider a more general class of planar graphs @xmath49 with horizontal and vertical edges , called _ se - graphs _ , and show that they satisfy the above properties ( i)(ii ) as well . our goal is to characterize quadratic identities just for the class of path matrices of se - graphs @xmath49 . since this class contains a generic @xmath5-matrix , the identities are automatically valid in @xmath6 . we take an advantage from the representation of @xmath5-minors of path matrices via systems of disjoint paths , or _ flows _ in our terminology , and the desired results are obtained by applying a combinatorial machinery of handling flows in se - graphs . our method of establishing or verifying one or another identity admits a rather transparent implementation and we illustrate the method by enlightening graphical diagrams . the paper is organized as follows . section [ sec : prelim ] contains basic definitions and backgrounds . section [ sec : flows ] defines flows and path matrices for se - graphs and states lindstrm s type theorem for them . section [ sec : double ] is devoted to crucial ingredients of the method . it describes _ exchange operations _ on _ double flows _ ( pairs of flows related to corteges @xmath23 ) and expresses such operations on the language of planar matchings . the main working tool of the whole proof , stated in this section and proved in appendix b , is theorem [ tm : single_exch ] giving a @xmath5-relation between double flows before and after an ordinary exchange operation . using this , section [ sec : q_relat ] proves the sufficiency in the main result : ( [ eq : q_ident ] ) is valid if the corresponding @xmath40 are @xmath5-balanced ( theorem [ tm : suff_q_bal ] ) . section [ sec : examples ] is devoted to illustrations of our method . it explains how to obtain , with the help of the method , rather transparent proofs for several representative examples of quadratic identities , in particular : ( a ) the pure commutation of @xmath14_q$ ] and @xmath60_q$ ] when @xmath61 and @xmath62 ; ( b ) a quasicommutation of _ flag @xmath5-minors _ @xmath63_q$ ] and @xmath64_q$ ] as in leclerc - zelevinsky s theorem @xcite ; ( c ) identities on flag @xmath5-minors involving triples @xmath65 and quadruples @xmath66 ; ( d ) dodgson s type identity ; ( e ) two general quadratic identities on flag @xmath5-minors from @xcite occurring in descriptions of quantized grassmannians and flag varieties . in section [ sec : necess ] we prove the necessity of the @xmath5-balancedness condition for validity of quadratic identities ( theorem [ tm : nec_q_bal ] ) ; here we rebuild a corresponding construction from @xcite to obtain , in case of the non-@xmath5-balancedness , an se - graph @xmath49 such that the identity for its path matrix is false ( in a special case of @xmath2 and @xmath5 ) . section [ sec : concl ] poses the problem : when an identity in the commutative case , such as ( [ eq : com_ident ] ) , can be turned , by choosing an appropriate @xmath39 , into the corresponding identity for the quantized case ? for example , this is impossible for the trivial identity @xmath63\,[j]=[j]\,[i]$ ] with usual flag minors when @xmath67 are not weakly separated , as is shown in @xcite . also this section gives a generalization of leclerc - zelevinsky s quasicommutation theorem to arbitrary ( non - flag ) quantum minors ( theorem [ tm : generallz ] ) and discusses additional results . finally , appendix a exhibits several auxiliary lemmas needed to us and proves the above - mentioned lindstrm s type result for se - graphs , and appendix b gives the proof of theorem [ tm : single_exch ] ( which is rather technical ) . throughout , by a _ graph _ we mean a directed graph . a _ path _ in a graph @xmath68 ( with vertex set @xmath69 and edge set @xmath70 ) is a sequence @xmath71 such that each @xmath72 is an edge connecting vertices @xmath73 . an edge @xmath72 is called _ forward _ if it is directed from @xmath74 to @xmath75 , denoted as @xmath76 , and _ backward _ otherwise ( when @xmath77 ) . the path @xmath78 is called _ directed _ if it has no backward edge , and _ simple _ if all vertices @xmath75 are different . when @xmath79 and @xmath80 , @xmath78 is called a _ cycle _ , and called a _ simple cycle _ if , in addition , @xmath81 are different . when it is not confusing , we may use for @xmath78 the abbreviated notation via vertices : @xmath82 , or edges @xmath83 . also , using standard terminology in graph theory , for a directed edge @xmath84 , we say that @xmath85 _ leaves _ @xmath86 and _ enters _ @xmath87 , and that @xmath86 is the _ tail _ and @xmath87 is the _ head _ of @xmath85 . it will be convenient for us to visualize matrices in the cartesian form : for an @xmath1 matrix @xmath88 , the row indices @xmath89 are assumed to increase upwards , and the column indices @xmath90 from left to right . as mentioned above , we deal with the _ quantized coordinate ring _ @xmath4 generated by indeterminates @xmath91 satisfying relations ( [ eq : xijkl ] ) , shortly called the algebra of @xmath1 _ quantum matrices_. a somewhat `` simpler '' object is the _ quantum affine space _ @xmath92 , the @xmath2-algebra generated by indeterminates @xmath47 ( @xmath93,\ , j\in[n]$ ] ) subject to relations ( [ eq : trelat ] ) . for an @xmath1 matrix @xmath88 , we denote by @xmath94 the submatrix of @xmath95 whose rows are indexed by @xmath15 $ ] , and columns by @xmath16 $ ] . let @xmath96 , and let @xmath97 consist of @xmath98 and @xmath99 consist of @xmath100 . then the @xmath5-_determinant _ of @xmath94 , or the @xmath5-_minor _ of @xmath95 for @xmath101 , is defined as @xmath102_{a , q}:=\sum_{\sigma\in s_k } ( -q)^{\ell(\sigma ) } \prod_{d=1}^{k } a_{i_dj_{\sigma(d)}},\ ] ] where , in the noncommutative case , the product under @xmath103 is ordered ( from left to right ) by increasing @xmath104 , and @xmath105 is the _ length _ ( number of inversions ) of a permutation @xmath106 . the terms @xmath95 and/or @xmath5 in @xmath14_{a , q}$ ] may be omitted when they are clear from the context . a graph @xmath68 of this sort ( also denoted as @xmath107 ) satisfies the following conditions : ( se1 ) @xmath49 is planar ( with a fixed layout in the plane ) ; ( se2 ) @xmath49 has edges of two types : _ horizontal _ edges , or _ h - edges _ , which are directed to the right , and _ vertical _ edges , or _ v - edges _ , which are directed downwards ( so each edge points to either _ south _ or _ east _ , justifying the term `` se - graph '' ) ; ( se3 ) @xmath49 has two distinguished subsets of vertices : set @xmath108 of _ sources _ and set @xmath109 of _ sinks _ ; moreover , @xmath110 are disposed on a vertical line , in this order upwards , and @xmath111 are disposed on a horizontal line , in this order from left to right ; the sources ( sinks ) are incident only with h - edges ( resp . v - edges ) ; ( se4 ) each vertex of @xmath49 belongs to a directed path from @xmath112 to @xmath42 . we denote by @xmath113 the set @xmath114 of _ inner _ vertices of @xmath49 . an example of se - graphs with @xmath115 and @xmath116 is drawn in the picture : a special case of se - graphs is formed by those corresponding to _ cauchon graphs _ introduced in @xcite ( which are associated with cauchon diagrams @xcite ) . in this case , @xmath117\}$ ] , @xmath118\}$ ] , and @xmath119\times [ n]$ ] . ( the correspondence with the definition in @xcite is given by @xmath120 and @xmath121 . ) when @xmath122\times [ n]$ ] ( equivalently : when the cauchon diagram has no black cells ) , we refer to such a graph as the _ extended @xmath123-grid _ and denote it by @xmath124 . each inner vertex @xmath125 is regarded as a _ generator _ , and we assign the weight @xmath126 to each edge @xmath127 in a way similar to that for cauchon graphs in @xcite , namely : [ eq : edge_weight ] * @xmath128 if @xmath129 ; * @xmath130 if @xmath85 is an h - edge and @xmath131 ; * @xmath132 if @xmath85 is a v - edge . this gives rise to defining the weight @xmath133 of a directed path @xmath83 ( written in the edge notation ) in @xmath49 , to be the ordered ( from left to right ) product @xmath134 then @xmath133 is a laurent monomial in elements of @xmath135 . note that when @xmath78 begins in @xmath112 and ends in @xmath42 , its weight can also be expressed in the following useful form ; cf . let @xmath136 be the sequence of vertices where @xmath78 makes turns ; namely , @xmath78 changes the horizontal direction to the vertical one at each @xmath137 , and conversely at each @xmath75 . then ( due to the `` telescopic effect '' caused by ( [ eq : edge_weight])(ii ) ) , @xmath138 we assume that the generators @xmath135 obey ( quasi)commutation laws somewhat similar to those in ( [ eq : trelat ] ) ; namely , for distinct @xmath131 , ( g1 ) if there is a directed _ horizontal _ path from @xmath86 to @xmath87 in @xmath49 , then @xmath139 ; ( g2 ) if there is a directed _ vertical _ path from @xmath86 to @xmath87 in @xmath49 , then @xmath140 ; ( g3 ) otherwise @xmath141 . as mentioned in the introduction , it is shown in @xcite that the path matrix associates with a cauchon graph @xmath49 has a nice property of lindstrm s type , saying that @xmath5-minors of this matrix correspond to appropriate systems of disjoint paths in @xmath49 . we will show that this property is extended to the se - graphs . let @xmath68 be an se - graph with sources @xmath142 and sinks @xmath143 , and let @xmath144 denote the edge weights in @xmath49 defined by ( [ eq : edge_weight ] ) . * definition . * the _ path matrix _ @xmath145 associated with @xmath49 is the @xmath1 matrix whose entries are defined by @xmath146\times [ n],\ ] ] where @xmath147 is the set of directed paths from @xmath51 to @xmath148 in @xmath49 . in particular , @xmath149 if @xmath150 . thus , the entries of @xmath151 belong to the @xmath2-algebra @xmath152 of laurent polynomials generated by the set @xmath135 of inner vertices of @xmath49 subject to relations ( g1)(g3 ) . * definition . * let @xmath153 denote the set of pairs @xmath154 such that @xmath155 $ ] , @xmath156 $ ] and @xmath57 . borrowing terminology from @xcite , for @xmath157 , a set @xmath158 of pairwise disjoint directed paths from the source set @xmath159 to the sink set @xmath160 in @xmath49 is called an @xmath154-_flow_. the set of @xmath154-flows @xmath158 in @xmath49 is denoted by @xmath161 . we usually assume that the paths forming a flow @xmath158 are ordered by increasing the source indices . namely , if @xmath97 consists of @xmath162 and @xmath99 consists of @xmath163 , then @xmath164-th path @xmath165 in @xmath158 begins at @xmath166 , and therefore , @xmath165 ends at @xmath167 ( which easily follows from the planarity of @xmath49 , the ordering of sources and sinks in the boundary of @xmath49 and the fact that the paths in @xmath158 are disjoint ) . we write @xmath168 and ( similar to path systems in @xcite ) define the weight of @xmath158 to be the ordered product @xmath169 then the desired @xmath5-analog of lindstrm s lemma expresses @xmath5-minors of path matrices via flows as follows . [ tm : linds ] for the path matrix @xmath145 of an @xmath123 se - graph @xmath49 and for any @xmath170 , there holds @xmath171_{\path , q}=\sum\nolimits_{\phi\in\phi(i|j ) } w(\phi).\ ] ] a proof of this theorem , which is close to that in @xcite , is given in appendix a. an important fact is that the entries of @xmath151 obey the ( quasi)commutation relations similar to those for the canonical generators @xmath91 of the quantum algebra @xmath6 given in ( [ eq : xijkl ] ) . it is exhibited in the following assertion , which is known for the path matrices of cauchon graphs due to @xcite ( where it is proved by use of the `` cauchon s deleting derivation algorithm in reverse '' @xcite ) . [ tm : path - a ] for an se - graph @xmath49 , the entries of its path matrix @xmath151 satisfy manin s relations . we will show this in section [ ssec : abcd ] as an easy application of our flow - matching method . this assertion implies that the map @xmath172 determines a homomorphism of @xmath6 to the subalgebra @xmath55 of @xmath152 generated by the entries of @xmath151 , i.e. , @xmath151 is a @xmath5-matrix for any se - graph @xmath49 . in an especial case of @xmath49 , a sharper result , attributed to cauchon and casteels , is as follows . [ tm : cach - cast ] if @xmath173 ( the extended @xmath1-grid defined in remark 1 ) , then @xmath151 is a generic @xmath5-matrix , i.e. , @xmath174 gives an injective map of @xmath6 to @xmath152 . due to this important property , the quadratic relations that are valid ( universally ) for @xmath5-minors of path matrices of se - graphs turn out to be automatically valid for the algebra @xmath6 of quantum matrices , and vice versa . quadratic identities of our interest in this paper involve products of quantum minors of the form @xmath14[i'|j']$ ] , where @xmath175 . this leads to a proper study of ordered pairs of flows @xmath176 and @xmath177 in an se - graph @xmath49 ( in light of theorem [ tm : linds ] ) . we need some definitions and conventions , borrowing terminology from @xcite . given @xmath178 as above , we call the pair @xmath179 a _ double flow _ in @xmath49 . let @xmath180 note that @xmath57 and @xmath181 imply that @xmath182 is even and @xmath183 we refer to the quadruple @xmath23 as above as a _ cortege _ , and to @xmath184 as the _ refinement _ of @xmath23 , or as a _ refined cortege_. it is convenient for us to interpret @xmath185 and @xmath186 as the sets of _ white _ and _ black _ elements of @xmath187 , respectively , and similarly for @xmath188 , and visualize these objects by use of a _ circular diagram _ @xmath189 in which the elements of @xmath187 ( resp . @xmath190 ) are disposed in the increasing order from left to right in the upper ( resp . lower ) half of a circumference @xmath191 . for example if , say , @xmath192 , @xmath193 , @xmath194 and @xmath195 , then the diagram is viewed as in the left fragment of the picture below . ( sometimes , to avoid a possible mess between elements of @xmath187 and @xmath190 , and when it leads to no confusion , we denote elements of @xmath190 with primes . ) let @xmath196 be a partition of @xmath197 into 2-element sets ( recall that @xmath32 denotes the disjoint union of sets @xmath33 ) . we refer to @xmath196 as a _ perfect matching _ on @xmath197 , and to its elements as _ couples_. more specifically , we say that @xmath198 is : an @xmath112-_couple _ if @xmath199 , a @xmath42-_couple _ if @xmath200 , and an @xmath201-_couple _ if @xmath202 ( as though @xmath203 `` connects '' two sources , two sinks , and one source and one sink , respectively ) . * definition . * a ( perfect ) matching @xmath196 as above is called a _ feasible _ matching for @xmath184 ( and for @xmath23 ) if : [ eq : feasm ] * for each @xmath204 , the elements @xmath205 have different colors if @xmath203 is an @xmath112- or @xmath42-couple , and have the same color if @xmath203 is an @xmath201-couple ; * @xmath196 is _ planar _ , in the sense that the chords connecting the couples in the circumference @xmath191 are pairwise non - intersecting . the set of feasible matchings for @xmath184 is denoted by @xmath206 and may also be denoted as @xmath207 . this set is nonempty unless @xmath208 . ( a proof : a feasible matching can be constructed recursively as follows . let for definiteness @xmath209 . if @xmath210 , then choose @xmath211 and @xmath212 with @xmath213 minimum , form the @xmath112-couple @xmath214 and delete @xmath205 . and so on until @xmath186 becomes empty . act similarly for @xmath215 and @xmath216 . eventually , in view of ( [ eq : balancij ] ) , we obtain @xmath217 and @xmath218 . then we form corresponding white @xmath201-couples . ) the right fragment of the above picture illustrates an instance of feasible matchings . return to a double flow @xmath179 as above . our aim is to associate to it a feasible matching for @xmath184 . to do this , we write @xmath219 and @xmath220 , respectively , for the sets of vertices and edges of @xmath49 occurring in @xmath158 , and similarly for @xmath221 . an important role will be played by the subgraph @xmath222 of @xmath49 induced by the set of edges @xmath223 ( where @xmath30 denotes @xmath31 ) . note that a vertex @xmath87 of @xmath222 has degree 1 if @xmath224 , and degree 2 or 4 otherwise . we slightly modify @xmath222 by splitting each vertex @xmath87 of degree 4 in @xmath222 ( if any ) into two vertices @xmath225 disposed in a small neighborhood of @xmath87 so that the edges entering ( resp . leaving ) @xmath87 become entering @xmath226 ( resp . leaving @xmath227 ) ; see the picture . the resulting graph , denoted as @xmath228 , is planar and has vertices of degree only 1 and 2 . therefore , @xmath229 consists of pairwise disjoint ( non - directed ) simple paths @xmath230 ( considered up to reversing ) and , possibly , simple cycles @xmath231 . the corresponding images of @xmath230 ( resp . @xmath231 ) give paths @xmath232 ( resp . cycles @xmath233 ) in @xmath222 . when @xmath222 has vertices of degree 4 , some of the latter paths and cycles may be self - intersecting and may `` touch '' , but not `` cross '' , each other . [ lm : p1pk ] ( i ) @xmath234 ; \(ii ) the set of endvertices of @xmath232 is @xmath235 ; moreover , each @xmath236 connects either @xmath237 and @xmath238 , or @xmath239 and @xmath240 , or @xmath237 and @xmath239 , or @xmath238 and @xmath240 ; \(iii ) in each path @xmath236 , the edges of @xmath158 and the edges of @xmath221 have different directions ( say , the former edges are all forward , and the latter ones are all backward ) . \(i ) is trivial , and ( ii ) follows from ( iii ) and the fact that the sources @xmath51 ( resp . sinks @xmath148 ) have merely leaving ( resp . entering ) edges . in its turn , ( iii ) easily follows by considering a common inner vertex @xmath87 of a directed path @xmath241 in @xmath158 and a directed path @xmath242 in @xmath221 . let @xmath243 ( resp . @xmath244 ) be the edges of @xmath241 ( resp . @xmath242 ) incident to @xmath87 . then : if @xmath245 , then @xmath87 vanishes in @xmath222 . if @xmath246 and @xmath247 , then either both @xmath248 enter @xmath87 , or both @xmath248 leave @xmath87 ; whence @xmath248 are consecutive and differently directed edges of some path @xmath236 or cycle @xmath249 . a similar property holds when @xmath250 , as being a consequence of splitting @xmath87 into two vertices as described . thus , each @xmath236 is represented as a concatenation @xmath251 of forwardly and backwardly directed paths which are alternately contained in @xmath158 and @xmath221 , called the _ segments _ of @xmath236 . we refer to @xmath236 as an _ exchange path _ ( by a reason that will be clear later ) . the endvertices of @xmath236 determine , in a natural way , a pair of elements of @xmath197 , denoted by @xmath252 . then @xmath253 is a perfect matching on @xmath197 . moreover , it is a feasible matching , since ( [ eq : feasm])(i ) follows from lemma [ lm : p1pk](ii ) , and ( [ eq : feasm])(ii ) is provided by the fact that @xmath230 are pairwise disjoint simple paths in @xmath229 . we denote @xmath196 as @xmath254 , and for @xmath198 , denote the exchange path @xmath236 corresponding to @xmath203 ( i.e. , @xmath255 ) by @xmath256 . [ cor : mphiphip ] @xmath257 . figure [ fig : phi ] illustrates an instance of @xmath179 for @xmath258 , @xmath259 , @xmath260 , @xmath261 . here @xmath158 and @xmath221 are drawn by solid and dotted lines , respectively ( in the left fragment ) , the subgraph @xmath262 consists of three paths and one cycle ( in the middle ) , and the circular diagram illustrates @xmath254 ( in the right fragment ) . and @xmath221 ( left ) ; @xmath263 ( middle ) ; @xmath254 ( right ) ] * flow exchange operation . * it rearranges a given double flow @xmath179 for @xmath264 into another double flow @xmath265 for some cortege @xmath266 , as follows . fix a submatching @xmath267 , and combine the exchange paths concerning @xmath268 , forming the set of edges @xmath269 ( where @xmath270 denotes the set of edges in a path @xmath78 ) . [ lm : phi - psi ] let @xmath271 . define @xmath272 then the subgraph @xmath273 induced by @xmath274 gives a @xmath275-flow , and the subgraph @xmath276 induced by @xmath277 gives a @xmath278-flow in @xmath49 . furthermore , @xmath279 , @xmath280 ( @xmath281 ) , and @xmath282 . consider a path @xmath283 for @xmath284 , and let @xmath78 consist of segments @xmath285 . let for definiteness the segments @xmath286 with @xmath104 odd concern @xmath158 , and denote by @xmath287 the common endvertex of @xmath286 and @xmath288 . under the operation @xmath289 the pieces @xmath290 in @xmath158 are replaced by @xmath291 . in its turn , @xmath292 replaces the pieces @xmath291 in @xmath221 by @xmath290 . by lemma [ lm : p1pk](iii ) , for each @xmath104 , the edges of @xmath293 incident to @xmath287 either both enter or both leave @xmath287 . also each intermediate vertex of any segment @xmath286 occurs in exactly one flow among @xmath294 . these facts imply that under the above operations with @xmath78 the flow @xmath158 ( resp . @xmath221 ) is transformed into a set of pairwise disjoint directed paths ( a flow ) going from @xmath295 to @xmath296 ( resp . from @xmath297 to @xmath298 ) . doing so for all @xmath256 with @xmath284 , we obtain flows @xmath299 from @xmath300 to @xmath301 and from @xmath302 to @xmath303 , respectively . the equalities in the last sentence of the lemma are easy . we call the transformation @xmath304 in this lemma the _ flow exchange operation _ for @xmath179 using @xmath267 ( or using @xmath305 ) . clearly the exchange operation applied to @xmath265 using the same @xmath268 returns @xmath179 . the picture below illustrates flows @xmath299 obtained from @xmath294 in fig . [ fig : phi ] by the exchange operations using the single path @xmath306 ( left ) and the single path @xmath307 ( right ) . so far our description has been close to that given for the commutative case in @xcite . from now on we will essentially deal with the quantum version . the next theorem will serve the main working tool in our arguments ; its proof appealing to a combinatorial techniques on paths and flows is given in appendix b. [ tm : single_exch ] let @xmath158 be an @xmath154-flow , and @xmath221 an @xmath308-flow in @xmath49 . let @xmath265 be the double flow obtained from @xmath179 by the flow exchange operation using a single couple @xmath309 . then : \(i ) when @xmath203 is an @xmath112- or @xmath42-couple and @xmath310 , @xmath311 \(ii ) when @xmath203 is an @xmath201-couple , @xmath312 . an immediate consequence from this theorem is the following [ cor : gen_exch ] for an @xmath154-flow @xmath158 and an @xmath308-flow @xmath221 , let @xmath265 be obtained from @xmath179 by the flow exchange operation using a set @xmath267 . then @xmath313 where @xmath314 ( resp . @xmath315 ) is the number of @xmath112- or @xmath42-couples @xmath316 such that @xmath310 and @xmath317 ( resp . @xmath318 ) . indeed , the flow exchange operation using the whole @xmath268 reduces to performing , step by step , the exchange operations using single couples @xmath319 ( taking into account that for any current double flow @xmath320 occurring in the process , the sets @xmath321 and @xmath322 , as well as the matching @xmath323 , do not change ; cf . lemma [ lm : phi - psi ] ) . then ( [ eq : gen_exch ] ) follows from theorem [ tm : single_exch ] . as before , we consider an se - graph @xmath324 and the weight function @xmath325 which is initially defined on the edges of @xmath49 by ( [ eq : edge_weight ] ) and then extends to paths and flows according to ( [ eq : wp ] ) and ( [ eq : w_phi ] ) . this gives rise to the minor function on the set @xmath326,\ ; j\in[n],\ ; |i|=|j|\}$ ] . in this section , based on corollary [ cor : gen_exch ] describing the transformation of the weights of double flows under the exchange operation , and developing a @xmath5-version of the flow - matching method elaborated for the commutative case in @xcite , we establish sufficient conditions on quadratic relations for @xmath5-minors of the matrix @xmath151 , to be valid independently of @xmath49 ( and some other objects , see remark 2 below ) . relations of our interest are of the form @xmath327[i'|j ' ] = \sum\nolimits_\kscr q^{\beta(k|l , k'|l ' ) } [ k|l][k'|l'],\ ] ] where @xmath328 are integer - valued , @xmath329 is a family of corteges @xmath330 ( with possible multiplicities ) , and similarly for @xmath331 . ( [ eq : q_ident ] ) . we usually assume that @xmath329 and @xmath331 are _ homogeneous _ , in the sense that for any @xmath332 and @xmath333 , @xmath334 moreover , we shall see that only the refinements @xmath184 and @xmath335 are important , whereas the sets @xmath336 and @xmath337 are , in fact , indifferent . ( as before , @xmath185 means @xmath338 , @xmath186 means @xmath339 , and so on . ) to formulate the validity conditions , we need some definitions and notation . @xmath340 we say that a tuple @xmath341 , where @xmath342 and @xmath343 ( cf . ( [ eq : feasm ] ) ) , is a _ configuration _ for @xmath329 . the family of all configurations for @xmath329 is denoted by @xmath344 . similarly , we define the family @xmath345 of configurations for @xmath331 . @xmath340 define @xmath346 to be the family of all matchings @xmath196 occurring in the members of @xmath344 , respecting multiplicities ( i.e. , @xmath346 is a multiset ) . define @xmath347 similarly . * definition . * families @xmath329 and @xmath331 are called _ balanced _ ( borrowing terminology from @xcite ) if there exists a bijection @xmath348 between @xmath344 and @xmath345 such that @xmath349 . in other words , @xmath329 and @xmath331 are balanced if @xmath350 . * definition . * we say that families @xmath329 and @xmath331 along with functions @xmath351 and @xmath352 are @xmath5-_balanced _ if there exists a bijection @xmath353 as above such that , for each @xmath354 and for @xmath355 , there holds @xmath356 ( in particular , @xmath357 are balanced . ) here @xmath358 are defined according to corollary [ cor : gen_exch ] . namely , @xmath314 and @xmath315 , where @xmath268 is the set of couples @xmath198 such that the white / black colors of the elements of @xmath203 in the refined corteges @xmath184 and @xmath335 are different . ( then @xmath359 ( @xmath360 ) is the number of @xmath112- and @xmath42-couples @xmath361 with @xmath310 and @xmath362 ( resp . @xmath363 . ) we say that @xmath335 is obtained from @xmath184 by the _ index exchange operation _ using @xmath268 , and may write @xmath364 for @xmath359 , and @xmath365 for @xmath360 . [ tm : suff_q_bal ] let @xmath329 and @xmath331 be homogeneous families on @xmath366 , and let @xmath351 and @xmath352 . suppose that @xmath367 are @xmath5-balanced . then for any se - graph @xmath324 , relation ( [ eq : gen_qr ] ) is valid for @xmath5-minors of @xmath151 . it is close to the proof for the commutative case in ( * ? ? ? * proposition 3.2 ) . we fix @xmath49 and denote by @xmath368 the set of double flows for @xmath369 in @xmath49 . a summand concerning @xmath370 in the l.h.s . of ( [ eq : gen_qr ] ) can be expressed via double flows as follows , ignoring the factor of @xmath371 : @xmath372[i'|j']=\left ( \sum\nolimits_{\phi\in\phi_g(i|j ) } w(\phi)\right ) \times\left(\sum\nolimits_{\phi'\in\phi_g(i'|j ' ) } w(\phi')\right ) \\ = \sum\nolimits_{(\phi,\phi')\in\dscr(i|j , i'|j ' ) } w(\phi)w(\phi ' ) \qquad\qquad\qquad\qquad\qquad\qquad\\ = \sum\nolimits_{m\in\mscr_{\iw,\ib,\jw,\jb } } \sum\nolimits_{(\phi,\phi')\in\dscr(i|j , i'|j')\,:\ , m(\phi,\phi')=m } w(\phi)w(\phi ' ) . \end{gathered}\ ] ] the summand for @xmath333 in the r.h.s . of ( [ eq : gen_qr ] ) is expressed similarly . consider a configuration @xmath373 and suppose that @xmath179 is a double flow for @xmath23 with @xmath374 ( if such a double flow in @xmath49 exists ) . since @xmath367 are @xmath5-balanced , @xmath375 is bijective to some configuration @xmath376 satisfying ( [ eq : q_balan ] ) . as explained earlier , the cortege @xmath377 is obtained from @xmath23 by the index exchange operation using some @xmath378 . then the flow exchange operation applied to @xmath179 using this @xmath268 results in a double flow @xmath265 for @xmath377 which satisfies relation ( [ eq : gen_exch ] ) in corollary [ cor : gen_exch ] . comparing ( [ eq : gen_exch ] ) with ( [ eq : q_balan ] ) , we observe that @xmath379 furthermore , such a map @xmath380 gives a bijection between all double flows concerning configurations in @xmath344 and those in @xmath345 . now the desired equality ( [ eq : gen_qr ] ) follows by comparing the last term in expression ( [ eq : ff ] ) and the corresponding term in the analogous expression concerning @xmath331 . as a consequence of theorems [ tm : cach - cast ] and [ tm : suff_q_bal ] , the following result is obtained . [ cor : qr_quant_matr ] if @xmath367 as above are @xmath5-balanced , then relation ( [ eq : gen_qr ] ) is valid for the corresponding minors in the algebra @xmath6 of quantum @xmath1 matrices . when speaking of a _ universal quadratic identity _ of the form ( [ eq : gen_qr ] ) with homogeneous @xmath329 and @xmath331 , abbreviated as a _ uq identity _ , we mean that it depends neither on the graph @xmath49 nor on the field @xmath2 and element @xmath381 , and that the index sets can be modified as follows . given @xmath332 , let @xmath382 , @xmath383 , @xmath384 and @xmath385 ( by the homogeneity , these sets do not depend on @xmath386 ) . take arbitrary @xmath387 and @xmath388 and replace @xmath389 by disjoint sets @xmath390 $ ] and disjoint sets @xmath391 $ ] such that @xmath392 , @xmath393 and @xmath394 . let @xmath395 and @xmath396 be the order preserving maps . transform each @xmath397 into @xmath398 , where @xmath399 forming a new family @xmath400 on @xmath401 . transform @xmath331 into @xmath402 in a similar way . one can see that if @xmath367 are @xmath5-balanced , then so are @xmath403 , keeping @xmath404 . therefore , if ( [ eq : gen_qr ] ) is valid for @xmath357 , then it is valid for @xmath405 as well . thus , the condition of @xmath5-balancedness is sufficient for validity of relation ( [ eq : gen_qr ] ) for minors of any @xmath5-matrix . in section [ sec : necess ] we shall see that this condition is necessary as well ( theorem [ tm : nec_q_bal ] ) . one can say that identity ( [ eq : gen_qr ] ) , where all summands have positive signs , is written in the _ canonical form_. sometimes , however , it is more convenient to consider equivalent identities having negative summands in one or both sides ( e.g. of the form ( [ eq : q_ident ] ) ) . also one may simultaneously multiply all summands in ( [ eq : gen_qr ] ) by the same degree of @xmath5 . a useful fact is that once we are given an instance of ( [ eq : gen_qr ] ) , we can form another identity by changing the white / black coloring in all refined corteges . more precisely , for a cortege @xmath27 , let us say that the cortege @xmath406 is _ reversed _ to @xmath375 . given a family @xmath329 of corteges , the _ reversed _ family @xmath407 is formed by the corteges reversed to those in @xmath329 . then the following property takes place . [ pr : reversed ] suppose that @xmath367 are @xmath5-balanced . then @xmath408 are q - balanced as well . therefore ( by theorem [ tm : suff_q_bal ] ) , @xmath409[i|j ] = \sum_{(k|l , k'|l')\in \kscr } q^{-\beta(k|l , k'|l ' ) } [ k'|l'][k|l].\ ] ] let @xmath410 be a bijection in the definition of @xmath5-balancedness . then @xmath353 induces a bijection of @xmath411 to @xmath412 ( also denoted as @xmath353 ) . namely , if @xmath413 for @xmath414 and @xmath415 , then we define @xmath416 . when coming from @xmath375 to @xmath417 , each @xmath112- or @xmath42-couple @xmath214 in @xmath196 changes the colors of both elements @xmath205 this leads to swapping @xmath359 and @xmath360 , i.e. , @xmath418 and @xmath419 ( where @xmath268 is the submatching in @xmath196 involved in the exchange operation ) . now ( [ eq : reverqr ] ) follows from relation ( [ eq : q_balan ] ) . another useful equivalent transformation is given by swapping row and column indices . namely , for a cortege @xmath27 , the _ transposed _ cortege is @xmath420 , and the family @xmath421 _ transposed _ to @xmath329 consists of the corteges @xmath422 for @xmath423 , and similarly for @xmath331 . one can see that the corresponding values @xmath359 and @xmath360 preserve when coming from @xmath329 to @xmath421 and from @xmath331 to @xmath424 , and therefore ( [ eq : q_balan ] ) implies the identity @xmath425[j'|i ' ] = \sum_{(k|l , k'|l')\in \kscr } q^{\beta(k|l , k'|l ' ) } [ l|k][l'|k'].\ ] ] ( note also that ( [ eq : transqr ] ) immediately follows from the known fact that any @xmath5-minor satisfies the symmetry relation @xmath426_q=[j|i]_q$ ] . ) we conclude this section with a rather simple algorithm which has as the input a corresponding quadruple @xmath367 and recognizes the @xmath5-balanced for it . therefore , in light of theorems [ tm : suff_q_bal ] and [ tm : nec_q_bal ] , the algorithm decides whether or not the given quadruple determines a uq identity of the form ( [ eq : gen_qr ] ) . * algorithm . * compute the set @xmath206 of feasible matchings @xmath196 for each @xmath332 , and similarly for @xmath331 . for each instance @xmath196 occurring there , we extract the family @xmath427 of all configurations concerning @xmath196 in @xmath344 , and extract a similar family @xmath428 in @xmath345 . if @xmath429 for at least one instance @xmath196 , then @xmath329 and @xmath331 are not balanced at all . otherwise for each @xmath196 , we seek for a required bijection @xmath430 by solving the maximum matching problem in the corresponding bipartite graph @xmath431 . more precisely , the vertices of @xmath431 are the tuples @xmath341 and @xmath432 occurring in @xmath427 and @xmath428 , and such tuples are connected by edge in @xmath431 if they obey ( [ eq : q_balan ] ) . find a maximum matching @xmath433 in @xmath431 . ( there are many fast algorithms to solve this classical problem ; for a survey , see , e.g. @xcite . ) if @xmath434 , then @xmath433 determines the desired @xmath435 in a natural way . taking together , these @xmath435 give a bijection between @xmath344 and @xmath345 as required , implying that @xmath367 are @xmath5-balanced and if @xmath436 for at least one instance @xmath196 , then the algorithm declares the non-@xmath5-balancedness . the flow - matching method described above is well adjusted to prove , relatively easily , classical or less known quadratic identities . in this section we give a number of appealing illustrations . instead of circular diagrams as in section [ sec : double ] , we will use more compact , but equivalent , _ two - level diagrams_. also when dealing with a flag pair @xmath154 , i.e. , when @xmath97 consists of the elements @xmath437 , we may use an appropriate _ one - level diagrams _ , which leads to no loss of generality . for example , the refined cortege @xmath438 with the feasible matching @xmath439 can be visualized in three possible ways as : a couple @xmath214 may be denoted as @xmath440 . also for brevity we write @xmath441 for @xmath442 , where @xmath7 and @xmath443 are disjoint . as before , we use notation @xmath14 $ ] for the corresponding @xmath5-minor of the path matrix @xmath151 ( defined in section [ sec : flows ] ) . in the flag case @xmath444 is usually abbreviated to @xmath64 $ ] ( in view of @xmath445 ) . we start with a simple illustration of our method by showing that @xmath5-minors @xmath14 $ ] and @xmath60 $ ] `` purely '' commute when @xmath61 and @xmath446 . ( this matches the known fact that a minor of a @xmath5-matrix commutes with any of its subminors , or that the @xmath5-determinant of a square @xmath5-matrix is a central element of the corresponding algebra . ) let @xmath447 consist of @xmath448 , and @xmath449 consist of @xmath450 . since @xmath451 and @xmath452 , there is only one feasible matching @xmath196 for @xmath184 ; namely , the one formed by the @xmath201-couples @xmath453 , @xmath454 . the index exchange operation applied to @xmath23 using the whole @xmath196 produces the cortege @xmath377 for which @xmath455 , @xmath456 , @xmath457 , @xmath458 ( and @xmath459 , @xmath460 ) . since @xmath196 consists of @xmath201-couples only , we have @xmath461 . so the ( one - element ) families @xmath462 and @xmath463 along with @xmath464 are @xmath5-balanced , and theorem [ tm : suff_q_bal ] gives the desired equality @xmath14[i'|j']=[i'|j'][i|j]$ ] . this is illustrated in the picture with two - level diagrams ( in case @xmath465 ) . hereinafter we indicate by crosses the couples that are involved in the index exchange operation that is applied ( i.e. , the couples where the colors of elements are changed ) . recall that two sets @xmath466 $ ] are called _ weakly separated _ if , up to renaming @xmath97 and @xmath99 , there holds : @xmath467 , and @xmath468 has a partition @xmath469 such that @xmath470 ( where we write @xmath471 if @xmath472 for any @xmath473 and @xmath474 ) . leclerc and zelevinsky proved the following [ tm : lz ] two flag minors @xmath63 $ ] and @xmath64 $ ] of a quantum matrix _ quasicommute _ , i.e. , satisfy @xmath475[j ] = q^c [ j][i]\ ] ] for some @xmath476 , if and only if the column sets @xmath67 are weakly separated . moreover , when @xmath467 and @xmath469 is a partition of @xmath468 with @xmath470 , the number @xmath477 in ( [ eq : quasiij ] ) is equal to @xmath478 . ( in case @xmath479 , `` if '' part is due to krob and leclerc @xcite ) . we explain how to obtain `` if '' part of theorem [ tm : lz ] by use of the flow - matching method . let @xmath480 , @xmath481 , and define @xmath482 one can see that @xmath483 has exactly one feasible matching @xmath196 ; namely , @xmath484 is coupled with the first @xmath485 elements of @xmath185 , @xmath486 is coupled with the last @xmath487 elements of @xmath185 ( forming all @xmath42-couples ) , and the rest of @xmath185 is coupled with @xmath488 ( forming all @xmath201-couples ) . observe that the index exchange operation applied to @xmath489 using the whole @xmath196 swaps @xmath490 and @xmath491 ( since it changes the colors of all elements in @xmath488 , @xmath185 and @xmath216 ) . also @xmath196 consists of @xmath492 @xmath42-couples and @xmath493 @xmath201-couples . moreover , the @xmath42-couples are partitioned into @xmath485 couples @xmath440 with @xmath494 and @xmath495 , and @xmath487 couples @xmath440 with @xmath494 and @xmath496 . this gives @xmath497 and @xmath498 . hence the ( one - element ) families @xmath499 and @xmath500 along with @xmath501 and @xmath502 are @xmath5-balanced . now theorem [ tm : suff_q_bal ] implies ( [ eq : quasiij ] ) with @xmath503 . the picture with two - level diagrams illustrates the case @xmath504 , @xmath505 , @xmath506 and @xmath507 . `` only if '' part of theorem [ tm : lz ] will be discussed in section [ sec : concl ] . also we will give there a generalization of this theorem that characterizes the set of all pairs of quasicommuting @xmath5-minors ( not necessarily flag ones ) . we prove theorem [ tm : path - a ] . \(a ) consider entries @xmath508 $ ] and @xmath509 $ ] with @xmath510 in @xmath151 . the cortege @xmath511 admits a unique feasible matching ; it consists of the single @xmath42-couple @xmath512 . the index exchange operation using @xmath203 transforms @xmath375 into @xmath513 ; see the picture with one - level diagrams : we observe that @xmath514 and @xmath515 along with @xmath516 and @xmath517 ( @xmath518 ) are @xmath5-balanced , and theorem [ tm : suff_q_bal ] yields @xmath508[i|j']=q[i|j'][i|j]$ ] , as required . \(b ) for a @xmath519 submatrix of @xmath151 , the argument is similar . \(c ) consider a @xmath520 submatrix @xmath521 of @xmath151 , where @xmath522 $ ] , @xmath523 $ ] , @xmath524 $ ] , @xmath525 $ ] ( then @xmath526 and @xmath510 ) . let @xmath329 consist of two corteges @xmath527 , @xmath528 , and @xmath331 consist of two corteges @xmath529 , @xmath530 ( note that @xmath531 ) . observe that @xmath532 admits 2 feasible matchings , namely , @xmath533 and @xmath534 , while @xmath535 admits only one feasible matching @xmath196 . in their turn , @xmath536 and @xmath537 . hence we can form the bijection between @xmath344 and @xmath345 that sends @xmath538 to @xmath539 , @xmath540 to @xmath541 , and @xmath542 to @xmath543 . this bijection is illustrated in the picture ( where , as before , we indicate the submathings involved in the exchange operations with crosses ) . assign @xmath544 , @xmath545 , @xmath546 and @xmath547 . one can observe from the above diagrams that @xmath367 are @xmath5-balanced . we obtain @xmath548[i'|j']+q^{-1}[i|j'][i'|j]=q[i|j'][i'|j]+[i'|j'][i|j],\ ] ] yielding @xmath549 , as required . finally , to see @xmath550 , take the 1-element families @xmath551 and @xmath552 ; then @xmath553 is the only feasible matching for each of @xmath554 . the above families along with @xmath464 are @xmath5-balanced , as is seen from the picture : this gives @xmath509[i'|j]=[i'|j][i|j']$ ] , or @xmath550 , as required . in the commutative case ( when dealing with the commutative coordinate ring of @xmath1 matrices over a field ) , the simplest examples of quadratic identities on flag minors are presented by the classical plcker relations involving 3- and 4-element sets of columns . more precisely , for @xmath555 $ ] , let @xmath556 denote the flag minor with the set @xmath95 of columns . then for any three elements @xmath65 in @xmath557 $ ] and a set @xmath558-\{i , j , k\}$ ] , there holds @xmath559 and for any @xmath66 and @xmath560-\{i , j , k,\ell\}$ ] , @xmath561 there are two quantized counterparts of ( [ eq : p3 ] ) ( concerning flag @xmath5-minors of the matrix @xmath151 ) . one of them is viewed as @xmath562[xik]=[xij][xk]+[xjk][xi],\ ] ] and the other as @xmath563[xj]=q^{-1}[xij][xk]+q[xjk][xi].\ ] ] to see ( [ eq : qp3a ] ) , associate to @xmath564 the white pair @xmath565 , and to @xmath566 the black pair @xmath567 , where @xmath568 is the last row index for @xmath569 $ ] ( i.e. , @xmath570 ) . then @xmath206 consists of two feasible matchings : @xmath571 and @xmath572 . now ( [ eq : qp3a ] ) is seen from the following picture with two - level diagrams , where we write @xmath375 for the cortege @xmath573\,|xj , [ p]\,|xik)$ ] , @xmath574 for @xmath575\,|xij , [ p-1]\,|xk)$ ] , and @xmath576 for @xmath575\,|xjk , [ p-1]\,|xi)$ ] : as to ( [ eq : qp3b ] ) , it suffices to consider one - level diagrams ( as we are not going to use @xmath201-couples in the exchange operations ) . now the `` white '' object is the column set @xmath577 and the `` black '' object is @xmath578 . then @xmath579 consists of two feasible matchings , one using the @xmath42-couple @xmath580 , and the other using the @xmath42-couple @xmath581 . now ( [ eq : qp3b ] ) can be seen from the picture , where we write @xmath375 for the flag cortege @xmath582 , @xmath574 for @xmath583 , and @xmath576 for @xmath584 . next we demonstrate the following quantized counterpart of ( [ eq : p4 ] ) : @xmath585[xj\ell]=q^{-1}[xij][xk\ell]+q[xi\ell][xjk].\ ] ] to see this , we use one - level diagrams and consider the column sets @xmath577 and @xmath586 . then @xmath587 consists of two feasible matchings : @xmath588 and @xmath589 . identity ( [ eq : qp4 ] ) can be seen from the picture , where @xmath590 , @xmath591 and @xmath592 . note that , if wished , one can produce more identities from ( [ eq : qp3a ] ) and ( [ eq : qp3b ] ) , using the fact that @xmath593 and @xmath594 ( as well as @xmath595 and @xmath596 ) are weakly separated , and therefore their corresponding flag @xmath5-minors quasicommute ( see section [ ssec : quasicommut ] ) . in contrast , @xmath564 and @xmath566 are not weakly separated . next , subtracting from ( [ eq : qp3b ] ) identity ( [ eq : qp3a ] ) multiplied by @xmath5 results in the identity of the form @xmath597[xj]=q[xj][xik]-(q - q^{-1})[xij][xk],\ ] ] which is in spirit of _ commutation relations _ for quantum minors studied in @xcite . as one more simple illustration of our method , we consider a @xmath5-analogue of the classical dodgson s condensation formula for usual minors @xcite . it can be stated as follows : for elements @xmath598 of @xmath599 $ ] , a set @xmath600-\{i , k\}$ ] , elements @xmath601 of @xmath557 $ ] , and a set @xmath602-\{i',k'\}$ ] ( with @xmath603 ) , @xmath604[xk|x'k']= q[xi|x'k'][xk|x'i']+[xik|x'i'k'][x|x'].\ ] ] in this case we deal with the cortege @xmath605 and its refinement @xmath184 of the form @xmath606 . the latter admits two feasible matchings : @xmath607 and @xmath608 . now ( [ eq : dodg ] ) can be concluded by examining the picture below , where @xmath574 stands for @xmath609 , and @xmath576 for @xmath610 : two representable quadratic identities of a general form were established for quantum flag minors in @xcite . the first one considers column subsets @xmath611 $ ] with @xmath612 and is viewed as @xmath613[j]= \sum_{\mu\subseteq j - i,\ , |\mu|=|j|-|i| } ( -q)^{inv(j-\mu,\,\mu)-inv(i,\,\mu ) } [ i\cup\mu][j-\mu],\ ] ] where @xmath614 denotes the number of pairs @xmath615 with @xmath616 . observe that ( [ eq : qp3a ] ) is a special case of ( [ eq : r1 ] ) in which the roles of @xmath97 and @xmath99 are played by @xmath564 and @xmath566 , respectively . indeed , in this case @xmath617 ranges over the singletons @xmath618 and @xmath619 , and we have @xmath620 and @xmath621 . ( for brevity , we write @xmath622 for @xmath623 . ) the second one considers @xmath611 $ ] with @xmath624 and is viewed as @xmath625[i - a ] = 0\ ] ] ( where we write @xmath626 for @xmath627 , and @xmath628 for @xmath629 ) . a special case is ( [ eq : qp4 ] ) ( with @xmath630 and @xmath631 ) . we explain how ( [ eq : r1 ] ) and ( [ eq : r2 ] ) can be proved for flag @xmath5-minors of @xmath151 by use of our flow - matching method . * proof of ( [ eq : r1 ] ) . * the pair @xmath632 corresponds to the cortege @xmath633\,|i,\;[p+k]\,|j)$ ] and its refinement @xmath634 , where @xmath635 and @xmath636 . in its turn , each pair @xmath637 occurring in the r.h.s . of ( [ eq : r1 ] ) corresponds to the cortege @xmath638\,|(i_\mu:=i\cup\mu),\ ; [ p]\,|(j_\mu:=j-\mu))$ ] and its refinement @xmath639 . so we deal with the set @xmath640 of corteges and the related set @xmath641 of configurations ( of the form @xmath642 or @xmath643 ) , and our aim is to construct an involution @xmath644 which is agreeable with matchings , signs and @xmath5-factors figured in ( [ eq : r1 ] ) . ( under reducing ( [ eq : r1 ] ) to the canonical form , @xmath645 splits into two families @xmath329 and @xmath331 , and @xmath353 determines the @xmath5-balancedness for @xmath357 with corresponding @xmath404 . ) consider a refined cortege @xmath646 and a feasible matching @xmath196 for it . note that @xmath196 consists of @xmath647 @xmath201-couples ( connecting @xmath648 and @xmath649 ) and @xmath650 @xmath42-couples ( connecting @xmath649 and @xmath651 ) . two cases are possible . : each @xmath42-couple connects @xmath651 and @xmath185 . then all @xmath201-couples in @xmath196 connect @xmath648 and @xmath617 . therefore , the exchange operation applied to @xmath652 using the set @xmath268 of all @xmath201-couples of @xmath196 produces the `` initial '' cortege @xmath375 ( corresponding to the refinement @xmath653 ) . clearly @xmath196 is a feasible matching for @xmath375 and the exchange operation applied to @xmath375 using @xmath268 returns @xmath652 . we link @xmath642 and @xmath643 by @xmath353 . note that for each @xmath42-couple @xmath654 and for each @xmath655 , either @xmath656 or @xmath657 ( otherwise the @xmath201-couple containing @xmath658 would `` cross '' @xmath203 , contrary to the planarity requirement ( [ eq : feasm])(ii ) for @xmath196 ) . this implies @xmath659 , whence the terms @xmath63[j]$ ] in the l.h.s . and @xmath660[j_\mu]$ ] in the r.h.s . of ( [ eq : r1 ] ) are @xmath5-balanced . : there is a @xmath42-couple in @xmath196 connecting @xmath651 and @xmath617 . among such couples , choose the couple @xmath661 with @xmath494 such that : ( a ) @xmath662 is minimum , and ( b ) @xmath21 is minimum subject to ( a ) . from ( [ eq : feasm ] ) and ( a ) it follows that [ eq : betw_ij ] if a couple @xmath663 has an element ( strictly ) between @xmath21 and @xmath664 , then @xmath665 connects @xmath185 and @xmath651 , and the other element of @xmath665 is between @xmath21 and @xmath664 as well . let @xmath666 be obtained by applying to @xmath652 the exchange operation using the single couple @xmath203 . then @xmath667 , @xmath668 and @xmath669 . the matching @xmath196 is feasible for @xmath666 , we are in case 2 with @xmath666 and @xmath196 , and one can see that the couple @xmath663 chosen for @xmath666 according to the above rules ( a),(b ) coincides with @xmath203 . based on these facts , we link @xmath670 and @xmath671 by @xmath353 . now we compute and compare the numbers @xmath672 and @xmath673 . let @xmath104 be the number of elements of @xmath185 between @xmath21 and @xmath664 ( recall that @xmath661 and @xmath494 ) . property ( [ eq : betw_ij ] ) ensures that the number of elements of @xmath651 ( as well as of @xmath674 ) between @xmath21 and @xmath664 is equal to @xmath104 too . consider two possibilities . : @xmath675 ( and @xmath676 ) . then @xmath677 and @xmath678 . this implies that @xmath679 and @xmath680 . : @xmath681 ( and @xmath682 ) . then @xmath683 and @xmath684 , yielding @xmath685 and @xmath686 . finally , let @xmath687 and @xmath688 be the multipliers to the terms @xmath689[j_\mu]$ ] and @xmath690[j_{\mu'}]$ ] in ( [ eq : r1 ] ) , respectively . then @xmath691 , which is equal to 1 in subcase 2a and @xmath692 in subcase 2b . in both cases this amounts to the value @xmath693 for the exchange operation applied to @xmath652 using @xmath203 , and validity of ( [ eq : r1 ] ) follows from theorem [ tm : suff_q_bal ] . sometimes it is useful to consider the identity formed by the corteges reversed to those in ( [ eq : r1 ] ) ; by proposition [ pr : reversed ] , it is viewed as @xmath694[i]= \sum_{\mu\subseteq j - i,\ , |\mu|=|j|-|i| } ( -q)^{inv(i,\,\mu)-inv(j-\mu,\,\mu ) } [ j-\mu][i\cup\mu].\ ] ] * proof of ( [ eq : r2 ] ) . * let @xmath695 , @xmath696 , @xmath697-[p+1]$ ] , @xmath698 and @xmath699 . for @xmath700 , the pair @xmath701 in ( [ eq : r2 ] ) corresponds to the cortege @xmath702\,|ja,\ ; [ p+k-1]\,|(i - a))$ ] and its refinement @xmath703 ( we use the fact that @xmath704 ) . we deal with the set @xmath705 of corteges and the related set @xmath641 of configurations @xmath706 , and like the previous proof , our aim is to construct an appropriate involution @xmath707 . consider a refined cortege @xmath708 and a feasible matching @xmath196 for it . take the couple in @xmath196 containing @xmath709 , say , @xmath710 . note that @xmath203 is a @xmath42-couple and @xmath711 ( since @xmath709 is white , and @xmath648 and @xmath712 are black ) . the exchange operation applied to @xmath713 using @xmath203 produces the member @xmath714 of @xmath645 , and we link @xmath713 and @xmath714 by @xmath353 . it remains to estimate the multipliers @xmath687 and @xmath688 to the terms @xmath715[i - a]$ ] and @xmath716[i - b]$ ] in ( [ eq : r2 ] ) , respectively . let @xmath104 be the number of elements of @xmath186 between @xmath709 and @xmath717 . it is equal to the number of elements of @xmath215 between @xmath709 and @xmath717 ( since , in view of ( [ eq : feasm ] ) , the elements of @xmath718 between @xmath709 and @xmath717 must be partitioned into @xmath42-couples in @xmath196 ) . this implies that if @xmath719 , then @xmath720 and @xmath721 . therefore , @xmath722 . and if @xmath616 , then @xmath723 and @xmath724 , whence @xmath725 . in both cases , @xmath726 coincides with the corresponding value of @xmath693 , and the result follows . in this section we show a converse assertion to theorem [ tm : suff_q_bal ] , thus obtaining a complete characterization for the uq identities on quantized minors . this characterization , given in terms of the @xmath5-balancedness , justifies the algorithm of recognizing uq identities described in the end of section [ sec : q_relat ] . as before , we deal with homogeneous families of corteges in @xmath366 . [ tm : nec_q_bal ] let @xmath2 be a field of characteristic zero and let @xmath3 be transcendental over @xmath41 . suppose that @xmath367 ( as in section [ sec : q_relat ] ) are not @xmath5-balanced . then there exists ( and can be explicitly constructed ) an se - graph @xmath49 for which relation ( [ eq : gen_qr ] ) is violated . we essentially use an idea and construction worked out for the commutative version in ( * ? ? ? * sec . 5 ) . recall that the homogeneity of @xmath727 means the existence of @xmath728 $ ] and @xmath729 $ ] such that any cortege @xmath730 satisfies @xmath731 ( cf . ( [ eq : homogen ] ) ) . for a perfect matching @xmath196 on @xmath732 , let us denote by @xmath733 the set of corteges @xmath414 for which @xmath196 is feasible ( see ( [ eq : feasm ] ) ) , and denote by @xmath734 a similar set for @xmath331 . the @xmath5-balancedness of @xmath367 would mean that , for any @xmath735 , there exists a bijection @xmath736 respecting ( [ eq : q_balan ] ) . that is , for any @xmath737 and for @xmath738 , there holds @xmath739 here : @xmath740 is the subset of @xmath196 such that the refined cortege @xmath335 is obtained from @xmath184 by the index exchange operation using @xmath268 , and @xmath741 ( resp . @xmath742 ) is the number of @xmath112- and @xmath42-couples @xmath743 with @xmath494 and @xmath744 ( resp . @xmath745 ) . the following assertion is crucial . [ pr : p1p2 ] let @xmath196 be a perfect planar matching on @xmath732 . then there exists ( and can be explicitly constructed ) an se - graph @xmath68 with the following properties : for each cortege @xmath746 satisfying ( [ eq : xryr ] ) , * if @xmath196 is feasible for @xmath375 , then @xmath49 has a unique @xmath154-flow and a unique @xmath308-flow ; * if @xmath196 is not feasible for @xmath375 , then at least one of @xmath747 and @xmath748 is empty . we will prove this proposition later , and now , assuming that it is valid , we complete the proof of the theorem . let @xmath367 be not @xmath5-balanced . then there exists a matching @xmath735 that admits no bijection @xmath435 as above between @xmath733 and @xmath734 ( in particular , at least one of @xmath733 and @xmath734 is nonempty ) . we fix one @xmath196 of this sort and consider a graph @xmath49 as in proposition [ pr : p1p2 ] for this @xmath196 . our aim is to show that relation ( [ eq : gen_qr ] ) is violated for @xmath5-minors of @xmath749 ( yielding the theorem ) . suppose , for a contradiction , that ( [ eq : gen_qr ] ) is valid . by ( p2 ) in the proposition , we have @xmath14[i'|j']=0 $ ] for each cortege @xmath750 , denoting @xmath751 . on the other hand , ( p1 ) implies that if @xmath752 , then @xmath753[i'|j']=w(\phi_{i|j})\,w(\phi_{i'|j'}),\ ] ] where @xmath754 ( resp . @xmath755 ) is the unique @xmath154-flow ( resp . @xmath308-flow ) in @xmath49 . thus , ( [ eq : gen_qr ] ) can be rewritten as @xmath756 for each cortege @xmath757 , the weight @xmath758 of the double flow @xmath759 is a monomial in weights @xmath126 of edges @xmath760 ( or a laurent monomial in inner vertices of @xmath49 ) ; cf . ( [ eq : wp]),([eq : edge_weight]),([eq : w_phi ] ) . for any two corteges in @xmath761 , one is obtained from the other by the index exchange operation using a submatching of @xmath196 , and we know from the description in section [ sec : double ] that if one double flow is obtained from another by the flow exchange operation , then the ( multi)sets of edges occurring in these double flows are the same ( cf . lemma [ lm : phi - psi ] ) . thus , the ( multi)set of edges occurring in the weight monomial @xmath762 is the same for all corteges @xmath375 in @xmath761 . fix an arbitrary linear order @xmath763 on @xmath70 . then the monomial @xmath764 obtained from @xmath762 by a permutation of the entries so as to make them weakly decreasing w.r.t . @xmath763 from left to right is the same for all @xmath765 . therefore , applying relations ( g1)(g3 ) on vertices of @xmath49 ( in sect . [ ssec : se ] ) , we observe that for @xmath765 , the weight @xmath762 is expressed as @xmath766 for some @xmath767 . using such expressions , we rewrite ( [ eq : qr_m ] ) as @xmath768 obtaining @xmath769 since @xmath5 is transcendental , the polynomials in @xmath5 in both sides of ( [ eq : q_rho ] ) are equal . then @xmath770 and there exists a bijection @xmath771 such that @xmath772 this together with relations of the form ( [ eq : qs - qsigma ] ) gives @xmath773 now , for @xmath737 , let @xmath774 and let @xmath775 . using relation ( [ eq : gen_exch ] ) from corollary [ cor : gen_exch ] , we have @xmath776 whence @xmath777 . thus , the bijection @xmath778 satisfies ( [ eq : baz ] ) . a contradiction . * proof of proposition [ pr : p1p2 ] . * we utilize the construction of a graph ( which need not be an se - graph ) with properties ( p1 ) and ( p2 ) from @xcite ; denote this graph by @xmath779 . we first outline essential details of that construction and then explain how to turn @xmath780 into an equivalent se - graph @xmath49 . a series of transformations of @xmath780 that we apply to obtain @xmath49 consists of subdividing some edges @xmath84 ( i.e. , replacing @xmath85 by a directed path from @xmath86 to @xmath87 ) and parallel shifting some sets of vertices and edges in the plane ( preserving the planar structure of the graph ) . such transformations maintain properties ( p1 ) and ( p2 ) , whence the result will follow . let @xmath781 and @xmath782 . denote the sets of @xmath112- , @xmath42- , and @xmath201-couples in @xmath196 by @xmath783 , and @xmath784 , respectively . an @xmath112-couple @xmath785 with @xmath494 is denoted by @xmath440 , and we denote by @xmath786 the natural partial order on @xmath112-couples where @xmath787 if @xmath788 is an @xmath112-couple with @xmath789 . and similarly for @xmath42-couples . when @xmath790 and there is no @xmath791 between @xmath203 and @xmath665 ( i.e. , @xmath792 ) , we say that @xmath665 is an immediate successor of @xmath203 and denote the set of these by @xmath793 . also for @xmath794 and @xmath795 , we say that @xmath104 is _ open _ for @xmath203 if @xmath796 and there is no @xmath797 with @xmath798 , and denote the set of these by @xmath799 . and similarly for couples in @xmath800 and elements of @xmath801 . a current graph and its ingredients are identified with their images in the plane , and any edge in it is represented by a ( directed ) straight - line segment . we write @xmath802 for the coordinates of a point @xmath87 , and say that an edge @xmath84 _ points down _ if @xmath803 . the initial graph @xmath780 has the following features ( seen from the construction in @xcite ) . \(i ) the `` sources '' @xmath804 ( `` sinks '' @xmath805 ) are disposed in this order from left to right in the upper ( resp . lower ) half of a circumference @xmath191 , and the graph @xmath780 is drawn within the circle ( disk ) @xmath806 surrounded by @xmath191 . ( strictly speaking , the construction of @xmath780 in @xcite is a mirror reflection of that we describe ; the latter is more convenient for us and does not affect the result . ) \(ii ) each couple @xmath807 is extended to a chord between the points @xmath21 and @xmath664 , which is subdivided into a path @xmath808 whose edges are alternately forward and backward . let @xmath809 denote the region in @xmath806 bounded by @xmath808 and the paths @xmath810 for @xmath811 . then each edge @xmath85 of @xmath780 ( regarded as a line - segment ) having a point in the interior of @xmath809 connects a vertex in @xmath808 with either a vertex in @xmath810 for some @xmath811 or some vertex @xmath812 . moreover , @xmath85 is directed to @xmath808 if @xmath813 , and from @xmath808 if @xmath814 . \(iii ) let @xmath815 be the region in @xmath806 bounded by the paths @xmath808 for all maximal @xmath112- and @xmath42-couples @xmath203 . then any edge @xmath85 of @xmath780 having a point in the interior of @xmath815 points down . also if such an @xmath85 has an incident vertex @xmath87 lying on @xmath808 for a maximal @xmath112-couple ( resp . @xmath42-couple ) @xmath203 , then @xmath85 leaves ( resp . enters ) @xmath87 . using these properties , we transform @xmath780 , step by step , keeping notation @xmath779 for a current graph , and @xmath806 for a current region ( which becomes a deformed circle ) containing @xmath780 . iteratively applied steps ( s1 ) and ( s2 ) are intended to obtain a graph whose all edges point down . ( s1 ) choose @xmath816 and let @xmath817 be the `` upper part '' of @xmath806 bounded by @xmath808 ( then @xmath817 contains @xmath808 , the paths @xmath810 for all @xmath787 , and the elements @xmath795 with @xmath796 ) . we shift @xmath818 upward by a sufficiently large distance @xmath819 . more precisely , each vertex @xmath820 lying in @xmath818 is replaced by vertex @xmath226 with @xmath821 and @xmath822 . each edge @xmath823 of the old graph induces the corresponding edge of the new one , namely : edge @xmath824 if both @xmath825 lie in @xmath818 ; edge @xmath826 if @xmath827 ; and edge @xmath828 if @xmath829 and @xmath830 . ( case @xmath831 and @xmath832 is impossible . ) accordingly , the region @xmath806 is enlarged by shifting the part @xmath817 by @xmath833 and filling the gap between @xmath808 and @xmath834 by the corresponding parallelogram . one can realize that upon application of ( s1 ) to all @xmath112-couples , the following property is ensured : for each @xmath813 , all initial edges incident to exactly one vertex on @xmath808 turn into edges pointing down . moreover , since @xmath808 is alternating and there is enough space ( from below and from above ) in a neighborhood of the current @xmath808 , we can deform @xmath808 into a zigzag path with all edges pointing down ( by shifting each inner vertex @xmath87 of @xmath808 by a vector @xmath835 with an appropriate ( positive or negative ) @xmath836 ) . ( s2 ) we choose @xmath814 and act similarly to ( s1 ) with the differences that now @xmath817 denotes the `` lower part '' of @xmath806 bounded by @xmath808 and that @xmath817 is shifted downward ( by a sufficiently large @xmath819 ) . upon termination of the process for all @xmath112- and @xmath42-couples , all edges of the current graph @xmath780 ( which is homeomorphic to the initial one ) point down , as required . moreover , @xmath780 has one more useful property : the sources @xmath804 are `` seen from above '' and the sinks @xmath837 are `` seen from below '' . hence we can add to @xmath780 `` long '' vertical edges @xmath838 entering the vertices @xmath839 , respectively , and `` long '' vertical edges @xmath840 leaving the vertices @xmath805 , respectively , maintaining the planarity of the graph . in the new graph one should transfer each source @xmath21 into the tail of @xmath841 , and each sink @xmath842 into the head of @xmath843 . one may assume that the new sources ( sinks ) lie within one horizontal line @xmath242 ( resp . @xmath844 ) , and that the rest of the graph lies between @xmath242 and @xmath844 . now we get rid of the edges @xmath845 such that @xmath846 ( i.e. `` pointing to the left '' ) , by making the linear transformation @xmath847 for the points @xmath87 in @xmath780 , defined by @xmath848 and @xmath849 with a sufficiently large @xmath819 . thus , we eventually obtain a graph @xmath780 ( homeomorphic to the initial one ) without edges pointing up or to the left . also the sources and sinks are properly ordered from left to right in the horizontal lines @xmath242 and @xmath844 , respectively . now it is routine to turn @xmath780 into an se - graph @xmath49 as required in the proposition . the transformation of @xmath780 into @xmath49 as in the proof is illustrated in the picture ; here @xmath850 , @xmath851 , @xmath852 , @xmath853 , and @xmath854 . it looks reasonable to ask : how narrow is the class of uq identities for minors of @xmath5-matrices compared with the class of those in the commutative version . we know that the latter class is formed by balanced families @xmath357 , whereas the former one is characterized via a stronger property of @xmath5-balancedness . so we can address the problem of characterizing the set of ( homogeneous ) balanced families @xmath855 that admit functions @xmath351 and @xmath352 such that the quadruple @xmath367 is @xmath5-balanced . in an algorithmic setting , we deal with the following problem @xmath856 : given @xmath357 ( as above ) , decide whether or not there exist corresponding @xmath404 ( as above ) . concerning algorithmic complexity aspects , note that the number @xmath857 of configurations for @xmath357 may be exponentially large compared with the number @xmath858 of corteges ( since a cortege of size @xmath433 may have @xmath859 feasible matchings ) . in light of this , it is logically reasonable to regard as the input of problem ( @xmath860 ) just the set @xmath861 rather than @xmath862 ( and measure the input size of ( @xmath860 ) accordingly ) . we conjecture that problem ( @xmath860 ) specified in this way is np - hard and , moreover , it remains np - hard even in the flag case . the simplest example of balanced @xmath357 for which problem ( @xmath860 ) has answer `` not '' arises in the flag case with @xmath357 consisting of single corteges . that is , we deal with quantized flag minors @xmath63=[a|i]$ ] and @xmath64=[b|j]$ ] , where @xmath480 and @xmath481 , and consider the ( trivially balanced ) one - element families @xmath863 and @xmath864 . by leclerc zelevinsky s theorem ( theorem [ tm : lz ] ) , @xmath63 $ ] and @xmath64 $ ] quasicommute if and only if the sets @xmath67 are weakly separated . we have explained how to obtain `` if '' part of this theorem by use of the flow - matching method , and now we explain how to use this method to show , relatively easily , `` only if '' part ( which has a rather sophisticated proof in @xcite ) . so , assuming that @xmath67 are not weakly separated , our aim is to show that there do not exist @xmath865 such that the equality @xmath866 holds for all feasible matching @xmath196 for @xmath375 . the crucial observation is that [ eq:1match ] @xmath611 $ ] are weakly separated if and only if @xmath375 has exactly one feasible matching ( where `` only if '' part , mentioned in [ ssec : quasicommut ] , is trivial ) . in fact , we need a sharper version of `` if '' part of ( [ eq:1match ] ) : when @xmath611 $ ] are not weakly separated , there exist @xmath867 such that @xmath868 then the fact that the exchange operation applied to @xmath375 using @xmath196 results in @xmath869 , and similarly for @xmath870 , implies that ( [ eq : btas ] ) can not hold simultaneously for both @xmath196 and @xmath870 . to construct the desired @xmath196 and @xmath870 , we argue as follows . let for definiteness @xmath467 and let @xmath871 and @xmath872 . from the property that @xmath67 are not weakly separated one can conclude that there are @xmath873 $ ] with @xmath719 such that the sets @xmath874 and @xmath875 satisfy @xmath876 , and @xmath877 has a partition into nonempty sets @xmath878 satisfying @xmath879 . let @xmath880 ( then @xmath881 and @xmath882 ) . choose an arbitrary matching @xmath883 , and consider the set @xmath268 of couples in @xmath196 containing elements of @xmath884 ; let @xmath885 , where @xmath886 . each @xmath887 is a @xmath42-couple ( since it can not be an @xmath201- couple , in view of @xmath888 ) , and the feasibility condition ( [ eq : feasm ] ) for @xmath196 implies that only two cases are possible : ( a ) @xmath568 couples in @xmath268 meet @xmath889 and the remaining @xmath890 couples meet @xmath891 , and ( b ) @xmath892 couples in @xmath268 meet @xmath889 and the remaining @xmath893 couples meet @xmath891 . in case ( a ) , we have @xmath894 for @xmath895 , and @xmath896 for @xmath897 . an especial role is played by the couple in @xmath196 containing the last element @xmath898 of @xmath891 , say , @xmath899 ( note that @xmath104 belongs to either @xmath900 or @xmath901 ) . we modify @xmath196 by replacing the couple @xmath203 by @xmath902 , and replacing @xmath903 by @xmath904 , forming matching @xmath870 . the picture illustrates the case @xmath905 , @xmath906 and @xmath907 . one can see that @xmath870 is feasible for @xmath375 . moreover , @xmath196 and @xmath870 satisfy ( [ eq : mmp ] ) . indeed , @xmath908 contributes one unit to @xmath909 while @xmath910 contributes one unit to @xmath911 , the contributions from @xmath203 and from @xmath665 are the same , and the rests of @xmath196 and @xmath870 coincide . thus , in case ( a ) , the one - element families @xmath514 and @xmath515 along with any numbers @xmath912 are not @xmath5-balanced . then relation ( [ eq : quasiij ] ) ( with any @xmath477 ) is impossible by theorem [ tm : nec_q_bal ] . in case ( b ) , the argument is similar . this yields the necessity ( `` only if '' part ) in theorem [ tm : lz ] . it is tempting to ask : can one characterize the set of quasicommuting quantum minors in a general ( non - flag ) case ? such a characterization can be obtained , without big efforts , by use of the flow - matching method , yielding a generalization of theorem [ tm : lz ] . [ tm : generallz ] let @xmath913 and let @xmath914 . the following statements are equivalent : \(i ) the minors @xmath14 $ ] and @xmath60 $ ] quasicommute , i.e. , @xmath14[i'|j ' ] = q^c [ i'|j'][i|j]$ ] for some @xmath476 ; \(ii ) the cortege @xmath27 admits exactly one feasible matching ; \(iii ) the sets @xmath915 are weakly separated , the sets @xmath916 are weakly separated , and for the refinement @xmath184 of @xmath375 , one of the following takes place : \(a ) @xmath917 ; or \(b ) both sets @xmath918 are nonempty , and either @xmath919 and @xmath920 , or @xmath921 and @xmath922 . also in case ( iii ) the number @xmath477 is computed as follows : if @xmath923 , @xmath924 and @xmath925 , then @xmath503 ; ( symmetrically ) if @xmath926 , @xmath927 and @xmath928 , then @xmath929 ; if @xmath919 and @xmath920 , then @xmath930 ; and ( symmetrically ) if @xmath921 and @xmath922 , then @xmath931 . implication ( ii)@xmath932(i ) is proved as in section [ ssec : quasicommut ] , and ( iii)@xmath932(ii ) is easy . to show ( i)@xmath932(iii ) , we use the fact that @xmath933 ( cf . ( [ eq : balancij ] ) ) and observe that a feasible matchings for @xmath375 can be constructed by the following procedure ( p ) consisting of three steps . first , choose an arbitrary maximal feasible set @xmath934 of @xmath112-couples in @xmath935 . here the feasibility means that the elements of each couple have different colors and there are neither couples @xmath214 and @xmath936 with @xmath937 , nor a couple @xmath214 and an element @xmath938 with @xmath796 ; cf . ( [ eq : feasm ] ) . second , choose an arbitrary maximal feasible set @xmath800 of @xmath42-couples in @xmath939 . third , when @xmath940 , the remaining elements of @xmath732 ( which are all white ) are coupled by a unique set @xmath784 of @xmath201-couples . then @xmath941 is a feasible matching for @xmath375 . suppose that ( iii ) is false and consider possible cases . \1 ) let @xmath916 be not weakly separated . then we construct @xmath942 by procedure ( p ) and work with the matching @xmath943 in a similar way as in the above proof for the flag case ( with non - weakly - separated column sets ) . this transforms @xmath944 into @xmath945 , and we obtain two different feasible matchings @xmath946 and @xmath947 for @xmath375 satisfying ( [ eq : mmp ] ) . this leads to a contradiction with ( i ) ( as well as ( ii ) ) in the theorem . when @xmath915 are not weakly separated , the argument is similar . \2 ) assuming that @xmath915 are weakly separated , and similarly for @xmath916 , let both @xmath948 be nonnempty . then @xmath949 are nonempty as well , and for the matching @xmath196 formed by procedure ( p ) , @xmath934 covers @xmath186 and @xmath800 covers @xmath216 . denote by @xmath950 ( resp . @xmath951 ) the minimal and maximal elements in @xmath187 ( resp . @xmath190 ) , respectively . suppose that both @xmath952 are black . then we can transform @xmath196 into @xmath870 by replacing the @xmath112-couple containing @xmath709 , say , @xmath953 , and the @xmath42-couple containing @xmath717 , say , @xmath954 , by the two @xmath201-couples @xmath955 and @xmath956 . it is easy to see that @xmath870 is feasible and @xmath957 satisfy ( [ eq : mmp ] ) ( since under the transformation @xmath958 the value @xmath693 decreases by two ) , whence ( i ) is false . when both @xmath959 are black , we act similarly . so we may assume that each pair @xmath960 and @xmath961 contains a white element . the case @xmath962 and @xmath963 is possible only if @xmath964 ( taking into account that @xmath965 and that @xmath966 , as well as @xmath967 , are weakly separated ) , implying @xmath968 . but then @xmath934 covers @xmath185 and @xmath800 covers @xmath215 ; so we can construct a feasible matching @xmath969 as in the previous case ( after changing the colors everywhere ) . and similarly when both @xmath959 are white . thus , we may assume that @xmath952 have different colors , and so are @xmath959 . suppose that @xmath970 and @xmath971 ( the case @xmath972 and @xmath973 is similar ) . this is possible only if @xmath964 ( since @xmath914 , and @xmath915 are weakly separated ) . then the feasible matching @xmath196 constructed by ( p ) consists of only @xmath112- and @xmath42-couples . take the @xmath112-couple in @xmath196 containing @xmath709 and the @xmath42-couple containing @xmath974 , say , @xmath975 and @xmath976 ; then both @xmath977 are white and both @xmath978 are black . replace @xmath979 by the @xmath201-couples @xmath980 and @xmath981 . this gives a feasible matching @xmath969 satisfying ( [ eq : mmp ] ) . the remaining situation is just as in ( a ) or ( b ) of ( iii ) , yielding ( i)@xmath932(iii ) . note that the situation when @xmath184 has only one feasible matching can also be interpreted as follows . let us change the colors of all elements in the upper half of the circumference @xmath191 ( i.e. , @xmath185 becomes black and @xmath186 becomes white ) . then the quantities of white and black elements in @xmath191 are equal and the elements of each color go in succession cyclically . when minors @xmath14 $ ] and @xmath60 $ ] quasicommute with @xmath982 , we obtain the situation of `` purely commuting '' quantum minors , such as those discussed in sect . [ ssec : com_min ] . the last assertion in theorem [ tm : generallz ] enables us to completely characterize the set of corteges @xmath23 determining commuting @xmath5-minors , as follows . [ pr : gen_commute ] @xmath14[i'|j']=[i'|j'][i|j]$ ] holds if and only if the refinement @xmath184 satisfies at least one of the following : ( c1 ) @xmath218 ( as well as @xmath983 ) and either @xmath919 and @xmath920 , or , symmetrically , @xmath921 and @xmath922 ; ( c2 ) assuming for definiteness that @xmath914 , either @xmath923 and @xmath216 has a partition @xmath469 such that @xmath984 and @xmath925 , or , symmetrically , @xmath926 and @xmath186 has a partition @xmath985 such that @xmath986 and @xmath928 . cases ( c1 ) and ( c2 ) are illustrated in the picture by two level diagrams . return to a general uq identity ( [ eq : gen_qr ] ) . in sect . [ sec : q_relat ] we demonstrated two transformations of @xmath5-balanced @xmath367 that preserve the @xmath5-balancedness ( namely , the ones of _ reversing _ and _ transposing _ , which result in @xmath408 and @xmath987 , respectively . ) now we demonstrate one more interesting ( and nontrivial ) transformation of @xmath367 ( in theorem [ tm : rotation ] ) . first , for corresponding @xmath988 $ ] and @xmath989 $ ] ( cf . ( [ eq : xryr ] ) ) , let @xmath990 and @xmath991 . choose @xmath992 such that @xmath993 assuming that the numbers @xmath994 , @xmath995 , @xmath996 , @xmath997 are large enough , we take sets @xmath998 $ ] and @xmath999 $ ] such that @xmath1000 , @xmath1001 , @xmath1002 , @xmath1003 , and [ eq : aapbbp ] ( a ) @xmath1004 and @xmath1005 if @xmath1006 ; ( a ) @xmath1007 and @xmath1008 if @xmath1009 ; \(b ) @xmath1010 and @xmath1011 if @xmath1012 ; ( b ) @xmath1013 and @xmath1014 if @xmath1015 . let @xmath763 ( @xmath1016 ) be the _ order - reversing _ bijection between @xmath95 and @xmath1017 ( resp . @xmath1018 and @xmath1019 ) , i.e. , @xmath164-th element of @xmath95 is bijective to @xmath1020-th element of @xmath1017 , and similarly for @xmath1016 . second , we transform each cortege @xmath1021 into cortege @xmath1022 such that @xmath1023 , @xmath1024 , and the refinement @xmath1025 of @xmath1026 is expressed via the refinement @xmath184 of @xmath375 as follows : * @xmath1027 and @xmath1028 ( where we write @xmath1029 for @xmath1030 in case @xmath1031 , and write @xmath1032 for @xmath1033 in case @xmath1034 ) ; * if @xmath211 ( @xmath1035 ) is not in @xmath1036 , then @xmath1037 ( resp . @xmath1038 ) , and symmetrically , if @xmath1039 ( @xmath1040 ) is not in @xmath1041 , then @xmath1042 ( resp . @xmath1043 ) ; * if @xmath211 ( @xmath1035 ) is in @xmath1036 , then the element bijective to @xmath21 ( by @xmath763 or @xmath1016 ) belongs to @xmath1044 ( resp . @xmath1045 ) ; and symmetrically , if @xmath1039 ( @xmath1040 ) is in @xmath1041 , then the element bijective to @xmath664 belongs to @xmath1046 ( resp . @xmath877 ) . ( in other words , @xmath763 and @xmath1016 change the colors of elements occurring in @xmath1047 ) . we call @xmath1048 the @xmath1049-_rotations _ of @xmath1050 , respectively . accordingly , we say that @xmath1051 is the @xmath1049-rotation of @xmath329 , and similarly for @xmath331 . ( this terminology is justified by the observation that if @xmath1052 , then each cortege @xmath375 is transformed as though being rotated ( by @xmath1053 positions clockwise or counterclockwise ) on the circular diagram on @xmath732 ; thereby each element moving across the middle horizontal line of the diagram changes its color . ) third , extend @xmath763 and @xmath1016 to the bijection @xmath1054 so that @xmath1055 be identical on @xmath1056 and on @xmath1057 . then a perfect matching @xmath196 on @xmath197 induces the perfect matching @xmath1058 on @xmath1059 , denoted as @xmath1060 . an important property ( which is easy to check ) is that [ eq : mgh ] if @xmath196 is a feasible matching for @xmath1061 , then @xmath1060 is a feasible matching for @xmath1026 , and vice versa . an example of rotation of @xmath375 with @xmath883 is illustrated in the picture where @xmath465 , @xmath1062 , @xmath1063 and @xmath1064 . fourth , for @xmath27 , define @xmath1065 , where @xmath1066 [ tm : rotation ] let @xmath367 be @xmath5-balanced and let @xmath1067 be as in ( [ eq : gh ] ) . define @xmath1068 for @xmath423 , and @xmath1069 for @xmath1070 . then @xmath1071 are @xmath5-balanced . let @xmath1072 be a bijection providing the @xmath5-balancedness of @xmath367 . by ( [ eq : mgh ] ) , @xmath353 induces a bijection @xmath1073 . more precisely , for configurations @xmath1074 and @xmath1075 , @xmath1076 maps the configuration @xmath1077 to @xmath1078 . we assert that @xmath1076 satisfies the corresponding equality of the form @xmath1079 ( cf . ( [ eq : q_balan ] ) ) , yielding the result ; here , as before , @xmath268 is the set of couples in @xmath196 having different colorings in the refinements of @xmath375 and @xmath869 . for additivity reasons , it suffices to show ( [ eq : balanc_gh ] ) when @xmath1080 . we will abbreviate corresponding @xmath1081 as @xmath1082 . ( so @xmath1083 is obtained from @xmath1084 by the exchange operation using @xmath1085 . ) let @xmath104 denote the ( only ) element of @xmath1086 that is not in @xmath1087 , and @xmath1088 the couple in @xmath196 containing @xmath104 . also we define @xmath1089 and @xmath1090 . our aim is to show that @xmath1091 ; then ( [ eq : balanc_gh ] ) would immediately follow from ( [ eq : q_balan ] ) . one can see that if @xmath1092 , then @xmath1093 , and @xmath1094 holds for @xmath1095 ( cf . ( [ eq : deltaab ] ) ) , implying @xmath1096 . so we may assume that @xmath284 . consider possible cases ( where @xmath27 and @xmath1097 ) . _ let @xmath1098 . then @xmath1099 . first suppose that @xmath1100 . then @xmath1101 and @xmath1102 ( since the exchange operation changes the color of @xmath104 , i.e. , @xmath1103 ) . if @xmath203 is an @xmath112-couple for @xmath375 , then @xmath203 contributes 1 to @xmath1104 ( since @xmath104 is white and @xmath1105 ) , and @xmath1106 contributes 0 to @xmath1107 ( since @xmath1106 is an @xmath201-couple for @xmath1084 ) . hence @xmath1108 , as required . and if @xmath203 is an @xmath201-couple for @xmath375 , then @xmath203 contributes 0 to @xmath1104 and @xmath1106 contributes @xmath692 to @xmath1107 ( since @xmath1106 is a @xmath42-couple for @xmath1084 , @xmath1109 is black , @xmath1110 is white , and @xmath1111 ) , giving again @xmath1112 . when @xmath1113 , we argue `` symmetrically '' ( as though the roles of @xmath375 and @xmath869 , as well as @xmath359 and @xmath360 , are exchanged ) . briefly , one can check that : @xmath1114 and @xmath1115 ; if @xmath203 is an @xmath112-couple , then @xmath203 contributes @xmath692 to @xmath1104 , and @xmath1106 contributes 0 to @xmath1107 ; and if @xmath203 is an @xmath201-couple then @xmath203 contributes 0 to @xmath1104 and @xmath1106 contributes 1 to @xmath1107 . thus , every time we obtain @xmath1116 , as required . _ let @xmath1117 . then @xmath1118 . suppose that @xmath1119 . then @xmath1120 and @xmath1121 . if @xmath203 is an @xmath112-couple for @xmath375 , then @xmath203 contributes @xmath692 to @xmath1104 ( since @xmath104 is white and @xmath1122 ) and @xmath1106 contributes 0 to @xmath1107 ( since @xmath1106 is an @xmath201-couple ) . and if @xmath203 is an @xmath201-couple for @xmath375 , then @xmath203 contributes 0 to @xmath1104 and @xmath1106 contributes 1 to @xmath1107 ( since @xmath1106 is a @xmath42-couple for @xmath1084 , @xmath1109 is black , and @xmath1123 ) . in both cases , we obtain @xmath1116 , as required . when @xmath1113 , we argue `` symmetrically '' . finally , the cases @xmath1124 and @xmath1064 are `` transposed '' to cases 1 and 2 , respectively , and ( [ eq : balanc_gh ] ) follows by using relation ( [ eq : transqr ] ) . _ acknowledgements . _ we thank gleb koshevoy for pointing out to us paper @xcite . 99 g. cauchon , spectre premier de @xmath1125 : image canonique et sparation normale , _ j. algebra _ * 260 * ( 2 ) ( 2003 ) 519569 . k. casteels , a graph theoretic method for determining generating sets of prime ideals in quantum matrices , _ j. algebra _ * 330 * ( 2011 ) 188205 . k. casteels , quantum matrices by paths , _ algebra and number theory _ * 8 * ( 8) ( 2014 ) 18571912 . v. danilov , a. karzanov , and g. koshevoy , planar flows and quadratic relations over semirings , _ j. algebraic combin . _ * 36 * ( 2012 ) 441474 . dodgson , condensation of determinants , _ proc . of the royal soc . of london _ * 15 * ( 1866 ) 150155 . faddeev , n. yu . reshetikhin , and l.a . takhtajan , quantization of lie groups and lie algebras , in : _ algebraic analysis _ , vol . * i * , pp . 129139 , academic press , boston , 1988 . r. fioresi , commutation relations among quantum minors in @xmath1125 , _ j. algebra _ * 280 * ( 2 ) ( 2004 ) 655682 . goodearl , commutation relations for arbitrary quantum minors , _ pacific j. of mathematics _ * 228 * ( 1 ) ( 2006 ) 63102 . d. krob and b. leclerc , minor identities for quasi - determinants and quantum determinants , _ commun . * 169 * ( 1995 ) 123 . v. lakshmibai and n. reshetikhin , quantum flag and schubert schemes , _ contemp . _ * 134 * ( 1992 ) 145181 . b. leclerc and a. zelevinsky , quasicommuting families of quantum plcker coordinates , _ _ a__mer . , ser . 2 , * 181 * ( 1998 ) 85108 . b. lindstrm , on the vector representations of induced matroids , _ bull . london math . * 5 * ( 1973 ) 8590 . manin , quantum groups and non commutative geometry , vol . * 49 * , centre de recherches mathematiques montreal , 1988 . e. taft and j. towber , quantum deformation of flag schemes and grassmann schemes . i. q - deformation of the shape - algebra for gl(n ) , _ j. algebra _ * 142 * ( 1991 ) 136 . a. schrijver , _ combinatorial optimization _ , vol . a , springer , berlin , 2003 . this appendix contains auxiliary lemmas that are used in the proof of theorem [ tm : linds ] given in this section as well , and in the proof of theorem [ tm : single_exch ] given in appendix b. these lemmas deal with special pairs @xmath1126 of paths in an se - graph @xmath324 and compare the weights @xmath1127 and @xmath1128 . similar or close statements for cauchon graphs are given in @xcite , and our method of proof is somewhat similar and rather straightforward as well . we first specify some terminology , notation and conventions . when it is not confusing , vertices , edges , paths and other objects in @xmath49 are identified with their corresponding images in the plane . we assume that the set @xmath108 of sources and the set @xmath109 of sinks lie on the coordinate rays @xmath1129 and @xmath1130 , respectively ( then @xmath49 is disposed within the nonnegative quadrant @xmath1131 ) . the coordinates of a point @xmath87 in @xmath1132 ( e.g. , a vertex @xmath87 of @xmath49 ) are denoted as @xmath1133 . it is convenient to assume that two vertices @xmath1134 have the same first ( second ) coordinate if and only if they belong to a vertical ( resp . horizontal ) path in @xmath49 , in which case @xmath1135 are called _ v - dependent _ ( resp . _ h - dependent _ ) ; for we always can slightly perturb @xmath49 to ensure such a property , without affecting the graph structure in essence . when @xmath1135 are v - dependent , i.e. , @xmath1136 , we say that @xmath86 is _ lower _ than @xmath87 ( and @xmath87 is _ higher _ than @xmath86 ) if @xmath1137 . ( in this case the commutation relation @xmath139 takes place . ) let @xmath78 be a path in @xmath49 . we denote : the first and last vertices of @xmath78 by @xmath1138 and @xmath1139 , respectively ; the _ interior _ of @xmath78 ( the set of points of @xmath1140 in @xmath1132 ) by @xmath1141 ; the set of horizontal edges of @xmath78 by @xmath1142 ; and the projection @xmath1143 by @xmath1144 . clearly if @xmath78 is directed , then @xmath1144 is the interval between @xmath1145 and @xmath1146 . for a directed path @xmath78 , the following are equivalent : @xmath78 is non - vertical ; @xmath1147 ; and @xmath1148 . we will refer to such a @xmath78 as a _ standard _ ( rather than non - vertical ) path . for a standard path @xmath78 , we will take advantage from a compact expression for the weight @xmath133 . we call a vertex @xmath87 of @xmath78 _ essential _ if either @xmath78 makes a turn at @xmath87 ( changing the direction from horizontal to vertical or back ) , or @xmath1149 and the first edge of @xmath78 is horizontal , or @xmath1150 and the last edge of @xmath78 is horizontal . if @xmath1151 is the sequence of essential vertices of @xmath78 in the natural order , then the weight of @xmath78 can be expressed as @xmath1152 where @xmath1153 if @xmath78 makes a -turn at @xmath137 or if @xmath1154 , while @xmath1155 if @xmath78 makes a -turn at @xmath137 or if @xmath1156 and @xmath1157 is the beginning of @xmath78 . ( compare with ( [ eq : telescop ] ) where a path from @xmath112 to @xmath42 is considered . ) it is easy to see that if @xmath78 does not begin in @xmath112 , then its essential vertices are partitioned into h - dependent pairs . throughout the rest of the paper , for brevity , we denote @xmath1158 by @xmath1159 , and for an inner vertex @xmath125 regarded as a generator , we may denote @xmath1160 by @xmath1161 . these lemmas deal with _ weakly intersecting _ directed paths @xmath78 and @xmath648 , which means that @xmath1162 in particular , @xmath1163 . for such @xmath1126 , we say that @xmath78 is _ lower _ than @xmath648 if there are points @xmath1164 and @xmath1165 such that @xmath1166 and @xmath1167 ( then there are no @xmath1168 and @xmath1169 with @xmath1170 and @xmath1171 ) . for paths @xmath1126 , we define the value @xmath1172 by the relation @xmath1173 obviously , @xmath1174 when @xmath78 or @xmath648 is a v - path . in the lemmas below we default assume that both @xmath1126 are standard . [ lm : varphi=1 ] let @xmath1175 . then @xmath1174 . consider an essential vertex @xmath86 of @xmath78 and an essential vertex @xmath87 of @xmath648 . then for any @xmath1176 , we have @xmath1177 unless @xmath1135 are dependent . suppose that @xmath1135 are v - dependent . from hypotheses of the lemma it follows that at least one of the following is true : @xmath1178 , or @xmath1179 . for definiteness assume the former . then there is another essential vertex @xmath1180 of @xmath78 such that @xmath1181 . moreover , @xmath78 makes a -turn an one of @xmath1182 , and a -turn at the other . since @xmath1183 ( in view of ( [ eq : pathspq ] ) ) , the vertices @xmath1182 are either both higher or both lower than @xmath87 . let for definiteness @xmath1182 occur in this order in @xmath78 ; then @xmath133 contains the terms @xmath1184 . let @xmath1185 contain the term @xmath1186 and let @xmath1187 , where @xmath1188 and @xmath1189 . then @xmath1190 , implying @xmath1191 . hence the contributions to @xmath1127 and @xmath1128 from the pairs using terms @xmath1192 ( namely @xmath1193 and @xmath1194 ) are equal . next suppose that @xmath1135 are h - dependent . one may assume that @xmath1195 . then @xmath648 contains one more essential vertex @xmath1196 with @xmath1197 . also @xmath1195 and @xmath1183 imply @xmath1198 . let for definiteness @xmath1199 . then @xmath1185 contains the terms @xmath1200 , and we can conclude that the contributions to @xmath1127 and @xmath1128 from the pairs using terms @xmath1201 are equal ( using the fact that @xmath1202 ) . these reasonings imply @xmath1174 . [ lm : asp = asq ] let @xmath1203 and @xmath1204 . let @xmath78 be lower than @xmath648 . then @xmath1205 . let @xmath86 and @xmath87 be the first essential vertices in @xmath78 and @xmath648 , respectively . then @xmath1206 ( in view of @xmath1203 ) . since @xmath78 is lower than @xmath648 , we have @xmath1207 . moreover , this inequality is strong ( since @xmath1208 is impossible in view of ( [ eq : pathspq ] ) and the obvious fact that @xmath1135 are the tails of first h - edges in @xmath1126 , respectively ) . now arguing as in the above proof , we can conclude that the discrepancy between @xmath1127 and @xmath1128 can arise only due to swapping the vertices @xmath1135 . since @xmath86 gives the term @xmath1209 in @xmath133 , and @xmath87 the term @xmath1161 in @xmath1185 , the contribution from these vertices to @xmath1127 and @xmath1128 are expressed as @xmath1210 and @xmath1211 , respectively . since @xmath1212 , we have @xmath1213 , and the result follows . [ lm : atp = atq ] let @xmath1214 and let either @xmath1215 or @xmath1216 . let @xmath78 be lower than @xmath648 . then @xmath1205 . we argue in spirit of the proof of lemma [ lm : asp = asq ] . let @xmath86 and @xmath87 be the last essential vertices in @xmath78 and @xmath648 , respectively . then @xmath1217 . also @xmath1212 ( since @xmath78 is lower than @xmath648 , and in view of ( [ eq : pathspq ] ) and the fact that @xmath1135 are the heads of h - edges in @xmath1126 , respectively ) . the condition on @xmath1145 and @xmath1218 imply that the discrepancy between @xmath1127 and @xmath1128 can arise only due to swapping the vertices @xmath1135 ( using reasonings as in the proof of lemma [ lm : varphi=1 ] ) . observe that @xmath133 contains the term @xmath86 , and @xmath1185 the term @xmath87 . so the generators @xmath1135 contribute @xmath1219 to @xmath1127 , and @xmath1220 to @xmath1128 . now @xmath1212 implies @xmath139 , and the result follows . [ lm:1atp = asq ] let @xmath1221 and @xmath1222 . then @xmath1205 . let @xmath86 be the last essential vertex in @xmath78 and let @xmath1223 be the first and second essential vertices of @xmath648 , respectively ( note that @xmath1180 exists because of @xmath1224 ) . then @xmath1225 . also @xmath1226 . let @xmath1227 and @xmath1228 be the parts of @xmath648 from @xmath1229 to @xmath1180 and from @xmath1180 to @xmath1230 , respectively . then @xmath1231 , implying @xmath1232 ( using lemma [ lm : varphi=1 ] when @xmath1228 is standard ) . hence @xmath1233 . to compute @xmath1234 , consider three possible cases . \(a ) let @xmath1235 . then @xmath1135 form the unique pair of dependent essential vertices for @xmath1236 . note that @xmath133 contains the term @xmath86 , and @xmath1237 contains the term @xmath1161 . since @xmath1235 , we have @xmath1238 , implying @xmath1239 . \(b ) let @xmath1240 and let @xmath86 be the unique essential vertex of @xmath78 ( in other words , @xmath78 is an h - path with @xmath1241 ) . note that @xmath1240 and @xmath1222 imply @xmath1242 . also @xmath1243 and @xmath1244 ; so @xmath1182 are dependent essential vertices for @xmath1236 and @xmath1245 . we have @xmath1246 and @xmath1247 ( in view of @xmath1240 ) . then @xmath1248 gives @xmath1239 . \(c ) now let @xmath1240 and let @xmath1249 be the essential vertex of @xmath78 preceding @xmath86 . then @xmath1242 , @xmath1250 , and @xmath1251 . hence @xmath1252 are dependent , @xmath133 contains @xmath1253 , and @xmath1254 . we have @xmath1255 again obtaining @xmath1239 . [ lm:2atp = asq ] let @xmath1221 and @xmath1256 . then @xmath1257 . let @xmath86 be the last essential vertex of @xmath78 , and @xmath87 the first essential vertex of @xmath648 . then @xmath1258 , and @xmath1256 together with ( [ eq : pathspq ] ) implies @xmath1212 . also @xmath133 contains @xmath86 and @xmath1185 contains @xmath1161 . now @xmath1259 implies @xmath1257 . it can be conducted as a direct extension of the proof of a similar lindstrm s type result given by casteels ( * ? ? ? 4 ) for cauchon graphs . to make our description more self - contained , we outline the main ingredients of the proof , leaving the details where needed to the reader . let @xmath157 , @xmath1260 and @xmath1261 . recall that an @xmath154-flow in an se - graph @xmath49 ( with @xmath1262 sources and @xmath1263 sinks ) consists of pairwise disjoint paths @xmath1264 from the source set @xmath1265 to the sink set @xmath1266 , and ( because of the planarity of @xmath49 ) we may assume that each @xmath1267 begins at @xmath1268 and ends at @xmath1269 . besides , we are forced to deal with an arbitrary _ path system _ @xmath1270 in which for @xmath1271 , @xmath1267 is a directed path in @xmath49 beginning at @xmath1268 and ending at @xmath1272 , where @xmath1273 are different , i.e. , @xmath1274 is a permutation on @xmath1275 $ ] . ( in particular , @xmath1276 is identical if @xmath1277 is a flow . ) we naturally partition the set of all path systems for @xmath49 and @xmath154 into the set @xmath1278 of @xmath154-flows and the rest @xmath1279 ( consisting of those path systems that contain intersecting paths ) . the following property easily follows from the planarity of @xmath49 ( cf . * lemma 4.2 ) ) : [ eq : pipi+1 ] for any @xmath1280 , there exist two _ consecutive _ intersecting paths @xmath1281 . the @xmath5-_sign _ of a permutation @xmath106 is defined by @xmath1282 where @xmath105 is the length of @xmath106 ( see sect . [ sec : prelim ] ) . now we start computing the @xmath5-minor @xmath14 $ ] of the matrix @xmath151 with the following chain of equalities : @xmath1283&= & \sum\nolimits_{\sigma\in s_k } \sign_q(\sigma ) \left ( \prod\nolimits_{d=1}^{k } \path_g(i(d)|j(\sigma(d))\right ) \\ & = & \sum\nolimits_{\sigma\in s_k } \sign_q(\sigma ) \left ( \prod\nolimits_{d=1}^{k } \left ( \sum(w(p)\;\colon p\in \phi_g(i(d)|j(\sigma(d))\right)\right ) \\ & = & \sum(\sign_q(\sigma_\pscr)w(\pscr)\;\colon \pscr\in\phi(i|j)\cup\psi(i|j ) ) \\ & = & \sum(w(\pscr)\;\colon \pscr\in\phi(i|j ) ) + \sum(\sign_q(\sigma_\pscr)w(\pscr)\;\colon \pscr\in\psi(i|j ) ) . \end{aligned}\ ] ] thus , we have to show that the second sum in the last row is zero . it will follow from the existence of an involution @xmath1284 without fixed points such that for each @xmath1285 , @xmath1286 to construct the desired @xmath1016 , consider @xmath1280 , take the minimal @xmath21 such that @xmath236 and @xmath1287 meet , take the last common vertex @xmath87 of these paths , represent @xmath236 as the concatenation @xmath1288 , and @xmath1287 as @xmath1289 , so that @xmath1290 , and exchange the portions @xmath1291 of these paths , forming @xmath1292 and @xmath1293 . then we assign @xmath1294 to be obtained from @xmath1277 by replacing @xmath1295 by @xmath1296 . it is routine to check that @xmath1016 is indeed an involution ( with @xmath1297 ) and that @xmath1298 assuming w.l.o.g . that @xmath1299 . on the other hand , applying to the paths @xmath1300 lemmas [ lm : asp = asq ] and [ lm:1atp = asq ] , one can obtain @xmath1301 whence @xmath1302 . this together with ( [ eq : ell+1 ] ) gives @xmath1303 yielding ( [ eq : invol ] ) , and the result follows . using notation as in the hypotheses of this theorem , we first consider the case when ( then @xmath1306 and @xmath1307 . ) we have to prove that @xmath1308 the proof is given throughout sects . [ ssec : seglink][ssec : degenerate ] . the other possible cases in theorem [ tm : single_exch ] will be discussed in sect . [ ssec : othercases ] . let @xmath1309 be the exchange path determined by @xmath203 ( i.e. , @xmath1310 in notation of sect . [ sec : double ] ) . it connects the sinks @xmath1311 and @xmath1312 , which may be regarded as the first and last vertices of @xmath1309 , respectively . then @xmath1309 is representable as a concatenation @xmath1313 , where @xmath1314 is even , each @xmath1315 with @xmath21 odd ( even ) is a directed path concerning @xmath158 ( resp . @xmath221 ) , and @xmath1316 stands for the path reversed to @xmath1315 . more precisely , let @xmath1317 , @xmath1318 , and for @xmath1319 , denote by @xmath1320 the common endvertex of @xmath1315 and @xmath1321 . then each @xmath1315 with @xmath21 odd is a directed path from @xmath1320 to @xmath1322 in @xmath1323 , while each @xmath1315 with @xmath21 even is a directed path from @xmath1322 to @xmath1320 in @xmath1324 . also we refer to the vertices @xmath1325 as the _ bends _ of @xmath1309 . a bend @xmath1320 is called a _ peak _ ( a _ pit _ ) if both path @xmath1326 leave ( resp . enter ) @xmath1320 ; then @xmath1327 are the peaks , and @xmath1328 are the pits . note that some peak @xmath1320 may coincide with some pit @xmath1329 ; in this case we say that @xmath1330 are _ twins_. the rests of flows @xmath158 and @xmath221 consist of directed paths that we call _ white _ and _ black links _ , respectively . more precisely , the white ( black ) links correspond to the connected components of the subgraph @xmath158 ( resp . @xmath221 ) from which the interiors of all snakes are removed . so a link connects either ( a ) a source and a sink ( being a component of @xmath158 or @xmath221 ) , or ( b ) a source and a pit , or ( c ) a peak and a sink , or ( d ) a peak and a pit . we say that a link is _ unbounded _ in case ( a ) , _ semi - bounded _ in cases ( b),(c ) , and _ bounded _ in case ( d ) . note that [ eq:4paths ] a bend @xmath1320 occurs as an endvertex in exactly four paths among snakes and links , namely : either in two snakes and two links ( of different colors ) , or in four snakes @xmath1331 ( when @xmath1330 are twins ) . the picture below illustrates an example . here @xmath1336 , the bends @xmath1337 are marked by squares , the white and black snakes are drawn by thin and thick solid zigzag lines , respectively , the white links ( @xmath1338 ) by short - dotted lines , and the black links ( @xmath1339 ) by long - dotted lines . the weight @xmath1340 of the double flow @xmath179 can be written as the corresponding ordered product of the weights of snakes and links ; let @xmath1341 be the string ( sequence ) of snakes and links in this product . the weight of the double flow @xmath265 uses a string consisting of the same snakes and links but occurring in another order ; we denote this string by @xmath1342 . we say that two elements among snakes and links are _ invariant _ if they occur in the same order in @xmath1341 and @xmath1342 , and _ permuting _ otherwise . in particular , two links of different colors are invariant , whereas two snakes of different colors are always permitting . for @xmath1345 , we write @xmath1346 ( resp . @xmath1347 ) if @xmath95 occurs in @xmath1341 ( resp . in @xmath1342 ) earlier than @xmath1018 . we define @xmath1348 if @xmath33 are invariant , and define @xmath1349 by the relation @xmath1350 if @xmath33 are permuting and @xmath1346 . note that @xmath1351 is defined somewhat differently than @xmath1352 in sect . [ ssec : aux ] . the proof of ( [ eq : pi = q ] ) subject to ( [ eq : nondegenerate ] ) will consist of three stages i , ii , iii where we compute the total contribution from the pairs of links , the pairs of snakes , and the pairs consisting of one snake and one link , respectively . as a consequence , the following three results will be obtained ( implying ( [ eq : pi = q ] ) ) . these propositions are proved in sects . [ ssec : prop1][ssec : prop3 ] . sometimes it will be convenient for us to refer to a white ( black ) snake / link concerning @xmath1332 as a @xmath158-snake / link ( resp . a @xmath221-snake / link ) , and similarly for @xmath1363 . under the exchange operation using @xmath1309 , any @xmath158-link becomes a @xmath273-link and any @xmath221-link becomes a @xmath276-link . the white links occur in @xmath1341 earlier than the black links , and similarly for @xmath1342 . therefore , if @xmath33 are permuting links , then they are of the same color . this implies that @xmath1364 . also each endvertex of any link either is a bend or belongs to @xmath1365 . then ( [ eq : nondegenerate ] ) implies that the sets @xmath1366 and @xmath1367 are disjoint . now lemma [ lm : varphi=1 ] gives @xmath1368 , and the proposition follows . consider two snakes @xmath1369 and @xmath1370 , and let @xmath1346 . if @xmath1371 then @xmath1364 and , moreover , @xmath1372 ( since @xmath1309 is simple and in view of ( [ eq : nondegenerate ] ) ) . this gives @xmath1368 , by lemma [ lm : varphi=1 ] . under the exchange operation using @xmath1309 , any snake changes its color ; so @xmath33 are permuting . applying to @xmath33 lemmas [ lm : asp = asq ] and [ lm : atp = atq ] , we obtain @xmath1378 in subcases 1a,2a , and @xmath1379 in subcases 1b,2b . it is convenient to associate with a bend @xmath1180 the number @xmath1380 which is equal to @xmath1381 if , for the corresponding pair @xmath1382 and @xmath1383 sharing @xmath1180 , @xmath95 is lower than @xmath1018 ( as in subcases 1a,2a ) , and equal to @xmath692 otherwise ( as in subcases 1b,2b ) . define @xmath1384 then @xmath1385 . thus , @xmath1386 is equivalent to @xmath1387 to show ( [ eq : gamma=1 ] ) , we are forced to deal with a more general setting . more precisely , let us turn @xmath1309 into simple cycle @xmath189 by combining the directed path @xmath1388 ( from @xmath1389 to @xmath1390 ) with the horizontal path from @xmath1311 to @xmath1312 ( to create the latter , we formally add to @xmath49 the horizontal edges @xmath1391 for @xmath1392 ) . the resulting directed path @xmath1393 from @xmath1389 to @xmath1394 is regarded as the new white snake replacing @xmath1388 . then @xmath1395 shares the end @xmath1396 with the black path @xmath1397 ; so @xmath1396 is a pit of @xmath189 , and @xmath1393 is lower than @xmath1397 . thus , compared with @xmath1309 , the cycle @xmath189 acquires an additional bend , namely , @xmath1396 . we have @xmath1398 , implying @xmath1399 . then ( [ eq : gamma=1 ] ) is equivalent to @xmath1400 . on this way , we come to a new ( more general ) setting by considering an arbitrary simple ( non - directed ) cycle @xmath189 rather than a special path @xmath1309 . moreover , instead of an se - graph as before , we can work with a more general directed planar graph @xmath49 in which any edge @xmath84 points arbitrarily within the south - east sector , i.e. , satisfies @xmath1401 and @xmath1402 . we call @xmath49 of this sort a _ weak se - graph_. so now we are given a colored simple cycle @xmath189 in @xmath49 , i.e. , @xmath189 is representable as a concatenation @xmath1403 , where each @xmath1404 is a directed path in @xmath49 ; a path ( _ snake _ ) @xmath1404 with @xmath21 odd ( even ) is colored white ( resp . let @xmath1405 be the sequence of bends in @xmath189 , i.e. , @xmath1406 is a common endvertex of @xmath1407 and @xmath1408 ( letting @xmath1409 ) . we assume that @xmath189 is oriented according to the direction of @xmath1404 with @xmath21 even . when this orientation is clockwise ( counterclockwise ) around a point in the open bounded region @xmath1410 of the plane surrounded by @xmath189 , we say that @xmath189 is _ clockwise _ ( resp . _ counterclockwise _ ) . then the cycle arising from the above path @xmath1309 is clockwise . we use induction on the number @xmath1412 of bends of @xmath189 . it suffices to consider the case when @xmath189 is clockwise ( since for a counterclockwise cycle @xmath1413 , the reversed cycle @xmath1414 is clockwise , and it is easy to see that @xmath1415 ) . \(a ) there exists a peak @xmath1406 such that the horizontal line through @xmath1406 meets @xmath189 on the left of @xmath1406 , i.e. , there is a point @xmath1423 in @xmath189 with @xmath1424 and @xmath1425 ; ( this can be seen as follows . let @xmath1426 be a peak with @xmath1427 maximum . then the clockwise orientation of @xmath189 implies that @xmath1428 lies on the right from @xmath1429 . if @xmath1430 , then , by easy topological reasonings , either the pit @xmath1431 is as required in ( b ) ( when @xmath1432 is on the right from @xmath1428 ) , or the peak @xmath1432 is as required in ( a ) ( when @xmath1432 is on the left from @xmath1428 ) , or both . and if @xmath1433 , then @xmath1434 is as in ( b ) . ) we may assume that case ( a ) takes place ( for case ( b ) is symmetric to ( a ) , in a sense ) . choose the point @xmath1423 as in ( a ) with @xmath1435 maximum and draw the horizontal line - segment @xmath242 connecting the points @xmath1423 and @xmath1406 . then the interior of @xmath242 does not meet @xmath189 . two cases are possible : since @xmath1423 can not be a bend of @xmath189 ( in view of @xmath1425 and @xmath1437 for any @xmath1438 ) , @xmath1423 is an interior point of some snake @xmath1429 ; let @xmath1439 and @xmath1440 be the parts of @xmath1429 from @xmath1441 to @xmath1423 and from @xmath1423 to @xmath1442 , respectively . using the facts that @xmath189 is oriented clockwise and this orientation is agreeable with the forward ( backward ) direction of each black ( resp . white ) snake , one can realize that [ eq : casesio ] ( a ) in case ( i ) , @xmath1429 is white and @xmath1443 ( i.e. , for the white snake @xmath1404 and black snake @xmath1408 that share the peak @xmath1406 , @xmath1408 is lower than @xmath1404 ) ; and ( b ) in case ( o ) , @xmath1429 is black and @xmath1444 ( i.e. , @xmath1404 is lower than @xmath1408 ) the points @xmath1423 and @xmath1406 split the cycle ( closed curve ) @xmath189 into two parts @xmath1445 , where the former contains @xmath1439 ( and @xmath1404 ) and the latter does @xmath1440 ( and @xmath1408 ) . we first examine case ( i ) . the line @xmath242 divides the region @xmath1410 into two parts @xmath1446 and @xmath1447 lying above and below @xmath242 , respectively . orienting the curve @xmath1448 from @xmath1423 to @xmath1406 and adding to it the segment @xmath242 oriented from @xmath1406 to @xmath1423 , we obtain closed curve @xmath1449 surrounding @xmath1446 . note that @xmath1449 is oriented clockwise around @xmath1446 . we combine the paths @xmath1439 , @xmath242 ( from @xmath1423 to @xmath1406 ) and @xmath1404 into one directed path @xmath95 ( going from @xmath1450 to @xmath1451 ) . then @xmath1449 turns into a correctly colored simple cycle in which @xmath95 is regarded as a white snake and the white / black snakes structure of the rest preserves ( cf . ( [ eq : casesio])(a ) ) . in its turn , the curve @xmath1452 oriented from @xmath1453 to @xmath1423 plus the segment @xmath242 ( oriented from @xmath1423 to @xmath1406 ) form closed curve @xmath1454 that surrounds @xmath1447 and is oriented clockwise as well . we combine @xmath242 and @xmath1408 into one black snake @xmath1018 ( going from @xmath1423 to @xmath1455 ) . then @xmath1454 becomes a correctly colored cycle , and @xmath1423 is a peak in it . ( the point @xmath1423 turns into a vertex of @xmath49 . ) we have @xmath1456 ( since the white @xmath1440 is lower than the black @xmath1018 ) . we observe that , compared with @xmath189 , the pair @xmath1457 misses the bend @xmath1406 ( with @xmath1443 ) but acquires the bend @xmath1423 ( with @xmath1456 ) . then @xmath1458 implying @xmath1459 therefore , we can apply induction . this gives @xmath1460 . now , by reasonings above , @xmath1461 as required . next we examine case ( o ) . the curve @xmath1448 ( containing @xmath1439 ) passes through the black snake @xmath1408 , and the curve @xmath1452 ( containing @xmath1440 ) through the white snake @xmath1404 . adding to each of @xmath1445 a copy of @xmath242 , we obtain closed curves @xmath1457 , respectively , each inheriting the orientation of @xmath189 . they become correctly colored simple cycles when we combine the paths @xmath1462 into one black snake ( from @xmath1434 to @xmath1455 ) in @xmath1449 , and combine the paths @xmath1463 into one white snake ( from the new bend @xmath1423 to @xmath1406 ) in @xmath1454 . let @xmath1464 be the bounded regions in the plane surrounded by @xmath1457 , respectively . it is not difficult topological exercise to see that two cases are possible : then in case ( o1 ) , @xmath1449 is clockwise and @xmath1454 is counterclockwise , whereas in case ( o2 ) the behavior is converse . also @xmath1444 and @xmath1465 . similar to case ( i ) , relation ( [ eq : ddd ] ) is true and we can apply induction . then in case ( o1 ) , we have @xmath1466 and @xmath1467 , whence @xmath1468 and in case ( o2 ) , we have @xmath1469 and @xmath1470 , whence @xmath1471 consider a link @xmath242 . by lemma [ lm : varphi=1 ] , for any snake @xmath78 , @xmath1472 is possible only if @xmath242 and @xmath78 have a common endvertex @xmath87 . note that @xmath1473 . in particular , it suffices to examine only bounded and semi - bounded links . first assume that @xmath1474 . then there are exactly two snakes containing @xmath1475 , namely , a white snake @xmath95 and a black snake @xmath1018 such that @xmath1476 . if @xmath242 is white , then @xmath95 and @xmath242 belong to the same path in @xmath158 ; therefore , @xmath1477 . under the exchange operation @xmath95 becomes black , @xmath1018 becomes white , and @xmath242 continues to be white . then @xmath1478 belong to the same path in @xmath273 ; this implies @xmath1479 . so both pairs @xmath1480 and @xmath1481 are permuting . lemma [ lm:1atp = asq ] gives @xmath1482 and @xmath1483 , whence @xmath1484 . the end @xmath1490 is examined in a similar way . assuming @xmath1491 , there are exactly two snakes , a white snake @xmath1017 and a black snake @xmath1019 , that contain @xmath1490 , namely : @xmath1492 . if @xmath242 is white , then @xmath1493 and @xmath1494 . therefore , @xmath1495 and @xmath1496 are invariant , yielding @xmath1497 . and if @xmath242 is black , then @xmath1498 and @xmath1499 . so both @xmath1500 and @xmath1501 are permuting , and we obtain from lemma [ lm:1atp = asq ] that @xmath1502 and @xmath1503 , yielding @xmath1504 . we have proved relation ( [ eq : pi = q ] ) in a non - degenerate case , i.e. , subject to ( [ eq : nondegenerate ] ) , and now our goal is to prove ( [ eq : pi = q ] ) when the set @xmath1505 contains distinct elements @xmath1135 with @xmath1136 . we say that such @xmath1135 form a _ defect pair_. a special defect pair is formed by twins @xmath1330 ( bends satisfying @xmath1506 , @xmath1507 and @xmath1508 ) . another special defect pair is of the form @xmath1509 when @xmath78 is a _ vertical _ snake or link , i.e. , @xmath1510 . let @xmath709 be the _ minimum _ number such that the set @xmath1511 contains a defect pair . we denote the elements of @xmath7 as @xmath1512 , where for each @xmath21 , @xmath74 is _ higher _ than @xmath75 , which means that either @xmath1513 , or @xmath73 are twins and @xmath74 is a pit ( and @xmath1514 is a peak ) in the exchange path @xmath1309 . the highest element @xmath1515 is also denoted by @xmath86 . in order to conduct induction , we deform the graph @xmath49 within a sufficiently narrow vertical strip @xmath1516\times \rset$ ] ( where @xmath1517 ) to get rid of the defect pairs involving @xmath86 in such a way that the configuration of snakes / links in the arising graph @xmath1518 remains `` equivalent '' to the initial one . more precisely , we shift the bend @xmath86 at a small distance ( @xmath1519 ) to the left , keeping the remaining elements of @xmath1520 ; then the bend @xmath1521 arising in place of @xmath86 satisfies @xmath1522 and @xmath1523 . the snakes / links with an endvertex at @xmath86 are transformed accordingly ; see the picture for an example . let @xmath268 and @xmath1524 denote the l.h.s . value in ( [ eq : pi = q ] ) for the initial and deformed configurations , respectively . under the deformation , the number of defect pairs becomes smaller , so we may assume by induction that @xmath1525 . thus , we have to prove that @xmath1526 we need some notation and conventions . for @xmath1527 , the set of ( initial ) snakes and links with an endvertex at @xmath87 is denoted by @xmath1528 . for @xmath1529 , @xmath1530 denotes @xmath1531 . corresponding objects for the deformed graph @xmath1518 are usually denoted with tildes as well ; e.g. : for a path @xmath78 in @xmath49 , its image in @xmath1518 is denoted by @xmath1532 ; the image of @xmath1528 is denoted by @xmath1533 ( or @xmath1534 ) , and so on . the set of standard paths in @xmath1530 ( resp . @xmath1535 ) is denoted by @xmath1536 ( resp . @xmath1537 ) . define @xmath1538 a similar product for @xmath1518 ( i.e. , with @xmath1539 instead of @xmath1540 ) is denoted by @xmath1541 . note that ( [ eq : varpi ] ) is equivalent to @xmath1542 this follows from the fact that for any paths @xmath1543 different from those involved in ( [ eq : u_x - u ] ) , the values @xmath1544 and @xmath1545 are equal . ( the only nontrivial case arises when @xmath1546 and @xmath648 is vertical ( so @xmath1547 becomes standard ) . then @xmath1548 . hence @xmath1549 , the pair @xmath1126 is involved in @xmath1550 , and the pair @xmath1551 in @xmath1541 . ) to simplify our description technically , one trick will be of use . suppose that for each standard path @xmath1552 , we choose a point ( not necessarily a vertex ) @xmath1553 in such a way that @xmath1554 , and the coordinates @xmath1555 for all such paths @xmath78 are different . then @xmath1556 splits @xmath78 into two subpaths @xmath1557 , where we denote by @xmath1558 the subpath connecting @xmath1138 and @xmath1556 when @xmath1559 , and connecting @xmath1556 and @xmath1139 when @xmath1560 , while @xmath1561 is the rest . this provides the following property : for any @xmath1562 , @xmath1563 ( in view of lemma [ lm : varphi=1 ] ) . hence @xmath1564 . also @xmath1565 . it follows that ( [ eq : varpix ] ) would be equivalent to the equality @xmath1566 for @xmath1570 , we denote by @xmath1571 , respectively , the white snake , black snake , white link , and black link that have an endvertex at @xmath75 . note that if @xmath73 are twins , then the fact that @xmath74 is a pit implies that @xmath1572 are the snakes entering @xmath74 , and @xmath1573 are the snakes leaving @xmath75 ; for convenience , we formally define @xmath1574 to be the same trivial path consisting of the single vertex @xmath75 . note that if @xmath1575 , then some paths among @xmath1576 vanish ( e.g. , both snakes and one link ) . when vertices @xmath75 and @xmath1577 are connected by a ( vertical ) path in @xmath1578 , we denote such a path by @xmath236 and say that the vertex @xmath75 is _ open _ ; otherwise @xmath75 is said to be closed . note that @xmath1579 can be connected by either one snake , or one link , or two links ( namely , @xmath1580 ) ; in the latter case , @xmath236 is chosen arbitrarily among them . in particular , if @xmath1579 are twins , then @xmath75 is open and the role of @xmath236 is played by any of the trivial links @xmath1580 . obviously , in a sequence of vertical paths @xmath1581 , the snakes and links alternate . one can see that if @xmath236 is a white snake , i.e. , @xmath1582 , then both black snakes @xmath1583 are standard , and we have @xmath1584 and @xmath1585 . see the left fragment of the picture : in its turn , if @xmath236 is a nontrivial white link , i.e. , @xmath1590 , then two cases are possible : either the black links @xmath1591 are standard , @xmath1592 and @xmath1593 , or @xmath1594 . and if @xmath236 is a black link , the behavior is symmetric . see the picture : indeed , suppose that @xmath1599 are white , and let @xmath648 and @xmath1227 be the components of the flow @xmath158 containing @xmath78 and @xmath1558 , respectively . since @xmath1599 are separated , the paths @xmath1600 are different . moreover , the fact that @xmath1558 is lower than @xmath78 implies that @xmath1227 is lower than @xmath648 ( since @xmath1600 are disjoint ) . then @xmath1227 precedes @xmath648 in @xmath158 , yielding @xmath1598 , as required . when @xmath1599 concern one of @xmath1601 , the argument is similar . in what follows we will use the abbreviated notation @xmath1602 for the paths @xmath1603 ( respectively ) having an endvertex at @xmath1604 . also for @xmath1605 , we denote the product @xmath1606 by @xmath1607 , and denote by @xmath1608 a similar product for the paths @xmath1609 ( concerning the deformed graph @xmath1518 ) . one can see that @xmath1550 ( resp . @xmath1610 ) is equal to the product of the values @xmath1607 ( resp . @xmath1611 ) over @xmath1605 . let @xmath1613 for @xmath1614 . observe that ( [ eq : assumption ] ) together with the fact that the vertex @xmath86 moves under the deformation of @xmath49 implies that @xmath1615 holds for any @xmath1616 . this gives @xmath1617 , by lemma [ lm : varphi=1 ] . note that @xmath1619 are as follows : either ( a ) @xmath1620 or ( b ) @xmath1621 , and either ( c ) @xmath1622 or ( d ) @xmath1623 . let us examine the possible cases when the combination of ( a ) and ( d ) takes place . \1 ) let @xmath112 be a white link , i.e. , @xmath1624 . since @xmath112 is white and lower than @xmath1602 , we have @xmath1625 ( cf . ( [ eq : monochpq ] ) ) . the exchange operation preserves the color of @xmath112 . then @xmath1626 . therefore , all pairs @xmath1627 with @xmath1567 are invariant , and @xmath1618 is trivial . \2 ) let @xmath1628 . since @xmath112 is black , we have @xmath1629 . the exchange operation changes the colors of @xmath33 and preserves the ones of @xmath1630 . hence @xmath1631 , giving the permuting pairs @xmath1632 and @xmath1633 . lemma [ lm : atp = atq ] applied to these pairs implies @xmath1634 and @xmath1635 . then @xmath1636 . \3 ) let @xmath1637 . then @xmath1625 and @xmath1631 ( since the exchange operation changes the colors of @xmath1638 ) . this gives the permuting pairs @xmath1633 and @xmath1639 . then @xmath1635 , by lemma [ lm : atp = atq ] , and @xmath1640 by lemma [ lm:2atp = asq ] , and we have @xmath1641 . \4 ) let @xmath1642 . ( in fact , this case is symmetric to the previous one , as it is obtained by swapping @xmath179 and @xmath265 . yet we prefer to give details . ) we have @xmath1629 and @xmath1626 , giving the permuting pairs @xmath1632 and @xmath1643 . then @xmath1634 , by lemma [ lm : atp = atq ] , and @xmath1644 , by lemma [ lm:2atp = asq ] , whence @xmath1618 . the other combinations , namely , ( a ) and ( c ) , ( b ) and ( c ) , ( b ) and ( d ) , are examined in a similar way ( where we appeal to appropriate lemmas from sect . [ sec : two_paths ] ) , and we leave this to the reader as an exercise . suppose that we replace @xmath1648 by a standard path @xmath1558 of the same color and type ( snake or link ) such that @xmath1649 ( and @xmath1650 ) . then the set @xmath1651 becomes as in case ( r1 ) , and by proposition [ pr : caser1 ] , the corresponding product @xmath1652 of values @xmath1653 over @xmath1654 is equal to 1 . ( this relies on the fact that @xmath112 is separated from @xmath1602 . ) now compare the effects from @xmath1558 and @xmath1655 . these paths have the same color and type , and both are separated from , and higher than @xmath112 . also @xmath1656 ( since @xmath1649 and @xmath1657 ) . then using appropriate lemmas from sect . [ sec : two_paths ] , one can conclude that @xmath1658 . therefore , @xmath1659 now let @xmath86 and @xmath1645 be connected by two paths , namely , by @xmath1660 . we can again appeal to proposition [ pr : caser1 ] . consider @xmath1661 , where @xmath1662 are standard links ( white and black , respectively ) with @xmath1663 . then @xmath1664 and @xmath1665 , and we obtain @xmath1666 as required . suppose that @xmath1648 and @xmath112 are contained in the same path of the flow @xmath158 ; equivalently , both @xmath1668 are white and @xmath1669 . then neither @xmath273 nor @xmath276 has a path containing both @xmath1668 ( this is easy to conclude from the fact that one of @xmath112 and @xmath1670 is a snake and the other is a link ) . consider four possible cases for @xmath1668 . \(a ) let both @xmath1668 be links , i.e. , @xmath1671 and @xmath1624 . then @xmath1672 and @xmath1673 ( since @xmath1674 is impossible by the above observation ) . this gives the permuting pairs @xmath1675 and @xmath1676 , yielding @xmath1677 . when @xmath1668 are contained in the same path in @xmath221 ( i.e. , @xmath1668 are black and @xmath1669 ) , we argue in a similar way . the cases with @xmath1668 contained in the same path of @xmath273 or @xmath276 are symmetric . a similar analysis is applicable ( yielding @xmath1647 ) when @xmath86 and @xmath1645 are connected by two vertical paths ( namely , @xmath1660 ) and exactly one relation among @xmath1693 , @xmath1694 , @xmath1695 and @xmath1696 takes place ( equivalently : either @xmath1697 or @xmath1698 are separated , not both ) . finally , let @xmath86 and @xmath1645 be connected by both @xmath1660 , and assume that @xmath1697 are not separated , and similarly for @xmath1698 . an important special case is when @xmath1699 and @xmath1700 are twins . note that from the assumption it easily follows that @xmath112 is a snake . if @xmath112 is the white snake @xmath1701 , then we have @xmath1702 and @xmath1703 . this gives the permuting pairs @xmath1681 and @xmath1704 , yielding @xmath1705 ( since @xmath1706 ) ) . the case with @xmath1642 is symmetric . in both cases , @xmath1647 . the equality @xmath1618 is trivial . to see @xmath1617 , consider possible cases for @xmath112 . if @xmath1708 , then @xmath1709 and @xmath1710 , giving the permuting pairs @xmath1711 and @xmath1712 ( note that @xmath1713 ) . if @xmath1714 , then @xmath1715 and @xmath1716 ; so all pairs involving @xmath1717 are invariant . if @xmath1718 , then @xmath1719 and @xmath1720 , giving the permuting pairs @xmath1721 and @xmath1722 ( note that @xmath1723 ) . and the case @xmath1724 is symmetric to the previous one . the equality @xmath1727 is trivial . to see @xmath1728 , observe that @xmath1729 and @xmath1730 . this gives the permuting pairs @xmath1711 and @xmath1731 . by lemma [ lm:1atp = asq ] , @xmath1732 and @xmath1733 , and the result follows . taken together , propositions [ pr : caser2_sep][pr : caser2_kl ] embrace all possibilities in case ( r2 ) . adding to them proposition [ pr : caser1 ] concerning case ( r1 ) , we obtain the desired relation ( [ eq : varpix ] ) in a degenerate case . let @xmath1734 and @xmath1304 be as in the hypotheses of theorem [ tm : single_exch ] . we have proved this theorem in case ( c ) , i.e. , when @xmath203 is a @xmath42-couple with @xmath310 and @xmath1735 ( see the beginning of sect . [ sec : exchange ] ) . in other words , the exchange path @xmath1310 , used to transform the initial double flow @xmath179 into the new double flow @xmath265 , connects the sinks @xmath1311 and @xmath1312 covered by the `` white flow '' @xmath158 and the `` black flow '' @xmath221 , respectively . case ( c1 ) is symmetric to ( c ) . this means that if double flows @xmath179 and @xmath265 are obtained from each other by applying the exchange operation using @xmath203 ( which , in particular , changes the `` colors '' of both @xmath1741 and @xmath1742 ) , and if one double flow is subject to ( c ) , then the other is subject to ( c1 ) . rewriting @xmath1743 as @xmath1744 , we just obtain the required equality in case ( c1 ) ( where @xmath265 and @xmath179 play the roles of the initial and updated double flows , respectively ) . to do this , we appeal to reasonings similar to those in sects . [ ssec : prop1][ssec : degenerate ] . more precisely , one can check that the descriptions in sects . [ ssec : prop1 ] and [ ssec : prop3 ] ( concerning link - link and snake - link pairs in @xmath1341 ) remain applicable and propositions [ pr : link - link ] and [ pr : seg - link ] are directly extended to cases ( c2 ) and ( c4 ) . the method of getting rid of degeneracies developed in sect . [ ssec : degenerate ] does work , without any troubles , for ( c2 ) and ( c4 ) as well . as to the method in sect [ ssec : prop2 ] ( concerning snake - snake pairs in case ( c ) ) , it should be modified as follows . we use terminology and notation from sects . [ ssec : seglink ] and [ ssec : prop2 ] and appeal to lemma [ lm : gammad ] . when dealing with case ( c2 ) , we represent the exchange path @xmath1310 as a concatenation @xmath1745 , where each @xmath1315 with @xmath21 odd ( even ) is a snake contained in the black flow @xmath221 ( resp . the white flow @xmath158 ) . then @xmath1388 begins at the source @xmath1746 and @xmath1397 begins at the source @xmath1747 . an example with @xmath1748 is illustrated in the left fragment of the picture : the common vertex ( bend ) of @xmath1315 and @xmath1321 is denoted by @xmath1320 . as before , we associate with a bend @xmath1180 the number @xmath1380 ( equal to 1 if , in the pair of snakes sharing @xmath1180 , the white snake is lower that the black one , and @xmath692 otherwise ) , and define @xmath1749 as in ( [ eq : gammaz ] ) . we turn @xmath1309 into simple cycle @xmath189 by combining the directed path @xmath1397 ( from @xmath1747 to @xmath1750 ) with the vertical path from @xmath1746 to @xmath1747 , which is formally added to @xmath49 . ( in the above picture , this path is drawn by a dotted line . ) then , compared with @xmath1309 , the cycle @xmath189 has an additional bend , namely , @xmath1746 . since the extended white path @xmath1751 is lower than the black path @xmath1388 , we have @xmath1752 , and therefore @xmath1399 . one can see that the cycle @xmath189 is oriented clockwise ( where , as before , the orientation is according to that of black snakes ) . so @xmath1400 , by lemma [ lm : gammad ] , implying @xmath1753 . this is equivalent to the `` snake - snake relation '' @xmath1386 , and as a consequence , we obtain the desired equality @xmath1754 finally , in case ( c4 ) , we represent the exchange path @xmath1309 as the corresponding concatenation @xmath1755 ( with @xmath1314 odd ) , where the first white snake @xmath1388 ends at the sink @xmath1312 and the last white snake @xmath1397 begins at the source @xmath1747 . see the right fragment of the above picture , where @xmath465 . we turn @xmath1309 into simple cycle @xmath189 by adding a new `` black snake '' @xmath1756 beginning at @xmath1747 and ending at @xmath1312 ( it is formed by the vertical path from @xmath1747 to @xmath1757 , followed by the horizontal path from @xmath1757 to @xmath1312 ; see the above picture ) . compared with @xmath1309 , the cycle @xmath189 has two additional bends , namely , @xmath1747 and @xmath1312 . since the black snake @xmath1756 is lower than both @xmath1388 and @xmath1397 , we have @xmath1758 , whence @xmath1759 . note that the cycle @xmath189 is oriented counterclockwise . therefore , @xmath1411 , by lemma [ lm : gammad ] , implying @xmath1760 . as a result , we obtain the desired equality @xmath312 .

we give a complete combinatorial characterization of homogeneous quadratic relations of `` universal character '' valid for minors of quantum matrices ( more precisely , for minors in the quantized coordinate ring @xmath0 of @xmath1 matrices over a field @xmath2 , where @xmath3 ) . this is obtained as a consequence of a study of quantized minors of matrices generated by paths in certain planar graphs , called _ se - graphs _ , generalizing the ones associated with cauchon diagrams . our efficient method of verifying universal quadratic identities for minors of quantum matrices is illustrated with many appealing examples . _ keywords _ : quantum matrix , quantum affine space , quadratic identity for minors , planar graph , cauchon diagram , lindstrm lemma _ msc - class _ : 16t99 , 05c75 , 05e99

over the last 15 years , far - red and near - infrared ground - based sky surveys have revealed the local - neighborhood populations of l and t dwarfs ( e.g. kirkpatrick 1995 ) . the 2-m - class telescopes of the two micron all sky survey ( 2mass , skrutskie et al . 2006 ) and the sloan digital sky survey ( sdss , york et al . 2000 ) extended the brown dwarf sequence to t8 spectral type with effective temperatures ( @xmath18 ) as low as 750 k ( e.g. burgasser et al . 2000 ) . the 4-m - class ukirt infrared deep sky survey ( ukidss , lawrence et al . 2007 ) and canada france brown dwarf survey ( cfbds , delorme et al . 2008 ) identified dwarfs as cool as @xmath19 k , with spectral types of t9 or t10 ( e.g. lucas et al . 2010 ) . in 2011 , the mid - infrared 0.4-m telescope wide - field infrared survey explorer ( _ wise _ , wright et al . 2010 ) revealed prototype y dwarfs ( cushing et al . 2011 ) , with @xmath20 300 450 k. the extension of the lower main - sequence to the brown dwarfs now spans a range in luminosity of @xmath21 to @xmath22 l@xmath23 , and in mass of @xmath24 to 10 m@xmath12 . the t and y dwarfs are more similar to the gas - giant planets than to the stars , and clouds complicate the analysis of their spectra . the spectral energy distributions ( seds ) of l dwarfs are reddened by clouds consisting of liquid and solid iron and silicates ( e.g. ruiz , leggett & allard 1997 ; ackerman & marley 2001 ; tsuji 2002 ; helling et al . the transition from l to t is associated with a clearing of these clouds , such that mid - type t dwarfs have clear atmospheres ( e.g. knapp et al . 2004 ; marley , saumon & goldblatt 2010 ) . we have recently shown ( morley et al . 2012 ) that chloride and sulfide clouds are important in the atmospheres of late - t and early - y dwarfs . first water and then ammonia clouds are expected to occur as dwarfs cool below @xmath25 k , as seen in jupiter . in this work we present new near - infrared @xmath0 photometry for the six objects identified by cushing et al . ( 2011 ) as y dwarfs . these data are more precise and cover a wider wavelength range than the @xmath26 photometry presented in cushing et al . and kirkpatrick et al . we compare the new and previously published data to the models which include the chloride and sulfide clouds , reconfirm the validity of the models , and use them to estimate temperature and gravity for the y dwarfs . we also present a far - red spectrum of one of the brightest y dwarfs , obtained as part of a search for nh@xmath7 absorption features in 400 k objects . photometry was obtained in some or all of the @xmath0 filters for the six objects identified by cushing et al . ( 2011 ) as y dwarfs : wisep j041022.71@xmath1150248.5 , wisepc j140518.40@xmath1553421.5 , wisep j154151.65@xmath3225025.2 , wisep j173835.52@xmath1273258.9 , wisep j182831.08@xmath1265037.8 and wisepc j205628.90@xmath1145953.3 . hereafter the source names are abbreviated to the first four digits of the ra and declination . all of @xmath0 were measured for the three brightest objects wisepc j0410@xmath11502 , wisepc j1738@xmath12732 and wisepc j2056@xmath11459 . @xmath2 only were obtained for wisepc j1405@xmath15534 and wisepc j1541@xmath272250 due to constraints on available telescope time the fainter @xmath17-band observations were omitted . wisepc j1828@xmath12650 was identified by cushing et al . as the coolest of the y dwarfs , and we obtained @xmath4 photometry for this particularly interesting object ; we could not improve on the accuracy of the @xmath28 magnitude given by kirkpatrick et al . ( 2012 ) in a reasonable amount of time , and so @xmath28-band data were not obtained . the @xmath2 photometry for wisepc j1405@xmath15534 and the @xmath29 photometry for wisepc j1541@xmath272250 was previously published by morley et al . ( 2012 ) . the near - infrared imager ( niri , hodapp et al . 2003 ) was used on gemini north in programs gn-2012a - dd-7 , gn-2012a - q-106 , gn-2012b - q-27 and gn-2012b - q-75 . the filter sets are on the mauna kea observatories ( mko ) system ( tokunaga , simons & vacca 2002 , tokunaga & vacca 2005 ) , although there is some variation in the @xmath30 filter bandpass between the cameras used on mauna kea ( see 3 and liu et al . 2012 ) . exposure times of 30 s or 60 s were used , with a 5- or 9-position telescope dither pattern . the total integration time is given in table 1 , together with the derived photometry . the data were reduced in the standard way using routines supplied in the iraf gemini package . ukirt faint standards were used for calibration , taking the @xmath30 data from the ukirt online catalog and the @xmath31 data from leggett et al . all data were taken on photometric nights with typical near - infrared seeing of @xmath32 . aperture photometry was carried out with apertures of radii 5 8 pixels , or diameters of @xmath33 @xmath34 ; aperture corrections were derived from stars in the field . sky levels were determined from concentric annular regions and uncertainties were derived from the sky variance . most of our measurements agree within the errors with the mko - system values presented in cushing et al . ( 2011 ) and kirkpatrick et al . however two of the dwarfs are much fainter in @xmath35 than the previously published values : wisepc j1405@xmath15534 is a magnitude fainter , and wisepc j1738@xmath12732 is 0.5 magnitudes fainter . also one dwarf is a magnitude fainter at @xmath28 , wisepc j0410@xmath11502 . our measurements are more consistent with the shape of the near - infrared spectra presented by cushing et al . , and it seems likely that the palomar wirc data are corrupted , perhaps by detector hot pixels or similar , and that the difference is not due to extreme variability . we obtained far - red spectra of wisepc j2056@xmath11459 using the gemini multi - object spectrograph ( gmos , hook et al . 2004 ) at gemini north , through director s discretionary time granted under program gn-2012a - dd-7 . the r150 grating was used with the rg610 blocking filter . the central wavelength was 1000 nm , with wavelength coverage of 600 1040 nm , though the y dwarf is only detected at wavelength longer than 830 nm . the 1@xmath365 slit was used with @xmath37 binning , and the resulting resolution was @xmath38 900 or 10 . four 3200 s frames were obtained on 2012 june 5 , for a total on - source time of 3.6 hours . flatfielding and wavelength calibration were achieved using lamps in the on - telescope calibration unit . the spectrophotometric standard bd + 28 4211 was used to determine the instrument response curve . the data were reduced using routines supplied in the iraf gemini package . the spectrum was flux calibrated using the measured niri @xmath30 photometry , extending the gmos spectrum from 1.04 @xmath5 m to 1.09 @xmath5 m using the cushing et al . ( 2011 ) @xmath30-band spectrum of wisepc j1541@xmath272250 as a template . figure 1 shows the gmos spectrum for wisepc j2056@xmath11459 , as well as the spectrum for ugps j072227.51@xmath27054031.2 ( ugps j0722@xmath270540 ) from leggett et al . the absorption features due to cs i that are apparent in the @xmath19 k dwarf ugps j0722@xmath270540 are not detected in the @xmath39 k dwarf wisepc j2056@xmath11459 . this is not unexpected cs should be predominantly in the form of cscl in atmospheres this cool ( e.g. lodders 1999 , leggett et al . what is unexpected is the lack of strong nh@xmath7 features at 1.02 1.04 @xmath5 m ( e.g. leggett et al . we return to this point below in 4.4 . kirkpatrick et al . ( 2011 , 2012 ) give trigonometric parallaxes for wisepc j0410@xmath11502 , wisepc j1405@xmath15534 , wisepc j1541@xmath272250 , wisepc j1738@xmath12732 and wisepc j1828@xmath12650 . table 2 lists these values , together with the niri photometry presented in 2.1 , and the @xmath28 band magnitude for wisepc j1828@xmath12650 presented in kirkpatrick et al . liu et al . ( 2012 , their appendix a ) show that small differences in the @xmath30 bandpass can lead to significant differences in t and y dwarf photometry . the niri @xmath30 filter is shifted blueward of the wfcam / ukidss mko-@xmath30 filter by 0.007 @xmath5 m or @xmath407% of the filter width ( the keck nirc2 filter is bluer than the niri filter by a similar amount ) . due to the rapidly rising flux to the red side of the filter , the blueward shift results in fainter niri magnitudes . liu et al . use spectra of t8 y0 dwarfs to synthesize photometry in the niri and ukidss systems , and derive @xmath41 . as the largest set of @xmath30-band photometry has been obtained via the ukidss project , it is assumed that the ukirt / wfcam filter defines the @xmath30-band of the mko photometric system , and in the plots shown in this paper we have reduced the niri @xmath30-band magnitudes by 0.17 . kirkpatrick et al . also give the all - sky data release _ wise _ photometry for the six y dwarfs , as well as _ spitzer _ warm - mission irac 3.6 @xmath5 m and 4.5 @xmath5 m photometry . irac data obtained by the _ wise _ team is now available in the spitzer science archive . the data were obtained between 2010 july and 2011 march , via cycle 7 go program 70062 and ddt program 551 , with pis kirkpatrick and mainzer . we have carried out aperture photometry on the archived data , using the mosaics produced by the spitzer pipeline s18.18.0 . for all but one source , apertures with radii of 6 pixels or diameter 7.2 " were used ; due to a nearby star , smaller apertures were used for wisepc j1541@xmath272250 of radius 3 and 4.5 pixels for the 3.6 @xmath5 m and 4.5 @xmath5 m channels respectively . aperture corrections were derived from stars in the field for wisepc j1541@xmath272250 , and from the irac handbook for the other brown dwarfs . in all cases sky levels were determined from concentric annular regions . uncertainties were derived from the sky variance . we find differences between our values and those of kirkpatrick et al . of typically 10% for the fainter 3.6 @xmath5 m data , and 5% for the brighter 4.5 @xmath5 m data . the differences are likely due to different data reduction techniques . table 2 lists the _ wise _ magnitudes as given by the wise all - sky data release and the irac magnitudes as determined by us . figure 2 plots the difference between the magnitudes measured in the similar passbands irac [ 3.6 ] and _ wise _ w1 , and irac [ 4.5 ] and _ wise _ w2 ; w1 extends the [ 3.6 ] bandpass to the blue , and w2 extends [ 4.5 ] to the red . the difference between w2 and [ 4.5 ] magnitudes is small for t and y dwarfs . w1 is significantly fainter than [ 3.6 ] for t dwarfs , because the bandpass extends into a region with very little flux . the single y dwarf detected in w1 is wise j1541@xmath272250 . the apparently low w1 @xmath27 [ 3.6 ] value for this object is likely to be spurious , as the _ wise _ images show no clear w1 source at the position of the w2 source , but does show other , bluer , nearby sources . the t8.5 dwarf wolf 940b ( burningham et al . 2009 ) also stands out in figure 2 . leggett et al . ( 2010b ) use near- and mid - infrared spectroscopy and photometry to show that wolf 940b is a fairly typical late - type t dwarf with @xmath42 k and @xmath43 ( cm s@xmath44 ) . most likely the _ wise _ photometry for this object is compromised by the presence of the close and infrared - bright primary , as is suggested by the _ wise _ images . the models used here are described in detail in morley et al . the model atmospheres are as described in saumon & marley ( 2008 ) and marley et al . ( 2002 ) , with updates to the line list of nh@xmath7 and of the collision - induced absorption of h@xmath6 as described in saumon et al . ( 2012 ) . to these models , morley et al . have added absorption and scattering by condensates of cr , mns , na@xmath6s , zns and kcl . these condensates have been predicted to be present by lodders ( 1999 ) and visscher , lodders & fegley ( 2006 ) . morley et al . use the ackerman & marley ( 2001 ) cloud model to account for these previously neglected clouds . the vertical cloud extent is determined by balancing upward turbulent mixing and downward sedimentation . a parameter @xmath45 describes the efficiency of sedimentation , and is the ratio of the sedimentation velocity to the convective velocity ; lower values of @xmath45 imply thicker ( i.e. more vertically extended ) clouds . we have found that our models which include iron and silicate grains and which have @xmath45 of typically 2 3 fit l dwarf spectra well , those with @xmath45 3 4 fit t0 to t3 spectral types well , and cloud - free models fit t4 t8 types well ( e.g. stephens et al . 2009 , saumon & marley 2008 ) . however for later spectral types significant discrepancies exist between our models and the observations in the near - infrared ( e.g. leggett et al . 2009 , 2012 ) . the new models with chloride and sulfide clouds show that these clouds are significant for dwarfs with @xmath46 900 k ( approximately t7 to y1 spectral types ) , with a peak impact at around 600 k ( or t9 types ) ; na@xmath6s is the dominant species by mass , however at 400 k kcl is also important . a suite of models was generated with @xmath47 k@xmath48 , @xmath49 and @xmath50 . below @xmath51 water clouds are expected to form , which have not yet been incorporated into the models ( although water condensation is accounted for by removal of water from the gas opacity ) . the chloride and sulfide clouds impact the 0.6 1.3 @xmath5 m wavelength region in particular , corresponding to the @xmath52 photometric passbands . this otherwise clear region of the photosphere becomes opaque , and the flux emitted at these wavelengths then arises from a higher atmospheric layer that is cooler by around 200 k. the 1 @xmath5 m flux is therefore significantly reduced by the presence of these clouds . morley et al . show that the new models which include the chloride and sulfide clouds reproduce the observations of late - type t dwarfs much better than is done by models without these clouds . in particular , for the well - studied 500 k brown dwarf ugps j0722@xmath270540 , with data that covers optical to mid - infrared wavelengths , the sed is fit remarkably well with a @xmath19 k , @xmath53 , @xmath54 model . the one region that is not fit well is the 5 @xmath5 m region , where the models are too bright by a factor of @xmath40 2 . this is because the cloudy models do not currently include departures from chemical equilibrium caused by vertical mixing . the mixing enhances the abundance of co and co@xmath6 and reduces the @xmath55 m flux ( morley et al . 2012 and references therein ) . it can be parametrized with an eddy diffusion coefficient of @xmath56 @xmath57 s@xmath58 , where values of log @xmath59 6 , corresponding to mixing timescales of @xmath60 yr to @xmath61 hr , respectively , reproduce the observations of t dwarfs ( e.g. saumon et al . leggett et al . ( 2012 ) find that ugps j0722@xmath270540 is undergoing vigorous mixing , with log @xmath62 6.0 , which results in an increase of w2 by @xmath63 magnitude ( their figures 6 and 7 ) . cloudless equilibrium and non - equilibrium models imply that the impact on the w2 or [ 4.5 ] magnitudes is around @xmath64 magnitudes for @xmath65 k @xmath66 and @xmath67 . given that the impact for the 500 k dwarf ugps j0722@xmath270540 is of a similar order , and that co is dredged up into atmospheres as cool as jupiter s ( e.g. noll et al . 1988 ) , for simplicity here we add 0.3 magnitudes to the w2 and [ 4.5 ] magnitudes computed for all the model sequences used in this paper , at all values of @xmath18 , to mimic the effect of vertical mixing . we find below that such a correction reproduces the observed trends in t and y dwarf w2 colors quite well ( see 4.3 ) . our next generation of models will incorporate water clouds and mixing . tables 3 and 4 list mko - system near - infrared photometry , and irac and _ wise _ photometry , generated by the morley et al . ( 2012 ) cloudy models and the saumon et al . ( 2012 ) cloud - free models , for brown dwarfs at 10 pc . the cloudy models include na@xmath6s , mns , zns , cr , kcl condensate clouds and do not include fe , mg@xmath6sio@xmath68 , or al@xmath6o@xmath7 condensate clouds . colors are calculated using saumon & marley ( 2008 ) cloud - free evolution model grids , and all are in the vega system . our sample for this work consists of brown dwarfs with spectral type t6 and later , that have been detected in the w2 band , and that have mko - system near - infrared photometry . eighty - three brown dwarfs satisfy these criteria at the time of writing . the appendix contains a data table which gives , for these 83 brown dwarfs : coordinates , spectral type , distance modulus , @xmath69[3.6][4.5][5.8][8.0]w1w2w3w4 photometry , uncertainties in distance modulus and photometry , and source references . the data table also flags objects that are close binary systems . recently , four new binary systems have been identified at or near the t / y dwarf boundary ( burgasser et al . 2012 ; liu et al . 2011 , 2012 ) . these systems are extremely useful for testing the models , as they will have the same metallicity , and the same age . the age constraint , together with the measured luminosity , translates into a maximum difference in @xmath70 of 0.4 dex . the binary systems are listed in table 5 . burgasser et al . and liu et al . give resolved mko - system near - infrared photometry for the binaries . the systems are not resolved by irac or wise . we have estimated w2 magnitudes for the individual components using the unresolved values , and the expected difference between the components based on the difference in spectral types ( figure 3 ) . the uncertainty in the estimate is around 0.3 magnitudes which is derived from the range of w2 values that will produce the measured unresolved w2 . figure 3 is a plot of various colors as a function of spectral type . spectral types are from kirkpatrick et al . ( 2011 , 2012 ) , and include the adjustment to the late - type t dwarf classification described in cushing et al . the reader should note that the definition of the end of the t sequence and the start of the y sequence is very preliminary at this time . it is likely that the classification scheme will need to be revised once more y dwarfs are known . @xmath71 for the y dwarfs shows a marked decline , while @xmath72 is mostly flat up to the y2 class . @xmath73 is also mostly flat , although there is more scatter , possibly reflecting a range in gravity , or age , for the sample ( see discussion of figure 8 below ) . @xmath74 w2 shows a steady and rapid increase with later spectral types . [ 3.6 ] @xmath27 w2 and w2 @xmath27 w3 also show an increase , however the object classified as y2 appears to turn over in the [ 3.6 ] @xmath27 w2 diagram , and the w2 @xmath27 w3 diagram shows a lot of scatter ( although the uncertainties are quite large ) . we come back to the y2 dwarf wisepc j1828@xmath12650 in 4.5 . figure 4 shows absolute @xmath35 magnitude as a function of the near - infrared colors @xmath71 , @xmath72 and @xmath75 . photometry and parallaxes are taken from the sources listed in the appendix . model sequences are shown , both cloud - free and cloudy , for @xmath76 and 5 , and for a range in gravity between @xmath77 and 5.0 . at the temperatures of primary interest here , @xmath78 k@xmath79 , @xmath77 corresponds to a mass around 5 m@xmath12 and age 0.1 1 gyr , @xmath53 to mass 12 m@xmath12 and age 1 10 gyr , and @xmath80 to mass 30 m@xmath12 and age @xmath81 6 gyr . given that the latest - type t dwarfs and the y dwarfs are necessarily nearby , we expect them to have an age similar to the sun , and therefore the @xmath53 model sequence should be most representative of the sample ( solid curves in figure 4 ) . it can be seen that the new cloudy models are required in order to reproduce the observed significant reddening of around 1 magnitude in @xmath72 for brown dwarfs with @xmath46 800 k. in the @xmath75 diagram the datapoints occupy a wider color envelope , possibly reflecting an intrinsic scatter in metallicity or gravity , which would impact the strong h@xmath6 opacity in the @xmath17 band . the cloudy models do not reproduce the @xmath71 colors at @xmath39 k , possibly due to changes associated with the formation of water clouds . figure 5 shows absolute w2 magnitude as a function of the mid - infrared colors @xmath82 w2 , @xmath74 w2 and w2 @xmath27 w3 . w3 is a wide filter spanning 7.5 16.5 @xmath5 m . only around 5 known late - type t dwarfs ( and no y dwarfs ) are detected in the w4 filter ( 19.8 25.5 @xmath5 m ) . in these plots , as explained above , we have added 0.3 magnitudes to the w2 magnitudes calculated by both the cloudy and cloud - free models , to mimic the affect of vertical mixing . with this adjustment , the models fit the data quite well , although the @xmath74 w2 diagram suggests that the correction should be larger for both cloudy and cloud - free models , and the @xmath82 w2 diagram suggests that the correction should be larger for the cloudy models . the two unusually red objects at m@xmath83 are both metal - poor high - gravity late - type t dwarfs , 2mass j09393548@xmath272448279 ( burgasser et al . 2008 , leggett et al . 2009 ) and sdss 1416@xmath11348b ( scholz 2010a , burningham et al . 2010b ) ; the former is likely to be a binary system . metal - poor t dwarfs are bright at 4.5 @xmath5 m ( e.g. leggett et al . 2010a ) ; non - solar metallicities have not yet been incorporated into these models . wise j1738@xmath12732 appears to be very red in w2 @xmath27 w3 . this object is also unusually blue in @xmath71 ( figures 4 and 7 ) and warrants further study . wisepc j1828@xmath12650 also stands out in figure 5 we return to this in 4.5 . @xmath71 as a function of @xmath72 and @xmath82 w2 is shown in figures 6 and 7 respectively . model sequences are also shown , as described above . the impact of the chloride and sulfide clouds is apparent in figure 6 , where cloud - free models extend to @xmath72 colors that are much bluer than observed . in figure 6 we have included data from the ukidss database to illustrate the location of the general stellar population . it will be extremely difficult to identify y dwarfs in sky surveys using near - infrared colors alone . these cold brown dwarfs have now wrapped around in @xmath2 colors to occupy a very similar region to that occupied by warm stars . allowing for uncertainties in the survey data at faint limits , it is not possible to extract even the earliest y dwarfs , with @xmath84 and @xmath85 . figure 7 shows , similarly to figure 4 , that the cloudy models do not reproduce the @xmath71 colors at @xmath39 k. figure 8 shows @xmath74 w2 as a function of @xmath72 , @xmath73 and [ 3.6 ] @xmath27 w2 . model sequences are also shown , as described above . again , the new cloudy models nicely account for the reddening of @xmath72 . there is a degeneracy between the effect of @xmath45 and @xmath70 in the @xmath72 diagram , where an increase in @xmath45 can be compensated by a decrease in gravity . the @xmath73 diagram , however , separates out the gravity and @xmath45 sequences , allowing us to differentiate between the two parameters . although @xmath17 is very faint for these objects it will be worth expending some telescope time to obtain such data in order to more fully understand this new class of objects . the model comparisons that we show in figures 4 , 5 , 7 and 8 allow us to estimate the properties of the five y0 y0.5 dwarfs for which we present new photometry in this paper . we discuss the y2 dwarf in 4.5 . we find that wisep j0410 + 1502 , wisep j1738@xmath12732 and wisepc j2056@xmath11459 have @xmath39 450 k , @xmath86 and @xmath11 ( based on the @xmath72 and @xmath73 colors ) . wisepc j1405@xmath15534 and wisepc j1541@xmath272250 are cooler than covered by the cloudy models , but the cloud - free models indicate @xmath87 k and 300 350 k respectively . trends in the figures indicate that wisepc j1405@xmath15534 has a higher gravity than wisepc j1541@xmath272250 , and we estimate that @xmath86 and 4.0 4.5 for wisepc j1405@xmath15534 and wisepc j1541@xmath272250 , respectively . these temperatures and gravities , with implied ranges in mass and age , are listed in table 6 . we have assumed solar metallicity , which for this sample within 10 pc of the sun is plausible . the comparisons can also be tested against the binary parameters given in table 5 . the model sequences are consistent with a single - age solution for the three pairs : wise j1217@xmath11626ab , cfbds 1458@xmath11013ab and wise j1711@xmath13500ab . wise j0458@xmath16434ab appears to be a very similar pair of t9s which we can not constrain . for all three binaries the models support the older 5 gyr solution and do not support the younger 1 gyr solution , based on agreement with the higher gravity sequences in figure 8 . the 5 gyr solution for wise j1217@xmath11626 implies @xmath80 and 4.7 for the primary and secondary , respectively ( table 5 ) . the middle panel of figure 8 shows that the observations are consistent with the @xmath54 ( i.e. relatively thin cloud layers ) @xmath80 and 4.5 sequences , and that @xmath73 is too blue to be consistent with with any lower - gravity solution . similarly cfbds 1458@xmath11013a constrains the system to the older solution where @xmath80 and 4.6 for the primary and secondary , respectively . the colors of the secondary are consistent with this , although they do not constrain the solution . for cfbds 1458@xmath11013a the clouds appear to be very thin to non - existent . the 5 gyr solution for wise j1711@xmath13500ab has @xmath88 and 4.8 for the primary and secondary , respectively . wise j1711@xmath13500b constrains the system to this older solution , as @xmath73 is too blue to be consistent with the alternative 1 gyr @xmath89 solution , for any cloud parameter . wise j1711@xmath13500b also has a thin cloud layer with @xmath90 . wise j1711@xmath13500a is too warm to constrain the gravity of the system . the color sequences indicate that wisepc j2056@xmath11459 has @xmath46 450 k , @xmath53 and @xmath76 . cushing et al . ( 2011 ) present @xmath35-band and @xmath28-band spectra for this y0 dwarf , which we have flux calibrated using our photometry . figure 9 combines this spectrum with the far - red spectrum obtained here , and compares these spectra to models with @xmath46 k and @xmath53 , with and without clouds . it can be seen that the cloudy model provides a superior fit in the red and at @xmath17 . the discrepancy at 1.0 @xmath5 m and 1.5 @xmath5 m , where the cloudy model flux is fainter than observed , can be explained by an overly - strong nh@xmath7 absorption in the models , due to the neglect of vertical mixing ( see 4.1 ) . the mixing enhances n@xmath6 while decreasing the abundance of nh@xmath7 . this effect can also explain the lack of a strong nh@xmath7 doublet at 1.03 @xmath5 m in figure 1 . the discrepancy at 1.60 1.65 @xmath5 m , where the model flux is too high , is most likely due to remaining incompleteness in the ch@xmath68 opacity line list ( ch@xmath68 is the dominant opacity at 1.6 @xmath5 m ) . the color sequences suggest that wisepc j1541@xmath272250 , the coldest dwarf in the sample ( apart from wisepc j1828@xmath12650 , see below ) , has colors approaching those of the cloud - free models ( figures 4 , 7 , 8) . in figure 10 we plot the near - infrared spectrum of this brown dwarf and compare it to a cloud - free model with @xmath91 k and @xmath53 . the cloudless synthetic spectrum reproduces the 1.0 1.6 @xmath5 m data quite well , with the caveats as before of overly strong nh@xmath7 due to the neglect of mixing . the fact that a cloudless model fits the spectrum reasonably well despite the fact that water clouds are expected by @xmath92 may not be surprising . for such a gravity near @xmath93 , the cloud base forms well above the 1 bar level where there is less mass available to condense , compared to still lower effective temperatures where the cloud base is much deeper ( e.g. in jupiter ) . furthermore , ice crystals or water drops with radii less than about @xmath94 do not interact strongly with near - infrared photons since the relevant mie absorption and scattering efficiencies are very low ( see figure 3 of zsom et al . 2012 for example ) . when water clouds are found deeper in the atmosphere or form with larger radii we would expect to see a greater effect . in the near - infrared this would likely first become noticeable in the @xmath30- and @xmath35-bands because of the longer atmospheric pathlength in these opacity windows , and the greater influence of scattering at the shorter wavelengths . burrows , sudarsky & lunine ( 2003 ) do include water condensates in their models of brown dwarfs with @xmath95 k@xmath66 . they derive condensate particle sizes of 20 to 150 @xmath5 m , and nevertheless still find that the absorptive opacity of the water clouds is small , such that water ice clouds only have a secondary influence on the spectra of the coolest isolated brown dwarfs . burrows et al . also calculate that nh@xmath7 clouds form when @xmath96 k , i.e. cooler than the _ wise _ y dwarfs . further modelling is required to investigate cloud formation for brown dwarfs with @xmath97 k. wisepc j1828@xmath12650 has the reddest @xmath74 w2 color of any known y dwarf ( @xmath74 w2 @xmath98 ) , and is therefore likely to be the coolest . cushing et al . classify this object as y2 . figures 4 and 5 show that this y2 dwarf is as intrinsically bright in the near - infrared as the y0.5 wisepc j1541@xmath272250 , and is actually brighter in the mid - infrared . wisepc j1828@xmath12650 is also similar in luminosity to the y0 wisepc j1405@xmath15534 . in order for a cooler brown dwarf to be more luminous than a warmer one , the radius must be significantly larger ( @xmath99 is compensated by the factor @xmath100 ) . in this particular case , with an estimated 10% difference in temperature , the radii would need to differ by @xmath101 20% . assuming the rest of the 350 450 k sample has a typical age of a few gyr and @xmath86 , then the radius of wisepc j1828@xmath12650 would need to be @xmath102r@xmath23 . this in turn implies an age younger than 50 myr , and mass smaller than 1 m@xmath12 . while the young age and low mass are possible , we have explored an alternative solution of binarity for wisepc j1828@xmath12650 . in this scenario , the cooler component would be almost as bright as the warmer component in the mid - infrared , but be significantly fainter in the near - infrared . we determined solutions that ( i ) reproduced the observed flux when the component fluxes were combined , ( ii ) produced absolute magnitudes generally consistent with our models ( as the distance is known , table 2 ) , and ( iii ) had @xmath18 and @xmath70 parameter pairs consistent with a coeval pair . table 7 gives our synthetically resolved photometry for the components ; the uncertainty is 0.3 magnitudes , constrained by the requirement to reproduce the observed flux . figure 3 shows the proposed components on spectral type : color diagrams trends with type are sensible if this brown dwarf is a binary . figures 4 to 8 show the proposed components on color diagrams . extrapolating from the models and the observed sequences , the proposed binary appears to consist of a @xmath103 k and @xmath14 primary , and a @xmath15 k and @xmath104 secondary . these imply a system age of around 2 gyr , and component masses of around 10 and 7 m@xmath12 . 300 k brown dwarfs appear to be significantly brighter than the cloud - free models in the @xmath17 and w2 bands ( figures 4 , 5 and 10 ) , a discrepancy to be addressed by the next generation of models . we have obtained @xmath0 , or a subset , for the six objects discovered in the _ wise _ database that have been identified as y dwarfs by cushing et al . we find large differences of 0.5 1.0 magnitudes between our mko - system magnitudes and those published by cushing et al . and kirkpatrick et al . ( 2012 ) for three of the brown dwarfs at @xmath35 or @xmath28 . as our photometry is consistent with the seds published by cushing et al . and kirkpatrick et al . we suspect that the previously published palomar wirc photometry is in error for these three objects at those particular wavebands . our results are otherwise consistent with the previously published data , where they overlap . we have also obtained a far - red spectrum of the y0 dwarf wisepc j2056@xmath11459 , which has @xmath39 k. the spectrum shows that the cs i lines seen in the 500 k brown dwarf ugps j0722@xmath270540 ( leggett et al . 2012 ) are not seen in this cooler object . this is not unexpected , as at these temperatures cs should exist predominantly in the form of cscl ( lodders 1999 ) . we confirm here that new models which include clouds of cr , mns , na@xmath6s , zns and kcl condensates reproduce the near - infrared colors of the latest - type t dwarfs , and the earliest - type y dwarfs , quite well , as previously shown by morley et al . the @xmath72:@xmath74 w2 and @xmath73:@xmath74 w2 color - color diagrams can be used to estimate @xmath18 , @xmath70 and sedimentation efficiency @xmath45 for the earliest y dwarfs ( figure 8) . we find that wisep j0410 + 1502 , wisep j1738@xmath12732 and wisepc j2056@xmath11459 have @xmath39 450 k , @xmath86 and @xmath11 . we also find that wisepc j1405@xmath15534 and wisepc j1541@xmath272250 are cooler than currently covered by the cloudy models , with @xmath87 k and 300 350 k and @xmath86 and 4.0 4.5 , respectively . we find that the 1.0 1.6 @xmath5 m spectrum of wisepc j1541@xmath272250 can be reproduced quite well with a cloud - free @xmath91 k @xmath53 model , which is somewhat surprising , as water clouds would be expected to form in such cool atmospheres . the lack of a strong cloud signature may be due to the fact that the cloud base for these relatively low - gravity dwarfs lies high in the atmosphere , and small droplets have low absorption and scattering efficiencies . improved water cloud models are clearly required . the temperatures and gravities of the five y0 y0.5 dwarfs imply a mass range of 5 15 m@xmath12 and an age around 5 gyr . we determine similar ages for three binary systems composed of late - t and early - y dwarfs ( liu et al . 2011 , 2012 ) . in each case comparison to model sequences indicates a value of @xmath70 which corresponds to an age around 5 gyr . the t9 y0 components of the three binaries wise j1217@xmath11626a and b , cfbds 1458@xmath11013a and b , and wise j1711@xmath13500b appear to have thinner clouds than the y0 dwarfs wisep j0410 + 1502 , wisep j1738@xmath12732 and wisepc j2056@xmath11459 : @xmath90 compared to @xmath11 . we have found that vertical mixing is likely to be important in the atmospheres of the early - type 400 k y dwarfs , as it is in the atmospheres of t dwarfs , and indeed as it is for jupiter ( e.g. noll , geballe & marley 1997 , leggett et al . 2002 , geballe et al . the mixing enhances the abundance of n@xmath6 at the expense of nh@xmath7 , explaining the lack of detection of what would otherwise be strong nh@xmath7 absorption at 1.03 @xmath5 m and 1.52 @xmath5 m ( figures 1 , 9 , 10 ) . the object wisepc j1828@xmath12650 is peculiar . assuming that the parallax for wisepc j1828@xmath12650 is not in error , its luminosity is not consistent with its colors , unless it is younger than around 50 myr . a more plausible solution may be that wisepc j1828@xmath12650 is a binary system . we find that a @xmath103 k and @xmath14 primary , and a @xmath15 k and @xmath104 secondary , would follow the observational trends seen with type , absolute magnitude and color ( figures 3 to 8) . the brightness and colors of the proposed components are also consistent with the cloud - free models , with the exception that the model fluxes at @xmath17 and w2 are too faint by 0.5 1.0 magnitudes . in future work we will incorporate water clouds into the models , as well as mixing . the atmospheres of these 300 500 k objects are extremely complex , but nevertheless the models , which now include chloride and sulfide clouds , have allowed us to estimate temperature and gravity , and hence mass and age , for these exciting _ wise _ discoveries . we look forward to the discovery of more 5 10 m@xmath12 objects in the solar neighborhood . ds is supported by nasa astrophysics theory grant nnh11aq54i . 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 science and technology facilities council ( united kingdom ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , ministrio da cincia , tecnologia e inovao ( brazil ) and ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) . skl s research is supported by gemini observatory . this publication makes use of data products from the wide - field infrared survey explorer , which is a joint project of the university of california , los angeles , and the jet propulsion laboratory / california institute of technology , funded by the national aeronautics and space administration . this research has made use of the nasa/ ipac infrared science archive , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . table 7 presents a sample of the data used in this paper . the full dataset is available online . the table gives , for 83 brown dwarfs which have spectral types of t6 or later , that have been detected in the w2 band , and that have mko - system near - infrared photometry : coordinates , spectral type , distance modulus , @xmath69[3.6][4.5][5.8][8.0]w1w2w3w4 photometry , uncertainties in distance modulus and photometry , and source references . close binary systems are also indicated . the coordinates are for equinox 2000 , please note that epoch varies and the dwarfs can have high proper motion . the @xmath105 photometry is on the ab system , while the rest of the photometry is on the vega magnitude system . the @xmath106 are on the mko photometric system , with @xmath30 data obtained using niri reduced by 0.17 magnitudes to put it on the ukidss @xmath30-system . the uncertainty in the spectral types are not given , but is typically 0.5 of a subclass , except for the spectral types latern than t8 . it is likely that the classification scheme for those objects will have to be revised as more y dwarfs are found , and we estimate an uncertainty of 1 subclass in type . all _ wise _ photometry is taken from the _ wise _ all - sky data release . the online table contains 48 columns : + 1 . name + 2 . other names(s ) + 3 . r.a . as hhmmss.ss + 4 . declination as sddmmss.s + 5 . spectral type + 6 . distance modulus as @xmath107 magnitudes + 7 . absolute @xmath35 magnitude + 8 . @xmath108 + 9 . @xmath109 + 10 . @xmath30 + 11 . @xmath35 + 12 . @xmath28 + 13 . @xmath17 + 14 . @xmath110 + 15 . @xmath111 + 16 . irac [ 3.6 ] + 17 . irac [ 4.5 ] + 18 . irac [ 5.8 ] + 19 . irac [ 8.0 ] + 20 . uncertainty in @xmath107 + 26 . uncertainty in @xmath108 + 27 . uncertainty in @xmath109 + 28 . uncertainty in @xmath30 + 29 . uncertainty in @xmath35 + 30 . uncertainty in @xmath28 + 31 . uncertainty in @xmath17 + 31 . uncertainty in @xmath110 + 31 . uncertainty in @xmath111 + 32 . uncertainty in [ 3.6 ] + 33 . uncertainty in [ 4.5 ] + 34 . uncertainty in [ 5.8 ] + 35 . uncertainty in [ 8.0 ] + 36 . uncertainty in w1 + 37 . uncertainty in w2 + 38 . uncertainty in w3 + 39 . uncertainty in w4 + 40 . reference for binarity + 41 . reference for discovery + 42 . reference for spectral type + 43 . reference for trigonometric parallax + 44 . reference for @xmath105 + 45 . reference for @xmath30 + 46 . reference for @xmath31 + 47 . reference for @xmath112 + 48 . reference for irac photometry + data are taken from this work , burningham et al . 2013 ( in preparation ) , data release 9 of both the ukidss and sdss catalogs , and the _ wise _ all - sky data release . brown dwarf discoveries and other data are taken from the following publications : burgasser et al . 1999 , 2000 , 2002 , 2003a and b , 2004 , 2006a and b , 2008 , 2012 ; burningham et al . 2008 , 2009 , 2010a and b , 2011 ; chiu et al . 2006 ; cushing et al . 2011 ; dahn et al . 2002 ; delorme et al . 2008a and b , 2010 ; dupuy & liu 2012 ; faherty et al . 2012 ; geballe et al . 2001 ; golimowski et al . 2004 ; harrington & dahn 1980 ; hewett et al . 2006 ; kirkpatrick et al . 2011 , 2012 ; knapp et al . 2004 ; leggett et al . 2002 , 2007 , 2009 , 2010 , 2012 ; liu et al . 2011 , 2012 ; lodieu et al . 2007 , 2009 ; lucas et al . 2010 ; marocco et al . 2010 ; patten et al . 2006 ; pinfield et al . 2008 , 2012 ; scholz 2010a and b ; strauss et al . 1999 ; tinney et al . 2003 , 2005 ; tsvetanov et al . 2000 ; vrba et al . 2004 ; warren et al . ackerman , a. s. & marley , m. s. 2001 , , 556 , 872 burgasser , a. j. et al . 1999 , , 522 , l65 burgasser , a. j. et al . 2000 , , 531 , l57 burgasser , a. j. et al . 2002 , , 564 , 421 burgasser , a. j. , mcelwain , m. w. & kirkpatrick , j. d. 2003a , , 126 , 2487 burgasser , a. j. , kirkpatrick , j. d. , reid , i. n. , brown , m . e. , miskey , c. l. & gizis , j. e. 2003b , , 586 , 512 burgasser , a. j. , mcelwain , m. w. , kirkpatrick , j. d. , cruz , k. l. , tinney , c. g. & reid , i. n. 2004 , , 127 , 2856 burgasser , a. j. , kirkpatrick , j. d. , cruz , k. l. , reid , i. n. , leggett , s. k. , liebert , j. , burrows , a. & brown , m. e. 2006a , , 166 , 585 burgasser , a. j. , geballe , t . r. , leggett , s. k. , kirkpatrick , j. d. & golimowski , d. a. 2006b , , 637 , 1067 burgasser , a. j. , tinney , c. g. , cushing , m. c. , saumon , d. , marley , m. s. , bennett , c. s. , & kirkpatrick , j. d. 2008 , , 689 , l53 burgasser , a. j. , gelino , c. r. , cushing , m. c. & kirkpatrick , j. d. 2012 , , 745 , 26 burningham , b. et al . 2008 , , 391 , 320 burningham , b. et al . 2009 , , 395 , 1237 burningham , b. et al . 2010a , , 406 , 1885 burningham , b. et al . 2010b , , 404 , 1952 burningham , b. et al . 2011 , , 414 , 3590 burrows , a. sudarsky , d. & lunine , j. i. 2003 , , 596 , 587 chiu , k. , fan , x. , leggett , s. k. , golimowski , d. a. , zheng , w. , geballe , t. r. , schneider , d. p. & brinkmann , j. 2006 , , 131 , 2722 cushing , m. c. et al . 2011 , , 743 , 50 dahn , c. c. et al . 2002 , , 124 , 1170 delorme p. 2008a , , 482 , 961 delorme p. et al . 2008b , , 484 , 469 delorme p. et al . 2010 , , 518 , 39 dupuy , t. j. & liu , m. c. 2012 , , 201 , 19 esa 1997 , the hipparcos and tycho catalogues ( esa sp-1200 ) ( noordwijk : esa ) faherty , j. k. et al . 2012 , , 752 , 56 geballe , t. r. , saumon , d. , leggett , s. k. , knapp , g. r. , marley , m. s. & lodders , k. 2001 , , 556 , 373 geballe , t. r. , saumon , d. , golimowski , d. a. , leggett , s. k. , marley , m. s. & noll , k. s. 2009 , , 695 , 844 golimowski , d. a. et al . 2004 , , 127 , 3516 harrington , r. s. & dahn , c. c. 1980 , , 85 , 168 helling , ch . 2008 , , 391 , 1854 hewett , p. c. , warren , s. j. , leggett , s. k. & hodgkin , s. t. 2006 , , 367 , 454 hodapp , k. w. et al . 2003 , , 115 , 1388 kirkpatrick , j.d . 1995 , , 43 , 195 kirkpatrick , j.d . 2011 , , 197 , 19 kirkpatrick , j. d. et al . 2012 , , 753 , 156 knapp , g. r. et al . 2004 , , 127 , 3553 lawrence a. et al . 2007 , , 379 , 1599 leggett , s. k. et al . 2002 , , 564 , 452 leggett , s. k. et al . 2006 , , 373 , 781 leggett , s. k. , saumon , d. , marley , m. s. , geballe , t. r. , golimowski , d. a. , stephens , d. c. & fan , x. 2007a , , 655 , 1079 leggett , s. k. , marley , m. s. , freedman , r. , saumon , d. , liu , m. c. , geballe , t. r. , golimowski , d. a. & stephens , d. c. 2007b , , 667 , 537 leggett , s. k. et al . 2009 , , 695 , 1517 leggett , s. k. et al . 2010a , , 710 , 1627 leggett , s. k. , saumon , d. , burningham , b. , cushing , m. c. , marley , m. s. & pinfield , d. j. 2010b , , 720 , 252 leggett , s. k. et al . 2012 , , 748 , 74 liu , m. c. , et al . 2011 , , 740 , 108 liu , m. c. , dupuy , t. j. , bowler , b. p. , leggett , s. k. & best , w. m. j. 2012 , , in press lodders , k. 1999 , , 519 , 793 lodieu , n. et al . 2007 , , 379 , 1423 lodieu , n. , burningham , b. , hambly , n. c. & pinfield , d. j. 2009 , , 397 , 258 looper , d , l. , kirkpatrick , j. d. & burgasser , a. j. 2007 , , 134 , 1162 luhman , k. l. et al . 2007 , , 654 , 570 lucas , p. w. et al . 2010 , , 408 , l56 marley , m. s. , seager , s. , saumon , d. , lodders , k. , ackerman , a. s. , freedman , r. s. & fan , x. 2012 , , 568 , 335 marley , m. s. ; saumon , d. & goldblatt , c. 2010 , , 723 , l117 marocco , f. et al . 2010 , , 524 , 38 morley , c. v. , fortney , j. j. , marley , m. s. , visscher , c. , saumon , d. & leggett , s. k. 2012 , , 756 , 172 noll , k. s. , knacke , r. f. , geballe , t. r. & tokunaga , a. t. 1988 , , 324 , 1210 noll , k. s. , geballe , t. r. & marley , m. s. 1997 , , 489 , l87 patten , b. m. et al . 2006 , , 651 , 502 pinfield , d. j. et al . 2008 , , 390 , 304 pinfield , d. j. et al . 2012 , , 422 , 1922 ruiz , m .- t , leggett , s. k. & allard , f. 1997 , , 491 , l107 saumon , d. et al . 2007 , , 656 , 1136 saumon , d. & marley , m. s. 2008 , , 689 , 1327 saumon , d. , marley , m. s. , abel , m. , frommhold , l. & freedman , r. s. 2012 , , 750 , 74 scholz , r. -d . 2010a , , 510 , l8 scholz , r. -d . 2010b , , 515 , 92 skrutskie m. f. et al . 2006 , , 131 , 1163 smart , r. l. et al . 2010 , , 511 , 30 stephens , d. c. et al . 2009 , , 702 , 154 strauss , m. a. et al . 1999 , , 522 , l61 tinney , c . g. , burgasser , a. j. & kirkpatrick , j. d. 2003 , , 126 , 975 tinney , c . g. , burgasser , a. j. , kirkpatrick , j. d. & mcelwain , m. w. 2005 , , 130 , 2326 tokunaga , a. t. , simons , d. a. & vacca , w. d. 2002 , , 114 , 180 tokunaga , a. t. & vacca , w. d. 2005 , , 117 , 421 tsuji , t. 2002 , , 575 , 264 tsvetanov et al . 2000 , , 531 , l61 visscher , c. lodders , k. & fegley , b. 2006 , , 648 , 1181 vrba , f. j. et al . 2004 , , 127 , 2948 warren . s. j. et al . 2007 , , 381 , 1400 wright , e. l. et al . 2010 , , 140 , 1868 york , d. g. et al . 2000 , , 120 , 1579 zsom , a. , kaltenegger , l. & goldblatt , c. 2012 , icarus , in press . magnitude as a function of @xmath71 , @xmath72 and @xmath75 . crosses without datapoints ( violet in the online version ) indicate the colors of wisepj 1828@xmath12650 if it consists of a 325 k and 300 k binary ( see 4.5 ) . the pair of lighter datapoints ( magenta in the online version ) represent cfbds 1458@xmath11013a and b. curves are model sequences , as described in the legend . in the lower panel large open circles along the sequences indicate where @xmath20 300 , 350 , 400 , 500 , 600 and 800 k. @xmath18 values on the vertical axes correspond to the @xmath54 and @xmath53 model for @xmath113 k@xmath114 and the cloud - free models for @xmath115 k@xmath116 . all data are on the mko photometric system . [ fig4 ] ] w2 , @xmath74 w2 and w2 - w3 . symbols and curves are as in figure 4 . the model w2 values have been increased by 0.3 magnitudes to mimic the effect of non - equilibrium chemistry as described in the text . [ fig5 ] ] as a function of @xmath71 . crosses ( violet in the online version ) indicate the colors of wisepj 1828@xmath12650 if it consists of a 325 k and 300 k binary ( see 4.5 ) . curves are model sequences with @xmath53 , black is cloud - free and the lighter curve ( orange in the online version ) is cloudy with @xmath54 . small dots are a representative sample of stars taken from the ukidss large area survey . medium - sized dots are m , l and early - type t dwarfs with data taken from the leggett et al . ( 2010 , and references therein ) . larger dots with error bars are the sample presented here . lighter datapoints ( cyan and green in the online version ) represent the binaries wise j1711@xmath13500a and b and wise j1217@xmath11626a and b. [ fig6 ] ] lrrrrrrr wisepc j0410@xmath11502 & 19.78 ( 0.04 ) & 19.44 ( 0.03 ) & 20.02 ( 0.05 ) & 19.91 ( 0.07 ) & 5 , 5 , 9 , 9 & 20120809/16 & gn-2012b - q-75 + wisepc j1405@xmath15534 & 21.41 ( 0.10 ) & 21.06 ( 0.06 ) & 21.41 ( 0.08 ) & & 9 , 9 , 58.5 , & 20120210 & gn-2012a - q-106 + wisepc j1541@xmath272250 & 21.63 ( 0.13 ) & 21.12 ( 0.06 ) & 22.17 ( 0.25 ) & & 18 , 18 , 45 , & 20120210 , 20120505 & gn-2012a - q-106 + wisepc j1738@xmath12732 & 20.03 ( 0.07 ) & 20.05 ( 0.09 ) & 20.45 ( 0.09 ) & 20.58 ( 0.10 ) & 5 , 2.5 , 9 , 18 & 20120503 , 20120702 & gn-2012a - q-106 , gn-2012b - q-75 + wisepc j1828@xmath12650 & 23.20 ( 0.17 ) & 23.48 ( 0.23 ) & & 23.48 ( 0.36 ) & 108 , 90 , , 117 & 20120503/04/07 , 20120703/07/10 & gn-2012a - q-106 , gn-2012b - q-27/75 + wisepc j2056@xmath11459 & 19.94(0.05 ) & 19.43 ( 0.04 ) & 19.96 ( 0.04 ) & 20.01 ( 0.06 ) & 5 , 2.5 , 9 , 13.5 & 20120517 , 20120609 & gn-2012a - dd-7 , gn-2012a - q-106 + lrrrrrrrrrrr wisepc j0410@xmath11502 & 164 ( 24 ) & 19.61 ( 0.04 ) & 19.44 ( 0.03 ) & 20.02 ( 0.05 ) & 19.91 ( 0.07 ) & 16.56 ( 0.01 ) & 14.12 ( 0.01 ) & @xmath8118.33 & 14.18 ( 0.06 ) & @xmath8111.86 & @xmath818.90 + wisepc j1405@xmath15534 & 207 ( 39 ) & 21.24 ( 0.10 ) & 21.06 ( 0.06 ) & 21.41 ( 0.08 ) & & 16.78 ( 0.01 ) & 14.02 ( 0.01 ) & @xmath8118.82 & 14.10 ( 0.04 ) & 12.43 ( 0.27 ) & @xmath819.40 + wisepc j1541@xmath272250 & & 21.46 ( 0.13 ) & 21.12 ( 0.06 ) & 22.17 ( 0.25 ) & & 16.92 ( 0.02 ) & 14.12 ( 0.01 ) & 16.74 ( 0.17 ) & 14.25 ( 0.06 ) & @xmath8112.31 & @xmath818.89 + wisepc j1738@xmath12732 & 111 ( 36 ) & 19.86 ( 0.08 ) & 20.05 ( 0.09 ) & 20.45 ( 0.09 ) & 20.58 ( 0.10 ) & 16.87 ( 0.01 ) & 14.42 ( 0.01 ) & @xmath8118.41 & 14.55 ( 0.06 ) & 11.93 ( 0.19 ) & @xmath818.98 + wisepc j1828@xmath12650 & 122 ( 13 ) & 23.03 ( 0.17 ) & 23.48 ( 0.23 ) & 22.85 ( 0.24 ) & 23.48 ( 0.36 ) & 16.84 ( 0.01 ) & 14.27 ( 0.01 ) & @xmath8118.47 & 14.39 ( 0.06 ) & @xmath8112.53 & @xmath818.75 + wisepc j2056@xmath11459 & & 19.77 ( 0.06 ) & 19.43 ( 0.04 ) & 19.96 ( 0.04 ) & 20.01 ( 0.06 ) & 15.90 ( 0.01 ) & 13.89 ( 0.01 ) & @xmath8118.25 & 13.93 ( 0.05 ) & 12.00 ( 0.27 ) & @xmath818.78 + lllrrrrrrrrrrrrrr 400.0 & 4.00 & 2.0 & 22.72 & 21.56 & 22.11 & 21.71 & 16.77 & 14.04 & 17.99 & 14.49 & 15.85 & 15.08 & 19.12 & 14.46 & 13.47 & 11.91 + 400.0 & 4.00 & 3.0 & 21.80 & 20.93 & 22.01 & 21.79 & 16.8 & 14.05 & 18.03 & 14.51 & 15.88 & 15.12 & 19.17 & 14.47 & 13.5 & 11.93 + 400.0 & 4.00 & 4.0 & 21.34 & 20.63 & 22.00 & 21.86 & 16.83 & 14.07 & 18.06 & 14.53 & 15.91 & 15.14 & 19.21 & 14.49 & 13.51 & 11.93 + 400.0 & 4.00 & 5.0 & 21.08 & 20.47 & 22.01 & 21.90 & 16.85 & 14.08 & 18.09 & 14.54 & 15.92 & 15.15 & 19.23 & 14.50 & 13.52 & 11.94 + 400.0 & 4.48 & 2.0 & 23.40 & 21.99 & 22.06 & 22.18 & 16.82 & 14.29 & 18.06 & 14.67 & 16.18 & 15.40 & 19.23 & 14.66 & 13.79 & 12.19 + 400.0 & 4.48 & 3.0 & 22.44 & 21.34 & 21.90 & 22.26 & 16.85 & 14.30 & 18.10 & 14.68 & 16.21 & 15.43 & 19.28 & 14.66 & 13.81 & 12.20 + 400.0 & 4.48 & 4.0 & 21.93 & 21.02 & 21.87 & 22.32 & 16.87 & 14.31 & 18.13 & 14.69 & 16.24 & 15.45 & 19.31 & 14.68 & 13.83 & 12.21 + 400.0 & 4.48 & 5.0 & 21.64 & 20.85 & 21.86 & 22.37 & 16.89 & 14.32 & 18.14 & 14.70 & 16.25 & 15.47 & 19.33 & 14.69 & 13.84 & 12.21 + 200.0 & 4.00 & 34.45 & 33.46 & 33.24 & 38.37 & 22.76 & 17.72 & 25.11 & 18.17 & 20.59 & 19.15 & 26.87 & 18.18 & 17.23 & 13.78 + 200.0 & 4.48 & 33.54 & 33.16 & 32.12 & 38.92 & 22.44 & 17.75 & 24.90 & 18.10 & 20.61 & 19.22 & 26.68 & 18.11 & 17.38 & 13.91 + 200.0 & 5.00 & 32.76 & 33.01 & 31.21 & 39.65 & 22.02 & 17.76 & 24.55 & 17.99 & 20.57 & 19.28 & 26.38 & 18.01 & 17.55 & 14.06 + 250.0 & 4.00 & 29.17 & 28.59 & 29.32 & 31.65 & 20.61 & 16.47 & 22.43 & 16.94 & 19.08 & 17.84 & 24.01 & 16.93 & 15.85 & 13.04 + 250.0 & 4.48 & 28.67 & 28.39 & 28.46 & 32.33 & 20.40 & 16.52 & 22.30 & 16.89 & 19.25 & 17.97 & 23.91 & 16.89 & 16.03 & 13.15 + 250.0 & 5.00 & 28.21 & 28.47 & 27.78 & 33.27 & 20.14 & 16.61 & 22.12 & 16.85 & 19.43 & 18.10 & 23.78 & 16.87 & 16.24 & 13.28 + 300.0 & 4.00 & 25.14 & 24.92 & 26.29 & 27.04 & 18.96 & 15.46 & 20.47 & 15.95 & 17.73 & 16.71 & 21.87 & 15.92 & 14.82 & 12.55 + 300.0 & 4.48 & 25.02 & 24.86 & 25.69 & 27.70 & 18.89 & 15.61 & 20.46 & 16.01 & 18.01 & 16.97 & 21.90 & 16.00 & 15.09 & 12.77 + 300.0 & 5.00 & 24.86 & 24.96 & 25.13 & 28.58 & 18.69 & 15.70 & 20.32 & 15.97 & 18.25 & 17.17 & 21.82 & 15.98 & 15.32 & 12.89 + lrrrrrrrrrrr wise j0458@xmath16434a & t8.5 & & & 17.50 ( 0.07 ) & 17.77 ( 0.11 ) & & 13.5 ( 0.3 ) & & & & b12 + wise j0458@xmath16434b & t9.5 & & & 18.48 ( 0.07 ) & 18.79 ( 0.11 ) & & 14.4 ( 0.3 ) & & & & b12 + wise j1217@xmath11626a & t9 & & 18.59 ( 0.04 ) & 17.98 ( 0.02 ) & 18.31 ( 0.05 ) & 18.94 ( 0.04 ) & 13.4 ( 0.3 ) & 13/33 & 550/650 & 4.5/5.0 & l12 + wise j1217@xmath11626b & y0 & & 20.26 ( 0.04 ) & 20.08 ( 0.03 ) & 20.51 ( 0.06 ) & 21.10 ( 0.12 ) & 14.5 ( 0.3 ) & 7/17 & 400/400 & 4.2/4.7 & l12 + cfbds 1458@xmath11013a & t9 & 34.0 ( 2.6 ) & & 19.86 ( 0.07 & 20.18 ( 0.10 ) & 20.63 ( 0.24 ) & 13.5 ( 0.3 ) & 12/35 & 550/600 & 4.5/5.0 & l12 + cfbds 1458@xmath11013b & y0 & 34.0 ( 2.6 ) & & 21.66 ( 0.34 ) & 22.51 ( 0.16 ) & 22.83 ( 0.30 ) & 14.4 ( 0.3 ) & 7/17 & 350/400 & 4.1/4.6 & l12 + wise j1711@xmath13500a & t8 & & 18.60 ( 0.03 ) & 17.67 ( 0.03 ) & 18.13 ( 0.03 ) & 18.30 ( 0.03 ) & 15.4 ( 0.3 ) & 20/45 & 750/850 & 4.7/5.2 & l12 + wise j1711@xmath13500b & t9.5 & & 21.31 ( 0.11 ) & 20.50 ( 0.06 ) & 20.96 ( 0.09 ) & 21.38 ( 0.15 ) & 16.0 ( 0.3 ) & 9/23 & 450/450 & 4.3/4.8 & l12 + llcccrr wisepc j0410@xmath11502 & y0 & 400 450 & 4.5 & 3 & 10 15 & 1 5 + wisepc j1405@xmath15534 & y0pec ? & 350 & 4.5 & cloud - free & 10 15 & 5 10 + wisepc j1541@xmath272250 & y0.5 & 300 350 & 4.0 4.5 & cloud - free & 5 13 & 1 10 + wisepc j1738@xmath12732 & y0 & 400 450 & 4.5 & 3 & 10 15 & 1 5 + wisepc j2056@xmath11459 & y0 & 400 450 & 4.5 & 3 & 10 15 & 1 5 + lrrrrrrrrrrr ab & 23.03 & 23.48 & 22.85 & 23.48 & 16.84 & 14.27 & 14.39 & & & & + a & 23.6 & 23.7 & 23.3 & 24.0 & 17.4 & 15.0 & 15.1 & 325 & 4.5 & 10 & 2 + b & 24.6 & 24.7 & 24.5 & 24.5 & 17.8 & 15.1 & 15.2 & 300 & 4.0 & 7 & 2 + llrrcrrrr ulas0034 - 0052 & & 00 34 02.77 & @xmath2700 52 06.7 & t8.5 & @xmath270.82 & 17.67 & & 22.00 + 2mass0034 + 0523 & & 00 34 51.57 & 5 23 05.0 & t6.5 & 0.11 & 15.22 & 24.47 & 18.93 + 2mass0050 - 3322 & & 00 50 19.94 & @xmath2733 22 40.2 & t7 & @xmath270.12 & 15.53 & & + cfbds0059 - 0114 & & 00 59 10.9 & @xmath2701 14 01.3 & t8.5 & 0.07 & 18.13 & & 21.73 + wisep0148 - 7202 & & 01 48 07.25 & @xmath2772 02 58.7 & t9.5 & & & & +

we present @xmath0 photometry , or a subset , for the six y dwarfs discovered in _ wise _ data by cushing et al .. the data were obtained using niri on the gemini north telescope ; @xmath0 were obtained for wisep j041022.71@xmath1150248.5 , wisep j173835.52@xmath1273258.9 and wisepc j205628.90@xmath1145953.3 ; @xmath2 for wisepc j140518.40@xmath1553421.5 and wisep j154151.65@xmath3225025.2 ; @xmath4 for wisep j182831.08@xmath1265037.8 . we also present a far - red spectrum obtained using gmos - north for wisepc j205628.90@xmath1145953.3 . we compare the data to morley et al . ( 2012 ) models , which include cloud decks of sulfide and chloride condensates . we find that the models with these previously neglected clouds can reproduce the energy distributions of t9 to y0 dwarfs quite well , other than near 5 @xmath5 m where the models are too bright . this is thought to be because the models do not include departures from chemical equilibrium caused by vertical mixing , which would enhance the abundance of co and co@xmath6 , decreasing the flux at 5 @xmath5 m . vertical mixing also decreases the abundance of nh@xmath7 , which would otherwise have strong absorption features at 1.03 @xmath5 m and 1.52 @xmath5 m that are not seen in the y0 wisepc j205628.90@xmath1145953.3 . we find that the five y0 to y0.5 dwarfs have @xmath8 k@xmath9 , @xmath10 and @xmath11 . these temperatures and gravities imply a mass range of 5 15 m@xmath12 and ages around 5 gyr . we suggest that wisep j182831.08@xmath1265037.8 is a binary system , as this better explains its luminosity and color . we find that the data can be made consistent with observed trends , and generally consistent with the models , if the system is composed of a @xmath13 k and @xmath14 primary , and a @xmath15 k and @xmath16 secondary , corresponding to masses of 10 and 7 m@xmath12 and an age around 2 gyr . if our deconvolution is correct , then the @xmath15 k cloud - free model fluxes at @xmath17 and w2 are too faint by 0.5 1.0 magnitudes . we will address this discrepancy in our next generation of models , which will incorporate water clouds and mixing .

it is being increasingly realized by those engaged in the search for supersymmetry ( susy)@xcite that the principle of r - parity conservation , assumed to be sacrosanct in the prevalent search strategies , is not inviolable in practice . the r - parity of a particle is defined as @xmath1@xcite and can be violated if either baryon ( b ) or lepton ( l ) number is not conserved . in recent years , the intensive studies of the supersymmetry that charactered by the bilinear r - parity violating terms in the superpotential and the nonzero vacuum expectation values ( vevs ) of sneutrinos@xcite have been undertaken . it stands as a simple supersymmetric ( susy ) model without r - parity which contains all particles as those in the standard model , and can be arranged in a way that there is no contradiction with the existing experimental data@xcite . an impact of the r - parity violation on the low energy phenomenology is twofold in the model . one leads the lepton number violation ( lnv ) explicitly . the other is that the bilinear r - parity violation terms in the superpotential and soft breaking terms generate nonzero vacuum expectation values for the sneutrino fields @xmath2 @xmath3 and cause the new type mixing , such as neutrinos with neutralinos , charged leptons with charginos and sleptons with higgs etc . the r - conserving superpotential for the minimal supersymmetric standard model ( mssm ) has the following form in superfields : @xmath4 where @xmath5 , @xmath6 are higgs superfields ; @xmath7 and @xmath8 are quark and lepton superfields respectively ( i=1 , 2 , 3 is the index of generation ) , and all of them are in su(2 ) weak - doublet . the rest superfields whereas @xmath9 and @xmath10 for quarks and @xmath11 for charged leptons are in su(2 ) weak - singlet . here the indices i , j are contracted in a general way for su(2 ) group and @xmath12 @xmath13 are the elements of the ckm matrix . however , when r - breaking interactions are considered , the superpotential is modified as the follows@xcite : @xmath14 with @xmath15 \nonumber \\ & & { \cal w}_{b } = \lambda_{ijk}^{\prime\prime } \hat{u}^{i } \hat{d}^{j } \hat{d}^ { k}. \label{eq-3}\end{aligned}\ ] ] since the proton decay experiments set down a very stringent limit on the byron number violation@xcite , we suppress the term @xmath16 totally . the first two terms in @xmath17 have received a lot of attention recently , and restrictions have been derived on them from existing experimental data@xcite . however , the term @xmath18 is also a viable agent for r - parity breaking . it is particularly interesting because it can result in observable effects that are not to seen with the trilinear terms alone . one of these distinctive effects is that , the lightest neutralino can decay invisibly into three neutrinos at the tree level , which is not possible if only the trilinear terms in @xmath17 are presented . the significance of such bilinear r - parity violating interaction is further emphasized by the following observations : * although it may seem possible to rotate away the @xmath19 terms by redefining the lepton and higgs superfields@xcite , their effect is bound to show up via the soft breaking terms . * even if one may rotate these terms away at one energy scale , they will reappear at another one as the couplings evolving radiatively@xcite . * the bilinear terms give rise to the trilinear terms at the one - loop level@xcite . * it has been argued that if one wants to subsume r - parity violation in a grand unified theory ( gut ) , then the trilinear r - parity violating terms come out to be rather small in magnitude ( @xmath20 or so)@xcite . however , the superrenormalizable bilinear terms are not subjected to such requirements . in this paper , we will keep @xmath21 as the only r - parity violating terms to study the phenomenology of the model . the plan of this paper is follows . in sect.ii , we will describe the basic ingredients of the supersymmetry with bilinear r - parity violation . the mass matrices of the cp - even , cp - odd and charged higgs are derived . some interesting relations for cp - even and cp - odd higgs masses are obtained . for completeness , we also give the mixing matrices of charginos with charged leptons and neutralinos with neutrinos . in sect.iii , we will give the feynman rules for the interaction of the higgs bosons ( sleptons ) with the gauge bosons , and the charginos , neutralinos with gauge bosons or higgs bosons ( sleptons ) . the self interactions of the higgs and the interactions of chargino ( neutralino)-squark - quark are also given . in sect.iv , we will analyze the particle spectrum by the numerical method under a few assumptions about the parameters in the model . we find that the possibility with large value for @xmath22 and @xmath23 still survives under strong experimental restrictions for the masses of @xmath24-neutrino : @xmath25 mev and @xmath24-lepton : @xmath26 gev . finally we will close our discussions with comments on the model . as stated above , we are to consider a superpotential of the form : @xmath27 with @xmath28 , @xmath29 are the parameters with units of mass , @xmath30 , @xmath31 and @xmath32 are the yukawa couplings as in the mssm with r - parity . in order to break the supersymmetry , we introduce the soft susy - breaking terms : @xmath33 where @xmath34 and @xmath35 are the parameters with units of mass squared while @xmath36 denote the masses of @xmath37 and @xmath38 , the @xmath39 gauginos . @xmath40 and @xmath41 are free parameters with units as mass . @xmath42 , @xmath43 @xmath44 are the soft breaking parameters that give the mass splitting between the quarks , leptons and their supersymmetric partners . the rest parts ( such as the part of gauge , matter and the gauge - matter interactions etc ) in the model are the same as the mssm with r - parity and we will not repeat them here . thus the scalar potential of the model can be written as @xmath45 where @xmath46 denote the scalar fields , @xmath47 is the usual d - terms , @xmath48 is the susy soft breaking terms given in eq . ( [ eq-5 ] ) . using the superpotential eq . ( [ eq-4 ] ) and the soft breaking terms eq . ( [ eq-5 ] ) , we can write down the scalar potential precisely . the electroweak symmetry is broken spontaneously when the two higgs doublets @xmath49 , @xmath50 and the sleptons acquire nonzero vacuum expectation values ( vevs ) : @xmath51 @xmath52 and @xmath53 where @xmath54 denote the slepton doublets and @xmath55 , @xmath28 , @xmath24 , the generation indices of the leptons . from eq . ( [ eq-5 ] ) , eq . ( [ eq-6 ] ) , we can find the scalar potential includes the linear terms as following : @xmath56 where @xmath57 here @xmath58 ( @xmath59 , @xmath60 , @xmath61 , @xmath62 , @xmath63 ) are tadpoles at the tree level and , the vevs of the neutral scalar fields should satisfy the conditions @xmath64 ( @xmath59 , @xmath60 , @xmath61 , @xmath62 , @xmath63 ) , therefore one can obtain : @xmath65 for convenience , we will call all of these scalar bosons ( @xmath49 , @xmath50 and @xmath54 ) as higgs below . now , we will give the higgs boson mass matrix explicitly . for the scalar sector , the mass squared matrices may be obtained by : @xmath66 here `` minimum '' means to evaluate the values at @xmath67 , @xmath68 , @xmath69 and @xmath70 ( @xmath46 represent for all other scalar fields ) . thus the squared mass matrices of the cp - even and the cp - odd scalar bosons both are @xmath71 , whereas matrix of the charged higgs is @xmath72 . from the scalar potential eq . ( [ eq-6 ] ) , we can find the mass terms : @xmath73 where `` current '' cp - even higgs fields @xmath74 , @xmath75 , @xmath76 , @xmath77 , @xmath78 . the mass matrix in eq . ( [ l - add1 ] ) is @xmath79 with notations @xmath80 and @xmath81 in order to obtain the above mass matrix , the eq . ( [ masspara ] ) is used . the physical cp - even higgs can be obtained by @xmath82 where @xmath83 ( @xmath84 , @xmath85 , @xmath60 , @xmath86 , @xmath87 , @xmath88 ) are the elements of the matrix that converts the mass matrix eq . ( [ matrix - even ] ) into a diagonal one i.e. translates the current fields into physical fields ( corresponding to the eigenstates of the mass matrix ) . in the current basis @xmath89 , @xmath90 , @xmath91 , @xmath92 , @xmath93 , the mass matrix for the cp - odd scalar fields can be written as : @xmath94 with @xmath95 from eq . ( [ massodd ] ) and eq . ( [ s12345 ] ) , one can find that the neutral goldstone boson ( with zero - mass ) can be given as@xcite : @xmath96 which is indispensable for spontaneous breaking the ew gauge symmetry . here the @xmath97 and the mass of @xmath98-boson @xmath99 is the same as r - parity conserved mssm . the other four physical neutral bosons can be written as : @xmath100 where @xmath101 ( @xmath84 , @xmath85 , @xmath60 , @xmath86 , @xmath87 , @xmath102 is the matrix that converts the current fields into the physical eigenstates . from the eigenvalue equations , one can find two independent relations : @xmath103^{2 } m_{z}^{2 } \prod_{i=2}^{5}m_{h_{5+i}^{0}}^{2 } . \label{massrelation}\end{aligned}\ ] ] if we introduce the following notations : @xmath104 the second relation of eq . ( [ massrelation ] ) can be written as : @xmath105 the first relation of eq . ( [ massrelation ] ) is also obtained in ref@xcite , whereas we consider the second relation of eq . ( [ massrelation ] ) is also important , the two equations are independent restrictions on the masses of neutral higgs bosons . for instance , from eq . ( [ massrelation ] ) and eq . ( [ massrelation1 ] ) , we have a upper limit on the mass of the lightest higgs at tree level in the model : @xmath106 where @xmath107 is the number of the cp - even higgs , @xmath108 is the mass of the lightest one among them and @xmath109 is the heaviest one . some points should be noted about eq . ( [ bound - masshiggs ] ) : * from eq . ( [ massrelation ] ) and eq . ( [ massrelation1 ] ) , we can find @xmath110 . * when @xmath111 or @xmath112 , @xmath113 , `` = '' is established . * in the case of mssm with r - parity ( n=2 ) , @xmath114 is recovered . so when @xmath115 , we can not imposed a upper limit on the @xmath108 as that for the r - parity conserved mssm at the tree level , namely , for the later it is just the case n=2@xcite : @xmath116 considering experimental data , one can not rule out large @xmath29 ( i=1 , 2 , 3)@xcite . further more , even if @xmath117 , we still have no reason to assume @xmath118 in general case . in the mssm with r - parity , the radiative corrections to mass of the lightest higgs are large@xcite , when complete one - loop corrections and leading two - loop corrections of @xmath119 are included , the ref@xcite gives the limit on the lightest higgs mass : @xmath120gev . in the mssm without r - parity , there is not so stringent restriction on the lightest higgs even at the tree level . with the `` current '' basis @xmath121 , @xmath122 , @xmath123 , @xmath124 , @xmath125 , @xmath126 , @xmath127 , @xmath128 and eq . ( [ eq-6 ] ) , we can find the following mass terms in lagrangian : @xmath129 and @xmath130 is given in appendix.a . diagonalizing the mass matrix for the charged higgs bosons , we obtain the zero mass goldstone boson state : @xmath131 together with the charge conjugate state @xmath132 , which are indispensable to break electroweak symmetry and give @xmath133 bosons masses . with the transformation matrix @xmath134 ( converts from the current fields into the physical eigenstates basis ) , the other seven physical eigenstates @xmath135 @xmath136 , @xmath86 , @xmath87 , @xmath88 , @xmath137 , @xmath138 , @xmath139 can be expressed as : @xmath140 due to the lepton number violation in the mssm without r - parity , fresh and interesting mixing of neutralinos with neutrinos and charginos with charged leptons may happen . the piece of lagrangian responsible for the mixing of neutralinos with neutrinos is : @xmath141 where @xmath142 is given by eq . ( [ eq-4 ] ) . @xmath143 and @xmath144 are the generators of the su(2)@xmath145u(1 ) gauge group and @xmath146 , @xmath46 stand for generic two - component fermions and scalar fields . writing down the eq . ( [ massneutra ] ) explicitly , we obtain : @xmath147 with the current basis @xmath148 , @xmath149 , @xmath150 , @xmath151 , @xmath152 , @xmath153 , @xmath154 and @xmath155 the mixing has the formulation : @xmath156 and transformation matrix @xmath157 has the property @xmath158 for convenience , we formulate all the neutral fermions into four component spinors as the follows : @xmath159 @xmath160 @xmath161 @xmath162 from eq . ( [ neutralino - matrix ] ) , we find that only one type neutrinos obtains mass from the mixing@xcite , we can assume it is the @xmath24-neutrino . one of the stringent restrictions comes from the bound that the mass of @xmath24-neutrino should be less than 20 mev@xcite . for convenience , sometimes we will call the mixing of neutralinos and neutrinos as neutralinos shortly late on . similar to the mixing of neutralinos and neutrinos , charginos mix with the charged leptons and form a set of charged fermions : @xmath163 , @xmath164 , @xmath165 , @xmath166 , @xmath167 . in current basis , @xmath168 , @xmath169 , @xmath170 , @xmath171 , @xmath172 and @xmath173 , @xmath174 , @xmath175 , @xmath176 , @xmath177 , the charged fermion mass terms in the lagrangian can be written as@xcite @xmath178 and the mass matrix : @xmath179 here @xmath180 and @xmath181 . two mixing matrices @xmath182 , @xmath183 can be obtained by diagonalizing the mass matrix @xmath184 i.e. the product @xmath185 is diagonal matrix : @xmath186 if we denote the mass eigenstates with @xmath187 : @xmath188 the four - component fermions are defined as : @xmath189 where @xmath166 , @xmath167 are the usual charginos and @xmath190 ( @xmath191 , @xmath87 , @xmath88 ) correspond to @xmath192 , @xmath28 and @xmath24 leptons respectively . for convenience , sometimes we will call the mixing of charginos with charged leptons as charginos shortly late on . from the above analyses , we have achieved the mass spectrum of the neutralino - neutrinos , chargino - charged leptons , neutral higgs - sneutrinos and charged higgs - charged sleptons . for the interaction vertices are also important , thus we will give the feynman rules which are different from those of the mssm with r - parity in next section . we have discussed the mass spectrum of the mssm with bilinear r - parity violation . now , we are discussing the feynman rules for the model that are different from those in mssm with r - parity . we are working in the t@xmath193hooft- feynman gauge@xcite which has the gauge fixed terms as : @xmath194 where @xmath195 and @xmath196 , @xmath197 were given as eq . ( [ ngold ] ) and eq . ( [ cgold ] ) . by inserting eq . ( [ gauge - fixed ] ) into interaction lagrangian , one obtains the desired vertices for the higgs bosons . if cp is conserved i.e. we assume the relevant parameters are real , one finds ( by analyzing the @xmath198 couplings ) that @xmath199 , @xmath200 , @xmath201 @xmath202 , @xmath203 are scalars and @xmath196 , @xmath204 , @xmath205 , @xmath206 , @xmath207 are pseudoscalar . let us compute the vertices of higgs ( slepton)- gauge bosons in the model . the original interaction terms of higgs bosons and gauge bosons are given as @xmath208 - \tilde{l}^{i\dag}\big(g\frac{\tau^{i}}{2}a_{\mu}^{i } \nonumber \\ & & - \frac{1}{2}g^{\prime}b_{\mu}\big ) \big(g\frac{\tau^{j}}{2}a^{j\mu } - \frac{1}{2}g^{\prime}b^{\mu}\big)\tilde{l}^{i } + \big ( ig^{\prime } b_{\mu}\tilde{r}^{i*}\partial^{\mu}\tilde{r}^{i } \nonumber \\ & & + h.c . \big ) - g^{\prime^{2}}\tilde{r}^{i*}\tilde{r}^{i}b_{\mu}b^{\mu } \bigg\ } + \bigg\ { h^{1\dag}\big(g\frac{\tau^{i}}{2}a_{\mu}^{i } - \frac{1}{2}g^{\prime}b_{\mu}\big)\partial^{\mu}h^{1 } \nonumber \\ & & + h.c . \bigg\ } - h^{1\dag}\big(g\frac{\tau^{i}}{2}a_{\mu}^{i } - \frac{1}{2}g^{\prime}b_{\mu}\big ) \big(g\frac{\tau^{j}}{2}a^{j\mu } - \frac{1}{2}g^{\prime}b^{\mu}\big)h^{1 } \nonumber \\ & & + \bigg\ { h^{2\dag}\big(g\frac{\tau^{i}}{2}a_{\mu}^{i } - \frac{1}{2}g^{\prime}b_{\mu}\big)\partial^{\mu}h^{2 } + h.c . \bigg\ } - h^{2\dag}\big(g\frac{\tau^{i}}{2}a_{\mu}^{i } \nonumber \\ & & - \frac{1}{2}g^{\prime}b_{\mu}\big ) \big(g\frac{\tau^{j}}{2}a^{j\mu } - \frac{1}{2}g^{\prime}b^{\mu}\big)h^{2 } \nonumber \\ & = & { \cal l}_{ssv } + { \cal l}_{svv } + { \cal l}_{ssvv}. \label{interaction1}\end{aligned}\ ] ] here @xmath209 , @xmath210 and @xmath211 represent the interactions in the physical basis , thus we have @xmath212 + h.c.\bigg\ } \nonumber \\ & & + \frac{i}{2}g\bigg\{w_{\mu}^{+ } \bigg[\phi_{1}^{0}\partial^{\mu}h_{2}^{1 * } - \partial^{\mu}\phi_{1}^{0}h_{2}^{1 } + \phi_{2}^{0}\partial^{\mu}h_{1}^{2 * } - \partial^{\mu}\phi_{2}^{0}h_{1}^{2 * } \nonumber \\ & & + \sum_{i}(\phi_{\tilde{\nu}_{i}}^{0}\partial^{\mu}\tilde{l}_{2}^{i } - \partial^{\mu}\phi_{\tilde{\nu}_{i}}^{0}\tilde{l}_{2}^{i } ) \bigg ] - h.c . \bigg\ } + \bigg\{\frac{1}{2}\sqrt{g^{2 } + g^{\prime^{2}}}\big(\cos 2\theta_{w}z_{\mu } \nonumber \\ & & - \sin 2\theta_{w}a_{\mu}\big)\bigg[\sum_{i}\big(\tilde{l}_{2}^{i*}\partial^{\mu } \tilde{l}_{2}^{i } - \partial^{\mu}\tilde{l}_{2}^{i*}\tilde{l}_{2}^{i}\big ) - h_{1}^{2*}\partial^{\mu}h_{1}^{2 } \nonumber \\ & & + \partial^{\mu}h_{1}^{2 * } h_{1}^{2 } + h_{2}^{1*}\partial^{\mu}h_{2}^{1 } - \partial^{\mu}h_{2}^{1*}h_{2}^{1}\bigg ] + \big(2\sin^{2}\theta_{w}z_{\mu } \nonumber \\ & & + 2\sin\theta_{w}\cos\theta_{w}a_{\mu}\big)\bigg[\sum_{i}\big(\tilde{r}^{i*}\partial^{\mu}\tilde{r}^{i } - \partial^{\mu}\tilde{r}^{i*}\tilde{r}^{i}\big)\bigg ] \bigg\ } \nonumber \\ & = & \frac{i}{2}\sqrt{g^{2 } + g^{\prime^{2}}}c_{eo}^{ij } \big(\partial^{\mu}h_{5+i}^{0}h_{j}^{0 } - h_{5+i}^{0}\partial^{\mu}h_{j}^{0}\big)z_{\mu } \nonumber \\ & & + \bigg\ { \frac{1}{2}g c_{ec}^{ij } \big(h_{i}^{0}\partial^{\mu}h_{j}^{- } -\partial^{\mu}h_{i}^{0}h_{j}^{-}\big)w_{\mu}^{+ } + h.c . \bigg\ } \nonumber \\ & & + \bigg\ { \frac{i}{2}gc_{co}^{ij } \big(h_{5+i}^{0}\partial^{\mu}h_{j}^{- } - \partial^{\mu}h_{5+i}^{0}h_{j}^{-}\big)w_{\mu}^{+ } \nonumber \\ & & + h.c . \bigg\ } + \bigg\{\frac{1}{2}\sqrt{g^{2 } + g^{\prime^{2}}}\bigg[\big ( \cos 2\theta_{w}\delta^{ij}\nonumber \\ & & -c_{c}^{ij}\big)z_{\mu } \big(h_{i}^{-}\partial^{\mu}h_{j}^{+ } - \partial^{\mu}h_{i}^{-}h_{j}^{+}\big ) \nonumber \\ & & - \sin 2\theta_{w}a_{\mu}\big(h_{i}^{-}\partial^{\mu}h_{i}^{+ } - \partial^{\mu}h_{i}^{-}h_{i}^{+}\big)\bigg]\bigg\ } , \label{vertex - ssv}\end{aligned}\ ] ] with @xmath213 where the transformation matrices @xmath214 , @xmath215 and @xmath216 are defined in section ii . @xmath217 + h.c . \bigg\ } \nonumber \\ & = & \frac{g^{2 } + g^{\prime^{2}}}{4}c_{even}^{i } \big(h_{i}^{0}z_{\mu}z^{\mu } + 2\cos^{2}\theta_{w}h_{i}^{0}w_{\mu}^{-}w^{+\mu}\big ) \nonumber \\ & & - \frac{g^{2 } + g^{\prime^{2}}}{2 } s_{w}c_{w}\upsilon\bigg[s_{w}z_{\mu}w^{+\mu}h_{1}^{- } + c_{w}a_{\mu}w^{+\mu}h_{1}^{- } + h.c . \bigg ] , \label{vertex - svv}\end{aligned}\ ] ] with @xmath218 the piece of @xmath211 is given as @xmath219 - \frac{g^{2 } + g^{\prime^{2}}}{4}\bigg[\frac{1}{2}\big(\phi_{1}^{0}\phi_{1}^{0 } \nonumber \\ & & + \phi_{2}^{0}\phi_{2}^{0 } + \sum_{i}\phi_{\tilde{\nu}_{i}}^{0}\phi_{\tilde{\nu}_{i}}^{0}\big)z_{\mu}z^{\mu } + \cos^{2}\theta_{w}\big(\phi_{1}^{0}\phi_{1}^{0 } + \phi_{2}^{0}\phi_{2}^{0 } + \phi_{\tilde{\nu}_{i}}^{0}\phi_{\tilde{\nu}_{i}}^{0}\big)w_{\mu}^{-}w^{+\mu } \bigg ] \nonumber \\ & & -\frac{g^{2 } + g^{\prime^{2}}}{4}\cos\theta_{w}\bigg[\big(-1 + \cos 2\theta_{w}\big)z_{\mu}w^{+\mu}\big(\chi_{1}^{0}h_{2}^{1 } - \chi_{2}^{0}h_{1}^{2 * } \nonumber \\ & & + \sum_{i}\chi_{\tilde{\nu}_{i}}^{0}\tilde{l}_{2}^{i}\big ) - \cos\theta_{w}\sin 2\theta_{w}a_{\mu}w^{+\mu}\big(\chi_{1}^{0}h_{2}^{1 } - \chi_{2}^{0}h_{1}^{2 * } \nonumber \\ & & + \sum_{i}\chi_{\tilde{\nu}_{i}}^{0}\tilde{l}_{2}^{i}\big ) + h.c.\bigg ] + \frac{i(g^{2 } + g^{\prime^{2}})}{4}\cos\theta_{w}\bigg[\big(-1 + \cos 2\theta_{w}\big)z_{\mu}w^{+\mu}\big(\phi_{1}^{0}h_{2}^{1 } \nonumber \\ & & -\phi_{2}^{0}h_{1}^{2 * } + \sum_{i}\phi_{\tilde{\nu}_{i}}^{0}\tilde{l}_{2}^{i}\big ) - \cos\theta_{w}\sin 2\theta_{w}a_{\mu}w^{+\mu}\big(\phi_{1}^{0}h_{2}^{1 } \nonumber \\ & & -\phi_{2}^{0}h_{1}^{2 * } + \sum_{i}\phi_{\tilde{\nu}_{i}}^{0}\tilde{l}_{2}^{i}\big ) + h.c.\bigg ] - \frac{1}{4}(g^{2 } + g^{\prime^{2}})\bigg[\sin^{2}2\theta_{w}a_{\mu}a^{\mu}\nonumber \\ & & \big(h_{2}^{1*}h_{2}^{1 } + h_{1}^{2*}h_{1}^{2 } + \sum_{i}\tilde{l}_{2}^{i * } \tilde{l}_{2}^{i}\big ) \nonumber \\ & & + \cos^{2}2\theta_{w}z_{\mu}z^{\mu}\big(h_{2}^{1*}h_{2}^{1 } + h_{1}^{2*}h_{1}^{2 } + \sum_{i}\tilde{l}_{2}^{i*}\tilde{l}_{2}^{i}\big ) \nonumber \\ & & -\sin 4\theta_{w}z_{\mu}a^{\mu}\big(h_{2}^{1*}h_{2}^{1 } + h_{1}^{2*}h_{1}^{2 } + \sum_{i}\tilde{l}_{2}^{i*}\tilde{l}_{2}^{i}\big ) \nonumber \\ & & + 2\cos^{2}\theta_{w}\big(h_{2}^{1*}h_{2}^{1 } + h_{1}^{2*}h_{1}^{2 } + \sum_{i}\tilde{l}_{2}^{i*}\tilde{l}_{2}^{i}\big ) \bigg ] \nonumber \\ & & -\sum_{i}g^{\prime^{2}}\tilde{r}^{i*}\tilde{r}^{i}b_{\mu}b^{\mu } \nonumber \\ & = & -\frac{1}{4}(g^{2 } + g^{\prime^{2}})\big(\frac{1}{2}h_{i}^{0}h_{i}^{0}z_{\mu}z^{\mu } + \cos^{2}\theta_{w}h_{i}^{0}h_{i}^{0}w_{\mu}^{-}w^{+\mu}\big ) \nonumber \\ & & -\frac{1}{4}(g^{2 } + g^{\prime^{2}})\big(\frac{1}{2}h_{5+i}^{0}h_{5+i}^{0}z_{\mu}z^{\mu } + \cos^{2}\theta_{w}h_{5+i}^{0}h_{5+i}^{0 } w_{\mu}^{-}w^{+\mu}\big ) \nonumber \\ & & + \frac{1}{4}(g^{2 } + g^{\prime^{2}})\sin 2\theta_{w}\bigg\{c_{ec}^{ij } \bigg[\sin\theta_{w}h_{i}^{0}z_{\mu}w^{+\mu}h_{j}^{- } \nonumber \\ & & + \cos\theta_{w}h_{i}^{0}a_{\mu}w^{+\mu}h_{j}^{-}\bigg ] + h.c . \bigg\ } \nonumber \\ & & -\frac{i}{4}(g^{2 } + g^{\prime^{2}})\sin 2\theta_{w}\bigg\{c_{co}^{ij } \bigg[\sin\theta_{w}h_{5+i}^{0}z_{\mu}w^{+\mu}h_{j}^{-}\nonumber \\ & & + \cos\theta_{w}h_{5+i}^{0}a_{\mu}w^{+\mu}h_{i}^{-}\bigg ] - h.c . \bigg\ } \nonumber \\ & & -\frac{1}{4}(g^{2 } + g^{\prime^{2}})\bigg\{2\cos^{2}\theta_{w}\big(\delta_{ij}- c_{c}^{ij}\big)h_{i}^{- } h_{j}^{+}w_{\mu}^{-}w^{+\mu } \nonumber \\ & & + \bigg[\cos^{2}2\theta_{w}\delta_{ij } - c_{c}^{ij}\big(4\sin^{3}\theta_{w } - \cos^{2}2\theta_{w}\big)\bigg]h_{i}^{-}h_{j}^{+}z_{\mu}z^{\mu } \nonumber \\ & & + \sin^{2}2\theta_{w}\delta_{ij}h_{i}^{-}h_{j}^{+}a_{\mu}a^{\mu } + \bigg[\sin 4\theta_{w } \delta_{ij } \nonumber \\ & & -c_{c}^{ij}\big(\sin 4\theta_{w } + 8\sin^{2}\theta_{w}\cos \theta_{w}\big)\bigg]z_{\mu}a^{\mu}h_{i}^{-}h_{j}^{+ } \bigg\ } \hspace{1 mm } , \label{vertex - ssvv}\end{aligned}\ ] ] where the @xmath220 , @xmath221 and @xmath222 are defined in eq . ( [ cdef ] ) . the relevant feynman rules may be summarized in fig . [ fig1 ] , fig . [ fig2 ] , fig . [ fig3 ] and fig . we would emphasize some features about them . first , the presence of the vertices @xmath223 @xmath224 , @xmath85 , @xmath60 , @xmath86 , @xmath87 , @xmath102 and the forbiddance of the vertices @xmath225 and @xmath226 ( @xmath84 , @xmath85 , @xmath60 , @xmath86 , @xmath87 , @xmath88 ) couplings are determined by cp nature . second , besides the @xmath227 ( @xmath132 is just the charged goldstone boson ) interaction , there are not vertices @xmath228 @xmath136 , @xmath86 , @xmath87 , @xmath88 , @xmath137 , @xmath138 , @xmath139 at the tree level , that is the same as the mssm with r - parity and general two - higgs doublet models@xcite . it is straightforward to insert eqs . ( [ masseven ] , [ ngold ] , [ oddhiggs ] , [ cgold ] , [ charhiggs ] ) into eqs . ( [ eq-6 ] ) to obtain the desired interaction terms . similar to the interaction of gauge - higgs ( slepton ) bosons , we split the lagrangian into pieces : @xmath229 where @xmath230 represents trilinear coupling terms , and @xmath231 represents four scalar boson coupling terms . the trilinear piece is most interesting . if the masses of the scalars are appropriate , the decays of one higgs boson into two other higgs bosons may be opened . after tedious computation , we have : @xmath232 and @xmath233 with @xmath234 the definitions of @xmath235 , @xmath236 , @xmath237 , @xmath238 , @xmath239 and @xmath240 can be found in appendix.c . the feynman rules are summarized in fig . [ fig5 ] and fig . note that the lepton number violation has led to very complicated form for the @xmath230 and @xmath231 . in this subsection , we compute the interactions of the higgs bosons with the supersymmetric partners of the gauge and higgs bosons ( the gauginos and higgsinos ) . after spontaneous breaking of the gauge symmetry su(2)@xmath145u(1 ) , the gauginos , higgsinos and leptons with the same electric charge will mix as we have described in section ii . let us proceed now to compute interesting interactions @xmath241 ( higgs - neutralinos - neutralinos interactions ) etc . the original interactions ( in two - component notations ) are@xcite : @xmath242 now we sketch the derivation for the vertices , such as @xmath241 etc . starting with the eq . ( [ snn - tcom ] ) , we convert the pieces from two - component notations into four - component notations first , then using the spinor fields defined by eq . ( [ define - eneutrino ] ) , eq . ( [ define - muneutrino ] ) , eq . ( [ define - tauneutrino ] ) , eq . ( [ define - neutralino ] ) and eq . ( [ define - chargino ] ) , we find : @xmath243 \nonumber \\ & & + \frac{g}{\sqrt{2}}\bigg[c_{skk}^{ij}h_{i}^{0}\bar{\kappa}_{m}^{+}p_{l}\kappa_{j}^{+ } + c_{skk}^{ij*}h_{i}^{0}\bar{\kappa}_{j}^{+}p_{r}\kappa_{m}^{+ } \bigg ] \nonumber \\ & & + i\frac{\sqrt{g^{2 } + g^{\prime^{2}}}}{2}\bigg [ c_{onn}^{ij}h_{5+i}^{0}\bar{\kappa}_{j}^{0}p_{r}\kappa_{m}^{0 } -c_{onn}^{ij*}h_{5+i}^{0}\bar{\kappa}_{m}^{0}p_{l}\kappa_{j}^{0}\bigg ] \nonumber \\ & & + i\frac{g}{\sqrt{2}}\bigg[c_{okk}^{ij}h_{5+i}^{0}\bar{\kappa}_{m}^{+}p_{l}\kappa_{j}^{+ } -c_{okk}^{ij*}h_{5+i}^{0}\bar{\kappa}_{m}^{+}p_{r}\kappa_{j}^{+ } \bigg ] \nonumber \\ & & + \sqrt{g^{2 } + g^{\prime^{2}}}\bigg[c_{lnk}^{ij } \bar{\kappa}_{j}^{+}p_{l}\kappa_{m}^{0}h_{i}^{+ } -c_{rnk}^{ij}\bar{\kappa}_{j}^{+}p_{r}\kappa_{m}^{0}h_{i}^{+}\bigg ] \label{vertex - snn}\end{aligned}\ ] ] with the definitions of @xmath244 , @xmath245 , @xmath246 and @xmath247 are given as @xmath248 \nonumber \\ & & \hspace{10 mm } + \frac{1}{2\sqrt{g^{2}+g^{\prime^{2}}}}\sum_{i=1}^{3}l_{i}\big(z_{c}^{i,5+i}z_{-}^{j,2+i}z_{n}^{m,3 } - z_{c}^{i,5+i}z_{-}^{j,2}z_{n}^{m,4+i}\big ) , \nonumber \\ & & c_{rnk}^{ij}=\bigg[z_{c}^{i,2}\bigg(\frac{1}{\sqrt{2 } } \big(\cos\theta_{w}z_{+}^{*j,2}z_{n}^{*m,2 } + \sin\theta_{w}z_{+}^{*j,2}z_{n}^{*m,1}\big ) \nonumber \\ & & \hspace{10mm}+\cos\theta_{w}z_{+}^{*j,1}z_{n}^{*m,4}\bigg ) + \sqrt{2}\sin\theta_{w}\sum_{i=1}^{3}z_{c}^{i,5+i}z_{+}^{*j,2+i } z_{n}^{*m,1}\bigg ] \nonumber \\ & & \hspace{10mm}+\frac{1}{2\sqrt{g^{2}+g^{\prime^{2 } } } } \sum_{i=1}^{3}l_{i}\big(z_{c}^{i,2+i}z_{n}^{*m,3}z_{+}^{*j,2+i } - z_{c}^{i,1 } z_{n}^{*m,3}z_{+}^{*j,2+i}\big ) . \label{def - coup}\end{aligned}\ ] ] here the project operators @xmath249 and the transformation matrices @xmath250 , @xmath157 defined in sect.ii . the corresponding feynman rules are summarized in fig . [ fig7 ] . as for @xmath251 being a majorana fermion , we note the useful identity : @xmath252 which holds for anticommuting four - component majorana spinors . this implies that the @xmath253 interactions can be rearranged in symmetry under the interchange of @xmath254 and @xmath255 . since @xmath256 @xmath257 , @xmath258 @xmath259 and @xmath260 @xmath261 should be identified with the three lightest neutralinos ( charginos ) in the model , there must be some interesting phenomena relevant to them , such as @xmath251 @xmath262 @xmath263,@xmath86 , @xmath264 , @xmath139 , @xmath251 @xmath265 @xmath266 , @xmath60 , @xmath264 , @xmath102 etc if the masses are suitable . namely , these interactions without r - parity conservation may induce interesting rare processes@xcite . in this subsection we will focus on the couplings of the gauge bosons ( @xmath267 , @xmath98 , @xmath268 ) to the charginos ( charged leptons ) and neutralinos ( neutrinos ) . since we identify the three type charged leptons ( three type neutrinos ) with the three lightest charginos ( neutralinos ) , the restrictions relating to them from the present experiments must be considered carefully . the relevant interactions come from the following pieces : @xmath269 with @xmath270 similar to the couplings in @xmath271 , we convert all spinors in eq . ( [ lang - gcn ] ) into four component ones and using eq . ( [ define - neutralino ] ) , eq . ( [ define - chargino ] ) , then we obtain : @xmath272\kappa_{j}^{+ } \bigg\ } \nonumber \\ & & + \bigg\ { g\bar{\kappa}_{j}^{+}\bigg [ \big(-z_{+}^{*i,1}z_{n}^{j,2 } + \frac{1}{\sqrt{2}}z_{+}^{*i,2 } z_{n}^{j,4}\big)\gamma^{\mu}p_{l } + \bigg(z_{n}^{*i,2}z_{-}^{j,1 } \nonumber \\ & & + \frac{1}{\sqrt{2}}\big(z_{n}^{*i,3}z_{-}^{j,2 } + \sum_{i=1}^{3}z_{n}^{*i,4+i}z_{-}^{j,2+i}\big)\bigg ) \gamma^{\mu}p_{r}\bigg]\kappa_{i}^{0}w_{\mu}^{+ } + h.c . \bigg\ } \nonumber \\ & & + \frac{\sqrt{g^{2}+g^{\prime^{2}}}}{2}\bar{\kappa}_{i}^{0}\gamma^{\mu } \bigg[\frac{1}{2}\bigg(z_{n}^{*i,4}z_{n}^{j,4 } - \big(z_{n}^{*i,3}z_{n}^{j,3 } + \sum_{\alpha=5}^{7}z_{n}^{*i,\alpha}z_{n}^{j,\alpha}\big)\bigg)p_{l } \nonumber \\ & & - \frac{1}{2}\bigg(z_{n}^{*i,4}z_{n}^{j,4 } - \big(z_{n}^{*i,3}z_{n}^{j,3 } + \sum_{\alpha=5}^{7}z_{n}^{*i,\alpha}z_{n}^{j,\alpha}\big)\bigg)p_{r}\bigg]\kappa_{j}^{0}z_{\mu } . \label{vertex - gcn}\end{aligned}\ ] ] the corresponding feynman rules are summarized in fig . [ fig8 ] . for we identify three lightest neutralinos ( charginos ) with three type neutrinos ( charged leptons ) , we want to emphasize some features about eq . ( [ vertex - gcn ] ) : * for the @xmath268 boson-@xmath273-@xmath273 vertices , there is not the lepton flavor changing current interaction at the tree level , that is same as the sm and mssm with r - parity . * for the tree level @xmath98 boson-@xmath273-@xmath273 vertices , there are the lepton flavor changing current interactions , this point is different from the mssm with r - parity . * similar to the @xmath98 boson-@xmath273-@xmath273 vertices , there are the tree level vertices such as @xmath274 which are forbidden in the mssm with r - parity . in this subsection , we will give the feynman rules for the interactions of quarks and squarks with charginos ( charged leptons ) and neutralinos ( neutrinos ) i.e. the @xmath275 vertices . because of lepton number violation so having the mixing of neutrinos ( charged leptons ) and original neutralinos ( charginos ) , the vertices may lead to interesting phenomenology , thus it is interesting to write them out . there are two pieces contributing to the above vertices . the first is the supersymmetric analogue of the @xmath276 and @xmath277 interaction . the second is the supersymmetric analogue of the @xmath278 interaction , which is proportional to quark mass and depends on the properties of the higgs bosons in the model . these two kinds of vertices correspond to the terms in eq . ( [ massneutra ] ) . to consider the @xmath279 interaction first , let us write down the interaction in two - component spinors for fermions as the follows : @xmath280 then convert the two - component spinors into four - component spinors as discussed above : @xmath281\psi_{u^{i}}\tilde{d}_{i , i}^{+ } \nonumber \\ & & + c^{ij*}\bar{\kappa}_{j}^{-}\bigg[\big(-gz_{u_{j}}^{i,1}z_{+}^{j,1 } + \frac{u^{j}}{2}z_{u_{j}}^{i,2}z_{+}^{j,2}\big)p_{l } \nonumber \\ & & -\frac{d^{i}}{2}z_{u_{j}}^{j,1*}z_{-}^{j,2*}p_{r}\bigg]\psi_{d^{j}}\tilde{u}_{ji}^{- } + h.c . \label{qqc - fcom}\end{aligned}\ ] ] now @xmath282 , @xmath283 are four - component quark spinors of the i - th generation . the @xmath284 ( c is the charge - conjugation matrix ) is a charged - conjugate state of @xmath285 , and @xmath285 is defined in eq . ( [ define - chargino ] ) . the feynman rules are summarized in fig . [ fig9 ] . for the @xmath286 interactions , we can write the pieces in two - component notations as : @xmath287 \nonumber \\ & & - \frac{u^{i}}{2}\bigg[\psi_{h^{2}}^{2}\psi_{q^{1}}^{i}\tilde{u}^{i } + \psi_{h^{2}}^{2}\psi_{u}^{i}\tilde{q}_{1}^{i } + h.c.\bigg ] \label{qqn - tcom}\end{aligned}\ ] ] after converting eq . ( [ qqn - tcom ] ) into four - component notations straightforwardly and using the definition for neutralino mass eigenstates , we have : @xmath288p_{l } + \bigg[\frac{2\sqrt{2}}{3}g^{\prime } z_{u^{i}}^{i,2*}z_{n}^{j,1 } - \frac{u^{i}}{2}z_{u^{i}}^{i,1*}z_{n}^{j,4*}\bigg]p_{r } \bigg\}\psi_{u^{i}}\tilde{u}_{i , i}^{- } \nonumber \\ & & + \bar{\kappa}_{j}^{0}\bigg\{\bigg[\frac{e}{\sqrt{2}\sin\theta_{w}\cos\theta_{w}}z_{d^{i}}^{i,1 } \big(-\cos\theta_{w}z_{n}^{i,2 } + \frac{1}{3 } \sin\theta_{w}z_{n}^{j,1}\big ) \nonumber \\ & & + \frac{d^{i}}{2}z_{d^{i}}^{i,2}z_{n}^{j,3}\bigg]p_{l } + \bigg[-\frac{\sqrt{2}}{3}g^{\prime } z_{d^{i}}^{i,2}z_{n}^{j,1 } \nonumber \\ & & + \frac{d^{i}}{2}z_{d^{i}}^{i,1*}z_{n}^{j,3*}\bigg]p_{r}\bigg\}\psi_{d^{i}}\tilde{d}_{i , i}^{+ } + h.c . , \label{qqn - fcom}\end{aligned}\ ] ] thus the feynman rules for the concerned interactions may be depicted exactly as the last two diagrams in fig . in this section , we will analyze the mass spectrum of neutral higgs , neutralinos and charginos numerically . we have obtained the mass matrices by set the three type sneutrinos with non - zero vacuum and @xmath289 @xmath290 , @xmath60 , @xmath291 . however , the matrices are too big to get the typical features . from now on , we will assume @xmath292 and @xmath293 . i.e. only @xmath24-lepton number is violated . we have two reasons to make the assumption : * under the assumption , we believe the key features will not be lost but the mass matrices will turn much simple . * according to experimental indications , the @xmath24-neutrino may be the heaviest among the three type neutrinos . in the numerical calculations below , the input parameters are chosen as : @xmath294 , @xmath295gev , @xmath296gev , @xmath297gev , for the unknown parameters @xmath298 , @xmath299 , we assume @xmath300gev and the upper limit on @xmath24-neutrino mass @xmath301mev is also taken into account seriously . now let us consider the mass matrix of neutralinos first . when @xmath302 and @xmath293 , the eq . ( [ neutralino - matrix ] ) is simplified as : @xmath303 as stated above , a strong restriction imposes on the matrix is from @xmath25mev . ref@xcite has discussed this limit impacting on the parameter space , the numerical result indicates the large value of @xmath304 can not be ruled out . in order to show the problem precisely , let us consider the equation for the eigenvalues of eq . ( [ rneutralino - matrix ] ) : @xmath305 \lambda^{2 } \nonumber \\ & & + \bigg[-m_{1}m_{2}(\mu^{2 } + \epsilon_{3}^{2 } ) + \frac{1}{16}(g^{2 } + g^{\prime^{2}})(\epsilon_{3}\upsilon_{1 } + \mu\upsilon_{\tilde{\nu}_{\tau}})^{2 } \nonumber \\ & & + \frac{1}{2}(g^{2}m_{1 } + g^{\prime^{2}}m_{2})(\mu\upsilon_{1}\upsilon_{2 } - \epsilon_{3}\upsilon_{2}\upsilon_{\tilde{\nu}_{\tau}})\bigg ] \lambda \nonumber \\ & & - \frac{1}{8}(g^{2}m_{1 } + g^{\prime^{2}}m_{2})\big(\mu\upsilon_{\tilde{\nu}_{\tau } } + \epsilon_{3}\upsilon_{1}\big)^{2 } \nonumber \\ & = & \lambda^{5 } + { \cal a}_{n}\lambda^{4 } + { \cal b}_{n}\lambda^{3 } + { \cal c}_{n}\lambda^{2 } + { \cal d}_{n}\lambda + { \cal e}_{n}. \label{eigen - neutralino}\end{aligned}\ ] ] for further discussions , let us introduce new symbols @xmath306 , @xmath307 as : @xmath308 thus we have the coefficients of eq . ( [ eigen - neutralino ] ) : @xmath309 if we fix the @xmath24-neutrino mass @xmath310 as an input parameter , so the equation @xmath311 can be written as : @xmath312 the coefficients @xmath313 , @xmath314 , @xmath315 , @xmath316 are related to the `` original '' ones @xmath317 , @xmath318 , @xmath319 , @xmath320 , and @xmath321 as : @xmath322 to obtain the masses of the other four neutralinos , let us solve the eq . ( [ eigen - neutralino1 ] ) by the numerical method . in fig . [ fig10 ] , we plot the mass of the lightest neutralino versus x. the three lines correspond to @xmath323mev , @xmath60mev and @xmath324mev respectively . from the figure , we find that the curve corresponding to @xmath325mev is the lowest and the second low one is correspond to @xmath326mev , so the tendency is that the curves are going `` up '' as the @xmath24-neutrino mass is decreasing . if the mass of the lightest neutralino is not too heavy ( such as @xmath327gev ) , the absolute value of x can not take very large ( for example @xmath328gev ) . as for the mass of the charginos , when @xmath329 , and @xmath330 , the eq . ( [ chargino - matrix ] ) becomes : @xmath331 because @xmath332 should be the lightest eigenvalue of the matrix @xmath333 , after taking the eigenvalue @xmath332 away , the surviving eigenvalue equation becomes : @xmath334 here , @xmath335 with the parameters @xmath306 , @xmath307 are defined by eq . ( [ define - xy ] ) . therefore the masses of the other two charginos are expressed as : @xmath336 the parameter @xmath337 can be fixed by the condition @xmath338 . in fig . [ fig11 ] , we plot the mass of the lightest chargino versus x. the three lines correspond to @xmath339mev , @xmath60mev and @xmath324mev respectively . similar to the neutralinos , we find that the curve corresponding to @xmath339mev is the lowest , the second low one is correspond to @xmath340mev and the tendency is very similar to the case for neutralinos . this can be understood as following : when the values of @xmath298 , @xmath299 , @xmath341 , @xmath342 and x are fixed , the value of y will be fixed by the mass of @xmath24-neutrino . in the numerical computation , we find that the absolute value of y turns small , as the @xmath310 changes large . this is the reason why the curve corresponding to @xmath339mev is the lowest among the three curves which we have computed here . now , we turn to discuss the mass matrix of the neutral higgs . under the same assumption , the mass matrix for cp - even higgs reduces to : @xmath343 with @xmath344 the mass matrix of cp - odd higgs reduces to : @xmath345 with @xmath346 introducing the following variables : @xmath347 the masses of the neutral higgs can be determined from the @xmath348 , @xmath349 , @xmath350 and @xmath341 , @xmath342 . for the masses of cp - odd higgs , we define : @xmath351 the masses of the two cp - odd higgs can be given as : @xmath352 in fig . [ fig12 ] , we plot the mass of the lightest cp - odd higgs versus the mass of the lightest cp - even higgs , where the ranges of the parameters are : @xmath353gev @xmath354 @xmath355gev@xmath354 and @xmath356 . from the fig . [ fig12 ] , we can find that there are no limit on the @xmath357 when we change those parameters in the above ranges . as for the lightest cp - even higgs , the difference from the mssm with r - parity is that @xmath108 can larger than @xmath358 at the tree level . this can be understood from eq . ( [ bound - masshiggs ] ) , under the assumptions , we have @xmath359 where the @xmath360 is the mass of the heaviest cp - even higgs in this case and we can not give the stringent limit on it as in the mssm with r - parity . in summary , we have analyzed the mass spectrum in the mssm with bilinear r - parity violation . from the restriction @xmath25mev , we can not rule out the possibilities with large @xmath304 and @xmath23 . we also derived the feynman rules in the @xmath361t - hooft feynman gauge , which are convenient when we study the phenomenology beyond the tree level in the model . recent experimental signals of neutrino masses and mixing may provide the first glimpses of the lepton number violation effects , ref@xcite have study the neutrino oscillations experiment constraint on the parameter space of the model . considering both the fermionic and scalar sectors , they find that a large area of the parameter space is allowed . here , we would also like to point out some references have analyzed the @xmath362-decay in the model@xcite and obtained new stringent upper limits on the first generation r - parity violating parameters , @xmath363 and @xmath364 ; whereas for the other two generations , there are not very serious restrictions on the upper limits of the r - parity violating parameters . as for other interesting processes in the model , they are discussed by ref@xcite . * acknowledgment * this work was supported in part by the national natural science foundation of china and the grant no . lwlz-1298 of the chinese academy of sciences . in the case of charged higgs , with the current basis @xmath121 , @xmath122 , @xmath123 , @xmath124 , @xmath125 , @xmath126 , @xmath127 , @xmath128 , the symmetric matrix @xmath130 is given as follows : @xmath365 note here that to obtain eq . ( [ eq-21 ] ) , eq . ( [ masspara ] ) is used sometimes . in a general case , the matrix of the squarks mixing should be 6@xmath1456 . under our assumptions , we do not consider the squarks mixing between different generations . from superpotential eq . ( [ eq-2 ] ) and the soft - breaking terms , we find the up squarks mass matrix of the i - th generation can be written as : @xmath366 where @xmath367 , @xmath60 , @xmath291 is the index of the generations . the current eigenstates @xmath368 and @xmath369 connect to the two physical ( mass ) eigenstates @xmath370@xmath371 , @xmath372 through @xmath373 and @xmath374 is determined by the condition : @xmath375 in a similar way , we can give the down squarks mass matrix of the i - th generation : @xmath376 the fields @xmath377 and @xmath378 relate to the two physical ( mass ) eigenstates @xmath379 @xmath371 , @xmath372 : @xmath380 in this appendix , we give precise expressions of the couplings that appear in the @xmath230 and @xmath231 . the method has been described clearly in text , the results are : @xmath381 + \frac{g^{2}}{4}\bigg(\upsilon_{\tilde{\nu}_{i}}z_{even}^{k,2 } + \upsilon_{2}z_{even}^{k,2+i}\bigg)\bigg(z_{c}^{i,2+i}z_{c}^{j,2 } \nonumber \\ & & + z_{c}^{i,2}z_{c}^{j,2+i}\bigg ) + \frac{g^{2}}{4}\bigg(\upsilon_{1}z_{even}^{k,2 } + \upsilon_{2}z_{even}^{k,1}\bigg)\bigg(z_{c}^{i,1}z_{c}^{j,2 } \nonumber \\ & & + z_{c}^{i,2}z_{c}^{j,1}\bigg ) + \frac{1}{\sqrt{2}}l_{i } \epsilon_{3}z_{even}^{k,1}\bigg(z_{c}^{i,4}z_{c}^{j,2 } \nonumber \\ & & + z_{c}^{i,2}z_{c}^{j,4 } \bigg ) + \frac{1}{\sqrt{2 } } \sum\limits_{i=1}^{3}l_{i}\epsilon_{i}z_{even}^{k,2}\bigg ( z_{c}^{i,5+i}z_{c}^{j,1 } + z_{c}^{i,1}z_{c}^{j,5+i } \bigg ) \nonumber \\ a_{oc}^{kij } & = & \sum\limits_{i=1}^{3}\bigg\{\bigg(\frac{g^{2}}{4 } - l_{i}^{2}\bigg ) \bigg[\upsilon_{\tilde{\nu}_{i}}z_{odd}^{k,1}\big(z_{c}^{i,1 } z_{c}^{j,2+i } - z_{c}^{i,2+i}z_{c}^{j,1}\big ) \nonumber \\ & & + \upsilon_{1}z_{odd}^{k,2+i}\big(z_{c}^{i,1}z_{c}^{j,2+i } - z_{c}^{i,2+i}z_{c}^{j,1}\big)\bigg ] \nonumber \\ & & + \frac{g^{2}}{4}\big(\upsilon_{\tilde{\nu}_{i}}z_{odd}^{k,2 } + \upsilon_{2}z_{odd}^{k,2+i}\big)\big(z_{c}^{i,2+i}z_{c}^{j,2 } - z_{c}^{i,2}z_{c}^{j,2+i}\big ) + \frac{g^{2}}{4}\big(\upsilon_{\tilde{\nu}_{i}}z_{odd}^{k,2 } \nonumber \\ & & + \upsilon_{2}z_{odd}^{k,2+i}\big)\big(z_{c}^{i,2+i}z_{c}^{j,2 } - z_{c}^{i,2}z_{c}^{j,2+i}\big ) \nonumber \\ & & \frac{1}{\sqrt{2}}l_{i}\epsilon_{i}z_{odd}^{k,1}\big(- z_{c}^{i,5+i}z_{c}^{j,2 } + z_{c}^{i,2}z_{c}^{j,5+i}\big ) \nonumber \\ & & -\frac{1}{\sqrt{2}}l_{i}\epsilon_{i}z_{odd}^{k,2}\big ( z_{c}^{i,5+i}z_{c}^{j,1 } - z_{c}^{i,1}z_{c}^{j,5+i}\big ) \bigg\ } \nonumber \\ { \cal a}_{ec}^{klij } & = & \frac{g^{2 } + g^{\prime^{2}}}{8}\bigg(z_{even}^{k,1}z_{even}^{l,1}z_{c}^{i,1}z_{c}^{j,1 } + z_{even}^{k,2}z_{even}^{l,2}z_{c}^{i,2}z_{c}^{j,2 } + \sum\limits_{i=1}^{3}z_{even}^{k,2+i}z_{even}^{l,2+i}z_{c}^{i,2+i}z_{c}^{j,2+i}\bigg ) \nonumber \\ & & + \sum\limits_{i=1}^{3}\bigg(\frac{g^{\prime^{2 } } - g^{2}}{8 } + \frac{1}{2}l_{i}^{2}\bigg ) \bigg(z_{even}^{k,1}z_{even}^{l,1}z_{c}^{i,2+i}z_{c}^{j,2+i } + z_{even}^{k,2+i}z_{even}^{l,2+i}z_{c}^{i,1}z_{c}^{j,1}\bigg ) \nonumber \\ & & + \frac{g^{\prime^{2 } } - g^{2}}{8}\bigg[z_{even}^{k,1}z_{even}^{l,1}z_{c}^{i,2}z_{c}^{j,2 } + z_{even}^{k,2}z_{even}^{l,2}z_{c}^{i,1}z_{c}^{j,1 } \nonumber \\ & & + \sum\limits_{i=1}^{3}\big(z_{even}^{k,2+i}z_{even}^{l,2+i}z_{c}^{i,2}z_{c}^{j,2 } + z_{even}^{k,2}z_{even}^{l,2 } z_{c}^{i,2+i}z_{c}^{j,2+i}\big)\bigg ] \nonumber \\ & & + \frac{g^{2}}{4}\bigg[z_{even}^{k,1}z_{even}^{l,2}\big(z_{c}^{i,2}z_{c}^{j,1 } + z_{c}^{i,1}z_{c}^{j,2 } ) + \sum\limits_{i=1}^{3}z_{even}^{k,1}z_{even}^{l,2+i}(z_{c}^{i,2+i}z_{c}^{j,1 } \nonumber \\ & & + z_{c}^{i,1}z_{c}^{j,2+i}\big ) + \sum\limits_{i=1}^{3}z_{even}^{k,2}z_{even}^{l,2+i } \big(z_{c}^{i,2+i}z_{c}^{j,2 } + z_{c}^{i,2}z_{c}^{j,2+i}\big ) \bigg ] \nonumber \\ & & -\sum\limits_{i=1}^{3}\bigg[\frac{l_{i}^{2}}{2}z_{even}^{k,1}z_{even}^{l,2+i}\big(z_{c}^{i,2+i}z_{c}^{j,1 } + z_{c}^{i,1}z_{c}^{j,2+i}\big ) - \frac{g^{\prime^{2}}}{4}\big ( z_{even}^{k,1}z_{even}^{l,1}z_{c}^{i,5+i}z_{c}^{j,5+i } \nonumber \\ & & - z_{even}^{k,2}z_{even}^{l,2}z_{c}^{i,5+i}z_{c}^{j,5+i } + z_{even}^{k,2+i}z_{even}^{l,2+i}z_{c}^{i,5+i}z_{c}^{j,5+i}\big ) \nonumber \\ & & + \frac{l_{i}^{2}}{2}z_{even}^{k,1}z_{even}^{l,1}z_{c}^{i,5+i}z_{c}^{j,5+i } \bigg ] \nonumber \\ { \cal a}_{oc}^{klij } & = & \frac{g^{2 } + g^{\prime^{2}}}{8}\big(z_{odd}^{k,1}z_{odd}^{l,1}z_{c}^{i,1}z_{c}^{j,1 } + z_{odd}^{k,2}z_{odd}^{l,2}z_{c}^{i,2}z_{c}^{j,2 } \nonumber \\ & & + \sum\limits_{i=1}^{3}z_{odd}^{k,2+i}z_{odd}^{l,2+i}z_{c}^{i,2+i}z_{c}^{j,2+i}\big ) + \sum\limits_{i=1}^{3}\bigg(\frac{g^{\prime^{2 } } - g^{2}}{8 } + \frac{1}{2}l_{i}^{2}\bigg ) \bigg(z_{odd}^{k,1}z_{odd}^{l,1}z_{c}^{i,2+i}z_{c}^{j,2+i } \nonumber \\ & & + z_{odd}^{k,2+i}z_{odd}^{l,2+i}z_{c}^{i,1}z_{c}^{j,1}\bigg ) + \frac{g^{\prime^{2 } } - g^{2}}{8}\bigg\{z_{odd}^{k,1}z_{odd}^{l,1}z_{c}^{i,2}z_{c}^{j,2 } \nonumber \\ & & + z_{odd}^{k,2}z_{odd}^{l,2 } z_{c}^{i,1}z_{c}^{j,1 } + \sum\limits_{i=1}^{3}\big(z_{odd}^{k,2+i}z_{odd}^{l,2+i}z_{c}^{i,2}z_{c}^{j,2 } \nonumber \\ & & + z_{odd}^{k,2}z_{odd}^{l,2}z_{c}^{i,2+i}z_{c}^{j,2+i}\big)\bigg\ } + \frac{g^{2}}{4}\bigg\{z_{odd}^{k,1}z_{odd}^{l,2}\big(z_{c}^{i,2}z_{c}^{j,1 } + z_{c}^{i,1}z_{c}^{j,2}\big ) \nonumber \\ & & + \sum\limits_{i=1}^{3}\bigg[z_{odd}^{k,1}z_{odd}^{l,3}\big(z_{c}^{i,3}z_{c}^{j,1 } + z_{c}^{i,1}z_{c}^{j,3}\big ) + z_{odd}^{k,2}z_{odd}^{l,3}\big(z_{c}^{i,3}z_{c}^{j,2 } + z_{c}^{i,2}z_{c}^{j,3}\big)\bigg ] \bigg\ } \nonumber \\ & & -\sum\limits_{i=1}^{3}\bigg\{\frac{l_{i}^{2}}{2}z_{odd}^{k,1}z_{odd}^{l,2+i } \big(z_{c}^{i,2+i}z_{c}^{j,1 } + z_{c}^{i,1}z_{c}^{j,2+i}\big ) - \frac{g^{\prime^{2}}}{4}\big(z_{odd}^{k,1}z_{odd}^{l,1}z_{c}^{i,5+i}z_{c}^{j,5+i } \nonumber \\ & & -z_{odd}^{k,2}z_{odd}^{l,2}z_{c}^{i,5+i}z_{c}^{j,5+i } + z_{odd}^{k,2+i}z_{odd}^{l,2+i}z_{c}^{i,5+i}z_{c}^{j,5+i}\big ) \nonumber \\ & & + \frac{l_{i}^{2}}{2}z_{odd}^{k,1}z_{odd}^{l,1}z_{c}^{i,5+i}z_{c}^{j,5+i}\bigg\ } \nonumber \\ { \cal a}_{eoc}^{klij } & = & \sum\limits_{i=1}^{3}\bigg[\big(\frac{g^{2}}{8 } + \frac{1}{2}l_{i}^{2}\big)\big(z_{even}^{k,2+i}z_{odd}^{l,1 } + z_{even}^{k,1}z_{odd}^{l,2+i}\big ) \big(z_{c}^{i,1}z_{c}^{j,2+i } - z_{c}^{i,2+i}z_{c}^{j,1}\big ) \nonumber \\ & & -\frac{g^{2}}{8}\big(z_{even}^{k,2}z_{odd}^{l,2+i } + z_{even}^{k,2+i}z_{odd}^{l,2}\big ) \big(z_{c}^{i,2+i}z_{c}^{j,2 } - z_{c}^{i,2}z_{c}^{j,2+i}\big)\bigg ] \nonumber \\ & & -\frac{g^{2}}{8}\big(z_{even}^{k,1}z_{odd}^{l,2 } + z_{even}^{k,2}z_{odd}^{l,1}\big ) \big(z_{c}^{i,1}z_{c}^{j,2 } - z_{c}^{i,2}z_{c}^{j,2}\big ) \nonumber \\ { \cal a}_{cc}^{ijkl } & = & \frac{g^{2 } + g^{\prime^{2}}}{8}\bigg [ \sum\limits_{m , n=1}^{5}z_{c}^{i , m}z_{c}^{j , m}z_{c}^{k , n}z_{c}^{l , n } + 2\sum\limits_{i=1}^{3}z_{c}^{i,2+i}z_{c}^{j,2+i}\big(z_{c}^{k,1}z_{c}^{l,1 } - z_{c}^{k,2}z_{c}^{l,2}\big ) \nonumber \\ & & -2z_{c}^{i,1}z_{c}^{j,1}z_{c}^{k,2}z_{c}^{l,2 } \bigg ] + \frac{g^{\prime^{2}}}{2}\bigg[\sum\limits_{i=1}^{3}z_{c}^{i,5+i}z_{c}^{j,5+i}\big ( - z_{c}^{k,5+i}z_{c}^{l,5+i } - z_{c}^{k,2+i}z_{c}^{l,2+i}\big ) \nonumber \\ & & -\sum\limits_{i=1}^{3}z_{c}^{i,5+i}z_{c}^{j,5+i}z_{c}^{k,1}z_{c}^{l,1}\bigg ] + \sum\limits_{i=1}^{3}l_{i}^{2}z_{c}^{i,5+i}z_{c}^{j,5+i}z_{c}^{k,1}z_{c}^{l,1 } \nonumber \end{aligned}\ ] ] where the mixing matrices @xmath214 , @xmath215 and @xmath216 are defined as in eq . 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the minimum supersymmetric standard model with bilinear r - parity violation is studied systematically . considering low - energy supersymmetry , we examine the structure of the bilinear r - parity violating model carefully . we analyze the mixing such as higgs bosons with sleptons , neutralinos with neutrinos and charginos with charged leptons in the model . possible and some important physics results such as the lightest higgs may heavy than the weak z - boson at tree level etc are obtained . the feynman rules for the model are derived in @xmath0 t hooft- feynman gauge , which is convenient if perturbative calculations are needed beyond the tree level . -5 mm

the results presented here are extracted from data collected at in the @xmath32 , data taking periods . the fixed - target experiment ran for more than @xmath33 years until @xmath34 at the electron - positron storage ring of hera at desy . the spectrometer @xcite was a forward - angle instrument consisting of two symmetric ( top , bottom ) halves above and below the horizontal plane defined by the lepton beam pipe . it was characterized by very high efficiency ( about @xmath35 ) in electron - hadron separation , provided by a transition radiation detector , a preshower scintillation counter and an electromagnetic calorimeter . in addition , a dual - radiator ring - imaging cherenkov ( rich ) detector provided hadron identification for momenta above 2 gev / c . in order to study the new structure functions @xmath10 and @xmath11 defined in eq . ( [ eq : noncol_csec ] ) , a measure of the azimuthal modulation of the unpolarized cross section is needed , which can be extracted via the so - called @xmath36-moments : @xmath37 with @xmath38 and @xmath39 defined in equation [ eq : noncol_csec ] . the extraction of these cosine moments from data is challenging because they couple to a number of _ experimental sources _ of azimuthal modulations , _ e.g. _ detector geometrical acceptance and higher - order qed effects ( _ radiative effects _ ) . moreover , in the typical case , the event sample is binned only in one variable ( @xmath40-dimensional analysis ) , and integrated over the full range of all the other ones , but the mentioned structure functions and the instrumental spurious contributions depend on all the kinematic variables @xmath26 , @xmath25 , @xmath27 and @xmath8 simultaneously . therefore a @xmath41-dimensional analysis is needed to take into account the correlations between the physical modulations and those spurious contributions , where the event sample is binned simultaneously in all the relevant variables . therefore , a detailed monte carlo simulation of the experimental apparatus including radiative effects is used to define a @xmath41-d unfolding procedure @xcite that corrects the extracted cosine moments for radiative and instrumenthal effects . the @xmath41-d unfolded yields are fit to the functional form : @xmath42 where @xmath43 and @xmath44 represent the desired moments . one moment pair ( @xmath45 , @xmath46 ) for each of the @xmath41-d kinematic bins is extracted , and the moment dependences on a single kinematic variable is obtained projecting the @xmath41-d results onto the variable under study by weighting the moment in each bin with the corresponding @xmath47 cross section obtained from a monte carlo calculation -d unfolding and extraction procedure as well as on the projection versus the single variable can be found in @xcite . ] . moments for positive ( upper panel ) and negative ( lower panel ) hadrons , extracted from hydrogen data projected versus the kinematic variables @xmath26 , @xmath25 , @xmath27 and @xmath8.,title="fig : " ] moments for positive ( upper panel ) and negative ( lower panel ) hadrons , extracted from hydrogen data projected versus the kinematic variables @xmath26 , @xmath25 , @xmath27 and @xmath8.,title="fig : " ] moments for positive ( upper panel ) and negative ( lower panel ) hadrons , extracted from hydrogen data projected versus the kinematic variables @xmath26 , @xmath25 , @xmath27 and @xmath8.,title="fig : " ] moments for positive ( upper panel ) and negative ( lower panel ) hadrons , extracted from hydrogen data projected versus the kinematic variables @xmath26 , @xmath25 , @xmath27 and @xmath8.,title="fig : " ] the cross section unintegrated over hadron transverse momentum gives access to new exciting aspects of the nucleon structure , which are currently under intense theoretical investigations . the cross section unintegrated over hadron transverse momentum gives access to new exciting aspects of the nucleon structure , which are currently under intense theoretical investigations . however , as the extraction of unpolarized cosine moments is experimentally challenging , very few measurements have been performed to date , and , with exception of compass @xcite , most of them average out any possible flavor dependence @xcite . to date , this analysis at represents the most complete data set on the subject , and allows access to flavor dependent information on the nucleon internal transverse degrees of freedom . pions moments projected in the relevant kinematic variables are shown in figure [ fig : pions ] for @xmath48 ( upper panel ) and @xmath49 ( middle panel ) moments . the @xmath48 moments are found to be negative for both charged pions , but larger in magnitude for positive ones , while the @xmath49 moments show opposite sign for positive and negative pions : both modulations are clearly charge dependent , and this feature is considered as an evidence of a non - zero boer - mulders effect @xcite . results for kaons are shown in figure [ fig : cosk ] for @xmath48 moments and figure [ fig : cos2k ] for @xmath49 moments . the upper panels show results for positive kaons ( stars ) compared to positive pions ( squares ) and unidentified hadrons ( circles ) results , the lower panels the same comparison for negatively charged kaons , pions and hadrons . due to the poorer statistics of kaon event samples , the kaon results have been extracted in a reduced kinematic region with respect to the pion moments discussed above resulting in a smaller number of bins shown in the pictures . the positive kaon @xmath48 moments ( figure [ fig : cosk ] ) are found to be negative and larger in magnitude than @xmath48 moments extracted for pions , while negative kaons behave similarly to negative pions , showing results compatible with zero . the absolute value of kaon @xmath1 modulations are found to be larger in magnitude than pions ones . while pion @xmath1 modulations change sign between differently charged pions , kaon s modulations are negative for both kaon charges . in general the hadrons have a similar trend as the pions but , particularly for the @xmath49 moments , the hadrons are shifted to lower values than the pions . the discrepancy between hadrons and pions is consistent with the observed kaon moments . + the cosine modulations have been extracted also for data collected with deuterium target , and they are found to be compatible with hydrogen results , both for unidentified hadrons , pions and kaons . this suggests that similar contributions arise from @xmath50 and @xmath51 quarks to the cosine modulations . 9 a. bacchetta and others , _ jhep _ 02:093 , 2007 . r. n. cahn , _ phys . _ b78:269 , 1978 . r. n. cahn , _ phys . _ d40:3107 , 1989 . d. boer and p. j. mulders , _ phys _ d57:5780 , 1998 . k. ackerstaff and others , collaboration , _ nucl . instrum . meth . _ a417:230 , 1998 . f. giordano , collaboration , proceedings of second international workshop on transverse polarisation phenomena in hard processes ( transversity 2008 ) ferrara , italy , may 28 - 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the @xmath0 and @xmath1 azimuthal modulations of the unpolarized hadron semi - inclusive deep inelastic scattering cross section are sensitive to the quark intrinsic transverse momentum and transverse spin . these modulations have been measured at in a fully differential way by means of a 4-dimensional unfolding procedure to correct for instrumental effects . results have been extracted for hydrogen and deuterium targets and separately for positively and negatively charged pions and kaons , to access flavor - dependent information about the nucleon internal transverse degrees of freedom . in lepton - nucleon deep - inelastic scattering ( dis ) , the structure of the nucleon is probed by the interaction of a high energy lepton with a target nucleon , via , at kinematics , the exchange of one virtual photon . if at least one of the produced hadrons is detected in coincidence with the scattered lepton , the reaction is called semi - inclusive deep - inelastic scattering ( sidis ) : @xmath2 where @xmath3 ( @xmath4 ) is the incident ( scattered ) lepton , @xmath5 is the target nucleon , @xmath6 is a detected hadron , @xmath7 is the target remnant and the quantities in parentheses in equation ( [ eq : one ] ) are the corresponding four - momenta . if unintegrated over the hadron momentum component transverse to the virtual photon direction @xmath8 ( fig . [ fig : evento ] ) , the cross section can be written as @xcite : @xmath9 where @xmath10 , @xmath11 , are azimuthally dependent structure functions , and are related respectively to @xmath12 and @xmath13 modulations , with @xmath14 the azimuthal angle of the hadron production plane around the virtual - photon direction ( fig . [ fig : evento ] ) . in equation [ eq : noncol_csec ] , the subscripts @xmath15 stand for unpolarized beam and target , @xmath16 ( @xmath17 ) indicates the transverse ( longitudinal ) polarization of the virtual photon , @xmath18 is the electromagnetic coupling constant , @xmath19 with @xmath20 the target mass , @xmath21 , @xmath22 , and @xmath23 . here @xmath24 and @xmath25 are respectively the negative squared four - momentum and the fractional energy of the virtual photon , @xmath26 the bjorken scaling variable and @xmath27 the fractional energy of the produced hadron . + between scattering plane , spanned by the in- and out - going lepton three - momenta ( @xmath28 , @xmath29 ) , and the hadron production plane , defined by the three - momenta of the virtual photon ( @xmath30 ) and produced hadron ( @xmath31 ) . ] among possible mechanisms , two are expected to give important contributions to the azimuthal dependence of the unpolarized cross section in the hadron transverse momentum range accessible at . the first one is called the _ cahn effect _ @xcite , a pure kinematic effect where the azimuthal modulations are generated by the non - zero intrinsic transverse motion of quarks . in the second mechanism , the _ boer - mulders effect _ @xcite , @xmath12 and @xmath13 modulations originate from the coupling of the quark intrinsic transverse momentum and intrinsic transverse spin , a kind of spin - orbit effect .

two of the most constraining observed properties of active galactic nuclei ( agn ) are their large luminosities over a broad range of energies ( @xmath1-ray through infrared ) and their rapid variability ( implying a small source size unless the emission is highly beamed ) . the inferred large energy densities have led to a standard model of the ultimate energy source being the release of gravitational potential energy of matter from an accretion disk surrounding a supermassive black hole ( e.g. , rees 1984 ) . although this general model has broad support , the specific physical processes that produce the complex , broadband spectral energy distributions ( seds ) observed from agn have not been clearly identified . it is believed that a mix of processes is important . the ultraviolet and optical emission may be primary radiation from an accretion disk ( shields 1978 ; malkan & sargent 1982 ; malkan 1983 ) . in low - luminosity objects starlight will contribute as well . thermal dust emission is an important ingredient of the infrared band ( barvainis 1987 ; sanders et al . the high - energy ( x - ray and @xmath1-ray)emission is not well understood . there are a variety of models for their origin ranging from electromagnetic cascades in an @xmath7 pair plasma ( zdziarski et al . 1990 ) to thermal comptonization models ( haardt & maraschi 1993 ; haardt , maraschi , & ghisellini 1994 ) . furthermore , gas near the central source may reprocess at least some of the primary radiation via compton scattering , absorption , and fluorescent processes ( guilbert & rees 1988 ; lightman & white 1988 ; george & fabian 1991 ; matt , fabian , & ross 1993 ) . determination of the mix of physical processes that produce these large , broadband luminosities is a major unresolved issue in agn research , and multi - waveband variability studies are potentially highly constraining . causality arguments imply that if emission in a secondary " band is produced when photons from a primary " band are reprocessed in material near the central engine , then variations in the secondary band could not be seen to lead those in the primary band . furthermore , if the emission in any given waveband is a combination of two independent components ( with presumably independent variability behavior ) , then measurement of broadband spectral variability might allow them to be separated . finally , if a characteristic variability time scale could be measured , it could be compared with those indicative of different physical processes ( e.g. , with the expected viscous , orbital , light - travel time scales ) . in spite of the potential power of this approach , it has not until recently been exploited because of the very large amount of telescope time required . in several experiments designed to measure the size of the broad - line region in the seyfert 1 galaxy ngc 5548 , variations at @xmath01400 were seen to track those at @xmath02800 and @xmath05000 to within @xmath212 day ( clavel et al . 1991 ; peterson et al . 1991 ; korista et al . this was taken to imply an ultraviolet - optical propagation time that is too short to be associated with any dynamics mediated by viscosity , such as variations in the mass inflow rate , in a standard @xmath8-disk ( krolik et al . 1991 ; but see 4.2 . similar problems were noted in ultraviolet and optical monitoring of ngc 4151 by ulrich et al . krolik et al . ( 1991 ) suggested that variation in the different wavebands in ngc 5548 were coordinated by a photon signal . nandra et al . ( 1992 ) suggested that this signal might be x - ray heating ( reprocessing ) . several authors constructed specific models of x - ray illuminated accretion disks to account for the ngc 5548 data ( collin - souffrin 1991 ; rokaki & magnan 1992 ; molendi , maraschi , & stella 1992 ; rokaki , collin - souffrin , & magnan 1993 ) as well as for ngc 4151 ( perola & piro 1994 ) . a strong test of the idea that the ultraviolet is produced by reprocessing x - ray photons could be made by measuring the time relationship between fluctuations in the ultraviolet and the x - rays , but previous attempts ( e.g. , clavel et al . 1992 ) lacked adequate temporal resolution . in order to attempt this test , an international consortium of agn observers undertook a campaign to intensively monitor a single seyfert 1 galaxy , ngc 4151 , at ultraviolet , x - ray , @xmath1-ray , and optical wavelengths for @xmath010 days in december 1993 . these data are described in detail in the three preceding papers ( paper i crenshaw et al . 1996 ; paper ii kaspi et al . 1996 ; and paper iii warwick et al . 1996 ) ; they are summarized in the following section . in this paper , the multi - wavelength data are analyzed in combination . the measurement of the multi - waveband variability , temporal correlations , phase lags , and the broadband optical through@xmath1-ray sed are analyzed in 3 and the scientific implications are briefly discussed in 4 . the aim of these observations was to monitor intensively ngc 4151 across the accessible optical - through-@xmath1-ray region during the period from mjd 22.5 to 32.2 . ( the modified julian date , mjd , is defined as mjd = jd 2,449,300 . all dates refer to the center points of observations unless otherwise noted . ) while scheduling difficulties , satellite malfunctions , and other minor problems did cause some gaps and perturbations in this plan , this campaign still produced the most intensive coordinated observations to date of an agn across these high - energy wavebands . along with daily observations before and after this period ( not analyzed in this paper ) , the _ international ultraviolet explorer _ ( _ iue _ ) observed ngc 4151 nearly continuously from mjd 22.6 to 31.9 . a pair of swp and lwp spectra , spanning the range 12002000 and 20003000 , respectively , was obtained every @xmath070 min , excluding a daily @xmath02 hr gap at about modulo mjd 0.40.5 . _ rosat _ observed the source twice per day , with nearly even sampling from mjd 22.5 to 28.0 . a satellite malfunction caused _ rosat _ to go into safe mode for most of the second half of the campaign , although it did make one final observation at mjd 32.2 . _ asca _ was also plagued by satellite problems ( this time in the first half of the campaign ) but four 10 ksec observations were successfully obtained at mjd 26.0 , 27.5 , 29.1 , and 31.6 . compton gamma - ray observatory ( _ cgro _ ) observations were made with the osse instrument during the period mjd 22.734.6 . ground - based optical spectra were obtained for a 2 month period that included these dates . figure 1 gives the light curves in nominal 100 kev , 1.5 kev , 1275 , 1820 , 2688 and 5125 wavebands . these bands are explicitly defined below , and further details of the sampling characteristics are given in table 1 . a total of 18 optical spectra were obtained during the intensive monitoring period ; 10 at perkins observatory ( osu ) and eight at wise observatory . the wise data covered 42106990 with @xmath05 resolution through a 10@xmath9 slit , while the osu data covered 44805660 at @xmath09 resolution through a 5@xmath9 slit . these data were intercalibrated to remove the effects of ( presumably non - variable ) extended emission and instrumental offsets from the light curves . continuum fluxes in the 4600@xmath1040 and 5125@xmath1025 bands and h@xmath11 line fluxes were measured from both sets of spectra , while 6200@xmath1030 and 6925@xmath1025 continuum fluxes and h@xmath8 line fluxes were measured only from the wise data . the wise spectrograph projects to @xmath12 on the sky , and approximately 25% of the light at a wavelength of 5125 is due to starlight from the host galaxy ( see 3.5 . ) . the uncertainties in measurements of the optical fluxes are at about the 1% level ( see paper ii for details ) . the 5125 data from both telescopes were combined in the correlation analyses and light curves . during the intensive monitoring period , _ iue _ obtained a total of 176 spectra with the swp camera and 168 with the lwp camera . observations were spaced as closely as possible , leading to fairly even sampling of @xmath018 spectra per day in each camera , with a @xmath02 hour period each day , during which no data were obtained because of earth occultation and high particle background . one - dimensional spectra were extracted using the tomsips package ( ayres 1993 ) . continuum fluxes were measured by summing over relatively line - free , @xmath030 wide bands centered at 1275 , 1330 , 1440 , 1820 , 1950 , 2300 , and 2688 . uncertainties , taken to be the standard error in the band , were typically 12% , consistent with the observed epoch - to - epoch dispersion in the measured fluxes . line fluxes ( c iv , he ii , and c iii ] ) were measured by fitting multiple gaussians . ( see paper i and penton et al . 1996 for details of the _ iue _ data reduction . ) the continuum light curves measured at 1275 , 1820 , 2688 , and 5125 were used in the light curves and time - series analyses , all of the ultraviolet / optical continuum data were used only in the variability amplitude and zero lag correlation analyses , and the emission line data were not used in this paper . the _ rosat _ pspc made a total of 13 observations of ngc 4151 between mjd 22.5 and mjd 28.0 . although the center points of the first 12 observations were almost evenly spaced ( every 0.5 day ) , the integration times varied from 0.8 ksec to 6.4 ksec . the two soft ( 0.10.4 kev and 0.50.9 kev ) bands , which showed no significant variability during the observations , are apparently dominated by emission from an extended component ( e.g. , elvis et al . 1983 ; morse et al . . however , the hardest _ rosat _ band ( 12 kev ) showed variations significantly in excess of the measured errors . the _ rosat _ spectra were well - fitted by a non - variable thermal bremsstrahlung ( @xmath13 kev ) component that contributes most of the flux below 1.4 kev and a heavily absorbed ( @xmath14 @xmath15 ) power - law with fixed slope ( @xmath16 ) and variable normalization ( @xmath17 ph @xmath15 s@xmath18 kev@xmath18 ) that dominates at medium x - ray energies . however , the proportional counter data have low resolution and thus a range of other models can not be ruled out . the four _ asca _ spectra have higher resolution , allowing more detailed spectral analysis . these data were compatible with a model that includes warm and cold absorbers , thermal bremsstrahlung , a power - law and a 6.4 kev iron line . again , this is not a unique solution . the iron line shows significant broadening to the redward wing , suggesting a gravitational redshift that would constrain the material to lie very near the black hole ( yaqoob et al . the 12 kev _ asca _ data were used in the light curves along with the _ rosat _ points , with a 10% uncertainty added to account for possible calibration differences . the harder ( 210 kev ) _ asca _ data were not used for variability analysis because only four epochs were obtained . ngc 4151 was continuously observed with the osse instrument on _ cgro_. the broadband 50150 kev count rate showed weak but significant variability . the raw data were calibrated by convolving the detector response function with a model assuming a power - law with an exponential cutoff . see paper iii for further details on the x - ray and @xmath1-ray data . ngc 4151 was near its historical peak brightness during this campaign . the average ( absorption - corrected ) flux in the 2 - 10 kev band was @xmath19 . this is bright compared to the compilation of results from _ exosat _ and _ ginga _ reported in yaqoob et al . ( 1993 ) , which ranged from 0.8 to @xmath20 . the mean 1275 flux of @xmath21 erg @xmath15 s@xmath18 @xmath18 is higher than the range of @xmath22 erg cm@xmath18 s@xmath18 @xmath18 seen in 19781990 ( edelson , krolik , & pike 1990 ; courvosier & palatini 1992 ) , although the ultraviolet flux was a bit higher during the march 1995 astro-2 campaign ( kriss et al . during the period mjd 22.532.3 , the light curve of ngc 4151 was densely sampled by _ iue _ , _ rosat _ , _ cgro _ , and ground - based telescopes . the initial analysis of these data involved comparing fractional variability amplitudes during this period as a function of observing frequency . the normalized variability amplitude ( nva , or @xmath24 ) was computed as follows . for each band , the mean ( @xmath25 ) and standard deviation ( @xmath26 ) of the flux points and the mean error level ( @xmath27 ) were measured . because the nva is intended to be free of instrumental effects , it was determined by subtracting in quadrature the measured mean error from the standard deviation , and then dividing by the mean flux . that is , @xmath28 note that this is essentially the same procedure used to derive @xmath24 in peterson et al . ( 1991 ) and @xmath29 in edelson ( 1992 ) . these quantities are given in table 2 , along with the observing band , number of observations ( @xmath30 ) and the difference in days between the first and last observation ( @xmath31 ) during this intensive period . figure 2 is a plot of nva as a function of observing waveband . the nva shows a strong dependence on photon energy , increasing from @xmath32% in the optical to @xmath04% in the lwp and @xmath33% in the swp . the variability in the swp is clearly stronger than at longer wavelengths , but because of the lower variability levels and stronger starlight corrections , it is impossible to say if the optical and lwp have significantly different non - stellar nvas . " this strong wavelength dependence has been seen previously in ngc 4151 and other agn ( e.g. , edelson , krolik & pike 1991 ) , and is suggestive of the superposition of a non - variable soft component that dominates in the optical / infrared and a variable hard component that dominates in the ultraviolet . the nva also shows interesting behavior at high energies . the soft _ rosat _ bands show no evidence for any significant variations , with nva formally undefined as the observed variation levels are smaller than the instrumental errors . the medium energy 1.5 kev x - ray band shows the strongest variability seen in any band , with @xmath34% . this large difference is apparently due to the superposition of two components : a soft , extended ( and therefore non - variable ) component seen in spectra and hri images ( elvis et al . 1983 ) , and a harder , strongly variable component . however , this strongly variable component must cut off at some higher energies , because the _ cgro _ data show weaker variability , with @xmath35% in the 50150 kev band . unfortunately , there are two important energy ranges in which the variability properties of ngc 4151 during this campaign were not measured : the hard x - ray 250 kev gap between _ rosat _ and _ cgro _ ( the 210 kev _ asca _ data can not be used to measure a meaningful nva with only four points ) and the extreme ultraviolet 1200 ( 10 ev ) to 100 ev gap in the mostly unobservable region between _ iue _ and _ rosat_. the former waveband could contain the bulk of the luminosity of the putative primary emission component , and the latter , the bulk of the disk luminosity . these gaps do limit the power of these data ( see 4 ) , although we note that this campaign has produced the most densely sampled grid in time and energy band obtained of any agn to date . table 2 also gives the monochromatic variable luminosity , defined as @xmath36 . ( @xmath37 is the observed monochromatic luminosity , as defined in 3.5 . ) this parameter behaves somewhat differently than the nva . in particular , the medium energy x - rays , which have the strongest nva ( @xmath38% ) actually show the lowest value of @xmath39 ( @xmath40 erg s@xmath18 ) . note however that the x - rays appear to be heavily absorbed ( @xmath09095% , see 3.5 . ) , so the intrinsic value of this quantity is actually much larger . these are the most densely sampled ultraviolet observations of any agn obtained to date , allowing examination of the short time scale variability properties . these data were used to measure the fluctuation power density spectra ( pds ) in four ultraviolet bands ( 1275 , 1330 , 1820 , and 2688 ) , on time scales of @xmath00.25 day . a regular grid of spacing 0.1 day was created , and the value at each point was taken to be the average of the nearest 12 points , weighted as @xmath41 . this resampling , which was necessary to mitigate problems introduced by the periodic @xmath00.1 day earth occultation gaps , destroys all information on shorter time scales . however , as seen below , the pds are dominated by noise on these shorter time scales , so this is not a problem . only the continuous data were used , and the mean was not subtracted . these resampled data were used to compute the pds in the four ultraviolet bands , the results of which are shown in figure 3 . the pds of agn show no discrete features that would indicate periodicity . instead , the fluctuation power is spread out over a wide range of temporal frequencies . it is common practice to paramterize the pds of agn by the function @xmath42 , where @xmath43 is the pds at temporal frequency @xmath44 , and @xmath45 is the power - law slope . these pds have @xmath46 , making them the reddest " agn pds yet measured . by comparison , the ultraviolet pds of ngc 5548 had a slope @xmath47 on time scales of weeks to months ( krolik et al . this indicates that the variability power is falling off rapidly at short time scales , and the bulk of the variability power is on time scales of days or longer . the x - ray pds ngc 4151 has @xmath47 time scales of hours to days and @xmath48 on longer time scales ( papadakis & mchardy 1995 ) . the fact that the ( rather noisy ) x - ray and ultraviolet pds appear to have different slopes over the same range of time scales may indicate that different processes power the variability at the two wavebands , although systematic differences in the data and reduction techniques for the two data sets make direct comparison difficult . the four pds in figure 3 look similar to within the noise . there do however appear to be some qualitative differences in the character of variations in different bands that can be discerned by direct examination of the light curves . in particular , the ultraviolet data give the impression that the most rapid variations occur at the shortest wavelengths , and are somewhat smeared out " to longer wavelengths . ( the @xmath00.2 day spikes " in the 2688 light curve may be due to instrumental effects . ) a similar effect has been seen at longer time scales in _ hst _ monitoring of ngc 5548 ( korista et al . such an effect would not be apparent in the above pds analysis . however , it is unclear how much of this is intrinsic and how much is due to instrumental differences , since the variability - to - noise " ratio ( that is , @xmath49 ) , decreases toward longer wavelengths in the ultraviolet , so that this experiment was less sensitive to rapid variations at longer ultraviolet wavelengths . because there was no measurable interband phase lag ( see 3.4 ) , a slightly different view may be obtained by measuring the zero lag correlations between variations in different wavebands . to investigate this , fluxes in a number of comparison bands ( 100 kev , 1.5 kev , 1330 , 1820 , 2688 , and 5125 ) were correlated with the 1275 fluxes . for each of the comparison bands , each flux point was paired with the average of the two 1275 flux points measured immediately before and after that point , and 1275 uncertainties were taken to be the average of the uncertainties measured for the two data points . the results are plotted in figure 4 . the number of points ( @xmath30 ) , linear correlation coefficient ( @xmath50 ) , spearman rank correlation coefficient ( @xmath51 ) , @xmath52-statistic , and y - intercept are tabulated in table 3 . the 100 kev fluxes ( @xmath53 ) show no correlation with 1275 ( or with any other band for that matter ) . this could suggest that the processes producing the @xmath1-rays are unrelated to those at work at lower energies , especially given the relatively low nva of the @xmath1-rays . the data are strongly correlated on these short time scales in all other cases , although the optical and x - ray correlations are a bit marginal , with @xmath54 for 18 points and @xmath55 for 17 points , respectively . note that perola et al . ( 1986 ) found a similar ultraviolet / x - ray zero lag correlation in previous data over longer ( @xmath01 year ) time scales , although it was claimed that corrrelation broke down at the high flux levels seen in this experiment . following the regression line to zero flux at 1275 yields a positive excess at all wavebands except 1.5 kev x - rays . this means that if one would model the flux at each waveband as a combination of a non - variable component and a variable component produced by reprocessing emission from the other waveband , the variable component is always at shorter wavelengths , and the non - variable component becomes a progressively larger fraction of the total flux as the wavelength increases . the most detailed analysis undertaken with these data was to measure temporal cross - correlation functions between wavebands . if emission in one band is reprocessed to another without feedback , measurement of an inter - band lag would provide an important confirmation , and indicate which were the primary and secondary bands . this intensive monitoring of ngc 4151 is clearly better suited for this test than any previous campaign . unfortunately , even in these data , the 1.5 kev x - rays , which show the strongest variations , are clearly undersampled , as they were observed at a temporal frequency only one - tenth that in the ultraviolet . the differences between the sampling , variability levels , and signal - to - noise ratios in the different wavebands present problems for measuring the interband lag . below , a detailed analysis is presented of two independent techniques used to measure the interband correlations and lags : the interpolation and discrete correlation functions . cross - correlation functions were computed using the interpolation cross - correlation function ( iccf ) method of gaskell & sparke ( 1986 ) , as subsequently modified by white & peterson ( 1994 ) . as described by gaskell & peterson ( 1987 ) , the cross - correlation function was determined by first cross - correlating the real observations from one time series with values interpolated from the second series . the calculation was then performed a second time , using the real values from the second series and interpolated values from the first series . the final cross - correlation function was then taken to be the mean of these two calculated functions . a sampling grid spacing of 0.05 day was used because that is approximately the mean interval between the ultraviolet observations . because the x - ray and optical continuum light curves are much less well - sampled than the ultraviolet light curves , the x - ray and optical cross - correlations with the 1275 ultraviolet light curve were performed by interpolating only in the ultraviolet light curve . that is , the computed cross - correlation functions are based on the real x - ray and optical points and interpolated ( or regularized ) ultraviolet points , and no interpolation of the x - ray or optical data was performed . the cross - correlation functions for the various continuum bands were similarly calculated by the @xmath56-transformed discrete correlation function ( zdcf ) method ( alexander 1996 ) , which is related to the discrete correlation function ( dcf ) of edelson & krolik ( 1988 ) . this method is rather more general than the iccf as ( a ) it does not require any assumptions about the continuum behavior between the actual observations , and ( b ) it is possible to reject data pairs at zero lag ( e.g. , two points in different wavebands , measured from one spectrum ) , and thus avoid false zero lag correlations that might arise from correlated flux errors . the zdcf differs in a number of ways from the dcf , a notable feature being that the data are binned by equal population rather than into time bins of equal width @xmath57 . one advantage of this technique is that it allows direct estimation of the uncertainties on the lag without the use of more assumption - dependent monte carlo simulations . the maximum likelihood error estimate on the position of the true cross - correlation function peak ( @xmath58 ) is a 68% fiducial confidence interval , meaning that it contains the peaks of 68% of the likelihood - weighted population of all possible cross - correlation functions ( alexander 1996 ) . phrased more loosely , the confidence interval is where 68% of the cross - correlation functions that are consistent with the zdcf points are likely to reach their peaks . the accuracy of this error estimate is limited by the assumption that the distributions of the true cross - correlation points around the zdcf points are independently gaussian , which is only approximately true , but it does not require any further _ a priori _ assumptions about the nature of the variations or transfer function between bands . the various columns in table 4 give the results obtained by cross - correlating the specified light curve with the 1275 light curve . the parameter @xmath59 is the maximum value of the cross - correlation function , which occurs at a lag @xmath60 . table 4 also gives the centroid of the cross - correlation function @xmath61 , which is based on all points with @xmath62 . the computed cross - correlation functions are shown in figure 5 ( iccf ) and figure 6 ( zdcf ) . the measured range of @xmath63 of the various bands with 1275 ( 0.20 to + 0.26 day for the 1.5 kev x - rays , 0.14 to + 0.01 day for 1820 , 0.05 to + 0.22 day for 2688 , 0.78 to 0.53 day for 5125 ) are all consistent with zero measurable lag . thus , we have assigned conservative upper limits on the lags of 1275 with 1.5 kev of @xmath64 day , with other ultraviolet bands of @xmath65 day , and with 5125 of @xmath66 day . the physical significance of these limits is discussed in 4 . figure 7 gives a broadband @xmath1-ray optical snapshot sed , constructed by combining nearly - simultaneous observations consisting of the wise spectrum taken on mjd 25.6 , _ spectra lwp 26907 and swp 49441 , taken near mjd 26.0 , the _ rosat _ observation centered at mjd 26.0 , and _ asca _ observation number 1 , centered near mjd 26.0 . because of the weak variability and low signal to noise in the @xmath1-ray band , the _ cgro _ spectrum was integrated over the entire campaign to produce the high - energy sed . additional ( non - simultaneous ) infrared data from edelson , malkan & rieke ( 1987 ) were used in the spectral fits ( 4.2 . ) but are not shown in figure 7 . the data plotted are monochromatic luminosities , @xmath37 , as a function of frequency , @xmath67 . the quantity @xmath37 is given by @xmath68 where @xmath69 is the flux density and @xmath70 mpc is assumed to be the distance to ngc 4151 ( tully 1989 ) . the sed of any seyfert 1 galaxy is of course a combination of a number of components , and in particular , we note that starlight from the underlying sab galaxy of ngc 4151 contributes a significant amount of the optical / infrared flux . peterson et al . ( 1995 ) estimated that the underlying galaxy produced a flux of 19 mjy at 5125 in the @xmath71 wise aperture , in good agreement with the independent result reached in paper ii of this series . this corresponds to approximately 25% of the 5125 flux at the light levels seen in this campaign , so the 5125 nva of the nuclear component ( excluding starlight ) is approximately 1.33 times the observed value quoted in table 2 . for a normal sab galaxy , approximately 42% of the light at 6925 , 31% of the light at 6200 , 21% of the light at 4600 , and 4% of the light at 2688 would be due to starlight . finally , we note one other important feature in the sed : the soft x - rays show significant absorption due to line - of - sight gas intrinsic to ngc 4151 ( evident below 4 kev ) , and in our own galaxy ( evident below 0.3 kev ) . spectral fits indicate that approximately 9095% of the 12 kev luminosity appears to have been absorbed by gas along the line of sight ( see paper iii for details ) , although this result is model - dependent . this would imply that the intrinsic 12 kev luminosity is 1020 times that observed . possible sources of the intrinsic absorption include partial covering by cold gas ( e.g. , holt et al . 1980 ; yaqoob , warwick & pounds 1989 ; yaqoob et al . 1993 ) or absorption in a warm , ionized medium ( weaver et al . 1994a , b ; warwick , done & smith 1996 ) . similar behavior has been seen in the sed of ngc 3783 obtained during the world astronomy day " campaign ( alloin et al . 1995 ) . a smaller fraction ( @xmath035% ) of the 210 kev flux is apparently absorbed , but unfortunately , the data are inadequate to characterize the variability in this band . ngc 4151 was monitored intensively with _ iue _ , _ rosat _ , _ asca _ , _ cgro _ and ground - based optical telescopes for @xmath010 days in december 1993 . these observations provided the most intensively sampled multi - wavelength light curve of an agn to date . the major new observational results are as follows : 1 . ngc 4151 showed strong variability from the optical through @xmath1-ray bands during this period . the optical / ultraviolet / x - ray light curves are similar but not identical , with no detectable lags between variations in the different bands . the upper limits on the lags are , between 1275 and 1.5 kev , @xmath20.3 day ; between 1275 and 5125 , @xmath21 day ; and between 1275 and the other ultraviolet bands , @xmath20.15 day . the 100 kev variations are not clearly related to those in any other band . the strongest variability was seen in the medium energy x - rays , with the 12 kev band showing an nva of 24% . the variations were systematically weaker at lower energies , with nvas of 9% , 5% , 4% and 0.5% at 1275 , 1820 , 2688 and 5125 , respectively . however , the 100 kev light curve showed an nva of only 6% , and the soft ( @xmath721 kev ) x - rays showed no detectable variability . the observed luminosity ( as opposed to fractional ) variability was seen to be lowest in the 1.5 kev band , apparently due to strong x - ray absorption along the line of sight . the pds of all four ultraviolet bands are similar , showing no evidence for periodicity , but is instead being well modeled as a power - law with @xmath73 . the broadband sed shows a strong deficit in the soft / mid - x - ray , which is well - fitted as a factor of @xmath015 absorption by gas on the line - of - sight . finally , the character of the multi - wavelength variability in the seyfert 1 ngc 4151 appears to differ markedly from the only other agn monitored with similar intensity . the bl lac object pks 2155304 was observed continually for @xmath03 days at ultraviolet ( urry et al . 1993 ) , x - ray ( brinkmann et al . 1994 ) , and optical ( courvoisier et al . 1995 ) wavebands . multi - wavelength analysis by edelson et al . ( 1995 ) found that the x - ray , ultraviolet and optical variations were almost identical in amplitude and shape , but that the x - ray variations led those in the ultraviolet and optical by @xmath023 hr . for these observations of ngc 4151 , and other seyfert 1s observed over longer time scales ( e.g. , clavel et al . 1991 ) , the light curves do show significant differences between wavebands , with nva a strong function of energy , but no lag has been measured between variations at different bands . this is an observed ( and therefore model - independent ) example of an intrinsic difference between seyfert 1s and bl lacs . these results have important implications for models that attempt to explain the ultraviolet emission from agn . there are currently two broad classes of such models . the first hypothesizes that the bulk of the ultraviolet luminosity is produced internally by viscosity in the inner regions of an accretion disk surrounding a central black hole , and the second , that the observed ultraviolet emission is produced in gas illuminated and heated by the source that we observe at high energies . of course , the true picture could be a combination of these processes , a hybrid in which both intrinsic emission from an accretion disk and reprocessing of x - ray emission are important ( and , indeed , may feed back upon each other ) , or conversely , it may be that neither of these models is relevant . although the specific processes responsible for agn emission have not been clearly identified , there is broad support for the general model of a black hole and accretion disk ( see 1 ) . in this model , the emission from normal ( non - blazar , radio - quiet ) seyfert 1s like ngc 4151 is relatively isotropic , not significantly doppler - boosted or beamed towards earth . in this case , it is possible to use variability to place relatively model - independent limits on the central black hole mass ( and therefore the size scale ) . the most general is the eddington limit , which requires only that the source be gravitationally bound and possess a high degree of spherical symmetry . the minimum central mass given by this limit is @xmath74 where @xmath75 and @xmath76 are the eddington luminosity and mass , @xmath77 is the thompson cross - section , and @xmath78 is the proton mass . as the integrated luminosity of ngc 4151 is @xmath79 erg s@xmath18 , the eddington mass is @xmath80 . for a source surrounding a black hole of mass @xmath81 , and schwarzschild radius @xmath82 , the minimum variability time scale ( @xmath83 ) can be estimated from the size ( @xmath84 ) of the smallest stable orbit , which is @xmath85 for a schwarzschild black hole . for @xmath86 , this implies @xmath87 cm or @xmath88 sec , which is not a significant constraint on data which were sampled every hour . a larger but more model - dependent constraint assumes keplerian orbits and uses the correlation between the widths of emission lines and the distances estimated from their lags to estimate the central mass . in the form in which this estimate is generally presented , the inferred mass is the true mass if the clouds travel on circular orbits . if the clouds are gravitationally bound , but non - gravitational forces ( e.g. radiation pressure or hydrodynamics ) affect cloud motions , the real mass is greater than this estimate ; if the motions are unbound , the real mass is smaller than this estimate . clavel et al . ( 1987 , 1990 ) used this method to derive a central mass of @xmath89 for ngc 4151 . this corresponds to a smallest stable orbit of @xmath90 lt - min . again , this relatively weak constraint is not significant for these data . if the ultraviolet / optical continuum is optically thick thermal emission , the variability amplitudes can be used to constrain the temperature distribution . the simplest non - trivial general case is a flat , azimuthally symmetric disk with a local blackbody temperature that drops with radius as some power - law : @xmath91 . when @xmath8 is 3/4 , this is a fair approximation of a standard accretion disk , except at the smallest radii . by contrast , in a disk that radiates predominantly by reprocessed energy ( see 4.3 . ) , @xmath8 can be smaller , depending on the geometry of both the disk and the source of the primary radiation . if the local emission is described by a blackbody , the contribution of an annulus to the total disk flux density at a given frequency @xmath67 is @xmath92 where @xmath93 , and the starting and ending points of the integral are defined by the temperatures at the inner and outer radii of the ring . the increasing amplitude of variations with observing frequency is then naturally attributed to changing emission from the hotter regions ( that is , the inner disk radii ) . there is a test of the simplest case that produces simultaneous multi - wavelength variations , in which the emission from the outer disk ( @xmath94 ) is constant , and all of the variability is produced by a complete , simultaneous modulation of the emission from the inner disk ( @xmath95 ) . the only free parameter is the radius that separates the variable and constant parts of the disk , and this can be determined from the nva at a single wavelength . for example , using the 1275 variability amplitude of 8.6% gives @xmath96 for @xmath97 or @xmath98 for @xmath99 . this corresponds to a boundary at the disk radius where the temperature is 215,000 or 78,000 k , respectively . the last two columns of table 2 show how the percentage flux would drop if all of the emission inside this radius ( the putative variable component ) went to zero , including the effects of starlight . there is good agreement between the simple thermal model and the observed wavelength dependence of the variability amplitude for the standard accretion disk case , @xmath100 . this is not a strong function of how the inner boundary is chosen ; for the @xmath99 case , truncating the integration to between @xmath101 and @xmath98 changes the result by only 20% . the success of this simple exercise motivated the fitting of the ultraviolet / optical sed with a standard model of a geometrically thin , optically thick accretion disk ( e.g. , following the formalism of sun & malkan 1989 ) . aside from the starlight , no additional long - wavelength component was included in the models , so no attempt was made to fit fluxes longward of 2 @xmath102 m . if the disk is assumed to be viewed face - on , and the black hole is spinning rapidly ( kerr metric ) , the best - fit model parameters are @xmath103 and @xmath104 yr@xmath18 . if on the other hand the black hole is assumed stationary ( schwarzschild case ) , the best - fit parameters are @xmath105 and @xmath106 yr@xmath18 . in either case , the accretion rate corresponds to 0.6% of the eddington rate . the black hole mass inferred from the schwarzschild fit agrees with the keplarian value obtained by clavel et al . ( 1987 , 1990 ) . these fits give higher weight to the higher signal - to - noise optical continuum than to the ultraviolet continuum where the disk light dominates , which in turn requires a hotter disk and consequently a lower black hole mass ( @xmath107 ) . all the disk fits would give significantly larger black hole masses if the disk had a non - zero inclination . integrating the multi - waveband disk emission out to a boundary radius of @xmath108 lt - day shows that approximately 30% , 20% and 5% of the total disk flux at 1275 , 2688 , and 5125 is produced in the inner disk . ( these numbers refer to the kerr model , but are not very model - dependent ; malkan 1991 . ) after correcting for the effects of galactic starlight in the @xmath109 aperture , the 5125 emission from the inner disk falls to @xmath04% . for @xmath110 , this radius corresponds to @xmath111 these fractions are approximately the same as the largest peak - to - peak variations observed in these bands during the intensive 10 day multi - waveband monitoring campaign . thus , one consistent explanation for the decline in variability with increasing wavelength is that the variations occur entirely inside the inner disk . at a distance @xmath112 lt - day from a @xmath113 black hole , the orbital time scale is @xmath01 day , whereas the dominant fluctuations clearly occur on longer time scales ( i.e. , the 1275 peak - to - peak variation was only 42% over the entire 10 day intensive campaign ) . thus , they could be associated with either orbital mechanics or thermal fluctuations in the inner disk . it must also be noted that this simple disk model does not produce any x - ray emission , although this is also likely to originate within the inner regions . these data also can be used to constrain models in which the ultraviolet radiation is produced in gas that is heated by the same x - ray continuum that we observe directly at higher energies . in this model , time variations in the ultraviolet and high energy fluxes should be closely coupled : the ultraviolet becomes stronger shortly after the high energy flux rises , and the high energy flux may itself respond to changes in the ultraviolet flux if the ultraviolet photons provide seeds for compton upscattering into the higher energy band . the delays in both cases are essentially due to the light travel time between the two source regions . consequently , the ( reprocessed ) ultraviolet emission should vary simultaneously with the ( primary ) x - rays on time scales longer than the round - trip light - travel time between the emitting regions ( e.g. , clavel et al . the strong correlation between the ultraviolet and x - ray variations therefore supports the reprocessing hypothesis . the lack of any lag within the ultraviolet could be explained in two ways : either the sub - regions responsible for variations in different wavelengths are likewise very close to each other , or , as with the accretion disk model , only the hottest part of the region is varying . in the context of this model , the lack of any detectable lag implies that the x - ray and ultraviolet emitting regions are separated by @xmath20.15 lt - day , ( @xmath114 lt - day , because the light must travel in both directions ) , so the bulk of the reprocessing must occur in the central regions . perola & piro ( 1994 ) applied a more detailed reprocessing model to earlier x - ray / ultraviolet observations and predicted that high time resolution monitoring would measure lags of order 0.030.1 day ( rather close to but still formally consistent with the measured limits of @xmath20.150.3 day ) . the reprocessing model is supported by the broad profile of the iron k@xmath8 line , which suggests relativistic effects associated with an origin very close to a central black hole ( yaqoob et al . however , the _ asca _ observations ( and previous medium x - ray observations ) found no evidence for any significant hard tail " in the x - ray spectrum ( paper iii ; maisack & yaqoob 1991 ) . while the presence of the iron line implies reprocessing by some material , the lack of a `` reflection hump '' suggests that the material is not optically thick to compton scattering , as would be expected in the putative reprocessing disk . an associated problem is the overall energy budget . the total x - ray/@xmath1-ray flux must be adequate to produce the observed ultraviolet / optical / infrared flux , and the variable high energy flux must also equal or exceed that at lower energies . the observed , integrated 0.11 @xmath102 m flux is @xmath115 erg @xmath15 s@xmath18 . as strong ly@xmath8 and c iv emission indicates that the ultraviolet bump extends to wavelengths substantially shorter than 1000 , the intrinsic , integrated flux is probably larger by a factor of @xmath03 , corresponding to a total ultraviolet / optical / infrared flux of order @xmath116 erg @xmath15 s@xmath18 . the observed , integrated 12 kev and 210 kev fluxes are @xmath117 erg @xmath15 s@xmath18 and @xmath118 erg @xmath15 s@xmath18 , respectively after correction for line - of - sight absorption ( which is particularly important at 12 kev ) , the intrinsic , integrated fluxes rise to @xmath119 erg @xmath15 s@xmath18 and @xmath120 erg @xmath15 s@xmath18 , for a total of @xmath121 erg @xmath15 s@xmath18 . this is not adequate to power the lower energies , but inclusion of the gro data and interpolating between asca and gro yields a ( rather uncertain ) integrated 1200 kev flux of order @xmath122 erg @xmath15 s@xmath18 . this would be of order the amount necessary to power the lower energies . however , the fact that the @xmath1-ray variations differ from those in all other wavebands suggests that the bulk of the 10200 kev emission ( which dominates the x - ray/@xmath1-ray flux ) arises in a component that does not fully participate in the reprocessing . in this case , the x - ray/@xmath1-ray luminosity would still be a factor of order @xmath03 too small , so at most @xmath01/3 of the observed infrared / optical / ultraviolet flux could be due to reprocessing . however , this can not be independently verified because the _ asca _ sampling above 2 kev is inadequate to characterize the variability , and indeed the entire 10 - 50 kev spectrum is interpolated , not observed . although the absolute x - ray luminosity changes are observed to be small compared to those in the ultraviolet / optical , spectral fits indicate that the x - rays are highly ( factor of @xmath01520 ) absorbed , and the intrinsic , absorption - corrected variations have more than enough power to drive the ultraviolet / optical variations . however , the nva , which measures the fractional ( as opposed to absolute ) variations is 24% in the x - rays , while it is only 9% , 5% , 4% , and 1% at 1275 , 1820 , 2688 , and 5125 , respectively . this indicates that at most 35% , 20% , 15% , and 4% of the emission at 1275 , 1820 , 2688 , and 5125 could be due to reprocessing , with the rest coming from a component with different ( slower ) variability . as with the previous analysis , this would indicate that at most @xmath01/3 of the ultraviolet / optical / infrared can be reprocessed emission from higher energies . 1 . ngc 4151 showed significant variability over time scales of days , so the emitting region must be smaller than of order a few light days across , assuming the emission is not beamed . the limits on the interband lags indicate that , if there is reprocessing of flux between bands , none of the emission regions could be larger than @xmath00.15 lt - day . the lower limit to the central black hole mass derived from the eddington limit is small ( @xmath123 ) , corresponding to a minimum variability time of only 10 sec , which is not a significant constraint . accretion disk fits to the sed yield a central black hole with a mass of @xmath124 , accreting well below the eddington limit . the fact that the low nvas decrease from medium energy x - ray to ultraviolet to optical wavebands is consistent with the accretion disk model if the bulk of the variable ultraviolet / optical emission originates in a region @xmath00.07 lt - day ( @xmath125 ) from the center . this model gives is no immediate explanation of the link between ultraviolet and x - ray variations . the reprocessing model predicts a strong correlation between ultraviolet and x - ray variability , which is observed . because the nvas become systematically smaller at longer wavelengths , and because the absorption corrected x - ray/@xmath1-ray luminosity is only much smaller than in the ultraviolet / optical / infrared , and at most @xmath01/3 of the lower energy emission could be produced by reprocessing . this suggests that perhaps the ultraviolet arises in a disk powered partially by illumination by an x - ray source and partially by internal viscosity and accretion . determining the exact mix of these emission components will require probing the ultraviolet / x - ray bands at the shortest accessible time scales . it is of particular importance to extend the coverage to the gap at x - ray energies harder than 2 kev . this is the goal of the upcoming coordinated _ xte / iue_/ground - based observations of ngc 7469 , scheduled for the middle of 1996 , and it is hoped that they will shed further light on this important question . the work of bmp was supported by nasa grants nagw-3315 and nag5 - 2477 . the work of dmc and the _ iue _ data reduction were supported by nasa adp grant s-30917-f . the work of avf and lch was supported by nsf grant ast-8957063 . this research has also been supported in part by nasa grants nag5 - 2439 , nag5 - 1813 , and nagw-3129 . a96 alexander , t. 1996 , preprint a95 alloin , d. et al . 1995 , a&a , 293 , 293 a93 ayres , t. 1993 , pasp , 105 , 538 b87 barvainis , r. 1987 , apj , 350 , 537 b94 brinkmann , w. et al . 1994 , a&a , 288 , 433 c87 clavel , j. et al . 1987 , apj , 321 , 251 c90 clavel , j. et al . 1990 , mnras , 246 , 668 c91 clavel , j. et al . 1991 , apj , 366 , 64 c92 clavel , j. et al . 1992 , apj , 393 , 113 c91 collin - 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ray data showed the strongest variability but the poorest sampling ; the interruption in the second half of the campaign was due to a spacecraft malfunction . the apparent short flares in the 2688 light curve are probably due to instrumental effects , and not intrinsic variability . * figure 2 * plot of nva as a function of observing frequency . note the strong correlation in the optical / ultraviolet band , where the nva rises from @xmath01% in the optical to @xmath09% in the ultraviolet . the open circles refer to the total variability ( @xmath126 ) , and the filled circles are the nva ( @xmath24 ) , which represent the variability after correction for instrumental uncertainties . the strongest nva is seen in medium energy ( 12 kev ) x - rays , with an nva of 24% , while there is no significant variability observed in the softer x - ray bands , and weaker variations ( nva = 6% ) at 100 kev . * figure 3 * pds of the resampled ultraviolet light curves in four bands . the units of the pds ( co - ordinate axis ) are erg@xmath127 @xmath128 @xmath129 , while the units of the ordinate are day@xmath18 . the pds , which all appear the same to within the errors , show no signs of periodic variability . instead , they are well - fitted by a power - law with slope @xmath46 . * figure 4 * plots of the correlation between continuum fluxes at bands centered on 1275 and 100 kev , 1.5 kev , 1330 , 1820 , 2668 and 5125 . the ultraviolet and @xmath1-ray data are not significantly correlated , but all of the other bands show a significant correlation . the solid lines are unbiased least squares fits to the data . the regression lines with the long - wavelength ultraviolet have a positive y - intercept , while the ultraviolet1.5 kev x - ray regression has a positive x - intercept . * figure 5 * interpolated cross - correlation functions between 1275 and 1275 ( top panel ; autocorrelation ) , 1.5 kev ( second panel ) , 1820 ( third panel ) , 2688 ( fourth panel ) , and 5125 ( bottom panel ) . in all cases , the peaks are consistent with zero lag . * figure 6 * discrete cross - correlation functions between 1275 and 1275 ( top panel ; autocorrelation ) , 1.5 kev ( second panel ) , 1820 ( third panel ) , 2688 ( fourth panel ) , and 5125 ( bottom panel ) . in all cases , the peaks are consistent with zero lag . * figure 7 * spectral energy distribution of ngc 4151 . plotted quantities are monochromatic luminosity ( @xmath130 ) as a function of frequency . optical , ultraviolet , x - ray , and @xmath1-ray data are taken from observations made near mjd 26 . the asterisks show the rms variability in the observing bands listed in table 2 , in terms of monochromatic luminosity . lcccccccc + + & & & & & & & & & & & & & & & & + 50 - 150 kev & 9 & 8.10 & 8.9 & 6.4 & 6.1 & 119 & & 1 - 2 kev & 17 & 9.72 & 26.5 & 11.4 & 23.9 & 6.7 & & 0.5 - 0.9 kev & 17 & 9.72 & 3.5 & 6.6 & & & & 0.1 - 0.4 kev & 17 & 9.72 & 3.0 & 5.0 & & & & 1275 & 176 & 9.20 & 8.6 & 0.9 & 8.6 & 235 & 8.6 & 8.6 1330 & 176 & 9.20 & 8.2 & 1.0 & 8.2 & 241 & 8.1 & 7.4 1440 & 176 & 9.20 & 6.7 & 1.5 & 6.6 & 179 & 7.2 & 6.5 1730 & 176 & 9.20 & 6.1 & 1.4 & 6.0 & 172 & 5.5 & 4.1 1820 & 176 & 9.20 & 5.0 & 1.3 & 4.8 & 124 & 5.1 & 3.6 1950 & 176 & 9.20 & 4.9 & 2.0 & 4.5 & 106 & 4.5 & 3.1 2300 & 168 & 9.15 & 5.3 & 2.0 & 4.9 & 108 & 3.6 & 1.9 2688 & 168 & 9.15 & 3.9 & 1.1 & 3.7 & 89 & 2.7 & 1.3 4600 & 17 & 7.94 & 1.3 & 0.9 & 0.9 & 23 & 1.0 & 0.3 5125 & 18 & 9.33 & 1.1 & 0.9 & 0.7 & 14 & 0.9 & 0.2 6200 & 8 & 8.91 & 1.5 & 0.6 & 1.4 & 33 & 0.6 & 0.2 6925 & 8 & 8.91 & 1.6 & 0.7 & 1.4 & 36 & 0.4 & 0.1 + + lrccrr + + & & & & & + 100kev & 9 & 0.33 & 0.37 & 1.04 & 1kev & 17 & 0.60 & 0.56 & 2.64 & @xmath135 1330 & 176 & 0.96 & 0.95 & 38.11 & @xmath136 1820 & 176 & 0.90 & 0.86 & 22.54 & @xmath137 2688 & 168 & 0.69 & 0.70 & 12.47 & @xmath138 5125 & 18 & 0.65 & 0.59 & 2.95 & @xmath139 + + lccccc + + & & & & & & & & & & + @xmath59 & iccf & 0.82 & 0.87 & 0.70 & 0.69 & zdcf & 0.71 & 0.87 & 0.65 & 0.57 @xmath60 ( days ) & iccf & @xmath140 & @xmath141 & @xmath142 & @xmath143 & zdcf & @xmath144 & @xmath143 & @xmath145 & @xmath143 @xmath61 ( days ) & iccf & @xmath146 & @xmath147 & @xmath148 & @xmath149 & zdcf & @xmath150 & @xmath151 & @xmath152 & @xmath153 @xmath63 ( days)&zdcf & @xmath154 & @xmath155 & @xmath156 & @xmath157 + +

this paper combines data from the three preceding papers in order to analyze the multi - waveband variability and spectral energy distribution of the seyfert 1 galaxy ngc 4151 during the december 1993 monitoring campaign . the source , which was near its peak historical brightness , showed strong , correlated variability at x - ray , ultraviolet , and optical wavelengths . the strongest variations were seen in medium energy ( @xmath01.5 kev ) x - rays , with a normalized variability amplitude ( nva ) of 24% . weaker ( nva = 6% ) variations ( uncorrelated with those at lower energies ) were seen at soft @xmath1-ray energies of @xmath0100 kev . no significant variability was seen in softer ( 0.11 kev ) x - ray bands . in the ultraviolet / optical regime , the nva decreased from 9% to 1% as the wavelength increased from 1275 to 6900 . these data do not probe extreme ultraviolet ( 1200 to 0.1 kev ) or hard x - ray ( 250 kev ) variability . the phase differences between variations in different bands were consistent with zero lag , with upper limits of @xmath20.15 day between 1275 and the other ultraviolet bands , @xmath20.3 day between 1275 and 1.5 kev , and @xmath21 day between 1275 and 5125 . these tight limits represent more than an order of magnitude improvement over those determined in previous multi - waveband agn monitoring campaigns . the ultraviolet fluctuation power spectra showed no evidence for periodicity , but were instead well - fitted with a very steep , red power - law ( @xmath3 ) . if photons emitted at a primary " waveband are absorbed by nearby material and reprocessed " to produce emission at a secondary waveband , causality arguments require that variations in the secondary band follow those in the primary band . the tight interband correlation and limits on the ultraviolet and medium energy x - ray lags indicate that the reprocessing region is smaller than @xmath00.15 lt - day in size . after correcting for strong ( factor of @xmath415 ) line of sight absorption , the medium energy x - ray luminosity variations appear adequate to drive the ultraviolet / optical variations . however , the medium energy x - ray nva is 24 times that in the ultraviolet , and the single - epoch , absorption - corrected x - ray/@xmath1-ray luminosity is only about 1/3 that of the ultraviolet / optical / infrared , suggesting that at most @xmath01/3 of the total low - energy flux could be reprocessed high - energy emission . the strong wavelength dependence of the ultraviolet nvas is consistent with an origin in an accretion disk , with the variable emission coming from the hotter inner regions and non - variable emission from the cooler outer regions . these data , when combined with the results of disk fits , indicate a boundary between these regions near a radius of order @xmath5 lt - day . no interband lag would be expected as reprocessing ( and thus propagation between regions ) need not occur , and the orbital time scale of @xmath01 day is consistent with the observed variability time scale . however , such a model does not immediately explain the good correlation between ultraviolet and x - ray variations . * multiwavelength observations of * + * short time - scale variability in ngc 4151 . * + * iv . analysis of multiwavelength continuum variability * + 0.4 cm r.a . edelson , t . alexander , d.m . crenshaw , s . kaspi , m.a . malkan , b.m . peterson , r.s . warwick , j . clavel , a.v . filippenko , k . horne , k.t . korista , g.a . kriss , j.h . krolik , d . maoz , k . nandra , p.t . obrien , s.v . penton , t . yaqoob , p . albrecht , d . alloin , t.r . ayres , t.j . balonek , p . barr , a.j . barth , r . bertram,@xmath6g.e . bromage , m . carini , t.e . carone,@xmath6f .- z . cheng , k.k . chuvaev,@xmath6 m . dietrich , d . dultzin - hacyan , c.m . gaskell , i.s . glass , m.r . goad , s . hemar , l.c . ho , j.p . huchra , j . hutchings , w.n . johnson , d . kazanas , w . kollatschny , a.p . koratkar , o . kovo , a . laor , g.m . macalpine , p . magdziarz , p.g . martin , t . matheson , b . mccollum , h.r . miller , s.l . morris , v.l . oknyanskij , j . penfold , e . prez , g.c . perola , g . pike,@xmath6r.w . pogge , r.l . ptak , b .- c . qian , m.c . recondo - gonzlez , g.a . reichert , j.m . rodrguez - espinoza , p.m . rodrguez - pascual , e.l . rokaki , j . roland , a.c . sadun , i . salamanca , m . santos - lle , j.c . shields,@xmath6j.m . shull,@xmath6d.a . smith , s.m . smith , m.a.j . snijders,@xmath6g.m . stirpe , r.e . stoner , w .- h . sun , m .- h . ulrich , e . van groningen , r.m . wagner,@xmath6s . wagner , i . wanders , w.f . welsh , r.j . weymann , b.j . wilkes , h . wu , j . wurster , s .- j . xue , a.a . zdziarski , w . zheng , and z .- l . zou 0.4 cm = 1.em = -1.em department of physics and astronomy , 203 van allen hall , university of iowa , iowa city , ia 52242 . electronic mail : edelson@spacly.physics.uiowa.edu school of physics and astronomy and the wise observatory , the raymond and beverly sackler faculty of exact sciences , tel - aviv university , tel - aviv 69978 , israel . astronomy program , computer sciences corporation , nasa goddard space flight center , code 681 , greenbelt , md 20771 . department of astronomy , university of california , math - science building , los angeles , ca 90024 . department of astronomy , the ohio state university , 174 west 18th avenue , columbus , oh 43210 . department of astronomy , university of leicester , university road , leicester le17rh , united kingdom iso project , european space agency , apartado 50727 , 28080 madrid , spain . department of astronomy , university of california , berkeley , ca 94720 . school of physics and astronomy , university of st . andrews , north haugh , st . andrews ky169ss , scotland , united kingdom . department of physics and astronomy , university of kentucky , lexington , ky 40506 . department of physics and astronomy , the johns hopkins university , baltimore , md 21218 . laboratory for high energy astrophysics , nasa goddard space flight center , greenbelt , md 20771 . center for astrophysics and space astronomy , university of colorado , campus box 389 , boulder , co 80309 . universitts - sternwarte gttingen , geismarlandstrasse 11 , d-37083 gttingen , germany . centre detudes de saclay , service dastrophysique , orme des merisiers , 91191 gif - sur - yvette cedex , france . department of physics and astronomy , colgate university , hamilton , ny 13346 . mailing address : lowell observatory , 1400 west mars hill road , flagstaff , az 86001 . computer sciences corporation , nasa goddard space flight center , code 684.9 , greenbelt , md 20771 . space sciences laboratory , university of california , berkeley , ca 94720 , and eureka scientific , inc . current address : 28740 w. fox river dr . , cary , il 60013 . center for astrophysics , university of science and technology , hefei , anhui , people s republic of china . crimean astrophysical observatory , p / o nauchny , 334413 crimea , ukraine . deceased , 1994 november 15 . landessternwarte , knigstuhl , d-69117 heidelberg , germany . universidad nacional autonoma de mexico , instituto de astronomia , apartado postal 70 - 264 , 04510 mexico d.f . , mexico . department of physics and astronomy , university of nebraska , lincoln , ne 68588 . south african astronomical observatory , p.o . box 9 , observatory 7935 , south africa . space telescope science institute , 3700 san martin drive , baltimore , md 21218 . harvard - smithsonian center for astrophysics , 60 garden street , cambridge , ma 02138 . dominion astrophysical observatory , 5071 west saanich road , victoria , b.c . v8x 4m6 , canada . naval research laboratory , code 4151 , 4555 overlook sw , washington , dc 20375 - 5320 . physics department , technion - israel institute of technology , haifa 32000 , israel . department of astronomy , university of michigan , dennison building , ann arbor , mi 48109 . astronomical observatory , jagiellonian university , orla 171 , 30 - 244 cracow , poland . canadian institute for theoretical astrophysics , university of toronto , toronto , on m5s 1a1 , canada . department of physics and astronomy , georgia state university , atlanta , ga 30303 . sternberg astronomical institute , university of moscow , universitetskij prosp . 13 , moscow 119899 , russia . department of physics and astronomy , university of calgary , 2500 university drive nw , calgary , ab t2n 1n4 , canada , and department of mathematics , physics , and engineering , mount royal college , calgary t3e 6k6 , canada . istituto astronomico delluniversit , via lancisi 29 , i-00161 rome , italy . mailing address : 816 s. lagrange rd . , lagrange , il 60525 . department of physics and astronomy , bowling green state university , bowling green , oh 43403 . shanghai observatory , chinese academy of sciences , people s republic of china . facultad de ciencias , dept . fsicas , universidad de oviedo , c/ calvo sotelo , s / n . oviedo , asturias , spain . nasa goddard space flight center , code 631 , greenbelt , md 20771 . instituto de astrofsica de canarias , e-38200 la laguna , tenerife , spain . esa iue observatory , p.o . box 50727 , 28080 madrid , spain . royal observatory edinburgh , university of edinburgh , blackford hill , edinburgh eh93hj , united kingdom . institut dastrophysique , 98 bis boulevard arago , f-75014 paris , france . department of physics and astronomy and bradley observatory , agnes scott college , decatur , ga 30030 . royal greenwich observatory , madingley road , cambridge cb3 0ez , united kingdom . laeff , apdo . 50727 , e-28080 madrid , spain . steward observatory , university of arizona , tucson , az 85726 . hubble fellow . jila ; university of colorado and national institute of standards and technology , campus box 440 , boulder , co 80309 . iram , 300 rue de la piscine , 38046 saint martin dheres , france . mailing address : rue elysee reclubus 1 bis . , 38100 grenoble , france . osservatorio astronomico di bologna , via zamboni 33 , i-40126 , bologna , italy . institute of astronomy , national central university , chung - li , taiwan 32054 , republic of china . european southern observatory , karl schwarzschild strasse 2 , 85748 garching , germany . astronomiska observatoriet , box 515 , s-751 20 uppsala , sweden . department of physics , keele university , keele st5 5bg , staffordshire , united kingdom . observatories of the carnegie institution of washington , 813 santa barbara street , pasadena , ca 91101 . beijing astronomical observatory , chinese academy of sciences , beijing 100080 , people s republic of china . n. copernicus astronomical center , bartycka 18 , 00 - 716 warsaw , poland . = 0em = 1.5em

as in the preceding paper @xcite ( paper i ) , we study maps of the phase plane , which have an unstable fixed point ( an -point ) and an associated homoclinic tangle of stable and unstable manifolds ( fig . [ fpsp ] ) . the stable and unstable manifolds intersect transversely , bounding a region of phase space ( called the `` complex '' ) from which a trajectory may or may not escape . we consider an initial distribution of points along a curve passing through the complex ( the line of initial conditions ) . the escape - time plot is the number of iterates of the map required for a point to escape the complex , plotted as a function along the line of initial conditions ( fig . [ fetp ] ) . for chaotic systems , such escape - time plots have a complicated set of singularities and structure at all levels of resolution @xcite . these fractal escape - time plots play a central role in a variety of classical decay and scattering problems ; we have been particularly motivated by the ionization of atoms , especially hydrogen in parallel electric and magnetic fields . the escape - time plot of a discrete map is divided into `` escape segments '' , intervals over which the escape time is constant . in paper i , we proved that there exist certain important sequences of consecutive escape segments , which we called epistrophes , at all levels of resolution . the epistrophes are characterized by the epistrophe theorem , whose core results are : ( 1 ) each endpoint of an escape segment spawns a new epistrophe which converges upon it ; ( 2 ) in the limit @xmath2 , every epistrophe converges geometrically , with rate equal to the liapunov factor @xmath3 of the -point ; ( 3 ) the asymptotic tails of any two epistrophes differ by a simple scaling . the focus of paper i was the asymptotic behavior of epistrophes . in the present paper , we address how epistrophes begin . we use the topological structure of the homoclinic tangle and the line of initial conditions to show that there is a certain minimal set of escape segments . for this minimal set , we prove the epistrophe start rule ( theorem [ t1 ] ) , which says that after a sufficiently large time , each epistrophe begins some number of iterates @xmath4 after the segment that spawned it ( i.e. the segment upon which the epistrophe converges . ) the number @xmath4 is the same for all epistrophes of a given map and is dependent on the topological structure of the tangle ; explicitly , @xmath5 , where @xmath1 is the minimum delay time of the complex , that is , the minimum number of iterates a scattering trajectory spends inside the complex . the bulk of our effort is devoted to developing an algebraic algorithm for constructing the minimal set of escape segments for a general line of initial conditions . this algorithm allows us to compute the initial structure of the escape - time plot for iterates before the epistrophe start rule sets in . the algorithm is then used to prove the epistrophe start rule itself . a critical aspect of this paper is our use of homotopy theory . we develop the necessary formalism in sect . [ shld ] , and prior knowledge of homotopy theory is not required . homotopy theory provides an algebraic framework for describing the topological structure of curves in the phase plane . as we shall explain , the phase plane has a set of `` holes '' into which the line of initial conditions can not pass . a symbol sequence can be used to describe how the line circumvents these holes . as the dynamics maps the line forward , there is an induced dynamics on the symbol sequence , representing a new kind of symbolic dynamics which we call `` homotopic lobe dynamics '' . from the symbol sequence , one can readily read off the structure of the minimal set of escape segments . lines with different symbol sequences may have different minimal sets ; however , at long enough times , these minimal sets always obey the epistrophe start rule . some escape segments , such as that marked with an asterisk in fig . [ fetp ] , are not within the minimal set guaranteed by the topology . these segments are `` surprises '' which , within the present topological analysis , we can not predict . since they break the regular structure and since they often have no obvious connection with any of the epistrophes , we call them `` strophes '' as in sect . iiib of paper i. strophes interfere with the self - similar structure of the fractal and typically do not go away in the asymptotic limit , resulting in what we called `` epistrophic self - similarity '' in sect . iiic of paper i. despite the presence of these strophes , the minimal set often seems to accurately describe the early and intermediate time structure of the escape - time plot . patterns similar to our epistrophe start rule have been seen in other work . in the numerical study of tiyapan and jaff@xcite , epistrophes and the epistrophe start rule are evident in the structure of the initial angle - final action plot ( analogous to the escape - time plot ) . similarly , jung and coworkers @xcite used symbolic dynamics to construct a tree - diagram that gives a comparable description of a scattering system . in each case , the authors consider a specific line of initial conditions that is far outside of the complex and is topologically simple . easton @xcite , followed by rom - kedar and others @xcite , showed that recursive patterns also apply to homoclinic intersections between the stable and unstable manifolds . thus , it may not be surprising that comparable patterns should apply to the intersections between the stable manifold and an arbitrary line of initial conditions , at least at sufficiently large iterate . but at what iterate does this pattern set in , and what is the minimal set before it sets in ? algorithm [ a1 ] answers both these questions , as well as giving a simple proof of the epistrophe start rule . an important observation is that the escape - time plot depends both on the topology of the tangle and on the topology of the line of initial conditions . the paper is organized as follows . section [ snumerics ] motivates our study by presenting numerical computations on a particular saddle - center map with a chosen line of initial conditions . section [ shld ] is the theoretical heart of the paper , in which we formally develop homotopic lobe dynamics . section [ salgorithm ] contains algorithm [ a1 ] for computing the minimal set of escape segments . section [ sesr ] contains theorem [ t1 ] , which includes the epistrophe start rule . in sect . [ sexamples ] we apply our techniques to the escape - time plots for two representative lines of initial conditions . conclusions are in sect . [ sconclusions ] . appendices [ salgproof ] and [ st1proof ] contain the proofs of algorithm [ a1 ] and theorem [ t1 ] respectively . table [ table1 ] summarizes the notation in this article . as an example we study the map @xmath6 defined by eqs . ( a1 ) ( a3 ) of paper i using parameter values @xmath7 , @xmath8 , @xmath9 . figure [ fpsp ] shows a phase space portrait for this map , along with the line of initial conditions @xmath10 considered here . the same map is plotted in fig . 1 , paper i , but with a different line of initial conditions . we review the basic picture of phase space transport described in paper i and refs . the map @xmath6 has an unstable fixed point ( -point ) denoted @xmath11 and having liapunov factor @xmath12 , which is the larger of the two eigenvalues of @xmath6 linearized about @xmath11 . the -point has an associated homoclinic tangle consisting of the branch @xmath13 of the stable manifold and the branch @xmath14 of the unstable manifold ( fig . [ fpsp ] ) . complex _ is the region of phase space bounded on the north by the segment of @xmath13 connecting the homoclinic intersection @xmath15 to @xmath11 and bounded on the south by the segment of @xmath14 connecting @xmath15 to @xmath11 . the forward and backward iterates @xmath16 are homoclinic intersections with the same sense as @xmath15 . the homoclinic intersection @xmath17 and its iterates @xmath18 have the opposite sense . the _ escape zone _ @xmath19 is the lobe bounded by the segments of @xmath13 and @xmath14 joining @xmath15 to @xmath17 . it maps forward to the lobes @xmath20 , @xmath21 , which all lie outside the complex , and backward to the lobes @xmath22 , @xmath23 , which all intersect the complex . similarly , the _ capture zone _ @xmath24 is the lobe bounded by the segments of @xmath13 and @xmath14 between @xmath25 and @xmath15 . it maps forward to the lobes @xmath26 , @xmath27 , which all intersect the complex , and backward to the lobes @xmath28 , @xmath21 , which all lie outside the complex . under one iterate of the map the escape zone @xmath29 maps from inside to outside the complex and the capture zone @xmath24 maps from outside to inside the complex ; the lobes @xmath29 and @xmath24 together form what is called a _ turnstile _ it is important to emphasize that all points which escape in @xmath30 iterates lie in the escape zone @xmath22 . in the escape - time plot shown in fig . [ fetp ] , the number of iterates @xmath31 to escape is plotted as a function along the line of initial conditions @xmath10 . figure [ fetp ] is analogous to fig . 2 of paper i , but for a different choice of @xmath10 . for a given @xmath31 , the set of escaping points is partitioned into open intervals called _ escape segments _ ; an escape segment is one connected component of @xmath32 . for example , the first three escape segments at @xmath33 in fig . [ fetp ] correspond to the three intersections of @xmath10 with the lobes @xmath22 , @xmath34 , shown in fig . [ fpsp ] . the epistrophe theorem of paper i says that the escape - time plot contains sequences of escape segments , called epistrophes . several epistrophes are denoted by arrows in fig . the first epistrophe starts at @xmath35 and converges monotonically upward , containing one escape segment for each @xmath36 . a second epistrophe begins at @xmath37 and converges downward upon the endpoint of the @xmath35 segment . we say that the @xmath35 segment `` spawns '' this epistrophe . two more epistrophes are spawned at @xmath38 and converge upon the two endpoints of the @xmath39 segment . similarly , the @xmath40 segment spawns two more epistrophes beginning at @xmath41 . the data in fig . [ fetp ] suggest the following epistrophe start rule : each endpoint of an escape segment at @xmath30 iterates spawns an epistrophe which begins at @xmath42 iterates , where in this case @xmath43 . this recursive rule is formulated precisely by theorem [ t1 ] in sect . [ sesr ] . in general , @xmath44 , where @xmath1 describes the global topology of the tangle ( sect . [ sgroupoid ] ) . the fact that @xmath43 in fig . [ fetp ] is a consequence of the fact that @xmath45 intersects @xmath46 ( and no earlier @xmath26 , @xmath47 ) in fig . [ fpsp ] . on the left of fig . [ fetp ] are plotted the winding numbers @xmath48 of the escaping trajectories , i.e. , the number of times a given trajectory winds around the central stable zone as it escapes to infinity . notice that all segments of the epistrophe beginning at @xmath35 have winding number @xmath49 . similarly , all segments of the epistrophes spawned by the @xmath50 segments have winding number @xmath51 . the data in fig . [ fetp ] thus suggests that all segments of an epistrophe have the same winding number and that this number is one greater than the winding number of the segment which spawned the epistrophe . this rule will be precisely formulated and proved in a separate publication . we introduce a new kind of symbolic dynamics , where the symbol sequences refer to paths in the plane ( rather than trajectories of the map . ) this symbolic dynamics allows us to identify and describe a minimal set of escape segments along an arbitrary line of initial conditions . the theory of homotopy is central to our development @xcite . homotopy theory allows us to ignore the detailed positions of the stable and unstable manifolds and concentrate instead on their global topological structure . homotopy theory also provides a natural algebraic framework for describing this global structure . we consider a `` saddle - center map '' @xmath6 , which has a simple homoclinic tangle , as seen in fig . [ fpsp ] and described precisely by assumptions 1 5 in paper i @xcite . we define the _ active region _ @xmath52 to be the union of the complex with all of its forward and backward iterates . by construction , it is an invariant region of the phase plane . the boundary of @xmath52 , denoted @xmath53 , contains alternating segments of @xmath13 and @xmath14 ( the outer boundaries of capture and escape zones ) as well as the -point @xcite . the boundary @xmath54 has a well - defined orientation determined by the orientations of @xmath13 and @xmath14 . let @xmath55 be the smallest integer such that @xmath56 intersects @xmath19 . considering all scattering trajectories which begin outside the complex , enter the complex , and eventually exit , @xmath1 is the smallest possible number of iterates spent inside the complex . for this reason , we call @xmath1 the _ minimum delay time _ of the complex or simply the _ delay time_. the delay time is equivalently defined by the first pre - iterate @xmath57 of @xmath19 which intersects @xmath24 . in fig . [ fpsp ] , the delay time @xmath1 is equal to 5 . ( the delay time @xmath1 agrees with easton s signature @xmath58 @xcite . ) for the case @xmath59 , shown in fig . [ fsosd1 ] , @xmath29 intersects @xmath60 , forming an open region @xmath61 , which we view as a _ hole _ in the active region @xmath52 . mapping this hole backwards and forwards gives an infinite set of holes @xmath62 . more generally , for arbitrary @xmath1 we define the holes @xmath63 , @xmath64 , where @xmath44 . ( see fig . [ fsosd3 ] for the case @xmath65 . ) the set @xmath66 is the active region minus all the holes @xmath62 . the @xmath1 holes @xmath67 are inside the complex ; all other holes are outside . the homoclinic intersections @xmath68 and @xmath69 , @xmath70 , form a subset @xmath3 of the boundary @xmath53 . two paths ( or directed curves ) having the same initial and final points @xmath71 are said to be _ homotopic _ if one can be continuously distorted into the other without passing through a hole @xmath62 and without moving their endpoints @xcite . the concept of homotopy defines equivalence classes of paths ; the path - class , or _ homotopy class _ , @xmath72 is the set of all paths homotopic to an arbitrary path @xmath73 . that is , two paths belong to the same homotopy class if they can be distorted one into the other without changing the endpoints or passing through any hole ; likewise , two homotopy classes @xmath72 and @xmath74 are equal if a path @xmath75 in class @xmath72 can be distorted into a path @xmath76 in class @xmath74 . we are particularly interested in the following paths . for each @xmath30 , we define @xmath77 to be the path along the @xmath13-boundary of capture zone @xmath26 , joining @xmath78 to @xmath68 , and we define @xmath79 to be the path along the @xmath14-boundary of escape zone @xmath20 , joining @xmath68 to @xmath69 , as shown in fig . [ fsosd1 ] . these paths lie in the boundary @xmath53 of the active region . similarly , for each @xmath30 , we define @xmath80 to be the path along the @xmath13-boundary of @xmath81 and @xmath82 to be the path along the @xmath14-boundary of @xmath83 . these paths bound the lobes @xmath20 and @xmath26 in the interior of @xmath52 . since each path @xmath80 , @xmath82 , @xmath79 , and @xmath77 has endpoints in @xmath3 and does not pass through any of the holes @xmath62 , each belongs to a well - defined homotopy class . these classes are distinct , since none of the curves can be distorted into any other , and we denote them by @xmath84 , @xmath85 , @xmath86 , and @xmath87 , respectively . these homotopy classes encode global topological information about the structure of the tangle . let @xmath88 be the collection of all homotopy classes of paths in @xmath89 having endpoints in @xmath3 . for a path - class @xmath90 joining @xmath91 to @xmath92 and a path - class @xmath93 joining @xmath92 to @xmath94 , their product @xmath95 joins @xmath91 to @xmath94 and is constructed by first traversing a representative path @xmath96 followed by a representative @xmath97 . the homotopy class of a constant path , i.e. one which remains at a given point @xmath98 for all times , is denoted @xmath99 ( with the endpoint @xmath98 being understood from context ) ; for all @xmath100 , @xmath101 . for a class @xmath100 , its inverse @xmath102 contains a representative path from @xmath72 , but traversed backwards ; clearly , @xmath103 . the set @xmath88 thus has most of the structure of a group ( multiplication , identity , and inverse ) except in one respect : the product @xmath95 is not defined between arbitrary elements @xmath104 and @xmath105 but only between elements such that the final point of @xmath104 equals the initial point of @xmath105 . a set with such a restricted product is called a groupoid @xcite , and @xmath88 is called the _ fundamental groupoid of path - classes in @xmath89 having base points in @xmath3_. the dynamical map @xmath6 , acting on points in the plane , induces a map on the path - classes , which forms a kind of symbolic dynamics on the symbols @xmath84 , @xmath85 , @xmath86 , and @xmath87 . when @xmath6 acts on these elements , it simply shifts their indices , @xmath106 [ r1 ] we now turn our attention to the line of initial conditions , which we assume is given by a path @xmath10 that ( 1 ) has endpoints @xmath107 , ( 2 ) does not self - intersect , and ( 3 ) does not intersect any hole @xmath62 or the -point @xmath108 @xcite . for the homotopy analysis , we must shift the endpoints of @xmath10 so that they lie in the set @xmath3 . for example , we shift the initial point by first traversing a path @xmath109 before traversing @xmath10 ; @xmath109 begins at one of the two points in @xmath3 on either side of @xmath110 , runs along @xmath54 , and finally terminates at @xmath110 . by thus shifting both endpoints , we assign to @xmath10 a well - defined homotopy class @xmath111 @xcite . the intersection of @xmath10 with escape zone @xmath22 is the set of points that escape on the @xmath30th iterate , and any connected component of this set is called an _ escape segment _ ; sometimes we will use the term _ @xmath112-segment _ to emphasize an intersection with @xmath22 . ( the index @xmath30 may , in fact , be either positive or negative . ) in this article , we answer the following two questions regarding a minimal set of escape segments . * question 1 * _ what is the minimum number of intersections possible between a representative path @xmath113 and a representative path @xmath114 ? _ the minimum number of @xmath112-segments is half the minimum number of intersections . * question 2 * _ let @xmath115 , @xmath116 , @xmath117 ( @xmath118 ) be paths which minimize all possible pairwise- and self - intersections . in particular , @xmath119 and @xmath120 do not intersect each other or themselves , and @xmath121 has the minimum number of escape segments at both @xmath122 and @xmath123 iterates . what are the positions of the escape segments at @xmath122 iterates relative to those at @xmath123 iterates ? @xcite _ in answering these questions we allow ourselves to distort @xmath10 and @xmath124 into paths @xmath121 and @xmath125 to minimize the number of intersections . thus we are constructing a `` distorted escape zone '' @xmath126 whose intersection with @xmath121 is a set of `` distorted escape segments '' . henceforth , we omit the descriptor `` distorted '' and leave it understood . the answer to the above two questions will be obtained from the algebraic algorithm in sect . [ salgorithm ] , which will lead in turn to a proof of the epistrophe start rule in sect . [ sesr ] . by a _ basis _ of a groupoid we mean a minimal set of elements that generate the entire groupoid . to construct a basis of the fundamental groupoid @xmath88 , we first include the path - classes @xmath127 along the boundary of the active region @xmath53 . we then select path - classes @xmath128 that encircle the holes in @xmath89 , so that the complete basis is @xmath129 shown schematically in fig . the elements @xmath130 are special in that they are the only basis elements which must enter the interior of the complex , encircling the @xmath1 holes @xmath67 . the representative paths @xmath82 , @xmath80 , @xmath77 , and @xmath79 for this basis satisfy ( see fig . [ fbc ] ) : ( 1 ) no path in the basis intersects itself or any other path in the basis ( except perhaps at the end points ) ; ( 2 ) each representative @xmath80 and @xmath82 in the basis encircles exactly one hole , and each hole is encircled exactly once . furthermore , ( 3 ) all homotopy classes of relevance to us , specifically @xmath131 , @xmath85 , and @xmath84 , @xmath64 , have a unique _ finite _ reduced expansion in the basis @xcite . ( a reduced expansion is a sequence of elements in which any two adjacent factors @xmath72 and @xmath102 have been canceled . ) because of these properties and the simple picture shown in fig . [ fbc ] , we call this basis the `` untangled basis '' . now we develop the symbolic dynamics that will describe the minimal set of escape segments . first , however , we must assign a direction to each escape segment . recall that the two endpoints of an @xmath84-segment ( @xmath64 ) are intersection points between a path @xmath132 and a path @xmath133 . using the orientation of @xmath134 , one of these endpoints occurs first . we define the direction of an escape segment to point along @xmath135 from the _ second _ endpoint to the _ first _ endpoint . ( see fig . [ fdir ] . ) recall that @xmath135 has an independent direction defined by its own parameterization . an escape segment is said to `` point forward '' if its direction is the same as @xmath135 and to `` point backward '' otherwise . a point on @xmath135 is said to lie on the `` positive '' side of an escape segment if the segment points toward it and on the `` negative '' side otherwise . we need the forward iterates of all untangled basis elements expressed in terms of the untangled basis . for most elements , this is given by eqs . ( [ r1 ] ) . only one additional equation is needed @xmath136 where @xmath137 is an abbreviation for the path - class @xmath138 notice that the right - hand side of eq . ( [ r15 ] ) [ after substituting in eq . ( [ r16 ] ) ] is expressed entirely in terms of the untangled basis ( [ r7 ] ) . equation ( [ r15 ] ) is proved by first observing @xmath139 which , though rather lengthy , can be directly verified from a figure such as [ fsosd1 ] or [ fsosd3 ] ; one simply concatenates the basis paths as shown on the right and then distorts the resulting path into @xmath140 . by applying @xmath6 to both sides of eq . ( [ r33 ] ) and solving for @xmath141 , one obtains eq . ( [ r15 ] ) . it is convenient to explicitly compute the forward iterate of @xmath137 from eq . ( [ r15 ] ) @xmath142 for the purpose of computing the minimal set of escape segments , the @xmath84 basis elements ( @xmath21 ) and the @xmath87 basis elements ( all @xmath30 ) can simply be omitted from any expression that contains them , resulting in significant computational simplification ; for example , eqs . ( [ r15 ] ) , ( [ r16 ] ) , and ( [ r9 ] ) above become eqs . ( [ r2 ] ) ( [ r3 ] ) below . this is explained more fully in appendix [ salgproof ] . we can now state the algorithm for constructing the minimal set of escape segments up to a given iterate @xmath143 . [ a1 ] let @xmath10 be the line of initial conditions and @xmath144 its homotopy class . \1 ) expand @xmath131 in the untangled basis , omitting any @xmath84-factors for @xmath21 and all @xmath87-factors for @xmath70 . \2 ) compute @xmath145 by iterating @xmath131 forward @xmath143 times using eqs . ( [ r8 ] ) , ( [ r6 ] ) , and @xmath146 where @xmath147 for convenience , one may also use the following formula , which explicitly maps @xmath137 forward , @xmath148 carry out any cancellations of factors using @xmath149 , so that @xmath145 is expressed as a reduced expansion in the untangled basis . then , \a ) each @xmath150 or @xmath151 factor ( @xmath21 ) in the expansion of @xmath145 corresponds to a segment that escapes in @xmath152 iterates and that points backwards or forwards , respectively . \b ) the relative positions of the @xmath150-factors in the expansion of @xmath145 are the same as the relative positions of their corresponding escape segments along @xmath10 . this algorithm is justified in appendix [ salgproof ] . we apply algorithm [ a1 ] to compute the minimal set of escape segments ( up to @xmath35 ) for the simple example @xmath59 , @xmath153 . carrying out step 2 , the first three iterates of @xmath131 are computed to be @xmath154 where the @xmath86-factors have been underlined for greater visibility . we now consider the consequences of results a and b in the algorithm . examining @xmath155 , it contains a single factor @xmath156 , which yields a single forward pointing escape segment at @xmath157 , as shown in fig . [ fetpq]a . iterating forward to @xmath158 , this @xmath157 escape segment corresponds to the factor @xmath159 in eq . ( [ r4 ] ) ; on either side of this factor are factors @xmath160 and @xmath156 , corresponding respectively to backward and forward pointing segments that escape at @xmath161 . iterating once more , @xmath162 has four @xmath160-factors ( either @xmath160 or @xmath156 ) corresponding to four escape segments at @xmath163 and with relative positions and directions shown in fig . [ fetpq]a . considering now an arbitrary @xmath1 , @xmath164 propagates forward as @xmath165 [ r19 ] the minimal set of escape segments for @xmath164 , as constructed from results a and b in the algorithm , is shown schematically in fig . [ fetpq]b . the set contains a left- and a right - converging epistrophe , with two additional segments at @xmath166 . these two segments are the beginnings of two new epistrophes spawned @xmath0 iterates after the first segment . this spawning behavior is also visible in fig . [ fetpq]a for @xmath167 . in the next section we show that all lines of initial conditions have a minimal set that eventually displays such spawning behavior . after a certain number of iterates , the minimal set for any @xmath10 has a simple recursive structure described by the following theorem , which is proved in appendix [ st1proof ] . [ t1 ] let @xmath6 be a `` saddle - center map '' satisfying assumptions 1 5 of paper i @xcite and having an arbitrary minimum delay time @xmath168 . let @xmath10 be the line of initial conditions . there exists some iterate @xmath169 such that the minimal set of escape segments at all @xmath170 iterates can be constructed from the following two recursive rules : \(i ) _ epistrophe continuation rule : _ every segment ( in the minimal set ) that escapes at @xmath171 iterates has on its immediate positive side a segment that escapes at @xmath143 iterates and which has the same direction . \(ii ) _ epistrophe start rule : _ every segment that escapes at @xmath172 iterates ( @xmath5 ) spawns immediately on both of its sides a segment that escapes at @xmath143 iterates and which points toward the spawning segment . explicitly , @xmath173 , where @xmath174 and @xmath175 are respectively the lowest indices of the @xmath85- and @xmath86-factors in the expansion of the path - class @xmath131 of @xmath10 in the untangled basis to say that an @xmath176-segment lies `` on the immediate positive / negative side of '' an @xmath177-segment means that in the minimal set there is no earlier @xmath178-segment , @xmath179 , between the two . notice that new epistrophes are spawned by the epistrophe start rule ; the epistrophe continuation rule simply propagates those epistrophes started earlier . notice also that segments of an epistrophe point in the direction of convergence of the epistrophe . the early structure of the minimal set ( before @xmath180 ) can be computed using the algorithm in sect . [ salgorithm ] . thus , the algorithm gives the early behavior of the minimal set , and the simpler recursive rules give the subsequent behavior . using the map @xmath6 discussed in sect . [ snumerics ] and illustrated by fig . [ fpsp ] , we consider the escape - time plots for two different lines of initial conditions . we consider the line of initial conditions @xmath10 in fig . first we determine the homotopy class of @xmath10 . since neither endpoint of @xmath10 is in @xmath3 , we must shift each endpoint as described in sect . [ sminimalset ] . since the initial ( southernmost ) endpoint is on the curve @xmath181 ( the southern boundary of @xmath45 ) , we can shift it either east to @xmath182 or west to @xmath183 ; we choose @xmath183 since this will guarantee that the beginning of @xmath10 still intersects @xmath45 . since the final ( northernmost ) endpoint is on the curve @xmath184 ( the northern boundary of @xmath46 ) , it does not matter whether we shift it east to @xmath185 or west to @xmath186 ; we choose @xmath186 . following step 1 in the algorithm , we scrutinize fig . [ fpsp ] to determine that the homotopy class @xmath187 of the adjusted curve is @xmath188 . after omitting @xmath84- and @xmath87-factors , this simplifies to @xmath189 following step 2 , we map @xmath131 forward using eqs . ( [ r8 ] ) , ( [ r6 ] ) , ( [ r2 ] ) , and ( [ r3 ] ) with @xmath190 , @xmath191 [ r60 ] for greater visibility , we have underlined each @xmath160-factor . mapping forward once more , we find @xmath192 below each @xmath86-factor in eq . ( [ r50 ] ) , we have recorded the number of iterates to escape ; the arrow indicates whether the segment is forward- or backward - pointing . the results of eqs . ( [ r60 ] ) and ( [ r50 ] ) are shown qualitatively in fig . [ fsetp]a ; they should be compared with the calculation in fig . we examine these results in detail . \(1 ) as stated in algorithm [ a1 ] , each @xmath86 or @xmath193 factor in @xmath145 corresponds to a segment of @xmath10 that escapes in @xmath194 iterates . equation ( [ r50 ] ) gives the minimal set of escape segments up to @xmath195 . ( 2 ) after a certain iterate @xmath180 , we can determine the minimal set using the epistrophe continuation rule and epistrophe start rule in theorem [ t1 ] . explicitly , @xmath196 ; examining eq . ( [ r61 ] ) we see @xmath197 , and since @xmath190 , @xmath198 . so , for all iterates @xmath199 , algorithm [ a1 ] and theorem [ t1 ] give identical results . ( 3 ) direct computation ( fig . [ fetp ] ) indicates that up to @xmath200 , there are no additional escape segments outside the minimal set . the first segment in the computation which is not in the minimal set is indicated by an asterisk in fig . [ fetp ] at @xmath201 ; it is an example of what we call a strophe . ( 4 ) no epistrophe converges upon the lower endpoint of the @xmath35 segment , either in the minimal set ( fig . [ fsetp]a ) or the numerical data ( fig . [ fetp ] ) , because this point is an intersection between @xmath10 and the _ unstable _ manifold . we consider the line of initial conditions @xmath10 in fig . 1 of paper i. in order to define the homotopy class of this line , it must first be adjusted . from fig . 1 , paper i , we see that @xmath10 intersects the holes @xmath202 and @xmath203 . we adjust @xmath10 within each of these holes so that it runs along the boundary of the hole , on either the east or west side , and not through the hole itself . for the northern hole @xmath204 , we adjust @xmath10 to run along the eastern boundary , so that it still passes through @xmath205 . for the southern hole @xmath206 , we adjust @xmath10 to run along the western boundary . as in sect . [ sexample1 ] , the endpoints of @xmath10 must also be adjusted , so that they lie in @xmath3 . we shift the southern endpoint to @xmath207 and the northern endpoint to @xmath208 . the homotopy class @xmath187 of the adjusted curve is @xmath209 , which simplifies to @xmath210 then @xmath131 maps forward as @xmath211 and @xmath212 the data from eq . ( [ r54 ] ) are summarized in fig . [ fsetp]b . this should be compared with the numerical calculation in fig . 2 of paper i. equation ( [ r54 ] ) gives the minimal set of escape segments up to @xmath41 . in this case , examining eq . ( [ r63 ] ) , @xmath213 and @xmath214 , yielding @xmath215 . therefore for @xmath216 , the minimal set can be generated from theorem [ t1 ] rather than the algorithm . the first numerically computed segment which is not in the minimal set ( a strophe ) does not occur until @xmath217 ; it is indicated by an asterisk in fig . 2 of paper i. as above , no epistrophe converges upon the lower endpoint of the @xmath218 segment because it is an intersection between @xmath10 and the unstable manifold . the results of the present paper combine with the results of paper i @xcite to create a detailed picture of escape - time plots . on the one hand , the present study predicts the existence of a minimal set of escape segments ( algorithm [ a1 ] ) . after some number of iterates , this set has a simple recursive pattern ( theorem [ t1 ] ) described by : ( 1 ) at each iterate , add new segments that perpetuate all earlier epistrophes ; ( 2 ) at @xmath44 iterates after a given segment , spawn two new epistrophes on either side of this segment . these results say nothing about the lengths of segments or the separation between segments , and in particular say nothing about convergence properties of epistrophes . on the other hand , the results of paper i do address such issues , and we find that epistrophes converge geometrically upon the endpoints of the segments that spawn them and furthermore that all epistrophes differ asymptotically by an overall scale factor ( epistrophe theorem , paper i ) . the minimal set of escape segments typically omits some segments ( strophes ) that appear in the actual numerically - computed escape - time plot . nevertheless , the results of paper i apply to such strophic segments as well . there will be an epistrophe which converges upon an endpoint of a strophe ( epistrophe theorem ) . however , we can not in general predict at which iterate such an epistrophe will begin . on the other hand , the numerical evidence of fig . [ fetp ] and of fig . 2 in paper i is suggestive that even in this case , the epistrophes often begin @xmath4 iterates beyond the strophe . strophes occur due to structure in the lobes @xmath22 that we have ignored in our simple topological picture of the tangle . for example , @xmath20 may develop additional `` fingers '' or `` branches '' as it is mapped backwards . these fingers spread out into the phase space , creating additional intersections with the line of initial conditions . ( in some cases , such fingers can be connected with the presence of an island chain inside the complex , such as the prominent period-5 chain in fig . [ fpsp ] . ) in general , a countable infinity of topological parameters are needed to completely describe the fingers@xcite , though we expect a finite number of parameters to suffice for the escape - time plot up to a given finite number of iterates . the homotopy formalism presented here can be generalized to incorporate these additional topological parameters , thereby predicting at least some of the strophe segments . we will address these issues in a future paper . in future work , we will also study winding numbers , explaining the patterns shown in fig . in addition , we will apply our results to the ionization of hydrogen in parallel electric and magnetic fields . the authors would like to thank prof . nahum zobin for many useful discussions . this work was financially supported by the national science foundation . we need only verify statements a and b in algorithm [ a1 ] . these are certainly true when all elements of the untangled basis are allowed in the expansion of @xmath145 ( i.e. , we do not omit the factors specified in step 1 . ) this fact is evident by simply considering how a path is constructed from a reduced product of the basis paths shown in fig . [ fbc ] ; at each occurence of a @xmath86-factor , the path must cross under the hole and hence through @xmath20 , thus creating an escape segment at the specified location ( and with the specified direction . ) the @xmath86-factors are thus the key to determining the minimal set of escape segments . the @xmath85 basis elements ( @xmath219 ) create new @xmath86-factors via eqs . ( [ r8 ] ) and ( [ r15 ] ) and are thus themselves critical in determining the minimal set . however , the @xmath84 ( @xmath21 ) and @xmath87 ( @xmath64 ) basis elements are `` inert '' , mapping forward via eqs . ( [ r11 ] ) and ( [ r12 ] ) , never producing any @xmath86-factor . we thus lose nothing by eliminating them altogether from any expression which might contain them , as we have done in eqs.([r2 ] ) ( [ r3 ] ) . ( one can verify that making these eliminations does not produce spurious cancellations of @xmath85- or @xmath86-factors . ) defining the two path - classes @xmath220 we have the following lemma ( recalling that all @xmath87 and @xmath84 basis elements are omitted from our formulas . ) _ lemma : for any @xmath170 , @xmath145 can be expressed as a product of elements in the set @xmath221 , assuming @xmath222 ; for @xmath59 , @xmath223 . ( the symbol @xmath224 emphasizes the absence of the classes @xmath160 , @xmath225 , @xmath226 . ) _ _ proof of lemma : _ it follows from the propagation formulas ( [ r8 ] ) , ( [ r6 ] ) , and ( [ r2 ] ) and the definitions of @xmath174 and @xmath175 that for @xmath227 , @xmath145 can be expressed as a product of the elements @xmath228 . thus , for @xmath170 , @xmath145 can be expressed as a product of the elements @xmath229 . since the elements @xmath230 are in the set @xmath231 , we need only verify that the elements @xmath232 can be expressed as products of elements in @xmath231 , a fact which follows from rewriting eqs . ( [ r22 ] ) ( [ r23 ] ) as for @xmath170 , we expand @xmath145 as a product of elements in @xmath231 . by using eqs . ( [ r20 ] ) and ( [ r21 ] ) to eliminate @xmath235 and @xmath236 , we obtain the expansion of @xmath145 in the untangled basis . it is easy to verify that when making these substitutions , there are no cancellations of any @xmath86-factors . ( here , we use the fact that powers of @xmath236 , such as @xmath237 , can not occur in the expansion of @xmath145 since this would imply that @xmath10 has a self - intersection . ) the theorem is now a trivial consequence of the representation of @xmath145 as a product of elements in the set @xmath231 . specifically , each occurrence of @xmath235 in the product yields a single segment which escapes at @xmath171 iterates , corresponding to the @xmath225-factor of @xmath235 in eq . ( [ r20 ] ) , and a single segment which escapes at @xmath143 iterates , corresponding to the @xmath160-factor of @xmath235 . the form of eq . ( [ r20 ] ) also implies that the directions and relative positions of these two segments obey the epistrophe continuation rule . similarly , eq . ( [ r21 ] ) implies that each occurrence of @xmath236 in the representation of @xmath145 yields a segment which escapes at @xmath172 iterates and two segments which escape at @xmath143 iterates ; the directions and relative positions of these segments obey the epistrophe start rule . since the basis elements @xmath160 , @xmath225 , and @xmath226 occur in the expansion of @xmath145 only within the @xmath235- and @xmath236-factors , these two rules completely determine the minimal set of escape segments at @xmath143 iterates . topologically , the boundary of @xmath52 would be defined as its closure minus its interior and would therefore include the untangled branches of the stable and unstable manifolds . here , when we say the boundary of @xmath52 we mean only the outer boundaries of the capture and escape zones as well as the -point . more formally , two ( continuous ) paths @xmath240 ( where @xmath241 $ ] ) are homotopic if there exists a continuous function @xmath242 such that @xmath243 , @xmath244 , @xmath245 and @xmath246 , @xmath247 , @xmath248 . the map @xmath249 is called a homotopy . in practice , the line of initial conditions typically extends beyond the active region ; here , we truncate it and consider only the piece inside @xmath52 . in practice , the line of initial conditions may also pass through a hole @xmath62 or @xmath250 ; here we assume for simplicity that it does not . in sect . [ sexample2 ] we consider an example in which @xmath10 does intersect a hole @xmath62 . each endpoint can be shifted either forward or backward along @xmath53 . if an endpoint lies on @xmath77 , for some @xmath30 , this choice does not affect the minimal set of escape segments . however , if an endpoint lies on @xmath251 , then that end of @xmath10 terminates inside @xmath20 , meaning that @xmath10 ends in an escape segment . depending upon the direction in which we shift this endpoint , we can either include or exclude this terminal segment from the minimal set . by convention , we shift the endpoint in the direction which includes the terminal segment . see the examples in sect . [ sexamples ] . one may ask whether it is even possible to choose curves @xmath121 , @xmath119 , and @xmath120 satisfying the conditions of question 2 ? that this is indeed possible is established by the following lemma , which is a simple corollary to a theorem of turaev ( theorem 2 of ref . @xcite . ) given a set of homotopy classes @xmath252 , @xmath253 , there exists a choice of representatives @xmath254 , @xmath255 , such that for each @xmath256 , the number of self - intersections of @xmath257 is minimized and for each pair @xmath258 , the number of pairwise - intersections between @xmath257 and @xmath259 is minimized . more generally , for a path @xmath75 that does not intersect @xmath260 and that has a well - defined homotopy class in @xmath88 , the homotopy class of @xmath75 has a ( unique ) finite reduced expansion in the basis .

we continue our study of the fractal structure of escape - time plots for chaotic maps . in the preceding paper , we showed that the escape - time plot contains regular sequences of successive escape segments , called epistrophes , which converge geometrically upon each endpoint of every escape segment . in the present paper , we use topological techniques to : ( 1 ) show that there exists a minimal required set of escape segments within the escape - time plot ; ( 2 ) develop an algorithm which computes this minimal set ; ( 3 ) show that the minimal set eventually displays a recursive structure governed by an `` epistrophe start rule '' : a new epistrophe is spawned @xmath0 iterates after the segment to which it converges , where @xmath1 is the minimum delay time of the complex . topological methods and symbolic dynamics have long been valuable tools for describing orbits of dynamical systems . for example , if a particle in the plane scatters from three fixed disks , labeled a , b , and c , its orbit can be characterized by a sequence of symbols , such as ... aba*bcbca ... , giving the sequence of collisions with the disks . the asterisk gives the location of the particle at the present time ; as time goes by the asterisk takes one step to the right . in this paper , we describe a new kind of symbolic dynamics , in which the symbol sequence describes the structure of a curve in the plane . the relevant curve is not the trajectory of a particle , but rather an ensemble of initial points in phase space the line of initial conditions . this line winds around `` holes '' in the plane in a manner described by the symbol sequence . the dynamical map applied to the line induces a map on the symbol sequence , which is more complicated than a simple shift . we use this symbolic dynamics to derive properties of the epistrophes introduced in the preceding paper . in particular , we use it to obtain a `` minimal set of escape segments '' and an `` epistrophe start rule . ''

we thank t. banks , d. gross , j. maldacena , d. marolf , and e. silverstein for comments and discussions . this work was supported in part by nsf grants phy05 - 51164 and phy07 - 57035 , and by fqxi grant rfp3 - 1017 . linearizing around a radial string in @xmath7 , the three bosonic fluctuations in the @xmath104 directions have @xmath105 in ads units , the five bosonic fluctuations along the @xmath51 are massless , and the eight fermions have @xmath106 @xcite . we write the action keeping only the unbroken @xmath107 symmetry manifest . in particular the fermion @xmath108 and supersymmetry transformation @xmath109 are in the majorana representation @xmath110 . we define gamma matrices for the respective factors , @xmath111 for @xmath112 , @xmath113 for @xmath114 , and @xmath115 for @xmath116 ; matrices from different sets are mutually commuting . building on the basic @xmath25 supermultiplet @xcite , the action and supersymmetry transformation are @xcite @xmath117 and @xmath118 we study the supersymmetry of the boundary conditions using the approach of ref . expanding the solutions of the field equation near the boundary gives @xmath119 where @xmath120 . the supersymmetry parameter is @xmath121 for arbitrary constant spinor @xmath122 , and the variations take the form @xmath123 for the massive @xmath124 , the only allowed quantization is the standard @xmath29 . the @xmath125 symmetry implies common boundary conditions on all fermionic components , and supersymmetry then requires @xmath126 , the dirichlet condition for the @xmath51 variables . the alternate @xmath127 quantization is not supersymmetric . y. m. makeenko , a. a. migdal , `` exact equation for the loop average in multicolor qcd , '' phys . * b88 * , 135 ( 1979 ) ; y. m. makeenko , a. a. migdal , `` quantum chromodynamics as dynamics of loops , '' nucl . * b188 * , 269 ( 1981 ) ; a. m. polyakov , `` string theory and quark confinement , '' nucl . phys . proc . * 68 * , 1 ( 1998 ) [ arxiv : hep - th/9711002 ] . i. r. klebanov and e. witten , `` ads / cft correspondence and symmetry breaking , '' nucl . phys . b * 556 * , 89 ( 1999 ) [ arxiv : hep - th/9905104 ] . e. witten , `` multi - trace operators , boundary conditions , and ads / cft correspondence , '' arxiv : hep - th/0112258 . v. s. dotsenko , s. n. vergeles , `` renormalizability of phase factors in the nonabelian gauge theory , '' nucl . phys . * b169 * , 527 ( 1980 ) .

the locally bps wilson loop and the pure gauge wilson loop map under ads / cft duality to string world - sheet boundaries with standard and alternate quantizations of the world - sheet fields . this implies an rg flow between the two operators , which we verify at weak coupling . many additional loop operators exist at strong coupling , with a rich pattern of rg flows . wilson loop renormalization group flows * joseph polchinski * _ kavli institute for theoretical physics _ _ university of california _ santa barbara , ca 93106 - 4030 _ * james sully * _ department of physics _ _ university of california _ _ santa barbara , ca 93106 _ = 17pt the wilson loop operator is a key observable in gauge theories . studies using ads / cft duality @xcite have largely focused on the locally bps loop operator , which couples with equal strength to the gauge and scalar fields . in euclidean signature , @xmath0 = \frac{1}{n } { \rm tr}\,pe^{\oint_c ds \,(i \dot x^\mu a_\mu + \theta^i \phi^i ) } \ , , \quad \theta^2 = \dot x^2 \ , . \label{bps}\ ] ] this is dual to the theory in the ads bulk with a string world - sheet bounded by the curve @xmath1 on @xmath2 . but what of the simple gauge holonomy @xmath3 = \frac{1}{n } { \rm tr}\,pe^{i \oint_c ds \ , \dot x^\mu a_\mu } \,?\ ] ] this is a natural observable in the gauge theory . does ads / cft duality allow us to calculate its correlators at strong coupling ? indeed , in ref . @xcite a simple prescription for the dual is given . in this paper we develop further the proposal of ref . @xcite . we point out that it implies an operator renormalization group flow , with the ordinary wilson loop in the uv and the bps loop in the . further , at strong coupling there is a much larger set of loop operators , with a rich set of rg flows . these operators do not have any simple weak coupling duals . we note that much recent work on scattering amplitudes deals with lightlike wilson loops , @xmath4 , for which the ordinary and bps loops coincide . however , the loop equations , at least in their usual form @xcite , take the string out of the space of locally bps configurations @xcite . making effective use of the loop equations requires a full understanding of the renormalization of loop operators @xcite , which is one of the motivations for the preset work . we first review and give further support to the prescription of ref . @xcite . consider the string world - sheet action in nambu - goto form , @xmath5 the vanishing of surface term in its variation requires that @xmath6 on the boundary of the world - sheet . in terms of the @xmath7 coordinates @xmath8 the boundary lies at @xmath9 , or perhaps on a regulating surface @xmath10 . for the bps loop the embedding is fixed at the boundary , @xmath11 and so the variation ( [ surf ] ) vanishes trivially . for the ordinary wilson loop , the dual theory is given by a world - sheet with boundary conditions @xcite @xmath12 replacing the dirichlet condition on the angular variables with a neumann condition .. ] these boundary conditions are conformally covariant , @xmath3 \to w[f(c ) ] \,,\ ] ] and @xmath13 invariant . at weak coupling the gauge holonomy is the unique operator with these properties : conformal covariance implies that it is constructed from the line integral of dimension one operators , and @xmath13 invariance excludes the scalars . as a further symmetry check , we verify in the appendix that the dirichlet condition is locally supersymmetric , and the neumann condition is not . one can give a formal derivation by introducing an independent world - line field @xmath14 with @xmath15 , and averaging the loop operator @xmath16 on the world - sheet the sum over dirichlet conditions produces a free boundary condition , giving the neumann condition as an equation of motion . expanding the loop operator in powers of @xmath17 , the linear term averages to zero , the quadratic term averages to a contour integral of @xmath18 , which is irrelevant , and so on , and only the holonomy survives . this is somewhat surprising , because it implies that at large t hooft coupling @xmath19 the expectation values for bps operators with fixed @xmath20 are the same as for the pure gauge operator . they are governed by the same saddle points , at constant @xmath21 , and differ only in the determinants which give an effect subleading in @xmath22 . thus the force between a fundamental and an antifundamental does not depend on whether they couple to a common scalar @xmath23 . this is certainly not true at weak coupling , where the scalar exchange is equal in magnitude to the gauge exchange . however , it is consistent with some earlier observations . in ref . @xcite it was noted that the bps loop satisfies zig - zag symmetry @xcite at strong coupling , whereas this is only expected for the simple loop . in ref . @xcite it was noted that the ads wilson loop satisfies the same loop equation as in pure gauge theory , and it was conjectured that this is a universal behavior at large @xmath19 . there is another way to think about these two boundary conditions . consider a string running radially in @xmath7 , e.g. along @xmath24 . its world - volume is an @xmath25 , and the world - sheet fluctuations @xmath26 for @xmath27 are described by massless fields in @xmath25 . near the ads boundary , one then has @xmath28 the dirichlet quantization sets @xmath29 , and the neumann quantization sets @xmath30 . thus , these are the standard and alternate quantizations in the sense of refs . @xcite . near the ads boundary every world - sheet is asymptotically @xmath25 near any smooth point of the loop @xmath31 , so this reasoning applies more generally . consequently , the insertion of @xmath32 into the loop , which should be dual to the boundary perturbation of @xmath26 , has dimension 0 in the ordinary loop ( up to a higher correction @xcite to be discussed below ) and dimension 1 in the bps loop . the latter can also be seen from the fact that the insertion of @xmath33 is just an infinitesimal @xmath13 rotation of the loop , and so marginal . using the conformal flatness of the @xmath25 metric @xmath34 we can immediately write down the bulk - to - bulk propagators @xmath35 with the upper sign for the standard quantization and the lower sign for the alternate quantization . taking the boundary limit @xmath36 gives the @xmath37 two - point function for the standard quantization and a logarithmic two - point function for the alternate quantization . in the latter case @xmath26 is not a good quantum field due to ir divergences , but its time derivative is . for a closed neumann wilson loop @xmath26 has a normalizable zero mode , whose integral enforces @xmath13 invariance . this interpretation immediately suggests an interesting rg flow . if instead of one of the pure boundary conditions we impose the mixed condition @xmath38 then the @xmath39 term will dominate near the boundary and the @xmath40 term near the horizon : we have an rg flow @xcite from the ordinary wilson loop in the uv to the bps loop in the . a more precise and @xmath13 invariant way to formulate this is to begin with the neumann theory and add a boundary perturbation @xmath41 on the world - sheet . since @xmath42 has dimension zero in this quantization , this is relevant : it is negligible in the uv and dominant in the . at low energy the path integral will be dominated by the configuration of minimum action ( [ j1pert ] ) , extending in the @xmath42 direction . the fluctuations around this configuration satisfy the boundary condition ( [ mixed ] ) . thus we identify the perturbed world - sheet with the family of operators @xmath43 = \frac{1}{n } { \rm tr}\,pe^{\oint_c d\tau \,(i \dot x^\mu a_\mu + \zeta |\dot x| \theta^i\phi^i)}\ , , \label{zeta}\ ] ] where here @xmath44 . these interpolate between the ordinary loop at @xmath45 and the bps loop at @xmath46 . that is , @xmath47 , increasing from 0 to 1 as @xmath48 decreases from @xmath49 to 0 ( the renormalization scale is included for dimensions ) . negative values of @xmath50 give a world - sheet extended in the opposite direction on @xmath51 , and correspond to the range @xmath52 where @xmath53 is again a bps loop . note that the double - trace interpretation @xcite of the flow for bulk fields is not relevant here , rather we are inserting additional scalars into the single wilson trace . one would expect this flow also to be evident at weak coupling . expanding perturbatively , power - counting gives a log divergence and so a possible contribution to the running of @xmath54 whenever a group of vertices ( on the wilson contour and/or in the volume ) approach one end of a scalar propagator attaching to the contour . at order @xmath55 the only graph is fig . 1 , as @xmath56 : the vertex correction is @xmath57 plus a similar piece from @xmath58 . one can evaluate this readily in dimensional regularization , but we will take a spatial regulator , requiring @xmath59 . defining @xmath60 , the contribution from the region near @xmath61 is then @xmath62 where we have linearized @xmath63 near @xmath61 . this exhibits the usual linear divergence proportional to the perimeter , plus a logarithm from the endpoint . gives a logarithm that must cancel those from the other two ranges . thus we need calculate only this ` connected ' term , but subtract from its group theory factor the group theory factor of the disconnected graphs . ] combined with the integral from @xmath58 , the logarithmic term is @xmath64 at renormalization scale @xmath65 , thus contributing @xmath66 to @xmath67 . there are a number of graphs of order @xmath68 , but we can deduce their contribution indirectly . the supersymmetric operator @xmath46 should be fixed under renormalization , and so , differing by a factor of 2 . ] @xmath69 more directly , one sees immediately that in feynman gauge the graphs in which the scalar 2 - 3 propagator in fig . 1 is replaced by a gauge propagator give precisely the order @xmath70 term in ( [ betazeta ] ) , and we have verified that in this gauge the graph of fig . 2 cancels against the scalar wavefunction renormalization , taking the latter from ref . @xcite . as argued at strong coupling , the @xmath45 simple loop is a uv attractor , and the @xmath71 bps loops are ir attractors . linearizing near @xmath45 , the dimension of a @xmath72 insertion into an ordinary wilson line is @xmath73 , as compared to 0 at infinite coupling . linearizing near @xmath46 , the dimension of an insertion of @xmath74 into a bps line in the 6-direction is @xmath75 . at infinite coupling this dimension is 2 ( the perturbation for the boundary condition flow has dimension @xmath76 at the uv end and @xmath77 at the ir end ) . the insertion of other @xmath32 into the bps loop has dimension 1 at both weak and strong coupling because it is exactly marginal . these dimensions have been discussed previously in ref . @xcite . note that the relation @xmath78 does not hold at weak coupling . for bulk fields this gets only @xmath79 corrections , but for the string world - sheet fields there are corrections in @xmath22 . we have not given a strong - coupling prescription for the loop with @xmath80 , but even at weak coupling this range is problematic : the flow ( [ betazeta ] ) leads to the bps loop in the ir , but diverges in the uv and there may be no continuum operator . note that we could also consider complex @xmath54 . almost all flows still go to the wilson loop in the uv and the bps loop in the ir , the exception being pure imaginary @xmath54 which diverges in the . the fields @xmath81 have @xmath82 @xcite , so @xmath83 and @xmath84 . the standard quantization corresponds to the insertion of @xmath85 @xcite . the alternate quantization should correspond to the fourier transform , the momentum loop of ref . @xcite , but the negative dimension is unacceptable : apparently the fourier transform does not exist at strong coupling . one could regulate this by inserting a factor of @xmath86 , producing a loop that flows to the momentum loop in the uv and to a smeared position - space loop in the . other such weighted sums over wilson loops have been considered in refs . @xcite . they may be useful in disentangling the loop equations . it is satisfying to have an ads interpretation for the cft operator @xmath87 $ ] , but there is still a mismatch in the other direction . one could satisfy the boundary equation ( [ surf ] ) by taking dirichlet conditions on some of the @xmath88 and neumann conditions on others , say @xmath89 ( note that the constraint @xmath90 is satisfied . ) this is different from any loop considered above , and the different world - sheet determinants will give a distinct amplitude . like the wilson and bps loops it is conformally covariant , but there is no candidate for a weak - coupling dual . to get some insight consider perturbing the neumann theory by the boundary operator @xmath91 the argument again has dimension 0 at strong coupling so this is relevant , and over long distances the loop will want to sit on the 2-sphere @xmath92 where the action is minimized ; the boundary conditions tangent to this @xmath93 remain neumann . thus , under this perturbation the ordinary wilson loop flows to the loop ( [ dn ] ) . at next order in @xmath22 , the operator @xmath94 with spherical harmonic @xmath95 has dimension @xmath96 @xcite . one way to see this is to think of @xmath94 as an open string vertex operator , for which the leading dimension is @xmath97 . this increases with decreasing coupling , and at zero coupling it reaches the dimension @xmath98 of the insertion @xmath99 . thus , at some coupling the dimension of @xmath100 passes through 1 , and the perturbation ( [ j2pert ] ) switches from relevant to irrelevant . the flow then reverses , and the wilson operator is the ir fixed point ( the @xmath101 symmetry prevents the flow from going to the bps loop ) . it is not clear whether there is any uv fixed point for this reverse flow the operator with insertion @xmath102 is perturbatively nonrenormalizable . general perturbations @xmath103 define a large set of loop operators at strong coupling . all flow to the neumann loop in the uv , and functions with a unique minimum flow to the dirichlet loop in the . potentials with continuous degenerate minima flow to other loops as above . potentials with discrete degenerate minima will flow to a sum over kinked loops . as the coupling is decreased the number of perturbatively renormalizable operators decreases in steps , leaving just the @xmath54-loops ( [ zeta ] ) at sufficiently weak coupling . finally , there should be a parallel story for t hooft loops and d - strings .

the feedback structure considered in this note is depicted in fig . 1 and the related transfer functions of the process @xmath0 and the controller @xmath1 are given by @xmath2 @xmath3 where @xmath4 is the plant steady - state gain , @xmath5 and @xmath6 the plant time constants , @xmath7 is the positive plant time delay and @xmath8 , @xmath9 and @xmath10 are the parameters of the pid controller . complete explicit expressions of the boundaries of the stability regions in first - order plants have been found in @xcite with a version of the hermite - biehler theorem derived by pontryagin , in @xcite with the nyquist criterion , and in @xcite with the root location method . moreover the second - order plants have been investigated in @xcite by means of a graphical approach ; the results obtained are correct , but the stability conditions are not all explicit and no finite number of required computation steps is specified . finally arbitrary - order plants have been studied with the nyquist criterion in @xcite , but p and pid controllers with a given @xmath8 separately are considered and no information about the set of the process parameters that allow stability is given . this note can be considered as an extension of @xcite to the arbitrary - order plants and is organized as follows . in section 2 all the analytical expressions that will be used in the next sections are evaluated in detail . in section 3 a process transfer function without zeros is considered and the stability regions are explicitly evaluated by means of a version of the hermite - biehler theorem derived by pontryagin , already used in @xcite . a second - order plant is studied as example and the related stability regions are determined and plotted in two figures . in section 4 a process transfer function with zeros is considered and the stability regions are found by means of a new theorem . the two procedures of the sections 3 and 4 are essentially equal , consist of a finite number of steps and yield the stability regions in both process and controller parameters planes . in section 5 some conclusive remarks are given . the importance of explicit expressions of the boundaries of the stability zones has been enhanced by the introduction of the controller tuning charts in @xcite ( used also in @xcite ) . the closed - loop transfer function @xmath11 of the system is given by @xmath12 according to the pontryagin s studies , presented in @xcite and summarized in @xcite , it is necessary that @xmath11 has a bounded number of poles with arbitrary large positive real part for stability . this holds if the denominator of @xmath11 has a principal term @xmath13 ( in our case , where @xmath14 and @xmath15 , it exists if @xmath16 ) and the function @xmath17 , coefficient of @xmath18 , ( in our case @xmath19 for @xmath20 and @xmath21 for @xmath22 ) has all the zeros in the open left half plane . this happens if one of the following conditions is satisfied : 1 . @xmath20 2 . @xmath22 and @xmath23 . the denominator of @xmath11 , given by ( [ eq:2.0a ] ) , divided by @xmath24 and hence named @xmath25 , can be written , according to ( [ eq:1.1 ] ) , as @xmath26 since all the poles of the closed - loop transfer function @xmath11 are zeros of @xmath25 and a system is stable if no pole of @xmath11 lies in the right half - plane , the above system is stable if no zero of @xmath25 lies in the right half - plane . for process transfer functions without zeros , examined in section 3 , @xmath25 is a quasi - polynomial and a version of the hermite - biehler theorem derived by pontryagin is employed . for process transfer functions with zeros , examined in section 4 , @xmath25 is a quasi - polynomial divided by a polynomial and a new theorem , proved by use of the principle of the argument , is employed . now , before the explanation of the proposed procedures , let us evaluate all the expressions that will be used in the next sections . it is convenient to introduce the normalized time referred to the plant time delay @xmath7 and the dimensionless parameters @xmath27 , @xmath28 , @xmath29 , @xmath30 , @xmath31 and @xmath32 , in order to obtain equations independent of the real values of the parameters . applying these simplifications , ( [ eq:2.0b ] ) becomes @xmath33 moreover , assuming @xmath34 and @xmath35 , the real and the imaginary components @xmath36 and @xmath37 of @xmath38 , calculated for @xmath39 , are given by @xmath40 @xmath41\ ] ] where @xmath42\ ] ] @xmath43 @xmath44 @xmath45 @xmath46 @xmath47 @xmath48 @xmath49 @xmath50 for sake of clarity , @xmath51 and @xmath52 are the symmetric expressions of the time constants @xmath53 and @xmath54 ; @xmath55 is the sum of the @xmath56 products of @xmath57 different @xmath53 selected among the total @xmath58 ( for example @xmath59 and @xmath60 ) . the derivative of @xmath37 with respect to @xmath61 is given by @xmath62 assuming @xmath63 and @xmath64 in ( [ eq:2.3 ] ) , one obtains @xmath65 where @xmath66 it is easy to check that @xmath67 exists for @xmath68 only if @xmath69 . the derivative of @xmath67 with respect to @xmath61 , evaluated at @xmath68 and named @xmath70 , is given by @xmath71 denoting by @xmath72 and @xmath73 the two branches of @xmath67 related respectively to the minus and plus signs , their derivatives @xmath74 and @xmath75 are higher than the derivative of @xmath76 , equal to 0.5 , depending on @xmath77 and @xmath78 , given by @xmath79 @xmath80 in detail , the number of the derivatives @xmath74 and @xmath75 higher than 0.5 are the following : zero if @xmath81 and @xmath82 , one if @xmath83 , and two if @xmath84 and @xmath82 . let us denote by @xmath85 @xmath86 where @xmath87 , corresponding to equal derivatives with respect to @xmath61 of @xmath76 and @xmath67 , is a root of @xmath88 . differentiating ( [ eq:2.5 ] ) with respect to @xmath61 once and twice , one obtains @xmath89 @xmath90 it is worthwhile to note that @xmath91 , @xmath92 , and @xmath93 , where @xmath78 is given by ( [ eq:2.17 ] ) . evaluating @xmath94 , where @xmath95 is a root of @xmath96 given by ( [ eq:2.20 ] ) , one obtains @xmath97 eliminating @xmath98 and @xmath99 from @xmath100 and @xmath63 , given by ( [ eq:2.2 ] ) and ( [ eq:2.3 ] ) , yields @xmath101 denote by @xmath102 and @xmath103 the two straight lines whose equations in the ( @xmath104)-plane are obtained introducing in ( [ eq:2.6 ] ) respectively @xmath105 , @xmath106 and @xmath107 , @xmath108 ( see figs . 2 and 5 ) ; denote further by @xmath109 , @xmath110 and @xmath111 the vertices of a triangle , whose sides are the axis @xmath112 and the two lines @xmath102 and @xmath103 . the coordinates of these vertices are given by @xmath113 @xmath114 considering ( [ eq:2.23 ] ) , the coordinates @xmath115 and @xmath116 of the points lying on @xmath102 and @xmath103 at @xmath117 are given by @xmath118 \frac{\sqrt{p^{2}(y_{bi})+q^{2}(y_{bi})-h^{2}}}{y_{bi}}\\ ] ] @xmath119 \frac{\sqrt{p^{2}(y_{ai})+q^{2}(y_{ai})-h^{2}}}{y_{ai}}\ .\ ] ] it is easy to check that , when @xmath120 , the absolute values of @xmath121 and @xmath122 , if @xmath20 , are equal to @xmath123 and , if @xmath22 , to @xmath124 given by @xmath125 when the process transfer function does not have zeros , @xmath126 and thus @xmath127 , @xmath128 , @xmath129 , @xmath130 hold ; therefore , the function @xmath38 , given by ( [ eq:2.1 ] ) , is a quasi - polynomial and the pontryagin s results are integrally applicable . the following two conditions derived from theorem 3.2 of @xcite and from theorem 13.7 of @xcite , respectively , must be satisfied in order to have a stable system : * condition no . 1 + consider that the principal term of @xmath38 , given by ( [ eq:2.1 ] ) , is @xmath131 , set @xmath132 and let @xmath133 be an appropriate constant such that the coefficient of @xmath134 in @xmath37 does not vanish at @xmath135 . the number @xmath136 of the real roots of @xmath37 in the interval @xmath137 for sufficiently large @xmath138 must be @xmath139 * condition no . 2 + for all the zeros @xmath140 of the function @xmath37 the inequality @xmath141 , that is @xmath142 , must hold . in order to study both stable and unstable free - delay plants , the following two cases , adopted also in @xcite , are considered : 1 . @xmath143 ( even number of negative plant time constants @xmath5 ) + @xmath144 and @xmath145 . 2 . @xmath146 ( odd number of negative plant time constants @xmath5 ) + @xmath147 and @xmath148 . from ( [ eq:2.3 ] ) , ( [ eq:2.9 ] ) and ( [ eq:2.10 ] ) it follows that the coefficient of the highest degree of @xmath61 in @xmath37 is @xmath149 when @xmath58 is even and @xmath150 when odd ; hence we assume @xmath151 if @xmath58 is even and @xmath152 if odd . and @xmath153 typical functions @xmath154 and @xmath155 are plotted in fig . 2 ; according to ( [ eq:2.3 ] ) there is one root of @xmath37 at @xmath68 and one for each intersection of @xmath155 with the horizontal line having the ordinate equal to a given @xmath156 . denoting by @xmath157 the number of the intersections between @xmath155 and @xmath64 corresponding to @xmath136 in fig . 2 and assuming that no local minimum or maximum of @xmath155 is equal to @xmath158 for @xmath159 , the relationship between @xmath157 and @xmath136 is given by @xmath160 where @xmath78 is according to ( [ eq:2.17 ] ) ; ( [ eq:3.2 ] ) can be easily checked considering that from ( [ eq:2.5 ] ) , ( [ eq:2.20 ] ) and ( [ eq:2.21 ] ) it follows @xmath91 , @xmath161 and @xmath162 , and also that @xmath144 must hold if @xmath143 and @xmath147 if @xmath146 . since @xmath64 is the common limit value of @xmath156 for the two considered cases ( @xmath146 and @xmath143 ) , the existence of the @xmath157 intersections given by ( [ eq:3.2 ] ) represents a prerequisite of a plant to be made stable . the number @xmath157 can be evaluated by counting the intersections of the plots of @xmath76 and @xmath67 , given by ( [ eq:2.14 ] ) . from ( [ eq:2.14 ] ) it follows that @xmath67 is an odd function of @xmath61 ; moreover , assuming @xmath163 $ ] and @xmath164 $ ] , @xmath165 and @xmath166 can be expressed as * @xmath58 even + @xmath167 and @xmath168 . * @xmath58 odd + @xmath169 for @xmath170 and @xmath171 for @xmath172 , + @xmath173 for @xmath170 and @xmath174 for @xmath172 . if @xmath67 has no pole , splitting @xmath157 into @xmath175 ( @xmath176 ) and @xmath177 and considering the above described behavior of @xmath67 at @xmath178 and also at @xmath68 , one obtains * @xmath176 + @xmath179 if @xmath81 and @xmath82 , + @xmath180 if @xmath83 , + @xmath181 if @xmath84 and @xmath82 , + where @xmath77 and @xmath78 are given by ( [ eq:2.16 ] ) and ( [ eq:2.17 ] ) . * @xmath182 ; @xmath183 ( see fig . 3 ( a ) ) + @xmath184 if @xmath58 is even , + @xmath185 if @xmath58 is odd and @xmath186 , + @xmath187 if @xmath58 is odd and @xmath188 , + where @xmath189=sign[-(-1)^{n}u(n , n-1)u(n , n)]$ ] . it is clear that , if @xmath67 has no pole , @xmath157 will be always lower than the value required by ( [ eq:3.2 ] ) for enough large @xmath58 . a positive solution can be reached only if @xmath67 is provided with a suitable number of poles , since @xmath177 is increased by one for each added pole ( see fig . 3 ) ; this happens if @xmath190 for case ( b1 ) , if @xmath191 for case ( b2 ) and without further condition for case ( b3 ) , where @xmath85 is given by ( [ eq:2.19 ] ) . since the denominator of @xmath67 is a polynomial of @xmath192 of degree @xmath193 , the maximum number of poles of @xmath67 is equal to @xmath194 and the actual number can be determined by means of the sturm theorem , as detailed in appendix a. and @xmath67 for @xmath195 the sought - after procedure can be detailed as follows : 1 . process parameters ( see fig . 4 ) + the stability region in the process parameters plane is that where ( [ eq:3.2 ] ) holds ; its boundary line is a proper set of the boundary lines of the zones with different numbers @xmath157 , that is , @xmath196 and @xmath197 according to ( [ eq:2.16 ] ) and ( [ eq:2.17 ] ) and @xmath198 , as explained in appendix a , and eventually @xmath199 given by ( [ eq:2.19 ] ) . since the expressions of these boundary lines are functions of @xmath28 , it is possible to evaluate the stability range of @xmath7 when the parameters @xmath5 are known and , conversely , of each @xmath5 when @xmath7 and the remaining @xmath5 are known . moreover , if one needs only to know if a given plant can be made stable , it is not necessary to determine the stability regions , but it is enough to examine the plots of @xmath67 and @xmath76 and to check whether the number of the intersections @xmath157 satisfies ( [ eq:3.2 ] ) . controller parameter @xmath156 ( see fig . 2 ) + the requirement , stated as condition no . 1 , is fulfilled if the selected value of @xmath156 is included in the interval from @xmath158 to @xmath200 for @xmath143 or from @xmath201 to @xmath158 for @xmath146 , where @xmath200 and @xmath201 are respectively the local maxima and minima of @xmath155 nearest to @xmath158 ; both can be calculated by introducing in ( [ eq:2.5 ] ) the related root of @xmath96 obtained from ( [ eq:2.20 ] ) . a finite number of these maxima or minima must be examined in order to find the nearest to @xmath158 , exactly up to the first value of @xmath95 higher than @xmath202 . this limit @xmath202 is the largest positive root of the derivative with respect to @xmath95 of @xmath203 , since @xmath203 , given by ( [ eq:2.22 ] ) , monotonically increases for @xmath204 . it is obvious that , if such root does not exist , only the first value must be considered . controller parameters @xmath205 , @xmath112 ( see fig . 5 ) + considering ( [ eq:2.2 ] ) , ( [ eq:2.6 ] ) and ( [ eq:2.13 ] ) , the requirement , stated as condition no . 2 , is fulfilled if the following inequalities @xmath206 hold for each root @xmath140 of @xmath37 , given by ( [ eq:2.3 ] ) , evaluated with a value of @xmath156 included in the above specified interval . + the stability region in the ( @xmath207)-plane consists of the intersection of a finite number of triangles ; each of them is related to a couple of roots @xmath208 and @xmath209 of @xmath37 , has the axis @xmath112 and the two straight lines given by ( [ eq:3.3 ] ) as sides and the points @xmath110 , @xmath109 and @xmath111 as vertices , whose coordinates are given by ( [ eq:2.24 ] ) and ( [ eq:2.25 ] ) ( see figs . 2 and 5 ) . since , as @xmath210 , @xmath121 and @xmath115 approach @xmath211 and @xmath122 and @xmath116 approach @xmath212 , each triangle includes definitely the first one when @xmath213 . this limit @xmath214 is the bigger among @xmath215 and the largest root of the derivatives with respect to @xmath209 of @xmath121 and @xmath115 , given by ( [ eq:2.25 ] ) and ( [ eq:2.26 ] ) . therefore , it is not necessary to examine the triangles for @xmath213 . a second - order plant , whose transfer function is without zeros , is considered as example and the stability regions of the plant and the controller parameters are depicted respectively in figs . 4 and 5 . in fig . 4 this region consists of the following : * @xmath216 : @xmath143 ; @xmath217 and @xmath218 * @xmath219 : @xmath143 ; @xmath220 and @xmath218 * @xmath221 : @xmath146 ; @xmath222 where @xmath223 and @xmath224 according to ( [ eq:2.16 ] ) and ( [ eq:2.17 ] ) . the required number @xmath157 of the intersections between @xmath67 and @xmath76 , equal to @xmath225 as per ( [ eq:3.2 ] ) , coincide with the actual number only in this zone . since @xmath226 @xmath227 , considering the point ( @xmath228 ; @xmath229 ) lying in @xmath216 of fig . 4 , one obtains @xmath230 from ( [ eq:2.20 ] ) for the first root @xmath231 of @xmath232 and @xmath233 from ( [ eq:2.5 ] ) . since @xmath234 must hold , let us assume @xmath156 equal to @xmath235 ; for the first two roots higher than zero of @xmath37 one obtains @xmath236 and @xmath237 from ( [ eq:2.3 ] ) and hence @xmath238 and @xmath239 from ( [ eq:2.4 ] ) . from ( [ eq:2.24 ] ) and ( [ eq:2.25 ] ) it follows @xmath240 , @xmath241 , @xmath242 and @xmath243 . similarly , for the third and fourth roots of @xmath37 one obtains @xmath244 and @xmath245 , @xmath246 , @xmath247 , @xmath248 and @xmath249 ; moreover @xmath250 and @xmath251 from ( [ eq:2.26 ] ) and ( [ eq:2.27 ] ) . and @xmath252 and @xmath112 for @xmath253 , @xmath254 , @xmath228 , @xmath229 , @xmath255 the function @xmath38 , given by ( [ eq:2.1 ] ) , can be rewritten as @xmath256 where @xmath257 @xmath258 @xmath20 or @xmath22 and @xmath259 , as explained in section 2 . since the denominator of @xmath38 is the numerator of the plant transfer function , the poles of @xmath38 are the zeros of the plant transfer function ( @xmath260 ) . in this case the proposed procedure will be in accordance with theorem 4.1 , a generalization of the theorems applied in section 3 and of theorem 13.5 of @xcite ; it will be here enunciated and proved by use of the principle of the argument . consider a function @xmath38 of the form in ( [ eq:4.1 ] ) , set @xmath132 and let @xmath133 be an appropriate constant such that the coefficient of @xmath261 in the numerator of @xmath37 does not vanish at @xmath135 . assume further that the @xmath262 poles with positive real part of @xmath38 lie all in the rectangle @xmath263 , described by the inequalities @xmath264 , @xmath265 ( @xmath266 ) . suppose finally that the function @xmath38 does not assume the value of zero on the imaginary axis , that is , @xmath267 . all the zeros of @xmath38 lie to the left of the imaginary axis if and only if : 1 . the vector @xmath268 for real @xmath61 ranging from @xmath269 to @xmath270 continually revolves in the positive direction at a positive velocity , that is , the inequality @xmath271 is satisfied for each root @xmath140 of @xmath37 . 2 . the number @xmath136 of the roots of @xmath37 in the interval @xmath272 for sufficiently large @xmath138 is @xmath273 denote by @xmath274 , @xmath275 , @xmath276 and @xmath277 the corners of the rectangle @xmath263 whose coordinates in the @xmath278-plane are respectively @xmath279 , @xmath280 , @xmath281 and @xmath282 ; the arguments @xmath283 and @xmath284 of @xmath38 are given by @xmath285 and @xmath286 , where @xmath287 if @xmath288 or @xmath289 if @xmath290 , and @xmath291 and @xmath292 simultaneously with @xmath293 . denote by @xmath294 the number of the zeros of @xmath38 lying in the rectangle @xmath263 , by @xmath295 the variation of the argument of @xmath38 in the counterclockwise direction as @xmath278 moves around the contour of @xmath263 from @xmath274 to @xmath277 through @xmath275 and @xmath276 , and by @xmath296 as @xmath278 moves directly from @xmath277 to @xmath274 . using the principle of the argument yields @xmath297 since @xmath298 for stability and , for @xmath299 , @xmath300 as per condition ( a ) and @xmath301 , ( [ eq:4.2 ] ) follows from ( [ eq:4.3 ] ) and the condition ( b ) is satisfied . the procedure detailed in section 3 for process transfer functions without zeros is fully applicable to process transfer functions with zeros ; it is only necessary to replace ( [ eq:3.1 ] ) with ( [ eq:4.2 ] ) and to consider @xmath302 and @xmath303 as functions given by ( [ eq:2.11 ] ) and ( [ eq:2.12 ] ) instead of @xmath127 and @xmath128 . moreover , for @xmath22 , the rectangle , according to ( [ eq:3.3 ] ) for @xmath304 and provided with horizontal sides symmetric with respect to the axis @xmath205 at a distance given by ( [ eq:2.28 ] ) , must be considered in the @xmath305-plane . in this note , both stable and unstable delay - free arbitrary - order plants , provided with one time delay and pid controller , have been examined and the related stability regions in process and controller parameter spaces have been determined by use of the pontryagin s studies . the proposed procedure , consisting of a finite number of steps , yields explicit expressions of the boundaries of the stability zone for the controller parameters . these results can be implemented in tuning charts , which become a complete tool for the design and the maintenance of control systems . the sturm theorem states that the number of real roots of an algebraic equation with real coefficients whose real roots are simple over an interval , the endpoints of which are not roots , is equal to the difference between the numbers of sign changes of the sturm chains formed for the interval ends . given a function @xmath306 of degree @xmath307 , assume @xmath308 and define the sturm functions by @xmath309,\ ] ] where @xmath310 $ ] is a polynomial quotient . these functions can be written as @xmath311 where @xmath312 depends on the coefficients of @xmath313 in @xmath314 . in our case @xmath315 and @xmath316 hold ; since the roots must be positive , the required interval is from @xmath317 to @xmath318 and , therefore , the signs of each sturm function at these ends are the signs of @xmath312 evaluated respectively for @xmath319 and @xmath320 .

the stability of feedback systems consisting of linear time - delay plants and pid controllers has been investigated for many years by means of several methods , of which the nyquist criterion , a generalization of the hermite - biehler theorem , and the root location method are well known . the main purpose of these researches is to determine the range of controller parameters that allow stability . explicit and complete expressions of the boundaries of these regions and computation procedures with a finite number of steps are now available only for first - order plants , provided with one time delay . in this note , the same results , based on pontryagin s studies , are presented for arbitrary - order plants . [ multiblock footnote omitted ]

computational algorithms are constructed on the basis of certain primitive operations . these operations manipulate data that describe `` numbers . '' these `` numbers '' are elements of a `` numerical domain , '' that is , a mathematical object such as the field of real numbers , the ring of integers , different semirings etc . in practice , elements of the numerical domains are replaced by their computer representations , that is , by elements of certain finite models of these domains . examples of models that can be conveniently used for computer representation of real numbers are provided by various modifications of floating point arithmetics , approximate arithmetics of rational numbers @xcite , interval arithmetics etc . the difference between mathematical objects ( `` ideal '' numbers ) and their finite models ( computer representations ) results in computational ( for instance , rounding ) errors . an algorithm is called _ universal _ if it is independent of a particular numerical domain and/or its computer representation @xcite . a typical example of a universal algorithm is the computation of the scalar product @xmath0 of two vectors @xmath1 and @xmath2 by the formula @xmath3 . this algorithm ( formula ) is independent of a particular domain and its computer implementation , since the formula is well - defined for any semiring . it is clear that one algorithm can be more universal than another . for example , the simplest newton cotes formula , the rectangular rule , provides the most universal algorithm for numerical integration . in particular , this formula is valid also for idempotent integration ( that is , over any idempotent semiring , see @xcite ) . other quadrature formulas ( for instance , combined trapezoid rule or the simpson formula ) are independent of computer arithmetics and can be used ( for instance , in the iterative form ) for computations with arbitrary accuracy . in contrast , algorithms based on gauss jacobi formulas are designed for fixed accuracy computations : they include constants ( coefficients and nodes of these formulas ) defined with fixed accuracy . ( certainly , algorithms of this type can be made more universal by including procedures for computing the constants ; however , this results in an unjustified complication of the algorithms . ) modern achievements in software development and mathematics make us consider numerical algorithms and their classification from a new point of view . conventional numerical algorithms are oriented to software ( or hardware ) implementation based on floating point arithmetic and fixed accuracy . however , it is often desirable to perform computations with variable ( and arbitrary ) accuracy . for this purpose , algorithms are required that are independent of the accuracy of computation and of the specific computer representation of numbers . in fact , many algorithms are independent not only of the computer representation of numbers , but also of concrete mathematical ( algebraic ) operations on data . in this case , operations themselves may be considered as variables . such algorithms are implemented in the form of _ generic programs _ based on abstract data types that are defined by the user in addition to the predefined types provided by the language . the corresponding program tools appeared as early as in simula-67 , but modern object - oriented languages ( like @xmath4 , see , for instance , @xcite ) are more convenient for generic programming . computer algebra algorithms used in such systems as mathematica , maple , reduce , and others are also highly universal . a different form of universality is featured by iterative algorithms ( beginning with the successive approximation method ) for solving differential equations ( for instance , methods of euler , euler cauchy , runge kutta , adams , a number of important versions of the difference approximation method , and the like ) , methods for calculating elementary and some special functions based on the expansion in taylor s series and continuous fractions ( pad approximations ) . these algorithms are independent of the computer representation of numbers . the concept of a generic program was introduced by many authors ; for example , in @xcite such programs were called ` program schemes . ' in this paper , we discuss universal algorithms implemented in the form of generic programs and their specific features . this paper is closely related to @xcite , in which the concept of a universal algorithm was defined and software and hardware implementation of such algorithms was discussed in connection with problems of idempotent mathematics , see , for instance , @xcite . the so - called _ idempotent correspondence principle _ , see @xcite , linking this mathematics with the usual mathematics over fields , will be discussed below . in a nutshell , there exists a correspondence between interesting , useful , and important constructions and results concerning the field of real ( or complex ) numbers and similar constructions dealing with various idempotent semirings . this correspondence can be formulated in the spirit of the well - known n. bohr s _ correspondence principle _ in quantum mechanics ; in fact , the two principles are closely connected ( see @xcite ) . in a sense , the traditional mathematics over numerical fields can be treated as a ` quantum ' theory , whereas the idempotent mathematics can be treated as a ` classical ' shadow ( or counterpart ) of the traditional one . it is important that the idempotent correspondence principle is valid for algorithms , computer programs and hardware units . in quantum mechanics the _ superposition principle _ means that the schrdinger equation ( which is basic for the theory ) is linear . similarly in idempotent mathematics the ( idempotent ) superposition principle ( formulated by v. p. maslov ) means that some important and basic problems and equations that are nonlinear in the usual sense ( for instance , the hamilton - jacobi equation , which is basic for classical mechanics and appears in many optimization problems , or the bellman equation and its versions and generalizations ) can be treated as linear over appropriate idempotent semirings , see @xcite . note that numerical algorithms for infinite dimensional linear problems over idempotent semirings ( for instance , idempotent integration , integral operators and transformations , the hamilton jacobi and generalized bellman equations ) deal with the corresponding finite - dimensional approximations . thus idempotent linear algebra is the basis of the idempotent numerical analysis and , in particular , the _ discrete optimization theory_. b. a. carr @xcite ( see also @xcite ) used the idempotent linear algebra to show that different optimization problems for finite graphs can be formulated in a unified manner and reduced to solving bellman equations , that is , systems of linear algebraic equations over idempotent semirings . he also generalized principal algorithms of computational linear algebra to the idempotent case and showed that some of these coincide with algorithms independently developed for solution of optimization problems . for example , bellman s method of solving the shortest path problem corresponds to a version of jacobi s method for solving a system of linear equations , whereas ford s algorithm corresponds to a version of gauss seidel s method . we briefly discuss bellman equations and the corresponding optimization problems on graphs , and use the ideas of carr to obtain new universal algorithms . we stress that these well - known results can be interpreted as a manifestation of the idempotent superposition principle . note that many algorithms for solving the matrix bellman equation could be found in @xcite . more general problems of linear algebra over the max - plus algebra are examined , for instance in @xcite . we also briefly discuss interval analysis over idempotent and positive semirings . idempotent interval analysis appears in @xcite , where it is applied to the bellman matrix equation . many different problems coming from the idempotent linear algebra , have been considered since then , see for instance @xcite . it is important to observe that intervals over an idempotent semiring form a new idempotent semiring . hence universal algorithms can be applied to elements of this new semiring and generate interval extensions of the initial algorithms . this paper is about software implementations of universal algorithms for solving the matrix bellman equations over semirings . in section [ s : sem ] we present an introduction to mathematics of semirings and especially to the tropical ( idempotent ) mathematics , that is , the area of mathematics working with _ idempotent semirings _ ( that is , semirings with idempotent addition ) . in section [ s : main ] we present a number of well - known and new universal algorithms of linear algebra over semirings , related to discrete matrix bellman equation and algebraic path problem . these algorithms are closely related to their linear - algebraic prototypes described , for instance , in the celebrated book of golub and van loan @xcite which serves as the main source of such prototypes . following the style of @xcite we present them in matlab code . the perspectives and experience of their implementation are also discussed . a broad class of universal algorithms is related to the concept of a semiring . we recall here the definition ( see , for instance , @xcite ) . a set @xmath5 is called a _ semiring _ if it is endowed with two associative operations : _ addition _ @xmath6 and _ multiplication _ @xmath7 such that addition is commutative , multiplication distributes over addition from either side , @xmath8 ( resp . , @xmath9 ) is the neutral element of addition ( resp . , multiplication ) , @xmath10 for all @xmath11 , and @xmath12 . let the semiring @xmath5 be partially ordered by a relation @xmath13 such that @xmath8 is the least element and the inequality @xmath14 implies that @xmath15 , @xmath16 , and @xmath17 for all @xmath18 ; in this case the semiring @xmath5 is called _ positive _ ( see , for instance , @xcite ) . an element @xmath19 is called _ invertible _ if there exists an element @xmath20 such that @xmath21 . a semiring @xmath5 is called a _ semifield _ if every nonzero element is invertible . a semiring @xmath5 is called _ idempotent _ if @xmath22 for all @xmath11 . in this case the addition @xmath6 defines a _ canonical partial order _ @xmath13 on the semiring @xmath5 by the rule : @xmath23 iff @xmath24 . it is easy to prove that any idempotent semiring is positive with respect to this order . note also that @xmath25 with respect to the canonical order . in the sequel , we shall assume that all idempotent semirings are ordered by the canonical partial order relation . we shall say that a positive ( for instance , idempotent ) semiring @xmath5 is _ complete _ if for every subset @xmath26 there exist elements @xmath27 and @xmath28 , and if the operations @xmath6 and @xmath7 distribute over such sups and infs . the most well - known and important examples of positive semirings are `` numerical '' semirings consisting of ( a subset of ) real numbers and ordered by the usual linear order @xmath29 on @xmath30 : the semiring @xmath31 with the usual operations @xmath32 , @xmath33 and neutral elements @xmath34 , @xmath35 , the semiring @xmath36 with the operations @xmath37 , @xmath38 and neutral elements @xmath39 , @xmath40 , the semiring @xmath41 , where @xmath42 , @xmath43 for all @xmath44 , @xmath45 if @xmath46 , and @xmath47 , and the semiring @xmath48 } = [ a , b]$ ] , where @xmath49 , with the operations @xmath50 , @xmath51 and neutral elements @xmath52 , @xmath53 . the semirings @xmath54 , @xmath55 , and @xmath48 } = [ a , b]$ ] are idempotent . the semirings @xmath55 , @xmath56}_{\mathrm{max , min}}$ ] , @xmath57 are complete . remind that every partially ordered set can be imbedded to its completion ( a minimal complete set containing the initial one ) . the semiring @xmath58 with operations @xmath59 and @xmath60 and neutral elements @xmath61 , @xmath62 is isomorphic to @xmath54 . the semiring @xmath54 is also called the _ max - plus algebra_. the semifields @xmath54 and @xmath63 are called _ tropical algebras_. the term `` tropical '' initially appeared in @xcite for a discrete version of the max - plus algebra as a suggestion of ch . choffrut , see also @xcite . many mathematical constructions , notions , and results over the fields of real and complex numbers have nontrivial analogs over idempotent semirings . idempotent semirings have become recently the object of investigation of new branches of mathematics , _ idempotent mathematics _ and _ tropical geometry _ , see , for instance @xcite . denote by @xmath64 a set of all matrices @xmath65 with @xmath66 rows and @xmath67 columns whose coefficients belong to a semiring @xmath5 . the sum @xmath68 of matrices @xmath69 and the product @xmath70 of matrices @xmath71 and @xmath72 are defined according to the usual rules of linear algebra : @xmath73 and @xmath74 where @xmath75 and @xmath76 . note that we write @xmath70 instead of @xmath77 . if the semiring @xmath5 is positive , then the set + @xmath64 is ordered by the relation @xmath78 iff @xmath79 in @xmath5 for all @xmath80 , @xmath81 . the matrix multiplication is consistent with the order @xmath13 in the following sense : if @xmath82 , @xmath83 and @xmath84 , @xmath85 , then @xmath86 in @xmath87 . the set @xmath88 of square @xmath89 matrices over a [ positive , idempotent ] semiring @xmath5 forms a [ positive , idempotent ] semi - ring with a zero element @xmath90 , where @xmath91 , @xmath92 , and a unit element @xmath93 , where @xmath94 if @xmath95 and @xmath96 otherwise . the set @xmath97 is an example of a noncommutative semiring if @xmath98 . in what follows , we are mostly interested in complete positive semirings , and particularly in idempotent semirings . regarding examples of the previous section , recall that the semirings @xmath56}_{\mathrm{max , min}}$ ] , @xmath99 , @xmath100 and @xmath101 are complete positive , and the semirings @xmath56}_{\mathrm{max , min}}$ ] , @xmath55 and @xmath102 are idempotent . @xmath103 is a completion of @xmath104 , and @xmath55 ( resp . @xmath102 ) are completions of @xmath54 ( resp . @xmath63 ) . more generally , we note that any positive semifield @xmath5 can be completed by means of a standard procedure , which uses dedekind cuts and is described in @xcite . the result of this completion is a semiring @xmath105 , which is not a semifield anymore . the semiring of matrices @xmath88 over a complete positive ( resp . , idempotent ) semiring is again a complete positive ( resp . , idempotent ) semiring . for more background in complete idempotent semirings , the reader is referred to @xcite . in any complete positive semiring @xmath5 we have a unary operation of _ closure _ @xmath106 defined by @xmath107 using that the operations @xmath6 and @xmath7 distribute over such sups , it can be shown that @xmath108 is the * least solution * of @xmath109 and @xmath110 , and also that @xmath111 is the the least solution of @xmath112 and @xmath113 . in the case of idempotent addition becomes particularly nice : @xmath114 if a positive semiring @xmath5 is not complete , then it often happens that the closure operation can still be defined on some `` essential '' subset of @xmath5 . also recall that any positive semifield @xmath5 can be completed @xcite , and then the closure is defined for every element of the completion . in numerical semirings the operation @xmath115 is usually very easy to implement : @xmath116 if @xmath117 in @xmath31 , or @xmath103 and @xmath118 if @xmath119 in @xmath103 ; @xmath120 if @xmath121 in @xmath54 and @xmath55 , @xmath122 if @xmath123 in @xmath55 , @xmath120 for all @xmath44 in @xmath48}$ ] . in all other cases @xmath124 is undefined . the closure operation in matrix semirings over a complete positive semiring @xmath5 can be defined as in : @xmath125 and one can show that it is the least solution @xmath126 satisfying the matrix equations @xmath127 and @xmath128 . equivalently , @xmath129 can be defined by induction : let @xmath130 in @xmath131 be defined by , and for any integer @xmath132 and any matrix @xmath133 where @xmath134 , @xmath135 , @xmath136 , @xmath137 , @xmath138 , by definition , @xmath139 d^ * a_{21 } a^*_{11 } & d^ * \end{pmatrix},\ ] ] where @xmath140 . defined here for complete positive semirings , the closure operation is a semiring analogue of the operation @xmath141 and , further , @xmath142 in matrix algebra over the field of real or complex mumbers . this operation can be thought of as * regularized sum * of the series @xmath143 , and the closure operation defined above is another such regularization . thus we can also define the closure operation @xmath144 and @xmath145 in the traditional linear algebra . to this end , note that the recurrence relation above coincides with the formulas of escalator method of matrix inversion in the traditional linear algebra over the field of real or complex numbers , up to the algebraic operations used . hence this algorithm of matrix closure requires a polynomial number of operations in @xmath67 , see below for more details . let @xmath5 be a complete positive semiring . the _ matrix ( or discrete stationary ) bellman equation _ has the form @xmath146 where @xmath147 , @xmath148 , and the matrix @xmath126 is unknown . as in the scalar case , it can be shown that for complete positive semirings , if @xmath129 is defined as in then @xmath149 is the least in the set of solutions to equation with respect to the partial order in @xmath150 . in the idempotent case @xmath151 consider also the case when @xmath152 is @xmath153 _ strictly upper - triangular _ ( such that @xmath154 for @xmath155 ) , or @xmath153 _ strictly lower - triangular _ ( such that @xmath154 for @xmath156 ) . in this case @xmath157 , the all - zeros matrix , and it can be shown by iterating @xmath127 that this equation has a unique solution , namely @xmath158 curiously enough , formula works more generally in the case of numerical idempotent semirings : in fact , the series converges there if and only if it can be truncated to . this is closely related to the principal path interpretation of @xmath129 explained in the next subsection . in fact , theory of the discrete stationary bellman equation can be developed using the identity @xmath159 as an axiom without any explicit formula for the closure ( the so - called _ closed semirings _ , see , for instance , @xcite ) . such theory can be based on the following identities , true both for the case of idempotent semirings and the real numbers with conventional arithmetic ( assumed that @xmath160 and @xmath161 have appropriate sizes ) : @xmath162 this abstract setting unites the case of positive and idempotent semirings with the conventional linear algebra over the field of real and complex numbers . suppose that @xmath5 is a semiring with zero @xmath8 and unity @xmath9 . it is well - known that any square matrix @xmath163 specifies a _ weighted directed graph_. this geometrical construction includes three kinds of objects : the set @xmath126 of @xmath67 elements @xmath164 called _ nodes _ , the set @xmath165 of all ordered pairs @xmath166 such that @xmath167 called _ arcs _ , and the mapping @xmath168 such that @xmath169 . the elements @xmath170 of the semiring @xmath5 are called _ weights _ of the arcs.conversely , any given weighted directed graph with @xmath67 nodes specifies a unique matrix @xmath147 . this definition allows for some pairs of nodes to be disconnected if the corresponding element of the matrix @xmath160 is @xmath8 and for some channels to be `` loops '' with coincident ends if the matrix @xmath160 has nonzero diagonal elements . recall that a sequence of nodes of the form @xmath171 with @xmath172 and @xmath173 , @xmath174 , is called a _ path _ of length @xmath175 connecting @xmath176 with @xmath177 . denote the set of all such paths by @xmath178 . the weight @xmath179 of a path @xmath180 is defined to be the product of weights of arcs connecting consecutive nodes of the path : @xmath181 by definition , for a ` path ' @xmath182 of length @xmath183 the weight is @xmath9 if @xmath95 and @xmath8 otherwise . for each matrix @xmath147 define @xmath184 ( where @xmath185 if @xmath95 and @xmath96 otherwise ) and @xmath186 , @xmath187 . let @xmath188}_{ij}$ ] be the @xmath189th element of the matrix @xmath190 . it is easily checked that @xmath191}_{ij } = \bigoplus_{\substack{i_0 = i,\ , i_k = j\\ 1 \leq i_1 , \ldots , i_{k - 1 } \leq n } } a_{i_0i_1 } \odot \dots \odot a_{i_{k - 1}i_k}.\ ] ] thus @xmath188}_{ij}$ ] is the supremum of the set of weights corresponding to all paths of length @xmath175 connecting the node @xmath192 with @xmath193 . let @xmath129 be defined as in . denote the elements of the matrix @xmath129 by @xmath194 , @xmath195 ; then @xmath196 the closure matrix @xmath129 solves the well - known _ algebraic path problem _ , which is formulated as follows : for each pair @xmath197 calculate the supremum of weights of all paths ( of arbitrary length ) connecting node @xmath198 with node @xmath199 . the closure operation in matrix semirings has been studied extensively ( see , for instance , @xcite and references therein ) . let @xmath200 , so the weights are real numbers . in this case @xmath201 if the element @xmath170 specifies the length of the arc @xmath197 in some metric , then @xmath194 is the length of the shortest path connecting @xmath198 with @xmath199 . let @xmath202 with @xmath37 , @xmath51 . then @xmath203 @xmath204 if the element @xmath170 specifies the `` width '' of the arc @xmath197 , then the width of a path @xmath205 is defined as the minimal width of its constituting arcs and the element @xmath194 gives the supremum of possible widths of all paths connecting @xmath198 with @xmath199 . let @xmath206 and suppose @xmath170 gives the _ profit _ corresponding to the transition from @xmath198 to @xmath199 . define the vector @xmath207 whose element @xmath208 gives the _ terminal profit _ corresponding to exiting from the graph through the node @xmath198 . of course , negative profits ( or , rather , losses ) are allowed . let @xmath66 be the total profit corresponding to a path @xmath209 , that is @xmath210 then it is easy to check that the supremum of profits that can be achieved on paths of length @xmath175 beginning at the node @xmath198 is equal to @xmath211 and the supremum of profits achievable without a restriction on the length of a path equals @xmath212 . note that in the formulas of this section we are using distributivity of the multiplication @xmath7 with respect to the addition @xmath6 but do not use the idempotency axiom . thus the algebraic path problem can be posed for a nonidempotent semiring @xmath5 as well ( see , for instance , @xcite ) . for instance , if @xmath213 , then @xmath214 if @xmath215 but the matrix @xmath216 is invertible , then this expression defines a regularized sum of the divergent matrix power series @xmath217 . we emphasize that this connection between the matrix closure operation and solutions to the bellman equation gives rise to a number of different algorithms for numerical calculation of the matrix closure . all these algorithms are adaptations of the well - known algorithms of the traditional computational linear algebra , such as the gauss jordan elimination , various iterative and escalator schemes , etc . this is a special case of the idempotent superposition principle ( see below ) . traditional interval analysis is a nontrivial and popular mathematical area , see , for instance , @xcite . an `` idempotent '' version of interval analysis ( and moreover interval analysis over positive semirings ) appeared in @xcite . rather many publications on the subject appeared later , see , for instance , @xcite . interval analysis over the positive semiring @xmath104 was discussed in @xcite . let a set @xmath5 be partially ordered by a relation @xmath13 . a _ closed interval _ in @xmath5 is a subset of the form @xmath218 = \{\ , x \in s \mid { \underline{{\mathbf x}}}\preceq x \preceq { \overline{{\mathbf x}}}\ , \}$ ] , where the elements @xmath219 are called _ lower _ and _ upper bounds _ of the interval @xmath220 . the order @xmath13 induces a partial ordering on the set of all closed intervals in @xmath5 : @xmath221 iff @xmath222 and @xmath223 . a _ weak interval extension _ @xmath224 of a positive semiring @xmath5 is the set of all closed intervals in @xmath5 endowed with operations @xmath6 and @xmath7 defined as @xmath225 $ ] , @xmath226 $ ] and a partial order induced by the order in @xmath5 . the closure operation in @xmath224 is defined by @xmath227 $ ] . there are some other interval extensions ( including the so - called strong interval extension @xcite ) but the weak extension is more convenient . the extension @xmath224 is positive ; @xmath224 is idempotent if @xmath5 is an idempotent semiring . a universal algorithm over @xmath5 can be applied to @xmath224 and we shall get an interval version of the initial algorithm . usually both versions have the same complexity . for the discrete stationary bellman equation and the corresponding optimization problems on graphs , interval analysis was examined in @xcite in details . other problems of idempotent linear algebra were examined in @xcite . idempotent mathematics appears to be remarkably simpler than its traditional analog . for example , in traditional interval arithmetic , multiplication of intervals is not distributive with respect to addition of intervals , whereas in idempotent interval arithmetic this distributivity is preserved . moreover , in traditional interval analysis the set of all square interval matrices of a given order does not form even a semigroup with respect to matrix multiplication : this operation is not associative since distributivity is lost in the traditional interval arithmetic . on the contrary , in the idempotent ( and positive ) case associativity is preserved . finally , in traditional interval analysis some problems of linear algebra , such as solution of a linear system of interval equations , can be very difficult ( more precisely , they are @xmath228-hard , see @xcite and references therein ) . it was noticed in @xcite that in the idempotent case solving an interval linear system requires a polynomial number of operations ( similarly to the usual gauss elimination algorithm ) . two properties that make the idempotent interval arithmetic so simple are monotonicity of arithmetic operations and positivity of all elements of an idempotent semiring . interval estimates in idempotent mathematics are usually exact . in the traditional theory such estimates tend to be overly pessimistic . there is a nontrivial analogy between mathematics of semirings and quantum mechanics . for example , the field of real numbers can be treated as a `` quantum object '' with respect to idempotent semirings . so idempotent semirings can be treated as `` classical '' or `` semi - classical '' objects with respect to the field of real numbers . let @xmath30 be the field of real numbers and @xmath31 the subset of all non - negative numbers . consider the following change of variables : @xmath229 where @xmath230 , @xmath231 ; thus @xmath232 , @xmath233 . denote by @xmath8 the additional element @xmath234 and by @xmath5 the extended real line @xmath235 . the above change of variables has a natural extension @xmath236 to the whole @xmath5 by @xmath237 ; also , we denote @xmath238 . denote by @xmath239 the set @xmath5 equipped with the two operations @xmath240 ( generalized addition ) and @xmath241 ( generalized multiplication ) such that @xmath236 is a homomorphism of @xmath242 to @xmath243 . this means that @xmath244 and @xmath245 , that is , @xmath246 and @xmath247 . it is easy to prove that @xmath248 as @xmath249 . @xmath31 and @xmath239 are isomorphic semirings ; therefore we have obtained @xmath54 as a result of a deformation of @xmath31 . we stress the obvious analogy with the quantization procedure , where @xmath250 is the analog of the planck constant . in these terms , @xmath31 ( or @xmath30 ) plays the part of a `` quantum object '' while @xmath54 acts as a `` classical '' or `` semi - classical '' object that arises as the result of a _ dequantization _ of this quantum object . in the case of @xmath63 , the corresponding dequantization procedure is generated by the change of variables @xmath251 . there is a natural transition from the field of real numbers or complex numbers to the idempotent semiring @xmath54 ( or @xmath63 ) . this is a composition of the mapping @xmath252 and the deformation described above . in general an _ idempotent dequantization _ is a transition from a basic field to an idempotent semiring in mathematical concepts , constructions and results , see @xcite for details . idempotent dequantization suggests the following formulation of the idempotent correspondence principle : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ there exists a heuristic correspondence between interesting , useful , and important constructions and results over the field of real ( or complex ) numbers and similar constructions and results over idempotent semirings in the spirit of n. bohr s correspondence principle in quantum mechanics . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ fig.2 ] thus idempotent mathematics can be treated as a `` classical shadow ( or counterpart ) '' of the traditional mathematics over fields . a systematic application of this correspondence principle leads to a variety of theoretical and applied results , see , for instance , @xcite . relations to quantum physics are discussed in detail , for instance , in @xcite . in this paper we aim to develop a practical systematic application of the correspondence principle to the algorithms of linear algebra and discrete mathematics . for the remainder of this subsection let us focus on an idea how the idempotent correspondence principle may lead to a unifying approach to hardware design . ( see @xcite for more information . ) the most important and standard numerical algorithms have many hardware realizations in the form of technical devices or special processors . these devices often can be used as prototypes for new hardware units resulting from mere substitution of the usual arithmetic operations by their semiring analogs ( and additional tools for generating neutral elements @xmath8 and @xmath9 ) . of course , the case of numerical semirings consisting of real numbers ( maybe except neutral elements ) and semirings of numerical intervals is the most simple and natural . note that for semifields ( including @xmath54 and @xmath63 ) the operation of division is also defined . good and efficient technical ideas and decisions can be taken from prototypes to new hardware units . thus the correspondence principle generates a regular heuristic method for hardware design . note that to get a patent it is necessary to present the so - called ` invention formula ' , that is to indicate a prototype for the suggested device and the difference between these devices . consider ( as a typical example ) the most popular and important algorithm of computing the scalar product of two vectors : @xmath253 the universal version of for any semiring @xmath160 is obvious : @xmath254 in the case @xmath255 this formula turns into the following one : @xmath256 this calculation is standard for many optimization algorithms , so it is useful to construct a hardware unit for computing . there are many different devices ( and patents ) for computing and every such device can be used as a prototype to construct a new device for computing and even . many processors for matrix multiplication and for other algorithms of linear algebra are based on computing scalar products and on the corresponding `` elementary '' devices . using modern technologies it is possible to construct cheap special - purpose multi - processor chips and systolic arrays of elementary processors implementing universal algorithms . see , for instance , @xcite where the systolic arrays and parallel computing issues are discussed for the algebraic path problem . in particular , there is a systolic array of @xmath257 elementary processors which performs computations of the gauss jordan elimination algorithm and can solve the algebraic path problem within @xmath258 time steps . in this section we discuss universal algorithms computing @xmath129 and @xmath149 . we start with the basic escalator and gauss - jordan elimination techniques in subsect . [ ss : gj ] and continue with its specification to the case of toeplitz systems in subsect . [ ss : toep ] . the universal ldm decomposition of bellman equations is explained in subsect . [ ss : ldm ] , followed by its adaptations to symmetric and band matrices in subsect . [ ss : ldmspec ] . the iteration schemes are discussed in subsect . [ ss : iter ] . in the final subsect . [ ss : impl ] we discuss the implementations of universal algorithms . algorithms themselves will be described in a language of matlab , following the tradition of golub and van loan @xcite . this is done for two purposes : 1 ) to simplify the comparison of the algorithms with their prototypes taken mostly from @xcite , 2 ) since the language of matlab is designed for matrix computations . we will not formally describe the rules of our matlab - derived language , preferring just to outline the following important features : * our basic arithmetic operations are @xmath259 , @xmath260 and @xmath108 . * the vectorization of these operations follows the rules of matlab . * we use basic keywords of matlab like ` for ' , ` while ' , if and end , similar to other programming languages like @xmath4 or java . let us give some examples of universal matrix computations in our language : + _ example 1 . _ @xmath261 means that the result of ( scalar ) multiplication of the first @xmath262 components of the @xmath175th column of @xmath160 by the closure of @xmath263 is assigned to the first @xmath262 components of @xmath264 . + _ example 2 . _ @xmath265 means that we add ( @xmath6 ) to the entry @xmath170 of @xmath160 the result of the ( universal ) scalar multiplication of the @xmath266th row with the @xmath262th column of @xmath160 ( assumed that @xmath160 is @xmath153 ) . + _ example 3 . _ @xmath267 means the outer product of the @xmath175th column of @xmath160 with the @xmath268th row of @xmath161 . the entries of resulting matrix @xmath269 equal @xmath270 , for all @xmath271 and @xmath272 . + _ example 4 . _ @xmath273 is the scalar product of vector @xmath44 with vector @xmath274 whose components are taken in the reverse order : the proper algebraic expression is @xmath275 . + _ example 5 . _ the following cycle yields the same result as in the previous example : @xmath276 + * for * @xmath277 + @xmath278 + * end * we first analyse the basic escalator method , based on the definition of matrix closures . let @xmath160 be a square matrix . closures of its main submatrices @xmath279 can be found inductively , starting from @xmath280 , the closure of the first diagonal entry . generally we represent @xmath281 as @xmath282 assuming that we have found the closure of @xmath279 . in this representation , @xmath283 and @xmath284 are columns with @xmath175 entries and @xmath285 is a scalar . we also represent @xmath286 as @xmath287 using we obtain that @xmath288 an algorithm based on can be written as follows . [ a : bordering ] escalator method for computing @xmath129 an @xmath153 matrix @xmath160 with entries @xmath289 , + also used to store the final result + and the intermediate results of the computation process . @xmath290 + * for * @xmath291 + @xmath292 + @xmath293 + @xmath294 + @xmath295 + @xmath296 + @xmath297 + @xmath298 + * end * in full analogy with its linear algebraic prototype , the algorithm requires @xmath299 operations of addition @xmath6 , @xmath299 operations of multiplication @xmath7 , and @xmath67 operations of taking algebraic closure . the linear - algebraic prototype of the method written above is also called the _ bordering method _ in the literature @xcite . alternatively , we can obtain a solution of @xmath300 as a result of elimination process , whose informal explanation is given below . if @xmath129 is defined as @xmath301 ( including the scalar case ) , then @xmath149 is the least solution of @xmath300 for all @xmath160 and @xmath161 of appropriate sizes . in this case , the solution found by the elimination process given below coincides with @xmath149 . for matrix @xmath152 and column vectors @xmath302 and @xmath303 ( restricting without loss of generality to the column vectors ) , the bellman equation @xmath304 can be written as @xmath305 @xmath306 after expressing @xmath307 in terms of @xmath308 from the first equation and substituting this expression for @xmath307 in all other equations from the second to the @xmath67th we obtain @xmath309 @xmath310 @xmath311 note that nontrivial entries in both matrices occupy complementary places , so during computations both matrices can be stored in the same square array @xmath312 . denote its elements by @xmath313 where @xmath175 is the number of eliminated variables . after @xmath314 eliminations we have @xmath315 after @xmath67 eliminations we get @xmath316 . taking as @xmath317 any vector with one coordinate equal to @xmath9 and the rest equal to @xmath8 , we obtain @xmath318 . we write out the following algorithm based on recursion . [ a : eliminate ] gauss - jordan elimination for computing @xmath129 . : an @xmath153 matrix @xmath160 with entries @xmath289 , + also used to store the final result + and intermediate results of the computation process . @xmath319 + @xmath320 + * for * @xmath321 + * if * @xmath322 + @xmath323 + * end * + * end * + * for * @xmath321 + * for * @xmath324 + * if * @xmath325 + @xmath326 + * end * + * end * + * for * @xmath324 + * if * @xmath327 + @xmath328 + * end * + * end * + * end * algorithm [ a : eliminate ] can be regarded as a `` universal floyd - warshall algorithm '' generalizing the well - known algorithms of warshall and floyd for computing the transitive closure of a graph and all optimal paths on a graph . see , for instance , @xcite for the description of these classical methods of discrete mathematics . in turn , these methods can be regarded as specifications of algorithm [ a : eliminate ] to the cases of max - plus and boolean semiring . algorithm [ a : eliminate ] is also close to yershov s `` refilling '' method for inverting matrices and solving systems @xmath329 in the classical linear algebra , see @xcite chapter 2 for details . we start by considering the escalator method for finding the solution @xmath330 to @xmath304 , where @xmath44 and @xmath317 are column vectors . firstly , we have @xmath331 . let @xmath332 be the vector found after @xmath333 steps , and let us write @xmath334 using we obtain that @xmath335 we have to compute @xmath336 . in general , we would have to use algorithm [ a : bordering ] . next we show that this calculation can be done very efficiently when @xmath160 is symmetric toeplitz . formally , a matrix @xmath337 is called _ toeplitz _ if there exist scalars @xmath338 such that @xmath339 for all @xmath266 and @xmath262 . informally , toeplitz matrices are such that their entries are constant along any line parallel to the main diagonal ( and along the main diagonal itself ) . for example , @xmath340 is toeplitz . such matrices are not necessarily symmetric . however , they are always _ persymmetric _ , that is , symmetric with respect to the inverse diagonal . this property is algebraically expressed as @xmath341 , where @xmath342 $ ] . by @xmath343 we denote the column whose @xmath266th entry is @xmath344 and other entries are @xmath345 . the property @xmath346 ( where @xmath347 is the @xmath153 identity matrix ) implies that the product of two persymmetric matrices is persymmetric . hence any degree of a persymmetric matrix is persymmetric , and so is the closure of a persymmetric matrix . thus , if @xmath160 is persymmetric , then @xmath348 further we deal only with symmetric toeplitz matrices . consider the equation @xmath349 , where @xmath350 , and @xmath351 is defined by the scalars @xmath352 so that @xmath353 for all @xmath266 and @xmath262 . this is a generalization of the yule - walker problem @xcite . assume that we have obtained the least solution @xmath354 to the system @xmath355 for some @xmath175 such that @xmath356 , where @xmath357 is the main @xmath358 submatrix of @xmath351 . we write @xmath359 as @xmath360 we also write @xmath361 and @xmath362 as @xmath363 using , and the identity @xmath364 , we obtain that @xmath365 denote @xmath366 . the following argument shows that @xmath367 can be found recursively if @xmath368 exists . @xmath369 \begin{pmatrix } e_{k-1}y^{(k-1)}\alpha_{k-1}\oplus y^{(k-1)}\\ \alpha_{k-1 } \end{pmatrix}\\ & = r_0\oplus r^{(k-1)t}y^{(k-1)}\oplus ( r^{(k-1)t } e_{k-1 } y^{(k-1)}\oplus \end{split}\ ] ] @xmath370 existence of @xmath368 is not universal , and this will make us write two versions of our algorithm , the first one involving , and the second one not involving it . we will write these two versions in one program and mark the expressions which refer only to the first version or to the second one by the matlab - style comments @xmath371 and @xmath372 , respectively . collecting the expressions for @xmath367 , @xmath373 and @xmath374 we obtain the following recursive expression for @xmath354 : @xmath375 recursive expression is a generalized version of the durbin method for the yule - walker problem , see @xcite algorithm 4.7.1 for a prototype . [ a : yw ] the yule - walker problem for the bellman equations with symmetric toeplitz matrix . @xmath376 : scalar , + @xmath377 : @xmath378 vector ; @xmath379 + @xmath380 + @xmath381 + * for * @xmath382 + @xmath383 + @xmath384 + @xmath385 + @xmath386 + @xmath387 + @xmath388 + * end * vector @xmath274 . in the general case , the algorithm requires @xmath389 operations @xmath6 and @xmath7 each , and just @xmath390 of @xmath6 and @xmath7 if inversions of algebraic closures are allowed ( as usual , just @xmath67 such closures are required in both cases ) . now we consider the problem of finding @xmath391 where @xmath351 is as above and @xmath392 is arbitrary . we also introduce the column vectors @xmath354 which solve the yule - walker problem : @xmath393 . the main idea is to find the expression for @xmath394 involving @xmath332 and @xmath354 . we write @xmath395 and @xmath396 as @xmath397 making use of the persymmetry of @xmath398 and of the identities @xmath399 and @xmath400 , we specialize expressions and obtain that @xmath401 the coefficient @xmath402 can be expressed again as @xmath403 , if the closure @xmath404 is invertible . using this we obtain the following recursive expression : @xmath405 expressions and yield the following generalized version of the levinson algorithm for solving linear symmetric toeplitz systems , see @xcite algorithm 4.7.2 for a prototype : [ a : levinson ] bellman system with symmetric toeplitz matrix @xmath376 : scalar , + @xmath377 : @xmath406 row vector ; + @xmath317 : @xmath407 column vector . @xmath379;@xmath408 ; + @xmath380 + @xmath381 + * for * @xmath382 + @xmath383 + @xmath384 + @xmath409 + @xmath410 + @xmath411 + @xmath412 + * if * @xmath413 + @xmath385 + @xmath386 + @xmath387 + @xmath388 + * end * + * end * vector @xmath44 . in the general case , the algorithm requires @xmath414 operations @xmath6 and @xmath7 each , and just @xmath415 of @xmath6 and @xmath7 if inversions of algebraic closures are allowed ( as usual , just @xmath67 such closures are required in both cases ) . factorization of a matrix into the product @xmath416 , where @xmath417 and @xmath418 are lower and upper triangular matrices with a unit diagonal , respectively , and @xmath419 is a diagonal matrix , is used for solving matrix equations @xmath420 . we construct a similar decomposition for the bellman equation @xmath421 . for the case @xmath420 , the decomposition @xmath416 induces the following decomposition of the initial equation : @xmath422 hence , we have @xmath423 if @xmath160 is invertible . in essence , it is sufficient to find the matrices @xmath417 , @xmath419 and @xmath418 , since the linear system @xmath424 is easily solved by a combination of the forward substitution for @xmath425 , the trivial inversion of a diagonal matrix for @xmath426 , and the back substitution for @xmath126 . using the ldm - factorization of @xmath424 as a prototype , we can write @xmath427 then @xmath428 a triple @xmath429 consisting of a lower triangular , diagonal , and upper triangular matrices is called an @xmath430-_factorization _ of a matrix @xmath160 if relations and are satisfied . we note that in this case , the principal diagonals of @xmath417 and @xmath418 are zero . our universal modification of the @xmath430-factorization used in matrix analysis for the equation @xmath424 is similar to the @xmath431-factorization of bellman equation suggested by carr in @xcite . if @xmath160 is a symmetric matrix over a semiring with a commutative multiplication , the amount of computations can be halved , since @xmath418 and @xmath417 are mapped into each other under transposition . we begin with the case of a triangular matrix @xmath432 ( or @xmath433 ) . then , finding @xmath126 is reduced to the forward ( or back ) substitution . note that in this case , equation @xmath300 has unique solution , which can be found by the obvious algorithms given below . in these algorithms @xmath161 is a vector ( denoted by @xmath317 ) , however they could be modified to the case when @xmath161 is a matrix of any appropriate size . we are interested only in the case of strictly lower - triangular , resp . strictly upper - triangular matrices , when @xmath434 for @xmath156 , resp . @xmath434 for @xmath155 . [ lower - triang ] forward substitution . strictly lower - triangular @xmath153 matrix @xmath268 ; + @xmath407 vector @xmath317 . @xmath435 + @xmath436 + * end * vector @xmath274 . [ upper - triang ] backward substitution . strictly upper - triangular @xmath153 matrix @xmath66 ; + @xmath407 vector @xmath317 . @xmath437 + @xmath438 + * end * vector @xmath274 . both algorithms require @xmath439 operations @xmath6 and @xmath7 , and no algebraic closures . after performing a ldm - decomposition we also need to compute the closure of a diagonal matrix : this is done entrywise . we now proceed with the algorithm of ldm decomposition itself , that is , computing matrices @xmath417 , @xmath419 and @xmath418 satisfying and . first we give an algorithm , and then we proceed with its explanation . [ a : ldm ] ldm - decomposition ( version 1 ) . : an @xmath153 matrix @xmath160 with entries @xmath289 , + also used to store the final result + and intermediate results of the computation process . @xmath440 + @xmath441 + @xmath442 + @xmath443 + @xmath444 + * end * the algorithm requires @xmath445 operations @xmath6 and @xmath7 , and @xmath446 operations of algebraic closure . the strictly triangular matrix @xmath417 is written in the lower triangle , the strictly upper triangular matrix @xmath418 in the upper triangle , and the diagonal matrix @xmath419 on the diagonal of the matrix computed by algorithm [ a : ldm ] . we now show that @xmath447 . our argument is close to that of @xcite . we begin by representing , in analogy with the escalator method , @xmath448 it can be verified that @xmath449 @xmath450 as the multiplication on the right hand side leads to expressions fully analogous to , where + @xmath451 plays the role of @xmath452 . here and in the sequel , @xmath453 denotes the @xmath454 matrix consisting only of zeros , and @xmath455 denotes the identity matrix of size @xmath268 . this can be also rewritten as @xmath456 where @xmath457 @xmath458 @xmath459 @xmath460 @xmath461 here we used in particular that @xmath462 and @xmath463 and hence @xmath464 and @xmath465 . the first step of algorithm [ a : ldm ] ( @xmath466 ) computes @xmath467 which contains all relevant information . we can now continue with the submatrix @xmath468 of @xmath469 factorizing it as in and , and so on . let us now formally describe the @xmath175th step of this construction , corresponding to the @xmath175th step of algorithm [ a : ldm ] . on that general step we deal with @xmath470 where @xmath471 @xmath472 like on the first step we represent @xmath473 where @xmath474 note that we have the following recursion for the entries of @xmath475 : @xmath476 this recursion is immediately seen in algorithm [ a : ldm ] . moreover it can be shown by induction that the matrix computed on the @xmath175th step of that algorithm equals @xmath477 in other words , this matrix is composed from @xmath478 , ... , @xmath479 ( in the upper triangle ) , @xmath480 , ... , @xmath481 ( in the lower triangle ) , @xmath482 ( on the diagonal ) , and @xmath483 ( in the south - eastern corner ) . after assembling and unfolding all expressions for @xmath475 , where @xmath484 , we obtain @xmath485 ( actually , @xmath486 and hence @xmath487 ) . noticing that @xmath488 and @xmath489 commute for @xmath490 we can rewrite @xmath491 consider the identities @xmath492 the first of these identities is evident . for the other two , observe that @xmath493 for all @xmath175 , hence @xmath494 and @xmath495 . further , @xmath496 for @xmath497 and @xmath498 for @xmath490 . using these identities it can be shown that @xmath499 which yields the last two identities of . notice that in we have used the nilpotency of @xmath500 and @xmath501 , which allows to apply . it can be seen that the matrices @xmath502 , @xmath503 and @xmath504 are contained in the upper triangle , in the lower triangle and , respectively , on the diagonal of the matrix computed by algorithm [ a : ldm ] . these matrices satisfy the ldm decomposition @xmath447 . this concludes the explanation of algorithm [ a : ldm ] . in terms of matrix computations , algorithm [ a : ldm ] is a version of ldm decomposition with outer product . this algorithm can be reorganized to make it almost identical with @xcite , algorithm 4.1.1 : [ a : ldmgvl ] ldm - decomposition ( version 2 ) . an @xmath153 matrix @xmath160 with entries @xmath289 , + also used to store the final result + and intermediate results of the computation process . @xmath324 + @xmath505 + * for * @xmath506 @xmath507 + * end * + * for * @xmath508 + @xmath509 + * end * + @xmath510 + * for * @xmath506 + @xmath511 + * end * + @xmath512 + @xmath513 + * end * this algorithm performs exactly the same operations as algorithm [ a : ldm ] , computing consecutively one column of the result after another . namely , in the first half of the main loop it computes the entries @xmath514 for @xmath515 , first under the guise of the entries of @xmath264 and finally in the assignment `` @xmath509 '' . in the second half of the main loop it computes @xmath516 . the complexity of this algorithm is the same as that of algorithm [ a : ldm ] . when matrix @xmath160 is symmetric , that is , @xmath517 for all @xmath518 , it is natural to expect that ldm decomposition must be symmetric too , that is , @xmath519 . indeed , going through the reasoning of the previous section , it can be shown by induction that all intermediate matrices @xmath475 are symmetric , hence @xmath520 for all @xmath175 and @xmath519 . we now present two versions of symmetric ldm decomposition , corresponding to the two versions of ldm decomposition given in the previous section . notice that the amount of computations in these algorithms is nearly halved with respect to their full versions . in both cases they require @xmath521 operations @xmath6 and @xmath7(each ) and @xmath446 operations of taking algebraic closure . [ a : ldl ] symmetric ldm - decomposition ( version 1 ) . an @xmath153 symmetric matrix @xmath160 with entries @xmath289 , + also used to store the final result + and intermediate results of the computation process . @xmath440 + @xmath441 + * for * @xmath522 + * for * @xmath523 + @xmath524 + * end * + * end * + @xmath525 + * end * the strictly triangular matrix @xmath417 is contained in the lower triangle of the result , and the matrix @xmath419 is on the diagonal . the next version generalizes @xcite algorithm 4.1.2 . like in the prototype , the idea is to use the symmetry of @xmath160 precomputing the first @xmath526 entries of @xmath264 inverting the assignment `` @xmath527 '' for @xmath528 . this is possible since @xmath529 belong to the first @xmath526 columns of the result that have been computed on the previous stages . [ a : ldlgvl ] symmetric ldm - decomposition + ( version 2 ) . @xmath160 is an @xmath153 symmetric matrix with entries @xmath289 , + also used to store the final result + and intermediate results of the computation process . @xmath324 + * for * @xmath508 + @xmath530 + * end * + @xmath531 + @xmath510 + * for * @xmath506 + @xmath511 + * end * + @xmath512 + @xmath513 * end * note that this version requires invertibility of the closures @xmath532 computed by the algorithm . in the case of idempotent semiring we have @xmath533 , hence + @xmath534 . when @xmath160 is symmetric we can write @xmath535 where @xmath536 . evidently , this * idempotent cholesky factorization * can be computed by minor modifications of algorithms [ a : ldl ] and [ a : ldlgvl ] . see also @xcite , algorithm 4.2.2 . @xmath152 is called a * band matrix * with upper bandwidth @xmath537 and lower bandwidth @xmath205 if @xmath434 for all @xmath538 and all @xmath539 . a band matrix with @xmath540 is called * tridiagonal*. to generalize a specific ldm decomposition with band matrices , we need to show that the band parameters of the matrices @xmath541 computed in the process of ldm decomposition are not greater than the parameters of @xmath542 . assume by induction that @xmath543 have the required band parameters , and consider an entry @xmath544 for @xmath539 . if @xmath545 or @xmath546 then @xmath547 , so we can assume @xmath548 and @xmath549 . in this case @xmath550 , hence @xmath551 and @xmath552 thus we have shown that the lower bandwidth of @xmath475 is not greater than @xmath205 . it can be shown analogously that its upper bandwidth does not exceed @xmath537 . we use this to construct the following band version of ldm decomposition , see @xcite algorithm 4.3.1 for a prototype . [ a : ldmband ] ldm decomposition of a band matrix . @xmath160 is an @xmath153 band matrix with entries @xmath289 , + lower bandwidth @xmath205 and upper bandwidth @xmath537 + also used to store the final result + and intermediate results of the computation process . @xmath440 + @xmath441 + * for * @xmath553 + @xmath554 + * end * + * for * @xmath555 + * for * @xmath553 + @xmath326 + * end * + * end * + * for * @xmath555 + @xmath556 + * end * + * end * when @xmath205 and @xmath537 are fixed and @xmath557 is variable , it can be seen that the algorithm performs approximately @xmath558 operations @xmath7 and @xmath6 each . _ there are important special kinds of band matrices , for instance , hessenberg and tridiagonal matrices . hessenberg matrices are defined as band matrices with @xmath559 and @xmath560 , while in the case of tridiagonal matrices @xmath540 . it is straightforward to write further adaptations of algorithm [ a : ldmband ] to these cases . _ we are not aware of any truly universal scheme , since the decision when such schemes work and when they should be stopped depends both on the semiring and on the representation of data . our first scheme is derived from the following iteration process : @xmath561 trying to solve the bellman equation @xmath300 . iterating expressions for all @xmath175 up to @xmath66 we obtain @xmath562 thus the result crucially depends on the behaviour of @xmath563 . the algorithm can be written as follows ( for the case when @xmath161 is a column vector ) . [ a : jacobi ] jacobi iterations @xmath153 matrix @xmath160 with entries @xmath289 ; + @xmath407 column vectors @xmath317 and @xmath44 situation@xmath564proceed + * while * situation@xmath565proceed + @xmath566 + situation@xmath564_newsituation _ ( ... ) + * if * situation@xmath565no convergence + * disp*(jacobi iterations did not converge ) + _ exit _ + * end * + * if * situation@xmath565convergence + * disp*(jacobi iterations converged ) + _ exit _ + * end * + * end * situation , @xmath44 . next we briefly discuss the behaviour of jacobi iteration scheme over the usual arithmetic with nonnegative real numbers , and over semiring @xmath54 . for simplicity , in both cases we restrict to the case of _ irreducible _ matrix @xmath160 , that is , when the associated digraph is strongly connected . over the usual arithmetic , it is well known that ( in the irreducible nonnegative case ) the jacobi iterations converge if and only if the greatest eigenvalue of @xmath160 , denoted by @xmath567 , is strictly less than @xmath568 . this follows from the behaviour of @xmath569 . in general we can not obtain exact solution of @xmath570 by means of jacobi iterations . in the case of @xmath54 , the situation is determined by the behaviour of @xmath569 which differs from the case of the usual nonnegative algebra . however , this behaviour can be also analysed in terms of @xmath567 , the greatest eigenvalue in terms of max - plus algebra ( that is , with respect to the max - plus eigenproblem @xmath571 ) . namely , @xmath572 and hence the iterations converge if @xmath573 . moreover @xmath574 and hence the iterations yield * exact * solution to bellman equation after a * finite * number of steps . to the contrary , @xmath575 and hence the iterations diverge if @xmath576 . see , for instance , @xcite for more details . on the boundary @xmath577 , the powers @xmath578 reach a periodic regime after a finite number of steps . hence @xmath579 also becomes periodic , in general . if the period of @xmath569 is one , that is , if this sequence stabilizes , then the method converges to a general solution of @xmath304 described as a superposition of @xmath580 and an eigenvector of @xmath160 @xcite . the vector @xmath580 may dominate , in which case the method converges to @xmath580 as `` expected '' . however , the period of @xmath579 may be more than one , in which case the jacobi iterations do not yield any solution of @xmath304 . see @xcite for more information on the behaviour of max - plus matrix powers and the max - plus spectral theory . in a more elaborate scheme of gauss - seidel iterations we can also use the previously found coordinates of @xmath581 . in this case matrix @xmath160 is written as @xmath582 where @xmath417 is the strictly lower triangular part of @xmath160 , and @xmath583 is the upper triangular part with the diagonal . the iterations are written as @xmath584 note that the transformation on the right hand side is unambiguous since @xmath417 is strictly lower triangular and @xmath585 is uniquely defined as @xmath586 ( where @xmath67 is the dimension of @xmath160 ) . in other words , we just apply the forward substitution . iterating expressions for all @xmath175 up to @xmath66 we obtain @xmath587 the right hand side reminds of the formula @xmath588 , see , so it is natural to expect that these iterations converge to @xmath149 with a good choice of @xmath589 . the result crucially depends on the behaviour of @xmath590 . the algorithm can be written as follows ( we assume again that @xmath161 is a column vector ) . [ a : seidel ] gauss - seidel iterations @xmath153 matrix @xmath160 with entries @xmath289 ; + @xmath407 column vectors @xmath317 and @xmath44 situation@xmath564proceed + * while * situation@xmath565proceed + * for * @xmath319 + @xmath591 + * end * + * for * @xmath592 + @xmath593 + * end * + situation@xmath564_newsituation _ ( ... ) + * if * situation@xmath565no convergence + * disp*(gauss - seidel iterations did not converge ) + _ exit _ + * end * + * if * situation@xmath565convergence + * disp*(gauss - seidel iterations converged ) + _ exit _ + * end * + * end * situation , @xmath44 . it is plausible to expect that the behaviour of gauss - seidel scheme in the case of max - plus algebra and nonnegative linear algebra is analogous to the case of jacobi iterations . software implementations for universal semiring algorithms can not be as efficient as hardware ones ( with respect to the computation speed ) but they are much more flexible . program modules can deal with abstract ( and variable ) operations and data types . concrete values for these operations and data types can be defined by the corresponding input data . in this case concrete operations and data types are generated by means of additional program modules . for programs written in this manner it is convenient to use special techniques of the so - called object oriented ( and functional ) design , see , for instance , @xcite . fortunately , powerful tools supporting the object - oriented software design have recently appeared including compilers for real and convenient programming languages ( for instance , @xmath4 and java ) and modern computer algebra systems . recently , this type of programming technique has been dubbed generic programming ( see , for instance , @xcite ) . _ @xmath4 implementation _ using templates and objective oriented programming , churkin and sergeev @xcite created a visual @xmath4 application demonstrating how the universal algorithms calculate matrix closures @xmath129 and solve bellman equations @xmath304 in various semirings . the program can also compute the usual system @xmath329 in the usual arithmetic by transforming it to the `` bellman '' form . before pressing `` solve '' , the user has to choose a semiring , a problem and an algorithm to use . then the initial data are written into the matrix ( for the sake of visualization the dimension of a matrix is no more than @xmath594 ) . the result may appear as a matrix or as a vector depending on the problem to solve . the object - oriented approach allows to implement various semirings as objects with various definitions of basic operations , while keeping the algorithm code unique and concise . _ examples of the semirings . _ the choice of semiring determines the object used by the algorithm , that is , the concrete realization of that algorithm . the following semirings have been realized : * @xmath595 and @xmath596 : the usual arithmetic over reals ; * @xmath597 and @xmath598 : max - plus arithmetic over @xmath599 ; * @xmath600 and @xmath598 : min - plus arithmetic over @xmath601 ; * @xmath597 and @xmath596 : max - times arithmetic over nonnegative numbers ; * @xmath597 and @xmath602 : max - min arithmetic over a real interval @xmath603 $ ] ( the ends @xmath604 and @xmath317 can be chosen by the user ) ; * @xmath605or and @xmath606and : boolean logic over the two - element set @xmath607 . _ algorithms . _ the user can select the following basic methods : * * gaussian elimination scheme * , including the universal realizations of escalator method ( algorithm [ a : bordering ] ) , floyd - warshall ( algorithm [ a : eliminate ] , yershov s algorithm ( based on a prototype from @xcite ch . 2 ) , and the universal algorithm of rote @xcite ; * * methods for toeplitz systems * including the universal realizations of durbin s and levinson s schemes ( algorithms [ a : yw ] and [ a : levinson ] ) ; * * ldm decomposition * ( algorithm [ a : ldm ] ) and its adaptations to the symmetric case ( algorithm [ a : ldl ] ) , band matrices ( algorithm [ a : ldmband ] ) , hessenberg and tridiagonal matrices . * * iteration schemes * of jacobi and gauss - seidel . as mentioned above , these schemes are not truly universal since the stopping criterion is different for the usual arithmetics and idempotent semirings . _ types of matrices . _ the user may choose to work with general matrices , or with a matrix of special structure , for instance , symmetric , symmetric toeplitz , band , hessenberg or tridiagonal . _ visualization . _ in the case of idempotent semiring , the matrix can be visualized as a weighted digraph . after performing the calculations , the user may wish to find an optimal path between a given pair of nodes , or to display an optimal paths tree . these problems can be solved using parental links like in the case of the classical floyd - warshall method computing all optimal paths , see , for instance , @xcite . in our case , the mechanism of parental links can be implemented directly in the class describing an idempotent arithmetic . _ other arithmetics and interval extensions . _ it is also possible to realize various types of arithmetics as data types and combine this with the semiring selection . moreover , all implemented semirings can be extended to their interval versions . such possibilities were not realized in the program of churkin and sergeev @xcite , being postponed to the next version . the list of such arithmetics includes integers , and fractional arithmetics with the use of chain fractions and controlled precision . _ matlab realization . _ the whole work ( except for visualization tools ) has been duplicated in matlab @xcite , which also allows for a kind of object - oriented programming . obviously , the universal algorithms written in matlab are very close to those described in the present paper . _ future prospects_. high - level tools , such as stl @xcite , possess both obvious advantages and some disadvantages and must be used with caution . it seems that it is natural to obtain an implementation of the correspondence principle approach to scientific calculations in the form of a powerful software system based on a collection of universal algorithms . this approach should ensure a working time reduction for programmers and users because of the software unification . the arbitrary necessary accuracy and safety of numeric calculations can be ensured as well . the system has to contain several levels ( including programmer and user levels ) and many modules . roughly speaking , it must be divided into three parts . the first part contains modules that implement domain modules ( finite representations of basic mathematical objects ) . the second part implements universal ( invariant ) calculation methods . the third part contains modules implementing model dependent algorithms . these modules may be used in user programs written in @xmath608 , java , maple , matlab etc . the system has to contain the following modules : * domain modules : * * infinite precision integers ; * * rational numbers ; * * finite precision rational numbers ( see @xcite ) ; * * finite precision complex rational numbers ; * * fixed- and floating - slash rational numbers ; * * complex rational numbers ; * * arbitrary precision floating - point real numbers ; * * arbitrary precision complex numbers ; * * @xmath205-adic numbers ; * * interval numbers ; * * ring of polynomials over different rings ; * * idempotent semirings ; * * interval idempotent semirings ; * * and others . * algorithms : * * linear algebra ; * * numerical integration ; * * roots of polynomials ; * * spline interpolations and approximations ; * * rational and polynomial interpolations and approximations ; * * special functions calculation ; * * differential equations ; * * optimization and optimal control ; * * idempotent functional analysis ; * * and others . this software system may be especially useful for designers of algorithms , software engineers , students and mathematicians . the authors are grateful to the anonymous referees for a number of important corrections in the paper . g. l. litvinov and v. p. maslov ( 1998 ) the correspondence principle for idempotent calculus and some computer applications . in j. gunawardena , editor , idempotency , cambridge univ . press , cambridge , . 420443 . http://www.arxiv.org/abs/math.gm/0101021 . g. l. litvinov , v. p. maslov , a. ya . rodionov , and a. n. sobolevski ( 2011 ) universal algorithms , mathematics of semirings and parallel computations . lecture notes in computational science and engineering , 75 : 6389 . http://www.arxiv.org/abs/1005.1252 . g. l. litvinov , a. ya . rodionov and a. v. tchourkin ( 2008 ) approximate rational arithmetics and arbitrary precision computations for universal algorithms . j. of pure and appl . math . , 45(2 ) : 193204 . http://www.arxiv.org/abs/math.na/0101152 . g. l. litvinov and a. n. sobolevski ( 2000 ) exact interval solutions of the discrete bellman equation and polynomial complexity in interval idempotent linear algebra . doklady mathematics , 62(2 ) : 199201 . http://www.arxiv.org/abs/math.la/0101041 . g. l. litvinov , maslov v. p , and a. ya . rodionov ( 2000 ) a unifying approach to software and hardware design for scientific calculations and idempotent mathematics . international sophus lie centre , moscow . http://www.arxiv.org/abs/math.sc/0101069 . s. sergeev ( 2011 ) universal algorithms for generalized discrete matrix bellman equations with symmetric toeplitz matrix . tambov university reports , ser . natural and technical sciences , 16(6 ) : 1751 - 1758 . http://www.arxiv.org/abs/math/0612309 . a. v. tchourkin and s. n. sergeev ( 2007 ) program demonstrating how universal algorithms solve discrete bellman equation over various semirings . in g. litvinov , v. maslov , and s. sergeev , editors , idempotent and tropical mathematics and problems of mathematical physics ( volume ii ) , moscow . french - russian laboratory j.v . http://www.arxiv.org/abs/0709.4119 . o. viro ( 2001 ) dequantization of real algebraic geometry on logarithmic paper . in 3rd european congress of mathematics : barcelona , july 10 - 14 , 2000 . birkhuser , basel , pp.135 http://www.arxiv.org/abs/math/0005163 .

this paper is a survey on universal algorithms for solving the matrix bellman equations over semirings and especially tropical and idempotent semirings . however , original algorithms are also presented . some applications and software implementations are discussed .

in 2d electron systems with high ( @xmath0 @xmath1/vs ) electron mobility , the magnetoresistance exhibits strong oscillations in `` pre - shubnikov '' range of magnetic fields under microwave irradiation @xcite . these oscillations are caused by electron transitions between landau levels due to the electric component of the microwave radiation . together with the oscillations of the diagonal components of the conductivity tensor , `` beats '' have been found experimentally @xcite in the interval of more weak magnetic fields . such beats are usually related with the manifestations of the interaction between kinetic and spin degrees of freedom of the conductivity electrons . such interaction is the spin - orbit interaction ( soi ) that is known to be the origin of numerous effects in transport phenomena observed in such systems @xcite , etc . soi also leads to the possibility of electron transitions between landau levels in the magnetic field at the combined resonance frequencies @xcite , thus transitions being possible both in antinodes of electric and magnetic fields @xcite . finally , operation of a spin transistor ( schemes of which have been considered in @xcite ) is based upon spin degrees of freedom . all of the above has determined elevated interest to investigations of the soi in 2d semiconductor structures . for the purpose of studying the soi , it appears interesting to investigate a model in which the role of the soi should manifest itself the strongest . since the soi depends upon both translational and spin degrees of freedom , then it is a channel over which energy ( both electric and magnetic ) can be absorbed from the ultra - high frequency field , thus causing transitions between landau levels . because of that , it is interesting to investigate the response of a non - equilibrium electron system to a dc weak ( `` measurement '' ) electric field for the case when the initial non - equilibrium state is created with a high - frequency ac magnetic field that leads to combined transitions . the question is how this perturbation affects transport coefficients , in particular , the conductivity tensor . the discussed model includes the contributions form landau quantization and ( in the long - wavelength limit ) from the microwave radiation . we consider impurity centers for the role of scatterers , treating the scattering process perturbatively . the hamiltonian of the system under consideration consists of the kinetic energy @xmath2 , zeeman energy @xmath3 in the magnetic field @xmath4 , the spin - orbit interaction @xmath5 , interactions of electrons with ac magnetic and dc electric fields and with impurities , and the hamiltonian of the impurities themselves : @xmath6 @xmath7 @xmath8 @xmath9 and @xmath10 are operators of the components of the spin and kinetic momentum of the @xmath11th electron , where @xmath12 = -i\delta_{ij } m \hbar\omega_c\varepsilon_{\alpha\beta z}$ ] , @xmath13 is the cyclotron frequency , and @xmath14 is bohr magneton . in this paper , we limit our consideration to the case when ac and dc magnetic fields are parallel to each other : @xmath15 . in this case , the hamiltonian of the interaction of electrons with the ac magnetic field has the form : @xmath16 we assume the specific form of the soi term , namely , rashba interaction , which is non - zero even in the linear order in momentum : @xmath17 @xmath18 here @xmath19 is the constant characterizing the soi , @xmath20 is the fully - antisymmetric levi chivita tensor . the spin - orbit interaction leads to correlation of spatial and spin motion of electrons , thus , the translational and spin - related subsystems are not well - defined . since the soi is in some sense small , then one can perform a momentum - dependent canonical transformation that decouples kinetic and spin degrees of freedom . all other terms in the hamiltonian , describing the interaction of electrons with the lattice and external fields also undergo the transformation . in this case , the effective interaction of electrons in the system with external fields appears , which leads to resonant absorption of the field energy not only at the frequency of the paramagnetic resonance @xmath21 or cyclotron resonance @xmath22 , but also at their linear combinations . the gauge - invariant theory describing such transitions has been developed in @xcite . assuming the soi to be small , we perform the canonical transformation of the hamiltonian . up to the terms linear in @xmath23 , we have : @xmath24.\ ] ] the operator of the canonical transformation @xmath25 has to be determined from the requirement that , after the transformation , the @xmath26 and @xmath27 subsystems become independent . this requirement , as one can easily see , is satisfied if one puts @xmath28 one can write the transformed hamiltonian in the following form : @xmath29,\ ] ] @xmath30 the effective interaction of the electrons and the ac magnetic field ( responsible for combined transitions ) can be found using the explicit expression for the operator @xmath23 : @xmath31 = \frac{i\alpha\omega_{1s}}{2(\omega_c - \omega_s ) } ( t^{+- } - t^{-+})\cos \omega t,\ ] ] @xmath32 where @xmath33 , @xmath34 is the intensity of the linearly polarized magnetic field , oscillating with the frequency @xmath35 according to the cosine law . it follows from eq . ( [ 10 ] ) that the effective interaction @xmath36 leads to combined transitions at frequency @xmath37 , while the interaction of the spin degrees of freedom of the conductivity electrons with the ac magnetic field @xmath38 leads to resonant transitions at the frequency @xmath21 . since , for our further calculations , the response of the non - equilibrium system to the measurement electric field is interesting , in which the contribution from the translational degrees of freedom dominates , we will restrict our consideration to the effective interaction solely . during the calculation of the non - equilibrium response of the electron system to the measurement electric field , there are technical difficulties cased by the dependence of the effective interaction @xmath36 upon time . one can avoid them by transferring the time dependence to the impurity subsystem . this can be done by means of a new canonical transformation . the explicit form of the canonical transformation @xmath39 is determined from the requirement that it excludes @xmath36 from the effective hamiltonian of a system without impurities : @xmath40 the operator @xmath39 is expressed in the following form : @xmath41 the operator @xmath42 is searched for in the form : @xmath43 where one has to determine the parameters @xmath44 . in the linear approximation in the constant of the spin - orbit interaction , we have : @xmath45 as a result of the canonical transformation @xmath39 , renormalization of the electron - impurity interaction happens . in the case of elastic scattering , for obtaining the renormalized hamiltonian of the electron - impurity interaction , it is sufficient to calculate @xmath46 . in the linear approximation in the constant of the spin - orbit interaction @xmath19 , we obtain : @xmath47 using the explicit form of @xmath44 , we have : @xmath48 the speed of the electron momentum change is : @xmath49 where @xmath50 as one can see from ( [ ei ] ) , the renormalized electron - impurity interaction hamiltonian acquired time dependence . in such canonically transformed system , impurities act as a coherent oscillating field that leads to resonant transitions . we assume that the initial non - equilibrium state of the system under consideration is created by the ultra - high frequency magnetic field and can be described with the distribution @xmath51 . if some additional perturbation acts upon the system , then a new non - equilibrium state is formed in the system , that requires an extended set of basis operators for its description . the new non - equilibrium distribution is described with the operator @xmath52 . the task is to find the response of a non - equilibrium system to a weak measurement field . using the technique of calculation the non - equilibrium statistical operator @xcite , we have for the momentum relaxation rate : @xmath53 @xmath54 \rho_q^ { 1-\lambda}.\ ] ] here @xmath55 is the common temperature of the kinetic and spin subsystem , that can be introduced when one neglects the `` heating '' effects , @xmath56 is the quasiequilibrium statistical operator . now we expand the formula ( [ t1 ] ) using the explicit expression for the renormalized electron - impurity interaction @xmath57 . inserting the explicit expression for the electron - impurity interaction and averaging over the system of scatterers , we obtain : @xmath58 \rho_q^{1-\lambda } \}\end{gathered}\ ] ] here @xmath59 is the impurity concentration . rewriting this expression using secondary quantization and averaging the fermi operators using wick s theorem , we have : @xmath60 where @xmath61 is the fermi dirac distribution . @xmath62 and @xmath63 are the electron states in the dc magnetic field , they are characterized by the landau level number @xmath64 , @xmath65 projection of the wave wector @xmath66 , and @xmath67 projection of the spin @xmath68 . then we should integrate over @xmath69 , @xmath70 and @xmath71 , and take the limit @xmath72 , @xmath73 ( because we are interested in the zero - frequncy response ) . for the correction to the momentum relaxation rate caused by the microwave radiation , we obtain : @xmath74 in the @xmath75 limit , we obtain : @xmath76 the equation ( [ t7 ] ) contains a singularity in its right hand side , which is removed , as usual , due to broadening of the landau levels by scattering electrons on impurities : @xmath77 the landau level width @xmath78 can be expressed via the electron mobility @xmath63 in zero magnetic field : @xmath79 integrating over the energy for the case @xmath80 , we have : @xmath81 calculating the matrix element in ( [ t7 ] ) on the wave functions @xmath82 we obtain : @xmath83 here @xmath84 is the cyclotron orbit center coordinate , @xmath85 is the magnetic length , @xmath86 denotes hermite polynomials , and @xmath87 is the eigenfunction of the @xmath67 spin projection . finally , integrating over @xmath88 , we have : @xmath89 thus , for the case of point scatterers , where @xmath90 does not depend on @xmath91 , the radiation - induced correction to the inverse relaxation time is : @xmath92 using the expression for the momentum relaxation rate , one can also write the formula for the diagonal components of the conductivity tensor @xmath93 . numerical calculations have been carried out with the following parameters : @xmath94 ( @xmath95 is the free electron mass ) , the fermi energy is @xmath96 mev , the mobility of the 2d electrons varies as @xmath97 @xmath1/vs , the electron density @xmath98 @xmath99 . the microwave radiation frequency is @xmath100 ghz , the temperature is @xmath101 k. the magnetic field varied as 0.02 0.3 t. the dependence of the 2d electron gas photoconductivity on the @xmath102 ration is presented in fig . 1 . .,width=377 ] one can see that the dependence of electron mobility upon the magnetic field has the oscillating character . the response of a non - equilibrium electron system to the dc electric measurement field has been studied for the case when the initial non - equilibrium state of the system is created by an ultra - high frequency magnetic field that leads to combined transitions . within the proposed theory , it has been shown that such perturbation of the electron system essentially influences the transport coefficients and leads to the oscillations of the diagonal components of the conductivity tensor . the discussed effect is analogous to the phenomenon observed in gaas / algaas heterostructures with ultra - high electron mobility @xcite . however , unlike that phenomenon , the manifestation of the oscillatory pattern is dictated by the spin - orbit interaction existing in the crystals under consideration .

the response of a 2d electron system to a dc measurement electric field has been investigated in the case when the system is driven out of the equilibrium by the magnetic ultra - high frequency field that leads to combined transitions involving the spin - orbit interaction . it has been shown that the method of non - equilibrium statistical operator in conjunction ith the method of canonical transformations allows one to build a theory of linear response of a non - equilibrium 2d electron gas to a weak `` measurement '' dc electric field . the proposed theory predicts that such perturbation of the electron system with high ( @xmath0 @xmath1/vs ) mobility leads to a new type of 2d electron gas conductivity oscillations controlled by the ratio of the radiation frequency to the cyclotron frequency .

star clusters provide important information for understanding the formation and evolution of galaxies . such systems are useful for highlighting substructures of their host galaxies and for revealing their merging history . in particular , the galaxy formation process can be traced through the ages , metallicities , and kinematics of star clusters . due to their proximity , galaxies in the local group provide us with ideal targets for detailed studies of star cluster properties . while the star cluster systems of the milky way and m31 have received close attention , the third spiral galaxy in the local group , m33 , has been less studied . at a distance of 870 kpc ( distance modulus = 24.69 ; * ) , m33 is the only nearby late - type spiral galaxy ( scd ) . with a large angular size and inclination of i=56@xmath5 @xcite , m33 is a suitable galaxy for studies of its stellar constituents . + there have been a number of m33 cluster catalogs published since the pioneering work of @xcite . an extensive and complete catalog can be found in the work of ( * ? ? ? * hereafter sm ) , which merged all of the modern catalogs compiled before 2007 . the catalogs that have appeared after the publication of sm have been incorporated into the web - based version of the sm catalog . this updated version of the catalog contains 595 candidates of which 349 are confirmed clusters based on _ hubble space telescope ( hst ) _ and high - resolution ground - based imaging . the most recent work in this field corresponds to ( * ? ? ? * hereafter zkh ) using the megacam camera on cfht . this study presents a catalog of 4,780 extended sources in a 1 deg@xmath6 region around m33 which includes 3,554 new candidate stellar clusters . + as pointed out by sm , the sample of clusters in m33 suffers from significant incompleteness . while _ hst _ and its several instruments have been successfully used in the search for star clusters , the small field of view permits surveys only over a limited region of the galaxy . with the most recent contribution of zkh , this area has been increased to 1 deg@xmath6 centered on m33 . however , as we discussed in @xcite , the zkh catalog has largely overestimated the number of clusters , due to a possible systematic misidentification , where only around 40@xmath7 of the 3554 proposed candidates are likely to be actual stellar clusters . for these reasons we have undertaken the present study . this paper is organized as follows : section 2 describes the observations and data reduction while section 3 discusses the adopted search method and the integrated photometry of the clusters . the analysis of the photometric properties and comparison with other galaxies are in section 4 . finally , section 5 presents a summary . the observations for the present study were obtained using the queue service observing mode at the 3.6 m canada france hawaii telescope ( cfht ) . the data are available on - line through _ the canadian astronomy data centre _ archive and were obtained as part of `` the m33 cfht variability survey '' @xcite . the images were taken using the megacam / megaprime wide - field mosaic imager which contains 36 individual ccds that combine to offer nearly a full @xmath0x @xmath0 field of view with a high angular resolution of 0.187pixel@xmath8 . megacam operates with a set of griz filters very similar to those of the sloan digital sky survey ( sdss ) but a slightly different near - uv filter called u@xmath9 . this filter was designed to maximize the capabilities of the instrument at short wavelengths and its effective wavelength is @xmath2 200 @xmath10 redder than the standard u filter . all of the archival images were pre - processed by the cfht s elixir project . this pipeline includes the standard steps of overscan and bias subtraction , flat - fielding , fringe correction , masking of bad pixels and merging of amplifiers . the elixir project also provides a preliminary photometric calibration for each image . + in order to facilitate the search for cluster candidates , only the best available images were analyzed consisting of 15 u@xmath9 , 15 g , 14 r , 28 i and 3 z. median seeing values of all analyzed images are @xmath1 0.8 in g , r and i filters and @xmath2 0.6 in u@xmath9 and z. prior to the data analysis , each field of 36 individual ccds were combined into a single master image . this process was done using the software module swarp 2.16.14 of the terapix pipeline which is mainly dedicated to the processing of megacam data . this specific module involves resampling of the individual images as well as co - adding the different exposures in an optimum way so that the point - spread function ( psf ) is not distorted ( * ? ? ? * for details ) . each final combined master image was divided into two sub - fields , including an overlapping area , to deal with the spatial variability of the psf . + the m33 images are extremely crowded making the construction of point - spread functions quite challenging . in order to perform accurate standard profile - fitting photometry , we used daophot / allstar routines @xcite in an iterative way . first , we found all of the stars on each image and produced small - aperture photometry for them . we then used the daophot / pick routine to select a set of 1000 reasonable candidates to be used as psf stars . after deleting those with bad pixels nearby , we subtracted the stars with surrounding neighbors to help isolate the psf stars . the resulting list of more than 500 stars , in all cases , was used to create a psf for each of the images . the shape of the psf was made to vary quadratically with position on the frame . to improve the psf , we created an image where all the neighbors and stars that do not fit the first psf were subtracted , obtaining an improved second - generation psf over this subtracted image . appropriate aperture corrections were calculated from isolated unsaturated bright stars with photometric errors smaller than 0.01 mag . since the correction varies with radius from the center of the images , a polynomial fit was applied to the aperture corrections in order to obtain the final instrumental magnitudes . all frames were matched using daomatch / daomaster routines to obtain common stars in all filters . following @xcite , the photometric calibration provided by the elixir pipeline was applied using the zero - point values . in addition and to deal with the differences between u@xmath9 and u , we applied the equations from @xcite to transform the photometry from @xmath11 to @xmath12 . + the integrated magnitudes and colors for each candidate cluster have been calculated using the aperture photometry routines in daophot @xcite . to be consistent with previous authors @xcite , we have adopted an aperture radius of 2.2 for the magnitude measurements and 1.5 for the colors . the background sky is always determined in an annulus with an inner radius of 3.5 and an outer radius of 5.0 . no aperture corrections have been applied to the extended objects , such as the star cluster candidates . once again , these magnitudes have been photometrically calibrated to the sdss standard system . to derive accurate positions of the clusters and to estimate properties such as ellipticity and full width at half maximum ( fwhm ) , we have applied the sextractor v2.5.0 @xcite image classification algorithm . our detection method is based on the fact that at the distance of m33 , non - stellar objects are expected to be more extended than the psf . after subtracting the stellar psf from all of the sources in our frames , extended objects leave a doughnut - shaped appearance , as they are under - subtracted in the wings and over - subtracted in the center . we have used daophot / allstar @xcite to produce residual images free of all psf sources . we have trained our eyes to recognize the residual pattern of candidate clusters . while most background galaxies show either a spiral arm structure or an elongated pattern , the candidate stellar clusters show some level of assembly . as an illustration , [ mosaic ] shows original and residual images of different types of extended objects . after visual inspection of the residual images as well as analysis of the original ones , this technique leaves us with a total of 2,990 extended objects : 803 candidate clusters , 1,969 galaxies and 218 unknown objects . from the total number of candidate clusters , 204 were previously identified clusters in the sm updated website and considered confirmed clusters based on hst and high - resolution ground - based imaging . + the 12 acs / hst fields examined in @xcite included 72 of the candidate clusters in the present catalog , where 51 turned out to be genuine star clusters . this suggests that around @xmath2 70% of the proposed candidates will be actual stellar clusters . however , from the 349 guaranteed clusters listed in the updated sm catalog , our catalog only recovers 204 objects implying missing objects mostly in the center of the galaxy , not surprising since the method is less effective in extremely crowded regions . + comparison with the similar study of zkh reveals unexpected discrepancies . from the total 1,752 common objects between both catalogs , only 124 sources were classified as candidate clusters by both authors . as @xcite argue , the total number of true cluster candidates in the zkh catalog is not likely to be larger than @xmath2 40% this suggests a systematic misidentification in the candidate object pattern or a defective psf subtraction . [ photometry ] shows the photometric differences between the two studies . we find a mean difference of @xmath1@xmath13g@xmath14 = 0.15 @xmath15 0.02 , @xmath1@xmath13r@xmath14 = 0.10 @xmath15 0.02 and @xmath1@xmath13i@xmath14 = 0.04 @xmath15 0.03 while the offsets for the colors are @xmath1@xmath13(g - r)@xmath14 = 0.018 @xmath15 0.012 and @xmath1@xmath13(r - i)@xmath14 = 0.037 @xmath15 0.030 . the disagreement in the magnitude offsets disappears in the color offsets , which indicates that the photometric variation corresponds to the different adopted apertures in each study . these photometric differences are not unexpected for integrated photometry of extended objects such as star clusters @xcite . we have obtained stellarity , full width at half - maximum ( fwhm ) , and ellipticity for the total sample of extended sources by applying the sextractor software to the target images . we have compared the photometric parameters of the known m33 star clusters in our sample with the parameters of candidate objects to find a suitable criterion to select highly probable clusters . [ subsample ] shows the distribution of the three sextractor parameters . the stellarity parameter object classification of sextractor allows us to examine our visual object classification with a more systematic algorithm . based on its definition , a stellarity of 1 corresponds to a point source ( star ) and a stellarity of 0 to a resolved object ( galaxy ) . considering the pixel scale of the ccd ( 0.187``@xmath16 ) , a typical seeing of 0.7 and the distance of m33 ( @xmath2 870 kpc ) , a mean cluster size of 4pc will appear in our images as a point source object of @xmath2 0.9 '' implying stellarities around 1 . the distribution of ellipticities peaks between e=0.05 - 0.2 with an extended tail reaching 0.5 in both samples . panel b ) shows the normalized fwhm assuming a mean seeing of 0.7 . the fwhm of our sample has two peaks : at fwhm @xmath2 1.5 that agrees with the confirmed cluster distribution and another peak at fwhm @xmath2 1.1 not associated with the confirmed cluster distribution [ subsample ] c ) shows that the distribution of stellarity for the confirmed clusters has a strong peak at @xmath2 1 with a weak peak @xmath1 0.2 . this distribution of stellarities suggests that a significant number of confirmed star clusters will be missing in the catalog if our main source of classification were the sextractor classification algorithm . although the stellarity is a very useful detection parameter to distinguish between point sources and non - point sources , the possibility of extended or partially resolved clusters in our images , means that the stellarity parameter must be used cautiously . + based on the properties of the confirmed star clusters , we selected a sample of highly probable clusters that satisfy the following criteria : a ) ellipticity @xmath1 0.4 ; b ) 1.1 @xmath1 fwhm(px/ seeing ) @xmath1 2.2 ; and c ) stellarity @xmath14 0.6 . we show in fig . [ subsample ] d ) the correlation between fwhm and ellipticity where the filled area corresponds to the specified selection criteria . a minimum condition has been applied to the fwhm in order to avoid stellar contamination and small stellar associations . no information on color or magnitude was used for selecting the candidates . this sub - sample of highly probable clusters contains 246 objects and has been designated as class 2 . analysis of the contamination of this subsample using a similar technique as above suggests that @xmath2 85% of highly probable clusters will be genuine stellar clusters . to illustrate the available data , table [ table1 ] shows an excerpt of the complete extended source catalog where the last column corresponds to our proposed classification of the objects : 1=galaxy , 0=unknown extended object , 1=candidate star cluster , 2=highly probable cluster and 3=confirmed cluster based on the sm catalog . based on this classification , the catalog contains 599 new candidate clusters ( 353 candidate clusters ( class 1 ) and 246 highly probable clusters ( class 2 ) ) . table [ table1 ] can be found in its entirety in the electronic edition of the journal . the sample of highly probable clusters as well as the guaranteed clusters in sm will be used as targets for future follow - up imaging and spectroscopic observations . intrinsic properties such as age , metallicity , and reddening govern the integrated magnitudes and colors of clusters . as described in the previous sections , we have performed aperture photometry of the candidate ( class 1 and 2 ) and confirmed ( class 3 ) star clusters . in addition , we have made use of the equations in @xcite to transform our photometry into the sdss _ ugriz _ standard filters . [ color_magnitude ] shows the color - magnitude diagrams ( cmds ) and color distributions of our sample ( class 1 and 2 ) as compared with the confirmed star clusters from the sm catalog ( class 3 ) . the magnitude distribution of our sample contains more faint star clusters than sm . the faintest clusters reach g @xmath2 22 that corresponds to m@xmath17 3.0 , assuming a distance modulus of @xmath18=24.69 @xcite and an average reddening correction of e(v i)=0.06 @xcite . the color range of our sample is significantly wider than the color range of the confirmed clusters : 0.4 @xmath1 ( g r ) @xmath1 1.5 and 1.0 @xmath1 ( r i ) @xmath1 1.0 . the lower panels show a unimodal distribution with a strong peak at ( g r ) @xmath2 0.1 and ( r i ) @xmath2 0.2 having extended tails redward in ( g r ) color and blueward in ( r i ) . [ galev_color_color ] shows color - color diagrams of the candidate star clusters . to compare with simple stellar populations ( ssp ) , two different sets of models have been used : ( * ? ? ? * bc03 ) and ( * ? ? ? * galev ) . bc03 models correspond to an evolutionary track for an instantaneous burst and a salpeter imf while galev models correspond to a customized set provided to us by ralf kotulla and the galev team . the galev ssp models were run assuming geneva evolutionary tracks with a minimum age and time resolution of 0.1 myrs until 100 myrs , and a time - step of 1 myrs for older ages . the models were run with a salpeter imf ( 1 - 120 m@xmath19 ) for different metallicities . it is important to note that galev models include contributions from nebular emission , considering the continuum nebular and also emission lines . all of the clusters have been shifted by a line - of - sight reddening value of e(v i)=0.06 @xcite and adopting an extinction relation from @xcite . + comparisons between the integrated cluster colors and the predictions of stellar population models can provide age estimates that are potentially useful for studies of galaxy evolution . however , uncertainties in ages derived from multi - color photometry come not only from the photometric errors , but also from reddening corrections and uncertainties in the metal abundance of each cluster . furthermore , the face - on view of the galaxy and the numerous spiral arms produce a broad range of reddening in m33 that can scatter the integrated colors of individual clusters . another effect that is important in this regard is the dispersion in the integrated colors due to stochastic effects and these can vary significantly along the age sequence @xcite . the adopted imf in the ssp also contributes to uncertainties in the models . given these points , we have not attempted to estimate ages based on the integrated photometry of the clusters . in any case , the color - color diagrams reveal a number of interesting features . + a significant fraction of ` bluish ' clusters that occupy a unique location in the diagram appear in both panels of fig . [ galev_color_color ] at colors ( r i ) @xmath1 0.2 and ( u i ) @xmath1 0.8 . these clusters represent a finger - like feature that deviates from the expected direction of evolution . at least five confirmed clusters from sm are associated with the feature , supporting the genuine cluster nature of these objects . the age estimates for three of them shows ages @xmath210@xmath20yrs . based on the lower panel in fig . [ galev_color_color ] , the finger feature could be associated with the presence of a significant population of very young clusters ( @xmath21yrs ) exhibiting nebular emission . the position of many of them below the theoretical line is consistent with internal reddening shifting their colors to redder values consistent with the dusty clouds in which they are born . fig . [ galex_finger ] shows the spatial distribution of these very young clusters on a galex fuv image . a close inspection of this image suggests that all the clusters are associated with regions of star formation activity . + the recent work of @xcite analyzes multi - wavelength observations of 32 young star clusters and associations in m33 . the sample was selected from catalogs of emission line objects based on their round shape and their position in regions of the galaxy that are not too crowded in the h@xmath22 map . all of the objects have oxygen abundances of 8 @xmath1 12 + log(o / h ) @xmath1 8.7 and have 24@xmath23 m counterparts in the _ spitzer / mips _ map . comparison of the @xcite catalog and ours reveals 10 common objects , all of them associated with the previously mentioned finger feature . table [ table2 ] provides a cross - identification of the common objects and includes the ages , extinctions and reddenings obtained from their spectral energy distribution ( sed ) fitting technique . this result confirms the young age of these clusters , younger than @xmath2 12 myr , and their relatively high extinction , a@xmath24 , between 0.5 and 0.9 . of the 10 common objects , 2 of them have been previously confirmed as genuine star clusters . + prior to this study , the m33 cluster system presented an age range of star clusters between 10 myrs 10 gyrs with the majority of clusters with ages around 100 400 myrs . [ galev_color_color ] reveals a wider age range of 1 myrs 10 gyrs with at least @xmath2 50@xmath7 of objects corresponding to young clusters ( @xmath1 100 myrs ) and @xmath2 10@xmath7 of the total corresponding to very young clusters with nebular emission ( finger - like feature ) . in spite of the redward tail at ( g r ) @xmath14 0.7 that is probably caused by reddening , the diagram suggests the presence of an old population at least as old as 10 gyr . + furthermore , the integrated colors of the nucleus of m33 ( ra= 01:33:51.02 ; dec= 30:39:36.68 ) have been plotted in the color - color diagrams of fig . [ galev_color_color ] and fig . [ galev_gal ] . after examining the curve of growth for different aperture diameters , we have adopted an aperture radius of 4.4 for the magnitude measurements and a background sky of 7.5 and 9 . the integrated light shows a blue nucleus with u=14.9 , ( u i)=0.92 , ( u g)=0.63 and ( r i)=0.33 . the color - color diagrams plotted herein reveal a gap in the distribution of star clusters centered at ( g r ) @xmath4 0.3 and ( u g ) @xmath4 0.8 . a similar anomaly was discovered by @xcite among lmc star clusters in the ( u b ) vs. ( b v ) diagram . the lmc gap was noticed at ( u b ) @xmath4 0.19 and ( b v ) @xmath4 0.47 with an approximate width of 0.1 mag in both colors . the upper panel in fig . [ gaps ] shows a small region of the lmc and m33 color - color diagrams for better visualization of the gaps . the lmc data are from @xcite and have been converted to the sdss _ ugriz _ system using relations published by @xcite . constant reddening values of e(v i)=0.06 @xcite and e(b v)=0.1 @xcite have been adopted for m33 and lmc , respectively . although the m33 gap appears bluer in the diagram , both gaps correspond to a similar range in age . the lower panel in fig . [ gaps ] shows the color distribution of the m33 and lmc clusters . to avoid contamination by clusters in the finger - like feature , which corresponds to a different age range , only clusters with ( u g ) @xmath14 0.4 have been considered in the construction of the color distribution . the gap in m33 looks smaller and slightly redder than the gap in the lmc . the offset could be a consequence of the reddening correction we have applied which does not account for dust internal to each galaxy . several authors @xcite have interpreted the lmc gap as being produced by the red giant branch phase transition . this transition would be produced by stars at the helium flash stage and would fit theoretical predictions . @xcite disagrees with this interpretation and argues that the lack of clusters in this region is determined by the natural dispersion of the colors . the stochastic effects on the mass distribution of stars could produce the dispersion in the colors so no additional peculiarities would be needed in the stellar models in order to reproduce this feature of the diagram . however , the discovery of the gap among m33 clusters supports the presence of an evolutionary effect at that particular age as the origin of both gaps . [ cmd_comparison ] presents the color - magnitude diagrams of the cluster system of our sample , the milky way ( mw ) , m31 and the large magellanic cloud ( lmc ) . the mw data belong to @xcite ( open clusters ) and @xcite ( globular clusters ) . the m31 data correspond to the candidate and confirmed clusters in @xcite . data for the lmc cluster system from @xcite have also been plotted . in order to compare the different cluster systems , we have plotted absolute magnitudes and reddening corrected colors . the mw absolute magnitudes are taken directly from the above - mentioned catalogs assuming a specific distance modulus and reddening for each cluster . we have adopted a lmc distance modulus of 18.50 @xcite and 24.36 for m31 @xcite . constant reddening values of e(v i)=0.06 @xcite , e(v i)=0.1 @xcite and e(b v)=0.1 @xcite have been adopted for m33 , m31 and the lmc , respectively . if needed , we have used the jester et al ( 2005 ) transformations to convert absolute magnitudes into the g - band . the dashed lines represent the division of galactic globular clusters at ( b v)@xmath25 = 0.5 . + no distinct cluster subpopulations can be identified within the m33 cluster system like in the lmc or mw . however , the integrated colors of the very young clusters are not necessarily a reflection of their ages because they could be affected by nebular emission . considering the significant number of this type of object in our sample , the color distribution could be distorted and appear unimodal when in fact it is not . when the nebular emission clusters are removed from the analysis ( see fig . [ gaps ] ) , the color - magnitude diagram shows a possible bimodality . the m33 and lmc systems are dominated by blue clusters , ( b v)@xmath25 @xmath1 0.5 , in contrast with the redder m31 system . however , while the red cluster subpopulation of lmc occupies a very narrow ( g r)@xmath25 region , m33 red clusters populate a significantly wider color range more similar to m31 red clusters . + when comparing the absolute magnitudes of the cluster systems , we see that the brightest clusters in the mw and lmc reach luminosities of m@xmath26 @xmath2 9.5 ; however , the brightest clusters in m33 correspond to m@xmath26 @xmath2 8 , more than one magnitude fainter . this effect could be explained by the relation between the star formation rate ( sfr ) of a galaxy and the maximum mass / luminosity of its star clusters @xcite . the empirical relation suggests that galaxies with high sfrs form proportionally more clusters , and as a consequence , the cluster mass function reaches higher masses . assuming a sfr of @xmath2 0.45 m@xmath19 yrs@xmath8 , the cluster system of m33 would fit onto this relation reasonably well ( see fig . 1 @xcite ) . with the slightly higher sfr for the mw , lmc and m31 ( eg . @xcite , @xcite ) these systems will produce brighter clusters . in addition , environmental variations , such as the mass / luminosity of the galaxy , can play a role in the color - magnitude diagram of a cluster system @xcite . [ galev_gal ] presents the color - color diagrams of the m33 cluster system using our sample , m31 and lmc . the sources of the data are the same as those given above . as a reference , ssp models from the galev team @xcite with a metallicity of z=0.0004 have been overplotted . to identify different time periods , the star symbols correspond to 10@xmath27 , 10@xmath28 , 10@xmath29 , 10@xmath30 and 10@xmath31 yrs . the same constant reddening value has been adopted for each sample as in the previous figure . + the wide color range of the m33 clusters , 0.4 @xmath1 ( g r ) @xmath1 1.5 , overlaps entirely with the young - intermediate age system of the lmc and with the older m31 system . the broad range of colors implies a large range of ages , suggesting a prolonged epoch of formation . based on this evidence , the majority of the clusters will be young - intermediate age objects although we would expect clusters older than 10 gyrs . the diagram also shows that a small group of m31 clusters occupies the unique area of the ` finger ' feature , however the region seems to be significantly more populated in m33 than in these two galaxies . + when comparing m33 with similar morphological type galaxies such as ngc 300 or m101 , m33 seems to posses a unique very young star cluster population . the color distribution of candidate clusters in m101 is similar to m33 candidate clusters but no evidence of very young clusters with nebular emission has been found @xcite . although ngc 300 is nearly a twin galaxy of m33 in terms of hubble type and mass , there are several differences between them @xcite . ngc 300 appears to have globular clusters similar to those of the milky way @xcite and a metallicity gradient consistent with stars formed prior to 6 gyrs ago @xcite . environmental factors may play a key role in the star formation history of m33 , as ngc 300 is isolated from other galaxies while m33 appears to be interacting with m31 @xcite . @xcite propose a plausible m31-m33 interaction model that reproduces with good agreement the observed distances , angular positions and radial velocities of these galaxies as well as the well - known hi warp in m33 . in this simulation , m33 starts its orbit around m31 about 3.4 gyrs ago reaching pericenter ( r @xmath2 56 kpc ) around 2.6 gyrs ago . after it passes apocenter ( r @xmath2 264 kpc ) about 900 myrs ago , m33 would be approaching m31 . this close encounter could have triggered an epoch of star formation in m33 . the significant population of very young clusters with nebular emission and their association with star formation regions are evidence in support of recent star formation activity in m33 . + many studies have shown that interacting / merger environments form large populations of clusters ( e.g.@xcite ; @xcite ) , especially very young clusters . we would expect to see very young clusters still embedded in their dust cocoons in these disturbed systems . yet , their color - color diagrams do not exhibit as prominent a finger - like feature due to nebular emission around very young clusters as compared with m33 ( see antenna @xcite ; stephan s quintet @xcite ) . in the unusual environment of hickson compact group 31 ( hgc 31 ) , @xcite found a large population of @xmath1 10 myr star clusters with strong nebular emission , similar to the one found in the present study . the main galaxies that make up hgc 31 are disrupted under the presence of strong gravitational interactions and show tidal structures . the star cluster candidates with nebular emission appear throughout hcg31 , specifically concentrated in the interaction regions . the existence of these very young star clusters seems to be the consequence of active recent and ongoing star formation in hgc 31 . the similarities between the m33 cluster system and that of hgc 31 , which is a strongly interacting environment , support two important assertions . first , the finger - like feature is a genuine characteristic and not an artificial effect due to the contamination of our cluster sample . second , the past interactions between m33 and m31 have likely had significant impact on the properties of the m33 cluster system , especially the youngest clusters . in order to analyze the spatial distribution of clusters with different ages , we have divided our sample into two groups based on comparisons with ssp models . based on bc03 models , clusters with ( r i)@xmath25 @xmath2 0 and ( g r)@xmath25 @xmath2 0.1 have ages of @xmath2 10@xmath29yrs . we are going to consider clusters with ( r i)@xmath25 @xmath32 0 and ( g r)@xmath25 @xmath14 0.1 as red or old clusters . the remaining clusters we categorize as blue or young objects . this partition minimizes the contamination of the old ( red ) clusters by the young clusters in the finger - like feature which exhibit integrated colors that are redder than expected . in order to place the cluster density distribution in the context of the field stars , we have made use of the @xcite star catalog constructed over the same megacam / cfht images used in the present study . [ cumulative ] shows the cumulative radial distributions of the young and old cluster populations as compared with the blue ( young ) and red ( old ) field star populations . blue clusters follow a spatial distribution similar to the blue field stars . the distribution also suggests that younger ( bluer ) clusters are more centrally concentrated as compared with older ( redder ) clusters . the red clusters are more dispersed in a wider region than the bluer ones , indicating that the majority of the red ( old ) clusters likely belong to the halo while the bluer ( younger ) clusters generally belong to the disk of m33 . analysis of these distributions using a kolmogorov - smirnov ( k s ) test shows that there is a greater than 99.9@xmath7 chance that the old cluster population is significantly different than the young cluster population . [ radial_distribution ] shows the radial density distribution of our entire cluster sample . the filled circles show the cluster density profile versus deprojected radius , assuming our adopted distance modulus of @xmath18=24.69 , while the open circles show the confirmed clusters from sm for comparison . the small dots correspond with the radial density distribution of the field stars where the solid line represents the best polynomial fit . the star density distribution has been scaled to match the cluster density in the region between r=0.6 2 kpc where both distributions are likely to have similar completeness levels . + inside @xmath2 0.8 kpc , the cluster profile presents a decrease in density , suggesting some level of incompleteness . the cluster profile outside of @xmath2 2.5 kpc could be reproduced by a power - law where the most distant clusters are located at @xmath2 15 kpc ( 63 arcmin ) from the center of the galaxy . confirmed clusters from sm show that the m33 cluster system seems to be more centrally concentrated than the field stars , however no other galaxy has been found with this characteristic @xcite . our radial profiles , shown in fig . [ radial_distribution ] , have reduced the discrepancy between the clusters and field stars but the former are still more centrally concentrated than the latter as shown by the cumulative distributions in fig . [ cumulative ] . the pronounced decrease at @xmath2 15 kpc in the cluster and field - star density distributions suggest that this distance may represent the outer edge of both distributions . for a given radial bin in the outer region of the galaxy , the density of clusters is significantly lower than the density of stars . we note the possibility that the cluster and stellar samples may have different completeness properties . in order to minimize the potential impact of incompleteness , we restrict the comparison of these samples to the region outside @xmath2 0.8 kpc from the center of the galaxy . if the incompleteness of our sample is the reason for the differences between the cluster and field star radial profiles , then we would expect a random bias or perhaps a larger incompleteness toward the center of the galaxy . however , the analysis shows a significantly lower density of clusters between r = 3 9 kpc than in the inner region between r=0.8 3 kpc . the ratio of stars to clusters is determined not only by the formation processes but also by the destruction processes . if we assume that the formation of star clusters and the formation of stars in a galaxy are correlated , then fig . [ radial_distribution ] suggests that the cluster system in m33 has suffered from destruction or depletion of clusters at specific radii . tidal interactions when passing through the disk or near massive objects such as giant molecular clouds could produce tidal shocks that lead to the ultimate destruction of a cluster ( e.g. @xcite ; @xcite ) . analysis of the dynamical evolution of these clusters is needed to reveal the level of influence of these interactions in the disruption process . other environment effects such as interactions between m33 and m31 can also play a role in the depletion or disruption of clusters at preferred galactocentric distances . @xcite discovered the presence of four new outlying star clusters in m33 which have large projected radii of 38 113 arcmin ( 9.6 28.5 kpc ) . based on the asymmetry in the distribution of these outer clusters , they suggest the possibility that interactions with m31 may have dramatically affected the population of m33 star clusters . regardless of the source of this anomaly , we would need an additional @xmath2 350 clusters between r=3 9 kpc in order to match the stellar density in the same region of the galaxy . if we rescale the density of stars to match the cluster density in the outer region , a notable excess of clusters occurs at r @xmath1 4 kpc . this scenario is highly unlikely since dynamical destruction processes are more effective near the central region of a galaxy . the short lifetime of such a young sample of clusters also makes the cluster migration scenario implausible . no case has been found in which the cluster density exceeds the star density in the inner region of a galaxy . future follow - ups of this sample will test the validity of the depletion phenomenon that could have widespread repercussions for our understanding of m33 s formation and evolution . we present a wide - field photometric survey of m33 extended objects using cfht/ megacam images . the resultant catalog contains 2,990 extended sources , including 599 new candidate stellar clusters and 204 previously identified clusters . we have investigated the photometric properties of the cluster sample , performing _ ugriz _ integrated photometry and using their morphological parameters . based on the properties of confirmed star clusters , we select a sub - sample of 246 highly probable objects . analysis of multicolor photometry of the candidate clusters reveals a wide range of colors including a finger - like feature in the color - color diagrams that deviates from the expected direction of evolution . color distributions of the cluster sample reveal a unimodal distribution . a comparison of the radial density distribution for the field stars and our cluster sample suggests that the m33 cluster system suffers from a depletion of clusters at all radii . color - color diagrams also reveal a gap in the distribution of star clusters similar to the gap detected among lmc clusters . we thank ralf kotulla for generously provide us with the customized galev models used in this work . we also appreciate the useful comments of rupali chandar . we are grateful for support from the united states national science foundation via grant number ast-0707277 . , e. , mellier , y. , radovich , m. , missonnier , g. , didelon , p. , & morin , b. 2002 , in astronomical society of the pacific conference series , vol . 281 , astronomical data analysis software and systems xi , ed . d. a. bohlender , d. durand , & t. h. handley , 228+ , w. l. , madore , b. f. , gibson , b. k. , ferrarese , l. , kelson , d. d. , sakai , s. , mould , j. r. , kennicutt , jr . , r. c. , ford , h. c. , graham , j. a. , huchra , j. p. , hughes , s. m. g. , illingworth , g. d. , macri , l. m. , & stetson , p. b. 2001 , , 553 , 47 , s. c. , durrell , p. r. , elmegreen , d. m. , chandar , r. , english , j. , charlton , j. c. , gronwall , c. , young , j. , tzanavaris , p. , johnson , k. e. , mendes de oliveira , c. , whitmore , b. , hornschemeier , a. e. , maybhate , a. , & zabludoff , a. 2010 , , 139 , 545 , s. m. , dalcanton , j. j. , williams , b. f. , rokar , r. , holtzman , j. , seth , a. c. , dolphin , a. , weisz , d. , cole , a. , debattista , v. p. , gilbert , k. m. , olsen , k. , skillman , e. , de jong , r. s. , karachentsev , i. d. , & quinn , t. r. 2010 , , 712 , 858 , s. , schneider , d. p. , richards , g. t. , green , r. f. , schmidt , m. , hall , p. b. , strauss , m. a. , vanden berk , d. e. , stoughton , c. , gunn , j. e. , brinkmann , j. , kent , s. m. , smith , j. a. , tucker , d. l. , & yanny , b. 2005 , , 130 , 873 , a. w. , irwin , m. j. , ibata , r. a. , dubinski , j. , widrow , l. m. , martin , n. f. , ct , p. , dotter , a. l. , navarro , j. f. , ferguson , a. m. n. , puzia , t. h. , lewis , g. f. , babul , a. , barmby , p. , bienaym , o. , chapman , s. c. , cockcroft , r. , collins , m. l. m. , fardal , m. a. , harris , w. e. , huxor , a. , mackey , a. d. , pearrubia , j. , rich , r. m. , richer , h. b. , siebert , a. , tanvir , n. , valls - gabaud , d. , & venn , k. a. 2009 , , 461 , 66 , d. l. , kent , s. , richmond , m. w. , annis , j. , smith , j. a. , allam , s. s. , rodgers , c. t. , stute , j. l. , adelman - mccarthy , j. k. , brinkmann , j. , doi , m. , finkbeiner , d. , fukugita , m. , goldston , j. , greenway , b. , gunn , j. e. , hendry , j. s. , hogg , d. w. , ichikawa , s. , ivezi , . , knapp , g. r. , lampeitl , h. , lee , b. c. , lin , h. , mckay , t. a. , merrelli , a. , munn , j. a. , neilsen , jr . , e. h. , newberg , h. j. , richards , g. t. , schlegel , d. j. , stoughton , c. , uomoto , a. , & yanny , b. 2006 , astronomische nachrichten , 327 , 821 1 & 1 31 33.13 & 31 04 05.65 & & & & & & 0.23 & 0.88 & 0.98 & & -1 + 2 & 1 31 33.39 & 30 38 07.97 & & & & & & 0.53 & 1.83 & 0.02 & & -1 + 3 & 1 31 33.88 & 30 58 40.34 & & & & & & 0.37 & 1.58 & 0.01 & & -1 + 4 & 1 31 33.96 & 31 09 09.60 & & & & & & 0.19 & 1.46 & 0.22 & & -1 + 5 & 1 31 34.05 & 31 07 12.55 & & & & & & 0.07 & 1.21 & 0.07 & & -1 + 6 & 1 31 34.07 & 30 40 58.22 & & & & & & 0.06 & 0.83 & 0.93 & & -1 + 7 & 1 31 34.21 & 30 36 09.61 & & & & & & 0.42 & 1.04 & 0.42 & & -1 + 8 & 1 31 34.36 & 31 00 37.44 & & & & & & 0.13 & 1.11 & 0.53 & & -1 + 9 & 1 31 34.52 & 30 53 13.78 & & & & & & 0.39 & 0.00 & 0.35 & & -1 + 10 & 1 31 34.62 & 30 39 10.50 & & & & & & 0.38 & 1.57 & 0.00 & & -1 + 11 & 1 31 34.67 & 30 20 06.59 & 18.464 & 0.950 & 0.750 & 0.932 & & 0.16 & 0.86 & 0.98 & & 2 + 12 & 1 31 34.78 & 31 01 35.22 & & & & & & 0.17 & 1.01 & 0.67 & & -1 + 13 & 1 31 35.13 & 30 17 42.01 & & & & & & 0.77 & 1.07 & 0.05 & & -1 + 14 & 1 31 35.20 & 30 48 11.67 & & & & & & 0.28 & 0.99 & 0.09 & & -1 + 15 & 1 31 35.28 & 30 32 39.49 & & & & & & 0.45 & 1.28 & 0.01 & & -1 + 16 & 1 31 35.49 & 30 45 21.35 & & & & & & 0.21 & 1.33 & 0.00 & & -1 + 17 & 1 31 35.57 & 30 27 44.69 & & & & & & 0.24 & 1.11 & 0.56 & & -1 + 18 & 1 31 35.59 & 30 27 55.07 & & & & & & 0.31 & 1.44 & 0.04 & & -1 + 19 & 1 31 35.67 & 31 00 10.35 & & & & & & 0.24 & 1.28 & 0.11 & & -1 + 20 & 1 31 35.69 & 31 07 25.79 & & & & & & 0.26 & 0.95 & 0.04 & & -1 + 21 & 1 31 35.70 & 30 17 19.51 & 20.107 & 0.957 & 0.453 & 0.625 & 0.704 & 0.19 & 0.89 & 0.93 & & 2 + 22 & 1 31 35.71 & 30 36 09.99 & & & & & & 0.15 & 1.24 & 0.07 & & -1 + 23 & 1 31 35.89 & 30 52 54.88 & & & & & & 0.22 & 1.22 & 0.38 & & -1 + 24 & 1 31 35.98 & 30 23 53.63 & & & & & & 0.28 & 2.57 & 0.06 & & -1 + 25 & 1 31 36.03 & 31 05 26.83 & & & & & & 0.43 & 1.55 & 0.00 & & -1 + 26 & 1 31 36.05 & 30 53 42.02 & & & & & & 0.32 & 1.29 & 0.29 & & -1 + 27 & 1 31 36.16 & 31 06 04.95 & & & & & & 0.21 & 0.89 & 0.89 & & -1 + [ table1 ] rrcccc 1742 & vghc 2 - 84 & 6.51@xmath150.14 & 0.58@xmath150.11 & 0.26@xmath150.03 & 3 + 735 & c400 & 6.65@xmath150.15 & 0.52@xmath150.17 & 0.23@xmath150.07 & 1 + 1144 & c129a & 6.35@xmath150.21 & 0.93@xmath150.09 & 0.39@xmath150.02 & 1 + 1084 & c121 & 6.36@xmath150.13 & 0.93@xmath150.06 & 0.41@xmath150.01 & 2 + 970 & b0221 & 6.99@xmath150.04 & 0.40@xmath150.02 & 0.16@xmath150.01 & 2 + 760 & lgc hii 3 & 6.43@xmath150.17 & 0.54@xmath150.08 & 0.22@xmath150.02 & 2 + 847 & b0261 & 6.53@xmath150.16 & 0.56@xmath150.10 & 0.24@xmath150.04 & 2 + 983 & mcm00em24 & 6.37@xmath150.09 & 0.61@xmath150.03 & 0.28@xmath150.01 & 2 + 1586 & b0013c & 7.16@xmath150.11 & 0.56@xmath150.11 & 0.27@xmath150.09 & 3 + 952 & c403 & 6.49@xmath150.14 & 0.88@xmath150.11 & 0.38@xmath150.03 & 2 + [ table2 ] 0.4 ( dashed line ) have been considered in the color distribution . _ lower panel : _ color distribution of our sample ( unfilled histogram ) and lmc ( filled histogram ) . m33 gap can be detected at ( g r ) @xmath4 0.3 and ( u g ) @xmath4 0.8 . , and the lmc gap at ( g r ) @xmath4 0.3 and ( u g ) @xmath4 1.3.,scaledwidth=80.0% ] , 10@xmath28 , 10@xmath29 , 10@xmath30 and 10@xmath31 yrs and the red cross corresponds to the integrated colors of the nucleus of m33 . [ _ see the electronic edition of the journal for a color version of this figure . _ ] , scaledwidth=80.0% ]

we present a catalog of 2,990 extended sources in a @xmath0x@xmath0 area centered on m33 using the megacam camera on the 3.6 m canada - france - hawaii telescope ( cfht ) . the catalog includes 599 new candidate stellar clusters , 204 previously confirmed clusters , 1969 likely background galaxies and 218 unknown extended objects . we present _ ugriz _ integrated magnitudes of the candidates and confirmed star clusters as well as full width at half maximum , ellipticity and stellarity . based on the properties of the confirmed star clusters , we select a sub - sample of highly probable clusters composed of 246 objects . the integrated photometry of the complete cluster catalog reveals a wide range of colors from 0.4 @xmath1 ( g r ) @xmath1 1.5 and 1.0 @xmath1 ( r i ) @xmath1 1.0 with no obvious cluster subpopulations . comparisons with models of simple stellar populations suggest a large range of ages some as old as @xmath210 gyrs . in addition , we find a sequence in the color - color diagrams that deviates from the expected direction of evolution . this feature could be associated with very young clusters ( @xmath3yrs ) possessing significant nebular emission . analysis of the radial density distribution suggests that the cluster system of m33 has suffered from significant depletion possibly due to interactions with m31 . we also detect a gap in the cluster distribution in the color - color diagram at ( g r ) @xmath4 0.3 and ( u g ) @xmath4 0.8 . this gap could be interpreted as an evolutionary effect . this complete catalog provides promising targets for deep photometry and high resolution spectroscopy to study the structure and star formation history of m33 .

the conjugate partition of @xmath5 is denoted @xmath69 . we write @xmath70 for the number of parts of @xmath5 equal to @xmath71 . the @xmath0-binomial coefficient is defined by @xmath72}{0pt}{}{a}{b}_{q } } = \frac{(1-q^a)(1-q^{a-1})\cdots(1-q^{a - b+1})}{(1-q^b)(1-q^{b-1})\cdots(1-q)}\ ] ] and is a polynomial in @xmath0 that gives @xmath73 when @xmath74 . for a partition @xmath5 , . given two partitions @xmath5 and @xmath63 , we say @xmath53 if @xmath76 for all @xmath77 , in which case we may consider the pair as a @xmath37 . we write @xmath78 $ ] for the cells @xmath79 . we say that @xmath37 is a ( respectively ) if @xmath78 $ ] contains no @xmath80 ( respectively @xmath81 ) block , equivalently , if @xmath82 ( respectively @xmath83 ) for all @xmath71 . we say that @xmath37 is a if @xmath78 $ ] is connected and if it contains no @xmath84 block , and that @xmath37 is a if @xmath78 $ ] contains no @xmath84 block , equivalently , if @xmath85 for @xmath86 . the young diagram of a broken ribbon is a disjoint union of @xmath87 number of ribbons . the @xmath88 ( respectively @xmath89 ) of a ribbon is the number of non - empty rows ( respectively columns ) of @xmath78 $ ] , minus @xmath90 . the height ( respectively width ) of a broken ribbon is the sum of heights ( respectively widths ) of the components . let us define some polynomials . for a horizontal strip @xmath37 , define @xmath91 if @xmath37 is not a horizontal strip , define @xmath92 . for a vertical strip @xmath37 , define @xmath93 if @xmath37 is not a vertical strip , define @xmath94 . for a broken ribbon @xmath37 , define @xmath95 if @xmath37 is not a broken ribbon , define @xmath96 . for any skew shape @xmath37 , define @xmath97 next , recall the _ @xmath0-binomial theorem_. for all @xmath98 , we have @xmath99}{0pt}{}{n}{k}_{q } } t^k.\ ] ] this may be proven by induction from the standard identity @xmath100}{0pt}{}{n}{k}_{q } } = q^k { \genfrac{[}{]}{0pt}{}{n-1}{k}_{q } } + { \genfrac{[}{]}{0pt}{}{n-1}{k-1}_{q}}$ ] . [ l : hs ] for fixed partitions @xmath52 satisfying @xmath101 , we have @xmath102 with the sum over all @xmath103 , @xmath104 , for which @xmath105 is a vertical strip . let @xmath106 . a partition @xmath103 , @xmath104 , for which @xmath105 is a vertical strip is obtained by choosing @xmath107 , @xmath108 , and removing @xmath107 bottom cells of column @xmath109 in @xmath5 . see figure [ fig1 ] for the example for @xmath110 and @xmath111 , where @xmath112 , @xmath113 , @xmath114 , @xmath115 and @xmath116 for all other @xmath71 . ( @xmath104 ) for which @xmath105 is a vertical strip within @xmath37 is built from @xmath5 by removing some number of the shaded cells of @xmath117 $ ] . , height=103 ] we have @xmath118 , @xmath119 . the choices of the @xmath107 are independent , which means that @xmath120}{0pt}{}{\nu^c_j - \mu ^c_{j+1}}{m_j(\mu)}_{t } } \prod_j { \genfrac{[}{]}{0pt}{}{\lambda^c_j - \lambda^c_{j+1}}{\lambda^c_j - \nu^c_j}_{t}}\ ] ] @xmath121}{0pt}{}{\lambda^c_j - k_j-\mu^c_{j+1}}{m_j(\mu)}_{t } } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{k_j}_{t}}.\ ] ] we analyze case - by - case , showing that it reduces to @xmath22 when @xmath37 is a horizontal strip and zero otherwise . assume first that @xmath37 is a horizontal strip . this means that @xmath122 for all @xmath109 . _ case 1 : @xmath123 . _ we have @xmath124 , so the inner sum in is equal to @xmath72}{0pt}{}{\lambda^c_j-\mu^c_{j+1}}{m_j(\mu)}_{t}}= { \genfrac{[}{]}{0pt}{}{\lambda^c_j-\mu^c_{j+1}}{\mu^c_j-\mu^c_{j+1}}_{t}}.\ ] ] if @xmath125 , this is @xmath90 , and if @xmath126 and @xmath127 , then @xmath128 and so the expression is also @xmath90 . _ case 2 : @xmath129 . _ this holds if and only if @xmath130 , @xmath131 , in which case the sum in is @xmath132}{0pt}{}{1+m_j(\mu)}{m_j(\mu)}_{t } } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{0}_{t } } + ( -t)^1 t^{\binom 0 2 } { \genfrac{[}{]}{0pt}{}{m_j(\mu)}{m_j(\mu)}_{t } } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{1}_{t}}\ ] ] @xmath133 indeed , @xmath134 and @xmath135 imply @xmath136 , while @xmath134 and @xmath137 imply @xmath138 and @xmath139 . thus equals @xmath22 whenever @xmath37 is a horizontal strip . now assume that @xmath37 is not a horizontal strip . let @xmath109 be the largest index for which @xmath140 . let us investigate two cases , when @xmath141 and when @xmath142 . _ case 1 : @xmath141 . _ we must have @xmath143 and @xmath128 , for otherwise @xmath144 , which contradicts the maximality of @xmath109 . so @xmath145 , @xmath146 , @xmath147 , @xmath148 and @xmath149}{0pt}{}{\lambda^c_j - k_j-\mu^c_{j+1}}{m_j(\mu)}_{t } } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{k_j}_{t } } = \sum_{k_j = 0}^{m_j(\lambda ) } ( -t)^{k_j } t^{\binom{m_j(\lambda)+1 -k_j } 2 } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{k_j}_{t } } \\ = & \sum_{k_j = 0}^{m_j(\lambda ) } ( -t)^{k_j } t^{\binom{m_j(\lambda)-k_j } 2 + m_j ( \lambda)- k_j } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{k_j}_{t } } = t^{m_j(\lambda ) } \sum_{k_j=0}^{m_j(\lambda ) } ( -1)^{k_j } t^{\binom{m_j(\lambda)-k_j } 2 } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{k_j}_{t}}. \end{aligned}\ ] ] using with @xmath150 , @xmath151 and @xmath152 , the above simplifies to @xmath153 _ case 2 : @xmath142 . _ we consider two further options . if @xmath154 , then @xmath155 and @xmath156}{0pt}{}{\lambda^c_j - k_j-\mu^c_{j+1}}{m_j(\mu)}_{t } } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{k_j}_{t } } \\ = & \sum_{k_j = 0}^{m_j(\lambda)-m_j(\mu ) } ( -t)^{k_j } t^{\binom{m_j(\lambda)-m_j(\mu ) -k_j } 2 } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)-k_j}{m_j(\mu)}_{t } } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{k_j}_{t } } \\ = & \sum_{k_j = 0}^{m_j(\lambda)-m_j(\mu ) } ( -t)^{k_j } t^{\binom{m_j(\lambda)-m_j(\mu ) -k_j } 2 } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)-m_j(\mu)}{k_j}_{t } } { \genfrac{[}{]}{0pt}{}{m_j(\lambda)}{m_j(\mu)}_{t}}. \end{aligned}\ ] ] if we use with @xmath157 , @xmath158 and @xmath152 , we get @xmath72}{0pt}{}{m_j(\lambda)}{m_j(\mu)}_{t } } \prod_{i=0}^{m_j(\lambda)-m_j(\mu ) - 1 } ( -t + t^i ) = 0.\ ] ] on the other hand , if @xmath159 , then @xmath160 and & _ k_j = 0^a_j ( -t)^k_j t^ 2 t t & + & = _ k_j = 0^m_j ( ) - m_j ( ) + 1 ( -t)^k_j t^ 2 t t & + & = _ k_j = 0^m_j ( ) - m_j ( ) + 1 ( -t)^k_j t^ 2 t t & & = t ( _ k_j = 0^m_j ( ) - m_j ( ) + 1 ( -t)^k_j t^ 2 t. & + & - . _ k_j = 0^m_j ( ) - m_j ( ) + 1 ( -1)^k_j t^ 2 t^m_j()+1 t ) . & we prove that the first ( respectively , second ) sum is @xmath161 by substituting @xmath162 , @xmath158 ( respectively , @xmath151 ) and @xmath152 in . this finishes the proof of the lemma . we give two applications of lemma [ l : hs ] , then prove some elementary properties on hall littlewood functions that will be useful in section [ s : skew proofs ] . the first application is a formula for the product of a hall littlewood polynomial with the schur function @xmath26 . the proof is by induction on @xmath13 . for @xmath163 , there is nothing to prove . for @xmath19 , we use the formula @xmath164 which is proven as follows . it is well - known and easy to prove ( see e.g. ( * ? ? ? * exercise 7.11 ) ) that @xmath165 the conjugate pieri rule then gives , for @xmath166 for @xmath30 , the coefficient of @xmath167 in @xmath168 reduces by induction , and to @xmath169 with the sum over all @xmath103 , @xmath170 , for which @xmath171 is a vertical strip of size at least @xmath90 . by lemma [ l : hs ] , this is equal to @xmath172 . recall that @xmath173 is the ( polynomial ) coefficient of @xmath174 in @xmath175 . [ cor : p.s ] the structure constants @xmath176 satisfy @xmath177 _ t^n ( ) f^_,(t ) = _ /(t ) . @xmath178 this follows from @xmath179 , which is ( 2 ) in @xcite and also theorem [ thm : p.s ] for @xmath180 . the second application of lemma [ l : hs ] is the following generalization of example 1 of ( * ? ? ? * , example 1 ) . [ thm : y ] for every @xmath52 , we have @xmath181 equivalently , for all @xmath182 , @xmath183}{0pt}{}{\ell(\sigma)}{m}_{t^{-1}}}.\ ] ] let us evaluate @xmath184 in two different ways . on the one hand , @xmath185 on the other hand , using example 1 on page 218 of @xcite , @xmath186 now follows by taking the coefficient of @xmath174 in both expressions . for , we use the @xmath0-binomial theorem and @xmath72}{0pt}{}{n}{k}_{t^{-1 } } } = t^{\binom k 2 + \binom { n - k}2 - \binom n 2 } { \genfrac{[}{]}{0pt}{}{n}{k}_{t}}.\ ] ] the theorem is indeed a generalization of ( * ? ? ? * , example 1 ) . for @xmath187 , @xmath188 , and the right - hand side of is non - zero only for @xmath189 , so the last equation on page 218 ( _ loc . _ ) follows . it also generalizes lemma [ l : hs ] : for @xmath190 , the right - hand side of is non - zero if and only if @xmath191 , and is therefore equal to @xmath22 . we finish the section with two more lemmas . given @xmath192 , we have @xmath193}{0pt}{}{\ell(\lambda)-1}{k}_{t } } p_\lambda.\ ] ] the lemma follows from a formula due to lascoux and sch " utzenberger . see ( * ? ? ? iii , ( 6.5 ) ) . in that terminology , we have to evaluate @xmath194 . we choose a semistandard young tableau @xmath195 of shape @xmath196 and type @xmath197 . clearly , such tableaux are in one - to - one correspondence with @xmath44-subsets of the set @xmath198 . for such a subset @xmath199 , write @xmath200 for the word with the elements of @xmath199 in increasing order , and write @xmath201 for the word with the elements of @xmath202 in decreasing order . the reverse reading word of the tableau corresponding to @xmath199 is @xmath203 . the subwords @xmath204 are all strictly decreasing , and @xmath205 . the charges of @xmath204 are @xmath206 , while the charge of @xmath207 is @xmath208 ( sum over @xmath209 , @xmath210 ) . we have @xmath211 and the formula @xmath212}{0pt}{}{\ell - 1}{k}_{t}}\ ] ] follows by induction on @xmath213 . this finishes the proof . [ l : omega ] let @xmath45 be the fundamental involution on @xmath1}\fi}$ ] defined by @xmath214 . we have @xmath215 where @xmath216 we have @xmath217}{0pt}{}{\ell(\lambda)-1}{k}_{t } } p_\lambda \right)= \\ & = \sum_{\lambda \vdash r } \left ( \sum_{k=0}^{\ell(\lambda)-1}(-t)^{r - k-1 } t^{\binom{\ell(\lambda)-k}2 + \sum_{i=2}^{\lambda_1 } \binom{\lambda_i^c}2 } { \genfrac{[}{]}{0pt}{}{\ell(\lambda)-1}{k}_{t } } \right ) p_\lambda.\end{aligned}\ ] ] now by the @xmath0-binomial theorem , @xmath218}{0pt}{}{\ell(\lambda)-1}{k}_{t } } \left ( - \frac 1 { t^2}\right)^k.\ ] ] simple calculations now show that the coefficient of @xmath174 in @xmath219 is indeed @xmath220 . recall that @xmath1}\fi}$ ] has another important basis @xmath221 , defined by @xmath222 , where @xmath223 . the ( extended ) hall scalar product on @xmath1}\fi}$ ] is uniquely defined by either of the ( equivalent ) conditions @xmath224 where , taking @xmath225 , @xmath226 see @xcite . the skew hall littlewood function @xmath227 is defined ( * ? ? ? iii , ( 5.1@xmath228 ) ) as the unique function satisfying @xmath229 for all @xmath230}\fi}$ ] . ( likewise for @xmath231 . ) if we choose to read @xmath227 as , `` @xmath232 skews @xmath174 , '' then we allow ourselves access to the machinery of hopf algebra actions on their duals . we introduce the basics in subsection [ s : hopf prelim ] and return to @xmath1}\fi}$ ] and hall littlewood functions in subsection [ s : hl setting ] . let @xmath233 be a graded algebra over a field @xmath234 . recall that @xmath235 is a hopf algebra if there are algebra maps @xmath236 , @xmath237 , and an algebra antimorphism @xmath238 , called the , , and , respectively , satisfying some additional compatibility conditions . see @xcite . let @xmath239 denote the graded dual of @xmath235 . if each @xmath240 is finite dimensional , then the pairing @xmath241 defined by @xmath242 is nondegenerate . this pairing naturally endows @xmath243 with a hopf algebra structure , with product and coproduct uniquely determined by the formulas : @xmath244for all homogeneous @xmath245 and @xmath246 . ( extend to all of @xmath243 by linearity , insisting that @xmath247 for @xmath248 . ) the finite dimensionality of @xmath240 ensures that the coproduct in @xmath243 is a finite sum of functionals , @xmath249 . here and below we use sweedler s notation for coproducts . we now recall some standard actions ( `` @xmath250 '' ) of @xmath235 and @xmath243 on each other . given @xmath251 and @xmath252 , put @xmath253 equivalently , @xmath254 and @xmath255 . we call these _ skew elements _ ( in @xmath235 and @xmath243 , respectively ) to keep the nomenclature consistent with that in symmetric function theory . our skew pieri rules ( theorems [ thm : e - pieri ] , [ thm : s - pieri ] and [ thm : q - pieri ] ) come from an elementary formula relating products of elements @xmath256 and skew elements @xmath257 in a hopf algebra @xmath235 : @xmath258 see ( @xmath259 ) in the proof of ( * ? ? ? * lemma 2.1.4 ) or ( * ? ? ? * lemma 1 ) . before turning to the proofs of these theorems , we first recall the hopf structure of @xmath1}\fi}$ ] . the ring @xmath1}\fi}$ ] is generated by the one - part power sum symmetric functions @xmath260 ( @xmath261 ) , so the definitions @xmath262 completely determine the hopf structure of @xmath1}\fi}$ ] . [ p : esq ] for @xmath261 , @xmath263 where @xmath264 is given by lemma [ l : omega ] . equalities for @xmath265 and @xmath26 are elementary consequences of and may be found in ( * ? ? ? * , example 25 ) . the coproduct formula for @xmath17 is ( 2 ) in ( * ? ? ? * , example 8) . the antipode formula for @xmath17 is identical to lemma [ l : omega ] , as the fundamental morphism @xmath45 and the antipode @xmath199 are related by @xmath266 on homogeneous elements @xmath256 of degree @xmath13 . it happens that @xmath1}\fi}$ ] is self - dual as a hopf algebra . this may be deduced from example 8 in @xcite , but we illustrate it here in the power sum basis for the reader not versed in hopf formalism . [ l : self - dual ] the hopf algebra @xmath1}\fi}$ ] is self - dual with the extended hall scalar product . write @xmath267 for @xmath268 . it is sufficient to check that @xmath269 for all partitions @xmath52 , and @xmath103 . _ products and coproducts in the power sum basis . _ given partitions @xmath270 and @xmath271 , we write @xmath272 for the partition @xmath273 . also , we write @xmath274 if @xmath275 for all @xmath77 . in this case , we define @xmath276 and otherwise define @xmath277 . since the power sum basis is multiplicative ( @xmath278 ) , we have @xmath279 since @xmath280 is an algebra map , the first formula in gives @xmath281 _ products and coproducts in dual basis . _ it is easy to see that @xmath282 whenever @xmath283 . using and the formulas for product and coproduct in the power sum basis , we deduce that @xmath284 _ checking the desired identities . _ using the preceding formulas , we get @xmath285 and @xmath286 this completes the proof of the lemma . after , and lemma [ l : self - dual ] , we see that @xmath287 and @xmath288 . we specialize to hall littlewood polynomials , putting @xmath289 . taking @xmath290 in , we get @xmath291 \label{e : eskew2 } & = \sum_{k=0}^r \left ( s(e_k)\harpoon q_\mu\right ) \harpoon \bigl(p_\lambda \cdot e_{r - k}\bigr ) \\[.25ex ] \label{e : eskew3 } & = \sum_{k=0}^r ( -1)^k \left ( s_k \harpoon q_\mu\right ) \harpoon \bigl(p_\lambda \cdot e_{r - k}\bigr ) \\[.25ex ] \label{e : eskew4 } & = \sum_{k=0}^r ( -1)^k \biggl ( \sum_\tau t^{n(\tau ) } q_{\mu/\tau } \biggr)\harpoon \bigl(p_\lambda \cdot e_{r - k}\bigr ) \\[.25ex ] \label{e : eskew5 } & = \sum_{k=0}^r ( -1)^k \biggl(\sum_{|\mu/\mu^-| = k } \biggl(\sum_\tau t^{n(\tau ) } f_{\mu^-,\tau}^{\,\mu}(t)\biggr ) q_{\mu^-}\biggr ) \harpoon \biggl(\sum_{|\lambda^+/\lambda| = r - k } \operatorname{vs}_{\lambda^+/\lambda}(t ) { p_{\lambda^+ } } \biggr)\\[.25ex ] \label{e : eskew6 } & = \sum_{\lambda^+,\mu^- } ( -1)^{|\mu/\mu^-| } \operatorname{sk}_{\mu/\mu^-}(t ) \operatorname{vs}_{\lambda^+/\lambda}(t ) p_{\lambda^+/\mu^- } \,.\end{aligned}\ ] ] for and , we used proposition [ p : esq ] . for , we expanded @xmath292 in the @xmath293 basis ( cf . the proof of corollary [ cor : p.s ] ) and used the hopf characterization of skew elements . explicitly , @xmath294 we use and to pass from to : the coefficient of @xmath295 in the expansion of @xmath296 is equal to the coefficient of @xmath297 in @xmath298 . finally , follows from corollary [ cor : p.s ] . taking @xmath299 in , we get @xmath300 \label{e : sskew2 } & = \sum_{k=0}^r \left ( s(s_k)\harpoon q_\mu\right ) \harpoon \bigl(p_\lambda \cdot s_{r - k}\bigr ) \\[.25ex ] \label{e : sskew3 } & = \sum_{k=0}^r ( -1)^k \left ( e_k \harpoon q_\mu\right ) \harpoon \bigl(p_\lambda \cdot s_{r - k}\bigr ) \\[.25ex ] \label{e : sskew4 } & = \sum_{k=0}^r ( -1)^k q_{\mu/1^k } \harpoon \bigl(p_\lambda \cdot s_{r - k}\bigr ) \\[.25ex ] \label{e : sskew5 } & = \sum_{k=0}^r ( -1)^k \biggl(\sum_{|\mu/\mu^-| = k } \operatorname{vs}_{\mu/\mu^-}(t ) q_{\mu^-}\biggr ) \harpoon \biggl(\sum_{|\lambda^+/\lambda| = r - k } \operatorname{sk}_{\lambda^+/\lambda}(t ) { p_{\lambda^+ } } \biggr ) \\[.25ex ] \label{e : sskew6 } & = \sum_{\lambda^+,\mu^- } ( -1)^{|\mu/\mu^-| } \operatorname{vs}_{\mu/\mu^-}(t ) \operatorname{sk}_{\lambda^+/\lambda}(t ) p_{\lambda^+/\mu^- } \,.\end{aligned}\ ] ] for and , the proof is the same as above . for , we used @xmath301 , while for , we used and . equation is obvious . we present two proofs . the first is along the lines of the preceding proofs of theorems [ thm : e - pieri ] and [ thm : s - pieri ] . taking @xmath302 in , we get @xmath303 \label{e : qskew2 } & = \sum_{k=0}^r \left ( s(q_k)\harpoon q_\mu\right ) \harpoon \bigl(p_\lambda \cdot q_{r - k}\bigr ) \\[.25ex ] \label{e : qskew3 } & = \sum_{k=0}^r \left ( \sum_{\tau \vdash k } c_\tau(t ) p_\tau \harpoon q_\mu\right ) \harpoon \bigl(p_\lambda \cdot q_{r - k}\bigr ) \\[.25ex ] \label{e : qskew4 } & = \sum_{k=0}^r \left ( \sum_{\tau \vdash k } c_\tau(t ) q_{\mu/\tau } \right)\harpoon \bigl(p_\lambda \cdot q_{r - k}\bigr ) \\[.25ex ] \label{e : qskew5 } & = \sum_{k=0}^r \left(\sum_{|\mu/\mu^-| = k } \left(\sum_\tau c_\tau(t ) f_{\mu^-,\tau}^{\,\mu}(t)\right ) q_{\mu^-}\right ) \harpoon \left(\sum_{|\lambda^+/\lambda| = r - k } \operatorname{hs}_{\lambda^+/\lambda}(t ) { p_{\lambda^+ } } \right)\\[.5ex ] \label{e : qskew6 } & = \sum_{\lambda^+,\mu^- } ( -1)^{|\mu/\mu^-| } ( -t)^{|\tau/\mu^-| } \operatorname{vs}_{\mu/\tau}(t ) \operatorname{sk}_{\tau/\mu^- } \operatorname{hs}_{\lambda^+/\lambda}(t ) p_{\lambda^+/\mu^- } \,.\end{aligned}\ ] ] the only line that needs a comment is . substitute @xmath304 , @xmath305 , @xmath306 and @xmath307 into theorem [ thm : y ] . we get @xmath308 and , after multiplying by @xmath309 , @xmath310 now @xmath311 and @xmath312 , which shows that @xmath313 with the sum over all @xmath314 satisfying @xmath315 . this completes the first proof . & = _ , , ^-,^+ ( -t)^|/^-|+|^+/| ( -1)^|/| + |/^-|_/(t ) _ ^+/(t ) p_^+/^- + & = _ , ^-,^+ ( -1)^|/^-| ( -t)^|/^-|_/(t)_/^-(t ) ( _ ( -t)^|^+/| _ ^+/(t ) _ /(t ) ) p_^+/^- + & = _ , ^-,^+ ( -1)^|/^-|(-t)^|/^-|_/(t)_/^-(t ) _ ^+/(t ) p_^+/^- , [ thm : unique ] let @xmath317 and @xmath318 be polynomials defined for @xmath319 , with @xmath320 . for fixed @xmath321 and @xmath27 , consider the expression @xmath322 1 ) if @xmath323 @xmath324 then @xmath325 and @xmath326 . + 2 ) if @xmath327 @xmath324 then @xmath328 and @xmath329 . + 3 ) if @xmath330 @xmath324 then @xmath331 and @xmath332 . we prove only the first statement , the others being similar . suppose that we have @xmath333 if we set @xmath187 , we get the expansion of @xmath334 over ( non - skew ) hall - littlewood polynomials , which is , of course , unique . therefore @xmath335 for all @xmath321 . we will prove by induction on @xmath336 that @xmath337 . for @xmath305 and @xmath163 , we get @xmath338 , so @xmath339 . suppose that @xmath337 for @xmath340 and that @xmath341 . take @xmath342 note that @xmath343 . also , the diagram of @xmath344 is a translation of the diagram of @xmath63 . that means there is only one lr - sequence @xmath199 ( see @xcite ) of shape @xmath344 , and it has type @xmath63 . this implies that @xmath345 , @xmath346 for @xmath347 ( see ( * ? ? ? * and 218 ) ) . therefore @xmath348 is a non - zero polynomial multiple of @xmath297 . now @xmath349 where we used theorem [ thm : e - pieri ] . by the induction hypothesis , @xmath350 if @xmath351 . after cancellations , we get @xmath352 where the sum on the left is over all @xmath353 such that @xmath354 . now take scalar product with @xmath355 . since @xmath356 is the coefficient of @xmath357 in @xmath348 , we see that @xmath358 . that is , @xmath337 . similar proofs show that the expansions of @xmath359 , @xmath360 and @xmath361 in terms of skew schur functions are also unique in the sense of theorem [ thm : unique ] , a fact that was not noted in either @xcite or @xcite . it would be preferable to have a simpler expression for the polynomial @xmath362 from theorems [ thm : q - pieri ] and [ thm : unique](3 ) , i.e. , one involving only the boxes of @xmath37 in the spirit of @xmath22 , so that we could write @xmath363 where the sum is over all @xmath29 , @xmath55 such that @xmath56 . toward this goal , we point out a hidden symmetry in the polynomials @xmath364(t ) . writing @xmath17 as @xmath365 before running through the second proof of theorem [ thm : q - pieri ] ( i.e. , applying theorems [ thm : e - pieri ] and [ thm : s - pieri ] in the reverse order ) reveals @xmath366 further toward this goal , note how similar is to the sum in lemma [ l : hs ] , which reduces to the tidy product of polynomials @xmath22 . basic computations suggest some hint of a polynomial - product description for @xmath318 , @xmath367{3,2,2,1 } * [ * ( white)]{4,3,3,1 } } } \ \ , : & \ \ -(t-1)^2 ( t+1 ) \left(t^3+t^2+t-1\right ) \\[.25ex ] { { \ytableausetup{aligntableaux = center , boxsize=6pt } \ydiagram[*(gray)]{3,2,2,1 } * [ * ( white)]{4,3,3,2 } } } \ \ , : & \ \ ( t-1)^2 ( t+1 ) \left(t^3+t^2+t-1\right)^2 \\[.25ex ] { { \ytableausetup{aligntableaux = center , boxsize=6pt } \ydiagram[*(gray)]{3,2,2,1 } * [ * ( white)]{5,3,3,2 } } } \ \ , : & \ \ t ( t-1)^2 ( t+1 ) \left(t^3+t^2+t-1\right)^2 \\[.25ex ] { { \ytableausetup{aligntableaux = center , boxsize=6pt } \ydiagram[*(gray)]{3,2,2,1 } * [ * ( white)]{5,3,3,2,1 } } } \ \ , : & \ \ t ( t-1)^2 ( t+1 ) \left(t^2+t-1\right ) \left(t^3+t^2+t-1\right)^2 , \intertext{but others suggest that such a description will not be tidy , } { { \ytableausetup{aligntableaux = center , boxsize=6pt } \ydiagram[*(gray)]{3,2,2,1 } * [ * ( white)]{5,3,3,2,2 } } } \ \ , : & \ \ -t^2 ( t-1)^2 ( t+1)^2 \left(t^3+t^2+t-1\right ) \left(t^7+t^6 + 2 t^5-t^3 - 2 t^2-t+1\right ) .\end{aligned}\ ] ] we leave a concise description of the @xmath318 as an open problem .

we produce skew pieri rules for hall littlewood functions in the spirit of assaf and mcnamara @xcite . the first two were conjectured by the first author @xcite . the key ingredients in the proofs are a @xmath0-binomial identity for skew partitions and a hopf algebraic identity that expands products of skew elements in terms of the coproduct and the antipode . let @xmath1}\fi}$ ] denote the ring of symmetric functions over @xmath2 , and let @xmath3 and @xmath4 denote its bases of schur functions and hall littlewood functions , respectively , indexed by partitions @xmath5 . the schur functions ( which are actually defined over @xmath6 ) lead a rich life making appearances in combinatorics , representation theory , and schubert calculus , among other places . see @xcite for details . the hall littlewood functions are nearly as ubiquitous ( having as a salient feature that @xmath7 under the specialization @xmath8 ) . see @xcite and the references therein for their place in the literature . a classical problem is to determine cancellation - free formulas for multiplication in these bases , @xmath9 the first problem was only given a complete solution in the latter half of the 20th century , while the second problem remains open . special cases of the problem , known as _ pieri rules , _ have been understood for quite a bit longer . the pieri rules for schur functions ( * ? ? ? * ch . i , ( 5.16 ) and ( 5.17 ) ) take the form @xmath10 with the sum over partitions @xmath11 for which @xmath12 is a vertical strip of size @xmath13 , and @xmath14 with the sum over partitions @xmath11 for which @xmath12 is a horizontal strip of size @xmath13 . ( see section [ s : prelim ] for the definitions of vertical- and horizontal strip . ) the pieri rules for hall littlewood functions ( * ? ? ? * ch . iii , ( 3.2 ) and ( 5.7 ) ) state that @xmath15 and @xmath16 with the sums again running over vertical strips and horizontal strips , respectively . here @xmath17 denotes @xmath18 for @xmath19 with @xmath20 , and @xmath21 , @xmath22 are certain polynomials in @xmath23 . ( see section [ s : prelim ] for their definitions , as well as those of @xmath24 and @xmath25 appearing below . ) in many respects ( beyond the obvious similarity of and ) , the @xmath17 play the same role for hall littlewood functions that the @xmath26 play for schur functions . still , one might ask for a link between the two theories . the following generalization of , which seems to be missing from the literature , is our first result ( section [ s : prelim ] ) . [ thm : p.s ] for a partition @xmath5 and @xmath27 , we have @xmath28 with the sum over partitions @xmath29 for which @xmath30 . the main focus of this article is on the generalizations of hall littlewood functions to skew shapes @xmath31 . our specific question about skew hall littlewood functions is best introduced via the recent answer for skew schur functions @xmath32 . in @xcite , assaf and mcnamara give a for schur functions . they prove ( bijectively ) the following generalization of : @xmath33 with the sum over pairs @xmath34 of partitions such that @xmath12 is a horizontal strip , @xmath35 is a vertical strip , and @xmath36 . this elegant gluing - together of an @xmath26-type pieri rule for the outer rim of @xmath37 with an @xmath38-type pieri rule for the inner rim of @xmath37 demanded further exploration . before we survey the literature that followed the assaf mcnamara result , we call attention to some work that preceded it . the skew schur functions do not form a basis ; so , from a strictly ring theoretic perspective ( or representation theoretic , or geometric ) , it is more natural to ask how the product in expands in terms of schur functions . this answer , and vast generalizations of it , was provided by zelevinsky in @xcite . in fact , provides such an answer as well , since @xmath39 and the coefficients @xmath40 are well - understood , but the resulting formula has an enormous amount of cancellation , while zelevinsky s is cancellation free . it is an open problem to find a representation theoretic ( or geometric ) explanation of . as an example of the type of explanation we mean , recall zelevinsky s realization @xcite of the classical jacobi trudi formula for @xmath41 ( @xmath42 ) from the resolution of a well - chosen polynomial representation of @xmath43 . see also @xcite . returning to the literature that followed @xcite , lam , sottile , and the second author @xcite found a hopf algebraic explanation for that readily extended to many other settings . a skew pieri rule for @xmath44-schur functions was given , for instance , as well one for ( noncommutative ) ribbon schur functions . within the setting of schur functions , it provided an easy extension of to products of arbitrary skew schur functions a formula first conjectured by assaf and mcnamara in @xcite . ( the results of this paper use the same hopf machinery . for the non - experts , we reprise most of details and background in section [ s : hopf ] . ) around the same time , the first author @xcite was motivated to give a skew murnaghan - nakayama rule in the spirit of assaf and mcnamara . along the way , he gives a bijective proof of the conjugate form of ( only proven in @xcite using the automorphism @xmath45 ) and a _ quantum _ skew murnaghan - nakayama rule that takes the following form . @xmath46 with the sum over pairs @xmath34 of partitions such that @xmath12 and @xmath35 are broken ribbons and @xmath36 . note that since @xmath47 , we recover the skew pieri rule for @xmath48 . also , since @xmath49 ( the @xmath13-th power sum symmetric function ) , we recover the skew murnaghan - nakayama rule @xcite if we divide the formula by @xmath50 and let @xmath51 . this formula , like that in theorem [ thm : p.s ] , may be viewed as a link between the two theories of schur and hall littlewood functions . one is tempted to ask for other examples of mixing , e.g. , swapping the rolls of schur and hall littlewood functions in . two such examples were found ( conjecturally ) in @xcite . their proofs , and a generalization of to the hall littlewood setting , are the main results of this paper . [ thm : e - pieri ] for partitions @xmath52 , @xmath53 , and @xmath27 , we have @xmath54 where the sum on the right is over all @xmath29 , @xmath55 such that @xmath56 . [ thm : s - pieri ] for partitions @xmath52 , @xmath53 , and @xmath27 , we have @xmath57 where the sum on the right is over all @xmath29 , @xmath55 such that @xmath56 . note that putting @xmath58 above recovers theorem [ thm : p.s ] . ( we offer two proofs of theorem [ thm : s - pieri ] ; one that rests on theorem [ thm : p.s ] and one that does not . ) [ thm : q - pieri ] for partitions @xmath52 , @xmath53 , and @xmath27 , we have @xmath59 where the sum on the right is over all @xmath29 , @xmath60 such that @xmath56 . we reiterate that the skew elements do not form a basis for @xmath1}\fi}$ ] , so the expansions announced in theorems [ thm : e - pieri][thm : q - pieri ] are by no means unique . however , if we demand that the expansions be over partitions @xmath61 and @xmath55 , and that the coefficients factor nicely as products of polynomials @xmath62 ( independent of @xmath63 ) and @xmath64 ( independent of @xmath5 ) , then they are in fact unique ( up to scalar ) . we make this remark precise in theorem [ thm : unique ] in section [ s : skew proofs ] . this paper is organized as follows . in section [ s : prelim ] , we prove some polynomial identities involving @xmath65 , @xmath66 and @xmath67 , prove theorem [ thm : p.s ] , and find @xmath68 . in section [ s : hopf ] , we introduce our main tool , hopf algebras . we conclude in section [ s : skew proofs ] with the proofs of our main theorems .

_ geometric crystals _ were invented by berenstein and kazhdan @xcite as birational analogues of kashiwara s crystal graphs . suppose @xmath2 is a geometric crystal . then the product @xmath3 is also a geometric crystal , and in certain cases @xcite one has a ( birational ) @xmath1-matrix @xmath4 giving rise to a birational action of the symmetric group @xmath5 on @xmath3 . since the @xmath1-matrix is an isomorphism of geometric crystals , the crystal structure of @xmath3 can be considered as @xmath5-invariants of @xmath3 . our investigations began with expressing the crystal structure in terms of the invariant rational functions @xmath6 in the special case that @xmath7 is the _ basic geometric crystal _ corresponding to symmetric powers of the standard representation of @xmath8 . equivalently , we construct the _ quotient crystal _ of @xmath9 by @xmath5 . it turns out that this quotient crystal can be constructed inside the unipotent loop group @xmath10 in an extremely natural way . the _ basic geometric crystal _ @xmath11 of type @xmath12 is the variety @xmath13 equipped with a distinguished collection of rational functions @xmath14 , @xmath15 , @xmath16 and a collection of rational @xmath17-actions @xmath18 , @xmath19 satisfying certain relations ( see section [ sec : geom ] ) . our @xmath11 is essentially the geometric crystal @xmath20 of kashiwara , nakashima , and okado ( * ? ? ? * section 5.2 ) , with the dependence on @xmath21 removed . it is a geometric analogue @xcite of a limit of perfect ( combinatorial ) crystals , the latter playing an important role in the theory of vertex models . now consider the product @xmath22 , where each @xmath23 . in this case , the birational @xmath1-matrix @xmath24 has been explicitly calculated @xcite . this same rational transformation appeared in the study of total positivity in loop groups . let @xmath25 denote the formal loop group , and let @xmath26 denote the maximal unipotent subgroup of @xmath27 . in @xcite , we defined certain elements @xmath28 ( see section [ sec : whirlchev ] ) , called _ whirls _ , depending on @xmath29 parameters @xmath30 . we showed that any totally nonnegative element @xmath31 whose matrix entries were polynomials , was a product of such whirls . it turns out that generically there is a unique non - trivial rational transformation of parameters @xmath32 such that @xmath33 up to a shift of indices , this transformation coincides with the birational @xmath1-matrices @xmath34 described above . let @xmath35 denote the closure of the set of elements @xmath36 which can be expressed as the product of @xmath37 whirls . we show that @xmath38 and that the geometric crystal structure of @xmath9 descends to @xmath39 , where it can be described explicitly in terms of left and right multiplication by one - parameter subgroups of @xmath10 ( theorems [ thm : ucrystal ] and [ thm : einv ] and corollary [ cor : same ] ) . this is reminiscent of the construction @xcite of a geometric crystal from a unipotent bicrystal , though @xmath39 is not closed under multiplication by @xmath10 . one intriguing observation we make is that the geometric crystal @xmath39 has a limit as @xmath40 , which morally one may think of as a quotient of the semi - infinite product @xmath41 . it gives rise to a `` geometric crystal '' structure on the whole unipotent loop group @xmath10 , where @xmath42 are no longer rational functions , but are asymptotic _ limit ratios _ of the matrix coefficients of @xmath10 ( theorem [ thm : infinity ] ) . these limit ratios played a critical role in the factorization of totally nonnegative elements @xcite . it points to a deeper connection which we have yet to understand . the matrix coefficients of @xmath39 give a natural set of ( algebraically independent ) generators of @xmath43 . these matrix coefficients are the _ loop elementary symmetric functions _ @xmath44 . the invariants also contain distinguished elements @xmath45 , called the _ loop schur functions_. it was observed in @xcite that the ( birational ) intrinsic energy function of @xmath9 can be expressed as a loop schur function of dilated staircase shape . in particular , energy is a polynomial , unlike the rational functions @xmath42 . we explicitly describe the crystal operator action @xmath46 on loop schur functions ( theorem [ thm : schuraction ] ) . we shall use @xcite as our main reference for affine geometric crystals . in this paper we shall consider affine geometric crystals of type @xmath0 . fix @xmath47 . let @xmath48 denote the @xmath12 cartan matrix . thus if @xmath49 then @xmath50 , @xmath51 for @xmath52 , and @xmath53 for @xmath54 . for @xmath55 , we have @xmath56 and @xmath57 . let @xmath58 be an affine geometric crystal for @xmath12 . thus , @xmath2 is a complex algebraic variety , @xmath59 and @xmath60 are rational functions , and @xmath61 @xmath62 is a rational @xmath63-action , satisfying : 1 . the domain of @xmath64 is dense in @xmath2 for any @xmath65 . 2 . @xmath66 for any @xmath67 . 3 . @xmath68 . 4 . for @xmath69 such that @xmath70 we have @xmath71 . 5 . for @xmath69 such that @xmath51 we have @xmath72 . we often abuse notation by just writing @xmath2 for the geometric crystal . we define @xmath73 , and sometimes define a geometric cyrstal by specifying @xmath74 and @xmath75 , instead of @xmath76 . if @xmath77 are affine geometric crystals , then so is @xmath78 @xcite . . then @xmath80 and @xmath81 where @xmath82 note that our notations differ from those in @xcite by swapping left and right in the product , and this agrees with @xcite . note also that the birational formulae we use should be tropicalized using @xmath83 ( rather than @xmath84 ) operations to yield the formulae for combinatorial crystals . we now introduce a geometric crystal @xmath11 which we call the _ basic geometric crystal _ of type @xmath12 . this is a geometric analogue of a limit of perfect crystals . it is essentially the geometric crystal @xmath20 of ( * ? ? ? * section 5.2 ) . we have @xmath85 and @xmath86 and @xmath87 that @xmath11 is an affine geometric crystal is shown in @xcite . let @xmath25 denote the formal loop group , consisting of non - singular @xmath88 matrices with complex formal laurent series coefficients . if @xmath89 where @xmath90 , we let @xmath91 denote the infinite periodic matrix defined by @xmath92 for @xmath93 . for the purposes of this paper we shall always think of formal loop group elements as infinite periodic matrices . the unipotent loop group @xmath94 is defined as those @xmath95 which are upper triangular , with 1 s along the main diagonal . we let @xmath35 denote the subset consisting of infinite periodic matrices supported on the @xmath37 diagonals above the main diagonal . thus if @xmath96 then @xmath97 if and only if @xmath98 for @xmath99 . it is clear that @xmath100 . for @xmath101 and @xmath102 , define the chevalley generator @xmath103 by @xmath104 these are standard one - parameter subgroups of @xmath10 corresponding to the simple roots . for @xmath105 , define the _ whirl _ @xmath106 @xcite to be the infinite periodic matrix with @xmath107 here and elsewhere the upper indices are to be taken modulo @xmath29 . let @xmath108 . for @xmath101 , define rational functions @xmath109 and @xmath110 on @xmath10 by @xmath111 ( these rational functions , and indeed the geometric crystal structure as well , extend to @xmath27 . however , we shall only consider the unipotent loop group . ) [ thm : ucrystal ] fix @xmath108 . the formal loop group @xmath10 , the rational functions @xmath112 , and the map @xmath113 form a geometric crystal . furthermore , the subvariety @xmath39 is invariant under @xmath46 , and thus is a geometric subcrystal . this result is not difficult to prove by direct calculation , but we omit the proof as it essentially follows from corollary [ cor : same ] ( which shows that one has a geometric crystal structure on @xmath39 ) . the formula is essentially the same formula as that used by berenstein and kazhdan ( * ? ? ? * ( 2.14 ) ) to define a geometric crystal from a unipotent crystal . indeed , @xmath10 clearly has a @xmath114 action ( though @xmath39 does not ) . however , despite the simplicity of our construction , we have been unable to put it inside their framework of unipotent crystals . ley @xmath96 . define the limit ratios @xcite @xmath115 assuming the limits exist . since @xmath109 and @xmath110 are not rational functions , we can not use them to construct an algebraic geometric crystal . however , treating @xmath109 and @xmath110 as functions with a restricted domain , we can still formally construct a geometric crystal using the formula . [ thm : infinity ] the functions @xmath112 , and the map @xmath116 satisfy the relations ( 2),(3),(4),(5 ) of a geometric crystal on @xmath10 ( see section [ sec : geom ] ) . note that if @xmath109 and @xmath110 are defined at @xmath96 , so is @xmath46 , and in addition @xmath117 also has this property . for simplicity we assume @xmath49 and suppose that @xmath110 and @xmath109 are defined for @xmath96 . we have @xmath118 and @xmath119 , and @xmath120 @xmath121 relations ( 2),(3),(4 ) are immediate . to obtain ( 5 ) , one uses the relation @xmath122 to get @xmath123 where @xmath124 . a similar equality for @xmath109 gives ( 5 ) . the last statement of the theorem is straightforward . in place of axiom ( 1 ) of a geometric crystal , we may note that the domain of definition of @xmath125 is dense in @xmath10 under ( matrix-)entrywise convergence . however , @xmath109 and @xmath110 are not continuous in this topology . the limit ratios @xmath109 and @xmath110 were introduced in @xcite for the study of the totally nonnegative part @xmath126 of the loop group . indeed , these limits always exist for totally nonnegative elements which are not supported on finitely many diagonals . they point to a deeper connection between total nonnegativity and crystals . let @xmath22 where each @xmath23 is a basic affine geometric crystal . we shall shift the indexing of the coordinates on @xmath127 , so that @xmath128 that is , @xmath129 , and so on . define @xmath130 in the variables @xmath131 , the birational @xmath1-matrix acts ( see ( * ? ? ? * proposition 3.1 ) ) via algebra isomorphisms @xmath132 of the field @xmath133 of rational functions in @xmath134 , given by @xmath135 and @xmath136 for @xmath137 . the birational @xmath1-matrix acts as geometric crystal isomorphisms , and thus the crystal structure of @xmath2 descends to the invariants @xmath138 . for a sequence @xmath139 of @xmath29-tuples of complex numbers , we define @xmath140 to be the product of whirls @xmath141 . the entries of @xmath142 have the following description . for @xmath143 and @xmath144 , define the _ loop elementary symmetric functions _ @xmath145 by convention we have @xmath146 if @xmath147 , and @xmath148 if @xmath149 . then by ( * ? ? ? * lemma 7.5 ) we have @xmath150 furthermore , it is shown in ( * ? ? ? * section 6 ) that @xmath151 the ring @xmath152 $ ] generated by the @xmath153 is called the ring of ( whirl ) loop symmetric functions . [ thm : einv ] we have @xmath154 . in particular , the @xmath153 are algebraically independent . the map @xmath155 given by @xmath156 identifies @xmath157 $ ] with the coordinate ring @xmath158 $ ] . it follows from that @xmath159 ( see also @xcite ) . now consider the map @xmath160 given by @xmath161 . it is shown in ( * ? ? ? * corollary 6.4 , proposition 8.2 ) that when @xmath162 are positive real numbers and the products @xmath163 are all distinct the map @xmath164 is @xmath165 to @xmath166 . since this is a zariski - dense subset of @xmath2 , we conclude that @xmath167 = m!$ ] . but then we must have @xmath154 . since the transcendence degree of @xmath138 is @xmath168 , it follows that the @xmath153 are algebraically independent . the last statement follows immediately from . in a future work we shall show the stronger result that @xmath169 = \c[e_r^{(s)}]$ ] . the aim of this section is to completely describe the geometric crystal structure of @xmath2 in terms of the invariants @xmath153 . [ lem : epph ] let @xmath170 . we have @xmath171 we proceed by induction on @xmath37 . for @xmath172 , we have @xmath173 and @xmath174 , agreeing with the theorem . let @xmath175 . supposing the result is true for @xmath176 , we calculate that @xmath177 the calculation for @xmath110 is similar . for @xmath178 we use @xmath179 . the next theorem completes the description of the geometric crystal structure in terms of the @xmath153 . [ thm : e ] the map @xmath180 induced by @xmath181 is given by @xmath182 in other words @xmath153 is sent to 1 . @xmath183 2 . @xmath184 3 . @xmath185 if @xmath186 4 . @xmath187 let @xmath188 . the action of @xmath189 on the crystal is induced by the transformation @xmath190 @xmath191 where @xmath192 the proof is by induction on @xmath37 . for @xmath193 the statement is easily verified from the definition of basic geometric crystal . assume now @xmath194 . we let @xmath195 , where @xmath196 . then @xmath197 @xmath198 plugging in @xmath199 and using we obtain @xmath200 @xmath201 @xmath202 if @xmath2 is a geometric crystal and @xmath203 is a group of crystal automorphisms of @xmath2 , we say that a geometric crystal @xmath204 is the quotient of @xmath2 by @xmath203 , if we have a rational map @xmath205 , commuting with the geometric crystal structure , and inducing @xmath206 . [ cor : same ] the map @xmath207 identifies the quotient of the product geometric crystal @xmath208 by the @xmath5 birational @xmath1-matrix action with the geometric crystal on @xmath39 of theorem [ thm : ucrystal ] . our whirls play a similar role to the @xmath209-matrices of @xcite , which have been studied for all affine types . this suggests that theorem [ thm : e ] and also our work on total positivity @xcite may have a generalization to other types in the same spirit . we recall the definition of the _ loop schur functions _ that were introduced in @xcite . let @xmath210 be a skew young diagram shape . a square @xmath211 in the @xmath212-th row and @xmath213-th column has _ we caution that our notion of content is the negative of the usual one . recall that a semistandard young tableaux @xmath215 with shape @xmath210 is a filling of each square @xmath216 with an integer @xmath217 so that the rows are weakly - increasing , and columns are increasing . for @xmath218 , the @xmath219-weight @xmath220 of a tableaux @xmath215 is given by @xmath221 . we shall draw our shapes and tableaux in english notation : @xmath222{\bl \times&\bl \times&\bl \times&\bl \times&\bl \times & \circ & & & \bullet \\\bl \times&\bl \times&\bl \times & \circ & & & & \bullet \\\bl \times&\bl \times&\bl \times & & & \bullet } \qquad \tableau[sy]{\bl & \bl&1&1&1&3\\1&2&2&3&4\\3&3&4}\ ] ] for @xmath223 the @xmath224-weight of the above tableau is @xmath225 we define the _ loop ( skew ) schur function _ by @xmath226 where the summation is over all semistandard young tableaux of ( skew ) shape @xmath210 . loop schur functions are of significance in the theory of geometric crystals because of the following result . let @xmath227 denote the staircase shape of size @xmath37 . for a skew shape @xmath210 we distinguish its nw corners and se corners ( in english convention ) . the figure above shows the nw corners marked with @xmath234 and the se corners marked with @xmath235 . let @xmath236 ( resp . @xmath237 ) denote the set of nw ( resp . se ) corners of @xmath210 of color @xmath238 , that is corner cells @xmath239 with @xmath240 . the proof of the following result follows from an examination of the effect of row and column operations on jacobi - trudi matrices , and is omitted .

we study products of the affine geometric crystal of type @xmath0 corresponding to symmetric powers of the standard representation . the quotient of this product by the @xmath1-matrix action is constructed inside the unipotent loop group . this quotient crystal has a semi - infinite limit , where the crystal structure is described in terms of limit ratios previously appearing in the study of total positivity of loop groups .

given the plethora of composition operations on graphs ( cartesian sum , tensor product , etc . ) , one is naturally led to the question of whether or not there is a sensible notion of the _ inverse _ of a graph . there is no shortage of possible definitions : a first attempt is to define two graphs to be inverses if they possess inverse adjacency matrices . this turns out to be overly restrictive , as under this definition only the graphs @xmath0 are invertible , with themselves as their own inverses ( @xcite ) . a second attempt , motivated by the observation that the eigenvalues of the sum and product of two graphs are the pairwise sums and products of the eigenvalues of the original graphs , is to call a graph @xmath1 invertible if there exists another graph @xmath2 such that for each eigenvalue @xmath3 of @xmath1 , @xmath4 is an eigenvalue of @xmath2 ( with the same multiplicity ) . this definition too allows some unfortunate phenomena : if @xmath5 and @xmath6 are cospectral and non - isomorphic , then @xmath7 and @xmath8 ( if such graphs exist ) both satisfy the criterion for being inverses to @xmath5 , and we are left with multiple non - isomorphic inverses . further , there would be no hope of attaining the obviously desirable property that @xmath9 be isomorphic to @xmath1 . it therefore behooves us to strengthen the condition defining the inverse . we begin by noting that since adjacency matrices are diagonalizable ( being real and symmetric ) , two such matrices are cospectral if and only if they are similar . the reciprocal eigenvalue condition described above is thus tantamount to asserting that the inverse @xmath10 of the adjacency matrix to @xmath1 is similar to the adjacency matrix of @xmath2 . a strengthening of the definition comes from a result of godsil ( @xcite ) that under certain conditions on @xmath1 ( described below ) , the inverse adjacency matrix @xmath10 is in fact _ signable _ to a non - negative symmetric integral matrix with zeros on the diagonal , i.e. , to the adjacency matrix of a graph . here we say @xmath11 is signable to @xmath12 if @xmath11 can be conjugated to @xmath12 by a diagonal matrix whose diagonal entries are all @xmath13 ( i.e. , by a _ signing matrix _ ) . we therefore adopt the following definition : given a graph @xmath1 , we say that a graph @xmath14 is an _ inverse _ of @xmath1 if they possess adjacency matrices @xmath15 and @xmath16 such that @xmath16 is signable to @xmath17 . we then say that @xmath1 is _ invertible _ , and say that @xmath1 is _ simply invertible _ if there exists a simple graph @xmath14 which is an inverse of @xmath1 . ( in particular , we emphasize that a simple graph can be invertible but not simply invertible . ) clearly this stronger condition defining invertibility implies the earlier reciprocal eigenvalue property , and it is thus easy to find non - invertible graphs namely , any graph with an eigenvalue of 0 , e.g. , bipartite graphs on an odd number of vertices . in fact , this is a convenient place to note that for an invertible graph @xmath1 with an inverse @xmath14 , we must have @xmath18 , and so @xmath19 for any invertible graph . this forces @xmath1 to admit a perfect matching ( or `` 1-factor '' ) , providing fairly compelling evidence that most graphs are not invertible . following godsil and the subsequent literature , we focus on graphs @xmath1 which are bipartite and have a _ unique _ perfect matching @xmath20 . the first significant invertibility result ( @xcite , theorem 2.2 ) gives that a simple graph @xmath1 ( bipartite with a unique perfect matching @xmath20 ) is invertible if the graph @xmath21 obtained by contracting each edge of @xmath20 is bipartite . the aim of the current paper is to extend results of this form in a variety of different directions . section 2 contains preliminaries on bipartite graphs with a unique perfect matching , focusing on inversion and extending previously well - known results for simple graphs to the context of multigraphs . in particular , we give a purely graph - theoretic construction of the inverse ( when it exists see theorem [ gottabe ] ) , which we dub the _ parity closure _ of the graph . we emphasize a graphical point of view ( as opposed to a poset - theoretical or linear - algebraic one ) , enough so that it is frequently possible to bypass any matrix - inversion calculations and `` eyeball '' both the invertibility and inverse of a given graph . further , we prove that the construction satisfies the desired properties of an inverse from the introduction ( i.e. , that @xmath22 see theorem [ g++ ] ) . in section 3 , we turn our attention to determining conditions for the inverse to exist . first , we extend a variety of known results on invertibility to the context of multigraphs , among them the result of godsil mentioned above and a related result of @xcite that a necessary condition for invertibility is the bipartiteness of a certain subgraph @xmath23 of @xmath21 . continuing , we note that the main result of @xcite gives much more , reducing the question of the invertibility of @xmath1 to the invertibility of a collection of subgraphs ( the `` undirected intervals '' ) of @xmath1 . their culminating necessary and sufficient condition for invertibility admits some curiosities , however . if @xmath1 is either : a. a simply invertible undirected interval graph with bipartite hasse diagram ( figure [ tkfigs ] , left ) ; or b. a non - invertible interval graph with bipartite hasse diagram all of whose proper sub - undirected intervals are invertible ( figure [ tkfigs ] , right ) , [ allunis ] every @xmath24 which is connected and unicyclic as an undirected graph can be constructed in this manner . let @xmath24 be such a graph , with @xmath20 vertices and a cycle of length @xmath25 . as above , without loss of generality we assume that the vertices contained in the unique cycle are labelled consecutively , say @xmath26 through @xmath27 . for each vertex @xmath28 in @xmath24 , let @xmath29 $ ] be the set of vertices edge - connected to @xmath28 and for @xmath30 $ ] , let @xmath31 correspond to those edges coming from other vertices in the cycle . then by connectedness and unicyclicity , the sets @xmath32 and @xmath33 form partitions as in the proposition , and we can follow the construction process in the proof of proposition [ uconst2 ] with adjacency sets @xmath34 and @xmath35 to reconstruct @xmath24 . if @xmath25 is even , the resulting unicyclic digraph is bipartite , and thus invertible . the requirement that @xmath36 for all @xmath37 forces @xmath38 and is equivalent to saying the undirected cycle subgraph @xmath39 induced by vertices @xmath40 must contain only one source and one sink vertex ( i.e. , @xmath41 in the language of theorem [ unicycleinvert ] ) , and that these two vertices must be adjacent . thus , the result follows directly from theorem [ unicycleinvert ] . 99 m. aigner . _ combinatorial theory_. 1979 , springer - verlag , new york . s. akbari and s. kirkland . on unimodular graphs . _ linear algebra and its applications _ , * 421 * ( 2001 ) pp . 3 - 15 . robert donaghey and louis w. shapiro . motzkin numbers . _ journal of combinatorial theory , series a. _ * 23 * , 201 - 301 . inverses of trees . _ combinatorica _ * 5 * ( 1 ) , pp . f. harary and h. minc . which nonnegative matrices are self - inverse ? _ mathematics magazine _ , vol . 2 ( mar . , 1976 ) , pp . 91 - 92 . mathematical association of america . the on - line encyclopedia of integer sequences , published electronically at http://oeis.org , 2011 . sequence a001006 . david speyer . partitions into 0,1 , and 2 with a partial sum condition , http://mathoverflow.net/questions/64802 ( version : 2011 - 05 - 12 ) . r. simion and d. cao . solution to a problem of c.d . godsil regarding bipartite graphs with unique perfect matching . _ combinatorica _ * 9 * ( 1 ) ( 1989 ) pp . r. tifenbach and s. kirkland . directed intervals and the dual of a graph . _ linear algebra and its applications _ , * 431 * ( 2009 ) pp . 792 - 807 .

extending the work of godsil and others , we investigate the notion of the inverse of a graph ( specifically , of bipartite graphs with a unique perfect matching ) . we provide a concise necessary and sufficient condition for the invertibility of such graphs and generalize the notion of invertibility to multigraphs . we examine the question of whether there exists a `` litmus subgraph '' whose bipartiteness determines invertibility . as an application of our invertibility criteria , we quickly describe all invertible unicyclic graphs . finally , we describe a general combinatorial procedure for iteratively constructing invertible graphs , giving rise to large new families of such graphs . [ section ] [ theo]proposition [ theo]lemma [ theo]corollary [ theo]definition [ theo]remark [ theo]example

galaxy clusters provide a special environment for their members . in contrast to the field , the number volume density of galaxies is high and the relative velocities are large . the gravitational potential of a cluster is filled by the intracluster medium ( icm ) , a hot x ray emitting gas , and the overall mass to light ratio is much larger than for the individual galaxies indicating the presence of vast amounts of dark matter . this environment exerts a strong influence on the evolution of the cluster galaxies superposed on the ( field ) evolution that arises from the hierarchical growth of objects and the declining starformation rates over cosmic epochs . besides tidal interactions between galaxies including merging as can also be observed in the field , cluster members are affected by clusterspecific phenomena related to the icm ( like ram - pressure stripping ) or the structure of the cluster ( like harassment ) . for a recent overview see third volume of the carnegie observatories astrophysics series . imprints of these interactions can not only be seen in present - day clusters , but also manifest themselves in a strong evolution of the population of cluster galaxies . one example is the photometric butcher oemler effect of an increasing fraction of blue galaxies with redshift ( e.g. * ? ? ? * ) implying a rising percentage of starforming galaxies . another example is the rapid decline of the abundance of lenticular galaxies ( s0 ) from the dominant population in local clusters to a few percent at a lookback time of @xmath5gyrs ( e.g. * ? ? ? these observations have led to the question whether field spirals falling into a cluster can be subject to such morphological transformations that they appear as s0 galaxies today . independent from whether this overall scenario is true or not , the observed tidal interactions ( either between galaxies or with the cluster potential ) may cause substantial distortions both on the structure and the kinematics of the galaxies involved . indeed , @xcite , for example , found that half of their sample of 89 disk galaxies in the virgo cluster exhibit kinematic disturbances ranging from modest ( e.g. asymmetric ) to severe ( e.g. truncated curves ) peculiarities . on the other hand , many local cluster galaxies , for which only hi velocity widths ( in contrast to spatially resolved velocity profiles ) were measured , follow a tight tully fisher relation similar to field spirals ( e.g. * ? ? ? this tfr connects the luminosity of the stellar population of a galaxy to its internal kinematics which are dominated by the presence of a dark matter halo @xcite . but it is not yet clear whether the halo of dark matter and , therefore , the total mass of a galaxy can also be affected by certain interaction phenomena . in numerical simulations of the evolution of substructure in clusters by @xcite , the dark matter halo of a galaxy that falls into the cluster is truncated via tidal interactions so that the mass - to - light ratio of a galaxy gets reduced during its passage to the cluster core . @xcite simulates the tidal field along galactic orbits in hierarchically growing clusters , and finds that about 40% of the dark halo of a massive galaxy ( @xmath6km / s ) is lost between @xmath7 and @xmath8 . but the rotational velocity at @xmath5 disk scale lengths is predicted to hardly change ( decrease by @xmath9% ) . since in models of hierarchically growing structure clusters are still in the process of forming at @xmath10 in the concordance cosmology , a higher infall rate and more interactions are expected at redshifts @xmath11 ( e.g. * ? ? ? * ) . with the availability of large telescopes , it is now feasible to conduct spatially resolved spectroscopy of the faint galaxies at these redshifts to observationally test these predictions . therefore , we have performed a large campaign at the vlt targeting seven distant rich clusters with @xmath12 . the clusters were chosen from a very limited list with existing hst / wfpc2 imaging ( mainly the core regions ) at the time the project started ( 1999 ) and that are accessible with the vlt . ms100812 has no hst imaging but was included since it was imaged extensively during fors science verification time . the main goal is to derive the two - dimensional internal kinematics of disk galaxies from emission lines . in combination with measurements of starformation rates , luminosities and structural parameters we aim at disentangling the effects of different interaction processes and find out about their respective actual effectiveness and importance for galaxy evolution . in this letter , we present the results for the first three clusters of our survey : ms100812 ( @xmath0 ) , cl0303@xmath117 ( @xmath2 ) , and cl041365 ( @xmath3 ) . they were observed with fors1 in mos mode , while the other four clusters ( cl0016@xmath11609 , ms0451.60305 , zwcl1447.2@xmath12619 & ms2137.32353 ) had mxu spectroscopic observations with fors2 requiring different reduction techniques that will be presented in future papers . while in paperii @xcite we give all the data that can be deduced from each single spectrum , we analyze in this letter only the late type galaxies which exhibit spatially resolved emission . one setup in the mos mode of fors1 provides 19 individual slitlets . for each cluster two setups have been designed with different rotation angles of the instrument . using grism 600r and slitwidths of 1 , the spectra have a dispersion of @xmath131.08 / pixel , a spectral resolution of @xmath14 and typical wavelengths of @xmath15 . in standard configuration fors has a field of view of @xmath16 with a spatial scale of 0.2/pixel . to achieve our signal to noise requirements of @xmath17 in the emission lines of an @xmath18 galaxy , the total integration time was set to @xmath132hrs . seeing conditions ranged between 0.7 and 1.3 fwhm . spatially resolved velocity profiles @xmath19 were determined from either the [ oii]3727 , h@xmath20 or [ oiii]5007 emission line . in eight cases , two lines with sufficient @xmath4 were visible which then were treated separately yielding consistent results . the spectral profile of an emission line was fitted by a gaussian ( in case of [ oii ] by a double gaussian ) after applying a median filter window of typically 0.6 to enhance the @xmath4 stepping along the spatial axis . since the apparent disk sizes of spirals at intermediate redshifts are only slightly larger than the slitwidth ( 1 ) , the slit covers a substantial fraction of the two dimensional velocity field . therefore , the spectroscopy is an integration perpendicular to the slit s spatial axis . because of this effect , the maximum rotation velocity @xmath21 can not be determined `` straightforward '' from the observed rotation . as described in @xcite and @xcite , we overcome this problem by simulating such longslit spectroscopy of each galaxy individually . in short , a two - dimensional velocity field is created assuming a specific rotation law that is weighted by the galaxy s luminosity profile and convolved to match the seeing at the time of our observations . taking into account the galaxy s inclination , position angle , and disk scalelength , a synthetic rotation curve is generated with @xmath21 as the only remaining free parameter . @xmath21 is then determined by matching the synthetic to the observed velocity profile . for the galaxies analyzed here , we used for the model rotation curve a simple parameterization with a linearily rising inner part that turns over into a flat outer part . but as is demonstrated by @xcite , @xmath21 is hardly changed when the universal rotation curve by @xcite is used instead . the inclinations , position angles and scalelengths were derived via 2-d @xmath22fits of exponential disks to the galaxies profiles in fors images . the fwhm was 0.59 ( ms100812 ) , 0.77 ( cl041365 ) and 1.0 ( cl0303@xmath117 ) , respectively . the fits accounted for the psf . spirals with too low an inclination ( @xmath23 ) have not been used for the @xmath21 derivation . galaxy luminosities were derived from total magnitudes of fors images in the @xmath24 ( ms100812 ) or @xmath25 ( cl041365 and cl0303@xmath117 ) band , respectively , as measured with sextractor @xcite . observed magnitudes were corrected for galactic ( from @xcite ) and intrinsic extinction ( following @xcite ) and transformed to restframe johnson @xmath26 according to their spectrophotometric type using model seds corresponding to sa , sb , sc and sdm , and calculated for a flat @xmath27 cosmology ( @xmath28kms@xmath29mpc@xmath29 ) . the average overall error in the photometry is estimated to be @xmath30 . the mos mode of fors1 , which was the only multiplex technique available for the first observations of our campaign , has some disadvantages for the spectroscopy of spiral galaxies in clusters , which are removed by the mxu mode of fors2 that we could use for the other four clusters of our survey . firstly , the 19 slitlets of fixed length ( @xmath31 ) have only one degree of freedom for placing a slitlet onto an object . this leads to somewhat inefficient coverage of cluster member candidates and many slitlets must be filled by strongly relaxing the ideal selection criteria . secondly , once a certain rotation of the field is chosen , all the 19 slitlets have the same orientation on the sky . ideally , the slit should be placed along the major axis of a galaxy to probe its rotation around the center . although we have targeted each cluster with two different setups , the deviation @xmath32 between slit angle and position angle was rather large in some cases leading to geometric distortions of the observed velocity profile that could not be corrected for . these galaxies are not used in the further analysis of the internal kinematics here but are valuable ingredients for future studies of e.g. starformation rates or structural properties of cluster members . as is specified in more detail in paperii , our method of selecting cluster spiral candidates was different for each cluster due to the limited information of published studies . the most comprehensive source was available for ms100812 , for which we exploited a catalog of @xmath33 cluster members with published spectral types @xcite . a list of candidates in cl041365 was prepared by comparing optical nearinfrared colors by @xcite to evolutionary stellar population models of an updated version of @xcite . for cl0303@xmath117 , we mainly utilized a spectroscopic catalog of @xcite . mos slitlets that could not be filled by a galaxy from our input lists were placed on objects selected according to their structural appearance and magnitudes as measured on our fors images . if no suitable candidate for a late type galaxy was available , the slitlet was placed on an elliptical candidate since understanding the phenomenon of galaxy transformation requires the analysis of the whole galaxy population of a cluster . redshifts and spectral types could be determined for 12/13 spiral / elliptical members of ms100812 ( plus 4/2 s / e field galaxies ) , 8/1 spiral / elliptical members of cl041365 ( plus 11/6 s / e field galaxies ) , and 10/7 spiral / elliptical members of cl0303@xmath117 ( plus 15/0 s / e field galaxies ) . while the early type galaxies will be discussed in a future paper , we here concentrate on the emission line galaxies . as was pointed out by @xcite only galaxies with a rotation curve that rises in the inner region and then clearly turns into a flat part should be used for a tully fisher diagram . in such a case , the measured @xmath21 is representative for the influence of the dark matter halo on the galaxy s kinematics and is indicative for the total dynamical mass of the galaxy . for our three clusters , we were able to determine @xmath21 for 7/5/1 different member galaxies , respectively , ( and 1/5/1 field spirals ) . the remaining cluster spirals either have too low @xmath4 of their emission lines to spatially analyse the internal kinematics ( in 0/0/4 cases ) , have too low inclination @xmath34 or too large mismatch angle @xmath32 ( in 1/3/5 cases ) , or exhibit intrinsic distortions ( in 4/0/0 cases ) . in fig.[fig : rc ] we show as an example position velocity diagrams for all member galaxies in ms100812 that have sufficient @xmath4 . of the 12 members , of which four were observed twice with different slit angles with respect to their major axis , seven exhibit the `` classical '' rotation curve shape rising in the inner part and turning over to a flat regime ( labeled `` tf '' ) . four members clearly show disturbed kinematics ( panels b , j , l , & m ) . the distortion seen in panel m ( & n ) most probably arises from a bar which is readily visible in the direct image . two `` double hits '' were observed with very big mismatch angles ( panels c & k ) so that their velocity profiles look peculiar . in one case ( panel h ) the object fell onto the slitlet by chance and is too weak ( small ) to derive its structural parameters . in the figure , the galaxies are ordered according to their projected distance to the cluster center ( ranging from 285kpc to 1230kpc ) . there is no trend visible of the rc form as a function of clustercentric distance , i. e. distorted velocities are not uniquely tied to the central region . but since our observations cover only the region within the virial radius of the cluster ( @xmath35kpc ) , this is in accordance with dynamical models in which the galaxy population of a cluster is well - mixed within that region @xcite . in particular , we most probably do not have any new arrivals from the field in our sample . in fig.2 , we present the tully fisher diagram for the distant cluster galaxies . only those galaxies for which @xmath21 could be determined enter the plot . the ordinate gives the restframe absolute johnson @xmath26-band magnitudes . we also show the position of those field galaxies that were serendipitously observed in the cluster fields and that have @xmath36 ( 3 spirals at @xmath37 may be members of a background cluster behind cl041365 ) . for comparison , we include the distant field galaxies from the fors deep field @xcite which were observed with exactly the same instrumental setup @xcite . the linear bisector fit to this sample is flatter than the slope fitted to the local sample of ( * ? ? ? * pt92 ) , which is given with its @xmath38 deviations . the distant cluster spirals are distributed very much alike the field population that covers similar cosmic epochs ( @xmath39 with the bulk of galaxies at @xmath40 and @xmath41 ) . no significant deviation from the distant field tfr is visible and the cluster sample has not any increased scatter , but the low number of cluster members prohibits any quantitative statistical analysis . nevertheless , we can conclude that the mass to light ratios of the observed distant cluster spirals cover the same range as the distant field population indicating that their stellar populations were not dramatically changed by possible clusterspecific interaction phenomena . in particular , we do not detect any significant overluminosities as would be expected in the @xmath26 band if strong starbursts had occured in the recent past of the examined cluster galaxies . with respect to the tully fisher relation obeyed by local galaxies ( e.g. pt92 ) , our cluster sample follows the same trend as the fdf galaxies . since mostly only the bright galaxies made it into the tf diagram , the cluster members occupy a region where no significant luminosity evolution is visible . tully fisher diagram of cluster spirals in ms100812 ( filled triangles ) , in cl041365 ( filled squares ) , and cl0303@xmath117(filled diamond ) . also shown are seven field objects ( open symbols ) that were serendipitously observed . in comparison to the fors deep field sample of 77 field galaxies @xcite with @xmath39 ( small open circles ) , the cluster galaxies are similarly distributed and do not deviate significantly from the linear fit to the fdf sample ( solid line ) . the cluster members follow the same trend with respect to the local tfr ( the fit @xmath38 to the @xcite field sample is given ) as the distant field galaxies with the brightest galaxies exhibiting the smallest luminosity evolution . but we emphasize that this conclusion is true only for those objects that enter the tf diagram . since a significant fraction of our cluster galaxies can not be used for a tf analysis due to their distorted kinematics the above conclusions are not generally valid for the whole cluster sample . the objects with intrinsically peculiar velocity curves may actually be subject to ongoing or may have experienced recent interactions . such processes most probably also influence the stellar populations of a galaxy changing its integrated luminosity as well . tidal interactions for example could distort the spiral arms while inducing starbursts at the same time leading to enhanced luminosities . since we do not know where the galaxies with peculiar kinematics lie in the tf plane , it is not possible to decide whether a particular galaxy has an increased or decreased luminosity . overall , these galaxies span a very similar range in apparent magnitudes to the tf cluster members . exploring the spatially resolved kinematics of disk galaxies in distant clusters has become feasible only recently . * mj03 ) studied a sample of 7 spiral galaxies in the cluster ms1054.40321 at @xmath42 with fors at the vlt in a similar configuration as our own spectroscopy . compared to a number of field spirals at corresponding redshifts , which were observed at the same time , they find that the cluster members have brighter @xmath26 luminosities by @xmath43 ( @xmath44 significance ) for their rotational velocities . the difference to the average brightening of our cluster members is hardly significant and may be due to a combination of low - number statistics and systematic deviations . but we also can not rule out that the differences are real and may be connected to the higher redshift of ms1054 or other characteristics of that cluster . @xcite et al . examined galaxies in the cluster cl0024@xmath11654 ( @xmath45 ) with the keck 10m - telescope . the ten galaxies that appear in their tf diagram have a larger scatter than the local pt92 sample , but show no evidence for an evolution of the zero - point . the authors argue that processes acting on the cluster galaxies some time before the lookback time of the observations involving either starbursts or a truncation of star formation may have caused a decreased or increased mass to light ratio , respectively , which is manifested in the increased scatter . in a future paper we will present our analysis for the other four clusters of our campaign . with more cluster member galaxies both in the tf diagram and those with peculiar kinematics , we may hopefully be able to quantitatively investigate the galaxy evolution in rich clusters and to give also statistical tests . we acknowledge the thorough comments by the referee . we are very grateful to drs . s. wagner ( heidelberg ) , u. hopp ( mnchen ) and r. h. mendez ( hawaii ) for performing part of the observations and thank eso and the paranal staff for efficient support . we also thank the pi of the fors project , prof . i. appenzeller ( heidelberg ) , and prof . k. j. fricke ( gttingen ) for providing guaranteed time for our project . we also acknowledge fruitful discussions with drs . b. milvang jensen ( mpe garching ) and m. verheijen ( potsdam ) . this work has been supported by the volkswagen foundation ( i/76520 ) and the deutsche forschungsgemeinschaft ( fr 325/461 and sfb 439 ) . metevier , a. j. : 2003 , in j. s. mulchaey , a. dressler , and a. oemler ( eds . ) , _ clusters of galaxies : probes of cosmological structure and galaxy evolution _ , carnegie observatories astrophysics series vol . 3 , p. in press , carnegie observatories , pasadena

we introduce our project on galaxy evolution in the environment of rich clusters aiming at disentangling the importance of specific interaction and galaxy transformation processes from the hierarchical evolution of galaxies in the field . emphasis is laid on the examination of the internal kinematics of disk galaxies through spatially resolved mos spectroscopy with fors at the vlt . first results are presented for the clusters ms1008.11224 ( @xmath0 ) , cl0303@xmath11706 ( @xmath2 ) , and cl04136559 ( f1557.19tc ) ( @xmath3 ) . out of 30 cluster members with emission - lines , 13 galaxies exhibit a rotation curve of the universal form rising in the inner region and passing over into a flat part . the other members have either intrinsically peculiar kinematics ( 4 ) , or too strong geometric distortions ( 9 ) or too low @xmath4 ( 4 galaxies ) for a reliable classification of their velocity profiles . the 13 cluster galaxies for which a maximum rotation velocity could be derived are distributed in the tully fisher diagram very similar to field galaxies from the fors deep field that have corresponding redshifts and do not show any significant luminosity evolution with respect to local samples . the same is true for seven galaxies observed in the cluster fields that turned out not to be members . the mass to light ratios of the 13 tf cluster spirals cover the same range as the distant field population indicating that their stellar populations were not dramatically changed by possible clusterspecific interaction phenomena . the cluster members with distorted kinematics may be subject to interaction processes but it is impossible to determine whether these processes also lead to changes in the overall luminosity of their stellar populations .

let @xmath7 be a partition of the integer @xmath8 , i.e. , @xmath9 and @xmath10 . the _ length _ @xmath11 of a partition @xmath0 is the number of nonzero parts of @xmath0 . the ( durfee or frobenius ) _ rank _ of @xmath0 , denoted rank(@xmath0 ) , is the length of the main diagonal of the diagram of @xmath0 , or equivalently , the largest integer @xmath12 for which @xmath13 . the rank of @xmath0 is the least integer @xmath14 such that @xmath0 is a disjoint union of @xmath14 border strips ( also called ribbons or rim hooks ) . denote by @xmath15 the character of the irreducible representation of @xmath5 indexed by @xmath0 evaluated at a permutation of cycle type @xmath16 . an easy application of the murnaghan - nakayama rule shows that if @xmath17 , we must have @xmath18 . as a corollary , the expansion of the schur function in terms of the power sum symmetric functions @xmath19 contains only terms @xmath20 such that @xmath21 rank @xmath22 . we generalize the notion of rank to shifted diagrams @xmath1 of a partition with distinct parts by counting the minimal number of bars in a bar tableau . using morris projective analogue of the murnaghan - nakayama rule , we show that the irreducible projective characters of @xmath5 vanish on conjugacy classes indexed by partitions with few parts . this enables us to give a lower bound on the length of the @xmath23 which appear in the expansion of the schur q - functions in terms of the @xmath24 . let @xmath25 be the set of all partitions of @xmath8 into distinct parts . the _ shifted diagram _ , @xmath1 , of shape @xmath0 is obtained by forming @xmath26 rows of nodes with @xmath27 nodes in the @xmath12th row such that , for all @xmath28 , the first node in row @xmath12 is placed underneath the second node in row @xmath29 . for instance figure [ fig : shiftedshape ] shows the shifted diagram of the shape 97631 . we follow the treatment of hoffman and humphreys @xcite to define bar tableaux . these occur in the inductive formula for the projective characters of @xmath5 , first proved by morris @xcite . let @xmath14 be an odd positive integer , and let @xmath30 have length @xmath26 . below we define : 1 . a subset , @xmath31 , of integers between @xmath32 and @xmath26 ; and 2 . for each @xmath33 , a strict partition @xmath34 in @xmath35 ( despite the notation , @xmath34 is a function of @xmath0 , as well as of @xmath36 ) . let @xmath37 in other words @xmath38 is the set of all rows of @xmath0 which we can remove @xmath14 squares from and still leave a composition with distinct parts . for example , if @xmath39 and @xmath40 , then @xmath41 . if @xmath42 , then @xmath43 , and we define @xmath34 to be the partition obtained from @xmath0 by removing @xmath27 and inserting @xmath44 between @xmath45 and @xmath46 . continuing our example above , @xmath47 . let @xmath48 which is empty or a singleton . for @xmath49 , remove @xmath27 from @xmath0 to obain @xmath34 . let @xmath50 equivalently @xmath51 is the set of all rows of @xmath0 for which there is some shorter row of @xmath0 such that the total number of squares in both rows is @xmath14 . for example , if @xmath52 and @xmath40 , then @xmath53 . if @xmath54 , then @xmath55 , and @xmath34 is formed by removing both @xmath27 and @xmath45 from @xmath0 . for each @xmath56 the associated _ r - bar _ is given as follows . if @xmath12 is in @xmath38 or @xmath57 , the @xmath14-bar consists of the rightmost @xmath14 nodes in the @xmath12th row of @xmath1 . we say the @xmath14-bar is of _ type 1 _ or _ type 2 _ respectively . for example , the squares in figure [ fig : bartableau ] labelled by 6 are a @xmath58-bar of type 1 . the squares labelled by 4 are a @xmath59-bar of type 2 . if @xmath12 is in @xmath51 , the @xmath14-bar consists of all the nodes in both the @xmath12th and @xmath60th rows , a total of @xmath14 nodes . we say the @xmath14-bar is of _ type 3_. the squares in figure [ fig : bartableau ] labelled by 3 are a @xmath58-bar of type 3 . define a _ bar tableau _ of shape @xmath0 to be an assignment of positive integers to the squares of @xmath1 such that 1 . the set of squares occupied by the biggest integer is an @xmath14-bar @xmath61 , and 2 . if we remove the @xmath14-bar @xmath61 and reorder the rows , the result is a bar tableau . equivalently we can define a _ bar tableau _ of shape @xmath0 to be an assignment of positive integers to the squares of @xmath1 such that 1 . the entries are weakly increasing across rows , 2 . each integer @xmath12 appears an odd number of times , 3 . @xmath12 can appear in at most two rows ; if it does , it must begin both rows ( equivalent to the bar being of type 3 ) , 4 . the composition remaining if we remove all squares labelled by integers larger than some @xmath12 has distinct parts . for example , figure [ fig : tableauchain ] shows the chain of partitions remaining if we remove all squares labelled by integers larger than some @xmath12 from the tableau in figure [ fig : bartableau ] . this demonstrates the legality of that tableau . we introduce an operation on minimal bar tableaux which preserves the number of bars , and prove some facts about tableaux resulting from this operation . a bar tableau of @xmath0 is _ minimal _ if the number of bars is minimized , i.e. there does not exist a bar tableau with fewer bars . [ l : noevenbdr ] there exists a minimal bar tableau @xmath62 such that there is no bar boundary an even number of squares along any row . for example figure [ fig : minbartableau ] shows a minimal bar tableau @xmath63 of shape 97631 and a minimal bar tableau @xmath62 of the same shape with no bar boundaries an even number of squares along any row ( we will verify later that these tableaux are minimal ) . let @xmath63 be a minimal bar tableau of @xmath0 . in each row @xmath64 of @xmath63 , at the last bar boundary an even number of squares along a row , let @xmath65 be the bar which begins to the right of the boundary . say that @xmath65 is labelled by @xmath60 . relabel the squares to the left of the boundary with @xmath60 . this preserves the ordering on labels and the parity of @xmath65 . the partitions remaining if we remove all squares labelled by integers larger than @xmath66 will be the same as before and have distinct parts . the partitions remaining if we remove all squares labelled by @xmath67 will not contain row @xmath64 but will otherwise have the same ( distinct ) parts as before . [ l : prepstruct ] let @xmath62 be a minimal bar tableau of @xmath0 such that there is no bar boundary an even number of squares along any row . then if row @xmath64 is odd , it is labelled entirely by one label @xmath60 . if row @xmath64 is even , it is labelled entirely by one label @xmath60 or it has exactly two labels each occurring an odd number of times . if row @xmath64 is odd and has more than one label , the second bar must be of type 1 . therefore the second bar must be odd , forcing the first bar to be even which is a contradiction . if row @xmath64 is even and has more than two labels , the final two bars are both of type 1 and so must be odd , forcing there to be a bar boundary an even number of squares along the row , a contradiction . we use the results from the previous section to give a count of how many bars are needed in a minimal bar tableau . define the _ shifted rank _ of a shape @xmath0 , denoted @xmath68 , to be the number of bars in a minimal bar tableau of @xmath0 . given an integer @xmath69 , define @xmath69 mod 2 to be 1 if @xmath69 is odd and 0 if @xmath69 is even . [ t : count ] given a shape @xmath0 , let @xmath3 be the number of odd rows of @xmath0 and @xmath4 be the number of even rows . then @xmath70 . for example , if @xmath71 , we have @xmath72 and @xmath73 . so @xmath74 which verifies that the tableaux shown in figure [ fig : minbartableau ] are indeed minimal . if @xmath75 , we have @xmath76 and @xmath77 . so @xmath78 . such a tableau is illustrated below . [ fig : minbartableau2 ] let @xmath63 be a minimal bar tableau of @xmath0 . preprocess @xmath63 into @xmath62 so that there are no bar boundaries an even number of squares along any row . this must preserve the number of bars . bars of type 3 consist of one even initial bar and one odd initial bar , and so by lemma [ l : prepstruct ] must be an entire even row and an entire odd row , or an entire even row and the initial odd bar of some other even row . first assume that @xmath79 . note that if @xmath80 , then @xmath81 . so when @xmath79 , @xmath82 . we claim that the bars of type 3 all consist of entire even row and entire odd row pairs , and that there are exactly @xmath4 of them . from the observations above , the number of bars of type 3 can not be larger than @xmath4 . suppose that there is a bar of type 3 consisting of an entire even row and the initial odd bar of some other even row . since @xmath79 , there must also be two other odd rows , not parts of bars of type 3 , each labelled entirely by some label ( by lemma [ l : prepstruct ] ) . the total number of bars in these 4 rows is 4 . so if we relabel ( with new large labels ) these four rows as two bars of type 3 , we save two bars , contradicting the minimality of @xmath62 ( note that relabelling entire rows with some integer larger than any current label preserves legality , provided the parities are correct ) . we illustrate this ( impossible ) situation below , and show the more economical version . thus there are no such bars of type 3 . [ fig : minbartableau3 ] now suppose that the number of bars of type 3 is smaller than @xmath4 ; thus there is some even row @xmath83 of the tableau which is not part of a bar of type 3 . also there is an odd row @xmath84 which is not part of a type 3 bar ( since @xmath85 ) . but we could relabel both these rows with some new large label saving at least one bar and contradict the minimality of @xmath62 . so there are exactly @xmath4 bars of type 3 , filling @xmath86 rows of @xmath0 . the remaining @xmath87 rows are odd and so must each be completely filled by a unique label . so the total number of bars is @xmath3 as required . 1.5em now assume that @xmath88 . first we show that we can relabel so that every odd row is part of a bar of type 3 . so suppose @xmath89 is an odd row which is not part of a bar of type 3 . _ there is an even row @xmath90 , completely filled by a label , which is part of a bar of type 3 with the initial part of some other even row @xmath91 . _ proof of claim . _ assume by way of contradiction that there is not , i.e. that the completely filled even rows are all parts of bars of type 3 with complete odd rows . but there must be at least one even row @xmath90 ( since @xmath88 and @xmath89 is not part of a bar of type 3 ) which is not part of a bar of type 3 with a complete odd row . so @xmath90 must not be part of a bar of type 3 at all ( by our claim assumption ) . but then we could relabel @xmath90 and @xmath89 entirely with some new large label and save a bar , a contradiction . this proves our claim . relabel @xmath90 and @xmath89 with some new large label . this leaves an odd number of squares in the initial part of row @xmath91 , and so preserves legality . these two rows are now a valid bar of type 3 , and this process did not cost us any bars . we illustrate one step of this process below . simply iterate this process until there are no odd rows which are not part of bars of type 3 . this proves that we can relabel so that every odd row is part of a bar of type 3 . [ fig : minbartableau4 ] so every odd row is part of a bar of type 3 , filling @xmath92 rows of @xmath0 . all ( except for possibly one ) of the even rows remaining must be paired up with another remaining even row , and each pair must contain one bar of type 3 ( filling one entire row and the odd initial part of the other row ) and one bar of type 1 ( filling the odd final part of the other row ) . if they were not , we would have two even rows costing 4 strips , and could reduce the number of strips by relabelling as above with large new numbers . the extra row exists only when @xmath11 is odd , and costs two bars ( i.e. one extra ) this situation is illustrated on the right hand side of the above figure . so we have @xmath93 strips as required . it remains an open problem to count how many minimal shifted tableau there are for a given shape @xmath0 . also , it would be natural to generalise theorem [ t : count ] to the skew shifted case . it is possible to define skew bar tableaux , but the minimal number of bars therein remains an open question ( see @xcite for more details ) . here we recall some facts about the projective representations of the symmetric group . we follow the treatment of stembridge @xcite . a projective representation of a group @xmath94 on a vector space @xmath95 is a map @xmath96 such that @xmath97 for suitable ( nonzero ) scalars @xmath98 . for the symmetric group , the associated coxeter presentation shows that a representation @xmath99 amounts to a collection of linear transformations @xmath100 ( representing the adjacent transpositions ) such that @xmath101 , and @xmath102 ( for @xmath103 ) are all scalars . the possible scalars that arise in this fashion are limited . of course , one possibility is that the scalars are trivial ; this occurs in any ordinary linear representation of @xmath5 . according to a result of schur @xcite , there is only one other possibility ( occurring only when @xmath104 ) ; namely @xmath105 all other possibilities can be reduced to this case or the trivial case by a change of scale . see @xcite , @xcite for details . it is convenient to regard @xmath106 as elements of an abstract group , and to take [ eq : sigmas ] as a set of defining relations . more precisely , for @xmath107 let us define @xmath108 to be the group of order @xmath109 generated by @xmath106 ( and @xmath110 ) , subject to the relations [ eq : sigmas ] , along with the obvious relations @xmath111 which force @xmath110 to be a central involution . by schur s lemma , an irreducible linear representation of @xmath108 must represent @xmath110 by either of the scalars @xmath112 or @xmath110 . a representation of the former type is a linear representation of @xmath5 , whereas one of the latter type corresponds to a projective representation of @xmath5 as in [ eq : sigmas ] . we will refer to any representation of @xmath108 in which the group element @xmath110 is represented by the scalar @xmath110 as a _ negative representation _ of @xmath108 . next we review the characters of the irreducible negative representations of @xmath108 . define @xmath113 to be the set of all partitions of @xmath8 . we say that a partition @xmath0 is _ odd _ if and only if the number of even parts in @xmath0 is odd , and is _ even _ if and only if it is not odd . thus , the parity of a permutation agrees with the parity of its cycle type . the parity of @xmath0 is also the parity of the integer @xmath114 . schur showed that the irreducible negative representations are indexed by partitions @xmath0 with distinct parts . recall that if @xmath99 is an irreducible negative representation indexed by @xmath0 that the character @xmath115 is a class function @xmath116 defined by @xmath117 . if @xmath118 , let @xmath119 be the cycle type ( in @xmath5 ) of @xmath120 . in the sequel we will evaluate @xmath121 instead of @xmath122 . define @xmath123 to be all partitions of @xmath8 such that all parts are odd . let @xmath124 have length @xmath125 , and let @xmath119 . 1 . suppose that @xmath0 is odd . if @xmath16 is neither in @xmath126 nor equal to @xmath0 then @xmath127 . [ it : odd ] suppose that @xmath0 is odd . if @xmath16 equals @xmath0 then @xmath128 3 . suppose that @xmath0 is even . if @xmath16 is not in @xmath129 then @xmath127 . for example we consider the situation when @xmath130 and @xmath131 . then @xmath132 when @xmath16 is @xmath133 or @xmath134 , as these partitions all have one even part . the second fact gives @xmath135 when @xmath136 . if @xmath137 then @xmath138 when @xmath16 is @xmath139 or @xmath134 . a combinatorial rule for calculating the characters not specified by schur s theorem was given by morris ; it is the projective analogue of the murnaghan - nakayama rule . let @xmath124 have length @xmath125 . suppose that @xmath140 and that @xmath16 contains @xmath14 at least once . define @xmath141 by removing a copy of @xmath14 from @xmath16 . then @xmath142 where @xmath143 n_i = \ { [ cols= " < , < " , ] . @xmath143 ( the integer @xmath60 is that occurring in the definitions of @xmath144 , and @xmath145 is the parity of @xmath0 ; i.e. 0 or 1 . ) for example if @xmath130 , @xmath146 and @xmath147 , we have @xmath148 and @xmath149 so @xmath150 , and we have @xmath151 expand this sum into a sum over all possible bar tableaux . define the _ weight _ of a tableau wt@xmath152 to be the product of all the powers of @xmath110 and @xmath153 which appear . then we have @xmath154 summed over all bar tableaux of shape @xmath0 and type @xmath16 . we know that the shifted rank of @xmath0 is the minimum number of bars needed in a bar tableau of shape @xmath0 . so we obtain the following result as a corollary to theorem [ t : count ] : given a shape @xmath0 of shifted rank @xmath155 and a shape @xmath16 such that @xmath156 , we have @xmath127 . @xmath157 we begin with schur s original inductive definition of the @xmath6 functions . denote the monomial symmetric functions by @xmath158 and define symmetric functions @xmath159 of degree @xmath155 by @xmath160 now we can state the base cases for the inductive definition . put @xmath161 and @xmath162 inductively we define @xmath163 and @xmath164 the @xmath6 may also be defined as the specialization at @xmath165 of one of the two equivalent defining formulae for hall - littlewood polynomials ; see ( * ? ? ? * iii ( 2.1 ) ( 2.2 ) ) . let @xmath166 act on @xmath167 by permuting the variables , so that , when @xmath168 , the young subgroup @xmath169 fixes each of @xmath170 . let @xmath0 be a strict partition of length @xmath125 . if @xmath168 , then @xmath171 \in s_r / s_1^{\ell } \times s_{r-\ell } } w \left \ { x_1^{\lambda_1 } \cdots x_{\ell}^{\lambda_{\ell } } \prod_{i=1}^{\ell } \prod_{j = i+1}^{r } \frac{x_i+x_j}{x_i - x_j } \right \}.\ ] ] if @xmath0 has length greater than @xmath14 , then @xmath172 . the @xmath6 symmetric functions are obtained by taking the limit as the number of variables becomes infinite ( for a mathematically precise definition of this limit see @xcite ) . schur @xcite defined these q - functions in order to study the projective representations of symmetric groups . the fundamental connection is given by the following theorem . let @xmath173 , the number of parts of @xmath0 equal to @xmath12 . define @xmath174 . denote the power sum symmetric functions by @xmath175 . again consider the example with @xmath130 and @xmath146 . we have @xmath177/2 } { \langle}51 { \rangle}(1 ^ 6 ) \frac{p_{1 ^ 6}}{z_{1 ^ 6}}+ 2^{[2 + 4 + 0]/2 } { \langle}51 { \rangle}(1 ^ 3 3 ) \frac{p_{1 ^ 3 3}}{z_{1 ^ 3 3}}+ \\ & & 2^{[2 + 2 + 0]/2 } { \langle}51 { \rangle}(15 ) \frac{p_{1 5}}{z_{15}}+ 2^{[2 + 2 + 0]/2 } { \langle}51 { \rangle}(3 ^ 2 ) \frac{p_{3 ^ 2}}{z_{3 ^ 2 } } \\ & = & 2^{4 } 16 \frac{p_{1 ^ 6}}{6!}+ 2 ^ 3 2 \frac{p_{1 ^ 3 3}}{3 ! 3 } - 2 ^ 2 1 \frac{p_{1 5}}{5}- 2 ^ 2 2 \frac{p_{3 ^ 2}}{18}\\ & = & \frac{16}{45 } p_{1 ^ 6 } + \frac{8}{9}p_{1 ^ 3 3 } -\frac{4}{5 } p_{1 5 } - \frac{4}{9}p_{3 ^ 2}\end{aligned}\ ] ] in our example srank@xmath182 and the @xmath183 satisfy @xmath184 . equivalently we can examine a specialization of the principal specialization of @xmath6 , i.e. @xmath185 since @xmath186 , we can rephrase the above result .

motivated by stanley s results in @xcite , we generalize the rank of a partition @xmath0 to the rank of a shifted partition @xmath1 . we show that the number of bars required in a minimal bar tableau of @xmath1 is max@xmath2 , where @xmath3 and @xmath4 are the number of odd and even rows of @xmath0 . as a consequence we show that the irreducible projective characters of @xmath5 vanish on certain conjugacy classes . another corollary is a lower bound on the degree of the terms in the expansion of schur s @xmath6 symmetric functions in terms of the power sum symmetric functions . * minimal bar tableaux * + _ cnri , _ _ dublin institute of technology , ireland _ + peter@cnri.dit.ie + : 05e05 , 05e10 , 20c25 , 20c30 + keywords : bar tableau , strip tableau , rank , shifted partition , shifted shape , projective character , negative character , schur q - functions

weak decays of heavy flavored hadrons are an excellent laboratory to study the standard model . in particular b meson decays provide a wealth of information on the quark mixing elements . while the next generation of high luminosity facilities have a good chance to measure @xmath2 asymmetries in these decays and perhaps shed some light on the puzzling and fundamental phenomenon of @xmath2 violation , good progress in measuring some parameters describing quark mixing has been achieved . in the framework of the standard model the gauge bosons , @xmath3 , @xmath4 and @xmath5 couple to mixtures of the physical @xmath6 and @xmath7 states . this mixing is described by the cabibbo - kobayashi - maskawa ( ckm ) matrix : @xmath8 a commonly used approximate parameterization was originally proposed by wolfenstein @xcite . it reflects the hierarchy between the magnitude of matrix elements belonging to different diagonals . the 3 diagonal elements and the 2 elements just above the diagonal are real and positive . it is defined as : @xmath9 @xmath0 decays probe several of these matrix elements . the study of semileptonic decays allows the measurement of @xmath10 and @xmath11 . in addition , the ratio @xmath12 is considered a promising avenue to measure the ratio @xmath13 @xcite . the standard model parameterization of the quark mixing via the ckm matrix element accomodates a complex phase , and therefore offers a natural way to model the intriguing phenomenon of @xmath2 violation . so far this violation has been measured only in neutral @xmath14 decays , and yet it may very well be at the origin of the matter dominated universe that exists now . the ckm matrix must be unitary and the relation between elements of different rows dictated by this property can be graphically represented as so called ` unitarity triangles ' . [ ckm_tri ] shows the most promising of the triangles : the angles @xmath15 , @xmath16 and @xmath4 are all related to the single phase in the @xmath17 matrix element . the study of @xmath0 decays will eventually allow the measurements of all the three angles . thus , it will pose a serious challenge to the standard model description of cp violation and perhaps shed some light on phenomenology beyond the standard model . the statistical accuracy corresponding to the present data sample accumulated with the cleo detector , about 3.5 @xmath18 for the results presented in this paper , is not yet sufficient to probe @xmath2 violation in @xmath0 decays or to measure rare decays like @xmath19 accurately , but is adequate to give extremely valuable information on the matrix elements @xmath20 and @xmath21 . these data allow us to put better constraints on the fundamental parameters in two major ways . on one hand , the measurements reported in this paper provide new information that reduces the experimental errors on the parameters . on the other hand , the experimental data provide constraints to the theoretical models that are needed to relate measured observables to the fundamental parameters of the standard model . in addition , heavy flavored meson decays are a laboratory to probe the strong interaction in different dynamical domains . @xmath0 meson decays probe a regime not fully amenable to perturbative qcd calculations , but suitable for the application of effective theories derived from qcd in some asymptotic conditions . in particular , an approach that has generated a lot of interest in the last few years is the so called ` heavy quark effective theory ' ( hqet ) @xcite , where the asymptotic behavior corresponds to taking the limit @xmath22 . new data shedding some light on the hadronic matrix element will be discussed . the ckm parameters and have been studied extensively by the cleo collaboration through the study of semileptonic decays @xmath23 . the experimental study of semileptonic decays has addressed both inclusive measurements , where the recoiling hadronic state is not identified , and exclusive measurements , where the recoiling hadron is reconstructed through one of its decay channels . inclusive decays have provided several interesting results . most notably the study of the end point of the lepton spectrum , where no charmed hadron final states are kinematically allowed , has provided the first conclusive evidence of a non zero value of @xcite . exclusive decays provide complementary information . in particular the semileptonic channel more extensively studied so far is @xmath24 . this channel is attractive from an experimental point of view because the slow @xmath25 emitted in the decay @xmath26 provides a unique signature of this hadronic final state . in addition , a sharpened attention to this decay has been prompted by the suggestion @xcite from hqet that its study provides a ` model independent ' method to determine . the arguments for this claim will be discussed below . the decay @xmath27 is interesting for several reasons . analyses similar those for the decay @xmath28 provide ways to understand the systematic uncertainty in the extraction of the parameter . the hadronic matrix element involved in this decay , the main source of uncertainty in extracting from the experimental data , is probed effectively by the differential decay width @xmath29 : @xmath30 where @xmath31 and @xmath32 are the @xmath0 and @xmath33 meson momenta respectively , @xmath34 is the invariant mass of the lepton - neutrino pair and @xmath14 is the @xmath33 momentum in the @xmath0 rest frame and is given by : @xmath35 - 4\frac{m_d^2q^2}{m_b^4}\right\}^{1/2}.\ ] ] @xmath36 is the form factor describing the hadronic interaction and must be extracted from theory . most quark model calculations assume a @xmath34 dependence for the form factor and predict the normalization at a given kinematic point . the normalization factor is calculated either at @xmath37 or @xmath38 . in general the arbitrariness of the assumed @xmath34 dependence is a reason for some concern , but in this specific decay the @xmath34 range is not very big and therefore we would not expect a strong model dependence . cleo has recently studied the decay @xmath39 with two different techniques @xcite . in the first method only the lepton and the @xmath33 candidates in the final state are found , using the decay @xmath40 . because the @xmath41 pairs are produced nearly at rest , the missing mass squared @xmath42 is calculated as : @xmath43 here the approximation consists of assuming @xmath44 , relying upon the low magnitude of the @xmath0 momentum because of the vicinity of the to the threshold for @xmath41 meson production . the second approach exploits the hermeticity of the cleo ii detector and infers the @xmath45 momentum from a full reconstruction of the semileptonic decay . stringent cuts need to be applied in this case to insure that no other sources of missing 4-momentum , like additional @xmath46 s or @xmath47 s , are present in the event . in the former analysis , the @xmath42 distribution of candidate events containing a @xmath48 and an opposite sign lepton is studied in 6 different @xmath34 bins gev@xmath49 , evenly spaced between 0 and 12 gev@xmath49 . [ mkpipi ] shows the @xmath50 invariant mass for the interval @xmath51 . one of the crucial elements of this analysis is an accurate background estimate . several sources are subtracted directly , using independent control samples . in particular , a major background is the decay @xmath52 , where the @xmath53 decays into the final state @xmath54 . this component is subtracted by measuring the @xmath42 distribution for identified @xmath52 decays , rescaled by the ratio in detection and reconstruction efficiency for the two channels . the data sample remaining after direct background subtraction has two components : the signal final state @xmath55 and final states of the kind @xmath56 , where @xmath57 is a hadronic final state not coming from the @xmath58 mode discussed above . this last background is subtracted by fitting its contribution with the shape given by a monte carlo simulation . the results of the fit in several @xmath34 bins is shown in fig . [ allqsq ] . note that in different @xmath34 bins the relative weight of signal and background is quite different . therefore the study of the @xmath42 in different @xmath34 regions is quite effective in isolating the signal from this last background contribution . the second technique reconstructs the @xmath46 four vector by summing all the charged track and photon momenta in the event . since the total energy in the event is equal to the center of mass energy and the total momentum is zero , the @xmath46 is assumed to have energy equal to the difference between center of mass energy and total reconstructed energy and momentum equal and opposite to the reconstructed momentum . in order to achieve adequate resolution , stringent event selection criteria are applied to avoid smearing due to additional undetected neutral particles or particles outside the detector acceptance . consistency between the reconstructed energy and momentum is required . once @xmath59 is estimated , the @xmath0 meson candidate can be reconstructed with the usual procedure for exclusive hadronic final states , as shown in fig . [ dplnunu ] . the @xmath42 technique gives a branching fraction of @xmath60% , the @xmath46 reconstruction technique gives @xmath61% with a combined ( preliminary ) branching fraction of @xmath62% . the statistical errors in these two techniques are essentially uncorrelated , while the systematic error is strongly correlated . the @xmath34 distribution for the @xmath42 method is shown in fig . [ qsq_dp ] . the intercept at @xmath63 is proportional to @xmath64 . the curve is fitted to the functional form : @xmath65 where @xmath66 is the normalization at @xmath37 , and @xmath67 is the mass of the pole . the quark model calculations predict @xmath66 , and the pole corresponds the vector meson exchanged in the t - channel , in this case the @xmath68 . the data are fitted including both @xmath66 and @xmath67 as fit parameters , except in the case of the model developed by demchuck _ @xcite , where the mass of the pole is a definite prediction of the theory . the results and a comparison with different models @xcite , @xcite , @xcite are given in table 1 . even if this study considers only a restricted set of models , the limited range of @xmath34 spanned by this decay makes the results quite general . the first errors in the average value of is the quadrature of the statistical and systematic errors in the data and the fact that the fraction of neutral @xmath69 final state produced at the is known only as @xmath70 @xcite . the second error is due only to model dependence . the same data can be used to extract with a different method , inspired by hqet . in this approach the relevant dynamical variable is the velocity transfer @xmath71 , related to @xmath34 by : @xmath72 the point @xmath73 corresponds to the situation where the @xmath0 decays to a @xmath33 at rest in the @xmath0 frame . [ dphqet ] shows the experimental data for @xmath74 . in particular it can be seen that it is difficult to ascertain the curvature of the isgur - wise universal function because of the low statistics at the points close to @xmath73 and thus the uncertainty in the extraction of must reflect this additional uncertainty @xcite . the fit to these data gives @xmath75 . in addition to the measurement discussed in this paper , the parameter has been studied experimentally with several different methods . many groups have reported their findings for the branching fraction @xmath76 @xcite and related their measurement to several different theoretical models to extract their estimate of . in addition this decay can be studied with the hqet method discussed above @xcite . this approach has a special interest from the point of view of hqet theorists because luke s theorem @xcite states that for @xmath73 , the mass dependent corrections vanish to first order and therefore the first non zero term occurs with power @xmath77 . this implies that for a well behaved perturbative expansion , these corrections should play a minor role and thus allow an extraction of prone to smaller theoretical uncertainty . lastly , the inclusive semileptonic decay can be related to @xcite . the average values of obtained with these techniques are shown in fig . [ vcb ] . the average value @xmath78 , corresponding to the value for the @xmath17 parameter @xmath79 , has been obtained by adding statistical and systematic errors in the various estimates linearly and then adding the different methods in quadrature . this procedure is by no means rigorous but it should give a conservative estimate of the final errors . the first evidence of a non - zero was obtained by cleo i @xcite , by studying inclusive semileptonic decays . they reported an excess of leptons beyond the kinematic endpoint for the decay @xmath39 . this result was quickly confirmed by argus @xcite and then studied in more detail and with better statistical accuracy by cleo ii @xcite . there are two crucial issues that make the extraction of from experimental data trickier than the extraction of . first of all , in this case we have the possibility of light hadronic systems recoiling against the lepton@xmath45 pair . therefore the @xmath34 domain spanned by these decays is much bigger and the assumed @xmath34 dependence of the form factors strongly affects the predicted rate @xmath80 and fraction of high momentum leptons @xmath81 . in addition , there is some uncertainty on the composition of the hadronic system recoiling against the lepton-@xmath45 pair . consequently models that focus on a few exclusive hadronic final states are not likely to give reliable predictions for @xmath80 , as it is quite unlikely that the whole dalitz plot is dominated by the low lying resonances . on the contrary , ` inclusive ' models , like the one proposed by altarelli et a. ( accmm ) @xcite , based upon a quark - hadron duality picture , become more relevant when several final states are involved . the importance of the theoretical uncertainty in the extraction of this parameter is illustrated by the fact that the cleo ii estimate of changed by a factor of two depending upon the model used @xcite . the theoretical uncertainty in the extraction of this parameter is closely related to the @xmath34 dependence of the form factors , as shown by artuso @xcite , by comparing the accmm and isgw @xcite predictions for the @xmath34 distributions . recently n. isgur and d. scora have revised the isgw model , aiming at making it more realistic at high and using some constraints on the form factors derived from hqet . this model , referred to as isgw ii @xcite , is now much closer to accmm in several respects . in particular , fig . [ qsq ] shows the the predicted distribution of events with leptons in the momentum range of 2.4 to 2.6 gev / c ( the interval adopted in the cleo ii analysis leading to the most precise value of available so far ) . it can be seen that the distribution predicted by isgw ii is much softer than the one expected by isgw . it is obvious that in order to reduce the errors on the estimate of the parameter it is necessary to enlarge the set of experimental observables . in particular the study of exclusive channels is the necessary first step to check in more detail different theoretical predictions . cleo has recently measured the branching fractions for exclusive charmless semileptonic decays involving a @xmath25 , @xmath82 or @xmath83 meson in the final state @xcite . both charged and neutral @xmath0 decays have been studied . the experimental technique adopted relies again upon the @xmath45 energy and momentum estimates from the rest of the event . the @xmath45 invariant mass squared is calculated from the reconstructed @xmath84 and @xmath85 as @xmath86 and the selection criterion @xmath87 is subsequently applied . this approach allows to reconstruct these decays with the techniques developed to study exclusive @xmath0 meson decays . [ xlnu ] shows the beam constrained invariant mass @xmath88 , evaluated as : @xmath89 in the study of hadronic final states involving @xmath90 in the final system it is often difficult to prove that this system is indeed prevalently @xmath82 . cleo addresses this question by plotting the @xmath91 summed mass spectrum , as shown in fig . [ rholnu ] . they also show the @xmath92 case , which can not be @xmath82 . in the latter case they see a flat background , while the former distribution show some peaking at the expected nominal resonance mass . on the other hand the @xmath93 spectrum show little evidence of resonant @xmath83 . there is clearly the need for more statistics , but this analysis is performed under the assumption that the resonant component is dominant . the five modes @xmath94 , @xmath95 , @xmath96 , @xmath97 , @xmath98 are fitted simultaneously , using the constraints following from isospin : @xmath99 the analysis yields the branching fractions @xmath100 and @xmath101 . the ratio between the partial widths to vector and pseudoscalar final state is interesting because it provides a useful consistency check of the soundness of the assumptions adopted by different phenomenological models . table 2 shows a comparison between the theoretical ratio @xmath102 predicted by a variety of quark model calculations @xcite , @xcite , @xcite , @xcite and the corresponding measured values , using the same model to estimate the efficiency corrections . it can be seen that for some models there is a quite significant discrepancy between the predicted and measured value . in particular the korner and schuler ( ks ) model has only a 0.5% likelihood to be consistent with the data . [ vub_cb ] summarizes our present knowledge of the parameter @xmath103 , both from the exclusive and inclusive analyses . for the inclusive analysis results from cleo i @xcite and argus @xcite have been included in the average . the ks model has been excluded from the average as their prediction of the pseudoscalar to vector ratio is inconsistent with the data . in addition , the isgw model has been replaced by the updated isgw ii model . the spread in the @xmath104 estimates related to model dependence is now narrowed compared to previous analyses . it should be noted that the models adopted are now much more similar in their estimate of the @xmath34 dependence of the form factors used and an experimental confirmation of the correctness of this predicted dependence is eagerly awaited . furthermore , new approaches , based either on qcd sum rule techniques @xcite , @xcite or on lattice qcd @xcite provide new theoretical perspectives on these decay . nonetheless at the present time the model dependence still dominates the errors . a conservative estimate gives : @xmath105 which corresponds to the constraint : @xmath106 fig . [ ckm_fig ] shows the constraints to the unitarity triangle deriving from the values of and reported in this paper together with the constraints coming from mixing and @xmath2 violation in the system ( @xmath107 ) @xcite . the width of the @xmath107 band is mostly affected by the uncertainties in the wolfenstein parameter @xmath108 , the top quark mass @xmath109 , the charm quark mass @xmath110 and the correction factor for the vacuum insertion approximation @xmath111 . the width of the mixing band is dominated by the uncertainty on the @xmath0 meson decay constant @xmath112 here taken to be in the range @xmath113 . the role of factorization in theoretical predictions on exclusive hadronic decays is multifaceted and has indeed been debated at length . in the early theoretical studies of hadronic heavy flavor decays @xcite , factorization was used as an _ ansatz _ , by assuming that in energetic two body decays direct formation of hadrons by a quark current is a useful approximation and that the two currents ` factorize ' , i.e. the hamiltonian can be expressed as a product of two currents , one which couples a meson in the final state with the decaying one and the other which produces the second meson out of the vacuum @xcite , @xcite . for instance , @xmath115 could be expressed as : @xmath116 there is no good reason why factorization should hold in all two body hadronic @xmath0 decays . however some arguments based on color transparency , originally proposed by bjorken @xcite and later by dugan and grinstein @xcite , make it plausible that decays involving one heavy and one light meson in the final state factorize . this is because the time scale of interaction between two mesons in the final state is too short to allow appreciable gluon exchange between them . dugan and grinstein base their arguments on perturbative qcd . the soundness of their approach has been questioned by aglietti @xcite . therefore precise experimental tests of factorization in hadronic @xmath0 decays is quite valuable to disentangle this very complex issue . a factorization test proposed by bjorken is based on the observation that if eq . 13 is valid , then we can related the amplitude @xmath117 to the corresponding matrix element in semileptonic @xmath0 decays . this implies : @xmath118 here @xmath119 is a calculable short distance qcd correction and @xmath120 is the light meson decay constant . we can use the measured differential decay width @xmath121 reported in this paper with the exclusive branching fractions for @xmath122 and @xmath123 @xcite and use the known @xmath124 and @xmath125 to extract an experimental estimate of the parameter @xmath119 . the experimental data can be compared with qcd calculations to assess the applicability of factorization to these hadronic decays . the results of this comparison are shown in table 3 . it can be seen that the agreement is quite satisfactory . @xmath0 decays continue to provide a wealth of information on the standard model . in particular , the detailed study of @xmath0 meson semileptonic decays has given new insights on the quark mixing parameters and on properties of the strong interactions when heavy quarks are involved in the decay . more detailed studies are forthcoming with the increased luminosity planned for the cleo upgrade and , later , with the @xmath0 factories and dedicated @xmath0 experiments at hadron machines . therefore most of the uncertainties related to model dependence will be disentangled and the standard model will have to withstand one of its more substantial challenges . the author would like to acknowledge the contribution of cleo and cesr scientists and technical staff to obtain the data presented in this paper . many thanks are due to c. sachrajda , g. martinelli and n. uraltsev for inspiring discussion during the beauty 96 conference . f. ferroni deserves gratitude for the impeccable organization and the unforgettable visit to the hadrian villa and p. schlein for his indefatigable work towards this conference series . lastly sheldon stone should be thanked for his insightful comments and julia stone should be thanked for nice breaks provided to my evening writing . n. isgur and m. b. wise , heavy quark symmetry , " in _ b decays _ , 2nd edition revised , ed . s. stone , world scientific , singapore ( 1994 ) ; 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weak decays of heavy flavored hadrons are sensitive probes of several facets of the standard model . in particular the experimental study of @xmath0 meson semileptonic decays is starting to pin down the quark mixing parameters in the cabibbo kobayashi maskawa matrix . in addition , some features of the non perturbative regime of the strong interaction are probed by these decays . new results from the cleo experiment at the cesr collider , based on a data sample of up to 3.5 fb@xmath1 provide crucial information on both of these aspects of heavy flavor phenomenology . psfig # 1#2#3mod . phys . lett . a * # 1 * , # 2 ( # 3 ) # 1#2#3nuovo cim . * # 1 * , # 2 ( # 3 ) # 1#2#3nucl . phys . * # 1 * , # 2 ( # 3 ) # 1#2#3#4pisma zh . eksp . . fiz . * # 1 * , # 2 ( # 3 ) [ jetp lett . * # 1 * , # 4 ( # 3 ) ] # 1#2#3phys . lett . * # 1 * , # 2 ( # 3 ) # 1#2#3phys . lett . b * # 1 * , # 2 ( # 3 ) # 1#2#3nucl . instr . & meth . * # 1 * , # 2 ( # 3 ) # 1#2#3phys . rev . * # 1 * , # 2 ( # 3 ) # 1#2#3phys . rev . d * # 1 * , # 2 ( # 3 ) # 1#2#3phys . rev . lett . * # 1 * , # 2 ( # 3 ) # 1#2#3phys . rep . * # 1 * , # 2 ( # 3 ) # 1#2#3prog . theor . phys . * # 1 * , # 2 ( # 3 ) # 1#2#3rev . mod . phys . * # 1 * , # 2 ( # 3 ) # 1 rp # 1 # 1#2#3#4yad . fiz . * # 1 * , # 2 ( # 3 ) [ sov . j. nucl . phys . * # 1 * , # 4 ( # 3 ) ] # 1#2#3#4#5#6zh . eksp . teor . fiz . * # 1 * , # 2 ( # 3 ) [ sov . phys . - jetp * # 4 * , # 5 ( # 6 ) ] # 1#2#3zeit . phys . c * # 1 * , # 2 ( # 3 ) # 1#1| # 1| # 1 4.0 cm * @xmath0 decay studies at cleo * 2.0 cm marina artuso department of physics , syracuse university , syracuse , new york 132441130 _ e - mail : artuso@physics.syr.edu_ 1.0 cm

we thank k. splittorff , m. stephanov and d. toublan for helpful discussions . this work was supported in part by grants de - fg-96er40945 and de - fg02 - 03er41241 from the department of energy ( doe ) . sc would also like to thank the institute of nuclear theory for hospitality where part of this work was done . the computations were performed on champ , a computer cluster funded in part by the doe and computers of dr . r. brown who graciously allowed us to use them when they were available .

we study strongly coupled lattice qcd with @xmath0 colors of staggered fermions in @xmath1 dimensions . while mean field theory describes the low temperature behavior of this theory at large @xmath0 , it fails in the scaling region close to the finite temperature second order chiral phase transition . the universal critical region close to the phase transition belongs to the 3d @xmath2 universality class even when @xmath0 becomes large . this is in contrast to gross - neveu models where the critical region shrinks as @xmath0 ( the number of flavors ) increases and mean field theory is expected to describe the phase transition exactly in the limit of infinite @xmath0 . our work demonstrates that close to second order phase transitions infrared fluctuations can sometimes be important even when @xmath0 is strictly infinite . mean field techniques provide a simple but powerful approach to gain qualitative insight of the underlying physics in a variety of field theories @xcite . the bardeen - cooper - schrieffer ( bcs ) solution to superconductivity is a well known application of such a technique . wilson s renormalization group shows that when correlation lengths @xmath3 , associated with the fluctuations of the field , become large compared to the microscopic length scale @xmath4 of the problem , mean field techniques become exact in dimensions greater than four . however , in lower dimensional systems infrared fluctuations can become important when @xmath5 and invalidate the mean field arguments . for this reason the mean field approach must be used with care close to second order phase transitions . the region close to the critical point where mean field theory fails is usually referred to as the ginsburg region @xcite . using field theoretic techniques sometimes it is possible to estimate the ginsburg region . in conventional superconductors the ginsburg region is known to be suppressed by some power of @xmath6 where @xmath7 is the critical temperature and @xmath8 is the fermi energy . mean field theory is believed to be reliable outside the ginsburg region . this is the reason why bcs is a good approach to understand the physics of superconductivity for all temperatures except very close to @xmath7 . there are certain limits in which the infrared fluctuations can be naturally suppressed even in low dimensional systems . for example consider a theory containing a field with @xmath0 components . when @xmath0 is large a saddle point approximation can be used to solve the theory and the leading term is nothing but the mean field solution . theoretical physicists often use this feature to solve a physical theory by increasing the number of components of the field artificially . by solving the theory in the limit of large number of components and computing the leading corrections they can sometimes estimate realistic answers in the physical theory . a variety of field theories can be studied in the large @xmath0 limit @xcite . although large @xmath0 approach to field theories has been very successful , in this article we show that not all large @xmath0 limits lead to the suppression of infrared fluctuations . we contrast two models , the gross - neveu ( gn ) model involving @xmath0 species of fermions studied recently @xcite and strongly coupled @xmath9 lattice qcd with staggered fermions ( sclqcd ) studied here . both these models contain a global symmetry which is spontaneously broken at low temperatures . further , the symmetry group and the breaking pattern is not affected by @xmath0 . these models undergo a second order phase transition to a symmetric phase at a finite critical temperature @xmath7 . in the large @xmath0 limit they can be solved exactly using mean field techniques at low temperatures . an interesting question then is whether the mean field description is valid even close to @xmath7 when @xmath0 becomes infinite . it was discovered in @xcite that indeed in the gn model the critical behavior near @xmath7 belongs to the landau - ginsburg mean field universality class at large @xmath0 . this implies that the ginsburg region , where the critical behavior is governed by a non - trivial universality class that depends on the symmetry group and the breaking pattern in a dimensionally reduced theory , has a zero width at large @xmath0 . it was later shown in @xcite that indeed the ginsburg region is suppressed by a factor @xmath10 . although a @xmath11 symmetric gn model was analyzed in @xcite , there are reasons to believe that the arguments would hold for continuous symmetries as well @xcite . as we will demonstrate here , in contrast to the gn model , the ginsburg region does not shrink in sclqcd in the large @xmath0 limit . while mean field theory is indeed a good approximation at low temperatures , the finite temperature phase transition is not described by mean field theory even at infinite @xmath0 . we believe our results should be of interest to a wide range of physicists since our model can be mapped into a theory of classical dimers . dimer models have a long history @xcite . in the 1960s these models attracted a lot of attention when it was shown that the ising model can be rewritten as a dimer model @xcite . in the late 1980s they gained popularity again in their quantum version @xcite as a promising approach to the famous resonating - valence - bond ( rvb ) liquid phase @xcite . more recently , this approach has gained momentum again since it was shown that the rvb phase was indeed realized on a triangular lattice but not on a cubic lattice @xcite . thus physicists attempting to use large @xmath0 techniques in dimer models can benefit from our results . the partition function of sclqcd is given by @xmath12 [ d\psi d\bar\psi]\ \exp\left(-s[u,\psi,\bar\psi]\right ) , \label{unpf}\ ] ] where @xmath13 $ ] is the haar measure over @xmath9 matrices and @xmath14 $ ] specify grassmann integration . the euclidean space action @xmath15 $ ] in the strong ( gauge ) coupling limit with staggered fermions is given by @xmath16 - m \sum_x \bar\psi_x\psi_x,\ ] ] where @xmath17 refers to the lattice site on a periodic four - dimensional hyper - cubic lattice of size @xmath18 along the three spatial directions and size @xmath19 along the euclidean time direction . the index @xmath20 refers to the four space - time directions , @xmath21 is the usual links matrix representing the gauge fields , and @xmath22 are the three - component staggered quark fields . the gauge fields satisfy periodic boundary conditions while the quark fields satisfy either periodic or anti - periodic boundary conditions . the factors @xmath23 are the well - known staggered fermion phase factors . using an asymmetry factor between space and time we introduce a temperature in the theory . we choose @xmath24 ( spatial directions ) and @xmath25 ( temporal direction ) , where the real parameter @xmath26 is a coupling that controls the temperature . by working on anisotropic lattices with @xmath27 at fixed @xmath19 and varying @xmath26 continuously one can study finite temperature phase transitions @xcite . in this article we fix @xmath28 for convenience . the partition function given in eq.([unpf ] ) can be rewritten as a partition function for a monomer - dimer system , which is given by @xmath29 } \ ; \prod_{x,\mu}\ ; ( z_{x,\mu})^{b_{x,\mu}}\frac{(n - b_{x,\mu})!}{b_{x,\mu } ! n ! } \ ; \prod_x \frac{n!}{n_x!}\;m^{n_x } , \label{pf}\ ] ] and is discussed in detail in @xcite . here @xmath30 refers to the number of monomers on the site @xmath17 , @xmath31 represents the number of dimers on the bond connecting @xmath17 and @xmath32 , @xmath33 is the monomer weight , @xmath34 are the dimer weights . note that while spatial dimers carry a weight @xmath35 , temporal dimers carry a weight @xmath36 . the sum is over all monomer - dimer configurations @xmath37 $ ] which are constrained such that at each site , @xmath38 = n$ ] . when @xmath39 , the action of sclqcd , eq . ( [ fact ] ) , is invariant under @xmath40 chiral transformations : @xmath41 and @xmath42 where @xmath43 for all even sites and @xmath44 for all odd sites . in the large @xmath0 limit mean field techniques can be used to show that this chiral symmetry is spontaneously broken at low temperatures @xcite . in @xcite a monte carlo method was developed to solve the problem from first principles and it was shown that mean field theory is indeed reliable at small temperatures @xcite . unfortunately , since the algorithm was inefficient at small quark masses , the finite temperature chiral phase transition was never studied in the large @xmath0 limit . recently a very efficient cluster algorithm was discovered to solve the model at any value of @xmath0 @xcite . using this algorithm it was shown with great precision that for @xmath45 the finite temperature phase transition belonged to the 3d @xmath2 universality class @xcite . here we extend that calculation to higher values of @xmath0 . the order parameter that signals chiral symmetry breaking is the chiral condensate , defined by @xmath46 for a fixed @xmath26 the large @xmath0 result for @xmath47 can easily be obtained by extending the calculation of @xcite . one gets @xmath48 which shows that the critical temperature , as we have defined it , is infinite . a calculation which includes the @xmath49 correction shows that @xmath50 . in the large @xmath51 ( spatial dimensions ) limit one obtains @xmath52 @xcite . in order to determine @xmath47 using monte carlo calculations we measure the chiral susceptibility in the chiral limit , @xmath53 the finite size scaling of this quantity is known from chiral perturbation theory @xcite and one expects @xmath54 . \label{chptchi}\ ] ] the constant @xmath55 is equal to @xmath56 ^ 2 + [ \sum_x j_{x,2}]^2 + [ \sum_x j_{x,3}]^2 \big\ } \bigg\rangle,\ ] ] at @xmath39 . the current @xmath57 is the conserved current associated with the @xmath40 chiral symmetry @xcite . by fitting the data for @xmath58 to the form given in eq . ( [ chptchi ] ) we can determine @xmath47 accurately . 0.3 in plot of @xmath47 vs. @xmath26 for various values of @xmath0 . the errors in the monte carlo data are less than the size of the symbols . the solid line is the mean field result given in eq . ( [ largen]).,scaledwidth=45.0% ] we have done extensive simulations for various values of @xmath18 and @xmath0 in order to extract @xmath47 as discussed above . in fig . ( [ fig1 ] ) we plot @xmath47 as a function of @xmath26 at @xmath59 . for comparison we also plot the mean field result ( eq.([largen ] ) ) . as the graph shows , at a fixed value of @xmath26 , our data approaches the mean field prediction quite nicely as @xmath0 becomes large . however , for every value of @xmath0 , as @xmath26 increases the order parameter approaches zero at some critical temperature @xmath7 . close to @xmath7 , the mean field theory is definitely not a good approximation . using our algorithm we can determine @xmath7 accurately for every value of @xmath0 ( see below ) . in the inset of fig . ( [ fig1 ] ) we plot @xmath7 as a function of @xmath0 . a fit to the form @xmath60 yields @xmath61 , @xmath62 and @xmath63 with a @xmath64 ( solid line in the inset ) . 0.3 in plot of @xmath47 vs. @xmath65 for various values of @xmath0 . the solid lines represent the fit to @xmath66 which is expected from 3d xy model . the values of @xmath67 for different @xmath0 are given in the text.,scaledwidth=45.0% ] one might think that it is quite easy to explain why the mean field theory breaks down close to @xmath7 . since @xmath7 grows as @xmath0 , the large @xmath0 theory does not know about the existence of a finite @xmath7 unless the @xmath49 corrections are included . we have computed these corrections and have found that they do not improve the situation much . perhaps one needs to develop a new mean field expansion where one holds @xmath68 fixed as @xmath0 becomes large . let us refer to this as the finite @xmath26 mean field theory ( ftmft ) . although we have not yet developed this mean field theory , we will argue that it is bound to fail close to @xmath7 since the ginsburg region where the 3d @xmath2 universality class is observed does not shrink with @xmath0 . indeed we find that @xmath69 close to @xmath7 where @xmath70 independent of @xmath0 @xcite . a fit of the data to this form was used to determine @xmath7 plotted in [ fig1 ] . in fig . [ fig2 ] we plot @xmath47 as a function of @xmath65 for @xmath71 for @xmath59 . the solid lines are fits to the form @xmath66 . we find @xmath72 for @xmath59 respectively all with a @xmath73 less than @xmath74 . 0.3 in plot of @xmath55 vs. @xmath65 for @xmath75 . the inset shows the same graph for a larger range of @xmath65 for @xmath76 . the text describes the physics of this plot.,scaledwidth=45.0% ] since @xmath67 decreases with increasing @xmath0 one might argue that @xmath67 will vanish in the limit @xmath77 . in that case some higher order term will become dominant in the large @xmath0 limit and it is not possible to rule out mean field behavior . however , we do not think this is the case and attribute the change in @xmath67 to a renormalization effect as a function of @xmath0 . in order to justify this , we next focus on a correlation length scale which does not need renormalization . if correlation lengths in the theory indeed scale with @xmath0 , we would expect the correlation lengths to be the same at any fixed @xmath65 . in our model @xmath55 can be defined as one such inverse correlation length scale . this implies @xmath78 , where @xmath79 in the 3d @xmath2 model @xcite . in fig . [ fig3 ] we plot @xmath55 as a function of @xmath65 for @xmath80 . when @xmath81 all the points at a fixed @xmath65 but at different @xmath0 fall on top of each other . further , when @xmath71 , all the data points fit extremely well to the form @xmath82 , with @xmath83 with a @xmath84 . we believe that this is strong evidence that even at infinite @xmath0 the phase transition belongs to the 3d @xmath2 universality class . outside the ginsburg region one would expect ftmft to be valid @xcite . interestingly , when @xmath85 we find that the data fits to the form @xmath86 with @xmath87 with a @xmath88 , suggesting @xmath89 . hence , we suspect that the ftmft in our model is similar to the mean field theory in 3d @xmath90 models which yields @xmath91 . the cross over to mean field theory occurs when @xmath92 so that the correlation length is @xmath93 in lattice units . one often hears the lore that in the large @xmath0 limit sclqcd is solvable using mean field theory . while this is true in certain cases , in this article we have demonstrated that the finite temperature chiral phase transition belongs to the 3d @xmath2 universality class even in the large @xmath0 limit . in an earlier study we found similar results in two spatial dimensions @xcite . in two dimensions continuous symmetries can not break at any finite temperature @xcite . however an @xmath40 symmetry is special and a phase transition in the berezinski - kosterlitz - thouless ( bkt ) universality class is possible @xcite . the bkt prediction for our model is that @xmath94 where @xmath95 is a function of temperature @xmath96 when @xmath97 . at the critical temperature @xmath98 . we found all this to be true even at large @xmath0 . interestingly , according to witten a large @xmath0 mean field theory would find @xmath99 @xcite . while our results agree with this observation at a fixed temperature @xmath26 , we find that if @xmath65 is held fixed then @xmath95 approaches a non - zero value in the predicted range showing that the mean field approach again breaks down in the large @xmath0 theory close to the phase transition . our study shows that the large @xmath0 limit may not always be able to suppress infrared fluctuations close to second order phase transitions . in a sense this is an indication that the perturbation expansion starting from a mean field solution breaks down . in other words a careful analysis of the ftmft should be able to reveal this . this has not yet been done and is a useful project for the future . it can help us classify the types of large @xmath0 models where mean field theory can break down .

in the last years there have been many attempts to develop non - perturbative methods for quantum field theoretical ( qft ) problems such as the infrared behaviour of qcd . many of the non - perturbative methods used so far , such as the coupled cluster expansion @xcite , originate from standard many - body theory and use states in the fock representation , thus they have the general problem of fock representations being disjunct to the hilbert space of qft @xcite . in this work we also adopt a successful method from standard many - body theory , which however is formulated in terms of green functions and therefore of a genuine field - theoretical nature . we propose an approach along the line of n - body correlation dynamics @xcite , which describes the propagation of the system in terms of equal - time green functions ( i.e. we use the density matrix formalism ) . the basic strategy of correlation dynamics is the truncation of the infinite hierarchy of equations of motion for n - point green functions via the use of cluster expansions , i.e. the expansion of green functions in terms of connected green functions ( correlation functions ) . in this work we go up to the connected 4-point level , i.e. we include the 2-,3- and 4-field correlation functions . for relativistic systems with a nonlocal interaction our method does not guarantee a covariant description , since retardation effects can not be included due to the equal - time formalism . there is , however , no objection against the use of an equal - time formalism in the case of a local relativistic field theory as long as one consistently considers green functions containing canonically conjugate field momenta as well as the fields themselves . this paper is organized as follows : in section [ correldyn ] we describe the derivation of the correlation dynamical equations of motion for @xmath6-theory in @xmath2 dimensions ; the final set of equations itself is shifted to appendix [ eqoms ] in view of its length . section [ application ] is devoted to the application of the method to the evaluation of the effective potential at zero temperature within various limits and thus to the investigation of ground state symmetry breaking . we consider the lagrangian density @xmath8 in one space and one time dimension , which corresponds to the hamiltonian @xmath9 \ ; , \label{hamiltonian}\end{aligned}\ ] ] where @xmath10 and @xmath11 is the bare mass of the scalar field @xmath12 . in order to evaluate the effective potential of the theory at zero temperature , i.e. the minimum of the energy density for a given magnetization @xmath13 , we decompose @xmath12 into a classical and a quantum part according to @xmath14 where @xmath15 is a real constant and assume @xmath16 since @xmath15 is a constant , we have @xmath17 the hamiltonian , expressed in terms of the classical part @xmath15 and the quantum part @xmath18 of @xmath12 , then reads @xmath19 \ ; . \label{splitoffhamiltonian}\end{aligned}\ ] ] for the time evolution of @xmath18 and @xmath20 we obtain with ( [ splitoffhamiltonian ] ) and the canonical equal - time commutation relations @xmath21 = \left [ \pi(x , t),\pi(y , t ) \right ] = 0 \ ; , \ ; \ ; \left [ \phi(x , t ) , \pi(y , t ) \right ] = i \delta ( x - y ) \label{commurel}\end{aligned}\ ] ] by means of the heisenberg equation : @xmath22 @xmath23 in analogy to ( [ heisenbergequations ] ) we get the equations of motion for equal - time operator products of @xmath18 and @xmath20 , e.g. @xmath24 @xmath25 \phi(x_2,t ) + \pi(x_1,t ) \pi(x_2,t ) \ ; , \ ; \ ; ... \ ; \ ; . \label{hierarchy}\end{aligned}\ ] ] after taking their expectation values , the equations of motion for all possible n - point equal - time products of @xmath18 and @xmath20 comprise an infinite hierarchy of equations of motion for equal - time green functions . the analogon of this in nonrelativistic many - body theory is the bbgky density matrix hierarchy @xcite . since we are considering a local field theory , the equations of motion for the green functions contain no retardation integrals , which implies that working in an equal - time limit is sufficient for describing the propagation of the system as long as one considers all green functions containing the fields as well as their conjugate momenta . for practical purposes , the infinite hierarchy of coupled differential equations of first order in time has to be truncated . this is done using the cluster expansions for n - point green functions @xcite , i.e. their decomposition into sums of products of connected green functions , which works as long as the system is in a pure phase @xcite . the explicit form of the cluster expansions can be derived from the generating functionals of full and connected green functions , @xmath26 $ ] and @xmath27 $ ] , given by @xmath28 = { \rm tr } \left\ { \rho \ ; t \left [ e^{i \int d^2 \hat{x } \left ( j(\hat{x } ) \phi(\hat{x } ) + \sigma(\hat{x } ) \pi(\hat{x } ) \right ) } \right ] \right\ } \ ; \ ; \ ; \ ; { \rm and } \ ; \ ; \ ; \ ; z[j , \sigma ] = e^ { w[j , \sigma ] } \ ; , \end{aligned}\ ] ] respectively , where @xmath29 is the time ordering operator , @xmath30 is the statistical density operator describing the pure or mixed state of the system ( @xmath31 ) and @xmath32 . we start with the cluster expansions for the time - ordered green functions with different time arguments : @xmath33 @xmath34 } } \nonumber \\ & & = \lim_{j , \sigma \to 0 } \frac{\delta}{i\delta j(\hat{x}_1 ) } \left\ { \left ( \frac{\delta}{i\delta j(\hat{x}_2 ) } w[j , \sigma ] \right ) e^{w[j , \sigma ] } \right\ } \nonumber \\ & & = \lim_{j , \sigma \to 0 } \left\ { \left ( \frac{\delta}{i\delta j(\hat{x}_1 ) } \frac{\delta}{i\delta j(\hat{x}_2 ) } w[j , \sigma ] \right ) + \left ( \frac{\delta}{i\delta j(\hat{x}_1 ) } w[j , \sigma ] \right ) \left ( \frac{\delta}{i\delta j(\hat{x}_2 ) } w[j , \sigma ] \right ) \right\ } e^{w[j , \sigma ] } \nonumber \\ \nonumber \\ & & = { \langle}t \phi(\hat{x}_1 ) \phi(\hat{x}_2){\rangle}_c + { \langle}\phi(\hat{x}_1){\rangle}{\langle}\phi(\hat{x}_2 ) { \rangle } \ ; , \end{aligned}\ ] ] where @xmath35 denotes the connected part of the expectation value . analogously we obtain @xmath36 @xmath37 @xmath38 @xmath39 the expressions for equal - time green functions are obtained by taking the well - defined equal - time limit which yields the appropriate operator ordering in the cluster expansions . we arrive at @xmath40 @xmath41 @xmath42 @xmath43 @xmath44 @xmath39 where all ( equal ) time arguments have been suppressed . in view of their length the cluster expansions for the other green functions required for our calculations are not explicitly given here , but e.g. can be found in @xcite . due to equations ( [ splitoffassumption ] ) and ( [ piassumption ] ) the 1-point functions in all cluster expansions are now assumed to vanish , which in the end leads to the same result as considering the cluster expansions for green functions containing the original fields @xmath12 and @xmath45 and setting @xmath46 and @xmath47 . the cluster expansions then are truncated by neglecting all connected n - point green functions with @xmath48 ; in our case with @xmath49 . inserting the truncated cluster expansions into the equations of motion of type ( [ hierarchy ] ) up to the equations for the 4-point functions leads to a closed system of coupled nonlinear equations for the connected green functions ( i.e. a system of correlation dynamical equations ) up to the 4-point level in analogy to the n - body correlation dynamics in nonrelativistic many - body theory @xcite . for this straightforward but somewhat tedious derivation , which in view of its length is not explicitly given here , all truncated cluster expansions up to the 6-point level are required , since the highest order green functions appearing in the hierarchy equations ( [ hierarchy ] ) up to the 4-point level are the 6-point functions . the resulting equations of motion for the connected green functions still have to be renormalized . @xmath6-theory in @xmath2 dimensions is superrenormalizable and only requires a mass renormalization . there is only one mass counterterm due to the divergent tadpole diagram comprising the contribution to the selfenergy of lowest order in the coupling constant . this mass counterterm can be evaluated analytically either in the framework of perturbation theory or , equivalently , by normal ordering the hamiltonian with respect to the perturbative vacuum . we obtain @xcite : @xmath50 with @xmath51 the logarithmically divergent mass counterterm ( [ counterterm ] ) can be analytically removed from the equations of motion by normal ordering all operator products with respect to the perturbative vacuum and thereby splitting off the short - distance singularities of the free equal - time green functions . the normal ordering within the connected green functions only takes place within the 2-point functions , since in the other connected n - point functions all field operators commute with each other . for the 2-point functions we have : @xmath52 @xmath53 @xmath54 @xmath55 after inserting ( [ counterterm ] ) and ( [ normalorder ] ) into the equations of motion for the connected green functions , all terms containing a factor @xmath56 mutually cancel out , leading to a closed set of renormalized equations for the normal ordered connected green functions . the final step is to transform the renormalized correlation dynamical equations of motion from coordinate space to an arbitrary single particle basis in order to simplify their numerical integration . in this respect we expand the field @xmath18 and its conjugate momentum @xmath20 according to @xmath57 for the corresponding equal - time green functions we then have @xmath58 @xmath59 @xmath60 the corresponding expressions for the other green functions are obtained analogously . by inserting the expansions ( [ greenfunctionexpansions ] ) into the renormalized equations of motion and projecting out the matrix elements with respect to the single particle basis we obtain the final result for the correlation dynamical equations of motion for the @xmath7-theory , which are denoted by @xmath61 ( @xmath6 @xmath62orrelation @xmath63ynamics ) furtheron , and can directly be used for a numerical integration . in view of their length the @xmath61 equations are shifted to appendix [ eqoms ] . in this section we apply the @xmath61 equations to the determination of the effective potential of the @xmath7-theory at zero temperature and thereby investigate the spontaneous breakdown of the symmetry under the discrete transformation @xmath64 for values of the coupling exceeding a critical value , which manifests itself in a nonzero ground state magnetization @xmath13 . since we also want to study the influence of the different connected n - point functions , we introduce 4 limiting cases of the correlation dynamical equations ; these are denoted as @xmath65 , @xmath66 , @xmath67 and @xmath68 , where the numbers in parantheses are the orders of connected n - point functions that are taken into account ( i.e. , for instance in the @xmath67 case the connected 3-point functions are set equal to zero and @xmath68 denotes the original @xmath61 approximation discussed in section [ correldyn ] ) . in order to integrate equations ( [ firsteqom ] ) - ( [ lasteqom ] ) numerically , we choose plane waves in a one - dimensional box with periodic boundary conditions as a single particle basis , i.e. we work in discretized momentum space . in order to select an appropriate box size for a given renormalized mass , we compare the numerically obtained gep solution for the ground state configuration within the discretized system to the analytically accessible gep solution in the continuum limit ( for the gep approximation , see appendix [ gep ] or @xcite ) . due to the large amount of computer time needed for @xmath61 calculations we have to compromise between a good momentum space resolution and the convergence with a minimum number of plane waves . it has proven to be most effective to choose a box size of 100 fm for a renormalized mass of 10 mev . in fig . [ picphstran0 ] we show the ground state magnetization @xmath46 as a function of the dimensionless coupling @xmath69 for these parameters , where different numbers of single particle states have been taken into account . the vertical line shows the position of the critical coupling for symmetry breaking in the continuum limit . for practical purposes we will always use the 15 lowest lying plane waves in the following calculations , unless explicitly stated otherwise . within the gep approximation we then obtain a critical coupling of @xmath5 for the discretized system as compared to a critical coupling of @xmath70 in the continuum limit . since the @xmath61 equations only describe the propagation of equal - time green functions for a given initial configuration , they can not directly be applied to the evaluation of static equilibrium properties of the system . as in the case of correlation dynamics for nonrelativistic many - body theory there is no easy access to the stationary solutions of the equations , and moreover it is not clear , in how far additional constraints have to be imposed on the subspace of stationary configurations in order to select only the physical solutions . we therefore use a different approach for the evaluation of equal - time ground state green functions within the @xmath61 approximation . starting with the trivial exact ground state configuration for @xmath71 and a given fixed value of @xmath15 as an initial condition , we continuously switch on the coupling while propagating the system in time . the time - dependence of the coupling is chosen to be linear with @xmath72 and @xmath73 . in fig . [ picconv ] the energy density obtained within this time - dependent method is shown as a function of the coupling for different values of @xmath74 for @xmath75 and all 4 limiting cases of the correlation dynamical equations . in all cases an asymptotic curve is approached with decreasing @xmath74 , i.e. when the coupling is switched on more slowly . this indicates that in the limit @xmath76 the whole process becomes fully adiabatic , i.e. the system will time - dependently follow the trajectory of the ground state as a function of the coupling . for the @xmath65 approximation , taking into account only the connected 2-point functions , which is the field theoretical analogon to time - dependent hartree - fock theory , we have direct access to the static ground state solutions since its stationary limit is simply given by the gep approximation . the corresponding curves are shown in the upper left part of fig . [ picconv ] ; the gep is displayed as the lower one of the two solid lines . indeed , the asymptotic curve of the time - dependent method is identical to the gep solution . in addition , we have checked numerically that for all 4 limiting cases of correlation dynamics the system increasingly equilibrates along its trajectory with decreasing @xmath74 , i.e. if we stop increasing @xmath77 at some point in time , the system will remain in its present state when propagated further . in choosing a finite value for @xmath74 we have to compromise between a good convergence of our time - dependent method and a minimum of computer time we want to invest ; the curves in the two following pictures have therefore been evaluated with @xmath78 . the ground state energy density for various values of @xmath15 is plotted versus the coupling @xmath69 in fig . [ picinccpl ] for all 4 limiting cases of correlation dynamics , where for the reasons mentioned above the @xmath65 approximation has been replaced by the gep approximation . the gep and the @xmath67 approximation each predict a first order phase transition , since the first curve to intersect the @xmath79 ( i.e. the symmetric phase ) energy density in the gep case has @xmath80 and in the @xmath67 case has @xmath81 ; i.e. they both lead to a finite value of @xmath15 , implying that the vacuum magnetization has to jump to that value discontinuously at the critical value of the coupling constant ( compare fig . [ picphstran0 ] for the gep ) . in contrast to that , the @xmath66 approximation and the @xmath68 approximation each predict a second order phase transition , since in both cases all curves with @xmath82 intersect the curve with @xmath79 in the correct order , thus enabling the vacuum magnetization to increase continuously once the critical coupling is exceeded . at this point it is useful to recall , that there is a rigorous mathematical proof of the statement that there can be no first order phase transition in the @xmath7-theory ( in the continuum limit ) @xcite . this proof is based on the simon - griffith theorem @xcite , which in turn is obtained by considering the @xmath6 field theory as a proper limit of a generalized ising model . thus we conclude , that in order to describe the order of the phase transition correctly , the inclusion of the connected 3-point function is required . a simple geometrical explanation of this fact will be given in the discussion below . in fig [ picveff ] the effective potential is plotted as a function of the magnetization @xmath15 for various values of the coupling . the same data have been used as for the previous figure . in this representation , the first order nature of the phase transition in the gep and the @xmath67 case can be seen from the fact , that for these approximations there is a maximum in the effective potential between the minimum at @xmath79 and the second minimum on the right hand side ( for the gep this minimum is not very pronounced ) ; thus at the critical coupling the second minimum has to be located at a finite value of @xmath15 . in the case of the @xmath66 and the @xmath68 approximation there is no such intermediate maximum , and as the critical coupling is approached from above , the second minimum is shifted to the left towards @xmath79 . for @xmath83 the system will approach the classical limit , i.e. with decreasing @xmath15 higher order correlations increasingly become important ; the inclusion of correlations always lowers the energy density , since a larger configuration space is opened up for the system . however , for obvious mathematical reasons the inclusion of the 2-point function alone can not lower the energy density at @xmath79 ; unlike the other n - point functions of even order the 2-point function therefore is most important in the region on the right hand side in the plots of fig . [ picveff ] . the 4-point function , as the highest order correlation included in this work , is most important around @xmath79 , i.e. for small classical field strengths . the intermediate maximum now appears in between the domains governed by the 2-point function on the right and the 4-point function on the left , an inclusion of higher correlations of even order would most probably lower the energy density in a region even more concentrated around @xmath79 and therefore not cure this problem . the inclusion of connected n - point functions of uneven order cures the problem , since for symmetry reasons these have to vanish as @xmath84 . the 3-point function therefore obviously assumes its main importance exactly in the region in between the domains of the 2- and the 4-point function , it bends down the intermediate maximum . finally , in table [ table ] we give the values for the critical coupling extracted from our calculations for different numerical parameters and the 4 different limiting cases of @xmath61 . the critical couplings @xmath85 obtained by other authors , that are also using non - perturbative techniques adopted from standard many - body theory , are as follows : @xcite obtain 1.829 or 1.375 , respectively , using the discretized light - front quantization method , @xcite obtain 1.72 using a varitional approximation with trial states which are quartic rather than gaussians ( quadratic exponentials ) causing an inclusion of the 3-field correlation amplitude in the language of the coupled cluster method @xcite , @xcite obtain the estimate @xmath86 using an improved version of the coupled cluster expansion method , and using second order perturbation theory in the residual interaction leads to 1.14 @xcite . all of these authors obtain a second order phase transition in agreement with the simon - griffith - theorem . .[table ] values for the critical coupling for different approximations and numerical parameters . [ cols="^,^,^,^,^",options="header " , ] this work comprises the first application of correlation dynamics for equal - time green functions to a field theoretical problem , i.e. to the determination of the effective potential in @xmath7-theory . after giving a derivation of the corresponding equations of motion , we showed that we are able to evaluate equal - time quantities in the interacting ground state for a given vacuum magnetization by time - dependently increasing the coupling in an adiabatic process . our numerical results predict a second order phase transition in agreement with the simon - griffith theorem , as soon as the connected 3-point function is included . however , since the connected 2-point function and the connected 3-point function alone are not able to lower the energy of the symmetric phase , the critical coupling obtained within the @xmath66 approximation is too low . going one step further in our expansion , we find that the connected 4-point function in the @xmath68 method increases the coupling to a value , which is only slightly lower than the value obtained by the gep approximation , where however the shape of the effective potential changes completely as compared to the gep . in general , the results of this work demonstrate the applicability of correlation dynamics to the description of low - energy ( ground state ) phenomena in local field theories . since in principle the equations are designed to describe the propagation of the system in time with arbitrary initial conditions , our method is also a potentially powerful tool for the investigation of non - equilibrium properties of relativistic field theories , e.g. the response to external perturbations . the present study can be viewed as a first step towards the application of correlation dynamics to su(n ) gauge theories , aiming at a non - perturbative description of the infrared behaviour of qcd ; we already presented the corresponding equations of motion in @xcite . in this appendix we present the renormalized correlation dynamical equations for the normal ordered connected green functions up to the 4-point level , formulated with respect to an arbitrary single particle basis set ( see section [ correldyn ] ) . in order to compactify the equations we introduce the following abbreviations : @xmath87 @xmath88 @xmath89 @xmath90 @xmath91 @xmath92 the permutation operator interchanging the indices @xmath93 and @xmath74 is denoted by @xmath94 . although normal ordering only affects the connected 2-point functions , we write out the normal ordering operation @xmath95 in all connected green functions in order to have a uniform notation . the equations then read : @xmath96 + @xmath97 + @xmath98 + @xmath99 + @xmath100 + @xmath101 + @xmath102 + @xmath103 + @xmath104 \rbrace \nonumber\\ \nonumber\\ & & -6\lambda\phi_0 \sum_{\lambda_1\lambda_2 } \langle\alpha|\lambda_1\lambda_2\rangle ( 1+{\cal p}_{\beta\gamma}+{\cal p}_{\beta\delta } ) ( \langle:\phi_{\lambda_1}\phi_\beta:\rangle_c + \delta_{\lambda_1\beta } ) \langle:\phi_{\lambda_2}\phi_\gamma\phi_\delta:\rangle_c \ ; , \end{aligned}\ ] ] + @xmath105 + @xmath106 + @xmath107 \rbrace \nonumber\\ \nonumber\\ & & -6\lambda\phi_0 ( 1+{\cal p}_{\alpha\beta}+{\cal p}_{\alpha\gamma}+ { \cal p}_{\alpha\delta } ) \sum_{\lambda_1\lambda_2 } \langle\alpha|\lambda_1\lambda_2\rangle ( 1+{\cal p}_{\beta\gamma}+{\cal p}_{\beta\delta } ) \langle:\phi_{\lambda_1}\pi_\beta:\rangle_c \langle:\phi_{\lambda_2}\pi_\gamma\pi_\delta:\rangle_c \nonumber\\ & & + \frac{3}{2 } \lambda ( 1+{\cal p}_{\alpha\beta}+{\cal p}_{\alpha\gamma}+ { \cal p}_{\alpha\delta } ) \sum_{\lambda } \langle\beta\gamma\delta|\lambda\rangle \langle:\phi_\lambda \pi_\alpha:\rangle_c \ ; . \label{lasteqom}\end{aligned}\ ] ] the hamiltonian for the @xmath7-system is given by @xmath108\end{aligned}\ ] ] with @xmath109 , @xmath110 ( see section [ correldyn ] ) . let furthermore @xmath111 , @xmath112 e^{ipx } \ ; , \ ] ] @xmath113 where @xmath114 is the perturbative vacuum . as in section [ correldyn ] , @xmath15 denotes the constant vacuum magnetization , i.e. the ground state expectation value of @xmath12 . the gep approximation consists in the ansatz @xmath115\end{aligned}\ ] ] for the variational hamiltonian , i.e. the interacting system is approximated by a free system with an effective mass @xmath116 , which serves as a variational parameter . one then has to minimize the expectation value of @xmath117 with respect to the ground state of @xmath118 . this method is equivalent to a bcs calculation , where the variational wavefunction is given by a boson pair condensate @xmath119 here @xmath120 denotes a normalization constant and @xmath121 is the variational parameter . @xmath122 is a quasiparticle vacuum , i.e. @xmath123 , where the @xmath124 can be obtained from the @xmath125 by means of a bogoliubov - transformation . we have @xmath126 e^{ipx } \label{bmodenentwicklung}\end{aligned}\ ] ] with @xmath127 . after minimizing the energy functional , the gep ansatz yields the equation @xmath128 with @xmath129 for the effective mass @xmath116 . equation ( [ gapgleichung ] ) is identical to the hfds equation and the gap equation following from the bcs ansatz @xcite . for known effective mass @xmath116 , the equal - time green functions can be evaluated via ( [ bmodenentwicklung ] ) as expectation values with respect to the quasiparticle vacuum @xmath122 . the value for the energy density functional as a function of @xmath15 ( obtained with the method described above ) is the gaussian approximation for the effective potential of the theory . blubs999a j. arponen , ann . of phys . * 151 * ( 1983 ) 311 . m. funke , u. kaulfuss , h. kmmel , phys . rev . * d35 * ( 1987 ) 621 . hsue , h. kmmel and p. ueberholz , phys . rev . * d32 * ( 1985 ) 1435 . f. strocchi , _ elements of quantum mechanics of infinite systems _ , world scientific , singapore ( 1985 ) . wang , w. cassing , ann . * 159 * ( 1985 ) 328 . r. balescu , _ equilibrium and nonequilibrium statistical mechanics _ , j. wiley ( 1975 ) . wang , w. zuo and w. cassing , nucl . phys . * a573 * ( 1994 ) 245 . brown , _ quantum field theory _ , university of washington , physics 520 . j. glimm , a. jaffe , _ quantum physics - a functional integral point of view _ , springer - verlag new york ( 1981 ) . wang , w. cassing , j.m . huser , a. peter and m.h . thoma , submitted to ann . thoma , z. phys . * c53 * ( 1992 ) 637 . chang , phys . rev . * d12 * ( 1975 ) 1071 . p.m. stevenson , phys . * d32 * ( 1985 ) 1389 . chang , phys . * d13 * ( 1976 ) 2778 . b. simon , r.g . griffith , commun . . phys . * 33 * ( 1973 ) 145 . a. harindranath , j.p . vary , phys . rev . * d36 * ( 1987 ) 1141 . a. harindranath , j.p . vary , phys . rev . * d 37 * ( 1988 ) 1076 . l. polley , u. ritschel , phys . lett . * b221 * ( 1989 ) 44 . m.h . thoma , z. phys . * c44 * ( 1989 ) 343 . [ picconv ] ground state energy density during the time - dependent process as a function of the coupling for a given vacuum magnetization of @xmath75 and different values of @xmath74 ; upper left : @xmath65 , upper right : @xmath66 , lower left : @xmath67 , lower right : @xmath68 ; in addition the static gep curve is displayed in the upper left plot . [ picinccpl ] ground state energy density as a function of the coupling for different vacuum magnetizations ; all curves except the one in gep approximation have been evaluated time - dependently with @xmath130 ; upper left : gep , upper right : @xmath66 , lower left : @xmath67 , lower right : @xmath68 . [ picveff ] effective potential as a function of the vacuum magnetization for different values of the coupling ; all curves except the one in gep approximation have been evaluated time - dependently with @xmath130 ; upper left : gep , upper right : @xmath66 , lower left : @xmath67 , lower right : @xmath68 .

using the cluster expansions for n - point green functions we derive a closed set of dynamical equations of motion for connected equal - time green functions by neglecting all connected functions higher than @xmath0 order for the @xmath1-theory in @xmath2 dimensions . we apply the equations to the investigation of spontaneous ground state symmetry breaking , i.e. to the evaluation of the effective potential at temperature @xmath3 . within our momentum space discretization we obtain a second order phase transition ( in agreement with the simon - griffith theorem ) and a critical coupling of @xmath4 as compared to a first order phase transition and @xmath5 from the gaussian effective potential approach .

a canonical form that fits to the gauge - fixing conditions ( gfc ) and is given by @xmath234,\\ r_n & = \mathbb{i } \quad \forall n \ge n_c,\\ r_n & = \mathrm{diag}(\lambda_{n,1}^2 \dots \lambda_{n , d}^2 ) \quad \forall n \in [ 0 , n_c - 1],\end{aligned}\ ] ] where @xmath235 for @xmath236 are the schmidt coefficients for the decomposition of the chain into two infinite halves by cutting between sites @xmath168 and @xmath237 . it corresponds to the gfc in the sense that changing the parameters as @xmath238 with @xmath94 satisfying the gfc does not alter @xmath239 or @xmath240 , which are constants in the above canonical form , to first order in @xmath241 . in practice , this means that the canonical form is approximately maintained when making finite steps in the tdvp algorithm . the above canonical form can be reached via a gauge - transformation @xmath242 where @xmath243 and @xmath244 are non - trivial ( see ) , such that the uniform bulk parameters @xmath30 are also transformed . since the overall state and also the left and right uniform bulk states are unaffected by these transformations , performing them does not affect evolution under the tdvp equations . for real - time evolution , numerical integration using the euler method is inefficient since small step sizes @xmath245 are required to keep the @xmath246 integration errors made with each finite step small . a well known integration method with more favorable error scaling is the 4th order runge - kutta method ( rk4 ) @xcite , which makes per - step errors @xmath247 at the cost of three extra evaluations of the derivative . it builds a final step by making three smaller steps and weighting the derivatives obtained at the visited points . given a differential equation @xmath248 , the rk4 method estimates @xmath249 with @xmath250 and @xmath251 the smps tdvp flow equations derived in the main part of this work provide the derivative function for the @xmath168th site @xmath252)$ ] , allowing us to implement the rk4 integrator without any additional tools . it is worth noting that @xmath253 , obtained by adding the tangent vector parameters from the various sub - steps , is not gauge - fixing . this is because each individual @xmath254 , although it is gauge - fixing for the sub - step point @xmath255 at which it was obtained , is not generally gauge - fixing when applied at the original point @xmath35 . additionally , each sub - step changes the gauge - choice slightly , since gauge - fixing only holds to first order in the step size . on the other hand , since the gauge - fixing flow equations do preserve the gauge choice when integrated exactly , gauge - fixing should improve with the accuracy of the numerical integration . we should thus expect the rk4 method to maintain the gauge choice up to errors of @xmath247 with each step . this is far better than the euler method , which incurs @xmath246 errors . the error can be quantified by the change in the energy expectation value , which is conserved under exact time evolution . we confirm the benefits of our rk4 implementation by comparing it to the euler method for the heisenberg model example described in the main text , which we simulate on a finite chain with open boundary conditions in order to avoid errors due to the interface with the bulk . to compare the efficiency of the two methods , we set the step sizes such that the computation time per unit simulated time is roughly the same and examine the overall change in the energy expectation value after a period of simulated time @xmath256 . since a single rk4 step requires roughly four times as much computation as an euler step , we choose @xmath257 . for @xmath258 , the energy errors after a time @xmath259 are @xmath260 and @xmath261 , showing a significant advantage for the rk4 method for the same computation time . the vast majority of the rk4 error comes from the first four steps , whereas the euler errors are uniformly distributed in time . excluding these steps from the rk4 error estimate results in @xmath262 . both @xmath263 and @xmath264 are in line with the theoretical global error estimates of @xmath265 and @xmath266 respectively . the comparatively large errors made by the rk4 method during the first few steps are caused by the presence of particularly small schmidt coefficients , indicating that the bond - dimension is higher than necessary . small schmidt coefficients lead to instability because the squares of the schmidt coefficients appear in the @xmath267 and @xmath268 matrices , which are inverted in the tdvp algorithm , amplifying errors on small values greatly . to mitigate this , the bond - dimension can be reduced dynamically ( and increased later if necessary ) , cutting off schmidt coefficients that are close to zero . alternatively , an integrator that is robust under low - rank conditions could be used @xcite . 25ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/revmodphys.82.277 [ * * , ( ) ] link:\doibase 10.1088/1742 - 5468/2007/08/p08024 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.97.157202 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1080/14789940801912366 [ * * , ( ) ] link:\doibase 10.1088/1751 - 8113/42/50/504004 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.69.2863 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.77.259 [ * * , ( ) ] link:\doibase 10.1103/physrevb.55.2164 [ * * , ( ) ] link:\doibase 10.1016/j.aop.2010.09.012 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.93.040502 [ * * , ( ) ] http://arxiv.org/abs/cond-mat/0407066 [ ( ) ] link:\doibase 10.1103/physrevlett.107.070601 [ * * , ( ) ] link:\doibase 10.1103/physrevb.87.075413 [ * * , ( ) ] link:\doibase 10.1063/1.3149556 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.102.240603 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.020407 [ * * , ( ) ] http://arxiv.org/abs/1210.7710 [ ( ) ] link:\doibase 10.1143/jpsj.62.1848 [ * * , ( ) ] link:\doibase 10.1143/jpsj.63.420 [ * * , ( ) ] link:\doibase 10.1103/physrevb.51.16115 [ * * , ( ) ] link:\doibase 10.1103/physrevb.53.40 [ * * , ( ) ] link:\doibase 10.1103/physrevb.53.r492 [ * * , ( ) ] http://arxiv.org/abs/1305.1894 [ ( ) ] link:\doibase 10.1088/1126 - 6708/2004/05/007 [ * * , ( ) ] http://arxiv.org/abs/1302.5582 [ ( ) ] https://github.com/amilsted/evomps [ ( ) ] , link:\doibase 10.1103/physrevb.86.245107 [ * * , ( ) ] http://arxiv.org/abs/1207.0678 [ ( ) ] http://arxiv.org/abs/1207.0862 [ ( ) ] in @noop _ _ ( , , ) ed . http://arxiv.org/abs/1301.1058 [ ( ) ]

we describe how to implement the time - dependent variational principle for matrix product states in the thermodynamic limit for nonuniform lattice systems . this is achieved by confining the nonuniformity to a ( dynamically expandable ) finite region with fixed boundary conditions . the suppression of nonphysical quasiparticle reflections from the boundary of the nonuniform region is also discussed . using this algorithm we study the dynamics of localized excitations in infinite systems , which we illustrate in the case of the spin-1 anti - ferromagnetic heisenberg model and the @xmath0 model . douglas adams ( nearly ) put it best : `` [ hilbert ] space is big . ... you just wo nt believe how vastly hugely mindbogglingly big it is . i mean , you may think it s a long way down the road to the chemist , but that s just peanuts compared to [ hilbert ] space . '' given said space s exponential growth with the size of a many - particle system , it is a little astounding that general techniques exist to allow efficient numerical calculations in a wide range of physically interesting cases . this is possible because physically relevant states have limited entanglement @xcite . this observation may be exploited to obtain an efficient parametrization of this _ physical corner _ of hilbert space . the class of _ matrix product states _ ( mps ) @xcite represents , in one dimension , a good parametrization of the physical corner . this is amply demonstrated by the unparalleled success of the _ density matrix renormalization group _ ( dmrg ) @xcite , which can be viewed as a variational method when formulated in the mps language @xcite . the mps class has served as the basis for many exciting generalizations , including the study of non - equilibrium dynamics @xcite and higher - dimensional systems @xcite . more recently , haegeman _ et al . _ have implemented the _ time - dependent variational principle _ ( tdvp framework for simulating dynamics , including finding ground states via imaginary time evolution , and an ansatz for studying excitations of one - dimensional lattice systems . [ tdvpbox ] the simulation of infinite quantum spin systems has mostly been confined to the translation invariant setting ( usually by restricting states to subsets of mps that are either fully translation invariant or invariant under translations by @xmath1 sites @xcite ) . however , the ability to explore locally nonuniform states on an infinite lattice is particularly attractive for studying the dynamics , e.g. scattering , of localized excitations in large systems . for example , this would provide a realistic setting in which to study quantum field excitations . there has been some prior work in this direction , building on previous light - cone results @xcite , where the dynamics of a local disturbance is ( partially ) studied in the heisenberg picture . these approaches can become expensive for systems with large local spin dimensions ( such as those appearing in lattice field theory ) . another direction that has been suggested @xcite , is to work completely in the schrdinger picture with infinite uniform mps and to add a finite nonuniform region . in this work we explore the locally optimal implementation of the tdvp for uniform mps with a dynamically expandable nonuniform segment . we derive the equations of motion for the variational parameters using a particular choice of gauge - fixing which allows us to integrate the variational dynamics with a complexity that scales as @xmath2 , where @xmath3 is the length of the nonuniform piece ( the number of sites ) , @xmath4 is the desired integration time , @xmath5 is the local spin dimension , and @xmath6 is the bond dimension . even though the ends of the nonuniform region can move , there may be some backscattering due to boundary effects ; we describe how to compensate for these with the addition of an _ optical potential _ term . these methods are illustrated in the case of local excitations of the spin-1 anti - ferromagnetic heisenberg model and for particles in @xmath0 theory . we assume throughout that our hamiltonian @xmath7 contains only nearest - neighbor terms . it is decomposed as @xmath8 , where @xmath9 with @xmath10 , @xmath11 , and @xmath12 with @xmath13 $ ] representing a contiguous region of the lattice and @xmath14 for @xmath15 , allowing us to also write @xmath16 $ ] . we consider two cases in particular : firstly , a non - trivial @xmath17 leads to a locally nonuniform ground state , which can be found using imaginary time evolution via our algorithm . secondly , given a purely uniform hamiltonian ( @xmath18 ) and an initial state that differs only locally ( in a region @xmath13 $ ] ) from an eigenstate of @xmath19 , our algorithm can be used to simulate the resulting locally non - trivial dynamics . to capture a locally nonuniform state using mps , we define a class of `` sandwich '' states ( smps ) , based on uniform mps , using two @xmath20 tensors @xmath21 and @xmath22 describing the ( asymptotic ) state either side of the nonuniform region @xmath13 $ ] , which is parametrized by @xmath3 further tensors . an smps state can be written as @xmath23 } = \!\!\sum_{\{s\}=1}^{d } \!\ ! v_l^\dagger \ ! \left[\prod_{i=-\infty}^0 \!\!\ ! a_l^{s_i}\right ] \!\ ! a_1^{s_1 } \dots a_n^{s_n } \!\ ! \left[\prod_{j = n+1}^\infty \!\!\ ! a_r^{s_j}\right ] \!\ ! v_r \ket{{\ensuremath{\bm{s}}}}\end{aligned}\ ] ] where @xmath24 and @xmath25 ( where @xmath26 $ ] ) . taking @xmath27 gives a completely uniform state . the vectors @xmath28 are , as with uniform mps @xcite , generically irrelevant to the tdvp algorithm and are not further specified . in principle , the dimensions of @xmath29 are subject only to the constraints of the matrix product , which can become important when maximizing numerical efficiency . however , for reasons of notational simplicity , we assume uniform dimensions here . @xmath30 represent the left and right asymptotic states : the reduced density matrix @xmath31}(a_l , a_r , a_{1 \dots n})$ ] of a piece of the lattice in the left or right region @xmath32 or @xmath33 tends to that of the uniform mps state @xmath31}(a_{l / r})$ ] as the distance from the nonuniform region increases . since @xmath30 represent infinite `` bulk '' regions of the lattice , their dynamics should not be affected by nonuniformities in the @xmath13 $ ] region , which spread at a finite speed . furthermore , if the left and right asymptotic states are eigenstates of @xmath19 , they are left completely unchanged by time evolution . assuming this , we restrict the variational parameters to the tensors @xmath34 and treat @xmath30 as boundary conditions . @xmath30 can be obtained for the ground state of @xmath19 using the existing tdvp algorithm for uniform mps @xcite . to accurately capture states with a nonuniform region @xmath13 $ ] in this way , @xmath3 should be sufficiently large so that the asymptotic states are already reached at the left and right boundaries with the bulk . the tensor network formed by the matrices @xmath35 can be visualized as with the nonuniform region marked in the center and the physical indices pointing upwards . expectation values of local operators can be calculated efficiently in terms of operators @xmath36 , with the `` transfer operators '' @xmath37 . for example , the expectation value of an operator @xmath38 that acts non - trivially on a pair of neighboring sites can be written as @xmath39 e_n^h \left [ \prod_{k = n+2}^{\infty } e_{n } \right ] |v_r},\end{aligned}\ ] ] with @xmath40 and @xmath41 as well as @xmath42 and @xmath43 and where @xmath44 . expressions for expectation values and for the norm of the state contain parts @xmath45 " and @xmath46 " that need not be well - defined , depending on the properties of @xmath47 and @xmath48 . to make these quantities finite , we must require that @xmath49 have spectral radius equal to 1 . to ensure that @xmath50 and @xmath51 remain irrelevant in calculations of bulk properties , we further demand that there is a single , non - degenerate ( so that @xmath52 are not block diagonalizable ) eigenvalue of largest magnitude that is equal to 1 , with all other eigenvalues having magnitude strictly less that 1 @xcite . the left and right eigenvectors corresponding to this eigenvalue , which are thus the unique left and right fixed points of @xmath49 , we name @xmath53 and @xmath54 , normalizing them such that @xmath55 . we can then write @xmath56 and @xmath57 , where @xmath58 is some vector that is not orthogonal to @xmath59 or @xmath60 . we now have a slightly simpler form for : @xmath61 e_n^h \left [ \prod_{k = n+2}^{n } e_{n } \right ] |r_r}$ ] . to further improve the notation , we define @xmath62 and @xmath63 , identifying @xmath64 and @xmath65 ( we will also use @xmath66 and @xmath67 ) . we then have @xmath68 : note that we are free to scale @xmath59 , @xmath60 and the tensors @xmath69 of the nonuniform region such that @xmath70 . for reasons of efficiency , when constructing numerical algorithms we work in the isomorphic setting where transfer operators are replaced by maps and vectors by matrices using the choi - jamiolkowski isomorphism . here , a @xmath71 transfer operator acting on a vector @xmath72 becomes @xmath73 with @xmath74 a @xmath75 matrix , so that expectation values can be computed using @xmath76 scalar multiplication operations : . we now determine the dimension of the sub - manifold @xmath77 of hilbert space defined by the smps variational class . naively , this is the number of complex entries of the parameter tensors @xmath78 , which is @xmath79 . however , an smps state is invariant under gauge transformations @xmath80 with @xmath81 . since @xmath30 are fixed , we restrict to @xmath82 leaving @xmath83 non - physical degrees of freedom corresponding to the gauge - transformation matrices @xmath84 , as well as a further one corresponding to the norm and phase . the dimension of the smps variational manifold is thus @xmath85 . the redundancy in the smps representation is familiar from other mps variational classes @xcite and is less inconvenient than it may appear , since the gauge - freedom in the representation of tangent vectors allows for significant simplification of the tdvp flow equations . to implement the tdvp ( see boxout ) , we must project exact infinitesimal time steps @xmath86}$ ] onto the tangent plane @xmath87}}$ ] to @xmath88 at the point @xmath89}$ ] . the tangent plane is spanned by tangent vectors @xmath90 } = \sum_{n=1}^{n } \sum_{i=1}^{dd^2 } b_{n , i } \ket{\partial_{n , i}\psi[a ] } \\ & \quad\includegraphics{picture3_tn_tangvec.eps } \nonumber\end{aligned}\ ] ] with @xmath91 } = \partial/\partial a_{n , i } \ket{\psi[a]}$ ] and the index @xmath92 running over all @xmath93 entries of each tensor @xmath69 or @xmath94 . the projection is achieved by finding a @xmath95}$ ] that satisfies @xmath96 } = \arg \min_{\ket{\phi[b']}}\| \mathrm{i}h\ket{\psi[a(t ) ] } + \ket{\phi[b']}\|^2.\end{aligned}\ ] ] expanding the rhs leaves terms @xmath97|\phi[b]}$ ] and @xmath97 | h | \psi[a ] } + \text{h.c.}$ ] , where the remaining @xmath98 term is a constant that can be ignored . the metric term @xmath97|\phi[b]}$ ] is at first glance very complicated , since it couples the tensors @xmath94 for different lattice sites in terms such as @xmath99 precluding a splitting of the problem into @xmath3 separate parts ( one for each @xmath94 ) . fortunately , these site - mixing terms can be eliminated by fixing the gauge - freedom in the tangent vector representation . if we impose the left gauge - fixing conditions ( gfc ) @xmath100 for sites @xmath101 and the right gauge - fixing conditions @xmath102 for sites @xmath103 , we eliminate all site - mixing terms like such that @xmath97|\phi[b ] } = \sum_{n=1}^n { \bra{l_{n-1 } } } e^{b_n}_{b_n } { \ket{r_n}}$ ] . note that , for some site @xmath104 in the nonuniform region , the tangent vector parameters @xmath105 are not constrained . for reasons of symmetry , we choose @xmath104 to be in the middle so that @xmath106 with odd @xmath3 . to see that the conditions and fix exactly the gauge degrees of freedom , we consider the one - parameter gauge transformation @xmath107 $ ] with @xmath108 . writing the transformed state as @xmath109}$ ] , the infinitesimal transformation has the form of a tangent vector @xmath110 } \right|_{\eta = 0 } = \ket{\phi[\mathcal{n}[x ] ] } = 0,\end{aligned}\ ] ] with @xmath111 = x_{n-1 } a_n^s - a_n^s x_n$ ] . tangent vector parameters of this form thus capture exactly the gauge freedom so that an arbitrary tangent vector fulfills @xmath95 } = \ket{\phi[b + \mathcal{n}[x]]}$ ] . using this freedom , we can always transform arbitrary @xmath112 as @xmath113 $ ] so that @xmath94 satisfies the gauge - fixing conditions and . to see this , we insert @xmath114 $ ] into to obtain @xmath115 which we can solve to fully determine @xmath116 given that @xmath117 has full rank and that @xmath118 is known . starting at @xmath119 with @xmath120 , this fixes all @xmath118 down to @xmath121 . we can perform the same trick with to get @xmath122 which determines the remaining @xmath118 ( up to @xmath123 ) given that @xmath124 and that @xmath125 has full rank . we can construct @xmath94 such that they automatically fulfill the gfc and . for @xmath103 we define the @xmath126 matrix @xmath127 to contain an orthonormal basis for the null space of @xmath128_{(\alpha , s ) ; \beta } = [ r_n^{1/2 } { a_n^s}^\dagger]_{\alpha , \beta}$ ] and set @xmath129,\end{aligned}\ ] ] with parameters @xmath118 . for @xmath101 , we define the @xmath130 matrix @xmath131 to contain an orthonormal basis for the null space of @xmath132_{\alpha ; ( s , \beta ) } = [ { a_n^s}^\dagger l_{n-1}^{1/2}]_{\alpha,\beta}$ ] and set @xmath133.\end{aligned}\ ] ] it is easy to check by insertion that and respectively satisfy the gfc and . note again that @xmath105 remains unconstrained . using the parametrizations , we obtain @xmath134|\phi[b ] } = \sum_{n\neq n_c } { \operatorname{tr}}[x_n^\dagger x_n ] + \braket{l_{n_c - 1}|e^{b_{n_c}}_{b_{n_c}}|r_{n_c}}.\end{aligned}\ ] ] having fixed the gauge , one non - physical degree of freedom remains , since @xmath135|\phi[b ] } = \braket{l_{n_c - 1}|e^{b_{n_c}}_{a_{n_c}}| r_{n_c } } \neq 0 $ ] , implying that the tangent plane contains infinitesimal changes to the norm and phase . we must thus explicitly eliminate norm and phase changes when implementing the tdvp , which can be done by replacing @xmath7 with @xmath136 in the tdvp flow equations , effectively projecting out the corresponding component of @xmath137 @xcite . with gauge - fixing , @xmath97 | \tilde{h } | \psi[a]}$ ] simplifies , but still contains terms mixing @xmath94 and @xmath138 for . each @xmath94 term contains a sum over @xmath139 extending into the left ( @xmath140 ) or right ( @xmath141 ) bulk or into both ( @xmath142 ) . this is understood by defining the right and left effective hamiltonians @xmath143 which also obey @xmath144 where @xmath145 and @xmath146 . for example , the terms containing @xmath94 with @xmath141 are : @xmath147 the sums over the uniform bulk @xmath148 and @xmath149 can be computed by exploiting the assumption that @xmath49 have a unique largest ( in magnitude ) eigenvalue equal to 1 , which allows us to rewrite the sum as a pseudo - inverse . for the right - hand bulk this gives @xmath150 or , equivalently , , which can then be solved for @xmath151 in the matrix representation using @xmath76 operations per iteration . @xmath152 can be computed analogously . note that the energy difference due to the nonuniformity is @xmath153 , where @xmath154 is the energy per - site of the uniform bulk state . we now have the ingredients needed to compute the hamiltonian term efficiently as @xmath155|\tilde{h}|\psi[a ] } & \label{eqn : ham_term } \\ = \sum_{n \neq n_c } & { \operatorname{tr}}\left [ x_n^\dagger f_n \right ] + \sum_{s=1}^d { \operatorname{tr}}\left [ l_{n_c-1 } g_{n_c}^s r_{n_c } { b_{n_c}^s}^\dagger \right ] , \nonumber\end{aligned}\ ] ] with @xmath156 @xmath157 @xmath158 , \nonumber\end{aligned}\ ] ] where @xmath159 $ ] and @xmath160 is the conjugate matrix representation of @xmath161 so that , for some vector @xmath162 , @xmath163 $ ] . having fixed the gauge , inserting and into the tdvp minimization problem and minimizing over the parameters @xmath164 and @xmath105 gives us @xmath165 independent matrix equations , @xmath166 ) \quad \text{and } \quad x_n = -{\ensuremath{\mathrm{i}}}f_n \;\ ; ( n \neq n_c),\end{aligned}\ ] ] representing the optimal time evolution for the variational parameters @xmath167 where we use the appropriate parametrization or for @xmath94 depending on the value of @xmath168 . with gauge - fixing , the independent terms to be minimized in , one for each @xmath94 , can be summarized diagrammatically as @xmath169 where the equations for @xmath118 are obtained again by replacing @xmath94 with or for @xmath170 as appropriate . the flow equations can be integrated numerically , for example with the following simple algorithm implementing the euler method : 1 . calculate @xmath171 . 2 . take a step by setting @xmath172 . 3 . restore a canonical form using a gauge transformation and normalize the state by rescaling @xmath173 . 4 . compute desired quantities , such as the energy expectation value , and adjust the step size @xmath174 as required . 5 . if needed , expand the nonuniform region to the left and/or right . normalization is necessary because the norm is only preserved to first order in @xmath174 . maintaining a canonical form ( for example , see appendix [ sec : app_can_form ] ) can simplify some parts of the tdvp calculations and improve the conditioning of the matrices involved . the last step allows for a small initial nonuniform region , which can be grown if the dynamics warrant changing the state significantly outside of it . this is done by `` absorbing '' sites from the uniform region(s ) into the nonuniform region , copying the @xmath21 and @xmath22 matrices as needed . whether it is necessary to grow the nonuniform region can be heuristically determined by observing the per - site contributions @xmath175 to the norm @xmath176 of the tdvp tangent vector @xmath95}$ ] . if @xmath177 and @xmath178 become significantly larger than the norm @xmath179 of the uniform mps tdvp tangent vector of the bulk state then the nonuniform region should be expanded until this is no longer the case . note also that the above algorithm is not well suited to simulating real - time dynamics because errors due to the simple integration method used are cumulative . instead , more sophisticated integrators such as the commonly used fourth - order explicit runge - kutta method ( see appendix [ sec : rk4 ] ) are preferable . the euler method is , however , still useful for finding ground states because imaginary time evolution is `` self - correcting '' | it will always take you towards the ground state , given that the starting point is not orthogonal to it . to test our algorithm , we use the antiferromagnetic spin-1 heisenberg model @xmath180 , with @xmath181 the uniform ground state respects the su(2 ) symmetry of the hamiltonian . having found a uniform mps approximation for the ground state , we use imaginary time evolution to find the ground state of a nonuniform model where one of the coupling terms has its sign flipped via the addition of @xmath182 , with all other @xmath183 , thus creating a ferromagnetic impurity . impurities have been studied in this model before @xcite however , to the best of our knowledge the case of a ferromagnetic bond has not yet been investigated . it appears to lead to localized su(2 ) symmetry - breaking , as can be seen in the relative distribution of the spin expectation values at each site , which we plot in fig . [ fig : heis_imp ] . this is expected , since the ground states of the uniform ferromagnetic model also break the symmetry . in this case , @xmath184 acts in the hamiltonian to approximately project the pair of sites @xmath185 and @xmath186 onto the spin 2 subspace , whose states are not invariant under su(2 ) . with @xmath187 and a nonuniform region @xmath188 $ ] . the initial uniform ground state used for the left and right bulk parts was converged to @xmath189 . ( color online . ) ] $ ] with @xmath187 and @xmath190 ) of the spin-1 anti - ferromagnetic heisenberg model with two localized entangled excitations generated by applying @xmath191 at @xmath192 with @xmath193 . the plots show the expectation value of @xmath194 with the top - right plot showing the excitation bouncing at the right boundary . for the bottom - right plot , we used a gaussian optical potential to suppress this reflection , albeit imperfectly , with @xmath195 . the uniform ground state was converged up to a state tolerance @xmath196 . for the time - integration we used a 4th order explicit runge - kutta algorithm . ( color online . ) ] of one half of the lattice , split at each site @xmath168 , for the same simulation as in fig . [ heis_real ] , except that dynamic expansion of the nonuniform region is used , as indicated by the `` staircase '' pattern | the bulk parts are shown in white . the inset shows a cross - section at site @xmath185 . ( color online . ) ] as a test of real - time evolution , we again use @xmath197 from , but without any local perturbations ( @xmath198 , @xmath199 ) . we begin with a uniform ground state approximation and introduce local excitations by applying the ( nonunitary ) operator @xmath191 with @xmath200 , which generates an entangled excitation , to two separated pairs of sites at @xmath201 inside a nonuniform region . by calculating the expectation value of an observable such as @xmath202 for a set of sites ( possibly extending into the left and right bulk regions ) after each step , the time evolution of the system can be visualized , for example by plotting the site spin expectation values as in fig . [ heis_real ] or the half - chain entropy for splittings at each site as in fig . [ heis_entr ] . for the latter , we use dynamic expansion of the nonuniform region to maximize numerical efficiency . note that the entropy for a splitting after site @xmath185 appears to tend to an asymptotic value of approximately @xmath203 . this suggests that a hybrid method whereby uniform matrices are reintroduced between the two excitations as they become separated could be used to study the asymptotics of entangled excitations for large times . to mitigate non - physical reflections that can occur when a traveling excitation meets a boundary with the uniform region , `` optical potential '' terms @xmath204 can be locally turned on near to the boundaries . this effectively carries out imaginary time evolution on a subsystem defined by the envelope function @xmath205 , where the magnitude of @xmath205 determines the rate of `` cooling '' at each site . if @xmath205 is a step function that turns on imaginary time evolution at a constant rate in a small part of the lattice , that part should ( in the absence of simultaneous real time evolution ) converge to the ground state of a finite chain with open boundary conditions . since we are working with gapped systems , the ground state of a smaller part should be the same as that of the uniform infinite system up to boundary effects . we find that choosing @xmath205 to be superposition of two gaussians , each localized near an edge of the nonuniform region , avoids significant boundary effects during evolution of the heisenberg model whilst successfully attenuating boundary reflections , as shown in fig . [ heis_real ] . note that the entanglement present in the excitations produced for this particular model mean that the boundary - absorption affects the evolution in the central region as well as at the boundaries themselves . further tuning of @xmath206 may help to more effectively dissipate the excitations heading out of the nonuniform region . as a final test of our approach we simulate the scattering of localized excitations in @xmath0 theory on a one - dimensional lattice . the hamiltonian is @xmath207 where @xmath208 = \mathrm{i}\delta_{nm}$ ] . the bare mass @xmath209 and coupling @xmath210 are dimensionless lattice parameters related to parameters with dimension @xmath211 ^ 2 $ ] by @xmath212 , @xmath213 , where @xmath214 is the lattice spacing . we fix @xmath214 for each set of parameters using the ground state correlation length in lattice sites @xmath215 , which is directly obtainable@xcite from the largest two eigenvalues of the uniform mps transfer operator @xmath216 . due to renormalization , @xmath209 is not equal to the particle mass and in fact diverges in the continuum limit . so that our parameters are well - defined in the limit , we separate out the divergent contribution @xmath217 to obtain the renormalized mass - squared parameter @xmath218 . for certain values of @xmath219 the ground state spontaneously breaks the global @xmath220 symmetry @xmath221 of such that @xmath222 . in fig . [ phi4 ] , we examine excitations of @xmath0 theory within a nonuniform region by applying the field operator to the ground state and simulating time - evolution . we do this for a sequence of parameters , approaching a continuum limit . more details about the application of mps to real scalar @xmath0 theory and its critical behavior are available elsewhere @xcite . , @xmath223 , @xmath224 ) of particle scattering in @xmath225-dimensional lattice @xmath0 real scalar field theory approaching a continuum limit ( from left to right ) , as determined by the ground - state correlation length in lattice sites @xmath215 . we created two excitations by applying the field operator @xmath226 at two different sites @xmath227 $ ] to an approximate ground state in the symmetry - broken phase . the coupling is @xmath228 for all three plots and the parameter ratio @xmath229 varies from left to right as @xmath230 . distance @xmath74 and time @xmath231 are scaled with the correlation length @xmath232 where @xmath214 is the lattice spacing . the field expectation value @xmath233 is left unscaled , although a more comprehensive treatment would scale it with the field strength renormalization factor , which can also be computed from the uniform mps approximate ground state . ( color online . ) ] in this paper , we have introduced an efficient means of simulating the dynamics of localized nonuniformities on spin chains in the thermodynamic limit using the time - dependent variational principle ( tdvp ) and a special class of matrix product states ( mps ) . as with the existing algorithms implementing the tdvp for mps in other settings @xcite , this algorithm approximates exact time evolution optimally given the restrictions of the variational class . our ( open source ) implementation evomps @xcite is available as python ( http://www.python.org ) source code , including example simulation scripts . during completion of this work , we learned of other independent results @xcite that use time - evolving block decimation to approximate the time evolution of a nonuniform window on an otherwise translation - invariant chain . our approach differs in that we define a variational class and apply the tdvp to obtain equations for locally optimal approximate time evolution . we are then able to apply standard numerical integration techniques . the idea of not only growing the nonuniform region , but also of ignoring the evolution of uninteresting parts of the nonuniformity for reasons of efficiency | say , to follow a wavefront @xcite can also be implemented in our scheme by restricting the variational parameters to a smaller part of the nonuniform region and leaving the rest constant ( up to gauge transformations ) . as mentioned above , another approach to studying entangled excitations may be to detect when the central region between two separating wavefronts becomes translation invariant over a sufficiently large region , taking this state as a new bulk state for one side of the system and restricting the nonuniform region to a single wavefront . _ acknowledgements _ | helpful discussions with florian richter , fabian transchel and fabian furrer are gratefully acknowledged . this work was supported by the erc grants qftcmps , querg and quevadis , the fwf sfb grants foqus and vicom and the cluster of excellence exc 201 quantum engineering and space - time research .

it has been well known for years @xcite that bose - einstein ( be ) correlations measured in hadron - hadron reactions at sufficiently high energies exhibit a pronounced dependence on multiplicity in the form of the strength parameter @xmath0 . in @xmath1 reactions , this behaviour is either absent @xcite or at least much weaker , especially with a 2-jet selection @xcite . heavy ion reactions , on the other hand , see @xmath0 values decreasing with multiplicity at lower multiplicity densities ( lower @xmath2 reactions ) but increasing again measurements of different experiments are not easily compared because of varying systematics such as the chosen fit function and fit range , the `` cleanness '' of the pion sample , different treatments of coulomb corrections etc . the described tendencies in @xmath0 are therefore more qualitative than quantitative in nature . ] at higher multiplicity densities @xcite . these findings may suggest a possible common explanation in terms of superposition of several sources or strings , where each of them symmetrizes separately according to the model of andersson and hofmann@xcite . no be correlations are then expected between decay products of different strings @xcite . the following arguments could be used : in the framework of the dual parton model for hadron hadron and heavy ion reactions @xcite , it is expected that higher multiplicities correspond to a higher number of strings or chains . this can explain the multiplicity dependence in hadron - hadron reactions . in @xmath1 reactions , with a selection of two - jet events , only one single string is formed and consequently one does not expect much dependence of be correlations on multiplicity . in heavy ion reactions , there is again an increasing number of strings with increasing multiplicity density , but eventually the densely - packed strings coalesce until they finally form a large single fireball . this picture can qualitatively explain why the be signal is decreasing at low @xmath2 and finally increasing again @xcite . a clean situation of a superposition of two strings occurs if two @xmath3-bosons are coproduced in @xmath1 reactions and both decay hadronically . if there are no be correlations between the pions from different strings , the @xmath0 values are expected to be diminished . unfortunately , these measurements are hampered by low statistics and experimental difficulties @xcite . it should be stressed , however , that this is not the only possible explanation of experimental behaviour . in the case of hadron - hadron reactions , for example , several alternative explanations do exist @xcite , among them the core - halo picture @xcite which connects consistently the multiplicity dependence of correlation functions of like - charge with those of opposite - charge pion pairs . from all these considerations , it should be clear that there is important information in the observed multiplicity dependencies , in particular when comparing different reaction types . this contribution concentrates on a discussion of possible origins which could lead in the case of high energy hadron - hadron reactions to the observed multiplicity dependence . we first adress in section 2 the question of the influence of jet production by using again @xmath4 collisions at @xmath5 gev measured in the ua1 detector @xcite . low-@xmath6 and high-@xmath6 subsamples are investigated . using the low-@xmath6 subsample where the influence of jets is removed to a large extent , we finally discuss in section 3 possible underlying physics . it is well known @xcite that in hadron - hadron reactions the probability of jet production rises with energy and multiplicity . the multiplicity dependence of correlation functions can be influenced by this hard subprocess . to investigate this influence , we used a data sample similar to that in @xcite but with larger statistics : it consists of 2,460,000 non - single - diffractive @xmath4 reactions at @xmath5 gev measured by the ua1 central detector @xcite . only vertex - associated charged tracks with transverse momentum @xmath7 gev / c and @xmath8 have been used . we restrict the azimuthal angle to @xmath9 ( `` good azimuth '' ) . these cuts define a region in momentum space which we call @xmath10 . since the multiplicity for the entire azimuthal range is the physically relevant quantity , we select events according to their uncorrected all - azimuth charged - particle multiplicity @xmath11 . the corrected multiplicity density is then estimated as twice @xmath12 , the charged - particle multiplicity in good azimuth : @xmath13 . the measured quantities are @xmath14 , the second order normalized cumulant correlation functions in several multiplicity intervals @xmath15 $ ] ; for a complete definition , refer to @xcite . they are measured for pairs of like - sign ( @xmath16 ) and opposite - sign ( @xmath17 ) charge separately as @xmath18 where @xmath19 , @xmath20 and @xmath21 are the mean numbers of positive or negative particles , @xmath17 pairs and @xmath16 pairs respectively in the whole interval @xmath10 and in the multiplicity range @xmath22 $ ] . the prefactors in front of the normalized density correlation functions @xmath23 correct for the bias introduced by fixing multiplicity @xcite . the functions @xmath24 and @xmath25 are defined for @xmath16 and @xmath17 pairs in the correlation integral description @xcite @xmath26 with @xmath27 the spacelike four - momentum difference of the pion pairs . internal cumulants ( [ se ] ) and ( [ se1 ] ) are analysed in three samples as a function of pion transverse momentum @xmath6 as follows : * an all-@xmath6 sample of all like - sign and opposite - sign pion pairs in @xmath10 . this is shown in fig . 1 for three representative multiplicity densities . * a low-@xmath6 subsample containing only charged particles with @xmath28 gev / c . also removed from the subsample were entire events containing either a jet with @xmath29 gev or at least one charged particle with @xmath30 gev / c . this is shown in fig . 2 with the same three multiplicity selections . the prefactors entering eqs . ( [ se ] ) , ( [ se1 ] ) are calculated in this case by using only charged particles @xmath31 gev / c . * a high-@xmath6 subsample where only charged particles with @xmath32 gev / c are considered . this is shown in fig . it has been demonstrated previously @xcite that all high-@xmath6 particles stem from jets or minijets . from figs . 13 , we see that whereas the low-@xmath6 subsample behaves very similarly to the all-@xmath6 sample , there is a pronounced increase in the strength of correlation functions for the high-@xmath6 case ( iii ) ( note the different scale on the plot ! ) . this can be interpreted as the influence of jets , which are inherently spiky in nature . in this case , be correlations are hence mixed up with correlations originating from jets , so that it would be difficult to measure them separately . we note further that @xmath16 functions in fig . 3 reveal a crossover in the region @xmath33 gev , making the determination of the multiplicity dependence of @xmath14 highly @xmath34-dependent . the fit to this high-@xmath6 sample using eq . ( [ ex1 ] ) gives `` radius parameters '' @xmath35 which decrease with increasing multiplicity , in contrast to the samples ( i ) @xcite and ( ii ) . 3 shows also a pronounced secondary peak after a minimum at @xmath36 gev which can be attributed to the onset of local @xmath6 compensation with two back - to - back particles with at least @xmath37 gev / c corresponding to the cut applied in this subsample @xcite . the fits performed to @xmath16 pair data in fig.1 fig.3 are exponential , @xmath38 the corresponding dependence of @xmath0 on multiplicity is used in subsequent figures . 4 compares the multiplicity dependence of the all-@xmath6 and low-@xmath6 samples in the small-@xmath34 region ( @xmath39 gev ) . the two samples differ only slightly in their respective cumulants as well as their @xmath0 values . 5 shows the corresponding multiplicity dependence in the high-@xmath6 sample , once again on a larger scale in @xmath14 . the influence of jets shows up dramatically : all correlation functions are increased in height , and the @xmath17 functions do not show a multiplicity dependence for particle densities @xmath40 . a decrease like in ( i ) and ( ii ) is probably compensated by increasing jet activity . the interplay of be correlations and resonance production with the onset of jet production and the transition from soft to hard interactions are interesting questions in their own right and can be studied with samples like the high-@xmath6 one . we will , however , concentrate in the following on the low-@xmath6 subsample . the results of a study with the all-@xmath6 sample ( i ) have been published in ref . @xcite . in fig . 6 , we plot for the low-@xmath6 sample ( ii ) the same ratio @xmath41 for like - sign and opposite - sign pairs respectively , while fig . 7 shows the behaviour of the low-@xmath6 cumulants for fixed @xmath34 but varying multiplicity . these figures show that the all-@xmath6 results found previously remain valid for the low-@xmath6 sample also , namely : * the like - sign and opposite - sign cumulants have very similar multiplicity dependence when compared in the same @xmath34 region . * cumulants behave distinctly differently at small and large @xmath34 : at small @xmath34 , the multiplicity dependence of both samples is weaker than @xmath42 , while at large @xmath43 gev ) the cumulants are negative and follow roughly a @xmath42 law instead of @xmath44 here and below ( @xmath45 ) . ] . * a third region around @xmath46 gev shows small and rapidly changing cumulants . in the following , we discuss four possible explanations of these phenomena . it should be stressed that the lund monte carlo model ( pythia ) can not reproduce the multiplicity dependence of correlation functions and in particular of the be effect @xcite . * * bose einstein correlations are the result of symmetrization within individual strings only * @xcite : when several strings are produced , each string symmetrizes separately and decay products of different strings would hence not contribute to be correlations @xcite . + because unnormalized cumulants of independent distributions combine additively , the independent superposition in momentum space of @xmath47 equal sources / strings , each with a @xmath48th order cumulant @xmath49 and each with some multiplicity distribution ( e.g. poisson ) results in an unnormalized combined cumulant @xmath50 of @xmath51 while the combined single particle spectrum is given in terms of individual sources spectra by @xmath52 so that the @xmath48th order normalized cumulants for @xmath47 superimposed sources is given by @xmath53 hence @xmath54 is inversely proportional to the number of sources . this remains true for the correlation integral @xmath55 also . + the above derivation is only for illustration . if @xmath56 , this would imply @xmath57 only if @xmath58 , i.e. for identical sources each of fixed multiplicity . in reality , the assumption of equal sources is probably not fulfilled . the following scenario might be more realistic : `` fixing multiplicity does not necessarily mean fixing the number of sources . the sources ( we will them define below ) probably do possess a whole multiplicity distribution rather than a single fixed multiplicity . our selected multiplicities range from 0.83 to 9.1 , varying over about a factor 10 . at small @xmath34 , however , the @xmath59 vary only by at most a factor 3 . this suggests that at the highest selected multiplicity we would observe the superposition of only 3 sources , from which we are sampling their high multiplicity tails . '' in the dual parton model approach @xcite , one source might be identified with the topology of one pomeron exchange . if we select low - multiplicity events , we expect to select the case of one pomeron exchange . multiparton collisions corresponding to multipomeron exchange are expected to contribute to higher - multiplicity events . estimates in ref . @xcite predict two- to three - pomeron exchanges at the highest multiplicities seen by ua1 . the number of sources would increase correspondingly . this could explain the suppression of @xmath16 ( bose - einstein ) functions in fig . however , additional assumptions are needed to explain the similar behaviour of @xmath16 and @xmath17 functions in figs . 6a and 7a and their @xmath60 dependence at large q ( figs 6b , 7b ) . resonance production and colour reconnection effects might be candidates ( see sect . * * quantum statistical approach * : a chaoticity parameter @xmath61 decreasing with multiplicity would , in the quantum statistical approach @xcite , decrease be correlations at higher multiplicities . in this picture , however , the question arises what the physical nature of the subprocess causing increased coherence at higher multiplicities would be . also , the similarity of the behaviour of @xmath16 and @xmath17 correlation functions in fig . 7 is not easily explained within this framework . * * be and resonance - induced correlations combined * : one could hypothesise that the observed multiplicity dependence of @xmath16 correlation functions is the result of two processes @xcite : + \a ) bose - einstein correlations in the classical sense which , being a global effect , are independent of multiplicity @xcite , ) and ( [ se1 ] ) . ] and b ) the production of higher mass - resonances or clusters decaying into two or more like - sign pions : @xmath62 ( as seen for example in @xmath63 decay ) . if the unnormalized cumulants @xmath64 and @xmath65 were wholly the result of resonance decays and if the number of resonances were proportional to the multiplicity @xmath66 , then @xmath67 . assuming @xmath68 gives @xmath69 , and hence after normalization , the resonance - inspired guess yields @xmath42 behaviour , @xmath70 a mixture of processes a ) and b ) would give @xmath71 in agreement with the behaviour of @xmath16 functions in fig . 7a ( straight line ) . the @xmath42 dependence in fig . 7b at large @xmath34 suggests the existence of a resonance / cluster component for both @xmath16 and @xmath17 pair production . the resonance contribution to the correlation functions would be concentrated mainly at the region around @xmath72 gev as in the case of @xmath73 production , which is visible as a peak in the @xmath17 functions in figs . this region is however difficult to investigate because there the @xmath74 are decreasing rapidly with increasing @xmath34 while the @xmath75 are already small . a @xmath42-dependence due to resonances or clusters in this dominant phase space region around 1 gev could presumably cause the large-@xmath34 region to follow suit via missing pairs , thus explaining the observed dependence there . + once again , however , the similarity of @xmath16 and @xmath17 functions in the small-@xmath34 region in fig . 7a can hardly be explained by assuming only the two components a ) and b ) . + one possible explanation could be the existence of global correlations for os pairs too . it would give a behaviour for @xmath76 similar to that of eq . ( [ resin ] ) and would explain its constant @xmath77 part by noting that the number of @xmath78 pairs that can be formed from @xmath79 positive pions and @xmath80 negative pions would be @xmath81 . if each of these pairs was correlated ( statistically speaking ) , i.e. if all @xmath82 pions were mutually correlated , the unnormalised @xmath83 would be proportional to @xmath84 and hence after normalisation @xmath76 constant in @xmath66 . such a constant term signalizes maximum possible correlations in some events . + more generally and following the arguments leading to eq . ( [ res1 ] ) , we could also say : `` if resonance production were to rise more quickly than @xmath85 , then @xmath86 would decrease more slowly than @xmath42 '' . * * the core - halo picture * @xcite . this picture is currently the only one which connects the multiplicity dependence of @xmath16 with that of @xmath17 correlation functions . the core - halo picture is based on the fact that bose - einstein correlations of decay products from long - lived resonances are not observable by experiments because they occur below experimental resolution and hence by definition belong to the `` halo '' of resonances that decay at large distances . examples are @xmath87 fm / c , @xmath88 fm / c , @xmath89 fm / c , which are all unresolvable within the ua1 experiment which can resolve decay products for distances @xmath90 fm only , corresponding to @xmath91 mev . because of the halo , the @xmath0 values of be fits are in general reduced @xcite , @xmath92 where the second factor describes the momentum dependence and @xmath93 is the smallest @xmath34 value accessible by the experiment mev and that the fit parameters ( of gaussian fits ) are insensitive to the exact value of @xmath93 in a certain restricted region . ] . the relation between @xmath0 and the fraction of pions emitted by the halo follows from eq . ( [ rla ] ) and from @xmath94 where @xmath95 is the mean multiplicity of directly produced `` core '' pions , @xmath96 is the mean multiplicity of halo pions and @xmath97 = @xmath98 + @xmath96 . this means that @xmath99 from eq . ( [ rla2 ] ) it is evident that @xmath0 remains independent of multiplicity only if @xmath100 and consequently @xmath101 . if however the number of halo - resonances increases faster than @xmath102 , then also the number of their decay particles @xmath103 . as a consequence the be parameter @xmath0 will decrease with multiplicity . + a second consequence emerges immediately ( see last sentence in sect . 3.3 ) : the fraction of @xmath86 stemming directly from the decay of halo resonances will decrease less rapidly than @xmath104 . a previous study @xcite revealed the @xmath34 region where two - body @xmath105 decay products of the halo resonances @xmath106 , @xmath107 and @xmath108 contribute , namely in @xmath109 gev . this is exactly the region where @xmath17 correlation functions indeed show a multiplicity dependence weaker than @xmath110 as shown in fig . 6b ( where the @xmath60 case is indicated by the dashed line ) . + so far , this discussion is purely qualitative . how the decrease of @xmath0 with @xmath66 would compare to an effective slower - than-@xmath42 decrease of @xmath76 is , of course , a quantitative question not answered by the above argument , both because @xmath0 refers to the minimum @xmath34 -value and its stability against shifts is not yet tested , and because the experimental fraction @xmath111 is known only sparsely , if at all . a more quantitative estimate has to be done in future , including the fact that the previously measured @xmath59 for the whole sample @xcite are already near unity at @xmath112 gev , even after subtracting the background contribution due to the non - poissonian overall multiplicity distribution . correcting for an additional halo contribution could finally cause @xmath59 for @xmath113 to be greater than 1 . the four different explanations considered above each have some merit . it is clearly desirable to shorten this list of candidates . we believe that higher order cumulants are suitable for this purpose : if symmetrization of individual strings is the right explanation , we can expect from eq . ( [ ss1 ] ) and ref . @xcite that the higher - order cumulants would decrease much faster with multiplicity than @xmath14 ( e.g. @xmath114 ) . since such a fast decrease is not predicted e.g. for the core - halo picture @xcite , the measurement of the multiplicity dependence of @xmath115 could probably decide between the two cases ( 3.1 ) and ( 3.4 ) . we thank b. andersson , a. biaas , t. csrg , k. fiakowski and w. kittel for useful discussions , and thank also the ua1 collaboration for freely providing the data . we gratefully acknowledge the technical support of g. walzel . hce thanks the institute for high energy physics in vienna for kind hospitality . this work was funded in part by the south african national research foundation .

the observed pronounced multiplicity dependence of correlation functions in hadron - hadron reactions and in particular of bose - einstein correlations provides information about underlying physics . we discuss in this contribution several interpretations , giving special attention to the string model for bose - einstein correlations of andersson and hofmann , as well as the core - halo picture of csrg , lrstad and zimanyi .

the galactic center source sagittarius a * ( sgr a * ) provides the best case for high - resolution , detailed observations of the accretion and outflow region surrounding the event horizon of a black hole . there are several compelling reasons to observe sgr a * with very long baseline interferometry ( vlbi ) at ( sub)millimeter wavelengths . the spectrum of sgr a * peaks in the millimeter ( * ? ? ? * and references therein ) . interstellar scattering , which varies as the wavelength @xmath3 , becomes less than the fringe spacing of the longest baseline available to vlbi in the millimeter - wavelength regime . indeed , vlbi on the longest baselines available at 345 ghz probes scales of twice the schwarzschild radius ( @xmath4 ) for a @xmath1 m@xmath2 black hole . from previous observations at 230 ghz , it is known that there are structures on scales smaller than a few @xmath4 @xcite . such high angular resolution , presently unattainable by any other method ( including facility instruments such as the very long baseline array ) , is necessary to match the expected spatial scales of the emitting plasma in the innermost regions surrounding the black hole and will be required to unambiguously determine the inflow / outflow morphology and permit tests of general relativity . this sensitivity to small spatial scales also makes millimeter polarimetric vlbi possible . although the linear polarization fraction of sgr a * integrated over the entire source is only a few percent ( e.g. , * ? ? ? * ) , the fractional polarization on small angular scales is likely much larger . in general , relativistic accretion flow models predict that the electric vector polarization angle ( evpa ) will vary along the circumference of the accretion disk @xcite , indicating that single - dish observations and connected - element interferometers probably underestimate linear polarization fractions due to beam depolarization . polarized synchrotron radiation coming from sgr a * was detected by @xcite at millimeter and submillimeter wavelengths . multiple observations since then have demonstrated that the polarized emission is variable on timescales from hours to many days @xcite . in one case , the timescale of variability and the trace of polarization in the stokes @xmath5 plane of the millimeter - wavelength emission are suggestive of the detection of an orbit of a polarized blob of material @xcite . near infrared observations by @xcite are also consistent with a hot spot origin for periodic variability . it is possible that connected - element interferometry may suffice to demonstrate polarization periodicity , but millimeter - wavelength vlbi , which effectively acts as a spatial filter on scales of a few to a few hundred @xmath4 , can be more sensitive to changing polarization structures . initial millimeter vlbi observations of sgr a * will necessarily utilize non - imaging analysis techniques , for reasons outlined in ( * ? ? ? * henceforth paper i ) . one way to do this is to analyze so - called interferometric `` closure quantities , '' which are relatively immune to calibration errors @xcite . in paper i , we considered prospects for detecting the periodicity signature of a hot spot orbiting the black hole in sgr a * via closure quantities in total - intensity millimeter - wavelength vlbi . in the single polarization case , it is necessary to construct closure quantities from at least three or four antennas in order to produce robust observables , since the timescales of atmospheric coherence and frequency standard stability do not permit standard nodding calibration techniques . closure quantities can be used in full - polarization observations as well , but it is also possible to construct robust observables on a baseline of two antennas by taking visibility ratios between different correlation products . in this work , we extend our techniques to explore polarimetric signatures of a variable source structure in sgr a * , with emphasis on ratios of baseline visibilities . we employ the same models discussed in paper i to describe the flaring emission of sgr a * , and shall only briefly review these here , directing the reader to paper i , and references therein , for more detail . these models consist of an orbiting hot spot , modelled by a gaussian over - density of power - law electrons , embedded in a radiatively inefficient accretion flow , containing both thermal and non - thermal electron populations . the primary emission mechanism for both components is synchrotron . we model the emission from the thermal and nonthermal electrons using the emissivities described in @xcite and @xcite , respectively , appropriately modified to account for relativistic effects ( see @xcite for a more complete description of polarized general relativistic radiative transfer ) . since we necessarily are performing the fully polarized radiative transfer , for the thermal electrons we employ the polarization fraction derived in @xcite . in doing so we have implicitly assumed that the emission due to the thermal electrons is locally isotropic , which , while generally not the case in the presence of ordered magnetic fields , is unlikely to modify our results significantly . for both electron populations the absorption coefficients are determined directly via kirchoff s law . as described in paper i , the assumed magnetic field geometry was toroidal , consistent with simulations and analytical expectations for magnetic fields in accretion disks , though other field configurations are possible ( e.g. , * ? ? ? * ) . while the overall flux of our models is relatively insensitive to the magnetic field geometry , the polarization is dependent on it . however , polarization light curves and maps with considerably different magnetic field geometries ( e.g. , poloidal ) are qualitatively similar , showing large swings in polarization angle and patches of nearly uniform polarization in the images . generally , synchrotron emission has both linearly and circularly polarized components . however , the circular polarization fraction is suppressed by an additional factor of the electron lorentz factor . for the electrons producing the millimeter emission , this corresponds to a reduction by a factor of @xmath6@xmath7 in stokes @xmath8 in comparison to stokes @xmath9 and @xmath10 . this is consistent with observations by @xcite , who obtain an upper limit of @xmath11 circular polarization at 340 ghz . therefore , we explicitly omitted the circular polarization terms in the computation of flaring polarization . in addition , we have neglected the potentially modest intrinsic faraday rotation . within @xmath12@xmath13 the accreting electrons are expected to be substantially relativistic , and thus not contribute significantly to the rotation measure within the millimeter - emitting region . this is consistent with the lack of observed faraday depolarization at these wavelengths ( e.g. , * ? ? ? * ; * ? ? ? * ) , which itself implies the absence of significant in situ faraday rotation . similarly , beam depolarization caused by variations within an external faraday screen on angular scales comparable to that of the emission region are empirically excluded . this leaves the possibility of a smoothly varying external faraday screen , which manifests itself in the vlbi data as an additional phase difference between right and left circularly polarized visibilities , but does not affect our analysis otherwise . model images are created in each of the stokes parameters @xmath14 , @xmath9 , and @xmath10 . six models differing in hot spot orbital period , black hole spin , and accretion disk inclination and major - axis orientation are produced at 230 and 345 ghz , as in paper i. model properties are summarized in table [ tab - models ] . source - integrated linear polarization fractions range from 0.8 to 26% for models including both a disk and a hot spot , depending on the model and hot spot orbital phase , with typical integrated quiescent polarization fractions ( of the disk alone ) of 10 to 15% . integrated evpa variation over the course of the hot spot orbit ranges from 4 to 57 , depending on the model . the integrated polarization fractions and evpa variations as well as the polarization traces in the stokes @xmath5 plane ( fig . [ fig - qu ] ) are all broadly consistent with the range of variability seen in the submillimeter array ( sma ) observations reported by @xcite . the local linear polarization fraction can be much higher , exceeding 70% in some parts of the accretion disk . simulated array data are produced by the astronomical image processing system ( aips ) task uvcon for each of the stokes parameters . the array is taken to consist of up to seven stations : the caltech submillimeter observatory , james clerk maxwell telescope , and six sma telescopes phased together into a single station ( hawaii ) ; the arizona radio observatory submillimeter telescope ( smt ) ; a phased array consisting of eight telescopes in the combined array for research in millimeter - wave astronomy ( carma ) ; the large millimeter telescope ( lmt ) ; the 30 m institut de radioastronomie millimtrique dish at pico veleta ( pv ) ; the plateau de bure interferometer phased together as a single station ( pdb ) ; and a site in chile , either a single 10 or 12 m class telescope ( chile 1 ) or a phased array of 10 dishes of the atacama large millimeter array ( chile 10 ) . details of the method as well as assumed parameters of the telescopes are given in paper i. llrcrcccccrcc a & 0 & 27.0 & 30 & 90 & 230 & 3.19 & 3.49 & 4.05 & 10 & @xmath1575 & 7.0 & 16 + & & & & & 345 & 3.36 & 3.63 & 5.28 & 11 & @xmath1581 & 6.0 & 24 + b & 0 & 27.0 & 60 & 90 & 230 & 3.03 & 3.05 & 4.03 & 14 & @xmath1584 & 6.9 & 20 + & & & & & 345 & 2.96 & 2.99 & 4.78 & 13 & @xmath1580 & 2.2 & 26 + c & 0 & 27.0 & 60 & 0 & 230 & 3.03 & 3.05 & 4.03 & 14 & 6 & 6.9 & 20 + & & & & & 345 & 2.96 & 2.99 & 4.78 & 13 & 10 & 2.2 & 26 + d & 0.9 & 27.0 & 60 & 90 & 230 & 2.98 & 2.99 & 4.05 & 15 & @xmath1586 & 0.8 & 21 + & & & & & 345 & 2.96 & 2.97 & 4.00 & 15 & @xmath1582 & 1.6 & 24 + e & 0.9 & 8.1 & 60 & 90 & 230 & 2.98 & 3.08 & 4.15 & 15 & @xmath1586 & 10 & 19 + & & & & & 345 & 2.96 & 3.04 & 6.07 & 15 & @xmath1582 & 9.7 & 24 + f & 0 & 166.9 & 60 & 90 & 230 & 3.07 & 3.08 & 3.38 & 15 & @xmath1584 & 9.8 & 19 + & & & & & 345 & 2.99 & 3.00 & 3.18 & 13 & @xmath1580 & 10 & 17 for ideal circularly - polarized feeds , the perfectly calibrated correlations are related to the complex stokes visibilities ( @xmath16 ) as follows : @xmath17 where @xmath18 and @xmath19 ( for example ) denotes the right circular polarized signal at one station correlated against the left circular polarized signal at another . we have used the convention of @xcite . other definitions , differing in sign or rotation of the @xmath19 and @xmath20 terms by factors of @xmath21 , are possible ( e.g. , * ? ? ? * ) , but do not affect the analysis . significant circular polarization is neither predicted in the hot spot models nor observed at the resolution of connected - element arrays @xcite . in the limit of no circular polarization ( @xmath22 ) , @xmath23 is a direct observable in the parallel - hand correlations , but @xmath24 and @xmath25 appear only in combination in the cross - hand correlations . @xmath19 and @xmath20 visibilities , which are direct observables , are constructed by appropriate complex addition of the stokes @xmath24 and @xmath25 visibilities . right- and left - circular polarized ( rcp and lcp ) feeds are preferable to linearly - polarized feeds for detecting linear polarization , since the latter mix stokes @xmath26 with @xmath24 in the parallel - hand correlations @xcite . for a point source , @xmath27 . however , for an extended distribution , the polarized stokes visibilities can exceed the amplitude of the stokes @xmath26 visibility . ( for instance , a uniform total intensity distribution with constant linear polarization fraction but a changing linear polarization angle will produce no power in stokes @xmath26 on scales small compared to the distribution , but the stokes visibilities @xmath24 and @xmath25 will be nonzero . ) analysis of polarimetric data is more complex than total intensity ( stokes @xmath14 ; we will henceforth drop the subscript on stokes visibilities ) data , but ratios of cross - hand ( @xmath19 and @xmath20 ) to parallel - hand ( @xmath28 and @xmath29 ) visibilities provide robust baseline - based observables immune to most errors arising from miscalibrated antenna complex gains . this stands in contrast to the single - polarization case in which robust observables can only be constructed from closure quantities on three or more telescopes . the procedure for referencing cross - hand data to parallel - hand data is explained in detail in @xcite and @xcite and has been used successfully in experiments ( e.g. , * ? ? ? * ) . several details warrant further discussion . we shall refer to the full expressions for the observed correlation quantities : @xmath30 , \\ ll = l_1l_2^ * = g_{1l}g_{2l}^ * & [ & ( i_{12}-v_{12 } ) e^{i(+\varphi_1-\varphi_2 ) } \\ & & + d_{1l } d_{2l}^ * ( i_{12}+v_{12 } ) e^{i(-\varphi_1+\varphi_2 ) } \\ & & + d_{1l } p_{12 } e^{i(-\varphi_1-\varphi_2 ) } \\ & & + d_{2l}^ * p_{21}^ * e^{i(+\varphi_1+\varphi_2 ) } ] , \\ rl = r_1l_2^ * = g_{1r}g_{2l}^ * & [ & p_{12 } e^{i(-\varphi_1-\varphi_2 ) } \\ & & + d_{1r } d_{2l}^ * p_{21}^ * e^{i(+\varphi_1+\varphi_2 ) } \\ & & + d_{1r } ( i_{12}-v_{12 } ) e^{i(+\varphi_1-\varphi_2 ) } \\ & & + d_{2l}^ * ( i_{12}+v_{12 } ) e^{i(-\varphi_1+\varphi_2 ) } ] , \\ lr = l_1r_2^ * = g_{1l}g_{2r}^ * & [ & p_{21}^ * e^{i(+\varphi_1+\varphi_2 ) } \\ & & + d_{1l } d_{2r}^ * p_{12 } e^{i(-\varphi_1-\varphi_2 ) } \\ & & + d_{1l } ( i_{12}+v_{12 } ) e^{i(-\varphi_1+\varphi_2 ) } \\ & & + d_{2r}^ * ( i_{12}-v_{12 } ) e^{i(+\varphi_1-\varphi_2)}],\end{aligned}\ ] ] where numeric subscripts refer to antenna number , letter subscripts refer to the polarization ( rcp or lcp ) , a star denotes complex conjugation , @xmath31 is the complex gain in polarization @xmath32 at antenna @xmath33 , @xmath34 , @xmath35 is the instrumental polarization , and @xmath36 is the parallactic angle ( equations reproduced from * ? ? ? * ) . the @xmath36 terms are constant for equatorial mount telescopes and can be incorporated into the @xmath37 and @xmath38 terms , while for alt - azimuth mount telescopes the @xmath36 terms vary predictably based on source declination , hour angle , and antenna latitude . it is likely that all of the telescopes in potential millimeter - wavelength vlbi arrays in the near future will have @xmath36 terms varying with parallactic angle . the ratio of cross - hand to parallel - hand data ( e.g. , @xmath39 ) contains an additional phase contribution @xmath40 equal to the phase difference of the complex gains of the right and left circular polarizations of antenna @xmath33 @xcite . these phase differences also enter into closure phases of cross - hand correlations as @xmath41 . fortunately , the right - left phase differences vary slowly with time ( e.g. , * ? ? ? * ) , since the atmospheric transmission is not significantly birefringent at millimeter wavelengths and both polarizations are usually tied to the same local oscillator . we will henceforth assume that the @xmath42 terms can be properly calibrated ( for instance by observations of an unpolarized calibrator source ) , although proper calibration may not be strictly necessary for periodicity detection , since the expected timescale of variation of source structure is significantly faster than the timescale of variation of @xmath42 . similarly , it is possible to determine the ratio of amplitudes of the real gains ( @xmath43 ) from observations of a suitable calibrator . in general , @xmath44 usually shows greater short - timescale variability than @xmath45 @xcite . provided that proper instrumental polarization calibration is done , the fluctuation in @xmath44 can be estimated from the @xmath46 visibility ratio , since sgr a * is expected to have no appreciable circular polarization enters the expressions for @xmath28 and @xmath29 only as @xmath47 , so even if circular polarization is detected , it will not prevent estimation of @xmath44 unless the circular polarization fraction on angular scales accessible to vlbi is large or highly variable . ] . even absent any complex gain calibration , it is probable that the contamination of the time series of cross - to - parallel amplitude ratios and ( especially ) phase differences by changes in @xmath44 and @xmath45 respectively will also be seen in the @xmath46 amplitude ratio and @xmath48 phase difference . thus , large deviations seen in the cross - to - parallel quantities but not in the parallel - to - parallel quantities will likely be due to source structure differences , not gain miscalibration . correcting for instrumental polarization ( the @xmath38-terms ) may be more difficult . effectively , the @xmath38-terms mix stokes @xmath14 into the @xmath19 and @xmath20 terms @xcite . observations of calibrators with the coordinated millimeter vlbi array ( cmva ) at @xmath49 mm found @xmath38-terms ranging from a few to 21% , with typical values slightly greater than 10% @xcite . polarimetric observations with carma and the sma in their normal capacity as connected - element interferometers have demonstrated that the instrumental polarization terms on some of the telescopes that will be included in future observations may be as low as a few percent @xcite . however , it is unknown how large the @xmath38-terms will be for potential vlbi arrays at @xmath50 and @xmath51 mm , as many of the critical pieces of hardware ( including feeds , phased - array processors , and even the antennas themselves ) do not yet exist for some of the elements of such arrays . in any case , contributions from the @xmath38-terms may be comparable to or larger than contributions from the source polarization , at least on the shorter baselines . the time scale of variations of @xmath38-terms is typically much longer than the time scale on which the source structure in sgr a * changes , so carefully - designed observations may allow for the @xmath38-terms to be calibrated . at the angular resolution of the sma , polarization fractions of sgr a * at 230 and 345 ghz range between 4 and 10% @xcite , although the polarization fraction may exceed this range during a flare @xcite . linear polarization fractions derived from single - dish and connected - element millimeter observations of sgr a * are likely underestimates of the linear polarization fractions that will be seen with vlbi , since partial depolarization from spatially separated orthogonal polarization modes may occur when observed with insufficient angular resolution to separate them . that is , the small - scale structure that will be seen by vlbi is likely to have a larger polarization fraction than that observed so far with connected - element interferometery . calibration of the electric vector polarization angle ( evpa ) may be difficult , at least in initial observations , due to the lack of known millimeter - wavelength polarization calibration sources ( see , e.g. , * ? ? ? evpa calibration will eventually be important for understanding the mechanism of linear polarization generation in sgr a * , assuming that the linear polarization can be unambiguously corrected for faraday rotation . however , the ability of cross - hand correlation data to detect _ changes _ in the evpa is unaffected by absolute evpa calibration . in the low signal - to - noise ( snr ) regime , the ratio of visibility amplitudes can be a biased quantity . visibility amplitudes are non - negative by definition , and the complex addition of a large noise vector to a small signal vector in the visibility plane will bias the visibility amplitude to higher values . nevertheless , even biased visibility amplitudes may be of some utility in detecting changing polarization structure . since the complex phase of noise is uniform random , phase differences are unbiased quantities . while lower - resolution observations of sgr a * find polarization of less than 10% , the effective fractional polarization on smaller scales can be much larger . figure [ fig - uvplt ] shows the amplitudes of the @xmath52 data and @xmath53 ) with the stokes parameters ( denoted by upper - case @xmath10 and @xmath8 ) . ] that would be produced by a disk and persistent , unchanging orbiting hot spot with parameters as given in model a at 230 ghz . the range in amplitudes reflects the changing flux density , both in total flux ( i.e. , the zero - spacing flux at @xmath54 ) as well as on smaller spatial scales , as would be sampled via vlbi . both total power ( stokes @xmath14 ) and polarization signatures fall off with baseline length , but on average the fractional polarization increases with longer baselines , and the ratio of stokes visibility amplitudes can exceed unity . all of our models produce much higher polarization fractions on small angular scales than at large angular scales , and all models except for model f at 345 ghz produce a substantial set of cross - to - parallel visibility amplitude ratios in excess of unity on angular scales of 4080 @xmath55as and smaller . we henceforth focus on ratios of cross - to - parallel baseline visibilities ( e.g. , @xmath56 ) . plots of the @xmath57 phase difference . ] are shown in figures [ fig - phase230 ] and [ fig - phase345 ] for models at 230 and 345 ghz , respectively . at a total data rate of 16 gbits@xmath58 , nearly all baselines exhibit signatures of changing polarization structure . due to the weak polarized signal on the longest baselines , a phased array of a subset of alma ( chile 10 , in the nomenclature of [ models ] ) may be required in order to confidently detect polarization changes on the long baselines , especially to europe . the pv - pdb baseline ( and to a lesser extent the smt - carma baseline at 230 ghz ) effectively tracks the orientation of the total linear polarization , since sgr a * is nearly unresolved on this short baseline , and the calibrated @xmath19 phase of a polarized point source at phase center is twice the evpa of the source ( e.g. , * ? ? ? . figures [ fig - amp230 ] and [ fig - amp345 ] show the @xmath56 visibility amplitude ratio for selected baselines at 230 and 345 ghz , respectively . the shortest baselines , pv - pdb and smt - carma , effectively track the large - scale polarization fraction as would be measured by the sma , for instance . because the short baselines resolve out several tens of percent of the total intensity emission ( as compared to the zero - spacing flux in fig . [ fig - uvplt ] ) but a much smaller fraction of the polarized emission , the variation in the @xmath56 and @xmath59 amplitude ratios is fractionally larger than in the large - scale polarization fraction . a bias can be seen in the amplitude ratios when the snr is small , as noted in [ polarimetric ] . ( for brevity , we have shown only plots of the @xmath57 phase difference and @xmath56 amplitude ratio . the @xmath60 phase difference and @xmath59 amplitude ratio exhibit similar behavior . ) closure phases of the cross - hand terms can be constructed in the same manner as for the parallel - hand terms , and these are robust observables . however , closure quantities are less necessary in the polarimetric case than for total - intensity observations because robust baseline - based observables can be constructed . as figure [ fig - uvplt ] shows , the visibility amplitude in the cross - hand correlations is much lower than that of the parallel - hand correlations on short baselines . the snr of the closure phase is lower by a factor of @xmath61 than the three constituent baseline snrs when the latter are all equal and is dominated by that of the weakest baseline when there is a large difference in the baseline snrs @xcite . in stokes @xmath14 , the mean baseline snrs ( averaged over multiple orbits ) are greater than or equal to 5 on virtually all baselines and all models at 16 gbits@xmath58 total bit rate ( 8 gbits@xmath58 each rr and ll ) in a 10 s coherence interval , provided that the chile 10 is used in lieu of chile 1 . the number of triangles with snrs greater than 5 on all baselines at 8 gbits@xmath58 in @xmath19 or @xmath20 is much smaller . depending on the model , the smt - carma - lmt and smt / carma - lmt - chile 10 triangles usually satisfy this condition , with hawaii - smt - carma also having sufficient snr . completion of the lmt , resulting in a system equivalent flux density of @xmath62 jy at 230 ghz , will allow for strong detections on the hawaii - lmt baseline and , importantly , significantly strengthen detections on the lmt - chile baseline . if the coherence time is significantly shorter than 10 s , or if the obseved flare flux density is substantially lower than assumed in our models , closure phases may not have a large enough snr to detect periodic changes . in any case , if polarimetric visibility ratios are successful in detecting periodicity , there may not be a need to appeal to closure quantities except insofar as they can be used to improve the array calibration . we have also simulated the effects of not correcting for parallactic angle terms and instrumental polarization by including gaussian random @xmath38-terms of @xmath63% with uniform random phases , based on the @xcite and @xcite cmva studies . this should be considered a worst - case scenario . @xmath38-terms at many of the telescopes will likely be substantially better : e.g. , 1 - 6% at the sma in observations by @xcite and about 5% at the 6 m antennas of the carma array @xcite . these quantities affect the observed correlation quantities @xmath28 , @xmath29 , @xmath19 , and @xmath20 as indicated in [ polarimetric ] . we have ignored terms of order @xmath64 , but we have included terms of the form @xmath65 , since the polarized visibility amplitudes can be larger than the stokes @xmath66 visibility amplitudes on long baselines ( fig . [ fig - uvplt ] ) . example data showing the effects of large uncalibrated @xmath38-terms is shown in figure [ fig - dterms ] . instrumental polarization adds a bias to the ratio of cross - hand to parallel - hand visibility amplitudes ( e.g. , @xmath67 ) as well as a phase slope and offset to the difference of cross - hand and parallel - hand phases ( e.g. , @xmath68 ) . these effects are much more pronounced on the short baselines , especially pv - pdb and smt - carma , because the fractional source polarization on large scales is small ( and thus @xmath69 ) . in most cases , the cross - to - parallel amplitude ratios and phase differences behave similarly whether instrumental polarization calibration is included or not simply by virtue of the fact that the polarized intensity is a large fraction of the total intensity . deviations in the cross - to - parallel phase difference response appear qualitatively large when the cross - hand amplitudes are near zero because small offsets from the source visibility , represented as a vector in the complex plane , can produce large changes in the angle ( i.e. , phase ) of the visibility . large instrumental polarization can affect the expected baseline - based signatures but do not obscure periodicity , since source structure changes in stokes @xmath14 and @xmath70 have the same period in our models . of course , proper @xmath38-term calibration is a sine qua non for modelling the polarized source structure ( but not for detecting periodicity ) . the @xmath38-terms can be measured by observing a bright unpolarized calibrator ( or polarized , unresolved calibrator ) , and the visibilities should be corrected for instrumental polarization if possible . as in paper i , we can define autocorrelation functions to test for periodicity . more optimal methods exist to extract the period of a time series of data @xcite , but the autocorrelation function is conceptually simple and suffices for our models . the amplitude autocorrelation function evaluated at lag @xmath71 on a time series of @xmath33 amplitude ratios @xmath72 ( or @xmath73 ) on a baseline is defined as @xmath74,\ ] ] where @xmath55 and @xmath75 are the mean and variance of the logarithm of the amplitude ratios , respectively . the phase autocorrelation function is defined as @xmath76 where @xmath77 denotes the @xmath57 or @xmath60 phase difference of point @xmath21 . by definition , @xmath78 . the largest non - trivial peak corresponds to the period , with the caveat that the changing baseline geometries caused by earth rotation can conspire to cause the autocorrelation function to be slightly greater at integer multiples of the true period . the phase autocorrelation function can suffer from lack of contrast when the visibility phase difference is not highly variable as may be the case for the shortest baselines depending on the model ( fig . [ fig - acfs ] ) , but the lack of contrast is not so severe as in the total - intensity case ( cf . paper i ) due to the sensitivity of short - baseline cross - to - parallel phase differences to the source - integrated evpa . an array consisting of hawaii , smt , and carma is sufficient to confidently detect periodicity at a total bit rate of 2 gbits@xmath58 ( i.e. , 0.25 ghz bandwidth per polarization ) over 4.5 orbits of data for models a - e . this contrasts with the total intensity case , in which a substantially higher bit rate is required on the same array , depending on the model ( paper i ) . the key point is that the cross - to - parallel visibility ratios on the short baselines trace the overall source polarization fraction and evpa , which are readily apparent even at the much coarser angular resolution afforded by connected - element interferometry ( e.g. , * ? ? ? long baselines are thus not strictly necessary to detect periodic polarimetric structural changes , although they will be important for modelling the small - scale polarimetric structure of the sgr a*. in contrast , significantly higher bit rates and 4-element arrays are usually required to detect periodic source structure changes in total intensity ( paper i ) . long - period models ( e.g. , model f ) are problematic for millimeter vlbi periodicity detection because it may not be possible to detect more than two full periods during the window of mutual visibility between most of the telescopes in a potential vlbi array . as with the total intensity case ( paper i ) , the most promising approach for millimeter vlbi is to observe with an array of four or five telescopes , since large changes in cross - to - parallel phase differences and visibility amplitude ratios tend to be episodic across most or all baselines . the lmt is usefully placed because it provides a long window of mutual visibility to chile and the us telescopes as well as a small time overlap with pv , thus enabling a large continuous time range over which sgr a * is observed . assuming that the flare source can survive for several orbital periods , connected - element interferometry may suffice to demonstrate periodicity . sgr a * is above 10 elevation for approximately 9 hr from hawaii and 12 hr from chile . vlbi polarimetry has several key advantages over single - dish and connected - element interferometry for understanding the polarization properties of sgr a*. first , even the shortest baselines likely to be included in the array will filter out surrounding emission . reliable single - dish extraction of polarization information requires subtracting the contribution from the surrounding dust , which can dominate the total polarized flux at 345 ghz and is significant even at 230 ghz @xcite . contamination by surrounding emission is much less severe for sma measurements , where the synthesized beamsize is on the order of an arcsecond , depending on configuration ( for instance , @xmath79 in the observations of * ? ? ? vlbi will do much better still , with the shortest baselines resolving out most of the emission on scales larger than @xmath80 mas ( @xmath81 ) , effectively restricting sensitivity to the inner accretion disk and/or outflow region . second , the resolution provided by millimeter vlbi will greatly reduce depolarization due to blending of emission from regions with different linear polarization directions . models of the accretion flow predict that linear polarization position angles and faraday rotation will be nonuniform throughout the source @xcite . for this reason , ratios of cross - hand to parallel - hand visibilities ( which are the visibility analogues of linear polarization fractions ) can greatly exceed the total linear polarization fraction integrated over the source ( cf . figure [ fig - uvplt ] and table [ tab - models ] ) . third , vlbi polarimetry has the potential to identify whether changes in detected polarization are due to intrinsic source variability or changes in the rotation measure at larger distances . the former would be expected to be variable on relatively short timescales ( minutes to tens of minutes ) , consistent with the orbital period of emission at a few gravitational radii . the latter would be expected to vary more slowly and affect only the polarized emission , not the total intensity . a cross - correlation between polarized and total - intensity data may allow the two effects to be disentangled . linear polarization at millimeter wavelengths can be used to estimate the accretion rate of sgr a * ( e.g. , * ? ? ? * ) . at frequencies below @xmath82 ghz , no linear polarization is detected due to faraday depolarization in the accretion region @xcite . linear polarization is detected toward sgr a * at higher frequencies ( * ? ? ? * e.g. , ) , where the effects of faraday rotation are smaller . ultimately , accretion rates are constrained by the lack of linear polarization at long wavelengths and its existence at short wavelengths . measurements of the faraday rotation exist , although it is unclear whether changes in detected polarization angles are due to changing source polarization structure or a variable rotation measure @xcite . longer term , imaging may be possible if all seven millimeter telescope sites heretofore considered ( and possibly others as well ) are used together as a global vlbi array ( e.g. , an event horizon telescope ; * ? ? ? imaging the quiescent polarization structure of sgr a * may allow the characteristics of the source emission region to be distinguished from those of the region producing faraday rotation ( which may overlap or be identical with the emission region ) . contemporaneous millimeter vlbi observations at two different frequencies would allow separate maps of the intrinsic polarization structure and the rotation measure to be produced . it may also then be possible to place strong constraints on the density ( @xmath83 ) and magnetic fields ( @xmath84 ) in sgr a*. briefly , the rotation measure is related to @xmath85 , while the total intensity is proportional to @xmath86 , where @xmath87 is the optically thin spectral index @xcite . obtaining these results will require the ability to fully calibrate the data for instrumental polarization terms and the absolute evpa . it may also require higher image fidelity than a seven - telescope vlbi array can provide @xcite . there are possibilities for extending a millimeter vlbi array beyond these seven sites by adding other existing ( e.g. , the south pole telescope ) or new telescopes @xcite , but full consideration of the scientific impact of potential future arrays is beyond the scope of this work . the inclusion of a screen of constant faraday rotation alters the phases of the cross - hand terms ( and therefore the cross - to - parallel phase differences ) but does not materially affect the detectability of changing polarization structure . the mean rotation measure of sgr a * averaged over multiple epochs is @xmath88 radm@xmath89 @xcite , which corresponds to a rotation of polarization vectors by @xmath90 at 230 ghz and @xmath91 at 345 ghz . persistent gradients of rotation measure across the source are virtually indistinguishable from intrinsic polarization structure in the case of a steady - state source , but it is possible that the source structure and rotation measure change on different timescales , which would allow the two effects to be disentangled @xcite . comparison of changes in the polarization data with total intensity data ( obtained from the parallel - hand correlations ) and total polarization fraction ( obtained from simultaneous connected - element interferometric data if available , else inferred from the shortest vlbi baselines ) may be useful for identifying whether observed polarization angle changes are due to a variable rotation measure @xcite . the physical mechanism that produces flares in total intensity and polarization changes is poorly understood . connected - element interferometry at millimeter wavelengths has not been conclusive as to whether orbiting hot spots are the underlying mechanism that produces flares in sgr a * ( cf . @xcite and @xcite ) , or even as to whether multiple mechanisms may be responsible for flaring . polarization variability can be decorrelated from total intensity variability ( e.g. , * ? ? ? * ) , and each shows variability on time scales ranging from tens of minutes to hours ( and possibly longer ) . spatial resolution will be key to deciphering the environment of sgr a * , and thus there is a critical need for polarimetric millimeter - wavelength vlbi . our results are generalizable to any mechanism producing changes in linear polarization , whether due to orbiting or spiralling hot spots , jets , disk instabilities , or any other mechanism in the inner disk of sgr a*. visibility ratios on baselines available for millimeter - wavelength vlbi will provide reasonably robust observables to detect changes in the polarization structure on relevant scales from a few to a few hundred @xmath4 , regardless of the cause of those changes . clearly , periodicity can only be detected if the underlying mechanism that produces polarization changes is itself periodic , but baseline visibility ratios will be sensitive to any changes that are rapid compared to the rotation of the earth . future millimeter - wavelength vlbi observations of sgr a * should clearly be observed in dual - polarization unless not allowed by telescope limitations . total - intensity analysis via closure quantities , as outlined in paper i , can be performed regardless of whether the data are taken in single- or dual - polarization mode , but the cross - hand correlations can only be obtained from dual - polarization data . the cross - hand correlation data provide additional chances to detect variability via changing source polarization structure . by virtue of its size and location , which produces medium - length baselines to chile and hawaii as well as a long window of mutual visibility with chile , the lmt is a very useful telescope . the sensitivities assumed in this work for first light on the lmt may lead to biased amplitude ratios on the hawaii - lmt baseline ( fig . [ fig - amp230 ] ) , but this will not prevent detection of periodicity ( fig . [ fig - acfs ] ) . thus , strong consideration should be given in favor of including the lmt in a millimeter - wavelength vlbi array observing sgr a * as soon as possible . if the parameters of the fully - completed lmt are assumed , the bias disappears and the scatter of points on the hawaii - lmt baseline in figure [ fig - amp230 ] is similar to that seen on the shorter hawaii - carma baseline , and baselines between continental north america and the lmt will be of comparably good snr to the lower - resolution pv - pdb baseline . eventually , the lmt and alma will be the most sensitive stations in a millimeter vlbi array and will enable sensitive modelling of the sgr a * system . if possible , it would be advantageous to obtain connected - element interferometric data of sgr a * simultaneously with vlbi data . while amplitude ratios and phase differences on the pv - pdb and ( to a lesser extent ) smt - carma baselines track the large - scale polarization fraction and evpa fairly well in these models , it is not known what fraction of the polarization structure arises from larger - scale emission in sgr a*. if interferometer stations can be configured to produce both cross - correlations betweeen telescopes as well as a phased output of all telescopes together , opportunities for simultaneous connected - element interferometry may exist with the pdb interferometer , the sma , carma , or alma . if system limitations prevent this , it may still be possible to acquire very - short - spacing data with those telescopes in carma or alma that are not phased together for vlbi . ( sub)millimeter - wavelength vlbi polarimetry is a very valuable diagnostic of emission processes and dynamics near the event horizon of sgr a*. we summarize the findings in this paper as follows : * millimeter - wavelength polarimetric vlbi can detect changing source structures . despite low polarization fractions seen with connected - element interferometry , the much higher angular resolution data provided by vlbi will be far less affected by beam depolarization and contamination from dust polarization . polarimetric vlbi provides an orthogonal way to detect periodic structural changes as compared with total intensity vlbi . * ratios of cross- to parallel - hand visibilities are robust baseline - based observables . short vlbi baselines approximately trace the integrated polarization fraction and position angle of the inner accretion flow of sgr a * , while longer vlbi baselines resolve smaller structures . * calibration of instrumental polarization terms is not necessary to detect a changing source structure , including periodicity , in sgr a*. * polarimetric vlbi may be able to disentangle the effects of rotation measure from intrinsic source polarization . initial results will likely come from observations of the timescale of polarimetric variability . if the initial array is expanded to allow high - fidelity imaging , polarimetric vlbi may be able to map the faraday rotation region and directly infer the density and magnetic field structure of the emitting region in sgr a*.

sagittarius a * is the source of near infrared , x - ray , radio , and ( sub)millimeter emission associated with the supermassive black hole at the galactic center . in the submillimeter regime , sgr a * exhibits time - variable linear polarization on timescales corresponding to @xmath0 schwarzschild radii of the presumed @xmath1 m@xmath2 black hole . in previous work , we demonstrated the potential for total - intensity ( sub)millimeter - wavelength very long baseline interferometry ( vlbi ) to detect time - variable and periodic source structure changes in the sgr a * black hole system using nonimaging analyses . here we extend this work to include full polarimetric vlbi observations . we simulate full - polarization ( sub)millimeter vlbi data of sgr a * using a hot - spot model that is embedded within an accretion disk , with emphasis on nonimaging polarimetric data products that are robust against calibration errors . although the source - integrated linear polarization fraction in the models is typically only a few percent , the linear polarization fraction on small angular scales can be much higher , enabling the detection of changes in the polarimetric structure of sgr a * on a wide variety of baselines . the shortest baselines track the source - integrated linear polarization fraction , while longer baselines are sensitive to polarization substructures that are beam - diluted by connected - element interferometry . the detection of periodic variability in source polarization should not be significantly affected even if instrumental polarization terms can not be calibrated out . as more antennas are included in the ( sub)mm - vlbi array , observations with full polarization will provide important new diagnostics to help disentangle intrinsic source polarization from faraday rotation effects in the accretion and outflow region close to the black hole event horizon .

a topological graph @xmath0 is a directed graph such that the vertex set @xmath1 and the edge set @xmath2 are both locally compact hausdorff spaces and the range and source maps @xmath7 satisfy appropriate continuity conditions ( see section [ sec : topologicalgraphs ] ) . a graph can be viewed as a generalization of a dynamical system where the graph @xmath6 is thought of as a partially defined , multi - valued , continuous map @xmath8 given by @xmath9 for every @xmath10 . in @xcite , katsura constructs a @xmath11-algebra @xmath3 from a topological graph @xmath6 and gives a very detailed analysis of these @xmath11-algebras . this construction generalizes both discrete graph algebras and homeomorphism @xmath11-algebras . as with discrete graphs , the structure of the graph algebra @xmath3 is closely related to the structure of the underlying graph @xmath6 . for instance , the ideal structure , @xmath12-theory , simplicity , and pure infiniteness of @xmath3 can all be described in terms of the graph @xmath6 . there are also generalizations of the cuntz - krieger and gauge invariant uniqueness theorems for topological graphs . although the class of @xmath11-algebras which are defined by discrete graphs is fairly limited , it seems that many interesting @xmath11-algebras ( especially those in the classification program ) appear as topological graph algebras . for instance , every crossed product @xmath13 , every uct kirchberg algebra , and every af - algebra appear as topological graph algebras . in fact , there does not seem to be any known nuclear uct @xmath11-algebras which do not arise as a topological graph algebras , although it seems unlikely that every nuclear uct @xmath11-algebra will have this form . for instance , in view of theorem [ thm : topgraphsafembedding ] below , it seems plausible that a finite @xmath11-algebra that is not stably finite will not be a topological graph algebra . we are interested in the finiteness properties of @xmath3 . in particular , when is @xmath3 af - embeddable , quasidiagonal , stable finite , or finite ? a @xmath11-algebra is called _ quasidiagonal _ if there is a net of completely positive contractive maps @xmath14 where each @xmath15 is a finite dimensional @xmath11-algebra and the @xmath16 are asymptotically isometric and multiplicative ; i.e. for every @xmath17 , we have @xmath18 it follows from arveson s extension theorem that every af - algebra is quasidiagonal . subalgebras of quasidiagonal @xmath11-algebras are certainly quasidiagonal and hence every af - embeddable @xmath11-algebra is quasidiagonal . moreover , it is well known that quasidiagonal @xmath11-algebras are stably finite ( ( * ? ? ? * proposition 7.1.15 ) ) . the converses are known to be false . in particular , by a result of choi ( ( * ? ? ? * theorem 7 ) ) , @xmath19 is quasidiagonal ( in fact the @xmath16 may be chosen to be multiplicative ) but @xmath19 is not af - embeddable since it is not exact ( @xcite ) . similarly , if @xmath20 is a non - amenable group , then @xmath21 is not quasidiagonal by a result of rosenberg ( @xcite ) , but it is well known that @xmath21 is stably finite since it has a faithful trace . blackadar and kirchberg conjectured in @xcite that the converses may be true with some amenability assumptions . in particular , they conjecture af - embeddability , quasidiagonality , and stable finiteness are equivalent for nuclear @xmath11-algebras . the main result of @xcite verifies this conjecture for graph @xmath11-algebras . also , the following result from pimsner verifies the conjecture for certain crossed products . [ thm : pimsner ] suppose @xmath22 is a compact metric space and @xmath23 is a homeomorphism of @xmath22 . then the following are equivalent : 1 . @xmath24 is af - embeddable ; 2 . @xmath24 is quasidiagonal ; 3 . @xmath24 is stably finite ; 4 . @xmath24 is finite ; 5 . every point in @xmath22 is pseudoperiodic for @xmath23 : given @xmath25 , there are points @xmath26 such that @xmath27 for all @xmath28 , where the subscripts are taken modulo @xmath29 . there is a similar theorem of n. brown for crossed products of af - algebras by @xmath30 and there are partial results for many other crossed products ( see @xcite ) . see ( * ? ? ? * chapters 7 and 8) for a survey of quasidiagonality and af - embeddability . our goal in this paper is to prove a version of theorem [ thm : pimsner ] for the @xmath11-algebra @xmath3 generated by a compact topological graph @xmath6 with no sinks ( theorem [ thm : topgraphsafembedding ] ) . we show that if @xmath3 is finite , then @xmath6 has no sources and every vertex in @xmath6 emits exactly one edge . in this case , @xmath4 , where @xmath5 is the space of all infinite paths in @xmath6 and the actions is given by the backward shift on @xmath5 ( theorems [ thm : finiteness ] and [ thm : infinitepathcrossedproduct ] ) . combining this with theorem [ thm : pimsner ] will give our result . pimsner s proof that @xmath31 in theorem [ thm : pimsner ] involves decomposing @xmath22 into its orbits ( viewed as discrete spaces ) and representing the elements of @xmath32 as weighted bilateral shifts on the @xmath33 spaces of these orbits . our techniques are similar but are more involved since a topological graph can be significantly more complicated than a dynamical system . given an infinite path @xmath34 in a topological graph @xmath6 , we define a directed tree @xmath35 which is thought of as the orbit of @xmath34 . in section [ sec : representations ] we modify a construction of katsura to represent @xmath3 on @xmath36 by viewing the elements of @xmath37 as diagonal operators and the elements of @xmath38 as weighted shifts on @xmath36 . weighted shifts on directed trees were defined and extensively studied in @xcite . in particular , it was shown that the fredholm theory of weighted shifts on directed a tree is closely related to the combinatorial structure of the tree ( see corollary [ cor : surjectiveshift ] ) . this allows us to relate the finiteness of @xmath3 to the `` combinatorial '' structure of the infinite path space @xmath5 and hence also to the graph @xmath6 . the paper is organized as follows . in sections [ sec : cuntzpimsner ] and [ sec : topologicalgraphs ] we recall the necessary definitions and results about cuntz - pimsner algebras and topological graph algebras . section [ sec : weightedshifts ] contains the necessary results from @xcite on weighted shifts on directed trees . in section [ sec : representations ] , we build representations of topological graph algebras on weighted trees , and finally section [ sec : finiteness ] contains our main results . we will adopt the following conventions throughout the paper . for graph algebras , we follow the conventions in @xcite . in particular , the partial isometries go in the same direction as the edges . we assume @xmath11-algebras are separable , spaces and second countable , and graphs are countable . if @xmath22 is a set , let @xmath39 denote the standard orthonormal basis of @xmath40 . that is , for @xmath41 , @xmath42 is given by @xmath43 and @xmath44 when @xmath45 . the author would like to thank his advisors allan donsig and david pitts for their encouragement and for pointing out several errors in earlier versions of this paper . cuntz - pimsner algebras were introduced by pimsner in @xcite and were expanded on by katsura in @xcite . this class of algebras was further studied in ( * ? ? ? * chapter 8) and ( * ? ? ? * section 4.6 ) . for the readers convenience , we recall the necessary definitions . a ( _ right _ ) _ hilbert module _ over a @xmath11-algebra @xmath46 is a right @xmath46-module @xmath22 together with an _ inner product _ @xmath47 such that for every @xmath48 , @xmath49 , and @xmath50 , 1 . @xmath51 with equality if and only if @xmath52 , 2 . @xmath53 , 3 . @xmath54 , and 4 . @xmath55 , and @xmath22 is complete with respect to the norm @xmath56 . an operator @xmath57 is called _ adjointable _ if there is an operator @xmath58 such that @xmath59 for every @xmath60 . if @xmath61 is adjointable , then @xmath62 is unique , and @xmath61 and @xmath62 are both @xmath46-linear and bounded . the collection @xmath63 of all adjointable operators on @xmath22 is a @xmath11-algebra . a _ @xmath11-correspondence _ over @xmath46 is a hilbert @xmath46-module @xmath22 together with a * -homomorphism @xmath64 . we often write @xmath65 for @xmath49 and @xmath66 . a _ toeplitz representation _ of @xmath22 on a @xmath11-algebra @xmath67 is a pair @xmath68 where @xmath69 is a * -homomorphism , @xmath70 is a linear map such that for every @xmath49 and @xmath60 , @xmath71 note that for every @xmath49 and @xmath66 , we have @xmath72 since @xmath73 moreover , the computation @xmath74 implies @xmath75 and if @xmath76 is injective , then @xmath77 is isometric . let @xmath78 be the universal @xmath11-algebra generated by a toeplitz representation of @xmath22 . more precisely , @xmath78 is a @xmath11-algebra together with a toeplitz representation @xmath79 of @xmath22 on @xmath78 such that for any other toeplitz representation @xmath68 on a @xmath11-algebra @xmath67 , there is a unique morphism @xmath80 such that @xmath81 and @xmath82 . @xmath83{ddr}[swap]{\pi } & \mathcal{t}(x ) \arrow[dotted]{dd}{\exists\hskip .5pt ! \,\psi } & x \arrow{l}[swap]{\widetilde{\tau } } \arrow[bend left]{ddl}{\tau } \\ & & \\ & b & \end{tikzcd}\ ] ] the @xmath11-algebra @xmath78 is called the _ toeplitz - pimsner algebra _ associated to @xmath22 . for @xmath60 , define @xmath84 and let @xmath85 be the norm closed span of the @xmath86 in @xmath63 . then @xmath85 is an ideal in @xmath63 and given a toeplitz representation @xmath68 of @xmath22 on @xmath67 , there is a unique * -homomorphism @xmath87 such that @xmath88 for every @xmath60 . moreover , for each @xmath49 , @xmath89 , and @xmath66 , @xmath90 if @xmath76 is injective , then @xmath91 is also injective . if @xmath22 is a @xmath11-correspondence over @xmath46 , define an ideal @xmath92 by @xmath93 a toeplitz representation @xmath68 is _ covariant _ if for every @xmath94 , @xmath95 . let @xmath96 denote the universal @xmath11-algebra generated by a covariant toeplitz representation of @xmath22 as with the toeplitz - pimsner algebra @xmath78 . the @xmath11-algebra @xmath96 is called the _ cuntz - pimsner algebra _ associated to @xmath22 . it can be shown that the cuntz - pimnser algebra @xmath96 always exists and the canonical maps @xmath97 and @xmath98 are injective and hence isometric . moreover , @xmath96 is unique up to a canonical isomorphism and is generated as a @xmath11-algebra by @xmath99 and @xmath100 . the same is true for the toeplitz - pimsner algebra @xmath78 . there are also concrete descriptions of @xmath96 and @xmath78 ( see ( * ? ? ? * section 4.6 ) for example ) , but for our purposes , the abstract definition is more helpful . in this section , we recall the necessary material from topological graphs . all the material is taken from @xcite . a _ topological graph _ @xmath0 consists of locally compact second countable spaces @xmath101 and continuous maps @xmath102 such that @xmath103 is a local homeomorphism ; i.e. for every @xmath104 , there is an open neighborhood @xmath105 of @xmath106 such that @xmath107 is a homeomorphism of @xmath108 onto @xmath109 and @xmath110 is an open neighborhood of @xmath111 . it can be shown that a local homeomorphism is always an open map . suppose @xmath6 is a topological graph . given @xmath112 and @xmath113 , define @xmath114 define @xmath115 to be the @xmath11-correspondence over @xmath37 obtained from completing @xmath38 and let @xmath116 . note that the sum in the definition of @xmath117 above is a finite sum . to see this , suppose @xmath118 . if @xmath104 is an accumulation point of @xmath119 , then there is a sequence @xmath120 such that @xmath121 . with @xmath122 for every @xmath29 . choose an open set @xmath105 such that @xmath123 . then there is an @xmath124 such that @xmath125 . now , @xmath126 and hence @xmath107 is not injective . this contradicts the fact that @xmath103 is a local homeomorphism . therefore @xmath127 has no accumulation points . thus if @xmath128 is compact , @xmath129 if finite for every @xmath118 . in particular , if @xmath113 , then @xmath130 and @xmath131 are finite sets . the claim follows . given @xmath113 , it is not immediately obvious that @xmath117 is a continuous function on @xmath1 . this is the content of lemma 1.5 in @xcite . the lemma relies heavily on the fact that @xmath103 is a local homeomorphism . when @xmath1 and @xmath2 are both discrete , then @xmath6 is a graph in the sense of @xcite and @xmath3 is the usual graph @xmath11-algebra . suppose @xmath22 is a locally compact hausdorff space and @xmath23 is a homeomorphism of @xmath22 . define a topological graph @xmath6 by @xmath132 , @xmath133 and @xmath134 . then @xmath135 . in general one can think of @xmath3 as a crossed product by a partially defined , multi - valued , continuous map @xmath8 given by @xmath9 for each @xmath136 . given @xmath118 , we say @xmath137 is _ regular _ if there is a neighborhood @xmath138 of @xmath137 such that @xmath139 is compact and @xmath140 . let @xmath141 denote the collection of regular vertices in @xmath1 . note that @xmath141 is the largest open subset @xmath142 such that @xmath143 restricts to a proper surjection from @xmath144 onto @xmath108 . moreover , for each @xmath145 , the set @xmath146 is compact and non - empty . the elements of @xmath147 are called the _ singular vertices _ of @xmath6 . a vertex @xmath118 is called a _ sink _ if @xmath148 and a _ source _ if @xmath149 . for a topological graph @xmath6 , @xmath150 . in particular , a toepltiz representation @xmath68 of @xmath115 is covariant if and only if @xmath151 for each @xmath152 . a _ path _ if @xmath6 is a list of edges @xmath153 in @xmath2 such that if @xmath154 , @xmath155 . if also @xmath156 , then @xmath34 is called a _ loop_. set @xmath157 and @xmath158 . the integer @xmath29 is called the _ length _ of @xmath34 and is written @xmath159 . let @xmath160 denote the collection of all paths in @xmath6 with length @xmath29 . equip @xmath160 with the product topology and let @xmath161 . then @xmath162 is continuous and @xmath163 is a local homeomorphism . the following is known as the gauge invariance uniqueness theorem . [ thm : giut ] suppose @xmath68 is a covariant representation of @xmath6 on a @xmath11-algebra @xmath46 and suppose @xmath46 has a gauge action @xmath164 such that @xmath165 for every @xmath112 , @xmath166 and @xmath167 for every @xmath168 , @xmath166 . then the induced map @xmath169 is injective if and only if @xmath76 is injective . all of the results in this section are taken from @xcite . a _ directed tree _ is a ( countable , discrete ) graph @xmath6 such that 1 . @xmath6 has no loops , 2 . @xmath143 is injective , and 3 . @xmath6 is connected ; i.e. given @xmath170 , there are vertices @xmath171 such that @xmath172 , @xmath173 , and there are edges @xmath174 such that either @xmath175 and @xmath176 or @xmath177 and @xmath178 . the connectedness condition in the above definition says any two vertices should be connected by an _ undirected _ path . if @xmath6 is a directed tree , then @xmath6 has at most one source . suppose @xmath6 is a directed tree and @xmath179 . given @xmath180 , define @xmath181 by @xmath182 now define a ( possibly unbounded ) operator @xmath183 on @xmath184 by @xmath185 and @xmath186 for @xmath187 . @xmath183 is called a _ weighted shift _ on @xmath6 . we are only interested in the case @xmath183 is bounded . fortunately , there is a simple characterization of the weights @xmath188 which define bounded operators . all the results stated in this section are also true when @xmath183 is an unbounded densely defined operator . if @xmath6 is a directed tree and @xmath179 , then the following are equivalent : 1 . @xmath189 , 2 . @xmath190 , 3 . @xmath191 . in this case , @xmath192 given the graph @xmath193 then @xmath183 is the weighted bilateral shift on @xmath194 given by @xmath195 for every @xmath196 . suppose @xmath6 is the graph below : & & 3 + 1 & 2 & + & & 4 for any @xmath197 , we have @xmath198 let @xmath183 be a bounded weighted shift on a directed tree @xmath6 . then @xmath183 has closed range if and only if @xmath199 let @xmath183 be a bounded weighted shift on a directed tree @xmath6 . then @xmath183 is injective if and only if @xmath200 for every @xmath118 . [ cor : boundedbelow ] let @xmath183 be a bounded weighted shift on a directed tree @xmath6 . then @xmath183 is bounded below if and only if @xmath201 in this case , @xmath202 for every @xmath180 . it turns out that the fredholm theory for the weighted shifts @xmath183 is closely related to the combinatorial structure of the underlying graph . the following theorem gives a simple formula for the index of a fredholm weighted shift . this result will be very important for understanding finite topological graph algebras since it gives an easy way of constructing left - invertible operators which are not right - invertible . let @xmath183 be a bounded weighted shift on a directed tree @xmath6 . . then 1 . @xmath204 , and 2 . @xmath205 . [ cor : surjectiveshift ] suppose @xmath183 is a bounded weighted shift on @xmath6 that is bounded below . then @xmath183 is surjective if and only if @xmath206 is surjective and @xmath207 is injective . our goal in this section is to build a family of representations of a topological graph on directed trees as defined in the previous section . roughly , given an infinite path @xmath34 in @xmath6 , we will define the `` orbit '' of @xmath34 to be a discrete directed tree @xmath35 inside the infinite path space @xmath5 . the elements of @xmath37 and @xmath115 will be represented , respectively , as diagonal operators and weighted shifts on the hilbert space @xmath36 . the construction is a slight modification of the representations built by katsura ( see remark [ remark : katsuraconstruction ] below ) . let @xmath6 be a topological graph and set @xmath208 and given @xmath209 , define @xmath210 . an element @xmath34 of @xmath5 is thought of as an infinite path as shown below @xmath211 in @xmath6 . given a path @xmath212 with @xmath213 , we define @xmath214 by @xmath215 we think of @xmath216 as being the path given by `` shifting forward '' by @xmath217 . there is also a notion of `` backward shift '' given by @xmath218 , @xmath219 . for each @xmath209 and @xmath196 , define @xmath220 set @xmath221 . note that for each @xmath209 and @xmath196 , @xmath222 . the elements of @xmath35 are all the infinite paths obtained by taking shifts of @xmath34 . an element of @xmath223 is thought of as an infinite path of `` length '' @xmath224 . the set @xmath35 can be thought of as a directed tree @xmath225 where @xmath226 , @xmath227 , and @xmath228 . [ remark : katsuraconstruction ] in ( * ? ? ? * section 12 ) katsura used a similar construction to study the primitive ideals in @xmath3 . in particular , given @xmath209 , define @xmath229 as with @xmath35 , we may think of @xmath230 as a ( discrete ) directed graph . when @xmath34 is periodic in the sense that there is a loop @xmath212 and a path @xmath231 with @xmath232 , then @xmath230 will have a loop and hence is distinct from @xmath35 . when @xmath34 is aperiodic , there is no difference between @xmath230 and @xmath35 . if @xmath104 is a loop ; i.e. @xmath233 , then we may form an infinite path @xmath234 . then in @xmath5 , @xmath235 , and hence @xmath235 in @xmath230 . however in @xmath35 , @xmath236 and @xmath34 are viewed as different elements of @xmath35 . in particular , @xmath237 and @xmath238 . similarly @xmath239 and @xmath34 are considered to be different elements in @xmath35 since @xmath240 . let @xmath6 be the graph shown below . @xmath241{}{e } \arrow[bend left]{r}{f } \arrow[bend right]{r}[swap]{g } & w \end{tikzcd}\ ] ] then @xmath242 . the graph @xmath243 is given by & ( f e^ , -1 ) & ( f e^ , 0 ) & ( f e^ , 1 ) & + & ( e^ , -1 ) & ( e^ , 0 ) & ( e^ , 1 ) & + & ( g e^ , -1 ) & ( g e^ , 0 ) & ( g e^ , 1 ) & in particular , @xmath244 . on the other hand , the graph @xmath245 is given by @xmath246 & & f e^\infty \\ e^\infty \arrow[loop left ] { } \arrow{rru } \arrow{rrd } & & \\ & & g e^\infty \end{tikzcd}\ ] ] for @xmath209 , define @xmath247 and @xmath248 . then @xmath249 and hence @xmath250 may be viewed as a @xmath30-graded hilbert space . the @xmath30-grading on @xmath250 , induces a natural gauge action on @xmath251 which we now describe . for each @xmath166 , define @xmath252 by @xmath253 for each @xmath254 . now set @xmath255 . it is easy to verify @xmath256 is a point - norm continuous group homomorphism . we will build a gauge invariant representation @xmath257 for each @xmath209 . fix @xmath209 . define maps @xmath258 and @xmath259 by @xmath260 for every @xmath112 , @xmath168 , and @xmath261 . although our orbit space @xmath35 is slightly different than katusra s orbit space @xmath230 ( see remark [ remark : katsuraconstruction ] ) , the definition of @xmath262 and the proof that this is a covariant toeplitz representation is identical to katusra s proof in ( * ? ? * section 12 ) . for the readers convenience we outline a proof . a routine ( but slightly tedious ) argument shows that @xmath262 is a toeplitz representation . let @xmath263 be the * -homomorphism given by @xmath264 . to show @xmath262 is covariant , we must show @xmath265 whenever @xmath266 . suppose @xmath266 . since @xmath267 is a compact subset of @xmath141 , the set @xmath268 is compact . choose @xmath269 and relatively compact open sets @xmath270 such that @xmath271 , @xmath272 is a homeomorphism , and @xmath273 . choose continuous functions @xmath274 such that @xmath275 , @xmath276 , and @xmath277 for each @xmath278 . for @xmath279 , set @xmath280 and @xmath281 . note that for @xmath104 and @xmath282 , @xmath283 hence @xmath284 . now , @xmath285 hence the toeplitz representation @xmath262 is covariant . let @xmath286 be the associated representation . since @xmath262 is covariant representation , there is an associated representation . if @xmath287 with @xmath288 , with the obvious modifications we can build @xmath286 as before ; in this case , @xmath289 for @xmath290 . for the details see ( * ? ? ? * section 12 ) . let @xmath291 . define @xmath292 then we have a representation of @xmath3 on @xmath293 given by @xmath294 we claim @xmath295 is injective . note that the map @xmath296 is injective since for every @xmath118 , there is an @xmath297 such that @xmath298 . moreover , @xmath299 is gauge invariant for each @xmath209 since for @xmath112 , @xmath300 and for @xmath168 , @xmath301 . hence @xmath295 is injective by the gauge invariant uniqueness theorem ( theorem [ thm : giut ] ) . [ thm : finiteness ] suppose @xmath0 is a compact topological graph with no sinks ; i.e. @xmath1 and @xmath2 are compact and @xmath103 is surjective . if @xmath3 is finite , then @xmath103 is a homeomorphism and @xmath143 is surjective . since @xmath2 is compact , @xmath302 . with the notation from the previous section , define @xmath303 for @xmath297 and define @xmath304 . recall that @xmath305 is a faithful representation of @xmath3 . identifying @xmath3 with the range of @xmath295 , we have @xmath306 and @xmath307 . fix @xmath297 . then @xmath35 can be viewed as a directed tree @xmath308 by setting @xmath226 , @xmath227 , and @xmath228 . then @xmath309 is a weighted shift on @xmath35 with constant weights 1 . for each @xmath261 , @xmath310 since @xmath6 has no sinks . therefore , by corollary [ cor : boundedbelow ] , @xmath309 is bounded below by 1 and hence there is an @xmath311 with @xmath312 and @xmath313 . set @xmath314 . then @xmath315 and @xmath61 is left - invertible . since @xmath3 is finite , we also have @xmath61 is right invertible . therefore , @xmath316 and @xmath309 is surjective for every @xmath297 . if @xmath118 is a source , then @xmath317 is also a source . but this contradicts corollary [ cor : surjectiveshift ] since @xmath309 is surjective . therefore , @xmath143 is surjective . similarly , if @xmath318 with @xmath319 , then choose @xmath297 with @xmath320 . now , @xmath321 and @xmath322 . since @xmath309 is surjective , @xmath323 is injective by corollary [ cor : surjectiveshift ] . hence @xmath324 and @xmath325 . thus @xmath103 is injective . the converse of theorem [ thm : finiteness ] fails . take @xmath326 with the usual topology and define @xmath327 and @xmath328 for each @xmath329 . then @xmath143 and @xmath103 are both homeomorphisms but @xmath3 is infinite as can be seen from theorem [ thm : pimsner ] or theorem [ thm : topgraphsafembedding ] below . [ lem : projectivelimit ] suppose @xmath6 is a compact topological graph with no sinks . if @xmath103 is injective and @xmath143 is surjective , then the maps @xmath331 given by @xmath332 induce a homeomorphism @xmath333 . it is clear that @xmath334 for every @xmath29 and hence @xmath335 is well - defined . since @xmath336 is surjective for every @xmath29 , @xmath335 is also surjective . since @xmath5 is compact and hausdorff , it suffices to show @xmath335 is injective . but if @xmath337 for every @xmath338 , then @xmath339 for every @xmath29 ; i.e. @xmath340 for every @xmath29 . now , @xmath341 for every @xmath29 and @xmath338 . [ thm : infinitepathcrossedproduct ] suppose @xmath6 is a compact topological graph with no sinks . if @xmath103 is injective and @xmath143 is surjective ( e.g. if @xmath3 is finite ) , then @xmath218 is a homeomorphism and there is a natural isomorphism @xmath342 . since @xmath143 is injective and @xmath103 is surjective , @xmath343 is a homeomorphism . let @xmath344 be a unitary in @xmath345 such that @xmath346 for each @xmath347 . define @xmath348 and @xmath349 by @xmath350 and @xmath351 for @xmath352 and @xmath353 , where @xmath354 is the evaluation map @xmath355 . a tedious but straight forward computation show @xmath68 is a covariant representation and hence induces a morphism @xmath356 . by the gauge invariance uniqueness theorem , @xmath295 is injective . it remains to show surjectivity . note that @xmath357 . hence , with the notation of lemma [ lem : projectivelimit ] , it suffices to show @xmath358 is in the range of @xmath295 for every @xmath352 and @xmath124 . but @xmath359 hence @xmath295 is surjective . using theorem [ thm : infinitepathcrossedproduct ] together with theorem [ thm : pimsner ] ) , we will be able to characterize the af - embeddability of @xmath3 for compact graphs @xmath6 with no sinks . first we need a simple lemma regarding pseduoperiodic points ( see condition ( 5 ) of theorem [ thm : pimsner ] for the definition ) . [ lem : pseudoperiodicpoints ] suppose @xmath22 is a compact metric space and @xmath360 is continuous and surjective . define @xmath361 and let @xmath362 denote the homeomorphism of @xmath363 induced by @xmath23 . then every point of @xmath364 is pseudoperiodic if and only if every point of @xmath365 is pseudoperiodic . suppose first every point of @xmath365 is pseduoperiodic . fix @xmath25 and @xmath373 . since @xmath23 is surjective , there is an @xmath374 with @xmath375 . choose @xmath376 such that @xmath377 for @xmath28 . define @xmath378 for @xmath379 . then @xmath380 hence @xmath381 is pseudoperiodic . now suppose every point of @xmath364 is pseudoperiodic . fix @xmath374 and @xmath373 . by the uniform continuity of @xmath23 , we may find a sequence such that @xmath382 whenever @xmath371 with @xmath383 . choose @xmath384 such that @xmath385 . since @xmath386 is pseduoperiodic point for @xmath364 , there are points @xmath387 such that @xmath388 for each @xmath28 . since @xmath23 is surjective , there are points @xmath389 such that @xmath390 . now it is to easy see @xmath391 for each @xmath28 and hence @xmath392 is pseudoperiodic . suppose @xmath6 is a topological graph and @xmath373 . let @xmath369 be a metric on @xmath1 compatible with the topology . @xmath393-pseudopath _ in @xmath6 is a finite sequence @xmath394 in @xmath2 such that for each @xmath395 , @xmath396 . we write @xmath157 and @xmath158 . an @xmath393-pseduopath @xmath34 is called an _ @xmath393-pseduoloop based at @xmath397 _ if @xmath398 . 1 . @xmath3 is af - embeddable ; 2 . @xmath3 is quasidiagonal ; 3 . @xmath3 is stably finite ; 4 . @xmath3 is finite ; 5 . @xmath103 is injective and for every @xmath118 and @xmath373 , there is an @xmath393-pseudoloop in @xmath6 based at @xmath137 . the implications @xmath399 are true for arbitrary ( * ? ? ? * propositions 7.1.9 , 7.1.10 , and 7.1.15 ) ) . if @xmath3 is finite , then by theorem [ thm : infinitepathcrossedproduct ] @xmath400 . by theorem [ thm : pimsner ] , every point in @xmath5 is pseudoperiodic for @xmath343 and hence every point of @xmath1 is pseduoperiodic for @xmath401 by lemmas [ lem : projectivelimit ] and [ lem : pseudoperiodicpoints ] . fix a metric @xmath369 on @xmath1 . given @xmath118 and @xmath373 , choose @xmath402 such that @xmath403 for every @xmath28 . set @xmath404 . then @xmath405 is a pseduoloop in @xmath6 based at @xmath137 . now suppose ( 5 ) holds . condition ( 5 ) implies @xmath143 has dense range and since @xmath2 is compact , @xmath143 is surjective . now , @xmath400 by theorem [ thm : infinitepathcrossedproduct ] . given @xmath118 and @xmath373 , choose an @xmath393-pseduoloop @xmath406 based at @xmath137 . set @xmath407 and note that @xmath408 . hence by theorem [ thm : pimsner ] and lemma [ lem : pseudoperiodicpoints ] , @xmath3 is af - embeddable . in theorem [ thm : topgraphsafembedding ] , the assumptions that @xmath6 is compact and has no sinks are necessary even when @xmath6 is discrete . consider the graphs below @xmath409 { } & \bullet & & & \bullet \arrow{r } \arrow[loop ] { } & \bullet \arrow{r } & \bullet \arrow{r } & \cdots \end{tikzcd}\ ] ] in both cases , the @xmath11-algebra defined by the graph is finite but @xmath103 is not injective . 99 b. blackadar , e. kirchberg , generalized inductive limits of finite dimensional @xmath11-algebras , _ math . ann . _ * 307*(1997 ) , 343380 . n. p. brown , af embeddability of crossed products of af algebras by the integers , _ j. funct . _ * 160*(1998 ) , 150175 . n. p. brown , n. ozawa , _ @xmath11-algebras and finite - dimensional approximations _ , graduate studies in mathematics , * 88*. amer . math . soc . , providence , ri , 2008 . , the full c-algebra of the free group on two generators , _ pacific journal of mathematics _ * 87*(1980 ) , 4148 . d. hadwin , strongly quasidiagonal @xmath11-algebras , _ j. operator theory _ * 18*(1987 ) , 3-18 . with an appendix by jonathan rosenberg . z. jaboski , i. jung , j. stochel , weighted shifts on directed trees , _ mem . _ , * 216 * ( 2012 ) , no . t. katsura , a construction of @xmath11-algebras from @xmath11-correspondences , _ advances in quantum dynamics , _ 173-182 , contemp . , * 335 * , amer . math . soc . , providence , ri , 2003 . , a class of @xmath11-algebras generalizing both graph algebra and homeomorphism @xmath11-algebras i , fundamental results , _ trans . * 356 * ( 2004 ) , 4287-4322 . , a class of @xmath11-algebras generalizing both graph algebra and homeomorphism @xmath11-algebras ii , examples , _ int . j. math . _ * 17 * ( 2006 ) , 791-833 . , a class of @xmath11-algebras generalizing both graph algebra and homeomorphism @xmath11-algebras iii , ideal structures , _ ergodic theory dynam . systems _ * 26 * ( 2006 ) , 1805-1854 . , a class of @xmath11-algebras generalizing both graph algebra and homeomorphism @xmath11-algebras iv , pure infinitenss , _ j. funct . anal . _ * 254 * ( 2008 ) , 1161-1187 . m. pimsner , embedding some transformation group @xmath11-algebras into af - algebras , _ ergodic theory dynam . systems _ * 3 * ( 1983 ) , 613626 . , a class of @xmath11-algebras generalizing both cuntz - krieger algebras and crossed products by @xmath30 . _ free probability theory , _ 189-212 , fields inst . , * 12 * , amer . soc . , providence , ri , 1997 . i. raeburn , _ graph algebras _ , cbms regional conference series in mathematics * 103 * published for the conference board of the mathematical sciences , washington d.c . by the ams , providence , ri ( 2005 ) . c. schafhauser , af - embeddings of graph @xmath11-algebras , _ j. operator theory _ , to appear ( arxiv:1406.7757 [ math.oa ] ) . s. wassermann , on tensor products of certain group @xmath11-algebras , _ j. functional analysis _ * 23*(1976 ) , 239254 .

let @xmath0 be a topological graph with no sinks such that @xmath1 and @xmath2 are compact . we show that when @xmath3 is finite , there is a natural isomorphism @xmath4 , where @xmath5 is the infinite path space of @xmath6 and the action is given by the backwards shift on @xmath5 . combining this with a result of pimsner , we show the properties of being af - embeddable , quasidiagonal , stably finite , and finite are equivalent for @xmath3 and can be characterized by a natural `` combinatorial '' condition on @xmath6 .

it was suggested long ago @xcite that solar flares , giant explosions on the sun , may cause acoustic waves traveling through the sun s interior , similar to the seismic waves on the earth . because the sound speed increases with depth the waves are reflected in the deep layers of the sun and appear back on the surface , forming expanding rings of the surface displacement . theoretical modeling @xcite predicted that the speed of the expanding seismic waves increases with distance because the distant waves propagate into the deeper interior where the sound speed is higher . first observations of the seismic waves caused by the x2.6 flare of july 9 , 1996 @xcite , proved these predictions . these observations also showed that the source of the seismic response was a strong shock - like compression wave propagating downwards in the photosphere . this wave was observed immediately after the hard x - ray impulse which produced by high - energy electrons hitting the low atmosphere . this led to a suggestion that the seismic response can be explained in terms of so - called `` thick - target '' models . in these models , a beam of high - energy is related to heating of the solar chromosphere , resulting in evaporation of the upper chromosphere and a strong compression of the lower chromosphere ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this high - pressure compression produces a downward propagating shock wave @xcite that hits the solar surface and causes sunquakes . this shock observed in soho / mdi dopplergrams as a localized large - amplitude velocity impulse of about 1 km / s or stronger represents the initial hydrodynamic impact resulting in the seismic response . in addition , @xcite found that the seismic wave was anisotropic , with a significant quadrupole component . the following observations of solar flares made by the michelson doppler imager ( mdi ) instrument on the nasa - esa mission soho did not show noticeable sunquake signals even for strong x - class flares . this search was carried out by calculating an `` egression '' power for high - frequency acoustic waves during the flares @xcite . it became clear that sunquakes are a rather rare phenomenon on the sun , which occurs only under some special conditions . surprisingly , seven years later several flares did show strong `` egression '' signals indicating new potential sunquakes @xcite ( for a list see http://www.maths.monash.edu.au/~adonea ) . it is interesting to note that the flare of july 9 , 1996 , was the last strong of the previous solar activity cycle , and the new strong sunquake events are observed in the declining phase of the current activity cycle after the maximum of 2000 - 2001 . it appears that during the rising phase of the solar cycle and during its peak the solar flares are rather a `` superficial '' coronal phenomenon not affecting much the solar surface and interior . this could happen if the topology of magnetic field of solar active regions which produce flares changes in such a way that the magnetic energy is released at lower altitudes in the declining phase of the solar cycle than in the rising and maximum phase . here , i present analysis of new observations of the seismic response to solar flares from the soho and rhessi space observatories , which show that the sunquakes were indeed caused by the hydrodynamic impacts of high - energy electrons accelerated in solar flares confirming the initial result of @xcite , and determine basic properties of the flare - generated seismic waves by investigating their time - distance characteristics . the mdi instrument on soho measures motions of the solar surface through the doppler shift of a photospheric absorption line ni i 6768 a. the measurements provide images of the line - of - sight velocity of the sun s surface every minute with the spatial resolution 2 arcsec per pixel . examples of the mdi dopplergrams obtained during the sunquake events are shown in the two right columns in figure 1 ( grey semitransparent images overlaying color images of sunspots ) . there are several types of motions on the solar surface , which contribute to the mdi signal . the largest contributions of about 500 m / s come from the solar convection and stochastic 5-min oscillations excited by convection ( they form the noisy granular - like pattern in fig.1 ) . the amplitude of the flare - generated seismic waves ( ring - like features identified in the middle column of fig.1 ) rarely exceeds 100 m / s . thus , because of the strong stochastic motions in the background , these waves are difficult to detect . however , these waves form an almost circular - shape expanding ring , velocity of which is determined by the sound speed inside the sun and can be calculated from solar models . this property is used to extract the seismic response signal from the noisy data . because the waves are close to circular the dopplergrams can be averaged over a range of the azimuthal angle around central points of the initial flare impact . these centers are identified during the flare impulsive phase as strong localized rapidly varying velocity perturbations of about 1 km / s ( light and dark features in left column of fig.1 ) . the azimuthally averaged dopplergrams are plotted as time - distance diagrams ( right columns of fig.1 ; the averaging angular range in the polar coordinates in indicated at the top ) , in which the seismic wave forms a continuous ridge corresponding the time - distance relation for acoustic propagating through the solar interior . the slope of this ridge is decreasing with distance , meaning that the waves accelerate . this happens because the seismic waves observed at longer distances travel through the deeper interior of the sun where the sound speed is higher because of higher plasma temperature . typically , the ring speed changes from 10 km / s to 100 km / s . in the `` egression power '' method @xcite the wave signal is integrated along the time - distance ridge , thus giving the total average of the seismic signal power for specific central points . the egression power can be calculated for the whole dopplergram revealing places of potential sunquakes . this method is useful as a search tool , but it does not provide characteristics of the seismic waves . because of the high solar noise , the seismic waves are not easily seen on individual dopplergrams . they are much easier recognized in dopplergram movies as expanding circular wave fronts . the typical oscillation frequency of the flare waves is higher than the mean frequency of the background fluctuations ( 4 - 5 mhz vs. 3 mhz ) . therefore , frequency filtering centered at 5 or 6 mhz helps to increase the signal - to - noise ratio . in most cases , a frequency filter centered at 6 mhz with the width of 2 mhz is used , and , in addition , the difference filter for consecutive images is applied . localized doppler perturbations during the flare impulsive phase , similar to shown in the fig.1 ( left panels ) and presumably associated with precipitation of high - energy particles are commonly observed . therefore , that one might expect that seismic waves are excited in most flares that affect the photosphere . however , in most cases the flare hydrodynamic impact in the photosphere and , thus , the seismic response appear to be weak . the main purpose of this paper is to investigate properties of strong seismic waves when their wave fronts can be observed explicitly in dopplergrams and time - distance diagrams . a list of 6 flares with such strong seismic waves , observed from soho / mdi between 1996 and 2005 in given in table 1 . figure 1 presents results for three strongest events so far , observed on 10/28/2003 , 07/16/2004 and 01/15/2005 . the first flare of october 28 , 2003 , was one of the strongest ever observed , having the soft x - ray class x17 . it is interesting that the two other flares had much weaker soft x - ray class , but produced higher amplitude seismic waves than this one . the analysis of these observations reveals new interesting features of the seismic response : 1 ) flares can produce multiple sunquakes almost simultaneously originating from separate positions ( as also found by @xcite for the 10/28/2003 flare ) ; 2 ) the seismic waves are highly anisotropic , their amplitude can vary significantly with angle ; 3 ) the strongest amplitude is commonly observed in the same direction as the direction of motion of flare ribbons ; 4 ) the wave fronts in most cases have elliptical shape , originating from elongated in one direction initial impulse ; 5 ) the centers of the expanding waves coincide very well with the places of hydrodynamic impacts in mdi dopplergrams ( confirming the initial observation of * ? ? ? * ) , however , not all impact sources produce strong seismic waves ; 6 ) the seismic waves are usually first observed 15 - 20 min after the initial impact , and reach the highest amplitude 20 - 30 min after the flare ; 7 ) the seismic waves can travel to large distances exceeding 120 mm , but , in some cases , decay more rapidly ; 8) the fronts of acoustic seismic waves propagate through sunspots without much distortion and significant decay , thus showing no evidence for conversion into other types of mhd waves ; 9 ) the time - distance diagrams for the waves propagating in sunspot regions show only small deviations of the order of 2 - 3 min from the wave travel times of the quiet sun ; these variations are consistent with the travel time measurements obtained by time - distance helioseismology using the cross - covariance function for random waves @xcite . for two of these flares , x17 of october 28 , 2003 , and january 15 , 2005 , x - ray data are available for analysis . the rhessi image reconstruction software was used to obtain locations of the x - ray sources in these flares and compare with the mdi doppler measurements of the hydrodynamic impulses and seismic responses . figure 2 shows a white - light image of the flaring active region ( noaa 10696 ) and the superimposed images of the doppler signal at the impulsive phase , 11:06 ut , ( blue and yellow spots show up and down photospheric motions with variations in the mdi signal stronger than 1 km / s ) , positions of three wave fronts at 11:37 ut , and also locations of the hard x - ray ( 50 - 100 kev ) sources ( yellow circles ) at 11:06 ut , and 2.2 mev gamma - ray sources ( green circles ) found by @xcite ( averaged for the whole flare duration ) . evidently , the x - ray and gamma - ray source are very close to the positions of the seismic sources , but there was no gamma - ray emission near source 3 . also , the gamma emission was not detected for other seismic events . this leads to the conclusion that the origin of the seismic response is the hydrodynamic impact ( shock ) , which is observed in the doppler signals at 11:06 ut and shows the best correspondence to the central positions of the wave fronts , contrary to the suggestion of @xcite that photospheric heating by high - energy protons is likely to be a major factor . this was verified by calculating the time - distance diagrams for various central positions and various angular sectors . when the central position of a time - distance diagram deviates from the seismic source position this deviation is immediately seen in the diagram as an off - set of the time - distance ridge . this approach provides effective source positions for complicated and distributed doppler signals . the flare of january 15 , 2005 , of moderate x - ray class , x1.2 , but it produced the strongest seismic wave observed so far by soho ( fig.1 , bottom row ) . its amplitude exceeded 100 m / s . this wave had an elliptical shape with the major axis in the se - nw direction . the elliptical shape corresponds very well to the linear shape of the seismic source extended in this case along the magnetic neutral line . this is illustrated in figure 3 . the left panel shows the grey - scale map the dopplergram difference at 0:40 ut , in which the long white feature near the center corresponds to strong downflows at the seismic source , and an image of the hard x - ray source ( color spot ) . the right panel shows the corresponding mdi magnetogram and an image of the soft x - ray emission ( in gray ) and contour line of the hard x - ray source . evidently , the region of the hydrodynamic impact was located just below the hard x - ray source , which was at a footpoint of the soft x - ray loop . figure 4 illustrates this sequence of events for the january 15 , 2005 , flare , from top to bottom . the high - energy electrons accelerated in the flare ( presumably , high in the corona ) produced hard x - ray impulse in the lower atmosphere and generated downward propagating shocks which hit the photosphere and generated the seismic waves . this picture corresponds very well to the standard thick target model of solar flares @xcite and the models of the hydrodynamic response ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the soft x - ray image indicates this flare was rather compact . one may suggest that the seismic response can be particularly strong in the case of a compact solar flare , but this needs to be confirmed by further observations . the new observations from soho and rhessi provide unique information about the interaction of the high - energy particles accelerated in solar flares with solar plasma and the dynamics of the solar atmosphere during solar flares . these data also provide unique information about the interaction of acoustic mhd waves with sunspots , showing explicitly propagation of wave fronts through sunspot regions . this opens opportunity for developing new methods of helioseismology analysis of flaring active regions , similar to the methods of earth - quake seismology .

the solar seismic waves excited by solar flares ( `` sunquakes '' ) are observed as circular expanding waves on the sun s surface . the first sunquake was observed for a flare of july 9 , 1996 , from the solar and heliospheric observatory ( soho ) space mission . however , when the new solar cycle started in 1997 , the observations of solar flares from soho did not show the seismic waves , similar to the 1996 event , even for large x - class flares during the solar maximum in 2000 - 2002 . the first evidence of the seismic flare signal in this solar cycle was obtained for the 2003 `` halloween '' events , through acoustic `` egression power '' by donea and lindsey . after these several other strong sunquakes have been observed . here , i present a detailed analysis of the basic properties of the helioseismic waves generated by three solar flares in 2003 - 2005 . for two of these flares , x17 flare of october 28 , 2003 , and x1.2 flare of january 15 , 2005 , the helioseismology observations are compared with simultaneous observations of flare x - ray fluxes measured from the rhessi satellite . these observations show a close association between the flare seismic waves and the hard x - ray source , indicating that high - energy electrons accelerated during the flare impulsive phase produced strong compression waves in the photosphere , causing the sunquake . the results also reveal new physical properties such as strong anisotropy of the seismic waves , the amplitude of which varies significantly with the direction of propagation . the waves travel through surrounding sunspot regions to large distances , up to 120 mm , without significant decay . these observations open new perspectives for helioseismic diagnostics of flaring active regions on the sun and for understanding the mechanisms of the energy release and transport in solar flares .

in @xcite , david gay and rob kirby introduced a beautiful decomposition of an arbitrary smooth , oriented closed @xmath0manifold , called _ trisection _ , into three handlebodies glued along their boundaries as follows . each handlebody is a boundary connected sum of copies of @xmath5 and has boundary a connected sum of copies of @xmath6 the triple intersection of the handlebodies is a closed orientable surface @xmath7 which divides each of their boundaries into two @xmath8dimensional handlebodies ( and hence is a heegaard surface ) . these @xmath8dimensional handlebodies are precisely the intersections of pairs of the 4dimensional handlebodies . in dimensions @xmath9 there is a bijective correspondence between isotopy classes of smooth and piecewise linear structures @xcite , but this breaks down in higher dimensions . this paper generalises gay and kirby s concept of a trisection to higher dimensions in the piecewise linear category , and hence all manifolds , maps and triangulations are assumed to be piecewise linear unless stated otherwise . our definition and results apply to any compact smooth manifold by passing to its unique piecewise linear structure @xcite . the definition of a multisection , which generalises both that of a heegaard splitting of a 3manifold and that of a trisection of a 4manifold , focuses on properties of spines . let @xmath10 be a compact manifold with non - empty boundary . the subpolyhedron @xmath11 is a _ spine _ of @xmath10 if @xmath12 and @xmath10 pl collapses onto @xmath13 [ def : multisection ] let @xmath14 be a closed , connected , piecewise linear @xmath1manifold . a _ multisection _ of @xmath14 is a collection of @xmath2 piecewise linear submanifolds @xmath15 where @xmath16 and @xmath4 or @xmath17 subject to the following four conditions : 1 . each @xmath18 has a single @xmath19handle and a finite number , @xmath20 , of @xmath21handles , and is homeomorphic to a standard piecewise linear @xmath1dimensional 1handlebody of genus @xmath22 2 . the handlebodies @xmath18 have pairwise disjoint interior , and @xmath23 3 . the intersection @xmath24 of any proper subcollection of the handlebodies is a compact , connected submanifold with boundary and of dimension @xmath25 moreover , it has a spine of dimension @xmath26 except if @xmath4 and @xmath27 then there is a spine of dimension @xmath28 4 . the intersection @xmath29 of all handlebodies is a closed , connected submanifold of @xmath30 of dimension @xmath31 and called the _ central submanifold_. it follows from our definitions that the first condition in the above definition is equivalent to * each @xmath18 has spine a graph with euler characteristic @xmath32 in the remainder of this introduction , we discuss some properties of multisections as well as directions for further research . it is our hope that this new structure and its derived invariants will help gain insights into new infinite families of piecewise linear manifolds , if not the realm of all such manifolds . * example . * ( the tropical picture of complex projective space . ) consider the map @xmath33 defined by @xmath34 \;\mapsto\ ; \frac{1}{\sum|z_k|}\;(\;|z_0|\;,\ ; \ldots\;,\ ; |z_n|\;).\ ] ] the _ dual spine _ @xmath35 in @xmath36 is the subcomplex of the first barycentric subdivision of @xmath36 spanned by the 0skeleton of the first barycentric subdivision minus the 0skeleton of @xmath37 this is shown for @xmath38 in figure[fig : pi2 ] and @xmath39 in figure[fig : pi3 ] . decomposing along @xmath35 gives @xmath36 a natural _ cubical structure _ with @xmath40 @xmath1cubes , and the lower - dimensional cubes that we will focus on are the intersections of non - empty collections of these top - dimensional cubes . each @xmath1cube pulls back to a @xmath41ball in @xmath42 and the collection of these balls is a multisection . for example , if @xmath43 the 2cubes pull back to 4balls , each 1cube pulls back to @xmath44 and the 0cube pulls back to @xmath45 as shown in figure[fig : cp2 ] . * existence ( [ sec : existence ] ) . * a multisection of a 1manifold is just the 1manifold . the study of multisections in dimension 2 is the study of separating , simple , closed curves . a multisection of a 3manifold is a heegaard splitting ( [ sec : construction dim 3 ] ) . a trisection in the sense of gay and kirby @xcite is a multisection of an orientable 4manifold with the additional property that the handlebodies @xmath46 have the same genus . after showing that every multisection of an orientable 4manifold can be modified to a trisection in the sense of @xcite ( [ sec : gkvsrt ] ) , we will start to use the term trisection to apply to all multisections in dimension four , so that we can talk about _ bisections _ ( @xmath47 ) , _ trisections _ ( @xmath48 ) , _ quadrisections _ ( @xmath49 ) , etc.without further qualification . every closed piecewise linear manifold has a multisection . * sketch of proof . * suppose @xmath14 is a closed , connected , piecewise linear manifold of dimension @xmath50 our strategy is to construct a piecewise linear map @xmath51 where @xmath52 is a @xmath53simplex for @xmath53 satisfying @xmath4 or @xmath17 and to obtain the multisection as the pull back of the cubical structure of @xmath52 to @xmath54 our map @xmath55 will have the property that each vertex of @xmath52 pulls back to a connected graph , and each top - dimensional cube pulls back to a regular neighbourhood of this graph , a 1handlebody . we use triangulations to define @xmath56 since a piecewise linear manifold admits a piecewise linear triangulation @xmath57 ( where the link of each simplex in the simplicial complex @xmath58 is equivalent to a standard piecewise linear sphere ) we can and will assume that such a triangulation of @xmath14 is fixed . since @xmath14 is closed , there is a finite number of simplices in the triangulation , and @xmath55 is uniquely determined by a partition of the vertices of the triangulation into @xmath2 sets , and a bijection between the sets in this partition and the vertices of @xmath59 we call such a map @xmath60 a _ partition map_. to ensure that the cubical structure of @xmath52 pulls back to submanifolds with the required properties , we determine suitable combinatorial properties on the triangulation . in the odd - dimensional case , we show that the first barycentric subdivision of any triangulation has a suitable partition ( see [ sec : construction dim 3 ] and [ sec : construction dim odd ] ) . moreover the @xmath61dimensional spine of the intersection of @xmath61 handlebodies meets each top - dimensional simplex in @xmath14 in exactly one @xmath61cube . in even dimensions , we obtain an analogous result after performing bistellar moves on this subdivision ( see [ sec : construction dim 4 ] and [ sec : construction dim even ] ) . we say that a triangulation _ supports a multisection _ if there is a partition of the vertices defining a partition map @xmath62 with the property that the pull back of the cubical structure is a multisection . special properties of triangulations may imply special properties of the supported multisections and vice versa . for instance , special properties of a heegaard splitting of a 3manifold are shown in @xcite to imply special properties of the dual triangulation . the cornerstone of the modern development of heegaard splittings is the work of casson and gordon @xcite , and it is a tantalising problem to generalise this to higher dimensions . we next discuss a number of general structure results . * non - positively curved cubings from multisections ( [ sec : cat(0 ) ] ) . * the partition map @xmath63 can be used to pull back the cubical structure of the target simplex . this gives a natural cell decomposition of the submanifolds in a multisection , with cells of very simple combinatorial types . in the case of the closed , central submanifold @xmath64 this is a cubing . we show that @xmath55 can be chosen such that the cubing of @xmath65 satisfies the gromov link conditions @xcite , and hence is non - positively curved : [ thm : cat(0 ) ] every piecewise linear manifold has a triangulation supporting a multisection , such that the central submanifold has a non - positively curved cubing . since a @xmath66manifold has a central submanifold of dimension @xmath67 this result produces manifolds with non - positively curved cubings in each dimension . we also remark that our construction yields cubings with precisely one top - dimensional cube in the central submanifold for each top - dimensional simplex in the triangulation of the manifold . what conditions does a non - positively curved cubed @xmath53manifold need to satisfy so that it is pl homeomorphic to the central submanifold in a multisection of a @xmath66manifold or a @xmath68manifold * structure of fundamental group ( [ sec : fundamental group ] ) . * a multisection of a manifold gives a decomposition of its fundamental group as a _ generalised _ graph of groups ( which is defined exactly as a graph of groups with the only modification that the homomorphisms from edge groups to vertex groups need not be monomorphisms ; cf.@xcite ) . in particular , since every finitely presented group is the fundamental group of a closed 4manifold , we obtain the following decomposition theorem , which may be of independent interest . [ pro : structure of fund gp ] every finitely presented group has a generalised graph of groups decomposition as shown in figure [ fig : structure of groups_intro ] , where the vertex groups @xmath69 are free of rank @xmath70 @xmath71 is the fundamental group of a closed orientable surface of genus @xmath72 all edge groups are naturally isomorphic with @xmath71 and the oriented edges represent epimorphisms . \(m ) [ matrix of math nodes , row sep=1em , column sep=1em ] & _ 0 & + & _ g & + _ 1&&_2 & + ; ( m-2 - 2 ) edge ( m-1 - 2 ) edge ( m-3 - 1 ) edge ( m-3 - 3 ) ; in higher dimensions , the role of the surface group is played by the fundamental group of a non - positively curved cube complex . * invariants . * the heegaard genus of a 3manifold has natural generalisations to multisections . for instance , each of @xmath73 and @xmath74 have trisections with three 4balls , but they are distinguished by the genera of the pairwise intersections . the _ genus of a trisection _ of a 4manifold , say @xmath75 can be defined as the minimum ( with respect to the lexicographic ordering ) of all 7tuples of the form @xmath76 the _ multisection genus _ of @xmath14 is then the minimum genus of any trisection of @xmath54 in higher dimensions , genera need to be replaced by other invariants . these invariants are also related to the associated generalised graph of groups decomposition of the fundamental group . are there interesting families of 4manifolds for which there exists an algorithm to compute a multisection of minimal genus for each member of the family ? * uniqueness . * there is a natural stabilisation procedure of multisections . in dimension 3 , this increases the genus of both 3dimensional handlebodies , whilst in higher dimensions , this increases the genus of just one of the top - dimensional handlebodies . the reidemeister - singer theorem @xcite states that any two heegaard splittings of a 3manifold have a common stabilisation . using in an essential way the uniqueness up to isotopy of genus @xmath77 heegaard splittings of @xmath78 due to waldhausen @xcite , gay and kirby @xcite show that any two trisections of a 4manifold have a common stabilisation up to isotopy . combined with [ sec : gkvsrt ] , this implies that any two multisections of a 4manifold also have a common stabilisation up to isotopy . we do not give an independent proof of this fact in this paper . under what conditions is there a common stabilisation for two given multisections of a manifold of dimension at least five ? our existence proof constructs multisections dual to triangulations . conversely , up to possibly stabilising the multisection , one can built triangulations dual to multisections . we expect that two multisections have a common stabilisation if and only if these dual triangulations are pl equivalent . * recursive structure and generalisations ( [ sec : recursive ] ) . * a multisection of @xmath14 induces a stratification of the boundary of each handlebody into lower dimensional manifolds . we will refer to this as the _ recursive structure _ of a multisection . for example , for a trisection ( @xmath79 ) the boundaries of each handlebody are divided into two submanifolds . for a quadrisection ( @xmath80 ) the boundaries of the handlebodies are divided into three submanifolds with connected pairwise intersections . these decompositions satisfy part of the definition of a multisection , namely the submanifolds have spines of small dimensions , but the top dimensional submanifolds are not necessarily handlebodies . from the viewpoint of the complexity theory of @xcite , such generalised multisections may be a fruitful approach to the study of classes of examples . for instance , a decomposition of a 4manifold into 4dimensional 1 or 2handlebodies is a decomposition into 4manifolds of complexity 0 . * constructions and examples ( [ sec : examples ] ) . * an extended set of examples of trisections of 4manifolds can be found in @xcite . the recent work of gay @xcite , meier , schirmer and zupan @xcite gives some applications and constructions arising from trisections of 4manifolds and relates them to other structures . in this section , we outline some further constructions , focussing on arbitrary dimensions . the above existence result shows that first barycentric subdivision leads to triangulations supporting multisections . however , more efficient triangulations can be identified using the symmetry representations of @xcite ( see [ sec : constructing with symmetric representations ] ) . we give a number of applications of this approach , including _ generalised multisections _ and _ twisted multisections_. we also discuss _ connected sums _ , a _ dirichlet construction _ , _ products _ and the case of _ manifolds with non - empty boundary . _ * acknowledgements * the authors are partially supported under the australian research council s discovery funding scheme ( project number dp130103694 ) . the authors would like to thank david gay for suggesting to the first author that there might be an approach to trisections via triangulations . the outline of the existence proof is given in the introduction . we give a motivation for both the definition of a multisection and the strategy of the existence proof in dimensions three ( [ sec : construction dim 3 ] ) and four ( [ sec : construction dim 4 ] ) , followed by the general arguments for arbitrary odd ( [ sec : construction dim odd ] ) and even ( [ sec : construction dim even ] ) dimensions . we also clarify the relationship between multisections of 4manifolds and the trisections of gay and kirby ( [ sec : gkvsrt ] ) . all manifolds , maps and triangulations are assumed to be piecewise linear ( pl ) unless stated otherwise ( in which case we will emphasise this by saying topological manifold , ) . our main reference on pl topology is rourke and sanderson @xcite . a primer can be found in thurston @xcite , and a collection of basic definitions and tools in martelli @xcite . we recall the classical existence proof of heegaard splittings ( see , for instance , @xcite ) , which motivates our definition in higher dimensions , and provides a model for the existence proofs . suppose that @xmath14 is a triangulated , closed , connected @xmath8manifold , and there is a partition @xmath81 of the set of all vertices in the triangulation , such that 1 . for each set @xmath82 every tetrahedron has a pair of vertices in the set ; and 2 . the union of all edges with both ends in @xmath83 is a connected graph @xmath84 in @xmath54 we can form regular neighbourhoods of each of these graphs @xmath85 which are handlebodies @xmath86 @xmath87 respectively , such that the handlebodies meet along their common boundary @xmath7 which is a normal surface consisting entirely of quadrilateral disks , one in each tetrahedron , separating the vertices in @xmath88 @xmath89 ( see figure [ fig:2-d - partition ] ) . hence @xmath65 is a heegaard surface in @xmath54 a triangulation with the desired properties is obtained as follows . suppose @xmath57 is a triangulation of @xmath90 and take the first barycentric subdivision @xmath91 of @xmath92 let @xmath93 be the set of all vertices of @xmath58 and barycentres of edges of @xmath94 and let @xmath89 be the set of all barycentres of the triangles and the tetrahedra of @xmath92 then @xmath81 is a partition of the vertices of @xmath91 satisfying @xmath95 and @xmath96 moreover , the vertices of the cubulated surface @xmath65 have degrees @xmath0 or @xmath97 and hence @xmath65 is a non - positively curved cube complex . examples of triangulations of manifolds that satisfy @xmath95 and @xmath98 but are not barycentric subdivisions , are the standard 2vertex triangulations of lens spaces . see [ sec : constructing with symmetric representations ] for a strategy to identify triangulations dual to multisections . the partition @xmath81 defines a piecewise linear map @xmath99 $ ] by @xmath100 and @xmath101 . this is often called a _ height function _ and we refer to it as a _ partition map_. the pre - image @xmath102 is a heegaard surface @xmath65 for @xmath14 as described above . the inverse image of any point in the interior of @xmath103 $ ] is a normal surface isotopic to @xmath65 . the intersection of this inverse image with any tetrahedron of @xmath104 is a quadrilateral disk ( @xmath105cube ) . the inverse image of either endpoint @xmath19 or @xmath21 is a graph and its intersection with any tetrahedron is an edge ( @xmath21-cube ) . the division of the closed interval ( @xmath21simplex ) into two half intervals is the dual decomposition into @xmath21cubes . an analogous decomposition is exactly what we will use in higher dimensions . let @xmath14 be a closed , connected @xmath0manifold with piecewise linear triangulation @xmath106 we assume that there is a partition @xmath107 of the set of all vertices of @xmath58 with the following properties : 1 . every 4simplex meets each of the sets @xmath93 and @xmath89 in two vertices and @xmath108 in a single vertex ; and 2 . the graph @xmath84 consisting of all edges connecting vertices in @xmath83 is connected for @xmath109 note that there are no edges between vertices in @xmath110 [ [ sec:4d - partition ] ] we first remark that @xmath14 has such a triangulation . if @xmath111 is any triangulation , pass to the first barycentric subdivision @xmath112 of @xmath113 label the vertices of @xmath114 as being barycentres of faces of dimension @xmath115 , for @xmath116 . this labelling is independent of the @xmath0simplex containing the vertex . now let @xmath93 be the set of all vertices of @xmath114 that are vertices or barycentres of edges in @xmath117 ; @xmath89 be the set of all vertices of @xmath114 that are barycentres of @xmath105faces or @xmath8faces in @xmath117 ; and @xmath108 be the set of all barycentres of @xmath0simplices . this is a partition of the desired form . ] [ [ sec:4d - stellar ] ] we now apply bistellar operations on @xmath58 as follows . each 4simplex @xmath52 in @xmath58 has a unique 3face @xmath118 not meeting @xmath119 and there is a unique 4simplex @xmath120 meeting @xmath52 in @xmath121 the double @xmath0simplex @xmath122 can be subdivided into four 4simplices by introducing a new edge @xmath123 between the two vertices in @xmath108 in @xmath122 ( see figure [ fig : stellar ] ) . this is a pachner move of type @xmath124 ( cf.@xcite ) . performing this move for every pair of such 4simplices in @xmath58 gives a new piecewise linear triangulation @xmath125 with the property that @xmath58 and @xmath126 have the same vertices , and we do not alter the partition of these vertices . moreover , all edges of @xmath58 are edges of @xmath127 and the only additional edges in @xmath126 are one edge introduced for each double 4simplex in @xmath92 the partition of the vertices of @xmath58 gives a partition of the vertices of @xmath128 we will show that this satisfies the following properties : 1 . each @xmath0simplex has two vertices in two of these sets and one vertex ( the _ isolated vertex _ ) in the third . 2 . for each @xmath129 the graph @xmath130 consisting of all edges connecting vertices in @xmath131 is connected and contains at least two vertices . 3 . for each @xmath132 each 3face with no vertex in @xmath83 has a 2face with the property that all but one vertex in the link of the 2face is in @xmath133 4 . the degree of each 2face that meets all three sets in the partition is at least 4 . , height=158 ] these properties have been singled out , since conditions @xmath134 and @xmath135 will imply all properties of a multisection except the drop of the dimension of the spine for @xmath136 , for which we use @xmath137 these conditions are sufficient but not necessary for obtaining a dual multisection . the last condition gives the additional property that the central submanifold has a non - positively curved cubing . [ [ sec:4d - circlelink ] ] the triangulation @xmath138 satisfies @xmath134 and @xmath135 by construction . for the next two properties , note that no vertex from the set @xmath108 is isolated since each 4simplex contains two vertices in @xmath108 due to the bistellar move . any 3face @xmath139 with no vertex in @xmath83 has a 1face @xmath140 with both vertices in @xmath119 and the remaing vertices in the remaining set @xmath141 let @xmath142 be a 2face of @xmath139 containing @xmath143 considering the four 4simplices incident with @xmath144 we see that the link of @xmath142 contains a circle @xmath145 triangulated with three 1simplices having one vertex in @xmath146 and two vertices in @xmath133 this shows @xmath137 to argue that @xmath147 holds , first notice that @xmath126 simplicial implies the degree of any 2face is at least 3 . if the 2face @xmath142 meets all three sets of the partition , then since no vertex from the set @xmath108 is isolated in any 4simplex of @xmath127 the degree of @xmath142 is at least 4 . hence assume that we have an arbitrary triangulation of @xmath14 with the property that there is a partition @xmath148 of the set of all vertices satisfying @xmath134@xmath137 define a simplicial map @xmath55 from @xmath14 to the @xmath105simplex @xmath149 by sending each vertex in @xmath131 to the vertex @xmath150 of @xmath151 and extending by affine linear mappings on each @xmath0simplex of @xmath152 then @xmath153 the dual 1skeleton of @xmath151 divides @xmath151 into three cubes @xmath154 where @xmath155 now @xmath156 is a regular neighbourhood of @xmath130 in @xmath90 and hence we have a decomposition @xmath157 into @xmath0dimensional 1handlebodies with pairwise disjoint interiors . thus , conditions ( 1 ) and ( 2 ) of definition [ def : multisection ] are satisfied . [ [ sec:4d - pl - sub ] ] we next verify that all intersections of the handlebodies are pl submanifolds . first consider @xmath158 since @xmath18 is a regular neighbourhood , it follows that @xmath159 is a 3dimensional pl submanifold . now @xmath160 and so the interior of @xmath161 is a 3dimensional pl submanifold . since @xmath162 is collared in @xmath163 it now follows that @xmath164 is a 3dimensional pl submanifold with boundary . whence @xmath165 is also a pl submanifold . [ [ sec:4d - connected ] ] we next show that these submanifolds are connected . the triple intersection @xmath65 is the pre - image under @xmath166 of the barycentre @xmath167 of @xmath168 we first consider the intersection with a single 4simplex , @xmath169 this is the cone on a loop passing through the barycentres of the four 2faces of @xmath170 that have one vertex in each set @xmath131 . so this is a @xmath105cube and is the intersection of @xmath65 with @xmath171 since by assumption , each of the three graphs @xmath130 is connected , we can follow a path in one of these graphs between any two @xmath0simplices . but the @xmath105cubes of @xmath65 have common edges along such a path and this establishes that @xmath65 is connected . each component of @xmath161 has boundary , and @xmath172 is connected . whence @xmath161 is also connected . [ [ sec:4d - cat0 ] ] we next show that @xmath147 implies that the cubing of @xmath65 is non - positively curved . a vertex @xmath173 of a 2cube in @xmath52 is contained on a 2face @xmath118 of the triangulation with the property that @xmath118 meets all three sets @xmath174 of the partition . hence if the degree of @xmath118 is at least 4 , then the degree of @xmath173 in @xmath65 is at least 4 , and so cubing of @xmath65 is non - positively curved . [ [ sec:4d - spinedim ] ] it remains to verify that each @xmath161 has 1dimensional spine . here we will use the link condition @xmath175 . the cubical structure of @xmath151 consists of the 2cubes @xmath154 the 1cubes @xmath176 and the 0-cube @xmath177 see figure [ fig : dual cubing 2d ] . we also denote @xmath178 the barycentre of the simplex spanned by @xmath150 and @xmath179 by the definition of @xmath166 , the natural collapse @xmath180 lifts to a collapse @xmath181 note that @xmath182 is a union of 1cubes and 2cubes . namely , each 4simplex with two vertices in @xmath131 and two vertices in @xmath146 meets @xmath182 in a 2cube , and each 4simplex with only one vertex in either @xmath131 or @xmath146 meets @xmath182 in a 1cube . the link condition implies that each 2cube in @xmath182 has at least one free edge ( cf.figure [ fig : facef_link ] ) , and hence the 2cube can be collapsed from this edge onto the complementary boundary 1cubes . whence @xmath182 has a 1dimensional spine , and therefore @xmath161 has a 1dimensional spine . we first recall the definition of a @xmath183trisection from @xcite , with the only modification that we state it in the piecewise linear ( instead of the smooth ) category . let @xmath184 with @xmath185 given an integer @xmath186 let @xmath187 be the standard genus @xmath77 heegaard splitting of @xmath188 obtained by stabilising the standard genus @xmath53 heegaard splitting @xmath189 times . given integers @xmath190 a @xmath183trisection of a closed , connected , oriented 4manifold @xmath14 is a decomposition of @xmath14 into three submanifolds @xmath191 satisfying the following properties : 1 . for each @xmath192 there is a piecewise linear homeomorphism @xmath193 2 . for each @xmath192 taking indices modulo 3 , @xmath194 and @xmath195 it follows immediately from the definitions that a @xmath183trisection is a multisection of a 4manifold . gay and kirby @xcite give two different existence proofs for @xmath183trisections , one using morse 2functions and one using handle decompositions . in the first proof , one arranges for the three handlebodies to have the same genus by a homotopy of the morse map , and in the second this is obtained by adding cancelling pairs of 1 and 2handles or 3 and 4handles . to connect our multisections to the @xmath183trisections , and hence to prepare for a third existence proof of @xmath183trisections using triangulations , we similarly require a stabilisation result . as noted in @xmath196 computing the euler characteristic of @xmath14 using the decomposition into handlebodies in a @xmath183trisection gives @xmath197 and hence each of the two constants @xmath77 and @xmath53 in the above definition determines the other . [ lem : stabilisation in dim4 ] let @xmath14 be a closed , connected , oriented 4manifold with multisection @xmath198 then the multisection can be modified to a @xmath183trisection with @xmath199 we first show that if @xmath191 is a multisection with @xmath200 then it is a @xmath183trisection with @xmath201 and @xmath202 indeed , each pairwise intersection @xmath203 is a compact connected 3manifold with 1dimensional spine , and hence is a 3dimensional 1handlebody . it has boundary the surface @xmath204 and hence each of the 3dimensional handlebodies has the same genus @xmath77 . any genus @xmath77 heegaard splitting of @xmath159 satisfies @xmath205 building on work of haken @xcite , waldhausen @xcite showed that there is a unique pl isotopy class of genus @xmath77 splittings of @xmath206 and hence there is a pl homeomorphism taking the splitting @xmath207 to the stabilised standard splitting @xmath208 in particular , we can now extend the map from @xmath209 to a piecewise linear homeomorphism @xmath210 by first applying radial extensions to the boundaries of a complete set of compression 3balls of @xmath18 and then radial extensions to the boundary 3spheres of the complementary 4balls in @xmath211 to prove the lemma , it now suffices to show that each multisection of @xmath14 can be modified to a multisection with all three 4dimensional 1handlebodies of the same genus @xmath199 to this end , it suffices to show that we can increase the genus of any one of the handlebodies , say @xmath86 whilst keeping the genera of the other two handlebodies unchanged . since @xmath212 this requires increasing the genus of @xmath213 i.e.increasing the genus of one of the handlebodies stabilises the heegaard splittings of the boundaries of the other two handlebodies . we now describe the required _ stabilisation move_. let @xmath214 be an arc properly embedded in the 3dimensional handlebody @xmath215 which is _ unknotted _ , i.e.isotopic into @xmath216 keeping its endpoints on @xmath65 fixed . take a 4dimensional 1handle @xmath217 based at @xmath218 running along @xmath214 and add this to @xmath86 creating a decomposition of @xmath14 into the three handlebodies @xmath219 @xmath220 and @xmath221 with pairwise disjoint interior . it follows from our construction that @xmath222 @xmath223 and @xmath224 is @xmath65 with the interior of @xmath225 ( two open discs ) deleted and the link of @xmath226 in @xmath227 ( an annulus ) added . we therefore have @xmath228 @xmath229 @xmath230 and @xmath231 to show that @xmath232 is again a multisection , it remains to verify the condition on the pairwise intersections . since @xmath233 is obtained from the 3dimensional handlebody @xmath227 by drilling out an unknotted ark , it is also a 3dimensional handlebody and hence has a 1dimensional spine . by our construction , the intersections @xmath234 and @xmath235 can be viewed as the handlebodies @xmath236 and @xmath237 with a 1handle attached to them , so are again 1handlebodies . this completes the proof . the proof of the above lemma shows that there is a natural _ stabilisation move _ , which allows us to increase the genus of any of the 4dimensional 1handlebodies in a multisection by one . ( this stabilisation generalises to non - orientable manifolds and higher dimensions . ) the lemma shows that the existence proof in dimension 4 of multisections ( in the sense of definition [ def : multisection ] ) in [ sec : construction dim 4 ] implies the existence of @xmath183trisections ( in the sense of gay and kirby ) . in light of this _ from now onwards , we will simply call a multisection in dimension 4 a trisection , _ even if the 4dimensional handlebodies do not have the same genus , or the manifold is non - orientable . [ thm : existence odd ] every closed , connected , odd dimensional pl manifold has a multisection . the proof is a generalisation of the construction for 3dimensional manifolds , using some of the arguments given in the 4dimensional case . assume @xmath14 has dimension @xmath238 and the piecewise linear triangulation @xmath106 denote @xmath91 the first barycentric subdivision of @xmath58 and partition the vertices of @xmath91 into sets @xmath239 as follows . the set @xmath131 contains all vertices of @xmath91 that are the barycentres of @xmath240simplices or @xmath241simplices in @xmath92 now define a simplicial map @xmath242 where @xmath52 is a @xmath53simplex , by mapping @xmath131 to the @xmath115^th^ vertex of @xmath59 this defines a piecewise linear map @xmath243 each @xmath1simplex in @xmath91 meets each set @xmath146 in precisely two vertices , and since @xmath14 is connected , the graph @xmath130 in the 1skeleton of @xmath91 spanned by all vertices in @xmath146 is connected . we identify @xmath14 with @xmath244 any regular neighbourhood of @xmath130 is an @xmath1dimensional 1handlebody in @xmath54 consider the cubical cell decomposition of @xmath52 arising from the dual spine @xmath245 this has @xmath2 @xmath53cubes , which meet in pairs along @xmath246cubes . the pull - back of this decomposition divides each @xmath1simplex in @xmath14 into regions by the inverse images of the @xmath53cubes , and , moreover , the pre - image of the cube @xmath247 containing the @xmath115^th^ vertex of @xmath52 is a regular neighbourhood of @xmath248 in particular , letting @xmath249 gives a decomposition of @xmath14 into @xmath2 1handlebodies satisfying ( 1 ) and ( 2 ) in definition [ def : multisection ] . we claim that ( 3 ) and ( 4 ) are also satisfied . first note that the arguments given in [ sec:4d - pl - sub ] and [ sec:4d - connected ] can be iterated to show that all intersections are connected pl submanifolds of the stated dimensions , and that all but the central submanifold have non - empty boundary . it remains to prove the claim in ( 3 ) about the codimension of the spine . the key is the partition map @xmath250 we first study the restriction @xmath251 , where @xmath151 is a @xmath66simplex of @xmath14 . the first claim is that @xmath252 is a cube of dimension @xmath253 , where @xmath254 is in the interior of a face of codimension @xmath255 in @xmath52 . we choose affine coordinates in the simplices @xmath151 and @xmath52 so that the coordinates are ordered in the following way . in @xmath151 , the first two coordinates @xmath256 are for vertices in @xmath93 , the second two @xmath257 for vertices in @xmath89 , and so forth . similarly in @xmath52 the coordinates are ordered to correspond to the vertices labelled @xmath258 . then @xmath259 using these coordinates . the @xmath260 satisfy @xmath261 and @xmath262 . if the image point @xmath263 lies in the interior of a codimension @xmath255 face , then @xmath255 of its coordinates are @xmath19 , and the remaining coordinates have fixed non - zero values . hence the inverse image is a @xmath264cube as claimed . in particular , the central submanifold @xmath265 has a natural cubing since it meets each @xmath1simplex @xmath266 in a single @xmath267cube , which is the pre - image of the barycentre @xmath268 it will be shown in the proof of theorem [ thm : cat(0 ) ] in [ sec : cat(0 ) ] that this cubing is in fact non - positively curved . now consider the intersection @xmath269 where @xmath270 by construction , @xmath271 is a @xmath272cube in @xmath273 the cube @xmath274 naturally collapses onto the barycentre @xmath275 of the subsimplex of @xmath52 with vertex set corresponding to @xmath276 this has codimension @xmath277 by construction of the partition map @xmath278 this collapse lifts to a collapse of @xmath279 onto the pre - image of @xmath280 this meets every top - dimensional simplex in @xmath14 in a cube of dimension @xmath281 whence @xmath279 has a spine with a cubing by @xmath61cubes , giving the claimed dimension . every closed , connected , pl manifold of dimension @xmath282 has a multisection with the property that each intersection @xmath24 for @xmath283 has a spine with a cubing by @xmath61cubes and the intersection of all @xmath2 handlebodies has a non - positively curved cubing by @xmath267cubes . [ thm : existence even ] every closed , connected , even dimensional pl manifold has a multisection . assume @xmath14 has dimension @xmath284 and the piecewise linear triangulation @xmath285 denote @xmath114 the first barycentric subdivision of @xmath117 and partition the vertices of @xmath114 into sets @xmath239 as follows . for @xmath286 the set @xmath131 contains all vertices of @xmath114 that are the barycentres of @xmath240simplices or @xmath241simplices in @xmath113 the set @xmath83 contains the barycentres of the @xmath68simplices . whence each @xmath1simplex has two vertices in the partition sets @xmath287 and a single vertex in @xmath83 . as in [ sec:4d - stellar ] , each @xmath1simplex @xmath52 of @xmath114 has a unique @xmath288face @xmath118 not meeting @xmath83 and there is a unique @xmath1simplex @xmath120 meeting @xmath52 in @xmath121 we again subdivide the double simplex @xmath289 by performing a pachner move of type @xmath290 which introduces a new edge between the two vertices of the double simplex that are in @xmath83 and replaces the two @xmath1simplices by @xmath1 @xmath1simplices . applying this to all such double simplices in @xmath114 gives a new pl triangulation @xmath57 of @xmath54 as before , there is a natural identification of the vertex sets of @xmath58 and @xmath114 , and we maintain the partition . whence the graphs @xmath130 spanned by @xmath131 in @xmath58 are all connected . now define a simplicial map @xmath291 where @xmath52 is a @xmath53simplex , by mapping @xmath131 to the @xmath115^th^ vertex of @xmath59 as before , we study the restriction of this to an @xmath1simplex @xmath151 in @xmath54 there is a unique partition set @xmath292 that @xmath151 meets in only one vertex . we can choose affine coordinates on @xmath151 such that @xmath293 the inverse image of a point @xmath294 is a cube whose dimension depends on the co - dimension of the face @xmath118 containing @xmath295 in its interior and on whether or not @xmath118 contains the @xmath296^th^ vertex of @xmath59 suppose @xmath118 has co - dimension @xmath297 if @xmath118 does not contain the @xmath296^th^ vertex of @xmath298 then @xmath252 is a @xmath264cube since @xmath299 coordinates in the preimage must equal zero . in contrast , if @xmath118 contains the @xmath296^th^ vertex of @xmath298 then @xmath252 is a @xmath300cube since @xmath301 coordinates in the preimage are zero and the coordinate corresponding to the singleton is a non - zero constant . in particular , the central submanifold @xmath265 has a natural cubing since it meets each @xmath1simplex @xmath266 in a single @xmath53cube , which is the pre - image of the barycentre @xmath268 as in the odd dimensional case , we now consider the intersection @xmath269 where @xmath270 by construction , @xmath271 is a @xmath272cube in @xmath273 the cube @xmath274 naturally collapses onto the barycentre @xmath275 of the subsimplex of @xmath52 with vertex set corresponding to @xmath276 this has codimension @xmath277 by construction of the partition map @xmath278 this collapse lifts to a collapse of @xmath279 onto the pre - image of @xmath280 this meets a top - dimensional simplex @xmath151 in @xmath14 either in a cube of dimension @xmath302 or in a cube of dimension @xmath303 depending on whether the singleton of @xmath151 is in a partition set corresponding to @xmath304 or not . whence @xmath279 has a spine with a cubing by @xmath61cubes and @xmath305cubes , giving the claimed dimension @xmath306 we now make the stronger , additional observation that each of the @xmath305cubes in the spine for @xmath279 is a boundary face of some @xmath61cube unless @xmath307 whence suppose @xmath151 is an @xmath1simplex in @xmath14 with the property that @xmath308 meets @xmath151 in an @xmath305cube . let @xmath292 be the partition set containing the singleton of @xmath309 so @xmath310 due to the barycentric subdivision and the pachner move , @xmath151 meets @xmath83 in precisely two vertices ; whence @xmath311 now @xmath151 is an @xmath1simplex obtained from a pachner move on a double @xmath1simplex , which contains exactly two vertices from each partition set . so @xmath151 contains all vertices of this double @xmath1simplex except for one vertex , say @xmath173 , that is in the partition set @xmath312 if there is a vertex @xmath313 of @xmath151 with @xmath314 and @xmath315 then there is an @xmath1simplex in @xmath58 with vertex set @xmath316 this meets @xmath308 in an @xmath61cube which has the @xmath305cube @xmath317 as a face . hence suppose there is no such vertex @xmath318 this implies that either @xmath319 or @xmath320 but then either @xmath321 or @xmath136 and @xmath322 to conclude the proof of the third property it suffices to show that if @xmath27 then each @xmath61cube in the spine of @xmath279 has one of its boundary @xmath305cubes as a free facet , and hence can be collapsed onto the union of its remaining top - dimensional facets . in this case , the set @xmath323 does not contain a single vertex of the target simplex @xmath59 suppose @xmath324 if the preimage of @xmath275 meets @xmath151 in an @xmath61cube , then the singleton of @xmath151 is in the set @xmath312 due to the barycentric subdivision and the pachner move , @xmath151 meets @xmath83 in precisely two vertices ; whence @xmath311 now @xmath151 is an @xmath1simplex obtained from a pachner move on a double @xmath1simplex , which contains exactly two vertices from each partition set . so @xmath151 contains all vertices of this double @xmath1simplex except for one vertex that is in the partition set @xmath312 let @xmath325 and consider the co - dimension two facet @xmath326 of @xmath151 which does not meet @xmath327 from the double @xmath1simplex and the fact that the triangulation is pl , it follows as in [ sec:4d - circlelink ] that the link of @xmath326 is a circle triangulated with three 1simplices having one vertex in @xmath292 and two vertices in @xmath327 so the @xmath328simplex @xmath326 is contained in precisely three @xmath1simplices : @xmath151 is obtained by adding the two vertices in @xmath329 and the other two are obtained by adding one vertex in @xmath330 and one vertex in @xmath312 whence the boundary @xmath305cube @xmath331 of the @xmath61cube @xmath332 is not a boundary face of another @xmath61cube . this completes the proof of the theorem . every closed , connected , pl manifold of dimension @xmath68 has a multisection with the property that each intersection @xmath24 for @xmath333 has a spine with a cubing by @xmath61cubes , each intersection of @xmath53 handlebodies has a spine with a cubing by @xmath246cubes and the intersection of all @xmath2 handlebodies has a cubing by @xmath53cubes . it is first shown that triangulations and partition maps can be chosen such that the central submanifold has a non - positively curved cubing . next , the structure result for finitely presented groups is given , and last the recursive structure of a multisection , as well as generalisations , are discussed . we work with the combinatorial definition of a non - positively curved cubing ( see @xcite ) . flag complex _ is a simplicial complex with the property that each subgraph in the 1skeleton that is isomorphic to the 1skeleton of a @xmath53dimensional simplex is in fact the 1skeleton of a @xmath53dimensional simplex . a cube complex is _ non - positively curved _ if the link of each vertex is a flag complex . here , the link of a vertex in a cube complex is the simplicial complex whose @xmath296simplices are the corners of @xmath334cubes adjacent with the vertex . the main facts we will need are that the barycentric subdivision of any complex is flag , and that the link ( in the sense of simplicial complexes ) of any simplex in a flag complex is a flag complex . thm : cat(0 ) every piecewise linear manifold has a triangulation supporting a multisection such that the central submanifold has a non - positively curved cubing . suppose the dimension is @xmath282 , so @xmath14 is mapped to the @xmath53simplex @xmath59 as in the existence proof , our starting point is a triangulation @xmath57 that is a first barycentric subdivision and the canonical partition map @xmath335 since @xmath58 is a first barycentric subdivision , it is a flag complex . the central submanifold @xmath336 is the pre - image of the barycentre @xmath337 of @xmath52 and meets each top - dimensional simplex @xmath151 in a @xmath267cube . a corner @xmath338 of such a @xmath267cube @xmath274 in @xmath65 is the barycentre of a @xmath53simplex @xmath339 in @xmath14 meeting all sets of the partition . so the @xmath340 vertices of @xmath341 correspond to the @xmath340 barycentres of the @xmath53faces of @xmath151 meeting each set of the partition . we claim that there is a simplicial isomorphism @xmath342 the latter is flag since it is the link of a simplex in a flag complex , and so this claim implies the conclusion of the theorem . suppose @xmath343 then the @xmath2 vertices of @xmath344 not in @xmath339 meet each set in the partition , and @xmath345 is spanned by these vertices as @xmath344 ranges over all top - dimensional simplices containing @xmath346 in particular , each @xmath53simplex in @xmath345 meets each set in the partition , and each simplex in @xmath345 has no two vertices in the same partition set . now each edge of such a @xmath344 between vertices in the same partition set corresponds to an edge in the central submanifold with an endpoint at @xmath338 and vice versa . hence there is a bijection between vertices in @xmath347 and @xmath348 the same holds for all higher dimensional cells in the links : an @xmath296simplex in @xmath345 corresponds to an @xmath296simplex @xmath349 in some @xmath350 then @xmath349 and @xmath339 span a subsimplex of @xmath351 which meets the central submanifold in an @xmath334cube . whence we obtain an injective simplicial map @xmath352 which is clearly bijective . this completes the proof for manifolds of odd dimension . now suppose the dimension of @xmath14 is @xmath353 for the triangulation constructed in the existence proof , it turns out that the cube complexes for @xmath354 do not satisfy the flag consition due to the @xmath355 pachner move . we therefore use a different construction , which results in a greater number of simplices . start with any piecewise linear triangulation @xmath285 denote @xmath114 the first barycentric subdivision of @xmath113 all dual 1cycles in @xmath114 have even length , and hence the dual 1skeleton is a bipartite graph . we therefore have a partition of the @xmath1simplices into two sets @xmath356 and @xmath357 such that @xmath1simplices in @xmath356 only meet @xmath1simplices in @xmath357 along their codimension - one faces and vice versa . now let @xmath358 be the second barycentric subdivision of @xmath113 we define a partition @xmath359 of the vertices of @xmath58 as follows . the set @xmath93 consists of all vertices of @xmath114 and all barycentres of @xmath1simplices in @xmath360 the set @xmath89 consists of all barycentres of edges in @xmath114 and all barycentres of @xmath1simplices in @xmath361 for each @xmath362 the set @xmath146 consists of all barycentres of @xmath363simplices and @xmath364simplices . we claim that the graph @xmath365 spanned by all vertices in @xmath146 is connected for each @xmath297 first consider the set @xmath146 for any @xmath366 then each vertex in @xmath146 is the barycentre of a @xmath363simplex or a @xmath364simplex in @xmath367 any @xmath363simplex in @xmath114 is in the boundary of a @xmath364simplex , and the dual graph of the @xmath364skeleton is connected . this shows that the graph @xmath365 is connected . now consider @xmath93 for @xmath368 or @xmath369 each element in @xmath93 is either a vertex in @xmath114 or the barycentre of some @xmath1simplex in @xmath367 any vertex of @xmath114 is connected in @xmath370 to the barycentre of some @xmath1simplex in @xmath367 any two barycentres of @xmath1simplices in @xmath114 that are in @xmath93 are connected by a path in the dual 1skeleton , and the vertices in this path alternate between vertices in @xmath93 and @xmath371 now any two adjacent @xmath1simplices share at least @xmath1 vertices , and any two @xmath1simplices at distance two in the dual 1skeleton share at least @xmath372 vertices . these vertices are all in @xmath93 and connect to the barycentre of any @xmath1simplex in @xmath114 that is in @xmath373 whence @xmath370 is connected . the argument that @xmath374 is connected is similar , by observing that any two @xmath1simplices at distance two in the dual 1skeleton share at least @xmath375 edges . this completes the argument that each graph @xmath365 is connected . as in the existence proof , to show that the associated partition function defines a multisection of @xmath90 we need to establish that the dimension of the spine drops by one when @xmath376 hence suppose @xmath377 from the computation of the dimensions of the pre - images in the proof of theorem [ thm : existence even ] , we know that if the singleton of the @xmath1simplex @xmath151 is mapped to a vertex in @xmath378 then we obtain a @xmath246cube in the spine . a singleton is either mapped to @xmath379 or @xmath369 so suppose the singleton of @xmath151 is mapped to @xmath296 and @xmath380 so that the spine for @xmath279 meets @xmath151 in a @xmath53cube . then the @xmath288facet @xmath381 with no vertex mapping to @xmath296 contains this @xmath53cube . due to the barycentric subdivision construction , the link of @xmath381 is a circle triangulated with four vertices , and precisely three of these map to @xmath296 and one maps to @xmath382 but then the @xmath53cube has a free @xmath246face and hence can be collapsed . ( in dimension four , this is precisely condition @xmath175 in [ sec:4d - stellar ] . ) whence we have shown that the intersection of any @xmath53 handlebodies has a spine of dimension @xmath383 having established that the given triangulation and partition function define a multisection , the proof given above for the odd - dimensional case now applies verbatim to show that the link of an arbitrary corner in the cubing is simplicially isomorphic to the link of the simplex in the triangulation @xmath384 having the corner as its barycentre . we analyse the structure of the fundamental group only in the case of a trisection of a 4-manifold , as this already gives a result about all finitely presented groups . the higher dimensional multisections give similar results . in particular , theorem [ thm : cat(0 ) ] implies that one may choose as the _ central _ group in the associated generalised graph of groups the fundamental group of a manifold with a non - positively curved cubing . pro : structure of fund gp every finitely presented group has a generalised graph of groups decomposition as shown in figure [ fig : structure of groups_intro ] , where the vertex groups @xmath69 are free of rank @xmath70 @xmath71 is the fundamental group of a closed orientable surface of genus @xmath72 all edge groups are naturally isomorphic with @xmath71 and the oriented edges represent epimorphisms . given a finitely presented group @xmath385 there is a closed , orientable pl 4manifold @xmath14 with @xmath386 choose a trisection @xmath191 of @xmath54 then the central submanifold @xmath65 is a closed , orientable surface . write @xmath387 the inclusions @xmath388 induce epimorphisms @xmath389 we now define a generalised graph of groups by choosing as vertex groups the four groups @xmath390 @xmath391 @xmath374 and @xmath392 and as edge groups three copies of @xmath71 with edge maps the identity map @xmath393 and the epimorphisms @xmath394 denote @xmath395 the fundamental group of the generalised graph of groups . we claim that @xmath396 to this end , we compute @xmath397 through two applications of the generalised van kampen theorem ( * ? ? ? * theorem 6.2.11 ) . first consider @xmath398 this is the push - out @xmath399 using the epimorphism @xmath400 we also obtain @xmath401 as the push out of the induced maps @xmath402 now @xmath403 gives the push - out @xmath404 and using the induced map @xmath405 gives desired isomorphism . an important part of multisections is their _ recursive structure_. by this we mean that inside a multisection of an @xmath1-dimensional manifold , we see a stratification of the boundary of each handlebody into lower dimensional manifolds . for example , for a trisection where @xmath48 , we see partition functions on the boundaries of each handlebody , dividing the boundary into two pieces . for a quadrisection , where @xmath49 , the boundaries of the handlebodies are divided into three pieces . however , the top dimensional pieces are not necessarily handlebodies , whereas all the pieces have spines of low dimension . so these are not multisections in the usual sense . the same works in all dimensions . namely for @xmath1manifolds , with @xmath4 or @xmath3 , the boundaries of the handlebodies have natural divisions into @xmath53 regions . each of these regions has a spine of dimension at most two . however , whereas these regions are 1handlebodies in the case @xmath406 this is not necessarily true for higher dimensions . in our construction , the vertices of each @xmath1simplex are partitioned into sets with at most two vertices in order to achieve spines of small dimension . other partitions allow the definition of more general types of multisections , which are decompositions of the @xmath1manifold into regions with spines of _ smallish _ dimension ( but possibly greater than one ) , and so they are more complicated than 1handlebodies . we first explain in [ sec : constructing with symmetric representations ] how the symmetric representations of @xcite can be used to construct multisections and give a number of applications of this approach : multisections of spheres and real projective spaces , and the three general constructions of _ products _ , _ generalised multisections _ and _ twisted multisections_. we also discuss _ connected sums _ in [ subsec : conn sums ] , a _ dirichlet construction _ in [ sec : dirichlet ] , and the case of _ manifolds with non - empty boundary _ in [ sec : multisections for non - closed ] . given a triangulated @xmath1manifold @xmath57 with the property that the degree of each @xmath328simplex is even , the authors defined a _ symmetric representation _ @xmath407 in @xcite as follows . pick one @xmath1simplex as a base , choose a bijection between its corners and @xmath408 and then _ reflect _ this labelling across its codimension - one faces to the adjacent @xmath1simplices . this induced labelling is propagated further and if one returns to the base simplex , one obtains a permutation of the vertex labels . since the dual 1skeleton carries the fundamental group , it can be shown that this gives a homomorphism @xmath409 see @xcite for the details . for example , the symmetric representation associated to any barycentric subdivision is trivial , since the labels correspond to the dimension of the simplex containing that vertex in its interior , but there may be more efficient even triangulations with this property . the symmetric representation can also be used to propagate partitions of the vertices of the base simplex ; this is done in @xcite for partitions into two sets , but extends to arbitrary partitions . one then obtains a _ induced representation _ , usually into a symmetric group of larger degree . the aim in @xcite was to obtain information on the topology of a manifold from a non - trivial symmetric representation arising from a triangulation with few vertices . our needs in this paper are opposite , and we wish to use the symmetric representations to identify triangulations to which we can apply our constructions without barycentric subdivision . so we either want the _ orbits _ of the vertices under the symmetric representation to give a partition satisfying the conditions in our constructions ; or we ask for partitions of the vertices with the property that the induced representation is trivial . the main properties to check for a given partition of the vertices are that the graphs spanned by the partition sets are connected , and , in even dimensions , that the dimension of the spine drops when intersecting all but one of the handlebodies . * spheres . * the @xmath1-sphere can be obtained by doubling an @xmath1-simplex . this is an even triangulation with trivial symmetric representation . in odd dimensions , this induces a multisection that is a division of a @xmath66dimensional sphere into @xmath2 balls , each corresponding to a pair of vertices of the @xmath1-simplex . in even dimensions this is a division of a @xmath410dimensional sphere into @xmath2 balls , one corresponding to a single vertex and each of the remaining ones corresponding to a pair of vertices ; here one obtains a multisection directly from the partition function without stellar subdivision . moreover , symmetries of the triangulation permuting the partition sets of the vertices interchange all handlebodies in odd dimensions ; in even dimensions they fix the handlebody corresponding to the singleton and act transitively on the remaining ones . * projective spaces . * a symmetric triangulation of the @xmath1sphere is obtained by the crosspolytope ( see @xcite ) , which is the set of vectors @xmath411 in @xmath412 satisfying @xmath413 . so there are @xmath414 @xmath1simplices . this is invariant under the antipodal map and so descends to an even triangulation of @xmath415 . we can label the vertices of the triangulation of @xmath416 by the position @xmath115 of the unique coordinate @xmath417 with @xmath418 . pairs of vertices with @xmath419 are interchanged by the antipodal map so this descends to a labelling of the @xmath1 vertices of the triangulation of @xmath415 . assume first that @xmath3 . then we can pair the vertices @xmath420 for @xmath16 . for @xmath421 , the multisection is the heegaard splitting of genus one , giving two solid tori . for @xmath422 , the multisection has @xmath2 handlebodies of genus @xmath21 , whence equivalent with @xmath423 in fact there are symmetries of the triangulation interchanging all the handlebodies , so they must all have the same genus . suppose next that @xmath4 . we can then pair up vertices @xmath424 for @xmath425 leaving @xmath19 as a singleton partition set . it is necessary to perform stellar subdivisions of facets with vertices labelled @xmath115 with @xmath426 . for the vertices labelled @xmath424 for @xmath425 , there is still a symmetry interchanging all the handlebodies corresponding to the partition sets . these handlebodies are equivalent with @xmath427 and hence have genus @xmath369 in contrast , the handlebody corresponding to the partition set labelled @xmath19 is of genus @xmath428 . * generalised multisections . * suppose that @xmath14 is a triangulated @xmath1manifold with an even triangulation with trivial symmetric representation . as above , given any triangulation , the first barycentric subdivision has this property . we can define further multisections as follows . suppose that @xmath429 . assume that we have partition sets @xmath430 where the sets have three vertices in every @xmath1-simplex . we then map each @xmath1simplex to the @xmath53simplex by mapping each partition set to a vertex of this @xmath53simplex . it is then easy to verify that we obtain a division of @xmath14 into @xmath2 regions , and each region has a @xmath105dimensional spine , given by the union of all the @xmath105simplices in each @xmath1simplex with all vertices in the same partition set . in this case , the manifold @xmath65 which is the intersection of all the handlebodies , is closed of dimension @xmath431 . again we can arrange that the induced cubing of @xmath65 is negatively curved and each intersection of a proper subcollection has a spine of low dimension . another interesting example is to have two partition sets of size @xmath432 of the vertices of each @xmath1simplex , so that @xmath433 . we assume that both @xmath434 . the induced decomposition is a bisection into two regions with spines of dimension @xmath432 . given a handle decomposition of @xmath14 , this is similar to a hypersurface which is the boundary of the region containing all the @xmath115handles for @xmath435 . finally a very specific example is a @xmath436manifold @xmath14 with three partition sets of respective sizes @xmath437 . this induces a trisection of @xmath14 into three regions , where two are handlebodies and the third has a spine of dimension @xmath105 . * twisted multisections . * suppose a closed pl @xmath1manifold has an even triangulation with a non - trivial symmetric representation . assume also that the symmetry preserves our standard partition of the vertices , i.e.every symmetry mapping produces a permutation of the partition sets of vertices . then there is an associated ` twisted ' multisection , which we illustrate with a simple example the general construction then becomes clear . assume @xmath14 is a 5manifold that admits an even triangulation with a symmetric representation with image @xmath438 . also assume this symmetry is a permutation of the form @xmath439 of the labelling of the vertices . in this case , we choose as partition sets @xmath440 then these are permuted under the action of the symmetric mapping . the edges joining these three pairs of vertex sets clearly form a connected graph @xmath441 . a regular neighbourhood of @xmath441 then forms a single handlebody @xmath442 whose boundary is glued to itself to form @xmath14 . the handlebody @xmath442 lifts to three handlebodies in a regular 3fold covering space @xmath443 of @xmath14 and these give a standard trisection of @xmath443 . the covering transformation group @xmath438 permutes the handlebodies and preserves the central submanifold . if the initial triangulation is flag , then the lifted triangulation is flag and hence the central submanifold has a non - positively curved cubing on which the covering transformation group acts isometrically . hence the quotient , which embeds in @xmath444 also has a non - positively curved cubing . suppose the @xmath1manifolds @xmath14 and @xmath445 have multisections . then we can define the connected sum of these by removing small open @xmath1disks @xmath446 @xmath447 from @xmath90 @xmath445 respectively , so that the closures meet the multisections in the standard multisection of the ball . we can then glue the multisections of @xmath448 and @xmath449 with a gluing map that matches the multisections along the boundary spheres to form the connected sum multisection , and the resulting connected sum depends on the way the handlebodies of @xmath14 and @xmath445 are matched up . the general construction uses the polyhedral metric defined by each simplex having the standard euclidean metric . such metrics arise in regge calculus ( see @xcite ) , which applies pl topology to general relativity and quantum gravity . suppose @xmath14 is given with a triangulation and partition function supporting a multisection . assume that @xmath130 is the graph of all edges corresponding to the @xmath115^th^ partition set of the vertices . then the handlebodies of a multisection are given by @xmath450 in other words , the multisection is defined by a dirichlet construction , where instead of considering distance from a set of distinct points , we use distance from a collection of disjoint graphs . in particular , each handlebody is the set of all points , which are not further from one of the partition edge graphs than any of the others . moreover the _ multisection submanifolds _ , i.e.the submanifolds arising from intersections , are then given by @xmath451 in particular , in some symmetric spaces we can use this dirichlet viewpoint to achieve that all multisection submanifolds are totally geodesic relative to the appropriate symmetric space metric . for these submanifolds are intersections of piecewise totally geodesic hypersurfaces , given by sets of points equidistant from two distinct points . this works in the case that the handlebodies are actually cells , so our dirichlet construction is the usual one . as an example , first consider @xmath416 with the standard round metric . first assume @xmath3 . we can pick equally spaced points @xmath452 from a totally geodesic embedded @xmath453 . these points form an orbit of the isometric action of the symmetric group @xmath454 on @xmath416 , where the action preserves the standard hopf link @xmath455 and the points lie in one of these copies of @xmath453 . then the standard dirichlet construction gives the standard multisection for @xmath416 . if @xmath4 , the same construction works by instead considering a totally geodesic @xmath456 in @xmath416 and choosing @xmath53 points symmetrically situated in @xmath456 to perform the dirichlet construction . note in both even and odd cases , all the components of the multisections are totally geodesic submanifolds . a similar construction applies to @xmath457 , so the multisection submanifolds are also piecewise totally geodesic in the standard symmetric space metric . in this case , the action of the symmetric group @xmath458 is given by permuting coordinates , so we can take our orbit of points as @xmath459 , [ 0,1,0 , \dots , 0 ] , \dots , [ 0,\dots,0 , 1].\ ] ] here is a simple example of a multisection dirichlet construction using graphs rather than points . start with the round metric on @xmath460 considered as the unit sphere in @xmath461 . choose three geodesic circles given by @xmath462 @xmath463 @xmath464 for @xmath465 . the dirichlet regions relative to these circles are given by @xmath466 for @xmath467 . the central submanifold is the flat 3-torus defined by @xmath468 . each handlebody is @xmath469 , and each intersection of pairs of handlebodies is a copy of @xmath470 . this multisection is in fact the pull back of the multisection of @xmath471 , viewing @xmath460 as a circle bundle over @xmath471 . the same construction works for all odd dimensional spheres they have multisections where all the handlebodies are a product of a circle and a ball . a natural product of multisections can be defined via the categorical product or the external join . however , both of these operations generally do not result in manifolds , and it is an interesting topic for investigation to relate them to a multisection of the direct product in a canonical way . the _ categorical product _ of @xmath10 and @xmath14 is a simplicial complex of dimension @xmath474 which is homotopy equivalent to the direct product @xmath475 ( * ? ? ? the set of vertices of a simplex in the categorical product is the direct product of the vertices of a simplex in @xmath10 with the set of vertices of a simplex in @xmath54 hence a partition of the vertices can be canonically defined from the partitions for @xmath10 and @xmath476 @xmath477 and @xmath478 are in the same partition set if and only if either @xmath479 and @xmath480 is a partition set of @xmath14 or @xmath481 is a partition set of @xmath10 and @xmath482 the _ join _ of @xmath10 and @xmath90 denoted @xmath483 is a simplicial complex of dimension @xmath484 which is equivalent to @xmath485 $ ] with @xmath486 collapsed to @xmath10 and @xmath487 collapsed to @xmath54 in this case , @xmath488 is equivalent with @xmath489 and has a natural cell decomposition . in the case @xmath10 and @xmath14 are standard spheres , @xmath490 is again a standard sphere ( * ? ? ? each top - dimensional simplex @xmath52 of @xmath490 is of the form @xmath491 with set of vertices the union of the vertices of @xmath492 and @xmath493 a partition of the vertices of @xmath52 can be defined by simply taking the partition sets of @xmath492 and @xmath494 unless both @xmath1 and @xmath167 are even , in which case the union of the two singletons is a partition set . to deal with compact manifolds with non - empty boundary , there are two different models in dimension @xmath495 which one can take as a launching point . one gives a decomposition of a @xmath8manifold into two compression bodies , which are obtained by attaching @xmath21handles to products of closed surfaces and intervals along one of the boundary surfaces in each such product . the other decomposes the manifold into two handlebodies , which are glued along subsurfaces of their boundaries . interesting examples of the former can be obtained as follows . birman @xcite shows that each closed orientable @xmath8manifold @xmath14 bounds a @xmath0manifold @xmath496 given by gluing two @xmath0balls together along a heegaard handlebody in their boundary @xmath8spheres . hence this is a trisection of @xmath496 with one component a collar of the boundary @xmath497 and the other two components @xmath0balls . a general theory , analogous to the one in this paper , can be developed by working with the interior of the manifold and using triangulations with ideal vertices . gay and kirby @xcite give a different construction for multisections of 4manifolds with boundary , motivated by their study of morse 2functions . here the boundary of the 4manifold has an induced fibration or open book decomposition . it is an interesting topic for further investigation to formulate a discrete version of this approach to all dimensions . daniel t. wise : _ from riches to raags : 3-manifolds , right - angled artin groups , and cubical geometry . _ cbms regional conference series in mathematics , 117 . published for the conference board of the mathematical sciences , washington , dc ; by the american mathematical society , providence , ri , 2012 .

recently gay and kirby described a new decomposition of smooth closed @xmath0manifolds called a trisection . this paper generalises heegaard splittings of 3-manifolds and trisections of 4-manifolds to all dimensions , using triangulations as a key tool . in particular , we prove that every closed piecewise linear @xmath1manifold has a _ multisection _ , i.e.can be divided into @xmath2 @xmath1dimensional 1handlebodies , where @xmath3 or @xmath4 , such that intersections of the handlebodies have spines of small dimensions . several applications , constructions and generalisations of our approach are given .

it is often considered that the evolution of protoplanetary disks and the consequent accretion of gas by the central protostar are driven by turbulent viscosity due to a magneto - rotational - instability ( mri ) ( e.g. , * ? ? ? @xcite and @xcite carried out fluid dynamical simulations of mri and found that the random torques due to the turbulent density fluctuations give rise to a random walk in semimajor axes of planetesimals . @xcite pointed out through model calculations that the random walk expands the effective feeding zone of protoplanets , and may lead to rapid formation of large cores for gas giants . through a fokker - planck treatment , @xcite also pointed out the importance of the random walk in planet accretion . adopting the semi - analytical formula for the random torque derived by @xcite , @xcite performed n - body simulations for the late stages of terrestrial planet accretion with a disk significantly depleted in gas , starting from mars - mass protoplanets . they found that the mri turbulence indeed helps to reduce the number of accreted terrestrial planets which is otherwise too large compared to our solar system . however , @xcite found through direct integrations of the orbits of protoplanets in a mri turbulent disk that orbital eccentricities are also excited . @xcite also found a similar feature in a self - gravitating disk . while the random walk itself is favorable to the growth of protoplanets by avoiding isolation , the excitation of their eccentricities , which had been neglected in @xcite and @xcite , is a threat for planetesimal accretion processes because of increased collision velocity . unfortunately , @xcite s orbital integrations were limited to 100150 keplerian times and neglected collision processes , so that it is not possible to conclude from that work whether planetesimals should grow or be eroded in the presence of turbulence . in the present article , we explore by which paths planetesimals may have grown to planet - sized bodies in turbulent disks . because the level of density fluctuations due to the mri turbulence is not well determined , we choose to study the qualitative effects of the turbulence on the accretion of planetesimals and their dependence on the key parameters of the problem , in particular the progressive removal of the gas disk . in 2 , we summarize the conditions for the accretion and destruction of planetesimals in terms of their orbital eccentricities . in 3 , we analytically derive the equilibrium eccentricity for which the excitation due to turbulence is balanced by damping due to tidal interactions with the disk gas , aerodynamic gas drag , and collisions . comparing the equilibrium eccentricities with critical eccentricities for accretion and destruction , we derive critical physical radii and masses of planetesimals for accretion or destruction . the results are applied to viscously evolving disks ( 4 ) . we then discuss possible solutions for the problem of the formation of planetesimals and planets ( 5 ) . we summarize the accretion and destruction conditions below . from energy conservation , the collision velocity ( @xmath2 ) between two planetesimals ( labeled 1 and 2 ) satisfies @xmath3 where @xmath4 and @xmath5 are the mass and physical radius of a planetesimal @xmath6 ( @xmath7 ) , and @xmath8 is their relative velocity when they are apart from each other . when the velocity dispersion of planetesimals @xmath9 is larger than the hill velocity that is given by @xmath10 , where @xmath11 is the mass of the host star and @xmath12 is the keplerian velocity , @xmath8 is approximated by @xmath9 ( e.g. , * ? ? ? * ; * ? ? ? the total energy then becomes : @xmath13 where @xmath14 is the ( two - body ) surface escape velocity defined by @xmath15 collisional dissipation decreases the energy by some fraction of @xmath16 . if @xmath17 , the collisional dissipation results in @xmath18 after collision . on the other hand , @xmath19 is likely to be still positive after a collision with @xmath20 . thus , for moderate dissipation , the condition for an accretional collision is @xmath21 ( e.g. , * ? ? ? * ) . since the orbital eccentricity @xmath22 , a collision should result in accretion for @xmath23 , where @xmath24 in the above relation , @xmath25 is the bulk density of the planetesimals and @xmath26 ( for simplicity , @xmath27 is assumed ) . the physical radius @xmath28 is given by @xmath29 a collision results in destruction if the collision velocity is such that the specific kinetic energy of a collision ( @xmath30 ) exceeds @xmath31 { \rm erg / g } , \label{eq : q_d}\ ] ] where @xmath32 is the material strength , @xmath332.1 , @xmath34 , and @xmath35 @xcite . for basalt rocks or water ice , @xmath36@xmath37 @xcite , but it can take a significantly smaller value for loose aggregates . is rather higher ( w. benz , private communication ) . ] we adopt @xmath38 as a nominal value . self - gravity ( the second term in the r.h.s . ) dominates the material strength when @xmath39 m. in this regime , adopting @xmath40 , a collision results in destruction for @xmath41 , where @xmath42 we first derive the equilibrium eccentricities of planetesimals at which the excitation by the mri turbulence is balanced by damping due to drag and/or collisions . comparing the estimated eccentricities with @xmath43 and @xmath44 , we then evaluate the outcome of collisions between planetesimals as a function of planetesimal size , turbulent strength , and surface density of disk gas . for an easy interpretation , we provide in this section analytical relations based on a model in which the gas and solid components of disk surface density are scaled with the multiplicative factors @xmath45 and @xmath46 : @xmath47 and @xmath48 where @xmath494 is an enhancement factor of @xmath50 due to ice condensation . if @xmath51 , @xmath52 and @xmath50 are 1.4 times those of the minimum mass solar nebula model @xcite . in this section , we also use the disk temperature distribution obtained in the optically thin limit @xcite , @xmath53 where @xmath54 and @xmath55 are the stellar and solar luminosities , respectively . the corresponding sound velocity is @xmath56 since the disk scale height is given by @xmath57 ( assuming that @xmath58 is vertically uniform in the disk ) , eqs . ( [ eq : sigma_g ] ) and ( [ eq : sound_velocity_hayashi ] ) yield the disk gas density at the midplane as @xmath59 the orbital eccentricities of planetesimals are pumped up both by the random gravitational perturbations from density fluctuations of disk gas , as well as by mutual gravitational scattering among planetesimals . assuming planetesimals have equal masses , their orbital eccentricities should be excited to at most @xmath60 by the mutual scattering ( e.g. , * ? ? ? as will be shown below , the value of this excentricity is smaller than that due to the turbulent excitation , except for very large planetesimals ( @xmath61 km or larger ) , and/or in the case of significantly depleted gas disks . for simplicity , in this work we choose to neglect the possibility that mutual scattering dominates over turbulent excitation . it is therefore important to note that our results may be slightly optimistic when concerning the possibility of accretion of massive planetesimals . the orbital eccentricities that result from the turbulent density fluctuations in the disk are provided by @xcite on the basis of orbital integrations with empirical formula by @xcite , as @xmath62 where @xmath63 is @xmath52 at 1au with @xmath64 ( eq . [ [ eq : sigma_g ] ] ) and @xmath65 is a non - dimensional parameter to express the disk turbulence . ) may be enhanced by a factor 10 by the inclusion of @xmath66 modes , the @xmath66 modes actually enhance only the amplitude of random walk in semimajor axis ( @xmath67 ) but not the eccentricity . since higher @xmath68 modes fluctuate over shorter timescales , they tend to cancel out on the orbital period of a planetesimal . for these modes , @xmath69 , which is due to time variation of the potential , is much smaller than @xmath70 , because the latter is also excited by the non - axisymmetric structure . the inclusion of slowly varying @xmath66 modes enhances @xmath69 up to the order of @xmath71 . on the other hand , the definition of @xmath72 in eqs . ( 5 ) and ( 34 ) of @xcite should be multiplied by @xmath73 . we use eq . ( [ eq : e_random ] ) for the eccentricity excitation , which is consistent with an orbital calculation including @xmath66 modes ( figure [ fig : e_evol ] ) . ] although @xcite showed the results only at @xmath74au , we here added a dependence on @xmath75 using scaling arguments ( see the appendix ) . orbital integration for other @xmath75 show a consistent dependence . note that @xmath76 , where @xmath67 is the amplitude of random walk in semimajor axis . since @xmath77 , the radial distance @xmath75 and semimajor axis @xmath78 are identified here . from the simulation results by @xcite , the value of @xmath65 may be @xmath79@xmath80 for mri turbulence . in this paper , we use @xmath81 as a fiducial value . interestingly , with a quite different approach , @xcite derived a similar formula for @xmath69 with the same dependences on @xmath75 , @xmath82 and @xmath83 . if @xmath84 , their formula is consistent with ours . they suggested that @xmath85 or @xmath86 where @xmath87 is disk scale height and @xmath88 is the parameter for the alpha prescription for turbulent viscosity @xcite . for @xmath89@xmath80 , their estimate is also similar to our fiducial value . the top panels in fig . [ fig : e_evol ] show the results of an orbital integration with 4-th order hermite scheme for the evolution of @xmath90 and @xmath69 with turbulent perturbations but without any damping . five independent runs with different random number seeds for the generation of turbulent density fluctuations @xcite are plotted in each panel . the initial @xmath90 and @xmath91 are @xmath92 . for @xmath64 , @xmath93 , and @xmath94au , as used in fig . [ fig : e_evol ] , eq . ( [ eq : e_random ] ) is reduced to @xmath95 . to highlight the effect of turbulence , we used a larger value of @xmath65 than the fiducial value . the evolution of the root mean squares of the five runs in fig . [ fig : e_evol ] agrees with eq . ( [ eq : e_random ] ) within a factor of @xmath96 . from eq . ( [ eq : e_random ] ) , the excitation timescale is @xmath97 the eccentricity damping processes are i ) tidal interaction with disk gas , ii ) aerodynamical gas drag , and iii ) inelastic collisions . the tidal damping timescale ( i ) is derived by @xcite as @xmath98 the gas drag damping timescale ( ii ) is derived by @xcite as @xmath99 for simplicity , we evaluate the damping timescale due to inelastic collision as the mean collision time of planetesimals , assuming that all the planetesimals have the same mass @xmath100 . since in the size distribution caused by collision cascade , collisions with comparable - sized bodies and those with smaller ones contribute similarly , the neglection of the size distribution may not be too problematic . since we look for the conditions in which collisions are non - accretional , we consider the case with @xmath101 . assuming that the gravitational focusing factor @xmath102 \sim 1 $ ] , the collision damping timescale is @xmath103 where @xmath104 is the spatial number density of planetesimals . note that @xmath105 . we now equate eq . ( [ eq : t_exc ] ) with eqs . ( [ eq : t_tidal ] ) , ( [ eq : t_drag ] ) , and ( [ eq : t_coll ] ) , respectively , to obtain an equilibrium eccentricity for each damping process . for simplicity and to a good approximation , the actual equilibrium eccentricity can be approximated as the minimum of the three equilibrium eccentricities . from eqs . ( [ eq : t_exc ] ) and ( [ eq : t_tidal ] ) , @xmath106 with eq . ( [ eq : t_drag ] ) , @xmath107 for @xmath64 , @xmath93 , and @xmath94au , eq . ( [ eq : e_drag ] ) predicts that @xmath108 for @xmath109 and @xmath110 for @xmath111 . an orbital integration in the middle and bottom panels in fig . [ fig : e_evol ] shows that the results agree with the analytical estimate within a factor @xmath112 . with eq . ( [ eq : t_coll ] ) , @xmath113 in figure [ fig : e_eq ] , the equilibrium eccentricity , @xmath114 , is plotted with solid lines as a function of the planetesimal radius @xmath28 , the corresponding planetesimal mass being @xmath115 , note again that the effect of mutual planetesimal scattering is neglected . for bodies with more than lunar to mars masses , tidal damping is dominant . this yields a decrease in the equilibrium eccentricity with increasing planetesimal radius for @xmath39 km . for smaller mass bodies , gas drag damping dominates tidal damping and the equilibrium eccentricity increases with increasing @xmath28 . for the smallest planetesimal sizes ( the regions with the slightly steeper positive gradient ) , collision damping is dominant , but with a significant contribution of gas drag damping . the limiting mass and radius at which @xmath116 ( eq . [ [ eq : e_drag ] ] ) and @xmath43 ( eq . [ [ eq : e_acc ] ] ) cross are @xmath117 the accretion of planetesimals is possible for @xmath118 ( @xmath119 ) . in the top panel of fig . [ fig : e_eq ] ( @xmath81 and @xmath64 ) , planetesimal accretion proceeds in a range of @xmath28 s in which the solid line ( @xmath120 ) is located below the dashed line ( @xmath43 ) , that is , only if a body is larger than ceres . such large planetesimals can be formed by a different mechanism than pairwise accretion such as self - gravitational instability in turbulent eddies ( e.g. , * ? ? ? when the disk gas is removed , accretion becomes possible for smaller planetesimals ( the 2nd panel of fig . [ fig : e_eq ] ) . on the other hand , if turbulence is stronger ( @xmath121 ) , planetesimal accretion requires more than 1000 km - sized bodies . this appears to be an insurmountable barrier to accretion , even for depleted gaseous disks ( @xmath122 ) , as shown in the the 3rd panel of fig . [ fig : e_eq ] . finally , at large orbital radii , planetesimal accretion is even more difficult ( the bottom panel ) . another critical mass ( radius ) is the point at which @xmath116 ( eq . [ [ eq : e_drag ] ] ) and @xmath44 in the gravity regime ( eq . [ [ eq : e_d ] ] ) cross , @xmath123 planetesimals with @xmath124 ( @xmath125 ) are disrupted by collisions down to the sizes for which material strength is dominant ( see below ) . for @xmath126 and @xmath127 ( the top panel of fig . [ fig : e_eq ] ) , planetesimals with sizes larger than several km radius survive but without growing , while smaller planetesimals are disrupted . when a planetesimal is smaller than @xmath128 m in size , it is bounded by material strength rather than self - gravity . in the regime of material strength , @xmath129 . the body is not disrupted if @xmath130 , which is equivalent to @xmath131 since in this regime , collision damping is slightly stronger than gas drag , actual values of @xmath132 and @xmath133 are determined by @xmath134 , so they are slightly larger than the above estimate ( see fig . [ fig : e_eq ] ) . when @xmath135 , the planetesimals motions are coupled to that of the gas . the collision velocity then can not be expressed in terms of orbital eccentricity . this limiting size is however much smaller than @xmath133 . the collision cascade would hence stop at @xmath136 ( @xmath137 . regions for which the dotted lines ( @xmath44 ) in fig . [ fig : e_eq ] have negative gradients correspond to the material strength regime . in the depleted disk case , @xmath44 is always larger than @xmath120 , so that the disruptive regions do not exist ( see the 2nd panel of fig . [ fig : e_eq ] ) . note that @xmath138 . if the planetesimals are loose aggregates so that @xmath139 ( the value for basalt rocks or water ice ) , the limiting size @xmath133 is smaller . we now put these various critical physical radii in the context of the evolution of the protoplanetary disk . in order to investigate the effect of departures from power - law relations of the surface density and temperature profiles in real disks , we also present in this section results obtained from a 1d disk model that includes an @xmath88-viscosity and photoevaporation ( see * ? ? ? * ; * ? ? ? the parameters used in the model presented here are a turbulent viscosity @xmath140 and an evaporation parameter @xmath141k ( the temperature of the evaporation part of the outer disk ) . another choice of the parameters would affect the results only marginally . in the numerical calculation in this section , we evaluate the equilibrium eccentricities @xmath120 as a function of planetesimal radius by solving the following relation : @xmath142 where the different timescales are given by eqs . ( [ eq : t_exc ] ) to ( [ eq : t_coll ] ) . the survival physical radius for accretion @xmath143 is then found , for each orbital radius in the protoplanetary disk and for each timestep , by solving the equation , @xmath144 where @xmath43 is given by eq . ( [ eq : e_acc ] ) . when the mean kinetic energy is larger than the strength of a planetesimal , the a collision is highly erosive . we then obtain the range @xmath125 corresponding to the highly erosive collisions by solving the equation , @xmath145 where @xmath146 is given by eq . ( [ eq : q_d ] ) . figures [ fig : accrete_gam1d-2 ] to [ fig : accrete_gam1d-4 ] show our results for three values of the turbulent excitation parameter , @xmath147 , @xmath148 ( our fiducial value ) , and @xmath149 . each figure shows , for three orbital distances , 1 , 5 and 30au , the planetesimal physical radii corresponding to accretive and erosive regions are plotted as a function of the total mass remaining in the disk . since the disk mass decreases with time as a result of viscous evolution and photoevaporation , a decrease in disk mass corresponds to evolution in time . as shown in the previous section , planetesimal accretion becomes easier as the disk becomes less massive simply because the turbulent excitation , directly proportional to the local surface density of the gas , becomes weaker . however , after some point , the disk becomes too light to provide a sufficient amount of gas to form jupiter - mass gas giants . the figures also show as thin black lines the values obtained for a disk that follows the slope in surface density versus orbital distance defined for the mmsn ( eq . [ eq : sigma_g ] ) as a function of a disk mass . the disk mass in this model is given by @xmath150 , where @xmath151 is the outer edge radius of the disk . although the original mmsn model by @xcite used @xmath152 au , we here adopt @xmath153 au ( for comparison , our fiducial alpha disk model with @xmath154k extends up to a maximum of 350au ) . these are found to be in excellent agreement with the analytical expressions derived in the previous section , with small differences arising from the simplifications inherent to the analytical approach . larger differences are found between the power - law disk and the @xmath88-disk models mostly because of the difference in slopes ( @xmath155 for the former , @xmath156 for the @xmath88-disk ) which implies that a given disk mass does not correspond to the same surface density with two models , the difference being larger at smaller orbital distances . however , the qualitative features of accretive and erosive regions are similar to each other . it should be noted that the models also differ in their temperature profiles , but this is found to be less important . in the highly turbulent regime presented in fig . [ fig : accrete_gam1d-2 ] , a self - sustained regime of accretion becomes possible only when planetesimals have become very large / massive , with sizes generally well over 100 km . this case also yields a sustained area of high erosion where the average kinetic energies of planetesimals are above their internal energies . it is difficult to imagine how planetary cores can form in this context especially if they have to grow large enough to form giant planets . with smaller perturbations from the turbulent disk ( fig . [ fig : accrete ] ) , planetesimals have more possibilities to accrete : with time , as the disk mass decreases , the inner disk rapidly moves out of the highly erosive regime , while erosion still remains important at large orbital distances . with a turbulence strength parameter @xmath157 , corresponding to a very weak turbulence ( fig . [ fig : accrete_gam1d-4 ] ) , the presence of a highly erosive regime centered around @xmath0 m planetesimals is limited only to the outer regions ( @xmath158 au ) , and the zone rapidly shrinks as the circumstellar gaseous disk disappears . in order to put these findings into context , we also show the mass of the disk when jupiter is believed to have started accreting its gaseous envelope . these values are calculated by assuming that the planet growth has been limited mostly by viscous diffusion in the disk , with the protoplanet capturing between 10% and 70% of the mass flux at its orbital distance in the disk evolution model by @xcite ( for details , see * ? ? ? * ) . if giant planets have to form , at some time corresponding to the disk mass interval defined by the hashed areas in figs . [ fig : accrete_gam1d-2 ] to [ fig : accrete_gam1d-4 ] , protoplanetary cores must be already large enough to start accreting the surrounding hydrogen and helium gas . we can now define three important disk masses , and their corresponding disk ages ( with the caution that ages are inherently model - dependent and are provided here for illustrative purposes only , on the basis of our particular model of @xmath88-disk evolution with photoevaporation ) : 1 . the maximum mass of the disk , following the collapse of the molecular cloud . this mass can vary quite significantly from one disk formation / evolution model to another . for the particular model shown here , it is of the order of @xmath159 , for an age of 0.6myrs . the disk mass necessary for jupiter to grow to its present mass if it captures 10% of the mass flux at its orbital distance . for realistic disk models , this depends weakly on parameters such as @xmath88 and the disk evaporation rate . in our case , it corresponds to @xmath160 and an age of 1.95myrs . 3 . the disk mass necessary for jupiter to grow to its present mass if it captures 70% of the mass flux at its orbital distance . for our model , @xmath161 ( about 5 times the mass of jupiter ) and an age of 2.85myrs . table [ tab : radii ] provides the values of the physical radii that define the accretive regime and the highly erosive ( disruptive ) regime of figs . [ fig : accrete_gam1d-2 ] to [ fig : accrete_gam1d-4 ] , namely @xmath143 and @xmath162 . in our solar system , the existence of jupiter implies that either turbulence was low , the planet grew from a protoplanetary core formed in the inner solar system , or a mechanism was able to lead to the rapid formation of embryos larger than 240 km in radius at 5 au ( 17 km in the low - turbulence case , 1080 km in the high - turbulence case ) by the time when the disk mass had decreased to @xmath163 . in the last case , it appears that a mechanism such as the standard gravitational instability ( e.g. , * ? ? ? * ; * ? ? ? * ) would not work because of the turbulence , but formation of relatively large protoplanets in eddies or vortexes ( e.g. , * ? ? ? * ; * ? ? ? * ) is a promising possibility . we have investigated the critical physical radii for collisions between planetesimals to be accretional ( @xmath119 ) or disruptive ( @xmath125 ) in turbulent disks , as functions of turbulent strength ( @xmath164 ) , disk gas surface density , and orbital radius . the results presented here highlight the fact that mri turbulence poses a great problem for the growth of planetesimals : generally , only those with sizes larger than a few hundred km are in a clearly accretive regime for a nominal value of @xmath126 . the others generally collide with velocities greater than their own surface escape velocities . for some of them , more severely in the kilometer - size regime , collisions are likely to be disruptive . the problem is greater when the disk is still massive and at large orbital distances . also , if turbulence is stronger than @xmath121 , planetesimal accretion becomes extremely difficult . however , the rate of occurrence of extrasolar giant planets around solar - type stars is inferred to be as large as @xmath165% @xcite , and depends steeply on the metallicity of the host star @xcite . this strongly suggests that the majority of extrasolar giant planets were formed by core accretion followed by gas accretion onto the cores @xcite . thus , planetesimals should commonly grow to planetary masses before the disappearance of gas in protoplanetary disks . the possibilities to overcome the barrier are in principle as follows ( their likelihood is commented below ) : 1 . large @xmath100 : large planetesimals with sizes of 100 to 1000 km are formed directly in turbulent environment by a mechanism other than collisional coagulation , jumping over the erosive regime for physical radii . small @xmath82 : planetesimals start their accretion to planet - size only after the disk surface density of gas has declined to sufficiently small values . small @xmath65 : planetesimals form in mri - inactive regions ( `` dead zones '' ) of protoplanetary disks . concerning point 1 , the first - born planetesimals with sizes larger than @xmath143 may be formed rapidly by an efficient capture of @xmath166meter - size boulders in vortexes @xcite . such large planetesimals may be consistent with the size distribution of asteroids ( morbidelli et al . even if the first - born planetesimals are not as large , a small fraction of them could continue to grow larger than @xmath143 by accreting smaller bodies , because accretion is not completely cut off as soon as @xmath167 ( there is always a small possibility for accretion ) and the large planetesimals would not be disrupted by smaller ones . this possibility , however , must be examined by a more detailed growth model taking into account the effect of fragmentation and the size distribution of planetesimals , which we neglected in this paper . concerning point 2 , we have shown that planetesimals are most fragile at early times , in massive disks , and at large orbital distances . we therefore suggest that the growth towards planet sizes may be delayed due to mri turbulence , and then proceed from inside out : planetesimals should start accretion first close to the star , then progressively at larger orbital distances , as the gas surface density declines . the possibility to delay planet formation while keeping non - migrating km - size planetesimals is noteworthy because it would help planetary systems resisting to type - i migration : they would grow in a gas disk that is less dense , and for which migration timescales may be considerably increased . @xcite , @xcite , and @xcite showed that type - i migration must be lowered by one to two orders of magnitude from the linear calculation @xcite to provide an explanation for the existence of a population of giant planets in agreement with observations . this `` late formation '' scenario is consistent with the noble gas enrichment in jupiter @xcite . however , in order to form gas giants , core accretion and gas accretion onto the cores must proceed fast enough to capture jupiter - mass amount of gas from the decaying gas disk . once the size of the largest planetesimals exceeds @xmath1 km , their eccentricities are damped by tidal drag and dynamical friction from small bodies . most of the other small bodies may be ground into sizes smaller than 1 km and their eccentricities could be kept very small by gas drag and collision damping . this could facilitate the runaway accretion of cores to become large enough ( @xmath168 ) for the onset of runaway gas accretion . this issue also has to be addressed by a detailed planetesimal growth model taking into account a size distribution . the likelihood of relatively rapid gas accretion without long `` phase 2 '' is discussed by @xcite . if planetesimal sizes are relatively small , gas drag damping opens up a gap in the planetesimal disk around the orbit of a core and truncates planetesimal accretion onto the core . the truncation of heating due to planetesimal bombardment enables the core to efficiently accrete disk gas . concerning point 3 , the mri inactive region ( `` dead zones '' ) may exist in inner disk regions in which the surface density is large enough to prevent cosmic and x rays from penetrating the disk @xcite . the preservation of a dead zone can also contribute to stall type - i migration by converting it to type - ii migration @xcite or by creating a local region with a positive radial gradient of disk pressure near the ice line @xcite . however , dead zones can be eliminated by turbulent mixing / overshoot @xcite , a self - sustaining mechanism @xcite , and dust growth @xcite . the last effect comes form the fact that small dust grains are the most efficient agents for charge recombination . according to grain growth , the ionization of the disk and its coupling with the magnetic field become stronger to activate mri turbulence . we remark that if mri turbulence is activated , collisions are disruptive and they re - produces small grains to decrease the ionization degree . this self - regulation process might maintain a marginally dead state and keep producing small dust grains . this might be related with relative chronological age difference ( @xmath96myr ) between chondrules and cais ( e.g. , * ? ? ? * ) . whether dead zones exist or not is one of the biggest issues in evolution of protoplanetary disks and planet formation a more detailed analysis of planetesimal accretion in turbulent disks could impose a constraint on this issue . at large orbital distances ( 10 s of au ) , the existence of a highly erosive regime that lasts until late in the evolution of the protoplanetary disk is an important feature of this scenario . it shows that the entire mass of solids is highly reprocessed by collisions , in qualitative agreement with the paucity of presolar grains ( intact remnants from the molecular cloud core ) found in meteorites . it also prevents the growth of large planetesimals and helps to maintain a large population of small grains in the disks . this is in qualitative agreement with observations that do not indicate a significant depletion of micron - sized grains with time , contrary to what would be predicted in the absence of turbulence @xcite . in conclusion , the existence of mri turbulence may be a threat to planetesimal accretion . given the uncertainties related to these explanations , we can not provide a definitive scenario for the formation of protoplanetary cores . however , it offers several promising hints to explain important features of planet formation as constrained by today s observations of protoplanetary disks , exoplanets and meteoritic samples in the solar system . this research was supported by the sakura program between japan and france , and by the cnrs interdisciplinary program _ `` origine des plantes et de la vie '' _ through a grant to t.g . here we derive the @xmath75-dependence in eq . ( [ eq : e_random ] ) . if the equation of motion is scaled by a reference radius @xmath169 and @xmath170 where @xmath170 is a keplerian period at @xmath169 , the only remaining non - dimensional parameter in the equations is @xmath171 ( see eqs . [ 4 ] , [ 5 ] , [ 6 ] in @xcite ) . consider the equation of motion scaled by @xmath169 and @xmath170 and that scaled by @xmath172 and @xmath173 . if @xmath65 is the same and @xmath174 , these two scaled - equations of motion are identical and evolution of eccentricity , which is a non - dimensional quantity , must be identical in terms of the scaled time . note that the magnitude of excited @xmath90 should be proportional to @xmath175 . since @xmath176 and ogihara et al.s eq . ( 34 ) derived for @xmath177au is proportional to @xmath178 , the formula for arbitrary @xmath75 is given by replacing a year by @xmath179 and @xmath65 by @xmath180 in their equation . as a result , @xmath181 where @xmath63 is @xmath52 at 1au with @xmath64 ( eq . [ [ eq : sigma_g ] ] ) and the numerical factor was corrected as explained in the footnote in 3.1 . assuming the simple power - law model defined by eq . ( 7 ) , @xmath182 & & & + @xmath183 & @xmath143 & @xmath162 & @xmath143 & @xmath162 & @xmath143 & @xmath162 + @xmath184 $ ] & [ km ] & [ km ] & [ km ] & [ km ] & [ km ] & [ km ] + + 0.25 & 280 . & @xmath185 $ ] & 1280 . & @xmath186 $ ] & 1680 . & @xmath187 $ ] + 0.035 & 86 . & @xmath188 & 440 . & @xmath189 $ ] & 850 . & @xmath190 $ ] + 0.0054 & 46 . & @xmath188 & 240 . & @xmath191 $ ] & 590 . & @xmath192 $ ] + + 0.25 & 16 . & @xmath188 & 150 . & @xmath193 $ ] & 590 . & @xmath194 $ ] + 0.035 & 3.9 & @xmath188 & 36 . & @xmath188 & 220 . & @xmath195 $ ] + 0.0054 & 1.7 & @xmath188 & 17 . & @xmath188 & 103 . & @xmath188 + + 0.25 & 2540 . & @xmath196 $ ] & 3920 . & @xmath197 $ ] & 4250 . & @xmath198 $ ] + 0.035 & 870 . & @xmath199 $ ] & 1590 . & @xmath200 $ ] & 2180 . & @xmath201 $ ] + 0.0054 & 510 . & @xmath202 $ ] & 1080 . & @xmath203 $ ] & 1580 . & @xmath204 $ ] + + 0.25 & 280 . & @xmath205 $ ] & 1280 . & @xmath206 $ ] & 1680 . & @xmath207 $ ] + 0.035 & 86 . & @xmath188 & 440 . & @xmath208 $ ] & 850 . & @xmath209 $ ] + 0.0054 & 46 . & @xmath188 & 240 . & @xmath210 $ ] & 590 . & @xmath211 $ ] + + 0.25 & 280 . & @xmath212 $ ] & 1280 . & @xmath213 $ ] & 1680 . & @xmath214 $ ] + 0.035 & 86 . & @xmath215 $ ] & 440 . & @xmath216 $ ] & 850 . & @xmath217 $ ] + 0.0054 & 46 . & @xmath188 & 240 . & @xmath218 $ ] & 590 . & @xmath219 $ ]

we study the conditions for collisions between planetesimals to be accretional or disruptive in turbulent disks , through analytical arguments based on fluid dynamical simulations and orbital integrations . in turbulent disks , the velocity dispersion of planetesimals is pumped up by random gravitational perturbations from density fluctuations of the disk gas . when the velocity dispersion is larger than the planetesimals surface escape velocity , collisions between planetesimals do not result in accretion , and may even lead to their destruction . in disks with a surface density equal to that of the `` minimum mass solar nebula '' and with nominal mri turbulence , we find that accretion proceeds only for planetesimals with sizes above @xmath0 km at 1au and @xmath1 km at 5au . we find that accretion is facilitated in disks with smaller masses . however , at 5au and for nominal turbulence strength , km - sized planetesimals are in a highly erosive regime even for a disk mass as small as a fraction of the mass of jupiter . the existence of giant planets implies that either turbulence was weaker than calculated by standard mri models or some mechanism was capable of producing ceres - mass planetesimals in very short timescales . in any case , our results show that in the presence of turbulence planetesimal accretion is most difficult in massive disks and at large orbital distances .

the observational goals of the sdss have remained stable since its early days : ( 1 ) @xmath5,@xmath6,@xmath7,@xmath8,@xmath9 ccd imaging of @xmath10 in the north galactic cap , to a depth @xmath11mag in the most sensitive bands , ( 2 ) a spectroscopic survey in this area of @xmath12 galaxies , @xmath13 quasars , and an assortment of stars and other targets , and ( 3 ) imaging of three @xmath14 stripes in the south galactic cap , with the equatorial stripe scanned repeatedly to allow variability studies and co - added imaging that goes @xmath15mag deeper than a single scan . the normal spectroscopic program is carried out on all three stripes in the south , and additional spectroscopy in the equatorial stripe will provide deeper samples of quasars and galaxies and more comprehensive coverage of stellar targets . the survey is carried out using a dedicated 2.5-m telescope on apache point , new mexico , equipped with a mosaic ccd camera ( gunn et al . 1998 ) and two fiber - fed double spectrographs that can obtain 640 spectra simultaneously ( uomoto et al . , in preparation ) . a technical overview of the survey appears in york et al . ( 2000 ) , and an updated but more focused technical summary appears in the paper by stoughton et al . ( 2002 ) that describes the early data release . the quasar survey is reviewed by schneider et al . ( these proceedings ) , and the quasar target selection is described by richards et al . there are two samples in the galaxy redshift survey , a magnitude limited sample to @xmath16 comprising 90% of the galaxy targets ( strauss et al . 2002 ) , and a sparser , deeper sample of luminous red galaxies ( eisenstein et al . 2001 ) . after a decade of preparatory work , the sdss formally began operations on the auspicious date of april 1 , 2000 . it is planned to run until summer , 2005 . the total area covered will depend on the weather between now and then , especially the amount of weather that satisfies the seeing and photometric requirements of the imaging survey . a reasonable guess is that the northern survey will cover @xmath17 of the original @xmath10 goal by summer , 2005 . as of january , 2002 , the sdss had obtained @xmath18 square degrees of imaging and @xmath19 spectra ( of which about 80% are galaxies ) , including both northern and southern observations . the quality of the spectra , which cover the wavelength range 3800 to 9200 at resolution @xmath20 , is spectacular . redshift completeness for spectroscopically observed galaxies is over 99% , and for most galaxies the spectra yield stellar velocity dispersions and valuable diagnostics of the stellar populations . the scientific analyses to date are only scratching the surface of what the spectroscopic data allow . in june , 2001 , the sdss released @xmath21 of imaging data ( @xmath22 million detected objects ) and @xmath23 follow - up spectra obtained during commissioning observations and the first phases of the survey proper , as documented by stoughton et al . in addition to providing data for the larger community , the early data release is a training exercise for the sdss collaboration . one lesson is that releasing data as complex as that in the sdss ( e.g. , radial profiles plus over 80 measured parameters for each photometric object , some of which are deblended , some imaged on more than one scan , some observed spectroscopically ) in a useful way is very challenging , even within the collaboration itself . the first data release will take place in january , 2003 , and subsequent releases will take place on a roughly annual basis ( see http://www.sdss.org/science/index.html ) . this paper is based on two talks that i gave at the _ new era in cosmology _ conference . section 2 reviews some of the recent sdss results on galaxies and large scale structure ; the topics mentioned are some of the ones that i have found interesting myself , and they by no means constitute a comprehensive list . all of these results are published or available on astro - ph , so my summaries are brief and do not include figures . in section 3 , i discuss what we can hope to learn from studying galaxy clustering in the sdss and the 2dfgrs , with focus on the halo occupation distribution as a way of thinking about the relation between galaxies and dark matter . this discussion is based on collaborative work with andreas berlind , zheng zheng , jeremy tinker , and others . the first science results from the sdss that really surprised me were the discoveries , made independently by yanny et al . ( 2000 ) using a - colored stars and by ivezi ' c et al . ( 2000 ) using rr lyrae candidates , of coherent , unbound structures in the milky way s stellar halo , stretching across tens of degrees . the idea that the stellar halo might be built by mergers of dwarf galaxies is an old one ( searle & zinn 1978 ) , and much of the recent theoretical modeling has focused on detecting fossil substructure through phase space studies of the local stellar distribution ( e.g. , johnston et al . 1995 ; helmi & white 1999 ; helmi & de zeeuw 2000 ) . even in the absence of kinematic data , the sdss is a powerful tool for detecting substructure in the outer halo because multi - color imaging allows the definition of samples of stars that are approximately standard candles . the two structures found by yanny et al . ( 2000 ) , in an area @xmath24 of the sky , may both be associated with the tidal stream of the sagittarius dwarf galaxy ( ibata et al . however , a recent study using f - stars ( newberg et al . 2001 ) appears to show several more substructures , and no clear indication that there is a smooth underlying halo at all . extending a model originally developed to investigate the dwarf satellite problem , bullock , kravtsov , & weinberg ( 2001 ) showed that the population of disrupted dwarfs expected in the cdm cosmological scenario could naturally account for the entire stellar halo . if this model is right , then the sdss should reveal ubiquitous substructure in the outer halo , where orbital times are long and the number of discrete streams is relatively small . in any event , the sdss imaging survey will answer fundamental questions about the origin of the milky way s stellar halo , and perhaps about the amount of power on sub - galactic scales in the primordial fluctuation spectrum . with multi - color data , one can define optimal filters to maximize the contrast between an object in the milky way or local group and the foreground and background stellar distributions . odenkirchen et al . ( 2001ab ) have developed this technique and applied it to great effect in studies of the globular cluster pal 5 and the dwarf spheroidal draco . pal 5 shows two well defined tidal tails that contain @xmath25 of the cluster s stars , demonstrating that the cluster is subject to heavy mass loss . the orientation of the tails reveals the projected direction of the cluster s orbit , and peaks within the tails may be a signature of disk shocking events . draco , on the other hand , shows no sign of tidal extensions even at surface densities @xmath26 of the central value , demonstrating that it is a bound , equilibrium system , and justifying standard kinematic estimates that yield a high mass - to - light ratio , @xmath27 . as the sdss covers more sky , we will get a more complete census of which objects are being tidally destroyed and which are still holding themselves together . tidal tails and tidal streams may also provide valuable constraints on the radial profile , shape , and clumpiness of the milky way s dark halo potential ( see , e.g. , johnston et al . 1999 ) . blanton et al . ( 2001 ) measured the galaxy luminosity function in @xmath5 , @xmath6 , @xmath7 , @xmath8 , and @xmath9 using a sample of 11,275 galaxies observed during sdss commissioning observations . the large sample size , accurate photometry , and use of the petrosian ( 1976 ) system for defining galaxy magnitudes yield small statistical errors and excellent control of systematic effects . this analysis confirms and quantifies previous indications that the galaxy luminosity function varies systematically with surface brightness , color , and morphology ; the first correlation implies that low surface brightness galaxies make only a small contribution to the mean luminosity density of the universe . the measured luminosity density exceeds the @xmath28-band estimate from the las campanas redshift survey ( lin et al . 1996 ) by a factor of two , and blanton et al . show that this difference arises from the isophotal magnitude definition adopted by the lcrs , which misses light in the outer parts of galaxies that have intrinsically low or cosmologically dimmed surface brightness . norberg et al . ( 2001b ) demonstrate convincingly that the high - precision measurement of the @xmath29-band luminosity function from the 2dfgrs is in good agreement with the luminosity function measured by the sdss . two main factors caused blanton et al . ( 2001 ) to reach a contrary conclusion : they used an inaccurate conversion from sdss bands to @xmath29 , and their maximum likelihood method effectively estimates the luminosity function at a redshift @xmath30 , while the method used by norberg et al . ( 2001b ) implicitly corrects for evolution to derive the @xmath31 luminosity function . using the sdss early release data , norberg et al . ( 2001b ) demonstrate excellent agreement in the mean between sdss and 2dfgrs galaxy magnitudes and redshifts , and they confirm earlier estimates of the completeness and stellar contamination of the 2dfgrs input catalog . while the optical luminosity function estimates are now in good agreement , there remains a puzzling discrepancy pointed out by wright ( 2001 ) between the luminosity density found in the optical bands by the sdss and the estimates of the @xmath32-band luminosity density from the 2dfgrs and 2mass data ( cole et al . 2001 ; see also kochanek et al . norberg et al . ( 2001b ) argue that the discrepancy is probably dominated by large scale structure fluctuations in the area used to normalize the @xmath32-band luminosity function . the sdss data are ideal for studying the correlations of galaxy properties , since the photometric and spectroscopic reduction pipelines measure many quantities automatically and one can create large samples with well understood selection effects . the main effort to date in this area is the comprehensive study of a sample of 9000 early type galaxies by bernardi et al . they determine the fundamental plane in the @xmath6 , @xmath7 , @xmath8 , and @xmath9 bands , and they measure bivariate correlations among luminosity , size , velocity dispersion , color , mass - to - light ratios , and spectral indices . the large sample size and high precision allow examination of relatively subtle effects , such as a slight difference in the fundamental plane of `` field '' and `` cluster '' galaxies . the evolution of the fundamental plane over the redshift range of the sample ( which extends to @xmath33 ) is consistent with passive evolution of old stellar populations . one of the most dramatic breakthroughs from the sdss has been the measurement of galaxy mass profiles and the galaxy - mass correlation function via galaxy - galaxy weak lensing . systematic effects are much easier to control for galaxy - galaxy lensing than for cosmic shear measurements because image distortion is measured perpendicular to the radial separation vector , which has a different orientation for each foreground - background pair . the large area gives the sdss great statistical power despite its rather shallow imaging ( by weak lensing standards ) . fischer et al.s ( 2000 ) analysis of two nights of sdss imaging data ( 225 deg@xmath34 ) was at the time the clearest detection of a galaxy - galaxy lensing signal , with extended shear profiles ( to @xmath35kpc ) offering direct evidence for the extended dark matter halos expected in standard models of galaxy formation . mckay et al . ( 2001 ) have since analyzed a sample of @xmath36 source galaxies around @xmath37 foreground galaxies in the spectroscopic sample , measuring the galaxy - mass correlation function and its dependence on galaxy luminosity , morphology , and environment . they find that the mass within an aperture of @xmath38kpc scales linearly with galaxy luminosity , that the excess mass density around galaxies in high density regions remains positive to @xmath39mpc while that around isolated galaxies is undetectable beyond @xmath40kpc , and that early type galaxies have a higher amplitude galaxy - mass correlation function , in part because of their preferential location in group environments . guzik & seljak ( 2002 ) have modeled the mckay et al . results to infer that @xmath41 galaxies have virial masses @xmath42 , implying that a large fraction of the baryons within the virial radius of an @xmath41 galaxy halo end up as stars in the central galaxy . by comparing to the tully - fisher relation , seljak ( 2002 ) concludes that circular velocities at the halo virial radius are typically a factor @xmath43 below the values measured at the galaxy optical radius , and in reasonably good agreement with predictions based on cdm halo profiles . early efforts to study galaxy clustering with the sdss have focused on the analysis of a @xmath14 stripe of imaging data that has been closely examined and reduced multiple times . scranton et al . ( 2001 ) carried out an exhaustive analysis of possible systematic effects associated with seeing variations , stellar density , galactic reddening , galaxy deblending , variations across the imaging camera , and so forth , showing that they have no significant impact on measurements of the angular correlation function at the obtainable level of statistical precision . these experiments demonstrate that star - galaxy separation in the sdss imaging works extremely well to @xmath44 . dodelson et al . ( 2001 ) modeled the measurements of the angular correlation function ( connolly et al . 2001 ) and the angular power spectrum ( tegmark et al . 2001 ) , to infer the 3-dimensional clustering of galaxies . their results , @xmath45 , @xmath46 , are consistent with those obtained by szalay et al . ( 2001 ) applying a different method , karhunen - loeve parameter estimation , to the same galaxy catalog . the statistical error bars from these analyses are not yet competitive with the highest precision analyses of the galaxy power spectrum ( e.g. , percival et al . 2001 ) , but they provide reassuring evidence that any systematic biases in the sdss imaging data , and thus in the input to the redshift survey , are well controlled . recently szapudi et al . ( 2001 ) have analyzed the higher order angular moments of this data set , finding agreement with the hierarchical scalings and values of skewness and kurtosis parameters predicted by @xmath47cdm models that incorporate mild suppression of galaxies in high mass halos . with the analysis tools now developed and tested and the systematic issues apparently well understood , the analyses of larger sky areas should soon yield precise measurements of angular clustering over a wide dynamic range . zehavi et al . ( 2001 ) carried out the first analysis of clustering in the sdss redshift survey , focusing on the real space correlation function @xmath48 and the pairwise velocity dispersion @xmath49 for different classes of galaxies , with a sample similar in size and geometry to the lcrs . galaxies in absolute magnitude bins centered on @xmath50 , @xmath51 , and @xmath52 have parallel power - law correlation functions with slopes @xmath53 , but their amplitudes are significantly different , with @xmath54 , 6.3 , and 7.4@xmath55mpc , respectively . the correlation function of red galaxies is both steeper and higher amplitude than that of blue galaxies . the pairwise dispersion for the full sample is @xmath56 at @xmath57mpc , but red galaxies have @xmath58 and blue galaxies only @xmath59 . the dependence of @xmath48 on galaxy properties resembles that found by norberg et al . ( 2001ac ) in the 2dfgrs ( and in earlier studies such as guzzo et al . 1997 ) , but there are significant differences of detail . the sdss data show a steady trend of correlation strength with luminosity , while norberg et al . ( 2001a ) find a transition from weak dependence below @xmath41 to strong dependence above @xmath41 . norberg et al . ( 2001c ) find similar @xmath48 slopes for galaxies of different spectral types , while the correlation function of blue galaxies in the sdss is clearly shallower than that of red galaxies . analysis of larger sdss samples should clarify the significance of these differences ; the first could reflect the difference between @xmath7-band and @xmath29-band selection , and the second could represent a difference between color and spectral type as a basis for galaxy classification . a consequence of the sdss observing strategy ( dictated largely by the instruments themselves ) is that the early redshift data had a 2-dimensional slice geometry , making it difficult to study the large scale power spectrum and statistics that require contiguous 3-d volumes , like void probabilities and topology . that situation is changing as the survey progresses , and first results on these topics should emerge over the next several months . the sdss collaboration involves hundreds of scientists , with eleven participating institutions on three continents . with such a large and far - flung collaboration , we spend a lot of energy just keeping ourselves organized . i have just finished my term as the sdss scientific publications coordinator , a position with one chief benefit : i was forced to pay attention as the scientific output of the sdss grew from a trickle to a flood , spreading rivulets into many different areas of astronomy . this development has been exciting to watch , and i have learned a lot of astronomy just by following it . while there are certainly challenges of communication in a collaboration this large , the process of going from data to science has worked , in my opinion , remarkably well . the ideal scenario is that each scientific analysis draws on the collective expertise of a very broad spectrum of astronomers ; i have been delighted to see how often we approach this ideal in practice . the richness of the sdss data is more than enough to keep us busy . indeed , while i am sure that we look enormous from the outside , it is constantly evident from the inside that we do nt have enough people to do all the science we would like to be doing . that , of course , is one of many good reasons for publishing the data . before completely shifting gears , let me pause to congratulate the members of the 2df galaxy and quasar redshift surveys for ( a ) obtaining more than 200,000 spectra , ( b ) publishing more than 100,000 spectra , and ( c ) writing a number of beautiful papers analyzing the results and implications . all three of these are great achievements . while the sdss and 2df teams can not help but see themselves in competition every now and then , the benefits to astronomy of having these independent data sets and independent analyses are already very clear . the above question is one has been pondered by many people over several decades . two developments that color recent considerations of this subject are the extraordinary improvements in the quantity and quality of the redshift survey data and the convergence of the cosmological community on a `` standard '' model , @xmath47cdm , that is supported by an impressive base of observational evidence . a third development that has deeply affected my own thinking is the emergence of a new way of describing galaxy bias , the halo occupation distribution ( hod ) . the hod characterizes the statistical relation between galaxies and mass in terms of the probability distribution @xmath0 that a halo of virial mass @xmath1 contains @xmath2 galaxies , together with prescriptions that specify the relative spatial and velocity distributions of galaxies and dark matter within these halos . note that `` halo '' here refers to a structure of typical overdensity @xmath60 , in approximate dynamical equilibrium ; higher density cores within a group or cluster are , in this description , treated as substructure , and characterized only in a statistical sense . since different types of galaxies have different space densities and different clustering properties , a given hod applies to a specific class of galaxies , e.g. , red galaxies brighter than @xmath41 , or late - type spirals with @xmath7 magnitudes @xmath61 to @xmath62 . the hod framework has roots in early analytic models that described galaxy clustering as a superposition of randomly distributed clusters with specified profiles and a range of masses ( neyman & scott 1952 ; peebles 1974 ; mcclelland & silk 1977 ) . a bevy of recent papers have shown that , when combined with numerical or analytic models of the clustering of the halos themselves , the hod is a powerful tool for analytic and numerical calculations of clustering statistics , for modeling observed clustering , and for characterizing the results of semi - analytic or numerical studies of galaxy formation ( e.g. , jing , mo , & brner 1998 ; benson et al . 2000 ; ma & fry 2000 ; peacock & smith 2000 ; seljak 2000 ; berlind & weinberg 2001 ; marinoni & hudson 2001 ; scoccimarro et al . 2001 ; yoshikawa et al . 2001 ; white , hernquist , & springel 2001 ; bullock , wechsler , & somerville 2002 ) . my own interest in this approach was spurred largely by the paper of benson et al . ( 2000 ) , who discussed the clustering predictions of their semi - analytic model of galaxy formation in these terms . a forthcoming paper by berlind et al . ( in preparation ; see also berlind 2001 ) compares the predictions of the benson et al . semi - analytic formalism to those of a large , smoothed particle hydrodynamics ( sph ) simulation ( murali et al . 2001 ; dav et al . , in preparation ) , for the same cosmological model . the agreement between the two approaches is remarkably good . if we select galaxies above a specified baryon mass threshold , chosen separately in the two calculations so that the space densities of the populations are equal , then the mean occupation @xmath3 is a non - linear function of mass with three basic features : a cutoff mass below which halos are not massive enough to host a galaxy above the threshold , a low occupancy regime ( @xmath63 ) in which the mean occupation grows slowly with increasing halo mass but the average galaxy mass itself increases , and a high occupancy regime in which @xmath3 grows more steeply with mass , though the growth is still sub - linear because larger , hotter halos convert a smaller fraction of their baryons into galaxies . in the low occupancy regime , the fluctuations about the mean are well below those of a poisson distribution a halo that is supposed to host one galaxy very rarely hosts two and the sub - poisson nature of these fluctuations has a crucial impact on some clustering statistics . the hod is strongly dependent on the age of the galaxies stellar populations ; old galaxies like to live together in massive , high occupancy halos , while young galaxies studiously avoid them . the sph simulations further show that the oldest , most massive galaxy in a halo usually resides near the halo center and moves at close to the center - of - mass velocity , while the remaining galaxies approximately trace the spatial and velocity distribution of the halo s dark matter . the agreement between the semi - analytic and sph calculations , despite some clear differences in the way that they treat radiative cooling and feedback from star formation , suggests that the hod emerges from fairly robust physics that both methods do right , given their common assumptions . whether these assumptions hold in the real universe is , of course , one of the things we hope to learn . figure 1 sketches the interplay between the `` cosmological model '' and the `` galaxy formation theory '' in determining galaxy clustering ( which i take to include the galaxy - mass correlations measured by weak lensing ) . the hod approach suggests a nice division of labor between these two theoretical inputs . the cosmological model , which specifies the initial conditions ( e.g. , scale - invariant fluctuations from inflation ) and the matter and energy contents ( e.g. , @xmath4 , @xmath64 , @xmath65 , @xmath66 , @xmath67 ) , determines the mass function , spatial correlations , and velocity correlations of the dark halo population . at our adopted overdensity threshold @xmath60 , these properties of the halo population are determined almost entirely by gravity , with no influence of complex gas physics . i have inserted a box in the path between cosmological model and dark halo population to indicate that the only features of the cosmological model that really matter in this context are @xmath4 , the fluctuation amplitude ( represented here by @xmath68 ) , and the power spectrum shape ( represented here by @xmath69 and @xmath70 , though it could , of course , be more interesting ) . other features of the cosmological model , such as the energy density and equation of state of the vacuum component , may have an important impact on other observables or on the _ history _ of matter clustering , but they have virtually no effect on the halo population at @xmath31 , if the shape of @xmath71 and the present day value of @xmath68 are held fixed . the galaxy formation theory incorporates the additional physical processes such as shock heating , radiative cooling , conversion of cold gas into stars , and feedback of star formation on the surrounding gas that are essential to producing distinct , dense , bound clumps of stars and cold gas . it further specifies what aspects of a galaxy s formation history determine its final mass , luminosity , diameter , color , morphology , and so forth . these physical processes operate in the background provided by the evolving halo population , so the predicted hod depends on both the theory of galaxy formation and the assumed cosmological model . as a description of bias , the crowning virtue of the hod is its completeness : given a dark halo population and a fully specified hod , one can predict the value of any galaxy clustering statistic , on any scale , using analytic approximations and/or numerical simulations . berlind & weinberg ( 2001 ) examined the influence of the hod on galaxy clustering and galaxy - mass correlations , for the halo population of a @xmath47cdm n - body simulation . we found that different clustering statistics , or even the same statistic at different scales , are sensitive to different aspects of the hod . for example , at large scales @xmath72 is proportional to the mass correlation function @xmath73 , with a bias factor equal to the average of the halo bias factor @xmath74 weighted by the halo number density and the mean occupation @xmath3 . on small scales , however , the explicit dependence on @xmath73 disappears , and @xmath72 depends on the halo mass function , on the mean number of pairs @xmath75 as a function of halo mass and virial radius , and ( to a lesser extent ) on the internal bias between galaxy profiles and mass profiles . connecting these pieces into a power - law galaxy correlation function is a rather delicate balancing act , and the success of sph simulations and semi - analytic models in reproducing the observed form of @xmath72 given a @xmath47cdm cosmology is entirely non - trivial ; the reduced efficiency of galaxy formation in high mass halos and the sub - poisson fluctuations in low mass halos are both crucial to this success . higher order correlation functions place greater weight on the high mass end of the halo population and on higher moments of @xmath0 . the void probability function , on the other hand , depends mainly on the low mass cutoff of the hod , which determines the probability of finding galaxies in the low mass halos that populate large scale underdensities . the pairwise velocity dispersion has distinct regimes much like @xmath72 , but it depends little on the low mass cutoff and strongly on the relative occupation of high and low mass halos , and the sub - poisson fluctuations that depress @xmath72 at small scales _ boost _ the pairwise dispersion by forcing those pairs that do exist at these separations to come from higher mass halos . the pairwise dispersion can also be influenced by velocity bias of galaxies within halos . the group multiplicity function bears a quite direct relation to the hod , to such an extent that one can `` read off '' @xmath3 if @xmath0 is reasonably narrow and one assumes an underlying halo mass function @xmath76 . peacock & smith ( 2000 ) and marinoni & hudson ( 2001 ) have applied variants of this approach to observational data and obtained results that agree rather well with the sph and semi - analytic predictions , assuming a @xmath47cdm halo mass function . berlind and i concluded that an empirical determination of the hod should be possible given high precision clustering measurements and the halo population of an assumed cosmological model . this , at a minimum , is what we can expect to learn from galaxy clustering : the halo occupation distributions of many different classes of galaxies , given a cosmological model motivated by independent observations . because the hod description of bias is complete , these hods encode everything that galaxy clustering has to teach us about galaxy formation . they encode it , moreover , in a physically informative way , allowing detailed tests of theoretical predictions and providing rather specific guidance when these predictions fail . if your theory of galaxy formation does almost everything right but puts too many blue - ish s0 galaxies in @xmath77 halos , then you might have some ideas on how to fix it . can we have our cake and eat it too ? in more precise words , if we find a combination of cosmological model and hod that matches all the galaxy clustering data , can we conclude that both are correct , or might there be other combinations that are equally successful ? to decide whether cosmology and bias are degenerate with respect to galaxy clustering , we must first know how changing the cosmology alters the halo population . this issue is the subject of a forthcoming paper by zheng et al . , where we investigate the effect of changing @xmath4 on its own , of changing @xmath4 and @xmath68 simultaneously while maintaining `` cluster normalization '' ( @xmath78constant ) , and of changing @xmath4 and @xmath68 in concert with @xmath69 or @xmath70 . the impact of a pure @xmath4 change is simple : the halo mass scale @xmath51 shifts in proportion to @xmath4 , pairwise velocities ( at fixed @xmath79 ) are proportional to @xmath80 , and halo clustering at fixed @xmath79 is unchanged . cluster normalized changes to @xmath4 and @xmath68 keep the space density of halos approximately constant near @xmath81 , and halo clustering and pairwise velocities remain similar at fixed @xmath1 . however , the shape of the halo mass function changes , with a decrease of @xmath4 from 0.3 to 0.2 producing a @xmath82 drop in the number of low mass halos . one can preserve the shape of the mass function over a large dynamic range by changing @xmath69 or @xmath70 , but the required changes are substantial e.g. , masking a decrease of @xmath4 from 0.3 to 0.2 requires @xmath83 or @xmath84 . these changes to the power spectrum significantly alter the halo clustering and halo velocities . the sensitivity of the halo population to the cosmological model parameters is encouraging , because these changes can not easily be masked by changing the hod . for a pure @xmath4 shift , one could keep the spatial clustering of galaxies the same by using the same hod as a function of @xmath79 , but the change would be detected by any dynamically sensitive clustering statistic , like large scale redshift - space distortions , the pairwise velocity dispersion , the galaxy - mass correlation function , or direct measurements of group and cluster masses . even velocity bias within halos could not hide all of these changes . a cluster - normalized change to @xmath68 and @xmath4 would require a change in galaxy occupation as a function of @xmath79 in order to maintain the galaxy space density and group multiplicity function , and this change would affect other measures of galaxy clustering . a simultaneous change to the power spectrum shape that preserved the halo mass function would change galaxy clustering by changing the clustering of the halos themselves . it remains to be seen just how well one can do quantitatively from realistic observations . the proof , ultimately , must await the pudding , but zheng has begun to investigate the question in a somewhat idealized context . as a starting point , he takes clustering measures predicted by a @xmath47cdm cosmology with the hod derived from the sph simulations , calculated by a variety of analytic approximations . he then changes the assumed cosmology , thus changing the halo mass function and halo clustering , and he allows the hod to change as well , using a parametrized form that gives flexibility in all of the essential features . he finds the hod that gives minimum @xmath85 for the original clustering `` measurements , '' which are assumed to have 10% fractional uncertainties , and the value of @xmath86 for the best - fit hod indicates the acceptability of the cosmological model . the preliminary results from this exercise are encouraging . for example , in the case of pure @xmath4 changes , the galaxy correlation function and group multiplicity function constrain the hod tightly enough that measurements of @xmath87 or the pairwise velocity dispersion impose useful constraints on @xmath4 . as the sdss and 2dfgrs measurements take shape , we can imagine taking a similar approach to the real data , albeit with careful attention to the accuracy of the clustering approximations . in terms of figure 1 , the surveys provide us with the entries in the lowest box , and using them , we search for maximum likelihood solutions for the parameters in the second boxes on the left and right hand sides . despite what might at first appear to be a lot of freedom , the degeneracies appear to be limited , and we can hope to do rather well . here , then , is my conjectural answer to the question posed in the section title : we can learn the hod of different classes of galaxies , gaining physical insight into their origin , and we can separately determine @xmath4 and the amplitude and shape of the linear theory power spectrum , from the largest scales probed by the surveys ( where perturbation theory describes the dark matter and the hod fixes the `` bias factors '' needed to connect galaxies to mass ) down to moderately non - linear scales ( below which information about the linear power spectrum may be effectively erased , at least as far as the halo mass function and halo clustering are concerned ) . we can also test for any departures from gaussian primordial fluctuations . we get these cosmological constraints _ without _ relying on a detailed theory of galaxy formation , only on the basic tenet that the hod formulation itself is valid . while we might be wary of relying on conclusions that involve complicated corrections for galaxy bias , the observed dependence of clustering on galaxy type allows powerful cross - checks . when we analyze different classes of galaxies , we should derive different hods , but we should always reach the same conclusions about the underlying cosmological model . if we do , then we have good reason to think that we are doing things right . given all the other methods that can constrain cosmology with tracers that are less physically complicated , one might wonder what galaxy clustering and galaxy - mass correlations have to contribute to cosmological tests , beyond a reassuring consistency check . after all , how important is the second decimal place on @xmath4 ? while i hear variants of this question often , i think it is a red herring , and that we should be relentless in our efforts to squeeze as much cosmological information as possible out of galaxy redshift surveys . even if we assume that there will be no major conceptual adjustments to the current leading model , there are at least two fundamental issues on which precision measurements from galaxy clustering can play a critical role : the contribution ( if any ) of gravity waves to cmb anisotropy , and the equation of state of dark energy . the first can be addressed by a precise comparison between the cmb fluctuation amplitude and the present day amplitude of matter fluctuations . evidence for or against gravity waves would take us much further in understanding the origin of density fluctuations , and perhaps even to understanding the mechanism ( inflation , colliding branes , ... ) that accounts for the size and homogeneity of the universe . galaxy clustering has no sensitivity to the equation of state on its own , but the sensitivity of other tests depends crucially on precise knowledge of @xmath4 , where the combination of galaxy clustering and galaxy - galaxy lensing may ultimately provide the best constraints . precise knowledge of today s fluctuation amplitude is also essential to some tests for the equation of state and its time dependence ( see , e.g. , the discussion of kujat et al . 2001 ) . constraining gravity waves , the dark energy equation of state , and neutrino masses are concrete goals that we can set for cosmological applications of galaxy clustering . but we should not assume that the simplest model consistent with the current data ( which already contains at least one very surprising element ) will remain consistent with improving observations . a break in the inflationary fluctuation spectrum , a relativistic background inconsistent with standard neutrino physics , a baryon density inconsistent with big bang nucleosynthesis , a small admixture of non - gaussian or isocurvature fluctuations all of these are departures from the standard model whose quantitative impact would be subtle but whose physical implications would be profound . what we will learn from the 2df and sdss galaxy surveys depends in large part on what the universe has to teach us , and that is something we can not yet know . finding out is an exciting task for the new era in cosmology . i am grateful to my numerous colleagues in the sdss for producing the exciting results that i have recapitulated in 2 , and for their efforts and progress in producing a data set that warrants the theoretical musings in 3 . i thank my collaborators on the work discussed in 3 , especially andreas berlind , zheng zheng , and jeremy tinker , whose contributions to the ideas and to the results have been central . i thank the nsf for its support of this research program and the institute for advanced study and the ambrose monell foundation for hospitality and support during the recent phases of this work . more details about the sdss , including links to the early data release , an ever - growing list of scientific publications based on the sdss data , and a list of the many participating institutions and funding agencies that have made the survey possible , can be found at the official sdss web site , http://www.sdss.org .

i review some of the recent results from the sdss related to galaxies and large scale structure , including : ( 1 ) discovery of coherent , unbound structures in the stellar halo of the milky way , ( 2 ) demonstration that the pal 5 globular cluster has tidal tails and that the draco dwarf spheroidal does not , ( 3 ) precise measurement of the galaxy luminosity function and its variation with galaxy surface brightness , color , and morphology , ( 4 ) detailed examination of the fundamental plane from a sample of 9000 early type galaxies , ( 5 ) measurement , via galaxy - galaxy lensing , of the extended dark matter distributions around galaxies and their variation with galaxy luminosity , morphology , and environment , ( 6 ) measurements of the galaxy angular power spectrum and of the spatial correlation function and pairwise velocity dispersion as a function of galaxy luminosity and color . i then turn to a more abstract discussion of what we can hope to learn , in the long run , from galaxy clustering in the sdss and the 2dfgrs . the clustering of a galaxy sample depends on the mass function and clustering of the dark halo population , and on the halo occupation distribution ( hod ) , which specifies the way that galaxies populate the halos . hydrodynamic simulations and semi - analytic models of galaxy formation make similar predictions for the probability @xmath0 that a halo of virial mass @xmath1 contains @xmath2 galaxies of a specified type : a non - linear form of the mean occupation @xmath3 , sub - poisson fluctuations about the mean in low mass halos , and a strong dependence of @xmath3 on the age of a galaxy s stellar population . different galaxy clustering statistics respond to different features of the hod , making it possible to determine the hod empirically given an assumed cosmological model . furthermore , changes to @xmath4 and/or the linear power spectrum produce changes in the halo population that would be difficult to mask by changing the hod . ultimately , we can hope to have our cake and eat it too , obtaining strong guidance to the physics of galaxy formation by deriving the hod of different classes of galaxies , while simultaneously carrying out precision tests of cosmological models . # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in

in a similar manner , the time - dependent quasi - particles @xmath16 are characterized by the time - dependent wave functions @xmath17 by @xmath18 . the time evolution of the quasi - particles under a one - body external perturbation @xmath19 are determined by the following tdhfb equation . @xmath20 , \ ] ] where the tdhfb hamiltonian is given by @xmath21 here and hereafter , the constant shift is neglected , since it does not play any role in the tdhfb equation ( [ motodist ] ) . @xmath22 and @xmath23 become time - dependent , since they depend on the densities , @xmath24 and @xmath25 , which are time - dependent . note that the static quasi - particles correspond to a quasi - static solution of eq . ( [ motodist ] ) , @xmath26 , with @xmath27 . let us assume that the nucleus is under a weak external field of a given frequency @xmath28 . @xmath29 where @xmath30 and @xmath31 . a small real parameter @xmath32 is introduced for the linearization . in the small - amplitude limit , the second term ( @xmath33-part ) in eq . ( [ fomega ] ) can be omitted , because it does nt contribute in the linear approximation . the bogoliubov transformation of the external fields ( @xmath34 and @xmath35 ) is given in appendix [ app - external ] . the external perturbation @xmath19 induces a density oscillation around the ground state with the same frequency @xmath28 . the density oscillation , then , produces the induced fields in the single - particle hamiltonian , @xmath36 and in the pair potential , @xmath37 . thus , the hamiltonian , eq . ( [ tdhfb_hamiltonian ] ) , is decomposed into the static and oscillating parts ; @xmath38 . @xmath39 here , the @xmath33-part is again neglected in eq . ( [ delta_h20 ] ) . see appendix [ app - dh ] for the derivation of @xmath40 . explicit expressions for @xmath41 and @xmath42 are found in eqs . ( [ dh20 ] ) and ( [ dh02 ] ) , respectively . the time - dependent quasi - particle operators are decomposed in a similar manner : @xmath43 where @xmath44 can be expanded in the quasi - particle creation operators : @xmath45 it should be noted that @xmath46 can be expanded only in terms of the creation operators , because the annihilation operators in the right - hand side of eq . ( [ deltaa ] ) simply represent the transformation among themselves , @xmath47 , and do not affect @xmath7 and @xmath48 . the amplitudes , @xmath49 and @xmath50 , must be anti - symmetric to satisfy the fermionic commutation relation , @xmath51 . keeping only linear terms in @xmath32 , eq . ( [ motodist ] ) becomes @xmath52+[\delta h(t)+f(t ) , a_{\mu}].\ ] ] substituting eqs . ( [ f])-([deltaa ] ) into eq . ( [ eqmot ] ) , we obtain the linear - response equations : @xmath53 in eq . ( [ seconda ] ) , setting the frequency complex , @xmath54 , we can introduce a smearing with a width @xmath55 . expanding @xmath56 and @xmath57 in terms of the forward and backward amplitudes , @xmath49 and @xmath50 , we obtain a familiar expression of the equation @xcite : @xmath58 \left ( \begin{array}{c } x(\omega ) \\ y(\omega ) \end{array } \right ) \ ! = \ ! \left ( \ ! \begin{array}{c } f^{20}(\omega ) \\ f^{02}(\omega ) \end{array } \ ! \right).\ ] ] this matrix formulation requires us to calculate the qrpa matrix elements , @xmath59 and @xmath60 . this is a tedious task and their dimension , which is equal to the number of two - quasi - particle excitations , becomes huge especially for deformed nuclei . instead , in the fam @xcite , we keep the form of eq . ( [ seconda ] ) and calculate the induced fields @xmath56 and @xmath57 using the numerical differentiation . we explain this trick in the next section . the expressions for @xmath61 and @xmath62 in eq . ( [ seconda ] ) are given in eqs . ( [ dh20 ] ) and ( [ dh02 ] ) , respectively . thus , we need to calculate @xmath63 and @xmath64 for given @xmath49 and @xmath50 . we perform this calculation following the spirit of the fam @xcite . from eqs . ( [ eq - a ] ) and ( [ deltaa ] ) , we obtain the time - dependent quasi - particle wave functions : @xmath65 where @xmath66 first , let us discuss how to obtain @xmath63 . the time - dependent single - particle hamiltonian @xmath22 depends on the densities which are determined by the wave functions @xmath17 . therefore , @xmath22 can be regarded as a functional of wave functions as @xmath67 = h\left[\mathcal{u}^*(t ) , \mathcal{v}^*(t ) ; \mathcal{u}(t),\mathcal{v}(t)\right ] .\ ] ] here , it should be noted that the phase factors , @xmath68 in eq . ( [ phase_factor ] ) , do not play a role . this is because @xmath5 is a functional of densities , @xmath7 , @xmath48 , and @xmath8 , which are given by products of one of @xmath69 and one of the complex conjugate @xmath70 , such as @xmath71 and @xmath72 . therefore , the time - dependent phases in eq . ( [ phase_factor ] ) are always canceled , thus can be omitted . now , we take the small - amplitude limit , keeping only the linear order in @xmath32 . @xmath73 \nonumber \\ & = & h\left[u^ * , v^ * ; u , v \right ] + \eta \left\ { \delta h(\omega ) e^{-i\omega t } + \mbox{h.c . } \right\ } .\end{aligned}\ ] ] here , @xmath63 can be obtained using eqs . ( [ ut ] ) and ( [ vt ] ) , expanding up to the first order in @xmath32 and collecting terms proportional to @xmath74 , as @xmath75 the calculation of the derivatives , such as @xmath76 , is a tedious task and requires a large memory capacity for their storage in the computation . in the fam , we avoid this explicit expansion , instead write the same quantity as follows : @xmath77 - h\left[u^ * , v^ * ; u , v \right]}{\eta } + \mathcal{o}(\eta^2 ) , \ ] ] where @xmath78 , @xmath79 , @xmath80 , and @xmath81 are given by @xmath82 this is the fam formula for the calculation of @xmath63 . all we need in the computer program is a subroutine to calculate the single - particle hamiltonian as a function of the wave functions , @xmath83 $ ] . for the pair potential , basically , the same arguments lead to the fam formulae for @xmath84 . the time - dependent pair potential @xmath23 can be written as @xmath85 \nonumber \\ & = & \delta\left[u^ * , v^ * ; u , v \right ] \nonumber \\ & & + \eta \left\ { \delta \delta^{(+)}(\omega ) e^{-i\omega t } + \delta\delta^{(-)}(\omega ) e^{i\omega t } \right\ } .\end{aligned}\ ] ] here , @xmath86 and @xmath87 are independent , since @xmath23 is non - hermitian in general . @xmath86 can be written in the same form as eq . ( [ dhomega_fam ] ) . @xmath88 - \delta\left[u^ * , v^ * ; u , v \right]}{\eta } \nonumber \\ & & + \mathcal{o}(\eta^2 ) , \end{aligned}\ ] ] where @xmath78 , @xmath79 , @xmath80 , and @xmath81 are given by eq . ( [ uv_eta_p ] ) . the expression for @xmath87 is also obtained from eq . ( [ delta_t ] ) , collecting terms proportional to @xmath89 . it is given by the same expression as eq . ( [ ddp_fam ] ) , @xmath90 - \delta\left[u^ * , v^ * ; u , v \right]}{\eta } \nonumber \\ & & + \mathcal{o}(\eta^2 ) .\end{aligned}\ ] ] however , @xmath91 here are different from eq . ( [ uv_eta_p ] ) and given by @xmath92 the essential trick of the fam is to calculate the induced fields , @xmath63 and @xmath84 , according to eqs . ( [ dhomega_fam ] ) , ( [ ddp_fam ] ) , and ( [ ddm_fam ] ) with a small but finite parameter @xmath32 . of course , the @xmath93 and higher - order terms bring some numerical errors , but they are negligible . therefore , for given @xmath49 and @xmath50 , we are able to calculate these induced fields , by using the static hfb code with some minor modifications . @xmath56 and @xmath57 of eq . ( [ seconda ] ) in the quasi - particle basis can be calculated with eqs . ( [ dh20 ] ) and ( [ dh02 ] ) , respectively . then , we may solve the qrpa linear - response equation ( [ seconda ] ) to obtain the self - consistent amplitudes , @xmath49 and @xmath50 , utilizing an iterative algorithm ( see sec . [ sec : fammethod ] ) . although the basic formulae of the fam has been provided in sec . [ secfam ] , we may need to modify them in the practical implementation of the fam . for instance , some hfb codes , such as hfbrad , contain subroutines to calculate mean fields as functions of densities , not of wave functions . in this subsection , we rewrite eqs . ( [ dhomega_fam ] ) , ( [ ddp_fam ] ) , and ( [ ddm_fam ] ) in terms of densities . the density @xmath94 is written up to linear order in @xmath32 as @xmath95 where @xmath96 this can be written in the fam form : @xmath97 where @xmath79 and @xmath81 are given in eq . ( [ uv_eta_p ] ) . the pair tensor @xmath98 , which is non - hermitian , can be expressed in a similar manner . @xmath99 here , @xmath100 can be given in the explicit form as @xmath101 and in the fam form as @xmath102 where @xmath79 and @xmath80 are given in eq . ( [ uv_eta_p ] ) for @xmath103 while they are given by eq . ( [ uv_eta_m ] ) for @xmath104 . now , let us present how to obtain the induced fields in terms of the densities . in general , @xmath22 and @xmath23 may depend on @xmath7 , @xmath48 , and @xmath8 . @xmath105 , \quad \delta(t)=\delta\left[\rho(t),\kappa(t),\kappa^*(t)\right ] .\ ] ] in order to obtain the induced fields , all we need to do is to replace @xmath7 by @xmath106 defined in eqs . ( [ eq - rho - eta ] ) , and @xmath48 by @xmath107 in eq . ( [ eq - kappa - eta ] ) , as follows : @xmath108 - h\left[\rho,\kappa,\kappa^ * \right]}{\eta } , \\ \delta \delta^{(+)}(\omega ) & = & \frac{\delta\left[\rho_\eta,\kappa_\eta^{(+)},\kappa_\eta^{(- ) * } \right ] - \delta\left[\rho,\kappa,\kappa^ * \right]}{\eta } , \\ \delta \delta^{(-)}(\omega ) & = & \frac{\delta\left[\rho_\eta,\kappa_\eta^{(-)},\kappa_\eta^{(+ ) * } \right ] - \delta\left[\rho,\kappa,\kappa^ * \right]}{\eta } , \end{aligned}\ ] ] where the terms of the second and higher orders in @xmath32 are neglected . here we provide a summary of the fam for the qrpa linear - response calculation for a prompt application . later , we discuss applications of the fam to the skyrme functionals , however , the fam formulated in this and previous sections is applicable to any kind of energy density functional ( mean - field ) models . the aim is to solve the linear - response equation ( [ seconda ] ) for a given external field @xmath109 . in order to obtain the forward and backward amplitudes , @xmath49 and @xmath50 , we resort to an iterative algorithm . namely , we start from the initial guess for @xmath110 , and calculate @xmath63 and @xmath64 according to the formulae , ( [ dhomega_fam ] ) , ( [ ddp_fam ] ) , and ( [ ddm_fam ] ) . then , they are converted into @xmath56 and @xmath57 , using eqs . ( [ dh20 ] ) and ( [ dh02 ] ) , respectively . in this way , we can evaluate the left and right hand sides of eq . ( [ seconda ] ) for a given @xmath111 . since eq . ( [ seconda ] ) is equivalent to eq . ( [ qrpa - standard ] ) , it is a linear algebraic equation for the vector @xmath112 , in the form of @xmath113 . many different algorithms are available for the solution of linear systems . in this paper , we resort to a procedure based on krylov spaces called generalized conjugate residual ( gcr ) method @xcite . within these kinds of methods , a succession of approximate solutions ( @xmath114 ) converging to the exact one is obtained by the iteration . the gcr algorithm consists in a series of steps each containing the operation of the matrix @xmath115 on a given vector , and sums and scalar products of two vectors . for the given @xmath116 , @xmath117 is equal to the left hand side of eq . ( [ seconda ] ) . therefore , the quantity @xmath117 can be calculated without the explicit knowledge of the qrpa matrix itself . here , we summarize the formulae . the linear response equation is given by @xmath113 , where @xmath118 and @xmath119 where @xmath120 denoting @xmath5 and @xmath6 collectively as @xmath121 , the induced fields @xmath122 are calculated by the fam formulae , @xmath123 -\mathcal{h}\left[u^ * , v^ * ; u , v \right]}{\eta } , \\\ ] ] where @xmath124 are given by @xmath125 for the calculation of @xmath63 and @xmath86 . for @xmath87 , they are @xmath126 the final result does not depend on the parameter @xmath32 , as far as it is in a reasonable range . the choice of @xmath32 is discussed in sec . [ sectcomparison ] . using the solution @xmath111 , we can calculate the strength function following the same procedure as ref . @xmath127 here , @xmath128 is obtained from the solution @xmath111 . for the operator in the form of eq . ( [ f - type ] ) , we may calculate @xmath128 as @xmath129 for the operator in the form of eq . ( [ g - type ] ) , we have @xmath130 for both cases , in the two - quasi - particle basis , eqs . ( [ sf - f - type ] ) and ( [ sf - g - type ] ) can be written in the unified expression . @xmath131 where @xmath132 and @xmath133 are given by eqs . ( [ f20-f - type ] ) and ( [ f02-f - type ] ) for the former case , and by eqs . ( [ f20-g - type ] ) and ( [ f02-g - type ] ) for the latter . in order to assess the validity of the fam , we install the fam in the hfbrad code @xcite . it has to be noted that the formalism of the hfbrad is slightly different from the one used in this paper which follows the notations in ref . in particular , the wave functions @xmath134 , the pairing tensor @xmath135 , and the pair potential @xmath136 are defined in a different manner ; @xmath137 , @xmath138 , @xmath139 , and @xmath140 , where @xmath141 is the time - reversal state of @xmath13 . a detailed discussion on the difference among the two notations can be found in ref . @xcite . the hfbrad @xcite is a well known code which solves the hfb in the radial coordinate space assuming the spherical symmetry . it has been designed to provide fast and reliable solutions for the ground state of spherical even - even nuclei . for these nuclei , the time - odd densities are identically zero and thus they have not been implemented in the code . in order to render the qrpa fully self - consistent , we have to add the time - odd terms in the calculation of the induced fields . this task can be simplified for a case of the presence of spherical and space - inversion symmetry , such as in the case of monopole excitations . for this case , the only time - odd terms with non - zero contribution are those due to the current density @xcite , moreover the only non - vanishing component of the current density is radial . we calculate the strength function of the isoscalar monopole for a neutron - rich nucleus , @xmath0sn . to check the self - consistency by looking at the spurious component , we also calculate the strength of the nucleon number operator . both operators are given by the form of eq . ( [ f - type ] ) with @xmath142 for the isoscalar monopole operator and @xmath143 for the number operator . in order to obtain the strength function , first , we have to solve the hfb equations to construct the ground - state wave functions @xmath69 . it is accomplished by using the hfbrad code . the parameters of the present calculation are adjusted to the values used by terasaki and co - workers in ref . @xcite ; the box size is @xmath144 fm , the quasi - particle energy cutoff is @xmath145 mev , the maximum angular momenta of the quasi - particle states are @xmath146 for neutrons , and @xmath147 for protons . we use the skyrme functional with the skm * parameter set @xcite in the ph - channel and a delta interaction of the volume type with the strength @xmath148 mev @xmath149 for the pp- and hh - channels . the next step is solving the linear - response equation for a given external field of the frequency @xmath28 . at first , we build the induced fields , @xmath63 and @xmath64 , starting from a guess choice of the qrpa amplitudes @xmath150 , according to eq . ( [ induced_fields ] ) . in the present calculation , we choose either @xmath151 or the values of @xmath49 and @xmath50 at the previous energy @xmath28 calculated . we resort to the iterative algorithm of the gcr method to solve the equation ( [ seconda ] ) . we include all the two - quasi - particle states @xmath152 within the hfb model space defined above ( @xmath153 mev ) . the two - quasi - particle space amounts to 12,632 states for @xmath154 . note that this number becomes much larger if we treat deformed systems . we set the accuracy of the convergence to be @xmath155 , where @xmath156 . the number of iterations needed depends on @xmath28 ; at low energies , about 50 - 60 iterations are enough to reach the convergence , while , close to the central peak at 12 mev , more than 300 iterations are needed . & & & + @xmath32 & @xmath157 & @xmath158 & @xmath157 & @xmath158 & @xmath157 & @xmath158 + @xmath159 & 0.44 & 1000 & 1.63 @xmath160 & 1000 & @xmath161 & 1000 + @xmath162 & @xmath163 & 1000 & 1.76 @xmath164 & 1000 & @xmath165 & 469 + @xmath166 & @xmath165 & 161 & @xmath165 & 439 & @xmath165 & 469 + @xmath167 & @xmath168 & 161 & @xmath168 & 439 & @xmath165 & 469 + @xmath169 & @xmath168 & 161 & @xmath165 & 439 & @xmath165 & 469 + @xmath170 & @xmath168 & 161 & 1.19 @xmath164 & 1000 & @xmath171 & 1000 + we studied the convergence quality of the solutions as a function of the parameter @xmath32 used for the numerical derivative . this is shown in table [ convergence2 ] . if @xmath32 is too big ( @xmath172 ) the derivative of the fam becomes inaccurate and the linearity of the procedure is partially broken . the residue @xmath173 reaches a plateau where increasing the number of iterations can not improve it anymore . for @xmath174 , the calculations converge well and the resulting strength function is stable . if @xmath32 becomes smaller than @xmath175 , the numerical precision limits are reached and the gcr procedure can no longer obtain the required precision . therefore , we may conclude that the parameter @xmath32 in the range of @xmath174 is appropriate to obtain the induced fields accurately . although the constant value @xmath176 is adopted in this paper , we may use a more sophisticated choice , such as the @xmath28-dependent @xmath32 values @xcite , we report the strength function of the isoscalar monopole mode . to smear the strengths at discrete eigenenergies , we add an imaginary term to the energy : @xmath177 , where @xmath178 mev . this procedure is almost equivalent to smearing the strength function with a lorentzian function with a width equal to @xmath55 . the calculated energy - weighted strengths are summed up to @xmath179 mev and we found that they exhaust 99.6 % of the theoretical sum - rule value given by @xmath180 . excitations in @xmath0sn ( solid red curve ) , compared with the result in @xcite with the cutoff ( iii ) ( green dashed curve ) . the transition strength associated to the number operator , magnified by a factor of 10,000 , in units of mev@xmath181 is shown by the blue dotted curve . see text for details.,width=321 ] in fig . [ fig : noi ] , we compare our results ( solid red curve ) with the one in ref . @xcite ( dashed green curve ) . the self - consistent result obtained by terasaki et al . @xcite also employs the hfb solutions calculated with the hfbrad . however , in ref . @xcite , the qrpa matrix is calculated in the canonical - basis representation and an additional truncation of the two - quasi - particle space is introduced for the construction of the qrpa matrix . in contrast , we introduce no additional truncation for our fam calculation . we compare our results with the one of the cutoff ( iii ) in ref . @xcite which takes into account the highest number of states for the construction of the qrpa matrix ; all the proton quasi - particles up to 200 mev and the neutron canonical levels with occupancy @xmath182 . in the first two peaks at @xmath183 and @xmath184 mev , the two curves are almost perfectly overlapping . the peaks between 11 mev and 18 mev occur at the same energy for the two calculations while their height is slightly different . the bump close to zero energy resulting in our calculations has to be attributed to the presence of a spurious mode . to check the position of the spurious mode related to the pairing rotation of the neutrons , we included in fig . [ fig : noi ] the transition strength associated to the number operator , by the blue dashed line . the spurious mode is well localized close to zero energy . the present result demonstrates the accuracy and usefulness of the fam for the superfluid systems . even if the two codes include some differences in the truncation of the two - quasi - particle space , the similarity of the results is very satisfying the finite amplitude method for the qrpa has been presented . the basic idea is identical to the original fam @xcite , that we resort to a numerical differentiation to calculate the induced fields and then solve the linear - response equation with an iterative algorithm such as the gcr . with the fam , a hfb code with simple modifications can be turned into a qrpa code . especially , it is very easy to construct the qrpa code which has the same symmetry of the parent hfb one whose subroutines are used to perform the numerical derivative . all the terms present in the tdhfb calculation , including the time - odd mean fields , should be taken into account to construct fully self - consistent codes . this requires us some effort to update the original hfb code . still , the necessary task for coding the fam is much less than that for the explicit calculation of the qrpa matrix elements for realistic energy functionals . in addition , it does not require a large memory capacity , since we do not construct the qrpa matrix . we have built a fully self - consistent qrpa code using the hfbrad @xcite . the iterative algorithm , for which we adopted the gcr method in this paper , may be replaced by a better algorithm in future . the resulting strength functions of the isoscalar @xmath185 mode of @xmath0sn show high similarity with the fully self - consistent calculations in ref . thus , this paper showed the first application of the fam for superfluid systems and demonstrated the usefulness of the fam for the construction of the qrpa code by modifying existing hfb codes . this work is supported by grant - in - aid for scientific research(b ) ( no . 21340073 ) and on innovative areas ( no . 20105003 ) . we thank the jsps core - to - core program `` international research network for exotic femto systems '' . we are thankful to j. terasaki for providing the numerical results of ref . p.a . thank k. matsuyanagi for the fruitful discussion on the linear expansion , c. losa and a. pastore for the suggestions on the hfbrad code and k. yoshida and t. inakura for the discussions on the qrpa and j. dobaczewski and v. nesterenko for the useful suggestions . thank m. matsuo for useful discussion and the support from the unedf scidac collaboration under doe grant de - fc02 - 07er41457 . the numerical calculations were performed in part on riken integrated cluster of clusters ( ricc ) . the tdhfb hamiltonian is given by eq . ( [ tdhfb_hamiltonian ] ) . we consider the small - amplitude limit , @xmath187 , where @xmath188 is the hfb hamiltonian of eq . ( [ hfb_hamiltonian ] ) and @xmath189 here , @xmath190 and @xmath191 are oscillating as @xmath192 note that @xmath64 are anti - symmetric but @xmath63 is not necessarily hermitian . the induced hamiltonian , eq . ( [ deltahdit ] ) , is now expressed in the form of eq . ( [ delta_h ] ) with @xmath40 given by @xmath193 hereafter , @xmath63 and @xmath64 are denoted by @xmath194 and @xmath84 , for simplicity . since the bogoliubov transformation can be written in terms of the unitary matrix @xmath195 @xcite as follows : @xmath196 we may rewrite eq . ( [ deltahomega ] ) in the quasi - particle basis : @xmath197 this transformation should provide @xmath61 and @xmath62 in eq . ( [ delta_h20 ] ) . @xmath198 we write here their explicit expression : @xmath199 @xmath200 the one - body field in general can be written in a form of eq . ( [ fomega ] ) in terms of the quasi - particle operators , neglecting a constant . suppose that @xmath201 in eq . ( [ f ] ) has a form @xmath202 where the difference of a constant shift is neglected . here , the matrix @xmath203 is a general complex matrix , since @xmath201 is non - hermitian in general . the bogoliubov transformation as in eq . ( [ deltahomega_3 ] ) , then , leads to @xmath132 and @xmath133 in eq . ( [ fomega ] ) , @xmath204 in case that @xmath201 has a form of pairing - type @xmath205 the same calculation provides @xmath132 and @xmath133 by @xmath206 99 p. ring and p. schuck : _ the nuclear many body problem _ , springer - verlag , berlin ( 1980 ) . m.bender , p - h . heneen , p - g . reinhard , rev . phys . * 75 * , 121 ( 2003 ) . h. imagawa and y. hashimoto , phys . rev * c 67 * , 037302 ( 2003 ) . n. paar , p. ring , t. niki , and d. vretenar , phys . rev * c 67 * , 034312 ( 2003 ) . t. nakatsukasa and k. yabana , phys . rev * c 71 * , 024301 ( 2005 ) . j.terasaki , j. engel , m. bender , j. dobaczewski , phys . rev * c 71 * , 034310 ( 2005 ) . s.fracasso and g. col , phys . rev * c 72 * , 064310 ( 2005 ) . t. sil , s. shlomo , b.k . agrawal , and p .- reinhard , phys . rev * c 73 * , 034316 ( 2006 ) . j.terasaki and j. engel , phys . rev * c 74 * , 044301 ( 2006 ) . d. p. artega and p. ring , phys . rev * c 77 * , 034317 ( 2008 ) . s. pru and h. goutte , phys . rev * c 77 * , 044313 ( 2008 ) . c. losa , a. pastore , t. dssing , e. vigezzi , r.a . broglia , phys . rev . * c 81 * , 064307 ( 2010 ) . j. terasaki , j. engel , phys . rev . * c 82 * , 034326 ( 2010 ) . k. yoshida and n. v. giai , phys . rev . c * 78 * , 064316 ( 2008 ) . k. yoshida and t. nakatsukasa , phys . c * 83 * , 021304(r ) ( 2011 ) . s. ebata , t. nakatsukasa , t. inakura , k. yoshida , y. hashimoto , and k. yabana , phys . c * 82 * , 034306 ( 2010 ) . t. nakatsukasa , t. inakura and k. yabana , phys . rev . * c 76 * 024318 ( 2007 ) . t. inakura , t. nakatsukasa and k. yabana , phys . rev . * c 80 * 044301 ( 2009 ) . j. toivanen , b. g. carlsson , j. dobaczewski , k. mizuyama , r.r . rodrguez - guzmn , p. toivanen , p. vesel , phys . rev . * c 81 * , 034312 ( 2010 ) . k. bennaceur and j. dobaczewski , comput . comm . * 168 * , 96 ( 2005 ) . y. saad : _ iterative methods for sparse linear systems _ siam , philadelphia , 2003 . rohozinsky , j. dobaczewski , w. nazarevic , phys . rev . * c 81 * , 014313 ( 2010 ) . j. bartel , p. quentin , m. brack , c. guet , and h.b . hkansson , nucl . phys . * a386 * , 79 ( 1982 ) . e. perlinska , s.g . rohozinsky , j. dobaczewski , w. nazarevic , phys . rev . * c 69 * , 014316 ( 2004 ) . j. dobaczewski , h. flocard and j. treiner , nucl . phys * a422 * , 103 ( 1984 ) .

we present the finite amplitude method ( fam ) , originally proposed in ref . @xcite , for superfluid systems . a hartree - fock - bogoliubov code may be transformed into a code of the quasi - particle - random - phase approximation ( qrpa ) with simple modifications . this technique has advantages over the conventional qrpa calculations , such as coding feasibility and computational cost . we perform the fully self - consistent linear - response calculation for a spherical neutron - rich nucleus @xmath0sn , modifying the hfbrad code , to demonstrate the accuracy , feasibility , and usefulness of the fam . introduction[intro ] elementary modes of excitation in nuclei provide valuable information about the nuclear structure . the random - phase approximation ( rpa ) based on energy density functionals ( edf ) is a leading theory applicable both to low - lying excited states and giant resonances @xcite . although the fully self - consistent treatment of the residual ( induced ) interactions for the realistic energy functionals is becoming more and more prevalent @xcite , the rpa calculations for deformed nuclei are still computationally demanding . at present , the quasi - particle random - phase approximation ( qrpa ) for deformed superfluid nuclei are limited only to axially deformed cases @xcite , except for ref . @xcite with an approximate treatment of the pairing interaction . recently , there has been a renewed interest in the solution of the rpa problem @xcite . in ref . @xcite , the finite amplitude method ( fam ) was proposed as a feasible method for a solution of the rpa equation . the fam allows to calculate all the induced fields as a finite difference , employing a computational program of the static mean - field hamiltonian . recently , the fam has been applied to the electric dipole excitations in nuclei using the skyrme energy functionals @xcite . there has been also a calculation making use the iterative arnoldi algorithm for a solution of the rpa equation @xcite . these newly developed technologies in conjunction with fast solving algorithms for linear systems open the possibility to explore systematically the nuclear excitations over the entire nuclear chart . so far , these new techniques @xcite have been developed for solutions of the rpa without the pairing correlations . it is well known , however , that almost all but magic nuclei display superfluid features . therefore , a further improvement is highly desirable to make these methods applicable to the qrpa equations including correlations in the particle - particle and hole - hole channels . the purpose of the present paper is to generalize the fam to superfluid systems , which enables us to perform a qrpa calculation utilizing a static hartree - fock - bogoliubov ( hfb ) code with minor modifications . our final goal would be the construction of a fast computer program for the fully self - consistent and triaxially deformed qrpa . this paper is a first step toward the goal , to present the basic equations of the fam for the qrpa and show the first result for spherical nuclei . we use the spherically symmetric hfb code called hfbrad @xcite to be converted into the qrpa code . this paper is organized as follows : in sec . [ sectdhfb ] , the qrpa equation is derived as the small - amplitude limit of the time - dependent hfb ( tdhfb ) equations . in sec . [ secfam ] , we obtain the fam formulae for the calculation of the induced fields . in sec . [ sec : fammethod ] , we summarize all the relevant formulae for practical application of the fam . in sec . [ sectcomparison ] , we apply the fam to the hfbrad and compare the result with that of another self - consistent calculation . sec . [ conclusioni ] is devoted to the conclusions .

materials with perpendicular magnetic anisotropy ( pma ) have recently received a lot of interest due to their use in spin - transfer - torque magnetic random access memory ( stt - mram ) and spin logic applications@xcite . magnetic tunnel junctions ( mtjs ) with pma are required for further scaling of the critical device dimension ( cd ) . the perpendicular mtjs ( p - mtj ) enable stt - mram devices with longer data retention time and lower switching current at a smaller cd when compared to the mtj s with in - plane magnetic anisotropy ( ima)@xcite . a typical p - mtj stack comprises an mgo tunnel barrier sandwiched between a synthetic antiferromagnet ( saf ) as fixed layer , and a magnetically soft layer as free layer . the material requirements for the fixed and free layer differ . whereas the free layer is aimed to have a high pma and low damping to ensure data retention and fast switching via stt , the saf requires high pma and has preferentially high damping to ensure that it remains fixed during stt writing and reading to avoid back hopping@xcite . as such , it is of high importance to control the pma strength and damping in pma materials . co - based pma multilayers [ co / x ] ( x = pt , pd ) have received a lot of attention for their potential application in stt - mram , especially as saf materials @xcite . the pma of these multilayers comes from the interface of [ co / x ] in each bilayer repeat@xcite . besides , [ co / ni ] has been researched as alternative pma material and has been employed in p - mtj because of its high spin polarization and low gilbert damping constant@xcite . also , [ co / ni ] has been incorporated in an ultrathin saf@xcite . recently , the use of [ co / ni ] in the free layer material was proposed to enable free layers with high thermal stability needed at cd below 20nm@xcite . next to stt - mram , [ co / ni ] has also been used as domain wall motion path in magnetic logic devices@xcite . briefly speaking , [ co / ni ] multilayers are being considered for various applications in next generation spintronic devices . first - principle calculations predicted that [ co / ni ] in _ fcc_(111 ) texture possesses pma . the maximum anisotropy is obtained when co contains just 1 monolayer and ni has 2 monolayers@xcite . experimental studies proved that prediction@xcite and reported on the pma in [ co / ni ] for various sublayer thickness , repetition number and deposition conditions@xcite . to get [ co / ni ] with the correct crystallographic orientation and good texture quality , a careful seed selection is required . various seed layers have been studied . as well as their impact on pma and damping , including cu@xcite , ti@xcite , au@xcite , pt@xcite , ru@xcite and ta@xcite . pma change in [ co / ni ] is commonly observed and attributed to interdiffusion of the co / ni bilayers@xcite . however , we observed earlier that also the diffusion of the seed material can strongly impact the [ co / ni ] magnetization reversal , especially after annealing@xcite . a more in - depth study on the impact of the seed after annealing on the pma and damping of [ co / ni ] is therefore required . certainly the thermal robustness is of high importance for cmos applications since the [ co / ni ] needs to be able to withstand temperatures up to 400@xmath0c that are used in back - end - of - line ( beol ) processes . in this paper , we study four sub-5 nm seed layers : pt(3 nm ) , ru(3 nm ) , ta(2 nm ) and hf(1 nm)/nicr(2 nm ) and present their impact on both the structural and magnetic properties of as - deposited and annealed [ co / ni ] . we show that a good lattice match to promote _ fcc_(111 ) texture and to avoid [ co / ni ] interdiffusion is not the only parameter that determines the choice of seed layer . the diffusion of the seed material in the [ co / ni ] is identified as a key parameter dominating the pma and damping of [ co / ni ] after annealing . the [ co / ni ] on various seed layers were deposited _ in - situ _ at room temperature ( rt ) on thermally oxidized si(100 ) substrates using physical vapor deposition system in a 300 mm canon anelva ec7800 cluster tool . prior to seed layer deposition , 1 nm tan is deposited to ensure adhesion and to reflect the bottom electrode material that is used in device processing@xcite . the detailed stack structure is si / sio@xmath3/tan(1.0)/seed layer/[ni(0.6)/co(0.3)]@xmath4/ni(0.6)/co(0.6)/ru(2.0)/ta(2.0 ) ( unit : nm ) . ru / ta on top serves as capping layer to protect [ co / ni ] from oxidation in air . the films were further annealed at 300@xmath0c for 30 min and 400@xmath0c for 10 min in n@xmath3 in a rapid thermal annealing ( rta ) set - up . the crystallinity of the [ co / ni ] films was studied via a @xmath5 - 2@xmath5 scan using the cu @xmath6 wavelength of @xmath7 nm in a bede metrixl x - ray diffraction ( xrd ) set - up . the degree of texture is evaluated in the same tool by measuring full - width at half - maximum ( fwhm ) of the rocking curve in an @xmath8 scan with 2@xmath5 fixed at the _ fcc_(111 ) peak position of [ co / ni ] . transmission electron microscopy ( tem ) is used to identify the microstructure of multilayers . time - of - flight secondary ion mass spectrometry ( tof - sims ) is used to study diffusion of the seed material in the [ co / ni ] . the measurements were conducted in a tofsims iv from ion - tof gmbh with dual beam configuration in interlaced mode , where o@xmath9 and bi@xmath10 are used for sputtering and analysis , respectively . x - ray photoelectron spectroscopy ( xps ) is used to quantify the diffusion amount of the seed layers . the measurements were carried out in angle resolved mode using a theta300 system from thermoinstruments . 16 spectra were recorded at an exit angle between 22@xmath0 and 78@xmath0 as measured from the normal of the sample . the measurements were performed using a monochromatized al @xmath6 x - ray source ( 1486.6 ev ) and a spot size of 400@xmath11 m . because of the surface sensitivity of the xps measurement ( depth sensitivity is @xmath125 nm ) , the xps analysis was carried out on [ co / ni ] films without ru / ta cap . a microsense vibrating samples magnetometer ( vsm ) is used to characterize the magnetization hysteresis loops and to determine the saturation magnetization ( @xmath13 ) . the effective perpendicular anisotropy field ( @xmath14 ) and the gilbert damping constant ( @xmath15 ) in [ co / ni ] was measured via vector - network - analyzer ferromagnetic resonance ( vna - fmr)@xcite and corresponding analysis@xcite . the effective perpendicular anisotropy energy ( @xmath16 ) is calculated by the equation @xmath17 in the unit of j / m@xmath2 . to obtain large pma in [ co / ni ] systems , highly textured , smooth _ fcc_(111 ) films are required@xcite . fig.1 shows the @xmath5 - 2@xmath5 xrd patterns of [ co / ni ] on different seed layers as - deposited and after annealing . fcc_(111 ) peaks of bulk co and ni are located at 44.1@xmath0 and 43.9@xmath0 , respectively@xcite . the presence of a peak around 2@xmath5 = 44@xmath0 confirms the _ fcc_(111 ) texture in as - deposited [ co / ni ] films on all seed layers@xcite . after annealing , the peak intensity increases , in particular for the [ co / ni ] sample on a pt seed . additionally , a shift in the peak position is observed for all seeds . the black arrows indicate the shift direction in fig.[f1](a)-(d ) . for [ co / ni ] on nicr and ta seed , the diffraction peak of [ co / ni ] shifts towards larger peak position of bulk co and ni , meaning that [ co / ni ] films on nicr and ta possess tensile stress after annealing . in contrast , compressive stress is induced in [ co / ni ] on pt seed , since the peak position moves towards the pt(111 ) peak due to lattice matching . on ru seed , the peak nearly does not shift after annealing . the quality of _ fcc_(111 ) texture of [ co / ni ] is further examined via rocking curves . the results of fwhm of the rocking curves are given in fig.[f2 ] , as well as the influence of post - annealing . the larger fwhm observed in [ co / ni ] on ru seed suggests a lower degree of texture , which is in agreement with the @xmath5 - 2@xmath5 pattern where [ co / ni ] on ru seed shows a peak with lower intensity . post - annealing at 300@xmath0c leads to further crystallization and enhanced texturing , as indicated by the decreased fwhm for [ co / ni ] on all seed layers . however , fwhm increases after 400@xmath0c annealing , which may be attributed to the intermixing of co and ni and hence the degradation in crystal quality . scan of as - deposited and post - annealed [ co / ni ] on different seed layers . ] fig.[f3 ] shows the microstructure of [ co / ni ] deposited on different seed layers imaged by tem after 300@xmath0c annealing . in all [ co / ni ] samples the grains extend from seed to cap . the interface between [ co / ni ] and seed layer is clear and smooth in the samples with pt and ru seed layer in fig.[f3](b ) and ( c ) , respectively . for [ co / ni ] on nicr , however , the interface between the multilayers and the seed layer can not be distinguished ( fig.[f3](a ) ) , but both show clearly crystalline and texture matched . for the [ co / ni ] on ta seed in fig.[f3](d ) , the interface is quite rough and a nanocrystalline structure at the interface between ta and [ co / ni ] can be spotted , which may indicate intermixing . the presence of pma in [ co / ni ] before and after annealing is firstly checked by vsm ( fig.[f4 ] ) . despite the presence of _ fcc_(111 ) peaks on all seeds , no pma was observed in the as - deposited films on nicr and ta seeds . in contrast , pma occurs in as - deposited [ co / ni ] on pt and ru seed . after 300@xmath0c annealing , pma appears in [ co / ni ] on nicr and ta seed ( see fig.4(a ) and ( c ) , respectively ) . simultaneously , an @xmath13 loss is observed . note that for the [ co / ni ] on nicr , the @xmath13 loss is large and the hysteresis loop becomes bow - tie like with coercivity ( @xmath18 ) increase ( fig.[f4](a ) ) . the large @xmath18 enables [ co / ni ] on nicr seed to function as hard layer in mtj stacks@xcite . fig.[f5](a ) and ( b ) summarizes @xmath13 and @xmath18 of the [ co / ni ] on different seed layers for various annealing conditions . the @xmath13 and @xmath18 of [ co / ni ] on nicr and ta seed decrease and increase , respectively , further after 400@xmath0c annealing . for the [ co / ni ] pt and ru seed , there is no change in @xmath13 and @xmath18(see fig.[f4](b ) and ( d ) , respectively ) , even after 400@xmath0c annealing , indicating a good thermal tolerance . on different seed layers as - deposited ( dashed lines ) and after 300@xmath0c annealing ( solid lines ) by vsm : ( a ) nicr , ( b ) pt , ( c ) ta and ( d ) ru . note that the _ x_-axis scale in ( a ) is different from the others . ] fig.5(c ) and ( d ) show the effective perpendicular anisotropy field ( @xmath14 ) values and the calculated @xmath16 of [ co / ni ] on each seed for different annealing conditions . as deposited , the @xmath14 of [ co / ni ] on nicr and ta seed is 0 , meaning that they have i m a , as shown in fig.[f4 ] , while the highest @xmath14 is found on pt seed as expected from the small lattice mismatch and the crystalline nature of pt buffers , i.e. _ fcc_(111 ) . after annealing at 300@xmath0c , @xmath14 significantly increases in [ co / ni ] on all seeds , except when the [ co / ni ] is grown on ru . on ru , @xmath14 is the lowest . after 400@xmath0c annealing , pma is maintained in all samples , though @xmath14 of [ co / ni ] on pt and ta seed slightly decrease . for the nicr seed , @xmath14 even increases further and becomes more than 2 times larger than the values found in the other samples . the large @xmath14 and @xmath16 of [ co / ni ] on nicr , especially after 400@xmath0c , will be analyzed in the following . on different seed layers under different annealing conditions : ( a ) @xmath13 , ( b ) @xmath18 , ( c ) @xmath14 and ( d ) @xmath16 . ] commonly , highly _ fcc_(111 ) textured [ co / ni ] films result in large pma . in our case , we have observed some anomalous behaviors . the as - deposited film on nicr and ta did not show pma . the pma occured and increased significantly after annealing , while @xmath13 loss was observed . on the other hand , only limited pma increase is observed on pt seed after annealing , despite the large increase in diffraction peak intensity shown in fig.[f1](b ) . in short , the improvement of [ co / ni ] film quality leads to limited increase in pma for [ co / ni ] on pt and ru seed , yet there is a huge increase in pma for [ co / ni ] on nicr and ta seed . in the further , we will discuss the mechanisms responsible for the observed trends . as shown in section iii.a , the _ fcc_(111 ) peak position of [ co / ni ] on the various seed layers differs from each other , indicating the existence of strain induced by the seed layer . because of the magneto - elastic effect , strain - induced magnetic anisotropy ( @xmath19 ) can be an important contribution of the total pma . @xmath19 is calculated as@xcite @xmath20 in this equation , @xmath21mj / m@xmath2 and @xmath22mj / m@xmath2 reflect the _ fcc_(111 ) cubic magneto - elastic coupling coefficients of bulk co and ni , respectively@xcite . possible thin film effects on the coefficients are beyond the scope of the paper . @xmath23 is the out - of - plane strain , which can be derived from the shift of _ fcc_(111 ) peak in xrd , with @xmath24 as the reference@xcite . and @xmath25 , where @xmath26 is poisson ratio . @xmath26 is calculated as weight - averaged values of bulk co and ni@xcite . fig.[f6 ] summarizes the strain - induced pma before and after annealing . it is clear that the strain from pt and ru seed result always in a negative contribution to the pma of [ co / ni ] , which may in both cases counteract with the increase in pma from improved film quality after annealing ( see the narrower peak of [ co / ni ] with larger intensity after annealing in fig.[f1 ] and decreased fwhm in fig.[f2 ] ) and results in little net @xmath16 improvement ( see fig.[f5](d ) ) . similarly observed in fig.[f5](d ) , as - deposited [ co / ni ] on ta seed shows low @xmath16 due to the negative strain - induced pma , even though it has the required texture ( see fig.[f1 ] ) . after annealing , strain - induced pma contributes positively to total @xmath16 of [ co / ni ] on ta seed . for [ co / ni ] on nicr seed , the strain after annealing promotes the increase in total @xmath16 . from this discussion , it is clear that the seed layer providing in - plane tensile strain to [ co / ni ] is desired for pma increase . however , only the strain contribution can not explain the large pma that is observed after annealing on nicr . moreover , as shown in fig.[f5](b ) , there is also a large increase in @xmath18 for [ co / ni ] on nicr and ta samples , while their @xmath13 reduce dramatically , which can not be explained by the previous strain - induced pma change . ) and strain - induced anisotropy energy ( @xmath19 ) . the arrows indicate the temperature of annealing conditions : as - deposited , 300@xmath0c , and 400@xmath0c . ] we have performed an advanced compositional analysis of the [ co / ni ] after annealing . fig.[f7 ] shows the tof - sims depth profiles of cr , pt , ru and ta as - deposited and after 400@xmath0c annealing . the ni signal is provided to indicate position of the [ co / ni ] multilayers . in the case of the nicr seed , cr is found throughout the whole layer of [ co / ni ] after 400@xmath0c annealing , since its signal appears at the same depth ( sputter time ) as ni . that means cr diffuses heavily in the [ co / ni ] . the same phenomenon is observed for pt seed , but the intensity of the signal from diffused pt is low when compared to the pt signal in the seed layer part , indicating that the diffusion amount of pt is limited . for ru shown in fig.[f7](c ) , the interface between the ru seed and [ co / ni ] remains sharp after annealing . on the contrary , the less steep increase in ta signal suggest intermixing between [ co / ni ] and ta after annealing at the interface , but does not suggest ta diffusion in the bulk of the [ co / ni ] films . to quantitatively study the diffusion of the seed in the [ co / ni ] , xps measurements are conducted and the apparent atomic concentration of co , ni and seed layer element in each sample with different annealing conditions are shown in fig.[f8 ] . note that the higher apparent concentration of co when compared to ni for the as - deposited sample is due to the surface sensitivity of the xps technique as explained in the figure caption . it is clear that cr diffuses the most among the four seed layers . the presence of cr in [ co / ni ] may also lead to the shift of the [ co / ni ] peak towards the nicr peak in the xrd pattern shown in fig.[f1](a ) , probably resulting in the formation of co - ni - cr alloy . furthermore , the observed change in magnetic properties can likely be attributed to the formation of the co - ni - cr alloy@xcite . indeed , the uniform diffusion of cr was reported to cause @xmath18 increase in co - ni film with in - plane anisotropy@xcite . and cr can be coupled antiferromagnetically with its co and ni hosts causing the @xmath13 drop@xcite and higher pma . less diffusion is observed for the pt seed , in agreement with the peak shift toward pt seed ( fig.1(b ) ) and for the lower increase in pma as well , since the [ co / pt ] system , alloys or multilayers , is a well - known pma system and so a change in pma due to pt atoms in the [ co / ni ] matrix is not necessarily detrimental@xcite . on the contrary , no significant diffusion of ru into [ co / ni ] layers has been observed in fig.[f8](c ) , so no impact on pma is to be expected . finally , a large increase in pma has been observed after annealing whereas the diffusion is limited for ta seed . possibly , the improvement of crystal structure and the formation of a co - ni - ta alloy at the interface of ta seed and [ co / ni ] part happens at higher temperature , which gives rise to pma and @xmath13 loss@xcite . apart from the @xmath13 and pma change , the diffusion of the seed material into the [ co / ni ] might also impact the magneto - elastic coefficients to be taken into account when calculating the strain - induced pma ( see section iii.c.1 ) . this impact is however not straightforward and requires further study . , are omitted in the figure . ] earlier studies reported on dopants that increase the damping when incorporated into a ferromagnetic film@xcite . in our case , it is natural to expect that an impact from the diffused seed layer element will be exerted on the dynamic magnetic properties of [ co / ni ] . therefore , the gilbert damping constant ( @xmath15 ) of each sample with different annealing conditions was derived from vna - fmr for study . fig.[f9 ] compares the permeabilities of the [ co / ni ] films as - deposited and after 400@xmath0c annealing when the fmr frequency is set to 15 ghz by a proper choice of the applied field . the broadening of the linewidths after annealing reflects the increase in damping for all cases , with the noticeable exception of ru . the nicr resonance was too broadened to be resolved after 400@xmath0c annealing , reflecting a very high damping or a very large inhomogenity in the magnetic properties . linear fits of the fmr linewidth versus fmr frequency were conducted ( not shown ) to extract the damping parameters , which are listed in table [ t1 ] . it should be noticed that in our vna - fmr measurement , the two - magnon contributions to the linewidth and hence its impact on damping derivation can be excluded due to the perpendicular geometry in the measurement@xcite . and the contribution of spin - pumping within the seed layer to the linewidth in our cases is always expected to be within the error bar@xcite . the lowest damping in the as - deposited [ co / ni ] films were obtained on ta and nicr seed , i.e. in the in - plane magnetized samples . the highest damping was found on pt seed , a fact that is generally interpreted as arising from the large spin - orbit coupling of pt . and the damping values increase with post annealing in [ co / ni ] on all seed layers except ru . though the damping of [ co / ni ] is larger than cofeb / mgo@xcite , it is equal to or smaller than [ co / pt ] and [ co / pd]@xcite , which makes it of interest to use as free layer material with high thermal stability in high density stt - mram applications@xcite . ccccc & seed & as - depo & rta 300@xmath0c & rta 400@xmath0c + ' '' '' & nicr & 0.020 & 0.040 & @xmath270.040 + & pt & 0.026 & 0.030 & 0.034 + & ru & 0.022 & 0.020 & 0.018 + & ta & 0.017 & 0.024 & 0.026 . ] fig.[f10](a ) and ( b ) plot the correlations between the dopant concentration and the @xmath15 and @xmath16 values , respectively . here the seed element concentration in the [ co / ni ] multilayer is reflected by the ratio of the apparent seed concentration ( at.% ) divided by the sum of the apparent co and ni concentrations ( at.% ) for each sample and annealing condition . there is a clear correlation between the damping and the concentration of the seed element in the [ co / ni ] system . the absence of evolution of the damping upon annealing for [ co / ni ] on ru can thus be explained by the non - diffusive character of the ru seed . in the case of the nicr seed , the damping is the largest after annealing . a similar trend was observed before on au seeds and attributed to the formation of superparamagnetic islands in thin [ co / ni ] multilayers@xcite . however , since we observe for the same [ co / ni ] system no @xmath13 loss and no @xmath18 increase on ru and pt , and since the pma on nicr after anneal is the highest , the formation of super paramagnetic islands can not be the reason to explain the magnetic behavior . it is more likely that the increased damping comes from the formation of a textured co - ni - cr alloy with high pma and high damping , as mentioned in section iii.c.2 . finally , in the case of pt and ta seeds , the damping on the pt seed is higher than on the ta seed for the same concentration . that is attributed to the spin - orbit coupling for pt dopants being substantially higher than for ta . in conclusion , it is clear that the seed diffusion can not be ignored when studying the impact of annealing on the structural and magnetic properties of [ co / ni ] . and ( b ) @xmath28 as a function of seed element concentration in [ co / ni ] . for each series , the annealing condition from left to right is as - deposited , rta 300@xmath0c and rta 400@xmath0c . the 400@xmath0c data of nicr are not shown ( cr concentration @xmath2960% ) . ] in summary , the structural and magnetic properties of [ co / ni ] on different seed layers are investigated . [ co / ni ] on pt and ru seed show _ fcc_(111 ) texture after deposition and have pma . i m a is observed for [ co / ni ] on nicr and ta seed , and pma appears and increases after post - annealing . further annealing improves the texture and hence increases pma . meanwhile , the shift of [ co / ni ] diffraction peaks in xrd curves indicates the presence of strain in the [ co / ni ] film , which can also influence pma . the strain - induced pma may have a positive effect on the total pma ( nicr and ta seed ) , or a negative impact ( pt and ru seed ) a dramatic reduction of @xmath13 and large increase in @xmath18 of [ co / ni ] on nicr and ta seed after annealing is observed . the damping property of [ co / ni ] on different seed layers evolves similarly as pma with post - annealing . the explanation for these phenomena is the diffusion property of seed layer materials . high pma and very large damping is obtained on nicr seed because of the dramatic diffusion of cr and the formation of co - ni - cr alloy . though non - diffusive ru seed results in low damping , low pma as - deposited and after annealing is obtained . pt seed can provide good pma , but its large spin - orbit coupling exerted from the interface with [ co / ni ] and diffused part increases damping , especially after annealing . pma as high as on pt has been observed for ta seed with lower damping due to its smaller spin - orbit coupling . fig.[f11 ] summarizes schematically the impact of seed layer and annealing on the structural and magnetic properties of [ co / ni ] on different seed layers . finally , by selecting the seed and post - annealing temperature , the [ co / ni ] can be tuned in a broad range from low damping to high damping while maintaining pma after annealing until 400@xmath0c . as such , the [ co / ni ] multilayer system is envisaged for various applications in spintronics , such as highly damped fixed layers , or low damping free layers in high density magnetic memory or domain wall motion mediated spin logic applications . + 59ifxundefined [ 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 http://www.itrs.net/ [ `` , '' ] ( ) in link:\doibase 10.1109/imw.2011.5873205 [ _ _ ] ( , ) pp . link:\doibase 10.1103/physrevb.93.024420 [ * * , ( ) ] link:\doibase 10.1063/1.3524230 [ * * , ( ) ] link:\doibase 10.1109/tmag.2014.2326731 [ * * , ( ) ] link:\doibase 10.7567/apex.8.063002 [ * * , ( ) ] link:\doibase 10.1088/0034 - 4885/59/11/002 [ * * , ( ) ] link:\doibase 10.1143/apex.4.013005 [ * * , ( ) ] link:\doibase 10.1116/1.3430549 [ * * , ( ) ] link:\doibase 10.1063/1.3358242 [ * * , ( ) ] link:\doibase 10.1063/1.4906843 [ * * , ( ) ] in link:\doibase 10.1109/iedm.2014.7047080 [ _ _ ] ( , ) pp . link:\doibase 10.1063/1.4923420 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1063/1.4945089 [ * * , ( ) ] in link:\doibase 10.1109/iedm.2014.7047159 [ _ _ ] ( , ) pp . link:\doibase 10.1103/physrevb.42.7270 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.68.682 [ * * , ( ) ] link:\doibase 10.1109/20.179619 [ * * , ( ) ] link:\doibase 10.1063/1.352048 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.014425 [ * * , ( ) ] link:\doibase 10.1063/1.4915106 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2007 - 00071 - 1 [ * * , ( ) ] , link:\doibase 10.1103/physrevb.86.184407 [ * * , ( ) ] link:\doibase 10.1063/1.4704184 [ * * , ( ) ] link:\doibase 10.1088/0022 - 3727/46/17/175001 [ * * , ( ) ] link:\doibase 10.1016/j.jmmm.2015.04.061 [ * * , ( ) ] link:\doibase 10.1063/1.3506688 [ * * , ( ) ] link:\doibase 10.1063/1.4799524 [ * * , ( ) ] link:\doibase 10.1063/1.4813542 [ * * , ( ) ] link:\doibase 10.1063/1.3481452 [ * * , ( ) ] link:\doibase 10.1063/1.3176901 [ * * , ( ) ] link:\doibase 10.1143/apex.3.113002 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.014401 [ * * , ( ) ] link:\doibase 10.7498/aps.64.097501 [ * * , ( ) ] link:\doibase 10.1063/1.4865212 [ * * , ( ) ] link:\doibase 10.1109/tmag.2011.2158082 [ * * , ( ) ] link:\doibase 10.1007/s12598 - 016 - 0782 - 8 [ ( ) ] link:\doibase 10.1063/1.2716995 [ * * , ( ) ] link:\doibase 10.1063/1.4775684 [ * * , ( ) ] @noop _ _ , asm specialty handbook ( , ) link:\doibase 10.1016/s0040 - 6090(00)01072 - 5 [ * * , ( ) ] link:\doibase 10.1016/s0304 - 8853(99)00310 - 8 [ * * , ( ) ] link:\doibase 10.1038/srep27774 [ * * , ( ) ] , @noop _ _ ( , ) link:\doibase 10.1109/tmag.1986.1064562 [ * * , ( ) ] link:\doibase 10.1109/tjmj.1990.4564145 [ * * , ( ) ] link:\doibase 10.3379/jmsjmag.13.s1445 [ * * , ( ) ] link:\doibase 10.1109/tmag.1978.1059928 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4930830 [ * * , ( ) ] link:\doibase 10.1109/20.50508 [ * * , ( ) ] link:\doibase 10.1361/105497101770339319 [ * * , ( ) ] link:\doibase 10.1109/tmag.2015.2438324 [ * * , ( ) ] link:\doibase 10.1063/1.2436471 [ * * , ( ) ] link:\doibase 10.1063/1.4892532 [ * * , ( ) ] link:\doibase 10.1103/physrevb.71.064420 [ * * , ( ) ] link:\doibase 10.1063/1.3615961 [ * * , ( ) ] link:\doibase 10.1109/tmag.2012.2198446 [ * * , ( ) ] link:\doibase 10.1109/tmag.2016.2517098 [ * * , ( ) ]

[ co / ni ] multilayers with perpendicular magnetic anisotropy ( pma ) have been researched and applied in various spintronic applications . typically the seed layer material is studied to provide the desired face - centered cubic ( _ fcc _ ) texture to the [ co / ni ] to obtain pma . the integration of [ co / ni ] in back - end - of - line ( beol ) processes also requires the pma to survive post - annealing . in this paper , the impact of nicr , pt , ru , and ta seed layers on the structural and magnetic properties of [ co(0.3 nm)/ni(0.6 nm ) ] multilayers is investigated before and after annealing . the multilayers were deposited _ in - situ _ on different seeds via physical vapor deposition at room temperature . the as - deposited [ co / ni ] films show the required _ fcc_(111 ) texture on all seeds , but pma is only observed on pt and ru . in - plane magnetic anisotropy ( i m a ) is obtained on nicr and ta seeds , which is attributed to strain - induced pma loss . pma is maintained on all seeds after post - annealing up to 400@xmath0c . the largest effective perpendicular anisotropy energy ( @xmath1j / m@xmath2 ) after annealing is achieved on nicr seed . the evolution of pma upon annealing can not be explained by further crystallization during annealing or strain - induced pma , nor can the observed magnetization loss and the increased damping after annealing . here we identify the diffusion of the non - magnetic materials from the seed into [ co / ni ] as the major driver of the changes in the magnetic properties . by selecting the seed and post - annealing temperature , the [ co / ni ] can be tuned in a broad range for both pma and damping .

the gravitational two - body problem is a fundamental issue in general relativity . this also attracts great interest in gravitational wave physics because binary inspirals are promising sources of gravitational waves which are expected to be detected directly by ongoing gravitational wave observatories in the world . understanding the dynamics of binary system is required to predict the emitted gravitational waveforms accurately for efficient searches of the signal in observed data . one of major approaches for this purpose is the gravitational self - force ( gsf ) picture in the black hole perturbation theory . in this picture , a binary is regarded as a point mass orbiting a black hole and the dynamics can be described by the equation of motion of the mass including the effect of the interaction with the self - field , that is , the gsf . after the formal expression of the gsf was presented by mino , sasaki and tanaka @xcite and quinn and wald @xcite , a lot of efforts have been devoted to develop practical formulations and methods to calculate the gsf ( for example , refer to @xcite for the formulation of gsf , @xcite for the recent progress in practical calculations of gsf ) . although a lot of progress has been made , however , it is still challenging to calculate the gsf directly for general orbits , especially in kerr spacetime . practical calculations of the gsf with high accuracy will require a huge amount of time and computer resources mainly because of the regularization problem induced by the point mass limit . therefore it is important to develop a way to reduce the cost of computing the gsf . the two - timescale expansion method @xcite gives a hint for it : assuming that a point mass does not encounter any transient resonances ( _ e.g. _ shown in @xcite ) , the orbital phase , which is the most important information to predict the waveform , can be expressed in the expansion with respect to the mass ratio , @xmath12 , as @xmath13,\ ] ] where @xmath14 and @xmath15 are quantities of order unity . the leading term , @xmath14 , can be calculated from the knowledge up to the time - averaged dissipative piece of the first order gsf , corresponding to the secular growth . the calculation of this secular contribution can be simplified significantly by using the radiative field defined as half the retarded solution minus half the advanced solution for the equation of the gravitational perturbation @xcite , _ i.e. _ the adiabatic approximation method , because the radiative field is the homogeneous solution free from the divergence induced by the point mass limit . this method allows us to calculate the leading term accurately without spending huge computational resources . on the other hand , the calculation of @xmath15 requires the rest of the first order gsf ( the oscillatory part of the dissipative gsf and the conservative gsf ) and the time - averaged dissipative piece of the second order gsf . there is no simplification in calculating these post-1 adiabatic pieces at present . since @xmath15 is subleading , however , the requirement of the accuracy is not so high compared to that of the leading term . this fact suggests that it is possible to reduce the computational cost by using a suitable method with an appropriate error tolerance to calculate each piece of the gsf ( for example , a hybrid approach is proposed in @xcite ) . in this work , we focus on the time - averaged dissipative part of the first order gsf , which has the dominant contribution to the evolution of inspirals , and present the analytic post - newtonian ( pn ) formulae . so far , several works in this direction had been done for two restricted classes of orbits : circular orbits and equatorial orbits . ( see @xcite and references therein for early works in 1990 s ) . recently , thanks to the progress of computer technology , the very higher order post - newtonian calculations can be possible for circular equatorial orbits : the 22pn calculation of the energy flux is demonstrated in schwarzschild case @xcite , and the 11pn calculation in kerr case @xcite . there is also the calculation of the secular gsf effects for slightly eccentric and slightly inclined ( non - equatorial ) orbits @xcite , and later it had been extended to orbits with arbitrary inclination @xcite , where the pn formulae of the secular gsf effects are presented in the expansion with respect to the orbital eccentricity . however , the calculation in @xcite has been done only up to the 2.5pn order with the second order correction of the eccentricity . also the absorption to the black hole is ignored there . the main purpose of this work is to update the results in @xcite up to the 4pn order and the sixth order correction of the eccentricity , including the effect of the absorption to the black hole . this paper is organized as follows . in sec.[sec : review ] , we give a brief review of the geodesic motion of a point particle in kerr spacetime , the gravitational perturbations induced by the particle , and the adiabatic approximation method of calculating the secular effect of the gsf . in sec.[sec : dote_formulae ] , we present the pn formulae of the secular changes of the the energy , azimuthal angular momentum , and carter parameter of the particle due to the gravitational radiation reaction in the expansion with respect to the orbital eccentricity . in sec.[sec : compare ] , we investigate the accuracy of our pn formulae by comparing to numerical results given by the method in @xcite , which can give each modal flux at the accuracy about 14 significant figures . in sec.[sec : exp - resum ] , we implement a resummation method to the pn formulae given in this work in order to improve the accuracy . in sec.[sec : delta_n ] , we discuss the convergence of the analytic formulae as the pn expansion and the expansion with respect to the eccentricity . finally , we summarize the paper in sec.[sec : summary ] . for the readability of the main text , we present the pn formulae for the orbital parameters , the fundamental frequencies , the orbital motion in appendices [ app : variable_exp ] and [ app : coeff_orbit ] , which are used in calculating the secular changes of the orbital parameters . we also present the pn formulae for the secular changes of an alternative set of the orbital parameters in appendix [ app : orbit_evolv ] . throughout this paper we use metric signature @xmath16 and `` geometrized '' units with @xmath17 . the orbital evolution of a point particle due to the time - averaged dissipative part of the gsf is often described in terms of the secular changes of the orbital parameters . in order to calculate the changes , we need the information on the first order gravitational perturbations induced by the particle when it moves along the background geodesics . in this section , we review the geodesic dynamics of a point particle in kerr spacetime , the gravitational perturbations induced by the particle , and the adiabatic evolution of the orbital parameters . the kerr metric in the boyer - lindquist coordinates , @xmath18 , is given by @xmath19 where @xmath20 , @xmath21 , @xmath22 and @xmath23 are the mass and angular momentum of the black hole , respectively . there are two killing vectors related to the stationarity and axisymmetry of kerr spacetime , which are expressed as @xmath24 and @xmath25 . in addition , it is known that kerr spacetime possesses a killing tensor , @xmath26 , where @xmath27 and @xmath28 are the kinnersley s null vectors given by @xmath29 for the geodesic motion of a particle in kerr geometry , there are three constants of motion related to the symmetries : @xmath30 where @xmath31 is the four velocity of the particle . @xmath32 and @xmath33 correspond to the specific energy and azimuthal angular momentum of the particle respectively . @xmath34 is called as the carter constant , which corresponds to the square of the specific total angular momentum in schwarzschild case . these specific variables are measured in units of the particle s mass , @xmath35 . one can recover the expressions in the standard units as @xmath36 there is another definition of the carter constant , @xmath37 , which vanishes for equatorial orbits . in this paper , we use @xmath38 as one of the orbital parameters , instead of @xmath39 . by using these constants of motion , the geodesic equations can be expressed in the following form as @xmath40 where we introduced a new parameter @xmath41 through the relation @xmath42 , and some functions as @xmath43 ^ 2-\delta[r^2+(a\hat{e}-\hat{l})^2+\hat{c } ] , \\ \theta(\cos\theta)&:= & \hat{c } - ( \hat{c}+a^2(1-\hat{e}^2)+\hat{l}^2)\cos^2\theta + a^2(1-\hat{e}^2)\cos^4\theta , \\ v_{tr}(r ) & : = & \frac{r^2+a^2}{\delta}p(r ) , \quad v_{t\theta}(\theta ) : = -a(a\hat{e}\sin^2\theta-\hat{l } ) , \\ v_{\varphi r}(r ) & : = & \frac{a}{\delta}p(r ) , \quad v_{\varphi\theta}(\theta ) : = -\left(a\hat{e}-\frac{\hat{l}}{\sin^2\theta}\right).\end{aligned}\ ] ] a generic geodesic orbit in kerr spacetime can be characterized by three parameters , @xmath44 . does not depend on the initial position . hence we do not need the information on the initial position to describe the secular evolution of the orbit at the order considered in this paper . ] in the case of a bound orbit , we can use an alternative set of parameters , @xmath45 , instead of @xmath44 , where @xmath46 and @xmath47 are the values of @xmath48 at the periapsis and apoapsis and @xmath49 is the minimal value of @xmath50 , respectively . using this set of parameters , we can describe the range in which the motion takes place as @xmath51 and @xmath52 . there is another useful choice of parameters used in @xcite , @xmath53 , defined by @xmath54 by analogy to the parametrization used in celestial mechanics , @xmath7 , @xmath8 , @xmath55 are referred as semi - latus rectum , orbital eccentricity , orbital inclination angle , respectively . for later convenience , we also introduce @xmath56 and @xmath57 . since @xmath58 corresponds to the magnitude of the orbital velocity , it can be used as the post - newtonian parameter . for example , we call the @xmath59-correction from the leading order as the fourth order post - newtonian ( 4pn ) correction . it is worth noting that , by introducing @xmath41 , the radial and longitudinal equations of motion in eq.([eq : eom_r_theta ] ) , are completely decoupled . for an bound orbit , therefore , the radial and longitudinal motions are periodic with the periods , @xmath60 , defined by @xmath61 this means that these motions can be expressed in terms of fourier series as @xmath62 where @xmath63 and @xmath64 are the radial and longitudinal frequencies given by @xmath65 and we choose the initial values so that @xmath66 and @xmath67 . approximately so that the radial and longitudinal oscillations reach the minima simultaneously at @xmath68 @xcite . on the other hand , it is not the case if the ratio is rational , _ i.e. _ the resonance case . this implies that the secular evolution of a resonant orbit can not be described only by the pn formulae derived in this work @xcite . ] since the temporal and azimuthal equations of motion in eq.([eq : eom_t_phi ] ) are divided into the @xmath48- and @xmath50-dependent parts , the solutions can be divided into three parts : the linear term with respect to @xmath41 , the oscillatory part with period of @xmath69 , and the oscillatory part with period of @xmath70 . they can be expressed as @xmath71 where the index @xmath72 runs over @xmath73 , and @xmath74 with @xmath75 , representing the time average along the geodesic . we choose the initial conditions as @xmath76 . @xmath77 corresponds to the frequency of the orbital rotation . in appendices [ app : variable_exp ] and [ app : coeff_orbit ] , we present the pn formulae of the orbital parameters , @xmath78 , the fundamental frequencies , @xmath79 , and the fourier coefficients of the motions in eqs . ( [ eq : r - fourier ] ) , ( [ eq : theta - fourier ] ) , ( [ eq : t - fourier ] ) and ( [ eq : phi - fourier ] ) . the gravitational perturbations in kerr spacetime can be described by the weyl scalar , @xmath80 , which satisfies the teukolsky equation @xcite . to solve the teukolsky equation , the method of separation of variables is often used , in which @xmath81 is decomposed in the form as @xmath82 where @xmath83 is the spin-2 spheroidal harmonics and @xmath84 represents a set of indices in the fourier - harmonic expansion , @xmath85 . the separated equation for the radial function is given by @xmath86 r_{\lambda}(r ) = t_{\lambda } , \label{eq : radial - teukolsky}\end{aligned}\ ] ] where @xmath87 @xmath88 is the source term constructed from the energy - momentum tensor of the point particle , and @xmath89 is the eigenvalue determined by the equation for @xmath90 ( to find the basic formulae for the teukolsky formalism used in this paper , refer to the section 2 in @xcite for example ) . the amplitudes of the partial waves at the horizon and at infinity are defined by the asymptotic forms of the solution of the radial equation as @xmath91 with @xmath92 and @xmath93 . since the spectrum with respect to @xmath94 gets discrete in the case of a bound orbit , @xmath95 take the form @xmath96 where @xmath97 denotes the set of indices , @xmath98 , and @xmath99 with these amplitudes , the secular changes of the orbital parameters , @xmath44 , can be expressed by @xmath100 where @xmath101 and @xmath102 is the starobinsky constant given by @xcite @xmath103 \left [ \bar{\lambda}^2 + 36a\omega m -36 a^2\omega^2 \right ] \nonumber \\ & & + ( 2\bar{\lambda}+3)(96a^2\omega^2 - 48a\omega m ) + 144\omega^2(m^2-a^2).\end{aligned}\ ] ] it should be noted that , in these formulae , the averaged rates of change are expressed with respect to the boyer - lindquist time , which can be related to those with respect to @xmath41 @xcite as @xmath104 for a function of time , @xmath105 . also it should be noted that each formula in eqs . ( [ eq : edot])-([eq : cdot ] ) can be divided into the infinity part and the horizon part : the former consists of the terms including the amplitudes of the partial waves at the infinity , @xmath106 , the latter consists of the terms including the amplitudes at the horizon , @xmath107 . as for the energy and azimuthal angular momentum , the infinity parts are balanced with the corresponding fluxes radiated to infinity and the horizon parts with the absorption of the gravitational waves into the central black hole @xcite . the practical calculation of @xmath108 involves solving the geodesic equations , calculating two independent homogeneous solutions of eq.([eq : radial - teukolsky ] ) and the spin-2 spheroidal harmonics , and the fourier transformation of functions consisting of them . in this work , we followed the same procedure proposed in @xcite to perform these calculations analytically . in performing the summation in eqs . ( [ eq : edot])-([eq : cdot ] ) practically , we need to truncate the summation to finite ranges of @xmath109 . to obtain the accuracy of the 4pn and @xmath110 , it is necessary to sum @xmath111 in the range @xmath112 ( @xmath113 ) , @xmath114 in the range @xmath115 ( @xmath116 ) and @xmath117 in the range @xmath118 ( @xmath119 ) for the infinity ( horizon ) part . the other modes out of these ranges are the higher pn corrections than the 4pn order or the higher order corrections than @xmath110 . in this work , we derived the analytic 4pn order formulae of eqs.([eq : edot])-([eq : cdot ] ) in the expansion with respect to the orbital eccentricity , @xmath8 , up to @xmath110 ( we simply call them as the 4pn @xmath110 formulae ) . since the full expressions of the 4pn @xmath110 formulae are too lengthy to show in the text , we show the infinity parts up to the 3pn order and the horizon parts up to the 3.5pn order ( while we keep the expansions with respect to @xmath8 up to @xmath110 ) . the complete expressions of the 4pn @xmath110 formulae will be publicly available online @xcite . the infinity parts of eqs.([eq : edot])-([eq : cdot ] ) are given by @xmath120 , \label{eq : dote8}\\ \left\langle \frac{dl}{dt } \right\rangle_t^\infty & = & \left ( \frac{dl}{dt } \right)_{\rm n } \biggl [ \left\{1+{\frac { 7}{8}}\,{e}^{2}\right\}\,y + \left\ { -{\frac { 1247}{336}}-{\frac { 425}{336}}\,{e}^{2}+{\frac { 10751}{2688}}\,{e}^{4 } \right\ } \,y\,{v}^{2 } \nonumber \\ \nonumber & & + \biggl\ { { \frac { 61}{24}}\,q-{\frac { 61}{8}}\,{y}^{2}q+4\,\pi \,y + \left ( { \frac { 63}{8}}\,q+{\frac { 97}{8}}\,\pi \,y-{\frac { 91}{4 } } \,{y}^{2}q \right ) { e}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \left ( { \frac { 95}{64}}\,q+{\frac { 49}{32}}\,\pi \,y-{\frac { 461}{64}}\,{y}^{2}q \right ) { e}^{4 } -{\frac { 49}{4608}}\,\pi \,y{e}^{6 } \biggr\ } { v}^{3 } \\ \nonumber & & + \biggl\ { -{\frac { 44711}{9072}}-{\frac { 57}{16}}{q}^{2}+{\frac { 45}{8}}\,{y}^{2}{q}^{2}+ \left ( -{\frac { 302893}{6048}}-{\frac { 201}{16}}{q}^{2}+{\frac { 37}{2}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \left ( -{\frac { 701675}{24192}}- { \frac { 351}{128}}{q}^{2}+{\frac { 331}{64}}\,{y}^{2}{q}^{2 } \right ) { e}^{4 } + { \frac { 162661}{16128}}{e}^{6 } \biggr\}\,y\,{v}^{4 } \\ \nonumber & & + \biggl\ { { \frac { 4301}{224}}\,{y}^{2}q-{\frac { 8191}{672}}\,\pi \,y-{\frac { 2633}{224}}\,q \\ \nonumber & & \hspace{0.5 cm } + \left ( -{\frac { 66139}{1344}}\,q-{\frac { 48361}{1344}}\,\pi \,y+{\frac { 18419}{448}}\,{y}^{2}q \right ) { e}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \left ( { \frac { 3959}{1792}}\,q+{\frac { 1657493}{43008}}\,\pi \,y-{\frac { 257605}{5376}}\,{y}^{2}q \right ) { e}^{4 } \\ \nonumber & & \hspace{0.5 cm } + \left ( { \frac { 19161}{3584}}\,q+{\frac { 5458969}{774144}}\,\pi \,y- { \frac { 52099}{1536}}\,{y}^{2}q \right ) { e}^{6}\biggr\ } { v}^{5 } \\ \nonumber & & + \biggl\ { { \frac { 145}{12}}\,\pi \,q+{\frac { 6643739519}{69854400}}\,y+\frac{16}{3}\,{\pi } ^{2}y-{\frac { 1712}{105}}\ , \gamma\,y-{\frac { 3424}{105}}\,\ln \left ( 2 \right ) y \\ \nonumber & & \hspace{0.5 cm } -{\frac { 171}{112}}\,y{q}^{2 } -{\frac { 145}{4}}\,\pi \,{y}^{2}q+{\frac { 1769}{112}}\ , { y}^{3}{q}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \biggl ( { \frac { 995}{12}}\,\pi \,q+{\frac { 229}{6}}\ , { \pi } ^{2}y+{\frac { 6769212511}{8731800}}\,y+{\frac { 1391}{30}}\,\ln \left ( 2 \right ) y-{\frac { 24503}{210}}\,\gamma\,y \\ \nonumber & & \hspace{1.0 cm } -{\frac { 78003}{280 } } \,\ln \left ( 3 \right ) y-{\frac { 46867}{1344}}\,y{q}^{2}-{\frac { 877}{4}}\,\pi \,{y}^{2}q+{\frac { 27997}{192}}\,{y}^{3}{q}^{2 } \biggr ) { e}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \biggl ( { \frac { 21947}{384}}\,\pi \,q+{\frac { 4795392143 } { 7761600}}\,y+{\frac { 3042117}{1120}}\,\ln \left ( 3 \right ) y-{\frac { 418049}{84}}\,\ln \left ( 2 \right ) y \\ \nonumber & & \hspace{1.0 cm } -{\frac { 11663}{140}}\,\gamma\,y + { \frac { 109}{4}}\,{\pi } ^{2}y-{\frac { 1481}{16}}\,y{q}^{2}-{\frac { 22403}{128}}\,\pi \,{y}^{2}q+{\frac { 267563}{1344}}\,{y}^{3}{q}^{2 } \biggr ) { e}^{4 } \\ \nonumber & & \hspace{0.5 cm } + \biggl ( { \frac { 38747}{13824}}\,\pi \,q+{\frac { 31707715321}{186278400}}\,y+{\frac { 23}{16}}\,{\pi } ^{2}y+{\frac { 94138279}{2160}}\,\ln \left ( 2 \right ) y \\ \nonumber & & \hspace{1.0 cm } -{\frac { 1044921875}{96768 } } \,\ln \left ( 5 \right ) y-{\frac { 42667641}{3584}}\,\ln \left ( 3 \right ) y-{\frac { 2461}{560}}\,\gamma\,y-{\frac { 68333}{3584}}\,y{q}^{2 } \\ \nonumber & & \hspace{1.0 cm } -{\frac { 59507}{4608}}\,\pi \,{y}^{2}q+{\frac { 183909}{3584}}\,{y}^ { 3}{q}^{2 } \biggr ) { e}^{6 } \\ & & \hspace{0.5 cm } - \left ( { \frac { 1712}{105}}+{\frac { 24503}{210}}{e}^{2}+{\frac { 11663}{140}}{e}^{4}+{\frac { 2461}{560}}{e}^{6 } \right ) \,y \ln v \biggr\ } { v}^{6 } \biggr],\label{eq : dotl8}\\ \left\langle \frac{dc}{dt } \right\rangle_t^\infty & = & \left ( \frac{dc}{dt } \right)_{\rm n } \biggl [ 1+{\frac { 7}{8}}\,{e}^{2}+ \left ( -{\frac { 743}{336}}+{\frac { 23}{42 } } \,{e}^{2}+{\frac { 11927}{2688}}\,{e}^{4 } \right ) { v}^{2 } \nonumber \\ \nonumber & & + \biggl\ { 4\,\pi-{\frac { 85}{8}}\,yq + \left ( { \frac { 97}{8}}\,\pi -{\frac { 211 } { 8}}\,yq \right ) { e}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \left ( { \frac { 49}{32}}\,\pi -{\frac { 517}{64 } } \,yq \right ) { e}^{4}-{\frac { 49}{4608}}\,\pi \,{e}^{6 } \biggr\ } { v}^{3 } \\ \nonumber & & + \biggl\ { -{\frac { 129193}{18144}}-{\frac { 329}{96}}\,{q}^{2}+{\frac { 53}{8}}\,{y}^{2}{q}^{2}+ \left ( -{\frac { 84035}{1728}}-{\frac { 929 } { 96}}\,{q}^{2}+{\frac { 163}{8}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \left ( -{\frac { 1030273}{48384}}-{\frac { 1051}{768}}\,{q}^{2}+{\frac { 387}{64}}\,{y}^{2}{q}^{2 } \right ) { e}^{4}+{\frac { 100103}{8064}}\,{e } ^{6 } \biggr\ } { v}^{4 } \\ \nonumber & & + \biggl\ { -{\frac { 4159}{672}}\,\pi + { \frac { 2553}{224}}\,yq + \left ( -{\frac { 21229}{1344}}\,\pi -{\frac { 553}{192}}\,yq \right ) { e}^{2}\\ \nonumber & & \hspace{0.5 cm } + \left ( { \frac { 2017013}{43008}}\,\pi -{\frac { 475541 } { 5376}}\,yq \right ) { e}^{4}+ \left ( { \frac { 6039325}{774144}}\,\pi - { \frac { 153511}{3584}}\,yq \right ) { e}^{6 } \biggr\ } { v}^{5 } \\ \nonumber & & + \biggl\ { { \frac { 11683501663}{139708800}}+\frac{16}{3}\,{\pi } ^{2 } -{\frac { 1712}{105}}\,\gamma-{\frac { 3424}{105}}\,\ln \left ( 2 \right ) + { \frac { 1277}{192}}\,{q}^{2}-{\frac { 193}{4}}\,\pi \,yq \\ \nonumber & & \hspace{0.5 cm } + { \frac { 2515}{48}}\,{y}^{2}{q}^{2 } + \biggl ( { \frac { 16319179321}{23284800}}+{\frac { 229}{6}}\,{\pi } ^{2 } -{\frac { 24503}{210}}\,\gamma+{\frac { 1391}{30}}\,\ln \left ( 2 \right ) \\ \nonumber & & \hspace{1.0 cm } -{\frac { 78003}{280}}\,\ln \left ( 3 \right ) + { \frac { 16979}{1344}}\,{q}^{2}-{\frac { 2077}{8}}\,\pi \,yq+{\frac { 118341}{448}}\,{y}^{2}{q}^{2 } \biggr ) { e}^{2 } \\ \nonumber & & \hspace{0.5 cm } + \biggl ( { \frac { 211889615389}{372556800}}+ { \frac { 109}{4}}\,{\pi } ^{2}+{\frac { 3042117}{1120}}\,\ln \left ( 3 \right ) -{\frac { 11663}{140}}\,\gamma-{\frac { 418049}{84}}\,\ln \left ( 2 \right ) \\ \nonumber & & \hspace{1.0 cm } -{\frac { 132193}{3584}}\,{q}^{2}-{\frac { 24543}{128 } } \,\pi \,yq+{\frac { 91747}{336}}\,{y}^{2}{q}^{2 } \biggr ) { e}^{4 } \\ \nonumber & & \hspace{0.5 cm } + \biggl ( { \frac { 33928992071}{186278400}}-{\frac { 1044921875}{96768}}\ , \ln \left ( 5 \right ) + { \frac { 23}{16}}\,{\pi } ^{2}-{\frac { 42667641 } { 3584}}\,\ln \left ( 3 \right ) \\ \nonumber & & \hspace{1.0 cm } + { \frac { 94138279}{2160}}\,\ln \left ( 2 \right ) -{\frac { 2461}{560}}\,\gamma-{\frac { 24505}{5376}}\,{q}^{2}- { \frac { 4151}{288}}\,\pi \,yq+{\frac { 718799}{10752}}\,{y}^{2}{q}^{2 } \biggr ) { e}^{6 } \\ & & \hspace{0.5 cm } - \left ( { \frac { 1712}{105}}+{\frac { 24503}{210}}\,{e}^{2}+{\frac { 11663}{140}}\,{e}^{4}+{\frac { 2461}{560}}\,{e}^{6 } \right ) \ln v \biggr\ } { v}^{6 } \biggl],\label{eq : dotc8}\end{aligned}\ ] ] where the leading contributions are given by @xmath121 the horizon parts of eqs.([eq : edot])-([eq : cdot ] ) are given by @xmath122 , \label{eq : doteh } \\ \left\langle \frac{dl}{dt } \right\rangle_t^{\rm h } & = & \left ( \frac{dl}{dt } \right)_{\rm n } \biggl [ -{\frac { \left ( 8 + 24\,{e}^{2}+3\,{e}^{4 } \right)}{1024 } } \left ( 16 + 33\,{q}^{2}+16\,{y}^{2}+18\,{y}^{2}{q}^{2}+45\,{y}^{4}{q}^{2 } \right ) q v^5 \nonumber \\ & & -\biggl\ { \frac{5}{4}+{\frac { 375}{128}}\,{q}^{2}-\frac{1}{4}\,{y}^{2 } -{\frac { 63}{64}}\,{y}^{2}{q}^{2}+{\frac { 15}{128}}\,{y}^{4}{q}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( 10+{\frac { 5955}{256}}\,{q}^{2}-\frac{5}{4}\,{y}^{2}-{\frac { 855}{128}}\,{y}^{2}{q}^{2}+{\frac { 675}{256}}\,{y}^{4}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( { \frac { 255}{32}}+{\frac { 18855}{1024}}\,{q}^{2}-{\frac { 15}{32}}\,{y}^{2}-{\frac { 2295}{512}}\,{y}^{2}{q}^{2}+{\frac { 3375}{1024}}\,{y}^{4}{q}^{2 } \right ) { e}^{4 } \nonumber \\ & & \hspace{0.5 cm } + \left ( { \frac { 15}{32}}+{\frac { 2205}{2048}}\,{q}^{2}-{\frac { 225}{1024}}\,{y}^{2}{q}^{2}+{\frac { 525}{2048}}\,{y}^{4}{q}^{2 } \right ) { e}^{6}\biggr\ } q v^7 \biggl ] , \label{eq : dotlh } \\ \left\langle \frac{dc}{dt } \right\rangle_t^{\rm h } & = & \left ( \frac{dc}{dt } \right)_{\rm n } \biggl [ -{\frac { 1}{1024 } } \left ( 8 + 24\,{e}^{2}+3\,{e}^{4 } \right ) \left ( 16 + 3\,{q}^{2}+45\,{y}^{2}{q}^{2 } \right ) q y v^5 \nonumber \\ & & + \biggl\ { { \frac{1}{16}+\frac { 93}{256}}\,{q}^{2}-{\frac { 165}{256}}\,{y}^{2}{q}^{2}+ \left(\frac{5}{8}+{\frac { 705}{256}}\,{q}^{2}-{\frac { 1125}{256}}\,{y}^{2}{q } ^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( { \frac { 27}{128}}+{\frac { 4131}{2048}}\ , { q}^{2}-{\frac { 8235}{2048}}\,{y}^{2}{q}^{2 } \right ) { e}^{4 } \nonumber \\ & & \hspace{0.5 cm } + \left ( - { \frac { 3}{128}}+{\frac { 27}{256}}\,{q}^{2}-{\frac { 165}{512}}\,{y}^{2 } { q}^{2 } \right ) { e}^{6 } \biggr\ } q y v^7 \biggr ] . \label{eq : dotch}\end{aligned}\ ] ] @xmath123 , @xmath124 and @xmath125 in eqs . ( [ eq : doteh])-([eq : dotch ] ) are new pn formulae derived in this paper . @xmath126 , @xmath127 and @xmath128 in eqs . ( [ eq : dote8])-([eq : dotc8 ] ) are consistent with those in ref . @xcite up to 2.5pn and @xmath129 . from the leading order expressions in eq . ( [ eq : dotfn ] ) , one will find that the carter parameter , @xmath38 , does not change due to the radiation of the gravitational waves when @xmath130 ( equatorial orbits ) because @xmath131 . in the schwarzschild case , the carter parameter corresponds to the square of the equatorial angular momentum ( the normal component to the rotational axis of the central black hole ) . then there is expected to exist the duality between @xmath132 and @xmath38 due to the spherical symmetry . in fact , from eqs . ( [ eq : dotl8 ] ) and ( [ eq : dotc8 ] ) , ( and also from ( [ eq : dotlh ] ) and ( [ eq : dotch ] ) ) , one can find that @xmath133 vanishes in @xmath134 ( polar orbits ) while @xmath135 for @xmath134 coincides with @xmath133 for @xmath130 . this can be also realized by seeing that the secular change of the total angular momentum , @xmath136 , is independent of @xmath137 . then , it might be possible to understand that @xmath138 becomes 1.5pn from the leading order when @xmath139 and @xmath134 ( polar orbits ) due to the spin - orbit coupling . from the expressions of the horizon parts shown in eqs . ( [ eq : doteh])-([eq : dotch ] ) , we find that the absorption of the gravitational waves to the central black hole contributes at @xmath140 from the leading order in eq . ( [ eq : dotfn ] ) for @xmath141 and at @xmath59 for @xmath142 . the @xmath140 and @xmath143 corrections in @xmath144 can be positive for @xmath145 , which means that the particle can gain the energy through a superradiance phenomenon . these observations are consistent with the results for circular , equatorial orbits shown in refs . @xcite . we also find that the superradiance terms in eq . ( [ eq : doteh ] ) vanish for @xmath134 , and that @xmath144 has only the 4pn and higher order corrections . the superradiance terms may come from the coupling between the black hole spin and the orbital angular momentum , like @xmath146 . hence , when the orbital inclination increases ( @xmath137 gets small correspondingly ) , the superradiance is suppressed @xcite . to investigate the accuracy of the 4pn @xmath110 formulae derived in this work , we compare them to the corresponding numerical results given by the method established in ref . @xcite , which enables one to compute the modal fluxes with the relative error of @xmath147 in double precision computations . in the practical computations , as well as in deriving the analytic expressions , we need to truncate the summation to finite ranges of @xmath109 in eqs . ( [ eq : edot])-([eq : cdot ] ) . in order to save the computation time in the numerical calculation , we sum @xmath111 up to @xmath148 . we can check that the error due to neglecting terms for @xmath149 is smaller than the relative error in the 4pn @xmath110 formulae from the corresponding numerical results up to @xmath150 . we also truncate @xmath114 and @xmath117 to achieve the relative error of @xmath151 in numerical results up to @xmath150 . for the parameters investigated in the comparison , the relative error of @xmath151 achieved by truncating @xmath114 and @xmath117 is again smaller than the relative error in the 4pn @xmath110 formulae from the numerical results up to @xmath150 . thus , we can regard numerical results as benchmarks to investigate the accuracy in our analytic formulae . here we define the relative error in the analytic formula of @xmath152 by @xmath153 where @xmath154 denotes the analytic formula in order to distinguish it from the corresponding numerical result , @xmath155 . we also define the relative errors in the analytic formulae of @xmath156 and @xmath135 in a similar manner and denote them as @xmath157 and @xmath158 respectively . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] shows several plots of @xmath159 for the 4pn @xmath110 formula as a function of @xmath7 for several sets of @xmath160 with @xmath161 . in the plots , we also show the relative errors in the 2.5pn @xmath129 and 3pn @xmath162 formulae for reference . from the plots for @xmath2 ( three on the top ) , one can find that @xmath159 falls off faster than @xmath163 for @xmath164 ( similarly , the relative errors in the 2.5pn @xmath129 and 3pn @xmath162 formulae fall off faster than @xmath165 and @xmath166 ) . noting @xmath57 , this would be a good indication that our pn formula has been derived correctly up to required order . @xmath159 is expected to contain not only higher order corrections than the 4pn order , but also the higher order corrections of eccentricity than @xmath110 in the lower pn terms , which will become dominant when @xmath7 and @xmath8 get larger . in fact , seeing the plots for @xmath6 in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , one can find that the relative error strays out of the expected power law line for large @xmath7 . this behavior is clearer in the plots of the relative error in the 2.5pn @xmath129 formula . from eqs . ( [ eq : dote8 ] ) and ( [ eq : doteh ] ) , we know that the relative error in the 2.5pn @xmath129 formula contains the @xmath162 correction in the @xmath167 term . the effect of this correction appears as large-@xmath7 plateaus in the plots ( also see fig . [ fig : dote_en_q0.9_y1_0 ] ) . this may motivate us to perform the higher order expansion with respect to the orbital eccentricity in the pn formulae or to derive the pn formulae without performing the expansion with respect to the orbital eccentricity @xcite . in addition , it might be noted that the behavior of the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 and @xmath8 . for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] + for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] + for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] for @xmath161 , @xmath169 and @xmath170 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . in addition to the error in the 4pn @xmath110 formula , those in the 2.5pn @xmath129 and the 3pn @xmath162 formulae are shown in each plot for reference . we truncated the plots at @xmath173 because the relative errors get too large ( nearly or more than unity ) in @xmath174 to be meaningful . one finds that the relative error becomes smaller with increasing orders of the pn approximation and the expansion with respect to the eccentricity . the relative error in our 4pn @xmath110 formula falls off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 . since @xmath57 , this would imply that our 4pn formula is correctly representing the secular change up to the 4pn order . note , however , that the relative error in the 4pn @xmath110 formula for @xmath6 falls off slower than @xmath163 when the semi - latus rectum becomes larger,_e.g . _ @xmath176 . this might be because of the higher order corrections of @xmath8 than @xmath110 , which will contain the lower pn terms than the 4pn order . we also note that changing the inclination angle , @xmath55 , does not change the dependence on @xmath7 of the relative error for fixed @xmath168 and @xmath8 so much . this might be checked more easily in contour plots in fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] , which show the relative error as a function of @xmath7 and @xmath8 for fixed @xmath168 and @xmath55 . , title="fig:",width=192 ] in fig . [ fig : dotf8h_q0.9_m0.9_e0.1_0.4_0.7_inc50 ] , we show the relative errors in the 4pn @xmath110 formulae for the secular changes of the three orbital parameters , @xmath78 , for several sets of @xmath177 and @xmath178 . as in the case of @xmath159 shown in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors , @xmath157 and @xmath158 , fall off faster than @xmath163 when @xmath179 , except for the large @xmath7 region ( @xmath180 ) in the case of @xmath6 . thus , the 4pn @xmath110 formulae for the secular changes of the orbital parameters are expected to be valid up to @xmath59 . from fig . [ fig : dotf8h_q0.9_m0.9_e0.1_0.4_0.7_inc50 ] , one might think that it is enough to investigate only @xmath181 to discuss the accuracy of our formulae since there are not large differences in the relative errors , @xmath159 , @xmath157 and @xmath158 . , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] + , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] + , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] + , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] , as functions of the semi - latus rectum @xmath7 for @xmath178 , @xmath182 and @xmath183 ( from top to bottom ) and @xmath169 and @xmath170 ( from left to right ) . we truncated the plots at @xmath184 , where @xmath185 is the value of @xmath7 at the `` separatrix '' ( the boundary between stable and unstable orbits ) , because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in our 4pn @xmath110 formulae fall off faster than @xmath163 when the eccentricity is small , _ e.g. _ @xmath175 , while the fall - off gets slower when @xmath7 is larger for @xmath6 . there are not large differences in the behaviors of @xmath159 , @xmath157 and @xmath158 . this suggests that it might be enough to focus only on @xmath187 to investigate the accuracy and convergence of our 4pn formulae . , title="fig:",width=192 ] fig . [ fig : dote8h_q0.9_m0.9_inc20_80 ] shows contour plots for @xmath159 as a function of @xmath7 and @xmath8 for several sets of @xmath188 . from these plots , one may be able to comprehend the accuracy of our pn formulae more easily than using figs . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] and [ fig : dotf8h_q0.9_m0.9_e0.1_0.4_0.7_inc50 ] . one will find that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 . moreover , it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as expected from fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . if one requires @xmath189 as an error tolerance , one can use the contour line with the label @xmath0 to find the region of validity in the figure . for example , one will find that @xmath190 for @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] + formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] + formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] + formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] formula for the secular change of the particle s energy , @xmath159 , as a function of the semi - latus rectum @xmath7 and the eccentricity @xmath8 for @xmath182 and @xmath183 ( from top to bottom ) and @xmath171 and @xmath172 ( from left to right ) . we truncated the plots at @xmath184 because the relative errors get too large in @xmath174 to be meaningful and the orbit is not stable for @xmath186 . from the figures , it is easily found that the relative error becomes smaller ( larger ) for larger ( smaller ) @xmath7 and smaller ( larger ) @xmath8 with fixed @xmath168 and @xmath55 . if one requires the relative error to be less than @xmath0 , the region in the semi - latus rectum @xmath7 and the eccentricity @xmath8 will be @xmath1 and @xmath2 , @xmath3 and @xmath4 , and @xmath5 and @xmath6 when @xmath161 . it might be noticed that the relative error does not strongly depend on the inclination angle @xmath55 for fixed @xmath168 as pointed out in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] . , title="fig:",width=192 ] in order to improve the accuracy in the analytic pn formulae , one may apply some resummation methods such as pad approximation @xcite , the factorized resummation @xcite and the exponential resummation @xcite . since the exponential resummation may be the simplest one to implement among them , we here choose to implement the exponential resummation . we apply it to our 4pn formulae and check how the accuracy is improved . to introduce the exponential resummation , we make use of the following identity @xmath191\right\ } , \label{eq : exp - ln - id}\ ] ] where @xmath192 . the exponential resummation can be obtained by replacing the exponent in ( [ eq : exp - ln - id ] ) to the expansion with respect to @xmath58 , @xmath193 \bigg|_{{\rm truncated \ after \ } n{\rm th \ order \ of } \ v},\ ] ] where we do not perform the expansion with respect to @xmath8 . since our pn formulae for @xmath194 are given at the 4pn order , we truncate @xmath195 after @xmath59 . finally , the exponential resummed form is expressed as @xmath196 fig . [ fig : dotf8h_q0.9_m0.9_e0.7_inc50_exp ] shows the relative errors in the exponential resummed forms of the secular changes of @xmath197 , @xmath198 and @xmath38 , estimated by using eq . ( [ eq : relative_error ] ) . we also show the relative errors in the taylor - type formulae in the same graphs for comparison . one will find that the relative errors in the exponential resummed forms are less than those in the taylor - type formulae in most cases , except for @xmath135 in the case of @xmath161 , @xmath199 . using the exponential resummation when @xmath161 and @xmath178 , the region to satisfy @xmath190 is extended to @xmath9 from @xmath1 for @xmath2 , @xmath10 from @xmath3 for @xmath4 , and @xmath11 from @xmath5 for @xmath6 . this might motivate us to use the resummation method to improve the accuracy of taylor - type formulae even in the case of general orbits . as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] + as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] + as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath178 and @xmath169 and @xmath170 ( from top to bottom ) . we truncated the plots at @xmath173 because the relative errors in the pn formulae get too large in @xmath174 to be meaningful . using the exponential resummation , the accuracy is improved in most cases . for example , the region to satisfy @xmath190 is improved from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . this would suggest us to try to apply resummation methods to the pn formulae even in the case of general orbits . , title="fig:",width=192 ] apart from comparisons to the numerical results , we may also discuss the convergence property in our pn formulae with respect to @xmath58 and @xmath8 by investigating the contribution of each order of @xmath58 and @xmath8 in the formulae although this is a rough estimation . first we assess the pn convergence of our formulae . for this purpose , we introduce @xmath200 as @xmath201 where @xmath202 and @xmath200 is the @xmath203 term in the pn formula of @xmath152 , _ e.g. _ @xmath204 , @xmath205 and @xmath206 . @xmath200 depends on @xmath207 in general although we omit the argument for simplicity . since @xmath200 shows the relative importance of the @xmath203 term in the pn formulae , it can be used to investigate the convergence property with respect to @xmath58 : it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . in fig . [ fig : dote_vn_q0.9_e0.1_0.9_y1_0 ] , we plot the relative contribution of each order , @xmath200 , as a function of @xmath7 for several sets of @xmath160 and @xmath161 . from this figure , one may find that @xmath200 does not strongly depend on the inclination angle , @xmath55 , as shown in sec . [ sec : compare ] , while it strongly depends on @xmath8 . the convergence gets worse when the orbital eccentricity becomes larger . this tendency is particularly evident in the small-@xmath7 region . fixing the value of @xmath7 , the orbit with larger @xmath8 passes by closer to the central black hole and will be affected by the stronger gravitational field . hence the pn convergence is expected to be worse when the eccentricity becomes larger . term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] + term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] + term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : delta_n ] ) . we plot the absolute value of @xmath200 as a function of the semi - latus rectum @xmath7 for @xmath169 and @xmath170 ( from left to right ) , and @xmath210 and @xmath211 ( from top to bottom ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath208 for moderately large @xmath209 if the pn formula converges . as shown in sec . [ sec : compare ] , @xmath200 does not strongly depend on @xmath55 for a fixed @xmath8 although @xmath200 strongly depends on @xmath8 . in fact , the convergence seems worse the orbital eccentricity becomes larger . this tendency is clear for small @xmath7 , _ e.g. _ @xmath212 . , title="fig:",width=192 ] next , in order to investigate the convergence of the expansion with respect to the orbital eccentricity in the pn formula , we introduce @xmath213 as @xmath214 , \label{eq : a_n}\end{aligned}\ ] ] where the term @xmath215 coincides with the energy flux for circular orbits and @xmath216 when @xmath209 is odd . one may ask whether the condition , @xmath217 , is satisfied for moderately large integer @xmath209 if the series converges . from fig . [ fig : dote_en_q0.9_y1_0 ] , it is found that the condition is satisfied in most cases . as expected , the convergence becomes slower when the eccentricity is larger . especially , the convergence gets worse when @xmath212 in @xmath6 case . the calculation of the higher pn corrections will be necessary to improve the bad convergence for small @xmath7 . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as in sec . [ sec : compare ] . term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] + term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] + term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] term in the pn formula for @xmath152 , defined in eq . ( [ eq : a_n ] ) . we plot the absolute value of @xmath218 as a function of the semi - latus rectum @xmath7 for @xmath219 and @xmath211 ( from left to right ) when @xmath161 . we truncated the plots at @xmath173 because the relative contributions get too large in @xmath174 to be meaningful . it is expected that @xmath217 for moderately large @xmath209 if the series with respect to @xmath8 converges . this condition is satisfied in most cases shown in this figure . the convergence becomes slower when the eccentricity is larger . especially , the convergence for @xmath212 is quite bad in @xmath6 case . we also note that @xmath213 does not strongly depend on @xmath55 for a fixed @xmath168 as mentioned in sec . [ sec : compare ] . , title="fig:",width=192 ] we have derived the secular changes of the orbital parameters , the energy , azimuthal angular momentum , and carter parameter of a point particle orbiting a kerr black hole , by using the post - newtonian approximation in the first order black hole perturbation theory . we have extended the previous work @xcite , which derived formulae up to the 2.5pn order with the second order correction with respect to the eccentricity , to the 4pn order with the sixth order correction with respect to the eccentricity . we have also included the contribution due to the black hole absorption , which has not been included in @xcite . as shown in the case of equatorial , circular orbits @xcite , we have found that the secular changes of the three orbital parameters due to the absorption , appear at the 2.5pn ( 4pn ) from the leading order in the kerr ( schwarzschild ) case , and that the 2.5pn and 3.5pn contributions of the absorption to the secular change of the particle s energy can be positive for @xmath145 , which implies that a superradiance can be realized in the kerr case . we have also found that the superradiant contributions in the secular change of the energy get smaller when the inclination angle becomes larger and they vanishes for polar ( @xmath134 ) orbits . this means that the superradiant scattering may be suppressed for inclined orbits @xcite . to investigate the accuracy in our 4pn formulae , we have compared the formulae to high - precision numerical results @xcite in sec . [ sec : compare ] . we have found that the accuracy gets worse when the orbital velocity and the orbital eccentricity become larger , as expected . if the relative error in the 4pn @xmath110 formula for the secular change of the energy is required to be less than @xmath0 , the parameter region to satisfy it might be @xmath1 for @xmath2 , @xmath3 for @xmath4 , and @xmath5 for @xmath6 when @xmath161 . the region does not strongly depend on the orbital inclination angle . from fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , one can clearly find the improvement of the accuracy in our pn formulae from the previous work at the 2.5pn order and the second order correction in the orbital eccentricity @xcite whose relative error is larger than @xmath220 for @xmath11 when @xmath221 since , in this region , the error due to the truncation of the expansion with respect to the orbital eccentricity is larger than the one of the pn expansion . one may improve the accuracy of our pn formulae by using resummation methods . in this paper , we have applied the exponential resummation @xcite to our 4pn formulae and confirmed that the resummation method improves the accuracy in most cases investigated here . for example , we found that the region in which the relative errors are less than @xmath0 can be extended from @xmath1 to @xmath9 for @xmath2 , @xmath3 to @xmath10 for @xmath4 , and @xmath5 to @xmath11 for @xmath6 . we also investigate the convergence properties of the pn expansion and the expansion with respect to the orbital eccentricity , respectively . both convergences get worse when the semi - latus rectum is smaller ; in other words , the gravitational field becomes stronger . this tendency gets clearer in the case of large eccentricity , in which the particle passes by closer to the central black hole . in order to improve the accuracy and convergence of the 4pn @xmath110 formulae near the central black hole and to obtain the physical information of the source in the strong - field region , it is necessary to derive the higher order corrections of the pn expansion and the expansion with respect to the eccentricity . it may be possible to avoid the expansion with respect to the eccentricity and to derive the pn formulae applicable to arbitrary eccentricity . so far the pn formulae of the rate of the energy loss without performing the expansion with respect to the eccentricity had been derived for equatorial orbits in @xcite . the extension of these results to the case of inclined orbits is challenging : we can obtain analytic expressions for general bound geodesic orbits in kerr spacetime without performing the expansion with respect to the eccentricity nor the inclination by using results in ref . @xcite , while we need to reformulate the source term of the teukolsky equation and the derivation of the partial waves constructed form the source term . we would like to leave it to the future work . we would like to thank takahiro tanaka and hiroyuki nakano for useful discussions and comments . we are also grateful to theoretical astrophysics group in kyoto university for hospitality during the intermediate stage of completing this paper . ns acknowledges the support of the grand - in - aid for scientific research ( no . 25800154 ) . rf s work was supported by the european union s fp7 erc starting grant `` the dynamics of black holes : testing the limits of einstein s theory '' grant agreement no . dybho256667 . some numerical computations were performed at the cluster `` baltasar - sete - sis '' in centra / ist . some analytic calculations were carried out on ha8000/rs440 at yukawa institute for theoretical physics in kyoto university . in this section , we present the pn formulae of the orbital parameters , @xmath222 , and the fundamental frequencies , @xmath79 . here we show the formulae up to the 3pn @xmath110 order to save space although it is possible to calculate them to the higher order . the higher order results will be publicly available online @xcite . @xmath223 , \\ \hat{c } & = & \left\{\frac{1}{y^2}-1\right\}\,l^2 , \nonumber \\ & = & p^2 v^2 ( 1-y^2 ) \bigl [ 1 + ( 3+e^2 ) v^2 - 2 q y \left ( 3+e^2 \right ) v^3 \nonumber \\ & & + \left\ { 9 + 2\,{y}^{2}{q}^{2}+ \left ( 6 + 2\,{y}^{2}{q}^{2 } \right ) { e}^{2 } + { e}^{4 } \right\ } v^4 - 4\,qy \left ( 2+{e}^{2 } \right ) \left ( 3+{e}^{2 } \right ) v^5 \nonumber \\ & & + \bigl\ { 27 - 8\,{q}^{2}+30\,{y}^{2}{q}^{2}+ \left ( 27 + 28\,{y}^{2}{q}^{2 } -8\,{q}^{2 } \right ) { e}^{2 } + \left ( 9 + 6\,{y}^{2}{q}^{2 } \right ) { e}^{4 } + { e}^{6 } \bigr\ } v^6 \bigr ] , \\ \omega_t & = & p^2 \biggl [ 1+\frac{3}{2}\,{e}^{2}+{\frac { 15}{8}}\,{e}^{4}+{\frac { 35}{16}}\,{e}^{6}+ \left\ { \frac{3}{2}-\frac{1}{4}\,{e}^{2}-{\frac { 15}{16}}\,{e}^{4}-{\frac { 45}{32}}\,{e}^{6 } \right\ } { v}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left\ { 2\,yq\,{e}^{2}+3\,yq\,{e}^{4}+{\frac { 15}{4}}\,yq\,{e}^{6 } \right\ } { v}^{3 } \nonumber \\ & & \hspace{0.5 cm } + \biggl\ { { \frac { 27}{8}}-\frac{1}{2}\,{y}^{2}{q}^{2 } + \frac{1}{2}\,{q}^{2}+ \left ( -{\frac { 99}{16}}+{q}^{2}-2\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{1.0 cm } + \left ( -{\frac { 567}{64}}+{\frac { 21}{16}}\,{q}^{2}- { \frac { 45}{16}}\,{y}^{2}{q}^{2 } \right ) { e}^{4}+ \left ( -{\frac { 1371 } { 128}}+{\frac { 25}{16}}\,{q}^{2}-{\frac { 55}{16}}\,{y}^{2}{q}^{2 } \right ) { e}^{6 } \biggr\ } { v}^{4 } \nonumber \\ & & \hspace{0.5 cm } + \left\ { -3\,yq+{\frac { 43}{2}}\,yq\,{e}^{2}+{\frac { 231}{8}}\,yq\,{e}^{4}+{\frac { 555}{16}}\,yq\,{e}^{6 } \right\ } { v}^{5 } \nonumber \\ & & \hspace{0.5 cm } + \biggl\ { { \frac { 135}{16}}-\frac{1}{4}\,{q}^{2}+\frac{3}{4}\,{y}^{2}{q}^{2}+ \left ( -{\frac { 1233}{32}}+{\frac { 47}{4}}\,{q}^{2}-{\frac { 75}{2}}\ , { y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{1.0 cm } + \left ( -{\frac { 6567}{128}}+{\frac { 499}{32}}\,{q}^{2}-{\frac { 1577}{32}}\,{y}^{2}{q}^{2 } \right ) { e}^{4 } \nonumber \\ & & \hspace{1.0 cm } + \left ( -{\frac { 15565}{256}}+{\frac { 75}{4}}\,{q}^{2}-{\frac { 1887 } { 32}}\,{y}^{2}{q}^{2 } \right ) { e}^{6 } \biggr\ } { v}^{6 } \biggr ] , \\ \omega_r & = & p\,v\biggl [ 1 + \left\ { -\frac{3}{2}+\frac{1}{2}\,{e}^{2 } \right\ } { v}^{2 } + \left\ { 3\,yq - yq\,{e}^{2 } \right\ } { v}^{3 } \nonumber \\ & & \hspace{0.5 cm } + \left\ { -{\frac { 45}{8}}+\frac{1}{2}\,{q}^{2}-2\,{y}^{2}{q}^{2 } + \left ( \frac{1}{4}\,{q}^{2}+\frac{1}{4}\,{y}^{2}{q}^{2 } \right ) { e}^{2}+ \frac{3}{8}\,{e}^{4 } \right\ } { v}^{4 } \nonumber \\ & & \hspace{0.5 cm } + \left\ { { \frac { 33}{2}}\,yq + 2\,yq\,{e}^{2}-\frac{3}{2}\,yq\,{e}^{4}\right\ } { v}^{5 } \nonumber \\ & & \hspace{0.5 cm } + \biggl\ { -{\frac { 351}{16 } } -{\frac { 51}{2}}\,{y}^{2}{q}^{2 } + { \frac { 33}{4}}\,{q}^{2}+ \left ( -{\frac { 135}{16}}+{\frac { 7}{8}}\,{q}^{2}-{\frac { 39}{8}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{1.0 cm } + \left ( { \frac { 21}{16}}+\frac{1}{8}\,{q}^{2}+{\frac { 13}{8}}\,{y}^{2}{q}^{2 } \right ) { e}^{4}+{\frac { 5}{16}}\,{e}^{6 } \biggr\ } { v}^{6 } \biggr ] , \\ \omega_\theta & = & p\,v\biggl [ 1 + \left\ { \frac{3}{2}+\frac{1}{2}\,{e}^{2}\right\ } { v}^{2}- \left\ { 3\,yq+yq\,{e}^{2 } \right\ } { v}^{3 } \nonumber \\ & & \hspace{0.5 cm } + \left\ { { \frac { 27}{8}}+\frac{7}{4}\,{y}^{2}{q}^{2}-\frac{1}{4}\,{q}^{2}+ \left ( \frac{9}{4}+\frac{1}{4}\,{q}^{2}+\frac{1}{4}\,{y}^{2}{q}^{2 } \right ) { e}^{2}+\frac{3}{8}\,{e}^{4 } \right\ } { v}^{4 } \nonumber \\ & & \hspace{0.5 cm } - \left\ { \frac{15}{2}\,yq+7\,yq\,{e}^{2}+\frac{3}{2}\,yq\,{e}^{4 } \right\ } { v}^{5 } \nonumber \\ & & \hspace{0.5 cm } + \biggl\ { { \frac { 135}{16}}+{\frac { 57}{8}}\,{y}^{2}{q}^{2 } -{\frac { 27}{8}}\,{q}^{2}+ \left ( { \frac { 135}{16}}-{\frac { 19}{4 } } \,{q}^{2}+{\frac { 45}{4}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{1.0 cm } + \left ( { \frac { 45}{16}}+\frac{1}{8}\,{q}^{2}+{\frac { 13}{8}}\,{y}^{2}{q}^{2 } \right ) { e}^{4}+{\frac { 5}{16}}\,{e}^{6 } \biggr\ } { v}^{6 } \biggr ] , \\ \omega_\varphi & = & p\,v\biggl [ 1 + \left\ { \frac{3}{2}+\frac{1}{2}\,{e}^{2 } \right\ } { v}^{2 } + \left\ { 2\,q-3\,yq -yq\,{e}^{2}\right\ } { v}^{3 } \nonumber \\ & & \hspace{0.5 cm } + \left\ { -\frac{3}{2}\,y{q}^{2}+\frac{7}{4}\,{y}^{2}{q}^{2}-\frac{1}{4}\,{q}^{2}+{\frac { 27}{8}}+ \left ( \frac{9}{4}+\frac{1}{4}\,{q}^{2}+\frac{1}{4}\,{y}^{2}{q}^{2 } \right ) { e}^{2}+\frac{3}{8}\,{e}^{4 } \right\ } { v}^{4 } \nonumber \\ & & \hspace{0.5 cm } + \left\ { 3\,q-\frac{15}{2}\,yq+ \left ( 4\,q-7\,yq \right ) { e}^{2}-\frac{3}{2}\,yq\,{e}^{4 } \right\ } { v}^{5 } \nonumber \\ & & \hspace{0.5 cm } + \biggl\ { -\frac{9}{4}\,y{q}^{2}+{\frac { 57}{8}}\,{y}^{2}{q}^{2}+{\frac { 135}{16 } } -{\frac { 27}{8}}\,{q}^{2}+ \left ( { \frac { 135}{16}}-{\frac { 19}{4 } } \,{q}^{2}-{\frac { 35}{4}}\,y{q}^{2}+{\frac { 45}{4}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{1.0 cm } + \left ( { \frac { 45}{16}}+\frac{1}{8}\,{q}^{2}+{\frac { 13}{8 } } \,{y}^{2}{q}^{2 } \right ) { e}^{4}+{\frac { 5}{16}}\,{e}^{6 } \biggr\ } { v}^{6 } \biggr].\end{aligned}\ ] ] here we show the pn formulae of the fourier coefficients in eqs . ( [ eq : r - fourier ] ) , ( [ eq : theta - fourier ] ) , ( [ eq : t - fourier ] ) and ( [ eq : phi - fourier ] ) up to the 3pn @xmath110 order . the 4pn @xmath110 results obtained in this work will be available online @xcite . in this work , we follow the same procedure as in @xcite to derive the amplitudes of the partial waves , @xmath95 in ( [ eq : partial - wave ] ) . in the formal expression , the dependence of @xmath224 appears in the form of the combination as @xmath225 , which can be expressed in the fourier series as @xmath226,\ ] ] therefore we show the fourier coefficients of @xmath227 instead of @xmath224 . @xmath228 @xmath229 @xmath230 @xmath231 @xmath232 @xmath233 an alternative set of the orbital parameters , @xmath234 , is also useful to specify the orbit . the secular changes of the parameters can be derived from those of @xmath192 , as @xmath235 where @xmath236 is the jacobian matrix for the transformation from @xmath78 to @xmath237 . , one need to calculate @xmath78 up to @xmath238 since the leading terms do not depend on @xmath8 and then the relative orders of accuracy of their first derivatives with the eccentricity of the jacobian matrix in eq . ( [ eq : trans_params ] ) are reduced by @xmath129 . for a similar reason , one also need to calculate @xmath197 up to 5pn order because the relative pn order of @xmath239 is reduced by @xmath240 compared to @xmath197 . ] substituting the 3pn @xmath110 formulae of @xmath241 shown in sec . [ sec : result ] into the above relation , we obtain the secular changes of @xmath237 associated with the flux of gravitational waves to infinity as @xmath242 , \label{eq : dotv8 } \\ \left < \frac{d{e}}{dt}\right>^\infty_t & = & \left(\frac{de}{dt}\right)_{\rm n } \biggl [ 1+{\frac { 121}{304}}\,{e}^{2}+ \left\ { -{\frac { 6849}{2128}}-{\frac { 2325}{2128}}\,{e}^{2}+{\frac { 22579}{17024}}\,{e}^{4 } \right\ } { v}^{2 } \nonumber \\ & & + \biggl\ { { \frac { 985}{152}}\,\pi -{\frac { 879}{76}}\,yq + \left ( { \frac { 5969}{608}}\,\pi -{\frac { 699}{76}}\,yq \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( { \frac { 24217}{29184}}\,\pi -{\frac { 1313}{608}}\,yq \right ) { e}^{4 } \biggr\ } { v}^{3 } \nonumber \\ & & + \biggl\ { - { \frac { 286397}{38304}}-{\frac { 3179}{608}}\,{q}^{2}+{\frac { 5869}{608 } } \,{y}^{2}{q}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( -{\frac { 2070667}{51072}}-{\frac { 8925}{1216 } } \,{q}^{2}+{\frac { 633}{64}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( -{\frac { 3506201}{306432}}-{\frac { 3191}{4864}}\,{q}^{2}+{\frac { 9009 } { 4864}}\,{y}^{2}{q}^{2 } \right ) { e}^{4 } \biggr\ } { v}^{4 } \nonumber \\ & & + \biggl\ { -{\frac { 1903}{304}}\,yq-{\frac { 87947 } { 4256}}\,\pi + \left ( -{\frac { 3539537}{68096}}\,\pi -{\frac { 93931 } { 8512}}\,yq \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( { \frac { 5678971}{817152}}\,\pi - { \frac { 442811}{17024}}\,yq \right ) { e}^{4 } \biggr\ } { v}^{5 } \nonumber \\ & & + \biggl\ { -{\frac { 82283}{1995 } } \,\gamma-{\frac { 11021}{285}}\,\ln \left ( 2 \right ) -{\frac { 234009 } { 5320}}\,\ln \left ( 3 \right ) + { \frac { 11224646611}{46569600}}+{\frac { 769}{57}}\,{\pi } ^{2 } \nonumber \\ & & \hspace{0.5 cm } + { \frac { 180255}{8512}}\,{q}^{2}-{\frac { 11809 } { 152}}\,\pi \,yq+{\frac { 598987}{8512}}\,{y}^{2}{q}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \biggl ( { \frac { 927800711807}{884822400}}-{\frac { 2982946}{1995}}\,\ln \left ( 2 \right ) + { \frac { 2782}{57}}\,{\pi } ^{2}+{\frac { 1638063}{3040}}\,\ln \left ( 3 \right ) \nonumber \\ & & \hspace{1.0 cm } -{\frac { 297674}{1995}}\,\gamma + { \frac { 536653}{8512 } } \,{q}^{2}-{\frac { 91375}{608}}\,\pi \,yq+{\frac { 356845}{8512}}\,{y } ^{2}{q}^{2 } \biggr ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \biggl ( { \frac { 190310746553}{262169600}}- { \frac { 1147147}{15960}}\,\gamma+{\frac { 10721}{456}}\,{\pi } ^{2}- { \frac { 1022385321}{340480}}\,\ln \left ( 3 \right ) \nonumber \\ & & \hspace{1.0 cm } + { \frac { 760314287}{47880}}\,\ln \left ( 2 \right ) -{\frac { 1044921875}{204288}}\,\ln \left ( 5 \right ) + { \frac { 56509}{9728}}\,{q}^{2 } \nonumber \\ & & \hspace{1.0 cm } -{\frac { 1739605 } { 29184}}\,\pi \,yq+{\frac { 3248951}{68096}}\,{y}^{2}{q}^{2 } \biggr ) { e}^{4 } \nonumber \\ & & \hspace{0.5 cm } - \left ( { \frac { 82283}{1995}}+{\frac { 297674}{1995}}\,{e}^{2 } + { \frac { 1147147}{15960}}\,{e}^{4 } \right ) \ln v \biggr\ } { v}^{6 } \biggr],\label{eq : dote8}\\ \left < \frac{dy}{dt}\right>^\infty_t & = & \left(\frac{dy}{dt}\right)_{\rm n } \biggl [ 1+{\frac { 189}{61}}\,{e}^{2}+{\frac { 285}{488}}\,{e}^{4 } \nonumber \\ & & + \biggl\ { -{\frac { 13}{244}}\,yq-{\frac { 277}{244}}\,yq { e}^{2}-{\frac { 1055}{1952}}\,yq{e}^{4 } \biggr\ } { v } \nonumber \\ & & + \left\ { - { \frac { 10461}{1708}}-{\frac { 83723}{3416}}\,{e}^{2}-{\frac { 21261 } { 13664}}\,{e}^{4}+{\frac { 49503}{27328}}\,{e}^{6 } \right\ } { v}^{2 } \nonumber \\ & & + \biggl\ { { \frac { 290}{61}}\,\pi -{\frac { 12755}{3416}}\,yq+ \left ( { \frac { 1990}{61}}\,\pi -{\frac { 27331}{1708}}\,yq \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( { \frac { 21947}{976}}\,\pi -{\frac { 540161}{27328}}\,yq \right ) { e}^{4}+ \left ( { \frac { 38747}{35136}}\,\pi -{\frac { 140001 } { 27328}}\,yq \right ) { e}^{6 } \biggr\ } { v}^{3 } \biggr],\label{eq : doty8}\end{aligned}\ ] ] where the leading contributions are given by @xmath243 in the same way , substituting the 3.5pn @xmath110 formulae of @xmath244 shown in sec . [ sec : result ] into eq . ( [ eq : trans_params ] ) , we obtain the secular changes of @xmath237 associated with the flux of gravitational waves to the horizon as @xmath245,\label{eq : dotvh}\\ \left < \frac{d{e}}{dt}\right>^{\rm h}_t & = & \left(\frac{de}{dt}\right)_{\rm n } \biggl [ -{\frac { 33}{4864 } } \left\ { 8 + 12\,{e}^{2}+{e}^{4 } \right\ } \left\ { 8 + 9\,{q}^{2}+15\,{y}^{2}{q}^{2 } \right\ } \,yq\,v^5 \nonumber \\ & & \biggl\ { -{\frac { 453}{152}}-{\frac { 8127}{1216}}\,{q}^{2}-{\frac { 45 } { 1216}}\,{y}^{2}{q}^{2}+ \left ( -{\frac { 2979}{304}}-{\frac { 28593 } { 1216}}\,{q}^{2}+{\frac { 1485}{608}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( -{\frac { 5649}{1216}}-{\frac { 111591}{9728}}\,{q}^{2}+ { \frac { 16515}{9728}}\,{y}^{2}{q}^{2 } \right ) { e}^{4 } \biggr\}\,yq\,v^7 \biggr],\label{eq : doteh}\\ \left < \frac{dy}{dt}\right>^{\rm h}_t & = & \left(\frac{dy}{dt}\right)_{\rm n } \biggl [ -{\frac { 3}{7808}}\ , \left\ { 8 + 24\,{e}^{2}+3\,{e}^{4 } \right\ } \left\ { 16 + 33\,{q}^{2}+15\,{y}^{2}{q}^{2 } \right\ } \,v^2 \nonumber \\ & & \biggl\ { -{\frac { 51}{122 } } + { \frac { 585}{1952}}\,{y}^{2}{q}^{2}-{\frac { 1953}{1952}}\,{q}^{2 } + \left ( -{\frac { 225}{61}}-{\frac { 16875}{1952}}\,{q } ^{2}+{\frac { 3375}{1952}}\,{y}^{2}{q}^{2 } \right ) { e}^{2 } \nonumber \\ & & \hspace{0.5 cm } + \left ( -{\frac { 2961}{976}}-{\frac { 109863}{15616}}\,{q}^{2}+{\frac { 16335 } { 15616}}\,{y}^{2}{q}^{2 } \right ) { e}^{4 } \nonumber \\ & & \hspace{0.5 cm } + \left ( -{\frac { 171}{976}}- { \frac { 3159}{7808}}\,{q}^{2}+{\frac { 405}{7808}}\,{y}^{2}{q}^{2 } \right ) { e}^{6 } \biggr\}\,v^4 \biggr].\label{eq : dotyh}\end{aligned}\ ] ] actually , we can obtain the higher pn results by using the 4pn @xmath110 formulae for the secular changes of @xmath78 , although we do not present them in the text . the full expressions of @xmath246 and @xmath247 for @xmath234 will be available online @xcite . here we make a comment on the reliable order of the expansion with respect to @xmath8 in @xmath248 . by using eq . ( [ eq : trans_params ] ) , @xmath248 can be calculated from the linear combination of the secular changes of @xmath78 . since the leading order of the @xmath249-component of the inverse jacobian matrix is @xmath250 , each term in the linear combination is apparently @xmath250 . however , the @xmath250 contribution turns out to vanish due to a cancellation in taking the combination , and hence @xmath248 is @xmath251 , which corresponds to the well - known fact that circular orbits remain circular @xcite , _ i.e. _ @xmath252 when @xmath253 . this cancellation reduces the reliable order in @xmath248 by @xmath129 , compared to the order of @xmath194 for @xmath192 . since we calculate @xmath194 up to @xmath110 in this paper , we can obtain @xmath248 correctly up to @xmath162 from the leading order . @xmath254 and @xmath255 in eqs . ( [ eq : dotv8 ] ) and ( [ eq : doty8 ] ) are consistent up to the 2.5pn @xmath129 order with the previous results in ref . @xcite , while we find inconsistency in the @xmath129 terms of the formula for @xmath256 in @xcite . this may be explained by the reduction in the reliable order mentioned above : the calculations of @xmath194 in @xcite are done up to @xmath129 , and therefore the resultant formula of @xmath248 is reliable only at the leading order . we can also confirm it numerically . in fig . [ fig : dedt8_q0.9_e0.1_0.7_inc50 ] we show the relative errors in the two analytic formulae by comparing to numerical results @xcite in a similar manner to eq . ( [ eq : relative_error ] ) . it can be found that the relative error in the previous 2.5pn @xmath129 formula strays out of the expected power law , @xmath166 , earlier than that in our 2.5pn @xmath129 formula . this trend is clearer for larger eccentricity . we can also confirm the validity of our formula by seeing that the relative errors in our 4pn @xmath162 formula falls off faster than @xmath163 . ( the leading pn order of the difference in the @xmath129 terms between the previous and our formulae is @xmath257 . if our formula contains any error in the @xmath258 term , the relative error will not fall off faster than @xmath163 for large @xmath7 . ) , defined in a similar manner to eq . ( [ eq : relative_error ] ) , as a function of the semi - latus rectum @xmath7 for @xmath161 , @xmath169 and @xmath170 ( from left to right ) and @xmath178 . we truncated the plots at @xmath173 because the relative errors get too large in @xmath174 to be meaningful . the relative error in the previous 2.5pn @xmath129 formula given in @xcite strays off the @xmath166 line earlier than the 2.5pn @xmath129 formula in this paper . this trend is clearer for larger @xmath8 . the relative errors in the 3pn @xmath162 and 4pn @xmath162 formulae fall off faster than @xmath166 and @xmath163 for small @xmath8 cases as expected , while this is not the case for @xmath6 because of the higher order correction of @xmath8 than @xmath162 . , title="fig:",width=192 ] , defined in a similar manner to eq . ( [ eq : relative_error ] ) , as a function of the semi - latus rectum @xmath7 for @xmath161 , @xmath169 and @xmath170 ( from left to right ) and @xmath178 . we truncated the plots at @xmath173 because the relative errors get too large in @xmath174 to be meaningful . the relative error in the previous 2.5pn @xmath129 formula given in @xcite strays off the @xmath166 line earlier than the 2.5pn @xmath129 formula in this paper . this trend is clearer for larger @xmath8 . the relative errors in the 3pn @xmath162 and 4pn @xmath162 formulae fall off faster than @xmath166 and @xmath163 for small @xmath8 cases as expected , while this is not the case for @xmath6 because of the higher order correction of @xmath8 than @xmath162 . , title="fig:",width=192 ] , defined in a similar manner to eq . ( [ eq : relative_error ] ) , as a function of the semi - latus rectum @xmath7 for @xmath161 , @xmath169 and @xmath170 ( from left to right ) and @xmath178 . we truncated the plots at @xmath173 because the relative errors get too large in @xmath174 to be meaningful . the relative error in the previous 2.5pn @xmath129 formula given in @xcite strays off the @xmath166 line earlier than the 2.5pn @xmath129 formula in this paper . this trend is clearer for larger @xmath8 . the relative errors in the 3pn @xmath162 and 4pn @xmath162 formulae fall off faster than @xmath166 and @xmath163 for small @xmath8 cases as expected , while this is not the case for @xmath6 because of the higher order correction of @xmath8 than @xmath162 . , title="fig:",width=192 ] a similar reduction in the pn order occurs in the calculation of @xmath259 : although each term in the linear combination of eq . ( [ eq : trans_params ] ) for @xmath260 is @xmath59 , the terms at the first two orders , @xmath59 and @xmath261 , vanish due to a cancellation in taking the combination . as a result , the leading order of @xmath262 is @xmath263 and hence the reliable order relative to the leading term is reduced to @xmath140 ( 2.5pn order ) when we have @xmath194 for @xmath192 up to @xmath59 ( 4pn order ) . in fig . [ fig : doti8h_q0.9_e0.1_0.4_0.7_inc50 ] , we show the relative errors in the analytic pn formulae for the secular changes of the orbital parameters , @xmath264 , derived from the 4pn @xmath110 formulae of @xmath194 for @xmath192 . similarly in fig . [ fig : dote8h_q0.9_e0.1_0.7_inc20_80 ] , the relative errors in the analytic formulae for @xmath265 and @xmath248 as functions of the semi - latus rectum @xmath7 fall off faster than @xmath163 when the eccentricity is small . observe , however , that the relative error in the analytic formula for @xmath262 falls off faster than @xmath165 , but slower than @xmath163 . . we plot the relative errors , @xmath266 defined in a similar manner to eq . ( [ eq : relative_error ] ) , as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath169 and @xmath170 ( from left to right ) and @xmath178 . we truncated the plots at @xmath173 because the relative errors get too large in @xmath174 to be meaningful . @xmath267 and @xmath268 fall off faster than @xmath163 , while @xmath269 approximately fall off as @xmath166 , slower than @xmath270 . this confirms that the relative order of the pn correction of the analytic formula for @xmath262 is reduced from 4pn to 2.5pn because of the cancellation of the low pn terms . , title="fig:",width=192 ] . we plot the relative errors , @xmath266 defined in a similar manner to eq . ( [ eq : relative_error ] ) , as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath169 and @xmath170 ( from left to right ) and @xmath178 . we truncated the plots at @xmath173 because the relative errors get too large in @xmath174 to be meaningful . @xmath267 and @xmath268 fall off faster than @xmath163 , while @xmath269 approximately fall off as @xmath166 , slower than @xmath270 . this confirms that the relative order of the pn correction of the analytic formula for @xmath262 is reduced from 4pn to 2.5pn because of the cancellation of the low pn terms . , title="fig:",width=192 ] . we plot the relative errors , @xmath266 defined in a similar manner to eq . ( [ eq : relative_error ] ) , as functions of the semi - latus rectum @xmath7 for @xmath161 , @xmath169 and @xmath170 ( from left to right ) and @xmath178 . we truncated the plots at @xmath173 because the relative errors get too large in @xmath174 to be meaningful . @xmath267 and @xmath268 fall off faster than @xmath163 , while @xmath269 approximately fall off as @xmath166 , slower than @xmath270 . this confirms that the relative order of the pn correction of the analytic formula for @xmath262 is reduced from 4pn to 2.5pn because of the cancellation of the low pn terms . , title="fig:",width=192 ] from the leading order expressions in eq . 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we calculate the secular changes of the orbital parameters of a point particle orbiting a kerr black hole , due to the gravitational radiation reaction . for this purpose , we use the post - newtonian ( pn ) approximation in the first order black hole perturbation theory , with the expansion with respect to the orbital eccentricity . in this work , the calculation is done up to the fourth post - newtonian ( 4pn ) order and to the sixth order of the eccentricity , including the effect of the absorption of gravitational waves by the black hole . we confirm that , in the kerr case , the effect of the absorption appears at the 2.5pn order beyond the leading order in the secular change of the particle s energy and may induce a superradiance , as known previously for circular orbits . in addition , we find that the superradiance may be suppressed when the orbital plane inclines with respect to the equatorial plane of the central black hole . we also investigate the accuracy of the 4pn formulae by comparing to numerical results . if we require that the relative errors in the 4pn formulae are less than @xmath0 , the parameter region to satisfy the condition will be @xmath1 for @xmath2 , @xmath3 for @xmath4 , and @xmath5 for @xmath6 almost irrespective of the inclination angle nor the spin of the black hole , where @xmath7 and @xmath8 are the semi - latus rectum and the eccentricity of the orbit . the region can further be extended using an exponential resummation method to @xmath9 for @xmath2 , @xmath10 for @xmath4 , and @xmath11 for @xmath6 . although we still need the higher order calculations of the pn approximation and the expansion with respect to the orbital eccentricity to apply for data analysis of gravitational waves , the results in this paper would be an important improvement from the previous work at the 2.5pn order , especially for large @xmath7 region .

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in the context of an extension of axelrod s model for social influence , we study the interplay and competition between the cultural drift , represented as random perturbations , and mass media , introduced by means of an external homogeneous field . unlike previous studies [ j. c. gonzlez - avella _ et al _ , phys . rev . e * 72 * , 065102(r ) ( 2005 ) ] , the mass media coupling proposed here is capable of affecting the cultural traits of any individual in the society , including those who do not share any features with the external message . a noise - driven transition is found : for large noise rates , both the ordered ( culturally polarized ) phase and the disordered ( culturally fragmented ) phase are observed , while , for lower noise rates , the ordered phase prevails . in the former case , the external field is found to induce cultural ordering , a behavior opposite to that reported in previous studies using a different prescription for the mass media interaction . we compare the predictions of this model to statistical data measuring the impact of a mass media vasectomy promotion campaign in brazil . .5 cm .5 cm 21truecm 15truecm _ keywords : _ sociophysics ; econophysics ; marketing ; advertising . the non - traditional application of statistical physics to many problems of interdisciplinary nature has been growing steadily in recent years . indeed , it has been recognized that the study of statistical and complex systems can provide valuable tools and insight into many emerging interdisciplinary fields of science @xcite . in this context , the mathematical modeling of social phenomena allowed to perform quantitative investigations on processes such as self - organization , opinion formation and spreading , cooperation , formation and evolution of social structures , etc ( see e.g. @xcite ) . in particular , a model for social influence proposed by axelrod @xcite , which aims at understanding the formation of cultural domains , has recently received much attention @xcite due to its remarkably rich dynamical behavior . in axelrod s model , culture is defined by the set of cultural attributes ( such as language , art , technical standards , and social norms @xcite ) subject to social influence . the cultural state of an individual is given by their set of specific traits , which are capable of changing due to interactions with their acquaintances . in the original proposal , the individuals are located at the nodes of a regular lattice , and the interactions are assumed to take place between lattice neighbors . social influence is defined by a simple local dynamics , which is assumed to satisfy the following two properties : ( a ) social interaction is more likely taking place between individuals that share some or many of their cultural attributes ; ( b ) the result of the interaction is that of increasing the cultural similarity between the individuals involved . by means of extensive numerical simulations , it was shown that the system undergoes a phase transition separating an ordered ( culturally polarized ) phase from a disordered ( culturally fragmented ) one , which was found to depend on the number of different cultural traits available @xcite . the critical behavior of the model was also studied in different complex network topologies , such as small - world and scale - free networks @xcite . these investigations considered , however , zero - temperature dynamics that neglected the effect of fluctuations . following axelrod s original idea of incorporating random perturbations to describe the effect of _ cultural drift _ @xcite , noise was later added to the dynamics of the system @xcite . with the inclusion of this new ingredient , the disordered multicultural configurations were found to be metastable states that could be driven to ordered stable configurations . the decay of disordered metastable states depends on the competition between the noise rate , @xmath0 , and the characteristic time for the relaxation of perturbations , @xmath1 . indeed , for @xmath2 , the perturbations drive the disordered system towards monocultural states , while , for @xmath3 , the noise rates are large enough to hinder the relaxation processes , thus keeping the disorder . since @xmath1 scales with the system size , @xmath4 , as @xmath5 , the culturally fragmented states persist in the thermodynamic limit , irrespective of the noise rate @xcite . more recently , an extension of the model was proposed , in which the role of _ mass media _ and other mass external agents was introduced by considering external @xcite and autonomous local or global fields @xcite , but neglecting random fluctuations . the interaction between the fields and the individuals was chosen to resemble the coupling between an individual and their neighbors in the original axelrod s model . according to the adopted prescription , the interaction probability was assumed to be null for individuals that do not share any cultural feature with the external message . in this way , intriguing , counterintuitive results were obtained : the influence of mass media was found to disorder the system , thus driving ordered , culturally polarized states towards disordered , culturally fragmented configurations @xcite . the aim of this work is to include the effect of cultural drift in an alternative mass media scenario . although still inspired in the original axelrod s interaction , the mass media coupling proposed here is capable of affecting the cultural traits of any individual in the society , including those who do not share any features with the external message . for noise rates below a given transition value , which depends on the intensity of the mass media interactions , only the ordered phase is observed . however , for higher levels of noise above the transition perturbation rate , both the ordered ( culturally polarized ) phase and the disordered ( culturally fragmented ) phase are found . in the latter case , we obtain an order - disorder phase diagram as a function of the field intensity and the number of traits per cultural attribute . according to this phase diagram , the role of the external field is that of inducing cultural ordering , a behavior opposite to that reported in ref . @xcite using a different prescription for the mass media interaction . in order to show the plausibility of the scenario considered here , we also compare the predictions of this model to statistical data measuring the impact of a mass media vasectomy promotion campaign in brazil @xcite . the model is defined by considering individuals located at the sites of an @xmath6 square lattice . the cultural state of the @xmath7th individual is described by the integer vector @xmath8 , where @xmath9 . the dimension of the vector , @xmath10 , defines the number of cultural attributes , while @xmath11 corresponds to the number of different cultural traits per attribute . initially , the specific traits for each individual are assigned randomly with a uniform distribution . similarly , the mass media cultural message is modeled by a constant integer vector @xmath12 , which can be chosen as @xmath13 without loss of generality . the intensity of the mass media message relative to the local interactions between neighboring individuals is controlled by the parameter @xmath14 ( @xmath15 ) . moreover , the parameter @xmath0 ( @xmath16 ) is introduced to represent the noise rate @xcite . the model dynamics is defined by iterating a sequence of rules , as follows : ( 1 ) an individual is selected at random ; ( 2 ) with probability @xmath14 , he / she interacts with the mass media field ; otherwise , he / she interacts with a randomly chosen nearest neighbor ; ( 3 ) with probability @xmath0 , a random single - feature perturbation is performed . the interaction between the @xmath7th and @xmath17th individuals is governed by their cultural overlap , @xmath18 , where @xmath19 is the kronecker delta . with probability @xmath20 , the result of the interaction is that of increasing their similarity : one chooses at random one of the attributes on which they differ ( i.e. , such that @xmath21 ) and sets them equal by changing the trait of the individual selected in first place . naturally , if @xmath22 , the cultural states of both individuals are already identical , and the interaction leaves them unchanged . the interaction between the @xmath7th individual and the mass media field is governed by the overlap term @xmath23 . analogously to the precedent case , @xmath24 is the probability that , as a result of the interaction , the individual changes one of the traits that differ from the message by setting it equal to the message s trait . again , if @xmath25 , the cultural state of the individual is already identical to the mass media message , and the interaction leaves it unchanged . notice that @xmath26 ; thus , the mass media coupling used here is capable of affecting the cultural traits of any individual in the society , including those who do not share any features with the external message . as commented above , this differs from the mass media interaction proposed in ref . @xcite , which was given by @xmath27 . = 4.2truein=3.1truein as regards the perturbations introduced in step ( 3 ) , a single feature of a single individual is randomly chosen , and , with probability @xmath0 , their corresponding trait is changed to a randomly selected value between 1 and @xmath11 . in the absence of fluctuations , the system evolves towards absorbing states , i.e. , frozen configurations that are not capable of further changes . for @xmath28 , instead , the system evolves continuously , and , after a transient period , it attains a stationary state . in order to characterize the degree of order of these stationary states , we measure the ( statistically - averaged ) size of the largest homogeneous domain , @xmath29 @xcite . the results obtained here correspond to systems of linear size @xmath30 and a fixed number of cultural attributes , @xmath31 , typically averaged over 500 different ( randomly generated ) initial configurations . = 4.2truein=3.1truein figure 1 shows the order parameter , @xmath32 , as a function of the noise rate , @xmath0 , for different values of the mass media intensity . the number of different cultural traits per attribute is @xmath33 . as anticipated , for small noise rates , the perturbations drive the decay of disordered metastable states , and thus the system presents only ordered states with @xmath34 . as the noise rate is gradually increased , the competition between characteristic times for perturbation and relaxation processes sets on , and , for large enough noise rates , the system becomes completely disordered . this behavior , which was already reported in the absence of mass media interactions @xcite , is here also observed for @xmath35 . as we consider plots for increasing values of @xmath14 , the transition between ordered and disordered states takes place for increasingly higher levels of noise . indeed , this is an indication of the competition between noise rate and external field effects , thus showing that the external field induces order in the system . figure 2 shows the order - disorder phase diagram as a function of the field intensity and the number of traits per cultural attribute , for the noise rate @xmath36 . the transition points correspond to @xmath37 . for the @xmath38 case , noise - driven order - disorder transitions were found to be roughly independent of the number of traits per cultural attribute , as long as @xmath39 @xcite . here , we observe a similar , essentially @xmath40independent behavior for @xmath35 as well . typical snapshot configurations of both regions are also shown in figure 2 , where the transition from the ( small-@xmath14 ) multicultural regime to the ( large-@xmath14 ) monocultural state is clearly observed . a majority of individuals sharing the same cultural state , identical to the external message , is found within the ordered phase . for smaller noise rates , @xmath41 , the system is ordered even for @xmath38 , and hence only the monocultural phase is observed . = 4.2truein=3.1truein in order to gain further insight into the interplay and competition between cultural drift and mass media effects , let us now consider the external message being periodically switched on and off . starting with a random disordered configuration and assuming a noise level above the transition value for the @xmath38 case , we observe a periodical behavior : the system becomes ordered within the time window in which the field is applied , while it becomes increasingly disordered when the message is switched off . a cycle representing this behavior is shown by the solid line in figure 3 , which corresponds to @xmath42 , @xmath43 , and @xmath33 . moreover , we can compare this behavior to statistical data measuring the impact of a mass media vasectomy promotion campaign in brazil @xcite . symbols in figure 3 correspond to the number of vasectomies performed monthly in a major clinic in so paulo , spanning a time interval of 2 years . the shaded region indicates the time window in which the mass media campaign was performed . the promotion campaign consisted of prime - time television and radio spots , the distribution of flyers , an electronic billboard , and public relations activities . in order to allow a comparison to model results , vasectomy data have been normalized by setting the maximal number of vasectomies measured equal to unity , while the relation between time scales has been chosen conveniently . in the model results , time is measured in monte carlo steps ( mcs ) , where 1 mcs corresponds to @xmath44 iterations of the set of rules ( 1)-(3 ) . for the comparison performed in figure 3 , we assumed that 1 month corresponds to 500 mcs . although the model parameters and scale units were arbitrarily assigned , it is reassuring to observe that a good agreement between observations and model results can be achieved . indeed , the steep growth in the number of vasectomies practiced during the promotion campaign , as well as the monotonic decrease afterwards , can be well accounted for by this model . in order to carry out a straightforward comparison between mass media effects in this model and the measured response within a social group , several simplifying assumptions were adopted . as commented above , the model parameters and scale units were conveniently assigned . moreover , no distinction was attempted between opinions ( as modeled , within axelrod s representation , by sets of cultural attributes in the mathematical form of vectors ) and actual choices ( as measured e.g. by the vasectomy data shown in figure 3 ) . in related contexts , analogous simplifying assumptions were adopted in the statistical physics modeling of political phenomena ( e.g. the distribution of votes in elections in brazil and india @xcite , italy and germany @xcite , four - party political scenarios @xcite , etc ) , marketing competition between two advertised products @xcite , and applications to finance @xcite . in summary , we have studied , in the context of an extension of axelrod s model for social influence , the interplay and competition between cultural drift and mass media effects . the cultural drift is modeled by random perturbations , while mass media effects are introduced by means of an external field . a noise - driven order - disorder transition is found . in the large noise rate regime , both the ordered ( culturally polarized ) phase and the disordered ( culturally fragmented ) phase can be observed , whereas in the small noise rate regime , only the ordered phase is present . in the former case , we have obtained the corresponding order - disorder phase diagram , showing that the external field induces cultural ordering . this behavior is opposite to that reported in ref . @xcite using a different prescription for the mass media field , which neglected the interaction between the field and individuals that do not share any features with the external message . the mass media coupling proposed in this work , instead , is capable of affecting the cultural traits of any individual in the society . in order to show the plausibility of the scenario considered here , we have compared the predictions of this model to statistical data measuring the impact of a mass media vasectomy promotion campaign in brazil . a good agreement between model results and measured data can be achieved . the observed behavior is characterized by a steep growth during the promotion campaign , and a monotonic decrease afterwards . we can thus conclude that the extension of axelrod s model proposed here contains the basic ingredients needed to explain the trend of actual observations . we hope that the present findings will contribute to the growing interdisciplinary efforts in the mathematical modeling of social dynamics phenomena , and stimulate further work .

to obtain a quasi - lagrangian field from its eulerian counterpart , we track a single lagrangian particle by using a bilinear - interpolation method [ 17 ] . if we replace the eulerian velocity in eq . ( 2 ) by its fourier - integral representation , we obtain @xmath104 d{\bf q } , \nonumber\ ] ] where @xmath105 is the wave vector . in the pseudospectral algorithm we use to solve eq . ( 1 ) , the quasi - lagrangian velocity is defined with respect to a lagrangian particle , which was at the point @xmath32 at time @xmath33 , and is at the position @xmath34 at time @xmath9 , such that @xmath35 $ ] . we calculate @xmath106 ; thus , the fourier integral above can be evaluated at each time step by an additional call to a fast - fourier - transform ( fft ) subroutine . the additional computational cost of obtaining @xmath37 at all collocation points is that of following a single lagrangian particle and an additional fft at each time step . to extract the integral time scale , of degree @xmath68 , from a time - dependent structure function , we have to evaluate the integral in eq . ( 4 ) numerically . in practice , because of poor statistics at long times , we integrate from @xmath107 to @xmath108 , where @xmath109 is the time at which @xmath110 ; we choose @xmath111 , but we have checked that our results do not change , within our error bars , for @xmath112 . this numerical integration is done by using the trapezoidal rule . equal - time eulerian structure functions have been discussed in our paper above . to obtain time - dependent , eulerian , vorticity structure functions we proceed as we did in the quasi - lagrangian case . we obtain the required vorticity increments and from these the purely isotropic part of the time - dependent , order-@xmath4 structure function @xmath113 . equations ( 4 ) and ( 5 ) now yield the order-@xmath4 , degree-@xmath68 integral and derivative eulerian time scales . for the former we should integrate @xmath114 from @xmath115 to @xmath116 ; in practice , because of poor statistics at long times , we integrate from @xmath107 to @xmath108 , where @xmath109 is the time at which @xmath117 ; we choose @xmath118 , but we have checked that our results do not change , within our error bars , for @xmath112 . slopes of linear scaling ranges of log - log plots of @xmath119 versus @xmath120 yield the dynamic multiscaling exponent @xmath121 . a representative plot for the eulerian case , @xmath96 , and @xmath97 is given in fig . ( [ figsupp1 ] a ) ; we fit over the range @xmath122 and obtain the local slopes @xmath99 with successive , non - overlapping sets of 3 points each . the mean values of these slopes yield our dynamic - multiscaling exponents ( column 4 in table [ table_eu ] ) and their standard deviations yield the error bars . we calculate the degree-@xmath68 , order-@xmath4 derivative time exponents by using a sixth - order finite difference scheme to obtain @xmath123 and thence the dynamic - multiscaling exponents @xmath124 ; data for the eulerian case and the representative value @xmath102 are given in column 6 of table [ table_eu ] . we find , furthermore , that both the integral and derivative bridge relations , eq . ( 6 ) , and eq . ( 7 ) . hold within our error bars , as shown for the representative values of @xmath4 and @xmath68 considered in table [ table_eu ] ( compare columns 3 and 4 for the integral relation and columns 5 and 6 for the derivative relation ) . the values of the integral and the derivative dynamic - multiscaling exponents are markedly different from each other ( compare columns 4 and 6 of table [ table_eu ] ) and the plots of these exponents versus @xmath4 in fig . ( [ figsupp1 ] b ) . in fig . ( [ figsupp1 ] c ) , we make the same comparison for the quasi - lagrangian case . furthermore , a comparison of the quasi - lagrangian and eulerian dynamic - multiscaling exponents given in tables i in the original paper and table [ table_eu ] , respectively , show that these are the same ( within our error bars ) . we have shown that in two dimensional turbulence with friction , the eulerian and the quasi lagrangian velocities have the same dynamical exponents . this is because the inverse cascade has a friction dependent infra - red cutoff . to illustrate the development of this cutoff scale , we have carried out dns studies of 2d fluid turbulence with @xmath125 , and @xmath126 , @xmath127 collocation points , and forcing at a wave - vector magnitude @xmath128 ; our dns studies resolve the inverse - cascade regime in the statistically steady state . the energy spectra from these dns studies , plotted in fig . ( [ figsupp1]d ) , show clearly that , as @xmath30 increases , the inverse cascade is cut off at ever larger values of @xmath129 . thus , the friction produces a regularization of the flow and suppresses infrared ( sweeping ) divergences .

we obtain , by extensive direct numerical simulations , time - dependent and equal - time structure functions for the vorticity , in both quasi - lagrangian and eulerian frames , for the direct - cascade regime in two - dimensional fluid turbulence with air - drag - induced friction . we show that different ways of extracting time scales from these time - dependent structure functions lead to different dynamic - multiscaling exponents , which are related to equal - time multiscaling exponents by different classes of bridge relations ; for a representative value of the friction we verify that , given our error bars , these bridge relations hold . the scaling properties of both equal - time and time - dependent correlation functions close to a critical point , say in a spin system , have been understood well for nearly four decades . by contrast , the development of a similar understanding of the multiscaling properties of equal - time and time - dependent structure functions in the inertial range in fluid turbulence still remains a major challenge for it requires interdisciplinary studies that must use ideas both from nonequilibrium statistical mechanics and turbulence @xcite . we develop here a complete characterization of the rich multiscaling properties of time - dependent vorticity structure functions for the direct - cascade regime of two - dimensional ( 2d ) turbulence in fluid films with friction , which we study via a direct numerical simulation ( dns ) . such a characterization has not been possible hitherto because it requires very long temporal averaging to obtain good statistics for _ quasi - lagrangian _ structure functions @xcite , which are considerably more complicated than their conventional , eulerian counterparts as we show below . our dns study yields a variety of interesting results that we summarize informally before providing technical details and precise definitions : ( a ) we calculate equal - time and time - dependent vorticity structure functions in eulerian and quasi - lagrangian frames @xcite . ( b ) we then show how to extract an infinite number of different time scales from such time - dependent structure functions . ( c ) next we present generalizations of the dynamic - scaling ansatz , first used in the context of critical phenomena @xcite to relate a diverging relaxation time @xmath0 to a diverging correlation length @xmath1 via @xmath2 , where @xmath3 is the dynamic - scaling exponent . these generalizations yield , in turn , an infinity of dynamic - multiscaling exponents @xcite . ( d ) a suitable extension of the multifractal formalism @xcite , which provides a rationalization of the multiscaling of equal - time structure functions in turbulence , yields linear bridge relations between dynamic - multiscaling exponents and their equal - time counterparts @xcite ; our study provides numerical evidence in support of such bridge relations . the statistical properties of fully developed , homogeneous , isotropic turbulence are characterized , _ inter alia _ , by the equal - time , order-@xmath4 , longitudinal - velocity structure function @xmath5^p \rangle$ ] , where @xmath6 $ ] , @xmath7 is the eulerian velocity at point @xmath8 and time @xmath9 , and @xmath10 . in the inertial range @xmath11 , @xmath12 , where @xmath13 , @xmath14 , and @xmath15 , are , respectively , the equal - time exponent , the dissipation scale , and the forcing scale . the pioneering work @xcite of kolmogorov ( k41 ) predicts simple scaling with @xmath16 for three - dimensional ( 3d ) homogeneous , isotropic fluid turbulence . however , experiments and numerical simulations show marked deviations from k41 scaling , especially for @xmath17 , with @xmath13 a nonlinear , convex function of @xmath4 ; thus , we have multiscaling of equal - time velocity structure functions . to examine dynamic multiscaling , we must obtain the order-@xmath4 , time - dependent structure functions @xmath18 , which we define precisely below , extract from these the time scales @xmath19 , and thence the dynamic - multiscaling exponents @xmath20 via dynamic - multiscaling anstze like @xmath21 . this task is considerably more complicated than its analog for the determination of the equal - time multiscaling exponents @xmath13 @xcite for the following two reasons : ( i ) in the conventional eulerian description , the sweeping effect , whereby large eddies drive all smaller ones directly , relates spatial separations @xmath22 and temporal separations @xmath9 linearly via the mean - flow velocity , whence we get trivial dynamic scaling with @xmath23 , for all @xmath4 . a quasi - lagrangian description @xcite eliminates sweeping effects so we calculate time - dependent , quasi - lagrangian vorticity structure functions from our dns . ( ii ) such time - dependent structure functions , even for a fixed order @xmath4 , do not collapse onto a scaling function , with a unique , order-@xmath4 , dynamic exponent . hence , even for a fixed order @xmath4 , there is an infinity of dynamic - multiscaling exponents @xcite ; roughly speaking , to specify the dynamics of an eddy of a given length scale , we require this infinity of exponents . statistically steady fluid turbulence is very different in 3d and 2d ; the former exhibits a direct cascade of energy whereas the latter shows an inverse cascade of kinetic energy from the energy - injection scale to larger length scales and a direct cascade in which the enstrophy goes towards small length scales @xcite ; in many physical realizations of 2d turbulence , there is an air - drag - induced friction . in this direct - cascade regime , velocity structure functions show simple scaling but their vorticity counterparts exhibit multiscaling @xcite , with exponents that depend on the friction . time - dependent structure functions have not been studied in 2d fluid turbulence ; the elucidation of the dynamic multiscaling of these structure functions , which we present here , is an important step in the systematization of such multiscaling in turbulence . we numerically solve the forced , incompressible , 2d navier - stokes ( 2dns ) equation with air - drag - induced friction , in the vorticity(@xmath24stream - function(@xmath25 representation with periodic boundary conditions : @xmath26 where @xmath27 , @xmath28 , and the velocity @xmath29 . the coefficient of friction is @xmath30 and @xmath31 is the external force . we work with both eulerian and quasi - lagrangian fields . the latter are defined with respect to a lagrangian particle , which was at the point @xmath32 at time @xmath33 , and is at the position @xmath34 at time @xmath9 , such that @xmath35 $ ] , where @xmath36 is the eulerian velocity . the quasi - lagrangian velocity field @xmath37 is defined @xcite as follows : @xmath38 ; \label{eq : qldef}\ ] ] likewise , we can define the quasi - lagrangian vorticity field @xmath39 in terms of the eulerian @xmath40 . to obtain this quasi - lagrangian field we use an algorithm developed in ref . @xcite , described briefly in the supplementary material . to integrate the navier - stokes equations we use a pseudo - spectral method with the @xmath41 rule for the removal of aliasing errors @xcite and a second - order runge - kutta scheme for time marching with a time step @xmath42 . we force the fluid deterministically on the second shell in fourier space . and we use @xmath43 , @xmath44 , and @xmath45 collocation points collocation points yield exponents that are consistent with those presented here . see s. s. ray , phd thesis , indian institute of science , bangalore ( 2010 ) , unpublished . ] we obtain a turbulent but statistically steady state with a taylor microscale @xmath46 , taylor - microscale reynolds number @xmath47 , and a box - size eddy - turn - over time @xmath48 . we remove the effects of transients by discarding data upto time @xmath49 . we then obtain data for averages of time - dependent structure functions for a duration of time @xmath50 . the energy spectrum averaged over the same time interval is shown in fig . ( [ fig1]a ) . the equal - time , order-@xmath4 , vorticity structure functions we consider are @xmath51^p \rangle \sim r^{\zeta^\phi_p } $ ] , for @xmath11 , where @xmath52 $ ] , the angular brackets denote an average over the nonequilibrium statistically steady state of the turbulent fluid , and the superscript @xmath53 is either @xmath54 , in the eulerian case , or @xmath55 , in the quasi - lagrangian case ; for notational convenience we do not include a subscript @xmath40 on @xmath56 and the multiscaling exponent @xmath57 . we assume isotropy here , but show below how to extract the isotropic parts of @xmath56 in a dns . we also use the time - dependent , order-@xmath4 vorticity structure functions @xmath58 \rangle ; \label{dynsp}\end{aligned}\ ] ] here @xmath59 are @xmath4 different times ; clearly , @xmath60 . we concentrate on the case @xmath61 and @xmath62 , with @xmath63 , and , for simplicity , denote the resulting time - dependent structure function as @xmath64 ; shell - model studies @xcite have shown that the index @xmath65 does not affect dynamic - multiscaling exponents , so we suppress it henceforth . given @xmath64 , it is possible to extract a characteristic time scale @xmath66 in many different ways . these time scales can , in turn , be used to extract the order-@xmath4 dynamic - multiscaling exponents @xmath20 via the dynamic - multiscaling ansatz @xmath67 . if we obtain the order-@xmath4 , degree-@xmath68 , _ integral _ time scale @xmath69^{(1/m ) } , \label{timp } \end{aligned}\ ] ] we can use it to extract the _ integral _ dynamic - multiscaling exponent @xmath70 from the relation @xmath71 . similarly , from the order-@xmath4 , degree-@xmath68 , _ derivative _ time scale @xmath72^{(-1/m ) } , \label{tdpm}\end{aligned}\ ] ] we obtain the _ derivative _ dynamic - multiscaling exponent @xmath73 via the relation @xmath74 . equal - time vorticity structure functions in 2d fluid turbulence with friction exhibit multiscaling in the direct cascade range @xcite . for the case of 3d homogeneous , isotropic fluid turbulence , a generalization of the multifractal model @xcite , which includes time - dependent velocity structure functions @xcite , yields linear bridge relations between the dynamic - multiscaling exponents and their equal - time counterparts . for the direct - cascade regime in our study , we replace velocity structure functions by vorticity structure functions and thus obtain the following bridge relations for time - dependent vorticity structure functions in 2d fluid turbulence with friction : @xmath75/m ; \label{zipm}\ ] ] @xmath76/m . \label{zdpm}\ ] ] the vorticity field @xmath77 can be decomposed into the time - averaged mean flow @xmath78 and the fluctuations @xmath79 about it . to obtain good statistics for vorticity structure functions it is important to eliminate any anisotropy in the flow by subtracting out the mean flow from the field . therefore , we redefine the order-@xmath4 , equal - time structure function to be @xmath80 , where @xmath81 has magnitude @xmath82 and @xmath83 is an origin . we next use @xmath84 , where the subscript @xmath83 denotes an average over the origin ( we use @xmath85 ) . these averaged structure functions are isotropic , to a good approximation for small @xmath82 , as can be seen from the illustrative pseudocolor plot of @xmath86 in fig . ( [ fig1]a ) . the purely isotropic parts of such structure functions can be obtained @xcite via an integration over the angle @xmath87 that @xmath81 makes with the @xmath88 axis , i.e. , we calculate @xmath89 and thence the equal - time multiscaling exponent @xmath57 , the slopes of the scaling ranges of log - log plots of @xmath90 versus @xmath82 . the mean of the local slopes @xmath91 in the scaling range yields the equal - time exponents ; and their standard deviations give the error bars . the equal - time vorticity multiscaling exponents , with @xmath92 , are given for eulerian and quasi - lagrangian cases in columns 2 and 3 , respectively , of table 1 ; they are equal , within error bars , as can be seen most easily from their plots versus @xmath4 in fig.([fig1]c ) . we obtain the isotropic part of @xmath93 in a similar manner . equations ( [ timp ] ) and ( [ tdpm ] ) now yield the order-@xmath4 , degree-@xmath68 integral and derivative time scales ( see the supplementary material ) . slopes of linear scaling ranges of log - log plots of @xmath94 versus @xmath82 yield the dynamic multiscaling exponent @xmath95 . a representative plot for the quasi - lagrangian case , @xmath96 and @xmath97 , is given in fig . ( [ fig1 ] d ) ; we fit over the range @xmath98 and obtain the local slopes @xmath99 with successive , nonoverlapping sets of 3 points each . the mean values of these slopes yield our dynamic - multiscaling exponents ( column 5 in table 1 ) and their standard deviations yield the error bars . we calculate the degree-@xmath68 , order-@xmath4 derivative time exponents by using a sixth - order , finite - difference scheme to obtain @xmath100 and thence the dynamic - multiscaling exponents @xmath101 . our results for the quasi - lagrangian case with @xmath102 are given in column 7 of table 1 . we find , furthermore , that both the integral and derivative bridge relations ( [ zipm ] ) and ( [ zdpm ] ) hold within our error bars , as shown for the representative values of @xmath4 and @xmath68 considered in table 1 ( compare columns 4 and 5 for the integral relation and columns 6 and 7 for the derivative relation ) . note also that the values of the integral and the derivative dynamic - multiscaling exponents are markedly different from each other ( compare columns 5 and 7 of table 1 ) . the eulerian structure functions @xmath103 also lead to nontrivial dynamic - multiscaling exponents , which are equal to their quasi - lagrangian counterparts ( see supplementary material ) . the reason for this initially surprising result is that , in 2d turbulence , the friction controls the size of the largest vortices , provides an infra - red cut - off at large length scales , and thus suppresses the sweeping effect . we have demonstrated this in the supplementary material . had the sweeping effect not been suppressed , we would have obtained trivial dynamic scaling for the eulerian case . the calculation of dynamic - multiscaling exponents has been limited so far to shell models for 3d , homogeneous , isotropic fluid @xcite and passive - scalar turbulence @xcite . we have presented the first study of such dynamic multiscaling in the direct - cascade regime of 2d fluid turbulence with friction by calculating both quasi - lagrangian and eulerian structure functions . our work brings out clearly the need for an infinity of time scales and associated exponents to characterize such multiscaling ; and it verifies , within the accuracy of our numerical calculations , the linear bridge relations ( [ zipm ] ) and ( [ zdpm ] ) for a representative value of @xmath30 . we find that friction also suppresses sweeping effects so , with such friction , even eulerian vorticity structure functions exhibit dynamic multiscaling with exponents that are consistent with their quasi - lagrangian counterparts . experimental studies of lagrangian quantities in turbulence have been increasing steadily over the past decade @xcite . we hope , therefore , that our work will encourage studies of dynamic multiscaling in turbulence . furthermore , it will be interesting to check whether the time scales considered here can be related to the persistence time scales for 2d turbulence @xcite . + we thank j. k. bhattacharjee for discussions , the european research council under the astro - dyn research project no . 227952 , national science foundation under grant no . phy05 - 51164 , csir , ugc , and dst ( india ) for support , and serc ( iisc ) for computational resources . pp and rp are members of the international collaboration for turbulence research ; rp , pp , and ssr acknowledge support from the cost action mp0806 . just as we were preparing this study for publication we became aware of a recent preprint @xcite on a related study for 3d fluid turbulence . we thank l. biferale for sharing the preprint of this paper with us .

the determination of the classical ground states of interacting many - particle systems ( minimum energy configurations ) is a subject of ongoing investigation in condensed - matter physics and materials science @xcite . while such results are readily produced by slowly freezing liquids in experiments and computer simulations , our theoretical understanding of classical ground states is far from complete . much of the progress to rigorously identify ground states for given interactions has been for lattice models , primarily in one dimension @xcite . the solutions in @xmath1-dimensional euclidean space @xmath2 for @xmath3 are considerably more challenging . for example , the ground state(s ) for the well - known lennard - jones potential in @xmath4 or @xmath5 are not known rigorously @xcite . recently , a collective - coordinate " approach has been employed to study and ascertain ground states in two and three dimensions for a certain class of interactions @xcite . a surprising conclusion of ref . @xcite is that there exist nontrivial _ disordered _ classical ground states without any long - range order @xcite , in addition to the expected periodic ones . despite these advances , new theoretical tools are required to make further progress in our understanding of classical ground states . in a recent letter , we derived duality relations for a certain class of soft pair potentials that can be applied to classical ground states whether they are disordered or not @xcite . soft interactions are considered because , as we will see , they are easier to treat theoretically and possess great importance in soft - matter systems , such as colloids , microemulsions , and polymers @xcite . these duality relations link the energy of configurations associated with a real - space pair potential @xmath6 to the energy associated with the dual ( fourier - transformed ) potential . duality relations are useful because they enable one to use information about the ground states of certain soft short - ranged potentials to draw conclusions about the nature of the ground states of long - ranged potentials and vice versa . the duality relations also lead to bounds on the zero - temperature energies in density intervals of phase coexistence . in the present paper , we amplify and extend the results of ref . @xcite . we also study in detail a one - dimensional system that torquato and stillinger claimed to possess an infinite number of structural phase transitions from bravais to non - bravais lattices at @xmath0 as the density is changed @xcite . a general set of potential functions that are self - similar under fourier transform are described and studied . we also derive the generalizations of the duality relations for three - body as well as higher - order interactions . a point process in @xmath2 is a distribution of an infinite number of points at number density @xmath7 ( number of points per unit volume ) with configuration @xmath8 ; see ref . @xcite for a precise mathematical definition . it is characterized by a countably infinite set of @xmath9-particle generic probability density functions @xmath10 , which are proportional to the probability densities of finding collections of @xmath9 particles in volume elements near the positions @xmath11 . for a general point process , it is convenient to introduce the _ @xmath9-particle correlation functions _ @xmath12 , which are defined by @xmath13 since @xmath14 for a completely uncorrelated point process , it follows that deviations of @xmath12 from unity provide a measure of the correlations between points in a point process . of particular interest is the pair correlation function , which for a translationally invariant point process of density @xmath7 can be written as @xmath15 closely related to the pair correlation function is the _ total correlation function _ , denoted by @xmath16 ; it is derived from @xmath17 via the equation @xmath18 since @xmath19 as @xmath20 ( @xmath21 ) for translationally invariant systems without long - range order , it follows that @xmath22 in this limit , meaning that @xmath16 is generally an @xmath23 function , and its fourier transform is well - defined . it is common in statistical mechanics when passing to reciprocal space to consider the associated _ structure factor _ @xmath24 , which for a translationally invariant system is defined by @xmath25 where @xmath26 is the fourier transform of the total correlation function , @xmath7 is the number density , and @xmath27 is the magnitude of the reciprocal variable to @xmath28 . the @xmath1-dimensional fourier transform of any integrable radial function @xmath29 is @xmath30 and the inverse transform of @xmath31 is given by @xmath32 here @xmath33 is the wavenumber ( reciprocal variable ) and @xmath34 is the bessel function of order @xmath35 . a special point process of central interest in this paper is a lattice . a _ lattice _ @xmath36 in @xmath2 is a subgroup consisting of the integer linear combinations of vectors that constitute a basis for @xmath2 , i.e. , the lattice vectors @xmath37 ; see ref . @xcite for details . in a lattice @xmath36 , the space @xmath2 can be geometrically divided into identical regions @xmath38 called _ fundamental cells _ , each of which corresponds to just one point as in figure [ bravaisfig ] . in the physical sciences , a lattice is equivalent to a bravais lattice . unless otherwise stated , for this situation we will use the term lattice . every lattice has a dual ( or reciprocal ) lattice @xmath39 in which the sites of that lattice are specified by the dual ( reciprocal ) lattice vectors @xmath40 , where @xmath41 . the dual fundamental cell @xmath42 has volume @xmath43 , where @xmath44 is the volume of the fundamental cell of the original lattice @xmath36 , implying that the respective densities @xmath7 and @xmath45 of the real and dual lattices are related by @xmath46 . a _ periodic _ point process , or non - bravais lattice , is a more general notion than a lattice because it is is obtained by placing a fixed configuration of @xmath47 points ( where @xmath48 ) within one fundamental cell of a lattice @xmath36 , which is then periodically replicated ( see figure [ bravaisfig ] ) . thus , the point process is still periodic under translations by @xmath36 , but the @xmath47 points can occur anywhere in the chosen fundamental cell . although generally a non - bravais lattice does not have a dual , certain periodic point patterns are known to possess _ formally dual _ non - bravais lattices . roughly speaking , two non - bravais lattices are formal duals of each other if their average pair sums ( total energies per particle ) obey the same relationship as poisson summation for bravais lattices for all admissible pair interactions ; for further details , the reader is referred to @xcite . for a configuration @xmath49 of @xmath50 particles in a bounded volume @xmath51 with stable pairwise interactions , the many - body function @xmath52 is twice the total potential energy per particle [ plus the self - energy " @xmath53 , where @xmath54 is a radial pair potential function and @xmath55 . a pair interaction @xmath6 is stable provided that @xmath56 for all @xmath57 and all @xmath58 . a nonnegative fourier transform @xmath59 implies stability , but this is a stronger condition than the former @xcite . a _ classical ground - state _ configuration ( structure ) within @xmath60 is one that minimizes @xmath61 . since we will allow for disordered ground states , then we consider the general ensemble setting that enables us to treat both disordered as well as ordered configurations . the _ ensemble average _ of @xmath62 for a statistically homogeneous and isotropic system in the thermodynamic limit is given by @xmath63 where @xmath64 is the number density and @xmath65 is the pair correlation function . in what follows , we consider those stable radial pair potentials @xmath6 that are bounded and absolutely integrable . we call such functions _ admissible _ pair potentials . therefore , the corresponding fourier transform @xmath59 exists , which we also take to be admissible , and @xmath66 * lemma . * for _ any ergodic configuration _ in @xmath2 , the following duality relation holds : @xmath67 if such a configuration is a ground state , then the left and right sides of ( [ plancherel ] ) are _ minimized_. * proof : * we assume _ ergodicity _ , i.e. , the macroscopic properties of any single configuration in the thermodynamic limit @xmath68 with @xmath69 constant are equal to their ensemble - average counterparts . the identity ( [ plancherel ] ) follows from plancherel s theorem , assuming that @xmath70 exists . it follows from ( [ h ] ) and ( [ plancherel ] ) that both sides of ( [ plancherel ] ) are minimized for any ground - state structure , although the duality relation ( [ plancherel ] ) applies to general ( i.e. , non - ground - state ) structures . * remarks : * 1 . the general duality relation ( [ plancherel ] ) does not seem to have been noticed or exploited before , although it was used for a specific pair interaction in ref . the reason for this perhaps is due to the fact that one is commonly interested in the total energy or , equivalently , the integral of ( [ g2 ] ) for which plancherel s theorem can not be applied because the fourier transform of @xmath65 does not exist . 2 . it is important to recognize that whereas @xmath71 always characterizes a point process @xcite , its fourier transform @xmath70 is generally not the total correlation function of a point process in reciprocal space . it is when @xmath71 characterizes a bravais lattice @xmath36 ( a special point process ) that @xmath70 is the total correlation function of a point process , namely the reciprocal bravais lattice @xmath39 . the ensemble - averaged structure factor is related to the collective density variable @xmath72 via the expression @xmath73 . 4 . on account of the uncertainty principle " for fourier pairs , the duality relation ( [ plancherel ] ) provides a computationally fast and efficient way of computing energies per particle of configurations for a non - localized ( long - ranged ) potential , say @xmath6 , by evaluating the equivalent integral in reciprocal space for the corresponding localized ( compact ) dual potential @xmath59 . * theorem 1 . * if an admissible pair potential @xmath6 has a bravais lattice @xmath36 ground - state structure at number density @xmath7 , then we have the following duality relation for the minimum @xmath74 of @xmath62 : @xmath75 where the prime on the sum denotes that the zero vector should be omitted , @xmath76 denotes the reciprocal bravais lattice @xcite , and @xmath59 is the dual pair potential , which automatically satisfies the stability condition , and therefore is admissible . moreover , the minimum @xmath74 of @xmath62 for any ground - state structure of the dual potential @xmath59 , is bounded from above by the corresponding real - space _ minimized _ quantity @xmath74 or , equivalently , the right side of ( [ duality ] ) , i.e. , @xmath77 whenever the reciprocal lattice @xmath39 at _ reciprocal lattice density _ @xmath78 is a ground state of @xmath59 , the inequality in ( [ bound ] ) becomes an equality . on the other hand , if an admissible dual potential @xmath59 has a bravais lattice @xmath39 at number density @xmath45 , then @xmath79 where equality is achieved when the real - space ground state is the lattice @xmath36 reciprocal to @xmath39 . * proof : * the radially averaged total correlation function for a bravais lattice , which we now assume to be a ground - state structure , is given by @xmath80 where @xmath81 is the surface area of a @xmath1-dimensional sphere of radius @xmath82 , @xmath83 is the coordination number ( number of points ) at the radial distance @xmath84 , and @xmath85 is a radial dirac delta function . substitution of this expression and the corresponding one for @xmath70 into ( [ plancherel ] ) yields @xmath86 where @xmath87 is the coordination number in the reciprocal lattice at the radial distance @xmath88 . recognizing that @xmath89 [ equal to twice the minimized energy per particle @xmath74 given by ( [ energy ] ) at its minimum in the limit @xmath90 and @xmath91 yields the duality relation ( [ duality ] ) . the fact that @xmath6 is stable @xcite means that the dual potential @xmath59 is stable since the left side of ( [ plancherel3 ] ) is nothing more than the sum given in ref . @xcite in the limit @xmath92 , which must be nonnegative . however , the minimum @xmath74 is generally not equal to the corresponding minimum @xmath93 associated with the ground state of the dual potential @xmath59 , i.e. , there may be periodic structures that have lower energy than the reciprocal lattice so that @xmath94 . to prove this point , notice that @xmath62 for any non - bravais lattice by definition obeys the inequality @xmath95 . however , because the corresponding fourier transform @xmath70 of total correlation function @xmath71 of the non - bravais lattice in real space generally does not correspond to a point process in reciprocal space ( see remark 2 under lemma 1 ) , we can not eliminate the possibilities that there are non - bravais lattices in reciprocal space with @xmath96 lower than @xmath74 . therefore , the inequality of ( [ bound ] ) holds in general with equality applying whenever the ground state structure for the dual potential @xmath59 is the bravais lattice @xmath39 at density @xmath45 . inequality ( [ bound2 ] ) follows in the same manner as ( [ bound ] ) when the ground state of the dual potential is known to be a bravais lattice . * remarks : * 1 . whenever equality in relation ( [ bound ] ) is achieved , then a ground state structure of the dual potential @xmath97 evaluated at the real - space variable @xmath82 is the bravais lattice @xmath39 at density @xmath78 . 2 . the zero - vector contributions on both sides of the duality relation ( [ duality ] ) are crucial in order to establish a relationship between the real- and reciprocal - space lattice " sums indicated therein . to emphasize this point , consider in @xmath5 the well - known yukawa ( screened - coloumb ) potential @xmath98 , which has the dual potential @xmath99 . at first glance , this potential would seem to be allowable because the real - space lattice sum , given on the left side of ( [ duality ] ) , is convergent . however , the reciprocal - space lattice sum on the right side does not converge . this nonconvergence arises because @xmath100 is unbounded . equality of infinities " is established , but of course this is of no practical value and is the reason why we demand that an admissible potential be bounded . 3 . can one identify specific circumstances in which the strict inequalities in ( [ bound ] ) and ( [ bound2 ] ) apply ? in addition to the theorem below that provides one such affirmative answer to this question , we will also subsequently give a specific one - dimensional example with unusual properties . * theorem 2 . * suppose that for admissible potentials there exists a range of densities over which the ground states are side by side coexistence of two distinct structures whose parentage are two different bravais lattices , then the strict inequalities in ( [ bound ] ) and ( [ bound2 ] ) apply at any density in this density - coexistence interval . * proof : * this follows immediately from the maxwell double - tangent construction in the @xmath62-@xmath101 plane , which ensures that the energy per particle in the coexistence region at density @xmath7 is lower than either of the two bravais lattices . as we will see , the duality relations of theorem 1 will enable one to use information about ground states of short - ranged potentials to draw new conclusions about the nature of the ground states of long - ranged potentials and vice versa . moreover , inequalities ( [ bound ] ) and ( [ bound2 ] ) provide a computational tool to estimate ground - state energies or eliminate candidate ground - state structures as obtained by annealing in monte carlo and molecular dynamics simulations . in the ensuing discussion , we will examine the ground states of several classes of admissible functions , focusing under what conditions the equalities or strict inequalities of the duality relations ( [ bound ] ) and ( [ bound2 ] ) apply . the aforementioned analysis can be extended to establish duality relations for many - particle systems interacting via three - body and higher - order interactions . for simplicity of exposition , we begin with a detailed construction of the three - body duality relations and then generalize to the higher - order case . we consider a statistically homogeneous @xmath47-particle interaction @xmath102 with one- , two- , and three - body contributions @xmath103 , @xmath104 , and @xmath105 , respectively . with the convention that @xmath106 and @xmath107 , we may write @xmath108 where @xmath109 is symmetric , bounded , and short - ranged . taking the ensemble average of this function implies @xmath110 involving averages over single particles , pairs , and triads . duality relations for the former two contributions have already been considered , and we therefore direct our attention to the last term in . since @xmath111 as @xmath112 , this function is generally not integrable , and we therefore introduce the associated three - body total correlation function @xmath113 . application of a double fourier transform and plancherel s theorem implies the following three - body analog of the lemma : @xmath114 where @xmath115 one can verify directly that the following relationship defines the three - body correlation function for any statistically homogeneous @xmath47-particle point pattern : @xmath116 for a bravais lattice , ergodicity should hold , and we can re - write as @xmath117 where the set @xmath118 in the summations includes all points of the lattice excluding the origin . the dual bravais lattice will possess a three - particle correlation function of the form @xmath119 , where @xmath7 is the _ real space _ number density . substituting and the corresponding @xmath120 for the dual bravais lattice into gives the following duality relation for three - particle interactions : @xmath121 where we have defined @xmath122 . the extension of this analysis to higher - order interactions is straightforward . specifically , we consider a @xmath9-particle bounded , symmetric , and short - ranged potential @xmath123 with a statistically homogeneous point distribution and the associated plancherel identity @xmath124 the @xmath9-particle correlation function of a bravais lattice is @xmath125 where @xmath126 denotes all sets of @xmath127 distinct vectors in a bravais lattice , excluding the origin , and @xmath128 indexes the lattice points . using this relationship , we find the following general @xmath9-particle duality relation : @xmath129,\label{hoduality}\ ] ] where @xmath130 . recently , the ground states have been studied corresponding to a certain class of oscillating real - space potentials @xmath6 as defined by the family of fourier transforms with compact support such that @xmath59 is positive for @xmath131 and zero otherwise @xcite . clearly , @xmath59 is an admissible pair potential . st @xcite showed that in three dimensions the corresponding real - space potential @xmath6 , which oscillates about zero , has the body - centered cubic ( bcc ) lattice as its unique ground state at the real - space density @xmath132 ( where we have taken @xmath133 ) . moreover , he demonstrated that for densities greater than @xmath134 , the ground states are degenerate such that the face - centered cubic ( fcc ) , simple hexagonal ( sh ) , and simple cubic ( sc ) lattices are ground states at and above the respective densities @xmath135 , @xmath136 , and @xmath134 . the long - range behavior of the real - space oscillating potential @xmath6 might be regarded to be unrealistic by some . however , since all of the aforementioned ground states are bravais lattices , the duality relation ( [ duality ] ) can be applied here to infer the ground states of real - space potentials with compact support . specifically , application of the duality theorem in @xmath5 and st s results enables us to conclude that for the real - space potential @xmath6 that is positive for @xmath137 and zero otherwise , the fcc lattice ( dual of the bcc lattice ) is the unique ground state at the density @xmath138 and the ground states are degenerate such that the bcc , sh and sc lattices are ground states at and below the respective densities @xmath139 , @xmath140 , and @xmath141 ( taking @xmath142 ) . specific examples of such real - space potentials , for which the ground states are not rigorously known , include the square - mound " potential @xcite [ @xmath143 for @xmath144 and zero otherwise ] and what we call here the overlap " potential , which corresponds to the intersection volume of two @xmath1-dimensional spheres of diameter @xmath145 whose centers are separated by a distance @xmath82 , divided by the volume of a sphere . the latter potential , which has support in the interval @xmath146 , remarkably arises in the consideration of the variance in the number of points within a spherical window " of diameter @xmath145 for point patterns in @xmath2 and its minimizer is an open problem in number theory @xcite . the @xmath1-dimensional fourier transforms of the square mound and overlap potentials are @xmath147 and @xmath148 , respectively , with @xmath142 . figure [ compact ] shows the real - space and dual potentials for these examples in three dimensions . the densities at which the aforementioned lattices are ground state structures are easily understood by appealing to either the square - mound or overlap potential . the fcc lattice is the unique ground state at the density @xmath138 because at this value ( where the nearest - neighbor distance is unity ) and lower densities the lattice energy is zero . at a slightly higher density , each of the 12 nearest neighbors contributes an amount of @xmath149 to the lattice energy . at densities lower than @xmath138 , there is an uncountably infinite number of degenerate ground states . this includes the bcc , sh and sc lattices , which join in as minimum - energy configurations at and below the respective densities @xmath139 , @xmath140 , and @xmath141 because those are the threshold values at which these structures have lattice energies that change discontinuously from some positive value ( determined by nearest neighbors only ) to zero . moreover , any structure , periodic or not , in which the nearest - neighbor distance is greater than unity is a ground state . however , at densities corresponding to nearest - neighbor distances that are less than unity , rigorous prediction of the possible ground - state structures is considerably more difficult . for example , it has been argued in ref . @xcite ( with good reason ) that real - space potentials whose fourier transforms oscillate about zero will exhibit polymorphic crystal phases in which the particles that comprise a cluster sit on top of each other . the square - mound potential is a special case of this class of potentials and the fact that it is a simple piecewise constant function allows for a rigorous analysis of the clustered ground states for densities in which the nearest - neighbor distances are less than the distance at which the discontinuity in @xmath6 occurs @xcite . and the three - dimensional overlap potential @xmath150 $ ] , where @xmath151 is the heaviside step function . right panel : corresponding dual potentials @xmath152 ( square - mound ; scaled by @xmath153 for clarity ) and @xmath154 ^ 2/k^3 $ ] ( overlap).,title="fig:",height=240 ] and the three - dimensional overlap potential @xmath150 $ ] , where @xmath151 is the heaviside step function . right panel : corresponding dual potentials @xmath152 ( square - mound ; scaled by @xmath153 for clarity ) and @xmath154 ^ 2/k^3 $ ] ( overlap).,title="fig:",height=240 ] another interesting class of admissible functions are those in which both @xmath6 and @xmath59 are nonnegative ( i.e. , purely repulsive ) for their entire domains . the overlap " potential discussed above is an example . . right panel : corresponding dual potential @xmath155.,title="fig:",height=240 ] . right panel : corresponding dual potential @xmath155.,title="fig:",height=240 ] here we examine the one - dimensional ground - state structures associated with the dual of the so - called overlap potential @xmath156 which is equal to the intersection volume , scaled by @xmath145 , of two rods of radius @xmath157 with centers separated by a distance @xmath82 . the dual potential is @xmath158 ^ 2;\ ] ] figure [ linear ] shows that both potentials are bounded and repulsive . however , while the overlap potential possesses the compact support @xmath159 $ ] , the dual potential is long - ranged with a countably infinite number of global minima determined by the zeros @xmath160 ( @xmath161 ) of @xmath162 . torquato and stillinger have shown @xcite that the unique ground state of the @xmath163 overlap potential is the integer lattice with density @xmath164 ; theorem 1 therefore implies that the integer lattice at reciprocal density @xmath165 is the unique ground state of the dual potential . this result intuitively corresponds to placing each point in an energy minimum of the dual potential , thereby driving the total potential energy to zero . this argument immediately implies that the integer lattice at reciprocal density @xmath166 for all @xmath161 is also a ground state of the dual potential ; however , the ground states at intermediate densities are generally non - bravais lattices and have heretofore been unexplored . based on these observations , previous work has suggested that the dual interaction undergoes an infinite number of structural phase transitions from bravais or simple non - bravais lattices to complex non - bravais lattices over the entire density range @xcite . we have characterized the ground states of the dual overlap potential numerically using the minop algorithm @xcite , which applies a dogleg strategy using a gradient direction when one is far from the energy minimum , a quasi - newton direction when one is close , and a linear combination of the two when one is at intermediate distances from a solution . the minop algorithm has been shown to provide more reliable results than gradient - based algorithms for similar many - body energy minimization problems @xcite . we fix the length @xmath167 of the simulation box and use a modified version of the dual potential @xmath168 ^ 2,\ ] ] where @xmath47 is the number of particles . note that @xmath169 provides the unit of length for the problem , allowing us to control the density of the resulting configuration by varying @xmath170 . for the case @xmath171 , we have numerically verified that the the integer lattice is the unique ground state ( up to translation ) of the dual potential ; indeed , direct calculation shows that the integer lattice minimizes the potential energy for all @xmath172 as expected from the arguments above . however , we have also identified degenerate ground states that are non - bravais lattices ; these systems are shown in figure [ gsconfigs ] . our results suggest that for @xmath173 the ground states are complex superpositions of bravais lattices with a minimum inter - particle spacing determined by @xmath170 . additionally , the conjectured infinite structural phase transitions are not observed in this density range owing to the high degeneracy of the ground state . we remark that although the integer lattice is a ground state for any @xmath172 , it is never observed in our numerical simulations because the energy landscape possesses a large number of global minima . furthermore , although the ground states for integral and non - integral values of @xmath170 are visually similar , we emphasize that the integer lattice is never a ground - state candidate for @xmath174 . for @xmath175 , the ground states are more difficult to resolve numerically because finite - size effects become more pronounced in this region ; justification for this behavior is provided below . nevertheless , we observe a `` clustered '' integer lattice structure in which several points occupy a single lattice site for @xmath176 . with @xmath177 and densities @xmath178 ( upper left ) , @xmath179 ( upper right ) , @xmath180 ( lower left ) , and @xmath181 ( lower right ) . the particles have been given a small but finite size for visual clarity . note that the @xmath181 configuration is a `` clustered '' integer lattice with more than one particle occupying certain lattice sites.,title="fig:",scaledwidth=45.0% ] with @xmath177 and densities @xmath178 ( upper left ) , @xmath179 ( upper right ) , @xmath180 ( lower left ) , and @xmath181 ( lower right ) . the particles have been given a small but finite size for visual clarity . note that the @xmath181 configuration is a `` clustered '' integer lattice with more than one particle occupying certain lattice sites.,title="fig:",scaledwidth=45.0% ] with @xmath177 and densities @xmath178 ( upper left ) , @xmath179 ( upper right ) , @xmath180 ( lower left ) , and @xmath181 ( lower right ) . the particles have been given a small but finite size for visual clarity . note that the @xmath181 configuration is a `` clustered '' integer lattice with more than one particle occupying certain lattice sites.,title="fig:",scaledwidth=45.0% ] with @xmath177 and densities @xmath178 ( upper left ) , @xmath179 ( upper right ) , @xmath180 ( lower left ) , and @xmath181 ( lower right ) . the particles have been given a small but finite size for visual clarity . note that the @xmath181 configuration is a `` clustered '' integer lattice with more than one particle occupying certain lattice sites.,title="fig:",scaledwidth=45.0% ] our numerical results suggest an exact approach to characterizing the ground states of the dual potential . for simplicity and without loss of generality , we will henceforth consider the scaled pair interaction @xmath182 corresponding to a normalized dual potential with @xmath177 . to facilitate the approach to the thermodynamic limit , we first examine a compact subset of @xmath183 subject to periodic boundary conditions . the entropy of this system for @xmath184 can be determined by relating the problem to the classic model of distributing @xmath47 balls into @xmath185 jars such that no more than one ball occupies each jar ( fermi - dirac statistics ) . specifically , choosing the parameter @xmath186 in is equivalent to choosing a density @xmath187 in the general problem . therefore , for any @xmath186 , there are @xmath188 `` jars '' for the @xmath47 particles ( `` balls '' ) . assuming that the particles are indistinguishable , the number of distinct ways of distributing the particles into the @xmath189 potential energy minima is the binomial coefficient @xmath190 . for @xmath47 large ( approaching the thermodynamic limit ) , stirling s formula implies that the entropy @xmath24 is @xmath191 where we have chosen units with @xmath192 . rearranging terms and substituting @xmath193 for the density , we find @xmath194 which is fixed in the thermodynamic limit and is plotted in figure [ enent ] . note that @xmath195 as @xmath196 , which is expected from the observation that the integer lattice is the unique ground state at unit density . this unusual residual entropy reflects the increasing degeneracy of the ground state with decreasing density and implies that in general the aforementioned infinite structural phase transitions from bravais to non - bravais lattices are not thermodynamically observed . instead , one finds an increasing number of countable coexisting ground - state structures as seen in our numerical energy minimizations . as a function of density @xmath7 for the dual potential @xmath197 in with @xmath177.,scaledwidth=38.0% ] for @xmath198 , determination of the ground states of the dual potential is nontrivial since it is no longer possible to distribute all of the particles into potential energy wells . therefore , fermi - dirac statistics are no longer applicable for the many - particle system . nevertheless , we can make some quantitative observations concerning the ground states in this density regime . we first consider the scenario of adding one particle to a local region , subject to periodic boundary conditions , of the integer lattice of unit spacing . since the potential energy minima of the pair interaction occur on the sites of the integer lattice , the total potential energy can not be driven to its global minimum . symmetry of the lattice implies that , without loss of generality , we can limit the location @xmath199 of the particle to the interval @xmath200 $ ] . since the energy of the underlying integer lattice is zero and the particle , by construction , will not interact with periodic images of itself , the total potential energy of the system after addition of the particle is exactly @xmath201}{\pi ( \xi+n)}\right)^2 + \sum_{n=0}^{+\infty } \left(\frac{\sin[\pi ( 1-\xi+n)]}{\pi(1-\xi+n)}\right)^2\label{esum}\\ & = \sin^2(\pi\xi)\left[\psi^{(1)}(\xi ) + \psi^{(1)}(1-\xi)\right]/\pi^2,\end{aligned}\ ] ] where @xmath202 is the trigamma function @xcite . from the reflection property of the trigamma function @xcite , the latter expression is exactly equal to unity [ @xmath203 for any value of the parameter @xmath199 . slightly greater than unity . lower : energy landscape associated with local perturbations of the integer lattice.,title="fig:",scaledwidth=75.0% ] slightly greater than unity . lower : energy landscape associated with local perturbations of the integer lattice.,title="fig:",scaledwidth=75.0% ] determination of the ground state then depends on `` relaxing '' the system by making a small perturbation @xmath204 in the underlying integer lattice ( see figure [ zrhog1 ] ) . the energy @xmath205 of this perturbed system is then parametrized by the displacements @xmath206 and @xmath199 as in figure [ zrhog1 ] and is given by @xmath207}{\pi(1-\gamma+n)}\right)^2 + \sum_{n=0}^{+\infty}\left(\frac{\sin[\pi(1+\gamma+n)]}{\pi(1+\gamma+n)}\right)^2 + \sum_{n=0}^{+\infty } \left(\frac{\sin[\pi(1-\xi+n)]}{\pi(1-\xi+n)}\right)^2\nonumber\\ & + \sum_{n=0}^{+\infty } \left(\frac{\sin[\pi(1+\gamma+n)]}{\pi(1+\gamma+n)}\right)^2+\left(\frac{\sin[\pi(\gamma+\xi+n)]}{\pi(\gamma+\xi+n)}\right)^2\\ & = 2+\tilde{v}(\gamma+\xi ) - \tilde{v}(\gamma)-\tilde{v}(\xi)\label{perturbe},\end{aligned}\ ] ] where we have utilized reflection and recurrence relations for the polygamma function @xcite with @xmath208 given by . figure [ zrhog1 ] illustrates that @xmath205 possesses a unique minimum value @xmath209 at @xmath210 . because the pair interaction is long - ranged , it is unclear if @xmath211 can be further decreased by additional local deformation of the structure . nevertheless , our analysis suggests that the ground state structures at densities slightly above unity are perturbed integer lattices with `` defects '' in the crystal structure . upon reaching @xmath212 , it is possible to `` stack '' two integer lattices with unit spacing for an energy per particle @xmath213 ; interestingly , the long - range nature of the pair potential implies that these lattices can be mechanically decoupled from each other without increasing the energy of the system . furthermore , the symmetry of this configuration implies that no local perturbation of the lattice structure can decrease the energy per particle , meaning that this `` stacked '' integer lattice and its translates within @xmath214 $ ] are at least local minima of the pair interaction ; a similar argument will hold for any @xmath215 . the energy of this stacked configuration is @xmath216 \qquad \delta = 1/\rho = 1/\alpha~~(\alpha \in \mathbb{n}).\ ] ] * remarks : * 1 . if the stacked integer lattices are global minima of the pair interaction for any number density @xmath215 , then the ground states are unique ( up to translation of layers ) at these densities , and the residual entropy will therefore vanish . however , for @xmath217 there is a combinatorial degeneracy associated with local deformations of the underlying integer lattice , implying that the residual entropy is _ nonanalytic _ over the full density range . this behavior in combination with the thermodynamic relation @xmath218 where @xmath219 is the thermal expansion coefficient and @xmath220 is the isothermal compressibility , suggests that there exist densities where the ground state exhibits negative thermal expansion as @xmath221 . one special case of the aforementioned `` stacked '' integer lattice configurations occurs when multiple particles occupy the same lattice sites ( i.e. , with no translation between layers ) . for these `` clustered '' integer lattices , pair interactions are _ localized _ to include only those particles on the same lattice site , meaning that there are no long - range interactions for these systems . however , we have seen that the inclusion of long - range pair interactions , such as with the @xmath212 integer lattice , does not affect the total energy of the system . since relative displacements between layers are uniformly distributed on @xmath222 $ ] , the average displacement of @xmath223 indeed corresponds to the @xmath212 integer lattice . our results imply that numerical methods are in general not appropriate for identifying the ground states for @xmath198 since truncation of the summation ( e.g. , with the minimal image convention ) breaks the translational degeneracy of the system . the ground states of the overlap potential @xmath224 also exhibit rich behavior for @xmath198 . since the interactions are localized to nearest neighbors , one can verify that addition of a particle to the unit density integer lattice increases the energy of the system by one unit , regardless of the position of the particle . however , unlike the dual potential , no local perturbation of the integer lattice can drive the system to lower energy , resulting in a large number of degenerate structures . the two - dimensional ground states of the generalized dual overlap potential @xmath225 ^ 2\ ] ] have also been numerically investigated @xcite ; the topology of the plane significantly increases the difficulty in analytically characterizing the ground - state configurations . another interesting example of nonnegative admissible functions is the gaussian core potential @xmath226 $ ] @xcite , which has been used to model interactions in polymers @xcite . the corresponding dual potentials are self - similar gaussian functions for any @xmath1 . the potential function pairs for the case @xmath227 with @xmath228 and @xmath229 are @xmath230 and @xmath231 . it is known from simulations @xcite that at sufficiently low densities in @xmath5 , the fcc lattices are the ground state structures for @xmath6 . it is also known that for the range @xmath232 , fcc is favored over bcc @xcite . if equality in ( [ bound ] ) is achieved for this density range , the duality theorem would imply that the bcc lattices in the range @xmath233 ( i.e. , high densities ) are the ground state structures for the dual potential . lattice - sum calculations and the aforementioned simulations for the gaussian core potential have verified that this is indeed the case , except in a narrow density interval of fcc - bcc coexistence @xmath234 around @xmath235 . in the coexistence interval , however , the corollary states the strict inequalities in ( [ bound ] ) and ( [ bound2 ] ) must apply . importantly , the ground states here are not only non - bravais lattices , they are not even periodic . the ground states are side - by - side coexistence of two macroscopic regions , but their shapes and relative orientations are expected to be rather complicated functions of density , because they depend on the surface energies of grain boundaries between the contacting crystal domains . proposition 9.6 of ref . @xcite enables us to conclude that the integer lattices are the ground states of the gaussian core potential for all densities in one dimension . note that in @xmath4 , the triangular lattices apparently are the ground states for the gaussian core potential at all densities ( even if there is no proof of such a conclusion ) , and therefore would not exhibit a phase transition . similar behavior has also been observed in four and eight dimensions , where the self - dual @xmath236 and @xmath237 lattices are the apparent ground states @xcite . cohn , kumar , and schrmann have recently identified _ non - bravais lattices _ in five and seven dimensions with lower ground - state energies than the densest known bravais lattices and their duals in these dimensions @xcite . interestingly , these non - bravais lattices , which are deformations of the @xmath238 and @xmath239 packings , possess the unusual property of _ formal self - duality _ , meaning that their average pair sums ( total energies per particle ) obey the same relation as poisson summation for bravais lattices for all admissible pair interactions . it is indeed an open problem to explain why formally - dual ground states exist for this pair potential . it is also instructive to apply our higher - order duality relations to the simple example of a three - body generalization of the aforementioned gaussian - core potential . specifically , we consider a three - body potential of the form @xmath240\label{threegauss}.\ ] ] applying a double fourier transform to this function shows that the dual potential , given by @xmath241,\ ] ] is self - similar to . as with the two - body version of the gaussian - core potential , this self - similarity implies that if a bravais lattice is the ground state of the three - body gaussian - core interaction at low density , then its dual lattice will be the ground state at high density with the exception of a narrow interval of coexistence around the self - dual density @xmath242 . however , we have been unable to find either numerical or analytical studies of the ground states of this higher - order interaction in the literature , and determining whether it shares ground states with its two - body counterpart is an open problem . a radial function @xmath29 is completely monotonic if it possesses derivatives @xmath243 for all @xmath244 and if @xmath245 . a radial function @xmath29 is completely monotonic if and only if it is the laplace transform of a finite nonnegative borel measure @xmath246 on @xmath247 $ ] , i.e. , @xmath248 @xcite . not all completely monotonic functions are admissible ( e.g. , the pure power - law potential @xmath249 in @xmath2 is inadmissible ) . examples of completely monotonic admissible functions in @xmath2 include @xmath250 for @xmath251 and @xmath252 for @xmath251 , @xmath253 . importantly , the fourier transform @xmath31 of a completely monotonic radial function @xmath29 is completely monotonic in @xmath254 @xcite . remarkably , the ground states of the pure exponential potential have not been investigated . here we apply the duality relations to the real - space potential @xmath255 in @xmath2 and its corresponding dual potential @xmath256 [ where @xmath257 , which has a slow power - law decay of @xmath258 for large @xmath33 . note that the dual potential is a completely monotonic admissible function in @xmath254 , and both @xmath6 and @xmath59 also fall within the class of nonnegative admissible functions . we have performed lattice - sum calculations for the exponential potential for a variety of bravais and non - bravais lattices in @xmath4 and @xmath5 . in @xmath4 , we found that the triangular lattices are favored at all densities ( as is true for the gaussian core potential ) . if equality in ( [ bound ] ) is achieved , then the triangular lattices are also the ground states for the slowly decaying dual potential @xmath259 at all densities . in @xmath5 , we found that the fcc lattices are favored at low densities ( @xmath260 ) and bcc lattices are favored at high densities ( @xmath261 ) . the maxwell double - tangent construction reveals that there is a very narrow density interval @xmath262 of fcc - bcc coexistence . we see that qualitatively the exponential potential appears to behave like the gaussian core potential . if equality in ( [ bound ] ) applies outside the coexistence interval , then the duality theorem would predict that the ground states of the slowly - decaying dual potential @xmath263 are the fcc lattices for @xmath264 and the bcc lattice for @xmath265 . note that in one dimension , it also follows from the work of cohn and kumar @xcite that since the integer lattices are the ground states of the gaussian potential , then these unique bravais lattices are the ground states of both the exponential potential and its dual evaluated at @xmath266 ( i.e. , @xmath267 ) . cohn and kumar @xcite have rigorously proved that certain configurations of points interacting with completely monotonic potentials on the surface of the unit sphere in arbitrary dimension were energy - minimizing . they also studied ways to possibly generalize their results for compact spaces to euclidean spaces and conjectured that the densest bravais lattices in @xmath2 for the special cases @xmath268 , 8 and 24 are the unique energy - minimizing configurations for completely monotonic functions . these particular lattices are self - dual and therefore phase transitions between different lattices is not possible . note that if the ground states for completely monotonic functions of squared distance in @xmath2 ( the gaussian function being a special case ) can be proved for any @xmath269 , it immediately follows from ref . @xcite that the completely monotonic functions of distance share the same ground states . thus , proofs for the gaussian core potential automatically apply to the exponential potential as well as its dual ( i.e. , @xmath270 ) because the latter is also completely monotonic in @xmath271 . based upon the work of cohn and kumar @xcite , it was conjectured that the gaussian core potential , exponential potential , the dual of the exponential potential , and any other admissible potential function that is completely monotonic in distance or squared distance share the same ground - state structures in @xmath2 for @xmath272 and @xmath273 , albeit not at the same densities @xcite . moreover , it was also conjectured for any such potential function , the ground states are the bravais lattices corresponding to the densest known sphere packings @xcite for @xmath274 and the corresponding reciprocal bravais lattices for @xmath275 , where @xmath276 and @xmath277 are the density limits of phase coexistence of the low- and high - density phases , respectively . in instances in which the bravais and reciprocal lattices are self - dual ( @xmath278 and @xmath279 ) @xmath280 , otherwise @xmath281 ( which occurs for @xmath282 and 7 ) . the second conjecture was recently shown by cohn and kumar to be violated for @xmath283 and @xmath284 . our discussion of the gaussian core model above suggests that one can exactly map the energy of a lattice at density @xmath7 to that of its dual lattice at reciprocal density @xmath45 for pair potentials that are _ self - similar _ ( defined below ) under fourier transform . here we provide additional examples of self - similar pair potentials , including radial functions that are eigenfunctions of the fourier transform . only some of these results are known in the mathematics literature @xcite , and this material has not previously been examined in the context of duality relations for classical ground states . pair potentials that are eigenfunctions of the fourier transform are unique in the context of the duality relations above since they preserve length scales for all densities ; i.e. , @xmath285 with no scaling factor @xmath246 in the argument . we therefore briefly review these eigenfunctions and the associated eigenvalues for radial fourier transforms . in order to simply the discussion , we will adopt a unitary convention for the fourier transform in this section @xmath286 which differs from our previous usage only by a scaling factor . the slight change in notation ( @xmath287 instead of @xmath288 ) is intended to clarify which convention is being used . the eigenfunctions of the fourier transform for @xmath163 can be derived from the generating function for the hermite polynomials , which , when scaled by a gaussian , is given by @xmath289 taking the fourier transform of both sides , one obtains @xmath290\right\ } dx = \sum_{n } \left(\frac{t^n}{n!}\right ) \mathfrak{f}\left\{\exp(-x^2/2 ) h_n(x)\right\}\label{thirteen},\end{aligned}\ ] ] implying @xmath291 h_n(k ) & = \sum_{n}\left(\frac{t^n}{n!}\right ) \mathfrak{f}\left\{\exp(-x^2/2 ) h_n(x)\right\}\label{fifteen}.\end{aligned}\ ] ] by collecting powers of @xmath292 in , we immediately conclude @xmath293 thereby identifying both the eigenfunctions and eigenvalues of the @xmath163 fourier transform . note that the eigenvalues are real when @xmath9 is even . we now seek eigenfunctions of the radially - symmetric fourier transform , defined here as @xmath294 dr\ ] ] for an isotropic function @xmath29 . direct substitution shows that @xmath295 is an eigenfunction for all @xmath1 with eigenvalue @xmath141 . other eigenfunctions of the fourier transform can be identified by noting that they are also eigenfunctions of the @xmath1-dimensional schr " odinger equation for the radial harmonic oscillator @xmath296 + \left(\frac{r^2}{2}\right ) \psi_n(r ) = e_n \psi_n(r),\ ] ] where we have used the relation @xmath297 for radially - isotropic functions in @xmath1 dimensions . the eigenvalues of the schrdinger equation are @xmath298 for some @xmath299 . the general solutions to are then given by @xmath300 where @xmath301 is the _ associated laguerre polynomial _ @xcite and @xmath302 is a dimension - dependent constant . note that for @xmath163 @xmath303 and we recover the even @xmath163 eigenfunctions of the harmonic oscillator . to determine the eigenvalues of the radial fourier transform , we note that if @xmath304 is an eigenfunction , then it must be true that @xmath305 for some eigenvalue @xmath306 . however , it is also true that @xmath307 equation implies that either @xmath308 or @xmath309 ; for a nontrivial solution we conclude that the eigenvalues of the radially - symmetric fourier transform are @xmath310 , which is in contrast to the general case on @xmath2 . this result is exactly consistent with the constraint that the index @xmath9 of an eigenstate of the radial schrdinger equation be even . note that when @xmath311 , the fourier transform changes the nature of the interaction ( i.e. , repulsive to attractive and vice - versa ) . the results above can be extended to include linear combinations of eigenfunctions of the fourier transform ; furthermore , we can generalize these functions to be simply _ self - similar _ under fourier transform , meaning that length scales are not preserved by the transformation . specifically , our interest is in functions for which : @xmath312 where @xmath313 and @xmath246 are constants . as an example , we consider the gaussian pair potential of the gaussian core model @xmath314 the corresponding fourier transform is given by @xmath315 now consider a pair potential @xmath6 that is a linear combination of two gaussians as follows : @xmath316 its fourier transform is @xmath317.\ ] ] in order for @xmath6 to be self - similar under fourier transformation , the constants @xmath246 and @xmath313 of must satisfy the following two equations for all @xmath318 : @xmath319 these equations will be satisfied by requiring @xmath320 leaving three independent parameters : @xmath246 , @xmath321 , and @xmath322 . the example extends to any even number of gaussian components . let @xmath323 where the @xmath324 are ordered by magnitude : @xmath325 the corresponding fourier transform is given by @xmath326 in order to ensure self - similarity , the terms can be paired and subject to the relations of the type ( [ cond1 ] ) . on account of the ordering condition ( [ order ] ) , we pair terms with indices @xmath327 and @xmath328 , @xmath329 , and hence require @xmath330 it is also possible to include an additional gaussian to make an odd number in total . this additional term must effectively pair with itself , so that @xmath331 where the corresponding parameter @xmath332 is uncontrained and the subscript 0 refers to the odd" term . these relations suggest an extension to the case of a _ continuous _ distribution of gaussian widths as follows : @xmath333 d\sigma.\ ] ] the corresponding fourier transform is given by @xmath334 d\sigma \nonumber \\ & \equiv & \lambda { v}(\mu k),\end{aligned}\ ] ] as required for self - similarity , where @xmath335 . in this work we have derived duality relations for interactions of arbitrarily high order that can be applied to help quantify and identify classical ground states for admissible potentials that arise in soft - matter systems . we have applied the duality relations for different classes of admissible potential functions , including potentials with compact support , nonnegative functions , and completely monotonic potentials . among these classes , the completely monotonic functions offer a new category of potentials for which the ground states might be identified rigorously . in particular , we seek a proof of the conjecture that functions in this class share the same ground - state structures in @xmath2 for @xmath336 and @xmath337 , albeit not at the same densities . no counterexample for this conjecture has been found to date . it should also be emphasized that the examples of admissible functions examined here are by no means complete . we have also identified a set of pair potentials on the line related to the overlap function that exhibit a `` stacking '' phenomenon at certain densities in the ground state . this behavior leads to an unusual mechanical decoupling between layers of integer lattices due entirely to the form of the interaction . these systems , previously thought to exhibit an infinite number of structural phase transitions from bravais to non - bravais structures @xcite , likely possess rich thermodynamic properties such as negative thermal expansion as @xmath221 . since overlap potentials arise in a variety of contexts , including the covering and quantizer problems @xcite and the identification and design of hyperuniform point patterns @xcite , further studies of these systems are warranted . toward this end , we plan to explore whether analogous duality relations can be established for positive but small temperatures by studying the properties of the phonon spectra of admissible potentials . the development of such relations would provide a unique and useful guide for mapping the phase diagrams of many - particle interactions , including those functions belonging to the class of `` self - similar '' potentials that we have introduced here . indeed , with the exception of the gaussian core model @xcite , little is known about the ground states and phase behaviors of self - similar functions . since most of these potentials contain both repulsive and attractive components , these interactions have direct implications for spatially inhomogeneous solvent compositions that simultaneously induce repulsion and attraction among macromolecules in solution . we expect that as the methodology continues to develop , duality relations of the type we have discussed here will play an invaluable role in understanding these complex physical systems . numerous investigations strongly suggest that the ground state in @xmath4 is the triangular lattice . in @xmath5 , slightly distorted hexagonal - close - packed crystals are believed to be the low - density ground states with a transition to fcc at high density . although it is not commonly regarded as such , the classical hard - sphere system exhibits disordered classical ground states from zero density up to the freezing point " because at any instant in time the total interaction energy is at its minimum of zero . however , because the total energy is either infinite or zero , we regard any disordered hard - sphere ground state as trivial . of course , at singular jammed states , the elastic moduli and , therefore , the strain energies are infinite ; see s. torquato , a. donev , and f. h. stillinger , int . j. solids structures * 40 * , 7143 ( 2003 ) . although it may seem unusual for a one - dimensional system to exhibit phase transitions , it is known that such can be the case if the potential has an infinite range . see , e.g. , _ mathematical physics in one dimension _ eds . e. h. lieb and d. c. mattis ( academic press , new york , 1966 ) . j. h. conway and n. j. a. sloane , _ sphere packings , lattices and groups _ ( springer - verlag , new york , 1998 ) . the self - dual @xmath237 and leech lattices in @xmath338 and @xmath339 , respectively , have remarkable properties . these highly symmetric lattices are almost surely the densest sphere packing arrangments in those dimensions [ see h. cohn and a. kumar , annal . math . , in press . ] and are of relevance in communications theory .

bounded interactions are particularly important in soft - matter systems , such as colloids , microemulsions , and polymers . we derive new duality relations for a class of soft potentials , including three - body and higher - order functions , that can be applied to ordered and disordered classical ground states . these duality relations link the energy of configurations associated with a real - space potential to the corresponding energy of the dual ( fourier - transformed ) potential . we apply the duality relations by demonstrating how information about the classical ground states of short - ranged potentials can be used to draw new conclusions about the ground states of long - ranged potentials and vice versa . the duality relations also lead to bounds on the @xmath0 system energies in density intervals of phase coexistence . additionally , we identify classes of `` self - similar '' potentials , for which one can relate low- and high - density ground - state energies . we analyze the ground state configurations and thermodynamic properties of a one - dimensional system previously thought to exhibit an infinite number of structural phase transitions and comment on the known ground states of purely repulsive monotonic potentials in the context of our duality relations .

the _ partially asymmetric simple exclusion process _ ( pasep ) is a physical model in which @xmath0 sites on a one - dimensional lattice are either empty or occupied by a single particle . these particles may hop to the left or to the right with fixed probabilities , which defines a markov chain on the @xmath1 states of the model . the explicit description of the stationary probability of the pasep was obtained through the matrix - ansatz @xcite . since then , the links between specializations of this model and combinatorics have been the subject of an important research ( see for example @xcite ) . a great achievment is the description of the stationary distribution of the most general pasep model through statistics defined on combinatorial objects called staircase tableaux @xcite . the objective of our work is to show how we can reveal an underlying tree structure in staircase tableaux and use it to obtain combinatorial properties . in this paper , we focus on some specializations of the _ fugacity partition function _ @xmath2 , defined in the next section as a @xmath3-analogue of the classical partition function of staircase tableaux . the present paper is divided into two sections . section [ sec : tlst ] is devoted to the definition of labeled tree - like tableaux , which are a new presentation of staircase tableaux , and to the presentation of a tree structure and of an insertion algorithm on these objects . section [ sec : app ] presents combinatorial applications of these tools . we get : * a new and natural proof of the formula for @xmath4 ; * a study of @xmath5 ; * a bijective proof for the formula of @xmath6 . a _ staircase tableau _ @xmath7 of size @xmath8 is a ferrers diagram of `` staircase '' shape @xmath9 such that boxes are either empty or labeled with @xmath10 , @xmath11 , @xmath12 , or @xmath13 , and satisfying the following conditions : * no box along the diagonal of @xmath7 is empty ; * all boxes in the same row and to the left of a @xmath11 or a @xmath13 are empty ; * all boxes in the same column and above an @xmath10 or a @xmath12 are empty . figure [ fig : st ] ( left ) presents a staircase tableau @xmath7 of size 5 . [ weight ] the _ weight _ @xmath14 of a staircase tableau @xmath7 is a monomial in @xmath15 and @xmath16 , which we obtain as follows . every blank box of @xmath7 is assigned a @xmath16 or a @xmath17 , based on the label of the closest labeled box to its right in the same row and the label of the closest labeled box below it in the same column , such that : * every blank box which sees a @xmath11 to its right gets a @xmath17 ; * every blank box which sees a @xmath13 to its right gets a @xmath16 ; * every blank box which sees an @xmath10 or @xmath12 to its right , and an @xmath10 or @xmath13 below it , gets a @xmath17 ; * every blank box which sees an @xmath10 or @xmath12 to its right , and a @xmath11 or @xmath12 below it , gets a @xmath16 . after filling all blank boxes , we define @xmath14 to be the product of all labels in all boxes . the weight of the staircase tableau @xmath18 on figure [ fig : st ] is @xmath19 . there is a simple correspondence between the states of the pasep and the diagonal labels of staircase tableaux : diagonal boxes may be seen as sites of the model , and @xmath10 and @xmath13 ( resp . @xmath11 and @xmath12 ) diagonal labels correspond to occupied ( resp . unoccupied ) sites . we shall use a variable @xmath3 to keep track of the number of particles in each state . to this way , we define @xmath20 to be the number of labels @xmath10 or @xmath13 along the diagonal of @xmath7 . for example the tableau @xmath7 in figure [ fig : st ] has @xmath21 . the _ fugacity partition function _ of the pasep is defined as @xmath22 we shall now define another class of objects , called labeled tree - like tableaux . they appear as a labeled version of tree - like tableaux ( tlts ) defined in @xcite . these tableaux are in bijection with staircase tableaux , and present two nice properties inherited from tlts : an underlying tree structure , and an insertion algorithm which provides a useful recursive presentation . in a ferrers diagram @xmath23 , the _ border edges _ are the edges that stand at the end of rows or columns . the number of border edges is clearly the half - perimeter of @xmath23 . for any box @xmath24 of @xmath23 , we define @xmath25 as the set of boxes placed in the same column and above @xmath24 in @xmath23 , and @xmath26 as the set of boxes placed in the same row and to the left of @xmath24 in @xmath23 . by a slight abuse , we shall use the same notations for any tableau @xmath18 of shape @xmath23 . these notions are illustrated at figure [ fig : bordedges - legarm ] . @xmath27{images / border_edges_leg_arm_1 } \end{array } $ ] @xmath27{images / border_edges_leg_arm_2 } \end{array } $ ] @xmath27{images / border_edges_leg_arm_3 } \end{array } $ ] [ def : ltlt ] a _ labeled tree - like tableau _ ( ltlt ) @xmath18 of size @xmath8 is a ferrers diagram of half - perimeter @xmath28 such that some boxes and all border edges are labeled with @xmath17 , @xmath10 , @xmath11 , @xmath12 , or @xmath13 , and satisfying the following conditions : * the northwestern - most box ( root box ) is labeled by @xmath17 ; * the labels in the first row and the first column are the only labels @xmath17 ; * in each row and column , there is at least one labeled box ; * for each box @xmath24 labeled by @xmath10 or @xmath12 , all boxes in @xmath29 are empty and at least one box in @xmath30 is labeled ; * for each box @xmath24 labeled by @xmath11 or @xmath13 , all boxes in @xmath30 are empty and at least one box in @xmath29 is labeled . [ prop : bij ] for @xmath31 , ltlts of size @xmath8 are in bijection with staircase tableaux of size @xmath32 . let us describe a correspondence @xmath33 that sends a staircase tableau @xmath7 of size @xmath32 to an ltlt @xmath18 . it consists in the following steps ( see figure [ fig : l - bij ] ) : * we add to @xmath7 a hook @xmath34 ; * in this hook , we label by @xmath17 : the root - cell , the border edges and the cells in the first row that see a @xmath11 or @xmath13 below , and the cells in the first column that see an @xmath10 or @xmath12 to their right ; * for each label @xmath10 or @xmath12 ( resp . @xmath11 or @xmath13 ) on the diagonal , we erase the corresponding column ( resp . row ) in the tableau , which has to be empty , and put the label on the vertical ( resp . horizontal ) border edge to its left ( resp . above it ) . the result of these operations is an ltlt @xmath35 of size @xmath8 . it is straightforward to construct the inverse of @xmath33 , since each operation may be reversed , thus proving that @xmath33 is a bijection . @xmath27{images / bijection_between_st_and_ltlt_1 } \end{array } \longleftrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / bijection_between_st_and_ltlt_4 } \end{array } \longleftrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / bijection_between_st_and_ltlt_5 } \end{array } $ ] a nice feature of the notion of ltlt is its underlying tree structure . let us consider an ltlt @xmath18 of size @xmath8 . we may see each label of @xmath18 as a node , and definition [ def : ltlt ] ensures that each node ( except the ne - most one , which appears as the root ) has either a node above it or to its left , which may be seen as its father . we refer to figure [ fig : tree ] which illustrates this property . crossing _ is a box @xmath24 such that * there is a label to the left and to the right of @xmath24 ; * there is a label above and below @xmath24 . in this way , we get a labeled binary tree with some additional information : crossings of edges . if we forget the crossings , we have a binary tree in which each internal node and leaf is labeled . as a consequence , any ltlt is endowed with an underlying binary tree structure , such that the size of the ltlt is equal to the number of internal nodes in its underlying binary tree . @xmath27{images / underlaying_tree_structure_1 } \end{array } $ ] @xmath27{images / underlaying_tree_structure_2 } \end{array } $ ] @xmath27{images / underlaying_tree_structure_3 } \end{array } $ ] given an ltlt @xmath18 and a border edge @xmath36 , a _ compatible bi - label _ is the choice of a ( new ) couple @xmath37 of labels such that * if @xmath36 is in the first row of @xmath18 ( thus vertical ) : @xmath38 and @xmath39 ; * if @xmath36 is in the first column of @xmath18 ( thus horizontal ) : @xmath40 and @xmath41 ; * otherwise : @xmath41 and @xmath39 . [ def : col_add ] given an ltlt @xmath18 , a vertical border edge @xmath36 ( thus at the end of a row @xmath42 ) with label @xmath43 , and a compatible bi - label @xmath37 , the _ column addition _ in @xmath18 at edge @xmath36 , with new label @xmath37 is defined as follows : * we add a cell to @xmath42 and to all rows above it ; * since vertical and horizontal border edges on the right of @xmath36 are shifted horizontally , we shift also the corresponding labels ; * we label the new box in row @xmath42 by @xmath43 ; * we label the two new vertical and horizontal border edges respectively by @xmath44 and @xmath3 . if the edge @xmath36 is horizontal , we define in the same way the _ row addition_. @xmath27{images / addition_of_row_1 } \end{array } \xrightarrow[(x , y ) = ( \gamma,\beta ) ] { } \begin{array}{c } \includegraphics[scale=\scalefigure]{images / addition_of_row_2 } \end{array } \xrightarrow[\text{substitution}]{\text{after $ x$ and $ y$ } } \begin{array}{c } \includegraphics[scale=\scalefigure]{images / addition_of_row_3 } \end{array } $ ] given two ferrers diagrams @xmath45 , we say that the set of cells @xmath46 ( set - theoretic difference ) is a _ ribbon _ if it is connected ( with respect to adjacency ) and contains no @xmath47 square . in this case we say that @xmath48 can be added to @xmath49 , or that it can be removed from @xmath50 . for our purpose , we shall only consider the addition of a ribbon to an ltlt @xmath18 between a vertical border edge @xmath51 and an horizontal border edge @xmath52 . as in the row / column insertion , we observe that vertical ( resp . horizontal ) border edges are shifted horizontally ( resp . vertically ) , thus we shift also the corresponding labels . figure [ fig : ribbon_insertion ] illustrates this operation . [ def : spec ] let @xmath18 be an ltlt . the _ special box _ of @xmath18 is the northeast - most labeled box among those that occur at the bottom of a column . this is well - defined since the bottom row of @xmath18 contains necessarily a labeled box . @xmath27{images / insertion_rubban_1 } \end{array } \xrightarrow[insertion]{ribbon } \begin{array}{c } \includegraphics[scale=\scalefigure]{images / insertion_rubban_2 } \end{array } $ ] @xmath27{images / special_box_1 } \end{array } $ ] an ltlt @xmath18 of size @xmath8 together with the choice of one of its border edges @xmath36 , and a compatible bi - label @xmath37 . [ etape_reperer_case_speciale ] find the special box @xmath53 of @xmath18 . [ etape_inserer_colonne ] add a row / column to @xmath18 at edge @xmath36 with new bi - label @xmath37 . [ etape_ajouter_ruban ] if @xmath36 is to the left of @xmath53 , perform a ribbon addition between @xmath36 and @xmath53 . a final ltlt @xmath54 of size @xmath28 . @xmath27{images / insertion_procedure_2 } \end{array } \longrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / insertion_procedure_3 } \end{array } \longrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / insertion_procedure_4 } \end{array } \longrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / insertion_procedure_5 } \end{array } $ ] [ prop : insertion ] the insertion algorithm [ algo : insertion ] induces a bijection between * ltlts of size @xmath8 together with the choice of a border edge and a compatible bi - label , * ltlts of size @xmath28 . we shall only give the key ingredient , which is analog to theorem 2.2 in @xcite : the insertion algorithm is constructed in such a way that the added labeled box becomes the special box of the new ltlt . this implies that the process may be inversed , thus proving proposition [ prop : insertion ] . an important feature of this algorithm is that it provides a recursive comprehension focused on the border edges of the ltlt , _ i.e. _ on the diagonal element of the staircase tableaux . up to now , the recursive approach was with respect to the first column , which is of course less significant since the state of the pasep is encoded by the labels on the diagonal . we give in the next section three examples of the use we can make of the tree structure and the insertion algorithm . proposition [ prop : insertion ] implies that given a ltlt @xmath18 of size @xmath8 , we can build through the insertion algorithm exactly @xmath58 different ltlts @xmath54 of size @xmath59 . using proposition [ prop : bij ] , let @xmath60 . the contribution to @xmath61 of the @xmath62 staircase tableaux @xmath63 is precisely @xmath64 whence ( [ eq : gen_fct ] ) . proposition [ prop : gen_fct ] corresponds to theorem 4.1 in @xcite . the insertion algorithm gives a trivial and natural explanation for this formula . moreover , proposition [ prop : gen_fct ] implies that the number of staircase tableaux of size @xmath8 is given by @xmath65 . it is clear that we may use the insertion algorithm to build a recursive bijection between staircase tableaux of size @xmath8 and objects enumerated by @xmath66 such as doubly signed permutations , _ i.e. _ triple @xmath67 where @xmath68 in a permutation of @xmath8 and @xmath69 , @xmath70 two vectors of @xmath71 . in this part , we consider staircase tableaux without any @xmath12 label . we denote by @xmath72 the number of such tableaux @xmath7 of size @xmath8 with @xmath73 labels @xmath10 or @xmath13 in the diagonal , _ i.e. _ such that @xmath74 . our goal is to get a recursive formula on @xmath72 numbers . we consider a staircase tableau @xmath7 of size @xmath32 with @xmath74 , and we let @xmath75 its associated ltlt of size @xmath8 . now we examine the possible insertions on @xmath18 to obtain an ltlt @xmath54 of size @xmath28 : let @xmath36 be the edge where the insertion occurs , @xmath76 be the label of @xmath36 ( which does not appear on the border edges of @xmath54 ) , and @xmath37 the compatible bi - label which appears on the border edges of @xmath54 . we denote by @xmath77 the staircase tableau @xmath78 of size @xmath8 . * @xmath80 when @xmath81 is either @xmath82 or @xmath83 or @xmath84 , which gives @xmath85 possibilities ; * @xmath17 when @xmath86 is either @xmath87 or @xmath88 or @xmath89 or @xmath90 or @xmath91 , which gives @xmath28 possibilities ; * @xmath92 when @xmath93 which gives @xmath94 possibilities ( the number of border edges labeled by @xmath11 ) . we let @xmath99 and use ( [ eq : stable ] ) to write @xmath100 with @xmath101 . then we check that the equality @xmath102 with @xmath103 implies by induction on @xmath8 that @xmath104 has at least @xmath8 distinct zeros in @xmath98 - 1,0[$ ] . since @xmath8 is the degree of @xmath104 , the conclusion follows . equation ( [ eq : stable ] ) , as well as proposition [ prop : stable ] are new . our insertion algorithm shows its strength when it comes to study recursively the diagonal in staircase tableaux , which is meaningful in the pasep model : it is by far more natural than the already studied @xcite recursion with respect to the first column of the tableau . we show how our tools lead to a bijection between staircase tableaux without any label @xmath13 or weight @xmath16 and a certain class of paths enumerated by the sequence a026671 of @xcite . this is an answer to problem 5.8 of @xcite . let us consider ltlts which are in bijection with staircase tableaux without @xmath13 or @xmath16 . the first observation is that these tableaux correspond to trees without any crossing , since a crossing sees an @xmath10 or a @xmath12 to its right and a @xmath11 below , which gives a weight @xmath16 ( _ cf . _ figure [ fig : no_cross ] ) . thus we have to deal with binary trees whose left sons only have one choice of label ( @xmath11 ) and whose right sons may have one ( @xmath10 ) or two choices ( @xmath10 or @xmath12 ) of labels . the restriction of no weight @xmath16 is equivalent to forbidding any point in the tree which sees an @xmath10 or a @xmath12 to its right and a @xmath12 below it . since we deal with binary trees ( without any crossing ) , we get that the only nodes @xmath112 where we may put a label @xmath12 are such that the path in the tree from the root to @xmath112 contains exactly one left son , followed by a right son . these nodes are illustrated on figure [ fig : pos_gamma ] . we may shift these labels to their father in the tree , and define the _ left depth _ of a node @xmath112 in a binary tree as the number of left sons in the path from the root to @xmath112 , and using the bijection @xmath33 , we get the following statement . [ lem : bij ] the set of staircase tableaux of size @xmath8 without any @xmath13 label or @xmath16 weight is in bijection with the set @xmath113 of binary trees of size @xmath8 whose nodes of left depth equal to @xmath17 are labeled by @xmath10 or @xmath12 . we shall now code the trees in @xmath113 by lattice paths . to do this , we use a deformation of the classical bijection @xcite between binary trees and dyck paths : we go around the tree , starting at the root and omitting the last external node , and we add to the path a step @xmath108 when visiting ( for the first time ) an internal node , or a step @xmath109 when visiting an external node . let us denote by @xmath114 the ( dyck ) path associated to the binary tree @xmath18 under this procedure . it is well - known that @xmath115 is a bijection between binary trees with @xmath8 internal nodes and dyck paths of length @xmath116 ( of size @xmath8 ) . if we use the same coding , but omitting the root and the last @xmath92 external nodes , we get a bijection between binary trees with @xmath8 internal nodes and _ almost - dyck _ paths ( whose ordinate is always @xmath117 ) of size @xmath32 . in the sequel , we shall call _ factor _ of a path a minimal sub - path starting from the axis and ending on the axis . we may replace the negative factors by steps @xmath118 to get a bijection @xmath119 between binary trees of size @xmath8 and _ positive _ lazy paths of size @xmath32 ( these objects appear under the name ( @xmath120 ) in @xcite ) . figure [ fig : bij_pi ] illustrates bijections @xmath115 and @xmath119 . we observe that the nodes with left depth equal to @xmath17 in a binary tree @xmath121 correspond to steps @xmath108 which start on the axis in @xmath122 , thus to strictly positive factors in @xmath122 . these nodes may be labeled with @xmath10 or @xmath12 . to translate this bijectively , we only have to leave unchanged a factor associated to a label @xmath10 , and to apply a mirror reflexion to a factor associated to a label @xmath12 . figure [ fig : bij ] illustrates this correspondence . thanks to proposition [ prop : bij ] and lemma [ lem : bij ] , we get a bijection denoted by @xmath123 , between staircase tableaux and lazy paths ( see figure [ fig : from_st_to_lazy_path ] ) . @xmath27{images / from_staircase_to_lazy_path_1 } \end{array } \longleftrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / from_staircase_to_lazy_path_2 } \end{array } \longleftrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / from_staircase_to_lazy_path_3 } \end{array } \longleftrightarrow $ ] @xmath27{images / from_staircase_to_lazy_path_4 } \end{array } \longleftrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / from_staircase_to_lazy_path_5 } \end{array } \longleftrightarrow \begin{array}{c } \includegraphics[scale=\scalefigure]{images / from_staircase_to_lazy_path_6 } \end{array } $ ] we recall the following definition from @xcite . a row _ indexed _ by @xmath11 or @xmath13 in a staircase tableau @xmath7 is a row such that its left - most label is @xmath11 or @xmath13 . in the same way , a column indexed by an @xmath10 or a @xmath12 is a column such that its top - most label is @xmath10 or @xmath12 . for example , the staircase tableau on the left of figure [ fig : st ] has 2 columns indexed by @xmath10 and 1 row indexed by @xmath13 . the application @xmath123 defines a bijection from the set of staircase tableaux @xmath7 without label @xmath13 and weight @xmath16 , of size @xmath8 , to the set of lazy paths of size @xmath8 . moreover , if we denote : @xmath124 the number of @xmath109 steps , @xmath125 the number of @xmath108 steps , @xmath126 the length of the initial maximal sequence of @xmath108 steps , @xmath127 the number of @xmath110 steps , @xmath128 the number of factors , and @xmath129 the number of negative factors in @xmath130 , then : * the number of @xmath12 labels in @xmath7 is given by @xmath129 ; * the number of @xmath10 labels in @xmath7 is given by @xmath131 ; * the number of @xmath11 labels in @xmath7 is given by @xmath132 ; * the number of columns indexed by @xmath11 in @xmath7 is given by @xmath127 ; * the number of rows indexed by @xmath10 or @xmath12 in @xmath7 is given by @xmath126 . we still have to check the assertions about the different statistics . we recall that , as defined in @xcite , rows indexed by @xmath11 or @xmath13 ( resp . columns indexed by @xmath10 or @xmath12 ) in a staircase tabelau @xmath7 correspond to non - root @xmath17 labels in the first column ( resp . first row ) of @xmath35 . we have : * the number of @xmath12 labels is by definition the number of negative factors in @xmath129 ; * the number of @xmath10 or @xmath12 labels is @xmath125 ; * the number of @xmath11 labels is the number of external nodes minus the number of nodes in the left branch of the tree , thus @xmath132 ; * columns indexed by @xmath11 correspond to nodes in the right branch of the tree , their number is @xmath127 ; * rows indexed by @xmath10 or @xmath12 correspond to nodes in the left branch of the tree , their number is @xmath126 . we may observe that among this class of staircase tableaux @xmath7 , those who have only @xmath11 or @xmath12 labels on the diagonal , _ i.e. _ such that @xmath133 are in bijection with binary trees whose internal nodes are of left depth at most @xmath17 , and such that we forbid the @xmath10 label on external nodes . the bijection @xmath123 sends these tableaux onto lazy paths of height and depth bounded by @xmath17 and whose factors preceding either a @xmath110 step or the end of the path are negative ( _ cf . _ figure [ fig : frob_path ] ) . let us denote by @xmath134 the number of such path of size @xmath8 . by decomposing the path with respect to its first two factors , we may write @xmath135 which corresponds ( _ cf . _ the entry a001519 in @xcite ) to the recurrence of odd fibonacci numbers @xmath136 , as claimed in corollary 3.10 of @xcite . another interesting special case concerns staircase tableaux of size @xmath8 without any @xmath13 or @xmath12 labels , and without weight @xmath16 . it is obvious that the bijection @xmath123 maps these tableaux onto binary trees , enumerated by catalan numbers @xmath137 . moreover , if we keep track of the number @xmath73 of external nodes labeled with @xmath11 , we get a bijection with dyck paths @xmath115 of size @xmath8 with exactly @xmath73 peaks , enumerated by narayana numbers @xmath138 . * forthcoming objective . * we are convinced that ltlts , because of their insertion algorithm , are objects that are both natural and easy to use , as shown in this paper on some special cases . since a nice feature of our insertion procedure is to work on the boundary edges , which encode the states in the pasep , an objective is to use these objects to describe combinatorially the general case of the pasep model @xcite . to do that , we have to find an alternate description of the weight on ltlts , hopefully simpler than the one defined on staircase tableaux in definition [ weight ] . * acknowledgements * [ sec : ack ] this research was driven by computer exploration using the open - source mathematical software ` sage ` @xcite and its algebraic combinatorics features developed by the ` sage - combinat ` community @xcite .

staircase tableaux are combinatorial objects which appear as key tools in the study of the pasep physical model . the aim of this work is to show how the discovery of a tree structure in staircase tableaux is a significant feature to derive properties on these objects .

translocation of protons over long distances has a key importance for biological and chemical systems . it is believed that proton migration along the chains of water molecules formed between the interior of proteins and the solvent , establishes electrochemical potential gradients playing an important functional role @xcite . experimental evidence indicates that the dominant mechanism responsible for proton transport in transmembrane proteins ( for instance , in bacteriorhodopsin of _ halobacterium halobium _ @xcite and in gramicidin a channels @xcite ) is the diffusion of h@xmath0 ions which is faster than the hydrodynamic flow of hydronium species ( h@xmath1o)@xmath0 . especially at low hydrogen concentrations in channels , proton conduction is determined by a two - stage grotthuss - type mechanism @xcite shown schematically in fig . the first stage involves the intrabond proton tunnelling along the hydrogen bridge which is connected with the formation and transfer of ionic positive ( h@xmath1o@xmath0 ) and negative ( oh@xmath2 ) charged defects . to sustain a flux of h@xmath0 in such proton wire , the inter - molecular proton transfer due to the reorientations of molecular group with proton is assumed . reorientation motion leads to the breaking of the hydrogen bond ( so - called orientational bjerrum l - defect ) and location of proton between another pair of molecular groups @xcite . consequently , the reorientation step in the presence of the second proton may induce high - energy configuration with both of the protons shared by two adjacent oxygen ions ( bjerrum d - defect ) . unlike the translocation of monovalent ions like cs@xmath0 , na@xmath0 or k@xmath0 via gramicidin requiring the net diffusion of the whole water column in the channel , the existence of the grotthuss - type selective migration of h@xmath0 through the h - bonded chain is supported by the absence of streaming potentials during h@xmath0 permeation @xcite . in contrast to the bulk water , the reorientation motion in one - dimensional water wire involving a migration of bjerrum defects with the period of reorientations about @xmath3 s , is much slower than the proton intra - bond hopping ( @xmath4 s)@xcite . this is also closely related to the fact that the mobility of bjerrum defects ( @xmath5 @xmath6v@xmath7s@xmath7 ) is much lower than that of the ionic defects ( @xmath8 @xmath6v@xmath7s@xmath7 ) @xcite . moreover , as appears from the results of molecular dynamics simulations @xcite , the translocation of the ionic defects in proton wires is almost activationless process , whereas the orientation defects involving an activation energy of about 5 kcal / mol in the chain containing eight water molecules , constitute a limiting step for the proton migration . besides the orientation defects , the recent experiments indicate that another possible rate - limiting step for the proton migration in gramicidin channels can be at the membrane - channel / solution interface@xcite . as was pointed in @xcite , the experimental analysis of the proton flow in bioenergetic proteins and the mechanism of proton translocation is very difficult because of its intrinsically transient nature . to shed more light on the microscopic nature of the proton transport and to analyze the influence of quantum effects and interaction with proton surrounding , theoretical modeling remains to be essentially important . recently , much attention has been focused on the theoretical studies of the dynamics of ionic defects using soliton models @xcite and molecular dynamics simulations @xcite . proton transfer in water was shown to be strongly coupled with the dynamics of local environment , and the density of ionic defects was found to increase exponentially with the increasing temperature@xcite . however , since the concentration of slow bjerrum defects in water solutions is much higher ( @xmath9 at @xmath10 c ) than that of fast ionic ( @xmath11)@xcite , the investigations of the reorientation step of proton migration are necessary for the better understanding of the proton transport process . the goal of the present work is to study proton wire containing a finite number of water molecules by the use of quantum statistical mechanics methods which are extremely effective in the description of the collective nature of the proton transfer and in the quantum treatment of the light h nuclei @xcite . to describe correctly the proton transport process , we employ here two - stage proton transport model @xcite incorporating quantum effects such as proton tunneling and zero - point vibration energy . in earlier papers @xcite we applied the two - stage model to analyze the effect of coupling between protons and molecular group vibrations on proton conductivity in infinite h - bonded chains and proton - conducting planes . in particular , it was shown that the proton - lattice vibration interactions induce structural instabilities and charge ordering in system @xcite , whereas the grotthuss - type transport mechanism manifests itself in nontrivial temperature- and frequency dependences of the proton conductivity @xcite . in this work we analyze the influence of proton - proton correlations , comparing two different protonated chains containing one and two excess protons respectively . we find that the reaction of protonation of water chain is extremely sensitive to the reorientation energy barrier of proton motion and the barrier for the chain protonation . we show that the increase of the reorientation energy results in the drastic decrease of the proton charge density at the boundary between the chain and surrounding with consequent localization of protons near the inner water molecules . as appears from our modeling , the application of external electric field induces the step - like threshold - type effects with the ordering of proton charge and stabilization of bjerrum d - defects in the wire . we analyze the temperature dependence of proton polarization and d - defect concentration , and examine the role of the interplay between different factors ( such as orientation energy , external field and temperature ) in the dynamics of bjerrum defects . to model a proton wire , we consider a linear chain containing @xmath12 hydrogen bonds and @xmath13 molecular groups . the outer left ( @xmath14 ) and right ( @xmath15 ) molecular complexes mimic the surrounding of the proton wire and differ from the inner ( @xmath16 ) water molecules . the transport of an excess proton through the wire proceeds via the following two steps : + ( i ) proton can be transferred within a h - bond ( process shown by short arrows in fig . 1(a ) ) which is modelled by a simple double - well potential , with the corresponding energy barrier @xmath17 for the proton transfer between the two minima : @xmath18 where @xmath19(@xmath20 ) are the operators of the proton creation(annihilation ) in the position ( @xmath21 , @xmath22 ) ( the index @xmath23 denotes the left / right position for the proton within the h - bond ) ; + ( ii ) a water molecule together with covalently bonded hydrogen ion can be rotated , and this process causes the breaking of the h - bond and location of h@xmath0 between two another nearest water molecules of the wire ( process depicted by long arrows in fig . 1(a ) ) : @xmath24 where @xmath25 is the effective energy barrier for the proton hopping between the states @xmath26 and @xmath27 ( reorientation of the @xmath21-th molecular group together with proton ) . as is shown in @xcite by the computation of the proton mean - force potential , this transition between donor - acceptor and acceptor - donor states reverses the chain dipole moment and requires a substantial energy barrier about @xmath28 kcal / mol for the whole chain containing up to eight water molecules . besides the transport process , we incorporate the following two types of interactions between protons in the chain : + ( iii ) different short - range configurations of the protons near an inner water molecule as well as an outer molecular group can appear due to the different nature of bonding ( shorter covalent or longer h - bond ) . the energies of possible configurations ( shown in fig . 1(b ) ) are described by the following terms : @xmath29 the parts @xmath30 and @xmath31 describe the energies of the boundary proton configurations near the left and the right surrounding molecular groups ( we assume for simplicity @xmath32 and @xmath33 in the boundary configurations shown in the upper scheme of fig . the terms @xmath34 ( @xmath35 ) contain the configuration energies for the water molecules in the interior of the wire ( the lower part in fig . 1(b ) ) . here the proton occupancy operators @xmath36 ; + ( iv ) a strong repulsion between two nearest protons shared by two neighboring oxygens ( so called bjerrum d - defect ) with a repulsion energy @xmath37 is represented by the term : @xmath38 in our following analysis we use the value of @xmath39 kcal / mol corresponding to the energy of relaxed @xmath40-defect estimated in @xcite on the basis of quantum chemical calculations . to model a field exerted by the surrounding , we apply an external electric field of a strength @xmath41 to the proton wire , which is described by the following term @xmath42 where @xmath43 is the coordinate of the proton position ( @xmath21 , @xmath22 ) with respect to the center of the chain , and @xmath44 denotes the proton charge . in order to analyze the dynamics of the proton wire embedded in the surrounding under the influence of the field , as well as the effect of rotational motions of covalent groups with proton , we will focus our attention on the polarization of the proton wire defined here as @xmath45 where @xmath46 denotes the statistical average with respect to the system energy ( [ h_t]-[h_e ] ) . the average probabilities @xmath47 of occupation of the position ( @xmath21 , @xmath22 ) by proton describe the distribution of the proton charge in the wire , and thus is another very important characteristics to track the proton migration . to calculate _ exactly _ the above - mentioned statistical averages , we need to know the quantum energy levels determined by the energy ( [ h_t]-[h_e ] ) . this can be done by a mapping of the proton states ( @xmath21,@xmath22 ) on the multi - site basis @xmath48 . then , using the projection operators @xmath49 acting on the new basis @xmath50 we rewrite the system energy ( [ h_t]-[h_e ] ) in a convenient form ( see appendix ) : @xmath51 each term @xmath52 in ( [ ham ] ) corresponds exactly to @xmath53 protons in the chain ( @xmath54 ) . this means in fact that the mapping on the states @xmath50 allows to decompose the terms ( [ h_t]-[h_e ] ) and analyze the cases of different number of protons in the wire separately . the energy barrier for the protonation of the chain is described by the parameter @xmath55 which appears in ( [ ham ] ) after the decomposition procedure ( see appendix ) . as follows from the definition ( [ delta ] ) , @xmath55 is the difference between the energies of the proton attraction to the boundary and to the inner water molecules . as the pmf - studies of the h - bonded chain dynamics @xcite show that the inner h - bonds are stronger ( shorter o - o separation distances ) than the outer h - bonds , it is reasonable to consider below the case @xmath56 ( we take @xmath57 kcal / mol in our numerical calculations ) , when the proton is attracted to the surrounding and needs to overcome the boundary energy barrier @xmath58 to protonate the water chain . the parameter @xmath59 ( see fig . 1(b ) ) is related to the effective short - range interactions between the protons near the water molecule . it describes , in fact , the energy of the formation of the pair of ionic defects : i@xmath60=h@xmath1o@xmath0 ( @xmath61 ) and i@xmath62=oh@xmath2 ( @xmath63 ) from two water molecules at the dissociation reaction ( 2h@xmath64o @xmath65 h@xmath1o@xmath0+oh@xmath2 ) . since the value of @xmath66 is about @xmath67 kcal / mol @xcite and is more than twice as much as @xmath37 , we exclude in our following analysis an appearance of the pair of ionic defects in the system . due to the two types of motions we have two different contributions to the proton dipole moment : the orientational part @xmath68 and the transfer part @xmath69 where @xmath70 denotes the distance h - h between the two nearest proton positions of the double - well h - bond . in our calculations , we use the values @xmath71 d and @xmath72 d corresponding to the moderately strong h - bond with an o - o distance @xmath73 and the covalent o - h bond of a length @xmath74 . as the starting point , in this section we mimic the situation when one proton is moved towards the water chain embedded into the solvent . to examine the behavior of the protonated chain with @xmath12 h - bonds and @xmath75 excess proton , we consider @xmath76 part of the energy given by ( [ ham_decomp ] ) . since the zero - point vibration energy for protonated chains is larger than the potential energy barrier for the proton transfer between two shared oxygens @xcite , the quantum tunneling is not required for the intra - bond h@xmath0 transfer . thus , in our modelling we set @xmath77 . with this assumption , the energy levels of @xmath76 can be found _ exactly _ : @xmath78 where @xmath79 . to analyze the role of @xmath25 we consider first the case without external field ( @xmath80 ) . depending on the value of @xmath25 , two different regimes may be stabilized in the system . in the first _ small-@xmath25 regime _ , the two lowest energy levels @xmath81 correspond to the superposition of the two boundary states @xmath82 with the proton located in the surrounding near the left or the right outer molecular groups . in the second _ large-@xmath25 regime _ the proton is shared between the inner water molecules of the chain and the ground state of the system corresponds to the superposition of the states @xmath83 with the energies @xmath84 . the `` critical '' value @xmath85 separating these two regimes , reflects the transition of the proton from the surrounding to the states where the proton is shared by the chain water molecules , which corresponds to the protonation chemical reaction . 2 shows the variation of the average occupancies of proton sites with @xmath25 for the chains containing @xmath86 and @xmath87 h - bonds . for low temperatures ( see the case @xmath88 ) the boundary proton occupancies @xmath89 drop drastically to zero at @xmath90 , whereas the occupation numbers of the central positions @xmath91 increase up to the value @xmath92 , reflecting the redistribution of collectivized proton between the inner sites in the wire . it should be noted here that @xmath93 reflects the change of the ground state of the system and is determined as the solution of the equation @xmath94 at @xmath80 which does not depend on temperature . however , as all statistical averages , the average proton occupancies ( for example , of the states @xmath95 and @xmath96 where @xmath97 and @xmath98 are the diagonalized states corresponding to @xmath99 and @xmath100 respectively ) , are temperature dependent . thus the value of @xmath101 where @xmath102 , for @xmath103 is not equal to @xmath93 ( see fig . 2 , case @xmath104 ) . this difference shows that the inner proton states ( [ inner ] ) are stabilized already at lower @xmath105 , although the occupancies of the inner positions at @xmath90 are still slightly lower then of the outer due to the temperature - induced fluctuations . as @xmath106 , the fluctuations decrease and @xmath107 . the effect of the proton localization inside the chain is supported by the results reported in @xcite showing that in h - bonded finite chains , the excess charge is best solvated by the central h - bonds . however , as results from the presented above analysis , the effect of proton localization is drastically influenced by the competition between two different tendencies : ( i ) for small @xmath25 , the proton is located near the surrounding / wire interface due to the nonzero protonation barrier @xmath55 ; ( ii ) to overcome the barrier between the surrounding and the wire , the reorientation energy should be sufficiently large ( @xmath108 ) in order to stabilize the inner proton configurations . these conclusions show that in general , these two different factors ( interface barrier and orientations ) can be rate - limiting for the charge translocation and proton conductivity of the wire . as was shown in @xcite , the effective reorientation barrier can be influenced by temperature factor or applied voltage ( for example , the reorientation rate of the wire decreases exponentially with @xmath109 decrease ) . thus , one can also expect that the increase of the temperature can result in the lower orientation barrier @xmath25 , delocalization of proton and consequently in higher values for the proton conductivity through the channel . however , more detailed theoretical analysis is needed to understand better the role of the interface in the behavior of the conductivity . we turn now to an analysis of the proton translocation directed by the external field ( the case @xmath110 ) . 3 shows the field - dependences of @xmath111 and @xmath47 for the chain containing @xmath86 h - bonds . we note that the behavior in the first small- and in the second large-@xmath25 regime differs drastically . in the small-@xmath25 regime the polarization increases smoothly with @xmath41 approaching finally its maximal saturation value @xmath112 ( fig . 3(a ) , inset ) . in contrast to this , in the large-@xmath25 regime the field dependence is rather nontrivial : first , for @xmath113 , @xmath111 changes very slowly , and than , at @xmath114 a strong increase of @xmath111 to @xmath115 is observed in fig . this rapid step - like change of the proton polarization reflects the threshold - type effect where the threshold electric field value at low @xmath109 is given by @xmath116 and does not depend on the chain size @xmath12 ( @xmath117 v / cm for the chains with @xmath86 as can be observed in fig . as we see in fig . 3(b ) , the proton charge translocation under the influence of the field in this case proceeds not smoothly , but has a step - like character . as results from ( [ thresh ] ) , the threshold value @xmath118 ( which is needed to overcome a barrier for pumping between the inner localized states ( [ inner ] ) and the boundary state @xmath119 in the direction of field ) increases for larger @xmath25 ( fig . 3(a ) , the cases with @xmath120 kcal / mol and @xmath121 kcal / mol ) . this implies that the conductivity of protonated chains can drop with an increasing @xmath25 which can occur in system for example due to the temperature - induced fluctuations of @xmath25 . however , as was shown in @xcite , the reorientation rate increases at increased voltage , which corresponds in our case to the smaller values of @xmath25 for @xmath110 . thus , we can expect that the external field - induced lowering of the orientation barrier for the proton translocation results in the increase of the proton conductivity in the wire . the drastic change of @xmath111 at @xmath122 leads to a strong qualitative difference in the temperature shapes of the polarization profiles shown in fig . 4(a ) for the large-@xmath25 regime . for @xmath113 and @xmath123 , the excess proton is located in the central sites , and the increase of @xmath109 results in a disorder - induced transfer of the proton from the inner positions to the chain boundary giving the increase of @xmath111 at @xmath124 k as compared to @xmath125 k ( fig . 4(a ) , cases @xmath126 v / cm and @xmath127 v / cm ) . as the proton is located in the outer state @xmath128 for @xmath129 ( corresponding to @xmath130 for @xmath106 ) , the dominant effect of @xmath109 in this case is the the disorder - induced proton redistribution between all sites in the chain leading to the lowering of total polarization ( fig . 4(a ) , cases @xmath131 v / cm and @xmath132 v / cm ) . the profiles of the polarization for @xmath25 below @xmath93 shown in fig . 4(b ) appear to be very similar to the high - field profiles in fig . since for @xmath133 the proton is located in the outer states ( [ outer ] ) near the chain boundary already at low @xmath41 , the increase of @xmath109 suppresses the polarization due to the increasing proton disorder . in order to examine the influence of proton correlations , we consider next the translocation of two excess protons in the wire which is described by the part @xmath134 of the total energy ( [ ham_decomp ] ) . since the presence of two protons in wire may lead to the formation of bjerrum d - defect , the energy @xmath134 for the chain with @xmath86 h - bonds given by the expression ( [ ham22 ] ) , contains the terms with the energy @xmath37 of the repulsion between two nearest - neighboring protons . analogously to the 1-proton wire , we analyze first the behavior of the chain without the electric field . for @xmath86 and @xmath135 the energy levels found from ( [ ham22 ] ) have the following form : @xmath136 and correspond to the following states of the wire : @xmath137 with @xmath138 . to study the influence of @xmath25 , we analyze ( [ u - levels ] ) and ( [ u - states ] ) for @xmath80 assuming @xmath139 and neglecting in this way by the formation of ionic defects described by the configuration @xmath140 . similarly to the 1-proton wire , the two different regimes can exist in the system depending on the value of the reorientation energy . in the first _ small-@xmath25 regime _ ( for @xmath141 ) , each proton is located near the outer molecular group and the state @xmath142 has the lowest energy @xmath143 . as @xmath25 increases and approaches the `` critical '' value @xmath144 , the transition to the _ large-@xmath25 regime _ occurs . in this regime ( for @xmath145 ) the lowest energy levels @xmath146 ( @xmath147 ) correspond to the states @xmath148 and @xmath149 in ( [ u - states ] ) , with one proton located in the interior of the wire . however , in contrast to section [ 1-prot ] , the transition between these two regimes is @xmath37-dependent , because @xmath144 contains the energy of the d - defect @xmath37 . 5(a ) shows the variation of the proton occupation numbers @xmath47 with @xmath25 for @xmath150 kcal / mol and @xmath151 kcal / mol ( plotted in the inset ) . the `` critical '' value @xmath152 kcal / mol for the repulsion energy @xmath150 kcal / mol is larger as compared with @xmath153 kcal / mol for @xmath151 kcal / mol . so far as the repulsion energy @xmath37 is significant , @xmath154 . however , as @xmath155 , @xmath144 approaches the `` critical '' value @xmath93 for the one - proton case . as we can see from ( [ u - states ] ) and ( [ sp2 ] ) , the ground states @xmath148 and @xmath149 in the large-@xmath25 regime are represented by the superpositions of the normal configurations @xmath156 and @xmath157 and the states @xmath158 and @xmath159 containing the d - defect . thus the transition to the large-@xmath25 regime stabilize d - defects inside the chain . the formation of the d - defect states is clearly observed in fig . 5(a ) where the occupation numbers of the d - defect states @xmath158 and @xmath159 significantly increase for @xmath145 . analogously to the one - proton case , the temperature fluctuations lead to the slight temperature - induced increase of the value @xmath160 ( corresponding to @xmath161 ) , as compared to @xmath144 where the states @xmath148 and @xmath149 are already stabilized . in fact , for @xmath162 the states corresponding to protonated chains with significant concentration of d - defects prevail ( @xmath163 ) , whereas the states with the protons located at the boundaries in the surrounding dominate for @xmath164 ( @xmath165 ) . for small @xmath166 , @xmath144 can be given by @xmath167\ ] ] since the line @xmath160 found from ( [ omr*2_t ] ) , is tilted with respect to @xmath109 in the state diagrams ( @xmath109 , @xmath160 ) ( fig . 5(b ) ) , the effect of temperature for the chain in the large-@xmath25 regime is crucial : with the increasing @xmath109 , the temperature fluctuations can destroy the d - defects and redistribute the proton charge between the other chain sites . see for example the case of @xmath168 kcal / mol and @xmath150 kcal / mol plotted in fig . 5(b ) where the d - defects annihilate at @xmath169 k. as @xmath170 , the weight constants for the d - defect states in ( [ u - states ] ) become smaller : @xmath171 . thus , the contribution of the d - defect - states to the stable wire configuration goes down as @xmath172 for the stronger proton repulsion @xmath37 ( see for the comparison @xmath173 for different @xmath37 plotted in fig . 5(a ) ) . the fact that the variation of temperature can lead to formation or annihilation of the d - defects is also observed in the @xmath109-dependence of the proton polarization . note that especially for weak external field @xmath41 , the behavior of @xmath174 in the small-@xmath25 ( fig . 6(a ) , @xmath175 ) and in the large-@xmath25 regime ( fig . 6(b ) , @xmath176 ) is drastically different . in the first case , at low @xmath109 , the predominantly occupied symmetric ground state @xmath177 has the total polarization @xmath178 . however , with @xmath109 increasing , protons tend to occupy the excited states with non - symmetric charge distribution that results in an increase of @xmath111 . 7(a ) demonstrates that the population of all excited states , in particular those containing d - defects ( fig . 7(a ) , inset ) , grows with @xmath109 . although the concentration of the d - defect states @xmath179 and @xmath180 is of 3 - 4 orders lower than that of the normal states ( see fig . 7(a ) , inset ) , it significantly increases up to 1 - 2 orders with the temperature increase from @xmath125 to @xmath124 k. in contrast to this , in the large-@xmath25 regime the protons and stable d - defects migrate in the direction of applied field @xmath41 for @xmath106 giving a non - zero @xmath111 ( fig . as @xmath109 increases , the population of the excited non - defect state with the protons redistributed at the boundaries grows ( see fig . 7(b ) ) which gives the lower chain polarization . the d - defects , located near the end of the chain for finite @xmath41 ( @xmath181 ) , are redistributed between another chain positions with @xmath109 , which is observed in fig . 7(b ) showing a decrease of @xmath179- together with a slight increase of the @xmath180-state population at @xmath182 k as compared to lower temperatures . we study now the electric field effect in correlated chains . 8 shows the variation of polarization and redistribution of protons with increasing @xmath41 . consider first the small-@xmath25 regime . in distinct to the 1-proton wire , where the polarization increases smoothly to its maximal value @xmath183 ( fig . 3(a ) , inset ) , we observe here two different threshold effects . the first transition from the state @xmath184 of ( [ sp2])(the ground state of the wire in the small-@xmath25 regime at @xmath80 ) to the state @xmath157 ( where both of the protons are ordered in the right position of each h - bond in the direction of the field ) occurs at the threshold field value @xmath185 the distribution of the occupation probabilities @xmath186 and @xmath187 for the states @xmath184 and @xmath157 is plotted in fig . we observe at @xmath188 the abrupt increase of @xmath187 , while at the same field value @xmath189 drops to zero . furthermore , we conclude from ( [ e1 ] ) that the value @xmath190 lowers with the number @xmath12 of the water molecules in the chain . this effect can be observed in fig . 9 where the jumps of the polarization are plotted for different @xmath12 . finally , for very long water chain ( @xmath191 ) @xmath192 . in contrast to the strong @xmath12-dependence of @xmath190 , the second threshold effect appears at @xmath193 essentially due to the proton correlations and does not depend on the chain length . the strong increase of @xmath111 at @xmath194 shown in fig . 8(a ) and fig . 9 is related to the second drastic redistribution of the proton charge in the wire . as can be observed in fig . 8(b ) , at @xmath194 the occupation probability @xmath195 of the d - defect - state @xmath159 drastically increases to 1 , whereas @xmath187 drops to zero . thus , as resulted from our model , the formation of d - defect in external electric field has a step - like character proceeding via the threshold mechanism . in the large-@xmath25 regime , where the protons are stabilized at the inner water molecules already at @xmath80 , the first threshold phenomenon at @xmath188 , observed for the small-@xmath25 case , does not occur . however , the transition at @xmath194 with the increase of the d - defect concentration appears in this regime similarly to the regime of small @xmath25 , that can be observed in the @xmath111-profile for @xmath196 kcal / mol shown in fig . note that the effect of the increasing double occupancy due to membrane potentials has been observed in the current / concentration plots in gramicidin channels @xcite , thus supporting our main conclusions about the role of the external electric field . the discussed above formation of the d - defects for @xmath123 in the high electric field results in the increase of @xmath111 for lower temperatures as shown in fig . basically , the essential effect of @xmath109 observed in the @xmath174-profiles in fig . 6 , is the suppression of the total polarization due to proton disorder however , the shapes of the polarization in fig . 6(a ) are drastically different for @xmath197 and @xmath198 . for low fields ( @xmath197 ) the polarization first increases ( reflecting the fluctuation - induced expansion of proton charge from the outer symmetric positions @xmath199 with @xmath178 to the inner positions of the chain accompanied by the formation / annihilation of d - defects ) , and then smoothly decreases due to the disorder effect . in contrast to this , as the increasing electric field induces the step - like formation of d - defects in the small-@xmath25 regime , the temperature behavior of @xmath111 in this case is similar to the the large-@xmath25 case ( compare fig . 6(a ) and fig . 6(b ) with @xmath200 v / cm ) showing the smooth disorder - induced decrease of @xmath111 with @xmath109 . we also note that the stable configurations with double proton occupancy require the additional reorientation steps for the proton translocation and can result in the smaller values for the proton conductivity . this fact has been observed in the measurements of the proton conductance in two different stereoisomers of the gramicidin @xcite , thus supporting a possibility of stabilization of the d - defect states in proton wires . in this work we studied the process of proton translocation in 1d - chains mimicking protonated water channels embedded in surrounding . we have analyzed the role of the reorientation motion of protons , as well as the effect of electric field and proton correlations on the chain dynamics . we have shown that the increase of the reorientation energy results in the transition to the large-@xmath25 regime characterized by the transfer of the proton charge from the surrounding to the inner water molecules in the chain . the process of proton migration along the chain in the external electric field has the step - like character leading to the appearance of the electric field threshold - type phenomena with drastic redistribution of proton charge . the correlations between protons in the chain increase the `` critical '' reorientation energy @xmath93 necessary for the transition into the large-@xmath25 regime , where the protonated chain contains a finite concentration of bjerrum defects . the temperature fluctuations induce a slight increase of @xmath201 separating the state with the protons located in surrounding near the outer groups , and the protonated state with d - defects . for the correlated chains , this temperature dependence of the `` critical '' reorientation energy can lead to the redistribution of proton charge and annihilation of d - defects with increasing @xmath109 . the electric field applied to the correlated chains induces first the formation of ordered dipole structures for the lower @xmath41 values , and than , with the further @xmath41 increase , the stabilization of the states with the bjerrum d - defects . generally , the increase of temperature suppresses the total polarization in the chain due to the increasing disorder . however , especially in the low electric fields , the shapes of the temperature profiles of the polarization appear to be drastically different in the small- and large-@xmath25 regimes demonstrating the complex interplay between the reorientation energy and temperature . finally , as follows from our analysis , the following factors strongly influence the formation of bjerrum defects : ( i ) the high electric fields can form the defects and pump them in the chain in the direction of field ; ( ii ) the increase of the orientational energy barrier leads to the stabilization of d - defects ; ( iii ) the increase of temperature in the large-@xmath25 regime results in the formation / annihilation of d - defects , whereas for small @xmath25 the concentration of d - defects significantly increases up to 1 - 2 orders at the room temperatures as compared to the low @xmath202 k. we demonstrate below the procedure of the mapping in the system with @xmath86 h - bonds on the multi - site states . for @xmath86 the basis @xmath50 includes @xmath203 states @xmath204 : @xmath205 and @xmath206 : @xmath207 where the expectation numbers @xmath208 can be found using the usual antisymmetric rules for fermi - operators @xcite . specifically , for the case @xmath86 the expressions ( [ ferm_hubb ] ) yield : @xmath209 using the relations ( [ c22 ] ) and the fact that @xmath210 ( due to the orthogonality of the states @xmath50 ) , we decompose ( [ h_t]-[h_e ] ) in terms of @xmath211 operators into the following 5 terms : @xmath212 where @xmath213 @xmath214 @xmath215 since the parameter @xmath216 in ( [ ham21])-([ham23 ] ) is the difference between proton configuration energies at the boundary ( @xmath14 or @xmath15 surrounding molecular groups ) , and at the inner ( @xmath217 ) water molecule , it describes , in fact , the energy barrier for the protonation of the water chain . for our analysis @xmath55 has the key importance , because the other energy constants in ( [ ham21])-([ham23 ] ) @xmath218 which appear due to the boundary effects , are independent of the proton location in the wire and thus do not influence the statistical characteristics like ( [ pp ] ) .

we present the results of the modeling of proton translocation in finite h - bonded chains in the framework of two - stage proton transport model . we explore the influence of reorientation motion of protons , as well as the effect of electric field and proton correlations on system dynamics . an increase of the reorientation energy results in the transition of proton charge from the surrounding to the inner water molecules in the chain . proton migration along the chain in an external electric field has a step - like character , proceeding with the occurrence of electric field threshold - type effects and drastic redistribution of proton charge . electric field applied to correlated chains induces first a formation of ordered dipole structures for lower field strength , and than , with a further field strength increase , a stabilization of states with bjerrum d - defects . we analyze the main factors responsible for the formation / annihilation of bjerrum defects showing the strong influence of the complex interplay between reorientation energy , electric field and temperature in the dynamics of proton wire .

in this paper , we will use probabilistic methods to solve the dirichlet boundary value problem for the semilinear second order elliptic pde of the following form : @xmath0 where @xmath1 is a bounded domain in @xmath2 . the operator @xmath3 is given by @xmath4 where @xmath5 ( @xmath6 ) is a measurable , symmetric matrix - valued function satisfying a uniform elliptic condition , @xmath7 , @xmath8 and @xmath9 are merely measurable functions belonging to some @xmath10 spaces , and @xmath11 is a nonlinear function . the operator @xmath3 is rigorously determined by the following quadratic form : @xmath12 we refer readers to @xcite and @xcite for details of the operator @xmath3 . probabilistic approaches to boundary value problems of second order differential operators have been adopted by many people . the earlier work went back as early as 1944 in @xcite . see the books @xcite and references therein . if @xmath13 ( i.e. , the linear case ) , and moreover @xmath14 , the solution @xmath15 to problem ( [ 0.1 ] ) can be solved by a feynman kac formula @xmath16 \qquad \mbox{for } x\in d,\ ] ] where @xmath17 , @xmath18 is the diffusion process associated with the infinitesimal generator @xmath19 @xmath20 is the first exit time of the diffusion process @xmath21 from the domain @xmath1 . very general results are obtained in the paper @xcite for this case . when @xmath22 , `` @xmath23 '' in ( [ 0.0 ] ) is just a formal writing because the divergence does not really exist for the merely measurable vector field @xmath24 . it should be interpreted in the distributional sense . it is exactly due to the nondifferentiability of @xmath24 , all the previous known probabilistic methods in solving the elliptic boundary value problems such as those in @xcite and @xcite could not be applied . we stress that the lower order term @xmath23 can not be handled by girsanov transform or feynman kac transform either . in a recent work @xcite , we show that the term @xmath24 in fact can be tackled by the time - reversal of girsanov transform from the first exit time @xmath20 from @xmath1 by the symmetric diffusion @xmath25 associated with @xmath26 , the symmetric part of @xmath3 . the solution to equation ( [ 0.1 ] ) ( when @xmath13 ) is given by @xmath27\\[-8pt ] & & \hphantom{e_x^0 \biggl[\varphi(x^0(\tau_d ) ) \exp\biggl\ { } { } -\frac{1}{2}\int_0^{\tau_d}(b-\hat{b})a^{-1}(b-\hat{b})^{\ast}(x^0(s))\,ds\nonumber \\ & & \hspace*{186pt } { } + \int_0^{\tau_d}q(x^0(s))\,ds \biggr\ } \biggr],\nonumber\end{aligned}\ ] ] where @xmath28 is the martingale part of the diffusion @xmath25 , @xmath29 denotes the reverse operator , and @xmath30 stands for the inner product in @xmath2 . nonlinear elliptic pdes [ i.e. , @xmath31 in ( [ 0.1 ] ) ] are generally very hard to solve . one can not expect explicit expressions for the solutions . however , in recent years backward stochastic differential equations ( bsdes ) have been used effectively to solve certain nonlinear pdes . the general approach is to represent the solution of the nonlinear equation ( [ 0.1 ] ) as the solution of certain bsdes associated with the diffusion process generated by the linear operator @xmath3 . but so far , only the cases where @xmath14 and @xmath32 being bounded were considered . the main difficulty for treating the general operator @xmath3 in ( [ 0.0 ] ) with @xmath22 , @xmath33 is that there are no associated diffusion processes anymore . the mentioned methods used so far in the literature ceased to work . our approach is to transform the problem ( [ 0.1 ] ) to a similar problem for which the operator @xmath3 does not have the `` bad '' term @xmath24 . see below for detailed description . there exist many papers on bsdes and their applications to nonlinear pdes . we mention some related earlier results . the first result on probabilistic interpretation for solutions of semilinear parabolic pde s was obtained by peng in @xcite and subsequently in @xcite . in @xcite , darling and pardoux obtained a viscosity solution to the dirichlet problem for a class of semilinear elliptic pdes ( through bsdes with random terminal time ) for which the linear operator @xmath3 is of the form @xmath34 where @xmath35 and @xmath36 . bsdes associated with dirichlet processes and weak solutions of semi - linear parabolic pdes were considered by lejay in @xcite where the linear operator @xmath3 is assumed to be @xmath37 for bounded coefficients @xmath38 and @xmath32 . bsdes associated with symmetric markov processes and weak solutions of semi - linear parabolic pdes were studied by bally , pardoux and stoica in @xcite where the linear operator @xmath3 is assumed to be symmetric with respect to some measure @xmath39 . bsdes and solutions of semi - linear parabolic pdes were also considered by rozkosz in @xcite for the linear operator @xmath3 of the form @xmath40 now we describe the contents of this paper in more details . our strategy is to transform the problem ( [ 0.1 ] ) by a kind of @xmath41-transform to a problem of a similar kind , but with an operator @xmath3 that does not have the `` bad '' term @xmath24 . the first step will be to solve ( [ 0.1 ] ) assuming @xmath14 . in section [ sec2 ] , we introduce the feller diffusion process @xmath42 whose infinitesimal generator is given by @xmath43 in general , @xmath17 , @xmath18 is not a semimartingale . but it has a nice martingale part @xmath44 , @xmath18 . in this section , we prove a martingale representation theorem for the martingale part @xmath44 , which is crucial for the study of bsdes in subsequent sections . in section [ sec3 ] , we solve a class of bsdes associated with the martingale part @xmath44 , @xmath18 : @xmath45 the random coefficient @xmath46 satisfies a certain monotonicity condition which is particularly fulfilled in the situation we are interested . the bsdes with deterministic terminal time were solved first and then the bsdes with random terminal time were studied . in section [ sec4 ] , we consider the dirichelt problem for the second order differential operator @xmath47 where @xmath48 for some @xmath49 and @xmath50 for some @xmath51 . we first solve the linear problem with a given function @xmath52 @xmath53 and then the nonlinear problem @xmath54 with the help of bsdes . finally , in section [ sec5 ] , we study the dirichlet problem @xmath55 where @xmath3 is a general second order differential operator given in ( [ 0.0 ] ) . we apply a transform we introduced in @xcite to transform the above problem to a problem like ( [ 0.8 ] ) and then a reverse transformation will solve the final problem . let @xmath3 be an elliptic operator of the following general form : @xmath56 where @xmath57 ( @xmath6 ) is a measurable , symmetric matrix - valued function which satisfies the uniform elliptic condition @xmath58 , @xmath59 and @xmath60 are measurable functions which could be singular and such that @xmath61 for some @xmath62 . here @xmath1 is a bounded domain in @xmath2 whose boundary is regular , that is , for every @xmath63 , @xmath64 , where @xmath65 is the first exit time of a standard brownian motion started at @xmath66 from the domain @xmath1 . let @xmath67 be a measurable nonlinear function . consider the following nonlinear dirichlet boundary value problem : @xmath68 let @xmath69 denote the usual sobolev space of order one : @xmath70 we say that @xmath71 is a continuous , weak solution of ( [ 1.01 ] ) if : * for any @xmath72 , @xmath73 * @xmath74 , * @xmath75 , @xmath76 . next we introduce two diffusion processes which will be used later . let @xmath77 be the feller diffusion process whose infinitesimal generator is given by @xmath78 where @xmath79 is the completed , minimal admissible filtration generated by @xmath80 , @xmath81 . the associated nonsymmetric , semi - dirichlet form with @xmath82 is defined by @xmath83\\[-8pt ] & = & { 1\over2}\sum_{i , j=1}^{d}\int_{r^d}a_{ij}(x)\,{\partial u\over { \partial x_i}}\,{\partial v\over{\partial x_j}}\,dx-\sum_{i=1}^{d}\int_{r^d}b_i(x)\,{\partial u\over{\partial x_i}}\,v(x)\,dx.\nonumber\end{aligned}\ ] ] the process @xmath17 , @xmath18 is not a semimartingale in general . however , it is known ( see , e.g. , @xcite and @xcite ) that the following fukushima s decomposition holds : @xmath84 where @xmath44 is a continuous square integrable martingale with sharp bracket being given by @xmath85 and @xmath86 is a continuous process of zero quadratic variation . later we also write @xmath87 , @xmath88 to emphasize the dependence on the initial value @xmath66 . let @xmath89 denote the space of square integrable martingales w.r.t . the filtration @xmath79 , @xmath18 . the following result is a martingale representation theorem whose proof is a modification of that of theorem a.3.20 in @xcite . it will play an important role in our study of the backward stochastic differential equations associated with the martingale part @xmath90 . for any @xmath91 , there exist predictable processes @xmath92 such that @xmath93 it is sufficient to prove ( [ 1.5 ] ) for @xmath94 , where @xmath95 is an arbitrary , but fixed constant @xmath95 . recall that @xmath89 is a hilbert space w.r.t . the inner product @xmath96 $ ] , where @xmath97 denotes the sharp bracket of @xmath98 and @xmath99 . let @xmath100 denote the subspace of square integrable martingales of the form ( [ 1.5 ] ) . let @xmath101 be the resolvent operators of the diffusion process @xmath17 , . fix any @xmath102 , we know that @xmath103 and @xmath104 . moreover , it follows from @xcite and @xcite that @xmath105 hence , @xmath106 is a bounded martingale that belongs to @xmath100 . the theorem will be proved if we can show that @xmath107 . take @xmath108 . since @xmath109 is stable under stopping , by lemma 2 in chapter iv in @xcite , we deduce @xmath110 for all @xmath111 . in particular , @xmath112 . from here , we can follow the same proof of theorem a.3.20 in @xcite to conclude @xmath113 . we will denote by @xmath114 the diffusion process generated by @xmath115 the corresponding fukushima s decomposition is written as @xmath116 for @xmath117 , the fukushima s decomposition for the dirichlet process @xmath118 reads as @xmath119 where @xmath120 , @xmath121 is a continuous process of zero energy ( the zero energy part ) . see @xcite for details of symmetric markov processes . let @xmath122 be the probability space carrying the diffusion process @xmath17 described in section [ sec2 ] . recall @xmath44 , @xmath18 is the martingale part of @xmath123 . in this section , we will study backward stochastic differential equations ( bsdes ) with singular coefficients associated with the martingale part @xmath44 . let @xmath124\times r\times r^d\times\omega \rightarrow r$ ] be a given progressively measurable function . for simplicity , we omit the random parameter @xmath125 . assume that @xmath126 is continuous in @xmath127 and satisfies : * @xmath128 , * @xmath129 , * @xmath130 , where @xmath131 , @xmath132 are a progressively measurable stochastic process and @xmath133 is a constant . let @xmath134 . let @xmath135 be the constant defined in ( [ 1.0 ] ) . [ thm3.1 ] assume @xmath136<\infty$ ] , @xmath137<\infty$ ] and @xmath138<\infty.\ ] ] then , there exists a unique ( @xmath79-adapted ) solution @xmath139 to the following bsde : @xmath140 where @xmath141 . we first prove the uniqueness . set @xmath142 . suppose @xmath143 and @xmath144 are two solutions to equation ( [ 2.1 ] ) . then @xmath145\\[-8pt ] & & \qquad\quad { } + 2\bigl(y^1(t)-y^2(t)\bigr)\langle z^1(t)-z^2(t),dm(t)\rangle \nonumber\\ & & \qquad\quad { } + \bigl\langle a(x(t))\bigl(z^1(t)-z^2(t)\bigr ) , z^1(t)-z^2(t)\bigr\rangle \,dt.\nonumber\end{aligned}\ ] ] by the chain rule , using the assumptions ( a.1 ) , ( a.2 ) and young s inequality , we get @xmath146 take expectation in above inequality to get @xmath147 \leq c_{\lambda}\int_t^te\bigl [ e^{\int_0^sd(u)\,du}|y^1(s)-y^2(s)|^2\bigr]\,ds.\ ] ] by gronwall s inequality , we conclude @xmath148 and hence @xmath149 by ( [ 2.3 ] ) . next , we prove the existence . take an even , nonnegative function @xmath150 with @xmath151 . define @xmath152 where @xmath153 . since @xmath126 is continuous in @xmath127 , it follows that @xmath154 as @xmath155 . furthermore , it is easy to see that for every @xmath156 , @xmath157 for some constant @xmath158 . consider the following bsde : @xmath159 in view of ( [ 2.4 ] ) and the assumptions ( a.2 ) , ( a.3 ) , it is known ( e.g. , @xcite ) that the above equation admits a unique solution @xmath160 . our aim now is to show that there exists a convergent subsequence @xmath161 . to this end , we need some estimates . applying it s formula , in view of assumptions ( a.1)(a.3 ) it follows that @xmath162 take expectation in ( [ 2.6 ] ) to obtain @xmath163+\frac{1}{2}e\biggl[\int_t^te^{\int _ 0^sd(u)\,du}\langle a(x(s))z_n(s ) , z_n(s)\rangle\,ds \biggr]\nonumber\\ & & \qquad \leq e\bigl[|\xi|^2e^{\int_0^td(s)\,ds}\bigr]+c_{\lambda}\int_t^t e\bigl[e^{\int_0^sd(u)\,du}y_n^2(s)\bigr]\,ds\\ & & \qquad \quad { } + e\biggl[\int_t^t e^{\int_0^sd(u)\,du}|f(s,0,0)|^2\,ds\biggr].\nonumber\end{aligned}\ ] ] gronwall s inequality yields @xmath164\nonumber\\[-8pt]\\[-8pt ] & & \qquad \leq c\biggl\{e\bigl[|\xi|^2e^{\int_0^td(s)\,ds}\bigr]+e\biggl[\int_0^t e^{\int_0^sd(u)\,du}|f(s,0,0)|^2\,ds\biggr]\biggr\}\nonumber\end{aligned}\ ] ] and also @xmath165<\infty.\ ] ] moreover , ( [ 2.6])([2.9 ] ) further imply that there exists some constant @xmath166 such that @xmath167\nonumber\\ & & \qquad \leq c+ce\biggl[\sup_{0\leq t\leq t}\int_0^te^{\int_0^sd(u)\,du}y_n(s)\langle z_n(s),dm(s)\rangle\biggr]\nonumber\\ & & \qquad \leq c+ce \biggl [ \biggl ( \int_0^te^{2\int_0^sd(u)\,du}y_n^2(s)\langle a(x(s))z_n(s ) , z_n(s)\rangle\,ds \biggr)^{{1/2 } } \biggr]\nonumber\\ & & \qquad \leq c+ce \biggl[\sup_{0\leq s\leq t}\bigl(e^{{(1/2)}\int_0^sd(u)\,du}|y_n(s)|\bigr)\nonumber\\ & & \qquad \quad \hphantom{c+ce \biggl [ } { } \times \biggl ( \int_0^te^{\int_0^sd(u)\,du}\langle a(x(s))z_n(s ) , z_n(s)\rangle\,ds \biggr)^{{1/2 } } \biggr]\\ & & \qquad \leq c+\frac{1}{2}e \bigl[\sup_{0\leq s\leq t}\bigl(e^{\int_0^sd(u)\,du}y_n^2(s)\bigr ) \bigr]\nonumber\\ & & \qquad \quad { } + c_1e \biggl [ \int_0^te^{\int_0^sd(u)\,du}\langle a(x(s))z_n(s ) , z_n(s)\rangle\,ds \biggr].\nonumber\end{aligned}\ ] ] in view of ( [ 2.9 ] ) , this yields @xmath168<\infty.\ ] ] by ( [ 2.9 ] ) and ( [ 2.11 ] ) , we can extract a subsequence @xmath169 such that @xmath170 converges to some @xmath171 in @xmath172)$ ] equipped with the weak star topology and @xmath173 converges weakly to some @xmath174 in @xmath175 , where @xmath176\times\omega$ ] . observe that @xmath177\\[-8pt ] & & \qquad \quad { } -\frac{1}{2}\int_t^te^{{(1/2)}\int_0^sd(u)\,du}y_{n_k}(s)d(s)\,ds\nonumber \\ & & \qquad \quad { } - \int_t^te^{{(1/2)}\int_0^sd(u)\,du}\langle z_{n_k}(s),dm(s)\rangle.\nonumber\end{aligned}\ ] ] letting @xmath178 in ( [ 2.12 ] ) , using the monotonicity of @xmath126 , following the same arguments as that in the proof of proposition 2.3 in darling and pardoux in @xcite , we can show that the limit @xmath179 satisfies @xmath180 set @xmath181 an application of it s formula yields that @xmath182 namely , @xmath139 is a solution to the backward equation ( [ 2.1 ] ) . the proof is complete . let @xmath183 satisfy ( a.1)(a.3 ) in section [ sec3.1 ] . in this subsection , set @xmath184 . the following result provides existence and uniqueness for bsdes with random terminal time . let @xmath185 be a stopping time . suppose @xmath186 is @xmath187-measurable . [ thm3.2 ] assume @xmath188<\infty$ ] , @xmath189<\infty$ ] and @xmath190<\infty,\ ] ] for some @xmath191 , where @xmath135 is the constant appeared in ( 2.1 ) . then , there exists a unique solution @xmath139 to the bsde @xmath192 furthermore , the solution @xmath139 satisfies @xmath193<\infty,\qquad e \biggl[\int_0^{\tau}e^{\int_0^{s}d(u)\,du}|z(s)|^2\,ds \biggr]<\infty,\hspace*{-35pt}\ ] ] and @xmath194<\infty.\ ] ] after the preparation of theorem [ thm3.1 ] , the proof of this theorem is similar to that of theorem 3.4 in @xcite , where @xmath195 , @xmath133 were both assumed to be constants . we only give a sketch highlighting the differences . for every @xmath196 , from theorem [ thm3.1 ] we know that the following bsde has a unique solution @xmath197 on @xmath198 : @xmath199+\int_{\tau\wedge t}^{\tau\wedge n}f(s , y_n(s ) , z_n(s))\,ds-\int_{\tau\wedge t}^{\tau\wedge n}\langle z_n(s ) , dm(s)\rangle.\hspace*{-35pt}\ ] ] extend the definition of @xmath197 to all @xmath18 by setting @xmath200,\qquad z_n(t)=0\qquad \mbox{for } t\geq n.\ ] ] then the extended @xmath197 satisfies a bsde similar to ( [ 2.18 ] ) with @xmath126 replaced by @xmath201 . let @xmath202 . by it s formula , we have @xmath203-e[\xi & & \qquad \quad { } -\int_{t\wedge\tau}^{n\wedge\tau } e^{\int_0^{s\wedge \tau}d(u)\,du}|y_n(s\wedge\tau)-y_m(s\wedge\tau)|^2d(s)\,ds\nonumber\hspace*{-35pt}\\ & & \qquad \quad { } + 2\int_{t\wedge \tau}^{n\wedge\tau}e^{\int_0^{s\wedge\tau}d(u)\,du}\bigl(y_n(s\wedge \tau)-y_m(s\wedge\tau)\bigr)\nonumber\hspace*{-35pt}\\ & & \qquad \quad\hphantom { { } + 2\int_{t\wedge\tau}^{n\wedge\tau } } { } \times \bigl ( f\bigl(s , y_n(s\wedge\tau ) , z_n(s\wedge \tau)\bigr)\nonumber\hspace*{-35pt } \\ & & \qquad \quad\hphantom { { } + 2\int_{t\wedge\tau}^{n\wedge\tau}\times \bigl ( } { } -f\bigl(s , y_m(s\wedge\tau ) , z_m(s\wedge\tau)\bigr ) \bigr)\,ds\nonumber\hspace*{-35pt}\\ & & \qquad \quad { } + 2\int_{m\wedge \tau}^{n\wedge\tau}e^{\int_0^{s\wedge\tau}d(u)\,du}\bigl(y_n(s\wedge \tau)-y_m(s\wedge\tau)\bigr)\nonumber \hspace*{-35pt}\\ & & \quad \qquad\hphantom{{}+2\int_{m\wedge\tau}^{n\wedge\tau } } { } \times f\bigl(s , y_m(s\wedge\tau ) , z_m(s\wedge \tau)\bigr)\,ds\nonumber\hspace*{-35pt}\\ & & \qquad \quad { } -2\int_{t\wedge \tau}^{n\wedge\tau}e^{\int_0^{s\wedge\tau}d(u)\,du}\bigl(y_n(s\wedge \tau)-y_m(s\wedge\tau)\bigr)\langle z_n(s\wedge\tau ) , dm(s)\rangle\nonumber\hspace*{-35pt}\\ & & \qquad \quad { } + 2\int_{t\wedge \tau}^{m\wedge\tau}e^{\int_0^{s\wedge\tau}d(u)\,du}\bigl(y_n(s\wedge \tau)-y_m(s\wedge\tau)\bigr)\langle z_m(s\wedge\tau ) , dm(s)\rangle.\nonumber\hspace*{-35pt}\end{aligned}\ ] ] choose @xmath204 , @xmath205 such that @xmath206 and @xmath207 . in view of the ( a.1 ) and ( a.2 ) , we have @xmath208\\[-8pt ] & & \qquad \quad { } + \delta_1d_2 ^ 2\int_{t\wedge \tau}^{n\wedge\tau}e^{\int_0^{s\wedge\tau}d(u)\,du}\bigl(y_n(s\wedge \tau)-y_m(s\wedge\tau)\bigr)^2\,ds\nonumber\\ & & \qquad \quad { } + \frac{1}{\lambda\delta_1}\int_{t\wedge\tau}^{n\wedge \tau}e^{\int_0^sd(u)\,du}\bigl\langle a(x(s))\bigl(z_n(s)-z_m(s)\chi_{\{s\leq m\wedge \tau\}}\bigr),\nonumber\\ & & \hspace*{186pt } z_n(s)-z_m(s)\chi_{\{s\leq m\wedge\tau\}}\bigr\rangle\,ds.\nonumber\end{aligned}\ ] ] on the other hand , by ( a.3 ) , it follows that @xmath209|\bigr)^2\,ds.\nonumber\hspace*{-35pt}\end{aligned}\ ] ] take expectation and utilize ( [ 2.19])([2.21 ] ) to obtain @xmath210\nonumber\hspace*{-35pt}\\ & & \quad { } + \biggl(1-\frac{1}{\lambda\delta_1}\biggr)e \biggl[\int_{t\wedge\tau}^{n\wedge \tau}e^{\int_0^sd(u)\,du}\bigl\langle a(x(s))\bigl(z_n(s)-z_m(s)\chi_{\{s\leq m\wedge \tau\}}\bigr ) , \nonumber\hspace*{-35pt}\\ & & \hspace*{207pt } z_n(s)-z_m(s)\chi_{\{s\leq m\wedge \tau\}}\bigr\rangle\,ds \biggr ] \nonumber\hspace*{-35pt}\\[-8pt]\\[-8pt ] & & \quad { } + ( \delta-\delta_1-\delta_2)d_2 ^ 2e \biggl[\int_{t\wedge\tau}^{n\wedge\tau}e^{\int_0^sd(u)\,du}\bigl(y_n(s\wedge \tau)-y_m(s\wedge \tau)\bigr)^2\,ds \biggr]\nonumber\hspace*{-35pt}\\ & & \qquad \leq e \bigl[e^{\int_0^{n\wedge\tau}d(s)\,ds } ( e[\xi|\mathcal { f}_n]-e[\xi|\mathcal { f}_m ] ) ^2 \bigr]\nonumber\hspace*{-35pt}\\ & & \qquad \quad { } + \frac{1}{\delta_2 d_2 ^ 2}e \biggl [ \int_{m\wedge \tau}^{n\wedge\tau}e^{\int_0^{s\wedge\tau}d(u)\,du}\bigl ( since the right - hand side tends to zero as @xmath211 , we deduce that @xmath212 converges to some @xmath213 in @xmath214 . furthermore , for every @xmath18 , @xmath215 converges in @xmath216 . we may as well assume @xmath217 for all @xmath218 . observe that for any @xmath219 , @xmath220 + \int_{\tau\wedge t}^{n\wedge\tau}e^{{(1/2)}\int_0^{s\wedge \tau}d(u)\,du}f(s , y_n(s ) , z_n(s))\,ds\nonumber\hspace*{-35pt}\\[-8pt]\\[-8pt ] & & \qquad \quad { } -\frac{1}{2}\int_{\tau\wedge t}^{n\wedge \tau}e^{{(1/2)}\int_0^{s\wedge\tau}d(u)\,du } y_n(s)d(s)\,ds\nonumber\hspace*{-35pt}\\ & & \qquad \quad { } -\int_{\tau\wedge t}^{n\wedge\tau}e^{{(1/2)}\int_0^{s\wedge \tau}d(u)\,du}\langle z_n(s ) , dm(s)\rangle.\nonumber\hspace*{-35pt}\end{aligned}\ ] ] letting @xmath221 yields that @xmath222 put @xmath223 an application of it s formula and ( [ 2.25 ] ) yield that @xmath224 hence , @xmath139 is a solution to the bsde ( [ 2.15 ] ) proving the existence . to obtain the estimates ( [ 2.16 ] ) and ( [ 2.17 ] ) , we proceed to get an uniform estimate for @xmath225 and then pass to the limit . let @xmath226 be chosen as before . similar to the proof of ( [ 2.8 ] ) , by it s formula , we have @xmath227|^2e^{\int_0^{n\wedge \tau}d(s)\,ds}-\int_{t\wedge\tau}^{n\wedge \tau } e^{\int_0^sd(u)\,du}|y_n(s)|^2d(s)\,ds\nonumber\\ & & \qquad \quad { } -2\int_{t\wedge\tau}^{n\wedge\tau } e^{\int_0^sd(u)\,du}d_1(s)y_n^2(s)\,ds\nonumber\\ & & \qquad \quad { } + 2\int_{t\wedge\tau}^{n\wedge \tau } e^{\int_0^sd(u)\,du}d_2|y_n(s)||z_n(s)|\,ds\nonumber\\ & & \qquad \quad { } + 2\int_{t\wedge\tau}^{n\wedge\tau } e^{\int_0^sd(u)\,du}|y_n(s)||f(s,0,0)|\,ds\nonumber\\ & & \qquad \quad { } -2\int_{t\wedge\tau}^{n\wedge \tau } e^{\int_0^sd(u)\,du}y_n(s)\langle z_n(s),dm(s)\rangle\\ & & \qquad \leq |e[\xi|\mathcal { f}_n]|^2e^{\int_0^{n\wedge \tau}d(s)\,ds}-\int_{t\wedge\tau}^{n\wedge\tau } e^{\int _ 0^sd(u)\,du}\delta d_2 ^ 2y_n^2(s)\,ds\nonumber\\ & & \qquad \quad { } + \int_{t\wedge\tau}^{n\wedge\tau } e^{\int_0^sd(u)\,du}\delta_1 d_2 ^ 2y_n^2(s)\,ds\nonumber\\ & & \qquad \quad { } + \frac{1}{\delta_1\lambda}\int_{t\wedge\tau}^{n\wedge\tau } e^{\int_0^sd(u)\,du}\langle a(x(s))z_n(s ) , z_n(s)\rangle\ , ds \nonumber\\ & & \qquad \quad { } + \int_{t\wedge\tau}^{n\wedge\tau}\delta_2d_2 ^ 2 e^{\int_0^sd(u)\,du}y_n^2(s)\,ds+\frac{1}{\delta_2 d_2 ^ 2}\int_{t\wedge \tau}^{n\wedge\tau } e^{\int_0^sd(u)\,du}|f(s,0,0)|^2\,ds\hspace*{-8pt}\nonumber\\ & & \qquad \quad { } -2\int_{t\wedge\tau}^{n\wedge\tau } e^{\int_0^sd(u)\,du}y_n(s)\langle z_n(s),dm(s)\rangle.\nonumber\end{aligned}\ ] ] recalling the choices of @xmath228 , @xmath204 and @xmath205 , using burkholder s inequality , we obtain from ( [ 2.27 ] ) that @xmath229\nonumber\\ \qquad & & \qquad \leq e\bigl[|\xi|^2e^{\int_0^{n\wedge\tau}d(s)\,ds}\bigr]+e \biggl[\int_{0}^ { \tau}e^{\int_0^sd(u)\,du}\frac{1}{\delta_2 d_2 ^ 2}|f(s,0,0)|^2\,ds \biggr]\\ \qquad & & \qquad \quad { } + 2ce \biggl [ \biggl(\int_{0}^{n\wedge\tau } e^{2\int_0^sd(u)\,du}y_n^2(s)\langle a(x(s))z_n(s ) , z_n(s)\rangle\,ds \biggr)^{{1/2 } } \biggr]\nonumber\\ \qquad & & \qquad \leq e\bigl[|\xi|^2e^{\int_0^{n\wedge\tau}d(s)\,ds}\bigr]+e \biggl[\int_{0}^ { \tau } e^{\int_0^sd(u)\,du}\frac{1}{\delta_2 d_2 ^ 2}|f(s,0,0)|^2\,ds \biggr]\nonumber\\ \qquad & & \qquad \quad { } + \frac{1}{2}e \bigl [ \sup_{0\leq t\leq n}|y_n(t\wedge \tau)|^2e^{\int_0^{t\wedge\tau}d(s)\,ds } \bigr]\nonumber\\ \qquad & & \qquad \quad { } + c_1e \biggl [ \int_{0}^{n\wedge\tau } e^{\int_0^sd(u)\,du}\langle a(x(s))z_n(s ) , z_n(s)\rangle\,ds \biggr].\nonumber\end{aligned}\ ] ] in view of ( [ 2.27 ] ) , as the proof of ( [ 2.9 ] ) , we can show that @xmath230<\infty.\ ] ] ( [ 2.29 ] ) and ( [ 2.28 ] ) together with our assumptions on @xmath126 and @xmath186 imply @xmath231<\infty.\ ] ] applying fatou lemma , ( [ 2.17 ] ) follows . let @xmath232 be a borel measurable function . assume that @xmath126 is continuous in @xmath127 and satisfies : * @xmath233 , where @xmath234 is a measurable function on @xmath2 . * @xmath235 . * @xmath236 let @xmath1 be a bounded regular domain . define @xmath237 given @xmath102 . consider for each @xmath238 the following bsde : @xmath239\\[-8pt ] & & { } -\int_{t\wedge\tau_d^x}^{\tau_d^x}\langle z_x(s),dm_x(s)\rangle,\nonumber\end{aligned}\ ] ] where @xmath240 is the martingale part of @xmath241 . as a consequence of theorem [ thm3.2 ] , we have the following theorem . [ thm3.3 ] suppose @xmath242 for @xmath62 , @xmath243<\infty,\ ] ] for some @xmath244 and @xmath245<\infty.\ ] ] the bsde ( [ 2.32 ] ) admits a unique solution @xmath246 . furthermore , @xmath247 as in previous sections , @xmath248 will denote the diffusion process defined in ( [ 1.3 ] ) . consider the second order differential operator @xmath249 let @xmath1 be a bounded domain with regular boundary ( w.r.t . the laplace operator @xmath250 ) and @xmath251 a measurable function satisfying @xmath252 [ thm4.1 ] assume ( [ 3.2 ] ) and that there exists @xmath253 such that @xmath254<\infty.\ ] ] then there is a unique , continuous weak solution @xmath15 to the dirichlet boundary value problem ( [ 3.3 ] ) which is given by @xmath255.\ ] ] write @xmath256,\ ] ] and @xmath257.\ ] ] we know from theorem 4.3 in @xcite that @xmath258 is the unique , continuous weak solution to the problem @xmath259 so it is sufficient to show that @xmath260 is the unique , continuous weak solution to the following problem : @xmath261 by lemma 5.7 in @xcite and proposition 3.16 in @xcite , we know that @xmath260 belong to @xmath262 . let @xmath263 denote the resolvent operators of the generator @xmath264 on @xmath1 with dirichlet boundary condition , that is , @xmath265.\ ] ] by the markov property , it is easy to see that @xmath266 since @xmath267 is strong continuous , it follows that @xmath268 in @xmath269 . this shows that @xmath270 and @xmath271 . the proof is complete . let @xmath272 be a borel measurable function that satisfies : * @xmath273 * @xmath274 , * @xmath275 where @xmath276 is a measurable function and @xmath277 are constants . consider the semilinear dirichlet boundary value problem @xmath278 where @xmath279 . [ thm4.2 ] assume @xmath280<\infty,\ ] ] for some @xmath244 . the dirichlet boundary value problem ( [ 3.6 ] ) has a unique continuous weak solution . set @xmath281 . according to theorem [ thm3.3 ] , for every @xmath238 the following bsde : @xmath282\\[-8pt ] & & { } -\int_{t\wedge\tau_d^x}^{\tau_d^x}\langle z_x(s),dm_x(s)\rangle,\nonumber\end{aligned}\ ] ] admits a unique solution @xmath283 . put @xmath284 and @xmath285 . by the strong markov property of @xmath123 and the uniqueness of the bsde ( [ 3.7 ] ) , it is easy to see that @xmath286 now consider the following problem : @xmath287 where @xmath82 is defined as in section [ sec2 ] . by theorem [ thm4.1 ] , problem ( [ 3.8 ] ) has a unique continuous weak solution @xmath288 . as @xmath71 , it follows from the decomposition of the dirichlet process @xmath289 ( see @xcite ) that @xmath290\\[-8pt ] & = & \varphi(x_x(\tau_d^x))+\int_{t\wedge\tau_d^x}^{\tau_d^x } f(x_x(s),y_x(s ) , z_x(s)))\,ds\nonumber\\ & & { } -\int_{t\wedge\tau_d^x}^{\tau_d^x}\bigl\langle\nabla u\bigl(x(s\wedge \tau_d^x)\bigr),dm_x(s)\bigr\rangle.\nonumber\end{aligned}\ ] ] take conditional expectation both in ( [ 3.9 ] ) and ( [ 3.7 ] ) to discover @xmath291.\end{aligned}\ ] ] in particular , let @xmath292 to obtain @xmath293 . on the other hand , comparing ( [ 3.7 ] ) with ( [ 3.9 ] ) and by the uniqueness of decomposition of semimartingales , we deduce that @xmath294 for all @xmath218 . by it s isometry , we have @xmath295\nonumber\\ & & \qquad = e \biggl [ \biggl(\int_{0}^{\infty}\bigl\langle\bigl(\nabla u(x(s ) ) -v_0(x_x(s))\bigr)\chi_{\{s<\tau_d^x\}},dm_x(s)\bigr\rangle \biggr)^2 \biggr]\nonumber\\[-8pt]\\[-8pt ] & & \qquad = e \biggl [ \int_{0}^{\infty}\bigl\langle a(x_x(s))\bigl(\nabla u(x(s ) ) -v_0(x_x(s))\bigr ) , \nonumber\\ & & \hspace*{122pt } \nabla u(x(s))-v_0(x_x(s ) ) \bigr\rangle \chi_{\{s<\tau_d^x\}}\,ds \biggr]=0.\nonumber\end{aligned}\ ] ] by fubini theorem and the uniform ellipticity of the matrix @xmath296 , we deduce that @xmath297=0\ ] ] a.e . in @xmath298 with respect to the lebesgue measure , where @xmath299 $ ] . the strong continuity of the semigroup @xmath300 implies that @xmath301 a.e . returning to problem ( [ 3.8 ] ) , we see that @xmath15 actually is a weak solution to the nonlinear problem : @xmath302 suppose @xmath303 is another solution to the problem ( [ 3.12 ] ) . by the decomposition of the dirichlet process @xmath304 , we find that @xmath305 is also a solution to the bsde ( [ 3.7 ] ) . the uniqueness of the bsde implies that @xmath306 . in particular , @xmath307 . this proves the uniqueness . in this section , we study the semilinear second order elliptic pdes of the following form : @xmath308 where the operator @xmath3 is given by @xmath309 as in section [ sec2 ] and @xmath279 . consider the following conditions : [ thm5.1 ] suppose that , hold and @xmath313\nonumber \\ & & \qquad < \infty\nonumber\end{aligned}\ ] ] for some @xmath238 , where @xmath25 is the diffusion generated by @xmath314 as in section [ sec2 ] and @xmath20 is the first exit time of @xmath25 from @xmath1 . then there exists a unique , continuous weak solution to equation ( [ 4.1 ] ) . set @xmath315 put @xmath316 let @xmath317 so that @xmath318 . by lemma 3.2 in @xcite ( see also @xcite ) , there exits a bounded function @xmath319 such that @xmath320 where @xmath321 is the zero energy part of the fukushima decomposition for the dirichlet process @xmath118 . furthermore , @xmath322 satisfies the following equation in the distributional sense : @xmath323 note that by sobolev embedding theorem , @xmath324 if we extend @xmath325 on @xmath326 . this implies that @xmath327 and @xmath321 are continuous additive functionals of @xmath25 in the strict sense ( see @xcite ) , and so is @xmath328 . thus , @xmath329 hence , @xmath330\\[-8pt ] & & \hphantom{{}\times\exp \biggl ( } { } + \int_0^{t } \biggl(q-\frac12 ( b-\hat{b}-a\nabla v ) a^{-1}(b-\hat{b}-a\nabla v)^ * \biggr)(x^0(s))\,ds\nonumber \\ & & \hspace*{95pt } { } + \int_0^{t } \biggl(\frac{1}{2 } ( \nabla v)a(\nabla v)^ * -\langle b-\hat{b } , \nabla v\rangle \biggr)(x^0(s))\,ds \biggr).\nonumber\end{aligned}\ ] ] note that @xmath331 is well defined under @xmath332 for every @xmath238 . set @xmath333 . introduce @xmath334\,{\partial\over{\partial x_i } } \\ & & { } -\langle b-\hat{b},\nabla v\rangle(x ) + { 1\over2}(\nabla v)a(\nabla v)^*(x)+q(x).\end{aligned}\ ] ] let @xmath335 be the diffusion process whose infinitesimal generator is given by @xmath336\,{\partial\over{\partial x_i}}.\ ] ] it is known from @xcite that @xmath337 is absolutely continuous with respect to @xmath338 and @xmath339 where @xmath340\\[-8pt ] & & \hphantom{\exp \biggl ( } { } -\int_0^{t } \biggl(\frac12 ( b-\hat{b}-a\nabla v ) a^{-1}(b-\hat{b}-a\nabla v)^ * \biggr)(x^0(s))\,ds \biggr).\nonumber\end{aligned}\ ] ] put @xmath341 then @xmath342 consider the following nonlinear elliptic partial differential equation : @xmath343 in view of ( [ e : zv ] ) , condition ( [ 4.0 ] ) implies that @xmath344\\[-8pt ] & & \hphantom{\hat{e}_x \biggl[\exp \biggl ( } { } + \int_0^{\tau_d } \biggl(\frac{1}{2 } ( \nabla v)a(\nabla v)^ * -\langle b-\hat{b } , \nabla v\rangle \biggr)(x^0(s))\,ds\biggr ) \biggr]<\infty,\nonumber\end{aligned}\ ] ] where @xmath345 indicates that the expectation is taken under @xmath337 . from theorem [ thm4.2 ] , it follows that equation ( [ 4.2 ] ) admits a unique weak solution @xmath346 . set @xmath347 . we will verify that @xmath15 is a weak solution to equation ( [ 4.1 ] ) . indeed , for @xmath348 , since @xmath349 is a weak solution to equation ( [ 4.2 ] ) , it follows that @xmath350\over{\partial x_i}}\,{\partial[h^{-1}(x)\psi]\over { \partial x_j}}\,dx\\ & & \quad { } -\sum_{i=1}^{d}\int_d[b_i(x)-\hat{b}_i(x)-(a\nabla v)_i(x)]\,{\partial[h(x)u(x)]\over{\partial x_i}}\,h^{-1}(x)\psi \,dx\\ & & \quad { } + \int_d\langle b-\hat{b},\nabla v(x)\rangle u(x)\psi(x)\,dx \\ & & \quad { } -{1\over2}\int_d(\nabla v)a(\nabla v)^*(x)u(x)\psi \,dx-\int _ dq(x)u(x)\psi(x ) \,dx\\ & & \qquad = \int_df(x , u(x))\psi(x ) \,dx.\end{aligned}\ ] ] denote the terms on the left of the above equality , respectively , by @xmath351 , @xmath352 , @xmath353 , @xmath354 , @xmath355 . clearly , @xmath356 using chain rules , rearranging terms , it turns out that @xmath357\,{\partial[\psi u(x)]\over{\partial x_i}}\,dx\\ & & { } -\sum_{i=1}^{d}\int_d(a\nabla v)_i(x)\,{\partial\psi\over{\partial x_i}}\,u(x ) \,dx+\sum_{i=1}^{d}\int _ d(a\nabla v)_i(x)\,{\partial v \over{\partial x_i}}\,u(x)\psi \,dx.\nonumber\end{aligned}\ ] ] in view of ( [ e : vv ] ) , @xmath358\,{\partial[\psi u(x)]\over{\partial x_i}}\,dx\nonumber\\[-8pt]\\[-8pt ] & & \qquad = \frac{1}{2}\sum _ { i=1}^{d}\int_d(a\nabla v)_i(x)\,{\partial[\psi u(x)]\over{\partial x_i}}\,dx.\nonumber\end{aligned}\ ] ] thus , @xmath359\over{\partial x_i}}\,dx\\ & & { } -\sum_{i=1}^{d}\int_d(a\nabla v)_i(x)\,{\partial\psi\over{\partial x_i}}\,u(x ) \,dx+\sum_{i=1}^{d}\int _ d(a\nabla v)_i(x)\,{\partial v \over{\partial x_i}}\,u(x)\psi \,dx.\nonumber\end{aligned}\ ] ] after cancelations , it is now easy to see that @xmath360\\[-8pt ] & & { } -\sum_{i}^{d}\int_d\hat{b}\,{\partial \psi\over \partial x_i}\,u(x)\,dx-\int_dq(x)u(x)\psi(x ) \,dx\nonumber \\ & = & \int_df(x , u(x))\psi(x ) \,dx.\nonumber\end{aligned}\ ] ] since @xmath361 is arbitrary , we conclude that @xmath15 is a weak solution of equation ( [ 4.1 ] ) . suppose @xmath15 is a continuous weak solution to equation ( [ 4.1 ] ) . put @xmath349 . reversing the above process , we see that @xmath346 is a weak solution to equation ( [ 4.2 ] ) . the uniqueness of the solution of equation ( [ 4.1 ] ) follows from that of equation ( [ 4.2 ] ) .

in this paper , we prove that there exists a unique solution to the dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations . our approach is probabilistic . the theory of dirichlet processes and backward stochastic differential equations play a crucial role . .

the presence of a flavor asymmetry in the light antiquark sea of the proton is now clearly established @xcite . it can be expressed either in terms of the difference , @xmath3 , or in terms of the ratio , @xmath4 . the fact that this difference is larger than zero ( or that the ratio is larger than one ) is usually referred to as @xmath5 flavor symmetry breaking in the proton sea . we will discuss in this paper the nonperturbative origin of the breaking of flavor symmetry , both at the @xmath5 and at the @xmath0 level . to this end , we will study the suppresion factor of @xmath6 antiquarks in the @xmath5 case , defined as _ ( 2 ) = , [ kappa2 ] and the suppression factor of strangeness in the @xmath0 case : _ ( 3 ) = . [ kappa3 ] we notice that in the limit of exact @xmath5 ( @xmath0 ) flavor symmetry @xmath7 ( @xmath8 ) . the ccfr collaboration has measured @xcite @xmath9 ( @xmath10 ) in a lo ( nlo ) qcd analysis . uncertainties apart , it is clear that there is a substantial violation of the @xmath0 flavor symmetry . in the nonstrange light antiquark sector , the use of the standard parametrizations leads to @xmath11 @xcite , indicating also a strong violation of the @xmath5 flavor symmetry in the proton sea . at the same time , the @xmath5 charge symmetry is believed to hold within the baryon octet , i.e. , @xmath12 in the proton is equal to @xmath13 in the neutron . an interesting question is how @xmath0 charge symmetry is broken within the baryon octet . if the symmetry were exact , it would mean , for instance , that @xmath14 in the proton should be equal to @xmath15 in the @xmath2 . however , as calculated by the authors of ref . @xcite , this is not the case , and in this work we also investigate the origins of the breaking of this symmetry . in qcd , exact @xmath0 symmetry implies that the @xmath16 , @xmath17 and @xmath18 quarks have the same mass . since the strange quark mass , @xmath19 , is significantly larger than the up and down quark masses , the symmetry is only approximate . at the hadronic level , exact @xmath0 symmetry also implies that the masses of baryons or mesons belonging to the same multiplets are all equal . clearly this is not the case and the masses within the baryon multiplets differ among themselves by more than @xmath20 . the mass discrepancy is even more pronounced in the meson octet . another consequence of the @xmath0 symmetry at the hadronic level is that the coupling constant in a generic baryon - baryon - meson ( @xmath21 ) vertex should be the same for all @xmath22 , @xmath23 and @xmath24 . since these three states together must form a @xmath0 singlet state , and the mesons are usually in octet states , it follows that the product of the two baryon representations must also be in a @xmath0 octet state . out of the ( @xmath25 ) product @xmath26 , we get two distinct octets and therefore two independent coupling constants . this is the origin of the two @xmath0 constants , @xmath27 and @xmath28 . when we consider some particular baryon - baryon - meson vertices , additional ( clebsch - gordan ) factors appear , so that the final couplings are different from each other . however , exact @xmath0 symmetry imposes a well defined connections between them . finally , analysis of experimental data determine the relation between @xmath27 and @xmath28 in terms of the parameter @xcite _ d = 0.64 [ alfa ] we can make use of qcd sum rules ( qcdsr ) to calculate the above mentioned coupling constants @xcite . in this approach we are able to identify the @xmath0 breaking sources affecting the couplings , which are mainly the quark and hadron mass differences . the different values of the condensates and other qcdsr parameters also play an important role . as for the origin os the asymmetry in the light antiquark distributions , there is now strong indications that part of the nucleon sea comes from fluctuations of the original nucleon into baryon - meson states , i.e. , from the meson cloud @xcite . the meson cloud model ( mcm ) is dominated by hadronic quantities like hadron masses and coupling constants . this bridge between the physics of parton distribution and the conventional hadron physics may also help us , by connecting one with the other , to understand both @xmath0 symmetry breaking at the hadron and parton levels . in what follows , we show the meson - baryon fock decomposition of the proton and of the @xmath29 . in the case of the proton , most of the material has been already presented elsewhere @xcite . we include it here just for completeness . parton distributions in the @xmath29 hyperon have been discussed in @xcite , and we will also address them in this work . this will enable us to make a close comparison between the proton and hyperon parton distributions . as usual , we decompose the proton in the following possible fock states : & & + |k^+ > + |^0 k^+ > + |^0 * k^+ > + |^+ k^0 > + |^+ * k^0 > ] [ fock ] where @xmath30 is the bare proton . we consider only light mesons . the relative normalization of these states is , in principle , fixed once the cloud parameters are given . the normalization constant @xmath31 measures the probability to find the proton in its bare state . in the @xmath32 state , the meson and the baryon have fractional momentum @xmath33 and @xmath34 , with distributions @xmath35 and @xmath36 , respectively . of course @xmath37 and these distributions are related by : f_m / mb(z ) = f_b / mb(1-z ) [ fmb ] the splitting function @xmath38 represents the probability density to find a meson with momentum fraction @xmath39 of the nucleon and is usually given by @xmath40}{[t - m_{m}^2]^2}\ , f_{m b p}^2 ( t)\ ; , \label{fpin}\ ] ] for baryons ( @xmath22 ) belonging to the octet , and @xmath41 ^ 2 [ ( m_{p } - m_b)^2 - t ] } { 12 m_b^2 m_{p}^2 [ t - m_{m}^2]^2}\ , f_{m b p}^2 ( t)\ ; \label{fpidel}\ ] ] for baryons belonging to the decuplet . in the calculations we need the baryon - meson - baryon form factors appearing in the splitting functions . following a phenomenological approach , we use the dipole form : @xmath42 where @xmath43 is the form factor cut - off parameter . in the above equations @xmath44 and @xmath45 are the four momentum square and the mass of the meson in the cloud state , @xmath46 is the maximum @xmath44 given by : @xmath47 where @xmath48 ( @xmath49 ) is the mass of the baryon ( proton ) . since the function @xmath38 has the interpretation of a flux of mesons inside the proton , the corresponding integral n_m / mb = _ mb _ 0 ^ 1 d y f_m / mb ( y ) , [ nmeson ] can be interpreted as the number of mesons in the proton , or the number of mesons in the air . in many works , the magnitude of the multiplicities @xmath50 has been considered as a measure of the validity of mcm in the standard formulation with @xmath51 states . if these multiplicities turn out to be large ( @xmath52 ) then there is no justification for employing a one - meson truncation of the fock expansion , as the expansion ceases to converge . this may happen for large cut - off values . once the splitting functions ( [ fpin ] ) and ( [ fpidel ] ) are known we can calculate the antiquark distribution in the proton coming from the meson cloud through the convolution : q_f ( x ) = _ mb _ x^1 f_m / mb ( y ) q^m_f ( ) [ quark ] where @xmath53 is the valence antiquark distribution of flavor @xmath54 in the meson . an analogous expression holds for the quark distributions . with the above formula we can compute the @xmath55 and @xmath6 distributions , their difference , @xmath56 , and hence the gottfried integral : s_g = - _ 0 ^ 1 [ d ( x ) - u(x ) ] dx [ gottfried ] since we are interested in determining the sources of @xmath0 symmetry breaking , we also study the parton distributions in the case where the @xmath0 symmetry is exact . in our case , this is the limit in which we take all the meson and baryon masses to be the same in the @xmath0 multiplets . all other ingredients are , from the start , compatible with @xmath0 symmetry , i.e. , all coupling constants follow @xmath0 relations @xcite , and the cut - off parameters are the same for a given multiplet . of course , the nonstrange subset of these couplings respects the @xmath5 ( isospin ) symmetry . the masses are @xmath57 , @xmath58 , @xmath59 , @xmath60 , @xmath61 , and @xmath62 . the octet coupling constants are given by the expressions in table i @xcite , where @xmath63 @xcite and @xmath64 was given in eq . ( [ alfa ] ) . for the decuplet coupling constants , in table ii , where @xmath65 @xcite , we also use the standard @xmath0 relations between the couplings @xcite . [ cols="^,^",options="header " , ] * table iv : * @xmath29 decuplet coupling constants . for the cut - off parameters , we will use the same values as given by eq . ( [ lambda ] ) . in fig . 5 we show the separate contributions from the octet and decuplet states for @xmath66 ( 5a ) , @xmath67 ( 5b ) , @xmath68 ( 5c ) . the total distributions are shown in the fig . 5d , and they should be compared with fig . we agree qualitatively with them . quantitative changes are noticeable , and they happen because of the inclusion of the decuplet states which play a significant role , as seen in figs . 5b and 5c . the fact that @xmath69 , was interpreted in @xcite as a violation of @xmath0 charge symmetry , and this really seems to be the case . even more indicative of this breaking is the direct comparison of @xmath70 in the proton ( dotted line ) with @xmath71 in the @xmath2 ( dot - dashed line ) , shown in fig . a huge discrepancy is seen between the two curves , a result in complete disagreement with naive expectations . as in the quark model the @xmath2 is a proton with the @xmath17 quark replaced by a @xmath18 quark , naively one would think that @xmath70 in the proton is equal to @xmath71 in the @xmath2 . as we saw in section 2.1 , the pb effect is important in describing the @xmath72 dependence of the light quark sea asymmetry . from the point of view of fermi statistics , the same effect should be present in the @xmath2 , with the @xmath18 quark here playing the role of the @xmath17 quark in the proton . because of the mass of the @xmath18 quark , the @xmath72 dependence of the pb in the @xmath2 may not be exactly the same as in the proton . however , to exemplify the size of the corrections from pb , we also plot in fig . 6 the distributions including the effect of the pb given by eq . ( [ pauli ] ) . the solid line is for @xmath70 in the proton , and the dashed line is for @xmath71 in the @xmath2 . it seems also appropriate to extend the comparisons to @xmath15 in the @xmath2 , and to @xmath14 in the proton . we show the @xmath15 in fig . 7 , where the decuplet and octet contributions are shown separately . 8 we show both differences and we see clearly the discrepancy between them , which is again an evidence of @xmath0 charge symmetry breaking . it is remarkable , however , that besides the small mass of the @xmath17 quark , the @xmath15 asymmetry in the @xmath2 is much larger than the @xmath14 asymmetry in the proton . finally , in order to compare the @xmath0 flavor breaking in sea parton distributions in the @xmath29 with the proton , we compute @xmath73 defined in eq . ( [ newkappa ] ) . the denominator in eq . ( [ newkappa ] ) is governed by the large perturbative contributions and is only slightly affected by the cloud component . it is therefore reasonable to assume that it is the same for the proton and for the @xmath29 . in the numerator we have approximated @xmath74^{np}$ ] by @xmath75^{np}$ ] in order to avoid uncertainties associated with @xmath76 in the hyperon . the resulting value for @xmath77 is then : _ 3 0.85 this value of @xmath77 indicates a violation of @xmath0 flavor inside the @xmath2 which is weaker than that inside the proton , whereas figs . 4 - 7 show a violation of the @xmath0 symmetry between the proton and the sigma . both symmetries are restored in the @xmath0 symmetry limit of eqs . ( [ su3lim ] ) , i.e. , @xmath78 and the curves in the figures assume their expected behaviour , with @xmath79 and @xmath80 . in the context of the meson cloud model this result is not surprising . the cloud expansion of the @xmath29 involves heavier states than those appearing in the proton expansion . as a consequence , the whole @xmath2 cloud will be suppressed with respect to the proton cloud . indeed , looking at the multiplicities we observe that the probabilities associated with the hyperon states are typically one order of magnitude smaller than those associated with the proton states . moreover , the strange states inside the proton are heavier and suppressed with respect to non - strange states , and therefore we expect ( and really observe ) @xmath81 . neglecting pauli blocking effects , ( which would slightly inhibit the @xmath82 production in comparison with the @xmath55 production in the @xmath2 ) we would expect the same behaviour for the @xmath2 and this is exactly what we find . quantitatively , the suppression of @xmath83 in @xmath2 ( with respect to @xmath84 or @xmath85 ) , happens because all the states in the cloud contain strangeness and are nearly equally suppressed . in the proton the suppression of @xmath83 ( always with respect to @xmath84 or @xmath85 ) is more pronounced because of the mass difference between strange and non - strange cloud states . in this work we have applied to meson cloud model to study the non - perturbative aspects of parton distributions , giving special emphasis to the strange sector . we have adjusted the cloud cut - off parameters to reproduce the e866 data on @xmath12 and @xmath86 . in this procedure the choices were not completely free . instead , the cut - off values had to be consistent with previous analises of other experimental information @xcite . having fixed the parameters we moved to the strange sector . in this sense , the results for the strange - anti - strange asymmetry and for @xmath87 can be considered as predictions . they are consistent with data . finally we have taken the @xmath0 limit in the meson cloud and found out that , in this limit , the parton distributions become @xmath0 flavor symmetric , i.e. , @xmath88 . we have thus presented additional experimental confirmation of the mcm . moreover we have concluded that the meson cloud is responsible for the @xmath0 flavor breaking in parton distributions . * \a ) @xmath12 calculated with eq . ( [ quark ] ) compared with e866 data ; b ) same as a ) for the ratio @xmath89 . the dashed lines represent our result without pauli blocking . * \a ) @xmath90 ( solid line ) and @xmath91 ( dashed line ) in the proton computed with the mcm ( using eq . ( [ quark ] ) ) ; b ) @xmath92 in the proton in the mcm . the octet and decuplet contributions are represented by the dashed and the dotted lines , respectively ; c ) same as b ) for the difference @xmath14 . the shaded area is the uncertainty range ofthe experimental data @xcite . * @xmath93 in the proton extracted from several parametrizations , and the resulting curves from the mcm ( solid line ) . the octet and decuplet contributions are the dashed and the dotted lines , respectively . * \a ) @xmath94 in the proton extracted from several parametrizations , and the result from the mcm ( solid line ) . the dashed line is the mcm in the @xmath0 limit . * \a ) @xmath95 in the @xmath2 calculated with the mcm ( solid line ) . the octet and decuplet contributions are the dashed and dotted lines , respectively ; b ) same as a ) for @xmath96 ; c ) same as a ) @xmath97 ; d ) all the curves together , where the decuplet and octet contributions were added . * @xmath95 in the proton , with ( solid line ) and without ( dotted line ) pauli blocking . @xmath98 in the @xmath2 , with ( dashed line ) and without ( dot - dashed line ) pauli blocking . all the curves were calculated in the mcm . * @xmath99 in the @xmath2 with the mcm . the octet and decuplet contributions are the dashed and the dotted lines , respectively . * \a ) @xmath100 in the proton ( solid line ) and @xmath99 in the @xmath2 ( dashed line ) . both curves were calculated in the mcm .

we apply the meson cloud model to the calculation of nonsinglet parton distributions in the nucleon sea , including the octet and the decuplet cloud baryon contributions . we give special attention to the differences between nonstrange and strange sea quarks , trying to identify possible sources of @xmath0 flavor breaking . a analysis in terms of the @xmath1 parameter is presented , and we find that the existing @xmath0 flavor asymmetry in the nucleon sea can be quantitatively explained by the meson cloud . we also consider the @xmath2 baryon , finding similar conclusions . + pacs numbers 14.20.dh 12.40.-y 14.65.-q + 0.3 cm 0.3 cm -1 cm = 10000

the study of kaon decays has played a pivotal role in formulating the standard model of electroweak interactions @xcite . in particular , the rare decay of @xmath9 was used to constrain the flavor changing neutral current @xcite as well as the top quark mass @xcite . however , there are ambiguities in extracting the short - distance contribution since the long - distance contribution dominated by the two - photon intermediate state is not well known because its dispersive part can not be calculated in a reliable way @xcite . to have a better understanding of this dispersive part , it is important to study the lepton pair decays of the @xmath0 meson such as @xmath10 and @xmath11 ( @xmath12 ) since they can provide us with information on the structure of the @xmath13 vertex @xcite . on the other hand , since these lepton pair decays are dominated by the long - distance physics , they can also be served as a testing ground for theoretical techniques such as chiral lagrangian or other non - perturbative methods that seek to account for the low - energy behavior of qcd . recently , several new measurements of the decay branching ratios of @xmath14 , @xmath15 , and @xmath16 have been reported @xcite . these decays proceed entirely through the @xmath17 vertex and provide the best opportunity for the study of its form factor . in ref . @xcite , since the assumption of neglecting the momentum dependence for the form factor was adopted , the results for the decays are only valid for those with only the electron - positron pair . in ref . @xcite , the decays were studied at the order @xmath18 in chiral perturbation theory ( chpt ) . however , all the results in ref . @xcite are smaller than the current experimental values . in this work , we consider another non - perturbative method in the lfqa to analyze the @xmath17 form factor . as is well known @xcite , the lfqa allows an exact separation in momentum space between the center - of - mass motion and intrinsic wave functions . a consistent treatment of quark spins and the center - of - mass motion can also be carried out . it has been successfully applied to calculate various form factors @xcite . the paper is organized as follows . in sec . ii , we derive the theoretical formalism for the decay constant and the @xmath17 vertex and use these formalism in the lfqa to extract the decay constant and the form factor . in sec . iii , we fix the parameters appearing in the wave functions and calculate the form factors and branching ratios . finally , conclusions are given in sec . we start with the @xmath19 meson decay constant @xmath20 , defined by 0|a^|k(p)=if_k p^,[decaydef ] where @xmath21 is the axial vector current . assuming a constant vertex function @xmath22 @xcite which is related to the @xmath23 bound state of the kaon . then the quark - meson diagram , depicted in fig . 1 ( a ) , yields 0|a^|k(p)=- _ k , [ decay1 ] where @xmath24 are the masses of @xmath25 and @xmath26 quark , respectively , and @xmath27 is the number of colors . we consider the poles in denominators in terms of the lf coordinates @xmath28 and perform the integration over the lf energy " @xmath29 in eq . ( [ amp ] ) . the result is 0|a^|k(p)= ( i^_1|_p^-_1=p^-_1on ) , [ decay2 ] where & = & dp^+_1 d^2p_1 , p^-_ion = m^2_i+p^2_i , + i^_1&=&tr[_5(+m_s)^_5(+m_u ) ] . for @xmath30 , with the assumption of @xmath31 conservation the amplitude is given by a(k_l^*(q_1,_1 ) ^*(q_2,_2 ) ) = if(q^2_1,q^2_2 ) _ ^_1 ^_2 q^_1 q^_2 , [ def ] where the form factor of @xmath32 in eq . ( [ def ] ) is a symmetric function under the interchange of @xmath33 and @xmath34 . in our model , by using the same procedure as above , from the quark - meson diagram depicted in fig . 2 we get a(k_l^ * ^*)&=&- _ k_l\{tr+(_1 _ 2 ) } , [ amp ] where @xmath35 , @xmath36 , and @xmath37 is the effective contribution to the inclusive @xmath38 decay . after integrating over @xmath29 , we obtain a(k_l^ * ^ * ) & = & \ { + & & + ( _ 1 _ 2 ) } , [ pole ] where @xmath39 and i_2&=&tr[_5(+m_s)(+m_s ) ( + m_d ) ] . we note that we do not expect that the absolute decay widths of @xmath40 and @xmath41 calculated from eq . ( [ pole ] ) can fit to the experimental values @xcite . however , we can estimate the relative form factors of these leptonic decays versus the two - photon decay , and compare the branching ratios with the experimental ones . recent works on both short - distance ( sd ) and long - distance ( ld ) contributions to @xmath42 can be found in ref . @xcite . as described in ref . @xcite , the vertex function @xmath43 and the denominators in eq . ( [ pole ] ) correspond to the @xmath0 meson bound state . in the lfqa , the internal structure of the meson bound state @xcite consists of @xmath44 , which describes the momentum distribution of the constituents in the bound state , and @xmath45 , which creates a state of definite spin ( @xmath46 ) out of lf helicity ( @xmath47 ) eigenstates and is related to the melosh transformation @xcite . a convenient approach relating these two parts is shown in ref . the interaction hamiltonian is assumed to be @xmath48 where @xmath49 is the quark field and @xmath50 is the meson field containing @xmath44 and @xmath45 . when considering the normalization of the meson state depicted in fig . 1 ( b ) in the lfqa , we obtain m ( p,s,s_z)|h_i h_i|m(p , s , s_z)&=&2(2)^3 ^ 3(p-p)_ss_s_zs_z + & & ^2 r^s , s_z__1,_2 r^s,s_z__1,_2 . + if we normalize the meson state and the momentum distribution function @xmath44 as @xcite m ( p,s,s_z)|h_i h_i|m(p , s , s_z)=2(2)^3p^+^3(p-p)_ss_s_zs_z , and the on - mass - shell momenta , we have that r^s , s_z__1,_2= . the wave function and the melosh transformation of the meson are related to the bound state vertex function @xmath51 by r^s , s_z__1,_2 _ m. we note that @xmath52 , @xmath53 and @xmath54 in the trace of @xmath55 must be on the mass shell for self - consistency . after taking the good " component @xmath56 , we use the definitions of the lf momentum variables @xmath57 @xcite and take a lorentz frame where @xmath58 to have @xmath59 and @xmath60 . the decay constant @xmath20 and the form factor @xmath61 can be extracted by comparing these results with eqs . ( [ decaydef ] ) and ( [ def ] ) , respectively , @xmath62 , f_k=2_k_l(x , k_)a , [ fp ] and f(q_1 ^ 2,q^2_2)&= & \{c_w(q^2_1 ) + & & + ( q_1 q_2;r_+ 1-r_- ) } , [ hi ] where a&=&m_u , dx+m_s(1-x ) , m_u = m_d , + r_&=&1 , [ y12 ] and @xmath63 is the momentum fraction carried by the spectator antiquark in the initial state . in principle , the momentum distribution amplitude @xmath64 can be obtained by solving the lf qcd bound state equation@xcite . however , before such first - principle solutions are available , we shall have to use phenomenological amplitudes . one momentum distribution function that has often been used in the literature for mesons is the gaussian - type , @xmath65 where @xmath66 and @xmath67 is of the internal momentum @xmath68 , defined through @xmath69 with @xmath70 . we then have m_0=e_1 + e_2 , k_z = xm_02-m_2 ^ 2+k_^2 2 xm_0 , [ kz ] and @xmath71 which is the jacobian of the transformation from @xmath72 to @xmath73 . to examine numerically the form factor derived in eq . ( [ hi ] ) , we need to specify the parameters appearing in @xmath74 . to fit the meson masses , in ref.@xcite @xmath75 gev and @xmath76 gev are obtained with some interaction potentials , while in ref . @xcite @xmath77 gev and @xmath78 gev in the invariant meson mass scheme . here we do not consider any potential form and scheme and just use the decay constant @xmath79 @xcite , charge radius @xmath80 @xcite , and the quark masses of @xmath81 to constrain the @xmath26 quark mass of @xmath82 and the scale parameter of @xmath83 in eq . ( [ gauss ] ) . by using @xmath84 @xcite , we find that @xmath85 and @xmath86 . we note that the lower mass of @xmath82 should not affect the meson masses once we choose a suitable potential @xcite or scheme @xcite . now , we use the momentum distribution functions @xmath87 to calculate the form factors @xmath61 in time - like region of @xmath88 and @xmath89 . in this low energy region , we neglect the momentum dependence of the effective vertex @xmath90 in eq . ( [ hi ] ) , that is , c_w(q^2 ) c_w(0 ) [ appi ] . we can use eqs . ( [ hi ] ) and ( [ appi ] ) to get the function @xmath91 , where @xmath92 , and the result for @xmath93 is shown in fig . 3 . from the figure , we see that our result with the assumption of eq . ( [ appi ] ) agrees well with experimental data @xcite , especially in the lower @xmath94 region . to get a better fit for a larger @xmath94 , we may use c_w(q^2 ) . [ appii ] as seen from fig . 3 , we find that the fit for @xmath95 is better than that for @xmath96 . in particular , a larger value of @xmath97 is preferred if we disregard the data from e845 at bnl @xcite in fig . the experimental result on @xmath98 from na48 at cern , which is currently being analyzed @xcite , should help to resolve this matter . to illustrate our results on the lepton pair decays , we shall take @xmath99 and @xmath100 , referring as ( i ) and ( ii ) , respectively . the function of @xmath101 is related to the differential decay rate of @xmath102 by = 2 ( ) |f(y)|^2 ^ 3/2 ( 1,q_1 ^ 2,0)g_l(q_1 ^ 2 ) , [ drate ] where ( a , b , c)=a^2+b^2+c^2 - 2(ab+bc+ca ) , and g_l(q^2)=(1 - 4m^2_l)^1/2(1 + 2m^2_l ) . integrating over @xmath33 in eq . ( [ drate ] ) , we get the branching ratios _ e^+e^-&&(k_l^0e^+e^- ) = 1.64 , 1.65 10 ^ -2 , + b_^+^-&&(k_l^0^+^- ) = 5.50,6.20 10 ^ -4 , for ( i ) and ( ii ) , respectively . these values agree well with the experimental data : @xmath103 @xcite and @xmath104 @xcite , where we have used @xcite ^(k_l^0)= [ ( 5.920.15)10 ^ -4]^(k_l^0 ) . [ gg ] on the other hand , our results are larger than @xmath105 and @xmath106 , respectively , obtained in ref . @xcite , where the momentum dependence of the form factor was neglected , i.e. , @xmath107 . this inconsistency is reasonable because the kinematic factor @xmath108 which leads the contribution at @xmath109 is important , and the electron mass is very small so that @xmath107 is only valid for the decay with an electron - positron pair . for the muonic pair case , since the mass of muon is not small , the effect of the deviation of neglecting the momentum dependence is evident . this situation also occurs in the decays with two lepton pairs . next , eq . ( [ hi ] ) can be also used to calculate the differential decay rates of @xmath2 by & = & 2 ( ) ^2|f(q_1 ^ 2,q_2 ^ 2 ) |^2 ^ 3/2 ( 1,q_1 ^ 2 , q_2 ^ 2)g_l(q_1 ^ 2)g_l(q_2 ^ 2 ) . after the integrations over @xmath33 and @xmath34 , for ( i ) and ( ii ) we obtain the branching ratios as follows : _ e^+e^-e^+e^-&&(k_l^0e^+e^-e^+e^- ) = 6.61,6.74 10 ^ -5 , + b_^+^-e^+e^-&&(k_l^0^+^-e^+e^- ) = 3.87 , 4.37 10 ^ -6 , + b_^+^-^+^-&&(k_l^0^+^-^+^- ) = 1.50 , 1.73 10 ^ -9 . [ bratios ] in table 1 , we summary the experimental and theoretical values of the decay branching ratios for the @xmath0 lepton pair modes . the results of ref . @xcite correspond a point - like form factor , while those in ref . @xcite are calculated at @xmath110 in the chpt . table 1 : summary of the lepton pair decays of @xmath0 . [ cols="^,^,^,^,^,^,^",options="header " , ] from table 1 , we may also combine the experimental values by assuming that they are uncorrelated and we find that _ k_l^+^-^exp & = & ( 5.930.26)10 ^ -4 , + b_k_le^+e^-e^+e^-^exp&=&(6.830.19)10 ^ -5 , + b_k_l^+^-e^+e^-^exp&= & ( 4.44^+0.84_-0.82)10 ^ -6 . [ cexp ] it is interesting to see that our results for @xmath111 are larger than those in refs . @xcite and agree very well with the experimental data . furthermore , as shown in eq . ( [ bratios ] ) , those for @xmath112 and @xmath113 also agree with the combined experimental values in eq . ( [ cexp ] ) . here , we do not consider the interference effect @xcite from the identical leptons in the final state . the reasons are given in the following . when we use the non - point - like form factor , this effect is about @xmath114 in the @xmath115 mode @xcite , which is beyond experimental access . for the @xmath116 mode , the relative size of the interference effect is larger , but it is outside the scope of future experiments because the total branching ratio is predicted to be about @xmath117 . we now use the form factor @xmath118 to calculate the decays of @xmath5 . the decay branching ratios of the modes can be generally decomposed in the following way _ l^+l^-= absorptive contribution and @xmath119 the dispersive one . the former can be determined in a model - independent form of |a_l|^2=^2 m_l^2 ^2 , [ imaginary ] where @xmath120 . the latter , however , can be rewritten as the sum of sd and ld contributions , a_l= a_l+ a_l . in the standard model , the sd part has been identified as the weak contribution represented by one - loop @xmath121-box and @xmath122-exchange diagrams @xcite , while the ld one is related to @xmath118 by |a_l|^2= 2 ^ 2m^2_l_l f(q^2,(p - q)^2 ) . [ rloop ] in general , an once - subtracted dispersion relation can be written for @xmath123 as @xcite r_l(p^2)= r_l(0)+p^2^_0 dp^2r_l(p^2 ) , [ rer ] where @xmath124 can be obtained by applying eq . ( [ hi ] ) in the soft limit of @xmath125 . for the @xmath126 decay , with @xmath99 and @xmath100 of ( i ) and ( ii ) in eq . ( [ appii ] ) we find that |a_e|^2&= & 5.60 , 6.5210 ^ -9 , respectively . since the sd part of @xmath127 can be neglected , we get _ e^+e^-^i & = & 1.09 10 ^ -8 , + b_e^+e^-^ii & = & 1.18 10 ^ -8 , [ bee ] where we have used @xmath128 . in terms of the total decay branching ratio @xmath129 , the numbers in eq . ( [ bee ] ) are about @xmath130 and @xmath131 , respectively . both results in eq . ( [ bee ] ) are consistent with the experimental value of @xmath132 measured by e871 at bnl @xcite , but they are lower than the value of @xmath133 [ @xmath134 given by the calculation in ref . @xcite with the chpt . it is interesting to note that @xmath135 slowly increases as @xmath97 and reaches @xmath136 for @xmath137 . clearly , our prediction is about @xmath7 smaller than that in the chpt @xcite . for the @xmath138 decay , by subtracting between the value of @xmath139 from the experimental data of @xmath140 @xcite , we obtain that |a_|^2 7.2 10 ^ -7 ( 90% c.l . ) . [ limit ] in the standard model , we have that @xcite |a_|^2b_k_l & = & 0.910 ^ -9(1.2-|)^2 ^ 3.1 ^ 4 , [ smf ] where @xmath141 . using the parameters of @xmath142 , @xmath143 and @xmath144 @xcite , from eqs . ( [ gg ] ) and ( [ smf ] ) we get a_-1.22 10 ^ -3 , [ sma ] which is larger than the limit in eq . ( [ limit ] ) . it is clear that the value of @xmath145 has to be either very small for the same sign as @xmath146 or the same order but the opposite sign . for the case of ( i ) , from eq . ( [ rer ] ) we find a^i_=-1.1110 ^ -3 , [ ldai ] which is very close to the sd value in eq . ( [ sma ] ) and clearly ruled out if the absolute sign in eqs . ( [ sma ] ) and ( [ ldai ] ) are the same . however , if the relative sign is opposite , the limit in eq . ( [ limit ] ) can be satisfied for certain values of @xmath147 . from eqs . ( [ smf ] ) , ( [ limit ] ) , and ( [ ldai ] ) , by taking @xmath148 and @xmath143 we extract that | > -0.37 or > -0.38 ( 90%c.l . ) . [ rhoi ] we note that the limit in eq . ( [ rhoi ] ) is close to that in eq . ( 41 ) of ref . this result is not surprising . if we fit @xmath118 in eq . ( [ hi ] ) with eq . ( 14 ) of ref . @xcite given by f(q_1 ^ 2,q_2 ^ 2)=f(q_1 ^ 2,q_2 ^ 2)f(0,0)= 1+(q_1 ^ 2q_1 ^ 2-m_^2+q_2 ^ 2q_2 ^ 2-m_^2)+ , we find that @xmath149 and @xmath150 and thus 1 + 2+=2.16 10 ^ -20 , which satisfies the bound of eq . ( 35 ) in ref . similarly , for ( ii ) we obtain a^ii_=-1.38 10 ^ -4 . [ ldaii ] it is very interesting to see that the value in eq . ( [ ldaii ] ) is much smaller than @xmath151 in eq . ( [ sma ] ) , which is exactly the case discussed in ref . @xcite . from eq . ( [ ldaii ] ) , with the same parameters as ( i ) , we find that | > 0.63 , 0.41 or > 0.65 , 0.42 ( 90%c.l . ) [ rhoii ] for the same and opposite signs between @xmath152 and @xmath153 , respectively . we note that the limits in eq . ( [ rhoii ] ) do not agree with the recent global fitted value of @xmath154 @xcite , which may not be unexpected since ( i ) we have not included various possible ranges of @xmath155 , @xmath156 , and quark masses in the calculation and ( ii ) we still need to fix @xmath97 in eq . ( [ appii ] ) and modify the form of @xmath90 @xcite . however , the important message here is that the ld dispersive contribution in @xmath9 is calculable in the lfqa . from our preliminary results , it seems that @xmath157 is indeed small as anticipated many years ago in ref . moreover , our approach here provides another useful tool for the decays beside the chpt . in this work , we have studied the @xmath0 lepton pair decays of @xmath1 and @xmath2 in the light - front qcd framework . in our calculations , we have adopted the gaussian - type wave function and assumed the form of the effective vertex @xmath90 in eq . ( [ appii ] ) to account for the momentum dependences in the low energy region . we have calculated the relative form factors of the leptonic decays vs. the two - photon decay , and have showed that our results on the decay branching ratios of @xmath111 and @xmath158 agree well with the experimental data . the remarkable agreements indicate that our form for @xmath90 is quite reasonable , but the number of @xmath97 still needs to be fixed . furthermore , all our predicted values for these decays are larger than those in the chpt @xcite , in particular for the modes of @xmath159 and @xmath160 for which the @xmath110 chpt results in ref . @xcite are ruled out by the new experimental data @xcite . on the other hand , for @xmath6 , we have found that @xmath161 is between @xmath162 and @xmath136 for @xmath163 , which are lower than @xmath164 in the chpt @xcite . for @xmath165 , we have demonstrated that the long - distance dispersive contribution is possibly small . however , to get a meaningful constraint on the ckm parameters , further theoretical studies @xcite as well as more precise experimental data such as those from na48 at cern @xcite on the spectra of the pair decays are needed . finally , we remark that our approach can not calculate the absolute decay widths of @xmath10 and @xmath166 . glashow , j. iliopoulos and l. maiani , , 1585 ( 1973 ) ; m.k . gaillard and b.w . lee , phys . rev . * d 10 * 897 ( 1974 ) , m.k . gaillard , b.w . lee and r.e . shrock , phys . rev . * d 13 * , 2674 ( 1976 ) ; r.e . shrock and m.b . voloshin , phys . lett . * b 87 * , 375 ( 1979 ) . l. bergstrom , e. masso , and p. singer , phys . * b 249 * , 141 ( 1990 ) ; g. blanger and c.q . geng , , 140 ( 1991 ) ; l. ritchie and s.g . wojcicki , rev . 65 , 1149 ( 1993 ) ; l. littenberg and g. valencia , annu . nucl . part . * 43 * 729 ( 1993 ) . 0.25 true cm * fig . 3 * the @xmath94-dependent behavior of @xmath93 , where the lines from bottom to top corresponding to @xmath169 are obtained by this work with @xmath170 and @xmath171 and the experimental data are taken from e799 at fnal @xcite , e845 at bnl @xcite , and na31 at cern @xcite , respectively .

we analyze @xmath0 lepton pair decays of @xmath1 and @xmath2 @xmath3 within the framework of the light - front qcd approach ( lfqa ) . with the @xmath4 form factors evaluated in a model with the lfqa , we calculate the decay branching ratios and find out that our results are all consistent with the experimental data . in addition , we study @xmath5 decays . we point out that our prediction on @xmath6 is about @xmath7 smaller than that in the chpt . we also discuss whether one could extract the short - distance physics from @xmath8 . 6.5 in 0.cm

recent years witness an avalanche investigation of complex networks @xcite . complex systems in diverse fields can be described with networks , the elements as nodes and the relations between these elements as edges . the structure - induced features of dynamical systems on networks attract special attentions , to cite examples , the synchronization of coupled oscillators @xcite , the epidemic spreading @xcite and the response of networks to external stimuli @xcite . synchronization is a wide - ranging phenomenon which can be found in social , physical and biological systems . recent works show that some structure features of complex networks , such as the small - world effect and the scale - free property , can enhance effectively the synchronizabilities of identical oscillators on the networks , i.e. , synchronization can occur in a much more wide range of the coupling strength . we consider a network of @xmath0 coupled identical oscillators @xcite . the network structure can be represented with the adjacent matrix @xmath1 , whose element @xmath2 is @xmath3 and @xmath4 if the nodes @xmath5 and @xmath6 are disconnected and connected , respectively . denoting the state of the oscillator on the node @xmath5 as @xmath7,the dynamical process of the system is governed by the following equations , @xmath8 where @xmath9 governs the individual motion of the @xmath5th oscillator , @xmath10 the coupling strength and @xmath11 the output function . the matrix @xmath12 is a laplacian matrix , which reads , @xmath13 where @xmath14 is the degree of the node @xmath5 , i.e. , the number of the nodes connecting directly with the node @xmath5 . the eigenvalues of @xmath12 are real and nonnegative and the smallest one is zero . that is , we can rank all the possible eigenvalues of this matrix as @xmath15 . herein , we consider the fully synchronized state , i.e. , @xmath16 as @xmath17 for any pair of nodes @xmath5 and @xmath6 . synchronizability of the considered network of oscillators can be quantified through the eigenvalue spectrum of the laplacian matrix @xmath12 . here we review briefly the general framework established in @xcite . the linear stability of the synchronized state is determined by the corresponding variational equations , the diagonalized @xmath0 block form of which reads , @xmath18z$ ] . @xmath19 is the different modes of perturbation from the synchronized state . for the @xmath5th block , we have @xmath20,@xmath21 . the synchronized state is stable if the lyapunov exponents for these equations satisfy @xmath22 for @xmath23 . detailed investigations @xcite show that for many dynamical systems , there is a single interval of the coupling strength @xmath24 , in which all the lyapunov exponents are negative . in this case , the synchronized state is linearly stable if and only if @xmath25 . while @xmath26 depends on the the dynamics , the eigenratio @xmath27 depends only on the topological structure of the network hence , this eigenratio represents the impacts of the network structure on the networks s synchronizability . this framework has stimulated an avalanche investigation on the synchronization processes on complex networks . it has been widely accepted as the quantity index of the synchronizability of networks . however , the eigenratio is a lyapunov exponent - based index . it can guarantee the linear stability of the synchronized state . it can not provide enough information on how the network structure impacts the dynamical process from an arbitrary initial state to the final synchronized state . how the structures of complex networks impact the synchronization is still a basic problem to be understood in detail . in this paper , by means of the random matrix theory ( rmt ) , we try to present a possible dynamical mechanism of the enhancement effect , based upon which we suggested a new dynamic - based index of the synchronizabilities of networks . the rmt was developed by wigner , dyson , mehta , and others to understand the energy levels of complex quantum systems , especially heavy nuclei @xcite . because of the complexity of the interactions , we can postulate that the elements of the hamiltonian describing a heavy nucleus are random variables drawn from a probability distribution and these elements are independent with each other . a series of remarkable predictions are found to be in agreement with the experimental data . the great successes of rmt in analyzing complex nuclear spectra has stimulated a widely extension of this theory to several other fields , such as the quantum chaos , the time series analysis @xcite , the transport in disordered mesoscopic systems , the complex networks @xcite , and even the qcd in field theory . for the complex quantum systems , the predictions represent an average over all possible interactions . the deviations from the universal predictions are the clues that can be used to identify system specific , non - random properties of the system under consideration . one of the most important concepts in rmt is the nearest neighbor level spacing ( nnls ) distribution @xcite . enormous experimental and numerical evidence tells us that if the classical motion of a dynamical system is regular , the nnls distribution of the corresponding quantum system behaves according to a poisson distribution . if the classical motion is chaotic , the nnls distribution will behave in accordance with the wigner dyson ensembles , i.e , @xmath28 . @xmath29 is the nnls . the nnls distribution of a quantum system can tell us the dynamical properties of the corresponding classical system . this fact is used in this paper to bridge the structure of a network with the dynamical characteristics of the dynamical system defined on it . from the state of the considered system , @xmath30 , we can construct the collective motion of the system as , @xmath31 where @xmath32 and @xmath33 are the phase and the amplitude of the oscillator @xmath5 . @xmath34 is the other oscillation - related parameters . @xmath35 describes the elastic wave on the considered network and @xmath36 presents the displacements at the positions @xmath21 at time @xmath37 . because of the identification of the oscillators , the individual motions should behave same except the phases and the amplitudes . the synchronization process can be described as the transition from an arbitrary initial collective state , @xmath38 , to the final fully synchronized state , @xmath39 . the probability of the transition should be the synchronizability of the considered network . the larger the transition probability , the easier for the system to achieve the fully synchronized state . the collective states are the elastic waves on the considered network . this kind of classical waves are analogous with the quantum wave of a tight - binding electron walking on the network . they obey exactly a same wave equation . in literature@xcite , this analogy is used to extend the concept of anderson localization state to the classical phenomena as elastic and optical waves . in this paper we will use it to find a quantitative description of the transition probability between the collective states . the tight - binding hamiltonian of an electron walking on the network reads , @xmath40 where @xmath41 is the site energy of the @xmath5th oscillator , @xmath42 the hopping integral between the nodes @xmath5 and @xmath6 . because of the identification of the oscillators , all the site energies are same , denoted with @xmath43 . generally , we can set @xmath44 and @xmath45 , which leads to the relation @xmath46 . ranking the spectrum of @xmath1 as @xmath47 , we denote the corresponding quantum states with @xmath48 . hence , the nnls distribution of the adjacent matrix @xmath1 can show us the dynamical characteristics of the collective motions . if the nnls obeys the poisson form , the transition probability between two eigenstates @xmath49 and @xmath50 will decrease rapidly with the increase of @xmath51 , and the transition occurs mainly between the nearest neighboring eigenstates . this state is called quantum regular state . if the nnls obeys wigner form , the transitions between all the states in the same chaotic regime the initial state belongs to can occur with almost same probabilities . the electron is in a quantum chaotic state . the corresponding collective states of the classical dynamical system to the quantum chaotic and regular sates are called collective chaotic and collective regular states , respectively . if the dynamical system is in a collective chaotic state , the collective motion modes in same chaotic regimes can transition between each other abruptly , while if the system is in a collective regular state only the neighboring collective motion modes can transition between each other . generally , a dynamical system may be in an intermediate state between the regular and the chaotic states , which is called soft chaotic state . the nnls distribution can be obtained by means of a standard procedure . the first step is the so - called unfolding . in the theoretical predictions for the nnls , the spacings are expressed in units of average eigenvalue spacing . generally , the average eigenvalue spacing changes from one part of the eigenvalue spectrum to the next . we must convert the original eigenvalues to new variables , called unfolded eigenvalues , to ensure that the spacings between adjacent eigenvalues are expressed in units of local mean eigenvalue spacing , and thus facilitates comparison with analytical results . define the cumulative density function as , @xmath52 , where @xmath53 is the density of the original spectrum . dividing @xmath54 into the smooth term @xmath55 and the fluctuation term @xmath56 , i.e. , @xmath57 , the unfolded energy levels can be obtained as , @xmath58 if the system is in a soft chaotic state , the nnls distribution can be described with the brody form @xcite , which reads , @xmath59.\ ] ] we can define the accumulative probability distribution as , @xmath60 . the parameter @xmath61 can be obtained from the linear relation as follows , @xmath62 = \beta lns - \beta ln\eta .\ ] ] for the special condition @xmath63 , the probability distribution function ( pdf ) @xmath64 degenerates to the poisson form and the system is in a regular state . for another condition @xmath65 , the pdf obeys the wigner - dyson distribution @xmath66 and the system is in a hard chaotic state . if the system is in an intermediate soft chaotic state , we have , @xmath67 . hence , from the perspective of random matrix theory , the synchronizability can be described with the parameter @xmath61 . the larger the value of @xmath68 , the easier for the system to become fully synchronized . by this way we find a possible dynamical mechanism for the enhancement effects of the network structures on the synchronization processes . in reference @xcite , the authors prove that the spectra of the erdos - renyi , the watts - strogatz(ws ) small - world , and the growing random networks ( grn ) can be described in a unified way with the brody distribution . herein , we are interested in the relation between the parameter @xmath61 and the eigenratio @xmath69 . detailed works show that @xmath69 is a good measure of the synchronizability of complex networks , especially the small world and scale - free networks @xcite . figure 1 shows the relation between @xmath61 and @xmath69 for ws small - world networks @xcite . we use the one - dimensional regular lattice - based model . in the regular lattice each node is connected with its @xmath70 right - handed neighbors . connecting the starting and the end of the lattice , with the rewiring probability @xmath71rewire the end of each edge to a randomly selected node . in this rewiring procedure self - edge and double edges are forbidden . numerical results for ws small - world networks with @xmath72 and @xmath73 are presented . we can find that the brody distribution can capture the characteristics of the pdfs of the nnls very well , as shown in the panel ( a ) in fig.1 . with the increase of @xmath71 , the parameter @xmath61 increases rapidly from @xmath74 to @xmath75 , while the parameter @xmath69 decreases rapidly from @xmath76 to @xmath77 . hence , there exists a monotonous relation between the two parameters @xmath68 and @xmath69 . figure 2 gives the results for barabasi - albert ( ba ) scale - free networks @xcite . starting from a seed of several connected nodes , at each time step connect a new node to the existing graph with @xmath78 edges . the preferential probability to create an edge between the new node and an existing node @xmath79 is proportional to its degree , i.e. , @xmath80 . numerical results for ba scale - free networks with @xmath72 and @xmath81 are presented . all the pdfs of the nnls obey the brody distribution almost exactly . with the increase of @xmath78 , the parameter @xmath61 increases from @xmath82 to @xmath83 , while the parameter @xmath69 decreases from @xmath84 to @xmath77 . we can find also a monotonous relation between the two parameters @xmath68 and @xmath69 . for @xmath85 , we have @xmath86 . that is , rather than the `` repulsions '' or un - correlations between the levels , there are a certain `` attractiveness '' between the levels . in the construction of the ba networks with @xmath85 , each time only one node is added to the existing network . the resulting network is a tree - like structure without loops at all . dividing the network into subnetworks , we can find that many of them have similar structures , which leads their corresponding level - structures being almost same . because of the weak coupling between the subnetworks , the total level structure can be produced just by put all the corresponding levels together . this kind of level - structure will lead many nnls tending to zero . hence , @xmath87 is an extreme case induced by tree - like structure . this special kind of tree - like ba networks can not enhance the synchronization at all . in summary , by means of the nnls distribution we consider the collective dynamics in the networks of coupling identical oscillators . for the two kinds of networks , we can find the monotonous relation between the two parameters @xmath61 and @xmath69 . this monotonous relation tells us that the high synchronizability is accompanied with a high extent of collective chaos . the collective chaos may increase significantly the transition probability of the initial random state to the final synchronized state . the collective chaotic processes may be the dynamical mechanism for the enhancement impacts of network structures on the synchronizabilities . the parameter @xmath61 in the nnls distribution can be a much more informative measure of the synchronizability of complex networks . it reveals the information of the dynamical processes from an arbitrary initial state to the final synchronized state . it can be regarded in a certain degree as the bridge between the structures and the dynamics of complex networks . one paradox may be raised about the argument in the present paper . the wigner distribution implies a larger correlation between the eigenstates of the network than does the poisson distribution . at the same time , one can reverse the argument that wigner distribution implies level repulsion and , therefore , different frequencies of oscillation of the normal modes , and therefore no synchronization when these modes are coupled . it should be emphasized that the eigenratio @xmath69 and the index @xmath68 should be used together to capture the impacts of the network structures on the synchronization processes . @xmath69 represents the linear stability of the synchronized state , but it can not tell us how the final synchronized state is reached from the initial state . on the other hand , @xmath68 provides us a possible mechanism for this dynamical processes , but it can not tell us the transition orientation . @xmath69 and @xmath68 reflect some features of the impacts of the network structures on the synchronization processes , but there may be some new important features to be found . this work was supported by the national science foundation of china under grant no.70571074 , no.70471033 and no.10635040 . it is also supported by the specialized research fund for the doctoral program of higher education ( srfd no . 20020358009 ) . one of the authors would like to thank prof . y. zhuo and j. gu in china institute of atomic energy for stimulating discussions . 1 r. albert , and a. -l . barabasi , rev . phys . * 74 * , 47(2002 ) . s. n. dorogovtsev , and j. f. f. mendes , adv . phys . * 51 * , 1079(2002 ) . m. e. j. newman , siam review * 45 * , 117(2003 ) . l. f. lago - 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[ 1]the relation of @xmath68 versus @xmath69 for the constructed ws small - world networks . ( a ) several typical results for pdf of the nnls . in the interested regions a brody distribution can capture the characteristics very well . ( b ) with the increase of the rewiring probability @xmath88 the eigenratio @xmath69 decreases rapidly . ( c ) with the increase of the rewiring probability @xmath88 the parameter @xmath68 increases rapidly . ( d ) the monotonous relation between the two parameters @xmath68 and @xmath69 . , title="fig : " ] [ 1]the relation of @xmath68 versus @xmath69 for the constructed ba scale - free networks . ( a ) results for pdf of the nnls . in the interested regions a brody distribution can capture the characteristics very well . ( b ) with the increase of @xmath78 the eigenratio @xmath69 decreases significantly . ( c ) with the increase of @xmath78 the parameter @xmath68 increases significantly . ( d ) the monotonous relation between the two parameters @xmath68 and @xmath69 . , title="fig : " ]

the random matrix theory is used to bridge the network structures and the dynamical processes defined on them . we propose a possible dynamical mechanism for the enhancement effect of network structures on synchronization processes , based upon which a dynamic - based index of the synchronizability is introduced in the present paper . * the impact of network structures on the synchronizability of the identical oscillators defined on them is an important topic both for theory and potential applications . from the view point of collective motions , the synchronization state is a special elastic wave occurring on the network , while the initial state is a abruptly assigned elastic state . the synchronizability should be the transition probability between the two states . by means of the analogy between the collective state and the motion of an electron walking on the network , we can use the quantum motion of the electron to find the motion characteristics of the collective states . the random matrix theory ( rmt ) tells us that the nearest neighbor level spacing distribution of the quantum system can capture the dynamical behaviors of the quantum system and the corresponding classical system . a poison distribution shows that the transition can occur only between successive eigenstates , while a wigner distribution shows that the transition can occur between any two eigenstates . a brody distribution , an intermediate between the two extreme conditions , can give us a quantitative description of the transition probability . hence , it can be used as an index to represent the synchronizability . as examples , the watts - strogatz ( ws ) small - world networks and the barabasi - albert(ba ) scale - free networks are considered in this paper . comparison with the widely used eigenratio index shows that this index can describe the synchronizability very well . it is a dynamic - based index and can be employed as a measure of the structures of complex networks . *

recent observations of type ia supernovae @xcite supported by wmap measurements of anisotropy of the angular temperature fluctuations @xcite indicate that our universe is spatially flat and accelerating . on the other hand , the power spectrum of galaxy clustering @xcite indicates that about @xmath7 of critical density of the universe should be in the form of non - relativistic matter ( cold dark matter and baryons ) . the remaining , almost two thirds of the critical energy , may be in the form of a component having negative pressure ( dark energy ) . although the nature of dark energy is unknown , the positive cosmological constant term seems to be a serious candidate for the description of dark energy . in this case the cosmological constant @xmath8 and energy density @xmath9 remain constant with time and the corresponding mass density @xmath10 @xmath11 , where @xmath12 is the hubble constant @xmath13 expressed in units of @xmath14 @xmath15 @xmath16 and @xmath17 . although the cold dark matter ( cdm ) model with the cosmological constant and dust provides an excellent explanation of the snia data , the present value of @xmath8 is @xmath18 times smaller than value predicted by the particle physics model . many alternative condidates for dark energy have been advanced and some of them are in good agreement with the current observational constraints @xcite . moreover , it is a natural suggestion that @xmath8-term has a dynamical nature like in the inflationary scenario . therefore , it is reasonable to consider the next simplest form of dark energy alternative to the cosmological constant @xmath19 for which the equation of state depends upon time in such a way that @xmath20 , where @xmath21 is a coefficient of the equation of state parametrized by the scale factor or redshift . it has been demonstrated @xcite that dynamics of such a system can be represented by one - dimensional hamiltonian flow @xmath22 where the overdot means differentiation with respect to the cosmological time @xmath23 and @xmath2 is a potential function of the scale factor @xmath24 given by @xmath25 where @xmath26 is the effective energy density which satisfies the conservation condition @xmath27 for example , for the @xmath28 model we have @xmath29 of course the trajectories of the system lie on the zero energy surface @xmath30 . hamiltonian ( [ eq:1 ] ) can be rewritten in the following form convenient for our current reconstruction of the equation of state from the potential function @xmath31 , namelly @xmath32 where @xmath33 , here overdot means differentiation with respect to some new reparametrized time @xmath34 , @xmath35 . for example , for mixture of non - interacting fluids potential @xmath36 takes the form @xmath37 where @xmath38 for @xmath39-th fluid and @xmath40 @xmath41 ( similar to the quiessence model of dark energy ) . due to the hamiltonian structure of friedmann - robertson - walker dynamics , with the general form of the equation of state @xmath1 , the dynamics is uniquely determined by the potential function @xmath2 ( or @xmath36 ) of the system . only for simplicity of presentation we assume that the universe is spatially flat ( in the opposite case trajectories of the system should be considered on the energy level @xmath42 ) . let us note that from the potential function we can obtain the equation of state coefficient @xmath43 , @xmath44 the term @xmath45 has a simple interpretation as an elasticity coefficient of the potential function with respect to the scale factor . thus from the potential function @xmath31 both @xmath46 and @xmath47 can be unambiguously calculated @xmath48 as it is well known in a flat frw cosmology the luminosity distance @xmath49 and the coordinate distance @xmath50 to an object at redshift @xmath51 are simply related as @xmath52 ( @xmath53 here and elsewhere ) . from equation ( [ eq:8 ] ) the hubble parameter is given by @xmath54 } = \bigg[\frac{d}{dz}\bigg(\frac{d_{l}(z)}{1+z}\bigg)\bigg ] . \label{eq:9}\ ] ] it is crucial that formula ( [ eq:9 ] ) is purely kinematic and depends neither upon a microscopic model of matter , including the @xmath8-term , nor on a dynamical theory of gravity . due to existance of such a relation it would be possible to calculate the potential function which is : @xmath55^{-2}}{2(1+z)^{2}}. \label{eq:10}\ ] ] this in turn allows us to reconstruct the potential @xmath56 from snia data . let us note that @xmath56 depends on the first derivative with respect to @xmath51 whereas @xmath57 is associated with the second derivative . let us also note that from of the potential function for a one - dimensional particle - universe moving in the configurational @xmath24 ( or @xmath58)-space can be reconstructed from recent measurements of angular size of high - z compact radio sources compiled by gurvits _ the corresponding formula is @xmath59^{-2}}{2(1+z)^{2 } } , \label{eq:10a}\ ] ] where the luminosity distance @xmath49 and the angular distance @xmath60 are related by the simple formula @xmath61 since the potential function is related to the luminosity function by relation ( [ eq:10 ] ) one can determine both the class of trajectories in the phase plane @xmath62 and the hamiltonian form as well as reconstruct the quintessence parameter @xmath57 provided that the luminosity function @xmath63 is known from observations . now we can reconstruct the form of the potential function ( [ eq:10 ] ) using a natural ansatz introduced by sahni _ _ @xcite . in this approach dark energy density which coincides with @xmath26 is given as a truncated taylor series with respect to @xmath64 @xmath65 this leads to @xmath66 and @xmath67 the values of three parameters @xmath68,@xmath69,@xmath70 can be obtained by applying a standard fitting procedure to snia observational data based on the maximum likelihood method . the potential function ( [ eq:10 ] ) written in terms of @xmath71 is @xmath72 \label{eq:13}\ ] ] or in dimensionless form @xmath73 , \label{eq:14}\ ] ] where @xmath74 , @xmath75 . our approach to the reconstruction of dynamics of the model is different from the standard approach in which @xmath57 is determined directly from the luminosity distance formula . it should be stressed out that the latter approach has an inevitable limitation because the luminosity distance dependence on @xmath57 is obtained through a multiple - integral relation that loses detailed information on @xmath57 @xcite . in our approach the reconstruction is simpler and more information on @xmath57 survives ( only a single integral is required ) . our approach is also different from the concept of reconstruction of potential of scalar fields considered in the context of quintessence @xcite . the key steps of our method are the following : * 1 ) * we reconstruct the potential function @xmath2 for the hamiltonian dynamics of the quintessential universe from the luminosity distances of supernovas type ia ; + * 2 ) * we draw the best fit curves and confidence levels regions obtained from the statistical analysis of snia data ; + * 3 ) * we set the theoretically predicted forms of the potential functions on the confidence levels diagram ; + * 4 ) * those theoretical potential which escape from the @xmath4 confidence level is treated as being unfitted to observations ; + * 5 ) * we choose this potential function which lie near the best fit curve . + our reconstruction is an effective statistical technique which can be used to compare a large number of theoretical models with observations . instead of estimating some revelant parameters for each model separately , we choose a model - independent fitting function and perform a maximum likelihood parameter estimation for it . the obtained confidence levels can be used to discriminate between the considered models . in this paper this technique is used to find the fitting function for the luminosity distance . the additional argument which is important when considering the potential @xmath2 is that it allows to find some modification in the friedmann equations along the `` cardassian expansion scenario '' @xcite . this proposition is very intriguing because of additional terms , which automatically cause the acceleration of the universe @xcite . these modifications come from the fundamental physics and these terms can be tested using astronomical observation of distant type ia supernovae . for this aim the recent measurements of angular size of high - redshift compact radio sources can also be used @xcite . the important question is the reliable data available . we expect that supernovae data would improve greatly over next few years . the ongoing snap mission should gives us about 2000 type ia supernovae cases each year . this satellite mission and the next planned ones will increase the accuracy of data compared to data from the 90s . in our analysis we use the availale data starting from the three perlmutter samples ( sample a is the complete sample of 60 supernovae , but in the analysis it is also used sample b and c in which 4 and 6 outliers were excluded , respectively ) . the fit for the sample c is more robust and this sample was accepted as the base of our consideration . for technical details of the metod the reader is referred to our previous two papers @xcite . in fig . 1 we show the reconstructed potential function obtained using the fitting values of @xmath76 as well as @xmath77 . the red line represents the potential function for the best fit values of parameters ( see tables [ resultsa ] , [ resultsc ] and [ resultspac ] ) . in each case the coloured areas cover the confidence levels @xmath78 ( @xmath79 ) and @xmath80 ( @xmath4 ) for the potential function . the different forms of the potential function which are obtained from the theory are presented in the confidence levels . here we consider one case , namely the cardassian model . in this case the standard frw equation is modified by the presence of an additional @xmath5 term , where @xmath81 is the energy density of matter and radiation . for simplicity we assume that density parameter for radiation is zero ( see table . [ pottab ] ) . the cardassian scenario is proposed as an alternative to the cosmological constant in explaining the acceleration of the universe . in this scenario the the universe automaticaly accelerates without any dark energy component . the additional term in the friedmann equation arises from exotic physics of the early universe ( i.e. , in the brane cosmology with randall - sundrum version @xmath82 ) . .the forms of the potential functions in dimensionless form for two cases : @xmath28 model and cardassian scenario . [ cols="^,^,^,^,^ " , ] [ resultspac ] the dynamics of the considered cosmological models is governed by the dynamical system @xmath83 with the first integral for ( [ eq:16 ] ) @xmath84 . the main aim of the dynamical system theory is the investigation of the space of all solutions ( [ eq:16 ] ) for all possible initial conditions , i.e. phase space @xmath85 . in the context of quintessential models with the equation of state @xmath0 there exists a systematic method of reducing einstein s field equations to the form of the dynamical system ( [ eq:16 ] ) @xcite . one of the features of such a representation of dynamics is the possibility of resolving of some cosmological problems like the horizon and flatness problems in terms of the potential function @xmath36 . the phase space @xmath85 ( or state space ) is a natural visualization of the dynamics of any model . every point @xmath86 corresponds to a possible state of the system . the r.h.s of the system ( [ eq:16 ] ) define a vector field @xmath87 $ ] belonging to the tangent space @xmath88 . integral curves of this vector field define one - parameter group of diffeomorphisms @xmath89 called the phase flow . in the phase space the phase curves ( orbits of the group @xmath89 ) represent the evolution of the system whereas the critical points @xmath90 , @xmath91 are singular solutions equilibria from the physical point of view . the phase curves together with critical points constitute the phase portrait of the system . now we can define the equivalence relation between two phase portraits ( or two vector fields ) by the topological equivalence , namely two phase portraits are equivalent if there exists an orientation preserving homeomorphism transforming integral curves of both systems into each other . following the hartman - grobman theorem , near hyperbolic critical points ( @xmath92 @xmath39 @xmath93 , where @xmath94 is the appropriate eigenvalue of linearization matrix @xmath95 of the dynamical system ) is equivalent to its linear part @xmath96 in our case the linearization matrix takes the form @xmath97_{(x_{0},0 ) } \label{eq:18}\ ] ] classification of critical points is given in terms of eigenvalues of the linearization matrix since the eigenvalues can be determined from the characteristic equation @xmath98 . in our case @xmath99 and eigenvalues are either real if @xmath100 or purely imaginary and mutually conjugated if @xmath101 . in the former case the critical points are saddles and in the latter case they are centres . the advantage of representing dynamics in terms of hamiltonian ( [ eq:1 ] ) is the possibility to discuss the stability of critical points which is based only on the convexity of the potential function . in our case the only possible critical points in a finite donain of phase space are centres or saddles . the dynamical system is said to be structurally stable if all other dynamical systems ( close to it in a metric sense ) are equivalent to it . two - dimensional dynamical systems on compact manifolds form an open and dense subsets in the space of all dynamical systems on the plane @xcite . structurally stable critical points on the plane are saddles , nodes and limit cycles whereas centres are structurally unstable . there is a widespread opinion among scientists that each physically realistic models of the universe should possess some kind of structural stability because the existence of many drastically different mathematical models , all in agreement with observations , would be fatal for the empirical method of science @xcite . basing on the reconstructed potential function one can conclude that : + * 1 ) * since the diagram of the potential function @xmath102 is convex up and has a maximum which corresponds to a single critical point the quantity @xmath103 at the critical point ( saddle point ) and the eigenvalues of the linearization matrix at this point are real with oposite signs ; + * 2 ) * the model is structurally stable , i.e. , small perturbation of it do not change the structure of the trajectories in the phase plane ; + * 3 ) * since @xmath104 one can easily conclude from the geometry of the potential function that in the accelerating region ( @xmath105 ) @xmath2 is a decreasing function of its argument ; + * 4 ) * the reconstructed phase portrait for the system is equivalent to the portrait of the model with matter and the cosmological constant . + ) reconstructed from the potential function ( [ eq:14 ] ) for the best fitted parameters ( table [ resultsa ] , @xmath106 ) . the coloured domain of phase space is the domain of accelerated expansion of the universe . the red curve represents the flat model trajectory which separates the regions with negative and positive curvature . ] transformed on the compact poincar sphere . non - physical domain of phase space is marked as pink . ] by an inspection of the phase portraits we can distinguish four characteristic regions in the phase space . the boundaries of region i are formed by a separatrix coming out from the saddle point and going to the singularity and another separatrix coming out of the singularity and approaching the saddle . this region is covered by trajectories of closed and recolapsing models with initial and final singularity . the trajectories moving in region iv are also confined by a separatrix and they correspond to the closed universes contracting from the unstable de sitter node towards the stable de sitter node . the trajectories situated in region iii correspond to the models expanding towards stable de sitter node at infinity . similarly , the trajectories in the symmetric region ii represent the universes contracting from the unstable node towards the singularity . the main idea of the qualitative theory of differential equations is the following : instead of finding and analyzing an indiwidual solution of the model one investigates a space of all possible solutions . any property ( for example acceleration ) is believed to be realistic if it can be attributed to a large subsets of models within the space of all solutions or if it possesses certain stability property which is shared also by all slightly perturbed models . the possible existence of the unknown form of matter called dark energy has usually been invoked as the simplest way to explain the recent observational data of snia . however , the effects arising from the new fundamental physics can also mimic the gravitational effects of dark energy through a modification of the friedmann equation . we exploited the advantages of our method to discriminate among different dark energy models . with the independently determined density parameter of the universe ( @xmath3 ) we found that the current observational results require the cosmological constant @xmath107 in the cardassian models . on fig . [ figpotw ] we can see that both in the case of sample a ( fig . [ figpotw]*c * ) and sample c ( fig . [ figpotw]*a * ) @xmath108 should be close to zero . similarly if we assume that the density parameter for barionic matter is @xmath109 then @xmath110 in the case of sample c ( fig . [ figpotw]*b * ) and @xmath111 for sample a ( fig . [ figpotw]*d * ) . moreover , we showed ( for the sample c of perlmutter snia data ) that a simple cardassian model as a candidate for dark energy is ruled out by our analysis if @xmath112 for @xmath3 and if @xmath113 for @xmath109 at the confidence level @xmath4 .

we demonstrate a model - independent method of estimating qualitative dynamics of an accelerating universe from observations of distant type ia supernovae . our method is based on the luminosity - distance function , optimized to fit observed distances of supernovae , and the hamiltonian representation of dynamics for the quintessential universe with a general form of equation of state @xmath0 . because of the hamiltonian structure of frw dynamics with the equation of state @xmath1 , the dynamics is uniquelly determined by the potential function @xmath2 of the system . the effectiveness of this method in discrimination of model parameters of cardassian evolution scenario is also given . our main result is the following , restricting to the flat model with the current value of @xmath3 , the constraints at @xmath4 confidence level to the presence of @xmath5 modification of the frw models are @xmath6 .

the family of 1212 cuprates has single block layers and involves capable materials for superconducting wires because their anisotropy is less than that of the 2212 and 2223 families , which have double block layers bonded by van der waals forces . the crystal structure of pbsr@xmath2cacu@xmath2o@xmath5 ( pb1212 ) is very similar to that of tlba@xmath2cacu@xmath2o@xmath14 ( tl1212 ) and hgba@xmath2cacu@xmath2o@xmath15 ( hg1212 ) , both of which have a critical temperature @xmath16 greater than 100 k. pb1212 is expected to be less anisotropic than tl1212 and hg1212 because the ionic radius of the element account for the block layers is the smallest in the series ( @xmath17 @xmath18 @xmath19 @xmath20 @xmath21 ) . among the pb1212 compounds , pb@xmath0bi@xmath1sr@xmath2y@xmath3ca@xmath4cu@xmath2o@xmath5 ( pbbi1212)@xcite and pb@xmath0cu@xmath1sr@xmath2y@xmath3ca@xmath4cu@xmath2o@xmath5 ( pbcu1212)@xcite are of specific importance for industry because they mainly consist of the ubiquitous elements pb and ca rather than the more exotic bi and y , which are currently used for high-@xmath16 superconducting wires . however , for pb1212 , the superconducting properties such as anisotropy and critical current are inadequately understood because of the difficulty of obtaining single - crystal samples of this material . this lack of reliable samples has even led to claims that pbbi1212 is nonsuperconducting and the reported superconductivity of pbbi1212 ( @xmath16 = 92 k@xcite ) is attributed to the secondary phase of bi@xmath22sr@xmath22cacu@xmath22o@xmath23 ( bi2212 ) . therefore , this study aims to verify the superconductivity of pbbi1212 by growing single - crystal pbbi1212 epitaxial films . growing thin films of cuprates that incorporate pb and bi is extremely challenging because their vapor pressure is very high to maintain the chemical composition of the film at the growth temperature of approximately 700 @xmath7c . karimoto and naito succeeded in growing an epitaxial film of pbsr@xmath2cuo@xmath24 by molecular beam epitaxy@xcite . however , they also reported that the method does not work well for the growth of pb1212 , because the growth temperature is limited due to the volatility of pb@xcite . although we attempted to grow pbbi1212 by conventional @xmath25 @xmath26 sputtering , we never obtained single - phase films of pbbi1212 because of the re - evaporation of pb and bi from the substrate . in the present paper , we describe a two - step growth technique@xcite that allows us to maintain the chemical composition of the film : pbbi1212 deposited on a srtio@xmath6 ( sto ) substrate at low temperature forms an amorphous film which , at high temperature , crystallizes with the crystallographic symmetry of the substrate . we demonstrated that this technique works well for fabricating single - phase pbbi1212 epitaxial films . it is found that the pbbi1212 system exhibits superconductivity at approximately 50 k. we also clarify various superconducting properties of the pb1212 system such as substitution effects , coherence lengths , and anisotropy . the two - step growth technique may provide a method to fabricate next - generation superconducting wires from ubiquitous elements . pbbi1212 epitaxial film was grown by a two - step technique consisting of a low temperature sputtering step and a high temperature @xmath27 @xmath26 growth step . sputtering targets were synthesized by the solid - state reaction method using high purity powders ( @xmath20 99.9% ) of pbo , bi@xmath2o@xmath6 , srco@xmath6 , y@xmath2o@xmath6 , caco@xmath6 , and cuo . these powders were mixed into compositions of ( pb@xmath28bi@xmath29)@xmath30sr@xmath2y@xmath3ca@xmath4cu@xmath31o@xmath32 ( @xmath33 ) and calcined two times : first at 860 @xmath7c for 10 h in air and then at 880 @xmath7c for 10 h in air . after calcination , the powders were pressed into cylindrical pellets 100 mm in diameter and 7 mm in height , and they were sintered at @xmath34 @xmath7c for 24 h in air , where @xmath8 is the ca concentration . for depositing the pbbi1212 amorphous films on sto ( 100 ) substrates , we used the following sputtering conditions : the sputtering gas pressure was 100 mtorr ( 60 sccm ar and 15 sccm o@xmath2 ) , the anode voltage was 1.4 kv , and the substrate temperature was approximately 200 @xmath7c ( the substrates were not heated ) . the deposition time was set to 1 2 h. the thickness of the pbbi1212 thin films measured by a stylus - based profilometer was 1800 3500 . pbbi1212 containers used for @xmath27 @xmath26 growth were made of polycrystalline pellets prepared in the same way as the sputtering targets at compositions of pb@xmath0bi@xmath1sr@xmath2y@xmath35ca@xmath36cu@xmath2o@xmath32 ( @xmath37 = 0 0.5 ) . the mixed powders were calcined three times at 880 @xmath7c for 10 h in air , pressed into two cylindrical pellets 26 mm in diameter and 5 mm in height , and sintered at 1007 @xmath7c for 3 h in air . for epitaxial growth , amorphous films on sto substrates were placed in a pit ( 8 @xmath38 8 @xmath38 2 mm@xmath39 ) formed at the center of one of the sintered pellets as shown in fig . the other pellet was used as a lid for the growth container . the container containing the amorphous film was heated in a muffle furnace at 970 @xmath7c for 6 h under an o@xmath2 atmosphere and cooled to room temperature at a rate of 200 @xmath7c / h . the concentration of the films was determined by energy dispersive x - ray spectroscopy ( eds ) , and it was found that the concentration of pb and bi drastically changes during the film growth . the concentration of pb and bi is dominated by the composition of the growth container , whereas the concentrations of the other elements in the films are dominated by the composition of the sputtering target . as - grown films are not superconducting and a subsequent quenching treatment is necessary to make them superconducting . in this treatment , the film was placed in a quartz tube and heated at 815 @xmath7c in air for 1 h. within two seconds after removing the quartz tube from the furnace , it was placed in liquid nitrogen . this procedure increases the hole concentration and makes the film superconducting , as for the case of bulk polycrystalline pbcu1212 , as reported by maeda @xmath40 @xmath41@xcite . the hall coefficient and temperature dependence of the resistivity under a magnetic field were measured by the ac four - probe method with a physical properties measurement system ( quantum design co. ltd . ) . the hall coefficient was determined at various temperatures by a linear fit to the transverse resistivity as a function of the external magnetic field ( @xmath42 @xmath43 @xmath13 ) between @xmath44 5 t. 2@xmath45 scan for single - phase pb1212 thin film ( @xmath8 = 0.32 , @xmath37 = 0.00 ) . ( c ) magnified @xmath452@xmath45 scan at pbbi1212 ( 005 ) with different ca and bi concentrations . red , blue , and black curves represent data for ( @xmath8 , @xmath37 ) = ( 0.32 , 0.00 ) , ( 0.32 , 0.21 ) , and ( 0.37 , 0.00 ) , respectively . @xmath13-axis lattice constants are 11.832 , 11.820 , and 11.861 , respectively . the dashed ( red ) line represents data for the as - grown film ( @xmath13 = 11.861 ) with ( @xmath8 , @xmath37 ) = ( 0.32 , 0.00 ) . ( d ) surface morphology for single - phase pbbi1212 thin film ( @xmath8 = 0.31 , @xmath37 = 0.06 ) . ( e ) in - plane xrd 2@xmath45@xmath46@xmath47 scan pattern around ( 200 ) peaks of pbbi1212 and sto . ( f ) @xmath47 scan pattern at pbbi1212 ( 200 ) ( 2@xmath45@xmath46 = 47.58@xmath7).,width=321 ] figure 1(b ) shows the out - of - plane x - ray diffraction ( xrd ) @xmath48 scan results for a thin film of a single - phase pbbi1212 . these data indicate the complete @xmath13-axis alignment of the film . in thin films of single - phase pbbi1212 , the peaks from other phases were completely absent or less than 0.5% of the peak magnitude for pbbi1212 ( 005 ) . it was found that a ca concentration of @xmath49 and a bi concentration of @xmath50 are required to obtain single - phase pbbi1212 films . impurity phases such as bi2212 were detected in samples whose concentrations of ca or bi exceeded these limits . to prevent the growth of the impurity phase , the ca concentration of the sputtering target has to be less than 0.5 and the bi concentration of the growth container has to be less than 0.25 . figure 1(c ) represents the magnified @xmath48 scan profiles at the peaks of pbbi1212 ( 005 ) . it is found that the quenching treatment reduces the @xmath13-axis lattice constant ( approximately 0.03 ) and fwhm of the peak . this indicates that the lattice strains along the @xmath13 axis are reduced by the quenching treatment . the o@xmath2 annealing at 500 @xmath7c for 12 h results in doping holes for quenched films . however , the annealing yields little effect for as - grown films before the quenching treatment . this suggests that a significant amount of oxygen deficiency is induced by the quenching treatment . hall effect measurements for samples before and after quenching revealed that the quenching treatment decreases the hall coefficient significantly . we consider that the decrease in oxygen content effects a reduction in the lattice strains and this is the reason why the hole concentration of the film increases by the quenching treatment . samples with low ca and bi concentrations often exhibit high crystallinity and the peaks of k@xmath51 and k@xmath52 are separately identified as shown in fig . increases in @xmath8 ( ca concentration ) and @xmath37 ( bi concentration ) lead to an increase and decrease in the @xmath13-axis lattice constant , respectively . a significant substitution of ca and bi tends to increase the fwhm of the peak and decrease the peak intensity , which are attributed to the lattice deformation . figure 1(d ) is an image of the surface of a thin film of single - phase pbbi1212 . the image was acquired by scanning electron microscopy . the surface morphology of the film is smooth except for a few steps . this morphology implies that film growth progresses along the surface of the substrate . note that no cracks , grain boundaries , or impurity phase precipitations were found on the entire surface of the film . we used in - plane xrd measurements to check whether the films grow epitaxially on the substrates . figure 1(e ) shows the result of a 2@xmath45@xmath46@xmath47 scan for a single - phase pbbi1212 film . the peaks of sto ( 200 ) and pbbi1212 ( 200 ) are identified . this result shows that lattice relaxation occurs from the 2% in - plane lattice mismatch between sto ( @xmath53 = 3.905 ) and pbbi1212 ( @xmath53 = 3.813.83 ) . the result of an in - plane @xmath47 scan at the peak of pbbi1212 ( 200 ) ( 2@xmath45@xmath46 = 47.58@xmath7 ) is displayed in fig . four peaks are found at 90@xmath7 between each peak , and the difference in the angle of the peak between sto and pbbi1212 is less than 0.7@xmath7 . these results indicate that the pbbi1212 thin films grow epitaxially on sto ( 100 ) substrates . , @xmath37 ) = ( 0.29 , 0.00 ) , ( 0.33 , 0.00 ) , and ( 0.37 , 0.00 ) , respectively . ( b ) black , blue , red , and green curves represent data for ( @xmath8 , @xmath37 ) = ( 0.35 , 0.17 ) , ( 0.35 , 0.12 ) , ( 0.37 , 0.00 ) , and ( 0.49 , 0.48 ) , respectively . ( c ) black diamonds , filled blue circles , and red squares represent data for ( @xmath8 , @xmath37 ) = ( 0.25 , 0.15 ) , ( 0.31 , 0.13 ) , and ( 0.36 , 0.17 ) , respectively . ( d ) black triangles , blue triangles , and red crosses represent data for ( @xmath8 , @xmath37 ) = ( 0.35 , 0.05 ) , ( 0.35 , 0.17 ) , and ( 0.50 , 0.65 ) , respectively . numbers in parentheses correspond to those in fig . 3.,width=321 ] [ ( red ) crosses and filled black circles , left axis ] for various bi concentrations @xmath37 and @xmath16@xmath54 [ ( blue ) triangles , right axis ] for @xmath11 as a function of ca concentration @xmath8 . , width=321 ] figures 2(a ) and 2(b ) show the temperature dependence of the resistivity for samples with different ca and bi concentrations , respectively . the composition of the film was determined by eds at three points ( 25 @xmath38 25 @xmath55m@xmath56 for each point ) . as shown in fig . 2(a ) , an increase in ca concentration @xmath8 leads to a decrease in resistivity and @xmath57 is necessary for the superconducting transition . notably , a very small increase in ca concentration induces a transition from an insulator to a superconductor . however , further increasing the ca concentration does not significantly affect @xmath16 . in contrast , replacing pb with bi up to @xmath37 = 0.17 drastically decreases @xmath16 , as shown in fig . 2(b ) . for @xmath37 @xmath20 0.20 , the films include bi2212 as a secondary phase and exhibit distinctly different properties than the single - phase films . the resistivity decreases with increasing bi concentration and even without the quenching treatment , a sharp superconducting transition takes place at 80 k. also , the xrd peaks of the bi2212 phase become more intense . therefore , we conclude that the superconducting transition observed for @xmath37 @xmath20 0.20 is due to the bi2212 impurity phase . the temperature dependence of the hall coefficient for different ca and bi concentrations is shown in figs . 2(c ) and 2(d ) . figure 2(c ) suggests that replacement of y with ca increases the hole concentration . however , as shown in fig . 2(d ) , no significant change in the hall coefficient as a function of the bi concentration is observed for the single - phase samples ( @xmath50 ) . the significant decrease in the hall coefficient for @xmath58 is attributed to the bi2212 impurity phase . we find that the slight increase in hole concentration , e.g. , from @xmath8 = 0.31 to 0.36 shown in fig . 2(c ) , causes superconductivity . all nonsuperconducting samples are insulators at low temperatures . this sudden change from insulator to superconductor is quite particular to this pbbi1212 system . in contrast , bi substitution does not introduce holes , but it introduces lattice deformation that strongly decreases @xmath16 , especially at @xmath8 @xmath18 0.32 . this is consistent with the @xmath59 data in fig . 2(b ) , where the residual resistivity [ determined by the intersection of the linear extrapolation of @xmath59 data above @xmath16 and @xmath60 = 0 axis ] increases with @xmath37 . the result of the xrd measurement in fig . 1(c ) also shows the influence of lattice deformation from the bi substitution . furthermore , the reduction in the @xmath13-axis lattice constant due to the bi substitution presumably causes the decrease in @xmath16 , because the @xmath16 of the 1212 family tends to decrease with decreasing distance between adjacent cuo@xmath22 double layers ( @xmath61 @xmath20 @xmath62 @xmath20 @xmath63 ) as seen in other families . the substitution effect found in this study is summarized in fig . this plot clearly illustrates that , for superconductivity in pbbi1212 , a ca concentration @xmath8 @xmath20 0.3 is essential and a lower bi concentration is preferable . the effective number of holes per cu atom ( @xmath64 ) as a function of the ca concentration @xmath8 for various bi concentrations @xmath37 is plotted in fig . 3(b ) , where @xmath65 is the unit cell volume for a cu atom . the quantity @xmath9 depends linearly on @xmath8 , irrespective of @xmath37 . among these samples ( @xmath66 ) , the superconductivity occurs at @xmath67 , which is between the hole number for optimum yba@xmath22cu@xmath68o@xmath14 ( ybco ) ( @xmath69 , @xmath70 ) and the hole number for the superconductor - insulator transition ( @xmath71)@xcite . this trend is the same as for la@xmath72sr@xmath73cuo@xmath74 ( lsco)@xcite . therefore , we expect a further increase in @xmath16 for our sample upon doping with more hole carriers . in fact , we find that , for samples with @xmath37 = 0 , @xmath16@xmath54 increases with @xmath8 up to @xmath8 = 0.37 , as shown in fig . the linear fit shown in fig . 3(b ) suggests that carrier doping is governed by the substitution of y@xmath75 for ca@xmath76 and 0.34 hole is introduced by the substitution of a ca@xmath76 ion . this is similar to the underdoped region in lsco@xcite and ybco@xcite , where the effective number of holes is well described as @xmath9 @xmath77 @xmath8 . the remarkable point of doping in pb1212 is that the fitting line is extrapolated to the zero point of @xmath9 for @xmath8 @xmath78 0 . this indicates that all the transport carriers originate from the ca substitution , and all the doped carriers contribute as the transport carriers . as @xmath8 is increased above 0.37 , @xmath16 remains constant , whereas at 300 k , @xmath79 decreases . this implies that increasing @xmath8 creates not only hole carriers but also lattice deformation that is more sensitive to the emergence of the superconductivity than to the normal - state conductivity . the broadened ( 005 ) peak shown in fig . 1(c ) elucidates that the ca substitution introduces crystallographic disorder as discussed above . the ( y , ca ) site of pb1212 is inside the cuo@xmath22 double layer , as is the case for ( y , ca)ba@xmath22cu@xmath68o@xmath80 with @xmath16 approximately equal to 90 k. eisaki @xmath40 @xmath41 . pointed out the strong influence of the ( y , ca ) site disorder on @xmath16@xcite . concerning ca substitution at @xmath11 , we conclude that chemical disorder prevents the increase in @xmath16 expected from hole doping . to obtain the highest @xmath16 for this material , the film properties must be further optimized . @xmath43 @xmath13 and ( b ) @xmath42 @xmath43 @xmath12 . ( c ) irreversibility fields and upper critical fields in a sample with ( @xmath8 , @xmath37 ) = ( 0.34 , 0.00 ) . ( c ) diamonds and filled circles represent data for @xmath42 @xmath43 @xmath13 and @xmath42 @xmath43 @xmath12 , respectively . ( d ) solid and dashed lines are obtained from the empirical formula with @xmath81 = 7.3 and 8.6 , respectively . , width=321 ] figures 4(a ) and 4(b ) show the temperature dependence of resistivity in a magnetic field of up to 9 t for @xmath42 @xmath43 @xmath13 and @xmath42 @xmath43 @xmath12 for the bi - free ( @xmath82 ) sample . the zero - field transition is significantly broadened by applying @xmath42 @xmath43 @xmath13 , whereas the transition is broadened only slightly for @xmath42 @xmath43 @xmath12 . no sharp transition , indicating the vortex lattice melting transition@xcite , is observed . this result is attributed to the distribution of the superconducting condensation energy accompanied by an inhomogeneous oxygen concentration and lattice disorder in the film . to discuss the superconducting anisotropy , the coherence lengths along the @xmath12 plane @xmath83 and the @xmath13 axis @xmath84 are estimated from the upper critical field @xmath85 , which is defined by the criterion of @xmath79/@xmath86 = 90% . this criterion excludes the influence of the flux flow on @xmath85 . @xmath85 for @xmath42 @xmath43 @xmath13 and @xmath42 @xmath43 @xmath12 is plotted in fig . 4(c ) as a function of temperature . the quantities @xmath87 and @xmath88 are estimated to be 54 and 492 t , respectively , using the werthamer - herfand - hohenberg theory@xcite , which gives @xmath89 . according to the relation @xmath90 , the coherence lengths @xmath91 were derived as @xmath83 = 25 and @xmath84 = 2.7 . for @xmath42 @xmath43 @xmath13 , the tangent @xmath92 t / k is slightly smaller than that of the ybco single - crystal ( 1.9 t / k)@xcite . this result indicates that the mean free path @xmath93 of pb1212 is slightly longer than that of ybco , because @xmath94 is roughly proportional to @xmath95 in the dirty limit . this suggests that pb1212 is slightly cleaner than ybco ( @xmath83=16 ) . meanwhile , the value of @xmath96 is close to the thickness of superconducting layers of pb1212 ( 3.2 ) . this is a common feature among cuprates except for the bi - family . consequently , the anisotropy , @xmath97 is estimated to be 9.2 . this anisotropy is similar to that of bulk tl1212 ( @xmath98 @xmath99 10)@xcite and hg1212 ( @xmath98 @xmath99 7.7)@xcite . a further reduction in anisotropy was observed for bi - doped samples ( e.g. , @xmath98 @xmath100 6.5 with @xmath37 @xmath100 0.12 ) , although the @xmath16 values of these samples are lower than those of bi - free samples as discussed above . the reduced anisotropy is presumably caused by the decrease in the distance between adjacent cuo@xmath22 double layers because the ionic radius of bi@xmath101 is smaller than that of pb@xmath102@xcite . these results reveal that the anisotropy of pbbi1212 is much less than that of the modulation - free pbbi2212 ( @xmath98 @xmath99 25)@xcite . this implies that replacing the double ( pb , bi)o block layers with monolayers results in a reduction in the superconducting anisotropy and an enhancement in the coupling between cuo@xmath22 layers . in fig . 4(c ) , the irreversibility field @xmath103 defined by the criterion @xmath79 @xmath100 @xmath104 @xmath105-cm is also plotted . unlike the case of @xmath85 , @xmath103 ( especially for @xmath42 @xmath43 @xmath13 ) is significantly shifted to the low temperature side in a high magnetic field . figure 4(d ) shows an enlarged plot of @xmath103 ( @xmath42 @xmath43 @xmath13 ) for @xmath42 @xmath106 1.6 t. it also shows the results of fitting the data to the empirical function + @xmath107=2\times 10^{3}{\rm exp}(-0.78$]@xmath108 ( @xmath109)@xcite , where @xmath81 ( )is the distance between adjacent cuo@xmath22 planes . the curve fit gives the value @xmath110 , which is less than the result of @xmath111 from crystal structure refinements based on xrd and neutron diffraction@xcite . based on eds analysis of the film , a considerable portion of pb ions in the block layer is possibly substituted by cu ions . assuming that the substituted sites , which form cuo@xmath112 octahedral sites , accumulate and form cuo@xmath22 single layers and thereby partly spread the film , the effective value of @xmath81 may be considerably reduced . in ybco , the empirical formula applies for @xmath81 = 4.15 which is close to the distance between the cuo@xmath22 plane and cuo chain@xcite . the superconductivity of pbbi1212 was vertified by measuring the anisotropic transport properties in epitaxial films prepared by the two - step growth technique . films of single - phase pbbi1212 were obtained with a ca concentration of less than 0.36 and a bi concentration of less than 0.20 . we found that a ca concentration greater than 0.3 and a quenching treatment are essential for making the epitaxial films superconducting . increasing the ca concentration @xmath8 leads to an increase in the hole density ( @xmath9 = @xmath113 ) , and increasing the bi concentration leads to a decrease in the critical temperature @xmath16 . a significant substitution of ca and bi induces the lattice deformation , which strongly decreases @xmath16 . the highest @xmath16@xmath54 found for single - phase pbbi1212 films is 65 k , and for the multiphase film , the drop in resistivity starting at a higher temperature is attributed to the bi2212 impurity phase . the coherence lengths of pb1212 estimated from the upper critical field are 25 and 2.7 along the @xmath12 plane and @xmath13 axis , respectively . 99 a. bauer , p. zoller , j. glaser , a. ehmann , w. wischert , and s. kemmler , physica c * 256 * , 177 ( 1996 ) . h. frank , r. stollmann , j. lethen , r. mller , l. v. gasparov , n. d. zakharov , d. hesse , and g. gntherodt , physica c * 268 * , 100 ( 1996 ) . p. zoller , a. ehmann , j. glaser , w.wischert , and s. kemmler - 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thin films of single - crystal pb@xmath0bi@xmath1sr@xmath2y@xmath3ca@xmath4cu@xmath2o@xmath5 ( pbbi1212 ) were grown on srtio@xmath6 ( 100 ) substrates by a two - step growth technique in which an amorphous film is annealed at 970 @xmath7c in a closed ceramic container prepared using the same material as the film . we find that pbbi1212 exhibits superconductivity when the ca concentration @xmath8 exceeds 0.3 . the effective number of holes per cu atom @xmath9 is well described as @xmath10 . the highest onset temperature for the superconducting transition attained in the present study is 65 k. the resistivity measurement in a magnetic field reveals that the coherence lengths of pb1212 ( @xmath11 ) are approximately 25 and 2.7 along the @xmath12 plane and the @xmath13 axis , respectively .

one of the present authors ( n.i . ) together with his collaborators has published series of papers on the calculations of the electrical and thermal conductivities of dense matter ( flowers & itoh 1976 , 1979 , 1981 ; itoh et al . 1983 ; mitake , ichimaru , & itoh 1984 ; itoh et al . 1984 ; itoh & kohyama 1993 ; itoh , hayashi , & kohyama 1993 ) . among these works , the calculation corresponding to the liquid metal case ( itoh et al . 1983 ; hereafter referred to as paper i ) appears to have been most widely used in various fields of stellar evolution studies . therefore , it is important to keep scrutinizing the accuracy of paper i , as this paper is in frequent use among the stellar evolution researchers . in paper i the coulomb scatterings of the electrons off the atomic nuclei have been calculated in the framework of the first born approximation . subsequently yakovlev ( 1987 ) has made an improvement on paper i by taking into account the contributions beyond the first born approximation . here we note that yakovlev ( 1987 ) also used the analytic approach by taking into account the second born term for the coulomb scattering cross section . however , his second born corrections did not include the screening effects . we shall consistently take into account the screening effects in our second born corrections . later works by his group ( potekhin , chabrier , & yakovlev 1997 ; potekhin et al . 1999 ) improved on yakovlev s ( 1987 ) original method by treating the first born term and the non - born term on the same footing , thereby taking into account the screening effects self - consistently . in these works they have made use of the fully numerical values of the cross section for the coulomb scattering of the electron by the atomic nucleus calculated by doggett & spencer ( 1956 ) . here we remark that the numerical calculation of the coulomb scattering cross section by doggett & spencer ( 1956 ) has been carried out for a limited number of @xmath1-values for atomic nucleus @xmath1=6 , 13 , 29 , 50 , 82 , and 92 , and for a limited number of electron energies , 0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , and 10mev . in this paper , we shall take a complementary semi - analytic approach by using the analytic expression for the second born cross section for the coulomb scattering of the electron off the atomic nucleus ( mckinley & feshbach 1948 ; feshbach 1952 ) . for nuclei @xmath0 , the inclusion up to the second born approximation is sufficiently accurate ( eby & morgan 1972 ) . in the following sections , however , we will confirm that the interpolations with respect to @xmath1 and the electron energies done by the previous authors are remarkably accurate . the basic formulae for the calculation of the electrical and thermal conductivities are presented in 2 by generalizing the formulation of paper i. the numerical results and the assessment of the contributions beyond the first born approximation are presented in 3 . the analytic formulae that fit the results of the numerical calculations are given in 4 . the case of the mixtures of nuclear species is dealt with in 5 . the last section is devoted to concluding remarks . in the appendix we evaluate the accuracy of the second born approximation by comparing with the exact results obtained by dogget & spencer ( 1956 ) . we shall closely follow the method described in paper i and generalize it in such a way that it include the second born term for the coulomb scattering of the electron off the atomic nucleus ( mckinley & feshbach 1948 ; feshbach 1952 ) . the reader is referred to paper i for the earlier references in this field of research . we shall consider the case that the atoms are completely pressure - ionized . we further restrict ourselves to the density - temperature region in which electrons are strongly degenerate . this condition is expressed as @xmath4^{1/2 } \ , - \ , 1 \right ] \ , [ \rm k ] \ , , \end{aligned}\ ] ] where @xmath5 is the fermi temperature , @xmath1 the atomic number of the nucleus , @xmath6 the mass number of the nucleus , and @xmath7 the mass density in units of 10@xmath8 g @xmath9 . the reader is referred to the paper by cassisi et al . ( 2007 ) for the case of the partial electron degeneracy . for the ionic system we consider the case that it is in the liquid state . the latest criterion corresponding to this condition is given by ( potekhin & chabrier 2000 ) @xmath10 where @xmath11^{1/3}$ ] is the ion - sphere radius , and @xmath12 the temperature in units of 10@xmath13 k. for the calculation of the electrical and thermal conductivities we use the ziman formula ( 1961 ) as is extended to the relativistically degenerate electrons ( flowers & itoh 1976 ) . on deriving the formula we retain the dielectric screening function due to the degenerate electrons . as to the explicit expressions for the dielectric function , we use the relativistic formula worked out by jancovici ( 1962 ) : @xmath14 where @xmath15 is the thomas - fermi wavenumber for the nonrelativistic electrons , @xmath16 is the momentum transfer measured in units of 2@xmath17 , @xmath18 is the dimensionless relativistic parameter @xmath19 and @xmath20 is the usual electron density parameter given by @xmath21 working on the transport theory for the relativistic electrons given by flowers & itoh ( 1976 ) and taking into account the finite - nuclear - size corrections ( itoh & kohyama 1983 ) and the second born term ( mckinley & feshbach 1948 ; feshbach 1952 ) , we obtain the expression for the electrical conductivity @xmath22 and the thermal conductivity @xmath23 : @xmath24 \ , , \\ \kappa & = & 2.363 \times 10^{17 } \frac { \rho_{6}t_{8}}{a } \frac{1 } { ( 1+b^{2})<s > } \ , \left [ { \rm ergs \ , \ , cm^{-1 } \ , s^{-1 } \ , k^{-1 } } \right ] \ , , \end{aligned}\ ] ] @xmath25^{2 } } \nonumber \\ & - & \frac{b^{2}}{1 + b^{2 } } \int_{0}^{1 } d \left(\frac{k}{2k_{f } } \right ) \left(\frac{k}{2k_{f } } \right)^{5 } \frac{s(k/2k_{f } ) \left|f(k/2k_{f } ) \right|^{2 } } { \left[(k/2k_{f})^{2 } \epsilon(k/2k_{f},0 ) \right]^{2 } } \nonumber \\ & \equiv & < s_{-1 } > \ , - \ , \frac{b^{2}}{1 + b^{2 } } < s_{+1 } > \ , , \\ < s>^{2b } & = & \pi z \alpha \frac{b}{(1 + b^{2})^{1/2 } } \nonumber \\ & \times & \left\ { \int_{0}^{1 } d \left(\frac{k}{2k_{f } } \right ) \left(\frac{k}{2k_{f } } \right)^{4 } \frac{s(k/2k_{f } ) \left|f(k/2k_{f } ) \right|^{2 } } { \left[(k/2k_{f})^{2 } \epsilon(k/2k_{f},0 ) \right]^{2 } } \right . \nonumber \\ & & - \left . \int_{0}^{1 } d \left(\frac{k}{2k_{f } } \right ) \left(\frac{k}{2k_{f } } \right)^{5 } \frac{s(k/2k_{f } ) \left|f(k/2k_{f } ) \right|^{2 } } { \left[(k/2k_{f})^{2 } \epsilon(k/2k_{f},0 ) \right]^{2 } } \right\ } \ , \nonumber \\ & \equiv & \pi z \alpha \frac{b}{(1 + b^{2})^{1/2 } } \left [ < s_{0 } > - < s_{+1 } > \right ] \ , , \end{aligned}\ ] ] where @xmath26 corresponds to the first born term and @xmath27 corresponds to the second born term . in the above , @xmath28 is the momentum transferred from the ionic system to an electron , @xmath29 the ionic liquid structure factor , and @xmath30 the static dielectric screening function due to degenerate electrons . for the ionic liquid structure factor we use the results of young , corey , & dewitt ( 1991 ) calculated for the classical one - component plasma ( ocp ) . in the above formulae we have also taken into account the finite - nuclear - size corrections through the use of the atomic form factor ( itoh & kohyama 1983 ) @xmath31 @xmath17 and @xmath32 being the electron fermi wave number and the charge radius of the nucleus , respectively . the electron fermi wave number is expressed as @xmath33 the charge radius of the nucleus is represented by @xmath34 the present method differs from that of baiko et al . these authors subtracted the contribution corresponding to the elastic scattering in the crystalline lattice phase from the total static structure factor in the liquid . the main motivation for the modification of the structure factor near the melting point by baiko et al . ( 1998 ) is the partial ordering of the coulomb liquid revealed by microscopic numerical simulations . this procedure was followed by potekhin et al . ( 1999 ) , gnedin et al . ( 2001 ) , and cassisi et al . ( 2007 ) . in the field of condensed matter physics , however , the correctness of the original ziman ( 1961 ) method with the use of the full liquid structure factor has long been established ( ashcroft & lekner 1966 ; rosenfeld & stott 1990 ) . part of the motivation for the introduction of baiko et al.s ( 1998 ) suggestion appears to be the finding by itoh , hayashi , & kohyama ( 1993 ) that the conductivity of astrophysical dense matter increases typically by 24 times upon crystallization . regarding this finding , we should note that simple metals in the laboratory show similar phenomena . the electrical conductivity of the simple metals in the laboratory shows significant ( 2 - 4 times ) jumps upon crystallization ( iida & guthrie 1993 ) . for these reasons we shall follow the method of paper i in which we use the full liquid structure factor , which is in accord with the method used in condensed matter physics ( ashcroft & lekner 1966 ; rosenfeld & stott 1990 ) . of course the analogy with simple metals should be examined with future full @xmath35 @xmath36 calculations . we have carried out integrations in equations ( 2.9 ) and ( 2.10 ) numerically for the cases of @xmath37h , @xmath38he , @xmath39c , @xmath40n , @xmath41o , @xmath42ne , @xmath43 mg , @xmath44si , @xmath45s , @xmath46ca , @xmath47fe by using the structure factor of the classical one - component plasma calculated by young , corey , & dewitt ( 1991 ) and jancovici s ( 1962 ) relativistic dielectric function for degenerate electrons . for the neutron star matter , the reader is referred to the paper by gnedin , yakovlev , & potekhin ( 2001 ) . we have made calculations for the parameter ranges , 0.1 @xmath48 , @xmath49 , which cover most of the density - temperature region of the dense matter in the liquid metal phase of astrophysical importance . note that for some elements such as the @xmath47fe matter these parameter ranges include the density - temperature region in which either the condition for the strong electron degeneracy or the condition for the complete pressure ionization does not hold . all of the considered elements are certainly unstable against nuclear reactions or electron captures at extremely high densities ( @xmath50g@xmath9 ) . we have chosen these wide parameter ranges in order to construct fitting formulae that have a wide applicable range . the reader should use our fitting formulae in the density - temperature region in which the conditions in the above are valid . corresponding to the parameter range @xmath51 , we have used the debye - h@xmath52ckel form for the structure factor @xmath53^{-1 } \ , .\end{aligned}\ ] ] here we remark that young , corey , & dewitt s ( 1991 ) calculation has been done for @xmath54 . we have made a smooth extrapolation to the debye - h@xmath52ckel regime @xmath55 . in figure 1 we show the results of the calculation for the case of @xmath39c . we find that the second born corrections amount to about 2% at @xmath56 and @xmath2=10@xmath8 g @xmath9 and about 5% at @xmath56 and @xmath57 g @xmath9 . in figure 2 we show the results of the calculation for the case of @xmath47fe . we find that the second born corrections amount to about 8% at @xmath56 and @xmath58 g @xmath9 and about 17% at @xmath56 and @xmath57 g @xmath9 . these values are in good quantitative agreement with those of potekhin , chabrier , & yakovlev ( 1997 ) . for the case of @xmath47fe at @xmath56 and @xmath57 g @xmath9 , the present second born corrections are significantly smaller than those of these authors who obtain about 22% non - born corrections for this case . significant part of this discrepancy appears to be due to the terms higher than the second born term . in table 1 we compare the present numerical results with the numerical results by potekhin et al . ( 1997 ) for the cases of @xmath59 g @xmath9 ; @xmath60=1 , 10 , 100 . we find generally good agreement between the present numerical results and the numerical results by potekhin et al . the present numerical results appear to underestimate the non - born effects for large values of @xmath1 ( @xmath61 ) . we have carried out the numerical integrations of equations ( 2.9 ) and ( 2.10 ) for @xmath37h , @xmath38he , @xmath39c , @xmath40n , @xmath41o , @xmath42ne , @xmath43 mg , @xmath44si , @xmath45s , @xmath46ca , @xmath47fe . for the convenience of application we have fitted the numerical results of the calculation by analytic formulae . we introduce the following variable @xmath62 the fitting has been carried out for the ranges @xmath63 g @xmath9 , @xmath64 . the fitting formulae are taken as follows : @xmath65 the coefficients are given in tables 25 . the accuracy of the fitting is better than 3% for most of the cases treated in this section . so far we have dealt with the case in which the matter consists of one species of atomic nucleus . in the actual application of the present calculation to the astrophysical studies , we often encounter the case in which the matter consists of more than one species of atomic nucleus . in this section we shall extend our calculation to the case of mixtures of nuclear species . the case of mixtures has been discussed by potekhin et al . ( 1999 ) and also by brown , bildsten , & chang ( 2000 ) and by cassisi et al . their formalism is based on the linear mixing rule . here we shall give expressions according to our notations . let us consider the case in which the mass fraction of the nuclear species @xmath66 is @xmath67 . the electrical resistivity @xmath68 due to the scattering by the nuclear species @xmath66 is given by @xmath69 \ , , \\ < s>_{j } & = & < s>_{j}^{1b } + < s>_{j}^{2b } \ , , \\ < s>_{j}^{1b } & = & < s_{-1}>_{j } \ , - \ , \frac{b^{2}}{1+b^{2 } } < s_{+1}>_{j } \ , , \\ < s>_{j}^{2b } & = & \pi z_{j } \alpha \frac{b}{(1+b^{2})^{1/2 } } \left [ < s_{0}>_{j } - < s_{+1}>_{j } \right ] \ , .\end{aligned}\ ] ] here for the mixture case the parameter @xmath20 in equation ( 2.5 ) is generalized as @xmath70 the total electrical resistivity @xmath71 is given by @xmath72 therefore , the electrical conductivity @xmath22 is given by @xmath73 \ , .\end{aligned}\ ] ] in the same manner , the thermal conductivity @xmath23 is given by @xmath74 \ , .\end{aligned}\ ] ] in the above , the scattering integral @xmath75 corresponding to the nuclear species @xmath66 should be calculated by using the coulomb coupling parameter ( itoh et al . 1979 ; potekhin et al . 1999 ; brown , bildsten , & chang 2002 ; itoh et al . 2004 ) @xmath76 where @xmath77 is the electron - sphere radius , and @xmath78 and @xmath79 are the number densities of the electrons and the @xmath80-th nuclear species @xmath81 , respectively . we have calculated the second born corrections to the electrical and thermal conductivities of the dense matter in the liquid metal phase for various elemental compositions of astrophysical importance by extending the calculations reported in paper i. we have used the semi - analytical approach which is in contrast to that of the previous authors ( yakovlev 1987 ; potekhin , chabrier , & yakovlev 1997 ; potekhin et al . 1999 ) , who made use of the fully numerical values of the cross section for the scattering of the electron by the atomic nucleus calculated by doggett & spencer ( 1956 ) . it should be noted that the numerical calculation of the coulomb scattering cross section by doggett & spencer ( 1956 ) has been carried out for a limited number of @xmath1-values for the atomic nucleus @xmath1=6 , 13 , 29 , 50 , 82 , and 92 , and for a limited number of electron energies 0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , and 10mev , and also for a limited number ( 13 ) of the scattering angles that are related to @xmath82 in equations ( 2.9 ) and ( 2.10 ) . the sparseness of data for light and medium nuclei ( only for @xmath1=6 , 13 , 29 ) is potentially vulnerable in order to obtain results with reliable @xmath1-dependence . however , our study has confirmed that the previous results have sufficiently accurate @xmath1-dependence and @xmath2-dependence , since they are recovered , within about 1% , if our second - born results are multiplied by the ratio of the full non - born @xmath83 to the second - born @xmath84 . the definitions of @xmath83 and @xmath84 are given in the appendix . we have found that our results are in general agreement with those of potekhin , chabrier , & yakovlev ( 1997 ) . our second born corrections are significantly smaller than the non - born corrections of these authors for the case of @xmath47fe at @xmath56 and @xmath57 g @xmath9 . significant part of this discrepancy appears to be due to the terms higher than the second born term . in the present calculation , in contrast to baiko et al . ( 1998 ) , we have used the full liquid structure factor , for the reasons explained in 2 . we have summarized our numerical results by accurate analytic fitting formulae . we have also presented the prescriptions to deal with the cases of mixtures of nuclear species . therefore , the present results should be readily applied to various studies in the field of stellar evolution . we wish to thank our referee for many useful comments that have greatly helped us in revising the manuscript . we also wish to thank d. g. yakovlev and a. y. potekhin for their very informative communication and providing us with the numerical data of their results in table 1 . one of the authors ( n.i . ) wishes to thank n. w. ashcroft , k. hoshino , and h. maebashi for their expert advice regarding the calculations of the conductivities of simple metals in the laboratory . he especially appreciates n. w. ashcroft s lucid reasoning regarding the correctness of ziman s original method with the use of the full liquid structure factor . he also wishes to thank h. e. dewitt and s. hansen for their valuable communication regarding the ocp structure factor . this work is financially supported in part by the grant - in - aid for scientific research of japanese ministry of education , culture , sports , science , and technology under the contract 16540220 . in this appendix we evaluate the accuracy of the second born approximation by comparing with the exact results obtained by doggett & spencer ( 1956 ) . the second born approximation gives a correction factor to the rutherford cross section ( mckinley & feshbach 1948 ; feshbach 1952 ) : @xmath85 where @xmath86^{1/2}}{1 + ( e_{kin}/0.5110{\rm mev } ) } \ , , \end{aligned}\ ] ] @xmath87 being the kinetic energy of the electron , and @xmath88 is the angle of scattering . the @xmath89 factor corresponding to the results by doggett & spencer ( 1956 ) is defined by @xmath90^{2 } } \ , r^{ds}(e_{kin } , k/2k_{f } ) \ , , \end{aligned}\ ] ] where @xmath91 is related to @xmath88 by @xmath92 in order to make the comparison self - consistent , in this appendix we define @xmath93^{2 } } \ , r^{1b+2b } \nonumber \\ & = & < s>^{1b } + < s>^{2b } \ , , \end{aligned}\ ] ] which of course coincides with our previous equations ( 2.8 ) , ( 2.9 ) , ( 2.10 ) . here we have used the relationship @xmath94^{1/2 } \ , - \ , 1 \right\ } \ , .\end{aligned}\ ] ] in table 6 we compare the results corresponding to the second born approximation with those corresponding to doggett & spencer ( 1956 ) for the cases of @xmath60=10 ; @xmath1=6 , 13 , 29 ; and @xmath87=0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , 10mev . we find the accuracy of the second born correction is better than 0.4% for @xmath1=6 , better than 1.4% for @xmath1=13 , and better than 6.0% for @xmath1=29 . ashcroft , n. w. , & lekner , j. 1966 , phys . , 145 , 83 baiko , d. a. , kaminker , a. d. , potekhin , a. y. , & yakovlev , d. g. 1998 , phys . letters , 81 , 5556 brown , e. f. , bildsten , l. , & chang , p. 2002 , apj , 574 , 920 cassisi , s. , potekhin , a. y. , pietrinferni , a. , catelan , m. , & salaris , m. 2007 , apj , 661 , 1094 doggett , j. a. , & spencer , l. v. 1956 , phys . , 103 , 1597 eby , p. b. , & morgan , s. h. , jr . 1972 , phys . a , 5 , 2536 feshbach , h. 1952 , phys . , 88 , 295 flowers , e. , & itoh , n. 1976 , apj , 206 , 218 flowers , e. , & itoh , n. 1979 , apj , 230 , 847 flowers , e. , & itoh , n. 1981 , apj , 250 , 750 gnedin , o. y. , yakovlev , d. g. , & potekhin , a. y. 2001 , mnras , 324 , 725 iida , t. , & guthrie , r. i. l. 1993 , the physical properties of liquid metals ( oxford univ . press ) itoh , n. , asahara , r. , tomizawa , n. , wanajo , s. , & nozawa , s. 2004 , apj , 611 , 1041 itoh , n. , hayashi , h. , & kohyama , y. 1993 , apj , 418 , 405 ; erratum 436 , 418 ( 1994 ) itoh , n. , & kohyama , y. 1983 , apj , 275 , 858 itoh , n. , & kohyama , y. 1993 , apj , 404 , 268 itoh , n. , kohyama , y. , matsumoto , n. , & seki , m. 1984 , apj , 285 , 758 ; erratum 404 , 418 ( 1993 ) itoh , n. , mitake , s. , iyetomi , h. , & ichimaru , s. 1983 , apj , 273 , 774 itoh , n. , totsuji , h. , ichimaru , s. , & dewitt , h. e. 1979 , apj , 234 , 1079 ; erratum 239 , 415 ( 1980 ) jancovici , b. 1962 , nuovo cimento , 25 , 428 mckinley , w. a. , jr . , & feshbach , h. 1948 , phys . , 74 , 1759 mitake , s. , ichimaru , s. , & itoh , n. , 1984 , apj , 277 , 375 potekhin , a. y. , baiko , d. a. , haensel , p. , & yakovlev , d. g. 1999 , a & a , 346 , 345 potekhin , a. y. , chabrier , g. 2000 , phys . e. , 62 , 8554 potekhin , a. y. , chabrier , g. , & yakovlev , d. g. 1997 , a & a , 323 , 415 rosenfeld , a. m. , & stott , m. j. 1990 , phys . rev . b , 42 , 3406 yakovlev , d. g. 1987 , sov . astron . , 31 , 347 young , d. a. , corey , e. m. , & dewitt , h. e. 1991 , phys . rev . a , 44 , 6508 ziman , j. 1961 , phil . mag . , 6 , 1013 cccc 1 & 6 & 1.0841 & 1.087 + & 7 & 1.1297 & 1.133 + & 8 & 1.1701 & 1.175 + & 10 & 1.2400 & 1.248 + & 12 & 1.2996 & 1.311 + & 14 & 1.3521 & 1.368 + & 16 & 1.3995 & 1.420 + & 20 & 1.4834 & 1.516 + & 26 & 1.5925 & 1.649 + 10 & 6 & 0.6490 & 0.651 + & 7 & 0.6975 & 0.701 + & 8 & 0.7407 & 0.745 + & 10 & 0.8159 & 0.823 + & 12 & 0.8806 & 0.891 + & 14 & 0.9379 & 0.953 + & 16 & 0.9898 & 1.009 + & 20 & 1.0819 & 1.113 + & 26 & 1.2017 & 1.256 + 100 & 6 & 0.5236 & 0.526 + & 7 & 0.5717 & 0.575 + & 8 & 0.6152 & 0.620 + & 10 & 0.6929 & 0.700 + & 12 & 0.7609 & 0.771 + & 14 & 0.8209 & 0.836 + & 16 & 0.8750 & 0.880 + & 20 & 0.9700 & 1.001 + & 26 & 1.0931 & 1.148 + crrrrrrrrrrr @xmath95 & 0.6496 & 0.7407 & 0.8981 & 0.9232 & 0.9457 & 0.9848 & 1.0181 & 1.0471 & 1.0729 & 1.1171 & 1.1690 + @xmath96 & 0.0471 & @xmath970.0007 & @xmath970.0666 & @xmath970.0781 & @xmath970.0884 & @xmath970.1065 & @xmath970.1221 & @xmath970.1357 & @xmath970.1477 & @xmath970.1684 & @xmath970.1970 + @xmath98 & @xmath970.0056 & @xmath970.0165 & @xmath970.0071 & @xmath970.0045 & @xmath970.0019 & 0.0031 & 0.0076 & 0.0117 & 0.0155 & 0.0222 & 0.0297 + @xmath99 & @xmath970.0284 & @xmath970.0376 & @xmath970.0558 & @xmath970.0588 & @xmath970.0615 & @xmath970.0663 & @xmath970.0703 & @xmath970.0737 & @xmath970.0767 & @xmath970.0818 & @xmath970.0869 + @xmath100 & 0.0054 & 0.0114 & 0.0247 & 0.0270 & 0.0291 & 0.0326 & 0.0356 & 0.0382 & 0.0404 & 0.0440 & 0.0481 + @xmath18 & 0.0921 & 0.1037 & 0.1068 & 0.1064 & 0.1059 & 0.1046 & 0.1032 & 0.1018 & 0.1004 & 0.0977 & 0.0946 + @xmath101 & 0.4531 & 0.3959 & 0.4040 & 0.4047 & 0.4053 & 0.4063 & 0.4069 & 0.4074 & 0.4078 & 0.4084 & 0.4017 + @xmath102 & 0.0268 & 0.2196 & 0.4347 & 0.4753 & 0.5166 & 0.5930 & 0.6520 & 0.6976 & 0.7358 & 0.8004 & 0.8856 + @xmath103 & 0.0012 & 0.0006 & 0.0084 & 0.0052 & 0.0006 & @xmath970.0094 & @xmath970.0176 & @xmath970.0243 & @xmath970.0304 & @xmath970.0423 & @xmath970.0612 + @xmath104 & 0.0051 & 0.0440 & 0.0741 & 0.0796 & 0.0854 & 0.0962 & 0.1035 & 0.1082 & 0.1116 & 0.1169 & 0.1224 + @xmath105 & @xmath970.0007 & @xmath970.0081 & @xmath970.0189 & @xmath970.0224 & @xmath970.0263 & @xmath970.0338 & @xmath970.0399 & @xmath970.0449 & @xmath970.0494 & @xmath970.0571 & @xmath970.0679 + @xmath106 & 0.0018 & 0.0155 & 0.0228 & 0.0241 & 0.0257 & 0.0287 & 0.0305 & 0.0316 & 0.0323 & 0.0333 & 0.0344 + @xmath107 & 0.0011 & 0.0056 & 0.0174 & 0.0184 & 0.0189 & 0.0194 & 0.0200 & 0.0208 & 0.0214 & 0.0223 & 0.0230 + @xmath108 & 0.0621 & 0.3641 & 0.3604 & 0.3670 & 0.3787 & 0.4011 & 0.4097 & 0.4097 & 0.4074 & 0.4036 & 0.3986 + crrrrrrrrrrr @xmath109 & 0.2781 & 0.3281 & 0.4042 & 0.4170 & 0.4286 & 0.4489 & 0.4662 & 0.4813 & 0.4946 & 0.5173 & 0.5452 + @xmath110 & 0.0357 & 0.0222 & @xmath970.0077 & @xmath970.0131 & @xmath970.0180 & @xmath970.0266 & @xmath970.0339 & @xmath970.0404 & @xmath970.0460 & @xmath970.0556 & @xmath970.0672 + @xmath111 & 0.0224 & 0.0249 & 0.0396 & 0.0423 & 0.0448 & 0.0491 & 0.0528 & 0.0560 & 0.0588 & 0.0635 & 0.0684 + @xmath112 & @xmath970.0072 & @xmath970.0134 & @xmath970.0245 & @xmath970.0264 & @xmath970.0280 & @xmath970.0309 & @xmath970.0333 & @xmath970.0353 & @xmath970.0370 & @xmath970.0397 & @xmath970.0426 + @xmath113 & 0.0059 & 0.0082 & 0.0152 & 0.0163 & 0.0174 & 0.0191 & 0.0205 & 0.0217 & 0.0227 & 0.0241 & 0.0254 + @xmath114 & 0.0303 & 0.0323 & 0.0269 & 0.0258 & 0.0247 & 0.0227 & 0.0209 & 0.0193 & 0.0179 & 0.0154 & 0.0127 + @xmath80 & 0.3087 & 0.2749 & 0.2790 & 0.2794 & 0.2797 & 0.2802 & 0.2805 & 0.2808 & 0.2810 & 0.2813 & 0.2773 + @xmath115 & 0.0225 & 0.1881 & 0.2913 & 0.3121 & 0.3353 & 0.3787 & 0.4085 & 0.4283 & 0.4436 & 0.4687 & 0.5053 + @xmath116 & 0.0007 & @xmath970.0008 & 0.0011 & @xmath970.0026 & @xmath970.0072 & @xmath970.0166 & @xmath970.0233 & @xmath970.0281 & @xmath970.0321 & @xmath970.0395 & @xmath970.0505 + @xmath117 & 0.0045 & 0.0385 & 0.0519 & 0.0551 & 0.0590 & 0.0664 & 0.0711 & 0.0736 & 0.0754 & 0.0781 & 0.0819 + @xmath118 & @xmath970.0006 & @xmath970.0071 & @xmath970.0133 & @xmath970.0156 & @xmath970.0182 & @xmath970.0229 & @xmath970.0263 & @xmath970.0288 & @xmath970.0307 & @xmath970.0339 & @xmath970.0383 + @xmath119 & 0.0016 & 0.0136 & 0.0164 & 0.0173 & 0.0185 & 0.0208 & 0.0221 & 0.0228 & 0.0231 & 0.0236 & 0.0243 + @xmath91 & 0.0007 & 0.0041 & 0.0095 & 0.0093 & 0.0087 & 0.0074 & 0.0066 & 0.0061 & 0.0056 & 0.0047 & 0.0033 + @xmath120 & 0.0557 & 0.3200 & 0.2571 & 0.2604 & 0.2694 & 0.2873 & 0.2923 & 0.2897 & 0.2856 & 0.2799 & 0.2763 + crrrrrrrrrrr @xmath121 & 0.1543 & 0.1881 & 0.2380 & 0.2466 & 0.2544 & 0.2679 & 0.2794 & 0.2893 & 0.2980 & 0.3126 & 0.3306 + @xmath122 & 0.0202 & 0.0137 & @xmath970.0068 & @xmath970.0105 & @xmath970.0138 & @xmath970.0196 & @xmath970.0244 & @xmath970.0287 & @xmath970.0323 & @xmath970.0384 & @xmath970.0451 + @xmath123 & 0.0205 & 0.0248 & 0.0367 & 0.0388 & 0.0406 & 0.0438 & 0.0465 & 0.0487 & 0.0506 & 0.0537 & 0.0569 + @xmath124 & @xmath970.0024 & @xmath970.0063 & @xmath970.0140 & @xmath970.0152 & @xmath970.0164 & @xmath970.0183 & @xmath970.0198 & @xmath970.0211 & @xmath970.0222 & @xmath970.0238 & @xmath970.0254 + @xmath125 & 0.0057 & 0.0074 & 0.0123 & 0.0131 & 0.0137 & 0.0149 & 0.0158 & 0.0165 & 0.0170 & 0.0178 & 0.0183 + @xmath126 & 0.0133 & 0.0135 & 0.0084 & 0.0074 & 0.0065 & 0.0049 & 0.0036 & 0.0024 & 0.0013 & @xmath970.0005 & @xmath970.0024 + @xmath127 & 0.2293 & 0.2069 & 0.2095 & 0.2097 & 0.2099 & 0.2102 & 0.2104 & 0.2105 & 0.2106 & 0.2108 & 0.2082 + @xmath128 & 0.0196 & 0.1636 & 0.2027 & 0.2148 & 0.2302 & 0.2597 & 0.2776 & 0.2872 & 0.2938 & 0.3049 & 0.3248 + @xmath129 & 0.0004 & @xmath970.0017 & @xmath970.0028 & @xmath970.0064 & @xmath970.0108 & @xmath970.0191 & @xmath970.0245 & @xmath970.0279 & @xmath970.0305 & @xmath970.0352 & @xmath970.0422 + @xmath130 & 0.0041 & 0.0339 & 0.0375 & 0.0397 & 0.0428 & 0.0487 & 0.0521 & 0.0536 & 0.0545 & 0.0561 & 0.0589 + @xmath131 & @xmath970.0006 & @xmath970.0064 & @xmath970.0099 & @xmath970.0115 & @xmath970.0134 & @xmath970.0168 & @xmath970.019 & @xmath970.0202 & @xmath970.0211 & @xmath970.0226 & @xmath970.0246 + @xmath132 & 0.0015 & 0.0121 & 0.0121 & 0.0129 & 0.0139 & 0.0159 & 0.0170 & 0.0174 & 0.0175 & 0.0177 & 0.0181 + @xmath133 & 0.0005 & 0.0032 & 0.0051 & 0.0044 & 0.0035 & 0.0017 & 0.0005 & @xmath970.0002 & @xmath970.0008 & @xmath970.0019 & @xmath970.0035 + @xmath134 & 0.0507 & 0.2834 & 0.1902 & 0.1933 & 0.2021 & 0.2188 & 0.2226 & 0.2192 & 0.2147 & 0.2089 & 0.2068 + crrrrrrrrrrr @xmath135 & 0.4288 & 0.1778 & 0.2634 & 0.2661 & 0.2619 & 0.2489 & 0.2461 & 0.2513 & 0.2583 & 0.2700 & 0.2760 + @xmath136 & @xmath970.5654 & @xmath970.5446 & @xmath970.5073 & @xmath970.5144 & @xmath970.5251 & @xmath970.5454 & @xmath970.5525 & @xmath970.5517 & @xmath970.5495 & @xmath970.5480 & @xmath970.5561 + @xmath137 & 0.0769 & 0.3337 & 0.2504 & 0.2474 & 0.2512 & 0.2638 & 0.2674 & 0.2632 & 0.2567 & 0.2453 & 0.2389 + @xmath138 & 0.0662 & 0.0467 & 0.0083 & 0.0155 & 0.0264 & 0.0472 & 0.0549 & 0.0546 & 0.0527 & 0.0512 & 0.0594 + @xmath139 & 0.5283 & 0.0090 & 0.3358 & 0.3352 & 0.3056 & 0.2147 & 0.1748 & 0.1905 & 0.2221 & 0.2725 & 0.2749 + @xmath140 & @xmath970.5546 & @xmath970.7314 & @xmath970.4548 & @xmath970.4723 & @xmath970.5089 & @xmath970.5916 & @xmath970.6101 & @xmath970.5802 & @xmath970.5437 & @xmath970.5001 & @xmath970.5121 + @xmath141 & @xmath970.0256 & 0.5152 & 0.1868 & 0.1849 & 0.2120 & 0.3007 & 0.3446 & 0.3334 & 0.3045 & 0.2539 & 0.2464 + @xmath142 & 0.0558 & 0.2428 & @xmath970.0421 & @xmath970.0245 & 0.0124 & 0.0973 & 0.1193 & 0.0915 & 0.0555 & 0.0106 & 0.0208 + @xmath143 & 0.6074 & @xmath974.1967 & 0.4740 & 0.4663 & 0.3642 & @xmath970.2350 & @xmath970.9551 & @xmath970.6418 & @xmath970.1656 & 0.2635 & 0.2676 + @xmath144 & @xmath970.5134 & @xmath972.8194 & @xmath970.1732 & @xmath970.2319 & @xmath970.3358 & @xmath970.7496 & @xmath971.0234 & @xmath970.6404 & @xmath970.3168 & @xmath970.1219 & @xmath970.1927 + @xmath145 & @xmath970.1064 & 4.9319 & 0.0621 & 0.0612 & 0.1539 & 0.7369 & 1.4929 & 1.2219 & 0.7513 & 0.2992 & 0.2587 + @xmath146 & 0.0142 & 2.4604 & @xmath970.3263 & @xmath970.2676 & @xmath970.1633 & 0.2629 & 0.5802 & 0.2094 & @xmath970.1265 & @xmath970.3460 & @xmath970.2894 + ccccc 6 & 0.05 & 0.9324 & 0.9299 & 0.9973 + & 0.1 & 0.8921 & 0.8897 & 0.9973 + & 0.2 & 0.8263 & 0.8239 & 0.9971 + & 0.4 & 0.7489 & 0.7465 & 0.9969 + & 0.7 & 0.6950 & 0.6929 & 0.9970 + & 1 & 0.6698 & 0.6677 & 0.9969 + & 2 & 0.6410 & 0.6386 & 0.9963 + & 4 & 0.6288 & 0.6268 & 0.9968 + & 10 & 0.6224 & 0.6202 & 0.9965 + 13 & 0.05 & 1.1805 & 1.1668 & 0.9883 + & 0.1 & 1.1506 & 1.1365 & 0.9878 + & 0.2 & 1.0914 & 1.0778 & 0.9876 + & 0.4 & 1.0182 & 1.0048 & 0.9868 + & 0.7 & 0.9658 & 0.9529 & 0.9867 + & 1 & 0.9412 & 0.9283 & 0.9863 + & 2 & 0.9125 & 0.8998 & 0.9860 + & 4 & 0.9008 & 0.8879 & 0.9857 + & 10 & 0.8915 & 0.8788 & 0.9858 + 29 & 0.05 & 1.5000 & 1.4295 & 0.9530 + & 0.1 & 1.4999 & 1.4239 & 0.9493 + & 0.2 & 1.4664 & 1.3875 & 0.9462 + & 0.4 & 1.4113 & 1.3321 & 0.9439 + & 0.7 & 1.3685 & 1.2900 & 0.9426 + & 1 & 1.3485 & 1.2695 & 0.9414 + & 2 & 1.3240 & 1.2453 & 0.9405 + & 4 & 1.3125 & 1.2343 & 0.9405 + & 10 & 1.2983 & 1.2204 & 0.9400 +

the second born corrections to the electrical and thermal conductivities are calculated for the dense matter in the liquid metal phase for various elemental compositions of astrophysical importance . inclusion up to the second born corrections is sufficiently accurate for the coulomb scattering of the electrons by the atomic nuclei with @xmath0 . our approach is semi - analytical , and is in contrast to that of the previous authors who have used fully numerical values of the cross section for the coulomb scattering of the electron by the atomic nucleus . the merit of the present semi - analytical approach is that this approach affords us to obtain the results with reliable @xmath1-dependence and @xmath2-dependence . the previous fully numerical approach has made use of the numerical values of the cross section for the scattering of the electron off the atomic nucleus for a limited number of @xmath1-values , @xmath1=6 , 13 , 29 , 50 , 82 , and 92 , and for a limited number of electron energies , 0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , and 10mev . our study , however , has confirmed that the previous results are sufficiently accurate . they are recovered , if the terms higher than the second born terms are taken into account . we make a detailed comparison of the present results with those of the previous authors . the numerical results are parameterized in a form of analytic formulae that would facilitate practical uses of the results . we also extend our calculations to the case of mixtures of nuclear species . the corresponding subroutine can be retrieved from http://www.ph.sophia.ac.jp/@xmath3itoh-ken/subroutine/subroutine.htm .

at present , graphite and its electronic properties attract considerable attention due to the discovery of novel carbon - based materials such as fullerenes and nanotubes constructed from wrapped graphite sheets . @xcite besides , thin films of graphite give promise of device applications . @xcite the attention to graphite is also caused by specific features of its electron energy spectrum which result in interesting physical effects . @xcite the electronic spectrum of graphite is described by the slonzewski - weiss - mcclure ( swm ) model , @xcite and values of the main parameters of this model were found sufficiently accurately from an analysis of various experimental data ; see , e.g. , the review of brandt _ et al_. @xcite and references therein . the fermi surface of graphite consists of elongated pockets enclosing the edge hkh of its brillouin zone ( see figures below ) . these pockets are formed by the two majority groups of electrons ( e ) and holes ( h ) which are located near the points k and h of the brillouin zone , respectively . there is also at least one minority ( m ) low - concentration group of charge carriers in graphite , and this group seems to be located near the point h. however , it is necessary to emphasize that in spite of the considerable attention attracted to graphite an unresolved problem concerned with its spectrum still exists . it is well known @xcite that in the edge hkh of the brillouin zone of graphite , two electron energy bands are degenerate , and in a small vicinity of the edge these bands split linearly in a deviation of the wave vector @xmath0 from the edge . in other words , the edge is the band - contact line . but , it was shown in our paper @xcite that if in the @xmath0-space a closed semiclassical orbit of a charge carrier surrounds a contact line of its band with some other band ( and lifting of the degeneracy is linear in @xmath0 ) , the wave function of this carrier after its turn over the orbit acquires the addition phase @xmath1 as compared to the case without the band - contact line . this @xmath2 is the so - called berry phase , @xcite and it modifies the constant @xmath3 in the well - known semiclassical quantization rule @xcite for the energy @xmath4 of a charge carrier in the magnetic fields @xmath5 : @xmath6 where @xmath7 is the cross - sectional area of the closed orbit in the @xmath0 space ; @xmath8 is the component of @xmath0 perpendicular to the plane of this orbit ; @xmath9 is a large integer ( @xmath10 ) ; @xmath11 is the absolute value of the electron charge , and the constant @xmath3 is now given by the formula:@xcite @xmath12 when the magnetic field is applied along the hkh axis , orbits of electrons and holes in the brillouin zone of graphite surround this axis . thus , one might expect to find @xmath13 for these orbits instead of the usual value @xmath14 ( the values @xmath13 and @xmath15 are equivalent ) . a value of @xmath3 can be measured using various oscillation effects and in particular , with the de haas - van alphen effect . @xcite for example , the first harmonic of the de haas - van alphen oscillations of the magnetic susceptibility has the form , @xcite @xmath16 where @xmath17 , @xmath18 is some extremal cross section of the fermi surface of a metal in @xmath8 , a positive @xmath19 is the amplitude of this first harmonic , and @xmath20 is its phase which is given by @xmath21 with @xmath22 for a minimum and maximum cross - section @xmath18 , respectively , and @xmath23 in the case of a two - dimensional fermi surface . @xcite it follows from eq . ( [ 4 ] ) that one has to obtain @xmath24 for the maximum cross - sections of the electron and hole majorities in graphite . however , the phases @xmath25 , @xmath26 measured long ago @xcite agree with the usual value @xmath27 ; see table i. recently a new method of determining the phase @xmath20 of the de haas - van alphen oscillations was elaborated , @xcite and the authors of that paper found @xmath13 for the cross section of the hole majority in graphite . however , in this determination they assumed the fermi surface of the holes to be two - dimensional ( @xmath28 ) ; see table i. besides this , they found @xmath27 for the maximum cross section of the electron majority , assuming the three - dimensional fermi surface for this majority ( @xmath29 ) . although the obtained value @xmath13 for the holes agrees with the above prediction , the results of ref . give rise to the following new problems : first , since the band - contact line in graphite penetrates both the electron and hole extremal cross sections , these cross sections must have the same @xmath3 . second , using the values of the parameters of swm model , @xcite one might expect that in graphite the electrons and holes of the extremal cross sections are both three - dimensional . in this paper we show that in graphite , apart from the band - contact line coinciding with the edge hkh , _ three additional _ band - contact lines exist near this edge . the existence of these lines leads to the usual value @xmath27 for the maximum cross sections of the electron and hole majority groups in graphite . in other words , we resolve the above - mention contradiction between the theoretical value of @xmath3 and the data of refs . . we also discuss the data of ref . . [ cols="<,^,>,>,^,^,>,>,^,^ , < " , ] when the magnetic field @xmath5 is directed along the @xmath30 axis , the maximum electron cross section in @xmath31 is located at @xmath32 , while the maximum cross section of the hole majority is between the points k and @xmath33 , viz . , at @xmath34 where @xmath35 is the fermi energy in graphite , see fig . 2 . thus , both these cross sections are penetrated by the four band - contact lines . however , an _ even _ number of the band - contact lines do not change @xcite the usual value @xmath14 . thus , we find @xmath14 for the maximum cross sections of the majority groups , which agrees with the experimental results of refs . . we now discuss briefly the value of @xmath3 for the minority group . for the parameters presented in table ii , the hole minority is located near the point h and it results from the band @xmath36 . at this point the minority and the hole majority produced by the band @xmath37 have equal cross sections when the magnetic field is along the hkh axis . since no contact lines of the bands @xmath36 and @xmath37 penetrate this common cross section , one might expect to find the usual value @xmath14 in this case . however , the semiclassical approximation which is used in deriving eqs . ( [ 1 ] ) and ( [ 2 ] ) fails for the hole orbits corresponding to this cross section since for this approximation to be valid , the orbits must be sufficiently far away from each other . the analysis carried out beyond the scope of the semiclassical approximation @xcite led to @xmath13 and @xmath23 for the `` degenerate '' orbit . in experiments this orbit is ascribed to the hole minority , and the phase @xmath38 measured in ref . agrees with these @xmath3 and @xmath39 , see table i. in ref . a new method was developed to determine the phase @xmath20 of the de haas -van alphen oscillations of the magnetic susceptibility . the appropriate results for @xmath20 and @xmath3 in graphite are presented in table i. however , authors of ref . implied in their analysis of @xmath3 that the sign of @xmath19 in formula ( [ 3 ] ) is positive in the case of electrons and negative for holes . this is not correct ; the sign is always positive . a re - examination of the derivation of the lifshits - kosevich formula @xcite proves this statement . @xcite with this in mind we have corrected @xmath39 and @xmath3 of ref . , and the obtained results are also presented in table i. for the hole minority and for the electron majority @xcite the corrected results coincide with those of williamson _ et al_.@xcite ( but @xmath40 can be caused by the above - mentioned degeneracy of the hole orbits rather than by the two - dimensional spectrum of the hole minority ) . for the hole majority the phases @xmath26 measured in refs . and the phase @xmath41 obtained by lukyanchuk and kopelevich @xcite means that either the spectrum of these carriers is two dimensional , or if @xmath42 , one obtains @xmath43 . however , in the semiclassical approximation , @xmath3 can be equal to @xmath44 or to @xmath45 only . @xcite intermediate values can occur in situations close to the magnetic breakdown . @xcite in principle , such the situation is possible for the swm model , but it does not occur for the parameters presented in table ii . the parameters of table ii correspond to three dimensional spectrum of graphite and lead to a consistent description of the experimental data @xcite obtained many yeas ago . however , lukyanchuk and kopelevich @xcite used the highly oriented pyrolytic graphite ( hopg ) with very high ratio of the out - of - plane to basal - plane resistivities ( @xmath46 ) , and in this sample , quantum - hall - effect features were observed which indicate a quasi two dimensional nature of this hopg . @xcite it was also argued @xcite that in similar samples of hopg an incoherent transport occurs in the direction perpendicular to the graphite layers , and the three dimensional spectrum of carriers seems to fail . if this conclusion is valid only for the hole majority , it could explain the above - mentioned disagreement . this also means that the parameters of swm model should be reconsidered to describe the spectrum of such hopg . to conclude , the phases of the de haas - van alphen oscillations in graphite were measured in refs . . the data of refs . can be completely explained in the framework of the known band structure of graphite @xcite if one takes into account that four band - contact lines exist near the hkh edge of its brillouin zone . the data of lukyanchuk and kopelevich @xcite obtained for hopg disagree with the experimental results of refs . for one of the two large cross sections and probably imply that a reconsideration of the energy - band parameters for such hopg is required .

we discuss the known experimental data on the phase of the de haas -van alphen oscillations in graphite . these data can be understood if one takes into account that four band - contact lines exist near the hkh edge of the brillouin zone of graphite .

the non - mesonic weak decay ( nmwd ) process of a @xmath11 hypernucleus , @xmath12 , gives a unique opportunity to study the weak interaction between baryons since this strangeness non - conserving process is purely attributed to the weak interaction . in the nmwd , there are two decay channels , @xmath13 @xmath14 ( @xmath9 ) and @xmath15 @xmath16 ( @xmath7 ) . the ratio of those decay widths , @xmath7/@xmath9 , is an important observable used to study the isospin structure of the nmwd mechanism . for the past 40 years , there has been a longstanding puzzle that the experimental @xmath7/@xmath9 ratio disagrees with that of theoretical calculations based on the most natural and simplest model , the one - pion exchange model ( ope ) . in this model , the @xmath12 reaction is expressed as a pion absorption process after the @xmath17 decay inside the nucleus . since the ope process is tensor - dominant and the tensor transition of the initial @xmath18 pair in the @xmath19-state requires the final @xmath20 pair to have isospin zero , the @xmath21 ratio in the ope process becomes close to 0 . however , previous experimental results have indicated a large @xmath21 ratio ( @xmath221 ) @xcite . this large discrepancy between the ope - model predictions and the experimental results has stimulated many theoretical studies : the heavy meson exchange model , the direct quark model and the two - nucleon ( 2@xmath23 ) induced model ( @xmath24 ) . after k. sasaki @xmath25 pointed out an error in the sign of the kaon exchange amplitudes in 2000 @xcite , those theoretical values of the @xmath21 ratio have increased to the level of 0.4@xmath220.7 @xcite . on the other hand , the experimental data still have large errors ( @xmath21 = 0.93 @xmath26 0.55 for @xmath0he @xcite ) , and it is hard to draw a definite conclusion on the @xmath21 ratio . when we compare the measured @xmath21 ratio with that obtained in theoretical calculations , the most serious technical problem was a treatment of the re - scattering effect in the residual nucleus , the so - called final state interaction ( fsi ) . moreover , the possible existence of a multi - nucleon induced process has been discussed theoretically ( such as 2@xmath23-induced process ) , though there has been no experimental evidence . several nucleon energy spectra from hypernuclear decay have been reported so far @xcite , in which it is however difficult to extract the @xmath21 ratio without theoretical assumptions on the effects of fsi and possible multi - nucleon induced processes . since the 1@xmath23-induced decay `` @xmath27 '' is two - body process , the outgoing nucleon - nucleon pair suffering no fsi effect must have about 180 degree opening angle and clear energy correlation . in the present experiment , we performed a coincident measurement of the two nucleons , @xmath4 and @xmath5-pairs , in the decay for the first time . the 1@xmath23-induced processes could be clearly observed by measuring yields of the back - to - back @xmath4- and @xmath5-pairs and confirming that the energy sums roughly correspond to their @xmath28-values ( @xmath22150 mev ) . the measured yields of the coincident back - to - back @xmath4- and @xmath5-pairs , @xmath29 , are represented as @xmath30 , where @xmath31 are the number of back - to - back @xmath4(@xmath5)-pair events from the decay ; @xmath32 , @xmath33 and @xmath34 stand for decay - counter acceptances and detection efficiencies and reduction factors ( due to the fsi or / and other non back - to - back processes ) for the @xmath4(@xmath5)-pair , respectively . it is noteworthy that the reduction factors are approximately canceled out with assumption of the charge symmetry , @xmath35 , when we take the ratio of the @xmath4- and @xmath5-pair yields , @xmath6 . in order to minimize the fsi effect , we selected a light @xmath19-shell hypernucleus , @xmath0he . in @xmath19-shell hypernucleus , initial relative @xmath18 states must be @xmath36 states , whereas in a @xmath37-shell hypernucleus they may be @xmath38 states . to investigate the @xmath37-wave effect , we also performed the same experiment for a typical light @xmath37-shell hypernucleus , @xmath1c . in this letter , we show the opening angle and the energy sum distributions of @xmath4- and @xmath5-pairs from the nmwd of @xmath0he and the @xmath39 ratio for both hypernuclei . the present experiments ( kek - ps e462/e508 ) were carried out at the 12-gev proton synchrotron ( ps ) in the high energy accelerator research organization ( kek ) . hypernuclei , @xmath0he and @xmath1c , were produced via the ( @xmath2,@xmath3 ) reaction at 1.05 gev/@xmath40 on @xmath41li and @xmath42c targets , respectively . since the ground state of @xmath43li is above the threshold of @xmath0he @xmath44 , it promptly decays into @xmath0he emitting a low - energy proton . the @xmath45li ( @xmath2,@xmath3 ) @xmath43li reaction was therefore employed to produce @xmath0he . the hypernuclear mass spectra were calculated by reconstructing the momenta of incoming @xmath2 and outgoing @xmath3 using a beam - line spectrometer composed of the qqdqq system and the superconducting kaon spectrometer ( sks ) @xcite , respectively . particles emitted from the decays of @xmath11 hypernuclei were detected by the decay - particle detection system installed symmetrically in the direction to the target in order to maximize acceptance of the back - to - back event for @xmath4- and @xmath5-pairs from the nmwd process , as shown in ref.@xcite ( fig . 1 ) . it was composed of plastic scintillation counters and multi - wire drift chambers . the decay particles were identified by the time - of - flight and the range . r0.65 the ground state yields of @xmath0he and @xmath1c are , respectively , about 4.6 @xmath46 10@xmath47 and 6.2 @xmath46 10@xmath47 events , which were one order - of - magnitude higher than those of previous experiments . the inclusive excitation - energy spectra of @xmath43li and @xmath1c are shown in ref.@xcite ( fig . 2 ) . upper figures of fig . [ coinfig ] , ( a ) and ( b ) , show opening angle distributions of @xmath4- and @xmath5-pairs at the energy threshold level of 30 mev for both of proton and neutron . they seem to have clear back - to - back correlations , though these are not corrected the angular dependent acceptance . the shaded histogram shows estimated nucleon contaminations due to the pion absorption process in which @xmath48 s from the mesonic decay of @xmath11 hypernucleus are absorbed by the materials around the target . the background was estimated by assuming that the shape of the angular distribution from this @xmath48 absorption process is the same as that from the @xmath48 decay of @xmath11 ( @xmath11 @xmath49 ) formed via the quasi - free formation process ( see ref.@xcite for the detail ) . the angular distributions of middle of fig . [ coinfig ] , ( c ) and ( d ) , are corrected for acceptances and efficiencies for @xmath4- and @xmath5-pairs , and normalized per nmwd . the estimated contamination due to the pion absorption stated above are subtracted . they still have back - to - back correlation , which indicates that the fsi effect is not so severe and 1@xmath23-induced nmwd ( two body process ) is the major one . lower figures of fig . [ coinfig ] , ( e ) and ( f ) , show energy sum distributions of the @xmath4- and @xmath5-pairs by gating back - to - back events as shown in the upper figures ( @xmath50 ) . we confirmed that those energy sum distributions have broad peak around these @xmath28-values as expected . the shaded histogram shows estimated contaminations due to pion absorption as described above , which distributes to lower energy region . also for @xmath1c , similar distributions of the angle and energy sum of the @xmath4- and @xmath5-pairs were obtained in a same way . we successfully observed @xmath4- and @xmath5-pairs from the nmwd of @xmath0he and @xmath1c . the ratio of the back - to - back @xmath4- and @xmath5-pair yields , @xmath51 , for @xmath0he and @xmath1c were obtained as @xmath52 where the quoted systematic errors mainly come from the neutron detection efficiency ( @xmath22 6 % ) . they can be approximately regarded as the @xmath21 with assumption of the charge symmetry . it is now revealed that the @xmath21 ratio is significantly less than unity , thus excluding the earlier claim that the ratio is close to unity @xcite . on the contrary , recent theoretical calculations seem to be supportive to our results being on the increase of the ratio toward 0.5 . the present results have finally given the answer to the longstanding @xmath21 ratio puzzle , and have made a significant contribution to the study of the nmwd .

we have measured both yields of neutron - proton and neutron - neutron pairs emitted from the non - mesonic weak decay process of @xmath0he and @xmath1c hypernuclei produced via the ( @xmath2,@xmath3 ) reaction for the first time . we observed clean back - to - back correlation of the @xmath4- and @xmath5-pairs in the coincidence spectra for both hypernuclei . the ratio of those back - to - back pair yields , @xmath6 , must be close to the ratio of neutron- and proton - induced decay widths of the decay , @xmath7(@xmath8)/@xmath9(@xmath10 ) . the obtained ratios for each hypernuclei support recent calculations based on short - range interactions .

* the sequence . * in blazars , radio loud active galactic nuclei ( agn ) with their relativistic jet axis pointing to our line of sight , the synchrotron peak frequency ( @xmath3 ) covers a wide range ( @xmath4 hz ) , with bllacs ( bll , lineless blazars ) spanning the entire range and fsrqs ( flat spectrum radio quasars , sources with strong broad emission lines ) having lower @xmath3 ( @xmath5 hz ) . following @xcite , we adopt the generic terms for low , intermediate , and high _ synchrotron - peaking _ ( lsp , isp , hsp ) blazars independently of the spectroscopic type . @xcite found that as the source synchrotron power @xmath6 increases , @xmath3 decreases , with predominantly fsrq sources at the low @xmath3 , high @xmath6 end through lsp , isp , and finally hsp bllacs at the low @xmath6 end . they also used the sparse _ egret _ data to argue that the same reduction of the peak frequency happens in the high energy - presumably inverse compton ( ic ) component - component and that the compton dominance ( the ratio of ic to synchrotron power ) increases with source power . @xcite suggested that more efficient cooling of particles in the jets of high luminosity blazars is responsible for the lower peak frequencies . * from sequence to envelope . * @xcite and @xcite identified relatively powerful sources with a radio to x - ray spectral index @xmath7 typical of weak sources with @xmath3 in the x - rays . such sources , if confirmed , challenge the sequence . upon close study , however , their x - ray emission was found not to be of synchrotron origin @xcite and as of now sources with high @xmath6 - high @xmath3 have not been found @xcite . sources below the blazar sequence are expected from jets less aligned to the line of sight . indeed , @xcite found that new sources they identified modify the blazar sequence to an _ envelope_. * challenges . * @xcite found several low @xmath6 - low @xmath3 sources that , because they have a high core dominance ( @xmath8 , ratio of core and therefore beamed to extended and therefore isotropic radio emission ) , are not intrinsically bright sources at a larger jet angle . these sources challenge the sequence because ( @xmath0 ) both intrinsically weak and intrinsically powerful jets can have similar @xmath3 and ( @xmath1 ) intrinsically weak jets can produce a wide range of @xmath3 from ( @xmath9 - @xmath10 hz ) . another challenge came from @xcite who showed that , contrary to what is anticipated by the sequence , high and low synchrotron peak frequency ( hsp and lsp ) bl lacertae objects ( blls , blazars with emission line ew @xmath11 ) have similar @xmath12 . these findings challenge the sequence , even after being extended to include the sources in the envelope as de - beamed analogs of the blazar sequence sources . @xcite argued that at a critical value of the accretion rate @xmath13 , the accretion switches from a standard radiatively efficient thin disk with accretion - related emission power @xmath14 for @xmath15 , to a radiatively inefficient mode where @xmath16 . this critical point may be connected to the transition between fanaroff riley ( fr ; * ? ? ? * ) ii to fr i radio galaxies ( rg ) : the level of the low frequency extended radio emission ( coming mostly from the radio lobes and considered to be isotropic ) that separates fr i and fr ii rg , has been shown to be a function of the host galaxy optical magnitude @xcite : the division between fr i and fr ii is at higher radio luminosities for brighter galaxies . @xcite argued that , because the optical magnitude of a galaxy is related to the central black home mass @xcite and the extended radio luminosity is related to the jet kinetic luminosity ( following the scaling of * ? ? ? * ) , this division can be casted as a division in terms of the fraction of the eddington luminosity carried by the jet : jets with kinetic luminosity @xmath17 give rise to fr i rg , while jets with @xmath18 are predominantly fr ii sources . interestingly , and in agreement with the unification scheme , @xcite and @xcite find that the same dichotomy applies to separating blls and fsrq , the aligned versions of fr i and fr ii respectively . finally , it is very intriguing that @xcite argue that there is a paucity of sources around @xmath19 . fr i , low line excitation fr ii and some high line excitation fr ii were found to occupy the low @xmath20 regime , while the high @xmath20 regime was occupied by high line excitation fr ii , broad line radio galaxies and powerful radio quasars . track ( a ) shows the path of a synchrotron peak for a single speed jet in an environment of radiatively efficient accretion and ( b ) for a decelerating jet of the type hypothesized to exist in fri sources as the jet orientation changes.__,width=268 ] recently , we ( * ? ? ? * heretofore m11 ) compiled the largest sample of radio loud agn for which sufficient data existed to determine variability - averaged @xmath3 @xmath6 , as well as the extended low frequency radio emission @xmath12 . this is an important quantity in our study , because it has been shown to be a good proxy for the jet kinetic power @xmath21 , as measured by the energy required to inflate the x - ray cavities seen to coincide with the radio lobes of a number of sources ( e.g. * ? ? ? * ; * ? ? ? * ) . the picture that emerges ( figure [ m11 ] ) exhibits some important differences with the blazar sequence . in particular , isp blls have @xmath21 comparable to that of hsp and lsp blls . also , although all the fr i galaxies were found to have similar @xmath21 with blls , no fr i galaxies were found with @xmath22 hz . because there is no obvious selection acting against the detection of fr i galaxies with core sed peaking at higher energies , we are lead to conclude that the un - aligned versions of hsp blazars have @xmath3 smaller by a factor of least @xmath23 compared to their aligned equivalent , something that agrees with the existence of velocity profiles in the emitting plasma , as supported by other investigations ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in m11 we suggested that _ extragalactic jets can be described in terms of two families_. the first is that of weak jets characterized by velocity profiles and weak or absent broad emission lines . hsps ( @xmath24 hz ) , isps ( @xmath25 hz ) , and fr i rg belong to this family . on the basis of having similar @xmath21 with hsps and fr i rg , the isp sources were argued to be somewhat un - aligned hsps . the second family is that of more powerful jets having a single lorentz factor emitting plasma and , in most cases , stronger broad emission lines . interestingly , the two families divide at @xmath26 erg s@xmath27 , which for @xmath28 , corresponds to @xmath29 , similar to the @xmath30 of @xcite . aligned sources are found along the broken power sequence depicted by the solid lines a and b with a ( b ) corresponding to jets in radiatively efficient ( inefficient ) accretion environments with @xmath31 ( @xmath32 ) . the broken lines a and b depict the tracks followed by two sources as they depart from the power sequence and their orientation angle @xmath33 increases . while in the first case a single velocity flow is assumed , in the second case emission from a decelerating flow is considered @xcite . . see text for the description of boxes a , b , c and zones @xmath34.__,width=268 ] we now discuss some predictions of the new unification scheme and their confirmation from the current data . * @xmath35 increases along the two branches of the broken power sequence . * we examine now if along the two branches of the power sequence , depicted schematically by the red and blue arrows in the figure [ prelim ] , @xmath21 increases . to do that , we select those sources that are close to the sequence of powerful aligned objects and split them in three groups a , b , c , as seen in figure [ prelim ] . in figure [ powerful ] we plot the @xmath21 distribution of sources in these three groups . as expected , the average @xmath21 increases from group c to a. running the same test for jets with inefficient accretion requires to use sources that are not aligned , because of the small number of sources . for this reason , we select all low power sources with @xmath36 to insure that we do not have any mixing with sources of the other branch and we separate them in the three groups @xmath37 ( figure [ prelim ] ) separated by the de - beaming tracks of a decelerating jet depicted also in figure [ m11 ] . as can be seen in figure [ weak ] , the average @xmath21 increases from group @xmath38 to @xmath39 , according to our expectations . * as @xmath35 increases , the fraction of the blls decreases along the powerful sequence . * as @xmath21 increases along the powerful sequence , @xmath6 increases , but @xmath3 decreases . at the same time , if we assume that @xmath21 scales with accretion power , we expect that the blr luminosity increases . if @xmath3 did not change , we would expect that the ratio of the blr to optical synchrotron emission would not change . but @xmath3 does decrease as @xmath21 increases , shifting the synchrotron component to lower frequencies and revealing more of the blr . thus we expect that the fraction of sources that is classified as blls will become smaller as @xmath21 increases along the powerful sequence . this is clearly seen in figure [ powerful ] , with the fraction of blls clearly decreases as @xmath21 decreases . * for powerful sources , the fraction of blls increases for less aligned sources . * in our scheme , we expect that for powerful sources of a given @xmath21 , as they become more un - aligned , the beamed synchrotron emission will decrease , while the blr luminosity will be much less affected , resulting to a decreasing fraction of blls for more un - aligned sources . to address this , we selected sources with @xmath40 erg s@xmath27 ( orange sources in figure [ prelim ] ) and we plotted the fraction of blls as a function of radio core dominance @xmath41 which is an orientation indicator . as can be seen in figure [ bll_fraction ] , as the core dominance decreases , the fraction of blls quickly decreases , in agreement with our expectations . * a given accretion power @xmath42 corresponds to a narrow @xmath35 range . * we collected black hole masses from the literature for most of the sources of m11 and used them to calculate the ratio of @xmath43 . we plot our results in figure [ mcrit ] : in blue sources with @xmath44 hz , almost exclusively blls , therefore radiatively inefficient accretors ; in red sources with @xmath45 hz and @xmath46 erg s@xmath27 , almost all fsrqs , therefore radiatively efficient accretors . the separation of red and blue sources at @xmath47 suggests that there is a transition at @xmath48 with radiatively efficient accretion at @xmath49 and that sources with a given accretion power do not produce jets with @xmath21 significantly smaller or larger than their accretion power .

we recently argued @xcite that the collective properties of radio loud active galactic nuclei point to the existence of two families of sources , one of powerful sources with single velocity jets and one of weaker jets with significant velocity gradients in the radiating plasma . these families also correspond to different accretion modes and therefore different thermal and emission line intrinsic properties : powerful sources have radiatively efficient accretion disks , while in weak sources accretion must be radiatively inefficient . here , after we briefly review of our recent work , we present the following findings that support our unification scheme : ( @xmath0 ) along the broken sequence of aligned objects , the jet kinetic power increases . ( @xmath1 ) in the powerful branch of the sequence of aligned objects the fraction of blls decreases with increasing jet power . ( @xmath2 ) for powerful sources , the fraction of blls increases for more un - aligned objects , as measured by the core to extended radio emission . our results are also compatible with the possibility that a given accretion power produces jets of comparable kinetic power .

the tremendous progress in the last decade has made it possible to pin down , with impressive accuracy , many of the fundamental parameters in the neutrino sector . a complete picture , however , is still not available . chief among the missing information is the determination of the @xmath1 element of the neutrino mixing matrix @xmath2 , which , in turn , is crucial in ascertaining the cp violation effects in the leptonic sector . given that direct cp violations in the quark sector @xcite have been well - established and accurately measured , it is imperative , from both the theoretical and experimental points of view , to assess the corresponding situation in the leptonic sector . another unsolved puzzle concerns the neutrino mass spectrum , in that there are the possibilities of either the normal " or inverted " orderings . it is certainly important to settle this question . while the fundamental parameters refer to those in vacuum , it has been well - established ( see , _ e.g. _ , ref.@xcite ) that they are modified when neutrinos propagate through matter , by giving the neutrino an induced mass , which is proportional to its energy and to the medium density . indeed , in the analyses of the solar neutrinos , certain features of the data , such as the modification of the energy spectra from the original , can only be understood by the inclusion of matter effects . with the advent of long baseline experiments ( lbl , for an incomplete list , see , _ e.g. _ , ref.@xcite ) , the induced mass can actually be tuned " by changing the neutrino energy ( @xmath3 ) . this provides a powerful tool which can be used to extract fundamental neutrino parameters from measurements . in this work , we will use a rephasing invariant parametrization which enables us to obtain simple formulas for the transition probabilities of neutrinos propagating through matter of constant density . it was shown earlier that these parameters obey evolution equations as a function of the induced mass . in addition , these equations preserve the approximate @xmath0 symmetry @xcite which characterizes the neutrino mixing in vacuum . incorporation of the @xmath0 symmetry for all induced mass values results in a set of very simple transition probabilities @xmath4 . in general , these formulas offer quick estimates of the various oscillation probabilities , using the known solutions obtained earlier . as an example , we will analyze @xmath5 in detail , emphasizing its dependence on the neutrino parameters . neutrino oscillations , being lepton - number conserving , are described in terms of a mixing matrix whose possible majorana phases are not observable . thus it behaves just like the ckm matrix under rephasing transformations , which leave physical observables invariant @xcite . to date , however , such observables are often given in terms of parameters which are not individually invariant . so it seems that the use of manifestly invariant parameters may be more physically relevant . two such sets are known to be @xmath6 @xcite and @xmath7 @xcite . recently , by imposing the condition @xmath8 ( without loss of generality ) , another set was found , given by @xcite @xmath9 where the common imaginary part can be identified with the jarlskog invariant @xmath10 @xcite . their real parts are labeled as @xmath11 the variables are bounded by @xmath12 with @xmath13 for any ( @xmath14 ) , and satisfy two constraints : @xmath15 @xmath16 eq . ( [ con2 ] ) , together with the relation @xmath17 follow @xcite from ( the imaginary and real parts of ) the identity @xmath18 . thus , flavor mixing is specified by the set @xmath19 plus a sign , according to @xmath20 . this sign arises since the transformation @xmath21 , corresponding to a cp conjugation , leaves the real part @xmath19 of @xmath22 invariant , but changes the sign of its imaginary part @xmath23 . note that , using @xmath24 , a complete parametrization also requires four @xmath24 elements plus a sign . the parameters @xmath19 are related to the rephasing invariant elements @xmath24 by @xmath25 = \left(\begin{array}{ccc } x_{1}-y_{1 } & x_{2}-y_{2 } & x_{3}-y_{3 } \\ x_{3}-y_{2 } & x_{1}-y_{3 } & x_{2}-y_{1 } \\ x_{2}-y_{3 } & x_{3}-y_{1 } & x_{1}-y_{2 } \\ \end{array}\right).\ ] ] one can readily obtain the parameters @xmath19 from @xmath26 by computing its cofactors , which form the matrix @xmath27 with @xmath28 , and is given by @xmath29 the relations between @xmath19 and @xmath30 are given by ( using @xmath31 ) : @xmath32 the second term in either expression is one of the @xmath33 s ( @xmath34 s ) defined in eq . ( [ eq : g ] ) . also , by using the constraint in eq . ( [ cons ] ) , @xmath35 can be expressed in terms of quadratics in @xmath19 , a result which will be used later in tables i and ii . for neutrinos in matter ( of constant density ) , it was shown @xcite that , as a function of the induced mass @xmath36 , the neutrino parameters satisfy a set of evolution equations which are greatly simplified by using the @xmath19 variables . it was found that @xmath37 where @xmath38 are the eigenvalues of the hamiltonian . also , the evolution equations for all @xmath39 can be obtained and are collected in table i of ref . @xcite . of particular interest for our purposes are the equations : @xmath40 and @xmath41 note that the quantities @xmath42 and @xmath43 form a closed system under the evolution equations , independent of other possible combinations of these variables . there remain two more independent evolution equations , which may be chosen as those for ( @xmath44 ) . we define @xmath45 @xmath46 then @xmath47.\end{aligned}\ ] ] it follows that @xmath48 if @xmath49 . this condition is equivalent to @xmath50 , @xmath51 , @xmath0 exchange symmetry . thus , the evolution equations preserve the @xmath0 symmetry , which was established ( approximately ) for neutrino mixing in vacuum . another useful property of the evolution equations is to establish matter invariants . for instance @xcite , @xmath52=0,\ ] ] where @xmath53 is defined in eq . ( [ eq : xi ] ) and @xmath54 ( also , @xmath55 , as mentioned before @xcite ) . in addition , there is a simple relation @xmath56=1.\ ] ] eqs . ( [ xd ] ) and ( [ sxd ] ) are three - flavor generalizations of the two - flavor results @xcite : @xmath57 @xmath58=-1,\ ] ] where @xmath59 , @xmath60 , @xmath61 , in the usual notation . the vacuum neutrino masses are known to be hierarchical , @xmath62 , @xmath63 , @xmath64 . there are two possibilities , the normal hierarchy ( @xmath65 ) , or the inverted hierarchy ( @xmath66 ) . in matter of constant density , @xmath67 , which are @xmath68-dependent . for the case of normal hierarchy , there are two @xmath68-values where the levels cross " , at the lower resonance , @xmath69 , @xmath70_{a_{l}}=0 $ ] , and at the higher resonance , @xmath71 , @xmath72_{a_{h}}=0 $ ] . from eqs . ( [ eq : ln ] ) , one finds that rapid variations occur only for @xmath68 to be near @xmath73 or @xmath74 . let us denote by @xmath75 the values of @xmath68 in vacuum @xmath76 , at the lower resonance @xmath77 , in the intermediate range @xmath78 , at the higher resonance @xmath79 , and in dense medium @xmath80 . then , the solutions for @xmath81 are well - approximated @xcite by two - flavor resonance solutions . for @xmath82 , @xmath83^{1/2 } , \nonumber \\ x_{1 } & = & \frac{1}{2}[p_{l}-(p^{2}_{l}a - q_{l}\delta_{0})/\delta_{21 } ] , \nonumber \\ x_{2 } & = & \frac{1}{2}[p_{l}+(p^{2}_{l}a - q_{l}\delta_{0})/\delta_{21 } ] , \nonumber \\ x_{3 } & \cong & ( x_{3})_{0},\end{aligned}\ ] ] where @xmath84 in matter , @xmath85 , @xmath86 , @xmath87 . note that @xmath88 , @xmath89 , and @xmath90 . for @xmath91 , @xmath92^{1/2 } , \nonumber \\ x_{1 } & \cong & ( x_{1})_{i } , \nonumber \\ x_{2 } & = & \frac{1}{2}[p_{h}-(p^{2}_{h } \bar{a}-q_{h } \delta_{i})/\delta_{32 } ] , \nonumber \\ x_{3 } & = & \frac{1}{2}[p_{h}+(p^{2}_{h } \bar{a}-q_{h } \delta_{i})/\delta_{32}].\end{aligned}\ ] ] here , @xmath93 , and @xmath94 , @xmath95 , @xmath96 are taken at @xmath97 . note that @xmath98 , @xmath99 , @xmath100 . also , @xmath101 is an invariant as @xmath68 varies . thus , the product @xmath102 has a resonance behavior near @xmath103 . note also that the minimum of @xmath104 is at @xmath105 . to obtain @xmath106 for @xmath91 and @xmath104 for @xmath82 , one first notes from eq . ( [ eq : di ] ) that @xmath107 for high @xmath68 . thus , a direct integration leads to @xmath108\ ] ] for @xmath91 , where @xmath109 . similarly , a direct integration of @xmath110 for low @xmath68 gives @xmath111.\ ] ] the solutions for @xmath112 in both regions of @xmath68 are obtained from @xmath113 . note that the solutions for @xmath82 and for @xmath91 should agree for @xmath114 . this condition leads to @xmath115 and @xmath116 . for inverted hierarchy , the behaviors of @xmath53 near @xmath73 are given by the same eq . ( [ low ] ) . however , for @xmath117 , there is no longer a resonance . instead , all @xmath53 change slowly , so that @xmath118 , @xmath119 , @xmath120 , for @xmath117 . the solutions for @xmath121 are obtained by @xmath122 . thus , there is a resonance behavior near @xmath74 , for the inverted hierarchy scenario . otherwise all the changes are small . the accuracy of the approximate formulas in eqs . ( [ low]-[high ] ) can be assessed by numerical integrations of the exact equations , eqs . ( [ eq : di ] ) and ( [ eq : ln ] ) . to do that we write @xmath123 where @xmath124 in vacuum , and @xmath26 reduces to the tribimaximal @xcite matrix when @xmath125 . it should be emphasized that the parameters @xmath126 carry quite distinct behaviors as @xmath68 varies , as shown in the following . ( [ eq : w ] ) and ( [ w0 ] ) give rise to @xmath127 and from @xmath128 , we have @xmath129 with the constancy of @xmath130 , one concludes that @xmath131 as @xmath68 varies . in addition , since @xmath132 , we have @xmath133,\ ] ] and @xmath134 furthermore , one obtains from @xmath135 that @xmath136,\ ] ] and @xmath137 thus , @xmath138 and @xmath139 can change considerably as functions of @xmath68 , but @xmath140 throughout . for numerical integrations , eqs . ( [ eq : w ] ) and ( [ w0 ] ) suggest the following initial values in vacuum : @xmath141 where @xmath142 is chosen and the terms in @xmath143 are ignored . we shall choose the initial values @xmath144 and @xmath145 , which correspond to the experimental bounds @xmath146 @xcite and an assumed cp violation phase @xmath147 , respectively . the numerical solutions for the @xmath19 parameters , the squared elements of the mixing matrix , and @xmath10 in matter follow directly and are shown in figs . 2 - 5 in ref . our choice of @xmath148 signifies a small @xmath0 symmetry breaking , the solutions verify that @xmath149 remain negligible for all a values . in addition , we show in fig . 1 both the numerical and the approximate solutions for @xmath150 in matter . note that the hierarchical relation among the @xmath150 s varies in matter and plays an important role in the oscillatory factor @xmath151 of the probability functions . it is seen that @xmath152 ( normal hierarchy ) and @xmath153 ( inverted hierarchy ) for @xmath154 . while in @xmath155 , the @xmath150 s are less hierarchical : @xmath156 ( normal ) and @xmath157 ( inverted ) . [ cols="^,^,^,^,^",options="header " , ] our results may be compared to formulas in terms of the standard parametrization " @xcite , given , @xmath158 , in kimura @xmath159 @xcite . the relations between @xmath19 and the standard parametrization " are given by @xmath160 where @xmath161 , @xmath162 , and @xmath163 is the dirac cp phase . it can be shown that the functions @xmath164 here in terms of @xmath19 are simply @xmath165 in eqs.(15 - 23 ) of ref . @xcite , and the resultant probability functions are identical . eq ( [ eq : kk ] ) also offers some insight on the @xmath68-independence of the approximate @xmath0 symmetry . it is seen that the conditions @xmath166 are fulfilled if 1 ) @xmath167 , and 2 ) @xmath168 . the behaviors of @xmath169 were given in fig . . @xcite . while @xmath170 is almost independent of @xmath68 , @xmath171 for low @xmath68 , and @xmath172 for high @xmath68 . they combine to validate conditions 1 ) and 2 ) , for all @xmath68 values . the other possibility is that @xmath173 . here , @xmath163 itself is largely @xmath68-independent because of the matter invariant @xmath174 @xcite . exact @xmath0 symmetry was studied earlier by harrison and scott @xcite . their formulation uses the mixing matrix @xmath2 ( with specific choice of phases ) , while our results are in terms of rephasing invariant ( and observable ) variables , making it possible to calculate transition probabilities directly . in addition , by comparing with the exact formulas in table i , one can quickly compute corrections to the presumed exact symmetry . the unique features of the @xmath19 parametrization can be used to facilitate , @xmath158 , the analyses of the lbl experiments . as an example , let us consider the probability @xmath5 explicitly . according to table i , with the approximation @xmath166 , @xmath175 \nonumber \\ & -&8j\sin\phi_{21}\sin\phi_{31}\sin\phi_{32},\end{aligned}\ ] ] with @xmath176 . using the solutions in eqs . ( [ low],[high ] ) , it is straightforward to infer the behaviors of @xmath5 . in the following , let us focus on the region of high @xmath68 values @xmath177 . here , @xmath178 so that ( excluding the case @xmath179 ) @xmath180 it is useful to examine the qualitative properties of @xmath181 and @xmath182 separately . if the mass hierarchy is normal , the solutions in eq . ( [ high ] ) suggest a higher resonance for @xmath181 at @xmath183 , where @xmath184 . with @xmath185[(e/\mbox{gev})]$ ] , @xmath186 @xmath187 , and @xmath188 @xmath189 , the location of resonance @xmath74 corresponds to an energy @xmath190 gev , which is independent of the baseline length . ( [ pemu ] ) shows that , in the high @xmath68 region , @xmath5 @xmath191 is two - flavor like . however , it does not mean that the three - flavor problem is reduced to a single two - flavor problem . this is because the probability @xmath192 , according to table i , would have contributions from all the @xmath193 s . as an illustration , we show @xmath194 , @xmath195 , and @xmath196 as functions of @xmath3 in fig . 2 , with @xmath197 km . it is seen that a resonance for @xmath194 occurs near @xmath198 gev as expected . however , the smallness of @xmath182 near @xmath198 gev suppresses the probability even if @xmath194 is at a resonance . on the other hand , the probability at the first peak of @xmath182 ( near @xmath199 gev ) also gets suppressed by the smallness of @xmath194 . as a result , a significant flavor transition only occurs when @xmath200 is adjusted so that the peak of @xmath195 is located near the resonance of @xmath194 . the first maximum of @xmath182 occurs if @xmath201 is properly chosen : @xmath202=\frac{\pi}{2}.\ ] ] for the first maximum to coincide with the resonance of @xmath181 , the value of @xmath104 is taken at @xmath74 : @xmath203 . it leads to @xmath204 using the current upper bound @xmath205 . one concludes that if the mass hierarchy is normal , an extra long baseline ( @xmath206 km ) can lead to a greatly enhanced probability for the neutrino beam near @xmath207 gev , at which energy both @xmath194 and @xmath182 reach the maximal values . the probability will be suppressed when @xmath200 starts to vary and @xmath182 moves away from the maximum . note that for the maxima of @xmath181 and @xmath182 to coincide near @xmath207 gev , the baseline @xmath200 and the undetermined @xmath208 are related by @xmath209 . on the other hand , since @xmath194 does not go through the higher resonance under the inverted hierarchy , the probability is in general suppressed even if @xmath182 reaches its maximum . one further concludes that under the inverted hierarchy , the transition probability remains small and is insensitive to variation of the baseline length @xmath200 . thus , if the mass hierarchy is normal , one would expect to observe sizable probability difference at high energy for experiments involving two baselines with sizable difference in length . on the other hand , the probability would be small and nearly independent of the baseline at high energy if the mass hierarchy is inverted . we show in fig . 3 the probability function under both hierarchies for two arbitrarily chosen baselines . note that the peak locations and the peak values vary as @xmath200 . it is seen that for the normal hierarchy , @xmath210 km ) @xmath211 km ) near the first peak is expected , while @xmath210 km ) @xmath212 km ) @xmath213 if the mass hierarchy is inverted . this result may provide useful hints to the determination of the mass hierarchy . note that the probabilities can be deduced if the details of the experiments are considered . if the neutrino energy can be reconstructed accurately from the secondary particles involved in an experiment , the observed spectrum will tell how the magnitude of the transition probability plays a role . on the other hand , if reliable measurement of the energy spectrum is not available , a collection of the event rates should also be useful in comparing the probabilities . another possible application is to look for both @xmath214 and @xmath215 for a single , but very long baseline . since the @xmath121 s only go through the higher resonance under the inverted hierarchy , one would expect to observe in the vicinity of the peak either @xmath216 if the hierarchy is normal , or @xmath217 if the hierarchy is inverted . we show an example in fig . 4 . note that although the peak value of the probability varies with the baseline length , the relative and qualitative features of the above observation remain valid for a chosen baseline . neutrino transition probabilities are usually given in terms of the simple expression @xmath218 , although the individual @xmath219 s are not directly observable . when one rewrites them using physical observables , such as those in the standard parametrization " , the resulting formulas are often very complicated . it is thus not easy to obtain general properties of these probabilities in experimental situations . in this paper we express the probabilities as functions of rephasing invariant parameters . in addition , we incorporate the @xmath0 symmetry , valid ( approximately ) for any value of the induced neutrino mass ( @xmath68 ) . the resulting formulas are very simple , and are listed in tables i and ii . they offer a quick quantitative assessment for any physical process at arbitrary @xmath68 values . as an illustration , we analyzed the probability @xmath214 , with emphasis on its dependence on @xmath3 , @xmath200 , and @xmath220 . by changing the value of @xmath3 and @xmath200 in various lbl experiments , one can hope not only to test the theory used to establish @xmath4 , but also to help in the efforts to determine the unknown parameter @xmath208 .

the vacuum neutrino mixing is known to exhibit an approximate @xmath0 symmetry , which was shown to be preserved for neutrino propagating in matter . this symmetry reduces the neutrino transition probabilities to very simple forms when expressed in a rephasing invariant parametrization introduced earlier . applications to long baseline experiments are discussed .

a spin glass ( sg ) is a complex system characterized by both quenched randomness and frustration , which lead to the irreversible freezing of spins to states without the long - range spatial order below the glass transition temperature ( @xmath0 ) @xcite . theoretical approaches for understanding sg transitions are generally concerned with the study of mean - field level calculations performed using infinite - range interaction models , of which the sherrington - kirkpatrick ( sk ) model @xcite is a prototype . some infinite - range interaction sg models have recently sparked interest in relation to the so - called inverse transitions . since tammann s hypothesis @xcite a century ago , there has been substantial interest in a different class of phase transitions known as inverse transitions ( melting or freezing ) . in these phase transitions , an ordered phase is more entropic than a disordered one , whereby the ordered phase may appear at a higher temperature than the disordered one . such inverse transitions have already been observed experimentally in physical systems such as of liquid crystals @xcite , polymers @xcite , high-@xmath1 superconductors @xcite , magnetic thin films @xcite , and organic monolayers @xcite . meanwhile , from a theoretical point of view , there have been various attempts to identify a suitable model for inverse transitions . spin - glass models have been suggested to be candidates for inverse freezing , wherein the sg phase becomes one with higher entropy . the ghatak - sherrington ( gs ) model @xcite is a spin-1 spin - glass model with a crystal field and it is especially well known as a prototypical sg model for inverse freezing @xcite . in ordinary sg systems , in general , the second - order phase transition from paramagnetic ( pm ) to sg occurs as temperature is decreased . however , according to crisanti and leuzzi @xcite , there seems to be a second reentrance as well as inverse freezing in the gs model . ( see fig . 2 in refs . this implies that phase transitions are likely when the phase is varied successively in the order pm @xmath2 sg @xmath2 pm @xmath2 sg as the temperature is reduced . in other words , there seems to exist two different sgs , i.e. , a sg in the higher - temperature region [ higher - temperature spin glass ( htsg ) ] and a sg in the lower - temperature region [ lower - temperature spin glass ( ltsg ) ] . the aim of this paper is to investigate the theoretical validity for the existence of such separated sgs using a simple gs - like model . for this purpose , we study a quantum version of the gs model by adding a transverse tunneling field , similar to the manner in which the quantum version of the sk model has been studied by considering quantum tunneling with a transverse field @xcite . we expect the quantum gs model to clarify the changes in the existence and features of the two sgs with respect to the transverse field . herein we use one - step replica symmetry breaking ( 1rsb ) for theoretical investigations instead of the replica symmetry ( rs ) @xcite and the full replica symmetry breaking ( frsb ) @xcite . we select the 1rsb because it provides more physically meaningful results than rs does and numerical values of order parameters more easily than frsb does . although 1rsb is approximated with respect to the exact frsb ansatz , it is a good approximation around transition lines because at criticality the thermodynamics is not very sensitive to the ansatz chosen , as shown in refs . the hamiltonian of the quantum gs model is @xmath3 where ( @xmath4 ) means all the distinct pairs of spins with the total number @xmath5 , @xmath6 are quenched random exchange interaction variables , @xmath7 is the crystal field , and @xmath8 is the transverse tunneling field . the spin-1 quantum spin operators @xmath9 and @xmath10 are defined by @xmath11 respectively . the distribution of @xmath6 is taken to be gaussian with a mean zero and a variance of @xmath12 . when @xmath13 and @xmath14 are the degeneracy of the filled or interacting states of @xmath9 and of the empty or noninteracting states of @xmath9 , respectively , we can define the relative degeneracy of the filled states as @xmath15 @xcite . by the imaginary - time formalism @xcite , the partition function of the system can be written as @xmath16 \mathcal{t } \exp \big[\int_{0}^{\beta } d\tau \nonumber\\ & & \big\ { \sum_{ij}^{n } j_{ij}s_{iz}(\tau)s_{jz}(\tau ) - d \sum_{i}^{n } ( s_{iz}(\tau))^{2 } \big\}\big]\end{aligned}\ ] ] where @xmath17 is the imaginary time , @xmath18 is the time - ordering operator , @xmath19 are the operators under the interaction representation introduced in the quantum physics , [ i.e. , @xmath20 where @xmath21 and @xmath22 ( where @xmath23 for simplicity ) . for this model , the free energy is calculated as @xmath24_{j } = \int \prod_{i , j}^{n } dj_{ij } p(j_{ij } ) \ln z ( \{j_{ij}\})$ ] , where @xmath25_{j}$ ] indicates an average over the quenched disorder of @xmath6 . for the quenched random system the free energy can be evaluated using the replica method @xmath26 $ ] . by averaging @xmath27 over @xmath28 , rearranging terms , and taking the method of steepest descent in the thermodynamic limit ( @xmath29 ) , the intensive free energy @xmath30 can be written as @xmath31 - \ln \textrm{tr } \exp ( \tilde{\mathcal{h } } ) \bigg\}\end{aligned}\ ] ] with the effective hamiltonian @xmath32 ~\mathcal{t } \exp \bigg\ { \int_{0}^{\beta } d\tau \int_{0}^{\beta } d\tau ' \big [ \frac{1}{2 } \sum_{(\alpha \beta)}^{n } q^{\alpha \beta}(\tau , \tau ' ) s_{z}^{\alpha}(\tau)s_{z}^{\beta}(\tau ' ) \nonumber\\ + \frac{1}{2 } \sum_{\alpha}^{n } r^{\alpha \alpha}(\tau , \tau ' ) s_{z}^{\alpha}(\tau)s_{z}^{\alpha}(\tau ' ) \big ] - d \int_{0}^{\beta } d\tau \sum_{\alpha}^{n } ( s_{z}^{\alpha}(\tau))^{2 } \bigg\}\end{aligned}\ ] ] where @xmath33 denotes a summation over replica indices @xmath34 and @xmath35 running from 1 to @xmath36 , and the trace @xmath37 is over @xmath36 replicas at a single spin site . here two order parameters are introduced : the spin - glass order parameter @xmath38 and the spin self - interaction @xmath39 , where @xmath40/\textrm{tr}~ e^{\tilde{\mathcal{h}}}$ ] . we take the static approximation @xcite by @xmath41 and @xmath42 . then the free energy @xmath43 is given by @xmath44~~\end{aligned}\ ] ] with the effective hamiltonian @xmath45 next , we use parisi s 1rsb scheme as in the case of the sk model @xcite : for the @xmath46 matrix @xmath47 in the replica spin space , the @xmath36 replicas of @xmath47 are divided into @xmath48 groups of @xmath49 replicas , assuming that @xmath36 must be a multiple of @xmath49 , so that @xmath47 consists of @xmath48 diagonal matrices of @xmath50 elements each ( in which all the diagonal elements are zero and off - diagonal elements are @xmath51 ) and @xmath52 matrices of @xmath50 elements ( in which all the elements are @xmath53 ) . then the free energy obtained by the 1rsb ansatz is given as follows : @xmath54^{m } \bigg]\end{aligned}\ ] ] we can complete phase diagrams of the present model from these equations . first , let us consider the @xmath61 case in order to check whether the result of crisanti and leuzzi @xcite is correct . the graphs in fig . 1 show the @xmath62 phase diagrams obtained for specific @xmath8 values . as shown in fig . 1(a ) , the @xmath62 phase diagram of the @xmath63 case ( gs model ) at @xmath61 is nearly the same as that of the model used by crisanti and leuzzi @xcite . the locations of the first - order phase boundary and tricritical point ( tcp ) , i.e. , the cross - point between first- and second - order phase boundaries , were determined by the same criteria proposed in ref . the tcp of fig . 1(a ) is located at ( 0.962 , 0.333 ) , as analytically obtained in ref . @xcite . in the region @xmath64 , the second - order phase transition from pm to sg occurs as the temperature is decreased , which is generally observed in ordinary sg systems . however , in the region @xmath65 , successive phase transitions occur for which the phase is varied in the order pm @xmath66 htsg @xmath67 pm @xmath67 ltsg , as the temperature is reduced . this result shows clearly the second reentrance that crisanti and leuzzi referred to previously @xcite . in the region @xmath68 , inverse freezing is shown through the phase transitions in the order pm @xmath66 sg @xmath67 pm , as the temperature is decreased . therefore , we have verified that inverse freezing , which many investigators of the gs model have focused upon , occurs only in a narrow region . figure . 1(b ) shows the @xmath62 phase diagrams for several values of @xmath8 , including the result of the @xmath63 case ( gs model ) . as @xmath8 is gradually increased , the glass transition temperatures decrease . in the range @xmath69 , only the second - order phase transition from pm to sg occurs as the temperature is reduced , and the glass transition temperatures decrease as @xmath8 is increased . however , when @xmath7 is larger than 0.7 , the first - order phase transitions occur and the position of each tcp depends on each @xmath8 value . the shapes of the phase boundaries in this range are rather complex , as can be checked in fig . 2(b ) . phase diagram for the @xmath63 case ( gs model ) and ( b ) @xmath62 phase diagrams for several values of @xmath8 . the solid - line ( dotted - line ) part of each phase boundary indicates the second - order ( first - order ) phase transition and each circle between the two kinds of lines denotes a tcp.,title="fig:",scaledwidth=50.0% ] phase diagram for the @xmath63 case ( gs model ) and ( b ) @xmath62 phase diagrams for several values of @xmath8 . the solid - line ( dotted - line ) part of each phase boundary indicates the second - order ( first - order ) phase transition and each circle between the two kinds of lines denotes a tcp.,title="fig:",scaledwidth=50.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] the graphs of fig . 2 show the @xmath71 phase diagrams obtained for specific @xmath7 values . figure 2(a ) represents the temperature - dependent variations in the phase boundaries , which are obtained for @xmath7 between 0.0 and 0.697 . in the ising spin - glass model with a transverse field , the glass transition temperature at @xmath63 is 1.0 @xcite , whereas in our model the transition temperature at @xmath63 is 0.86 . the difference between the two values can be attributed to the fact that our model includes the eigenvalues of @xmath72 as well as @xmath73 and @xmath74 . when @xmath75 is increased , the glass transition temperature gradually increases to 1.0 . as expected , the phase boundary is shifted to a lower temperature with the increase in @xmath7 . according to our detailed numerical calculation , the first - order phase transition first arises at @xmath76 , where the tcp is located at @xmath77 . when @xmath7 is larger than 0.697 , the shift becomes more complex , as shown in fig . as @xmath7 is larger than 0.697 , one tcp is separated into two new tcps and a first - order phase transition lies between these two tcps @xcite . as @xmath7 is gradually increased , the phase boundary is kinked in the direction of the dashed arrow of fig . 2(b ) and the region of the first - order phase transition simultaneously broadens . as @xmath7 increases further , one of the tcps collapses with the @xmath8 axis . when @xmath7 becomes 0.879 , the phase boundary starts to split . the second reentrance of the gs model [ fig . 1(a ) ] is a zero-@xmath8 case reflecting this splitting of the phase boundary . the two sg phases generated by the splitting are the htsg and the ltsg . the htsg is inside the extremely narrow region of @xmath8 and surrounded by the @xmath78 axis , the second - order phase boundary , one tcp , and the first - order phase boundary . however , the ltsg is spread along the @xmath8 axis and is surrounded only by the axis and the first - order phase boundary . in the case of @xmath70 of fig . 2(c ) , two types of sgs ( htsg and ltsg ) exist between @xmath79 . however , for values of @xmath8 greater than 0.045 , only one type of sg ( ltsg ) exists under the pm phase . the case of @xmath80 in fig . 2(d ) is characterized by the clear occurrence of inverse freezing in the extremely narrow region of @xmath81 . however , for @xmath82 , there is no other phase except the pm phase at any temperature . for the @xmath83 region , the ltsg exists under the pm phase . when @xmath7 reaches the value of 0.962 , the htsg converges to one point @xmath84 , which is the tcp of the gs model . therefore , as @xmath7 increases , one tcp corresponding to the @xmath7 value greater than 0.697 gradually shifts to the tcp of the gs model , and the area of the htsg reduced throughout this process , until the htsg converges to the tcp of the gs model . during the same process , the area of the ltsg also decreases gradually . when @xmath7 reaches the value of 1.024 , the ltsg converges to a point @xmath85 . when @xmath7 is larger than 1.024 , no sg phase exists for any temperature or @xmath8 field . the appearance and disappearance of the htsg and ltsg thus depend on the value of @xmath7 . note that the two sg phases ( htsg and ltsg ) originate from the @xmath7 field , irrespective of the @xmath8 field . as shown in fig . 1(a ) , in the region @xmath65 , the two sg phases occur even when @xmath86 . the role of the @xmath8 field is to lower the glass transition temperature through quantum tunneling in proportion to the @xmath8 value , as already checked in refs . @xcite . in particular , in our model , the @xmath8 field plays a role in the sudden lowering of the second - order transition temperature of the sg ( at @xmath87 ) or htsg ( at @xmath88 ) . thus even a small value of the @xmath8 field ( about 0.05 ) makes the htsg disappear in the region @xmath88 . and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] our previous results can be directly checked by numerical analysis of the free energy @xmath59 , @xmath53 , and @xmath51 . all values of @xmath53 and @xmath51 shown in fig . 3 are obtained for @xmath70 , which is given for comparison with fig . 2(c ) . for @xmath89 , as shown in fig . 3(a ) , phase transitions occur in the order pm @xmath66 htsg @xmath67 pm @xmath67 ltsg as the temperature is reduced . here the first - order phase transitions can be easily confirmed as sudden changes in the free energy @xmath59 or discontinuities of the entropy @xmath93 , which is the temperature - derivative of the free energy @xmath59 . the pm phase gap between htsg and ltsg , i.e. , the difference between the first - order transition temperature of the htsg - to - pm transition and that of the pm - to - ltsg transition , is an extremely small value of 0.04 . in fig . 3(b ) , when @xmath8 is increased to 0.04 , the pm phase gap between the htsg and the ltsg widens to 0.1 , and the phase transitions occur in the order pm @xmath66 htsg @xmath67 pm @xmath67 ltsg as the temperature is reduced . when @xmath8 is increased to 0.1 , as shown in fig . 3(c ) , the htsg disappears and a first - order phase transition occurs from pm to ltsg as the temperature is decreased . this feature is maintained even when @xmath8 is increased to 0.55 , which is shown in fig . 3(d ) . for @xmath94 and @xmath70.,scaledwidth=50.0% ] for @xmath63 , @xmath70 , and several @xmath75 values.,scaledwidth=50.0% ] in the inverse freezing among pm - sg - pm phases in the gs model , there has been a discovery that the higher - temperature pm phase is characterized by a low density of empty states , whereas the lower - temperature pm phase has a higher density of empty states @xcite . here the density of empty states @xmath95 plays a crucial role in distinguishing the two pm phases . similarly , in order to clarify a difference between two sg phases , we draw a graph of the spin self - interaction @xmath58 [ eq.(10 ) ] , which signifies the density of filled states . as shown in fig . 4 , @xmath58 shows the difference between two sg phases clearly : the htsg has lower @xmath58 values than the ltsg does . thus , we can infer that the ltsg is characterized by a higher density of filled states . we finally examine whether the second reentrance or the splitting between the htsg and the ltsg occur at @xmath96 . as shown in fig . 5 , at @xmath97 , the pm phase gap between the htsg and the ltsg is wider than that of the @xmath61 case . at @xmath98 , there is an extremely narrow gap near @xmath99 . when @xmath75 is larger than 1.015 , there exists only one sg phase , instead of the two separated sg phases . since schupper and shnerb @xcite focused on the inverse freezing of the gs model , they selected large values of @xmath75 ( e.g. , 6.0 ) . in order to observe sg splitting , however , it is better to select @xmath75 values smaller than 1.0 because when the degeneracy of the empty states of @xmath9 ( @xmath14 ) is larger than one of the filled states of @xmath9 ( @xmath13 ) , the pm phase gap generating the sg splitting becomes wider . in the present work , we proposed an expanded spin - glass model , the quantum gs model , in order to obtain more meaningful evidence for the second reentrance observed in the gs model . by obtaining the 1rsb solutions of the quantum gs model , we could check the detailed pm - sg phase boundaries depending on the crystal field @xmath7 and the transverse field @xmath8 . we first confirmed that a second reentrance occurs in the gs model ( @xmath89 case ) , as reported by crisanti and leuzzi @xcite . we can thus describe the gs model as a prototypical model that can be used to verify the second entrance as well as inverse freezing . furthermore , there exist first - order phase transitions and tcps for @xmath100 and large values of @xmath7 . this is clearly observable from the @xmath71 phase diagrams for @xmath101 , which are shown in fig . 2(b ) . in particular , when @xmath7 is larger than 0.879 , one sg phase is split into two sg phases ( htsg and ltsg ) . we can distinguish the two sg phases by the spin self - interaction @xmath58 . the htsg and ltsg show certain differences in shape and phase boundaries . such sg splitting becomes more distinctive when @xmath75 is less than 1 . we verified that the empty states of @xmath9 are thus crucial for the occurrence of sg splitting . it is well known that the sk model with a transverse field @xcite has been successfully applied to the quantum spin glass @xmath102 @xcite , a site - diluted and isostructural derivative of the dipolar - coupled ising ferromagnet @xmath103 ( @xmath1=1.53k ) . in the absence of a magnetic field , @xmath102 is a conventional spin glass with the glass transition temperature @xmath104 . when an externally tunable magnetic field is induced transverse to the magnetic easy axis , quantum tunneling occurs . provided we can identify a suitable candidate spin - glass material with @xmath105 and crystal field and provided quantum tunneling by an externally tunable transverse magnetic field occurs in the material , we may be able to observe and verify sg splitting through experimental results . in contrast , it would be of interest to extend our theory beyond the static approximation used in this work in order to obtain analytic solutions for free energy and order parameters . it would also be interesting theoretically to search for other sg models for sg splitting . we believe that these topics will extend our viewpoint on sg systems . two tcps and a first - order phase transition lying between the two tcps were also found in the sk model under a bimodal random field . see e. nogueira , jr . , f. d. nobre , f. a. da costa , and s. coutinho , phys . e * 57 * , 5079 ( 1998 ) .

we propose an expanded spin - glass model , called the quantum ghatak - sherrington model , which considers spin-1 quantum spin operators in a crystal field and in a transverse field . the analytic solutions and phase diagrams of this model are obtained by using the one - step replica symmetry - breaking ansatz under the static approximation . our results represent the splitting within one spin - glass ( sg ) phase depending on the values of crystal and transverse fields . the two separated sg phases , characterized by a density of filled states , show certain differences in their shapes and phase boundaries . such sg splitting becomes more distinctive when the degeneracy of the empty states of spins is larger than one of their filled states .

a result of this work is that there is a simple and illuminating formula for the rate . there is a quantity @xmath2 , given by the flux of the surrounding particles or excitations , and the @xmath3 matrix for the interaction of our system ( e.g. the chiral molecule ) with these surroundings : @xmath4 the imaginary part gives the rate or loss of phase coherence per unit time @xmath5 : @xmath6 ( the real part also has a significance , a level shift induced by the surroundings . this turns out to be a neat way to find the index of refraction formula for a particle in a medium @xcite . ) the labels ( l , r ) on the @xmath3 refer to which state of the molecule ( or other system ) is doing the interacting with the surroundings . here with ( l , r ) we have taken the case of the simplest non - trivial system , the two - level system . these equations may be derived@xcite by thinking of the s - matrix as the operator which transforms the initial state of an incoming object into the final state . if the different states ( l , r ) of our system scatter the object differently , a `` lack of overlap '' or `` unitarity deficit '' as given by eq [ [ lam ] ] arises . these intuitive arguments can also supported by more formal manipulations @xcite . an important point that we see here , in eq [ [ lam ] ] , is that the environment `` chooses a direction in hilbert space''@xcite . that is , there is some direction ( here l , r ) in the internal space of the system under study ( the molecule ) that is left unchanged is not `` flipped'' by the interaction with the surroundings . such states however get a phase factor by the interaction , and this is the . if the interaction did not distinguish some direction , if we had @xmath7 then the formula tells us there would be no . this is intuitively correct in accord with one s ideas about `` measurement '' . if the probe does not distinguish any state there are no `` wavefunction collapses '' and no takes place . ( this is not meant to imply sanctioning of `` wavefunction collapses '' in any way . ) another simple limit for the formula occurs when only one state interacts , say no interaction for l , or @xmath8 . then one finds that the rate is 1/2 the scattering rate for the interacting component @xcite . thus eqs [ [ lam],[d ] ] have two interesting limits : @xmath9 and @xmath10 the latter followed from an application of the optical theorem . with appropriate evaluation of the s - matrices , eqs [ [ lam],[d ] ] can be applied to many types of problems , like quantum dots @xcite or neutrinos @xcite , or even gravity @xcite . eq [ [ no ] ] is quite interesting in that it says the system can interact but nevertheless retain its internal coherence . a lesson here is that one should nt think that every interaction or disturbance `` decoheres '' or `` reduces '' the system . the system can interact quite a bit as long as the interactions do nt distinguish the different internal states . the fact that the interaction responsible for the must `` choose a direction in hilbert space '' has some interesting implications . one of these has to do with the of a free particle in some background environment . eq [ [ lam ] ] was for a two - state system , and the extension to a larger number of states , as long as it is a finite number , can be easily envisioned as following the logic@xcite used in finding eq [ [ lam ] ] . however if we go to the continuum , that is if we have a infinite number of states , the problem becomes more subtle . the most common example of this is the free particle which , say in the limit of an infinitely large `` box '' , is described as system of continuous , dense , levels . a number of authors , in talking about this system , have automatically assumed , as indeed first seems plausible , that at long times the particle under the influence of some continually interacting environment becomes totally `` decohered '' ; in the sense that the density matrix of the particle @xmath11 approaches the situation of no off - diagonal elements , that @xmath12 approaches a @xmath13 function . although this may seem plausible , that under the repeated bombardment by the surroundings the particle becomes more and more `` decohered '' , it is in fact wrong @xmath14 consider the simplest case , that of a thermal environment . on general grounds we expect the particle in a thermal environment to be described by the boltzmann factor , to be given by a density matrix operator @xmath15 , where t is the temperature and h the hamiltonian , say @xmath16 for a non - relativistic particle . now evaluate this operator in the position representation : @xmath17 this is the stationary , long time value of @xmath12 . it applies for nearly any state we care to initially throw into the medium . evidently it shows no signs of changing and certainly no sign of turning into a @xmath18 function . of course at high temperature our expression will resemble a delta function . the practical importance of this will depend on the other length scales in the problem at hand . the point we wish to make , however , is of a conceptual nature , namely that repeated interactions with the environment do nt necessarily lead to more `` decoherence '' . indeed eq [ [ boltz ] ] says if we were initially to put @xmath19 or some other `` highly incoherent '' density matrix into the medium , the density matrix of the particle would become _ more coherent _ with time until it reached the value eq [ [ boltz ] ] . apparently the medium can `` give coherence '' to a state that never had any to start with . `` creating coherence '' by an outside influence is not as mysterious as it may sound , there are familiar cases where we know this already . for example , using a high resolution detector can `` create a long wavepacket '' @xcite or in particle physics neutral @xmath0 oscillations and the like may be enhanced or `` created '' by using some subset of our total event sample , such as a `` flavor tag '' . where did the seemingly plausible argument or feeling about the indefinitely increasing go wrong ? it s the question of the `` direction chosen in hilbert space '' . the feeling is right , but we must know where to apply it . as we can see from the boltzmann factor , thermodynamics likes to work in momentum ( actually energy ) space . the intuition would have been right there , in momentum space but this then means something non - trivial in position space . the lesson here is that the notion of `` by the environment '' must be understood to include a statement about the `` direction chosen in hilbert space '' by that environment @xcite . the interest in these issues has had a revival with the advances made possible by the technologies of mesoscopic systems . in one such system , the `` quantum dot observed by the qpc '' , one has a complete model of the measurement process , including the `` observer '' , `` who '' in this case is a quantum point contact ( qpc ) @xcite . in a slight generalization of the original experiment @xcite one can see how not only the density matrix of the object being observed is `` reduced '' by the observing process , but also see how the readout current the `` observer '' responds . in particular one may see how effects looking very much like the `` collapse of the wavefunction '' , that is sequences of repeated or `` telegraphic '' signals indicating one or another of the two states of the quantum dot , arise . all this without putting in any `` collapses '' by hand @xcite . we should stress that what we are not only talking about a reduction of fringe contrast due to `` observing '' or disturbing an interference experiment , as in @xcite ; and also in interesting experiments in quantum optics where an environment is simulated @xcite or different branches of the interferometer @xcite interact differently and adjustably with the radiation in a cavity ( like our two s - matrices ) . by the `` collapses '' however , we are referring not so much to the interferometer itself as to the signal from some `` observing '' system , like the current in the qpc . with repeated probing of the _ same _ object ( say electron or atom ) , in the limit of strong `` observation '' this signal repeats itself -this is the `` collapse '' . for not too strong observation there is an intermediate character of the signal , and so on . all this may be understood by considering the amplitude for the interference arrangement and the readout procedure to give a certain result @xcite . the properties of the readout signal naturally stand in some relation to the loss of coherence or `` fringe contrast '' of the interference effect under study . following this line of thought we come to the idea that there should be some relation between the fluctuations of a readout signal and decoherence . indeed the decoherence rate , the imaginary part of eq [ [ lam ] ] is a dissipative parameter in some sense ; it characterizes the rate of loss of coherence . now there is the famous `` dissipation- fluctuation theorem '' , which says that dissipative parameters are related to fluctuations in the system . is there some such relationship here ? indeed , one is able to derive a relation between the fluctuations of the readout current and the value of @xmath5@xcite . the interesting and perhaps practical lesson here is that the parameter can be observed in two ways . one is the direct way , just observe the damping out of the coherent oscillations of the system in question . experimentally , this involves starting the system in a definite , selected state . however , as just explained , there is a second way ; namely observe the fluctuations of the readout . this can be done even if the system is in the totally `` decohered '' @xmath20 state . another mesoscopic system , the squid and in particular the rf squid , has been long discussed@xcite as a candidate for showing that even macroscopic objects are subject to the rules of quantum mechanics . the rf squid , a josephson device where a supercurrent goes around a ring , can have two distinct states , right- or left- circulation of the current . these two conditions apparently differ greatly , since a macroscopic number of electrons change direction . it would be a powerful argument for the universality of the quantum rules if one could demonstrate the meaningfulness of quantum linear combinations of these two states . such linear combinations can in principle be produced since there is some amplitude for a tunneling between the two configurations . in fact this was recently manifested through the observation of the `` repulsion of levels '' to be anticipated if the configurations of opposite current do behave as quantum states@xcite . another approach , where we would directly `` see '' the meaningfulness of the relative quantum phase of the two configurations , is the method of `` adiabatic inversion '' @xcite . this method also offers the possibility of a direct measurement of the time . in adiabatic inversion the `` spin '' representing a two - level system @xcite , @xcite is made to `` follow '' a slowly moving `` magnetic field '' ( meant symbolically , as an analogy to spin precession physics ) , which is swept from `` up '' to `` down '' . in this way the system can be made to invert its direction in `` spin space '' , that is to reverse states and go from one direction of circulation of the current to the other . this inversion is an intrinsically quantum phenomenon . if it occurs it shows that the phases between the two configurations were physically meaningful and that they behave quantum mechanically . this may be dramatically manifested if we let destroy the phase relation between the two configurations . now the configurations act classically and the inversion is blocked . we thus predict that when the rate is low the inversion takes place , and when it is high it does not . figs 1 and 2 show the idea of this procedure . .3 cm since in such an experiment we have the sweep speed at our disposal , we have a way of determining the time . it is simply the slowest sweep time for which the inversion is successful . we must only be sure that for the sweep speeds in question the the conditions remain adiabatic . setting up the adiabatic condition and taking some estimates for the time , it appears that the various requirements can be met @xcite , @xcite when operating at low temperature . hence it may be realistically possible to move between the classical and quantum mechanical worlds to turn quantum mechanics `` on and off '' in one experiment . this would be a beautiful experiment , the main open question being if the estimates of the rate are in fact realistic , since we are entering a realm which has not been explored before . a two - state system behaving quantum mechanically can serve as the physical embodiment of a quantum mechanical bit , the `` qbit '' . furthermore , the adiabatic inversion procedure just described amounts to a quantum realization of one of the basic elements of computer logic : the not . if one configuration is identified as 1 and the other as 0 , then the inversion turns a linear combination of 1 and 0 into a linear combination of 0 and 1 with reversed weights . we can try to push this idea of `` adiabatic logic '' a step further . not was a one bit operation . the next most complicated logic operation is a two bit operation , which we may take to be `` controlled not '' or cnot . in cnot the two bits are called the control bit and the target bit , and the operation consists of performing or not performing a not on the target bit , according to the state of the control bit . to realize cnot , an idea which suggests itself @xcite as a generalization of adiabatic inversion is the following . we have a two bit operation and so two squids . these are devices with magnetic fields . now if one squid , the target bit , is undergoing a not operation , it can be influenced by the control bit , a second nearby squid , through its linking flux . we could imagine that this linking flux can be arranged so that it helps or hinders the not operation according to the state of the second squid . this would amount to a realization of `` controlled not '' , again by means of an adiabatic sweep . to analyze this proposal we must set up the two - variable schroedinger equation describing the two devices and their interaction . the result is a hamiltonian with the usual kinetic energy terms and a potential energy term in the two variables , which in this case are the fluxes in the squids , @xmath21 : @xmath22+\beta_1f(\phi_1)+\beta_2f(\phi_2 ) \bigr\ } \ ; .\ ] ] the @xmath23 are external biases which in general will be time varying . the @xmath24 are dimensionless inductances and @xmath25 represents the coupling between the two devices . the @xmath26 are symmetric functions starting at one and decreasing with increasing @xmath27 so as to produce a double well potential when combined with the quadratic term ; in the squid @xmath28 . 3 shows this `` potential landscape '' for some typical values of the parameters . given the hamiltonian , we must search for values of the control parameters @xmath23 , the `` external fields '' , which can be adiabatically varied in such a way as to produce cnot . preliminary analysis indicates favorable regimes of the rather complex parameter space where this can in fact be done @xcite . finally we would like to recall that there are still some fundamental and beautiful experiments waiting to be done in these areas . \a ) one is the demonstration of the large effects of parity violation for appropriately chosen and contained handed molecules @xcite . because of what we now call this seemed very remote at the time . but now with the existence of single atom / molecule traps and related techniques , perhaps it s not so hopeless . \b ) another , concerned with fundamentals of quantum mechanics , could be called the `` adjustable collapse of the wavefunction '' where the `` strength of observing '' can be varied , leading to effects like washing out of interferences , as already seen in @xcite and a number of further predictions where we vary the qualities of the `` observer '' @xcite , or slowing down of relaxation according to the rate of probing of the object @xcite . many of the questions we have briefly touched upon had their origins in an unease with certain consequences of quantum mechanics , often as `` paradoxes '' and `` puzzles '' . it is amusing to see how , as we get used to them , the `` paradoxes '' fade and yield to a more concrete understanding , sometimes even with consequences for practical physics or engineering . if we avoid overselling and some tendency to an inflation of vocabulary , we can anticipate a bright and interesting future for `` applied fundamentals of quantum mechanics '' . on the time dependence of optical activity , r.a . harris and , j. chem . phys . * 74 * ( 4 ) , 2145 ( 1981).[decoherence by environment , formula for decoherence rate , application to chiral molecules . ] the notion that handed molecules could `` decohere '' or be stabilized by the environment somehow was raised by h.d.zeh , found . phys . 1,69 ( 1970 ) , m. simonius , phys rev . 40 , 980 ( 1978 ) . two level systems in media and ` turing s paradox ' , r.a . harris and , phys . b 116(1982)464.[decoherence by environment , formula for decoherence rate , quantitative explanation of `` zeno '' , prediction of anti - intuitive relaxation , application to neutrinos . ] on the treatment of neutrino oscillations in a thermal environment , , phys d 36(1987)2273 .[method for decoherence in neutrino oscillations . spin precession picture for neutrinos . ] see chapter 9 of g.g . raffelt , _ stars as laboratories for fundamental physics _ ( univ . chicago press , 1996 ) measurement process in a variable - barrier system , , phys . * b459 * pages 193 - 200 , ( 1999 ) . [ formalism for quantum dot - qpc system . prediction of novel phase effect . prediction of `` collapse - like '' behavior of readout . ] decoherence - fluctuation relation and measurement noise , , physics reports * 320 * 51 - 58 ( 1999 ) , quant - ph/9903075 . [ suggestion of decoherence - fluctuation relation connecting decoherence rate and fluctuations of readout signal . ] see a. j. leggett , les houches , session xlvi ( 1986 ) _ le hasard et la matiere _ ; north -holland ( 1987 ) , references cited therein , and introductory talk , conference on macroscopic quantum coherence and computing , naples , june 2000 , proceedings published by academic - plenum . ) j. friedman , v. patel , w.chen , s.k . tolpygo and j.e . lukens , nature * 406 * , 43 ( 2000 ) . van der wal , a.c . ter haar , f. k. wilhelm , r. n. schouten , c.j.p.m . harmans , t.p . orlando , seth lloyd , and j.e . mooij , science * 290 * 773 ( 2000 ) and in mqc2 : conference on macroscopic quantum coherence and computing , naples , june 2000 , proceedings published by academic - plenum . study of macroscopic coherence and decoherence in the squid by adiabatic inversion , paolo silvestrini and , physics letters * a280 * 17 - 22 ( 2001).[linear combinations of macroscopic states . measuring decoherence time . relation to not operation . ] cond - mat/0004472 adiabatic inversion in the squid , macroscopic coherence and decoherence , paolo silvestrini and , _ macroscopic quantum coherence and quantum computing _ , pg.271 , eds . d. averin , b. ruggiero and p. silvestrini , kluwer academic / plenum , new york ( 2001 ) . [ linear combinations of macroscopic states . measuring decoherence time.]cond - mat/0010129 . averin , solid state communications * 105 * , 659 ( 1998 ) , has discussed related ideas using adiabatic operations on the charge states of small josephson junctions . adiabatic methods for a quantum cnot gate , valentina corato , paolo silvestrini , , and jacek wosiek , cond - mat/0205514 , and contribution to _ macroscopic quantum coherence and quantum computing 2002_. [ principles of quantum gates using adiabatic inversion . design parameters for cnot . ]

we indicate some of the lessons learned from our work on coherence and decoherence in various fields and mention some recent work with solid state devices as elements of the `` quantum computer '' , including the realization of simple logic gates controlled by adiabatic processes . we correct a commonly held misconception concerning decoherence for a free particle . = -0.35 cm = 0.3 cm presented at the xxii solvay conference , the physics of information delphi , nov 2001 2.0pc the subject of `` quantum information '' and in particular its realization in terms of real devices revolves in large measure around the problems of coherence and . thus it may be of interest here to review the origins of the subject and see what has been learned in applications to various areas . we first got involved in these issues through the attempt to see the effects of parity violation ( `` weak neutral currents '' ) in handed molecules @xcite . the method we found an analogy to the famous neutral @xmath0 meson behavior with chiral molecules seemed too good to be true : we had a way of turning @xmath1 ev into a big effect ! there must be some difficulty , we felt . indeed there was ; it turned out to be what we called `` quantum damping '' and what now - a - days is called `` decoherence '' . the lessons from this work were several and interesting . first , concerning parity violation , we realized that this could solve hund s `` paradox of the optical isomers '' as to why we observe handed molecules when the true ground state should be parity even- or- odd linear combinations . we realized that for molecules where tunneling between chiral isomers is small , parity violation dominates and the stationary state of the molecule becomes a handed or chiral state , and not a 50 - 50 linear combination of chiral states . this holds for a perfectly isolated molecule , and in itself has nothing to do with . however , and this is very related , even a very small interaction with the surroundings suffices to destroy the coherence necessary for the aforesaid linear combination , in effect the environment can stabilize the chiral states . this now goes under the catch - word `` by the environment '' . the limit of strong damping or stabilization is often called the zeno or `` watched pot '' effect , an idea which as far as i can tell , goes back to turing . we were able to show how this just arises as the strong damping limit of some simple `` bloch - like '' equations @xcite .

mixtures of water and organic solutes are of fundamental importance for understanding biological and chemical processes as well as transport properties of fluids . even though the simplicity , of these solutions some of them show a complex behavior of their thermodynamic and structural properties @xcite . for example , close to the ambient conditions , around @xmath0 , @xmath1 , the excess volume in binary mixtures of water and alcohols @xcite and of water and alkanolamines @xcite is negative and it exhibits a minimum as the fraction of the solute is increased . in the case of water - ionic liquids , however , the excess volume depend on the hydrophobicity of the solute . simulation results suggest that for hydrophilic solutes as the 1,3-dimethylimidazolium chloride the excess volume has a minimum as in the case of the alcohols , whereas for the case of more hydrophobic liquids as the 1,3-dimethylimidazolium hexafluorophosphate the excess volume is positive @xcite . the excess enthalpy of the aqueous organic mixtures also show a distinct behavior . while the mixtures of water with small alcohol molecules as methanol @xcite and ethanol @xcite exhibit a negative excess enthalpy , the mixtures of water with large alcohol molecules as propanol and butanol isomers show a positive excess enthalpy @xcite . similarly to the small alcohol - water mixtures the excess enthalpy for the water - alkanolamine solutions also show a minimum @xcite . in the case of ionic liquids the excess enthalpy also show two types of behavior . for the less hydrophobic ionic liquids in which the excess volume is negative , the excess enthalpy is negative and shows a minimum at the same solute fraction of the minimum of the excess volume . for the hydrophobic ionic liquids the excess enthalpy is positive and shows a maximum for the same fraction of the solute of the maximum of the excess volume @xcite . the excess isobaric specific heat for the methanol at ambient conditions increases with the fraction of the solute and exhibits a maximum value around the solute concentration @xmath2 @xcite . the excess free energy presents a harmonic dependence on the methanol fraction @xcite and the excess entropy of mixing , differently from the ideal mixtures @xcite , assumes negative values and decrease its value as the increasing methanol concentrations @xcite . in the case of the ionic liquids the constant pressure heat capacity also shows an oscillatory behavior but the peak occurs at higher concentrations of the solute @xmath3 @xcite . the description of this complex behavior of the organic solutes in water in can be made , in principle , in the framework of the frank and evans @xcite iceberg theory . these authors proposed that water is able to form microscopic icebergs around solute molecules depending on their size and the water - solute interactions . recent experiments @xcite using neutron diffraction support frank and evans @xcite scenario for the methanol . the diffraction of a concentrated alcohol - water mixture ( @xmath4 ) suggests that at these conditions most of the water molecules ( @xmath5 ) are organized in water clusters bridging methanol hydroxyl groups through hydrogen bonds . in the same direction an experimental result from x - ray emission spectroscopy for an equimolar mixture of methanol and water carried out by guo _ et.al . _ @xcite suggests that in the mixture the hydrogen bonding network of the pure components would persist to a large extent , with some water molecules acting as bridges between methanol chains . consistent with these results , recent experimental work for the methanol @xcite suggests that the negative excess the entropy of mixing arises due to a relatively small degree of the interconnection between the hydrogen bonding networks of the different components rather than from a water restructuring @xcite . motivated by these experimental results and by the huge number of applications , water - methanol mixtures have been intensively studied by computer simulations . in these simulations , water molecules are represented by one of well known classical models spc@xmath6e @xcite , st4 @xcite , tip5p @xcite and methanol molecules are frequently modeled by opls force field @xcite . using molecular dynamics simulation , bako _ et.al . _ @xcite found that on increasing the methanol fraction in the mixture , water essentially maintains its tetrahedral structure , whereas the number of hydrogen - bonds is substantially reduced . allison _ et.al . _ @xcite showed that not only the number hydrogen - bonds decreases , but the water molecules become eventually distributed in rings and clusters in accordance with the experimental results @xcite . analyzing the spatial distribution function of the water , laaksonen _ et.al . _ @xcite observed that the system is highly structured around the hydroxyl groups and that the methanol molecules are solvated by water molecules , in accordance with well known iceberg theory @xcite . in addition to the atomistic approaches , water - methanol mixture has been modeled by continuous potentials in which the water is represented by a spherical symmetric two length scale potential while the methanol is represented by a dimer in which the methyl group is characterized by a hard sphere and the hydroxyl is a water - like group @xcite . numerical simulations for this system displays good qualitative agreement with the response functions for different temperatures @xcite but fails to produce the heat capacity behavior and does not provide the structural network observed in experiments and predicted by the iceberg theory . due to the variety and complexity of the ionic liquids , very few theoretical studies have been made for analyzing the ionic liquids aqueous solutions . for example , there is no clear picture explaining why the excess volume of some ionic liquids is negative while for others is positive . in addition , it is not clear why for large alcohols the excess enthalpy is positive while for the methanol is negative . the explanation for these different behaviors both in the alcohols and in the ionic liquids might rely in the disruption of the iceberg theory as the solute is large of hydrophobic . in order to test this idea , here we explore how the excess properties of the water - solute mixture is affected by the change of the water - solute interaction from attractive to repulsive . in order to allow for the water to form a structure not present in the continuous effective potentials , our model exhibits a tetrahedral structure . in this work the water and the solute are modeled following the associating lattice gas model ( alg ) @xcite scheme . the two molecules are specified by adapting the hydrogen bond and the attractive interactions for each molecule . the excess volume and enthalpy are computed for various types of water - solute interactions . the remaining of the paper goes as follows . in the section [ sec : model ] the models for water , solute and mixture are outlined and the ground state behavior is presented . the technical details about the calculations of ground state are presented in the appendix [ ap : entropy ] . in the section [ sec : methods ] the computational methods are described and the technical aspects can be found in appendix [ sec.isop ] and [ sec.isopmix ] . in the section [ sec : results ] results are presented . section [ sec : conclusions ] ends the paper with the conclusions . we consider three systems : pure water , pure solute and water - solute mixture . in the three cases the system is defined on a body - centered cubic ( bcc ) lattice . sites on the lattice can be either empty or occupied by a water or by a solute molecule . particles representing both water and solute molecules carry four arms that point to four of the nearest neighbor ( nn ) sites on the bcc lattice as illustrated by the figure [ fig : model ] . the interactions between nn molecules are described in the framework of the lattice patchy models @xcite . the particles carry eight patches ( four of them corresponding to the arms in the alg model ) , and each of the patches points to one of the nn sites in the bcc lattice as illustrated in the figure [ fig : model ] . the water molecules have two patches of the type @xmath7 ( acceptors ) , two patches of the type @xmath8(donors ) and four patches of the type @xmath9 ( which do not participate in bonding interactions ) . since the patches of the types @xmath7 and @xmath8 participate in the hydrogen bonding , a water molecule can participate in up to four hydrogen bonds . the structure of the solute is similar to the structure of the water , but it has only one patch of type @xmath7 , the other patch @xmath7 is replaced by a patch of the type @xmath10 that represents the anisotropic group which makes water and the solute different . in the case in which the solute is the methanol @xmath10 is the methyl group while for other alcohols and ionic liquids it does represent larger chains . and @xmath8 represent the acceptors and donors arms respectively . the red sphere represent the solute particle and the arms @xmath7 and @xmath8 represent the acceptor and donors , and the patch @xmath10 represents the anisotropic group , title="fig:",width=264 ] and @xmath8 represent the acceptors and donors arms respectively . the red sphere represent the solute particle and the arms @xmath7 and @xmath8 represent the acceptor and donors , and the patch @xmath10 represents the anisotropic group , title="fig:",width=264 ] and @xmath11 represent particle on its respective positions @xmath12 and @xmath13 . blue sphere represent a water and red , a solute particle . @xmath14 and @xmath15 represent the patches b of water and solute respectively . the patch @xmath9 is not represented here for the clarity of the image.,width=302 ] the distinction between patches implies @xmath16 possible orientations for the water molecules and @xmath17 possible orientations for the solute molecules . the potential energy is defined as a sum of interactions between pairs of particles located at sites which are nn on the bcc lattice . the interaction between particles @xmath12 and @xmath13 , which are nn , only depends on the type of patch of particle @xmath12 that points to particle @xmath13 , and on the type of patch of particle @xmath13 that points to particle @xmath12 . the values of the interaction as a function of the types of the two interacting patches are summarized in the table [ tabla1 ] . the interaction between occupied neighbor sites is repulsive with an increase of energy by @xmath18 with the exception of three cases . for patch - patch interaction of type @xmath19 the energy interaction is taken as : @xmath20 . if the interaction is of type @xmath21 , with the @xmath8 patch belonging to a solute molecule there is also an attractive interaction @xmath22 ( with @xmath23 , whereas if the patch @xmath8 belongs to a water molecule the interaction energy is given by @xmath24 . we have considered @xmath25 , and three cases for the @xmath8-@xmath10 water - solute interaction : attraction with @xmath26 , non - interacting with @xmath27 and repulsion with @xmath28 . the first case represents systems dominated by the water - solute attraction . this is the case of the methanol in which it is assumed that the methyl group shows a small but attractive interaction with the water . this also represents the ionic liquids in which the anions groups are hydrophilic and the cationic chains are not too long @xcite . the second case represents alcohols with larger non - polar alkyl substituents @xcite . the third case represents the ionic liquids in which the combination of the anions and cations lead to an hydrophobic interaction @xcite . due to the simplicity of our model solute , size and hydrophobicity effects are not taken into account independently , but both are considered through the @xmath29 parameter . .interactions between nn particles of the same type ( solute or water ) . the interaction depends on the patches of both particles involved in the interparticle bond . the interaction between patches of type c and b depends on the type of molecule : water ( w ) or solute ( s ) that provides the patch b. we consider @xmath30 ; and @xmath31 . patches of types a , b , and c correspond to the four arms of the standard alg model . [ cols="^,^,>,>,>,>,>,>,>,>,>,>",options="header " , ] [ table - s0 ] in order to estimate the value of @xmath32 in the thermodynamic limit we have considered the scaling relations used by berg _ et al . _ @xcite , @xmath33 the fitting of the simulation results given in table [ table - s0 ] to eq . ( [ eq - s0 m ] ) , with @xmath34 , @xmath35 , and @xmath36 being adjustable parameters leads to : @xmath37 where the label @xmath38 refers to water . considering the quantities @xmath39 $ ] , and fitting the results to @xmath40 we get @xmath41 the values of the exponent @xmath36 agree within statistical uncertainty with the results of berg et al . @xcite . for the residual entropy of the ordinary ice . interestingly , our estimate of @xmath42 for our model defined over a system with cubic symmetry and the estimate of for the ordinary ice of berg et al . @xcite : @xmath43 ; @xmath44 , seem to coincide ( at least within error bars ) in spite of the different structures of the underlying lattices . in principle , we could apply the same simulation techniques used for the water in the determination of the residual entropy of the lattice gas model of the solute . however , the value of @xmath32 for methanol can be deduced directly from the water results . given a ground state , the configuration of the water for a system with @xmath45 molecules ( occupied positions ) one can build up @xmath46 directly related ground states for the methanol model , since the two ( undistinguishable ) @xmath7 patches of each particle in the water model correspond to two distinguishable ( @xmath7 and @xmath10 ) patches in the methanol model . therefore , we get : @xmath47 the excess properties of binary mixtures are usually measured experimentally at fixed conditions of temperature and pressure @xcite . for lattice gas models it is neither straightforward not practical the use of simulation in the npt ensemble . the usual alternative is to carry out simulations in the grand canonical ensemble and compute the pressure by means of thermodynamic integration . since we are interested in analyzing the excess properties at fixed pressure , we have developed a procedure to build up the lines @xmath48 for pure components , i.e. we fix the pressure and compute the chemical potential as a function of temperature at fixed pressure . the objective is to apply this to the ordered phases : ldl and hdl . the pressure at ( very ) low temperature for these phases can be computed from the ground state analysis . in the gce the change of the pressure for transformations at constant @xmath49 and @xmath50 , is given by @xmath51 . the density of the condensed phases at very low temperature hardly changes with @xmath52 , therefore , we can integrate the pressure to get . @xmath53 where the values of @xmath54 , @xmath55 , and @xmath56 can be taken as those corresponding to the phase coexistence at low temperature ( eqs . [ eq.gsw]-[eq.coexm ] ) . once we now how to compute the chemical potential for a given pressure @xmath57 at a ( low ) temperature @xmath58 , we will develop the integration scheme to move on the @xmath59 plane at the fixed pressure @xmath57 . imposing @xmath60 in the differential form for the thermodynamic potential of the gce we get : @xmath61 we typically considered systems with @xmath62 . the excess properties of mixing are usually defined as the differences between the values of the property of the mixture at a given composition , @xmath63 , and the value of the same property for an _ ideal _ mixture of the components at the same conditions of @xmath63 , @xmath49 , and @xmath57 . it is , therefore , desirable to develop simulation strategies to sample in an efficient way different compositions of a given mixture for fixed conditions of temperature and pressure . in order to achieve this aim for our lattice model we have borrowed ideas to form the gibbs - duhem integration procedures , as we did for computing isobars of pure components . the differential form for the grand canonical potential of a binary mixture can be written as : @xmath64 where @xmath65 is the number of molecules of component @xmath12 , and @xmath66 is the chemical potential of component @xmath12 . if we fix @xmath49 , @xmath57 , and @xmath50 , the chemical potential of the two components can not vary independently when modifying the composition . it should be fulfilled : @xmath67 using activities @xmath68 $ ] to carry out the integration of eq . ( [ gdi ] ) we get : @xmath69 let us assume that for some values of @xmath49 , and @xmath57 , we know the values of the activities of the pure components @xmath70 , and @xmath71 . we can integrate numerically ( using simulation results ) the differential equation : @xmath72 for instance , using as starting point @xmath73 and considering @xmath74 as the independent variable and integrating eq . ( [ eq.gdi3 ] ) up to @xmath75 , we should reach @xmath76 . this condition provides a powerful consistency check of the thermodynamic integration schemes at constant pressure . the numerical integration of ( [ eq.gdi3 ] ) can be carried out using the same numerical procedures as in sec . [ sec.isop ] . there is still , a minor technical problem , that appears in the limits @xmath77 ; where @xmath78 , and therefore the ratio @xmath79 can not be directly computed from the simulation . this problem can be solved by applying the widom - insertion test technique@xcite to compute the activity of the minority component ( which actually has mole fraction @xmath80 ) as a function of its density . the result can be written as : @xmath81 } \right\rangle } ; \label{eq.int - mix}\ ] ] where @xmath82 } \right\rangle}$ ] represents the average of the boltzmann exponential over attempts of insertion of a test particle of type @xmath12 with random position and random orientation on a pure component system of the other component and @xmath83 is the number of possible orientations for molecules of type @xmath12 . results were obtained from simulations of systems with @xmath62 . a. p. furlan and m. c. barbosa acknowledge the brazilian agency capes ( coordenao de aperfeicoamento de pessoal de nvel superior ) for the financial support and centro de fsica computacional - cfcif ( if - ufrgs ) for computational support . partial financial support from the direccin general de investigacin cientfica y tcnica ( spain ) under grant no . fis2013 - 47350-c5 - 4-r is acknowledged . in the course of writing this article , no g. almarza unexpectedly passed away . the authors would like to dedicate this work to his memory .

a lattice model for the study of mixtures of associating liquids is proposed . solvent and solute are modeled by adapting the associating lattice gas ( alg ) model . the nature of interaction solute / solvent is controlled by tuning the energy interactions between the patches of alg model . we have studied three set of parameters , resulting on , hydrophilic , inert and hydrophobic interactions . extensive monte carlo simulations were carried out and the behavior of pure components and the excess properties of the mixtures have been studied . the pure components : water ( solvent ) and solute , have quite similar phase diagrams , presenting : gas , low density liquid , and high density liquid phases . in the case of solute , the regions of coexistence are substantially reduced when compared with both the water and the standard alg models . a numerical procedure has been developed in order to attain series of results at constant pressure from simulations of the lattice gas model in the grand canonical ensemble . the excess properties of the mixtures : volume and enthalpy as the function of the solute fraction have been studied for different interaction parameters of the model . our model is able to reproduce qualitatively well the excess volume and enthalpy for different aqueous solutions . for the hydrophilic case , we show that the model is able to reproduce the excess volume and enthalpy of mixtures of small alcohols and amines . the inert case reproduces the behavior of large alcohols such as , propanol , butanol and pentanol . for last case ( hydrophobic ) , the excess properties reproduce the behavior of ionic liquids in aqueous solution .

it is known that lipid bilayers ( abound in living cells membranes ) exhibit a ripple phase of the bilayer - water interface , in a narrow temperature range @xcite . the ripple phase is characterized by permanent wave - like deformations of the interfaces , the origin of which is still a widely debated topic . however , it was observed that a giant dielectric dispersion can occur in the radiofrequency range , signified by a large dielectric increment in that frequency range @xcite . in practice , these membrane systems are under intensive research because they are responsible for delivery and retention of drugs @xcite . we aim to obtain some results on the dielectric dispersion spectrum of corrugated membranes , by using our recently developed green s function formalism of periodic interfaces @xcite . in that formalism , we obtained an analytic expression for the greenian that includes the effect of periodicity . in this work , we extend the green s function formalism to compute the local field distribution for a lipid bilayer membrane of arbitrary shape , separating two media of different dielectric constants . we will calculate the effective dielectric constant of the membrane subject to an ac applied field . the green s function formalism has been published recently @xcite . here we re - iterate the formalism to establish notation . we will apply the formalism to a single interface and then extend to a bilayer membrane . the electrostatic potential satisfies the laplace s equation : @xmath0 = -4\pi \rho({\bf r } ) , \label{electrostatics}\end{aligned}\]]with standard boundary conditions on the interface , where @xmath1 is the free charge density , @xmath2 equals @xmath3 in the host and @xmath4 in the embedding medium . let @xmath5 and @xmath6 be the volume of the embedding and host medium , separated by an interface @xmath7 . denoting @xmath8 if @xmath9 and 0 otherwise , leads to an integral equation @xcite : @xmath10\phi({\bf r } ) = \phi_0({\bf r } ) + { u\over 4\pi } \oint_s ds'\left [ \hat{\bf n } ' \cdot \nabla ' g({\bf r } , { \bf r } ' ) \right ] \phi({\bf r } ' ) , \label{integral}\end{aligned}\]]where @xmath11 , @xmath12 is unit normal to @xmath7 , @xmath13 and @xmath14 is the solution of @xmath15 . accordingly , our approach aims to solve a surface integral equation for the potential at the expense of a two - step solution @xcite : 1 . step 1 : determine @xmath16 for all * r * @xmath17 by solving eq.([integral ] ) , and then 2 . step 2 : obtain @xmath16 for all * r * by using eq.([integral ] ) and the results of step 1 . in step 1 , we encounter a singularity when the integration variable @xmath18 approaches the point of observation * r*. to circumvent the problem , we take an infinitesimal volume around * r * and perform the surface integral analytically , we find @xcite @xmath19 \phi({\bf r } ' ) , \ \ \ { \bf r } \in s , \label{r_in_s}\end{aligned}\]]where `` prime '' denotes a restricted integration which excludes @xmath20 . the ( surface ) integral equation ( [ r_in_s ] ) can be solved for @xmath21 . here we apply the integral equation formalism to a periodic interface . suppose the interface profile depends only on @xmath22 , described by @xmath23 , where @xmath24 is a periodic function of @xmath22 with period @xmath25 : @xmath26 . without loss of generality , we will let @xmath27 in subsequent studies . thus medium 1 occupies the space @xmath28 while medium 2 occupies the space @xmath29 separated by the interface at @xmath23 . the external field is @xmath30 and @xmath31 is the potential . for a periodic system , @xmath16 is a periodic function of the lattice vector @xmath32 . in what follows , we adopt similar treatment as the korringa , kohn and rostoker ( kkr ) method @xcite and rewrite the integral equation as : @xmath33where the integration is performed within a _ unit cell_. the structure green s function ( greenian ) is given by @xcite : @xmath34we were able to evaluate the greenian analytically @xcite : @xmath35 \over \cos 2\pi ( x - x ' ) - \cosh 2\pi ( y - f(x ' ) ) } . \label{greenian}\end{aligned}\]]eq.([greenian ] ) is a truly remarkable result the analytic expression is valid for an arbitrary interface profile . if the point of observation @xmath36 is located at the interface , the greenian has a finite limit as @xmath37 : @xmath38we first solve eq.([unit - cell ] ) for the potential @xmath39 right at the interface : @xmath40then we use eq.([integral ] ) to find the potential at any arbitrary point @xmath41 , using the potential at the interface . @xmath42for @xmath43 and @xmath44 respectively . here we extend the formalism to a bilayer membrane . consider two interface profiles described by @xmath45 , where @xmath46 denote the lower and upper interface profiles respectively . again @xmath47 is a periodic function of @xmath22 . thus medium 1 occupies the space @xmath48 while medium 2 occupies the space @xmath49 and @xmath50 . for the upper ( lower ) profile , @xmath51 ( @xmath52 ) , thus the greenian becomes @xmath53 \over \cos 2\pi(x - x ' ) - \cosh2\pi(f_t(x ) - f_{t'}(x'))}.\end{aligned}\]]the effective dielectric constant @xmath54 of the bilayer membrane satisfies the relation : @xmath55 where @xmath5 is the volume of the embedded medium . as @xmath56 , the volume integration can be converted into a surface integration by the green s theorem . moreover , for the upper ( lower ) profile , @xmath51 ( @xmath52 ) , thus the effective dielectric constant becomes @xmath57.\end{aligned}\ ] ] to solve the integral equation , we express the potential at an arbitrary point into a mode expansion : @xmath58where @xmath59 and @xmath60 are mode functions . the potential on the interfaces suffices : @xmath61where @xmath62 . here we make a few remarks on the choice of the mode functions . the choice of the mode function is somewhat arbitrary in theory . in practice , these functions should be simple and easy to use . common choice ranges from extended mode functions like the fourier series expansions to localized mode functions like the step and triangular functions @xcite . substituting the mode expansion eq.([mode - a ] ) into eq.([unit - cell ] ) , the coefficients @xmath63 satisfy the matrix equation : @xmath64 { \bf a } = -e_0 { \bf v},\end{aligned}\]]where @xmath65it should be remarked that the mode functions need not be orthonormal and the matrix b is non - diagonal in general . as a model bilayer membrane , we adopt the interface profiles : @xmath66 where @xmath67 is the amplitude of corrugation , and the sine function is added to upset the reflection symmetry about @xmath68 @xcite . the width of the bilayer membrane is thus unity . we adopt the step functions for the mode expansions : @xmath69 where @xmath70 is the width of the step function . in what follows , we adopt 100 step functions both for the lower and upper profiles , equally spaced in the unit interval @xmath71 $ ] . the integrals eqs.(16)(18 ) can be readily performed . to study the dielectric behavior , we apply an ac field at a frequency @xmath72 . the embedded medium inside the membrane has a complex dielectric constant @xmath73 where @xmath4 and @xmath74 are the dielectric constant and conductivity of the embedded medium respectively , with @xmath72 being the frequency of the applied field . we adopt the following parameters in the calculation : @xmath75 , while @xmath76 . also let @xmath77 . the maxwell - wagner relaxation time of a planar interface is given by : @xmath78 in fig.[fig1 ] , we plot ( a ) the real and ( b ) imaginary parts of the complex effective dielectric constant @xmath54 normalized to @xmath3 as function of frequency for various amplitude of corrugation @xmath67 ranging from 0.1 to 1.0 . as is evident from fig.[fig1 ] , there is a giant dielectric dispersion as the amplitude of corrugation becomes large ( @xmath79 ) . in summary , we have employed the green s function formalism to study the dielectric behavior of a corrugated membrane . the integral equation is solved and the dielectric dispersion spectrum is obtained for a periodic corrugated membrane . we should remark that the present formalism can readily be generalized to multi - layers systems as well as corrugations in two dimensions @xcite . this work was supported by the research grants council of the hong kong sar government under grant cuhk 4245/01p . k. w. yu acknowledges useful conversation with professor hong sun . g. s. smith , e. b. sirota , g. r. safinya and n. a. clark , phys . lett . * 60 * , 813 ( 1988 ) ; m. p. hentschel and f. rustichelli , phys . lett . * 66 * , 903 ( 1991 ) . a. raudino , f. castelli , g. briganti and c. cametti , j. chem . phys . * 115 * , 8238 ( 2001 ) . k. sugano , h. hamada , m. machida , et al . * 228 * , 181 ( 2001 ) . k. w. yu and jones t. k. wan , _ proceedings of the conference on computational physics ( ccp2000 ) _ , comput . . commun . * 142 * , 368 ( 2001 ) ; see also cond - mat/0102059 . k. w. yu , hong sun and jones t. k. wan , _ proceedings of the 5th international conference on electrical transport and optical properties of inhomogeneous media _ , physica b * 279 * , 78 ( 2000 ) . j. korringa , physica * 13 * , 392 ( 1947 ) ; w. kohn and n. rostoker , phys . rev . * 94 * , 1111 ( 1954 ) . f. c. mackintosh , current opinion in colloid and interface science * 2 * , 382 ( 1997 ) .

we have employed our recently developed green s function formalism to study the dielectric behavior of a model membrane , formed by two periodic interfaces separating two media of different dielectric constants . the maxwell s equations are converted into a surface integral equation ; thus it greatly simplifies the solutions and yields accurate results for membranes of arbitrary shape . the integral equation is solved and dielectric dispersion spectrum is obtained for a model corrugated membrane . we report a giant dielectric dispersion as the amplitude of corrugation becomes large .

consider any system of chemical reactions , in which certain molecule types catalyse reactions and where there is a pool of simple molecule types available from the environment ( a ` food source ' ) . one can then ask whether , within this system , there is a subset of reactions that is both self - sustaining ( each molecule can be constructed starting just from the food source ) and collectively autocatalytic ( every reaction is catalysed by some molecule produced by the system or present in the food set ) @xcite,@xcite . this notion of ` self - sustaining and collectively autocatalytic ' needs to be carefully formalised ( we do so below ) , and is relevant to some basic questions such as how biochemical metabolism began at the origin of life @xcite , @xcite , @xcite . a simple mathematical framework for formalising and studying such self - sustaining autocatalytic networks has been developed so - called ` raf ( reflexively - autocatalytic and f - generated ) theory ' . this theory includes an algorithm to determine whether such networks exists within a larger system , and for classifying these networks ; moreover , the theory allows us to calculate the probability of the formation of such systems within networks based on the ligation and cleavage of polymers , and a random pattern of catalysis . however , this theory relies heavily on the system being closed and finite . in certain settings , it is useful to consider polymers of arbitrary length being formed ( e.g. in generating the membrane for a protocell @xcite ) . in these and other unbounded chemical systems , interesting complications arise for raf theory , particularly where the catalysis of certain reactions is possible only by molecule types that are of greater complexity / length than the reactants or product of the reactions in question . in this paper , we extend earlier raf theory to deal with unbounded chemical reaction systems . as in some of our earlier work , our analysis ignores the dynamical aspects , which are dealt with in other frameworks , such as ` chemical organisation theory ' @xcite ; here we concentrate instead on just the pattern of catalysis and the availability of reactants . in this paper , a _ chemical reaction system _ ( crs ) consists of ( i ) a set @xmath0 of molecule types , ( ii ) a set @xmath1 of reactions , ( iii ) a pattern of catalysis @xmath2 that describes which molecule(s ) catalyses which reactions , and ( iv ) a distinguished subset @xmath3 of @xmath0 called the _ food set_. we will denote a crs as a quadruple @xmath4 , and encode the pattern of catalysis @xmath2 by specifying a subset of @xmath5 so that @xmath6 precisely if molecule type @xmath7 catalyses reaction @xmath8 . see fig . [ fig1 ] for a simple example ( from @xcite ) . and seven reactions . dashed arrows indicate catalysis ; solid arrows show reactants entering a reaction and products leaving . in this crs there are exactly four rafs ( defined below ) , namely @xmath9 , @xmath10 , @xmath11 , and @xmath12 . ] in certain applications , @xmath0 often consist of or at least contain a set of polymers ( sequences ) over some finite alphabet @xmath13 ( i.e. chains @xmath14 , @xmath15 , where @xmath16 ) , as in fig . [ fig1 ] ; such polymer systems are particularly relevant to rna or amino - acid sequence models of early life . reactions involving such polymers typically involve cleavage and ligation ( i.e. cutting and/or joining polymers ) , or adding or deleting a letter to an existing chain . notice that if no bound is put on the maximal length of the polymers , then both @xmath0 and @xmath1 are infinite for such networks , even when @xmath17 . in this paper we do not necessarily assume that @xmath0 consists of polymers , or that the reactions are of any particular type . thus , a reaction can be viewed formally as an ordered pair @xmath18 consisting of a multi - set @xmath19 of elements from @xmath0 ( the reactants of @xmath8 ) and a multi - set @xmath20 of elements of @xmath0 ( the products of @xmath8 ) ; but we will mostly use the equivalent and more conventional notation of writing a reaction in the form : @xmath21 where the @xmath22 s ( reactants of @xmath8 ) and @xmath23 s ( products of @xmath8 ) are elements of @xmath0 , and @xmath24 ( e.g. @xmath25 and @xmath26 are reactions ) . in this paper , we extend our earlier analysis of rafs to the general ( finite or infinite ) case and find that certain subtleties arise that are absent in the finite case . we will mostly assume the following conditions ( a1 ) and ( a2 ) , and sometimes also ( a3 ) . * @xmath3 is finite ; * each reaction @xmath27 has a finite set of reactants , denoted @xmath28 , and a finite set of products , denoted @xmath29 ; * for any given finite set @xmath30 of molecule types , there are only finitely many reactions @xmath8 with @xmath31 . given a subset @xmath32 of @xmath1 , we say that a subset @xmath33 of molecule types is _ closed _ relative to @xmath32 if @xmath34 satisfies the property @xmath35 in other words , a set of molecule types is closed relative to @xmath32 if every molecule that can be produced from @xmath34 using reactions in @xmath32 is already present in @xmath34 . notice that the full set @xmath0 is itself closed . the _ global closure _ of @xmath3 relative to @xmath32 , denoted here as @xmath36 , is the intersection of all closed sets that contain @xmath3 ( since @xmath0 is closed , this intersection is well defined ) . thus @xmath36 is the unique minimal set of molecule types containing @xmath3 that is closed relative to @xmath32 . we can also consider a _ constructive closure _ of @xmath3 relative to @xmath32 , denoted here as @xmath37 , which is union of the set @xmath3 and the set of molecule types @xmath7 that can be obtained from @xmath3 by carrying out any finite sequence of reactions from @xmath32 where , for each reaction @xmath8 in the sequence , each reactant of @xmath8 is either an elements of @xmath3 or a product of a reaction occurring earlier in the sequence , and @xmath7 is a product of the last reaction in the sequence . note that @xmath36 always contains @xmath37 ( and these two sets coincide when the crs is finite ) but , for an infinite crs , @xmath37 can be a strict subset of @xmath36 , even when ( a1 ) holds . to see this , consider the system @xmath38 where @xmath39 , @xmath40 , where @xmath41 is defined as follows : @xmath42 @xmath43 @xmath44 then @xmath45 . in this example , notice that @xmath46 has infinitely many reactants , which violates ( a2 ) . by contrast , when ( a2 ) holds , we have the following result . [ lem1 ] suppose that ( a2 ) holds . then @xmath47 . moreover , under ( a1 ) and ( a2 ) , if @xmath32 is countable , then this ( common ) closure of @xmath3 relative to @xmath32 is countable also . suppose the condition of lemma [ lem1 ] holds but that @xmath37 is not closed ; we will derive a contradiction . lack of closure means there is a molecule @xmath7 in @xmath48 which is the product of some reaction @xmath49 that has all its reactants in @xmath37 . by ( a2 ) , the set of reactants of @xmath8 is finite , so we may list them as @xmath50 , and , by the definition of @xmath37 , for each @xmath51 , either @xmath52 or there is a finite sequence @xmath53 of reactions from @xmath32 that generates @xmath54 starting from reactants entirely in @xmath3 and using just elements of @xmath3 or products of reactions appearing earlier in the sequence @xmath53 . by concatenating these sequences ( in any order ) and appending @xmath8 at the end , we obtain a finite sequence of reactions that generate @xmath7 from @xmath3 , which contradicts the assumption that @xmath37 is not closed . if follows that @xmath37 is closed relative to @xmath32 , and since it is clearly a minimal set containing @xmath3 that is closed relative to @xmath32 , it follows that @xmath55 . that @xmath37 is countable under ( a1 ) and ( a2 ) follows from the fact that any countable union of finite sets is countable . @xmath56 in view of lemma [ lem1 ] , whenever ( a2 ) holds , we will henceforth denote the ( common ) closure of @xmath3 relative to @xmath32 as @xmath57 . * definition [ raf , and related concepts ] * suppose we have a crs @xmath58 , which satisfies condition ( a2 ) . an raf for @xmath59 is a non - empty subset @xmath32 of @xmath1 for which * for each @xmath60 , @xmath61 ; and * for each @xmath60 , at least one molecule type in @xmath57 catalyses @xmath8 . in words , a non - empty set @xmath32 of reactions forms an raf for @xmath59 if , for every reaction @xmath8 in @xmath32 , each reactant of @xmath8 and at least one catalyst of @xmath8 is either present in @xmath3 or able to be constructed from @xmath3 by using just reactions from within the set @xmath32 . an raf @xmath32 for @xmath59 is said to be a _ finite raf _ or an _ infinite raf _ depending on whether or not @xmath62 is finite or infinite . the concept of an raf is a formalisation of a ` collectively autocatalytic set ' , pioneered by stuart kauffman @xcite and @xcite . since the union of any collection of rafs is also an raf , any crs that contains an raf necessarily contains a unique maximal raf . irrraf _ is an ( infinite or finite ) raf that is minimal i.e. it contains no raf as a strict subset . in contrast to the uniqueness of the maximal raf , a finite crs can have exponentially many irrrafs @xcite . the raf concept needs to be distinguished from the stronger notion of a _ constructively autocatalytic and f - generated _ ( caf ) set @xcite which requires that @xmath32 can be ordered @xmath63 so that all the reactants and at least one catalyst of @xmath64 are present in @xmath65 for all @xmath66 ( in the initial case where @xmath67 , we take @xmath68 ) . this condition essentially means that in a caf , a reaction can only proceed if one of its catalysts is already available , whereas an raf could become established by allowing one or more reactions @xmath8 to proceed uncatalysed ( presumably at a much slower rate ) so that later , in some chain of reactions , a catalyst for @xmath8 is generated , allowing the whole system to ` speed up ' . notice that although the crs in fig . [ fig1 ] has four rafs it has no caf . , and @xmath69 are products ) , reactions are hollow squares , and dashed arrows indicate catalysis . ] the raf concept also needs to be distinguished from the weaker notion of a _ pseudo - raf _ @xcite , which replaces condition ( ii ) with the relaxed condition : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( ii)@xmath70 : for all @xmath60 , there exists @xmath71 or @xmath72 for some @xmath60 such that @xmath6 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in other words , a pseudo - raf that fails to be an raf is an autocatalytic system that could continue to persist once it exists , but it can never form from just the food set @xmath3 , since it is not @xmath3-generated . these two alternatives notions to rafs are illustrated ( in the finite setting ) in fig . [ fig1b ] . notice that every caf is an raf and every raf is a pseudo - raf , but these containments are strict , as fig . [ fig1b ] shows . while the notion of a caf may seem reasonable , it is arguably too conservative in comparison to an raf , since a reaction can still proceed if no catalyst is present , albeit it at a much slower rate , allowing the required catalyst to eventually be produced . however relaxing the raf definition further to a pseudo - raf is problematic ( since a reaction can not proceed at all , unless all its reactants are present , and so such a system can not arise spontaneously just from @xmath3 ) . this , along with other desirable properties of rafs ( their formation requires only low levels of catalysis in contrast to cafs @xcite ) , suggests that rafs are a reasonable candidate for capturing the minimal necessary condition for self - sustaining autocatalysis , particularly in models of the origin of metabolism . as in the finite crs setting , the union of all rafs is an raf , so any crs that contains an raf has a unique maximal one . it is easily seen that an infinite crs that contains an raf need not have a maximal finite raf , even under ( a1)(a3 ) , but in this case , the crs would necessarily also contain an infinite raf ( the union of all the finite rafs ) . a natural question is the following : if an infinite crs contains an infinite raf , does it also contain a finite one ? it is easily seen that even under conditions ( a1 ) and ( a2 ) , the answer to this last question is ` no ' . we provide three examples to illustrate different ways in which this can occur . this is in contrast to cafs , for which exactly the opposite holds : if a crs contains an infinite caf , then it necessarily contains a sequence of finite ones . moreover , two of the infinite rafs in the following example contain no irrrafs ( in contrast to the finite case , where every raf contains at least one irrraf ) . * example 1 : * let @xmath73 , @xmath74 and @xmath75 . let @xmath76 . we will specify particular crs s by describing @xmath77 , and the pattern of catalysis as follows . * @xmath78 has a reaction @xmath79 for each @xmath80 and @xmath64 is catalysed by @xmath81 for each @xmath82 . * @xmath83 has a reaction @xmath84 \rightarrow x_i)$ ] for each @xmath80 and @xmath64 is catalysed by @xmath81 for each @xmath82 . * @xmath85 has the same reactions as @xmath83 but @xmath64 is now catalysed by every @xmath86 . [ figx ] illustrates the three crs s . each of @xmath87 satisfy ( a1 ) and ( a2 ) , but only @xmath78 satisfies ( a3 ) . all three crss contain infinite rafs , but no finite raf , and no caf . more precisely : * @xmath78 has @xmath1 as its unique raf ( which is therefore an irrraf ) . * the rafs of @xmath83 consist precisely of all subsets of @xmath88 for some @xmath89 . thus @xmath83 has a countably infinite number of rafs but no irrraf . * the rafs of @xmath85 consist precisely of all infinite subsets of @xmath1 . thus , the set of rafs for @xmath85 in uncountably infinite , and it contains no irrraf . in this section , we assume that both ( a1 ) and ( a2 ) hold . given a crs @xmath58 , consider the following nested decreasing sequence of reactions : @xmath90 defined by @xmath91 and for each @xmath80 : @xmath92 thus , @xmath93 is obtained from @xmath94 by removing any reaction that fails to have either all its reactants or at least one catalyst in the closure of @xmath3 relative to @xmath94 . let @xmath95 . it is easily shown that any raf @xmath32 present in @xmath59 is necessarily a subset of @xmath96 ( since @xmath97 for all @xmath82 by induction on @xmath98 ) . thus if @xmath99 then @xmath59 does not have an raf . in the finite case there is a strong converse if @xmath100 then @xmath59 has an raf , and @xmath96 is the unique maximal raf for @xmath59 ( this is the basis for the ` raf algorithm ' @xcite and @xcite ) . however , in contrast , this result can fail for an infinite crs , as we now show with a simple example , which also satisfies ( a1)(a3 ) . * example 2 : * consider the following infinite crs , @xmath101 , where @xmath74 , and @xmath102 where @xmath103 ( this set can be thought of as all polymers of @xmath104 ) . the reaction set is @xmath105 , where , for all @xmath106 @xmath107 @xmath108 the pattern of catalysis is defined as follows : @xmath109 catalyses @xmath110 and @xmath111 catalyses @xmath112 , and for all @xmath80 @xmath113 catalyses @xmath64 and @xmath54 catalyses @xmath114 . this crs is illustrated in fig [ figy ] . which has no raf even though @xmath115 is non - empty ( equal to @xmath116 ) . this crs satisfies ( a1)(a3 ) and ( a5 ) , but not ( a4 ) . ] notice that @xmath117 satisfies ( a1 ) , ( a2 ) and ( a3 ) . however , if we construct the sequence @xmath94 described above , then as the sole catalyst ( @xmath111 ) of @xmath112 is neither in the food set , nor generated by any other reaction , it follows that @xmath112 will be absent from @xmath118 , and so @xmath119 will also be absent from @xmath120 ( since the only catalyst of @xmath119 is produced by @xmath112 ) . continuing in this way , we obtain @xmath121 , but this set is not an raf , since the sole catalyst @xmath109 of @xmath110 does not lie lie in the closure of @xmath3 relative to @xmath122 it was produced by the @xmath123 reactions and in these have all disappeared in the limit ; moreover it is clear that no subset of @xmath117 is an raf . @xmath56 thus , we require slightly stronger hypotheses than just ( a1)(a3 ) in order to ensure that @xmath59 has an raf when @xmath100 . this , is provided by the following result . [ infp ] let @xmath58 satisfy ( a1 ) and ( a2 ) . the following then hold : * @xmath96 contains every raf for @xmath59 ; in particular , if @xmath99 , then @xmath59 has no raf . * suppose that @xmath59 satisfies both of the following further conditions : * * @xmath124 , for the sequence @xmath94 defined in ( [ r1eqx ] ) . * * each reaction @xmath27 is catalysed by only finitely many molecule types . + then @xmath59 contains an raf if and only if @xmath96 is non - empty ( in which case , @xmath96 is the maximal raf for @xmath59 ) . before proving this result , we pause to make some comments and observations concerning the new conditions ( a4 ) and ( a5 ) . regarding condition ( a4 ) , containment in the opposite direction is automatic ( by virtue of the fact that @xmath125 for any function @xmath104 and sets @xmath126 ) , so ( a4 ) amounts to saying that the two sets described are equal . notice also that @xmath117 in example 2 ( fig . [ figy ] ) satisfies ( a5 ) but it violates ( a4 ) , as it must , since @xmath117 does not have an raf . to see how @xmath117 violates ( a4 ) , notice that @xmath127 , while @xmath128 . condition ( a5 ) is quite strong , but proposition [ infp ] is no longer true if it is removed . to see why , consider the following modification @xmath129 of @xmath117 in which the only product of @xmath130 ( for @xmath80 ) is @xmath54 , and @xmath54 catalyses @xmath110 for all @xmath80 ( in addition to @xmath114 ) , as shown in fig . then @xmath131 so ( a4 ) holds ; however @xmath132 which , as before , is not an raf for @xmath129 since there is no catalyst of @xmath110 in @xmath133 . notice that ( a5 ) fails for @xmath129 since @xmath110 has infinitely many catalysts . nevertheless , it is possible to obtain a result that dispenses with ( a5 ) at the expense of a strengthening ( a4 ) , which we will do shortly in proposition [ infpro ] . which has no raf even though @xmath134 is non - empty ( equal to @xmath116 ) . this crs satisfies ( a1)(a3 ) and ( a4 ) , but not ( a5 ) , nor ( a4)@xmath70 . ] _ proof of proposition [ infp ] : _ suppose @xmath32 is any raf for @xmath59 . induction on @xmath135 shows that @xmath97 for all @xmath98 , so that @xmath136 ; in particular , if @xmath99 , then @xmath59 has no raf . the proof of part ( ii ) of proposition [ infp ] relies on a simple lemma . [ simlem ] suppose that @xmath137 is any nested decreasing sequence of subsets and @xmath20 is a finite set for which @xmath138 for all @xmath82 . then some element of @xmath20 is present in every set @xmath139 . _ proof of lemma : _ suppose , to the contrary , that for every element @xmath140 , there is some set @xmath141 in the sequence that fails to contain @xmath142 ( we will show this is not possible by deriving a contradiction ) . let @xmath143 . since @xmath20 is a finite set , @xmath144 is a finite integer , and since the sequence @xmath137 is a nested decreasing sequence , it follows that @xmath145 , a contradiction . @xmath56 returning to the proof of part ( ii ) , suppose that @xmath100 ; we will show that @xmath96 is an raf for @xmath59 ( and so , by part ( i ) , the unique maximal raf for @xmath59 ) . for @xmath146 for each @xmath98 ( otherwise @xmath8 would not be an element of @xmath93 and thereby fail to lie in @xmath96 ) . thus @xmath147 by ( a4 ) . it remains to show that @xmath8 is catalysed by at least one element of @xmath148 . let @xmath149 . by ( a5 ) , @xmath150 is finite . moreover , for each @xmath135 , @xmath151 ( otherwise @xmath8 would fail to be in @xmath93 and thereby not lie in @xmath96 ) . by lemma [ simlem ] , there is a molecule type @xmath152 that lies in @xmath153 and this latter set is contained in @xmath154 by ( a4 ) . in summary , every reaction in @xmath96 has all its reactants and at least one catalyst present in @xmath148 and so @xmath96 is an raf for @xmath59 , as claimed . @xmath56 suppose we now remove condition ( a5 ) in proposition [ infp ] . in this case , by a slight strengthening of ( a4 ) , we obtain a positive result ( proposition [ infpro ] ) . to describe this , we first require a further definition . recall that @xmath2 is the set of pairs @xmath155 where molecule type @xmath7 catalyses reaction @xmath8 . given a subset @xmath156 of @xmath2 , let @xmath157 = \{r \in { { \mathcal r } } : ( x , r ) \in c ' \mbox { for some } x\in x\}.\ ] ] define a nested decreasing sequence of subsets @xmath158 by @xmath159 and for each @xmath82 , @xmath160}(f)\},\ ] ] and let @xmath161 . [ infpro ] let @xmath59 satisfy ( a1 ) and ( a2 ) , as well as the following property : @xmath162}(f ) \subseteq { \rm cl}_{r[c_\infty]}(f ) , \mbox { for the sequence $ c_i$ defined in ( \ref{cieqx})}.\ ] ] then @xmath59 has an raf if and only if @xmath163 , in which case @xmath164 $ ] is a maximal raf for @xmath59 . suppose that @xmath165 . then for any @xmath166 $ ] there exists @xmath167 such that @xmath168 . it follows that @xmath169 for all @xmath98 . by definition , this means that @xmath170}(f)$ ] for all @xmath98 , and so @xmath171}(f).$ ] now , by ( a4)@xmath70 , this means that @xmath172}(f)$ ] . in summary , every reaction in the non - empty set @xmath164 $ ] has all its reactants and at least one catalyst in the closure of @xmath3 with respect to @xmath164 $ ] and so @xmath164 $ ] forms an raf for @xmath59 . conversely , suppose that @xmath59 contains an raf @xmath32 ; we will show that @xmath165 . for each @xmath49 , select a catalyst @xmath173 for @xmath8 for which @xmath174 . let @xmath175 . we use induction on @xmath98 to show that @xmath176 for all @xmath135 . clearly @xmath177 , so suppose that @xmath176 and select an element @xmath178 . by definition , @xmath179}(f ) \subseteq { \rm cl}_{{{\mathcal r}}[c_i]}(f),\ ] ] which means that @xmath180 , establishing the induction step . it follows that @xmath181 and so @xmath165 as claimed . @xmath56 notice that , just as for condition ( a4 ) , the condition ( a4)@xmath70 is equivalent to requiring that the two sets described be identical . notice also that , although condition ( a4 ) applies to the crs @xmath182 , condition ( a4)@xmath70 fails , since @xmath183 and so @xmath184}(f ) = { { \mathcal f}}= \{f , ff , fff , \ldots\}$ ] , while @xmath185}(f)$ ] for all @xmath82 , and so @xmath186}$ ] is not a subset of @xmath187}(f)$ ] . in summary , a single application of @xmath188 allows us to determine when @xmath59 has an raf , provided the additional condition ( a4)@xmath70 holds . example 2 showed that some additional assumption of this type is required , however one could also consider other approaches for determining the existence rafs that do not assume a further condition like ( a4)@xmath70 , but instead iterate the map @xmath188 . in other words , consider the following ` higher level ' sequence of subsets of @xmath1 : @xmath189 where @xmath190 for each @xmath191 . again , this forms a decreasing nested sequence of subsets of @xmath1 and so we can consider the set : @xmath192 in the example above for @xmath117 where @xmath100 , notice that @xmath193 ( and so @xmath194 ) . it follows from proposition [ infp ] that if @xmath195 for any @xmath196 then @xmath59 has no raf . however , just because @xmath197 , this does not imply that @xmath59 contains an raf as the next example shows . * example 3 : * consider the infinite crs @xmath198 which is obtained by taking a countably infinite number of ( reaction and molecule disjoint ) copies of @xmath117 ( from example 2 ) and letting the molecule type @xmath109 in the @xmath98-th copy of @xmath117 play the role of the molecule @xmath111 in the @xmath199th copy of @xmath117 . in addition , let @xmath46 be the reaction @xmath200 ( where @xmath201 is an additional molecule ) catalysed by the @xmath109-products of all the copies of @xmath117 . now @xmath202 contains all but the first @xmath203 copies of @xmath117 , plus @xmath46 . consequently , @xmath204 but , as before , this is not an raf . notice , however that this example violates condition ( a3 ) . we have seen from the last section that applying @xmath188 , even infinitely often , does not seem to provide a way to determine whether a crs possesses an raf . however , in most applications , the main interest will generally be in finite rafs . from the earlier theory it is clear that if @xmath202 is finite for some integer @xmath196 then any rafs that may exist for @xmath59 are necessarily finite , and finite in number . moreover , if @xmath205 and this set is finite , then @xmath202 is the unique ( and necessarily finite ) maximal raf for @xmath59 . however , it is also quite possible that a crs might contain both finite and infinite rafs , and in this section we describe a characterisation of when an raf contains a finite raf . given a crs @xmath59 define a sequence @xmath206 of subsets of @xmath1 as follows : @xmath207 @xmath208 in words , @xmath209 is the set of reactions that have all their reactants in @xmath3 , and for @xmath80 @xmath94 is the set of reactions for which each reactant is either an element of @xmath3 or products of some reaction in @xmath210 for @xmath211 . [ finiteraf ] suppose a crs @xmath59 satisfies ( a1)(a3 ) . let @xmath212 for all @xmath135 , where @xmath213 is as defined above . then : * @xmath214 is a nested increasing sequence of finite sets . * @xmath59 has a finite raf if and only if @xmath215 for some @xmath135 . * if @xmath216 for some @xmath98 , then @xmath217 is a finite raf for @xmath59 for all @xmath218 . * every finite raf for @xmath59 is contained in @xmath219 for some @xmath220 . by ( a1 ) and ( a3 ) , it follows that @xmath221 is finite , and , by induction , that @xmath213 is finite for all @xmath80 . moreover , if @xmath222 then @xmath223 and so @xmath224 ( i.e. @xmath225 ) and so the sets @xmath226 form an increasing nested sequence . this establishes ( i ) . for parts ( ii ) and ( iii ) , suppose that @xmath59 contains a finite raf @xmath32 . since ( a1 ) and ( a2 ) hold , we can apply lemma [ lem1 ] to deduce that every reaction @xmath49 is an element of @xmath213 for some @xmath98 . thus , since @xmath32 is finite , and the sequence @xmath213 is a nested increasing sequence of finite sets , it follows that @xmath227 for some fixed @xmath203 , in which case @xmath228 . conversely , if @xmath216 , then it is clear from the definitions that @xmath229 is an finite raf for @xmath59 ; moreover , so also is @xmath217 for all @xmath230 . part ( iv ) also follows easily from the definitions , since if @xmath32 is a finite raf for @xmath59 then @xmath231 for some @xmath232 , and since @xmath32 is finite we have @xmath233 and so @xmath234 . this completes the proof . @xmath56 theorem [ finiteraf ] provides an algorithm to search for finite rafs in any infinite crs that satisfies ( a1)(a3 ) . given @xmath59 , construct @xmath221 and run the ( standard ) raf algorithm @xcite and @xcite on @xmath221 . if it fails to find an raf , then construct @xmath235 and run the algorithm on this set , and continue in the same manner . if @xmath59 contains a finite raf , then this process is guaranteed to find it , however , there is no assurance in advance of how long this might take ( if not constraint is placed on the size of the how large the smallest finite raf might be ) . finally , we show how proposition [ infpro ] can be reformulated more abstractly in order to makes clear the underlying mathematical principles ; the added generality may also be useful for settings beyond chemical reaction systems . this uses the notion of `` @xmath236-compatibility '' from @xcite , which we now explain . suppose we have an arbitrary set @xmath30 and an arbitrary partially ordered set @xmath34 , together with some functions @xmath237 consider the function @xmath238 , where @xmath239 we are interested in the non - empty subsets of @xmath30 fixed points of @xmath240 , particularly , when @xmath104 is _ monotonic _ ( i.e. , where @xmath241 ) . a subset @xmath19 of @xmath30 is said to be _ @xmath236-compatible _ if @xmath19 is non - empty and @xmath242 . the notion of an raf can be captured in this general setting as follows . given a crs @xmath243 satisfying ( a2 ) , take @xmath244 and @xmath245 ( partially ordered by set inclusion ) , and define @xmath246 as follows : @xmath247}(f ) \mbox { and } g((x , r ) ) = \{x\ } \cup \rho(r),\ ] ] where , as earlier , @xmath248 $ ] is the set of reactions @xmath27 for which there is some @xmath249 with @xmath250 . notice that @xmath104 is monotonic and when @xmath59 is finite , the set @xmath251 can be computed in polynomial time in the size of @xmath59 . [ lemcom ] suppose we have a crs @xmath59 satisfying ( a2 ) , and with @xmath104 and @xmath252 defined as in ( [ fgeq ] ) . if @xmath19 is @xmath236-compatible , then @xmath248 $ ] is an raf for @xmath59 . conversely , if @xmath32 is an raf for @xmath59 , then a @xmath236-compatible set @xmath19 exists with @xmath248 = { { \mathcal r}}'$ ] . in particular , @xmath59 has an raf if and only if @xmath30 contains a @xmath236-compatible set . if @xmath19 is @xmath236-compatible subset of @xmath30 , then for @xmath253 $ ] , each reaction @xmath27 has at least one molecule type @xmath167 for which @xmath254 . @xmath236-compatibility ensures that @xmath255 , in other words , @xmath256 for some catalyst @xmath7 of @xmath8 . this holds for every @xmath49 , so @xmath32 is an raf for @xmath59 . conversely , if @xmath32 is an raf , then for each reaction @xmath49 , we can choose an associated catalyst @xmath173 so that @xmath257 . then @xmath258 is a @xmath236-compatible subset of @xmath30 , with @xmath248 = { { \mathcal r}}'$ ] . @xmath56 the problem of finding a @xmath236-compatible set ( if one exists ) in a general setting ( arbitrary @xmath30 , and @xmath34 , not necessarily related to chemical reaction networks ) can be solved in general polynomial time when @xmath30 is finite and @xmath104 is monotonic and computable in finite time . this provides a natural generalization of the classical raf algorithm . in @xcite , we showed how other problems ( including a toy problem in economics ) could by formulated within this more general framework . however , if we allow the set @xmath30 to be infinite , then monotonicity of @xmath104 needs to be supplemented with a further condition on @xmath104 . we will consider a condition ( ` @xmath201-continuity ' ) , which generalizes ( a4)@xmath70 , and that applies automatically when @xmath30 is finite we say that @xmath259 is ( weakly ) @xmath201-continuous if , for any nested descending chain @xmath260 of sets , we have : @xmath261 recall that an element in a partially ordered set need not have a greatest lower bound ( glb ) ; but if it does , it has a unique one . notice that when @xmath30 is finite , this property holds trivially , since then @xmath262 for the last set @xmath263 in the ( finite ) nested chain . for a subset @xmath19 of @xmath30 and @xmath264 , define @xmath265 to be the result of applying function @xmath240 iteratively @xmath203 times starting with @xmath19 . thus @xmath266 and for @xmath264 , @xmath267 . taking the particular interpretation of @xmath104 and @xmath252 in ( [ fgeq ] ) , the sequence @xmath268 is nothing more than the sequence @xmath269 from ( [ cieqx ] ) . notice that the sequence @xmath270 is a nested decreasing sequence of subsets of @xmath30 , and so we may define the set : @xmath271 which is a ( possibly empty ) subset of @xmath30 ( in the setting of proposition [ infpro ] , @xmath272 ) . given ( finite or infinite ) sets @xmath273 , where @xmath34 is partially ordered , together with functions @xmath274 , it is routine to verify that the following properties hold : * the @xmath236-compatible subsets of @xmath30 are precisely the non - empty subsets of @xmath30 that are fixed points of @xmath240 ; * if @xmath104 is monotonic then @xmath275 contains all @xmath236-compatible subsets of @xmath30 ; in particular , if @xmath276 , then there is no @xmath236compatible subset of @xmath30 . * if @xmath104 is @xmath201-continuous then @xmath275 is @xmath236-compatible , provided it is non - empty ; in particular , if @xmath104 is monotonic and @xmath201-continuous then ( by ( ii ) ) there a @xmath236-compatible subset of @xmath30 exists if and only if @xmath275 is nonempty . * without the assumption that @xmath104 is weakly @xmath201-continuous in part ( iii ) , it is possible for @xmath275 to fail to be @xmath236-compatible when @xmath30 is infinite , even if @xmath104 is monotone . the proof of parts ( i)(iii ) proceeds exactly as in @xcite , with the addition of one extra step required to justify part ( iii ) , assuming @xmath201-continuity . namely , condition ( [ glbeq ] ) ensures that @xmath277 is also @xmath201-continuous in the sense that for any nested descending chain @xmath260 of sets , we have : @xmath278 and so @xmath279 . the proof of ( [ psieq ] ) from ( [ glbeq ] ) is straightforward : firstly , @xmath280 holds for _ any _ function @xmath240 , while if @xmath281 , then , by definition of @xmath240 , @xmath282 for all @xmath98 and @xmath283 for all @xmath82 and so @xmath284 , and @xmath283 for all @xmath82 . now , since @xmath285 is a glb of @xmath286 , we have @xmath287 for all @xmath98 ( i.e. @xmath288 ) and so @xmath289 . part ( iv ) follows directly from parts ( ii ) and ( iii ) . for part ( vi ) , consider the infinite crs @xmath117 in example 2 . as above , take @xmath290 and , for @xmath291 , with @xmath104 and @xmath252 defined as in ( [ fgeq ] ) . then @xmath292 , where @xmath293 however , @xmath19 is not @xmath236-compatible , since @xmath294 and @xmath295 but this is not a subset of @xmath296}(f ) = { { \mathcal f}}$ ] since @xmath297 . in this example , @xmath104 fails to be weakly @xmath201-continuous , and the argument is analogous to where we showed earlier that @xmath129 fails to satisfy ( a4)@xmath70 . more precisely , for each @xmath82 , let @xmath298 , where @xmath94 is defined in ( [ r1eqx ] ) and where , for each reaction @xmath299 , @xmath300 is the unique catalyst of @xmath8 . then @xmath301 and so @xmath302 . however , @xmath303 and so @xmath304 , which differs from the glb of @xmath305 , namely @xmath302 . the examples in this paper are particularly simple indeed mostly we took the food set to consist of just a single molecule , and reactions often had only one possible catalyst . in reality more ` realistic ' examples can be constructed , based on polymer models over an alphabet , however the details of those examples tends to obscure the underlying principles so we have kept with our somewhat ` toy ' examples in order that the reader can readily verify certain statements . section [ finitesec ] describes a process for determining whether an arbitrary infinite crs ( satisfying ( a1)(a3 ) ) contains a finite raf . however , from an algorithmic point of view , proposition [ finiteraf ] is somewhat limited , since the process described is not guaranteed to terminate in any given number of steps . if no further restriction is placed on the ( infinite ) crs , then it would seem difficult to hope for any sort of meaningful algorithm ; however , if the crs has a ` finite description ' ( as do our main examples above ) , then the question of the algorithmic decidability of the existence of an raf or of a finite raf arises . more precisely , suppose an infinite crs @xmath306 consists of ( i ) a countable set of molecule types @xmath307 , where we may assume ( in line with ( a1 ) ) that @xmath308 , for some finite value @xmath309 , and ( ii ) a countable set @xmath310 of reactions , where @xmath64 has a finite set @xmath311 of reactants , a finite set @xmath312 of products , and a finite or countable set @xmath313 of catalysts , where @xmath314 and @xmath315 are computable ( i.e. partial recursive ) set - valued functions defined on the positive integers . given this setting , a possible question for further investigation is whether ( and under what conditions ) there exists an algorithm to determine whether or not @xmath59 contains an raf , or more specifically a finite raf ( i.e. when is this question decidable ? ) . the author thanks the allan wilson centre for funding support , and wim hordijk for some useful comments on an earlier version of this manuscript . i also thank marco stenico ( personal communication ) for pointing out that @xmath201-consistency is required for part ( iii ) of the @xmath236-compatibility result above when @xmath30 is infinite , and for a reference to a related fixed - point result in domain theory ( theorem 2.3 in @xcite ) , from which this result can also be derived . p. dittrich , p. speroni di fenizio , chemical organisation theory . bull . math . biol . * 69 * , 11991231 ( 2007 ) p. g. higgs , n. lehman , the rna world : molecular cooperation at the origins of life . genet . * 16*(1 ) , 717 ( 2015 ) w. hordijk , m. steel , autocatalytic sets extended : dynamics , inhibition , and a generalization . . chem . * 3*:5 ( 2012 ) w. hordijk , m. steel , autocatalytic sets and boundaries . j. syst . 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( special issue : origin of life 2011 ) * 12 * , 30853101 ( 2011 ) w. hordijk , m. steel , s. kauffman , the structure of autocatalytic sets : evolvability , enablement , and emergence . acta biotheor . * 60 * , 379392 ( 2012 ) s. a. kauffman , autocatalytic sets of proteins . . biol . * 119 * , 124 ( 1986 ) s. a. kauffman , the origins of order ( oxford university press , oxford 1993 ) e. mossel , m. steel , random biochemical networks and the probability of self - sustaining autocatalysis . j. theor . biol . * 233*(3 ) , 327336 ( 2005 ) j. smith , m. steel , w. hordijk , autocatalytic sets in a partitioned biochemical network . chem . * 5*:2 ( 2014 ) m. steel , w. hordijk , j. smith , minimal autocatalytic networks . journal of theoretical biology * 332 * : 96107 ( 2013 ) v. stoltenberg - hansen , i. lindstrm , e. r. griffor , mathematical theory of domains . cambridge tracts in theoretical computer science 22 ( cambridge university press , cambridge 1994 ) v. vasas , c. fernando , m. santos , s. kauffman , e. szathmry , evolution before genes . . dir . * 7*:1 ( 2012 ) m. villani , a. filisetti , a. graudenzi , c. damiani , t. carletti , r. serra , growth and division in a dynamic protocell model . life * 4 * , 837864 ( 2014 )

given any finite and closed chemical reaction system , it is possible to efficiently determine whether or not it contains a ` self - sustaining and collectively autocatalytic ' subset of reactions , and to find such subsets when they exist . however , for systems that are potentially open - ended ( for example , when no prescribed upper bound is placed on the complexity or size / length of molecules types ) , the theory developed for the finite case breaks down . we investigate a number of subtleties that arise in such systems that are absent in the finite setting , and present several new results .

the statistical mechanics of systems with long - range interactions has recently attracted a lot of attention @xcite . typical systems with long - range interactions include self - gravitating systems @xcite , two - dimensional vortices @xcite , non - neutral plasmas @xcite , free electrons lasers @xcite and toy models such as the hamiltonian mean field ( hmf ) model @xcite . unusual properties of systems with long - range interaction such as negative specific heats or ensembles inequivalence have been evidenced and linked with lack of additivity @xcite . in addition , a striking property of these systems is the rapid formation of quasi stationary self - organized states ( coherent structures ) such as galaxies in the universe @xcite , large scale vortices in geophysical and astrophysical flows @xcite or quasi - stationary states in the hmf model @xcite . these qsss can be explained in terms of statistical mechanics using the theory developed by lynden - bell @xcite for the vlasov equation or by miller @xcite and robert & sommeria @xcite for the 2d euler equation . two - dimensional vortices interact via a logarithmic potential . interaction of vortices in 3d turbulence is weaker than in 2d turbulence , but still long - range . due to dissipative anomaly and vortex stretching , statistical mechanics of 3d turbulence has so far eluded theories . recent progress was recently made considering 3d inviscid axisymmetric flows @xcite that are intermediate between 2d and 3d flows : they are subject to vortex stretching like in 3d turbulence , but locally conserve a scalar quantity in the ideal limit , like in 2d turbulence . it is therefore interesting to study whether these systems obey the peculiarities observed in other systems with long - range interactions such as violent relaxation , existence of long - lived quasi - stationary states , negative specific heats and ensembles inequivalence . the general study of the stability of axisymmetric flows , and the possible occurrence of phase transitions , is difficult due to the presence of an infinite number of casimir invariants linked with the axisymmetry of the flow . in a previous paper @xcite , hereafter paper i , we have considered a simplified axisymmetric euler system characterized by only three conserved quantities : the fine - grained energy @xmath0 , the helicity @xmath1 and the angular momentum @xmath2 . we have developed the corresponding statistical mechanics and shown that equilibrium states of this system have the form of beltrami mean flows on which are superimposed gaussian fluctuations . we have shown that the maximization of entropy @xmath3 at fixed helicity @xmath1 , angular momentum @xmath2 and microscopic energy @xmath0 ( microcanonical ensemble ) is equivalent to the maximization of free energy @xmath4 at fixed helicity @xmath1 and angular momentum @xmath2 ( canonical ensemble ) . these variational principles are also equivalent to the minimization of macroscopic energy @xmath5 at fixed helicity @xmath1 and angular momentum @xmath2 . this provides a justification of the minimum energy principle ( selective decay ) from statistical mechanics . we have furthermore discussed the analogy with the simplified thermodynamical approach of 2d turbulence developed in @xcite based on only three conserved quantities : the fine - grained enstrophy @xmath6 , the energy @xmath7 and the circulation @xmath8 . we have shown that equilibrium states of this system have the form of beltrami mean flows ( linear vorticity - stream function relationship ) on which are superimposed gaussian fluctuations . we have shown that the maximization of entropy @xmath3 at fixed energy @xmath7 , circulation @xmath8 and microscopic enstrophy @xmath6 ( microcanonical ensemble ) is equivalent to the maximization of grand potential @xmath9 at fixed energy @xmath7 and circulation @xmath8 ( grand microcanonical ensemble ) . these variational principles are also equivalent to the minimization of macroscopic enstrophy @xmath10 at fixed energy @xmath7 and circulation @xmath8 . this provides a justification of a minimum enstrophy principle ( selective decay ) from statistical mechanics . in the analogy between 2d turbulence and 3d axisymmetric turbulence , the energy plays the role of the enstrophy . in the present paper , we study more closely the equilibrium states of axisymmetric flows and explore their stability . we show that all critical points of macroscopic energy at fixed helicity and angular momentum are _ saddle points _ , so that they are unstable in a strict sense . indeed , there is no minimum ( macroscopic ) energy state at fixed helicity and angular momentum ( either globally or locally ) because we can always decrease the energy by considering a perturbation at smaller scales . this is reminiscent of the richardson energy cascade in 3d turbulence . inversely , in 2d turbulence , there exists minimum enstrophy states that develop at large scales ( inverse cascade ) . therefore , our system is intermediate between 2d and 3d turbulence : there exists equilibrium states in the form of coherent structures ( that are solutions of a mean field differential equation ) like in 2d turbulence , but they are saddle points of macroscopic energy and are expected to cascade towards smaller and smaller scales like in 3d turbulence . however , we give arguments showing that saddle points can be robust in practice and play a role in the dynamics . indeed , they are unstable only for some particular ( optimal ) perturbations and can persist for a long time if the system does not spontaneously generate these perturbations . therefore , these large - scale coherent structures can play a role in the dynamics and they have indeed been observed in experiments of von krmn flows @xcite . in order to make this idea more precise , we have explored their stability numerically using phenomenological relaxation equations derived in @xcite . we have found some domains of robustness in the parameter space . in particular , the one cell structure is highly robust for large values of the angular momentum @xmath11 and becomes weakly robust for low values of the angular momentum . in that case , we expect a phase transition ( bifurcation ) from the one - cell structure to the two - cells structure . we have also found that the value of the critical angular momentum @xmath12 changes depending whether we use relaxation equations associated with a canonical ( fixed temperature ) or microcanonical ( fixed microscopic energy ) description . at low temperatures @xmath13 , we have evidence a new kind of `` ensembles inequivalence '' characterizing the robustness of saddle points with respect to random perturbations . the paper is organized as follows : in sec . [ setup ] , we set - up the various notations and hypotheses we are going to use . the computation and characterization of equilibrium states is done in sec . [ computation ] . the stability analysis of these equilibrium states is performed in sec . [ analytics ] where we show analytically that all states are unstable with respect to large wavenumber perturbations . we evidence a process of energy condensation at small scales that is reminiscent of the richardson cascade . we explore numerically the robustness of the equilibria in both canonical and microcanonical ensembles in sec . [ stability ] . our numerical method is probabilistic and rather involved . a discussion of our results is done in sec . [ discussion ] where a bifurcation scenario relevant to the turbulent experimental von krmn flow is suggested . we consider a system with a cylindrical geometry enclosed in the volume delimited above and below by surfaces @xmath14 and @xmath15 , and radially by @xmath16 . like in paper i , we consider an axisymmetric euler - beltrami system characterized by a velocity field @xmath17 , with axisymmetric time averaged @xmath18 . we furthermore assume that the only relevant invariants of the axisymmetric euler equations for our problem are the averaged energy @xmath19 , the averaged helicity @xmath20 and the averaged angular momentum @xmath21 where @xmath22 . we introduce the potential vorticity @xmath23 and the stream function @xmath24 such that @xmath25 and @xmath26 . they are related to each other by the generalized laplacian operator @xmath27 in actual turbulent von krmn experiments , we have been able to observe that the largest part of the kinetic energy is contained in the toroidal motions . it is therefore natural , as a first elementary step , to consider a model in which only toroidal fluctuations are considered , and suppose that the fluctuations in the other ( poloidal ) directions are simply frozen . with such an assumption , poloidal vorticity fluctuations are allowed , but toroidal vorticity fluctuations are excluded . we therefore only include a fraction of the vorticity fluctuations , that presumably become predominant at small scale , due to the existence of vortex stretching . as shown below and in the next paper @xcite , this simplification however still allows for vortex stretching and energy cascades towards smaller scales , and leads to predictions that are in good agreement with experiments . moreover , our hypotheses lead to a model that is self - contained and analytically tractable . according to our hypotheses , neither @xmath28 nor @xmath24 fluctuates in time : @xmath29 and @xmath30 . in that case , the conserved quantities can be rewritten @xmath31 @xmath32 @xmath33 where @xmath34 denotes the spatial average . ] @xmath35 the helicity and the angular momentum are _ robust constraints _ because they can be expressed in terms of coarse - grained quantities @xmath36 and @xmath37 . by contrast , the energy is a _ fragile constraint _ because it can not be expressed in terms of coarse - grained quantities . indeed , it involves the fluctuations of angular momentum @xmath38 . to emphasize that point , we have introduced the notation @xmath39 to designate the fine - grained ( microscopic ) energy . splitting @xmath40 into a mean part @xmath37 and a fluctuating part @xmath41 , we define the coarse - grained ( macroscopic ) energy by @xmath42 then , the energy contained in the fluctuations is simply @xmath43 where @xmath44 is the local centered variance of angular momentum . we stress that the microscopic energy @xmath45 is conserved while the macroscopic energy @xmath5 is _ not _ conserved and is likely to decrease ( see below ) . in paper i , we have developed a simplified thermodynamic approach of axisymmetric flows under the above - mentioned hypothesis . let @xmath46 denote the pdf of @xmath40 and let us recall the expression of the entropy @xmath47 we have proven the equivalence between the _ microcanonical _ ensemble @xmath48\ , | \ , e^{f.g . } , \ , h,\ , i , \ , \int \rho d\eta=1 \rbrace , \label{bes1}\ ] ] and the _ canonical _ ensemble @xmath49=s-\beta e^{f.g.}\ , | \ , h,\ , i , \ , \int \rho d\eta=1 \rbrace . \label{bes2}\ ] ] in each ensemble , the critical points are determined by the first order condition @xmath50 . the equilibrium distribution is gaussian @xmath51 the mean flow is a beltrami state @xmath52 @xmath53 and the centered variance of angular momentum is @xmath54 these equations determine _ critical points _ of the variational problems ( [ bes1 ] ) and ( [ bes2 ] ) that cancel the first order variations of the thermodynamical potential . clearly , ( [ bes1 ] ) and ( [ bes2 ] ) have the same critical points . furthermore , it is shown in paper i that ( [ bes1 ] ) and ( [ bes2 ] ) are equivalent for the _ maximization _ problem linked with the sign of the second order variations of the thermodynamical potential : a critical point determined by eqs . ( [ vp1])-([vp4 ] ) is a maximum of @xmath3 at fixed microscopic energy , helicity and angular momentum iff it is a maximum of @xmath55 at fixed helicity and angular momentum . this equivalence is not generic . we always have the implication ( [ bes2 ] ) @xmath56 ( [ bes1 ] ) but the reciprocal may be wrong . here , the microcanonical and canonical ensembles are equivalent due to the quadratic nature of the microscopic energy @xmath57 . we note that , according to eq . ( [ vp4 ] ) , @xmath58 is positive . in the canonical ensemble , @xmath58 is prescribed . in the microcanonical ensemble , @xmath58 is a lagrange multiplier that must be related to the energy @xmath45 . according to eqs . ( [ fluct ] ) and ( [ vp4 ] ) , we find that @xmath59 is determined by the condition @xmath60 this relation shows that @xmath61 plays the role of a temperature associated with the fluctuations of angular momentum . in 3d is the counterpart of the chemical potential @xmath62 associated with the conservation of the fine - grained enstrophy in 2d ( see introduction ) . ] finally , we have proven in paper i that the two variational problems ( [ bes1 ] ) and ( [ bes2 ] ) are equivalent to @xmath63=-\beta e^{c.g.}\ , | \ , h,\ , i \rbrace,\label{res13}\ ] ] or equivalently @xmath64\ , | \ , h,\ , i \rbrace,\label{res13b}\ ] ] in the sense that the solution of ( [ bes1 ] ) or ( [ bes2 ] ) is given by eq . ( [ vp1 ] ) where @xmath65 are the solutions of ( [ res13 ] ) or ( [ res13b ] ) . this justifies a _ selective decay principle _ from statistical mechanics . indeed , it is often argued that an axisymmetric turbulent flow should evolve so as to minimize energy at fixed helicity and angular momentum . in general , this phenomenological principle is motivated by viscosity or other dissipative processes . in our approach , it is justified by the maximum entropy principle ( [ bes1 ] ) of statistical mechanics when a coarse - graining is introduced . in the sequel , we shall study the maximization problem ( [ res13b ] ) since it is simpler than ( [ bes1 ] ) or ( [ bes2 ] ) , albeit equivalent . remark : _ although the variational problems ( [ bes1 ] ) and ( [ bes2 ] ) determining equilibrium states are equivalent , this does not mean that the relaxation equations associated with these variational problems are equivalent . to take an analogy , the boltzmann ( microcanonical ) and the kramers ( canonical ) equations have the same equilibrium states -the maxwell distribution- but a different dynamics . in the following , we will show that the equilibrium variational problems ( [ bes1 ] ) and ( [ bes2 ] ) have no solution . indeed , there is no maximum of entropy at fixed @xmath0 , @xmath1 and @xmath2 and no minimum of free energy at fixed @xmath1 and @xmath2 . all the critical points of ( [ bes1 ] ) and ( [ bes2 ] ) are saddle points of the thermodynamical potentials . then , the idea is to consider the out - of - equilibrium problem , introduce relaxation equations and study the robustness of saddle points with respect to random perturbations . for what concerns the out - of - equilibrium problem , the microcanonical and canonical ensembles may be inequivalent . we will see that they are indeed inequivalent . in this section , we shall study the minimization problem @xmath64\ , | \ , h,\ , i \rbrace.\label{res13c}\ ] ] the critical points of macroscopic energy at fixed helicity and angular momentum are determined by the condition @xmath66 where @xmath67 ( helical potential ) and @xmath68 ( chemical potential ) are lagrange multipliers . introducing the notations @xmath69 and @xmath70 , the variations on @xmath71 and @xmath72 lead to @xmath73 @xmath74 which are equivalent to eqs . ( [ vp2])-([vp3 ] ) up to a change of notations . in the following , it will be convenient to work with the new field @xmath75 . it is easy to check that @xmath76 where @xmath77 is the usual laplacian . therefore , eq . ( [ su1 ] ) becomes @xmath78 and the previous equations can be rewritten @xmath79 @xmath80 where @xmath81 is solution of @xmath82 with @xmath83 on the boundary . this is the fundamental differential equation of the problem . note that a particular solution of this differential equation is @xmath84 but it does not satisfy the boundary conditions . using eqs . ( [ dh ] ) and ( [ di ] ) , the helicity and the angular momentum are given by @xmath85 @xmath86 these equations are relationships between @xmath87 and @xmath88 . _ remark : _ we have not taken into account the conservation of circulation @xmath89 because this would lead to a term @xmath90 in the r.h.s . of eq . ( [ phi ] ) that diverges as @xmath91 . to construct the different solutions of eq . ( [ phi ] ) and study their stability , we shall follow the general procedure developed by chavanis & sommeria @xcite for the 2d euler equation . we first introduce an eigenmode decomposition to compute all critical points of ( [ res13c ] ) . then , we investigate their stability by determining whether they are ( local ) minima of macroscopic energy or saddle points . we first assume that @xmath92 in that case , the differential equation ( [ phi ] ) becomes @xmath93 with @xmath83 on the domain boundary . we introduce the eigenfunctions @xmath94 of the operator @xmath95 . they are defined by @xmath96 with @xmath97 on the domain boundary . it is easy to show that the eigenvalues @xmath98 of @xmath99 are positive ( hence the notation @xmath100 ) . indeed , we have @xmath101 and @xmath102 , which proves the result . it is also easy to show that the eigenfunctions are orthogonal with respect to the scalar product @xmath103 finally , we normalize them so that @xmath104 . the eigenvalues and eigenfunctions of the operator @xmath99 can be determined analytically . the differential equation ( [ e3 ] ) can be rewritten @xmath105 we look for solutions in the form @xmath106 . this yields @xmath107 where the sign of the constant has been chosen in order to satisfy the boundary condition @xmath83 in @xmath14 and @xmath15 . the differential equation for @xmath108 is readily solved and we obtain @xmath109 with @xmath110 where @xmath111 is a strictly positive integer . on the other hand , the differential equation for @xmath112 is @xmath113 if we define @xmath114 and @xmath115 , the foregoing equation can be rewritten @xmath116 this is a bessel equation whose solution is @xmath117 now , the boundary condition @xmath118 implies @xmath119 so that @xmath120 where @xmath121 is the @xmath122-th zero of bessel function @xmath123 . in conclusion , the eigenvalues are @xmath124 and the eigenfunctions are @xmath125 with the normalization constant @xmath126 the mode @xmath127 corresponds to @xmath122 cells in the @xmath128-direction and @xmath111 cells in the @xmath129-direction . we shall distinguish two kinds of modes , according to their properties regarding the symmetry @xmath130 with respect to the plane @xmath131 . the _ odd eigenmodes _ denoted @xmath132 are such that @xmath133 and correspond to @xmath111 even . they have zero mean value in the @xmath129 direction ( @xmath134 ) . for example , the mode @xmath135 is a two - cells solution in the vertical direction . the _ even eigenmodes _ denoted @xmath136 are such that @xmath137 and correspond to @xmath111 odd . they have non zero mean value in the vertical direction ( @xmath138 ) . in particular , the mode @xmath139 is a one - cell solution . returning to eq . ( [ e2 ] ) , this differential equation has solutions only for quantized values of @xmath140 ( eigenvalues ) and the corresponding solutions ( eigenfunctions ) are @xmath141 where we have used the helicity constraint ( [ de20 ] ) to determine the normalization constant . note that eq . ( [ de20 ] ) implies that @xmath142 and @xmath1 have the same sign , so that the square root is always defined . substituting this result in eq . ( [ de21 ] ) , and introducing the control parameter @xmath143 we find that these solutions exist only for @xmath144 with @xmath145 for the odd eigenmodes @xmath132 , we have @xmath146 and for the even eigenmodes @xmath136 , we have @xmath147 . we now assume that @xmath148 and define @xmath149 in that case , the fundamental differential equation ( [ phi ] ) becomes @xmath150 with @xmath151 on the domain boundary . we also assume that @xmath152 . in that case , eq . ( [ c2 ] ) admits a unique solution that can be obtained by expanding @xmath153 on the eigenmodes . using the identity @xmath154 we get @xmath155 of course , @xmath153 can also be obtained by solving the differential equation ( [ phi ] ) numerically . note that this solution is even since only the even modes are `` excited '' . substituting eq . ( [ c1 ] ) in eq . ( [ de21 ] ) , we obtain @xmath156 then , substituting eqs . ( [ c1 ] ) and ( [ c4 ] ) in eq . ( [ de20 ] ) , we get @xmath157 this equation gives a relationship between @xmath158 and @xmath159 . then , @xmath160 is determined by eq . ( [ c4 ] ) . these equations can therefore be viewed as the equations of state of the system . they determine the branch formed by the solutions of the continuum . using @xmath161 and @xmath162 we obtain @xmath163 this implies that @xmath158 is of the same sign as @xmath159 , hence @xmath1 . furthermore , @xmath159 is an odd function of @xmath158 . in the sequel , we shall consider only cases with @xmath164 , i.e. @xmath165 and @xmath166 for illustration and figures . note that eq . ( [ c5 ] ) involves the important function @xmath167 for @xmath146 , the inverse helical potential is @xmath168 or @xmath169 where @xmath170 is any zero of @xmath171 , i.e. @xmath172 for simplicity , we shall call @xmath173 the first zero of @xmath171 . this first zero is always between the first and the second even eigenmodes ( see appendix a ) . its location with respect to the first odd eigenmode @xmath174 depends on the aspect ratio of the cylinder : for @xmath175 , we have @xmath176 ( case l - for large aspect ratio ) while for @xmath177 , @xmath178 ( case s - for small aspect ratio ) . we now consider the case where @xmath148 and @xmath140 . for @xmath179 , we recover the eigenfunction @xmath180 as a limit case . therefore , the even eigenmodes are limit points of the main branch . on the other hand , for @xmath181 , the solution of eq . ( [ c2 ] ) is not unique . indeed , we can always add to the solution ( [ c3 ] ) an eigenmode @xmath182 . this leads to the mixed solution @xmath183 the `` proportion '' @xmath184 of the eigenmode present in the mixed solution is determined by the control parameter @xmath159 . taking the norm of @xmath185 and its scalar product with @xmath128 , we get @xmath186 substituting these results in eqs . ( [ de20 ] ) and ( [ de21 ] ) , we find that @xmath184 is determined by @xmath159 according to @xmath187 with @xmath181 . these mixed solutions exist in the range @xmath188 and they form a plateau at constant @xmath181 . for @xmath189 , we recover the odd eigenmode @xmath132 at @xmath146 and for @xmath190 , the plateau connects the branch of continuum solutions . the mixed solutions are therefore symmetry breaking solutions . they can be seen as a mixture of a continuum solution and an eigenmode solution , like in situations with different phase coexistence . in this section , we plot @xmath158 as a function of @xmath159 . for given @xmath191 , this curve determines the inverse helical potential @xmath192 as a function of the inverse helicity @xmath193 ( conjugate variables ) . it is represented in figs . [ blaml ] and [ blams ] for the cases l and s respectively . one sees that , for a given value of the control parameter @xmath159 , there exists multiple solutions with different values of @xmath158 . we will see in sec . [ coarsegrainedenergy ] . that , for a given value of @xmath159 , the macroscopic energy @xmath5 decreases as @xmath158 increases . therefore , low values of @xmath158 correspond to high energies states and high values of @xmath158 correspond to low energies states . as a function of @xmath159 for case l ( we have taken @xmath194 and @xmath195 ) . for a given value of @xmath159 ( we have taken @xmath196 ) , the solutions of the continuum are denoted by red circles and the mixed solutions by green circles . the mixed solution branches are drawn using dotted lines . one observes multiplicity of solutions : at given @xmath159 correspond several solutions with different @xmath158 . ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] + along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] + along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] _ case l : _ in this case @xmath176 and the curve @xmath206 looks typically like in fig . [ blaml ] . for a given value of @xmath159 , we have different solutions as represented in fig . [ fig : sel_ex2 ] . the highest energy solution is a one cell solution ( continuum branch ) , that we choose to call `` direct monopole '' . the second one is a two vertical cells solution ( mixed branch ) . the cells are symmetric for @xmath146 but one of the two cells grows for increasing @xmath159 . the third highest energy solution is another one - cell solution ( continuum branch ) rotating in a direction opposite to that of the highest energy solution . we therefore call it a `` reversed monopole '' . we call these three respective branches of solutions `` branch 1 '' and `` branch 2 '' for the continuum solutions , and `` mixed branch '' for the mixed solutions . the branches 1 and 2 connect each other at @xmath207 , the location of the first even eigenmode . a typical sequence of variation of the stream function with increasing @xmath159 on these three branches is given in fig . [ fig : branches ] . one sees that , as we increase @xmath159 on the mixed branch , the two cells solution , with a mixing layer at @xmath131 continuously transforms itself into a one cell solution , via a continuous shift of the mixing layer towards the vertical boundary . _ case s : _ in this case @xmath178 and the curve @xmath206 looks typically like in fig . [ blams ] . the highest energy solution is a one cell solution ( continuum branch ) , ( direct monopole ) . the second solution is another one - cell solution ( continuum branch ) rotating in the opposite direction ( reversed monopole ) . the third solution is a two horizontal cells solutions ( continuum branch ) . some stream functions are represented in fig . [ bifu04 ] . as a function of @xmath159 for case s ( here @xmath194 and @xmath208 ) . the solutions of the continuum are denoted by red circles and the mixed solutions by green circles . the mixed solution branches are drawn using dotted lines . ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] _ remark : _ there is a maximum value of @xmath214 above which there is no critical point of energy at fixed helicity and angular momentum . in that case , the system is expected to cascade towards smaller and smaller scales since there is no possibility to be blocked in a `` saddle point '' . this is a bit similar to the antonov instability in stellar dynamics due to the absence of critical point of entropy at fixed mass and energy below a critical value of energy @xcite . in that case , the system is expected to collapse ( gravothermal catastrophe ) . it is not yet clear whether a similar process can be achieved in experiments of turbulent axisymmetric flows . for @xmath215 , the system could become non - axisymmetric ruling out the theoretical analysis . in the previous section , we have found several solutions with different values of @xmath158 for each value of the control parameter @xmath216 . according to the variational principle ( [ res13c ] ) , we should select the solution with the minimum macroscopic energy . combining eqs . ( [ dh ] ) , ( [ di ] ) , ( [ e_cg ] ) , ( [ bel1 ] ) and ( [ bel2 ] ) , we obtain the relation @xmath217 for the eigenmodes ( @xmath218 ) , we find that @xmath219 let us consider the odd eigenmodes @xmath132 that exist for @xmath146 only . they are in competition with each other . we see that there is no minimum energy state since the energy decreases when @xmath127 increase , i.e. when the eigenmodes develop smaller and smaller scales . therefore , the minimum energy state corresponds to the structure concentrated at the smallest accessible scale . for the solutions of the continuum , using eq . ( [ c4 ] ) , the macroscopic energy is @xmath220 we can easily plot it as a function of @xmath158 ( see figs . [ fig : sel_ex ] and [ evsb04 ] ) . combining figs [ blaml ] , [ blams ] , [ fig : sel_ex ] and [ evsb04 ] , we see that , for a given value of @xmath159 , the solution with the smallest macroscopic energy corresponds to the highest @xmath158 , i.e. to small - scale structures . this is in complete opposition to what happens in 2d turbulence . in that case , the counterpart of the macroscopic energy @xmath5 is the macroscopic enstrophy @xmath10 and the minimum enstrophy state corresponds to structures spreading at the largest scale . strikingly , the bifurcation diagram in 2d turbulence @xcite is reversed with respect to the present one . as a function of @xmath158 for case l. the energy of the even eigenmodes are denoted by red circles and the energy of the odd eigenmodes by green circles . ] as a function of @xmath158 for case s. the energy of the even eigenmodes are denoted by red circles and the energy of the odd eigenmodes by green circles . ] in conclusion , there is no global minimum of macroscopic energy at fixed helicity and angular momentum . we can always decrease the macroscopic energy by considering structures at smaller and smaller scales . since ( [ bes1 ] ) , ( [ bes2 ] ) and ( [ res13b ] ) are equivalent , we also conclude that there is no global maximum of entropy at fixed microscopic energy , helicity and angular momentum . we may note a similar fact in astrophysics . it is well - known that a stellar system has no global entropy maximum at fixed mass and energy @xcite . this is associated to gravitational collapse ( called the gravothermal catastrophe in the microcanonical ensemble ) leading to the formation of binary stars . however , in the astrophysical problem , there exists local entropy maxima ( metastable states ) at fixed mass and energy if the energy is sufficiently high ( above the antonov energy ) . similarly , we could investigate the existence of metastable states in the present problem . however , we will show in sec . [ analytics ] that there is no local minimum of macroscopic energy at fixed helicity and angular momentum . all the critical points ( [ bel1])-([bel2 ] ) of the variational problem ( [ res13c ] ) are saddle points ! in our system , the chemical potential is @xmath221 . for given @xmath1 , we have to plot @xmath68 as a function of @xmath2 ( conjugate variables ) . the chemical potential is zero for the eigenmodes . using the equation of state ( [ c4 ] ) , we can express @xmath68 for the continuum solutions as @xmath222 where @xmath159 is expressed as a function of @xmath158 by eq . ( [ c5 ] ) . therefore , eq . ( [ chemical ] ) gives @xmath223 as a function of @xmath158 . eliminating @xmath158 between eqs . ( [ chemical ] ) and ( [ c5 ] ) , we obtain @xmath223 as a function of @xmath159 for the continuum . for the mixed solutions , we have @xmath224 corresponding to straight lines as a function of @xmath225 . the chemical potential curve @xmath223 as a function of @xmath225 is represented in fig . [ csursqrth - casel ] for case l and in fig . [ csursqrth - cases ] for case s. for fixed @xmath1 , this gives @xmath68 as a function of @xmath2 . if we come back to the initial variational problem ( [ bes1 ] ) , the caloric curve should give @xmath58 as a function of the microscopic energy @xmath45 ( conjugate variables ) for fixed values of @xmath1 and @xmath2 . now , the temperature is determined by the expression @xmath226 for given @xmath1 and @xmath2 , we can determine the _ discrete _ values of @xmath227 and the corresponding _ discrete _ values of @xmath228 as explained previously . then , for each discrete value , the temperature is related to the energy by eq . ( [ et ] ) . therefore , the mean flow ( beltrami state ) is fully determined by @xmath1 and @xmath2 and , for a given mean flow , the variance of the fluctuations ( temperature ) is determined by the energy @xmath45 according to @xmath229 in conclusion , the caloric curve @xmath230 , or more properly the series of equilibria , is formed by a a discrete number of straight lines with value at the origin @xmath231 and with constant specific heats @xmath232 . the specific heat is positive since the microcanonical and canonical ensembles are equivalent in our problem . in this section , we prove that the critical points of macroscopic energy at fixed helicity and angular momentum are all saddle points . a critical point of macroscopic energy at fixed helicity and angular momentum is a minimum ( resp . maximum ) iff the second order variations @xmath233 are definite positive ( resp . definite negative ) for all perturbations that conserve helicity and angular momentum at first order , i.e. @xmath234 and @xmath235 . adapting the procedure of chavanis & sommeria @xcite to the present context , we shall determine sufficient conditions of _ \(i ) let us prove that there is no local maximum of macroscopic energy at fixed angular momentum and helicity . consider first the even solutions , including the continuum solutions and the even eigenmodes . we choose a perturbation such that @xmath236 is odd and @xmath237 . for symmetry reason , this perturbation does not change @xmath2 nor @xmath1 at first order . on the other hand , for this perturbation @xmath238 . consider now the odd eigenmodes . we choose a perturbation of the form @xmath237 and @xmath239 , where @xmath240 is the first continuum solution such that @xmath241 . for this perturbation , we have @xmath242 , @xmath243 and @xmath244 since @xmath132 is orthogonal to @xmath240 . therefore , this perturbation does not change the helicity and the angular momentum at first order . on the other hand , for this perturbation @xmath238 . as a result , the critical points of macroscopic energy at fixed helicity and angular momentum can not be energy maxima since we can always find particular perturbations that increase the energy while conserving the constraints . \(ii ) let us prove that there is no local minimum of macroscopic energy at fixed angular momentum and helicity . to that purpose , we consider perturbations of the form @xmath245 and @xmath246 . the corresponding stream function is @xmath247 . consider first the even solutions , including the continuum solutions and the even eigenmodes . in that case , we have @xmath248 , @xmath249 and @xmath250 since @xmath251 is orthogonal to @xmath153 . the preceding relations remain valid for the odd eigenmodes @xmath127 provided that @xmath252 . therefore , these perturbations do not change the helicity and the angular momentum at first order . on the other hand , for these perturbations , we have @xmath253 thus , for given @xmath158 and @xmath254 sufficiently large , if the critical point is an odd eigenmode . ] i.e. @xmath255 , we have @xmath256 . as a result , the critical points of macroscopic energy at fixed helicity and angular momentum can not be energy minima since we can always find particular perturbations that decrease the energy while conserving the constraints . in conclusion , the critical points of macroscopic energy at fixed helicity and angular momentum are saddle points since we can find perturbations making @xmath257 positive and perturbations making @xmath257 negative . this analysis shows that all beltrami solutions are unstable . however , saddle points may be characterized by very long lifetimes as long as the system does not explore dangerous perturbations that destabilize them . this motivates the numerical stability analysis of sec . [ stability ] . _ remark : _ let us consider the odd eigenmode @xmath135 . we have seen that it can be destabilized by a perturbation @xmath258 or by a perturbation @xmath251 at smaller scale . let us now consider the effect of a perturbation of the form @xmath259 and @xmath239 , where @xmath240 is the first continuum mode such that @xmath241 . the corresponding stream function is @xmath260 . for this perturbation , we have @xmath242 , @xmath261 and @xmath262 since @xmath263 is orthogonal to @xmath240 . therefore , this perturbation does not change the helicity and the angular momentum at first order . for this perturbation , we have in addition @xmath264 this quantity is negative when @xmath265 corresponding to case l. this implies that the eigenmode @xmath135 is also destabilized by the perturbation @xmath260 which is at larger scale than the perturbations @xmath258 . the stability analysis performed in sec . [ analytics ] has shown that all the critical points of entropy at fixed microscopic energy , helicity and angular momentum are saddle points . we shall now investigate their robustness by using the relaxation equations derived in paper i ( for a review of relaxation equations in the context of 2d hydrodynamics , see @xcite ) . these relaxation equations can serve as numerical algorithms to compute maximum entropy states or minimum energy states with relevant constraints . their study is interesting in its own right since these equations constitute non trivial dynamical systems leading to rich bifurcations . although these relaxation equations do not provide a parametrization of turbulence ( we have no rigorous argument for that ) , they may however give an idea of the true dynamical evolution of the flow . in that respect , it would be interesting to compare these relaxation equations with navier - stokes simulations . this will , however , not be attempted in the present paper . by construction , the relaxation equations monotonically increase entropy , or decrease energy , with relevant constraints . different generic evolutions are possible : ( i ) they can relax towards a fully stable state ( global maximum of entropy or global minimum of energy ) ; ( ii ) they can relax towards a metastable state ( local maximum of entropy or local minimum of energy ) ; ( iii ) they do not relax towards a steady state and develop structures at smaller and smaller scales . in the present situation , we have seen that there are no stable and metastable states . therefore , the stability analysis of sec . [ analytics ] predicts that the system should cascade towards smaller and smaller scales without limit ( except the one fixed by the finite resolution of the simulations ) . this is a possible regime ( see top of fig . [ fig : ex_relaxinst ] ) but this is not what is generically observed in the experiments where long - lived structures at large scales are found ( like at the bottom of fig . [ fig : ex_relaxinst ] ) . here , we explore the possibility that these long - lived structures are saddle points of entropy or energy with relevant constraints . these saddle points are steady states of the relaxation equations . although they are unstable ( strictly speaking ) , we argue that these saddle points can be long - lived and relatively robust ( this idea was previously developed for 2d flows in @xcite ) . indeed , they are unstable only for certain ( dangerous ) perturbations , but not for all perturbations . therefore , they can be stable as long as the system does not explore dangerous perturbations that destabilize them . of course , the rigorous characterization of this form of stability is extremely complex . in order to test this idea in a simple manner , we shall use the relaxation equations and study the robustness of the saddle points with respect to them . our stability analysis is based on the numerical integration of the relaxation equations @xmath266 where @xmath267 and @xmath268 are given functions of @xmath128 and @xmath129 , and @xmath58 , @xmath67 and @xmath68 evolve in time ( see below ) so as to guarantee the conservation of the invariants . in the canonical ensemble , the temperature @xmath58 is fixed and the conserved quantities are the helicity and the angular momentum . the lagrange multipliers @xmath269 and @xmath270 are computed at each time so as to guarantee the conservation of @xmath1 and @xmath2 . one may check that they are solutions of the system of algebraic equations ( see paper i ) @xmath271 these relaxation equations are associated with the maximization problem ( [ bes2 ] ) provided that , at any given time , the distribution of angular momentum is given by eq . ( [ vp1 ] ) with constant @xmath58 ( see paper i for details ) . by properly redefining the lagrange multipliers , they are also associated with the minimization problem ( [ res13b ] ) . in the microcanonical ensemble , the conserved quantities are @xmath7 , @xmath1 and @xmath2 . in the sequel , it will be convenient to fix the time dependence of @xmath58 by imposing @xmath272 at each time . taking into account the two other invariants , one may check that @xmath273 , @xmath270 and @xmath274 are solution of the system of algebraic equations @xmath275 these relaxation equations are associated with the maximization problem ( [ bes1 ] ) provided that , at any given time , the distribution of angular momentum is given by eq . ( [ vp1 ] ) with @xmath276 ( see paper i for details ) . in the sequel we focus on the special case @xmath277 and @xmath278 , where @xmath279 and @xmath280 are constants , that allows a simple numerical treatement of the relaxation equations by projection along the beltrami eigenmodes : @xmath281 where @xmath282 , @xmath283 , @xmath284 label the modes and @xmath285 is the number of modes . in that case , eqs . ( [ rel1 ] ) and ( [ eq : canorelax ] ) can be transformed into a set of @xmath286 odes : @xmath287 , \label{ode1 } \\ & & \dot{x_{\bf n } } = -\chi _ * \left [ \beta p_{\bf n } + \mu s_{\bf n}\right],\label{ode2}\\ & & p_{\bf n } = b_{\bf n}^{-2 } x_{\bf n } , \label{eq : relax_simp_proj}\end{aligned}\ ] ] where @xmath288 is such that @xmath289 . note that the constraints couple eqs . ( [ ode1])-([eq : relax_simp_proj ] ) through the parameters @xmath58 , @xmath68 and @xmath67 . to investigate the robustness of a given stationary solution , we first perturb it with a suitable perturbation ( see below ) , and then follow its dynamics thanks to the relaxation equations . two typical time evolutions are provided in fig [ fig : ex_relaxinst ] : if the solution is fragile with respect to the perturbation , it will cascade to another solution ( usually the solution of smallest scale permitted by our resolution ) ; if the solution is robust with respect to this perturbation , it will eventually return to its initial unperturbed state . to quantify the robustness of a given solution , we define a probabilistic stability criterion by computing the probability for the solution `` to escape '' from its basin of attraction . to that purpose , we select a threshold @xmath290 and compute at each time the probability of escape @xmath291,\ ] ] using @xmath292 realizations with perturbations drawn at random at @xmath293 from a suitable ensemble ( see below ) . this allows us to define `` statistically fragile '' solutions as those for which @xmath294 when @xmath295 , the others being referred to as `` statistically robust '' . in practice , the limit @xmath295 is not accessible . we thus generalize this notion to a `` finite time '' , by considering the asymptotic value of @xmath296 reached at the largest time of the simulation , @xmath297 . in addition , the asymptotic value of @xmath298 provides a mean to quantify the degree of robustness of a solution . examples are given in fig . [ fig : criteres ] , for a fragile and for a robust solution . as can be seen , the fragile solution is fragile whatever the threshold @xmath290 . however , the degree of robustness of a solution depends on the threshold @xmath290 . quite naturally , the larger the threshold , the more robust the solution . made with 200 perturbations around two beltrami states . top : fragile solution ; bottom : robust solution . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] made with 200 perturbations around two beltrami states . top : fragile solution ; bottom : robust solution . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] _ remark _ : although the variational problems ( [ bes1 ] ) , ( [ bes2 ] ) and ( [ res13b ] ) are equivalent , and all lead to the absence of stable equilibrium state , the corresponding relaxation equations described previously are different . therefore , the robustness of the saddle points will be different in the canonical and microcanonical settings . this can be viewed as a form of `` ensembles inequivalence '' for an out - of - equilibrium situation . the stability must be investigated using perturbations that rigorously conserve the integral constraints . this puts some conditions regarding the shape of the possible perturbations that we can use . in the canonical ensemble , the integral constraints are @xmath1 and @xmath2 . given an initial stationary solution @xmath301 , the perturbations @xmath302 must obey @xmath303 one can check that this set of constraints is satisfied by any perturbation of the form @xmath304\\ & & \delta \xi = { \epsilon } r^{-1 } { s_{\bf i_2}^\star}^{-1}\left[\displaystyle\frac{\langle r\phi_{\bf i_1}\rangle } { \langle r\phi_{\bf i_0}\rangle } x^\star_{\bf i_0}-x^\star_{\bf i_1}\right]\phi_{\bf i_2}\\ & & \delta \psi = { \epsilon } r { s_{\bf i_2}^\star}^{-1 } b_{\bf i_2}^{-2}\left[\displaystyle\frac{\langle r\phi_{\bf i_1}\rangle } { \langle r\phi_{\bf i_0}\rangle } x^\star_{\bf i_0}-x^\star_{\bf i_1}\right]\phi_{\bf i_2 } \label{eq : perturb}\end{aligned}\ ] ] where @xmath305 is the amplitude of the perturbation , @xmath306 labels an even mode while @xmath307 and @xmath308 label two different modes different from @xmath306 such that @xmath309 . following eqs . ( [ dec1 ] ) and ( [ dec2 ] ) , we have set @xmath310 and @xmath311 . in the sequel , we fix the amplitude of the perturbation @xmath305 through the norm @xmath312 by imposing @xmath313^{-\frac{1}{2}}.\ ] ] the modes @xmath306 , @xmath307 and @xmath308 are chosen randomly according to the following procedure : i ) we draw @xmath306 following a uniform law among the @xmath314 even modes ; ii ) we draw @xmath308 following a uniform law among the @xmath314 or @xmath315 modes of the set of allowed @xmath316 , excluding @xmath306 . this mode is therefore necessarily even for solution of continuum , and often even for mixed solutions ; iii ) we draw @xmath307 following a uniform law among the @xmath317 even and odd modes , excluding @xmath306 and @xmath308 . this choice allows the generation of @xmath292 random perturbations with the same amplitude @xmath318 . in the microcanonical ensemble , the relaxation equations conserve in addition the energy . to satisfy this additional constraint , we choose the perturbations according to the same procedure as in the canonical case , and then determine the initial value of the temperature @xmath319 in order to guarantee the conservation of the energy . and angular momentum @xmath2 determine the mean flow while the energy @xmath7 determines the temperature . ] as explained previously , this amounts to taking @xmath320 in the following , we shall group the perturbations into subclasses such that perturbations of the same class have the same temperature @xmath321 or , equivalently , the same macroscopic energy @xmath322 . note that the initial temperature of the perturbation differs from the temperature of the equilibrium state which is given by @xmath323 in the sequel , we focus on the stability analysis in the case l , for the first three branches of solutions , relevant for comparison with experiments , see paper iii @xcite . our parameters are as follows : * the number of modes is @xmath324 with @xmath325 ( radial modes ) and @xmath326 ( vertical modes ) corresponding to 120 even modes and 120 odd modes . the radial and vertical lengths are @xmath327 and @xmath328 . * the amplitude of the perturbations is @xmath329 . we consider @xmath330 realizations for each given stationary solution . * the parameters @xmath279 and @xmath280 are both taken equal to @xmath331 . the relaxation equations are integrated using an implicit heun scheme . the time step is empirically chosen proportional to @xmath332 . for @xmath333 , the time step is 0.02 . we have checked that this time step is small enough to guarantee the numerical conservation of @xmath2 and @xmath1 ( canonical case ) or @xmath7 , @xmath2 and @xmath1 ( microcanonical case ) . for any value of @xmath159 on a given branch of solutions , we proceed as follows : [ [ canonical - ensemble ] ] canonical ensemble * we fix the value of the temperature @xmath58 ( it remains constant during the evolution ) . in the sequel , we focus on five arbitrary values , @xmath334 , chosen so as to span a wide range . * we compute the beltrami solution @xmath335 corresponding to a prescribed value of @xmath159 on the given branch . * we generate @xmath292 perturbed initial conditions leaving unchanged the helicity and the angular momentum of @xmath335 . * we evolve the perturbed initial conditions through eqs . ( [ ode1])-([eq : relax_simp_proj ] ) and eq . ( [ eq : coefrelaxcano1],[eq : coefrelaxcano2 ] ) for a certain amount of time @xmath297 . [ [ microcanonical - ensemble ] ] microcanonical ensemble * we fix the value of the energy @xmath7 ( it remains constant during the evolution ) . in the sequel , it is fixed after arbitrary choice of five values of the temperature , @xmath336 , chosen so as to span a wide range . once @xmath321 has been fixed , the total energy is then fixed . it can vary from one realization to the other , but does not vary along the evolution . * we compute the stationary beltrami solution @xmath335 corresponding to a prescribed value of @xmath159 on the given branch . * we generate @xmath292 perturbed initial conditions leaving unchanged the helicity and the angular momentum of @xmath335 . * we group together the perturbations that have the same initial temperature @xmath337 measuring the initial energy of the fluctuations ( equivalently , these perturbations have the same value of macroscopic energy @xmath5 ) . * we evolve the perturbed initial condition through eqs . ( [ ode1])-([eq : relax_simp_proj ] ) and ( [ eq : coefrelaxmicro1]-[eq : coefrelaxmicro3 ] ) for a certain amount of time @xmath297 . on the three branches , we computed the value @xmath338 ( computed at @xmath297 ) as a function of @xmath159 for different temperatures @xmath339 . the results are displayed on fig . [ fig : inequivalence_cano ] . the mixed branch ( vertical dipoles ) and branch 2 ( reversed monopoles ) are found very robust for high threshold , and still retain a certain degree of robustness for a small threshold , with a 40 per cent probability of escape . there is no clear dependence on the temperature . this is natural , since temperature can be eliminated by a suitable rescaling of time ( or of coefficients @xmath267 and @xmath268 ) and redefinition of lagrange parameters . the behavior on branch 1 ( direct monopoles ) is more contrasted and provides a very clear transition around the critical value @xmath340 . for @xmath341 , the probability to escape is close to 1 , meaning large fragility of the branch . for @xmath342 , the branch becomes much more robust , reaching a larger degree of robustness than the two other branches for high threshold , while reaching the same robustness for small threshold . the different behaviors are summarized on fig . [ fig : stabcano ] . as function of @xmath159 on the three branches ( left : branch 1 , middle : branch 2 , right : mixed branch ) at different temperatures @xmath343 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig:"][fig : micro_branche1 ] as function of @xmath159 on the three branches ( left : branch 1 , middle : branch 2 , right : mixed branch ) at different temperatures @xmath343 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig:"][fig : micro_branche2 ] as function of @xmath159 on the three branches ( left : branch 1 , middle : branch 2 , right : mixed branch ) at different temperatures @xmath343 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig:"][fig : micro_mixte ] . the lines are increasingly fat with increasing @xmath338 , i.e. robustness . note that the value of @xmath349 increases with increasing @xmath297 . ] note that the value @xmath350 is somewhat arbitrary . indeed , increasing @xmath297 further , we observed the same qualitative scenario , with an increased value of @xmath349 . we also observed that over sufficiently long time , the branch 2 tends to become unstable , past a value of the order @xmath351 . on the three branches , we computed the value @xmath338 ( computed at @xmath297 ) as a function of @xmath159 for classes of perturbations with different initial temperature @xmath352 . note that the initial temperature fixes the amplitude of the velocity fluctuations . the results are displayed on fig . [ fig : inequivalence_micro ] . for the mixed branch ( vertical dipoles ) and branch 2 ( reversed monopoles ) , the microcanonical results do not noticeably differ from the canonical results : the two branches are found very robust for large threshold , and still retain a certain degree of robustness for a small threshold , with a 40 per cent probability of escape . there is no clear dependence on the initial temperature . there is therefore no ensembles inequivalence for these two branches . this is not true anymore for branch 1 ( direct monopoles ) . indeed , one still observes a transition from robustness to fragility around a critical value @xmath349 but this quantity depends on the initial temperature @xmath353 : it takes a value @xmath340 at large initial temperatures ( large velocity fluctuations ) and then decreases to @xmath354 for small initial temperatures ( small velocity fluctuations ) . the difference of robustness observed between the two ensembles may be seen as a kind of inequivalence of ensembles at small initial temperatures . the different behaviors are summarized on fig . [ fig : stabmicro ] . like in the canonical case , we checked that an increase of @xmath297 results in a larger fragility of the branch 1 and 2 towards small @xmath159 , at a given temperature . _ remark : _ note that perturbations with small initial temperature have large macroscopic energies corresponding to perturbations at large scales . according to sec . [ analytics ] such perturbations are less destabilizing than perturbations at small scales ( associated with small macroscopic energies hence large temperatures ) . this may explain the numerical results . as function of @xmath159 on the three branches ( left : branch 1 ; middle : branch 2 ; right : mixed branch ) at different initial temperatures @xmath355 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] as function of @xmath159 on the three branches ( left : branch 1 ; middle : branch 2 ; right : mixed branch ) at different initial temperatures @xmath355 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] as function of @xmath159 on the three branches ( left : branch 1 ; middle : branch 2 ; right : mixed branch ) at different initial temperatures @xmath355 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] . left : for a low initial temperature ; right : for a high initial temperature . the lines are increasingly fat with increasing @xmath338 , i.e. robustness . note that the value of @xmath349 increases with increasing @xmath297.,title="fig:"][fig : stab0 ] . left : for a low initial temperature ; right : for a high initial temperature . the lines are increasingly fat with increasing @xmath338 , i.e. robustness . note that the value of @xmath349 increases with increasing @xmath297.,title="fig:"][fig : stab1 ] we have studied the thermodynamics of axisymmetric euler - beltrami flows and proved the coexistence of several equilibrium states for the same values of the control parameters . all these states are saddle points of entropy but they can have very long lifetime as long as the system does not spontaneously develop dangerous perturbations . we have numerically explored the robustness of some of these states by using relaxation equations in the canonical and microcanonical ensembles . the dipoles ( mixed branch ) and the reversed monopoles ( branch 2 ) were found to be rather robust in both ensembles . furthermore , in the microcanonical ensemble there is no dependence on the initial temperature on these branches . by contrast , the direct monopoles ( branch 1 ) display a sharp transition around a critical value @xmath349 . the value of @xmath349 increases with increasing integration time . in the microcanonical ensemble , this value also decreases with decreasing initial temperature , resulting in a difference of robustness in the canonical and microcanonical ensembles . this difference may be seen as a kind of `` ensembles inequivalence '' . this is , however , a very unconventional terminology since it concerns here the robustness of _ saddle points _ with respect to random perturbations that keep the energy or the temperature fixed , over a finite amount of time . the simulations have shown that the dipole ( two - cells solution ) is relatively robust for any value of the angular momentum . on the other hand , the direct monopole ( one - cell solution ) is very fragile at low angular momentum but becomes robust at high angular momentum . in that case , it is even more robust than the dipole . therefore , increasing the total angular momentum of the flow , one expects to observe a transition from the two - cells solution ( antisymmetric with respect to the middle plane ) to the one - cell solution ( symmetric with respect to the middle plane ) . this bifurcation scenario is sketched in fig . [ fig : scenarhyst ] . it is reminiscent of the turbulent transition reported in the von krmn flow @xcite in which the initial two - cells flow observed at zero global rotation suddenly bifurcates when the rotation is large enough . once the bifurcation has taken place , the level of fluctuation is experimentally observed to decrease strongly , resulting in a decrease of the statistical temperature . in our scenario , this means that the monopole branch is suddenly stabilized with respect to redecrease of the total angular momentum of the flow , resulting in a hysteresis that has also been observed experimentally . it would therefore be interesting to investigate more closely the relevance of our scenario to the experimental system . this is done in the next paper @xcite . , on the mixed branch , in a two - cells topology ( vertical dipole ) ; middle : increasing @xmath159 up to @xmath356 ( red arrow ) , the system follows the mixed branch . one cell grows at the expense of the other , resulting in a shift of the mixing layer upwards ; right : at @xmath349 , the system bifurcates towards branch 1 ( more stable ) , resulting in a one - cell topology ( direct monopole ) . in this new state , the fluctuations are much milder ( empirical fact from experiments ) , thereby allowing a stabilization of the branch towards lower @xmath159 . therefore , the monopole subsists beyond this point , even after a redecreasing of @xmath159 , depicted by the black arrow.,title="fig : " ] , on the mixed branch , in a two - cells topology ( vertical dipole ) ; middle : increasing @xmath159 up to @xmath356 ( red arrow ) , the system follows the mixed branch . one cell grows at the expense of the other , resulting in a shift of the mixing layer upwards ; right : at @xmath349 , the system bifurcates towards branch 1 ( more stable ) , resulting in a one - cell topology ( direct monopole ) . in this new state , the fluctuations are much milder ( empirical fact from experiments ) , thereby allowing a stabilization of the branch towards lower @xmath159 . therefore , the monopole subsists beyond this point , even after a redecreasing of @xmath159 , depicted by the black arrow.,title="fig : " ] , on the mixed branch , in a two - cells topology ( vertical dipole ) ; middle : increasing @xmath159 up to @xmath356 ( red arrow ) , the system follows the mixed branch . one cell grows at the expense of the other , resulting in a shift of the mixing layer upwards ; right : at @xmath349 , the system bifurcates towards branch 1 ( more stable ) , resulting in a one - cell topology ( direct monopole ) . in this new state , the fluctuations are much milder ( empirical fact from experiments ) , thereby allowing a stabilization of the branch towards lower @xmath159 . therefore , the monopole subsists beyond this point , even after a redecreasing of @xmath159 , depicted by the black arrow.,title="fig : " ] an interesting outcome of our study lies in the fate of the solutions when they are destabilized by a dangerous perturbation : due to the energy minimization principle , the unstable solution tends to `` cascade '' towards a higher wavenumber solution in a way reminiscent to the richardson energy cascade of 3d turbulence ( see fig . [ fig : ex_relaxinst ] ) . the cascade stops when the largest available wavenumber is reached , since dangerous perturbations are necessarily at smaller scale than the achieved state . this form of energy condensation at the smallest scale may be seen as an interesting counterpart ( in the opposite sense ) of the large scale energy condensation observed in 2d turbulence via the inverse energy cascade process . this is the signature of the 2d and a half nature of our system , intermediate between 2d and 3d turbulence . we have characterized the thermodynamical equilibrium states of axisymmetric euler - beltrami flows and proved the coexistence of several equilibrium states for a given value of the control parameter like in 2d turbulence @xcite . we further showed that all states are saddle points of entropy and can , in principle , be destabilized by a perturbation with a larger wavenumber , resulting in a structure at the smallest available scale . this mechanism is therefore reminiscent of the 3d richardson energy cascade towards smaller and smaller scales . therefore , our system is truly intermediate between 2d turbulence ( coherent structures ) and 3d turbulence ( energy cascade ) . through a numerical exploration of the robustness of the equilibrium states with respect to random perturbations using a relaxation algorithm in both canonical and microcanonical ensembles , we showed however that these saddle points of entropy can be very robust and therefore play a role in the dynamics . we evidenced differences in the robustness of the solutions in the canonical and microcanonical ensembles leading to a theoretical scenario of bifurcation between two different equilibria ( with one or two cells ) that resembles a recent observation of a turbulent bifurcation in a von krmn experiment @xcite . this work was supported by european contract wallturb . we show that the first zero of @xmath357 , denoted @xmath358 , is always between the first @xmath359 and the second @xmath360 even eigenmode . to that purpose , we note that if @xmath361 then @xmath362 for any @xmath127 so that @xmath363 . there is no discontinuity of @xmath171 in the interval @xmath364 so that there is no zero in that interval . consider now the interval @xmath365b''_1,b''_2[$ ] . in that interval , @xmath171 is also continuous and increasing since @xmath366 moreover , for @xmath367 , @xmath368 . similarly , @xmath369 when @xmath370 . therefore , there exists a unique value of @xmath358 in the range @xmath365b''_1,b''_2[$ ] , such that @xmath371 . this shows that the first zero of @xmath171 lies in between the first two even eigenmodes . this property remains true for the successive values of @xmath170 and the successive even eigenmodes .

we characterize the thermodynamical equilibrium states of axisymmetric euler - beltrami flows . they have the form of coherent structures presenting one or several cells . we find the relevant control parameters and derive the corresponding equations of state . we prove the coexistence of several equilibrium states for a given value of the control parameter like in 2d turbulence [ chavanis & sommeria , j. fluid mech . * 314 * , 267 ( 1996 ) ] . we explore the stability of these equilibrium states and show that all states are saddle points of entropy and can , in principle , be destabilized by a perturbation with a larger wavenumber , resulting in a structure at the smallest available scale . this mechanism is therefore reminiscent of the 3d richardson energy cascade towards smaller and smaller scales . therefore , our system is truly intermediate between 2d turbulence ( coherent structures ) and 3d turbulence ( energy cascade ) . we further explore numerically the robustness of the equilibrium states with respect to random perturbations using a relaxation algorithm in both canonical and microcanonical ensembles . we show that saddle points of entropy can be very robust and therefore play a role in the dynamics . we evidence differences in the robustness of the solutions in the canonical and microcanonical ensembles . a scenario of bifurcation between two different equilibria ( with one or two cells ) is proposed and discussed in connection with a recent observation of a turbulent bifurcation in a von krmn experiment [ ravelet _ et al . _ , phys . rev . lett . * 93 * , 164501 ( 2004 ) ] .

the formation and evolution of galaxies is one of the fundamental problems in astrophysics . the recent deep imaging of very faint galaxies made with _ hubble space telescope _ ( williams et al . 1996 ) and the detection of co emission from a high-@xmath7 quasar br 1202@xmath40725 at @xmath7 = 4.69 ( ohta et al . 1996 ; omont et al . 1996 ) have encouraged us to study the problem mentioned above . since the galaxies should form from gaseous system , it is important to investigate the major epoch of star formation in the gas system and to study how stars have been made during the course of galaxy evolution . when we study evolution of galaxies , we usually use stellar lights as the tracer of evolution ( cf . tinsley 1980 ; arimoto & yoshii 1986 , 1987 ; bruzual & charlot 1993 ) . however , much data of interstellar medium ( ism ) of galaxies from x - ray emitting hot gas through warm hi gas to cold molecular gas and dust have been accumulated for these decades ( cf . wiklind & henkel 1989 ; lees et al . 1991 ; fabbiano , kim , & trinchier 1992 ; kim , fabbiano , & trinchier 1992 ; wang , kenney , & ishizuki 1992 ) . therefore , the time is ripe to begin the study of evolution of ism of galaxies from the epoch of galaxy formation to the present day for both elliptical and disk galaxies . in this _ paper _ , appreciating the recent detection of co emission from the high-@xmath7 quasar br 1202@xmath40725 ( ohta et al . 1996 ; omont et al . 1996 ) , we discuss the evolution of molecular gas content in galaxies . since active galactic nuclei ( agn ) are associated with their host galaxies , the co luminosity of agn depends on the gaseous content of their host galaxies . therefore , any observations of molecular - line emission from high-@xmath7 objects are very useful in studying evolution of molecular gas content in galaxies.0725 , there are two more successful detections of co from high-@xmath7 objects ; 1 ) the hyperluminous infrared galaxy iras f10214 + 4725 at @xmath7 = 2.286 ( brown & vanden bout 1991 ; solomon , downes , & radford 1992 ; tsuboi & nakai 1992 ; radford et al . 1996 ) , and 2 ) the cloverleaf quasar h1413 + 135 at @xmath7 2.556 ( barvanis et al . 1994 ) . since , however , these two sources are gravitationally amplified ones ( elston , thompson , & hill 1994 ; soifer et al . 1995 ; trentham 1995 ; graham & liu 1995 ; broadhurst & lehr 1995 ; serjeant et al . 1995 ; close et al . 1995 ) , we do not use these data in this study because there may be uncertainty in the amplification factor . ] in spite of the successful co detection from br 1202@xmath40725 , evans et al . ( 1996 ) reported the negative co detection from 11 high-@xmath7 ( @xmath6 ) powerful radio galaxies ( prgs ) and gave the upper limits of order @xmath8 ( hereafter @xmath9 = 50 km s@xmath10 mpc@xmath10 and @xmath11 ) , being comparable to or larger than that of br 1202@xmath40725 ( ohta et al . 1996 ; omont et al . 1996 ) . here a question arises as why co emission was detected from the high-@xmath7 quasar at @xmath12 while not detected from the high-@xmath7 ( @xmath6 ) prgs . there may be two alternative answers : 1 ) the host galaxies are different between quasars and prgs in terms of molecular gas contents . or , 2 ) although the host galaxies are basically similar between quasars and prgs , their evolutionary stages are different and thus the molecular gas contents are systematically different between the two classes . provided that the current unified model for quasars and radio galaxies ( barthel 1989 ) is also applicable to high-@xmath7 populations , it is unlikely that their host galaxies are significantly different . since it is usually considered that luminous agns like quasars and prgs are associated with either massive ellipticals or bulges of disk galaxies as well as merger nuclei , the evolution of ism would be rapidly proceeded during the era of spheroidal component formation therefore , we investigate the latter possibility ( i.e. , evolutionary effect ) and discuss some implications on the evolution of ism in galaxies . assuming that the luminous agns are harbored in giant elliptical galaxies and/or in bulges of spiral galaxies , we investigate the evolution of co luminosity based on a galactic wind model for elliptical galaxies proposed by arimoto & yoshii ( 1987 ) and a bulge - disk model for spiral galaxies by arimoto & jablonka ( 1991 ) . the so - called _ infall _ model of galaxy chemical evolution is adopted for both spheroidals and disks and time variations of gas mass and gas metallicity , in particular @xmath13 , are calculated numerically by integrating usual differential equations for chemical evolution without introducing the instantaneous recycling approximation for stellar lifetime . model parameters , such as star formation rate ( sfr ) @xmath14 , a slope of initial mass function ( imf ) @xmath15 , and gas accretion rate ( acr ) @xmath16 , are taken from arimoto & yoshii ( 1987 ) and arimoto & jablonka ( 1991 ) . the lower and upper stellar mass limits are set to be @xmath17 m@xmath18 and @xmath19 m@xmath18 , respectively . according to jablonka , martin , & arimoto ( 1996 ) , who found that the @xmath20 relation of bulges are exactly identical to that of elliptical galaxies , we consider that bulges are small ellipticals of equivalent luminosity and that both spheroidal systems share the similar history of star formation . thus , for ellipticals and bulges , we assume that the remaining gas is expelled completely after the onset of galactic wind , which takes place once the thermal energy released from supernovae exceeds the binding energy of the gas . the wind times , @xmath21 gyr for giant ellipticals ( m@xmath22 m@xmath18 ) and @xmath23 gyr for bulges ( m@xmath24 m@xmath18 ) , are taken from arimoto & yoshii ( 1987 ) . for spiral galaxies , assuming that the bulge and disk evolve independently , we construct a model by combining the bulge and disk models with m@xmath24 m@xmath18 and @xmath25 m@xmath18 , respectively . this model gives @xmath26 mag for the bulge and @xmath27 mag for the disk at the age of 15 gyr old ( arimoto & jablonka 1991 ) . the bulge - to - disk light ratio in v - band is @xmath28 , nearly twice of typical values for early type spirals ( simien & de vaucouleurs 1986 ) . the @xmath29 refers to that of co(@xmath30=1 - 0 ) . note that the @xmath29 of high-@xmath7 galaxies are measured by using much higher transitions such as @xmath30=3 - 2 , 4 - 3 , and so on . however , it is known that that local co - rich galactic nuclei and starburst nuclei have @xmath29(@xmath30=3 - 2)@xmath31(@xmath30=1 - 0 ) @xmath32 ( devereux et al . 1994 ; israel & van der werf 1996 ) . therefore high-@xmath7 analogs may have the similar properties . in fact , two high-@xmath7 objects iras f10214 + 4724 and h1413 + 117 have @xmath29(@xmath30=4 - 3)@xmath31(@xmath30=3 - 2 ) @xmath32 and @xmath29(@xmath30=6 - 5)@xmath31(@xmath30=3 - 2 ) @xmath33 ( see table 1 of israel & van der werf ) . thus , the uncertainty due to use of higher transition data may be 50 percent at most , when we compare model @xmath29(@xmath30=1 - 0 ) and the observed @xmath29 at higher transitions . ] of a model galaxy can be calculated from molecular hydrogen mass by using the empirical co to h@xmath34 conversion factor ( @xmath35 ) . arimoto , sofue & tsujimoto ( 1995 ) showed that @xmath35 strongly depends on the gas metallicity and derived the following relationship valid for nearby spirals and irregular galaxies : @xmath36 where @xmath37 h@xmath34/ k km s@xmath10 = @xmath38 and o / h is the oxygen abundance of hii regions . we introduce a fractional mass of hydrogen molecule to that of atomic hydrogen , @xmath39 , and write the co luminosity in k km s@xmath10 pc@xmath40 as follows : @xmath41 where @xmath42 and @xmath43 are in @xmath44 . chemical evolution model gives @xmath45 and o / h as a function of time and the co luminosity evolution can be traced with a help of eq.(2 ) provided that @xmath46 is known _ a priori_. we assume time invariant @xmath46 throughout the course of galaxy evolution . in principle , @xmath46 itself should evolve as well , since the hydrogen molecule is newly produced on the surface of dust ejected from evolving stars and/or formed in expanding shells of supernovae remnants while at the same time a part of molecules are dissociated by uv photons emitted from young hot stars . the mass of dust and the number of uv photons should also evolve as a result of galactic chemical evolution ( honma , sofue , & arimoto 1995 ) . detailed evolution of @xmath46 will be shown in our subsequent paper ( ikuta et al . 1997 ) , instead in this _ paper _ we assume @xmath47 . recent studies of nearby ellipticals suggest @xmath48 ( wiklind & rydbeck 1986 ; sage & wrobel 1989 ; lees et al . 1991 ; eckart , cameron , & genzel 1991 ) . the contribution of helium to the gas mass is entirely ignored for simplicity , but our conclusions change little even if the evolution of helium gas is precisely taken into account . the formation epoch of galaxies is assumed to be @xmath49 . although the choice of @xmath50 is rather arbitrary , @xmath51 has some supports from recent studies on the metallicity of broad emission - line regions of high-@xmath7 quasars ( hamann & ferland 1992 , 1993 ; kawara et al . 1996 ; taniguchi et al . 1997 ) . figure 1 shows the result for elliptical galaxies . the thick solid line represents the galactic wind model , and the dotted line a model with continuous star formation ( the wind is suppressed even after the wind criterion is satisfied ) . the dashed line shows a case for a wind model , but the gas ejected from evolving giants after the wind is bound and accumulated in the galaxy to form neutral gas ( bound - wind model ; arimoto 1989 ) . the co luminosity , @xmath52 , of elliptical galaxies increases prominently soon after their birth , and attains the maximum at an epoch of about 0.85 gyr since the birth , or at @xmath53 . then , it suddenly decreases when the galactic wind has expelled the ism from the galaxy . the extremely luminous phase in co observed for the high @xmath7 quasar br 1202@xmath40725 ( ohta et al . 1996 ; omont et al . 1996 ) can be well explained , if it is in the star forming phase of the whole elliptical system . moreover , the non - detection of the smaller redshift galaxies as observed by evans et al . ( 1996 ) and van ojik et al . ( 1997 ) is also naturally understood by the present model : it is because of the fact that elliptical galaxies at @xmath54 contains little ism . in the figure , we also superpose co observational data for lower redshift elliptical galaxies ( wiklind & rydbeck 1986 ; sage & wrobel 1989 ; wiklind & henkel 1989 ; eckart et al . 1991 ; lees et al . 1991 ; sage & galletta 1993 ; sofue & wakamatsu 1993 ; wiklind , combes , & henkel 1995 ) . the theoretical curve for the bound - wind model is clearly inconsistent with the observations for galaxies at @xmath55 . this suggests that the gas has been expelled continuously after the galactic wind ( @xmath7 @xmath56 4 ) and has not been bound to the system . this , in turn , is consistent with the idea that the intracluster hot gas with high metallicity , as observed in x - rays , may have been supplied by the winds from early type galaxies ( ishimaru & arimoto 1997 ) . although it is not clarified how the gas has been expelled out of the galaxies , without being bound to the system , recent studies suggest that it is probably due to the energy supply from either the type ia supernovae ( renzini et al . 1994 ) or the intermittent agn activities ( ciotti & ostriker 1997 ) . figure 2 shows the result for a spiral galaxy , where the initial masses of bulge and disk are taken to be @xmath57 and @xmath58 , respectively . the co luminosity of the bulge evolves in almost the same fashion as an elliptical galaxy as above : @xmath52 increases rapidly after the birth , attains the maximum within 0.36 gyr , and , then , suddenly decreases because of the strong wind from the star - forming bulge . the co luminosity of the thus - calculated forming bulge seems insufficient to be detected as the observed luminosity of br 1202@xmath40725 , unless the bulge is much heavier than @xmath57 . moreover , we emphasize that the duration of this bright phase in @xmath52 is shorter than that obtained for ellipticals by a factor of two , and therefore , the probability to detect such co - bright phase for a bulge would be much smaller than that for elliptical galaxies . on the other hand , formation of the gaseous disk due to gas infall and star formation then proceeds mildly , and , therefore , the metal pollution of ism in the disk is slower , which results in a slower increase of the co luminosity . as a consequence , the co luminosity increases gradually and monotonically until today . also , the less - luminous phase due to the disk , following the wind phase of the bulge , is in agreement with the upper - limit observations of evans et al . ( 1996 ) and van ojik et al . ( 1997 ) . we also plot co observations for more other nearby spiral galaxies , as plotted by filled circles ( braine et al . the evolution of the co luminosity of these galaxies can be traced back by adjusting the present - day luminosity of the calculated track . the most luminous nearby spirals in co is ngc 4565 ( @xmath59 k km s@xmath10 pc@xmath40 ) . it is interesting to mention that , if the model is normalized to this galaxy , the peak co luminosity corresponding to the forming bulge phase can be still sufficient to explain the luminosity of br 1202@xmath40725 . the present study has shown that the current radio telescope facilities are capable of detecting co emission from high - redshift galaxies which experience their initial starbursts if the following two conditions are satisfied ; 1 ) the masses of systems should exceed @xmath60 , and 2 ) their evolutionary phases should be prior to the galactic wind . therefore , the detectability of co emission from high-@xmath7 galaxies is severely limited by the above two conditions . our study suggests that co emission can be hardly detected from galaxies with redshift @xmath3 without an amplification either by galaxy mergers and/or by gravitational lensing . this prescription is consistent with the observations ; co emission was detected from the high - redshift quasar br 1202@xmath40725 at @xmath5 ( ohta et al . 1996 ; omont et al . 1996 ) while not detected from the radio galaxies with @xmath6 ( evans et al . 1996 and van ojik et al . 1997 ) and quasars with redshift @xmath0 2 ( takahara et al . further , the two convincing detections of co emission from the high-@xmath7 objects at @xmath61 , iras f10214 + 4724 ( cf . radford et al . 1996 ) and the cloverleaf quasar h1413 + 135 ( barvanis et al . 1994 ) , are actually gravitationally amplified sources . the striking non - detection of high-@xmath7 galaxies in co at @xmath6 implies that most elliptical galaxies and bulges of spiral galaxies were formed before @xmath62 , or high-@xmath7 galaxies with @xmath6 observed in the optical and infrared studies may be galaxies after the epoch of galactic wind . this implication is consistent with the formation epoch ( @xmath63 ) of high-@xmath7 quasars studied by chemical properties of the broad emission - line regions ( hamann & ferland 1992 , 1993 ; hill , thompson , & elston 1993 ; elston et al . 1994 ; kawara et al . 1996 ; taniguchi et al . therefore it is strongly suggested that most host galaxies of high-@xmath7 agn were formed before @xmath62 . according to our model , it would be worth noting that quasar nuclei are hidden by the dusty clouds unless the galactic wind could expel them from the host galaxies . we also mention that any quasar nuclei are not necessarily to associate with gas - rich circumnuclear environment though this implication is in contradiction to what suggested for low-@xmath7 agn ( yamada 1994 ) . therefore , it seems very lucky that the co emission was detected from br 1202@xmath40725 at @xmath12 . finally , we revisit the important question : what is br 1202@xmath40725 ? as shown in section 3 , the unambiguous co detection from br 1202@xmath40725 is interpreted as an initial starburst galaxy which is forming either an elliptical or a bulge with mass larger than @xmath64 . the elongated ( ohta et al . 1996 ) or the double - peaked ( omont et al . 1996 ) co distribution may be understood as possible evidence for galactic wind in terms of our scenario . if it is an elliptical galaxy , its formation epoch is estimated to be @xmath65 . however , if it were a bulge former , the mass of bulge should be comparable with that of typical ellipticals . since such massive bulges are rarer by two orders of magnitude than elliptical with similar masses ( e.g. , woltier 1990 ) , the host of br 1202@xmath40725 may be an elliptical from a statistical ground . we gratefully acknowledge t. kodama and o. nakamura for kindly providing us chemical evolution program packages . our special thanks to k. ohta , t. yamada , and r. mcmahon for fruitful discussions . we also thank t. hasegawa and m. honma for useful comments . this work was financially supported in part by a grant - in - aid for the scientific research by the japanese ministry of education , culture , sports and science ( nos . 07044054 and 09640311 ) .

we present co luminosity evolution of both elliptical and spiral galaxies based on a galactic wind model and a bulge - disk model , respectively . we have found that the co luminosity peaks around the epoch of galactic wind caused by collective supernovae @xmath0 0.85 gyr after the birth of the elliptical with @xmath1 while @xmath0 0.36 gyr after the birth of the bulge with @xmath2 . after these epochs , the co luminosity decreases abruptly because the majority of molecular gas was expelled from the galaxy system as the wind . taking account of typical masses of elliptical galaxies and bulges of spiral galaxies , we suggest that co emission can be hardly detected from galaxies with redshift @xmath3 unless some amplification either by galaxy mergers and/or by gravitational lensing is working . therefore , our study explains reasonably why co emission was detected from the high - redshift quasar br 1202@xmath40725 at @xmath5 while not detected from the powerful radio galaxies with @xmath6 .

an important diagnostic of the physical state of the interstellar medium is its large - scale velocity dispersion . this parameter is however very difficult to derive , since it is in general dominated by the contribution of the systematic velocity gradients in the beam , which are not well - known . exactly face - on galaxies are ideal objects for this study , since the line - width can be attributed almost entirely to the z - velocity dispersion @xmath0 . indeed , the systematic gradients perpendicular to the plane are expected negligible ; for instance no systematic pattern associated to spiral arms have been observed in face - on galaxies ( e.g. shostak & van der kruit 1984 , dickey et al 1990 ) , implying that the z - streaming motions at the arm crossing are not predominant . in an inclined galaxy on the contrary , it is very difficult to obtain the true velocity dispersion , since the systematic motions in the plane @xmath1 ( rotation , arm streaming motions ) widen the spectra due to the finite spatial resolution of the observations ( e.g. garcia - burillo et al 1993 , vogel et al 1994 ) . nearly face - on galaxies have already been extensively studied in the atomic gas component , in order to derive the true hi velocity dispersion ( van der kruit & shostak 1982 , 1984 , shostak & van der kruit 1984 , dickey et al 1990 ) . the evolution of @xmath0 as a function of radius was derived : the velocity dispersion is remarkably constant all over the galaxy @xmath0 = 6 = @xmath2 , and only in the inner parts it increases up to 12 . the constancy of @xmath0 in the plane , and in particular in the outer parts of the galaxy disk , is not yet well understood ; it might be related to the large - scale gas stability and to the linear flaring of the plane , as is observed in the milky - way ( merrifield 1992 ) and m31 ( brinks & burton 1984 ) . in the isothermal sheet model of a thin plane , where the z - velocity dispersion @xmath3 is independent of z , the height @xmath4(r ) of the gaseous plane , if assumed self - gravitating , is @xmath5 where @xmath6 is the gas velocity dispersion , and @xmath7 the gas surface density . the density profile is then a sech@xmath8 law . but to have the gas self - gravitating , we have to assume that either there is no dark matter component , or the gas is the dark matter itself ( e.g. pfenniger et al 1994 ) . since in general the hi surface density decreases as 1/r in the outer parts of galaxies ( e.g. bosma 1981 ) , a linear flaring ( @xmath9 ) corresponds to a constant velocity dispersion with radius . on the contrary hypothesis of the gas plane embedded in an external potential of larger scale height , where the gravitational acceleration close to the plane can be approximated by @xmath10 , the z - density profile is then a gaussian : @xmath11 and the characteristic height , or gaussian scale height of the gas is : @xmath12 and @xmath13 is @xmath14 , where @xmath15 is the density in the plane of the total matter , stellar component plus dark matter component , in which the gas is embedded . if the dark component is assumed spherical , the density in the plane is dominated by the stellar component , which is distributed in an exponential disk . this hypothesis would predict an exponential flare in the gas , while the gas flares appear more linear than exponential ( e.g. merrifield 1992 , brinks & burton 1983 ) . the knowledge of their true shape is however hampered by the presence of warps . also , the flattening of the dark matter component , and its participation to the density @xmath16 in the plane , is unknown . as for the stability arguments , let us assume here the z - velocity dispersion comparable to the radial velocity dispersion , or at least their ratio constant with radius . the velocity dispersion of the gas component is self - regulated by dynamical instabilities . if the toomre q parameter for the gas @xmath17 is lower than 1 , instabilities set in , heat the medium and increase @xmath18 until @xmath19 is 1 . the critical velocity dispersion @xmath20 depends on the epicyclic frequency @xmath21 and on the gas surface density @xmath7 ; assuming again an hi surface density decreasing as 1/r in the outer parts and a flat rotation curve , where @xmath21 also varies as 1/r , then @xmath20 is constant . to maintain @xmath22 all over the outer parts , @xmath6 should also remain constant . however , the gas density gradient appears often steeper than @xmath23 and the @xmath19 parameter is increasing towards the outer parts . this has been noticed by kennicutt ( 1989 ) , who concluded that there exists some radius in every galaxy where the gas density reaches the threshold of global instability ( @xmath24 ) ; he identifies this radius to the onset of star formation in the disk . in fact , this threshold does not occur exactly at @xmath19 = 1 , but at a slightly higher value , around 1.4 , which could be due to the fact that the @xmath25 criterion is a single - fluid one , which does not take into account the coupling between gas and stars . the determination of the z - velocity dispersion in the molecular component has not yet been done . it could bring complementary insight to the hi results , since in general the center of galaxies is much better sampled through co emission ( a central hi depletion is frequent ) , and also the thickness of the h@xmath26 plane can be lower by a factor 3 or 4 than the hi layer ( case of mw , m31 , boulanger et al 1981 ) . in the case of m51 , an almost face - on galaxy ( i=20@xmath27 ) , the estimated @xmath0 determined from the co lines is surprisingly large ( up to @xmath0 = 25 in the southern arm ) once the rotation field , and even streaming - motions are taken into account , at the beam scale . an interpretation could be that the co lines are broadened by macroscopic opacity , i.e. cloud overlapping ( garcia - burillo et al 1993 ) , since such large line - widths are not observed in galaxies with less co emission . however , one could also suspect turbulent motions , generated at large - scale by gravitational instabilities or viscous shear . the level of star formation could be another factor : as for turbulence , it generally affects the molecular component more than the hi , except for very violent events like sne . but the finite inclination ( 20@xmath27 ) of m51 makes the discrimination between in - plane and z - dispersion very delicate . it is therefore necessary to investigate in more details this problem in exactly face - on galaxies , and determine whether there exist spatial variations of @xmath0 over the galaxy plane . in this paper we report molecular gas observations of two face - on galaxies ngc 628 ( m74 ) and ngc 3938 , in the co(1 - 0 ) , co(2 - 1 ) and @xmath28co lines , using the iram 30m telescope . after a brief description of the galaxy parameters in section 2 , and the observational parameters in section 3 , we derive the amplitude and the spatial variations of @xmath0 perpendicular to the plane in ngc 628 and ngc 3938 . section 5 summarises and discusses the physical interpretations . from these mass models , we have derived the epicyclic frequency as a function of radius ( this does not depend on the precise model used , as long as the rotation curve is fitted ) , and the critical velocity dispersion required for axisymmetric stability , for the stellar and gaseous components ( figures [ vrot628 ] and [ vrot3938 ] ) . the comparison with the observed vertical velocity dispersions for hi and co is clear : the observed values are most of the time larger , in particular for ngc 3938 . this means that , if the gas velocity dispersion can be considered isotropic , the toomre stability parameter in the galaxy plane is always @xmath31 , and most of the time @xmath32 2 - 3 , for ngc 3938 . for ngc 628 , @xmath19 is near 1 between 3 and 20kpc , and the threshold for star formation , @xmath33 according to kennicutt ( 1989 ) is reached at 23 kpc . this is far in the outer parts of the galaxy , since r@xmath30 = 15.5 kpc . if the vertical dispersion is lower than in the plane , as could be the case ( e.g. olling 1995 ) , than @xmath19 is even larger . the gas appears then to be quite stable , unless the coupling gas - stars has a very large effect . figures [ vrot628 ] and [ vrot3938 ] also plot the critical velocity dispersion for the stellar component , together with a fit to the observed stellar velocity dispersions , from van der kruit & freeman ( 1984 ) for ngc 628 and from bottema ( 1988 , 1993 ) for ngc 3938 . from a sample of 12 galaxies where such data are available , bottema ( 1993 ) concludes that the stellar velocity dispersion is declining exponentially as @xmath34 , as expected for an exponential disk of scale - length @xmath35 and constant thickness , as found by van der kruit & searle ( 1981 ) . since mostly the vertical stellar dispersion @xmath36 is measured , it is assumed that there is a constant ratio between the radial dispersion @xmath37 , comparable to that observed in the solar neighbourhood @xmath38 = 0.6 . this is already well above the minimum ratio required for vertical stability , i.e. @xmath38 = 0.3 ( araki 1985 , merritt & sellwood 1994 ) . within these assumptions , it can be derived that the toomre parameter for the stars @xmath39 is about constant with radius , within the optical disk ; it depends of course on the mass - to - light ratio adopted for the luminous component , and is in the range @xmath40 1 for m(stars)/l@xmath41 = 3 . figures [ vrot628 ] and [ vrot3938 ] confirm the result of almost constant @xmath39 , but with low values , especially for ngc 3938 . this could be explained , if the vertical dispersion is indeed much lower than the radial one . the minimum value for the ratio @xmath38 is 0.3 ( for stability reasons ) , so that the derived @xmath39 values displayed in figures [ vrot628 ] and [ vrot3938 ] could be multiplied by @xmath42 2 . the idea of stellar velocity dispersion regulated by gravitational instabilities appears therefore supported by the data , within the uncertainties . the most intriguing result is the large gas vertical dispersion observed for ngc 3938 , and its distribution with radius . the large corresponding @xmath19 values , that will mean comfortable stability , are difficult to reconcile with the observed large and small - scales gas instabilities : clear spiral arms are usually observed in the outer hi disks , with small - scale structure as well ( see e.g. van der hulst & sancisi 1988 , richter & sancisi 1994 ) . this is also the case here for ngc 628 showing all signs of gravitational instabilities in its outer hi disk ( kamphuis & briggs 1992 ) , and for ngc 3938 ( van der kruit & shostak 1982 ) . a possibility to reduce @xmath19 is that also the gas dispersion is anisotropic , this time the vertical one being larger than in the plane . however we will see , through comparison with gas dispersion in the plane of the galaxy ( cf next section ) that the anisotropy of gas dispersion does not appear so large . another explanation could be that the present rough calculations of the @xmath25-parameter concern only a simplified one - component stability analysis , and could be significantly modified by multi - components analysis . it has been shown ( jog & solomon 1984 , romeo 1992 , jog 1992 & 1996 ) that the coupling between several components de - stabilises every dynamical component . the apparent stability ( @xmath43 ) of the gas component might therefore not be incompatible with an instability - regulated velocity dispersion for the gas . but then , in the vertical direction , the dispersion is much higher than the minimum required for vertical stability . could this large velocity dispersion be powered by star formation ? this is not likely , at least for the majority of the hi gas well outside the optical disk , where no stellar activity is observed . a possible explanation would be to suppose that the hi is tracing a much larger amount of gas , in the form of molecular clouds , which will then be self - gravitating , with @xmath44 ( pfenniger et al 1994 ; pfenniger & combes 1994 ) . with a flat rotation curve , and a gas surface density decreasing as @xmath23 , the critical dispersion would then be constant with radius . another puzzle is the similarity of the co and hi vertical velocity dispersions . if the gas layers are indeed isothermal in z , we can deduce that both atomic and molecular layers have also similar heights . this means that the atomic and molecular components can be considered as a unique dynamical component , which can be observed under two phases , according to the local physical conditions ( density , excitation temperature , etc .. ) . the amplitudes of z - oscillations of the molecular and atomic gas are the same , only we see the gas as molecular when it is at heights lower than @xmath42 50pc . at these heights , the molecular fraction is @xmath45 ( imamura & sofue 1997 ) , which means that almost all clouds are molecular , taking into account their atomic envelope . in fact it is not clear whether we see the co or h@xmath26 formation and destruction , since we can rely only on the co tracer . also , it is possible that the density of clouds at high altitude is not enough to excite the co molecule , which means that the limit for observing co will not be coinciding with the limit for molecular presence itself . the latter is strongly suggested by the observed vertical density profiles of the h@xmath26 and hi number density : there is a sharp boundary where the apparent @xmath46 falls to zero , while we expect a smoother profile for a unique dynamical gas component . that the gas can change phase from molecular to atomic and vice - versa several times in one z - oscillation is not unexpected , since the time - scale of molecular formation and destruction is smaller than the z - oscillation period , of @xmath42 10@xmath47 yrs at the optical radius : the chemical time - scale is of the order of 10@xmath48 yrs ( leung et al 1984 , langer & graedel 1989 ) . morever , as discussed in the previous section ( _ 5.1 _ ) , the key factor controlling the presence of molecules is photodestruction , which explains why there is a column density threshold above which the gas phase turns to molecular ( elmegreen 1993 ) . this threshold could be reached at some particular height above the plane . should we expect the existence of several layers of gas at different tmperatures , and therefore different thicknesses , in galaxy planes ? in the very simple model of a diffuse and homogeneous gas , unperturbed by star - formation , we can compute the mixing time - scale of two layers at different temperatures , through atomic or molecule collisions : this is of the order of the collisional time - scale , @xmath42 10@xmath49 yrs for an average volumic density of 1 @xmath50 , and a thermal velocity of 0.3 . this is very short with respect to the z - oscillation time scale of @xmath42 10@xmath47 yrs , and therefore mixing should occur , if differential dissipation or gravitational heating is not taken into account . this simple model is of course very far from realistic . we know that the interstellar medium , atomic as well as molecular , is distributed in a hierachical ensemble of clouds , similar to a fractal . let us then consider another simple modelisation of an ideal gas where the particles are in fact the interstellar clouds , undergoing collisions ( cf oort 1954 , cowie 1980 ) . for typical clouds of 1pc size , and 10@xmath51 @xmath50 volumic density , the collisional time - scale is of the order of 10@xmath47 yrs , comparable with the vertical oscillations time - scale . this figure should not be taken too seriously , given the rough simplifications , but it corresponds to what has been known for a long time , i.e. the ensemble of clouds can not be considered as a fluid in equilibrium , since the collisional time - scale is comparable to the dynamical time , like the spiral - arm crossing time ( cf bash 1979 , kwan 1979 , casoli & combes 1982 , combes & gerin 1985 ) . if the collisions were able to redistribute the kinetic energy completely , there should be equipartition , i.e. the velocity dispersion would decrease with the mass @xmath52 of the clouds like @xmath53 . in fact the cloud - cloud relative velocities are roughly constant with mass ( between clouds of masses 100 m@xmath54 and gmcs of 10@xmath55 m@xmath54 , a ratio of 100 would be expected in velocity dispersions , which is not observed , stark 1979 ) . towards the galactic anticenter , where streaming motions should be minimised , the one - dimensional dispersion for the low - mass and giant clouds are found to be about 9.1 and 6.6 respectively , with near constancy over several orders of magnitude , and therefore no equipartition of energy ( stark 1984 ) . the almost constancy of velocity dispersions with mass requires to find other mechanisms responsible for the heating . if relatively small clouds can be heated by star - formation , supernovae , etc ... (e.g . chize & lazareff 1980 ) , the largest clouds could be heated by gravitational scattering ( jog & ostriker 1988 , gammie et al 1991 ) . in the latter mechanism , encounters between clouds with impact parameters of the order of their tidal radius in a differentially rotating disk are equivalent to a gravitational viscosity that pumps the rotational energy into random cloud kinetic energy . a 1d velocity dispersion of 5 - 7is the predicted result , independent of mass . this value is still slightly lower than the observed 1d dispersion of clouds observed in the milky way . stark & brand ( 1989 ) find 7.8from a study within 3 kpc of the sun . but collective effects , gravitational instabilities forming structures like spiral arms , etc ... have not yet been taken into account . given the high degree of structure and apparent permanent instability of the gas , they must play a major role in the heating , the source of energy being also the global rotational energy . dissipation lowering the gas dispersion continuously maintains the gas at the limit of instability , closing the feedback loop of the self - regulation ( lin & pringle 1987 , bertin & romeo 1988 ) . in the external parts of galaxies , where there is no star formation , gravitational instabilities are certainly the essential heating mechanism this again will tend to an isothermal , or more exactly isovelocity , ensemble of clouds , since the gravitational mechanism does not depend on the particle mass . the molecular or atomic gas are equivalent in this process , and should reach the same equilibrium dispersion . in the milky way , although the kinematics of gas is much complicated due to our embedded perspective , we have also the same puzzle . the velocity dispersion has been estimated through several methods , with intrinsic biases for each method , but essentially the dispersion has been estimated in the plane . only with high - latitude molecular clouds , can we have an idea of the local vertical velocity dispersion . magnani et al ( 1996 ) have recently made a compilation of more than 100 of these high - latitude clouds . the velocity dispersion of the ensemble is 5.8if seven intermediate velocity objects are excluded , and 9.9 otherwise . this is interestingly close to the values we find for ngc 628 ( 6 ) and ngc 3938 ( 8.5 ) . unfortunately there is always some doubt in the galaxy that all molecular clouds are taken into account , due to many selection effects , while the measurement is much more direct at large scale in external face - on galaxies . in fact , it has been noticed by magnani et al ( 1996 ) that there were an inconsistency between the local measured scale - height of molecular clouds ( about 60pc ) and the vertical velocity dispersion . however , they conclude in terms of a different population for the local high - latitude clouds ( hlc ) . indeed , the total mass of observed hlc is still a small fraction of the molecular surface density at the solar radius . the local gaussian scale height of the molecular component has been derived to be 58pc ( at r@xmath54 = 8.5kpc ) through a detailed data modelling by malhotra ( 1994 ) ; this is also compatible with all previous values ( dame et al 1987 , clemens et al 1988 ) . the local hi scale height is 220pc ( malhotra 1995 ) . we therefore would have expected a ratio of 3.8 between the dispersions of the h@xmath26 and hi gas , but these are very similar , within the uncertainties , which come mainly from the clumpiness of the clouds for the h@xmath26 component . if we believe the more easily determined hi dispersion of 9(malhotra 1995 ) , then the h@xmath26 dispersion is expected to be 2.4 , clearly outside of the error bars or intrinsic scatter : the value at the solar radius is estimated at 7.8by malhotra ( 1994 ) . of course , all this discussion is hampered by the fact that we discuss mainly horizontal dispersions in the case of the milky way , while the gas dispersions could well be anisotropic . this is why the present results on external face - on galaxies are more promising . the vertical gas velocity dispersion in spiral galaxies is an important parameter required to determine the flattening of the dark matter component , combined with the observation of the gas layer thickness ( cf olling 1995 , becquaert & combes 1997 ) . we have shown here that the gas dispersion does not appear very anisotropic , in the sense that the vertical dispersion is not much smaller that what has been derived in the plane of our galaxy ( for instance by the terminal velocity method , burton 1992 , malhotra 1994 ) . such vertical dispersion data should be obtained in much larger samples , to consolidate statistically this result . adler d.s . , liszt h.s . : 1989 , ap.j . 339 , 836 araki s. : 1985 , phd thesis , massachussetts institute of technology bash f.h . : 1979 , apj 233 , 524 becquaert j - f . , combes f. : 1997 , a&a in press bertin g. , romeo a. : 1988 , a&a 195 , 105 binney , j. & tremaine , s. 1987 , `` galactic dynamics '' , princeton university press , princeton , new jersey bosma a. : 1981 , aj 86 , 1971 bottema r. : 1988 , a&a 197 , 105 bottema r. : 1993 , a&a 275 , 16 boulanger f. , stark a.a . , combes f. : 1981 , a&a 93 , l1 braine j. , combes f. , casoli f. et al : 1993 a&as 97 , 887 brinks e. , burton w.b . : 1984 , a&a 141 , 195 briggs f.h . , wolfe a.m. , krumm n. , salpeter e.e . : 1980 , apj 238 , 510 burton w.b . : 1992 , in `` the galactic interstellar medium '' , saas - 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we present co(1 - 0 ) and co(2 - 1 ) observations of the two nearly face - on galaxies ngc 628 and ngc 3938 , in particular cuts along the major and minor axis . the contribution of the beam - smeared in - plane velocity gradients to the observed velocity width is quite small in the outer parts of the galaxies . this allows us to derive the velocity dispersion of the molecular gas perpendicular to the plane . we find that this dispersion is remarkably constant with radius , 6 for ngc 628 and 8.5 for ngc 3938 , and of the same order as the hi dispersion . the constancy of the value is interpreted in terms of a feedback mechanism involving gravitational instabilities and gas dissipation . the similarity of the co and hi dispersions suggests that the two components are well mixed , and are only two different phases of the same kinematical gas component . the gas can be transformed from the atomic phase to the molecular phase and vice - versa several times during a z - oscillation . psfig = + 2.0 cm

over the past decade or so two separate developments have occurred in computer science whose intersection promises to open a vast new area of research , an area extending far beyond the current boundaries of computer science . the first of these developments is the growing realization of how useful it would be to be able to control distributed systems that have little ( if any ) centralized communication , and to do so `` adaptively '' , with minimal reliance on detailed knowledge of the system s small - scale dynamical behavior . the second development is the maturing of the discipline of reinforcement learning ( rl ) . this is the branch of machine learning that is concerned with an agent who periodically receives `` reward '' signals from the environment that partially reflect the value of that agent s private utility function . the goal of an rl algorithm is to determine how , using those reward signals , the agent should update its action policy to maximize its utility @xcite . ( until our detailed discussions below , we will use the term `` reinforcement learning '' broadly , to include any algorithm of this sort , including ones that rely on detailed bayesian modeling of underlying markov processes @xcite . intuitively , one might hope that rl would help us solve the distributed control problem , since rl is adaptive , and , in particular , since it is not restricted to domains having sufficient breadths of communication . however , by itself , conventional single - agent rl does not provide a means for controlling large , distributed systems . this is true even if the system @xmath0 have centralized communication . the problem is that the space of possible action policies for such systems is too big to be searched . we might imagine as a variant using a large set of agents , each controlling only part of the system . since the individual action spaces of such agents would be relatively small , we could realistically deploy conventional rl on each one . however , now we face the central question of how to map the world utility function concerning the overall system into private utility functions for each of the agents . in particular , how should we design those private utility functions so that each agent can realistically hope to optimize its function , and at the same time the collective behavior of the agents will optimize the world utility ? we use the term `` collective intelligence '' ( coin ) to refer to any pair of a large , distributed collection of interacting computational processes among which there is little to no centralized communication or control , together with a ` world utility ' function that rates the possible dynamic histories of the collection . the central coin design problem we consider arises when the computational processes run rl algorithms : how , without any detailed modeling of the overall system , can one set the utility functions for the rl algorithms in a coin to have the overall dynamics reliably and robustly achieve large values of the provided world utility ? the benefits of an answer to this question would extend beyond the many branches of computer science , having major ramifications for many other sciences as well . section [ sec : back ] discusses some of those benefits . section [ sec : lit ] reviews previous work that has bearing on the coin design problem . section [ sec : math ] section constitutes the core of this chapter . it presents a quick outline of a promising mathematical framework for addressing this problem in its most general form , and then experimental illustrations of the prescriptions of that framework . throughout , we will use italics for emphasis , single quotes for informally defined terms , and double quotes to delineate colloquial terminology . there are many design problems that involve distributed computational systems where there are strong restrictions on centralized communication ( `` we ca nt all talk '' ) ; or there is communication with a central processor , but that processor is not sufficiently powerful to determine how to control the entire system ( `` we are nt smart enough '' ) ; or the processor is powerful enough in principle , but it is not clear what algorithm it could run by itself that would effectively control the entire system ( `` we do nt know what to think '' ) . just a few of the potential examples include : \i ) designing a control system for constellations of communication satellites or for constellations of planetary exploration vehicles ( world utility in the latter case being some measure of quality of scientific data collected ) ; \ii ) designing a control system for routing over a communication network ( world utility being some aggregate quality of service measure ) \iii ) construction of parallel algorithms for solving numerical optimization problems ( the optimization problem itself constituting the world utility ) ; \iv ) vehicular traffic control , _ e.g. _ , air traffic control , or high - occupancy toll - lanes for automobiles . ( in these problems the individual agents are humans and the associated utility functions must be of a constrained form , reflecting the relatively inflexible kinds of preferences humans possess . ) ; \v ) routing over a power grid ; \vi ) control of a large , distributed chemical plant ; \vii ) control of the elements of an amorphous computer ; \viii ) control of the elements of a ` noisy ' phased array radar ; \ix ) compute - serving over an information grid . such systems may be best controlled with an artificial coin . however , the potential usefulness of deeper understanding of how to tackle the coin design problem extends far beyond such engineering concerns . that s because the coin design problem is an inverse problem , whereas essentially all of the scientific fields that are concerned with naturally - occurring distributed systems analyze them purely as a `` forward problem . '' that is , those fields analyze what global behavior would arise from provided local dynamical laws , rather than grapple with the inverse problem of how to configure those laws to induce desired global behavior . ( indeed , the coin design problem could almost be defined as decentralized adaptive control theory for massively distributed stochastic environments . ) it seems plausible that the insights garnered from understanding the inverse problem would provide a trenchant novel perspective on those fields . just as tackling the inverse problem in the design of steam engines led to the first true understanding of the macroscopic properties of physical bodes ( aka thermodynamics ) , so may the cracking of the coin design problem may improve our understanding of many naturally - occurring coins . in addition , although the focuses of those other fields are not on the coin design problem , in that they are related to the coin design problem , that problem may be able to serve as a `` touchstone '' for all those fields . this may then reveal novel connections between the fields . as an example of how understanding the coin design problem may provide a novel perspective on other fields , consider countries with capitalist human economies . although there is no intrinsic world utility in such systems , they can still be viewed from the perspective of coins , as naturally occurring coins . for example , one can declare world utility to be a time average of the gross domestic product ( gdp ) of the country in question . ( world utility per se is not a construction internal to a human economy , but rather something defined from the outside . ) the reward functions for the human agents in this example could then be the achievements of their personal goals ( usually involving personal wealth to some degree ) . now in general , to achieve high world utility in a coin it is necessary to avoid having the agents work at cross - purposes . otherwise the system is vulnerable to economic phenomena like the tragedy of the commons ( toc ) , in which individual avarice works to lower world utility @xcite , or the liquidity trap , where behavior that helps the entire system when employed by some agents results in poor global behavior when employed by all agents @xcite . one way to avoid such phenomena is by modifying the agents utility functions . in the context of capitalist economies , this kind of effect can be achieved via punitive legislation that modifies the rewards the agents receive for engaging in certain kinds of activity . a real world example of an attempt to make just such a modification was the creation of anti - trust regulations designed to prevent monopolistic practices . in designing a coin we usually have more freedom than anti - trust regulators though , in that there is no base - line `` organic '' private utility function over which we must superimpose legislation - like incentives . rather , the entire `` psychology '' of the individual agents is at our disposal when designing a coin . this obviates the need for honesty - elicitation ( ` incentive compatible ' ) mechanisms , like auctions , which form a central component of conventional economics . accordingly , coins can differ in certain crucial respects from human economies . the precise differences the subject of current research seem likely to present many insights into the functioning of economic structures like anti - trust regulators . to continue with this example , consider the usefulness , as far as the world utility is concerned , of having ( commodity , or especially fiat ) money in the coin . formally , from a coin perspective , the use of ` money ' for trading between agents constitutes a particular class of couplings between the states and utility functions of the various agents . for example , if one agent s ` bank account ' variable goes up in a ` trade ' with another agent , then a corresponding ` bank account ' variable in that other agent must decrease to compensate . in addition to this coupling between the agents states , there is also a coupling between their utilities , if one assume that both agents will prefer to have more money rather than less , everything else being equal . however one might formally define such a ` money ' structure , we can consider what happens if it does ( or does not ) obtain for an arbitrary dynamical system , in the context of an arbitrary world utility . for some such dynamical systems and world utilities , a money structure will improve the value of that world utility . but for the same dynamics , the use of a money structure will simultaneously induce _ low levels _ of other world utilities ( a trivial example being a world utility that equals the negative of the first one ) . this raises a host of questions , like how to formally specify the most general set of world utilities that benefits significantly from using money - based private utility functions . if one is provided a world utility that is not a member of that set , then an `` economics - like '' configuration of the system is likely to result in poor performance . such a characterization of how and when money helps improve world utilities of various sorts might have important implications for conventional human economics , especially when one chooses world utility to be one of the more popular choices for social welfare function . ( see @xcite and references therein for some of the standard economics work that is most relevant to this issue . ) there are many other scientific fields that are currently under investigation from a coin - design perspective . some of them are , like economics , part of ( or at least closely related to ) the social sciences . these fields typically involve rl algorithms under the guise of human agents . an example of such a field is game theory , especially game theory of bounded rational players . as illustrated in our money example , viewing such systems from the perspective of a non - endogenous world utility , _ i.e. _ , from a coin - design perspective , holds the potential for providing novel insight into them . ( in the case of game theory , it holds the potential for leading to deeper understanding of many - player inverse stochastic game theory . ) however there are other scientific fields that might benefit from a coin - design perspective even though they study systems that do nt even involve rl algorithms . the idea here is that if we viewed such systems from an `` artificial '' teleological perspective , both in concentrating on a non - endogenous world utility and in casting the nodal elements of the system as rl algorithms , we could learn a lot about the form of the ` design space ' in which such systems live . ( just as in economics , where the individual nodal elements _ are _ rl algorithms , investigating the system using an externally imposed world utility might lead to insight . ) examples here are ecosystems ( individual genes , individuals , or species being the nodal elements ) and cells ( individual organelles in eukaryotes being the nodal elements ) . in both cases , the world utility could involve robustness of the desired equilibrium against external perturbation , efficient exploitation of free energy in the environment , etc . the following list elaborates what we mean by a coin : \1 ) there are many processors running concurrently , performing actions that affect one another s behavior . \2 ) there is little to no centralized personalized communication , _ i.e. _ , little to no behavior in which a small subset of the processors not only communicates with all the other processors , but communicates differently with each one of those other processors . any single processor s `` broadcasting '' the same information to all other processors is not precluded . \3 ) there is little to no centralized personalized control , _ i.e. _ , little to no behavior in which a small subset of the processors not only controls all the other processors , but controls each one of those other processors differently . `` broadcasting '' the same control signal to all other processors is not precluded . \4 ) there is a well - specified task , typically in the form of extremizing a utility function , that concerns the behavior of the entire distributed system . so we are confronted with the inverse problem of how to configure the system to achieve the task . the following elements characterize the sorts of approaches to coin design we are concerned with here : \5 ) the approach for tackling ( 4 ) is scalable to very large numbers of processors . \6 ) the approach for tackling ( 4 ) is very broadly applicable . in particular , it can work when little ( if any ) `` broadcasting '' as in ( 2 ) and ( 3 ) is possible . \7 ) the approach for tackling ( 4 ) involves little to no hand - tailoring . \8 ) the approach for tackling ( 4 ) is robust and adaptive , with minimal need to `` get the details exactly right or else , '' as far as the stochastic dynamics of the system is concerned . \9 ) the individual processors are running rl algorithms . unlike the other elements of this list , this one is not an _ a priori _ engineering necessity . rather , it is a reflection of the fact that rl algorithms are currently the best - understood and most mature technology for addressing the points ( 8) and ( 9 ) . there are many approaches to coin design that do not have every one of those features . these approaches constitute part of the overall field of coin design . as discussed below though , not having every feature in our list , no single one of those approaches can be extended to cover the entire breadth of the field of coin design . ( this is not too surprising , since those approaches are parts of fields whose focus is not the coin design problem per se . ) the rest of this section consists of brief presentations of some of these approaches , and in particular characterizes them in terms of our list of nine characteristics of coins and of our desiredata for their design . of the approaches we discuss , at present it is probably the ones in artificial intelligence and machine learning that are most directly applicable to coin design . however it is fairly clear how to exploit those approaches for coin design , and in that sense relatively little needs to be said about them . in contrast , as currently employed , the toolsets in the social sciences are not as immediately applicable to coin design . however , it seems likely that there is more yet to be discovered about how to exploit them for coin design . accordingly , we devote more space to those social science - based approaches here . we present an approach that holds promise for covering all nine of our desired features in section [ sec : math ] . there is an extensive body of work in ai and machine learning that is related to coin design . indeed , one of the most famous speculative works in the field can be viewed as an argument that ai should be approached as a coin design problem @xcite . much work of a more concrete nature is also closely related to the problem of coin design . as discussed in the introduction , the maturing field of reinforcement learning provides a much needed tool for the types of problems addressed by coins . because rl generally provides model - free and `` online '' learning features , it is ideally suited for the distributed environment where a `` teacher '' is not available and the agents need to learn successful strategies based on `` rewards '' and `` penalties '' they receive from the overall system at various intervals . it is even possible for the learners to use those rewards to modify _ how _ they learn @xcite . although work on rl dates back to samuel s checker player @xcite , relatively recent theoretical @xcite and empirical results @xcite have made rl one of the most active areas in machine learning . many problems ranging from controlling a robot s gait to controlling a chemical plant to allocating constrained resource have been addressed with considerable success using rl @xcite . in particular , the rl algorithms @xmath1 ( which rates potential states based on a _ value function _ ) @xcite and @xmath2learning ( which rates action - state pairs ) @xcite have been investigated extensively . a detailed investigation of rl is available in @xcite . although powerful and widely applicable , solitary rl algorithms will not perform well on large distributed heterogeneous problems in general . this is due to the very big size of the action - policy space for such problems . in addition , without centralized communication and control , how a solitary rl algorithm could run the full system at all , poorly or well , becomes a major concern . for these reasons , it is natural to consider deploying many rl algorithms rather than a single one for these large distributed problems . we will discuss the coordination issues such an approach raises in conjunction with multi - agent systems in section [ sec : mas ] and with learnability in coins in section [ sec : math ] . the field of distributed artificial intelligence ( dai ) has arisen as more and more traditional artificial intelligence ( ai ) tasks have migrated toward parallel implementation . the most direct approach to such implementations is to directly parallelize ai production systems or the underlying programming languages @xcite . an alternative and more challenging approach is to use distributed computing , where not only are the individual reasoning , planning and scheduling ai tasks parallelized , but there are _ different modules _ with different such tasks , concurrently working toward a common goal @xcite . in a dai , one needs to ensure that the task has been modularized in a way that improves efficiency . unfortunately , this usually requires a central controller whose purpose is to allocate tasks and process the associated results . moreover , designing that controller in a traditional ai fashion often results in brittle solutions . accordingly , recently there has been a move toward both more autonomous modules and fewer restrictions on the interactions among the modules @xcite . despite this evolution , dai maintains the traditional ai concern with a pre - fixed set of _ particular _ aspects of intelligent behavior ( _ e.g. _ reasoning , understanding , learning etc . ) rather than on their _ cumulative _ character . as the idea that intelligence may have more to do with the interaction among components started to take shape @xcite , focus shifted to concepts ( _ e.g. _ , multi - agent systems ) that better incorporated that idea @xcite . the field of multi - agent systems ( mas ) is concerned with the interactions among the members of such a set of agents @xcite , as well as the inner workings of each agent in such a set ( _ e.g. _ , their learning algorithms ) @xcite . as in computational ecologies and computational markets ( see below ) , a well - designed mas is one that achieves a global task through the actions of its components . the associated design steps involve @xcite : 1 . decomposing a global task into distributable subcomponents , yielding tractable tasks for each agent ; 2 . establishing communication channels that provide sufficient information to each of the agents for it to achieve its task , but are not too unwieldly for the overall system to sustain ; and 3 . coordinating the agents in a way that ensures that they cooperate on the global task , or at the very least does not allow them to pursue conflicting strategies in trying to achieve their tasks . step ( 3 ) is rarely trivial ; one of the main difficulties encountered in mas design is that agents act selfishly and artificial cooperation structures have to be imposed on their behavior to enforce cooperation @xcite . an active area of research , which holds promise for addressing parts the coin design problem , is to determine how selfish agents `` incentives '' have to be engineered in order to avoid the tragedy of the commons ( toc ) @xcite . ( this work draws on the economics literature , which we review separately below . ) when simply providing the right incentives is not sufficient , one can resort to strategies that actively induce agents to cooperate rather than act selfishly . in such cases coordination @xcite , negotiations @xcite , coalition formation @xcite or contracting @xcite among agents may be needed to ensure that they do not work at cross purposes . unfortunately , all of these approaches share with dai and its offshoots the problem of relying excessively on hand - tailoring , and therefore being difficult to scale and often nonrobust . in addition , except as noted in the next subsection , they involve no rl , and therefore the constituent computational elements are usually not as adaptive and robust as we would like . because it neither requires explicit modeling of the environment nor having a `` teacher '' that provides the `` correct '' actions , the approach of having the individual agents in a mas use rl is well - suited for mas s deployed in domains where one has little knowledge about the environment and/or other agents . there are two main approaches to designing such mas s : + ( i ) one has ` solipsistic agents ' that do nt know about each other and whose rl rewards are given by the performance of the entire system ( so the joint actions of all other agents form an `` inanimate background '' contributing to the reward signal each agent receives ) ; + ( ii ) one has ` social agents ' that explicitly model each other and take each others actions into account . both ( i ) and ( ii ) can be viewed as ways to ( try to ) coordinate the agents in a mas in a robust fashion . * solipsistic agents : * mas s with solipsistic agents have been successfully applied to a multitude of problems @xcite . generally , these schemes use rl algorithms similar to those discussed in section [ sec : control ] . however much of this work lacks a well - defined global task or broad applicability ( _ e.g. _ , @xcite ) . more generally , none of the work with solipsistic agents scales well . ( as illustrated in our experiments on the `` bar problem '' , recounted below . ) the problem is that each agent must be able to discern the effect of its actions on the overall performance of the system , since that performance constitutes its reward signal . as the number of agents increases though , the effects of any one agent s actions ( signal ) will be swamped by the effects of other agents ( noise ) , making the agent unable to learn well , if at all . ( see the discussion below on learnability . ) in addition , of course , solipsistic agents can not be used in situations lacking centralized calculation and broadcast of the single global reward signal . * social agents : * mas s whose agents take the actions of other agents into account synthesize rl with game theoretic concepts ( _ e.g. _ , nash equilibrium ) . they do this to try to ensure that the overall system both moves toward achieving the overall global goal and avoids often deleterious oscillatory behavior @xcite . to that end , the agents incorporate internal mechanisms that actively model the behavior of other agents . in section [ sec : bar ] , we discuss a situation where such modeling is necessarily self - defeating . more generally , this approach usually involves extensive hand - tailoring for the problem at hand . some human economies provides examples of naturally occurring systems that can be viewed as a ( more or less ) well - performing coin . the field of economics provides much more though . both empirical economics ( _ e.g. _ , economic history , experimental economics ) and theoretical economics ( _ e.g. _ , general equilibrium theory @xcite , theory of optimal taxation @xcite ) provide a rich literature on strategic situations where many parties interact . in fact , much of the entire field of economics can be viewed as concerning how to maximize certain constrained kinds of world utilities , when there are certain ( very strong ) restrictions on the individual agents and their interactions , and in particular when we have limited freedom in setting either the utility functions of those agents or modifying their rl algorithms in any other way . in this section we summarize just two economic concepts , both of which are very closely related to coins , in that they deal with how a large number of interacting agents can function in a stable and efficient manner : general equilibrium theory and mechanism design . we then discuss general attempts to apply those concepts to distributed computational problems . we follow this with a discussion of game theory , and then present a particular celebrated toy - world problem that involves many of these issues . often the first version of `` equilibrium '' that one encounters in economics is that of supply and demand in single markets : the price of the market s good is determined by where the supply and demand curves for that good intersect . in cases where there is interaction among multiple markets however , even when there is no production but only trading , one can not simply determine the price of each market s good individually , as both the supply and demand for each good depends on the supply / demand of other goods . considering the price fluctuations across markets leads to the concept of ` general equilibrium ' , where prices for each good are determined in such a way to ensure that all markets ` clear ' @xcite . intuitively , this means that prices are set so the total supply of each good is equal to the demand for that good . the existence of such an equilibrium , proven in @xcite , was first postulated by leon walras @xcite . a mechanism that calculates the equilibrium ( _ i.e. _ , ` market - clearing ' ) prices now bears his name : the walrasian auctioner . in general , for an arbitrary goal for the overall system , there is no reason to believe that having markets clear achieves that goal . in other words , there is no _ a priori _ reason why the general equilibrium point should maximize one s provided world utility function . however , consider the case where one s goal for the overall system is in fact that the markets clear . in such a context , examine the case where the interactions of real - world agents will induce the overall system to adopt the general equilibrium point , so long as certain broad conditions hold . then if we can impose those conditions , we can cause the overall system to behave in the manner we wish . however general equilibrium theory is not sufficient to establish those `` broad conditions '' , since it says little about real - world agents . in particular , general equilibrium theory suffers from having no temporal aspect ( _ i.e. _ , no dynamics ) and from assuming that all the agents are perfectly rational . another shortcoming of general equilibrium theory as a model of real - world systems is that despite its concerning prices , it does not readily accommodate the full concept of money @xcite . of the three main roles money plays in an economy ( medium of exchange in trades , store of value for future trades , and unit of account ) none are essential in a general equilibrium setting . the unit of account aspect is not needed as the bookkeeping is performed by the walrasian auctioner . since the supplies and demands are matched directly there is no need to facilitate trades , and thus no role for money as a medium of exchange . and finally , as the system reaches an equilibrium in one step , through the auctioner , there is no need to store value for future trading rounds @xcite . the reason that money is not needed can be traced to the fact that there is an `` overseer '' with global information who guides the system . if we remove the centralized communication and control exerted by this overseer , then ( as in a real economy ) agents will no longer know the exact details of the overall economy . they will be forced to makes guesses as in any learning system , and the differences in those guesses will lead to differences in their actions @xcite . such a decentralized learning - based system more closely resembles a coin than does a conventional general equilibrium system . in contrast to general equilibrium systems , the three main roles money plays in a human economy are crucial to the dynamics of such a decentralized system @xcite . this comports with the important effects in coins of having the agents utility functions involve money ( see background section above ) . even if there exists centralized communication so that we are nt considering a full - blown coin , if there is no centralized walras - like control , it is usually highly non - trivial to induce the overall system to adopt the general equilibrium point . one way to try to do so is via an auction . ( this is the approach usually employed in computational markets see below . ) along with optimal taxation and public good theory @xcite , the design of auctions is the subject of the field of mechanism design . more generally , mechanism design is concerned with the incentives that must be applied to any set of agents that interact and exchange goods @xcite in order to get those agents to exhibit desired behavior . usually that desired behavior concerns pre - specified utility functions of some sort for each of the individual agents . in particular , mechanism design is usually concerned with incentive schemes which induce ` ( pareto ) efficient ' ( or ` pareto optimal ' ) allocations in which no agent can be made better off without hurting another agent @xcite . one particularly important type of such an incentive scheme is an auction . when many agents interact in a common environment often there needs to be a structure that supports the exchange of goods or information among those agents . auctions provide one such ( centralized ) structure for managing exchanges of goods . for example , in the english auction all the agents come together and ` bid ' for a good , and the price of the good is increased until only one bidder remains , who gets the good in exchange for the resource bid . as another example , in the dutch auction the price of a good is decreased until one buyer is willing to pay the current price . all auctions perform the same task : match supply and demand . as such , auctions are one of the ways in which price equilibration among a set of interacting agents ( perhaps an equilibration approximating general equilibrium , perhaps not ) can be achieved . however , an auction mechanism that induces pareto efficiency does not necessarily maximize some other world utility . for example , in a transaction in an english auction both the seller and the buyer benefit . they may even have arrived at an allocation which is efficient . however , in that the winner may well have been willing to pay more for the good , such an outcome may confound the goal of the market designer , if that designer s goal is to maximize revenue . this point is returned to below , in the context of computational economics . ` computational economies ' are schemes inspired by economics , and more specifically by general equilibrium theory and mechanism design theory , for managing the components of a distributed computational system . they work by having a ` computational market ' , akin to an auction , guide the interactions among those components . such a market is defined as any structure that allows the components of the system to exchange information on relative valuation of resources ( as in an auction ) , establish equilibrium states ( _ e.g. _ , determine market clearing prices ) and exchange resources ( _ i.e. _ , engage in trades ) . such computational economies can be used to investigate real economies and biological systems @xcite . they can also be used to design distributed computational systems . for example , such computational economies are well - suited to some distributed resource allocation problems , where each component of the system can either directly produce the `` goods '' it needs or acquire them through trades with other components . computational markets often allow for far more heterogeneity in the components than do conventional resource allocation schemes . furthermore , there is both theoretical and empirical evidence suggesting that such markets are often able to settle to equilibrium states . for example , auctions find prices that satisfy both the seller and the buyer which results in an increase in the utility of both ( else one or the other would not have agreed to the sale ) . assuming that all parties are free to pursue trading opportunities , such mechanisms move the system to a point where all possible bilateral trades that could improve the utility of both parties are exhausted . now restrict attention to the case , implicit in much of computational market work , with the following characteristics : first , world utility can be expressed as a monotonically increasing function @xmath3 where each argument @xmath4 of @xmath3 can in turn be interpreted as the value of a pre - specified utility function @xmath5 for agent @xmath4 . second , each of those @xmath5 is a function of an @xmath4-indexed ` goods vector ' @xmath6 of the non - perishable goods `` owned '' by agent @xmath4 . the components of that vector are @xmath7 , and the overall system dynamics is restricted to conserve the vector @xmath8 . ( there are also some other , more technical conditions . ) as an example , the resource allocation problem can be viewed as concerning such vectors of `` owned '' goods . due to the second of our two conditions , one can integrate a market - clearing mechanism into any system of this sort . due to the first condition , since in a market equilibrium with non - perishable goods no ( rational ) agent ends up with a value of its utility function lower than the one it started with , the value of the world utility function must be higher at equilibrium than it was initially . in fact , so long as the individual agents are smart enough to avoid all trades in which they do not benefit , any computational market can only improve this kind of world utility , even if it does not achieve the market equilibrium . ( see the discussion of `` weak triviality '' below . ) this line of reasoning provides one of the main reasons to use computational markets when they can be applied . conversely , it underscores one of the major limitations of such markets : starting with an arbitrary world utility function with arbitrary dynamical restrictions , it may be quite difficult to cast that function as a monotonically increasing @xmath3 taking as arguments a set of agents goods - vector - based utilities @xmath5 , if we require that those @xmath5 be well - enough behaved that we can reasonably expect the agents to optimize them in a market setting . one example of a computational economy being used for resource allocation is huberman and clearwater s use of a double blind auction to solve the complex task of controlling the temperature of a building . in this case , each agent ( individual temperature controller ) bids to buy or sell cool or warm air . this market mechanism leads to an equitable temperature distribution in the system @xcite . other domains where market mechanisms were successfully applied include purchasing memory in an operating systems @xcite , allocating virtual circuits @xcite , `` stealing '' unused cpu cycles in a network of computers @xcite , predicting option futures in financial markets @xcite , and numerous scheduling and distributed resource allocation problems @xcite . computational economics can also be used for tasks not tightly coupled to resource allocation . for example , following the work of maes @xcite and ferber @xcite , baum shows how by using computational markets a large number of agents can interact and cooperate to solve a variant of the blocks world problem @xcite . viewed as candidate coins , all market - based computational economics fall short in relying on both centralized communication and centralized control to some degree . often that reliance is extreme . for example , the systems investigated by baum not only have the centralized control of a market , but in addition have centralized control of all other non - market aspects of the system . ( indeed , the market is secondary , in that it is only used to decide which single expert among a set of candidate experts gets to exert that centralized control at any given moment ) . there has also been doubt cast on how well computational economies perform in practice @xcite , and they also often require extensive hand - tailoring in practice . finally , return to consideration of a world utility function that is a monotonically increasing function @xmath9 whose arguments are the utilities of the agents . in general , the maximum of such a world utility function will be a pareto optimal point . so given the utility functions of the agents , by considering all such @xmath9 we map out an infinite set @xmath10 of pareto optimal points that maximize _ some _ such world utility function . ( @xmath10 is usually infinite even if we only consider maximizing those world utilities subject to an overall conservation of goods constraint . ) now the market equilibrium is a pareto optimal point , and therefore lies in @xmath10 . but it is only one element of @xmath10 . moreover , it is usually set in full by the utilities of the agents , in concert with the agents initial endowments . in particular , it is independent of the world utility . in general then , given the utilities of the agents and a world utility @xmath9 , there is no _ a priori _ reason to believe that the particular element in @xmath10 picked out by the auction is the point that maximizes that particular world utility . this subtlety is rarely addressed in the work on using computational markets to achieve a global goal . it need not be uncircumventable however . for example , one obvious idea would be to to try to distort the agents _ perceptions _ of their utility functions and/or initial endowments so that the resultant market equilibrium has a higher value of the world utility at hand . game theory is the branch of mathematics concerned with formalized versions of `` games '' , in the sense of chess , poker , nuclear arms races , and the like @xcite . it is perhaps easiest to describe it by loosely defining some of its terminology , which we do here and in the next subsection . the simplest form of a game is that of ` non - cooperative single - stage extensive - form ' game , which involves the following situation : there are two or more agents ( called ` players ' in the literature ) , each of which has a pre - specified set of possible actions that it can follow . ( a ` finite ' game has finite sets of possible actions for all the players . ) in addition , each agent @xmath4 has a utility function ( also called a ` payoff matrix ' for finite games ) . this maps any ` profile ' of the action choices of all agents to an associated utility value for agent @xmath4 . ( in a ` zero - sum ' game , for every profile , the sum of the payoffs to all the agents is zero . ) the agents choose their actions in a sequence , one after the other . the structure determining what each agent knows concerning the action choices of the preceding agents is known as the ` information set . ' games in which each agent knows exactly what the preceding ( ` leader ' ) agent did are known as ` stackelberg games ' . ( a variant of such a game is considered in our experiments below . see also @xcite . ) in a ` multi - stage ' game , after all the agents choose their first action , each agent is provided some information concerning what the other agents did . the agent uses this information to choose its next action . in the usual formulation , each agent gets its payoff at the end of all of the game s stages . an agent s ` strategy ' is the rule it elects to follow mapping the information it has at each stage of a game to its associated action . it is a ` pure strategy ' if it is a deterministic rule . if instead the agent s action is chosen by randomly sampling from a distribution , that distribution is known a ` mixed strategy ' . note that an agent s strategy concerns @xmath11 possible sequences of provided information , even any that can not arise due to the strategies of the other agents . any multi - stage extensive - form game can be converted into a ` normal form ' game , which is a single - stage game in which each agent is ignorant of the actions of the other agents , so that all agents choose their actions `` simultaneously '' . this conversion is acieved by having the `` actions '' of each agent in the normal form game correspond to an entire strategy in the associated multi - stage extensive - form game . the payoffs to all the agents in the normal form game for a particular strategy profile is then given by the associated payoff matrices of the multi - stage extensive form - game . a ` solution ' to a game , or an ` equilibrium ' , is a profile in which every agent behaves `` rationally '' . this means that every agent s choice of strategy optimizes its utility subject to a pre - specified set of conditions . in conventional game theory those conditions involve , at a minimum , perfect knowledge of the payoff matrices of all other players , and often also involve specification of what strategies the other agents adopted and the like . in particular , a ` nash equilibrium ' is a a profile where each agent has chosen the best strategy it can , _ given the choices of the other agents_. a game may have no nash equilibria , one equilibrium , or many equilibria in the space of pure strategies . a beautiful and seminal theorem due to nash proves that every game has at least one nash equilibrium in the space of mixed strategies @xcite . there are several different reasons one might expect a game to result in a nash equilibrium . one is that it is the point that perfectly rational bayesian agents would adopt , assuming the probability distributions they used to calculate expected payoffs were consistent with one another @xcite . a related reason , arising even in a non - bayesian setting , is that a nash equilibrium equilibrium provides `` consistent '' predictions , in that if all parties predict that the game will converge to a nash equilibrium , no one will benefit by changing strategies . having a consistent prediction does not ensure that all agents payoffs are maximized though . the study of small perturbations around nash equilibria from a stochastic dynamics perspective is just one example of a ` refinement ' of nash equilibrium , that is a criterion for selecting a single equilibrium state when more than one is present @xcite . in cooperative game theory the agents are able to enter binding contracts with one another , and thereby coordinate their strategies . this allows the agents to avoid being `` stuck '' in nash equilibria that are pareto inefficient , that is being stuck at equilibrium profiles in which all agents would benefit if only they could agree to all adopt different strategies , with no possibility of betrayal . characteristic function _ of a game involves subsets ( ` coalitions ' ) of agents playing the game . for each such subset , it gives the sum of the payoffs of the agents in that subset that those agents can guarantee if they coordinate their strategies . an @xmath12 is a division of such a guaranteed sum among the members of the coalition . it is often the case that for a subset of the agents in a coalition one imputation @xmath13 another , meaning that under threat of leaving the coalition that subset of agents can demand the first imputation rather than the second . so the problem each agent @xmath4 is confronted with in a cooperative game is which set of other agents to form a coalition with , given the characteristic function of the game and the associated imputations @xmath4 can demand of its partners . there are several different kinds of solution for cooperative games that have received detailed study , varying in how the agents address this problem of who to form a coalition with . some of the more popular are the ` core ' , the ` shapley value ' , the ` stable set solution ' , and the ` nucleolus ' . in the real world , the actual underlying game the agents are playing does not only involve the actions considered in cooperative game theory s analysis of coalitions and imputations . the strategies of that underlying game also involve bargaining behavior , considerations of trying to cheat on a given contract , bluffing and threats , and the like . in many respects , by concentrating on solutions for coalition formation and their relation with the characteristic function , cooperative game theory abstracts away these details of the true underlying game . conversely though , progress has recently been made in understanding how cooperative games can arise from non - cooperative games , as they must in the real world @xcite . not surprisingly , game theory has come to play a large role in the field of multi - agent systems . in addition , due to darwinian natural selection , one might expect game theory to be quite important in population biology , in which the `` utility functions '' of the individual agents can be taken to be their reproductive fitness . as it turns out , there is an entire subfield of game theory concerned with this connection with population biology , called ` evolutionary game theory ' @xcite . to introduce evolutionary game theory , consider a game in which all players share the same space of possible strategies , and there is an additional space of possible ` attribute vectors ' that characterize an agent , along with a probability distribution @xmath14 across that new space . ( examples of attributes in the physical world could be things like size , speed , etc . ) we select a set of agents to play a game by randomly sampling @xmath14 . those agents attribute vectors jointly determine the payoff matrices of each of the individual agents . ( intuitively , what benefit accrues to an agent for taking a particular action depends on its attributes and those of the other agents . ) however each agent @xmath4 has limited information concerning both its attribute vector and that of the other players in the game , information encapsulated in an ` information structure ' . the information structure specifies how much each agent knows concerning the game it is playing . in this context , we enlarge the meaning of the term `` strategy '' to not just be a mapping from information sets and the like to actions , but from entire information structures to actions . in addition to the distribution @xmath14 over attribute vectors , we also have a distribution over strategies , @xmath15 . a strategy @xmath16 is a ` population strategy ' if @xmath15 is a delta function about @xmath16 . intuitively , we have a population strategy when each animal in a population `` follows the same behavioral rules '' , rules that take as input what the animal is able to discern about its strengths and weakness relative to those other members of the population , and produce as output how the animal will act in the presence of such animals . given @xmath14 , a population strategy centered about @xmath16 , and its own attribute vector , any player @xmath4 in the support of @xmath14 has an expected payoff for any strategy it might adopt . when @xmath4 s payoff could not improve if it were to adopt any strategy other than @xmath16 , we say that @xmath16 is ` evolutionary stable ' . intuitively , an evolutionary stable strategy is one that is stable with respect to the introduction of mutants into the population . now consider a sequence of such evolutionary games . interpret the payoff that any agent receives after being involved in such a game as the ` reproductive fitness ' of that agent , in the biological sense . so the higher the payoff the agent receives , in comparison to the fitnesses of the other agents , the more `` offspring '' it has that get propagated to the next game . in the continuum - time limit , where games are indexed by the real number @xmath17 , this can be formalized by a differential equation . this equation specifies the derivative of @xmath18 evaluated for each agent @xmath4 s attribute vector , as a montonically increasing function of the relative difference between the payoff of @xmath4 and the average payoff of all the agents . ( we also have such an equation for @xmath15 . ) the resulting dynamics is known as ` replicator dynamics ' , with an evolutionary stable population strategy , if it exists , being one particular fixed point of the dynamics . now consider removing the reproductive aspect of evolutionary game theory , and instead have each agent propagate to the next game , with `` memory '' of the events of the preceding game . furthermore , allow each agent to modify its strategy from one game to the next by `` learning '' from its memory of past games , in a bounded rational manner . the field of learning in games is concerned with exactly such situations @xcite . most of the formal work in this field involves simple models for the learning process of the agents . for example , in ` ficticious play ' @xcite , in each successive game , each agent @xmath4 adopts what would be its best strategy if its opponents chose their strategies according to the empirical frequency distribution of such strategies that @xmath4 has encountered in the past . more sophisticated versions of this work employ simple bayesian learning algorithms , or re - inventions of some of the techniques of the rl community @xcite . typically in learning in games one defines a payoff to the agent for a sequence of games , for example as a discounted sum of the payoffs in each of the constituent games . within this framework one can study the long term effects of strategies such as cooperation and see if they arise naturally and if so , under what circumstances . many aspects of real world games that do not occur very naturally otherwise arise spontaneously in these kinds of games . for example , when the number of games to be played is not pre - fixed , it may behoove a particular agent @xmath4 to treat its opponent better than it would otherwise , since @xmath4 @xmath19 have to rely on that other agent s treating it well in the future , if they end up playing each other again . this framework also allows us to investigate the dependence of evolving strategies on the amount of information available to the agents @xcite ; the effect of communication on the evolution of cooperation @xcite ; and the parallels between auctions and economic theory @xcite . in many respects , learning in games is even more relevant to the study of coins than is traditional game theory . however it suffers from the same major shortcoming ; it is almost exclusively focused on the forward problem rather than the inverse problem . in essence , coin design is the problem of @xmath20 game theory . the `` el farol '' bar problem and its variants provide a clean and simple testbed for investigating certain kinds of interactions among agents @xcite . in the original version of the problem , which arose in economics , at each time step ( each `` night '' ) , each agent needs to decide whether to attend a particular bar . the goal of the agent in making this decision depends on the total attendance at the bar on that night . if the total attendance is below a preset capacity then the agent should have attended . conversely , if the bar is overcrowded on the given night , then the agent should not attend . ( because of this structure , the bar problem with capacity set to @xmath21 of the total number of agents is also known as the ` minority game ' ; each agent selects one of two groups at each time step , and those that are in the minority have made the right choice ) . the agents make their choices by predicting ahead of time whether the attendance on the current night will exceed the capacity and then taking the appropriate course of action . what makes this problem particularly interesting is that it is impossible for each agent to be perfectly `` rational '' , in the sense of correctly predicting the attendance on any given night . this is because if most agents predict that the attendance will be low ( and therefore decide to attend ) , the attendance will actually high , while if they predict the attendance will be high ( and therefore decide not to attend ) the attendance will be low . ( in the language of game theory , this essentially amounts to the property that there are no pure strategy nash equilibria @xcite . ) alternatively , viewing the overall system as a coin , it has a prisoner s dilemma - like nature , in that `` rational '' behavior by all the individual agents thwarts the global goal of maximizing total enjoyment ( defined as the sum of all agents enjoyment and maximized when the bar is exactly at capacity ) . this frustration effect is similar to what occurs in spin glasses in physics , and makes the bar problem closely related to the physics of emergent behavior in distributed systems @xcite . researchers have also studied the dynamics of the bar problem to investigate economic properties like competition , cooperation and collective behavior and especially their relationship to market efficiency @xcite . properly speaking , biological systems do not involve utility functions and searches across them with rl algorithms . however it has long been appreciated that there are many ways in which viewing biological systems as involving searches over such functions can lead to deeper understanding of them @xcite . conversely , some have argued that the mechanism underlying biological systems can be used to help design search algorithms @xcite . these kinds of reasoning which relate utility functions and biological systems have traditionally focussed on the case of a single biological system operating in some external environment . if we extend this kind of reasoning , to a set of biological systems that are co - evolving with one another , then we have essentially arrived at biologically - based coins . this section discusses some of how previous work in the literature bears on this relationship between coins and biology . the fields of population biology and ecological modeling are concerned with the large - scale `` emergent '' processes that govern the systems that consist of many ( relatively ) simple entities interacting with one another @xcite . as usually cast , the `` simple entities '' are members of one or more species , and the interactions are some mathematical abstraction of the process of natural selection as it occurs in biological systems ( involving processes like genetic reproduction of various sorts , genotype - phenotype mappings , inter and intra - species competitions for resources , etc . ) . population biology and ecological modeling in this context addresses questions concerning the dynamics of the resultant ecosystem , and in particular how its long - term behavior depends on the details of the interactions between the constituent entities . broadly construed , the paradigm of ecological modeling can even be broadened to study how natural selection and self - regulating feedback creates a stable planet - wide ecological environment gaia @xcite . the underlying mathematical models of other fields can often be usefully modified to apply to the kinds of systems population biology is interested in @xcite . ( see also the discussion in the game theory subsection above . ) conversely , the underlying mathematical models of population biology and ecological modeling can be applied to other non - biological systems . in particular , those models shed light on social issues such as the emergence of language or culture , warfare , and economic competition @xcite . they also can be used to investigate more abstract issues concerning the behavior of large complex systems with many interacting components @xcite . going a bit further afield , an approach that is related in spirit to ecological modeling is ` computational ecologies ' . these are large distributed systems where each component of the system s acting ( seemingly ) independently results in complex global behavior . those components are viewed as constituting an `` ecology '' in an abstract sense ( although much of the mathematics is not derived from the traditional field of ecological modeling ) . in particular , one can investigate how the dynamics of the ecology is influenced by the information available to each component and how cooperation and communication among the components affects that dynamics @xcite . although in some ways the most closely related to coins of the current ecology - inspired research , the field of computational ecologies has some significant shortcomings if one tries to view it as a full science of coins . in particular , it suffers from not being designed to solve the inverse problem of how to configure the system so as to arrive at a particular desired dynamics . this is a difficulty endemic to the general program of equating ecological modeling and population biology with the science of coins . these fields are primarily concerned with the `` forward problem '' of determining the dynamics that arises from certain choices of the underlying system . unless one s desired dynamics is sufficiently close to some dynamics that was previously catalogued ( during one s investigation of the forward problem ) , one has very little information on how to set up the components and their interactions to achieve that desired dynamics . in addition , most of the work in these fields does not involve rl algorithms , and viewed as a context in which to design coins suffers from a need for hand - tailoring , and potentially lack of robustness and scalability . the field of ` swarm intelligence ' is concerned with systems that are modeled after social insect colonies , so that the different components of the system are queen , worker , soldier , etc . it can be viewed as ecological modeling in which the individual entities have extremely limited computing capacity and/or action sets , and in which there are very few types of entities . the premise of the field is that the rich behavior of social insect colonies arises not from the sophistication of any individual entity in the colony , but from the interaction among those entities . the objective of current research is to uncover kinds of interactions among the entity types that lead to pre - specified behavior of some sort . more speculatively , the study of social insect colonies may also provide insight into how to achieve learning in large distributed systems . this is because at the level of the individual insect in a colony , very little ( or no ) learning takes place . however across evolutionary time - scales the social insect species as a whole functions as if the various individual types in a colony had `` learned '' their specific functions . the `` learning '' is the direct result of natural selection . ( see the discussion on this topic in the subsection on ecological modeling . ) swarm intelligences have been used to adaptively allocate tasks in a mail company @xcite , solve the traveling salesman problem @xcite and route data efficiently in dynamic networks @xcite among others . despite this , such intelligences do not really constitute a general approach to designing coins . there is no general framework for adapting swarm intelligences to maximize particular world utility functions . accordingly , such intelligences generally need to be hand - tailored for each application . and after such tailoring , it is often quite a stretch to view the system as `` biological '' in any sense , rather than just a simple and _ a priori _ reasonable modification of some previously deployed system . the two main objectives of artificial life , closely related to one another , are understanding the abstract functioning and especially the origin of terrestrial life , and creating organisms that can meaningfully be called `` alive '' @xcite . the first objective involves formalizing and abstracting the mechanical processes underpinning terrestrial life . in particular , much of this work involves various degrees of abstraction of the process of self - replication @xcite . some of the more real - world - oriented work on this topic involves investigating how lipids assemble into more complex structures such as vesicles and membranes , which is one of the fundamental questions concerning the origin of life @xcite . many computer models have been proposed to simulate this process , though most suffer from overly simplifying the molecular morphology . more generally , work concerned with the origin of life can constitute an investigation of the functional self - organization that gives rise to life @xcite . in this regard , an important early work on functional self - organization is the _ lambda calculus _ , which provides an elegant framework ( recursively defined functions , lack of distinction between object and function , lack of architectural restrictions ) for studying computational systems @xcite . this framework can be used to develop an artificial chemistry `` function gas '' that displays complex cooperative properties @xcite . the second objective of the field of artificial life is less concerned with understanding the details of terrestrial life per se than of using terrestrial life as inspiration for how to design living systems . for example , motivated by the existence ( and persistence ) of computer viruses , several workers have tried to design an immune system for computers that will develop `` antibodies '' and handle viruses both more rapidly and more efficiently than other algorithms @xcite . more generally , because we only have one sampling point ( life on earth ) , it is very difficult to precisely formulate the process by which life emerged . by creating an artificial world inside a computer however , it is possible to study far more general forms of life @xcite . see also @xcite where the argument is presented that the richest way of approaching the issue of defining `` life '' is phenomenologically , in terms of self-@xmath22similar scaling properties of the system . cellular automata can be viewed as digital abstractions of physical gases @xcite . formally , they are discrete - time recurrent neural nets where the neurons live on a grid , each neuron has a finite number of potential states , and inter - neuron connections are ( usually ) purely local . ( see below for a discussion of recurrent neural nets . ) so the state update rule of each neuron is fixed and local , the next state of a neuron being a function of the current states of it and of its neighboring elements . the state update rule of ( all the neurons making up ) any particular cellular automaton specifies the mapping taking the initial configuration of the states of all of its neurons to the final , equilibrium ( perhaps strange ) attractor configuration of all those neurons . so consider the situation where we have a desired such mapping , and want to know an update rule that induces that mapping . this is a search problem , and can be viewed as similar to the inverse problem of how to design a coin to achieve a pre - specified global goal , albeit a `` coin '' whose nodal elements do not use rl algorithms . genetic algorithms are a special kind of search algorithm , based on analogy with the biological process of natural selection via recombination and mutation of a genome @xcite . although genetic algorithms ( and ` evolutionary computation ' in general ) have been studied quite extensively , there is no formal theory justifying genetic algorithms as search algorithms @xcite and few empirical comparisons with other search techniques . one example of a well - studied application of genetic algorithms is to ( try to ) solve the inverse problem of finding update rules for a cellular automaton that induce a pre - specified mapping from its initial configuration to its attractor configuration . to date , they have used this way only for extremely simple configuration mappings , mappings which can be trivially learned by other kinds of systems . despite the simplicity of these mappings , the use of genetic algorithms to try to train cellular automata to exhibit them has achieved little success @xcite . equilibrium statistical physics is concerned with the stable state character of large numbers of very simple physical objects , interacting according to well - specified local deterministic laws , with probabilistic noise processes superimposed @xcite . typically there is no sense in which such systems can be said to have centralized control , since all particles contribute comparably to the overall dynamics . aside from mesoscopic statistical physics , the numbers of particles considered are usually huge ( _ e.g. _ , @xmath23 ) , and the particles themselves are extraordinarily simple , typically having only a few degrees of freedom . moreover , the noise processes usually considered are highly restricted , being those that are formed by `` baths '' , of heat , particles , and the like . similarly , almost all of the field restricts itself to deterministic laws that are readily encapsulated in hamilton s equations ( schrodinger s equation and its field - theoretic variants for quantum statistical physics ) . in fact , much of equilibrium statistical physics is nt even concerned with the dynamic laws by themselves ( as for example is stochastic markov processes ) . rather it is concerned with invariants of those laws ( _ e.g. _ , energy ) , invariants that relate the states of all of the particles . trivially then , deterministic laws without such readily - discoverable invariants are outside of the purview of much of statistical physics . one potential use of statistical physics for coins involves taking the systems that statistical physics analyzes , especially those analyzed in its condensed matter variant ( _ e.g. _ , spin glasses @xcite ) , as simplified models of a class of coins . this approach is used in some of the analysis of the bar problem ( see above ) . it is used more overtly in ( for example ) the work of galam @xcite , in which the equilibrium coalitions of a set of `` countries '' are modeled in terms of spin glasses . this approach can not provide a general coin framework though . in addition to the restrictions listed above on the kinds of systems it considers , this is due to its not providing a general solution to arbitrary coin inversion problems , and to its not employing rl algorithms . another contribution that statistical physics can make is with the mathematical techniques it has developed for its own purposes , like mean field theory , self - averaging approximations , phase transitions , monte carlo techniques , the replica trick , and tools to analyze the thermodynamic limit in which the number of particles goes to infinity . although such techniques have not yet been applied to coins , they have been successfully applied to related fields . this is exemplified by the use of the replica trick to analyze two - player zero - sum games with random payoff matrices in the thermodynamic limit of the number of strategies in @xcite . other examples are the numeric investigation of iterated prisoner s dilemma played on a lattice @xcite , the analysis of stochastic games by expressing of deviation from rationality in the form of a `` heat bath '' @xcite , and the use of topological entropy to quantify the complexity of a voting system studied in @xcite . other quite recent work in the statistical physics literature is formally identical to that in other fields , but presents it from a novel perspective . a good example of this is @xcite , which is concerned with the problem of controlling a spatially extended system with a single controller , by using an algorithm that is identical to a simple - minded proportional rl algorithm ( in essence , a rediscovery of rl ) . much of the theory of physics can be cast as solving for the extremization of an actional , which is a functional of the worldline of an entire ( potentially many - component ) system across all time . the solution to that extremization problem constitutes the actual worldline followed by the system . in this way the calculus of variations can be used to solve for the worldline of a dynamic system . as an example , simple newtonian dynamics can be cast as solving for the worldline of the system that extremizes a quantity called the ` lagrangian ' , which is a function of that worldline and of certain parameters ( _ e.g. _ , the ` potential energy ' ) governing the system at hand . in this instance , the calculus of variations simply results in newton s laws . if we take the dynamic system to be a coin , we are assured that its worldline automatically optimizes a `` global goal '' consisting of the value of the associated actional . if we change physical aspects of the system that determine the functional form of the actional ( _ e.g. _ , change the system s potential energy function ) , then we change the global goal , and we are assured that our coin optimizes that new global goal . counter - intuitive physical systems , like those that exhibit braess paradox @xcite , are simply systems for which the `` world utility '' implicit in our human intuition is extremized at a point different from the one that extremizes the system s actional . the challenge in exploiting this to solve the coin design problem is in translating an arbitrary provided global goal for the coin into a parameterized actional . note that that actional must govern the dynamics of the physical coin , and the parameters of the actional must be physical variables in the coin , variables whose values we can modify . the field of active walker models @xcite is concerned with modeling `` walkers '' ( be they human walkers or instead simple physical objects ) crossing fields along trajectories , where those trajectories are a function of several factors , including in particular the trails already worn into the field . often the kind of trajectories considered are those that can be cast as solutions to actional extremization problems so that the walkers can be explicitly viewed as agents optimizing a private utility . one of the primary concerns with the field of active walker models is how the trails worn in the field change with time to reach a final equilibrium state . the problem of how to design the cement pathways in the field ( and other physical features of the field ) so that the final paths actually followed by the walkers will have certain desirable characteristics is then one of solving for parameters of the actional that will result in the desired worldline . this is a special instance of the inverse problem of how to design a coin . using active walker models this way to design coins , like action extremization in general , probably has limited applicability . also , it is not clear how robust such a design approach might be , or whether it would be scalable and exempt from the need for hand - tailoring . this subsection presents a `` catch - all '' of other fields that have little in common with one another except that they bear some relation to coins . an extremely well - researched body of work concerns the mathematical and numeric behavior of systems for which the probability distribution over possible future states conditioned on preceding states is explicitly provided . this work involves many aspects of monte carlo numerical algorithms @xcite , all of markov chains @xcite , and especially markov fields , a topic that encompasses the chapman - kolmogorov equations @xcite and its variants : liouville s equation , the fokker - plank equation , and the detailed - balance equation in particular . non - linear dynamics is also related to this body of work ( see the synopsis of iterated function systems below and the synopsis of cellular automata above ) , as is markov competitive decision processes ( see the synopsis of game theory above ) . formally , one can cast the problem of designing a coin as how to fix each of the conditional transition probability distributions of the individual elements of a stochastic field so that the aggregate behavior of the overall system is of a desired form . unfortunately , almost all that is known in this area instead concerns the forward problem , of inferring aggregate behavior from a provided set of conditional distributions . although such knowledge provides many `` bits and pieces '' of information about how to tackle the inverse problem , those pieces collectively cover only a very small subset of the entire space of tasks we might want the coin to perform . in particular , they tell us very little about the case where the conditional distribution encapsulates rl algorithms . the technique of iterated function systems @xcite grew out of the field of nonlinear dynamics @xcite . in such systems a function is repeatedly and recursively applied to itself . the most famous example is the logistic map , @xmath24 for some @xmath25 between 0 and 4 ( so that @xmath26 stays between 0 and 1 ) . more generally the function along with its arguments can be vector - valued . in particular , we can construct such functions out of affine transformations of points in a euclidean plane . iterated functions systems have been applied to image data . in this case the successive iteration of the function generically generates a fractal , one whose precise character is determined by the initial iteration-1 image . since fractals are ubiquitous in natural images , a natural idea is to try to encode natural images as sets of iterated function systems spread across the plane , thereby potentially garnering significant image compression . the trick is to manage the inverse step of starting with the image to be compressed , and determining what iteration-1 image(s ) and iterating function(s ) will generate an accurate approximation of that image . in the language of nonlinear dynamics , we have a dynamic system that consists of a set of iterating functions , together with a desired attractor ( the image to be compressed ) . our goal is to determine what values to set certain parameters of our dynamic system to so that the system will have that desired attractor . the potential relationship with coins arises from this inverse nature of the problem tackled by iterated function systems . if the goal for a coin can be cast as its relaxing to a particular attractor , and if the distributed computational elements are isomorphic to iterated functions , then the tricks used in iterated functions theory could be of use . although the techniques of iterated function systems might prove of use in designing coins , they are unlikely to serve as a generally applicable approach to designing coins . in addition , they do not involve rl algorithms , and often involve extensive hand - tuning . a recurrent neural net consists of a finite set of `` neurons '' each of which has a real - valued state at each moment in time . each neuron s state is updated at each moment in time based on its current state and that of some of the other neurons in the system . the topology of such dependencies constitute the `` inter - neuronal connections '' of the net , and the associated parameters are often called the `` weights '' of the net . the dynamics can be either discrete or continuous ( _ i.e. _ , given by difference or differential equations ) . recurrent nets have been investigated for many purposes @xcite . one of the more famous of these is associative memories . the idea is that given a pre - specified pattern for the ( states of the neurons in the ) net , there may exist inter - neuronal weights which result in a basin of attraction focussed on that pattern . if this is the case , then the net is equivalent to an associative memory , in that a complete pre - specified pattern across all neurons will emerge under the net s dynamics from any initial pattern that partially matches the full pre - specified pattern . in practice , one wishes the net to simultaneously possess many such pre - specified associative memories . there are many schemes for `` training '' a recurrent net to have this property , including schemes based on spin glasses @xcite and schemes based on gradient descent @xcite . as can the fields of cellular automata and iterated function systems , the field of recurrent neural nets can be viewed as concerning certain variants of coins . also like those other fields though , recurrent neural nets has shortcomings if one tries to view it as a general approach to a science of coins . in particular , recurrent neural nets do not involve rl algorithms , and training them often suffers from scaling problems . more generally , in practice they can be hard to train well without hand - tailoring . packet routing in a data network @xcite presents a particularly interesting domain for the investigation of coins . in particular , with such routing : + ( i ) the problem is inherently distributed ; + ( ii ) for all but the most trivial networks it is impossible to employ global control ; + ( iii ) the routers have only access to local information ( routing tables ) ; + ( iv ) it constitutes a relatively clean and easily modified experimental testbed ; and + ( v ) there are potentially major bottlenecks induced by ` greedy ' behavior on the part of the individual routers , which behavior constitutes a readily investigated instance of the tragedy of the commons ( toc ) . many of the approaches to packet routing incorporate a variant on rl @xcite . q routing is perhaps the best known such approach and is based on routers using reinforcement learning to select the best path @xcite . although generally successful , q routing is not a general scheme for inverting a global task . this is even true if one restricts attention to the problem of routing in data networks there exists a global task in such problems , but that task is directly used to construct the algorithm . a particular version of the general packet routing problem that is acquiring increased attention is the quality of service ( qos ) problem , where different communication packets ( voice , video , data ) share the same bandwidth resource but have widely varying importances both to the user and ( via revenue ) to the bandwidth provider . determining which packet has precedence over which other packets in such cases is not only based on priority in arrival time but more generally on the potential effects on the income of the bandwidth provider . in this context , rl algorithms have been used to determine routing policy , control call admission and maximize revenue by allocating the available bandwidth efficiently @xcite . many researchers have exploited the noncooperative game theoretic understanding of the toc in order to explain the bottleneck character of empirical data networks behavior and suggest potential alternatives to current routing schemes @xcite . closely related is work on various `` pricing''-based resource allocation strategies in congestable data networks @xcite . this work is at least partially based upon current understanding of pricing in toll lanes , and traffic flow in general ( see below ) . all of these approaches are particularly of interest when combined with the rl - based schemes mentioned just above . due to these factors , much of the current research on a general framework for coins is directed toward the packet - routing domain ( see next section ) . traffic congestion typifies the toc public good problem : everyone wants to use the same resource , and all parties greedily trying to optimize their use of that resource not only worsens global behavior , but also worsens _ their own _ private utility ( _ e.g. _ , if everyone disobeys traffic lights , everyone gets stuck in traffic jams ) . indeed , in the well - known braess paradox @xcite , keeping everything else constant including the number and destinations of the drivers but opening a new traffic path can _ increase _ everyone s time to get to their destination . ( viewing the overall system as an instance of the prisoner s dilemma , this paradox in essence arises through the creation of a novel ` defect - defect ' option for the overall system . ) greedy behavior on the part of individuals also results in very rich global dynamic patterns , such as stop and go waves and clusters @xcite . much of traffic theory employs and investigates tools that have previously been applied in statistical physics @xcite ( see subsection above ) . in particular , the spontaneous formation of traffic jams provides a rich testbed for studying the emergence of complex activity from seemingly chaotic states @xcite . furthermore , the dynamics of traffic flow is particular amenable to the application and testing of many novel numerical methods in a controlled environment @xcite . many experimental studies have confirmed the usefulness of applying insights gleaned from such work to real world traffic scenarios @xcite . finally , there are a number of other fields that , while either still nascent or not extremely closely related to coins , are of interest in coin design : * amorphous computing : * amorphous computing grew out of the idea of replacing traditional computer design , with its requirements for high reliability of the components of the computer , with a novel approach in which widespread unreliability of those components would not interfere with the computation @xcite . some of its more speculative aspects are concerned with `` how to program '' a massively distributed , noisy system of components which may consist in part of biochemical and/or biomechanical components @xcite . work here has tended to focus on schemes for how to robustly induce desired geometric dynamics across the physical body of the amorphous computer issue that are closely related to morphogenesis , and thereby lend credence to the idea that biochemical components are a promising approach . especially in its limit of computers with very small constituent components , amorphous computing also is closely related to the fields of nanotechnology @xcite and control of smart matter ( see below ) . * control of smart matter:*. as the prospect of nanotechnology - driven mechanical systems gets more concrete , the daunting problem of how to robustly control , power , and sustain protean systems made up of extremely large sets of nano - scale devices looms more important @xcite . if this problem were to be solved one would in essence have `` smart matter '' . for example , one would be able to `` paint '' an airplane wing with such matter and have it improve drag and lift properties significantly . * morphogenesis : * how does a leopard embryo get its spots , or a zebra embryo its stripes ? more generally , what are the processes underlying morphogenesis , in which a body plan develops among a growing set of initially undifferentiated cells ? these questions , related to control of the dynamics of chemical reaction waves , are essentially special cases of the more general question of how ontogeny works , of how the genotype - phenotype mapping is carried out in development . the answers involve homeobox ( as well as many other ) genes @xcite . under the presumption that the functioning of such genes is at least in part designed to facilitate genetic changes that increase a species fitness , that functioning facilitates solution of the inverse problem , of finding small - scale changes ( to dna ) that will result in `` desired '' large scale effects ( to body plan ) when propagated across a growing distributed system . * self organizing systems * the concept of self - organization and self - organized criticality @xcite was originally developed to help understand why many distributed physical systems are attracted to critical states that possess long - range dynamic correlations in the large - scale characteristics of the system . it provides a powerful framework for analyzing both biological and economic systems . for example , natural selection ( particularly punctuated equilibrium @xcite ) can be likened to self - organizing dynamical system , and some have argued it shares many the properties ( _ e.g. _ , scale invariance ) of such systems @xcite . similarly , one can view the economic order that results from the actions of human agents as a case of self - organization @xcite . the relationship between complexity and self - organization is a particularly important one , in that it provides the potential laws that allow order to arise from chaos @xcite . * small worlds ( 6 degrees of separation ) : * in many distributed systems where each component can interact with a small number of `` neighbors '' , an important problem is how to propagate information across the system quickly and with minimal overhead . on the one extreme the neighborhood topology of such systems can exist on a completely regular grid - like structure . on the other , the topology can be totally random . in either case , certain nodes may be effectively ` cut - off ' from other nodes if the information pathways between them are too long . recent work has investigated `` small worlds '' networks ( sometimes called 6 degrees of separation ) in which underlying grid - like topologies are `` doped '' with a scattering of long - range , random connections . it turns out that very little such doping is necessary to allow for the system to effectively circumvent the information propagation problem @xcite . * control theory : * adaptive control @xcite , and in particular adaptive control involving locally weighted rl algorithms @xcite , constitute a broadly applicable framework for controlling small , potentially inexactly modeled systems . augmented by techniques in the control of chaotic systems @xcite , they constitute a very successful way of solving the `` inverse problem '' for such systems . unfortunately , it is not clear how one could even attempt to scale such techniques up to the massively distributed systems of interest in coins . the next section discusses in detail some of the underlying reasons why the purely model - based versions of these approaches are inappropriate as a framework for coins . summarizing the discussion to this point , it is hard to see how any already extant scientific field can be modified to encompass systems meeting all of the requirements of coins listed at the beginning of section [ sec : lit ] . this is not too surprising , since none of those fields were explicitly designed to analyze coins . this section first motivates in general terms a framework that is explicitly designed for analyzing coins . it then presents the formal nomenclature of that framework . this is followed by derivations of some of the central theorems of that framework . finally , we present experiments that illustrate the power the framework provides for ensuring large world utility in a coin . what mathematics might one employ to understand and design coins ? perhaps the most natural approach , related to the stochastic fields work reviewed above , involves the following three steps : \1 ) first one constructs a detailed stochastic model of the coin s dynamics , a model parameterized by a vector @xmath27 . as an example , @xmath27 could fix the utility functions of the individual agents of the coin , aspects of their rl algorithms , which agents communicate with each other and how , etc . \2 ) next we solve for the function @xmath28 which maps the parameters of the model to the resulting stochastic dynamics . \3 ) cast our goal for the system as a whole as achieving a high expected value of some `` world utility '' . then as our final step we would have to solve the inverse problem : we would have to search for a @xmath27 which , via @xmath9 , results in a high value of e(world utility @xmath29 ) . let s examine in turn some of the challenges each of these three steps entrain : \i ) we are primarily interested in very large , very complex systems , which are noisy , faulty , and often operate in a non - stationary environment . moreover , our `` very complex system '' consists of many rl algorithms , all potentially quite complicated , all running simultaneously . clearly coming up with a detailed model that captures the dynamics of all of this in an accurate manner will often be extraordinarily difficult . moreover , unfortunately , given that the modeling is highly detailed , often the level of verisimilitude required of the model will be quite high . for example , unless the modeling of the faulty aspects of the system were quite accurate , the model would likely be `` brittle '' , and overly sensitive to which elements of the coin were and were not operating properly at any given time . \ii ) even for models much simpler than the ones called for in ( i ) , solving explicitly for the function @xmath9 can be extremely difficult . for example , much of markov chain theory is an attempt to broadly characterize such mappings . however as a practical matter , usually it can only produce potentially useful characterizations when the underlying models are quite inaccurate simplifications of the kinds of models produced in step ( i ) . \iii ) even if one can write down an @xmath9 , solving the associated inverse problem is often impossible in practice . \iv ) in addition to these difficulties , there is a more general problem with the model - based approach . we wish to perform our analysis on a `` high level '' . our thesis is that due to the robust and adaptive nature of the individual agents rl algorithms , there will be very broad , easily identifiable regions of @xmath27 space all of which result in excellent e(world utility @xmath29 ) , and that these regions will not depend on the precise learning algorithms used to achieve the low - level tasks ( cf . the list at the beginning of section [ sec : lit ] ) . to fully capitalize on this , one would want to be able to slot in and out different learning algorithms for achieving the low - level tasks without having to redo our entire analysis each time . however in general this would be possible with a model - based analysis only for very carefully designed models ( if at all ) . the problem is that the result of step ( 3 ) , the solution to the inverse problem , would have to concern aspects of the coin that are ( at least approximately ) invariant with respect to the precise low - level learning algorithms used . coming up with a model that has this property while still avoiding problems ( i - iii ) is usually an extremely daunting challenge . fortunately , there is an alternative approach which avoids the difficulties of detailed modeling . little modeling of any sort ever is used in this alternative , and what modeling does arise has little to do with dynamics . in addition , any such modeling is extremely high - level , intented to serve as a decent approximation to almost any system having `` reasonable '' rl algorithms , rather than as an accurate model of one particular system . we call any framework based on this alternative a * descriptive framework*. in such a framework one identifies certain * salient characteristics * of coins , which are characteristics of a coin s entire worldline that one strongly expects to find in coins that have large world utility . under this expectation , one makes the assumption that if a coin is explicitly modified to have the salient characteristics ( for example in response to observations of its run - time behavior ) , then its world utility will benefit . so long as the salient characteristics are ( relatively ) easy to induce in a coin , then this assumption provides a ready indirect way to cause that coin to have large world utility . an assumption of this nature is the central leverage point that a descriptive framework employs to circumvent detailed modeling . under it , if the salient characteristics can be induced with little or no modeling ( e.g. , via heuristics that are nt rigorously and formally justified ) , then they provide an indirect way to improve world utility without recourse to detailed modeling . in fact , since one does not use detailed modeling in a descriptive framework , it may even be that one does not have a fully rigorous mathematical proof that the central assumption holds in a particular system for one s choice of salient characteristics . one may have to be content with reasonableness arguments not only to justify one s scheme for inducing the salient characteristics , but for making the assumption that characteristics are correlated with large world utility in the first place . of course , the trick in the descriptive framework is to choose salient characteristics that both have a beneficial relationship with world utility and that one expects to be able to induce with relatively little detailed modeling of the system s dynamics . there exist many ways one might try to design a descriptive framework . in this subsection we present nomenclature needed for a ( very ) cursory overview of one of them . ( see @xcite for a more detailed exposition , including formal proofs . ) this overview concentrates on the four salient characteristics of intelligence , learnability , factoredness , and the wonderful life utility , all defined below . intelligence is a quantification of how well an rl algorithm performs . we want to do whatever we can to help those algorithms achieve high values of their utility functions . learnability is a characteristic of a utility function that one would expect to be well - correlated with how well an rl algorithm can learn to optimize it . a utility function is also factored if whenever its value increases , the overall system benefits . finally , wonderful life utility is an example of a utility function that is both learnable and factored . after the preliminary definitions below , this section formalizes these four salient characteristics , derives several theorems relating them , and illustrates in some computer experiments how those theorems can be used to help the system achieve high world utility . * 1 ) * we refer to an rl algorithm by which an individual component of the coin modifies its behavior as a * microlearning * algorithm . we refer to the initial construction of the coin , potentially based upon salient characteristics , as the coin * initialization*. we use the phrase * macrolearning * to refer to externally imposed run - time modifications to the coin which are based on statistical inference concerning salient characteristics of the running coin . * 2 ) * for convenience , we take time , @xmath17 , to be discrete and confined to the integers , _ z_. when referring to coin initialization , we implicitly have a lower bound on @xmath17 , which without loss of generality we take to be less than or equal to @xmath30 . * 3 ) * all variables that have any effect on the coin are identified as components of euclidean - vector - valued * states * of various discrete * nodes*. as an important example , if our coin consists in part of a computational `` agent '' running a microlearning algorithm , the precise configuration of that agent at any time @xmath17 , including all variables in its learning algorithm , all actions directly visible to the outside world , all internal parameters , all values observed by its probes of the surrounding environment , etc . , all constitute the state vector of a node representing that agent . we define @xmath31 to be a vector in the euclidean vector space @xmath32 , where the components of @xmath31 give the state of node @xmath33 at time @xmath17 . the @xmath4th component of that vector is indicated by @xmath34 . * observation 3.1 : * in practice , many coins will involve variables that are most naturally viewed as discrete and symbolic . in such cases , we must exercise some care in how we choose to represent those variables as components of euclidean vectors . there is nothing new in this ; the same issue arises in modern work on applying neural nets to inherently symbolic problems . in our coin framework , we will usually employ the same resolution of this issue employed in neural nets , namely representing the possible values of the discrete variable with a unary representation in a euclidean space . just as with neural nets , values of such vectors that do not lie on the vertices of the unit hypercube are not meaningful , strictly speaking . fortunately though , just as with neural nets , there is almost always a most natural way to extend the definitions of any function of interest ( like world utility ) so that it is well - defined even for vectors not lying on those vertices . this allows us to meaningfully define partial derivatives of such functions with respect to the components of @xmath35 , partial derivatives that we will evaluate at the corners of the unit hypercube . * 4 ) * for notational convenience , we define @xmath36 to be the vector of the states of all nodes at time @xmath17 ; @xmath37 to be the vector of the states of all nodes other than @xmath33 at time @xmath17 ; and @xmath38 to be the entire vector of the states of all nodes at all times . @xmath39 is infinite - dimensional in general , and usually assumed to be a hilbert space . we will often assume that all spaces @xmath40 over all times @xmath17 are isomorphic to a space @xmath41 , i.e. , @xmath39 is a cartesian product of copies of @xmath41 . also for notational convenience , we define gradients using @xmath42-shorthand . so for example , @xmath43 is the vector of the partial derivative of @xmath44 with respect to the components of @xmath45 . also , we will sometimes treat the symbol `` @xmath17 '' specially , as delineating a range of components of @xmath35 . so for example an expression like `` @xmath46 '' refers to all components @xmath45 with @xmath47 . * 5 ) * to avoid confusion with the other uses of the comma operator , we will often use @xmath48 rather than @xmath49 to indicate the vector formed by concatenating the two ordered sets of vector components @xmath50 and @xmath51 . for example , @xmath52 refers to the vector formed by concatenating those components of the worldline @xmath35 involving node @xmath33 for times less than 0 with those components involving node @xmath53 that have times greater than 0 . * 6 ) * we take the universe in which our coin operates to be completely deterministic . this is certainly the case for any coin that operates in a digital system , even a system that emulates analog and/or stochastic processes ( _ e.g. _ , with a pseudo - random number generator ) . more generally , this determinism reflects the fact that since the real world obeys ( deterministic ) physics , @xmath54 real - world system , be it a coin or something else , is , ultimately , embedded in a deterministic system . the perspective to be kept in mind here is that of nonlinear time - series analysis . a physical time series typically reflects a few degrees of freedom that are projected out of the underlying space in which the full system is deterministically evolving , an underlying space that is actually extremely high - dimensional . this projection typically results in an illusion of stochasticity in the time series . * 7 ) * formally , to reflect this determinism , first we bundle all variables we are not directly considering but which nonetheless affect the dynamics of the system as components of some catch - all * environment node*. so for example any `` noise processes '' and the like affecting the coin s dynamics are taken to be inputs from a deterministic , very high - dimensional environment that is potentially chaotic and is never directly observed @xcite . given such an environment node , we then stipulate that for all @xmath55 such that @xmath56 , @xmath45 sets @xmath57 uniquely . * observation 7.1 : * when nodes are `` computational devices '' , often we must be careful to specify the physical extent of those devices . such a node may just be the associated cpu , or it may be that cpu together with the main ram , or it may include an external storage device . almost always , the border of the device @xmath33 will end before any external system that @xmath33 is `` observing '' begins . this means that since at time @xmath17 @xmath33 only knows the value of @xmath31 , its `` observational knowledge '' of that external system is indirect . that knowledge reflects a coupling between @xmath31 and @xmath58 , a coupling that is induced by the dynamical evolution of the system from preceding moments up to the time @xmath17 . if the dynamics does not force such a coupling , then @xmath33 has no observational knowledge of the outside world . * 8) * we express the dynamics of our system by writing @xmath59 . ( in this paper there will be no need to be more precise and specify the precise dependency of @xmath60 on @xmath17 and/or @xmath61 . ) we define @xmath62 to be a set of constraint equations enforcing that dynamics , and also , more generally , fixing the entire manifold @xmath63 of vectors @xmath64 that we consider to be ` allowed ' . so @xmath63 is a subset of the set of all @xmath65 that are consistent with the deterministic laws governing the coin , _ i.e. _ , that obey @xmath66 . we generalize this notation in the obvious way , so that ( for example ) @xmath67 is the manifold consisting of all vectors @xmath68 that are projections of a vector in @xmath63 . * observation 8.1 : * note that @xmath67 is parameterized by @xmath69 , due to determinism . note also that whereas @xmath60 is defined for any argument of the form @xmath70 for some @xmath17 ( _ i.e. _ , we can evolve any point forward in time ) , in general not all @xmath71 lie in @xmath72 . in particular , there may be extra restrictions constraining the possible states of the system beyond those arising from its need to obey the relevant dynamical laws of physics . finally , whenever trying to express a coin in terms of the framework presented here , it is a good rule to try to write out the constraint equations explicitly to check that what one has identified as the space @xmath40 contains all quantities needed to uniquely fix the future state of the system . * observation 8.2 : * we do not want to have @xmath73 be the phase space of every particle in the system . we will instead usually have @xmath73 consist of variables that , although still evolving deterministically , exist at a larger scale of granularity than that of individual particles ( _ e.g. _ , thermodynamic variables in the thermodynamic limit ) . however we will often be concerned with physical systems obeying entropy - driven dynamic processes that are contractive at this high level of granularity . examples are any of the many - to - one mappings that can occur in digital computers , and , at a finer level of granularity , any of the error - correcting processes in the electronics of such a computer that allow it to operate in a digital fashion . accordingly , although the dynamics of our system will always be deterministic , it need not be invertible . * observation 8.3 : * intuitively , in our mathematics , all behavior across time is pre - fixed . the coin is a single fixed worldline through @xmath39 , with no `` unfolding of the future '' as the die underlying a stochastic dynamics get cast . this is consistent with the fact that we want the formalism to be purely descriptive , relating different properties of any single , fixed coin s history . we will often informally refer to `` changing a node s state at a particular time '' , or to a microlearner s `` choosing from a set of options '' , and the like . formally , in all such phrases we are really comparing different worldlines , with the indicated modification distinguishing those worldlines . * observation 8.4 : * since the dynamics of any real - world coin is deterministic , so is the dynamics of any component of the coin , and in particular so is any learning algorithm running in the coin , ultimately . however that does not mean that those deterministic components of the coin are not allowed to be `` based on '' , or `` motivated by '' stochastic concepts . the _ motivation _ behind the algorithms run by the components of the coin does not change their underlying nature . indeed , in our experiments below , we explicitly have the reinforcement learning algorithms that are trying to maximize private utility operate in a ( pseudo- ) probabilistic fashion , with pseudo - random number generators and the like . more generally , the deterministic nature of our framework does not preclude our superimposing probabilistic elements on top of that framework , and thereby generating a stochastic extension of our framework . exactly as in statistical physics , a stochastic nature can be superimposed on our space of deterministic worldlines , potentially by adopting a degree of belief perspective on `` what probability means '' @xcite . indeed , the macrolearning algorithms we investigate below implicitly involve such a superimposing ; they implicitly assume a probabilistic coupling between the ( statistical estimate of the ) correlation coefficient connecting the states of a pair of nodes and whether those nodes are in the one another s `` effect set '' . similarly , while it does not salient characteristics that involve probability distributions , the descriptive framework does not preclude such characteristics either . as an example , the `` intelligence '' of an agent s particular action , formally defined below , measures the fraction of alternative actions an agent could have taken that would have resulted in a lower utility value . to define such a fraction requires a measure across the space of such alternative actions , even if only implicitly . accordingly , intelligence can be viewed as involving a probability distribution across the space of potential actions . in this paper though , we concentrate on the mathematics that obtains before such probabilistic concerns are superimposed . whereas the deterministic analysis presented here is related to game - theoretic structures like nash equilibria , a full - blown stochastic extension would in some ways be more related to structures like correlated equilibria @xcite . * 9 ) * formally , there is a lot of freedom in setting the boundary between what we call `` the coin '' , whose dynamics is determined by @xmath63 , and what we call `` macrolearning '' , which constitutes perturbations to the coin instigated from `` outside the coin '' , and which therefore is @xmath74 reflected in @xmath63 . as an example , in much of this paper , we have clearly specified microlearners which are provided fixed private utility functions that they are trying to maximize . in such cases usually we will implicitly take @xmath63 to be the dynamics of the system , microlearning and all , _ for fixed private utilities _ that are specified in @xmath35 . for example , @xmath35 could contain , for each microlearner , the bits in an associated computer specifying the subroutine that that microlearner can call to evaluate what its private utility would be for some full worldline @xmath35 . macrolearning overrides @xmath63 , and in this situation it refers ( for example ) to any statistical inference process that modifies the private utilities at run - time to try to induce the desired salient characteristics . concretely , this would involve modifications to the bits \{@xmath75 } specifying each microlearner @xmath4 s private utility , modifications that are @xmath74 accounted for in @xmath63 , and that are potentially based on variables that are not reflected in @xmath39 . since @xmath63 does not reflect such macrolearning , when trying to ascertain @xmath63 based on empirical observation ( as for example when determining how best to modify the private utilities ) , we have to take care to distinguish which part of the system s observed dynamics is due to @xmath63 and which part instead reflects externally imposed modifications to the private utilities . more generally though , other boundaries between the coin and macrolearning - based perturbations to it are possible , reflecting other definitions of @xmath39 , and other interpretations of the elements of each @xmath64 . for example , say that under the perspective presented in the previous paragraph , the private utility is a function of some components @xmath16 of @xmath35 , components that do not include the \{@xmath75}. now modify this perspective so that in addition to the dynamics of other bits , @xmath63 also encapsulates the dynamics of the bits \{@xmath75}. having done this , we could still view each private utility as being fixed , but rather than take the bits \{@xmath75 } as `` encoding '' the subroutine that specifies the private utility of microlearner @xmath4 , we would treat them as `` parameters '' specifying the functional dependence of the ( fixed ) private utility on the components of @xmath35 . in other words , formally , they constitute an extra set of arguments to @xmath4 s private utility , in addition to the arguments @xmath16 . alternatively , we could simply say that in this situation our private utilities are time - indexed , with @xmath4 s private utility at time @xmath17 determined by \{@xmath76 } , which in turn is determined by evolution under @xmath63 . under either interpretation of private utility , any modification under @xmath63 to the bits specifying @xmath4 s utility - evaluation subroutine constitutes dynamical laws by which the parameters of @xmath4 s microlearner evolves in time . in this case , macrolearning would refer to some further removed process that modifies the evolution of the system in a way not encapsulated in @xmath63 . for such alternative definitions of @xmath63/@xmath39 , we have a different boundary between the coin and macrolearning , and we must scrutinize different aspects of the coin s dynamics to infer @xmath63 . whatever the boundary , the mathematics of the descriptive framework , including the mathematics concerning the salient characteristics , is restricted to a system evolving according to @xmath63 , and explicitly does not account for macrolearning . this is why the strategy of trying to improve world utility by using macrolearning to try to induce salient characteristics is almost always ultimately based on an assumption rather than a proof . * 10 ) * we are provided with some von neumann * world utility * @xmath77 that ranks the various conceivable worldlines of the coin . note that since the environment node is never directly observed , we implicitly assume that the world utility is not directly ( ! ) a function of its state . our mathematics will not involve @xmath78 alone , but rather the relationship between @xmath78 and various sets of * personal utilities * @xmath79 . intuitively , as discussed below , for many purposes such personal utilities are equivalent to arbitrary `` virtual '' versions of the private utilities mentioned above . in particular , it is only private utilities that will occur within any microlearning computer algorithms that may be running in the coin as manifested in @xmath63 . personal utilities are external mathematical constructions that the coin framework employs to analyze the behavior of the system . they can be involved in learning processes , but only as tools that are employed outside of the coin s evolution under @xmath63 , _ i.e. _ , only in macrolearning . ( for example , analysis of them can be used to modify the private utilities . ) * observation 10.1 : * these utility definitions are very broad . in particular , they do not require casting of the utilities as discounted sums . note also that our world utility is not indexed by @xmath17 . again reflecting the descriptive , worldline character of the formalism , we simply assign a single value to an entire worldline of the system , implicitly assuming that one can always say which of two candidate worldlines are preferable . so given some `` present time '' @xmath80 , issues like which of two `` potential futures '' @xmath81 , @xmath82 is preferable are resolved by evaluating the relevant utility at two associated points @xmath35 and @xmath83 , where the @xmath84 components of those points are the futures indicated , and the two points share the same ( usually implicit ) @xmath85 `` past '' components . this time - independence of @xmath78 automatically avoids formal problems that can occur with general ( _ i.e. _ , not necessarily discounted sum ) time - indexed utilities , problems like having what s optimal at one moment in time conflict with what s optimal at other moments in time .. the effects of the actions by the nodes , adn therefore whether those actions are `` optimal '' or not , depends on the future actions of the nodes . however if they too are to be `` optimal '' , according to their world - utility , those future actions will depend on _ their _ futures . so we have a potentially infinite regress of differing stipulations of what `` optimal '' actions at time @xmath17 entails . ] for personal utilities such formal problems are often irrelevant however . before we begin our work , we as coin designers must be able to rank all possible worldlines of the system at hand , to have a well - defined design task . that is why world utility can not be time - indexed . however if a particular microlearner s goal keeps changing in an inconsistent way , that simply means that that microlearner will grow `` confused '' . from our perspective as coin designers , there is nothing _ a priori _ unacceptable about such confusion . it may even result in better performance of the system as a whole , in whic case we would actually want to induce it . nonetheless , for simplicity , in most of this paper we will have all @xmath86 be independent of @xmath17 , just like world utility . world utility is defined as that function that we are ultimately interested in optimizing . in conventional rl it is a discounted sum , with the sum starting at time @xmath17 . in other words , conventional rl has a time - indexed world utility . it might seem that in this at least , conventional rl considers a case that has more generality than that of the coin framework presented here . ( it obviously has less generality in that its world utility is restricted to be a discounted sum . ) in fact though , the apparent time - indexing of conventional rl is illusory , and the time - dependent discounted sum world utilty of conventional rl is actually a special case of the non - time - indexed world utility of our coin framework . to see this formally , consider any ( time - independent ) world utility @xmath87 that equals @xmath88 for some function @xmath89 and some positive constant @xmath90 with magnitude less than 1 . then for any @xmath91 and any @xmath83 and @xmath92 where @xmath93 , @xmath94 = sgn[\sum_{t=0}^{\infty } \gamma^t r(\underline{\zeta}'_{,t } ) - \sum_{t=0}^{\infty } \gamma^t r(\underline{\zeta''}_{,t})]$ ] . conventional rl merely expresses this in terms of time - dependent utilities @xmath95 by writing @xmath94 = sgn[u_{t'}(\underline{\zeta } ' ) - u_{t'}(\underline{\zeta}'')]$ ] for all @xmath61 . since utility functions are , by definition , only unique up to the relative orderings they impose on potential values of their arguments , we see that conventional rl s use of a time - dependent discounted sum world utility @xmath96 is identical to use of a particular time - independent world utility in our coin framework . * 11 ) * as mentioned above , there may be variables in each node s state which , under one particular interpretation , represent the `` utility functions '' that the associated microlearner s computer program is trying to extremize . when there are such components of @xmath35 , we refer to the utilities they represent as * private utilities*. however even when there are private utilities , formally we allow the personal utilities to differ from them . the personal utility functions \{@xmath97 } do not exist `` inside the coin '' ; they are not specified by components of @xmath35 . this separating of the private utilities from the \{@xmath97 } will allow us to avoid the teleological problem that one may not always be able to explicitly identify `` the '' private utility function reflected in @xmath35 such that a particular computational device can be said to be a microlearner `` trying to increase the value of its private utility '' . to the degree that we can couch the theorems purely in terms of personal rather than private utilities , we will have successfully adopted a purely behaviorist approach , without any need to interpret what a computational device is `` trying to do '' . despite this formal distinction though , often we will implicitly have in mind deploying the personal utilities onto the microlearners as their private utilities , in which case the terms can usually be used interchangeably . the context should make it clear when this is the case . we will need to quantify how well the entire system performs in terms of @xmath78 . to do this requires a measure of the performance of an arbitrary worldline @xmath35 , for an arbitrary utility function , under arbitrary dynamic laws @xmath63 . formally , such a measure is a mapping from three arguments to @xmath98 . such a measure will also allow us to quantify how well each microlearner performs in purely behavioral terms , in terms of its personal utility . ( in our behaviorist approach , we do not try to make specious distinctions between whether a microlearner s performance is due to its level of `` innate sophistication '' , or rather due to dumb luck all that matters is the quality of its behavior as reflected in its utility value for the system s worldline . ) this behaviorism in turn will allow us to avoid having private utilities explicitly arise in our theorems ( although they still arise frequently in pedagogical discussion ) . even when private utilities exist , there will be no formal need to explicitly identify some components of @xmath35 as such utilities . assuming a node s microlearner is competent , the fact that it is trying to optimize some particular private utility @xmath99 will be manifested in our performance measure s having a large value at @xmath35 for @xmath63 for that utility @xmath99 . the problem of how to formally define such a performance measure is essentially equivalent to the problem of how to quantify bounded rationality in game theory . some of the relevant work in game theory , for example that involving ` trembling hand equilibria ' or ` @xmath100 equilibria ' @xcite is concerned with refinements or modifications of nash equilibria ( see also @xcite ) . rather than a behaviorist approach , such work adopts a strongly teleological perspective on rationality . in general , such work is only applicable to those situations where the rationality is bounded due to the precise causal mechanisms investigated in that work . most of the other game - theoretic work first models ( ! ) the microlearner , as some extremely simple computational device ( _ e.g. _ , a deterministic finite automaton ( dfa ) . one then assumes that the microlearner performs perfectly for that device , so that one can measure that learner s performance in terms of some computational capacity measure of the model ( _ e.g. _ , for a dfa , the number of states of that dfa ) @xcite . however , if taken as renditions of real - world computer - based microlearners never mind human microlearners the models in this approach are often extremely abstracted , with many important characteristics of the real learners absent or distorted . in addition , there is little reason to believe that any results arising from this approach would not be highly dependent on the model choice and on the associated representation of computational capacity . yet another disadvantage is that this approach concentrates on perfect , fully rational behavior of the microlearners , within their computational restrictions . we would prefer a less model - dependent approach , especially given our wish that the performance measure be based solely on the utility function at hand , @xmath35 , and @xmath63 . now we do nt want our performance measure to be a `` raw '' utility value like @xmath101 , since that is not invariant with respect to monotonic transformations of @xmath102 . similarly , we do nt want to penalize the microlearner for not achieving a certain utility value if that value was impossible to achieve not due to the microlearner s shortcomings , but rather due to @xmath63 and the actions of other nodes . a natural way to address these concerns is to generalize the game - theoretic concept of `` best - response strategy '' and consider the problem of how well @xmath33 performs _ given the actions of the other nodes_. such a measure would compare the utility ultimately induced by each of the possible states of @xmath33 at some particular time , which without loss of generality we can take to be 0 , to that induced by the actual state @xmath103 . in other words , we would compare the utility of the actual worldline @xmath35 to those of a set of alternative worldlines @xmath83 , where @xmath104 , and use those comparisons to quantify the quality of @xmath33 s performance . now we are only concerned with comparing the effects of replacing @xmath35 with @xmath83 on @xmath105 contributions to the utility . but if we allow arbitrary @xmath106 , then in and of themselves the difference between those past components of @xmath83 and those of @xmath35 can modify the value of the utility , regardless of the effects of any difference in the future components . our presumption is that for many coins of interest we can avoid this conundrum by restricting attention to those @xmath83 where @xmath106 differs from @xmath107 only in the internal parameters of @xmath33 s microlearner , differences that only at times @xmath108 manifest themselves in a form the utility is concerned with . ( in game - theoretic terms , such `` internal parameters '' encode full extensive - form strategies , and we only consider changes to the vertices at or below the @xmath109 level in the tree of an extensive - form strategy . ) although this solution to our conundrum is fine when we can apply it , we do nt want to restrict the formalism so that it can only concern systems having computational algorithms which involve a clearly pre - specified set of extensive strategy `` internal parameters '' and the like . so instead , we formalize our presumption behaviorally , even for computational algorithms that do not have explicit extensive strategy internal parameters . since changing the internal parameters does nt affect the @xmath110 components of @xmath111 _ that the utility is concerned with _ , and since we are only concerned with changes to @xmath35 that affect the utility , we simply elect to not change the @xmath110 values of the internal parameters of @xmath111 at all . in other words , we leave @xmath112unchanged . the advantage of this stipulation is that we can apply it just as easily whether @xmath33 does or does nt have any `` internal parameters '' in the first place . so in quantifying the performance of @xmath33 for behavior given by @xmath35 we compare @xmath35 to a set of @xmath83 , a set restricted to those @xmath83 sharing @xmath35 s past : @xmath113 , @xmath114 , and @xmath115 . since @xmath116 is free to vary ( reflecting the possible changes in the state of @xmath33 at time 0 ) while @xmath106 is not , @xmath117 , in general . we may even wish to allow @xmath118 in certain circumstances . ( recall that @xmath63 may reflect other restrictions imposed on allowed worldlines besides adherence to the underlying dynamical laws , so simply obeying those laws does not suffice to ensure that a worldline lies on @xmath63 . ) in general though , our presumption is that as far as utility values are concerned , considering these dynamically impossible @xmath83 is equivalent to considering a more restricted set of @xmath83 with `` modified internal parameters '' , all of which are @xmath119 . we now present a formalization of this performance measure . given @xmath63 and a measure @xmath120 demarcating what points in @xmath121 we are interested in , we define the ( @xmath122 ) * intelligence * for node @xmath33 of a point @xmath35 with respect to a utility @xmath99 as follows : @xmath123 \cdot \delta(\underline{\zeta}'_{\;\hat{}\eta,0 } - \underline{\zeta}_{\;\hat{}\eta,0})\ ] ] where @xmath124 is the heaviside theta function which equals 0 if its argument is below 0 and equals 1 otherwise , @xmath125 is the dirac delta function , and we assume that @xmath126 . intuitively , @xmath127 measures the fraction of alternative states of @xmath33 which , if @xmath33 had been in those states at time 0 , would either degrade or not improve @xmath33 s performance ( as measured by @xmath99 ) . sometimes in practice we will only want to consider changes in those components of @xmath103 that we consider as `` free to vary '' , which means in particular that those changes are consistent with @xmath63 and the state of the external world , @xmath128 . ( this consistency ensures that @xmath33 s observational information concerning the external world is correct ; see observation 7.1 above . ) such a restriction means that even though @xmath129 may not be consistent with @xmath63 and @xmath107 , by itself it is still consistent with @xmath63 ; in quantifying the quality of a particular @xmath103 . so we do nt compare our point to other @xmath129 that are physically impossible , no matter what the past is . any such restrictions on what changes we are considering are reflected implicitly in intelligence , in the measure @xmath130 . as an example of intelligence , consider the situation where for each player @xmath33 , the support of the measure @xmath131 extends over all possible actions that @xmath33 could take that affect the ultimate value of its personal utility , @xmath97 . in this situation we recover conventional full rationality game theory involving nash equilibria , as the analysis of scenarios in which the intelligence of each player @xmath33 with respect to @xmath97 equals 1 . whose components need not all equal 1 . many of the theorems of conventional game theory can be directly carried over to such bounded - rational games @xcite by redefining the utility functions of the players . in other words , much of conventional full rationality game theory applies even to games with bounded rationality , under the appropriate transformation . this result has strong implications for the legitimacy of the common criticism of modern economic theory that its assumption of full rationality does not hold in the real world , implications that extend significantly beyond the sonnenschein - mantel - debreu theorem equilibrium aggregate demand theorem @xcite . ] as an alternative , we could for each @xmath33 restrict @xmath131 to some limited `` set of actions that @xmath33 actively considers '' . this provides us with an `` effective nash equilibrium '' at the point @xmath35 where each @xmath132 equals 1 , in the sense that _ as far it s concerned _ , each player @xmath33 has played a best possible action at such a point . as yet another alternative , we could restrict each @xmath131 to some infinitesimal neighborhood about @xmath129 , and thereby define a `` local nash equilibrium '' by having @xmath133 for each player @xmath33 . in general , competent greedy pursuit of private utility @xmath99 by the microlearner controlling node @xmath33 means that the intelligence of @xmath33 for personal utility @xmath99 , @xmath127 , is close to 1 . accordingly , we will often refer interchangeably to a capable microlearner s `` pursuing private utility @xmath99 '' , and to its having high intelligence for personal utility @xmath99 . alternatively , if the microlearner for node @xmath33 is incompetent , then it may even be that `` by luck '' its intelligence for some personal utility \{@xmath102 } exceeds its intelligence for the different private utility that it s actually trying to maximize , @xmath134 . say that we expect that a particular microlearner is `` smart '' , in that it is more likely to have high rather than low intelligence . we can model this by saying that given a particular @xmath135 , the conditional probability that @xmath136 is a monotonically increasing function of @xmath137 . since for a given @xmath135 the intelligence @xmath138 is a monotonically increasing function of @xmath102 , this modelling assumption means that the probability that @xmath136 is a monotonically increasing function of @xmath139 . an alternative weaker model is to only stipulate that the probability of having a particular pair @xmath140 with @xmath138 equal to @xmath141 is a monotonically increasing function of @xmath141 . ( this probability is an integral over a joint distribution , rather than a conditional distribution , as in the original model . ) in either case , the `` better '' the microlearner , the more tightly peaked the associated probability distribution over intelligence values is . any two utility functions that are related by a monotonically increasing transformation reflect the same preference ordering over the possible arguments of those functions . since it is only that ordering that we are ever concerned with , we would like to remove this degeneracy by `` normalizing '' all utility functions . in other words , we would like to reduce any equivalence set of utility functions that are monotonic transformations of one another to a canonical member of that set . to see what this means in the coin context , fix @xmath142 . viewed as a function from @xmath143 , @xmath144 is itself a utility function , one that is a monotonically increasing function of @xmath99 . ( it says how well @xmath33 would have performed for all vectors @xmath111 . ) accordingly , the integral transform taking @xmath99 to @xmath144 is a ( contractive , non - invertible ) mapping from utilities to utilities . applied to any member of a utility in @xmath99 s equivalence set , this mapping produces the same image utility , one that is also in that equivalence set . it can be proven that any mapping from utilities to utilities that has this and certain other simple properties must be such an integral transform . in this , intelligence is the unique way of `` normalizing '' von neumann utility functions . for those conversant with game theory , it is worth noting some of the interesting aspects that ensue from this normalizing nature of intelligences . at any point @xmath35 that is a nash equilibrium in the set of personal utilities \{@xmath97 } , all intelligences @xmath132 must equal 1 . since that is the maximal value any intelligence can take on , a nash equilibrium in the \{@xmath97 } is a pareto optimal point in the associated intelligences ( for the simple reason that no deviation from such a @xmath35 can raise any of the intelligences ) . conversely , if there exists at least one nash equilibrium in the \{@xmath97 } , then there is not a pareto optimal point in the \{@xmath132 } that is not a nash equilibrium . now restrict attention to systems with only a single instant of time , _ i.e. _ , single - stage games . also have each of the ( real - valued ) components of each @xmath145 be a mixing component of an associated one of @xmath33 s potential strategies for some underlying finite game . then have @xmath146 be the associated expected payoff to @xmath33 . ( so the payoff to @xmath33 of the underlying pure strategies is given by the values of @xmath146 when @xmath35 is a unit vector in the space @xmath147 of @xmath33 s possible states . ) then we know that there must exist at least one nash equilibrium in the \{@xmath97}. accordingly , in this situation the set of nash equilibria in the \{@xmath97 } is identical to the set of points that are pareto optimal in the associated intelligences . ( see eq . 5 in the discussion of factored systems below . ) intelligence can be a difficult quantity to work with , unfortunately . as an example , fix @xmath33 , and consider any ( small region centered about some ) @xmath35 along with some utility @xmath99 , where @xmath35 is not a local maximum of @xmath99 . then by increasing the values @xmath99 takes on in that small region we will increase the intelligence @xmath127 . however in doing this we will also necessarily @xmath148 the intelligence at points outside that region . so intelligence has a non - local character , a character that prevents us from directly modifying it to ensure that it is simultaneously high for any and all @xmath35 . a second , more general problem is that without specifying the details of a microlearner , it can be extremely difficult to predict which of two private utilities the microlearner will be better able to learn . indeed , even @xmath149 the details , making that prediction can be nearly impossible . so it can be extremely difficult to determine what private utility intelligence values will accrue to various choices of those private utilities . in other words , macrolearning that involves modifying the private utilities to try to directly increase intelligence with respect to those utilities can be quite difficult . fortunately , we can circumvent many of these difficulties by using a proxy for ( private utility ) intelligence . although we expect its value usually to be correlated with that of intelligence in practice , this proxy does not share intelligence s non - local nature . in addition , the proxy does not depend heavily on the details of the microlearning algorithms used , _ i.e. _ , it is fairly independent of those aspects of @xmath63 . intuitively , this proxy can be viewed as a `` salient characteristic '' for intelligence . we motivate this proxy by considering having @xmath150 for all @xmath33 . if we try to actually use these \{@xmath102 } as the microlearners private utilities , particularly if the coin is large , we will invariably encounter a very bad signal - to - noise problem . for this choice of utilities , the effects of the actions taken by node @xmath33 on its utility may be `` swamped '' and effectively invisible , since there are so many other processes going into determining @xmath78 s value . this makes it hard for @xmath33 to discern the echo of its actions and learn how to improve its private utility . it also means that @xmath33 will find it difficult to decide how best to act once learning has completed , since so much of what s important to @xmath33 is decided by processes outside of @xmath33s immediate purview . in such a scenario , there is nothing that @xmath33 s microlearner can do to reliably achieve high intelligence . in addition to this `` observation - driven '' signal / noise problem , there is an `` action - driven '' one . for reasons discussed in observation 7.1 above , we can define a distribution @xmath151 reflecting what @xmath33 does / doesnt know concerning the actual state of the outside world @xmath152 at time 0 . if the node @xmath33 chooses its actions in a bayes - optimal manner , then @xmath153 $ ] , where @xmath141 runs over the allowed action components of @xmath33 at time 0 . since this will differ from @xmath154 $ ] in general , this bayes - optimal node s intelligence will be less than 1 for the particular @xmath35 at hand , in general . moreover , the less @xmath99 s ultimate value ( after the application of @xmath63 , etc . ) depends on @xmath135 , the smaller the difference in these two argmax - based @xmath141 s , and therefore the higher the intelligence of @xmath33 , in general.s ultimate value to not depend on @xmath135 . ] we would like a measure of @xmath99 that captures these efects , but without depending on function maximization or any other detailed aspects of how the node determines its actions . one natural way to do this is via the * ( utility ) learnability * : given a measure @xmath131 restricted to a manifold @xmath63 , the ( @xmath155 ) utility learnability of a utility @xmath99 for a node @xmath33 at @xmath35 is : @xmath156 * intelligence learnability * is defined the same way , with @xmath157 replaced by @xmath158 . note that any affine transformation of @xmath99 has no effect on either the utility learnability @xmath159 or the associated intelligence learnability , @xmath160 . the integrand in the numerator of the definition of learnability reflects how much of the change in @xmath99 that results from replacing @xmath129 with @xmath161 is due to the change in @xmath33 s @xmath122 state ( the `` signal '' ) . the denominator reflects how much of the change in @xmath99 that results from replacing @xmath35 with @xmath83 is due to the change in the @xmath162 states of nodes other than @xmath33 ( the `` noise '' ) . so learnability quantifies how easy it is for the microlearner to discern the `` echo '' of its behavior in the utility function @xmath99 . our presumption is that the microlearning algorithm will achieve higher intelligence if provided with a more learnable private utility . intuitively , the ( utility ) * differential learnability * of @xmath99 at a point @xmath35 is the learnability with @xmath130 restricted to an infinitesimal ball about @xmath35 . we formalize it as the following ratio of magnitudes of a pair of gradients , one involving @xmath33 , and one involving @xmath152 : @xmath163 note that a particular value of differential utility learnability , by itself , has no significance . simply rescaling the units of @xmath103 will change that value . rather what is important is the ratio of differential learnabilities , at the same @xmath35 , for different @xmath99 s . such a ratio quantifies the relative preferability of those @xmath99 s . one nice feature of differential learnability is that unlike learnability , it does not depend on choice of some measure @xmath164 . this independence can lead to trouble if one is not careful however , and in particular if one uses learnability for purposes other than choosing between utility functions . for example , in some situations , the coin designer will have the option of enlarging the set of variables from the rest of the coin that are `` input '' to some node @xmath33 at @xmath162 and that therefore can be used by @xmath33 to decide what action to take . intuitively , doing so will not affect the rl `` signal '' for @xmath33 s microlearner ( the magnitude of the potential `` echo '' of @xmath33 s actions are not modified by changing some aspect of how it chooses among those actions ) . however it _ will _ reduce the `` noise '' , in that @xmath33 s microlearner now knows more about the state of the rest of the system . in the full integral version of learnability , this effect can be captured by having the support of @xmath164 restricted to reflect the fact that the extra inputs to @xmath33 at @xmath122 are correlated with the @xmath122 state of the external system . in differential learnability however this is not possible , precisely because no measure @xmath164 occurs in its definition . so we must capture the reduction in noise in some other fashion . occurring in the definition of differential learnability with something more nuanced . for example , one may wish to replace it with the maximum of the dot product of @xmath165 with any @xmath39 vector @xmath166 , subject not only to the restrictions that @xmath167 and @xmath168 * 0 * , but also subject to the restriction that @xmath166 must lie in the tangent plane of @xmath63 at @xmath35 . the first two restrictions , in concert with the extra restriction that @xmath169 * 0 * , give the original definition of the noise term . if they are instead joined with the third , new restriction , they will enforce any applicable coupling between the state of @xmath33 at time 0 and the rest of the system at time 0 . solving with lagrange multipliers , we get @xmath170 , where @xmath171 is the normal to @xmath63 at @xmath35 , @xmath172 , and @xmath173 while @xmath174 . as a practical matter though , it is often simplest to assume that the @xmath135 can vary arbitrarily , independent of @xmath103 , so that the noise term takes the form in eq . 3 . ] alternatively , if the extra variables are being input to @xmath33 for all @xmath108 , not just at @xmath162 , and if @xmath33 `` pays attention '' to those variables for all @xmath108 , then by incorporating those changes into our system @xmath63 itself has changed , @xmath175 . hypothesize that at those @xmath17 the node @xmath33 is capable of modifying its actions to `` compensate '' for what ( due to our augmentation of @xmath33 s inputs ) @xmath33 now knows to be going on outside of it . under this hypothesis , those changes in those external events will have less of an effect on the ultimate value of @xmath97 than they would if we had not made our modification . in this situation , the noise term has been reduced , so that the differential learnabiliity properly captures the effect of @xmath33 s having more inputs . another potential danger to bear in mind concerning differential learnability is that it is usually best to consider its average over a region , in particular over points with less than maximal intelligence . it is really designed for such points ; in fact , at the intelligence - maximizing @xmath35 , @xmath176 . whether in its differential form or not , and whether referring to utilities or intelligence , learnability is not meant to capture all factors that will affect how high an intelligence value a particular microlearner will achieve . such an all - inclusive definition is not possible , if for no other reason the fact that there are many such factors that are idiosyncratic to the particular microlearner used . beyond this though , certain more general factors that affect most popular learning algorithms , like the curse of dimensionality , are also not ( explicitly ) designed into learnability . learnability is not meant to provide a full characterization of performance that is what intelligence is designed to do . rather ( relative ) learnability is ony meant to provide a _ guide _ for how to improve performance . a system that has infinite ( differential , intelligence ) learnability for all its personal utilities is said to be `` perfectly '' ( differential , intelligence ) learnable . it is straight - forward to prove that a system is perfectly learnable @xmath177 iff @xmath178 can be written as @xmath179 for some function @xmath180 . ( see the discussion below on the general condition for a system s being perfectly factored . ) with these definitions in hand , we can now present ( a portion of ) one descriptive framework for coins . in this subsection , after discussing salient characteristics in general , we present some theorems concerning the relationship between personal utilities and the salient characteristic we choose to concentrate on . we then discus how to use these theorems to induce that salient characteristic in a coin . the starting point with a descriptive framework is the identification of `` salient characteristics of a coin which one strongly expects to be associated with its having large world utility '' . in this chapter we will focus on salient characteristics that concern the relationship between personal and world utilities . these characteristics are formalizations of the intuition that we want coins in which the competent greedy pursuit of their private utilities by the microlearners results in large world utility , without any bottlenecks , toc , `` frustration '' ( in the spin glass sense ) or the like . one natural candidate for such a characteristic , related to pareto optimality @xcite , is * weak triviality*. it is defined by considering any two worldlines @xmath35 and @xmath83 both of which are consistent with the system s dynamics ( _ i.e. _ , both of which lie on @xmath63 ) , where for every node @xmath33 , @xmath181 . , and require only that both of the `` partial vectors '' @xmath182 and @xmath183 obey the relevant dynamical laws , and therefore lie in @xmath184 . ] if for any such pair of worldlines where one `` pareto dominates '' the other it is necessarily true that @xmath185 , we say that the system is weakly trivial . we might expect that systems that are weakly trivial for the microlearners private utilities are configured correctly for inducing large world utility . after all , for such systems , if the microlearners collectively change @xmath35 in a way that ends up helping all of them , then necessarily the world utility also rises . more formally , for a weakly trivial system , the maxima of @xmath78 are pareto - optimal points for the personal utilities ( although the reverse need not be true ) . as it turns out though , weakly trivial systems can readily evolve to a world utility @xmath186 , one that often involves toc . to see this , consider automobile traffic in the absence of any traffic control system . let each node be a different driver , and say their private utilities are how quickly they each individually get to their destination . identify world utility as the sum of private utilities . then by simple additivity , for all @xmath35 and @xmath83 , whether they lie on @xmath63 or not , if @xmath187 it follows that @xmath185 ; the system is weakly trivial . however as any driver on a rush - hour freeway with no carpool lanes or metering lights can attest , every driver s pursuing their own goal definitely does not result in acceptable throughput for the system as a whole ; modifications to private utility functions ( like fines for violating carpool lanes or metering lights ) would result in far better global behavior . a system s being weakly trivial provides no assurances regarding world utility . this does not mean weak triviality is never of use . for example , say that for a set of weakly trivial personal utilities each agent can guarantee that _ regardless of what the other agents do _ , its utility is above a certain level . assume further that , being risk - averse , each agent chooses an action with such a guarantee . say it is also true that the agents are provided with a relatively large set of candidate guaranteed values of their utilities . under these circumstances , the system s being weakly trivial provides some assurances that world utility is not too low . moreover , if the overhead in enforcing such a future - guaranteeing scheme is small , and having a sizable set of guaranteed candidate actions provided to each of the agents does not require an excessively centralized infrastructure , we can actually employ this kind of scheme in practice . indeed , in the extreme case , one can imagine that every agent is guaranteed exactly what its utility would be for every one of its candidate actions . ( see the discussion on general equilibrium in the background section above . ) in this situation , nash equilibria and pareto optimal points are identical , which due to weak triviality means that the point maximizing @xmath78 is a nash equilibrium . however in any less extreme situation , the system may not achieve a value of world utility that is close to optimal . this is because even for weakly trivial systems a pareto optimal point may have poor world utility , in general . situations where one has guarantees of lower bounds on one s utility are not too common , but they do arise . one important example is a round of trades in a computational market ( see the background section above ) . in that scenario , there is an agent - indexed set of functions \{@xmath188 } and the personal utility of each agent @xmath189 is given by @xmath190 , where @xmath191 is the end of the round of trades . there is also a function @xmath192 @xmath193 that is a monotonically increasing function of its arguments , and world utility @xmath78 is given by @xmath194 . so the system is weakly trivial . in turn , each @xmath195 is determined solely by the `` allotment of goods '' possessed by @xmath33 , as specified in the appropriate components of @xmath196 . to be able to remove uncertainty about its future value of @xmath197 in this kind of system , in determining its trading actions each agent @xmath33 must employ some scheme like inter - agent contracts . this is because without such a scheme , no agent can be assured that if it agrees to a proposed trade with another agent that the full proposed transaction of that trade actually occurs . given such a scheme , if in each trade round @xmath17 each agent @xmath33 myopically only considers those trades that are assured of increasing the corresponding value of @xmath197 , then we are guaranteed that the value of the world utility is not less than the initial value @xmath198 . the problem with using weak triviality as a general salient characteristic is precisely the fact that the individual microlearners @xmath199 greedy . in a coin , there is no system - wide incentive to replace @xmath35 with a different worldline that would improve everybody s private utility , as in the definition of weak triviality . rather the incentives apply to each microlearner individually and motivate the learners to behave in a way that may well hurt some of them . so weak triviality is , upon examination , a poor choice for the salient characteristic of a coin . one alternative to weak triviality follows from consideration of the stricture that we must ` expect ' a salient characteristic to be coupled to large world utility in a running real - world coin . what can we reasonably expect about a running real - world coin ? we can not assume that all the private utilities will have large values witness the traffic example . but we @xmath200 assume that if the microlearners are well - designed , each of them will be doing close to as well it can _ given the behavior of the other nodes_. in other words , within broad limits we can assume that the system is more likely to be in @xmath35 than @xmath83 if for all @xmath33 , @xmath201 . we define a system to be * coordinated * iff for any such @xmath35 and @xmath83 lying on @xmath63 , @xmath185 . ( again , an obvious variant is to restrict @xmath202 , and require only that both @xmath183 and @xmath182 lie in @xmath184 . ) traffic systems are @xmath74 coordinated , in general . this is evident from the simple fact that if all drivers acted as though there were metering lights when in fact there were nt any , they would each be behaving with lower intelligence given the actions of the other drivers ( each driver would benefit greatly by changing its behavior by no longer pretending there were metering lights , etc . ) . but nonetheless , world utility would be higher . like weak triviality , coordination is intimately related to the economics concept of pareto optimality . unfortunately , there is not room in this chapter to present the mathematics associated with coordination and its variants . we will instead discuss a third candidate salient characteristic of coins , one which like coordination ( and unlike weak triviality ) we can reasonably expect to be associated with large world utility this alternative fixes weak triviality not by replacing the personal utilities \{@xmath97 } with the intelligences \{@xmath203 } as coordination does , but rather by only considering worldlines whose difference at time 0 involves a single node . this results in this alternative s being related to nash equilibria rather than pareto optimality . say that our coin s worldline is @xmath35 . let @xmath83 be any other worldline where @xmath204 , and where @xmath205 . now restrict attention to those @xmath83 where at @xmath162 @xmath35 and @xmath83 differ only for node @xmath33 . if for all such @xmath83 @xmath206 = sgn[g(\underline{\zeta } ) - g(\underline{\zeta}_{,t<0 } \bullet c(\underline{\zeta}'_{,0 } ) ) ] \ ; , \ ] ] and if this is true for all nodes @xmath33 , then we say that the coin is * factored * for all those utilities \{@xmath102 } ( at @xmath35 , with respect to time 0 and the utility @xmath78 ) . for a factored system , for any node @xmath33 , _ given the rest of the system _ , if the node s state at @xmath162 changes in a way that improves that node s utility over the rest of time , then it necessarily also improves world utility . colloquially , for a system that is factored for a particular microlearner s private utility , if that learner does something that improves that personal utility , then everything else being equal , it has also done something that improves world utility . of two potential microlearners for controlling node @xmath33 ( _ i.e. _ , two potential @xmath145 ) whose behavior until @xmath122 is identical but which differ there , the microlearner that is smarter with respect to @xmath14 will always result in a larger @xmath14 , by definition of intelligence . accordingly , for a factored system , the smarter microlearner is also the one that results in better @xmath78 . so as long as we have deployed a sufficiently smart microlearner on @xmath33 , we have assured a good @xmath78 ( given the rest of the system ) . formally , this is expressed in the fact @xcite that for a factored system , for all nodes @xmath33 , @xmath207 one can also prove that nash equilibria of a factored system are local maxima of world utility . note that in keeping with our behaviorist perspective , nothing in the definition of factored requires the existence of private utilities . indeed , it may well be that a system having private utilities \{@xmath134 } is factored , but for personal utilities \{@xmath102 } that differ from the \{@xmath134}. a system s being factored does @xmath74 mean that a change to @xmath103 that improves @xmath146 can not also hurt @xmath208 for some @xmath209 . intuitively , for a factored system , the side effects on the rest of the system of @xmath33 s increasing its own utility do not end up decreasing world utility but can have arbitrarily adverse effects on other private utilities . ( in the language of economics , no stipulation is made that @xmath33 s `` costs are endogenized . '' ) for factored systems , the separate microlearners successfully pursuing their separate goals do not frustrate each other _ as far as world utility is concerned_. in addition , if @xmath210 is factored with respect to @xmath78 , then a change to @xmath211 that improves @xmath212 improves @xmath213 . but it may @xmath214 some @xmath215 and/or @xmath216 . ( this is even true for a discounted sum of rewards personal utility , so long as @xmath217 . ) an example of this would be an economic system cast as a single individual , @xmath33 , together with an environment node , where @xmath78 is a steeply discounted sum of rewards @xmath33 receives over his / her lifetime , @xmath217 , and @xmath218 , @xmath219 . for such a situation , it may be appropriate for @xmath33 to live extravagantly at the time @xmath61 , and `` pay for it '' later . as an instructive example of the ramifications of eq . 5 , say node @xmath33 is a conventional computer . we want @xmath220 to be as high as possible , i.e. , given the state of the rest of the system at time 0 , we want computer @xmath33 s state then to be the best possible , as far as the resultant value of @xmath78 is concerned . now a computer s `` state '' consists of the values of all its bits , including its code segment , i.e. , including the program it is running . so for a factored personal utility @xmath102 , if the program running on the computer is better than most others as far as @xmath102 is concerned , then it is also better than most other programs as far as @xmath78 is concerned . our task as coin designers engaged in coin initialization or macrolearning is to find such a program and such an associated @xmath102 . one way to approach this task is to restrict attention to programs that consist of rl algorithms with private utility specified in the bits \{@xmath75 } of @xmath33 . this reduces the task to one of finding a private utility \{@xmath75 } ( and thereby fully specifying @xmath103 ) such that our rl algorithm working with that private utility has high @xmath138 , i.e. , such that that algorithm outperforms most other programs as far as the personal utility @xmath97 is concerned . perhaps the simplest way to address this reduced task is to exploit the fact that for a good enough rl algorithm @xmath221 will be large , and therefore adopt such an rl algorithm and fix the private utility to equal @xmath97 . in this way we further reduce the original task , which was to search over all personal utilities @xmath97 and all programs @xmath222 to find a pair such that both @xmath97 is factored with respect to @xmath78 and there are relatively few programs that outperform @xmath222 , as far as @xmath97 . the task is now instead to search over all private utilities \{@xmath75 } such that both \{@xmath75 } is factored with respect to @xmath78 and such that there are few programs ( _ of any sort _ , rl - based or not ) that outperform our rl algorithm working on \{@xmath75 } , as far as that self - same private utility is concerned . the crucial assumption being leveraged in this approach is that our rl algorithm is `` good enough '' , and the reason we want learnable \{@xmath75 } is to help effect this assumption . in general though , we ca nt have both perfect learnability and perfect factoredness . as an example , say that @xmath223 , and that the dynamics is the identity operator : @xmath218 , @xmath224 . then if @xmath225 and the system is perfectly learnable , it is not perfectly factored . this is because perfect learnability requires that @xmath226 for some function @xmath180 . however any change to @xmath103 that improves such a @xmath97 will either help or @xmath214 @xmath87 , depending on the sign of @xmath135 . for the `` wrong '' sign of @xmath135 , this means the system is actually `` anti - factored '' . due to such incompatibility between perfect factoredness and perfect learnability , we must usually be content with having high degree of factoredness and high learnability . in such situations , the emphasis of the macrolearning process should be more and more on having high degree of factoredness as we get closer and closer to a nash equilibrium . this way the system wo nt relax to an incorrect local maximum . in practice of course , a coin will often not be perfectly factored . nor in practice are we always interested only in whether the system is factored at one particular point ( rather than across a region say ) . these issues are discussed in @xcite , where in particular a formal definition of of the * degree of factoredness * of a system is presented . if a system is factored for utilities @xmath227 , then it is also factored for any utilities @xmath228 where for each @xmath33 @xmath229 is a monotonically increasing function of @xmath97 . more generally , the following result characterizes the set of all factored personal utilities : * theorem 1 : * a system is factored at all @xmath230 iff for all those @xmath35 , @xmath231 , we can write @xmath232 for some function @xmath233 such that @xmath234 for all @xmath230 and associated @xmath78 values . ( the form of the \{@xmath97 } off of @xmath63 is arbitrary . ) * proof : * for fixed @xmath103 and @xmath107 , any change to @xmath103 which keeps @xmath235 on @xmath63 and which at the same time increases @xmath236 must increase @xmath237 , due to the restriction on @xmath238 . this establishes the backwards direction of the proof . for the forward direction , write @xmath239 . define this formulation of @xmath97 as @xmath240 , which we can re - express as @xmath241 . now since the system is factored , @xmath242 , @xmath243 @xmath244 @xmath245 so consider any situation where the system is factored , and the values of @xmath78 , @xmath246 , and @xmath135 are specified . then we can find _ any _ @xmath103 consistent with those values ( _ i.e. _ , such that our provided value of @xmath78 equals @xmath247 ) , evaluate the resulting value of @xmath248 , and know that we would have gotten the same value if we had found a different consistent @xmath103 . this is true for all @xmath249 . therefore the mapping @xmath250 is single - valued , and we can write @xmath251 . * qed . * by thm . 1 , we can ensure that the system is factored without any concern for @xmath63 , by having each @xmath252 . alternatively , by only requiring that @xmath253 does @xmath254 ( _ i.e. _ , does @xmath255 ) , we can access a broader class of factored utilities , a class that @xmath0 depend on @xmath63 . loosely speaking , for those utilities , we only need the projection of @xmath256 onto @xmath257 to be parallel to the projection of @xmath258 onto @xmath257 . given @xmath78 and @xmath63 , there are infinitely many @xmath259 having this projection ( the set of such @xmath260 form a linear subspace of @xmath39 ) . the partial differential equations expressing the precise relationship are discussed in @xcite . as an example of the foregoing , consider a ` team game ' ( also known as an ` exact potential game ' @xcite ) in which @xmath261 for all @xmath33 . such coins are factored , trivially , regardless of @xmath63 ; if @xmath97 rises , then @xmath78 must as well , by definition . ( alternatively , to confirm that team games are factored just take @xmath262 in thm . 1 . ) on the other hand , as discussed below , coins with ` wonderful life ' personal utilities are also factored , but the definition of such utilities depends on @xmath63 . due to their often having poor learnability and requiring centralized communication ( among other infelicities ) , in practice team game utilities often are poor choices for personal utilities . accordingly , it is often preferable to use some other set of factored utilities . to present an important example , first define the ( @xmath122 ) * effect set * of node @xmath33 at @xmath35 , @xmath263 , as the set of all components @xmath264 for which @xmath265 . define the effect set @xmath266 with no specification of @xmath35 as @xmath267 . ( we take this latter definition to be the default meaning of `` effect set '' . ) we will also find it useful to define @xmath268 as the set of components of the space @xmath39 that are not in @xmath266 . intuitively , @xmath33 s effect set is the set of all components @xmath264 which would be affected by a change in the state of node @xmath33 at time 0 . ( they may or may not be affected by changes in the @xmath122 states of the other nodes . ) note that the effect sets of different nodes may overlap . the extension of the definition of effect sets for times other than 0 is immediate . so is the modification to have effect sets only consist of those components @xmath34 that vary with with the state of node @xmath33 at time 0 , rather than consist of the full vectors @xmath31 possessing such a component . these modifications will be skipped here , to minimize the number of variables we must keep track of . next for any set @xmath269 of components ( @xmath270 ) , define @xmath271 as the `` virtual '' vector formed by clamping the @xmath269-components of @xmath35 to an arbitrary fixed value . ( in this paper , we take that fixed value to be @xmath272 for all components listed in @xmath269 . ) consider in particular a * wonderful life * set @xmath269 . the value of the * wonderful life utility * ( wlu for short ) for @xmath269 at @xmath35 is defined as : @xmath273 in particular , the wlu for the effect set of node @xmath33 is @xmath274 , which for @xmath249 can be written as @xmath275 . we can view @xmath33 s effect set wlu as analogous to the change in world utility that would have arisen if node @xmath33 `` had never existed '' . ( hence the name of this utility - cf . the frank capra movie . ) note however , that @xmath276 is a purely `` fictional '' , counter - factual operation , in the sense that it produces a new @xmath35 without taking into account the system s dynamics . indeed , no assumption is even being made that @xmath277 is consistent with the dynamics of the system . the sequence of states the node @xmath33 is clamped to in the definition of the wlu need not be consistent with the dynamical laws embodied in @xmath63 . this dynamics - independence is a crucial strength of the wlu . it means that to evaluate the wlu we do _ not _ try to infer how the system would have evolved if node @xmath33 s state were set to 0 at time 0 and the system evolved from there . so long as we know @xmath35 extending over all time , and so long as we know @xmath78 , we know the value of wlu . this is true even if we know nothing of the dynamics of the system . an important example is effect set wonderful life utilities when the set of all nodes is partitioned into ` subworlds ' in such a way that all nodes in the same subworld @xmath278 share substantially the same effect set . in such a situation , all nodes in the same subworld @xmath278 will have essentially the same personal utilities , exactly as they would if they used team game utilities with a `` world '' given by @xmath278 . when all such nodes have large intelligence values , this sharing of the personal utility will mean that all nodes in the same subworld are acting in a coordinated fashion , loosely speaking . the importance of the wlu arises from the following results : * theorem 2 : * i ) a system is factored at all @xmath230 iff for all those @xmath35 , @xmath231 , we can write @xmath279 for some function @xmath280 such that @xmath281 for all @xmath230 and associated @xmath78 values . ( the form of the \{@xmath97 } off of @xmath63 is arbitrary . ) \ii ) in particular , a coin is factored for personal utilities set equal to the associated effect set wonderful life utilities . * proof : * to prove ( i ) , first write @xmath282 . for all @xmath249 , @xmath283 is independent of @xmath103 , and so by definition of @xmath60 it is a single - valued function of @xmath135 for such @xmath35 . therefore @xmath284 for some function @xmath285 . accordingly , by thm . 1 , for \{@xmath97 } of the form stipulated in ( i ) , the system is factored . going the other way , if the system is factored , then by thm . 1 it can be written as @xmath237 . since both @xmath107 and @xmath286 , we can rewrite this as @xmath287_{,t<0 } , [ \hat{}c^{eff}_{\eta}]_{\;\hat{}\eta,0 } , g(\underline{\zeta}))$ ] . * qed . * part ( ii ) of the theorem follows immediately from part ( i ) . for pedagogical value though , here we instead derive it directly . first , since @xmath288 is independent of @xmath264 for all @xmath289 , so is the @xmath290 vector @xmath291 , _ i.e. _ , @xmath292_{\eta',t } = \vec{0 } \;\ ; \forall ( \eta ' , t ) \in c^{eff}_\eta$ ] . this means that viewed as a @xmath107-parameterized function from @xmath293 to @xmath290 , @xmath294 is a single - valued function of the @xmath295 components . therefore @xmath296 can only depend on @xmath107 and the non-@xmath33 components of @xmath129 . accordingly , the wlu for @xmath266 is just @xmath78 minus a term that is a function of @xmath107 and @xmath295 . by choosing @xmath233 in thm . 1 to be that difference , we see that @xmath33 s effect set wlu is of the form necessary for the system to be factored . * qed . * as a generalization of ( ii ) , the system is factored if each node @xmath33 s personal utility is ( a monotonically increasing function of ) the wlu for a set @xmath297 that contains @xmath266 . for conciseness , except where explicitly needed , for the remainder of this subsection we will suppress the argument `` @xmath107 '' , taking it to be implicit . the next result concerning the practical importance of effect set wlu is the following : * theorem 3 : * let @xmath269 be a set containing @xmath298 . then @xmath299 * proof : * writing it out , @xmath300 the second term in the numerator equals 0 , by definition of effect set . dividing by the similar expression for @xmath301 then gives the result claimed . * qed . * so if we expect that ratio of magnitudes of gradients to be large , effect set wlu has much higher learnability than team game utility while still being factored , like team game utility . as an example , consider the case where the coin is a very large system , with @xmath33 being only a relatively minor part of the system ( _ e.g. _ , a large human economy with @xmath33 being a `` typical john doe living in peoria illinois '' ) . often in such a system , for the vast majority of nodes @xmath302 , how @xmath78 varies with @xmath303 will be essentially independent of the value @xmath103 . ( for example , how gdp of the us economy varies with the actions of our john doe from peoria , illinois will be independent of the state of some jane smith living in los angeles , california . ) in such circumstances , thm . 3 tells us that the effect set wonderful life utility for @xmath33 will have a far larger learnability than does the world utility . for any fixed @xmath269 , if we change the clamping operation ( _ i.e. _ , change the choice of the `` arbitrary fixed value '' we clamp each component to ) , then we change the mapping @xmath304 , and therefore change the mapping @xmath305 . accordingly , changing the clamping operation can affect the value of @xmath306 evaluated at some point @xmath129 . therefore , by thm . 3 , changing the clamping operation can affect @xmath307 . so properly speaking , for any choice of @xmath269 , if we are going to use @xmath308 , we should set the clamping operation so as to maximize learnability . for simplicity though , in this paper we will ignore this phenomenon , and simply set the clamping operation to the more or less `` natural '' choice of * 0 * , as mentioned above . next consider the case where , for some node @xmath33 , we can write @xmath309 as @xmath310 . say it is also true that @xmath33 s effect set is a small fraction of the set of all components . in this case it often true that the values of @xmath311 are much larger than those of @xmath312 , which means that partial derivatives of @xmath311 are much larger than those of @xmath312 . in such situations the effect set wlu is far more learnable than the world utility , due to the following results : * theorem 4 : * if for some node @xmath33 there is a set @xmath269 containing @xmath313 , a function @xmath314 , and a function @xmath315 , such that @xmath316 , then @xmath317 * proof : * for brevity , write @xmath318 and @xmath319 both as functions of full @xmath320 , just such functions that are only allowed to depend on the components of @xmath35 that lie in @xmath269 and those components that do not lie in @xmath269 , respectively . then the @xmath269 wlu for node @xmath33 is just @xmath321 . since in that second term we are clamping all the components of @xmath35 that @xmath312 cares about , for this personal utility @xmath322 . so in particular @xmath323 . now by definition of effect set , @xmath324 , since @xmath325 does not contain @xmath298 . so @xmath326 . * qed . * the obvious extensions of thm.s 3 and 4 to effect sets with respect to times other than 0 can also be proven @xcite . an important special case of thm . 4 is the following : * corollary 1 : * if for some node @xmath33 we can write \i ) @xmath327_{t\ge0 } ) + g_3(\underline{\zeta}_{,t<0})$ ] for some set @xmath269 containing @xmath313 , and if \ii ) @xmath328_{\sigma } ) ||$ ] , then @xmath329 . in practice , to assure that condition ( i ) of this corollary is met might require that @xmath269 be a proper superset of @xmath266 . countervailingly , to assure that condition ( ii ) is met will usually force us to keep @xmath269 as small as possible . one can often remove elements from an effect set and still have the results of this section hold . most obviously , if ( @xmath53 , t ) @xmath330 but @xmath331 = * 0 * , we can remove ( @xmath53 , t ) from @xmath266 without invalidating our results . more generally , if there is a set @xmath332 such that for each component ( @xmath333 the chain rule term @xmath334 \;\cdot \ ; [ \partial_{\underline{\zeta}_{\eta,0;i } } [ c(\underline{\zeta}_{,0})]_{\eta',t}]$ ] = 0 , then the effects on @xmath78 of changes to @xmath103 that are `` mediated '' by the members of @xmath335 cancel each other out . in this case we can usually remove the elements of @xmath335 from @xmath266 with no ill effects . usually the mathematics of a descriptive framework a formal investigation of the salient characteristics will not provide theorems of the sort , `` if you modify the coin the following way at time @xmath17 , the value of the world utility will increase . '' rather it provides theorems that relate a coin s salient characteristics with the general properties of the coin s entire history , and in particular with those properties embodied in @xmath63 . in particular , the salient characteristic that we are concerned with in this chapter is that the system be highly intelligent for personal utilities for which it is factored , and our mathematics concerns the relationship between factoredness , intelligence , personal utilities , effect sets , and the like . more formally , the desideratum associated with our salient characteristic is that we want the coin to be at a @xmath35 for which there is some set of \{@xmath97 } ( not necessarily consisting of private utilities ) such that ( a ) @xmath35 is factored for the \{@xmath97 } , and ( b ) @xmath132 is large for all @xmath33 . now there are several ways one might try to induce the coin to be at such a point . one approach is to have each algorithm controlling @xmath33 explicitly try to `` steer '' the worldline towards such a point . in this approach @xmath33 need nt even have a private utility in the usual sense . ( the overt `` goal '' of the algorithm controlling @xmath33 involves finding a @xmath35 with a good associated extremum over the class of all possible @xmath97 , independent of any private utilities . ) now initialization of the coin , _ i.e. _ , fixing of @xmath129 , involves setting the algorithm controlling @xmath33 , in this case to the steering algorithm . accordingly , in this approach to initialization , we fix @xmath129 to a point for which there is some special @xmath97 such that both @xmath336 is factored for @xmath97 , and @xmath337 is large . there is nothing peculiar about this . what is odd though is that in this approach we do not know what that `` special '' @xmath97 is when we do that initialization ; it s to be determined , by the unfolding of the system . in this chapter we concentrate on a different approach , which can involve either initialization or macrolearning . in this alternative we deploy the \{@xmath97 } as the microlearners private utilities at some @xmath338 , in a process not captured in @xmath63 , so as to induce a factored coin that is as intelligent as possible . ( it is with that `` deploying of the \{@xmath97 } '' that we are trying to induce our salient characteristic in the coin . ) since in this approach we are using private utilities , we can replace intelligence with its surrogate , learnability . so our task is to choose \{@xmath97 } which are as learnable as possible while still being factored . solving for such utilities can be expressed as solving a set of coupled partial differential equations . those equations involve the tangent plane to the manifold @xmath63 , a functional trading off ( the differential versions of ) degree of factoredness and learnability , and any communication constraints on the nodes we must respect . while there is not space in the current chapter to present those equations , we can note that they are highly dependent on the correlations among the components of @xmath31 . so in this approach , in coin initialization we use some preliminary guesses as to those correlations to set the initial \{@xmath97}. for example , the effect set of a node constitutes all components @xmath339 that have non - zero correlation with @xmath103 . furthermore , by thm . 2 the system is factored for effect set wlu personal utilities . and by coroll . 1 , for small effect sets , the effect set wlu has much greater differential utility learnability than does @xmath78 . extending the reasoning behind this result to all @xmath35 ( or at least all likely @xmath35 ) , we see that for this scenario , the descriptive framework advises us to use wonderful life private utilities based on ( guesses for ) the associated effect sets rather than the team game private utilities , @xmath340 . in macrolearning we must instead run - time estimate an approximate solution to our partial differential equations , based on statistical inference . as an example , we might start with an initial guess as to @xmath33 s effect set , and set its private utility to the associated wlu . but then as we watch the system run and observe the correlations among the components of @xmath35 , we might modify which components we think comprise @xmath33 s effect set , and modify @xmath33 s personal utility accordingly . as implied above , often one can perform reasonable coin initialization and/or macrolearning without writing down the partial differential equations governing our salient characteristic explicitly . simply `` hacking '' one s way to the goal of maximizing both degree of factoredness and intelligibility , for example by estimating effect sets , often results in dramatic improvement in performance . this is illustrated in the experiments recounted in the next two subsections . even if we do nt exactly know the effect set of each node @xmath33 , often we will be able to make a reasonable guess about which components of @xmath35 comprise the `` preponderance '' of @xmath33 s effect set . we call such a set a * guessed effect set*. as an example , often the primary effects of changes to @xmath33 s state will be on the future state of @xmath33 , with only relatively minor effects on the future states of other nodes . in such situations , we would expect to still get good results if we approximated the effect set wlu of each node @xmath33 with a wlu based on the guessed effect set @xmath341 . in other words , we would expect to be able to replace wlu@xmath342 with wlu@xmath343 and still get good performance . this phenomenon was borne out in the experiments recounted in @xcite that used coin initialization for distributed control of network packet routing . in a conventional approach to packet routing , each router runs what it believes ( based on the information available to it ) to be a shortest path algorithm ( spa ) , _ i.e. _ , each router sends its packets in the way that it surmises will get those packets to their destinations most quickly . unlike with an approach based on our coin framework , with spa - based routing the routers have no concern for the possible deleterious side - effects of their routing decisions on the global performance ( _ e.g. _ , they have no concern for whether they induce bottlenecks ) . we performed simulations in which we compared such a coin - based routing system to an spa - based system . for the coin - based system @xmath78 was global throughput and no macrolearning was used . the coin initialization was to have each router s private utility be a wlu based on an associated guessed effect set generated _ a priori_. in addition , the coin - based system was realistic in that each router s reinforcement algorithm had imperfect knowledge of the state of the system . on the other hand , the spa was an idealized `` best - possible '' system , in which each router knew exactly what the shortest paths were at any given time . despite the handicap that this disparity imposed on the coin - based system , it achieved significantly better global throughput in our experiments than did the perfect - knowledge spa - based system , and in particular , avoided the braess paradox that was built - in to some of those systems @xcite . the experiments in @xcite were primarily concerned with the application of packet - routing . to concentrate more precisely on the issue of coin initialization , we ran subsequent experiments on variants of arthur s famous `` el farol bar problem '' ( see section [ sec : lit ] ) . to facilitate the analysis we modified arthur s original problem to be more general , and since we were not interested in directly comparing our results to those in the literature , we used a more conventional ( and arguably `` dumber '' ) machine learning algorithm than the ones investigated in @xcite . in this formulation of the bar problem @xcite , there are @xmath344 agents , each of whom picks one of seven nights to attend a bar the following week , a process that is then repeated . in each week , each agent s pick is determined by its predictions of the associated rewards it would receive . these predictions in turn are based solely upon the rewards received by the agent in preceding weeks . an agent s `` pick '' at week @xmath17 ( _ i.e. _ , its node s state at that week ) is represented as a unary seven - dimensional vector . ( see the discussion in the definitions subsection of our representing discrete variables as euclidean variables . ) so @xmath33 s zeroing its state in some week , as in the cl@xmath345 operation , essentially means it elects not to attend any night that week . the world utility is @xmath346 where : @xmath347 ; @xmath348 is the total attendance on night @xmath349 at week @xmath17 ; @xmath350 ; and @xmath351 and each of the \{@xmath352 } are real - valued parameters . intuitively , the `` world reward '' @xmath222 is the sum of the global `` rewards '' for each night in each week . it reflects the effects in the bar as the attendance profile of agents changes . when there are too few agents attending some night , the bar suffers from lack of activity and therefore the global reward for that night is low . conversely , when there are too many agents the bar is overcrowded and the reward for that night is again low . note that @xmath353 reaches its maximum when its argument equals @xmath351 . in these experiments we investigate two different @xmath354 s . one treats all nights equally ; @xmath355 $ ] . the other is only concerned with one night ; @xmath356 $ ] . in our experiments , @xmath357 and @xmath344 is chosen to be 4 times larger than the number of agents necessary to have @xmath351 agents attend the bar on each of the seven nights , _ i.e. _ , there are @xmath358 agents ( this ensures that there are no trivial solutions and that for the world utility to be maximized , the agents have to `` cooperate '' ) . as explained below , our microlearning algorithms worked by providing a real - valued `` reward '' signal to each agent at each week @xmath17 . each agent s reward function is a surrogate for an associated utility function for that agent . the difference between the two functions is that the reward function only reflects the state of the system at one moment in time ( and therefore is potentially observable ) , whereas the utility function reflects the agent s ultimate goal , and therefore can depend on the full history of that agent across time . we investigated three agent reward functions . one was based on effect set wlu . the other two were `` natural '' rewards included for comparison purposes . with @xmath359 the night selected by @xmath33 , the three rewards are : @xmath360 \mbox{global ( g ) : } \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; & r_{\eta}(\underline{\zeta}_{,t } ) & \equiv r(\underline{\zeta}_{,t } ) = \sum_{k=1}^7 \gamma_k(x_k(\underline{\zeta}_{,t } ) ) \\ [ -.05 in ] \mbox{wonderful life ( wl ) : } \ ; \ ; \ ; & r_{\eta}(\underline{\zeta}_{,t } ) & \equiv r(\underline{\zeta}_{,t } ) - r(\mbox{cl}_{\underline{\zeta}_{\eta , t}}(\underline{\zeta}_{,t } ) ) \\ [ -.05 in ] & & = \gamma_{d_\eta } ( x_{d_\eta } ( \underline{\zeta}_{,t } ) ) - \gamma_{d_\eta } ( x_{d_\eta } ( \mbox{cl}_{\underline{\zeta}_{\eta , t } } ( \underline{\zeta}_{,t})))\end{aligned}\ ] ] the conventional ud reward is a natural `` naive '' choice for the agents reward ; the total reward on each night gets uniformly divided among the agents attending that night . if we take @xmath361 ( _ i.e. _ , @xmath33 s utility is an undiscounted sum of its rewards ) , then for the ud reward @xmath362 , so that the system is weakly trivial . the original version of the bar problem in the physics literature @xcite is the special case where ud reward is used but there are only two `` nights '' in the week ( one of which corresponds to `` staying at home '' ) ; @xmath354 is uniform ; and @xmath363 for some vector @xmath364 , taken to equal ( .6 , .4 ) in the very original papers . so the reward to agent @xmath33 is 1 if it attends the bar and attendance is below capacity , or if it stays at home and the bar is over capacity . reward is 0 otherwise . ( in addition , unlike in our coin - based systems , in the original work on the bar problem the microlearners work by explicitly predicting the bar attendance , rather than by directly modifying behavior to try to increase a reward signal . ) in contrast to the ud reward , providing the g reward at time @xmath17 to each agent results in all agents receiving the same reward . this is the team game reward function , investigated for example in @xcite . for this reward function , the system is automatically factored if we define @xmath365 . however , evaluation of this reward function requires centralized communication concerning all seven nights . furthermore , given that there are 168 agents , g is likely to have poor learnability as a reward for any individual agent . this latter problem is obviated by using the wl reward , where the subtraction of the clamped term removes some of the `` noise '' of the activity of all other agents , leaving only the underlying `` signal '' of how the agent in question affects the utility . so one would expect that with the wl reward the agents can readily discern the effects of their actions on their rewards . even though the conditions in coroll . 1 do nt hold elements of @xmath366 are just @xmath367 , but the contributions of @xmath368 to @xmath78 can not be written as a sum of a @xmath367 contribution and a @xmath369 contribution . ] , this reasoning accords with the implicit advice of coroll . 1 under the approximation of the @xmath162 effect set as @xmath370 . in other words , it agrees with that corollary s implicit advice under the identification of @xmath371 as @xmath33 s @xmath122 guessed effect set . in fact , in this very simple system , we can explicitly calculate the ratio of the wl reward s learnability to that of the g reward , by recasting the system as existing for only a single instant so that @xmath372 exactly and then applying thm . so for example , say that all @xmath373 , and that the number of nodes @xmath344 is evenly divided among the seven nights . the numerator term in thm . 3 is a vector whose components are some of the partials of g evaluated when @xmath374 . this vector is @xmath375 dimensional , one dimension for each of the 7 components of ( the unary vector comprising ) each node in @xmath152 . for any particular @xmath302 and night @xmath4 , the associated partial derivative is @xmath376 $ ] , where as usual `` @xmath377 '' indicates the @xmath4th component of the unary vector @xmath378 . since @xmath379 , for any fixed @xmath4 and @xmath53 , this sum just equals @xmath380 . since there are @xmath375 such terms , after taking the norm we obtain @xmath381 \ ; \sqrt{7(n-1})|$ ] . the denominator term in thm . 3 is the difference between the gradients of the global reward and the clamped reward . these differ on only @xmath382 terms , one term for that component of each node @xmath302 corresponding to the night @xmath33 attends . ( the other @xmath383 terms are identical in the two partials and therefore cancel . ) this yields @xmath381 \ ; [ 1 - e^{1/c } ( 1 - \frac{7}{n-7c } ) ] \ ; \sqrt{n-1}$ ] . combining with the result of the previous paragraph , our ratio is @xmath384 . in addition to this learnability advantage of the wl reward , to evaluate its wl reward each agent only needs to know the total attendance on the night it attended , so no centralized communication is required . finally , although the system wo nt be perfectly factored for this reward ( since in fact the effect set of @xmath33 s action at @xmath17 would be expected to extend a bit beyond @xmath31 ) , one might expect that it is close enough to being factored to result in large world utility . each agent keeps a seven dimensional euclidean vector representing its estimate of the reward for attending each night of the week . at the end of each week , the component of this vector corresponding to the night just attended is proportionally adjusted towards the actual reward just received . at the beginning of the succeeding week , the agent picks the night to attend using a boltzmann distribution with energies given by the components of the vector of estimated rewards , where the temperature in the boltzmann distribution decays in time . ( this learning algorithm is equivalent to claus and boutilier s @xcite independent learner algorithm for multi - agent reinforcement learning . ) we used the same parameters ( learning rate , boltzmann temperature , decay rates , etc . ) for all three reward functions . ( this is an _ extremely _ primitive rl algorithm which we only chose for its pedagogical value ; more sophisticated rl algorithms are crucial for eliciting high intelligence levels when one is confronted with more complicated learning problems . ) figure [ fig : barfig ] presents world reward values as a function of time , averaged over 50 separate runs , for all three reward functions , for both @xmath355 $ ] and @xmath356 $ ] . the behavior with the g reward eventually converges to the global optimum . this is in agreement with the results obtained by crites @xcite for the bank of elevators control problem . systems using the wl reward also converged to optimal performance . this indicates that for the bar problem our approximations of effects sets are sufficiently accurate , _ i.e. _ , that ignoring the effects one agent s actions will have on future actions of other agents does not significantly diminish performance . this reflects the fact that the only interactions between agents occurs indirectly , via their affecting each others reward values . however since the wl reward is more learnable than than the g reward , convergence with the wl reward should be far quicker than with the g reward . indeed , when @xmath356 $ ] , systems using the g reward converge in 1250 weeks , which is 5 times worse than the systems using wl reward . when @xmath355 $ ] systems take 6500 weeks to converge with the g reward , which is more than _ 30 times _ worse than the time with the wl reward . in contrast to the behavior for reward functions based on our coin framework , use of the conventional ud reward results in very poor world reward values , values that deteriorated as the learning progressed . this is an instance of the toc . for example , for the case where @xmath356 $ ] , it is in every agent s interest to attend the same night but their doing so shrinks the world reward `` pie '' that must be divided among all agents . a similar toc occurs when @xmath354 is uniform . this is illustrated in fig . [ fig : attend ] which shows a typical example of daily attendance figures ( \{@xmath385 } ) for each of the three reward functions for @xmath386 . in this example optimal performance ( achieved with the wl reward ) has 6 agents each on 6 separate nights , ( thus maximizing the reward on 6 nights ) , and the remaining 132 agents on one night . figure [ fig : numagents ] shows how @xmath387 performance scales with @xmath344 for each of the reward signals for @xmath356 $ ] . systems using the ud reward perform poorly regardless of @xmath344 . systems using the g reward perform well when @xmath344 is low . as @xmath344 increases however , it becomes increasingly difficult for the agents to extract the information they need from the g reward . ( this problem is significantly worse for uniform @xmath354 . ) because of their superior learnability , systems using the wl reward overcome this signal - to - noise problem ( _ i.e. _ , because the wl reward is based on the _ difference _ between the actual state and the state where one agent is clamped , it is much less affected by the total number of agents ) . in the experiments recounted above , the agents were sufficiently independent that assuming they did not affect each other s actions ( when forming guesses for effect sets ) allowed the resultant wl reward signals to result in optimal performance . in this section we investigate the contrasting situation where we have initial guesses of effect sets that are quite poor and that therefore result in bad global performance when used with wl rewards . in particular , we investigate the use of macrolearning to correct those guessed effect sets at run - time , so that with the corrected guessed effect sets wl rewards will instead give optimal performance . this models real - world scenarios where the system designer s initial guessed effect sets are poor approximations of the actual associated effect sets and need to be corrected adaptively . in these experiments the bar problem is significantly modified to incorporate constraints designed to result in poor @xmath78 when the wl reward is used with certain initial guessed effect sets . to do this we forced the nights actually attended by some of the agents ( followers ) to agree with those attended by other agents ( leaders ) , regardless of what night those followers `` picked '' via their microlearning algorithms . ( for leaders , picked and actually attended nights were always the same . ) we then had the world utility be the sum , over all leaders , of the values of a triply - indexed reward matrix whose indices are the nights that each leader - follower set attends : @xmath388 where @xmath389 is the night the @xmath390 leader attends in week @xmath17 , and @xmath391 and @xmath392 are the nights attended by the followers of leader @xmath4 , in week @xmath17 ( in this study , each leader has two followers ) . we also had the states of each node be one of the integers \{0 , 1 , ... , 6 } rather than ( as in the bar problem ) a unary seven - dimensional vector . this was a bit of a contrivance , since constructions like @xmath393 are nt meaningful for such essentially symbolic interpretations of the possible states @xmath103 . as elaborated below , though , it was helpful for constructing a scenario in which guessed effect set wlu results in poor performance , _ i.e. _ , a scenario in which we can explore the application of macrolearning . to see how this setup can result in poor world utility , first note that the system s dynamics is what restricts all the members of each triple @xmath394 to equal the night picked by leader @xmath4 for week so @xmath395 and @xmath392 are both in leader @xmath4 s actual effect set at week @xmath17 whereas the initial guess for @xmath4 s effect set may or may not contain nodes other than @xmath389 . ( for example , in the bar problem experiments , the guessed effect set does not contain any nodes beyond @xmath389 . ) on the other hand , @xmath78 and @xmath222 are defined for all possible triples ( @xmath396 ) . so in particular , @xmath222 is defined for the dynamically unrealizable triples that can arise in the clamping operation . this fact , combined with the leader - follower dynamics , means that for certain @xmath222 s there exist guessed effect sets such that the dynamics assures poor world utility when the associated wl rewards are used . this is precisely the type of problem that macrolearning is designed to correct . as an example , say each week only contains two nights , 0 and 1 . set @xmath397 and @xmath398 . so the contribution to @xmath78 when a leader picks night 1 is 1 , and when that leader picks night 0 it is 0 , independent of the picks of that leader s followers ( since the actual nights they attend are determined by their leader s picks ) . accordingly , we want to have a private utility for each leader that will induce that leader to pick night 1 . now if a leader s guessed effect set includes both of its followers ( in addition to the leader itself ) , then clamping all elements in its effect set to 0 results in an @xmath222 value of @xmath398 . therefore the associated guessed effect set wlu will reward the leader for choosing night 1 , which is what we want . ( for this case wl reward equals @xmath399 if the leader picks night 1 , compared to reward @xmath400 for picking night 0 . ) however consider having two leaders , @xmath401 and @xmath402 , where @xmath401 s guessed effect set consists of @xmath401 itself together with the two followers of @xmath402 ( rather than together with the two followers of @xmath401 itself ) . so neither of leader @xmath401 s followers are in its guessed effect set , while @xmath401 itself is . accordingly , the three indices to @xmath401 s @xmath222 need not have the same value . similarly , clamping the nodes in its guessed effect set wo nt affect the values of the second and third indices to @xmath401 s @xmath222 , since the values of those indices are set by @xmath401 s followers . so for example , if @xmath402 and its two followers go to night 0 in week 0 , and @xmath401 and its two followers go to night 1 in that week , then the associated guessed effect set wonderful life reward for @xmath401 for week 0 is @xmath403 $ ] . this equals @xmath404 . simply by setting @xmath405 we can ensure that this is negative . conversely , if leader @xmath401 had gone to night 0 , its guessed effect wlu would have been 0 . so in this situation leader @xmath401 will get a greater reward for going to night 0 than for going to night 1 . in this situation , leader @xmath401 s using its guessed effect set wlu will lead it to make the wrong pick . to investigate the efficacy of the macrolearning , two sets of separate experiments were conducted . in the first one the reward matrix @xmath222 was chosen so that if each leader is maximizing its wl reward , but for guessed effect sets that contain none of its followers , then the system evolves to @xmath406 world reward . so if a leader incorrectly guesses that some @xmath269 is its effect set even though @xmath269 does nt contain both of that leader s followers , and if this is true for all leaders , then we are assured of worst possible performance . in the second set of experiments , we investigated the efficacy of macrolearning for a broader spectrum of reward matrices by generating those matrices randomly . we call these two kinds of reward matrices _ worst - case _ and _ random _ reward matrices , respectively . in both cases , if it can modify the initial guessed effect sets of the leaders to include their followers , then macrolearning will induce the system to be factored . the microlearning in these experiments was the same as in the bar problem . all experiments used the wl personal reward with some ( initially random ) guessed effect set . when macrolearning was used , it was implemented starting after the microlearning had run for a specified number of weeks . the macrolearner worked by estimating the correlations between the agents selections of which nights to attend . it did this by examining the attendances of the agents over the preceding weeks . given those estimates , for each agent @xmath33 the two agents whose attendances were estimated to be the most correlated with those of agent @xmath33 were put into agent @xmath33 s guessed effect set . of course , none of this macrolearning had any effect on global performance when applied to follower agents , but the macrolearning algorithm can not know that ahead of time ; it applied this procedure to each and every agent in the system . figure [ fig : worstreward ] presents averages over 50 runs of world reward as a function of weeks using the worst - case reward matrix . for comparison purposes , in both plots the top curve represents the case where the followers are in their leader s guessed effect sets . the bottom curve in both plots represents the other extreme where no leader s guessed effect set contains either of its followers . in both plots , the middle curve is performance when the leaders guessed effect sets are initially random , both with ( right ) and without ( left ) macrolearning turned on at week 500 . the performance for random guessed effect sets differs only slightly from that of having leaders guessed effect sets contain none of their followers ; both start with poor values of world reward that deteriorates with time . however , when macrolearning is performed on systems with initially random guessed effect sets , the system quickly rectifies itself and converges to optimal performance . this is reflected by the sudden vertical jump through the middle of the right plot at 500 weeks , the point at which macrolearning changed the guessed effect sets . by changing those guessed effect sets macrolearning results in a system that is factored for the associated wl reward function , so that those reward functions quickly induced the maximal possible world reward . figure [ fig : randomreward ] presents performance averaged over 50 runs for world reward as a function of weeks using a spectrum of reward matrices selected at random . the ordering of the plots is exactly as in figure [ fig : worstreward ] . macrolearning is applied at 2000 weeks , in the right plot . the simulations in figure [ fig : randomreward ] were lengthened from those in figure [ fig : worstreward ] because the convergence time of the full spectrum of reward matrices case was longer . in figure [ fig : randomreward ] the macrolearning resulted in a transient degradation in performance at 2000 weeks followed by convergence to the optimal . without macrolearning the system s performance no longer varied after 2000 weeks . combined with the results presented in figure [ fig : worstreward ] , these experiments demonstrate that macrolearning induces optimal performance by aligning the agents guessed effect sets with those agents that they actually do influence the most . many distributed computational tasks can not be addressed by direct modeling of the underlying dynamics , or are at best poorly addressed that way due to robustness and scalability concerns . such tasks should instead be addressed by model - independent machine learning techniques . in particular , reinforcement learning ( rl ) techniques are often a natural choice for how to address such tasks . when as is often the case we can not rely on centralized control and communication , such rl algorithms have to be deployed locally , throughout the system . this raises the important and profound question of how to configure those algorithms , and especially their associated utility functions , so as to achieve the ( global ) computational task . in particular we must ensure that the rl algorithms do not `` work at cross - purposes '' as far as the global task is concerned , lest phenomena like tragedy of the commons occur . how to initialize a system to do this is a novel kind of inverse problem , and how to adapt a system at run - time to better achieve such a global task is a novel kind of learning problem . we call any distributed computational system analyzed from the perspective of such an inverse problem a collective intelligence ( coin ) . as discussed in the literature review section of this chapter , there are many approaches / fields that address aspects of coins . these range from multi - agent systems through conventional economics and on to computational economics . ( human economies are a canonical model of a functional coin . ) they range onward to game theory , various aspects of distributed biological systems , and on through physics , active walker models , and recurrent neural nets . unfortunately , none of these fields seems appropriate as a general approach to understanding coins . after this literature review we present a mathematical theory for coins . we then present experiments on two test problems that validate the predictions of that theory for how best to design a coin to achieve a global computational task . the first set of experiments involves a variant of arthur s famous el farol bar problem . the second set instead considers a leader - follower problem that is hand - designed to cause maximal difficulty for the advice of our theory on how to initialize a coin . this second set of experiments is therefore a test of the on - line learning aspect of our approach to coins . in both experiments the procedures derived from our theory , procedures using only local information , vastly outperformed natural alternative approaches , even such approaches that exploited global information . indeed , in both problems , following the theory summarized in this chapter provides good solutions even when the exact conditions required by the associated theorems hold only approximately . there are many directions in which future work on coins will proceed ; it is a vast and rich area of research . we are already successfully applying our current understanding of coins , tentative as it is , to internet packet routing problems . we are also investigating coins in a more general optimization context where economics - 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this paper surveys the emerging science of how to design a `` collective intelligence '' ( coin ) . a coin is a large multi - agent system where : \i ) there is little to no centralized communication or control . \ii ) there is a provided world utility function that rates the possible histories of the full system . in particular , we are interested in coins in which each agent runs a reinforcement learning ( rl ) algorithm . the conventional approach to designing large distributed systems to optimize a world utility does not use agents running rl algorithms . rather , that approach begins with explicit modeling of the dynamics of the overall system , followed by detailed hand - tuning of the interactions between the components to ensure that they `` cooperate '' as far as the world utility is concerned . this approach is labor - intensive , often results in highly nonrobust systems , and usually results in design techniques that have limited applicability . in contrast , we wish to solve the coin design problem implicitly , via the `` adaptive '' character of the rl algorithms of each of the agents . this approach introduces an entirely new , profound design problem : assuming the rl algorithms are able to achieve high rewards , what reward functions for the individual agents will , when pursued by those agents , result in high world utility ? in other words , what reward functions will best ensure that we do not have phenomena like the tragedy of the commons , braess s paradox , or the liquidity trap ? although still very young , research specifically concentrating on the coin design problem has already resulted in successes in artificial domains , in particular in packet - routing , the leader - follower problem , and in variants of arthur s el farol bar problem . it is expected that as it matures and draws upon other disciplines related to coins , this research will greatly expand the range of tasks addressable by human engineers . moreover , in addition to drawing on them , such a fully developed science of coin design may provide much insight into other already established scientific fields , such as economics , game theory , and population biology . 0.2 in 5.8 in -0.5 in 9.0 in 0.2 in

the discovery of gliese 229b ( oppenheimer et al . 1995 ) and the successes of the 2mass ( reid 1994 ; stiening , skrutskie , and capps 1995 ) , sloan ( strauss et al . 1999 ) , and denis ( delfosse et al . 1997 ) surveys have collectively opened up a new chapter in stellar astronomy . the l and t dwarfs ( kirkpatrick et al . 1999,2000 ; martn et al . 1999 ; burgasser et al . 1999 , 2000a , b , c ) that have thereby been discovered and characterized comprise the first new stellar " types to be added to the stellar zoo in nearly 100 years . the lower edge of the solar - metallicity main sequence is an l dwarf not an m dwarf , with a near 1700 kelvin ( k ) , and more than 200 l dwarfs spanning a range from @xmath02200 k to @xmath01300 k are now inventoried . the coolest l dwarfs are also brown dwarfs , objects too light ( @xmath4 m@xmath5 ) to ignite hydrogen stably on the main sequence ( burrows et al . similarly , to date approximately 40 t dwarfs have been discovered spanning the range from @xmath01200 k to @xmath0750 k. these are all brown dwarfs and are the coldest stars " currently known . however , the edge of the stellar " mass function in the field , in the solar neighborhood , or in star clusters has not yet been reached and it is strongly suspected that in the wide mass and gap between the currently known t dwarfs and jovian - like planets there resides a population of very cool ( @xmath6 k ) brown dwarfs . such objects could be too dim in the optical and near - infrared to have been seen with current technology , but might be discovered in the not - too - distant future by the ngss / wise infrared space survey ( wright et al . 2001 ) , sirtf ( space infrared telescope facility ; werner and fanson 1995 ) , and/or jwst ( james webb space telescope ; mather and stockman 2000 ) . in this paper , we calculate the spectra and colors of such a population in order to provide a theoretical underpinning for the future study of these coolest of brown dwarfs . dwelling as they do at beyond those of the currently - known t dwarfs , these stars " emit strongly in the near- and mid - infrared . consequently , we highlight their fluxes from 1 to 30 microns and compare these fluxes with the putative sensitivities of instruments on sirtf and planned instruments on jwst . we include the effects of water clouds that form in the coolest of these objects . the presence of clouds of any sort emphasizes the kinship of this transitional class with solar system planets , in which clouds play a prominent role . ( note , however , that on jupiter itself water clouds are too deep below the ammonia cloud layer to have been unambiguously detected . ) since we focus on isolated free - floaters or wide binary brown dwarfs , we do not include external irradiation by companions . the of this model set ( @xmath7 k ) are such that silicate and iron clouds are deeply buried . hence , unlike for l dwarfs and early t dwarfs ( marley et al . 2002 ; burrows et al . 2002 ) , the effect of such refractory clouds on emergent spectra can be ignored . in [ approach ] , we discuss our numerical approaches and inputs . we go on in [ models ] to describe our mass - age model set and our use of the burrows et al . ( 1997 ) evolutionary calculations to provide the mapping between ( , gravity [ @xmath8 ) and ( mass , age ) . in [ profiles ] , we present a representative sample of derived atmospheric temperature(t)/pressure(p ) profiles and their systematics . this leads in [ sense ] to a short discussion of the sirtf and jwst point - source sensitivities . section [ spectra ] concerns the derived spectra and is the central section of the paper . in it , we discuss prominent spectral features from the optical to 30 microns , trends as a function of age , , and mass , diagnostics of particular atmospheric constituents , and detectability with instruments on sirtf and jwst . we find that these platforms can in principle detect brown dwarfs cooler than the current t dwarfs out to large distances . we also explore the evolution of @xmath9 and its eventual return to the red " ( marley et al . 2002 ; stephens , marley , and noll 2001 ) , reversing the blueward trend with decreasing that roughly characterizes the known t dwarfs . furthermore , we make suggestions for filter sets that may optimize their study with nircam on jwst . finally , we present physical reasons for anticipating the emergence of a new stellar type beyond the t dwarfs . in [ conclusion ] , we summarize what we have determined about this coolest - dwarf family and the potential for their detection . to calculate model atmospheres of cool brown dwarfs requires 1 ) a method to solve the radiative transfer , radiative equilibrium , and hydrostatic equilibrium equations , 2 ) a convective algorithm , 3 ) an equation of state that also provides the molecular and atomic compositions , 4 ) a method to model clouds that may form , and 5 ) an extensive opacity database for the constituents that arise in low - temperature , high - pressure atmospheres . the computer program we use to solve the atmosphere and spectrum problem in a fully self - consistent fashion is an updated version of the planar code tlusty ( hubeny 1988 ; hubeny & lanz 1995 ) , which uses a hybrid of complete linearization and accelerated lambda iteration ( hubeny 1992 ) . to handle convection , we use mixing - length theory ( with a mixing length equal to one pressure - scale height ) . the equation of state we use to find the p / t / density(@xmath10 ) relation is that of saumon , chabrier , and van horn ( 1995 ) and the molecular compositions are calculated using a significantly updated version of the code solgasmix ( burrows and sharp 1999 ) . the latter incorporates a rainout algorithm for refractory silicates and iron ( burrows et al . the most important molecules are h@xmath1 , h@xmath1o , ch@xmath2 , co , n@xmath1 , and nh@xmath3 and the most important atoms are na and k. we determine when water condenses by comparing the water ice condensation curve ( the total pressure at which the partial pressure of water is at saturation ) with the object s t / p profile . for pressures lower than that near the associated intercept , we deplete the vapor phase through the expected rainout and embed an absorbing / scattering water - ice cloud with a thickness of one pressure - scale - height in the region above . note that the total gas pressures at which the partial pressure of water is at the triple - point pressure of water are generally higher than the intercept pressures we find hence , the water gas to water ice ( solid ) transition is the more relevant . note also that the optical properties of water ice and water droplets are not very different . the ice particles are assumed to be spherical and their modal particle radii are derived using the theory of cooper et al . they vary in size from @xmath020 ( higher-@xmath11/lower- ) to @xmath0150 ( lower-@xmath11/higher- ) and we assume that the particle size is independent of altitude . a canonical super - saturation factor ( cooper et al . 2003 ; ackermann and marley 2001 ) of 0.01 ( 1.0% ) is used . curiously , with such large particles and such a small super - saturation , the absorptive opacity of our baseline water - ice clouds , when they do form , is not large . in fact , the consequences for the emergent spectrum of the associated drying of the upper atmosphere , and the corresponding diminution of the water vapor abundance there , are comparable to the effects on the spectrum of the clouds themselves . without an external flux source , and the scattering of that flux back into space by clouds , water - ice clouds seem to have only a secondary influence on the spectra of the coolest isolated brown dwarfs . we use the constantly - updated opacity database described in burrows et al . ( 1997,2001,2002 ) . this includes rayleigh scattering , collision - induced absorption ( cia ) for h@xmath1 ( borysow and frommhold 1990 ; borysow , jrgensen , and zheng 1997 ) , and t / p - dependent absorptive opacities from 0.3 to 300 for h@xmath1o , ch@xmath2 , co , and nh@xmath3 . the opacities of the alkali metal atoms are taken from burrows , marley , and sharp ( 2000 ) , which are similar in the line cores and near wings to those found in burrows and volobuyev ( 2003 ) . the opacities are tabulated in t/@xmath10/frequency space using the abundances derived for a solar - metallicity elemental abundance pattern ( anders and grevesse 1989 ; grevesse and sauval 1998 ; allende - prieto , lambert , and asplund 2002 ) . during the tlusty iterations , the opacity at any thermodynamic point and for any wavelength is obtained by interpolation . the absorptive opacities for the ice particles are derived using mie theory with the frequency - dependent spectrum of the complex index of refraction of water ice . ammonia clouds form in the upper atmospheres of the coldest exemplars of the late brown dwarf family ( @xmath12 160 k ; [ profiles ] ) . nevertheless , since the scattering of incident radiation that gives them their true importance in the jovian context is absent , we ignore them here . we have chosen for this study a set of models with the masses and ages given in table 1 . also shown in table 1 are the corresponding gravities and . these models span an effective temperature range from @xmath0800 k to @xmath0150 k that allows us to probe the realm between the known t dwarfs and the known jovian planets . to establish the mapping between mass / age pairs and the /@xmath11 pairs that are needed for atmospheric calculations , we use the evolutionary models of burrows et al . ( 1997 ) . while this procedure does not ensure that the atmospheres we calculate are fully consisitent with those evolutionary tracks , the errors are not large . figure [ fig:1 ] depicts evolutionary trajectories and isochrones in /@xmath11 space for models in the realm beyond the t dwarfs . the depicted isochrones span the range from 10@xmath13 to @xmath14 years and the masses cover the range from 0.5 to 25 . the large dots denote the models found in table 1 for which we have calculated spectra and atmospheres . for contrast , the approximate region in which the currently known t dwarfs reside is also shown . in addition , we provide the demarcation lines that separate ( in a rough sense ) the cloud - free models from those with water clouds and ammonia clouds . the clouds form to the left of the corresponding condensation lines . figure [ fig:1 ] emphasizes the transitional and as - yet - unstudied character of this family of objects . it also provides at a glance a global summary of family properties . figure [ fig:2 ] is a companion figure to fig . [ fig:1 ] , but shows iso - lines in mass / age space . for a given mass , fig . [ fig:2 ] allows one to determine the evolution of and at what age a given is achieved . it also makes easy the determination of the combination of mass and age for which clouds form , as well as the minimum mass for which a given is reached after approximately the galactic disk s or the sun s age ( @xmath15 and @xmath16 years , respectively ) . for instance , fig . [ fig:2 ] shows that it takes @xmath17 myr for a 2-object to reach a of 400 k , that it takes the same object 1 gyr to reach a of @xmath0250 k , and that in the age of the solar system a 2-object can reach the nh@xmath3 condensation line . similarly , fig . [ fig:2 ] indicates that a 10-object takes @xmath01 gyr to reach a of @xmath0400 k , and that it has water - ice clouds in its upper atmosphere . figures [ fig:1 ] and [ fig:2 ] are , therefore , useful maps of the model domain to which the reader may want often to return . to calculate absolute fluxes at 10 parsecs one needs the radius of the object . we determine this for each model in table 1 by using a fit to the results of burrows et al . ( 1997 ) that works reasonably well below @xmath025 and after deuterium burning has ended : @xmath18 where @xmath19 is jupiter s radius ( @xmath20 cm ) . shown in fig . [ fig:3 ] are representative temperature - pressure profiles at 300 myr ( blue ) for models with masses of 1 , 2 , 5 , 7 , and 10 and at 5 gyr ( red ) for models with masses of 2 , 5 , 7 , 10 , 15 , 20 , and 25 . superposed are the water ice and ammonia condensation lines at solar metallicity . the radiative - convective boundary pressures are near 0.1 - 1.0 bars for the lowest - mass , oldest models and are near 10 - 30 bars for the youngest , most massive models . at a given temperature , lower - mass objects have higher pressures ( at a given age ) . similarly , an object with a given mass evolves to higher and higher pressures at a given temperature . this trend is made clear in fig . [ fig:4 ] , in which the evolving t / p profiles for 1-and 5-models are depicted , and is not unexpected ( marley et al . 1996,2002 ; burrows et al . note that fig . [ fig:4 ] implies that a 5-object takes @xmath0300 myr to form water clouds , but that a 1-object takes only @xmath0100 myr . after @xmath01 gyr , a 1-object forms ammonia clouds , signature features of jupiter itself . these numbers echo the information also found in fig . [ fig:2 ] . the appearance of a water - ice cloud manifests itself in figs . [ fig:3 ] and [ fig:4 ] by the kink in the t / p profile near the intercept with the associated condensation line . generally , the higher the intercept of the t / p profile with the condensation line ( the lower the intercept pressure ) the smaller the droplet size ( cooper et al . 2003 ) . note that after an age of @xmath0300 myr a 7-object is expected to form water clouds high up in its atmosphere and that after @xmath05 gyr even a 25-object will do so . the higher the atmospheric pressure at which the cloud forms the greater the column thickness of the cloud . this results in a stronger cloud signature for the lower - mass models than for the higher - mass models . however , given the generally large ice particle sizes derived with the cooper et al . ( 2003 ) model , the low assumed supersaturation ( [ approach ] ) , the tendency for larger particle radii to form for larger intercept pressures , and the modest to low imaginary part of the index of refraction for pure water ice , the effect of water clouds in our model set is not large . this translates into a small cloud effect on the corresponding flux spectra ( [ spectra ] ) . figure [ fig:5 ] portrays the evolution of the t / p profiles for 10-and 20-objects . this figure is provided to show , among other things , the position of the forsterite ( mg@xmath1sio@xmath2 ) condensation line relative to that of water ice . mg@xmath1sio@xmath2 clouds exist in these brown dwarfs , but at significantly higher pressures and temperatures and are , therefore , buried from view . hence , unlike in l dwarfs , such clouds have very little effect on the emergent spectra of the coolest brown dwarfs that are the subject of this paper . finally , the high pressures achieved at low temperatures for the lowest mass , oldest objects shown in figs . [ fig:3 ] , [ fig:4 ] , and [ fig:5 ] suggest that the cia ( pressure - induced ) opacity of h@xmath1 might for them be important . this is indeed the case at longer wavelengths and is discussed in [ spectra ] . we mention this because cia opacity is yet another characteristic signature of the jovian planets in our own solar system and to emphasize yet again that our cold brown dwarf model suite is a bridge between the realms of the planets and the stars . " before we present and describe our model spectra , we discuss the anticipated point - source sensitivities of the instruments on board the sirtf and jwst space telescopes . sirtf has a 0.84-meter aperture and is to be launched in mid - april of 2003 . jwst is planned to have a collecting area of @xmath025 square meters over a segmented 6-meter diameter mirror and is to be launched at the beginning of the next decade . while sirtf is the last of the great observatories , " and will view the sky with unprecedented infrared sensitivity , jwst will in turn provide a two- to four - order - of - magnitude gain in sensitivity through much of the mid - infrared up to 27 microns . while their fields of view are limited and missions like wise ( formerly ngss ; wright et al . 2001 ) are more appropriate for large - area surveys , the extreme sensitivity of both sirtf and jwst will bring the coolest brown dwarfs and isolated giant planets into the realm of detectability and study . sirtf / irac has four channels centered at 3.63 , 4.53 , 5.78 , and 8.0 that are thought to have 5-@xmath21 point - source sensitivities for 200-second integrations of @xmath02.5 , @xmath04.5 , @xmath015.5 , and @xmath025.0 microjanskys , respectively . hst / nicmos achieves a bit better than one microjansky sensitivity at 2.2 , but does not extend as far into the near ir . the short - wavelength , low - spectral resolution module ( short - low " ) of sirtf / irs extends from @xmath05.0 to @xmath014.0 and has a 5-@xmath21 point - source sensitivity for a 500-second integration of @xmath0100 microjanskys . the other three modules on irs cover other mid - ir wavelength regimes at either low- or high - spectral resolution , but will have smaller brown dwarf detection ranges . the @xmath020.5 to @xmath026 channel on sirtf / mips is the most relevant channel on mips for brown dwarf studies and has a suggested 1-@xmath21 point - source sensitivity at @xmath024 of @xmath070 microjanskys . this is @xmath01000 times better in imaging mode than for the pioneering iras . all these sirtf sensitivities are derived from various sirtf web pages and are pre - launch estimates ( ` http://sirtf.caltech.edu ` ) . furthermore , for all three sirtf instruments , one can estimate the point - source sensitivities for different values of the signal - to - noise and integration times . these signals - to - noise and integration times are the nominal combinations for each instrument and the quoted sensitivities serve to guide our assessment of sirtf s capabilities for cool brown dwarf studies in advance of real on - orbit calibrations and measurements . the capabilities of jwst are even more provisional , but the design goals for its instruments are impressive ( ` http://ngst.gsfc.nasa.gov ` ) . jwst / nircam is to span @xmath00.6 to @xmath05.0 in various wavelength channels / filters , though the final design has not been frozen . the seven so - called b " filters have widths of 0.51.0 microns centered at @xmath00.71 , @xmath01.1 , @xmath01.5 , @xmath02.0 , @xmath02.7 , @xmath03.6 , and @xmath04.4 microns and are expected to have 5-@xmath21 point - source sensitivities in imaging mode , for an assumed exposure time of @xmath22 seconds , of @xmath01.6 , @xmath00.95 , @xmath01.0 , @xmath01.2 , @xmath00.95 , @xmath01.05 , and @xmath01.5 nanojanskys ( nj ) , respectively . in addition , a set of so - called i " filters , with about half to one quarter the spectral width of the b filters , and sensitivites comparable to that of the b filters , are available in the 1.55.0 region . furthermore , jwst / nircam may have a tunable filter to examine selected spectral regions beyond 2.5 at a resolution ( @xmath23 ) of @xmath0100 , though at the time of this writing the availability of such a capability remained uncertain . hence , with jwst / nircam we enter the world of _ nano_jansky sensitivity . this is greater than one hundred times more sensitive than hst / nicmos at 2.2 and enables one to probe deeply in space , as well as broadly in wavelength . jwst / miri spans the mid - ir wavelength range from @xmath05.0 to @xmath027.0 and will have in imaging mode a 10-@xmath21 point - source sensitivity for a 10@xmath24-second integration of from @xmath063 nj at the shortest wavelength to @xmath010 microjanskys at the longest . this is orders of magnitude more sensitive than any previous mid - ir telescope in imaging mode . ( in spectral mode with an @xmath25 near 1000 , jwst / miri will be @xmath0100 times less sensitive than in imaging mode . ) given the importance of the mid - ir for understanding those brown dwarfs that may exist in relative abundance at cooler than those of the currently known t dwarfs , miri provides what is perhaps a transformational capability . as with sirtf , the quoted jwst sensitivities are taken from the associated web pages and , hence , should be considered tentative . we now turn to a discussion of the spectra , spectral evolution , defining features , systematics , and diagnostics for the cool brown dwarf models listed in table 1 and embedded in figs . [ fig:1 ] and [ fig:2 ] . on each of figs . [ fig:6 ] to [ fig:11 ] in [ spectra ] , we plot for the sirtf ( red ) and jwst ( blue ) instruments the broadband sensitivities we have summarized in this section . using the numerical tools and data referred to in [ approach ] , and the mapping between /@xmath11 and mass / age found in table 1 , we have generated a grid of spectral and atmospheric models for cool brown dwarfs that reside in the low - sector of /@xmath11 space ( fig . [ fig:1 ] ) . some of the associated t / p profiles were given in figs . [ fig:3 ] , [ fig:4 ] , and [ fig:5 ] . in figs . [ fig:6 ] to [ fig:11 ] , we plot theoretical flux spectra ( f@xmath26 , in millijanskys ) from the optical to 30 at a distance of 10 parsecs . these figures constitute the major results of our paper . for comparision , superposed on each figure are the estimated point - source sensitivities of the instruments on board sirtf and jwst ( [ sense ] ) . in addition , included at the top of figs . [ fig:8 ] through [ fig:11 ] are the rough positions of the major atmospheric absorption features . ( the full model set is available from the first author upon request . ) figures [ fig:6 ] and [ fig:7 ] portray the mass dependence of a cool brown dwarf s flux spectrum at 10 parsecs for ages of one and five gyr , respectively . the model masses are 25 , 20 , 15 , 10 , 7 , 5 , 2 , and 1 . the top panels depict the most massive four , while the bottom panels depict the least massive four ( three for fig . [ fig:7 ] ) . together they show the monotonic diminution of flux with object mass at a given age that parallels the associated decrease in with mass ( from @xmath0800 k to @xmath0130 k ) seen in table 1 and fig . [ fig:2 ] . figures [ fig:6 ] through [ fig:11 ] show the peaks due to enhanced flux through the water vapor absorption bands that define the classical terrestrial photometric bands ( @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31 ) and that have come to characterize brown dwarfs since the discovery of gliese 229b ( oppenheimer et al . 1995 ; marley et al . 1996 ) . for the more massive models , the near - ir fluxes are significantly above black - body values . at @xmath0800 k , the 25-/1-gyr model shown in fig . [ fig:6 ] could represent the known late t dwarfs , but all other models in this model set are later " and , hence , represent as yet undetected objects . apart from the distinctive water troughs , generic features are the hump at 4 - 5 microns ( @xmath31 band ) , the broad hump near 10 microns , the methane features at 2.2 , 3.3 , 7.8 , and in the optical ( particularly at 0.89 ) , the ammonia features at @xmath01.5 , @xmath01.95 , @xmath02.95 , and @xmath010.5 , and the na - d and k i resonance lines at 0.589 and 0.77 , respectively . however , as figs . [ fig:6]-[fig:11 ] indicate , the strengths of each of these features are functions of mass and age . for lower masses or greater ages , the centroid of the @xmath31 band hump shifts from @xmath04.0 to @xmath05.0 . in part , this is due to the swift decrease with at the shorter wavelengths of the wien tail . even after the collapse of the flux in the optical and near - ir after @xmath01 gyr for masses below 5 or after @xmath05 gyr for masses below 10 , the @xmath31 band flux persists as a characteristic marker and will be sirtf s best target . moreover , irac s filters are well - positioned for this task . as one would expect , the relative importance of the mid - ir fluxes , in particular between 10 and 30 microns , grows with decreasing mass and increasing age . since this spectral region is near the linear rayleigh - jeans tail , fluxes here persist despite decreases in from @xmath0800 k to @xmath0130 k. figure [ fig:11 ] depicts this clearly for the older 2-models . the rough periodicity in flux beyond 10 is due predominantly to the presence of pure rotational bands of water and , for cooler models , methane as well . for the coldest models depicted in figs . [ fig:6 ] , [ fig:7 ] , and [ fig:11 ] , this behavior subsides , but is replaced with long - period undulations due to cia absorption by h@xmath1 . such a signature is characteristic of jovian planets and is expected for low - t , high - p atmospheres . its appearance marks yet another transition , seen first in this model set for the old 5-and middle - aged 2-objects , between t - dwarf - like and planet "- like behavior . as figs . [ fig:6]-[fig:11 ] imply , sirtf / mips should be able to detect at 10 parsecs the @xmath024-flux of objects more massive than 2 - 4 at age 1 gyr or more massive than 10 at 5 gyr . methane forms at low temperatures and high pressures and makes its presence felt in older and less massive objects . hence , its features at 0.89 , 2.2 , 3.3 , and 7.8 deepen with age and decreasing mass . an example of such strengthening at 7.8 and 2.2 can be seen in fig . [ fig:10 ] by comparing the 100-myr and 5-gyr models with a mass of 5 . clear indications of the strengthening of the methane absorption feature at 0.89 with decreasing mass can be seen in the upper panel of fig . [ fig:7 ] . this trend is accompanied by a corresponding weakening of the cs i feature on top of it . however , due to its presence in the @xmath32 band at relatively short wavelengths , the methane feature at 0.89 may be difficult to detect for all but the youngest and/or most massive models . the actual strength of the 7.8-feature depends on the t / p profile in the upper layers of the atmosphere , which in turn might be affected by ambient uv ( disfavored for free - floating brown dwarfs ) or processes that could create a stratosphere and a temperature inversion . hence , the filling in or reshaping of the 7.8-feature might signal the presence of a stratosphere . such a temperature inversion could also affect the depths of the water troughs . as can be seen by comparing the top panels of figs . [ fig:8]-[fig:11 ] , the alkali metal features at 0.589 and 0.77 diminish in strength with decreasing mass and increasing age . these features are signatures of the known t dwarfs ( burrows , marley , and sharp 2000 ; burrows et al . 2002 ; tsuji , ohnaka , and aoki 1999 ) , so their decay signals a gradual transformation away from standard t - dwarf behavior . for the 10-model older than 1 gyr and the 2-model older than 100 myr , these alkali resonance features cease to be primary signatures . this happens near a of 450 k. ammonia makes an appearance at even lower temperatures than methane and due to the relatively high abundance of nitrogen its absorption features are generally strong , particularly for the cool objects in our model set . for the higher in the mid - t - dwarf range , ammonia may have been seen , but is weak ( saumon et al . 2000 ) . figs . [ fig:10 ] and [ fig:11 ] evince strong ammonia features in the upper panels at @xmath01.5 , @xmath01.95 , and @xmath02.95 and in figs . [ fig:8]-[fig:11 ] in the lower panels at @xmath010.5 . as figs . [ fig:6]-[fig:11 ] imply , the short - low module on sirtf / irs should be able to study the 10.5-ammonia feature . even for the 25-/1gyr model , the @xmath010.5 feature is prominent . for the more massive objects ( 10 - 25 ) , the strength of the 10.5-feature increases with age . for the lowest mass objects ( 2 - 7 ) , the strength of the 10.5-ammonia feature actually decreases with age , even though the strengths of the other ammonia lines increase . as the more massive objects age , their atmospheric pressures increase , shifting the n@xmath1/nh@xmath3 equilibrium towards nh@xmath3 . for the less massive models , pressured - induced absorption by h@xmath1 grows with increasing atmospheric pressure ( fig . [ fig:3]-[fig:4 ] ) and partially flattens an otherwise strengthening 10.5-ammonia feature . below of @xmath0160 k , figs . [ fig:1 ] and [ fig:2 ] demonstrate that ammonia clouds form . however , given that we are studying isolated objects that have no reflected component ( unlike jupiter and saturn ) , and given that realistic supersaturations are only @xmath01% , we have determined that ammonia clouds do not appreciably affect the emergent spectra . as a consequence , we ignore them in the three relevant models ( fig . [ fig:1 ] ) . as with the known t and l dwarfs , water vapor absorptions dominate and sculpt the flux spectra of the cooler brown dwarfs and these features generally deepen with increasing age and decreasing mass . the latter trend is in part a consequence of the increase with decreasing gravity of the column depth of water above the ( roughly - defined ) photosphere . at below @xmath0400 - 500 [ fig:1]-[fig:5 ] ) , water condenses in brown dwarf atmospheres . the appearance of such water - ice clouds constitutes yet another milestone along the bridge from the known t dwarfs to the giant planets . associated with cloud formation is the depletion of water vapor above the tops of the water cloud , with the concommitant decrease at altitude in the gas - phase abundance of water . within @xmath0100 myr , water clouds form in the atmosphere of an isolated 1-object and within @xmath05 gyr they form in the atmosphere of a 25-object . in fact , approximately two - thirds of the models listed in table 1 incorporate water - ice clouds . however , at supersaturations of 1% and for particle sizes above 10 microns ( [ approach]-[profiles ] ; cooper et al . 2003 ) , such clouds ( and the corresponding water vapor depletions above them ) only marginally affect the calculated emergent spectra . even though we see in figs . [ fig:3]-[fig:5 ] the associated kinks in the t / p profiles , these do not translate into a qualitative change in the emergent spectra at any wavelength . for wavelengths longward of 1 micron , the cloudy spectra differ from the no - cloud spectra by at most a few tens of percent . for a representative 2-model at 300 myr ( @xmath0280 k ) , if we increase the supersturation factor by a factor of ten from 1% to 10% , the flux at 5 microns decreases by approximately a factor of two , while the flux from 10 to 30 microns increases by on average @xmath050% . these are not large changes , given the many orders of magnitude covered by the fluxes in figs . [ fig:6]-[fig:11 ] . the prominence of water features provides a guide to the optimal placement of nircam filters for the detection and characterization of brown dwarfs . for example , the water feature near 0.93 is missed by the b filters , while those features at @xmath01.4 and @xmath01.8 are not centered on the respective adjacent filters and , hence , are diluted by the adjoining continuum . the i filters on nircam would partially overcome these limitations . even so , as fig . [ fig:12 ] shows , the broadband fluxes in the nircam filters provide useful diagnostics of the differences among brown dwarfs and extrasolar giant planets ( here expressed as mass at a given age ) , with particular sensitivity to the large flux differences between the 5-window and the region shortward . a tunable filter could provide even greater diagnostic capability by permitting in and around the 5-window a spectral resolution near 100 to more definitively characterize the effective temperature and , hence , the mass of detected objects ( for a given age and composition ) . nevertheless , fig . [ fig:7 ] indicates that at 10 parsecs even a 7-object at 5 gyr should easily be detected in imaging mode in the @xmath28 and @xmath29 bands . in the @xmath31 band , a 2-object could be seen by nircam out to @xmath0100 parsecs . furthermore , a 25-object at 5 gyr and a distance of 1000 parsecs should be detectable by nircam in a number of its current broadband filters . figure [ fig:13 ] shows predicted spectra of a 20-/5-gyr model in the mid - infrared for the sirtf / irs and jwst / miri instruments . to generate the sirtf / irs curve in fig . [ fig:13 ] , we multiplied the theoretical spectra by the irs response curves for the entire wavelength range , not just the 5 - 14 of the short - low " module . the irs spectral resolution has been assumed to be 100 , while that of jwst / miri is @xmath01000 . we find that the irs spectra are useful at 10 parsecs only for the warmer brown dwarfs ( @xmath33 ) , but for these brown dwarfs even at this modest spectral resolution one can clearly identify the various dominant molecular bands . in its broadband detection ( imaging ) mode , jwst / miri will be @xmath0100 times more capable than sirtf from @xmath05 to @xmath027 ( [ sense ] ) . since the mid - ir is one of the spectral regions of choice for the study of the coolest brown dwarfs , miri will assume for their characterization a role of dramatic importance . at wavelengths longward of 15- , miri will be able to detect objects 10 parsecs away down to 2 or lower . in addition , it could detect an object just 10 times the mass of jupiter with an age of 5 gyr out to a distance of one kiloparsec . furthermore , jwst / miri provides 10 times better spectral resolution than sirtf / irs for objects down to 10 . [ fig:6 ] through [ fig:13 ] collectively summarize the flux spectra and evolution of the cool brown dwarfs yet to be discovered , as well as the extraordinary capabilities of the various instruments on board both sirtf and jwst for the diagnosis and characterization of their atmospheres . these figures highlight the prominent molecular features of h@xmath1o , ch@xmath2 , nh@xmath3 , in particular , that are pivotal in the evolution of the differences between the coolest brown dwarfs and the known t dwarfs , most of which are at higher and gravities . the latest known t dwarf has been typed a t8 ( burgasser et al . 2000a ) , but its effective temperature is near 750 - 800 k ( geballe et al . 2001 ; burrows et al . this does not leave much room for the expansion of the t dwarf subtypes to the lower and masses discussed in this paper , and suggests that yet another spectroscopic class beyond the t dwarfs might be called for . many of the spectral trends described in this paper are gradual , but the near disappearance of the alkali features below = 500 k , the onset of water cloud formation below = 400 - 500 k , the collapse below @xmath0350 k of the optical and near - ir fluxes relative to those longward of @xmath05 , and the growing strengths of the nh@xmath3 features all suggest physical reasons for such a new class . figure [ fig:14 ] depicts isochrones from 100 myr to 5 gyr on the @xmath34 versus @xmath9 color - magnitude diagram and demonstrates that the blueward trend in @xmath9 that so typifies the t dwarfs stops and turns around ( marley et al . 2002 ; stephens , marley , and noll 2001 ) between effective temperatures of 300 and 400 k. this is predominantly due not to the appearance of water clouds , but to the long - expected collapse of flux on the wien tail . note that the at which the @xmath9 color turns around is not the same for all the isochrones . this is because the colors are not functions of just , but of gravity as well . the decrease in , that for the t dwarfs squeezes the @xmath30-band flux more than the @xmath28-band flux , finally does to @xmath9 what people had expected such a decrease to do before the discovery of t dwarfs , i.e. , redden the color . we remind the reader that unlike m dwarfs , the @xmath9 colors of t dwarfs actually get bluer with decreasing ( for a given surface gravity ) . this may be counterintuitive , but it is a result of the increasing role of methane and h@xmath1 collision - induced absorption with decreasing temperature , as well as the positive slope of the opacity / wavelength curve of water and its gradual steepening with decreasing temperature . were it not for the extremely low fluxes at such low shortward of 4 microns , we might have suggested the use of this turnaround to mark the beginning of a new spectroscopic class . moreover , clearly the optical can not be used and with the diminishing utility of the near infrared as drops , that leaves the mid - ir longward of @xmath04 as the most logical part of the spectrum with which to characterize a new spectroscopic class . as is usual , this will be determined observationally , and it might be done arbitrarily to limit the growth of the t sequence . nevertheless , we observe that the region between 300 k and 500 k witnesses a few physical transitions that might provide a natural break between stellar " types . we have generated a new set of brown dwarf spectral models that incorporate state - of - the - art opacities and the effects of water clouds . our focus has been on the low - branch of the brown dwarf tree beyond the known t dwarfs . to this end , we have investigated the range from @xmath0800 k to @xmath0130 k and the low - mass range from 25 to 1 . as fig . [ fig:1 ] indicates , this is mostly unexplored territory . our calculations have been done to provide a theoretical foundation for the new brown dwarf studies that will be enabled by the launch of sirtf and the eventual launch of jwst , as well as for the ongoing ground - based searches for the coolest substellar objects . we provide spectra from @xmath00.4 to 30 , investigate the dependence on age and mass of the strengths of the h@xmath1o , ch@xmath2 , and nh@xmath3 molecular features , address the formation and effect of water clouds , and compare the calculated fluxes with the suggested sensitivities of the instruments on board sirtf and jwst . from the latter , detection ranges can be derived , which for jwst can exceed a kiloparsec . we find that the blueward trend in near - infrared colors so characteristic of the t dwarfs stops near a of 300 - 400 k and we identify a few natural physical transitions in the low - realm which might justify the eventual designation of at least one new spectroscopic type after the t dwarfs . these include the formation of water clouds ( @xmath0400 - 500 k ) , the strengthening of ammonia bands , the eventual collapse in the optical , the shift in the position of the @xmath31 band peak , the turnaround of the @xmath9 color , the near disappearance of the strong na - d and k i resonance lines ( @xmath0500 k ) , and the increasing importance with decreasing of the mid - ir longward of 4 . for these cooler objects , the mid - infrared assumes a new and central importance and first mips and irs on sirtf , then miri on jwst , are destined to play pivotal roles in their future characterization and study . finally , the formation of ammonia clouds below @xmath0160 k suggests yet another natural breakpoint , and a second new stellar " class . therefore , there are reasons to anticipate that perhaps two naturally defined , yet uncharted , spectral types reside beyond the t dwarfs at lower . the current filter set for jwst / nircam from 0.6 to 5.0 is good , but not yet fully optimized for cool brown dwarf detection . placing filters on the derived spectral peaks and troughs ( robustly defined by the water bands ) would improve its already good performance for substellar research . in any case , our theoretical spectra are meant to bridge the gap between the known t dwarfs and those cool , low - mass free - floating brown dwarfs with progressively more planetary features which may inhabit the galaxy in interesting , but as yet unknown , numbers . the authors thank ivan hubeny , bill hubbard , john milsom , christopher sharp , jim liebert , curtis cooper , and jonathan fortney for fruitful conversations and help during the course of this work , as well as nasa for its financial support via grants nag5 - 10760 , nag5 - 10629 , and nag5 - 12459 . ackermann , a. and marley , m.s . 2001 , , 556 , 872 allende - prieto , c. , lambert , d.l . , and asplund , m. 2002 , , 573 , l137 anders , e. and grevesse , n. 1989 , geochim . cosmochim . acta , 53 , 197 bessell , m.s . , and brett , j. m. 1988 , , 100 , 1134 borysow , a. and frommhold , l. 1990 , , 348 , l41 borysow , a. , jrgensen , u.g . , and zheng , c. 1997 , , 324 , 185 burgasser , a.j . , et al . 1999 , , 522 , l65 burgasser , a.j . , et al . 2000a , , 531 , l57 burgasser , a.j . , kirkpatrick , j. d. , reid , i. n. , liebert , j. , gizis , j. e. , & brown , m. e. 2000b , , 120 , 473 burgasser , a.j . , et al . 2000c , , 120 , 1100 burrows , a. , marley m. , hubbard , w.b . lunine , j.i . , guillot , t. , saumon , d. freedman , r. , sudarsky , d. and sharp , c.m . 1997 , , 491 , 856 burrows , a. and sharp , c.m . 1999 , , 512 , 843 burrows , a. , marley , m. s. , and sharp , c. m. 2000 , , 531 , 438 burrows , a. , hubbard , w.b . , lunine , j.i . , and liebert , j. 2001 , rev . phys . , 73 , 719 burrows , a. , burgasser , a.j . , kirkpatrick , j. d. , liebert , j. , milsom , j.a . , sudarsky , d. , and hubeny , i. 2002 , , 573 , 394 burrows , a. and volobuyev , m. 2003 , , 583 , 985 cooper , c.s . , sudarsky , d. , milsom , j.a . , lunine , j.i . , & burrows , a. 2003 , , 586 , 1320 delfosse , x. , tinney , c.g . , forveille , t. , epchtein , n. , bertin , e. , borsenberger , j. , copet , e. , de batz , b. , fouqu , p. , kimeswenger , s. , le bertre , t. , lacombe , f. , rouan , d. , and tiphne , d. 1997 , , 327 , l25 geballe , t.r . , saumon , d. , leggett , s.k . , knapp , g.r . , marley , m.s . , and lodders , k. 2001 , , 556 , 373 grevesse , n. , sauval , a.j . 1998 , space sci . , 85 , 161 hubeny , i. 1988 , computer physics comm . , 52 , 103 hubeny , i. 1992 , in _ the atmospheres of early - 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427 ( sirtf ) wright , e. and the ngss team 2001 , bull . a.a.s . , 198 , 407 @xmath35 & @xmath36 & @xmath37 & @xmath38 + & @xmath39 & @xmath40 & @xmath41 + & @xmath42 & @xmath43 & @xmath44 + @xmath45 & @xmath36 & @xmath46 & @xmath47 + & @xmath39 & @xmath48 & @xmath49 + & @xmath42 & @xmath50 & @xmath51 + & @xmath52 & @xmath53 & @xmath54 + & @xmath55 & @xmath56 & @xmath57 + @xmath58 & @xmath36 & @xmath59 & @xmath60 + & @xmath39 & @xmath61 & @xmath62 + & @xmath42 & @xmath63 & @xmath64 + & @xmath52 & @xmath65 & @xmath66 + & @xmath55 & @xmath67 & @xmath68 + @xmath69 & @xmath36 & @xmath70 & @xmath71 + & @xmath39 & @xmath72 & @xmath73 + & @xmath42 & @xmath74 & @xmath75 + & @xmath52 & @xmath76 & @xmath77 + & @xmath55 & @xmath78 & @xmath79 + @xmath80 & @xmath36 & @xmath81 & @xmath82 + & @xmath39 & @xmath83 & @xmath84 + & @xmath42 & @xmath85 & @xmath86 + & @xmath52 & @xmath87 & @xmath88 + & @xmath55 & @xmath89 & @xmath90 + @xmath91 & @xmath42 & @xmath92 & @xmath93 + & @xmath52 & @xmath94 & @xmath95 + & @xmath55 & @xmath96 & @xmath97 + @xmath98 & @xmath42 & @xmath99 & @xmath100 + & @xmath52 & @xmath101 & @xmath102 + & @xmath55 & @xmath103 & @xmath104 + @xmath105 & @xmath42 & @xmath106 & @xmath107 + & @xmath52 & @xmath108 & @xmath109 + & @xmath55 & @xmath101 & @xmath110 +

we explore the spectral and atmospheric properties of brown dwarfs cooler than the latest known t dwarfs . our focus is on the yet - to - be - discovered free - floating brown dwarfs in the range from @xmath0800 k to @xmath0130 k and with masses from 25 to 1 . this study is in anticipation of the new characterization capabilities enabled by the launch of sirtf and the eventual launch of jwst . in addition , it is in support of the continuing ground - based searches for the coolest substellar objects . we provide spectra from @xmath00.4 to 30 , highlight the evolution and mass dependence of the dominant h@xmath1o , ch@xmath2 , and nh@xmath3 molecular bands , consider the formation and effects of water - ice clouds , and compare our theoretical flux densities with the putative sensitivities of the instruments on board sirtf and jwst . the latter can be used to determine the detection ranges from space of cool brown dwarfs . in the process , we determine the reversal point of the blueward trend in the near - infrared colors with decreasing ( a prominent feature of the hotter t dwarf family ) , the at which water and ammonia clouds appear , the strengths of gas - phase ammonia and methane bands , the masses and ages of the objects for which the neutral alkali metal lines ( signatures of l and t dwarfs ) are muted , and the increasing role as decreases of the mid - infrared fluxes longward of 4 . these changes suggest physical reasons to expect the emergence of at least one new stellar class beyond the t dwarfs . furthermore , studies in the mid - infrared could assume a new , perhaps transformational , importance in the understanding of the coolest brown dwarfs . our spectral models populate , with cooler brown dwarfs having progressively more planet - like features , the theoretical gap between the known t dwarfs and the known giant planets . such objects likely inhabit the galaxy , but their numbers are as yet unknown .

in the standard model ( sm ) , lepton - flavor - violating ( lfv ) decays of charged leptons are forbidden ; even if neutrino mixing is taken into account , they are still highly suppressed . however , lfv is expected to appear in many extensions of the sm . some such models predict branching fractions for @xmath0 lfv decays at the level of @xmath15 @xcite , which can be reached at the present b - factories . observation of lfv will then provide evidence for new physics beyond the sm . in this paper , we report on a search for lfv in @xmath16 decays into neutrinoless final states with one charged lepton @xmath17 and one vector meson @xmath2 : @xmath18 , @xmath19 , @xmath20 , @xmath21 , @xmath22 , @xmath23 , @xmath24 and @xmath25 @xcite . a search for the @xmath26 , @xmath27 and @xmath28 modes was performed for the first time at the cleo detector , where 90% confidence level ( cl ) upper limits ( ul ) for the branching fractions in the range @xmath29 were obtained using a data sample of 4.79 fb@xmath7 @xcite . later we carried out a search for these modes in the belle experiment using 158 fb@xmath7 of data and set upper limits in the range @xmath30 @xcite . here we present results of a new search based on a data sample of 543 fb@xmath7 corresponding to @xmath31 @xmath0-pairs collected with the belle detector @xcite at the kekb asymmetric - energy @xmath32 collider @xcite . the belle detector is a large - solid - angle magnetic spectrometer that consists of a silicon vertex detector , a 50-layer central drift chamber , an array of aerogel threshold cherenkov counters , a barrel - like arrangement of time - of - flight scintillation counters , and an electromagnetic calorimeter comprised of 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 @xmath33 mesons and identify muons . the detector is described in detail elsewhere @xcite . two inner detector configurations were used . a 2.0 cm radius beam - pipe and a 3-layer silicon vertex detector were used for the first sample of 158 fb@xmath7 , while a 1.5 cm radius beam - pipe , a 4-layer silicon detector and a small - cell inner drift chamber were used to record the remaining 385 fb@xmath7 @xcite . we search for @xmath34 , @xmath10 , @xmath13 and @xmath14 candidates in which one @xmath0 decays into a final state with a @xmath1 , two charged hadrons ( 3-prong decay ) , and the other @xmath0 decays into one charged particle ( 1-prong decay ) , any number of @xmath35 s and missing particle(s ) . we reconstruct @xmath3 candidates from @xmath36 , @xmath4 from @xmath37 , @xmath5 from @xmath38 and @xmath6 from @xmath39 . the selection criteria described below are optimized from studies of monte carlo ( mc ) simulated events and the experimental data in the sideband regions of the @xmath40 and @xmath41 distributions described later . the background ( bg ) mc samples consist of @xmath42 ( 1524 fb@xmath7 ) generated by kkmc @xcite , @xmath43 continuum , and two - photon processes . the signal mc events are generated assuming a phase space distribution for @xmath0 decay . the transverse momentum for a charged track is required to be larger than 0.06 gev/@xmath44 in the barrel region ( @xmath45 , where @xmath46 is the polar angle relative to the direction opposite to that of the incident @xmath47 beam in the laboratory frame ) and 0.1 gev/@xmath44 in the endcap region ( @xmath48 and @xmath49 ) . the energies of photon candidates are required to be larger than 0.1 gev in both regions . to select the signal topology , we require four charged tracks in an event with zero net charge , and a total energy of charged tracks and photons in the center - of - mass ( cm ) frame less than 11 gev . we also require that the missing momentum in the laboratory frame be greater than 0.6 gev/@xmath44 , and that its direction be within the detector acceptance ( @xmath50 ) , where the missing momentum is defined as the difference between the momentum of the initial @xmath8 system , and the sum of the observed momentum vectors . the event is subdivided into 3-prong and 1-prong hemispheres with respect to the thrust axis in the cm frame . these are referred to as the signal and tag side , respectively . we allow at most two photons on the tag side to account for initial state radiation , while requiring at most one photon for the @xmath12 , @xmath13 , @xmath14 modes , and two photons except for @xmath51 daughters for the @xmath10 modes on the signal side to reduce the @xmath52 bg . we require that the muon likelihood ratio @xmath53 be greater than 0.95 for momentum greater than 1.0 gev/@xmath44 and the electron likelihood ratio @xmath54 be greater than 0.9 for momentum greater than 0.5 gev/@xmath44 for the charged lepton - candidate track on the signal side . here @xmath55 is the likelihood ratio for a charged particle of type @xmath56 ( @xmath57 , @xmath58 , @xmath59 or @xmath60 ) , defined as @xmath61 , where @xmath62 is the likelihood for particle type @xmath56 , determined from the responses of the relevant detectors @xcite . the efficiencies for muon and electron identification are 92% for momenta larger than 1.0 gev/@xmath44 and 94% for momenta larger than 0.5 gev/@xmath44 . candidate @xmath3 mesons are selected by requiring the invariant mass of @xmath36 daughters to be in the range @xmath63 . we require that both kaon daughters have kaon likelihood ratios @xmath64 and electron likelihood ratios @xmath65 to reduce the background from @xmath32 conversions . candidate @xmath4 mesons are reconstructed from @xmath37 with the invariant mass requirement @xmath66 . the @xmath51 candidate is selected from @xmath35 pairs with invariant mass in the range , @xmath67 . in order to improve the @xmath4 mass resolution , the @xmath51 mass is constrained to be 135 mev/@xmath68 for the @xmath4 mass reconstruction . candidate @xmath5 and @xmath6 mesons are selected with @xmath69 invariant mass in the range @xmath70 , and requiring that the kaon daughter have @xmath64 and both daughters have @xmath65 . [ fig : vmass](a , b , c ) show the invariant mass distributions of the @xmath3 , @xmath4 and @xmath5 candidates for @xmath71 , @xmath72 and @xmath73 , respectively . the estimated bg distributions agree with the data . the main bg contribution is due to @xmath52 events with @xmath3 mesons for the @xmath74 mode , @xmath75 with the pion misidentified as a lepton for the @xmath76 mode , and @xmath77 with one pion misidentified as a kaon and another misidentified as a lepton for the @xmath78 and @xmath28 modes . for @xmath71 , ( b ) @xmath79 for @xmath72 and ( c ) @xmath80 for @xmath73 after muon identification . the points with error bars are data . the open histogram shows the expected @xmath42 bg mc and the hatched one @xmath52 mc and two - photon mc . the regions between the vertical red lines are selected.,title="fig:",scaledwidth=23.0% ] for @xmath71 , ( b ) @xmath79 for @xmath72 and ( c ) @xmath80 for @xmath73 after muon identification . the points with error bars are data . the open histogram shows the expected @xmath42 bg mc and the hatched one @xmath52 mc and two - photon mc . the regions between the vertical red lines are selected.,title="fig:",scaledwidth=23.0% ] for @xmath71 , ( b ) @xmath79 for @xmath72 and ( c ) @xmath80 for @xmath73 after muon identification . the points with error bars are data . the open histogram shows the expected @xmath42 bg mc and the hatched one @xmath52 mc and two - photon mc . the regions between the vertical red lines are selected.,title="fig:",scaledwidth=23.0% ] to reduce the remaining bg from @xmath42 and @xmath52 , we require the relations between the missing momentum @xmath81 ( gev/@xmath44 ) and missing mass squared @xmath82 ( ( gev/@xmath68)@xmath83 ) summarized in table [ tbl : vcut ] . .selection criteria using @xmath81(gev/@xmath44 ) and @xmath82((gev/@xmath68)@xmath83 ) where @xmath81 is missing momentum and @xmath82 is missing mass squared . [ cols="^,^",options="header " , ] we have searched for lfv decays @xmath74 , @xmath84 , @xmath27 and @xmath28 using a 543 fb@xmath7 data sample from the belle experiment . no evidence for a signal is observed and upper limits on the branching fractions are set in the range @xmath85 at the 90% confidence level . this analysis is the first search for @xmath86 modes . the results for the @xmath74 , @xmath27 and @xmath28 modes are @xmath11 times more restrictive than our previous results obtained using 158 fb@xmath7 of data . the sensitivity improvement includes a factor of 3.4 in data statistics and an optimized analysis with higher efficiency and much improved bg suppression . the improved upper limits can be used to constrain the parameter spaces of various scenarios beyond the sm . 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 and the national institute of informatics for valuable computing and super - sinet network support . we acknowledge support from the ministry of education , culture , sports , science , and technology of japan and the japan society for the promotion of science ; the australian research council and the australian department of education , science and training ; the national science foundation of china and the knowledge innovation program of the chinese academy of sciences under contract no . 10575109 and ihep - u-503 ; the department of science and technology of india ; the bk21 program of the ministry of education of korea , the chep src program and basic research program ( grant no . r01 - 2005 - 000 - 10089 - 0 ) of the korea science and engineering foundation , and the pure basic research group program of the korea research foundation ; the polish state committee for scientific research ; the ministry of education and science of the russian federation and the russian federal agency for atomic energy ; the slovenian research agency ; the swiss national science foundation ; the national science council and the ministry of education of taiwan ; and the u.s.department of energy .

we have searched for neutrinoless @xmath0 lepton decays into @xmath1 and @xmath2 , where @xmath1 stands for an electron or muon , and @xmath2 for a vector meson ( @xmath3 , @xmath4 , @xmath5 or @xmath6 ) , using 543 fb@xmath7 of data collected with the belle detector at the kekb asymmetric - energy @xmath8 collider . no excess of signal events over the expected background is observed , and we set upper limits on the branching fractions in the range @xmath9 at the 90% confidence level . these upper limits include the first results for @xmath10 as well as new limits that are @xmath11 times more restrictive than our previous results for @xmath12 , @xmath13 and @xmath14 .

high energy particle acceleration and magnetic field amplification appear to be tightly related phenomena in many astrophysical environments . for instance , x - ray observations of synchrotron emission from ultrarelativistic electrons in young supernova remnants ( snrs ) @xcite suggest that electrons are efficiently accelerated in these environments , and that the ambient magnetic field in the dowstream medium of snr forward shocks is amplified by a factor of @xmath5 compared to its typical value in the interstellar medium . also , cosmic rays ( crs ) with energies up to @xmath6ev are believed to originate in snrs . calculations based on the diffusive shock acceleration mechanism @xcite show that such high energies can only be reached if crs are efficiently confined to the remnant @xcite , a condition that would be eased by a substantial magnetic field amplification . while it is supported by direct observations of snrs , the process by which the field amplification takes place is still a mystery . @xcite suggested the possibility of magnetic amplification driven by non - resonant crs propagating in the upstream region of shocks . this instability is different from the alfvn wave amplification due to cyclotron resonance with crs @xcite . instead , it only requires positively charged crs propagating along a background magnetic field , @xmath7 , with larmor radii much larger than the wavelength of the wave . this condition is expected to be satisfied by diffusively shock - accelerated crs in the upstream region of both relativistic and non - relativistic shocks . while individual crs are relativistic , on average they stream with respect to the upstream plasma with a drift velocity , @xmath8 , that depends on the distance from the shock . the lowest energy crs are efficiently confined to the shock vicinity by the upstream turbulence , and drift with the shock with @xmath9 , where @xmath10 is the shock speed . the higher energy crs are less affected by the upstream scattering , so they tend to escape more easily from the shock . thus , as the distance from the shock increases , @xmath8 will increase , asymptotically approaching @xmath11 for non - relativistic shocks ( for relativistic shocks , @xmath12 everywhere ) . also , considering that electrons are more affected by radiative losses than ions , it is reasonable to think that at some distance from the shock crs will be mainly positively charged particles . their drift will drive a constant current , @xmath13 , through the upstream plasma . this current will be compensated by the opposite return current , @xmath14 , provided by the background plasma . if there is a small magnetic field perturbation @xmath15 , perpendicular to @xmath7 , a @xmath16 force will push the background plasma transversely . a force of the same magnitude will push the crs in the opposite direction , so that the net force acting on the background plasma - crs system vanishes . however , given the high rigidity of the crs , only the background plasma will experience a significant transverse motion . for a helical magnetic perturbation @xmath15 , this transverse motion will stretch the magnetic field lines , producing an amplification of @xmath15 . using analytic mhd analysis , @xcite showed that for a right - handed circularly polarized electromagnetic wave , this amplification would be exponential , and faster than the resonant instability and @xmath13 are parallel . in the antiparallel case , the polarization is left - handed ] . the linear dispersion relation of this cosmic ray current - driven ( crcd ) instability has been calculated using both mhd @xcite and kinetic treatments @xcite . these works found that , if @xmath13 is kept constant , the instability will grow at a prefered wavelength @xmath17 , with a growth rate @xmath18 , where @xmath19 is the mass density of the background plasma . the non - linear evolution of the crcd instability has been studied making use of both mhd and particle - in - cell ( pic ) simulations . the mhd studies @xcite have shown a substantial amplification of the ambient magnetic field and the formation of turbulence , which is characterized by prominent density fluctuations in the plasma . they have established a saturation criterion that depends on the wavelength , @xmath20 , of each mode and that is given by @xmath21 . this saturation criterion would imply that , if @xmath22 is constant , the field never stops growing but only migrates into longer wavelengths . this migration would be such that the dominant wavelength , @xmath0 , goes roughly as @xmath23 , where b is the mean value of the magnetic field @xcite . the saturation in mhd simulations with constant cr current is either not observed @xcite or is due to the size of magnetic fluctuations reaching the size of the box @xcite . one important limitation of the mhd simulations is that they can not follow the instability at arbitrarily low densities , which makes it difficult to model large plasma density fluctuations properly . also , the mhd simulations do not include the back - reaction on the crs , which must play a fundamental role in the saturation of the instability . both difficulties can be potentially resolved by fully kinetic pic simulations . a first attempt to use pic simulations for this problem was made by @xcite . even though their results show magnetic amplification , it accurs at a significantly lower rate and through a kind of turbulence that essentially differs from the circularly polarized , growing waves predicted by @xcite . this raised a question about the existence of the crcd instability beyond the mhd approximation . in this work we confirm the existence of the crcd instability using pic simulations and establish the conditions under which it is present . in order to understand the saturation mechanisms of the instability , we separate our study in three parts , presented in [ sec : thewaves ] , [ sec : multidimensional ] , and [ freecrs ] . in [ sec : thewaves ] , we show the main non - linear properties of the crcd waves , focusing on their intrinsic saturation mechanism in the presence of constant cr current , @xmath13 . we do this using an one - dimensional , analytic model , and check our results with one - dimensional pic simulations . we calculate a non - linear dispersion relation that includes the time evolution of the phase velocity of the waves . also , our model quantifies all the plasma motions induced by the waves , which is needed to understand the wave behavior in the multidimensional context . in [ sec : multidimensional ] , we present the multidimensional properties of the instability using two- and three - dimensional simulations , also with constant @xmath13 . first , we determine the conditions under which the crcd instability can grow without being affected by plasma filamentation as in the case of @xcite . then , combining the multidimensional simulations with our results from [ sec : thewaves ] , we determine the main properties of the instability in its non - linear stage . we confirm the generation of turbulence as suggested by previous mhd studies , and reexamine its main properties . we estimate the typical turbulence velocity and length scale as a function of the magnetic amplification , finding a faster migration to longer wavelengths than predicted by mhd simulations . we find that the acceleration of background plasma along the direction of motion of the crs causes the intrinsic saturation of the crcd instability at constant @xmath13 . in [ freecrs ] , we study the effect of the back - reaction on the crs as a second saturation mechanism for the instability . [ conclusions ] presents our conclusions and an application to the case of snr environments . in this section we present the one - dimensional analysis of the crcd waves at constant @xmath13 , i.e. , without considering the back - reaction on the crs . in 2.1 we show an analytic , kinetic model for the crcd waves , valid in the non - linear regime . after that , in 2.2 , we check our model making use of one - dimensional pic simulations . we consider a piece of upstream plasma through which positively charged crs flow , providing the current @xmath13 . we focus on the situation where the initial magnetic field @xmath7 , @xmath13 , and the wave vector of the crcd mode @xmath24 , are parallel . local charge neutrality is assumed as initial condition , so @xmath25 , where @xmath26 , @xmath27 , and @xmath28 correspond to the density of ions , electrons , and crs , respectively . in this section we study the time evolution of crcd waves of different wavevectors @xmath24 , which are characterized by their growth rate @xmath29 and phase velocity @xmath30 , where @xmath31 . the derivation is made for a right - handed polarized wave , and is based on the calculation of the drift velocities of plasma components in the presence of a wave of arbitrary amplitude . if we know these drift velocities and the number densities of the different species , we can calculate the total current provided by the background plasma as a function of space and time . adding this total plasma current to the constant @xmath13 contributed by the crs , the time evolution of the wave can be directly obtained from the ampere s and faraday s laws . the details of the calculation are presented in appendix [ app : analytic ] . here we describe its main results , which are summarized in the dispersion relation given by equation ( [ eq : dispersion ] ) . this dispersion relation assumes a constant @xmath29 and allows @xmath30 to evolve in time . we will see below that @xmath29 is indeed constant as long as @xmath32 , where @xmath2 is the alfvn velocity of the backgroud plasma . this condition not only puts a limit to the validity of the derivation , but also sets a saturation criterion for the crcd waves . also , our derivation is in the low plasma temperature limit and assumes that @xmath33 , where @xmath34 is the initial alfvn velocity of the plasma . using two - dimensional pic simulations , we show below that this second condition is actually a requirement for the crcd waves not to be quenched by weibel - like plasma filamentation . from the real part of equation ( [ eq : dispersion ] ) , we see that the growth rate , @xmath29 , is maximized when the wavenumber @xmath35 , which corresponds to the same wavenumber of maximum growth found in the linear regime @xcite . from the imaginary part , we obtain the following differential equation for @xmath36 as a function of the amplification factor of the waves , @xmath37 ( defined as the ratio between the magnitude of the transverse magnetic field , @xmath38 , and @xmath39 ) , @xmath40 the solution for equation ( [ eq : diferencial ] ) is @xmath41 which , when @xmath42 , can be approximated as @xmath43 . this implies that , although in the linear regime the crcd waves are almost purely growing ( @xmath44 ) , if @xmath45 gets close to @xmath3 , their phase velocity can also become comparable to @xmath3 . taking the real part of equation ( [ eq : dispersion ] ) and evaluating at @xmath46 , we obtain that @xmath47 where we have kept terms only to first order in @xmath48 and @xmath49 . we see that in the regime @xmath50 , the instability grows exponentially with a maximum growth rate , @xmath51 that is constant and has the same value as obtained in the previous linear studies @xcite . even though equation ( [ eq : growthrate ] ) shows that our assumption of constant @xmath52 is only valid when @xmath50 , it also indicates that , as @xmath45 approaches @xmath3 , the growth rate will be substantially reduced , suggesting an intrinsic saturation limit for the crcd waves at @xmath53 . in appendix [ app : analytic ] we also show that the presence of the crcd waves induces bulk motions of the plasma particles both parallel and transverse to @xmath13 . the parallel motion has a velocity @xmath54 , while the transverse motion has a velocity @xmath55 , which always points perpendicular to @xmath15 ( and to @xmath13 ) . the parallel plasma motion implies that , when @xmath53 , the entire plasma will move at a speed close to @xmath3 , which , from the point of view of the plasma , substantially reduces @xmath13 . this reduction in @xmath13 explains the intrinsic saturation of the waves at @xmath53 . the tranverse motion , on the other hand , becomes of the order of the alfvn velocity of the plasma when the crcd waves become non - linear ( @xmath56 ) . we will see below that this increasing transverse velocity is related to turbulence formation in the non - linear regime . we checked our analytical results with one - dimensional pic simulations . we use the pic code tristan - mp @xcite , which can run in one , two , and three dimensions . in these simulations , like in our analytic model , all plasma properties depend only on one spatial direction ( @xmath57 ) , but both the velocities of the particles and the electromagnetic fields keep their three - dimensional components . we set up a periodic box that contains an initially cold background plasma ( with typical particle thermal velocity of @xmath58 ) composed of ions and electrons , and a small population of relativistic ions ( crs ) . the driving current is given by the crs that move along @xmath59 with a mean velocity @xmath3 that we vary between runs . these crs are not allowed to change their velocities , as if they had an infinite lorentz factor , @xmath60 . such locked " crs allow us to study the non - linear evolution of the instability considering a constant @xmath13 , i.e. , eliminating the back - reaction on the crs . we give electrons a small velocity along @xmath59 such that the background plasma carries a current @xmath61 . this way the net current is zero . the initial magnetic field , @xmath7 also points along @xmath59 . since we want to simulate a situation where @xmath62 , having good cr statistics would imply a large number of particles per cell . in order to overcome this difficulty , we have initialized the same number of macroparticles for crs , ions , and electrons , but modified their charges so that @xmath63 and @xmath64 , where @xmath65 . we change the mass of the particles accordingly in order to keep the right charge to mass ratios . particles are initially located randomly in the box such that at the position of each ion we also have an electron and a cr , so the initial charge density is zero . this initialization is also used in our two- and three - dimensional runs . the common numerical parameters for the simulations are @xmath66 @xmath67 ( where @xmath68 is the electron plasma frequency and @xmath67 is the grid cell size ) , ion - electron mass ratio @xmath69 , speed of light @xmath70 @xmath67/@xmath71 ( where @xmath71 is the time step ) , and 12.5 particles per species per cell . here we test the non - linear dispersion relation found in [ sec : analytical ] in the relativistic regime ( @xmath72 ) , which would be appropriate for the upstream medium of a relativistic shock front . we ran one - dimensional simulations in boxes of different sizes @xmath73 , set up to probe the growth rate of different wavelengths , @xmath20 . as an initial condition we used a right - handed , circularly polarized , growing wave of amplitude 0.1@xmath74 , whose fields and particle velocities were determined from our analytic model ( appendix [ app : analytic ] ) . we put only one period of the wave in a box , so that the only other modes that could be excited are shorter or equal to @xmath75 . we choose the density of crs such that the corresponding maximum growth rate of the instability , @xmath52 , is 0.2 @xmath76 , so we are in the regime where the background plasma is well magnetized . simulations were run for @xmath20 equal to 0.5 , 0.75 , 1 , 1.25 , 1.5 , 2 , and 3@xmath77 . we used several values for the initial alfvn velocities , @xmath78 , in the range @xmath79 to @xmath80 . this implies that the initial gyrotime @xmath81 ranges from 189 to 1456 @xmath71 , so it is resolved with about 20 @xmath71 even when @xmath1 , and @xmath82 ranges from 1 to 8 . the results for the cases @xmath83 and @xmath80 are presented in fig . [ fg : figura1 ] . we observe that for @xmath84 there is practically no growth . for @xmath85 1 , 1.25 , and 1.5@xmath77 we obtain nearly the same growth rate , which is very close to the analytic @xmath52 . for longer wavelengths , the growth rate gradually decreases . in all our experiments , for @xmath86 , the exponential growth continues until @xmath45 becomes close to @xmath87 , which confirms our analytical saturation criterion for fixed crs . at later times we see that , depending on @xmath73 , the amplitude of the wave either oscillates or keeps growing but at a much lower rate . while individual crs near a non - relativistic shock move at almost the speed of light , on average they move with respect to the upstream at a drift velocity , @xmath3 , that is less than c. in order to study this case , we ran a series of simulations where , besides not allowing crs to alter their trajectories , we make them drift along @xmath59 at a velocity @xmath88 and @xmath89 . we do this using a box size @xmath90 . in this case we do not seed the instability with a small amplitude , growing wave , as done in [ sec : rel ] . instead , we only put the initial magnetic field , @xmath7 , such that @xmath91 , forming an angle @xmath92 with @xmath93 . we use this set - up to show that the instability can develop from any kind of noise . we tilt @xmath7 by a small angle to inject a small amount of magnetic and kinetic energy perpendicular to @xmath13 , so it acts as an initial seed that does not favor any particular @xmath20 ( experiments with @xmath94 were also run , showing no difference besides requiring a longer initial time for the wave to appear ) . other numerical parameters are the same as in the relativistic experiments . is plotted as a function of time , @xmath95 , for experiments similar to the ones depicted in fig . [ fg : figura1 ] , but for @xmath96 1/10 and a box of @xmath97 , where @xmath77 is the wavelength of the theoretically determined fastest growing mode . the instability is seeded by tilting @xmath7 by @xmath98 with respect to @xmath13 . a series of @xmath3 is tested : @xmath99 ( solid black ) , 0.9 ( solid green ) , 0.8 ( solid red ) , 0.6 ( dotted black ) , 0.4 ( dotted green ) , and 0.2 ( dotted red ) . @xmath52 is the maximum theoretical growth rate for the case @xmath99 . the results are consistent with the theoretical @xmath52 and @xmath77 , and with the intrinsic saturation criterion , @xmath53.,width=292 ] fig . [ fg : lesscurr ] shows the magnetic energy evolution for the six @xmath3 tested . we observe that the growth rate is @xmath100 , where @xmath52 is the maximum growth rate for @xmath101 . the amplitude at which the exponential growth stops is such that @xmath53 , confirming our results from [ sec : analytical ] . ( red line ) and @xmath102(green line ) , are plotted as a function of distance , @xmath57 , at two different times for two of the runs described in fig . [ fg : lesscurr ] ( @xmath99 and 0.6 ) . the black line represents @xmath103 . the two left plots show the case @xmath104 at @xmath105 and 50 . the two right plots show the case @xmath106 at @xmath107 and 50 . the results are consistent with the instability appearing initially as right - handed circularly polarized wave with a preferred wavelength @xmath108 , and with a migration into longer wavelengths as the instability grows.,width=321 ] fig . [ fg : sequenceapj ] shows the different components of the magnetic field as a function of position , @xmath57 , at different times for @xmath101 , and @xmath109 . we can see that in both cases the instability appears as a right - handed polarized wave and at an initial wavelength @xmath110 , where @xmath77 is the wavelength of maximum growth for @xmath101 . after the wave reaches saturation , there is a migration into longer wavelengths . this migration appears because the modes with wavelengths greater than @xmath77 grow more slowly , but still grow and saturate at @xmath53 , as can be seen in fig . [ fg : figura1 ] for the relativistic regime ( @xmath101 ) . it means that , as the instability reaches @xmath53 , the spectrum of the waves gradually receives more contribution from wavelengths longer than @xmath77 . as we saw in [ sec : analytical ] , crcd waves induce plasma motions both parallel and perpendicular to @xmath59 . the left panel in fig . [ parperfig ] shows the mean velocity of plasma particles along @xmath57 ( i.e. , parallel to @xmath13 ) , and the analytic estimate for this velocity , @xmath111 . this velocity comes from the @xmath112 drift of background particles , which in a well magnetized plasma ( @xmath113 ) is much larger than other plasma drifts ( see appendix [ app : analytic ] ) . the velocities in fig . [ parperfig ] are computed for two simulations from the right panel of fig . [ fg : figura1 ] : @xmath114 with @xmath115 and @xmath116 with @xmath117 . these wavelengths correspond to the fastest growing waves for the two @xmath3 . the right panel in fig . [ parperfig ] , shows the mean magnitude of the _ transverse _ velocity of plasma particles for the same simulations , and the analytic estimate , @xmath118 . considering that these two cases keep growing exponentially until @xmath119 and 12 , respectively ( see fig . [ fg : figura1 ] ) , we see that , in the @xmath50 regime , our analytical estimates for both longitudinal and transverse motions are in good agreement with our numerical results . so far we have studied the properties of the crcd waves assuming an ideal one - dimensional geometry and constant cr current . in this section , we relax the first of these conditions and use two- and three - dimensional pic simulations to study the crcd instability , still keeping @xmath13 constant . we identify two main differences with respect to the one - dimensional case . the first has to do with the possibility of plasma filamentation that happens before the crcd instability sets in , as suggested by previous works @xcite . we will see below that this filamentation does not occur if the plasma is sufficiently magneitzed , @xmath120 . the second multidimensional effect is the interference between crcd waves generated in different regions of space . since typically the instability starts from random noise , different regions will give rise to crcd waves which in general are out of phase with each other . during the non - linear stage , this non - coherence makes the transverse plasma motions from adjacent regions interfere with each other , giving rise to density fluctuations and turbulence in the plasma . motivated by previous pic studies by @xcite , we studied the possibility of an initial plasma filamentation that could suppress the formation of the crcd instability . we ran a series of high space resolution ( @xmath121 @xmath67 ) two - dimensional simulations whose numerical parameters and results are described in table [ table : fil2 ] . all our two - dimensional simulations are set up in the @xmath122 plane , with @xmath13 and @xmath7 parallel to the @xmath59 axis . also , as in some of our one - dimensional simulations , there is a small component of @xmath7 pointing along @xmath123 , working as a seed for the instability . we identified three regimes , represented in figs . [ fg : filamentationc ] , [ fg : filamentationb ] and [ fg : filamentationa ] . [ fg : filamentationc ] shows the plasma density and three components of the magnetic field for a simulation with @xmath124 ( run m5 in table [ table : fil2 ] ) at two times @xmath125 and 11 . even though some crcd field is observed , especially in @xmath126 , the dominant instability corresponds to a transverse filamentation that appears initially on the scale of @xmath127 times the electron skin depth . as time goes on , the filaments merge , creating prominent holes in the plasma that preclude the growth of the instability . [ fg : filamentationa ] , on the other hand , shows the same quantities for a simulation where the relative number of crs was decreased by a factor of 10 ( run m7 ) , implying that @xmath128 . in this case , a crcd wave of the size of the box does form ( we have chosen the @xmath57-size of the box to be @xmath129 ) . finally , fig . [ fg : filamentationb ] shows the case @xmath130 ( run m6 ) , in which both the crcd instability and the initial filamentation coexist ( we call these cases transitional " and indicate them with the letter t " in table [ table : fil2 ] ) . these three examples indicate that the crcd instability will develop as long as @xmath120 , which is equivalent to having a well magnetized plasma in the sense that @xmath113 ( see appendix [ app : analytic ] ) . as shown in table [ table : fil2 ] , we tested the dependence of this criterion on both the magnetization of the plasma ( using @xmath131 and 31.5 ) and the mass ratio , @xmath132 ( using @xmath133 and 100 ) . we see no difference in our results except that the runs with @xmath134 require a slightly higher value of the ratio @xmath135 ( a factor of 2 larger ) for the transverse filamentation to dominate , but the qualitative criterion remains the same . also , varying @xmath132 allows us to determine the physical length scale , @xmath136 , at which the transverse filaments appear . this scale shows no dependence on @xmath132 and corresponds to @xmath127 times the electron skin depth . another @xmath69 simulation was run with zero initial magnetic field ( m8 ) , showing that the filaments appear at practically the same scale as in the finite magnetic field case , suggesting a similarity between this filamentation and the weibel instability . finally , these results were also tested for a non - relativistic case , @xmath137 , obtaining the same conclusions . thus , the crcd instability will grow if the condition @xmath120 ( or , equivalently , @xmath113 ) is satisfied . for comparison , the smallest @xmath138 factor used by @xcite is 1.31 , which is close to the regime where the transverse filaments appear . this fact would explain their plasma filamentation , which may have suppressed the appearance of the crcd instability . + + + [ cols= " < , < , < , < , < , < , < , < , < , < , < , < , < " , ] we studied crcd instability with a series of two- and three - dimensional simulations whose numerical parameters are summarized in table [ table : interference ] . as we will see below , when multidimensional effects are considered , the dominant wavelength of the instability , @xmath0 , is initially equal to @xmath77 but then rapidly grows as the field is amplified . this can make @xmath0 equal to the size of the box @xmath73 before the instability reaches saturation , which can make sufficiently large three - dimensional simulations challenging . we discuss here the results of our three - dimensional runs , and check them in appendix [ sec:2d ] with large two - dimensional simulations , for which @xmath0 is always significantly smaller than the size of the box . in this section we present the results of three three - dimensional simulations that test saturation in the non - relativistic and relativistic regimes , and the dependence of the amplification on the initial magnetic field and the cr drift velocity . two of the simulations have the same @xmath139 , but @xmath101 and @xmath11 ( runs i1 and i2 in table [ table : interference ] ) . the third simulation has @xmath91 and @xmath101 ( run i3 in table [ table : interference ] ) . as in all our simulations so far , @xmath13 and @xmath7 point along @xmath59 ( apart from a small component of the magnetic field along @xmath123 of magnitude @xmath140 ) , and the back - reaction on the crs is not included . the rest of the numerical parameters are specified in table [ table : interference ] . , but with overplotted arrows showing the magnetic field projection on the @xmath141 plane . the clock - wise orientation of the magnetic field lines around the plasma holes shows the presence of crcd waves driving the turbulence.,width=302 ] the evolution of the plasma density and the three components of the magnetic field for simulation i1 can be seen in figs . [ fg : cortes1 ] , [ fg : cortes2 ] , [ fg : cortes3 ] , and [ fg : cortes4 ] . these figures show two slices of the simulation box . one is longitudinal and corresponds to the plane @xmath142 ( top panels ) , and the other is transverse and corresponds to @xmath143 ( bottom panels ) , where @xmath144 . the magnetic energy evolution for the same run is depicted in fig . [ fg : departure3d ] . [ fg : cortes1 ] shows the early moments of the instability ( @xmath145 ) . the longitudinal slice shows how the crcd waves form independently in different regions of the box . this is also seen in the transverse slice , which shows how the phases of the waves differ between different points of the plane @xmath143 . [ fg : cortes2 ] shows the beginning of the non - linear regime ( @xmath146 ) , which corresponds to @xmath147 . in this case , the phases of the waves are transversely more correlated compared to fig . [ fg : cortes1 ] , as can be seen in the plots of the transverse slice for @xmath126 and @xmath102 . this increased spatial correlation indicates that , until this moment , the adjacent waves were merging without significantly interfering with each other . also , fig . [ fg : cortes2 ] shows the appearance of prominent density fluctuations ( @xmath148 , where @xmath149 is the plasma density ) . as we saw in [ sec : thewaves ] , when the crcd waves are in the exponential growth regime ( @xmath50 ) , the background plasma will move transversely at @xmath150 . so , when the instability gets non - linear , the transverse velocity of the plasma becomes close to @xmath45 . in a low temperature regime , this velocity corresponds to the magnetosonic sound speed of the plasma . thus , as soon as @xmath151 , the transverse motions will produce moderate shocks , giving rise to significant density fluctuations in the plasma . at this point the transverse plasma motions develop into isotropic turbulence with velocities of the order of @xmath152 . , for which ( @xmath153 ) = ( 1/40,1 ) , ( 1/20,0.5 ) , and ( 1/10,1 ) , respectively . time is normalized in terms of the @xmath52 of each simulation . in all the runs , the departure from exponential growth occurs after @xmath151 , but saturation happens at @xmath1.,width=292 ] what happens after the density fluctuations appear can be seen in fig . [ fg : cortes3 ] ( @xmath154 ) . the longitudinal slice shows how the magnetic fluctuations get distorted and increase rapidly in size . as already mentioned in [ sec : nonrel ] , even in one - dimensional geometry the instability is expected to evolve into wavelengths longer than @xmath77 . however , in a multidimensional set - up , the evolution into magnetic fluctuations of larger size gets accelerated after the appearance of the density fluctuations and turbulence in the plasma . we will quantify this migration in [ sec : migration ] . the transverse slice of fig . [ fg : cortes3 ] shows how the underdense regions ( or holes ) have merged and increased their size with respect to fig . [ fg : cortes2 ] . it is also interesting to see from fig . [ fg : zoomcorte ] how the holes are separated by plasma walls " through which the transverse magnetic field reverses direction . we see that the magnetic field has a clockwise orientation around the holes , which is consistent with the presence of right - handed waves producing the expansion of the holes . finally , fig . [ fg : cortes4 ] ( @xmath155 ) shows essentially no difference with respect to fig . [ fg : cortes3 ] besides the growth of the size of both the magnetic fluctuations and the plasma holes , which at this point are close to @xmath73 . [ fg : departure3d ] shows the magnetic energy evolution for the three - dimensional simulations . we can see that , in the three cases , the departure from the exponential growth occurs shortly after the wave becomes non - linear ( which coincides with the generation of significant density fluctuations and turbulent motions in the plasma ) . we also see that the final saturation satisfies the @xmath53 condition , which suggests that , when multidimensional effects are included , the intrinsic saturation of the crcd instability is still given by the @xmath53 criterion . unfortunately , in our three - dimensional simulations , this saturation happens when the dominant sizes of the holes and magnetic fluctuations have already become close to @xmath156 . saturation still happens at @xmath53 because , when @xmath157 , the three - dimensional simulations behave more like the one - dimensional simulations presented in [ sec : rel ] , in the sense that there is only one dominant mode that saturates at @xmath53 . after @xmath157 , the density fluctuations almost disappear and the turbulent motions transform into more coherent transverse plasma motions . in any case , the @xmath53 saturation criterion is confirmed by two - dimensional simulations presented in appendix [ sec:2d ] for which @xmath0 is always smaller than @xmath73 . fig . [ fg : departure3d ] also shows that , in the three - dimensional simulations , the magnitude of the magnetic component along @xmath59 is comparable to the transverse one , suggesting a rather isotropic orientation of the crcd field . , as a function of the amplification factor , @xmath37 , for three - dimensional simulations i1 ( dot - dashed ) , i2 ( dashed ) , and i3 ( dotted ) . for comparison , our semi - analytical formula , @xmath158 , is shown as solid line.,width=302 ] even though migration to longer wavelengths is already observed in one - dimensional simulations , it becomes faster when multidimensional effects are considered . in this section we propose a semi - analytic model that quantifies this migration in terms of the amplification factor of the field , @xmath37 . as we saw in [ sec : threed ] , the motions associated with the turbulence tend to distort the crcd waves , producing a damping of the shortest wavelength modes . thus , the dominant wavelength , @xmath0 , will correspond to the fastest growing mode that can be amplified without being strongly affected by the turbulence . considering that a crcd wave of wavelength @xmath20 grows in a time scale comparable to the inverse of its growth rate @xmath159 from equation ( [ eq : dispersion ] ) , and that the turbulence will kill it in a time scale comparable to @xmath160 , where @xmath161 is the typical turbulent velocity , then @xmath0 will be such that @xmath162 . since the turbulence is due to the transverse plasma motions produced by non - coherent crcd waves , then @xmath161 must be comparable to the transverse velocity of the waves , which we already determined to be @xmath163 . so , @xmath0 will be such that @xmath164 , where @xmath165 is an unknown constant that quantifies the relative importance of the two time scales . if we get @xmath166 from the real part of equation ( [ eq : dispersion ] ) , we can obtain @xmath165 by fitting the evolution of @xmath0 in our three - dimensional simulations , obtaining @xmath167 . this way we find @xmath0 as a function of @xmath37 , @xmath168/2 , \label{eq : wavelength}\ ] ] which is intended to be valid after the turbulence becomes significant ( @xmath169 ) . in fig . [ fg : wavelength ] we show a comparison between this formula and the evolution of @xmath0 as a function of @xmath37 for our three - dimensional simulations . we computed @xmath0 by performing fourier transforms of @xmath126 and @xmath102 along lines of constant @xmath170 and @xmath171 coordinates , and then finding the mean wavelength of the peak of the fourier transform . as the dominant wavelengths approach @xmath73 ( @xmath172 ) , the determination of @xmath0 becomes quite noisy . due to this reason , we have plotted our simulation results only until @xmath173 . we see from fig . [ fg : wavelength ] that equation ( [ eq : wavelength ] ) appears to provide an acceptable fit for the evolution of @xmath0 . our result shows a growth of @xmath0 substantially faster than the direct proportionality between @xmath0 and @xmath37 suggested by @xcite . we will see below that , when the back - reaction on the crs is considered , this difference has important implications to the saturation of the crcd instability . and @xmath174 normalized in terms of @xmath175 , are plotted as a function of time ( normalized using @xmath52 ) for one- and three - dimensional runs in order to study the effect on the magnetic energy evolution due to the back - reaction on the cr . in all cases crs are monoenergetic and have a semi - isotropic momentum distribution such that @xmath137 . the two one - dimensional simulations have numerical parameters : @xmath176 , @xmath177 , @xmath178 , @xmath179 @xmath67 , @xmath180 , @xmath69 , @xmath181 , @xmath182 @xmath71 , @xmath183 , and @xmath184 . their cr lorentz factor @xmath60 is 20 ( solid , red line ) and 40 ( solid , green line ) , respectively . the three - dimensional simulation , whose transverse magnetic energy is represented by the solid , blue line , has the same parameters as run i2 in table [ table : interference ] , but with @xmath185 . the dotted , blue line represents the longitudinal magnetic energy ( @xmath186 ) . the dashed , black line shows the constant magnetic energy along @xmath59 for the two one - dimensional simulations.,width=302 ] in this section we use one- and three - dimensional simulations to study the effect of the dynamic evolution of the crs on the saturation of the crcd instability . we will concentrate on the case of a beam of monoenergetic crs drifting at half the speed of light ( @xmath187 ) in the @xmath59 direction , which is parallel to @xmath188 ( except for a small magnetic component along @xmath123 ) . this drift is obtained by sampling the cr velocities from an isotropic , monoenergetic momentum distribution , but only keeping the velocities in the positive @xmath57 direction . this choice for the cr momentum distribution has a direct application to the most energetic crs that propagate in the upstream medium of snr shocks ( see [ conclusions ] ) . the red and green curves in fig . [ fg : unlocked ] represent the magnetic energy evolution for two one - dimensional simulations that only differ in their lorentz factors , @xmath60 , taken to be 20 and 40 ( the rest of the numerical parameters are specified in the caption of fig . [ fg : unlocked ] ) . these simulations saturate at @xmath189 and @xmath190 , when the larmor radius of crs is close to the dominant wavelength of the instability ( @xmath191 and @xmath192 for two runs ) . to obtain this result we use that initially @xmath193 , and consider that at saturation the dominant wavelength has grown ( @xmath194 and @xmath195 , respectively ) and crs have lost part of their energy ( the mean @xmath60 of crs is 18.9 and 37.6 , respectively ) . these results are confirmed by a three - dimensional simulation whose magnetic energy evolution is represented by the blue lines in fig . [ fg : unlocked ] . the numerical parameters of this three - dimensional simulation are the same as in run i2 ( see table [ table : interference ] ) but includes the back - reaction on the crs , whose @xmath185 . if we consider that at saturation @xmath196 and the mean @xmath60 of the crs is 27.6 , we obtain that the cr deflection saturates the instability when @xmath197 . note that in this simulation @xmath0 is always a factor of 4 smaller than the size of the box , so the saturation is not affected by box effects . for the semi - isotropic distribution of monoenergetic crs presented here , we find that the saturation due to cr back - reaction will happen when @xmath198 . although this result is valid for our particular choice of cr momentum distribution , we expect that in general the saturation of the crcd instability will be determined either by the intrinsic limit @xmath53 , or by the strong cr deflection when @xmath199 . as will be discussed in [ conclusions ] , achieving @xmath53 requires a very high cr energy density , a condition that is not expected for non - relatistic shocks environments . also , our simulations show that , at saturation , many crs have negative @xmath57 velocity . this suggests that , besides the field amplification , the crcd instability can provide an efficient scattering mechanism for crs upstream of shocks . using fully kinetic pic simulations , we confirmed the existence of the crcd instability predicted by bell ( 2004 ) . combining one- , two- , and three - dimensional simulations with an analytic , kinetic model we studied the non - linear properties of the instability and its possible saturation mechanisms . in the first part , we studied non - linear crcd waves under idealized conditions , namely : _ i _ ) ignoring multidimensional effects , and _ ii _ ) assuming a constant cr current without back - reaction on the crs . we confirm that the crcd waves can grow exponentially at the wavelengths and rates predicted by the analytic dispersion relation @xcite . we find that the exponential growth can continue into the very nonlinear regime , until the alfven velocity in the amplified field is comparable to the cr drift velocity , @xmath53 . this saturation is due to plasma acceleration along the direction of motion of the crs , which reduces the cr current observed by the plasma particles . the plasma moves at the velocity @xmath200 , where @xmath37 is the amplification factor of the field ( @xmath201 ) . at saturation , when @xmath202 , the plasma moves together with crs , decreasing the net driving current . the waves also induce transverse plasma motions with velocities @xmath203 . these motions generate plasma turbulence when multidimensional effects are included . in the second part , we considered more realistic conditions by including the multidimensional effects using two- and three - dimensional simulations with constant @xmath13 . our main results are : _ i ) _ in the linear regime , if the plasma is well magnetized ( @xmath113 , or , equivalently , @xmath120 ) , the crcd waves grow at the rate and preferred wavelength close to the ones obtained in the one - dimensional analysis . if this condition is not met , weibel - like filaments form in the plasma , supressing the appearance of the waves . in this case , the streaming crs can still amplify the magnetic field to non - linear values , but at a rate significantly lower than that of the crcd waves @xcite . this regime , however , might be relevant to the upstream medium of relativistic shocks in grbs , where @xmath28 could exceed @xmath26 @xcite . _ ii ) _ in the non - linear crcd regime , the transverse plasma motions associated with the instability create significant density fluctuations and turbulence in the plasma . these turbulent motions suppress the growth of the shortest crcd waves , producing a fast evolution into longer wavelengths that can be approximated by @xmath204/2 $ ] . also , even though the field will continue to be amplified until @xmath53 , the nonlinear growth will be slower than in the linear regime . in the third part , we include the back - reaction on the crs and find that the cr deflection by the amplified field constitutes another possible saturation mechanism . we tested this effect for a semi - isotropic distribution of monoenergetic crs propagating at @xmath187 with respect to the upstream medium , which would be appropriate for the most energetic crs that escape from snrs . we find that the field is amplified until the larmor radii of the crs becomes approximately equal to the size of the dominant magnetic fluctuations . when that happens , the crs get strongly deflected by the magnetic field , which decreases their current and stops the growth of the field . ignoring the migration to longer wavelengths , for a generic cr momentum distribution with @xmath205 , saturation due to cr deflection will happen when @xmath206 , where @xmath207 is the typical lorentz factor of current - carrying cr . on the other hand , saturation due to plasma acceleration to @xmath3 velocity occurs when @xmath208 . thus , the deflection of crs will dominate if the cr energy density is such that @xmath209 . if the migration to longer wavelengths is included , saturation due to cr deflection would happen at even smaller magnetic amplification . in the upstream medium of snr forward shocks , we expect the cr energy density to be low enough so that the maximum crcd amplification is determined by the back - reaction on the crs . in order to make an estimate of typical magnetic amplification in these environments , let us consider a piece of upstream whose distance from the shock is such that it can only feel the most energetic crs that escape from the remnant . we use only the most energetic particles because they are the only ones whose larmor radii are much larger than the typical wavelength of the crcd waves , which is an essential condition of the instability . also , our estimate is based on the following assumptions . first , all the escaping particles have positive charge , which is reasonable considering the much shorter cooling time of electrons compared to ions , and that ions are presumably more efficiently injected into the acceleration process in shocks . second , we assume that the escaping crs have the same energy , @xmath210 , which is roughly the minimum energy required for them to run away from the remnant . third , there is a fixed ratio , @xmath211 , between the flux of cr energy emitted by the shock , @xmath212 , and the flux of energy coming from the upstream medium as seen from the frame of the shock , @xmath213 , where @xmath214 is the shock velocity and @xmath19 is the mass density of the upstream plasma . finally , we assume a plane geometry and that all the ions are protons . under these conditions , the time scale of growth of the instability , @xmath215 , is @xmath216 and the initial length scale of maximum growth is @xmath217 the ratio @xmath218 is @xmath219 which confirms that in the case of snrs the crcd instability will not be affected by the weibel - like filamentation studied in [ sec : magnetization ] . considering the migration into longer wavelengths given by equation ( [ eq : wavelength ] ) , the amplification factor , @xmath37 , in snrs will satisfy @xmath220 \approx 130\bigg(\frac{10 \textrm{km / sec}}{v_{a,0}}\bigg)^2 \bigg(\frac{\eta_{esc}}{0.05}\bigg)\bigg(\frac{v_{sh}}{10 ^ 4\textrm{km / sec}}\bigg)^3 , \label{eq : estimate}\ ] ] which , for typical parameters would imply @xmath221 . we see that , even though run - away crs can significantly amplify the ambient magnetic field , the upstream amplification alone is not enough to explain the factors of @xmath222 inferred from observations of forward shocks in young snrs @xcite . also , note that , if @xmath223 and @xmath224 , the distance swept by the shock in a time @xmath215 is @xmath225 , which is comparable to the typical size of a snr . this means that the advection of the upstream fluid into the shock may happen faster than the growth of the field , and may put further restrictions on the amplification . it has been suggested that a further crcd amplification could be provided by the current of lower energy crs that are confined closer to the shock and move diffusively at drift velocity @xmath226 @xcite . we believe , however , that this possibility requires a more detailed study . such lower energy crs can be magnetized in the sense that their larmor radii are smaller than the typical size of magnetic fluctuations , @xmath0 , violating the conditions for the crcd instability . although on large scales these crs will still produce a current of magnitude @xmath227 parallel to the shock normal , on scales of the crcd wavelength the local cr current may get significantly affected by the amplified field because of the deflection of crs . field amplification may then proceed in essentially different way . the magnetization of crs could be less of an issue if the wavelength of the instability due to low energy crs is shorter than the cr larmor radius . indeed , since lower energy crs are more numerous than the most energetic ones , their larger current will generate shorter crcd waves ( remember that @xmath17 ) . however , the crcd turbulence generated further upstream by the highest energy crs may modify the condition @xmath228 and may suppress the growth of the small wavelength modes closer to the shock . this suggests that other non - linear mechanisms , such as the cyclotron resonance of crs with alfvn waves @xcite , may still be important components in the amplification of the field . the full effect of the low - energy cr contribution needs to be investigated using a fully kinetic treatment of crs that includes their reacceleration by the shock , the presence of pre - existing turbulence , and the eventual contribution of cr electrons to the cancelation of ion current . although we have applied our results only to the non - relativistic case of snrs ( @xmath205 ) , the crcd instability may also play an important role in relativistic shocks in jets and gamma ray bursts , where @xmath72 . our simulations show that , at constant cr current , the evolution of the instability is the same as in the non - relativistic case . in particular , the intrinsic saturation criterion due to plasma acceleration is valid , implying a maximum magnetic fiel such that @xmath229 . however , if the back - reaction on the crs is considered , the non - linear evolution of the field and the saturation due to cr deflection may be dominated by cr beam filamentation @xcite . also , since in the upstream of grb shocks the density of crs might be close to the density of upstream ions @xcite , the magnetization requirement ( @xmath113 ) may not be satisfied in these environments . in this case , a non - linear magnetic amplification is still expected , but through an instability that is characterized by weibel - like filamentation of the plasma and whose properties may be different to the crcd instability described here @xcite . detailed analysis of the relativistic shock case in application to grbs will be presented elsewhere . in conclusion , we have shown that the crcd instability is a viable mechanism for the non - linear amplification of magnetic field upstream of both non - relativistic and relativistic shocks , and that it can provide an efficient scattering mechanism for crs in these environments . this research is supported by nsf grant ast-0807381 and us - israel binational science foundation grant 2006095 . a.s . acknowledges the support from alfred p. sloan foundation fellowship . we thank yury lyubarsky and ehud nakar for useful discussions . in this appendix we calculate a dispersion relation for the crcd waves for the case where @xmath230 , @xmath13 , and the wave vector of the electromagnetic mode , @xmath24 , are all parallel and point along the @xmath59 axis . we will separate the fields and currents into components that are transverse and parallel to @xmath59 , and will identify them with the subscripts @xmath231 ( standing for transverse " ) and @xmath232 , respectively . thus , the magnetic field perpendicular to @xmath59 will be given by @xmath233 which corresponds to a right - handed polarized wave , where the phase @xmath234 is an unknown function of time , @xmath95 , the growth rate @xmath235 is a constant , and @xmath74 is the magnitude of the initial background field @xmath230 . note that the time is chosen so that the wave is in the linear regime for @xmath236 . then , from the ampere s and faraday s laws , we get that the electric field and the current perpendicular to @xmath59 are given by @xmath237 and @xmath238 where @xmath36 and @xmath239 are the time and second time derivative of @xmath240 , respectively . in order to obtain a dispersion relation , we need another expression connecting @xmath241 with @xmath242 and @xmath15 . we find it by making the following assumptions . first , the thermal velocities of the particles in the background plasma will not give rise to any significant drift velocity . second , we will assume that @xmath29 , @xmath243 and @xmath244/dt^n)/(d^{(n-1)}[\omega]/dt^{(n-1)})$ ] are constant and much smaller than @xmath245 , where @xmath246 , @xmath247 , and @xmath248 are the cyclotron frequency , the charge , and the mass of the @xmath249 species , respectively . ( we will see at the end of this appendix that , in order to satisfy the last three conditions , we need @xmath250 , where @xmath34 is the initial alfvn velocity of the plasma . ) third , the electric and magnetic fields are perpendicular , which is a reasonable assumption in the case of a quasineutral plasma . and finally , @xmath32 , where @xmath2 is the alfvn velocity of the plasma and @xmath3 is the drift velocity of the crs . considering this , given @xmath242 and @xmath15 , we can find @xmath241 as follows . if a particle @xmath251 experiences electric and magnetic fields , @xmath252 and @xmath253 , its velocity perpendicular to @xmath253 has two components , @xmath254 and @xmath255 , that satisfy the equations @xmath256 and @xmath257 . in the case of constant and uniform @xmath252 and @xmath253 , @xmath254 represents the classical gyration around @xmath253 , while @xmath255 corresponds to the drift of the particle , which is @xmath258 . when the fields change both in time and space , we can still decompose the velocity perpendicular to the field into @xmath259 [ again , satisfying @xmath256 and @xmath257 ] . in this case , the space and time variations of @xmath253 can also produce drift velocities due to the @xmath254 motion . the space variations will give rise to a drift due to the curvature of the magnetic field lines ( curvature drift ) . the curvature drift velocity , however , is of the order of the thermal speed of the particles times the ratio between their larmor radii and the curvature radius of the lines . the time variations of the field , on the other hand , can also give rise to drift velocities . to first order , these velocities will also be proportional to the thermal speed of the particles times the ratio between the rate of change of the field ( determined by the quantities @xmath29 and @xmath261 ) and @xmath246 . we will neglect these possible drift velocities using our first assumption that the thermal velocities of the particles are low enough not to produce any important drift velocity . on the other hand , in the case of a non - uniform and time - changing fields , the @xmath255 velocity is given by the series , @xmath262 where @xmath263 for @xmath264 . we see from equations ( [ eq : b ] ) , ( [ eq : e ] ) , and ( [ eq : velocities ] ) that @xmath265 as long as @xmath29 , @xmath243 , and @xmath244/dt^n)/(d^{(n-1)}[\omega]/dt^{(n-1)})$ ] are constant and much smaller than @xmath245 , which is our second assumption . notice that , even if @xmath266 , the currents produced by these two velocities can be comparably important . this is because @xmath267 is independent of @xmath248 and @xmath247 , thus it has the same value for ions and electrons ( so from now we will just drop the subscript @xmath251 and will refer to this velocity as simply @xmath268 ) . thus , when considering both species , the current produced by @xmath269 , @xmath270 , will be due to the tiny excess of electrons in the background required to compensate the crs charge , so it will be proportional to @xmath28 . on the other hand , since @xmath271 is proportional to @xmath272 , it will be much larger for ions than for electrons . so the corresponding @xmath273 will be proportional to the total density of ions in the background , @xmath26 , which is typically much larger than @xmath28 . since @xmath274 for @xmath275 will also affect mainly the ions , their contribution to the current in the plasma will be much smaller than the one of @xmath271 provided that @xmath276 , so we will just neglect them . thus the currents @xmath270 and @xmath273 can be calculated considering equations ( [ eq : b ] ) , ( [ eq : e ] ) , and ( [ eq : velocities ] ) , finding that @xmath277 , \label{eq : jotazero}\ ] ] and @xmath278 , \end{array } \label{eq : jotauno}\end{aligned}\ ] ] where we have defined the field amplification factor @xmath279 . this way we have calculated all the currents in the plasma that are perpendicular to @xmath253 , but we still have to determine the ones that are parallel to @xmath253 . we do that using our third assumption , @xmath280 , which implies that @xmath281 using the ampere s law and the fact that in a one dimensional problem @xmath282 , we have that @xmath283 where @xmath284 is the @xmath57 component of the plasma current parallel to @xmath253 , @xmath285 . given this , the component of @xmath285 perpendicular to @xmath59 is just @xmath286 then , using equations ( [ eq : b ] ) , ( [ eq : e ] ) , ( [ eq : perpendicularity ] ) , ( [ eq : jotapax ] ) and ( [ eq : jotapape ] ) we get that @xmath287 now we have the expressions for all the components of the current perpendicular to @xmath59 , @xmath241 , so we can use equation ( [ eq : jotaperp ] ) to find the dispersion relation , @xmath288 from this derivation we can also obtain an estimate of the plasma velocities due to the crcd waves . we know that the motion of particles perpendicular to @xmath253 is dominated by @xmath289 ( since @xmath268 affects ions and electrons in the same way and @xmath290 ) , and that the motion of particles parallel to @xmath253 is mainly given by electrons moving at @xmath291 . thus , by looking at the expressions for @xmath270 and @xmath285 given by equations ( [ eq : jotazero ] ) , ( [ eq : jotapape ] ) , and ( [ eq : jotapa ] ) , we find that the dominant plasma motion will be given by @xmath268 and will imply a velocity of ions and electrons that can be decomposed into a component along @xmath57 , @xmath54 ( where the subscript @xmath292 stands for @xmath293 ) , and a transverse component , @xmath294 , that is always perpendicular to @xmath15 ( and to @xmath59 ) . in [ sec : analytical ] we use the dispersion relation given by equation ( [ eq : dispersion ] ) to calculate the wavenumber , @xmath295 , and growth rate , @xmath52 , of the fastest growing mode , as well as @xmath36 as a function of the amplitude of the wave . we will use these results here in order to check the consistency of assuming that @xmath29 , @xmath243 , and @xmath244/dt^n)/(d^{(n-1)}[\omega]/dt^{(n-1)})$ ] are constant and much smaller than @xmath76 , which are necessary for neglecting @xmath296 when @xmath297 . we know from equation ( [ eq : phasevelocity ] ) that when @xmath50 , @xmath43 . it means that @xmath298/dt^n)/ ( d^{(n-1)}[\omega]/dt^{(n-1 ) } ) \approx 2\gamma$ ] ( except when @xmath299 and @xmath300 , in which case @xmath298/dt^n)/ ( d^{(n-1)}[\omega]/dt^{(n-1 ) } ) \ll \gamma$ ] ) . so , in order to neglect @xmath296 for @xmath297 , we only require @xmath29 and @xmath243 to be constant and much smaller than @xmath76 . it is possible to show from equation ( [ eq : growthrate ] ) that the first condition , which is equivalent to @xmath113 , where @xmath52 is given by equation ( [ eq : growthrate ] ) , is satified if @xmath250 . from equation ( [ eq : phasevelocity ] ) we see that , if we approximate @xmath301 , @xmath302 becomes approximately equal to @xmath303 , which is also constant and much smaller than @xmath52 . this way we see that our analytical results are valid if the plasma is well magnetized in the sense that @xmath113 , which is equivalent to @xmath304 . using two - dimensional pic simulations , we show in [ sec : magnetization ] that this condition is actually a requirement for the crcd not to be quenched by the weible - like filamentation . we saw in [ sec : threed ] that , when multidimensional effects are considered , the dominant wavelength of the crcd instability , @xmath0 , grows according to equation ( [ eq : wavelength ] ) . this makes it numerically expensive to run three - dimensional simulations that could amplify the field substantially without making @xmath0 too close to the size of the simulation box , @xmath73 . in order to overcome this difficulty , in this section we present the results of two - dimensional simulations whose @xmath73 is always bigger than @xmath0 . despite some artifacts related to the two - dimensional geometry , these simulations help us confirm the main results obtained from the three - dimensional analysis presented above . figures [ fg : shocks1 ] , [ fg : shocks2 ] , and [ fg : shocks3 ] show the results at three different times ( @xmath305 , 9 , and 11 , respectively ) for one of the simulations ( run i4 of table [ table : interference ] ) , which corresponds to crs drifting at the speed of light and without considering their back - reaction . we see that initially the instability is produced independently in different regions of the simulation box ( as seen in fig . [ fg : shocks1 ] ) . in this linear stage of evolution , the waves produced in adjacent regions of space seem to grow without interfering with each other . however , when the waves become non - linear , strong density fluctuations appear on scales of a few @xmath77 ( as shown in fig . [ fg : shocks2 ] ) . the beginning of this stage is shown in fig . [ fg : shocks2 ] . it is also apparent from fig . [ fg : shocks3 ] that the magnetic fluctuations get distorted and evolve into larger scales right after the density fluctuations and turbulence form . the formation of density fluctuations also affects the growth rate of the instability . [ fg : departure ] presents the magnetic energy evolution for the two - dimensional simulations i5 and i6 , whose @xmath91 and @xmath306 , respectively . the rest of their numerical parameters are specified in table [ table : interference ] . we see that , as in the three - dimensional case , the exponential growth stops shortly after @xmath151 . after that , the crcd instability grows at a lower rate , reaching saturation when @xmath53 . this result had already been obtained in the three - dimensional case , but in this case we allow the instability to evolve into larger scales as the magnetic field grows . in two dimensions , the formation of density fluctuations produces a clear differentiation between the @xmath170 and @xmath171 components of the field ( as can be seen in figs . [ fg : shocks3 ] and [ fg : departure ] ) . this is because in the low density regions the plasma can not generate the return current necessary to compensate @xmath13 . thus , the uncompensated cr current produces a toroidal " magnetic field around the underdense regions that , in the two - dimensional case , manifests itself as an amplification of the out of the plane component of the field , @xmath102 . even though , as seen in fig . [ fg : shocks3 ] , both the toroidal " and the crcd field coexist , the two - dimensional simulations can still give us information about the point when the crcd instability stops amplifying the field . , represented by black and red lines , respectively . the @xmath57 , @xmath170 , and @xmath171 components are shown using dotted , dashed , and solid lines , respectively . time is normalized in terms of the @xmath52 of each simulation . the differentiation between the @xmath170 and @xmath171 components of the field as well as the departure from exponential growth after @xmath151 can be seen for both runs . saturation still happens when @xmath53.,width=292 ] axford , w. i. , leer , e. , & skadron , g. , 1977 , 15th int . cosmic ray conf . , 11 , 132 ballet , j. , 2006 , adv . in space res . , 37 , 1902 bell , a. r. , 1978 , , 182 , 147 bell , a. r. , 2004 , , 353 , 550 bell , a. r. , 2005 , , 358 , 181 blandford , r. d. , & ostriker , j. p. , 1978 , , 221 , l29 blasi , p. , & amato , e. , 2008 , arxiv:0806.1223v1 buneman , o. , 1993 , `` computer space plasma physics '' , terra scientific , tokyo , 67 couch , s. , milosavljevi , m. , & nakar , e. , 2008 , arxiv:0807.4117v1 krymsky , g. f. , 1977 , sov . , 23 , 327 kulsrud , r. , & pearce , w. p. , 1969 , , 156 , 445 lagage , p. o. , & cesarsky , c. j. , 1983 , , 125 , 249 mckenzie , j. f. , & volk , h. j. , 1982 , , 116 , 191 niemiec , j. , pohl , m. , stroman , t. , & nishikawa , k. , 2008 , , 684 , 1189 reville , b. , kirk , j. g. , & duffy , p. , 2006 , plasma phys . fusion , 48 , 1741 - 1747 riquelme , m. a. , & spitkovsky , a. , 2008 , int . j. mod d , in press spitkovsky , a. , 2005 , aip conf . proc , 801 , 345 , astro - ph/0603211 uchiyama , y. , aharonian , f. a. , tanaka , t. , takahashi , t. , & maeda , t. , 2007 , nature , 449 volk , h. j. , berezhko , e. g. , & ksenofontov , l. t. , 2005 , 29th int . cosmic ray conf . , 3 , 233 - 236 zirakashvili , v. n. , ptuskin , v. s. , & volk , h. j. , 2008 , , 678 , 255

the cosmic ray current - driven ( crcd ) instability , predicted by @xcite , consists of non - resonant , growing plasma waves driven by the electric current of cosmic rays ( crs ) that stream along the magnetic field ahead of both relativistic and non - relativistic shocks . combining an analytic , kinetic model with one- , two- , and three - dimensional particle - in - cell simulations , we confirm the existence of this instability in the kinetic regime and determine its saturation mechanisms . in the linear regime , we show that , if the background plasma is well magnetized , the crcd waves grow exponentially at the rates and wavelengths predicted by the analytic dispersion relation . the magnetization condition implies that the growth rate of the instability is much smaller than the ion cyclotron frequency . as the instability becomes non - linear , significant turbulence forms in the plasma . this turbulence reduces the growth rate of the field and damps the shortest wavelength modes , making the dominant wavelength , @xmath0 , grow proportional to the square of the field . at constant cr current , we find that plasma acceleration along the motion of crs saturates the instability at the magnetic field level such that @xmath1 , where @xmath2 is the alfvn velocity in the amplified field , and @xmath3 is the drift velocity of crs . the instability can also saturate earlier if crs get strongly deflected by the amplified field , which happens when their larmor radii get close to @xmath0 . we apply these results to the case of crs propagating in the upstream medium of the forward shock in supernova remnants . if we consider only the most energetic crs that escape from the shock , we obtain that the field amplification factor of @xmath4 can be reached . this confirms the crcd instability as a potentially important component of magnetic amplification process in astrophysical shock environments .

low - mass galaxies are arguably the best places to test dark matter ( dm ) models since they are dynamically dominated by the dm haloes they are embedded in well within their inner regions . the kinematical information that is inferred from low surface brightness galaxies ( e.g. * ? ? ? * ) , nearby field dwarf galaxies ( e.g. * ? ? ? * ; * ? ? ? * ) and milky way ( mw ) dwarf spheroidals ( dsphs ) ( e.g. * ? ? ? * ; * ? ? ? * ) , seem to favour the presence of @xmath6 dark matter cores with different degrees of certainty . the former two cases are more strongly established while the latter is still controversial ( e.g. * ? ? ? * ) , which is unfortunate since the mw dsphs have the largest dynamical mass - to - light ratios and are thus particularly relevant to test the dm nature . although not necessarily related to the existence of cores , it has also been pointed out that the population of dark satellites obtained in cdm @xmath7body simulations , are too centrally dense to be consistent with the kinematics of the mw dsphs @xcite . this problem possibly also extends to isolated galaxies @xcite . the increasing evidence of lower than expected central dm densities among dm - dominated systems is a lasting challenge to the prevalent collisionless cold dark matter ( cdm ) paradigm . on the other hand , the low stellar - to - dm content of dwarf galaxies represents a challenge for galaxy formation models since these have to explain the low efficiency of conversion of baryons into stars in dwarf galaxies . it is possible that these two outstanding issues share a common solution rooted in our incomplete knowledge of processes that are key to understand how low - mass galaxies form and evolve : gas cooling , star formation and energetic feedback from supernovae ( sne ) . in particular , episodic high - redshift gas outflows driven by sne have been proposed as a mechanism to suppress subsequent star formation and lower , irreversibly , the central dm densities ( e.g. * ? ? ? * ; * ? ? ? although such mechanism seemingly produces intermediate mass galaxies ( halo mass @xmath8 ) with realistic cores and stellar - to - halo mass ratios @xcite , it is questionable if it is energetically viable for lower mass galaxies @xcite . even though environmental effects such as tidal stripping might alleviate this stringent energetic condition in the case of satellite galaxies @xcite , the issue of low central dm densities seems relevant even for isolated galaxies @xcite . this seems to indicate that sne - driven outflows can only act as a solution to this problem if they occur very early , when the halo progenitors of present - day dwarfs were less massive @xcite . it remains unclear if such systems can avoid regenerating a density cusp once they merge with smaller , cuspier , haloes . it is also far from a consensus that the implementation of strong `` bursty '' star formation recipes in simulations , a key ingredient to reduce central dm densities , is either realistic or required to actually produce consistent stellar - to - halo mass ratios ( e.g. , * ? ? ? * ) , and other observed properties . it is therefore desirable , but challenging , to identify observables that could unambiguously determine whether bursty star formation histories with a strong energy injection efficiency ( into the dm particles ) are realistic or not . an exciting alternative solution to the problems of cdm at the scale of dwarfs is that of self - interacting dark matter ( sidm ) . originally introduced by @xcite , it goes beyond the cdm model by introducing significant self - collisions between dm particles . the currently allowed limit to the self - scattering cross section is imposed more stringently by observations of the shapes and mass distribution of elliptical galaxies and galaxy clusters @xcite , and is set at : @xmath9 . dm particles colliding with roughly this cross section naturally produce an isothermal core with a @xmath6 size in low - mass galaxies , close to what is apparently observed . sidm is well - motivated by particle physics models that introduce new force carriers in a hidden dm sector ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which predict velocity - dependent self - scattering cross sections . in the case of massless bosons for instance , the cross section scales as @xmath10 as in rutherford scattering . the renewed interest in sidm has triggered a new era of high resolution dm - only sidm simulations : velocity - dependent in @xcite ( vzl hereafter ) , and velocity - independent in @xcite , that hint at a solution to the cdm problems in low - mass galaxies . in particular for the mw dsphs , it has been established that the resultant dark satellites of a mw - size halo are consistent with the dynamics of the mw dsphs , have cores of @xmath6 and avoid cluster constraints only if @xmath11 , or if the cross section is velocity - dependent @xcite . recently , simulations of sidm models with new light mediators have shown that is possible to also suppress the abundance of dwarf galaxies due to the modified early - universe power spectrum caused by the interactions of the dm with the dark radiation @xcite . given the recent success of sidm , a natural step is to elevate its status to that of cdm by studying the synergy between baryonic physics and dm collisionality in a suitable galaxy formation model . so far , this has been studied only analytically @xcite , with a focus in more massive galaxies where baryons dominate the central potential . interestingly , in this case , the dm core size is reduced and the central densities are higher compared to sidm simulations without the effect of baryons . in this paper we concentrate on the regime of dwarf galaxies by pioneering cosmological hydrodynamical simulations that include the physics of galaxy formation within a sidm cosmology . we compare them with their counterparts ( under the same initial conditions ) in the cdm model with the main objective of understanding the impact of sidm on the formation and evolution of dwarf galaxies . .dm models considered in this paper . cdm is the standard collisionless model without any self - interaction . sidm10 is a reference model with a constant cross section an order of magnitude larger than allowed by current observational constraints . we note that such a model could still be realized in nature if this large cross section would only hold over a limited relative velocity range . sidm1 is also a model with constant cross section , which is potentially in the allowed range . vdsidma and vdsidmb have a velocity - dependent cross section motivated by the particle physics model presented in @xcite . these two models are allowed by all astrophysical constraints , and solve the `` too big to fail '' problem ( see * ? ? ? * ) as demonstrated in vzl . [ cols="^,^,^,^",options="header " , ] in this case the fit is poor in the inner regions ( @xmath12 ) , and thus , we use only the exponential gas profile instead of the two - component model as in the other cases . the lower right panel of figure [ fig : halo_profiles ] demonstrates that the feedback associated with sne does not alter the dm density distribution in our model . this is not surprising since we do not employ a very bursty star formation model , but a rather smooth star formation prescription . as a consequence , the dm density profile is not affected at all by the formation of the baryonic galaxy and the related feedback processes for the cdm case . the sidm models lead to core formation due to self - interactions of dm particles . such core makes it easier for sne feedback to drive gas outwards , which should cause some effect on the dm distribution . in fact , the lower right panel of figure [ fig : halo_profiles ] demonstrates that the dm density is slightly reduced in the cored region even with a smooth feedback model like ours . however , this effect is rather small and at maximum @xmath13 relative to the sidm10 simulation without baryons . this effect is therefore small compared to the effect of self - interactions , which reduce the central dm density much more significantly . so far we have discussed the relative differences between the different profiles . to quantify the spatial distribution of the dm and the baryons , gas and stars , in more detail , we now find analytical fits to the spherically averaged density distributions . we have found that the different dm models require different density profiles profiles to achieve a reasonable quality of the fits . , are stated for each fit . the fits were performed over the full radial range . all models lead to essentially perfect exponential profiles with no significant bulge components . the largest cross section models ( sidm1 , sidm10 ) produce a stellar core in the center.,scaledwidth=49.0% ] we start with the dm profile for the cdm case . it is well - known that cdm haloes have spherically averaged density profiles that are well described by nfw @xcite or einasto profiles @xcite . we therefore fit the dm profile of the cdm model with the two - parameter nfw profile : @xmath14 on the other hand , the sidm haloes are well fitted by cored - like profiles that vary according to the amplitude of the self - scattering cross section at the typical velocities of the halo . in the case of the strongest cross section , sidm10 , a good fit is obtained with the following three - parameter profile : @xmath15 while for intermediate cross sections , sidm1 and vdsidma , a burkert - like three - parameter formula provides a better fit : @xmath16 finally , for the weakest cross section , vdsidmb , a good fit is given by : @xmath17 next we consider the profiles of the baryonic components . for the stars and the gas , we use a two component density profile : an exponential profile in the outer region , which is a good approximation except for the gas beyond @xmath18 , and a cored profile in the inner region , analogous to eq . ( [ sidm10_rho ] ) : @xmath19 where we find that @xmath20 provides a good fit in all cases except for the gas distribution in the sidm1 case . for each profile ( dm , gas , and stars ) , we find the best fit parameters by minimising the following estimate of the goodness of the fit : @xmath21 where the sum goes over all radial bins . we summarise the best fit parameters for each component in table [ table : fits ] . we stress again that we need distinct parametric density profiles to better describe the spatial dm structure of the halo for the different dm models . for instance , in the case of sidm10 , the value of @xmath22 for the best fit using eq . ( [ sidm10_rho ] ) is @xmath23 , whereas using eqs . ( [ burkert]-[burkert_2 ] ) is @xmath24 and @xmath25 , respectively . on the other hand , for sidm1 , the values of @xmath22 using eqs . ( [ sidm10_rho]-[burkert_2 ] ) , are , respectively : 0.020 , 0.003 , 0.021 . clearly , in this case , eq . ( [ burkert ] ) is the best fit . for the stars we can also inspect the stellar surface density profiles , which are closely related to the measured stellar surface brightness profiles . the stellar surface density profiles of the da dwarfs for the different dm models are shown in figure [ fig : stellar_profiles ] . the exponential scale length , @xmath26 , of the different models is quoted for each model , and the dashed lines show the actual exponential fits for each model . for the cdm case , we find over a large radial range an exponential profile and no significant bulge contribution , similar to what is observed for most dwarfs . we have checked that the surface density profiles do not vary much if the orientation of the galaxy changes . the reason for this is that the dwarfs do not form thin disks , but rather extended puffed up ellipsoidal distributions similar to , for example , the stellar population of the isolated dwarf wlm . the scale length values we find are in reasonable agreement with other recent simulation of dwarf galaxies at this mass scale ( e.g. , * ? ? ? in the case of sidm1 and sidm10 , the presence of a small stellar core is visible in figure [ fig : stellar_profiles ] . the scale length does not change significantly as a function of the underlying dm model . however , it can clearly be seen that dm self - interactions lead to slightly larger exponential scale radii . we note that , contrary to previous studies , we achieve exponential stellar surface density profiles without a bursty star formation model or a high density thresholds for star formation . we therefore find that our quiescent , smooth star formation model leads to non - exponential star formation histories , and to exponential stellar surface density profiles . it has been argued that these characteristics are intimately connected to `` bursty '' star formation rates ( see e.g. * ? ? ? as a corollary , it was argued that the formation of a dm core is then naturally expected . however , we find that this is is not necessarily the case . we should note that @xcite simulated an isolated dwarf of a similar halo mass and stellar mass as our dwarf da but with a considerably bursty star formation model that produced a @xmath27 core . this is in clear contrast to our simulation where baryonic effects are unable to create a dm core despite of the high global efficiency of star formation . the key is then , once more , in the time scales and efficiency of energy injection during sne - driven outflows . it remains to be seen if star formation histories in real dwarf galaxies occur in bursts with a timescale much shorter than the local dm dynamical timescale , and with an effective energy injection into the dm particles that is sufficient to significantly alter the dm distribution . as we have shown above , halo da is in relative isolation and has a quiet merger history . we therefore expect that the final stellar and dm configuration is nearly in equilibrium . in the case of sidm , once the isothermal core forms , further collisions are not relevant anymore in changing the dm phase - space distribution . we can then ignore the collisional term in the boltzmann equation and test the equilibrium hypothesis by solving the jeans equation for the radial velocity dispersion profile using as input the density and anisotropy profiles : @xmath28 where @xmath29 is the total enclosed mass . we solve eq . ( [ jeans_eq ] ) independently for the collisionless components , dm and stars , using the fits to the density profiles with the analytic formulae introduced above . in addition , we also fit the corresponding radial anisotropy profiles for both the dm and the stars with the following five - parameter formula : @xmath30 the best fit parameters for this relation for each dm model are listed in table [ table : fits ] . the result obtained by solving the jeans equation for the cdm and sidm10 cases is seen in figure [ fig : sigma_jeans ] . here we show the predicted dispersion profiles with dashed lines for dm ( thick lines ) and stars ( thin lines ) . the solid lines show the actual simulation results . although the agreement between the velocity dispersion predicted by the jeans analysis and the simulation is not perfect , the comparison still indicates that halo da is roughly in equilibrium and that the spherical approximations assumed above are partially correct . in the sidm10 case , this would suggest that the dark matter core formed in the past and that any subsequent scattering does not affect the final equilibrium configuration once the galaxy forms . this would justify the use of the jeans equation without considering a collisional term . we will consider a more detailed dynamical analysis in a subsequent paper analysing the different sidm cases , having a closer look at the velocity anisotropies , and also investigating departures from spherical symmetry ( zavala & vogelsberger , in prep ) . for dm ( top ) and stellar mass ( bottom ) for halo da . the enclosed dm mass is for all times and for all models significantly larger than the stellar mass , and therefore dynamically dominates the center of the dwarf . the central dm mass is substantially reduced for the sidm1 and sidm10 models , but only slightly for the vdsidm models . similarly , the stellar mass is only reduced for the models with constant cross section , whereas the stellar mass growth of vdsidm closely follows that of the cdm case.,title="fig:",scaledwidth=49.0% ] for dm ( top ) and stellar mass ( bottom ) for halo da . the enclosed dm mass is for all times and for all models significantly larger than the stellar mass , and therefore dynamically dominates the center of the dwarf . the central dm mass is substantially reduced for the sidm1 and sidm10 models , but only slightly for the vdsidm models . similarly , the stellar mass is only reduced for the models with constant cross section , whereas the stellar mass growth of vdsidm closely follows that of the cdm case.,title="fig:",scaledwidth=49.0% ] in this section we study in more detail the matter content and structure of the simulated dwarf da within the central region , @xmath31 , which roughly encloses the dm core size for all models . we start with figure [ fig : histories1000 ] , which shows the mass buildup of dm ( top ) and stars ( bottom ) within @xmath4 as a function of time . in the cases with a constant scattering cross section , it is clear that there is a significant amount of dark matter mass expelled from the central kiloparsec . in the case of sidm1 for example , about @xmath32 have been removed by @xmath2 . for the vdsidm models however , there is only a minimal deviation from the evolution of the base cdm model . in fact , the vdsidmb model mass evolution follows the cdm result very closely and shows a nearly constant central mass after early times @xmath33 . the vdsidma model leads to a small depletion of dm in the central @xmath4 of about @xmath34 . the largest depletion can be seen for the sidm10 model , where the central mass is reduced by nearly a factor @xmath35 . the central stellar mass on the other hand grows steadily with time but it is at all times , and for all dm models , sub - dominant compared to the inner dm mass . for all models the central stellar mass is below @xmath36 at @xmath2 , which is a factor @xmath37 lower than the central dm mass at that time . the stellar mass in sidm1 and sidm10 grows more slowly than in the cdm and vdsidm cases . the vdsidm models behave very similar to the cdm case , where the stellar mass grows nearly linearly with time reaching a mass of about @xmath38 . the stellar mass within @xmath4 grows initially similar sidm10 ( sidm1 ) , however , after @xmath39 ( @xmath40 ) the stellar mass growth is slowed down for sidm10 ( sidm1 ) . after that time the growth is still linear but with a significantly shallower slope compared to the cdm and vdsidm cases . we note that sidm1 is an allowed model , and it is striking how different its stellar mass is growing compared to the other allowed vdsidm models . to quantify this in more detail we present a closer look of the density profiles of dm ( solid lines ) and stars ( dashed lines ) in figure [ fig : rho_inner ] . this reveals a tight correlation between the shape of the dm and stellar density distributions . the stars within the core react to the change in the potential of the dominant dm component due to self - interactions . the size of the stellar core is therefore tied , to certain degree , to the core sizes of the dm distribution . in the cases where the scattering cross section has a velocity dependence , although the creation of a dm core is evident , the impact is minimal in the stellar distribution compared to the models with a constant cross section . this is mainly because even in the cdm case , the stellar distribution forms a core which is roughly the size of the dm core observed in the vdsidm cases . we conclude that self - interactions drive the sizes of the cores in dm and stars to track each other . for sidm1 , the density within the core is a factor of @xmath41 smaller than in cdm . the central distribution of stars can therefore probe the nature of dm and can potentially be used to distinguish different sidm models . the strong correlation between dm and stars that we are finding is similar to the one suggested recently by @xcite using analytical arguments , but the regimes and interpretations are quite different . whereas these authors investigated the response of sidm to a dominant stellar component , we are investigating a system where dm still dominates dynamically . thus in the former , the dm cores sizes are reduced relative to expectations from dm - only simulations due to the formation of the galaxy , while in the latter , the stellar distribution of the galaxy responds to the formation of the sidm core by increasing its own stellar core relative to the cdm case . this regime is therefore more promising to derive constraints for the nature of dm . for the different dm models . the stars trace the evolution of dm and also form a core . the size of the stellar core is closely related to the size of the dm core . this can be seen most prominently for the sidm1 and sidm10 models.,scaledwidth=49.0% ] next we are interested in the time evolution of the core radii . it was already obvious from figure [ fig : histories1000 ] that for the largest cross section cases , the core should already be present early on during the formation history of the galaxy . this is indeed the case as we demonstrate more clearly in figure [ fig : coresize ] , where the evolution of the core sizes are shown as a function of time . as a measure of core radius , we fit burkert profiles @xcite at each time , for each of the models , to extract the core size @xmath42 : @xmath43 we note that we use this two - parameter fit for simplicity to fit all sidm models and give a measure of the core size . as we explored in detail above , the different sidm models are actually better fitted by different radial profiles . however , our purpose here is not to rigorously define a core size but simply to present an evolutionary trend for the different models . this trend is clearly visible in the figure as well as the dependence of the amplitude of the core size on the scattering cross section . figure [ fig : coresize ] shows the core radii determined by these two - parametric burkert fits for all dm models with ( solid lines ) and without ( dashed lines ) the effects baryons . , in the dm - only simulations ( dashed ) with the simulations including baryons ( solid ) . baryons have only a tiny effect on the evolution and size of the cores . the largest effect can be seen for sidm10 , where the shallow dm profile allows sne feedback to expand the core a bit more compared to the dm - only case.,scaledwidth=49.0% ] figure [ fig : coresize ] also demonstrates that the actual impact of baryons on the dm distribution relative to the dm - only case is minor , as we discussed already above ( see lower right panel of figure [ fig : halo_profiles ] ) . in the case of cdm this is not surprising since : ( i ) our star formation model is less bursty compared to models where the cusp - core transformation is efficient and ( ii ) for the mass scale we are considering , halo mass @xmath3 for halo da , the energy released by sne is not expected to be sufficient to create sizeable dm cores @xcite , although see @xcite . figure [ fig : coresize ] demonstrates that our star formation and feedback model creates only a slightly larger core for the sidm10 model . this is because expelling gas in this case is easier due to the reduced potential well caused by dm collisions . we stress again that these results are sensitive to the model used for sne - driven energy injection into the dm particles ( both efficiency and time scales ) . larger efficiencies of energy injection into shorter timescales would result in a larger removal of dm mass from the inner halo . according to figure [ fig : coresize ] a sizeable core is already present very early on . by @xmath44 all the models already have cores more than half of their present day size . furthermore , figure [ fig : coresize ] also demonstrates , that none of our sidm models lead to the gravothermal catastrophe where the core collapses following the outward flux of energy caused by collisions . this is consistent with the findings in vzl , where only one subhalo , with similar total dark matter mass as halo da , of the analogous sidm10 mw - size simulation was found to enter that regime towards @xmath2 . . the vdsidm models have a weaker impact.,scaledwidth=49.0% ] as a consequence of the dm core settling early on in the formation history of the galaxy , the star formation rate within the central @xmath4 is reduced significantly at late times in the cases with constant cross section . this results in a stellar population that is in average older than in the case of cdm . this is clearly shown in figure [ fig : metals ] , where we plot the time evolution of the ratio of the metallicity averaged within the central @xmath4 , relative to the cdm case . the difference today is @xmath45 . interestingly , in the vdsidm cases , there is an excess in star formation within @xmath4 in the last stages of the evolution resulting in a younger stellar population since the last @xmath46 ( see also figure [ fig : coresize ] ) . we will investigate this issue , and in general the properties of the central @xmath47 region , in a follow - up paper using simulations with increased resolution ( zavala & vogelsberger , in prep ) . as a function of total stellar mass . bottom panel : dm density slope at @xmath48 as a function of total stellar mass . the different dm models lead to significantly different slopes and masses at and within @xmath48 . at this radius even the vdsidm models clearly deviate from the cdm case . both the mass and the slope clearly scale with the cross section and allow to disentangle the different dm models . observational estimates from a combined sample of dwarf galaxies @xcite and from the things survey @xcite are also shown in the top and bottom panels , respectively.,title="fig:",scaledwidth=49.0% ] as a function of total stellar mass . bottom panel : dm density slope at @xmath48 as a function of total stellar mass . the different dm models lead to significantly different slopes and masses at and within @xmath48 . at this radius even the vdsidm models clearly deviate from the cdm case . both the mass and the slope clearly scale with the cross section and allow to disentangle the different dm models . observational estimates from a combined sample of dwarf galaxies @xcite and from the things survey @xcite are also shown in the top and bottom panels , respectively.,title="fig:",scaledwidth=49.0% ] in figure [ fig : encmass500 ] we focus on a region even closer to the halo centre and show the total mass within @xmath48 ( top ) and the slope of the density profile measured at this radius ( bottom ) . we compare both to observational estimates using samples of dwarf galaxies compiled in @xcite ( top ) and from the things survey ( bottom , * ? ? ? * ) . at these small radii , the change in the enclosed mass is still more dramatic for the constant cross section sidm models having a deficit in mass by a factor @xmath49 relative to the cdm case , while the vdsidm cases , although close to cdm , still deviate visibly . the logarithmic slope of the density profile at this radius varies between @xmath50 ( sidm10 ) and @xmath51 ( cdm ) . figure [ fig : encmass500 ] shows that given the large dispersion in the data , all dm models are essentially consistent with observations . there is however some tension with the cdm simulation of halo da having a slightly too large total mass , and a slightly too steep dm density slope at @xmath52 . on the other hand , the sidm10 case might be to cored for the stellar mass of halo db ( @xmath53 ) . taking both haloes into account , and looking at the two relations of figure [ fig : encmass500 ] only , it seems that sidm1 agrees best with these observations . we stress however , that our dwarf sample is far to small to draw any conclusions based on this result and these observations are in any case , too uncertain to use them as constraints . self - interacting dark matter ( sidm ) is one the most viable alternatives to the prevailing cold dark matter ( cdm ) paradigm . current limits on the elastic scattering cross section between dm particles are set at @xmath54 @xcite . at this level , the dm phase space distribution is altered significantly relative to cdm in the centre of dm haloes . the impact of dm self - interactions on the baryonic component of galaxies that form and evolve in sidm haloes has not been explored so far . recently , @xcite analytically estimated the dm equilibrium configuration that results from a stellar distribution added to the centre of a halo in the case of sidm . these authors studied the regime where the stellar component dominates the gravitational potential and concluded that the dm core sizes ( densities ) are smaller ( higher ) than observed in dm - only sidm simulations . this might have important consequences on current constraints of sidm models since they have been derived precisely in the baryon - dominated regime . in this paper we explore the opposite regime , that of dwarf galaxies where dm dominates the gravitational potential even in the innermost regions . our analysis is based on the first hydrodynamical simulations performed in a sidm cosmology . we focus most of the analysis on a single dwarf with a halo mass @xmath55 . we study two cases with a constant cross section : sidm1 and sidm10 , @xmath56 , respectively , and two cases with a velocity - dependent cross section : vdsidma - b , that were also studied in detail in vzl and @xcite . except for sidm10 , all these models are consistent with astrophysical constraints , solve the `` too big to fail '' problem and create @xmath57(1 kpc ) cores in dwarf - scale haloes . our simulations include baryonic physics using the implementation described in @xcite employing the moving mesh code arepo @xcite . we use the same model that was set up to reproduce the properties of galaxies at slightly larger mass - scales . our intention in this first analysis is not to match the properties of dwarf galaxies precisely , but rather to compare sidm and cdm with a single prescription for the baryonic physics , which has been thoroughly tested on larger scales . our most important findings are : * impact of sidm on global baryonic properties of dwarf galaxies : * the stellar and gas content of our simulated dwarfs agree reasonably well with various observations including the stellar mass as a function of halo mass , the luminosity metallicity relation , the neutral hydrogen content , and the cumulative star formation histories . the latter are similar to those of local isolated group dwarf galaxies with similar stellar masses . we find that the stellar mass , the gas content , the stellar metallicities and star formation rates are only minimally affected by dm collisions in allowed sidm models . the allowed elastic cross sections are too small to have a significant global impact on these quantities , and the relative differences between the different dm models are typically less than @xmath58 . in most cases these changes are not systematic as a function of the employed dm model . the modifications in the global baryonic component of the galaxies can therefore not be used to constrain sidm models since the effects are too small and not systematic . * impact of sidm on the inner halo region : * within @xmath31 , we find substantial differences driven by the collisional nature of sidm . besides the well - known effect of sidm on the dm density profiles , we also find that at these scales the distribution of baryons is significantly affected by dm self - interactions . both stars and gas show relative differences up to @xmath59 in the density , the velocity dispersion , and the gas temperature . most of the effects increase with the size of the cross section in the central region . the strongest correlation with the cross section can be found for the stellar profiles , where the central stellar density profile clearly correlates with the central cross section leading to lower central densities for dm models with larger central cross sections . * impact of baryons on the inner halo region : * we find that the impact of baryons on the dm density profile is small for the dm - dominated dwarf ( @xmath60 ) studied here . however , this result is also connected to our smooth star formation model , which is not as bursty as models where a significant core formation is observed due to baryonic feedback . the size of the dm core and the central density are therefore essentially the same as in our simulations that have no baryons , although the core size is slightly larger in the former than in the latter . * disentangling different sidm models : * for the cases where the scattering cross section is constant , the combination of two key processes : ( i ) an early dm core formation such that by @xmath44 , the dm cores already have half of their size today ; and ( ii ) a star formation history dominated by the period after the formation of the dm core , result in the following characteristics of the stellar distribution of sidm galaxies : ( a ) the development of a central stellar core with a size that correlates with the amplitude of the scattering cross section . for instance , for the sidm1 case with @xmath61 , the density within the stellar core is a factor of @xmath41 smaller than for the cdm case . ( b ) a reduced stellar mass in the sub - kpc region ( @xmath62 ) as a byproduct of the reduced dm gravitational potential due to self - scattering . ( c ) a reduced central stellar metallicity ; by @xmath45 at @xmath2 compared to the cdm case . around @xmath63 the metallicity can be reduced by up to @xmath64 . for the cases where the scattering cross section is velocity - dependent , even though a sizeable dm core can still be created ( @xmath65 ) , the effect in the stellar distribution at all scales is minimal relative to cdm . this is likely because the amplitude of the cross section within the inner region of the dwarf is not large enough to produce a dm core that is larger than the stellar core that forms in the cdm case . whether the latter could be the result of numerical resolution is something we will investigate in a forthcoming paper . any changes that we found in the vdsidm cases seem to be only related to the stochastic nature of the simulated star formation and galactic wind processes . these conclusions are key predictions of sidm that can in principle be tested to either constrain currently allowed models , particularly constant cross section models , or to find signatures of dm collisions in the properties of the central stellar distributions of dwarf galaxies . in future works we will explore these possibilities in more detail . we thank daniel weisz for providing cumulative star formation histories of local group dwarfs to us , and michael boylan - kolchin for help with the initial conditions . we further thank volker springel for useful comments and giving us access to the arepo code . the dark cosmology centre is funded by the dnrf . jz is supported by the eu under a marie curie international incoming fellowship , contract piif - ga-2013 - 627723 . the initial conditions were made using the dirac data centric system at durham university , operated by the institute for computational cosmology on behalf of the stfc dirac hpc facility ( www.dirac.ac.uk ) . the dirac system is funded by bis national e - infrastructure capital grant st / k00042x/1 , stfc capital grant st / h008519/1 , stfc dirac operations grant st / k003267/1 , and durham university . dirac is part of the re : green card uk national e - infrastructure .

we present the first cosmological simulations of dwarf galaxies , which include dark matter self - interactions and baryons . we study two dwarf galaxies within cold dark matter , and four different elastic self - interacting scenarios with constant and velocity - dependent cross sections , motivated by a new force in the hidden dark matter sector . our highest resolution simulation has a baryonic mass resolution of @xmath0 and a gravitational softening length of @xmath1 at @xmath2 . in this first study we focus on the regime of mostly isolated dwarf galaxies with halo masses @xmath3 where dark matter dynamically dominates even at sub - kpc scales . we find that while the global properties of galaxies of this scale are minimally affected by allowed self - interactions , their internal structures change significantly if the cross section is large enough within the inner sub - kpc region . in these dark - matter - dominated systems , self - scattering ties the shape of the stellar distribution to that of the dark matter distribution . in particular , we find that the stellar core radius is closely related to the dark matter core radius generated by self - interactions . dark matter collisions lead to dwarf galaxies with larger stellar cores and smaller stellar central densities compared to the cold dark matter case . the central metallicity within @xmath4 is also larger by up to @xmath5 in the former case . we conclude that the mass distribution , and characteristics of the central stars in dwarf galaxies can potentially be used to probe the self - interacting nature of dark matter . [ firstpage ] cosmology : dark matter galaxies : halos methods : numerical

the parent compounds of cuprate superconductors are identified as mott insulators @xcite , in which the lack of conduction arises from the strong electron - electron repulsion . superconductivity then is obtained by adding charge carriers to insulating parent compounds with the superconducting ( sc ) transition temperature @xmath0 has a dome - shaped doping dependence @xcite . since the discovery of superconductivity in cuprate superconductors , the search for other families of superconductors that might supplement what is known about the sc mechanism of doped mott insulators has been of great interest . fortunately , it has been found @xcite that there is a class of cobaltate superconductors na@xmath3coo@xmath4h@xmath5o , which displays most of the structural and electronic features thought to be important for superconductivity in cuprate superconductors : strong two - dimensional character , proximity to a magnetically ordered nonmetallic state , and electron spin 1/2 . in particular , @xmath0 in cobaltate superconductors has the same unusual dome - shaped dependence on charge - carrier doping @xcite . however , there is one interesting difference : in the cuprate superconductors @xcite , cu ions in a square array are ordered antiferromagnetically , and then spin fluctuations are thought to play a crucial role in the charge - carrier pairing , while in the cobaltate superconductors @xcite , co ions in a triangular array are magnetically frustrated , and therefore this geometric frustration may suppress @xmath0 to low temperatures . it has been argued that the triangular - lattice cobaltate superconductors are probably the only system other than the square - lattice cuprate superconductors where a doped mott insulator becomes a superconductor . the heat - capacity measurement of the specific - heat can probe the bulk properties of a superconductor , which has been proven as a powerful tool to investigate the low - energy quasiparticle excitations , and therefore gives information about the charge - carrier pairing symmetry , specifically , the existence of gap nodes at the fermi surface @xcite . in conventional superconductors @xcite , the absence of the low - energy quasiparticle excitations is reflected in the thermodynamic properties , where the specific - heat of conventional superconductors is experimentally found to be exponential at low temperatures , since conventional superconductors are fully gaped at the fermi surface . however , the situation in the triangular - lattice cobaltate superconductors is rather complicated , since the experimental results obtained from different measurement techniques show a strong sample dependence @xcite . thus it is rather difficult to obtain conclusive results . the early specific - heat measurements @xcite showed that the specific - heat in the triangular - lattice cobaltate superconductors reveals a sharp peak at @xmath0 , and can be explained phenomenologically within the bardeen - cooper - schrieffer ( bcs ) formalism under an unconventional sc symmetry with line nodes . however , by contrast , the latest heat - capacity measurements @xcite indicated that among a large number of the gap symmetries that have been suggested @xcite , the sc - state with d - wave ( @xmath6 pairing ) symmetry without gap nodes at the fermi surface is consistent with the observed specific - heat data . furthermore , by virtue of the magnetization measurement technique , the value of the upper critical field and its temperature dependence have been observed for all the temperatures @xmath7 @xcite , where the temperature dependence of the upper critical field follows qualitatively the bcs type temperature dependence . on the theoretical hand , there is a general consensus that superconductivity in the triangular - lattice cobaltate superconductors is caused by the strong electron correlation @xcite . using the resonating - valence - bond mean - field approach , it has been suggested that the spin fluctuation enhanced by the dopant dynamics leads to a d - wave sc - state @xcite . based on the mean - field variational approach with gutzwiller approximation , a d - wave sc - state is realized in the parameter region close to the triangular - lattice cobaltate superconductors @xcite . within the framework of the kinetic - energy - driven sc mechanism @xcite , it has been demonstrated that charge carriers are held together in d - wave pairs at low temperatures by the attractive interaction that originates directly from the kinetic energy by the exchange of spin excitations @xcite . moreover , superconductivity with the d - wave symmetry has been explored by a large - scale dynamical cluster quantum monte carlo simulation on the triangular - lattice hubbard model @xcite . in particular , using the diagram technique in the atomic representation , the sc phase with the d - wave symmetry in an ensemble of the hubbard fermions on a triangular lattice has been discussed @xcite , where the domelike shape of the doping dependence of @xmath0 is obtained . however , to the best of our knowledge , the thermodynamic properties of the triangular - lattice cobaltate superconductors have not been treated starting from a microscopic sc theory , and no explicit calculations of the doping dependence of the upper critical field have been made so far . in this case , a challenging issue for theory is to explain the thermodynamic properties of the triangular - lattice cobaltate superconductors . in our recent study @xcite , the electromagnetic response in the triangular - lattice cobaltate superconductors is studied based on the kinetic - energy - driven sc mechanism @xcite , where we show that the magnetic - field - penetration depth exhibits an exponential temperature dependence due to the absence of the d - wave gap nodes at the charge - carrier fermi surface . moreover , in analogy to the dome - shaped doping dependence of @xmath2 , the superfluid density increases with increasing doping in the lower doped regime , and reaches a maximum around the critical doping , then decreases in the higher doped regime . in this paper , we start from the theoretical framework of the kinetic - energy - driven superconductivity , and then provide a natural explanation to the thermodynamic properties in the triangular - lattice cobaltate superconductors . we evaluate explicitly the internal energy , and then qualitatively reproduced some main features of the heat - capacity and magnetization measurements on the triangular - lattice cobaltate superconductors @xcite . in particular , we show that a sharp peak in the specific - heat of the triangular - lattice cobaltate superconductors appears at @xmath0 , and then the specific - heat varies exponentially as a function of temperature for the temperatures @xmath1 due to the absence of the d - wave gap nodes at the charge - carrier fermi surface , which is much different from that in the square - lattice cuprate superconductors @xcite , where the characteristic feature is the existence of the gap nodes on the charge - carrier fermi surface , and then the specific - heat in the square - lattice cuprate superconductors decreases with decreasing temperatures as some power of the temperature in the temperature range @xmath1 . moreover , the upper critical field follows qualitatively the bcs type temperature dependence , and has the same dome - shaped doping dependence as @xmath2 . the rest of this paper is organized as follows . we present the basic formalism in section [ framework ] , and then the quantitative characteristics of the thermodynamic properties in the triangular - lattice cobaltate superconductors are discussed in section [ thermodynamic ] , where we show that although the pairing mechanism is driven by the kinetic energy by the exchange of spin excitations @xcite , the sharp peak of the specific - heat in the triangular - lattice cobaltate superconductors at @xmath0 can be described qualitatively by the kinetic - energy - driven d - wave bcs - like formalism . finally , we give a summary in section [ conclusions ] . in the triangular - lattice cobaltate superconductors , the characteristic feature is the presence of the two - dimensional coo@xmath5 plane @xcite . in this case , a useful microscopic model that has been widely used to describe the low - energy physics of the doped coo@xmath5 plane is the @xmath8-@xmath9 model on a triangular lattice @xcite . this @xmath8-@xmath9 model is defined through only two competing parameters : the nearest - neighbor ( nn ) hopping integral @xmath8 in the kinetic - energy term , which measures the electron delocalization through the lattice , and the nn spin - spin antiferromagnetic ( af ) exchange coupling @xmath9 in the magnetic - energy part , which describes af coupling between localized spins . in particular , the nn hopping integral @xmath8 is much larger than the af exchange coupling constant @xmath9 in the heisenberg term , and therefore the spin configuration is strongly rearranged due to the effect of the charge - carrier hopping @xmath8 on the spins , which leads to a strong coupling between the charge and spin degrees of freedom of the electron . since the triangular - lattice cobaltate superconductors are viewed as an electron - doped mott insulators @xcite , this @xmath8-@xmath9 model is subject to an important local constraint @xmath10 to avoid zero occupancy , where @xmath11 ( @xmath12 ) is the electron creation ( annihilation ) operator . in the hole - doped side , the local constraint of no double electron occupancy has been treated properly within the fermion - spin approach @xcite . however , for an application of the fermion - spin theory to the electron - doped case , we @xcite should make a particle - hole transformation @xmath13 , where @xmath14 ( @xmath15 ) is the hole creation ( annihilation ) operator , and then the local constraint @xmath10 without zero occupancy in the electron - doped case is replaced by the local constraint of no double occupancy @xmath16 in the hole representation . this local constraint of no double occupancy now can be dealt by the fermion - spin theory @xcite , where the hole operators @xmath17 and @xmath18 are decoupled as @xmath19 and @xmath20 , respectively , with the charge degree of freedom of the hole together with some effects of spin configuration rearrangements due to the presence of the doped charge carrier itself that are represented by the spinful fermion operator @xmath21 , while the spin degree of freedom of the hole is represented by the spin operator @xmath22 . the advantage of this fermion - spin approach is that the local constraint of no double occupancy is always satisfied in actual calculations . based on the @xmath8-@xmath9 model in the fermion - spin representation , the kinetic - energy - driven sc mechanism has been developed for the square - lattice cuprate superconductors in the doped regime without an af long - range order ( aflro ) @xcite , where the attractive interaction between charge carriers originates directly from the interaction between charge carriers and spins in the kinetic energy of the @xmath8-@xmath9 model by the exchange of spin excitations in the higher powers of the doping concentration . this attractive interaction leads to the formation of the charge - carrier pairs with the d - wave symmetry , while the electron cooper pairs originated from the charge - carrier d - wave pairing state are due to the charge - spin recombination @xcite , and they condense into the d - wave sc - state . furthermore , within the framework of the kinetic - energy - driven superconductivity , the doping dependence of the thermodynamic properties in the square - lattice cuprate superconductors has been studied @xcite , and then the striking behavior of the specific - heat in the square - lattice cuprate superconductors are well reproduced . the triangular - lattice cobaltate superconductors on the other hand are the second known example of superconductivity arising from doping a mott insulator after the square - lattice cuprate superconductors . although @xmath0 in the triangular - lattice cobaltate superconductors is much less than that in the square - lattice cuprate superconductors , the strong electron correlation is common for both these materials , which suggest that these two oxide systems may have the same underlying sc mechanism . in this case , the kinetic - energy - driven superconductivity developed for the square - lattice cuprate superconductors has been generalized to the case for the triangular - lattice cobaltate superconductors @xcite . the present work of the discussions of the thermodynamic properties in the triangular - lattice cobaltate superconductors builds on the kinetic - energy - driven sc mechanism developed in refs . @xcite and @xcite , and only a short summary of the formalism is therefore given in the following discussions . in our previous discussions in the doped regime without aflro , the full charge - carrier diagonal and off - diagonal green s functions of the @xmath8-@xmath9 model on a triangular lattice in the charge - carrier pairing state have been obtained explicitly as @xcite , [ bcsgf ] @xmath23 where the charge - carrier quasiparticle coherent weight @xmath24 , the charge - carrier quasiparticle coherence factors @xmath25 and @xmath26 , the charge - carrier quasiparticle energy spectrum @xmath27 , and the charge - carrier excitation spectrum @xmath28 . in the early days of superconductivity in the triangular - lattice cobaltate superconductors , some nmr and nqr data are consistent with the case of the existence of a pair gap over the fermi surface @xcite , while other experimental nmr and nqr results suggest the existence of the gap nodes @xcite . in particular , it has been argued that only involving the pairings of charge carriers located at the next nn sites can give rise to the nodal points of the complex gap appearing inside the brillouin zone @xcite . moreover , the nodal points of the complex gap has been obtained theoretically by considering the interaction between the hubbard fermions @xcite . however , although the recent experimental results @xcite obtained from the specific - heat measurements do not give unambiguous evidence for either the presence or absence of the nodes in the energy gap , the experimental data of the specific - heat @xcite are consistent with these fitted results obtained from phenomenological bcs formalism with the d - wave symmetry without gap nodes . furthermore , some theoretical calculations based on the numerical simulations indicate that the d - wave state without gap nodes is the lowest state around the electron - doped regime where superconductivity appears in triangular - lattice cobaltate superconductors @xcite . in particular , the recent theoretical studies based on a large - scale dynamical cluster quantum monte carlo simulation @xcite and a combined cluster calculation and renormalization group approach @xcite show that the d - wave state naturally explains some sc - state properties as indicated by experiments . in this case , we only consider the case with the d - wave pairing symmetry as our previous discussions @xcite , and then the d - wave charge - carrier pair gap @xmath29 in eq . ( [ bcsgf ] ) has been given in ref . @xcite . since the spin part in the @xmath8-@xmath9 model in the fermion - spin representation is anisotropic away from half - filling @xcite , two spin green s functions @xmath30 and @xmath31 have been defined to describe properly the spin part , and can be obtained explicitly as , [ sgf ] @xmath32 where the function @xmath33 and spin excitation spectrum @xmath34 in the spin green s function ( [ mfsgf ] ) have been given in ref . @xcite , while the function @xmath35 , and the spin excitation spectrum @xmath36 in the spin green s function ( [ mfsgfz ] ) is obtained as , @xmath37(\gamma_{\bf k } -1),\end{aligned}\ ] ] where @xmath38/3 $ ] , the parameters @xmath39 , @xmath40 , @xmath41 , the decoupling parameter @xmath42 , and the spin correlation function @xmath43 have been also given in ref . @xcite . in particular , the charge - carrier quasiparticle coherent weight @xmath24 , the charge - carrier pair gap parameter @xmath44 , all the other order parameters , and the decoupling parameter @xmath42 have been determined by the self - consistently calculation @xcite . in spite of the pairing mechanism driven by the kinetic energy by the exchange of spin excitations , the results in eq . ( [ bcsgf ] ) are the standard bcs expressions for a d - wave charge - carrier pair state . now we turn to evaluate the the internal energy of the triangular - lattice cobaltate superconductors . the internal energy in the charge - spin separation fermion - spin representation can be expressed as @xcite @xmath45 , where @xmath46 and @xmath47 are the corresponding contributions from charge carriers and spins , respectively , and can be obtained in terms of the charge - carrier spectral function @xmath48 , and the spin spectral functions @xmath49 and @xmath50 . following the previous work for the case in the square - lattice cuprate superconductors @xcite , it is straightforward to find the internal energy of the triangular - lattice cobaltate superconductors in the sc - state as , @xmath51\nonumber\\ & + & { z_{\rm af}\over n}\sum_{\bf k}\bar{\xi}_{\bf k } + 6j_{\rm eff}(\chi+\chi^{\rm z}),\end{aligned}\ ] ] where @xmath52 with the doping concentration @xmath53 , while the spin correlation function @xmath54 has been given in ref . @xcite . in the normal - state , the charge carrier pair gap @xmath55 , and in this case , the sc - state internal energy ( [ es ] ) can be reduced to the normal - state case as , @xmath56\nonumber\\ & + & { z_{\rm af}\over n}\sum_{\bf k}\bar{\xi}_{\bf k } + 6j_{\rm eff}(\chi+\chi^{\rm z}).\end{aligned}\ ] ] we are now ready to discuss the thermodynamic properties in the triangular - lattice cobaltate superconductors . the charge - carrier pair gap parameter @xmath44 is one of the characteristic parameters in the triangular - lattice cobaltate superconductors , which incorporates both the pairing force and charge - carrier pair order parameter , and therefore measures the strength of the binding of two charge carriers into a charge - carrier pair . in particular , the charge - carrier pair order parameter and the charge - carrier pair macroscopic wave functions in the triangular - lattice cobaltate superconductors are the same within the framework of the kinetic - energy - driven sc mechanism @xcite , i.e. , the charge - carrier pair order parameter is a _ magnified _ version of the charge - carrier pair macroscopic wave functions . for the convenience in the following discussions , we plot the charge - carrier pair gap parameter @xmath44 as a function of temperature at the doping concentration @xmath57 for parameter @xmath58 in fig . [ pair - gap - parameter - temp ] . it is shown clearly that the charge - carrier pair gap parameter follows qualitatively a bcs - type temperature dependence , i.e. , it decreases with increasing temperatures , and eventually vanishes at @xmath0 . for @xmath58 . [ pair - gap - parameter - temp ] ] for @xmath58 and @xmath59mev . the dashed line is obtained from a numerical fit @xmath60 $ ] , with @xmath61 and @xmath62 . inset : the corresponding experimental data of na@xmath3coo@xmath5@xmath63h@xmath5o taken from ref [ specific - heat ] ] one of the characteristics quantites in the thermodynamic properties is the specific - heat , which can be obtained by evaluating the temperature - derivative of the internal energy as , [ heat ] @xmath64 in the sc - state and normal - state , respectively , where @xmath65 and @xmath66 are the temperature dependence of the specific - heat coefficients in the sc - state and normal - state , respectively . in fig . [ specific - heat ] , we plot the specific - heat @xmath67 ( solid line ) as a function of temperature at @xmath68 for @xmath58 and @xmath59mev . for comparison , the corresponding experimental result @xcite of na@xmath3coo@xmath5@xmath63h@xmath5o is also shown in fig . [ specific - heat ] ( inset ) . apparently , the main feature of the specific - heat observed experimentally on the triangular - lattice cobaltate superconductors @xcite is qualitatively reproduced . as can be seen from fig . [ specific - heat ] , the specific - heat anomaly ( a jump ) at @xmath0 appears . the sc transition is reflected by a sharp peak in the specific - heat at @xmath0 , however , the magnitude of the specific - heat decreases dramatically with decreasing temperatures for the temperatures @xmath1 . moreover , the calculated result of the specific - heat difference @xmath69/c^{(\rm n)}_{\rm v}(t_{\rm c})=4.7 $ ] for the discontinuity in the specific - heat at @xmath0 , which is roughly consistent with the experimental data @xcite @xmath70 observed on na@xmath3coo@xmath5@xmath63h@xmath5o . for a better understanding of the physical properties of the specific - heat in the triangular - lattice superconductors , we have fitted our present theoretical result of the specific - heat for the temperatures @xmath1 , and the fitted result is also plotted in fig . [ specific - heat ] ( dashed line ) , where we found that @xmath71 varies exponentially as a function of temperature ( @xmath60 $ ] with @xmath61 and @xmath62 ) , which is an expected result in the case without the d - wave gap nodes at the charge - carrier fermi surface , and is in qualitative agreement with experimental data @xcite . however , this result in the triangular - lattice superconductors is much different from that in the square - lattice cuprate superconductors , where the characteristic feature is the existence of the gap nodes on the charge - carrier fermi surface , and then the specific - heat of the square - lattice cuprate superconductors decreases with decreasing temperatures as some power of the temperature for the temperatures @xmath1 . for @xmath58 and @xmath59mev . [ econd ] ] in the framework of the kinetic - energy - driven sc mechanism @xcite , the exchanged bosons are spin excitations that act like a bosonic glue to hold the charge - carrier pairs together , and then these charge - carrier pairs ( then electron pairs ) condense into the sc - state . as a consequence , the charge - carrier pairs in the triangular - lattice cobaltate superconductors are always related to lower the total free energy . the condensation energy @xmath72 on the other hand is defined as the energy difference between the normal - state free energy , extrapolated to zero temperature , and the sc - state free energy , @xmath73\nonumber\\ & -&[u^{(\rm s)}(t)-ts^{(\rm s)}(t)],\end{aligned}\ ] ] where the related entropy of the system is evaluated from the specific - heat coefficient in eq . ( [ heat ] ) as , @xmath74 where @xmath75 , @xmath76 referring to the sc - state and normal - state , respectively . we have made a calculation for the condensation energy ( [ condensation - energy ] ) , and the result of @xmath72 as a function of temperature at @xmath77 for @xmath58 and @xmath59mev is plotted in fig . [ econd ] . in comparison with the result of the temperature dependence of the charge - carrier pair gap parameter shown in fig . [ pair - gap - parameter - temp ] , we therefore find that in spite of the pairing mechanism driven by the kinetic energy by the exchange of spin excitations , the condensation energy of the triangular - lattice cobaltate superconductors follows qualitatively a bcs type temperature dependence . for @xmath58 and @xmath59mev . [ bc - doping ] ] a quantity which is directly related to the condensation energy @xmath72 in eq . ( [ condensation - energy ] ) is the upper critical field @xmath78 , @xmath79 this upper critical field @xmath78 is a fundamental parameter whose variation as a function of doping and temperature provides important information crucial to understanding the details of the sc - state . in fig . [ bc - doping ] , we plot the upper critical field @xmath78 as a function of doping with @xmath80 for @xmath58 and @xmath59mev . it is shown clearly that the upper critical field takes a dome - shaped doping dependence with the underdoped and overdoped regimes on each side of the optimal doping , where @xmath78 reaches its maximum . moreover , the calculated upper critical field at the optimal doping is @xmath81 , which is not too far from the range @xmath82 estimated experimentally for different samples of na@xmath3coo@xmath4h@xmath5o @xcite . for a superconductor , the upper critical field is defined as the critical magnetic field that destroys the sc - state at zero temperature , which therefore means that the upper critical field also measures the strength of the binding of charge carriers into the charge - carrier pairs . in this case , the domelike shape of the doping dependence of @xmath78 is a natural consequence of the domelike shape of the doping dependence of @xmath44 and @xmath0 as shown in ref . @xcite . to further understand the intrinsic property of the upper critical field @xmath78 in the triangular - lattice cobaltate superconductors , we have also performed a calculation for @xmath78 at different temperatures , and the result of @xmath78 as a function of temperature at @xmath77 for @xmath58 and @xmath59mev is plotted in fig . [ bc - temp ] in comparison with the corresponding experimental result @xcite of na@xmath3coo@xmath4h@xmath5o ( inset ) . it is thus shown that @xmath78 varies moderately with initial slope . in particular , as in the case of the temperature dependence of the condensation energy shown in fig . [ econd ] , the upper critical field @xmath78 also follows qualitatively the bcs type temperature dependence , i.e. , it decreases with increasing temperature , and vanishes at @xmath0 , which is also qualitatively consistent with the experimental results @xcite . for @xmath58 and @xmath59mev . insets : the corresponding experimental data of na@xmath3coo@xmath5@xmath63h@xmath5o taken from ref . [ bc - temp ] ] the coherence length @xmath83 also is one of the basic sc parameters of the triangular - lattice cobaltate superconductors , and is directly associated with the upper critical field as @xmath84 $ ] , where @xmath85 is the magnetic flux quantum . in fig . [ coherence ] , we plot the coherence length @xmath83 as a function of doping with @xmath80 for @xmath58 and @xmath59mev . since the coherence length @xmath83 is inversely proportional to the upper critical field @xmath78 , the coherence length @xmath83 in the triangular - lattice cobaltate superconductors reaches a minimum around the optimal doping , then grows in both the underdoped and overdoped regimes . in particular , at the optimal doping , the anticipated coherence length @xmath86 nm approximately matches the coherence length @xmath87 nm observed in the optimally doped na@xmath3coo@xmath5@xmath63h@xmath5o @xcite . this coherence length @xmath86 nm at the optimal doping estimated from the upper critical field using the ginzburg - landau expression also is qualitatively consistent with that obtained based on the microscopic calculation @xmath88 nm , where @xmath89 is the charge carrier velocity at the fermi surface . this relatively short coherence length is surprising for a superconductor with such a low @xmath0 , but is consistent with the narrow bandwidth in the triangular - lattice superconductors @xcite , since the charge - carrier quasiparticle spectrum @xmath90 in the full charge - carrier diagonal green s function ( [ bcsdgf ] ) and off - diagonal green s function ( [ bcsodgf ] ) has a narrow bandwidth @xmath91 . for @xmath58 and @xmath59mev . [ coherence ] ] within the framework of the kinetic - energy - driven sc mechanism , we have discussed the doping dependence of the thermodynamic properties in the triangular - lattice cobaltate superconductors . we show that the specific - heat anomaly ( a jump ) appears at @xmath0 , and then the specific - heat varies exponentially as a function of temperature for the temperatures @xmath1 due to the absence of the d - wave gap nodes at the charge - carrier fermi surface , which is much different from that in the square - lattice cuprate superconductors @xcite , where the characteristic feature is the existence of the gap nodes on the charge - carrier fermi surface , and then the specific - heat of the square - lattice cuprate superconductors decreases with decreasing temperatures as some power of the temperature in the temperature range @xmath1 . on the other hand , both the condensation energy and the upper critical field in the triangular - lattice cobaltate superconductors follow qualitatively the bcs type temperature dependence . in particular , in analogy to the dome - shaped doping dependence of @xmath0 , the maximal upper critical field occurs around the optimal doping , and then decreases in both underdoped and overdoped regimes . incorporating the present result @xcite with that obtained in the square - lattice cuprate superconductors , it is thus shown that the dome - shaped doping dependence of the upper critical field is a universal feature in a doped mott insulator , and it does not depend on the details of the geometrical spin frustration . since the knowledge of the thermodynamic properties in the triangular - lattice cobaltate superconductors is of considerable importance as a test for theories of superconductivity , the qualitative agreement between the present theoretical results and experimental data also provides an important confirmation of the nature of the sc phase of the triangular - lattice cobaltate superconductors as a conventional bcs - like with the d - wave symmetry , although the pair mechanism is driven by the kinetic energy by the exchange of spin excitations . yoshihiko ihara , kenji ishida , hideo takeya , chishiro michioka , masaki kato , yutaka itoh , kazuyoshi yoshimura , kazunori takada , takayoshi sasaki , hiroya sakurai , and eiji takayama - muromachi , j. phys . . jpn . * 75 * , 013708 ( 2006 ) .

the study of superconductivity arising from doping a mott insulator has become a central issue in the area of superconductivity . within the framework of the kinetic - energy - driven superconducting mechanism , we discuss the thermodynamic properties in triangular - lattice superconductors . it is shown that a sharp peak in the specific - heat appears at the superconducting transition temperature @xmath0 , and then the specific - heat varies exponentially as a function of temperature for the temperatures @xmath1 due to the absence of the d - wave gap nodes at the charge - carrier fermi surface . in particular , the upper critical field follows qualitatively the bardeen - cooper - schrieffer type temperature dependence , and has the same dome - shaped doping dependence as @xmath2 .

many samples of uv ceti type stars posses the stellar spot activity known as by dra syndrome . the by dra syndrome among the uv ceti stars was first found by @xcite . he reported the existence of sinusoidal - like variations at out - of - eclipses of the eclipsing binary star yy gem . @xcite explained this sinusoidal - like variation at out - of - eclipse as a heterogeneous temperature on star surface , which was called by dra syndrome by @xcite . this interpretation of by dra syndrome in terms of dark regions of the surface of rotating stars was confirmed , based on more rigorous arguments , by later works of @xcite . since the most of solar flares occur over solar spot regions , in the stellar case it is also expected to find a correlation between the frequency of flares and the effects caused by spots in the light curve . in order to determine a similar relation among the stars , lots of studies have been made using uv ceti type stars showing stellar spot activity such as by dra . one of these studies was reported by @xcite on yy gem . @xcite did not find a clear correlation between the location and extent in longitude of flares and spots on yy gem . moreover , he notes that the longitudinal extent he derived for a flare - producing region is in good agreement with the longitudinal extents of starspots previously calculated for by dra and cc eri by @xcite . in another study , @xcite compared longitudes of the stellar spots obtained from two years observations of yz cmi and ev lac with longitudes of flare events and distributions of flare energy and frequencies along the longitude obtained in the same work . direct comparisons and statistical tests are not able to reveal positive relationships between flare frequency or flare energy and the position of the spotted region . in another extended work , @xcite looked for whether there is any relation between stellar spots and flares observed from 1967 to 1977 in the observations of ev lac . the authors were able to find a relation in the year 1970 . they could not find any relation in other observing seasons because of the higher threshold of the system used for flare detection . in the last years , @xcite found some flares occurring in the same active area with other activity patterns with using simultaneous observations . since no correlation is found between stellar spots and flares , a hypothesis about fast and slow flares was put forward . the hypothesis is based on the work named as fast electron hypothesis . according to this hypothesis , the shape of a flare light variation depends on the location of the event on the star surface in respect to direction of observer . if the flaring area is on the front side of the star according to the observer , the light variation shape looks as a fast flare . if the flaring area is on the opposite side of the star according to the observer , the light variation shape looks as a slow flare @xcite . in addition , @xcite described two types of flares to model flare light curves . @xcite indicated that thermal processes are dominant in the processes of slow flares , which are 95@xmath0 of all flares observed in uv ceti type stars . non - thermal processes are dominant in the processes of fast flares , which are classified as `` other '' flares . according to @xcite , there is a large energy difference between these two types of flares . moreover , @xcite developed a rule to the classifying of fast and slow flares . when the ratios of flare decay times to flare rise times are computed for two types of flares , the ratios never exceed 3.5 for all slow flares . on the other hand , the ratios are always above 3.5 for fast flares . it means that if the decay time of a flare is 3.5 times longer than its rise time at least , the flare is a fast flare . if not , the flare is a slow flare . in this paper , the results obtained from johnson ubvr observations of ad leo , ev lac and v1005 ori will be discussed . @xcite and @xcite reported that v1005 ori is a flare star that exhibits rotational modulation due to stellar spots . the authors found an amplitude variation of @xmath1 with a period of @xmath2 in v band . besides , @xcite examined photometric data in the time series analyses and found 4 periods for rotational modulation . the period of @xmath3 is suggested as the most probable period among them . on contrary , in b band observations of 1981 , a @xmath4 day period variation with an amplitude of @xmath5 was found @xcite . no important light curve changes are seen in the years 1996 and 1997 , while the minimum phases of rotation modulation is varied from @xmath6 to @xmath7 . the amplitude of the curves is @xmath8 in the year 1996 , but in the year 1997 it gets larger than the previous ones @xcite . in the case of ad leo , it is a debate issue whether ad leo has any stellar spot activity , or not . @xcite and @xcite show that ad leo does not exhibit any rotational modulation caused by stellar spots . besides , @xcite found no variations at the @xmath9 magnitude level during the period 1978 , may 10 to 17 . however , @xcite reveals that ad leo demonstrates by dra syndrome with a period of @xmath10 days . in addition , @xcite confirmed this period of ad leo for by dra variation . on the other hand , ev lac is a well known active star with both high level flare and stellar spot activities . @xcite indicated that the star has no variation caused by rotational modulation in the observations in b band from 1972 to 1976 . however , @xcite showed a rotational modulation with a period of @xmath11 and an amplitude of @xmath12 . @xcite , based on continuous observations from 1979 to 1981 , renewed the ephemerides of the variation as a period of @xmath13 and an amplitude of @xmath1 . this indicates that the light curves of ev lac were almost constant for 2.5 years because the spot groups on the star are stable during these 2.5 years . using the renewed ephemerides , @xcite found the amplitude of light curve enlarging from @xmath1 to @xmath5 in the year 1986 . on the other hand , no variation was seen in the light curve of the year 1987 . @xcite showed that the spotted area is located in the same semi - sphere on ev lac for 10 years . comparing the phases of the light curve minima caused by rotational modulation with the flare frequencies and the distribution of the flare equivalent durations for yz cmi and ev lac , @xcite showed that there is no relation between the flare activity and stellar spot activity on these stars . eq peg is classified as a metal - rich star and it is a member of the young disk population in the galaxy @xcite . eq peg is a visual binary @xcite . both components are flare stars @xcite . angular distance between components is given as a value between 3@xmath14.5 and 5@xmath14.2 @xcite . one of the components is 10.4 mag and the other is 12.6 mag in v band @xcite . observations show that flares on eq peg generally come from the fainter component @xcite . @xcite proved that 65@xmath0 of the flares come from faint component and about 35@xmath0 from the brighter component . the fourth star in this study is v1054 oph , whose flare activity was discovered by @xcite . @xcite demonstrated that eq peg has a variability with the period of @xmath15 . v1054 oph (= wolf 630abab , gliese 644abab ) is a member of wolf star group @xcite . wolf 630abab , wolf 629ab (= gliese 643ab ) and vb8 (= gliese 644c ) , are the members of the main triplet system , whose scheme is shown in fig.1 given by @xcite . the masses were derived for each components of wolf 630abab by @xcite . the author showed that the masses are 0.41 @xmath16 for wolf 629a , 0.336 @xmath16 for wolf 630ba and 0.304 @xmath16 for wolf 630bb . in addition , @xcite demonstrated that the age of the system is about 5 gyr . in this study , for each program stars , we analyse the variations at out - of - flare for each light curves obtained in johnson ubvr observations , or not . although all of them show high flare activity , ev lac , v1005 ori and eq peg exhibit stellar spot activity . on the other hand , the spot activity is not obvious for ad leo . it is discussed whether ad leo has any stellar spot activity , or not . finally , this work do not demonstrate any variation from rotational modulations . to perform this kind of studies we would require a long term observing program . as a part of this study , the phase distributions for both fast and slow flares are examined in terms of the minimum phases of rotational modulation . thus , hypothesis developed by @xcite is tested . the observations were acquired with a high - speed three channel photometer attached to the 48 cm cassegrain type telescope at ege university observatory . observations were grouped in two schedules . using a tracking star in second channel of the photometer , flare observations were only continued in standard johnson u band with exposure times between 2 and 10 seconds . the same comparison stars were used for all observations . the second observation schedule was used for determining whether there was any variation out - of - flare . pausing flare patrol of program stars , we observed them once or twice a night , when they were close to the celestial meridian . using a tracking star in second channel of the photometer , the observations in this schedule were made with the exposure time of 10 seconds in each band of standard johnson ubvr system , respectively . there were any delay between the exposure in different filters due to the high - speed three channel photometer . although the program and comparison stars are so close on the sky , differential atmospheric extinction corrections were applied . the atmospheric extinction coefficients were obtained from the observations of the comparison stars on each night . moreover , the comparison stars were observed with the standard stars in their vicinity and the reduced differential magnitudes , in the sense variable minus comparison , were transformed to the standard system using procedures outlined by @xcite . the standard stars are listed in the catalogues of @xcite and @xcite . and also , the de - reddened colour of the systems were computed . heliocentric corrections were also applied to the times of observations . the mean averages of the standard deviations are @xmath17 , @xmath18 , @xmath19 and @xmath19 for the observations acquired in standard johnson ubvr bands , respectively . to compute the standard deviations of observations , we use the standard deviations of the reduced differential magnitudes in the sense comparisons ( c1 ) minus check ( c2 ) stars for each night . there is no variation in the standard brightness comparison stars . [ cols="<,^,^,^",options="header " , ] ( 130mm,60mm)figure12.ps all the flares of ad leo detected in three seasons were again combined for this analysis . the same histograms were derived for both the fast and slow flares of ad leo . they are shown in figure 12 . as it is seen from the analyses of the histogram in the figure , the phase of mfor is @xmath20 , while it is @xmath21 for the slow flares . there is a difference of @xmath22 between two types . according to @xcite , it is expected that there should be a difference of @xmath23 between them . ( 120mm,60mm)figure13.ps in the case of ev lac , it was seen that the phase distribution of the fast flares is not enough to compare it with slow flares for the season 2005 . this is must be because the frequency of the fast flares is not as high as that of the slow flares , as mentioned by @xcite . we only compared them for the season 2004 and 2006 . all histograms of ev lac are shown in figure 13 . the phase of mfor for the fast flares is @xmath24 , while it is @xmath25 for the slow flares in the season 2004 . the difference between the phases of mfor is @xmath26 for two types in this season . the phase of mfor is @xmath27 for the fast flares , while it is @xmath28 for the slow flares in the season 2006 . the difference between the phases of mfor is @xmath29 for both types in the season 2006 , as expected . ( 130mm,60mm)figure14.ps in the case of v1005 ori , comparison could be done for the season 2005/2006 . the histograms of v1005 ori are shown in figure 14 . as it is seen from the analyses of the histogram in the figure , the phase of mfor for the fast flares is @xmath30 , while it is @xmath31 for the slow flares . there is a difference of @xmath32 between two types . ( 130mm,60mm)figure15.ps the same comparison was done for the seasons 2004 and 2005 for eq peq . the histograms of eq peg are shown in figure 15 . as it is seen from the analyses of the histogram in the figure , the phase of mfor for the fast flares is about @xmath33 , while it is @xmath7 for the slow flares . there is a difference of @xmath6 between two types . this value is an acceptable value and close to the expected value according to the hypothesis discussed by @xcite . most of the uv ceti type stars are full convective red dwarfs with sudden - high energy emitting . as it can be seen in the literature , by dra syndrome at out - of - flares is seen in a few stars among 463 flare stars catalogued by @xcite . ev lac and v1005 ori can be given as two examples because the studies in the literature and this study indicate that both stars show the variation due to rotational modulation at out - of - flares . in the case of ev lac , the time series analyses show that the period of rotational modulation found for each data set is range from @xmath34 to @xmath11 . the periods found are similar to those found by @xcite and @xcite . although the periods found for each season are a little bit different , this difference is relatively small . when the amplitudes of the light curves are examined for ev lac , the amplitude of this variation was dramatically decreasing from the year 2004 to 2005 , while the amplitude was clearly larger than ever in this study . however , the mean average of brightness in the light curves was slowly decreasing from the year 2004 to 2006 . the minima phases of the light curves for the three seasons were computed and , it was found as @xmath35 for the season 2004 , @xmath36 for 2005 and @xmath37 for 2006 . in the case of v1005 ori , the periods of the rotational modulation for each season are range from @xmath38 to @xmath39 . in the literature , @xcite found four possible periods varied from @xmath40 to @xmath41 . on the other hand , @xcite found a period of @xmath42 . as it is seen , the periods found in this study are close to the period found by @xcite . when the amplitudes of the light curves were examined , the amplitude observed in the season of 2004/2005 was so smaller than the ones observed in the previous and later seasons that there was no minimum in the light curve . although the mean average of brightness in the light curves was not changing , the minimum phases of the light curves were varying . the minimum phase of the curve for the season 2004/2005 was about @xmath43 and about @xmath44 for the season 2006/2007 . it is hard to say that the minimum phase of the light curves for the season 2005/2006 was about @xmath45 . the case of ad leo is different from the other two stars . the time series analyses do not show any regular variation over the @xmath46 level in one season . on the other hand , the mean brightness levels were increased a value of @xmath47 from the first season to the second and a value of @xmath48 from the second to the last season . this can be because of the stellar polar spots . if the literature is considered , the stellar spots can be carried to polar regions in the case of rapid rotation in the young stars @xcite . according to @xcite , ad leo is at the age of 200 myr . the range of equatorial rotational velocity ( @xmath49 ) given in the literature is between @xmath50 - @xmath51 and @xmath52 @xmath53 for ad leo @xcite . besides , considering these values of @xmath49 , the real rotational velocities must be larger than these values . if both the age and equatorial rotational velocity value parameters in these papers are considered , according to @xcite and @xcite , some spots might be located on the polars for ad leo . in fact , @xcite indicate that by dra had spotted area near polar region , which was stable for 14 years and ev lac has a similar area for 10 years . if the studies made by @xcite and @xcite are considered , ad leo might sometimes show rotational modulation due to the spotted area occurring near the equatorial regions . on the other hand , there is another probability . if the colour index of v - r is considered , it is seen that the star gets bluer from a season to next one , when the star get brighter . besides , no amplitude is seen in the light curves . these can be some indicators that all the surface of the star is covered by cool spots and the efficiency of the spots gets weaker from one season to next one . the colour curves of both ev lac and v1005 ori sometimes exhibit a clear colour excess around the minimum phases of the light curves for some observing seasons . this can be an indicator of some bright areas such as faculae on the surface of these young stars . the effects of the bright areas such as faculae can be seen in the variations of b - v and sometimes v - r colour , while these effects are not seen in the variations in the light curves of bvr bands due to cool spots . the cool spots are more efficient in the light curves of b , especially v and r bands . the same effect is seen in the variations caused by the flare activity . although there is some small effects or no effect of the flare activity in v and r bands , but there is some clear variations in u band light and u - b colour curves . in this study , we observed the stars in u band to investigate the variations out - of - flares . in a sense , u band observations is used to control whether there is any flare activity in the observing durations . if there is some variations in u band light and u - b colour , we did not used the observation to investigate the variation out - of - flares . @xcite showed that eq peg has a variability with the period of @xmath15 . in this study , the time series analyses supported this period . according to our analyses , eq peg exhibits short - term variability with the period of @xmath54 . however , it is seen that there is not any variability in the colour indexes . analysing the light curve of eq peg , it was found as @xmath55 for the minimum phase of the rotational modulation . there are many studies about whether the flares of uv ceti type stars showing by dra syndrome are occurring at the same longitudes of stellar spots , or not . having the same longitudes of flare and spots is an expected case for these stars , because solar flares are mostly occurring in the active regions , where spots are located on the sun @xcite . in the respect of stellar - solar connection , a result of the @xmath56 @xmath57 @xmath58 project of mount wilson observatory @xcite , if the areas of flares and spots are related on the sun , the same case might be expected for the stars . in fact , @xcite have found some evidence to demonstrate this relations . besides , @xcite have found a variations of both the rotational modulation and the phase distribution of flare occurence rates in the same way for the observations in the year 1970 . on the other hand , no clear relation between stellar flares and spots has been found by @xcite . however , @xcite did not draw firm conclusions because of being a non - uniqueness problem . in this study , the flare occurence rates , the ratio of flare number to monitoring time , were computed in intervals of 0.10 phase length as the same method used by @xcite with just one difference . the flare maximum times were used to compute the phases due to main energy emitting in this part of the flare light curves . we observed ad leo for 79.61 @xmath59 and detected 119 flares in three seasons . ev lac was observed 109.63 @xmath59 and 93 flares were detected in three seasons . v1005 ori was observed for 44.75 @xmath59 and 44 flares were detected in two seasons . eq peg was observed for 100.26 @xmath59 and 73 u band flare were detected . since no rotational modulation was found to compare for ad leo , all the flares detected in three season were combined in order to just find whether there is any phase , in which the flare occurence rate gets a peak . on the other hand , we examined flare phase distributions for each season for both ev lac and v1005 ori . in the case of these stars , if the distribution of flares did not cover almost all phases in an observing season of a star , the season is neglected for the comparison of flare and spot activity . consequently , for both ev lac and v1005 ori , we chose the seasons , in which the best flare distributions were obtained . thus , we only used the seasons , in which there is enough data to get reliable conclusions about flare occurrence distributions . in addition , to determine the phases of mfor , all the distributions were modelled with the polynomial function . resolving these models , maximum flare occurrence rates and their phase were found for all program stars . in the case of ev lac , no relation is seen between the minimum phase of the rotational modulation and the phase , in which flare activity reaches the mfor . the minimum phase of the rotational modulation observed in the season 2004 is @xmath35 , while the phase of mfor is @xmath60 . the minimum phase of rotational modulation is @xmath36 , while the flare occurrence rate reaches maximum level in about the phase of @xmath24 for the season 2005 . in the last season of ev lac , rotational modulation minimum is seen in @xmath37 , as mfor is in @xmath61 . in the case of v1005 ori , there is enough data in only one season to compare . as it is seen , the minimum phase of rotational modulation is @xmath45 , while phase of mfor is about @xmath27 for the season 2005/2006 . in this study , the time series analyses indicated that ad leo does not have any rotational modulation . therefore , any minimum time could not have been determined from the observations of three seasons for ad leo . because of this , we could not compare the rotational modulation with flare activity in the case of ad leo . on the other hand , using combined data of three seasons , we found that the mfor is seen in @xmath24 . this phases was computed with using the ephemeris given in equation ( 1 ) taken from @xcite . the time series analyses do not show any short - term variation in the light curves of ad leo . because of this , we waited that there is no any phase , in which the flare activity gets higher levels . on the other hand , as it is seen from the histogram and its normal gaussian model for ad leo , there is a phase for mfor . considering the phase of mfor , the active region(s ) in some particular part of the surface can be more active than the others on the surface of the star . considering the light and colour curves of ad leo , almost all surface of the star may be covered by stellar spots , while it is seen that some region(s ) in the surface of the star can be more active than the remainder of the surface . in the case of eq peg , the minimum phase of the rotational modulation is @xmath55 , while the phase of mfor is @xmath31 . the results acquired from ev lac and v1005 ori demonstrated that flare activity can reach high levels at almost the same longitudes , in which stellar spots occur . on the other hand , there is a considerable difference between the phases of stellar spot and mfor for the observing season 2007 of ev lac . in conclusion , it is seen that there is a longitudinal relation between stellar spot and flare activities in general manner . nevertheless , there are some differences and this makes difficult to do a definite conclusion . moreover , in the case of eq peg , the mfor gets the minimum towards the minimum phase of the rotational modulation . all these cases can be because of a dynamo which is working in the red dwarf stars . in spite of the sun , red dwarf stars are mostly known to have a different dynamo because of full convective outer atmosphere . however , in the last years , some studies showed that flares on the sun do not have to be located upon the spotted areas on the sun @xcite . in addition , it should be kept in mind that most of the studies have been done with using the data obtained from white - light flare observations , but a white - light flare does not have to occur in a flare process . recent studies have shown that non white - light flares may be so common in uv ceti - type stars as they are in the sun @xcite . in this point , it can be mentioned that the analyses of data obtained from only white - light flare observations are not sufficiently qualified . for instance , @xcite found some flares occurring in the same active area with other activity patterns with using simultaneous observations . using the inverse compton event , @xcite developed a hypothesis called fast electron hypothesis , in which red dwarfs generate only fast flares on their surface . on the other hand , according to the flare region on the surface of the star in respect to direction of observer , the shapes of the flare light variations can be seen like a slow flare @xcite . if the scenario in this hypothesis is working , it is expected that the fast and slow flares should collected into two phases in the light curves of uv ceti type stars showing by dra syndrome . it is also expected that these two phases are separated from each other with intervals of @xmath23 in phase . in this study , according to the rule described by @xcite , the flares are classified as fast and slow flares . then the phase distributions of fast flares were compared with the phases of slow flares in order to find out whether there is any separation as expected in this respect . when the phases of both fast and slow flares are examined one by one , it is clear that both of them can occur in any phase . to reach a definite result , the phase distributions of both fast and slow flares are statistically investigated . as it is stated in the previous section , if the distribution of flares did not cover almost all phases in an observing season of a star , the season is neglected for that star . consequently , we chose the seasons , in which there is enough data to get reliable conclusions about flare occurrence distributions for both fast and slow flares . in the case of ad leo and eq peg , we combined all the fast flares of three seasons as we made for the slow flares . for both fast and slow flares , using equation ( 5 ) , the number of flares occurring per an hour in intervals of 0.10 phase length was computed . the obtained occurrence rates for both fast and slow flares are shown by histograms in figures 12 , 13 , 14 and 15 . once again , all the distributions were modelled with the polynomial function . resolving these models , maximum flare occurrence rates and their phase of both slow and fast flares were found for all program stars . in the case of ad leo , the analyses show that both fast and slow flares have a difference of @xmath22 between the phases , in which flare occurrence rates in intervals of 0.10 phase length reach maximum amplitudes . the same difference is @xmath62 for ev lac in the season of 2004 . although these differences are acceptable as low values according to fast electron hypothesis , the difference seen in the season of 2006 is @xmath23 for ev lac . this value is the expected value in respect of fast electron hypothesis . in the case of v1005 ori , slow and fast flares could be compared only for the season of 2005/2006 . the result is that both fast and slow flares have a difference of @xmath63 between the phases of maximum flare occurrence rates . in the case of eq peg , the phase difference between mfors of slow and fast flares is about @xmath6 . the value obtained from eq peg is also the expected value in respect of fast electron hypothesis . it should be noted that in the case of eq peg , it is seen just one clear peak for the distribution of mfor for the fast flares , while there are several peaks for the slow flares . as it is seen from the analyses , both the fast and the slow flares sometimes the same longitudinal distributions and sometimes different . this makes difficult to say that there is a regular longitudinal division between these two types of flares as expected according to @xcite . this means that , when a slow flare is observed , it does not have to be a fast flare occurred on the opposite side of the star in respect to observer direction . the authors acknowledge generous allotments of observing time at the ege university observatory . we thank both dr . hayal boyaciolu , who gave us important suggestions about statistical analyses , and professor m. can akan , who gave us valuable suggestions that improved the language of the paper . we also thank the referee for useful comments that have contributed to the improvement of the paper . we finally thank the ege university research found council for supporting this work through grant no . 2005/fen/051 . 63 amado , p. j. , zboril , m. , butler , c. j. & byrne , p. b. , 2001 , coska , 31 , 13 anderson , c. m. , 1979 , , 91 , 202 baliunas , s.l . , donahue , r.a . , soon , w.h . , horne , j.h . , frazer , j. , woodard - 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in this study , we discuss stellar spots , stellar flares and also the relation between these two magnetic proccess that take place on uv ceti stars . in addition , the hypothesis about slow flares described by @xcite will be discussed . all these discussions are based on the results of three years of observations of the uv ceti type stars ad leo , ev lac , v1005 ori , eq peg and v1054 oph . first of all , the results show that the stellar spot activity occurs on the stellar surface of ev lac , v1005 ori and eq peg , while ad leo does not show any short - term variability and v1054 oph does not exhibits any variability . we report new ephemerides , for ev lac , v1005 ori and eq peg , obtained from the time series analyses . the phases , computed in intervals of 0.10 phase length , where the mean flare occurence rates get maximum amplitude , and the phases of rotational modulation were compared to investigate whether there is any longitudinal relation between stellar flares and spots . although , the results show that flare events are related with spotted areas on the stellar surfaces in some of the observing seasons , we did not find any clear correlation among them . finally , it is tested whether slow flares are the fast flares occurring on the opposite side of the stars according to the direction of the observers as mentioned in the hypothesis developed by @xcite . the flare occurence rates reveal that both slow and fast flares can occur in any rotational phases . the flare occurence rates of both fast and slow flares are varying in the same way along the longitudes for all program stars . these results are not expected based on the case mentioned in the hypothesis .

recently five - dimensional ( 5d ) supergravity ( sugra ) on the orbifold @xmath1 has been studied as an interesting theoretical framework for physics beyond the sm . it has been noted that 5d orbifold sugra with a @xmath0 symmetry gauged by the @xmath2-odd graviphoton can provide the supersymmetric randall - sundrum ( rs ) model @xcite in which the weak to planck scale hierarchy can arise naturally from the geometric localization of 4d graviton @xcite , and/or yukawa hierarchy can be generated by the quasi - localization of the matter zero modes in extra dimension where we generically have an interesting correlation between the flavor structure in the sparticle spectra and the hierarchical yukawa couplings @xcite . in the former case , the bulk cosmological constant and brane tensions which are required to generate the necessary ads@xmath4 geometry appear in the lagrangian as a consequence of the @xmath0 fi term with @xmath2-odd coefficient . in this talk we consider a more generic orbifold sugra which contains a @xmath2-even 5d gauge field @xmath5 participating in the @xmath0 gauging @xcite . if 4d @xmath6 susy is preserved by the compactification , the 4d effective theory of such model will contain a gauged @xmath0 symmetry associated with the zero mode of @xmath5 , which is not the case when the 5d @xmath0 is gauged only through the @xmath2-odd graviphoton . based on the known off - shell formulation @xcite , we formulate a gauged @xmath0 sugra on @xmath1 in which both @xmath5 and the graviphoton take part in the @xmath0 gauging and then analyze the structure of fi terms allowed in such model . as expected , introducing a @xmath2-even @xmath0 gauge field accompanies new bulk and boundary fi terms in addition to the known integrable boundary fi term which could be present in the absence of any gauged @xmath0 symmetry @xcite . as we will see , those new fi terms can have interesting implications to the quasi - localization of the matter zero modes in extra dimension and the susy breaking @xcite and also to the radion stabilization . for a minimal setup , we introduce two vector multiplets and two hypermultiplets in the off - shell formulation of 5d ( conformal ) sugra @xcite : @xmath7 and @xmath8 with the norm function @xmath9 and the hypermultiplet gauging @xmath10 where we adopt the @xmath11 matrix notations omitting @xmath12 index and @xmath13 indices @xmath14 , and the hyperscalars satisfy the reality condition @xmath15 , @xmath16 . the @xmath2-even bosonic ( non - auxiliary ) components are @xmath17 , @xmath18 , @xmath19 , @xmath20 and @xmath21 , and @xmath22 , @xmath23 are the graviphoton vector multiplet and the compensator hypermultiplet respectively . the @xmath2-odd coefficient @xmath24 in the hypermultiplet gauging is consistently introduced by the mechanism proposed in @xcite . the nonzero value of the charge @xmath25 corresponds to the @xmath0 symmetry gauged by @xmath2-even vector field @xmath19 . the bosonic part of the lagrangian is given by @xmath26 @xmath27 \nonumber \\ & & -{\textstyle \frac{1}{2 } } { \rm tr}\big [ { \cal n}_{ij } y^{i\dagger}y^j -4y^{i\dagger } \big ( { \cal a}^\dagger { t}_i { \cal a } -\phi^\dagger { t}_i \phi \big)\big ] , \nonumber \\ % % e_{_{(4)}}^{-1}{\cal l}_{\partial \epsilon } & = & -2\alpha \big ( 3k + { \textstyle \frac{3}{2}}k\ , { \rm tr } \left [ \phi^\dagger \phi \right ] + c\ , { \rm tr } \left [ \phi^\dagger \sigma_3 \phi \sigma_3 \right ] \big ) \nonumber \\ & & \qquad \times \left ( \delta(y)-\delta(y-\pi r ) \right ) , \nonumber \\ e_{_{(4)}}^{-1 } { \cal l}_{n=1 } & = & m_{_{(4)}}^2 \big [ \ , -2r \big(\ , 2y^{x(3)}-e^{-1}e_{_{(4)}}\partial_y\beta \ , \big ) -{\textstyle \frac{1}{2}}r^{(4 ) } \big ] \nonumber \\ & & \qquad \times \left ( \lambda_0 \delta(y ) + \lambda_\pi \delta(y-\pi r ) \right ) , \nonumber\end{aligned}\ ] ] where the matrix notations are employed again , @xmath28 , @xmath29 , @xmath30 \big)^{2/3}$ ] and @xmath31 . here we have included only 4d @xmath6 _ pure _ sugra action at the orbifold fixed points without any khler and superpotentials for simplicity . we remark that after the superconformal gauge fixing , @xmath32/2 } , \nonumber\end{aligned}\ ] ] we find the bulk fi term @xmath33 in @xmath34 and the boundary fi term @xmath35 in @xmath36 for the auxiliary fields @xmath37 in the vector multiplets . we are interested in the 4d poincar invariant background geometry , @xmath38 and the gravitino- , hyperino- and gaugino - killing parameters on this background are given respectively by @xmath39 where @xmath40 , \nonumber\end{aligned}\ ] ] and @xmath41 is the physical gauge scalar field parameterizing the ( very special ) manifold of vector multiplet determined by @xmath42 with the metric @xmath43 . we choose @xmath44 and @xmath45 in the following . the real and diagonal component of the quaternionic hyperscalar field @xmath46 is represented by @xmath47 in the killing parameters , and zero vacuum values are assumed for the other components for simplicity . in terms of these killing parameters , the 4d energy density is found to be @xmath48 and it is obvious that the killing condition @xmath49 determines a stationary point of the 4d scalar potential if the solution exists . now we examine some physical consequences of the 5d gauged @xmath0 supergravity on @xmath1 which can have the bulk and the boundary fi term , for the supersymmetric vacuum configurations , @xmath49 . first we consider the case that we have a charged hypermultiplet @xmath46 with the charge satisfying @xmath50 . for @xmath51 that results in @xmath52 , the vacuum values of the scalar fields are given by @xmath53 for @xmath54 , and @xmath55 for @xmath56 , where @xmath57 and @xmath58 . we find a nontrivial @xmath59-dependent vacuum values for the latter case due to the boundary fi term . notice that the vacuum value of the gauge scalar @xmath60 gives the @xmath59-dependent mass for the charged hypermultiplets which results in nontrivial zero - mode wavefunctions for them . we will show the zero - mode profile in the next more simple but interesting case . next we consider the case there are charged chiral multiplets @xmath61 with the charge @xmath62 at the orbifold fixed points @xmath63 respectively , but no hypermultiplets with the charge @xmath50 in bulk . we introduce minimal khler potential and no superpotential for them at the fixed points . for @xmath51 , the vacuum values of the scalar fields are given by @xmath64 where @xmath65 . we find a linear profile of @xmath41 in the @xmath59-direction due to the bulk fi term , which results in the gaussian form of the zero - mode wavefunction for the charged hypermultiplet , @xmath66 the ratio of the wavefunction values between two fixed points are then shown to be @xmath67 . some numerical plots are shown in fig . [ fig:1 ] for @xmath68 but @xmath69 and in fig . [ fig:2 ] for both @xmath70 . from these figures we find that the nonvanishing @xmath25 ( i.e. , gauging @xmath0 by @xmath2-even vector field ) as well as the bare kink mass @xmath71 affects the zero - mode profiles of the charged hypermultiplets significantly . the nonvanishing charge @xmath72 changes the linear profile of @xmath41 resulting in a more / less severe localization of the charged hypermultiplet zero - mode , depending on the sign of @xmath73 . @xmath60 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 @xmath60 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 + @xmath81 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 @xmath81 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 + ( a ) @xmath82 , @xmath69 ( b ) @xmath83 , @xmath69 @xmath60 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 @xmath60 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 + @xmath81 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 @xmath81 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 + ( a ) @xmath82 , @xmath86 ( b ) @xmath83 , @xmath87 we have studied a 5d gauged @xmath0 supergravity on @xmath1 in which both a @xmath2-even @xmath3 gauge field and the @xmath2-odd graviphoton take part in the @xmath0 gauging . based on the off - shell 5d supergravity of ref . @xcite , we examined the structure of fayet - iliopoulos ( fi ) terms allowed by such theory . as expected , introducing a @xmath2-even @xmath0 gauging accompanies new bulk and boundary fi terms in addition to the known integrable boundary fi term which could be present in the absence of any gauged @xmath0 symmetry . the new ( non - integrable ) boundary fi terms originate from the @xmath6 boundary supergravity , and thus are free from the bulk supergravity structure in contrast to the integrable boundary fi term which is determined by the bulk structure of 5d supergravity @xcite . we have examined some physical consequences of the @xmath2-even @xmath0 gauging in several simple cases . it is noted that the fi terms of gauged @xmath2-even @xmath0 can lead to an interesting deformation of vacuum structure which can affect the quasi - localization of the matter zero modes in extra dimension and also the susy breaking and radion stabilization . thus the 5d gauged @xmath0 supergravity on orbifold has a rich theoretical structure which may be useful for understanding some problems in particle physics such as the yukawa hierarchy and/or the supersymmetry breaking @xcite . for such phenomenological study and for the analysis of the radion stabilization , the @xmath6 superfield description @xcite will be useful . when one tries to construct a realistic particle physics model within gauged @xmath0 supergravity , one of the most severe constraint will come from the anomaly cancellation condition . in some cases the green - schwarz mechanism might be necessary to cancel the anomaly , which may introduce another type of fi term into the theory @xcite . these issues will be studied in future works . k. choi , d. y. kim , i. w. kim and t. kobayashi , eur . j. c * 35 * , 267 ( 2004 ) [ hep - ph/0305024 ] ; 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we discuss a gauged @xmath0 supergravity on five - dimensional orbifold ( @xmath1 ) in which a @xmath2-even @xmath3 gauge field takes part in the @xmath0 gauging , and show the structure of fayet - iliopoulos ( fi ) terms allowed in such model . some physical consequences of the fi terms are examined . address = department of physics , kyoto university , kyoto 606 - 8502 , japan

in this section , we would like to give an intuitive picture as to why impurity sites in @xmath1 and @xmath2-wave superconductors can bind an impurity states ( the shiba states ) and why those sites can experience a topological transition , at which the parity of the many - body ground state changes . to this end , we consider the simplest noninteracting model of a single site and a pair of sites , respectively , but work in its many - body hilbert space . in symmetry class @xmath154 ( applies to chiral @xmath2-wave superconductors ) the @xmath56 topological index of a 1d superconductor denotes the change in fermion parity of the ground - state as a flux @xmath155 is inserted through a system with periodic boundary conditions . this parity change can even be observed in zero - dimensional models of isolated impurities and provides basic intuition whether and how a 1d chain of scalar impurities has the potential to undergo a topological phase transition . here , we discuss this minimal model for a parity changing transition for one and two sites populated with spinless fermions and for a single site populated with spinful fermions . these trivial models provide insight into why in - gap shiba states can be present in scalar impurities placed in superconductors , complementing the more well known situation of shiba states occurring in magnetic impurities . a single spinless fermion can not exhibit superconducting pairing . irrespective of that , we see that an on - site chemical potential @xmath90 can change the parity @xmath156 of the ground state , for the latter is given by @xmath157 . in the occupation basis @xmath158 , the hamiltonian reads h= 0&0 + 0&- . [ eq : hamham ] any other terms in the hamiltonian violate the conservation of fermion parity . the ground state is given by @xmath159 and @xmath160 for @xmath161 and @xmath162 , respectively , with opposite parity . the minimal extension to this model that includes superconducting pairing includes two sites with spinless fermions . in this case , we can have triplet but not singlet superconducting pairing . in the basis @xmath163 , the hamiltonian reads h= 0&0&0 & + 0&-&t&0 + 0&t&-&0 + & 0&0&-2 , where @xmath144 is the hopping integral between the two sites . the energies are ^=- , ^=t- . whenever @xmath164 ( commonly referred to as the weak pairing phase ) , we can induce a parity change of the ground state ( protected crossing ) by changing the chemical potential at ^2=t^2-^2 . for smaller @xmath165 , the ground state has odd parity , for larger @xmath165 , it has even parity . this is in line with the behavior of bound states of two impurities in @xmath2-wave superconductors : they exhibit a protected crossing in the bound state spectrum . the presence of this protected crossing implies the existence of sub - gap shiba states in the energy spectrum : since a protected crossing has to occur upon varying @xmath90 , which could be considered as modeling the scalar impurity strength , it must be that sub - gap @xmath166 states exist - these are the shiba states . a single site with a spinful fermion degree of freedom we exhibit singlet superconducting pairing @xmath167 . we also apply a zeeman field @xmath168 in the direction of the spin - quantization axis . in the basis @xmath169 , the hamiltonian reads h= 0&0&0 & + 0&-+b&0&0 + 0&0&--b&0 + & 0&0&-2 + . the energies are ^=- , ^=b- . we observe a level crossing protected by parity symmetry at b^2=^2+^2 . for smaller @xmath170 , the ground state has even parity , for larger @xmath170 , it has odd parity . this is congruent with the behavior of a ferromagnetic shiba chain on an @xmath1-wave superconductor . this model also shows that a density impurity can not induce a subgap bound state deep in an @xmath1-wave superconducting gap , for @xmath90 does not induce any phase transition in this model for @xmath171 . starting from eq . , we want to consider the limit of small @xmath46 and @xmath47 . we rewrite e^2_;= , where _ , |@xmath172| : = 2m(^ _ + ^ _ |@xmath172| ) . we will need the following two integrals . we are only interested in the lowest order nonvanishing terms in @xmath173 , which is order @xmath174 for the first integral and @xmath175 for the second integral . i_1,-(k_1 ) : = & + = & _ = + = & _ = [ eq : i1 ] we compute the integral ( for @xmath176 ) & + & = _ = _ = -^=+ + & = _ = + ( ) + & = _ = + ( ) , [ eq : nr integral 1 ] with @xmath177 the inverse of the hyperbolic tangent . notice that @xmath167 , that will be substituted by @xmath178 $ ] with @xmath179 , is a dimensionless number . here we used ( ) = + ( b)()+(b ) and = ( b)+(b ) . we now use eq . to determine eq . with @xmath180 , @xmath181 , and @xmath182 $ ] for the case @xmath183 i_1,-(k_1 ) = & + ( 1 ) , where @xmath184\right|^0\right)$ ] . in contrast , if @xmath185 the integral in eq . is completely regular in the limit @xmath186 , so that we conclude that the leading order in @xmath187 is given by i_1,-(k_1 ) = & + ( 1),&|k_1|<k _ , + ( 1),&|k_1|>k_. the second integral that we need is i_2,-,s(k_1):= & ( 1+s ) . we can compute the leading contributions @xmath175 of this integral simply by setting @xmath188 yielding i_2,-,s(k_1 ) = & ( 1+s ) + ( _ -/k_1 ^ 2 ) + = & _ = \ { } + ( _ -/k_1 ^ 2 ) . we will need two more equalities . they can be solved by manipulations similar to eq . , namely & + & = i_1,-(k_1 ) + _ = + ( 1 ) + & = i_1,-(k_1 ) + ( 1 ) [ eq : i1 variant 1 ] and & + & = k _ i_1,-(k_1 ) + _ = + ( 1 ) + & = i_1,-(k_1 ) + ( 1 ) . [ eq : i1 variant 2 ] we can now approximate the functions entering the chain hamiltonian using eqs . and f__1;s_2 & = 2m^2_n _ = + & = 2m^2_n _ = + ( 1 ) + & = 2m^2_n _ = ( 1-s_2 ) i_1,(k_1 ) + ( 1 ) and _ _ 1;s_2 & = 2m^2_n _ = + & = 2m^2_n _ = + ( 1 ) + & = 2m^2_n _ = ( -k_+s_2 k_1 ) i_1,(k_1 ) + ( 1 ) as well as g__1;s_2 = & m _ n_= ( |@xmath172|+k _ ) ( |@xmath172|-k_- ) + = & m _ n_= + ( ) + & m_n_= i_2,,-s_2(k_1 ) . [ eq : f , g , tilde g app ] in particular , the linear combination _ f__1;s_2 + _ _ _ 1;s_2 & 2m^2 _ n_= i_1,-(k_1 ) + & = m _ n_= i_1,-(k_1 ) ( 1-s_2 ) _ -,k _ + & m _ n_= where it is understood that only terms with real square root contribute . here we provide an analytic argument that in the limit where the lattice spacing @xmath12 of the impurity chain is much smaller than the inverse fermi momenta @xmath189 of the underlying superconductor it is possible to construct a nontrivial 1d superconducting chain on a trivial 2d superconductor . to obtain a nontrivial chain in this limit , we further require that the triplet gap @xmath47 is nonvanishing and that the strength @xmath13 of the scalar impurities can be appropriately tuned . in the limit @xmath190 we can discard the summation over @xmath86 entering the eq . ( i.e. , only take the @xmath119 term ) and find the leading behavior of the quantities entering @xmath191 as ( note that @xmath81 is manifestly positive ) note that the first function is odd while the second function is even in @xmath128 to leading order for large @xmath192 as long as @xmath193 . as a consequence , the term multiplying @xmath194 changes sign at some @xmath26 in the limit @xmath195 and for the function to be periodic is has to change sign twice . let us denote the zeros of this term by @xmath106 and @xmath107 ( in general there could be any even number of zeros , but we shall focus on the simplest case of two zeros here ) . the function @xmath88 entering the hamiltonian in the term proportional to @xmath196 will take two in general different values at @xmath106 and @xmath107 , say @xmath197 . then , for impurity strengths @xmath13 such that the hamiltonian for a chiral @xmath2-wave superconductor is readily obtained by evaluating hamiltonian from the main text for one spin species ( say @xmath150 ) in the limit @xmath60 and @xmath61 . in this case , @xmath198 defined in eq . [ eg : defintion d ] is odd in @xmath26 and therefore @xmath199 , while @xmath200 is even in @xmath26 . we can thus conclude that the winding number @xmath73 is odd ( and thus the phase topologically nontrivial ) , whenever @xmath200 changes sign between @xmath201 and @xmath202 , i.e. , if where the sum is only taken over the values of @xmath86 for which the summand is real and @xmath205 . observe that @xmath206 and @xmath207 diverges when @xmath208 approaches from below an integer and half - integer value , respectively . we conclude that for sufficiently small impurity potentials @xmath13 , there exists a series of topological phases for values of @xmath12 slightly below half - integer or integer multiples of the fermi wavelength @xmath149 ( see fig . [ fig : phase diag ] in the main text ) . the red and blue phase boundaries in [ fig : phase diag ] are the values of @xmath209 and @xmath210 defined in eq . and border the region where @xmath151 according to condition .

we show that a chain of _ nonmagnetic _ impurities deposited on a fully gapped two- or three - dimensional superconductor can become a topological one - dimensional superconductor with protected majorana bound states at its end . a prerequisite is that the pairing potential of the underlying superconductor breaks the spin - rotation symmetry , as it is generically the case in systems with strong spin - orbit coupling . we illustrate this mechanism for a spinless triplet - superconductor ( @xmath0 ) and a time - reversal symmetric rashba superconductor with a mixture of singlet and triplet pairing . for the latter , we show that the impurity chain can be topologically nontrivial even if the underlying superconductor is topologically trivial . majorana bound states are a distinctive feature of topological superconductors . to verify the exotic properties that theory attributes to them , in particular their non - abelian braiding statistics , @xcite majorana modes have to be obtained as controllable , localized excitations . for that reason , significant research efforts are focused on one - dimensional ( 1d ) topological superconductors , in which majorana states naturally appear as zero - dimensional , i.e. , fully localized , end states . @xcite one of the ways to create a topological 1d superconductor is through the deposition of a chain of magnetic adatoms on the surface of a thin - film or bulk superconductor . if these adatoms order magnetically either as a helical magnetic spiral or as a ferromagnet the chain can become a 1d topological superconductor , even if the underlying superconductor is of @xmath1-wave spin - singlet type . @xcite ferromagnetic chains of adatoms also require the presence of spin - orbit coupling in the underlying superconductor in order to become topological . the key point in this construction is that magnetic impurities in @xmath1-wave superconductors feature so - called shiba midgap bound states , @xcite which can hybridize along the chain and experience band inversion . the appearance of shiba states can be intuitively understood from anderson s theorem " , @xcite stating that @xmath1-wave superconductivity is ( locally ) suppressed by magnetic impurities . @xcite here , we exploit a second implication of anderson s theorem " , namely the fact that _ _ un__conventional superconducting pairing is suppressed by _ _ non__magnetic impurities . @xcite we ask the question under which conditions bound states of a chain of nonmagnetic scalar impurities can hybridize in such a way that they form a 1d topological superconductor . in particular , such a construction involving nonmagnetic impurities could allow for time - reversal symmetric ( trs ) 1d superconductors with kramers pairs of majorana end states . we consider two examples for the underlying unconventional superconductor : ( i ) a trs superconductor with rashba spin - orbit interaction that mixes @xmath1-wave singlet and @xmath2-wave triplet pairing and ( ii ) a spinless trs breaking @xmath2-wave superconductor . while the latter is per se a bulk topological superconductor that in itself supports majorana edge modes , the former can either be in a topological or in a trivial phase , depending on whether the triplet or the singlet pairing dominates . we find that the chain of nonmagnetic impurities can support majorana end modes , both in the topological and trivial phase of the underlying superconductor , provided that the triplet pair potential is sufficiently large . despite the fact that the required fully gapped unconventional or strong - rashba superconductors are much less abundant than conventional @xmath1-wave superconductors , several promising examples have been discovered and examined recently . this includes thin - film and interface superconductors such as laalo@xmath3/srtio@xmath3 @xcite and single - layer of fese on srtio@xmath3 @xcite as well as bulk materials such as sr@xmath4ruo@xmath5 @xcite and cept@xmath3si @xcite . the observation of majorana bound states at the end of non - magnetic adatom chains deposited on these superconductors would also confirm their unconventional pairing . before we consider a specific model system , let us outline the general derivation of an effective hamiltonian for the chain of scalar impurities following ref . @xcite . we consider a two - dimensional ( 2d ) superconductor with hamiltonian @xmath6 where @xmath7 is the 4-spinor - valued annihilation operator of a bogoliubov quasiparticle at momentum @xmath8 and @xmath9 is the @xmath10 bogoliubov de - gennes ( bdg ) hamiltonian including the spin and particle - hole grading . @xcite on this superconductor , impurities are deposited at positions @xmath11 , i.e. , with distance @xmath12 along the 1-direction . for a delta - function density impurity of strength @xmath13 , the impurity hamiltonian reads @xmath14 with @xmath15 , where we use the notation @xmath16 for the tensor product of pauli matrices @xmath17 and @xmath18 , @xmath19 , that act on spin - space and on particle - hole space , respectively , with @xmath20 and @xmath21 being the identity matrices . here , @xmath22 is the fourier transform of @xmath23 and the 4-spinor @xmath24 is defined in as @xmath25 notice that @xmath24 is periodic in @xmath26 with period @xmath27 , while the momentum @xmath28 . the schroedinger equation for the superconducting electrons in presence of the chain of impurity potentials , governed by the hamiltonian @xmath29 , reads @xmath30 where @xmath31 , @xmath32 . we note that @xmath26 , but not @xmath8 , is a good quantum number which can be used to label the energies . after multiplication with @xmath33 from the left , integration over @xmath34 and summation over all momenta @xmath35 on both sides , it can be transformed into an eigenvalue equation for @xmath36 @xmath37 where we defined @xmath38 as a @xmath39-periodic matrix - valued function of @xmath40 . analytically , the only solution possible is in the case where the impurity bound states hybridize into a band of energies @xmath41 well below the bulk gap of the superconductor so that we can perform the expansion @xmath42 in particular , this expansion is justified as we are primarily interested in topological phase transitions of the shiba wire at which the energy eigenvalue @xmath41 vanishes . inserting eq . in eq . yields the effective hamiltonian of the 1d chain of impurity bound states @xmath43 , \label{eq : effective chain hamiltonian}\ ] ] provided that @xmath44 is invertible for all @xmath15 . so far , we have outlined the general derivation without assuming anything about the band structure or pairing potential of the underlying superconductor . we now particularize to the continuum limit of a 2d system with rashba spin - orbit coupling of strength @xmath45 and a superconducting pairing potential that mixes an @xmath1-wave spin - singlet component @xmath46 with a @xmath2-wave spin - triplet component @xmath47 . the hamiltonian @xmath48 has the eigenvalues @xmath49 with @xmath50 . the trs represented by @xmath51 and the particle hole symmetry @xmath52 square to @xmath53 and @xmath54 , respectively ( @xmath55 is complex conjugation ) . this places the model in class diii in the classification of refs . @xcite with a @xmath56 topological classification in 2d . if @xmath57 , a gap - closing phase transition between the trivial and topological phase of the superconductor occurs ( see fig . [ fig : band structure 2d trs sc ] ) . the trivial phase is the one that is adiabatically connected to the limit @xmath58 . here , @xmath59 are the fermi momenta of the normal state band structure . in the limit @xmath60 and @xmath61 , this model reduces to two copies of spinless chiral @xmath2-wave superconductors . following our analysis of the full model , we will discuss this limit more specifically . we emphasize that this is a simplified , low energy effective model . the strong - coupling from of the gap can be quite different and influence our results . ( green dots ) with respect to the fermi momentum @xmath62 . for the topological properties of a scalar shiba chain , the relative position of these nodes with respect to the fermi momentum in the chain , rather than that of the 2d superconductor , is crucial . ] in passing from the underlying 2d superconductor hamiltonian to the effective 1d shiba - chain hamiltonian , the integration over @xmath34 will eliminate all terms odd in @xmath34 . these will be the terms proportional to the pauli matrix @xmath63 . to understand this , we observe that the 2d model has a mirror - symmetry @xmath64 . for the hamiltonian , this mirror - symmetry acting in spin and orbital space simultaneously , is represented by @xmath65 . ( the impurity hamiltonian is also explicitly invariant under the mirror operation . ) this representation of the mirror symmetry also holds for the inverse of @xmath66 that enters the effective hamiltonian @xmath67 via eq . . since the effective hamiltonian is obtained by integration over @xmath34 , the action of the mirror operation on it reduces to @xmath68 . this is equivalent to the conservation of spin in the 2-direction . the effective hamiltonian then only contains terms proportional to @xmath21 and @xmath69 , restoring a @xmath70 spin - rotation symmetry around the @xmath69 axis . it is a consequence of this extra @xmath70 symmetry that @xmath67 is also invariant under the time - reversal symmetry of spinless fermions @xmath71 which squares to @xmath54 . this extra symmetry elevates @xmath67 from the symmetry class diii of the underlying system to symmetry class bdi , which features a @xmath72 classification in 1d . at the same time , it provides a simple way to determine the topological index of the model . the index of class bdi is an integer winding number @xmath73 that determines the number of kramers pairs of majorana end states . in class diii , the @xmath56 index is the parity of this winding number . a similar shift of the symmetry class also appears in magnetic shiba chains realized experimentally in ref . @xcite . in the eigenspaces of @xmath69 with eigenvalue @xmath74 the effective hamiltonian reads @xmath75 ( a @xmath20-term would not respect @xmath76 and @xmath77 ) , where @xmath78 is defined in terms of the scalar functions @xmath79 and we have implied the notation @xmath80 on the righthand sides of the equations . notice that the factor @xmath81 , is manifestly positive for all @xmath26 . [ eq : hamiltonian s+p - wave chain ] to simplify matters , we would like to consider the hamiltonian in the limit of infinitesimal superconducting gaps @xmath46 and @xmath47 . to lowest order , the hamiltonian only depends on the sign @xmath82 of the pairing potential on the two fermi surfaces at momenta @xmath62 , for ( see @xcite for the derivation ) [ eq : g and f in generality ] @xmath83 \biggr\ } \nonumber\end{aligned}\ ] ] where it is understood that only the terms with @xmath84 contribute to the sum for @xmath85 . the @xmath86-dependence of the sum rests in @xmath87 . formally , the sum in@xmath88 is ultraviolet divergent . introducing a debye - frequency cutoff for the superconducting interaction energy scale much larger than @xmath46 , and @xmath89 and much smaller than @xmath90 , is the physical cure to this divergence . the square - root singularities in eqs . and are regulated if higher - order contributions in @xmath47 are considered . let us now explore the topological properties of the 1d chain . as the form of hamiltonian suggests , the winding number @xmath73 equals the number of windings of @xmath91 around the origin of the 1 - 3-plane as @xmath26 changes from @xmath92 to @xmath93 . a prerequisite for a nonvanishing winding number is that both components @xmath85 and @xmath94 change sign as a function of @xmath15 . in particular , the expansions in eq . can be analyzed in this light . if @xmath95 , which is precisely the condition for the underlying 2d superconductor to be topologically trivial , the lowest order expansion of @xmath85 , eq . , does not change sign as function of @xmath26 . in contrast , when @xmath96 , i.e. , if the underlying 2d superconductor is topologically nontrivial , the summand in eq . diverges to @xmath97 , when @xmath98 . if we include the folding of the momentum to the brillouin zone @xmath15 , this means that @xmath85 diverges to @xmath99 at @xmath100 and to @xmath101 at @xmath102 . thus , @xmath103 has to change sign between @xmath104 and @xmath105 if @xmath96 . due to the @xmath27 periodicity in @xmath26 , @xmath85 has thus at least two zeros in the interval @xmath15 . denote these zeros by @xmath106 and @xmath107 , and let us consider the case where these are the only zeros of @xmath85 . for the winding number @xmath73 to be nontrivial , @xmath94 has to change sign between these two zeros . generically , the function @xmath108 that enters @xmath94 will take different values at the two zeros . let us assume , without loss of generality , @xmath109 . then , according to eq . , if @xmath110 the component @xmath94 changes sign between the two zeros and the winding number is one . our considerations thus show how the impurity chain becomes a nontrivial 1d superconductor , if the underlying superconductor is topological , @xmath13 lies in the appropriate range and @xmath85 has only two zeros . if one of these conditions is not met , however , the chain may remain in a topologically trivial state . a natural question that follows from these observations is whether the nontrivial topology of the underlying superconductor is a _ prerequisite _ for the impurity chain to become topological . we will now argue that this is not the case . first , we show analytically that eqs . allow for this scenario in a certain limit . second , we numerically solve finite systems and show the existence of majorana bound states in a chain of scalar impurities on a trivial superconductor . for the analytical argument , we consider the simplified case @xmath111 and the limit @xmath112 . by treating the underlying 2d superconductor in the continuum limit , we have assumed that its lattice constant @xmath113 is much smaller than that of the chain @xmath114 . in a lattice description of the 2d superconductor , its brillouin zone @xmath115 is thus much larger than that of the chain @xmath116 . the constraint that the fermi momenta lie in the smaller chain brillouin zone @xmath112 is thus satisfied if the chemical potential of the superconductor is sufficiently low . then , the expansion of @xmath85 to lowest ( zeroth ) order in the gap functions , as given by eq . , vanishes identically for @xmath117 ( while it has definite sign @xmath95 otherwise ) . we thus have to appeal to the next ( linear ) order corrections in the superconducting gaps to this result , and in particular ask whether they allow for a sign change of @xmath85 for @xmath26 with @xmath117 . in the limit @xmath118 , due to the denominator , it is sufficient to restrict the sums over @xmath86 in eqs . to the @xmath119 term . further , if we also restrict us to small @xmath120 , we can expand the terms in @xmath26 , with the understanding that @xmath121 , and obtain @xmath122 , as well as @xmath123 . we observe that @xmath85 indeed changes sign at momentum @xmath124 . ( note that the condition to have a trivial underlying superconductor @xmath95 implies @xmath125 . ) one can take advantage of this sign change in @xmath126 by choosing @xmath13 in such a way that the vector @xmath91 winds . thus , the chain of scalar impurities can be topologically nontrivial , hosting a kramers pair of majorana bound states at its end , even if the underlying superconductor is topologically trivial . this mathematical result has an intuitive understanding . if the 2d superconductor has a sizeable triplet pair potential , there is a circle at @xmath127 in the @xmath128-@xmath34-plane along which the gap function is nodal . if the 2d superconductor is topologically trivial , the fermi surfaces of the metallic state before pairing lie inside this nodal circle , i.e. , the fermi momenta satisfy @xmath129 . the impurity chain , on the other hand , will have its own independent fermi momentum @xmath130 . the chain is in a nontrivial state if the paring potential is of opposite sign on the two fermi points of the chain , that is , if the nodal line of the order parameter lies between the 2d and 1d fermi momenta @xmath131 . we note that this is a one - body consideration beyond the weak pairing limit ( requires the superconducting gaps to be well formed for a larger @xmath132-space region than just the superconductor fermi surfaces ) , and might not survive a self - consistent calculation . , @xmath133 , @xmath134 , @xmath135 . between @xmath136 and @xmath137 the chain is in a topological and the underlying superconductor are in a trivial phase . ( b ) the kramers pair of majorana modes at each end of the chain is exponentially localized ( c ) phase diagram as a function of the triplet pairing potential @xmath47 and the strength of the scalar impurities @xmath13 , with the color scale representing the lowest energy eigenvalue of hamiltonian . thus , blue regions in the phase diagram either indicate the presence of majorana end states or a gapless bulk superconductor . the shaded region between the solid lines marks the topological phase transition of the 2d bulk superconductor . the dashed line shows an analytical estimate for the topological phase transition in the chain . it is obtained by restricting hamiltonian to the lattice sites one the chain only , with the inclusion of second order virtual processes that correspond to hopping one site away from the chain and back . ] we have obtained the analytical result as a proof of principle in the limit @xmath120 in order to make the calculation tractable . however , the general statement that nontrivial chain topology can emerge out of a trivial substrate ( in bdg formalism ) is not restricted to this limit , as we now show numerically . to that end , consider the bdg hamiltonian @xmath138 \\ & + \sum_{{\ensuremath{\boldsymbol{r}}}\in\lambda } \left[\sum_{s=\pm } \frac{u_{{\ensuremath{\boldsymbol{r}}}}-\mu}{2}\,c^\dagger_{{\ensuremath{\boldsymbol{r}}},s}c^{\ } _ { { \ensuremath{\boldsymbol{r}}},s } + \delta_{\mathrm{s}}\,c^\dagger_{{\ensuremath{\boldsymbol{r}}},\uparrow}c^{\dagger}_{{\ensuremath{\boldsymbol{r}}},\downarrow } \right ] + \mathrm{h.c.}. \end{split } \label{eq : lattice hamiltonian}\ ] ] defined on a square lattice @xmath139 , spanned by the vectors @xmath140 and @xmath141 , with periodic boundary conditions here , @xmath142 creates an electron of spin @xmath1 at site @xmath143 , while @xmath144 and @xmath90 are the nearest neighbor hopping integrals and the chemical potential , respectively . the local potential @xmath145 takes the nonzero value @xmath13 along the shiba chain of length @xmath146 , say for @xmath147 and @xmath148 , while it vanishes otherwise . figure [ fig : nontrivial on trivial ] shows that for a choice of @xmath144 and @xmath90 , for which the 2d fermi surface is small , there exists a kramers pair of exponentially localized majorana bound states at each end of the chain for values of the pairing potential @xmath47 that lie below the bulk trivial to topological superconductor phase transition . -wave superconductor as a function of the strength of the impurity potential @xmath13 and the spacing @xmath12 of the impurity atoms in units of the fermi wavelength @xmath149 of the chiral superconductor . the colored regions correspond to different values of the topological index @xmath73 of the 1d chain , which has been evaluated numerically in presence of a small regulating gap @xmath47 . physically , @xmath73 equals the number of majorana end modes of the chain . ] before closing , let us discuss topological properties of a chain of scalar impurities on a chiral @xmath2-wave superconductor of spineless fermions . the hamiltonian for this system is readily obtained by evaluating hamiltonian for one spin species ( say @xmath150 ) in the limit @xmath60 and @xmath61 . in this limit , one obtains a series of transitions between phases @xmath151 and with even @xmath73 , maked by the phase transition lines in see fig . [ fig : phase diag ] as a function of the impurity spacing @xmath12 . @xcite an explicit numerical evaluation of the winding number @xmath73 reveals an even richer structure including regions in the @xmath13@xmath152 phase diagram in which @xmath153 . we explicitly checked that the hamiltonian is gapped for all the phases presented in fig . [ fig : phase diag ] . in summary , we have shown that a chain of scalar impurities that is deposited on fully gapped unconventional superconductors can be a topologically nontrivial 1d superconductor . we have exemplified this for the case of a chiral @xmath2-wave superconductor and for a rashba spin - orbit coupled superconductor with both singlet and triplet pairing potential . in the latter case , the impurity chain can be in a nontrivial phase , even if the underlying superconductor is topologically trivial . however , for this to happen , the oder parameter must be nodal for momenta outside the fermi sea not necessarily a generic situation . we hope our results will stimulate work on nonmagnetic impurities on sr@xmath4ruo@xmath5 . the system we considered may naturally occur at step edges @xcite and related linear defects on the surface of such chiral superconductors . our results suggest that the ends of such linear defects could carry interesting bound states . this work was supported by nsf career dmr-095242 , onr - n00014 - 11 - 1 - 0635 , aro muri w911nf-12 - 1 - 0461 , nsf - mrsec dmr-0819860 , packard foundation and keck grant . 99 n. read , and d. green , phys . rev . b * 61 * , 10267 ( 2000 ) . d. a. ivanov , phys . rev . lett . * 86 * , 268 ( 2001 ) . a.y . kitaev , phys . usp . * 44 * , 131 ( 2001 ) . r.m . lutchyn , j.d . sau , and s. das sarma , phys . rev.lett . * 105 * , 077001 ( 2010 ) . y. oreg , g. refael , and f. von oppen , phys . rev . lett.*105 * , 177002 ( 2010 ) . j. d. sau , r. m. lutchyn , s. tewari , and s. das sarma , phys . rev . lett . * 104 * , 040502 ( 2010 ) . j. alicea , phys . rev . b * 81 * , 125318 ( 2010 ) . a. c. potter and p. a. lee , phys . rev . lett . * 105 * , 227003 ( 2010 ) . j. alicea , y. oreg , g. refael , f. von oppen , and m.p.a.fisher , nature phys . * 7 * , 412 ( 2011 ) . b. i. halperin , y. oreg , a. stern , g. refael , j. alicea , and f. von oppen , phys . rev . b * 85 * , 144501 ( 2012 ) . t. d. stanescu and s. tewari , j. phys . : condens . matter * 25 * , 233201 ( 2013 ) . t .- p . choy , j. m. edge , a. r. akhmerov , and c. w. j. beenakker , phys . rev . b * 84 * , 195442 ( 2011 ) . s. nadj - perge , i.k . drozdov , b.a . bernevig , and a. yazdani , phys . rev . b * 88 * , 020407(r ) ( 2013 ) . s. nakosai , y. tanaka , and n. nagaosa , phys . rev . b * 88 * , 180503(r ) ( 2013 ) . j. klinovaja , p. stano , a. yazdani , and d. loss , phys.rev . lett . * 111 * , 186805 ( 2013 ) . b. braunecker and p. simon , phys . rev . lett . * 111 * , 147202 ( 2013 ) . m.m . vazifeh and m. franz , phys . rev . lett . * 111 * , 206802 ( 2013 ) . f. pientka , l. glazman , and f. von oppen , phys . rev.b * 88 * , 155420 ( 2013 ) . k. pyhnen , a. weststrm , j. rntynen , and t. ojanen , arxiv:1308.6108 . f. pientka , l. i. glazman , and f. von oppen , phys . rev . b * 89 * , 180505(r ) ( 2014 ) . j. li , t. neupert , b.a . bernevig , a. yazdani , arxiv:1404.4058 . s. nadj - perge , i.k . drozdov , j. li , h. chen , s. jeon , j. seo , a.h . macdonald , b.a . bernevig , and a. yazdani , science * 346 * 602 - 607 ( 2014 ) . r. pawlak , m. kisiel , j. klinovaja , t. meier , s. kawai , t. glatzel , d. loss , and e. meyer , arxiv:1505.06078 . m. ruby , f. pientka , y. peng , f. von oppen , b. w. heinrich , and k. j. franke , phys . rev . lett . * 115 * , 197204 ( 2015 ) . l. yu , acta phys . sin . 21 , * 75 * ( 1965 ) . h. shiba , prog . theor . phys . * 40 * , 435 ( 1968 ) . a. i. rusinov , sov . phys . jetp * 29 * , 1101 ( 1969 ) . v. kaladzhyan , c. bena , and p. simon , arxiv:1512.05575 . p. w. anderson , j. phys . chem . sol . * 11 * , 26 ( 1959 ) . j. byers , m. flatte and d. j. scalapino , phys . rev . lett . * 71 * , 3363 ( 1993 ) . c. h. choi , j. korean phys . soc . * 44 * , 355359 ( 2004 ) . j. d. sau and e. demler , phys . rev . b * 88 * , 205402 ( 2013 ) . h. hu , l. jiang , h. pu , y. chen , x .- j . liu , phys . rev . lett . * 110 * , 020401 ( 2013 ) . y. kim , j. zhang , e. rossi , and r. m. lutchyn , phys . rev . lett . * 114 * , 236804 ( 2015 ) . n. reyren et al . , science * 317 * , 1196 ( 2007 ) . j .- f . ge et al . , arxiv:1406.3435 . a. p. mackenzie and y. maeno , rev . mod . phys . 75 , 657 ( 2003 ) . e. bauer et al . , phys . rev . lett . * 92 * , 027003 ( 2004 ) . the same derivation would hold for a 3d superconductor , with quantitatively different results . a. p. schnyder , s. ryu , a. furusaki , and a. w. w. ludwig , phys . rev . b * 78 * , 195125 ( 2008 ) . a. kitaev , aip conf . proc . * 1134 * , 22 ( 2009 ) . s. misra et al . , phys . rev . b * 66 * , 100510(r ) ( 2002 ) . b.b . zhou et al . , nature physics * 9 * , 474 ( 2013 ) . see supplemental material .

the discovery of iron - based high - temperature superconductors has ignited intensive studies.@xcite the superconducting transition temperature ( @xmath4 ) has risen up to 56 k in @xmath5feaso@xmath6f@xmath7 ( @xmath5=sm , nd , pr , ... ) .@xcite on the other hand , the sibling ni - based compounds only possess relatively low @xmath4 , i.e. , laonip ( @xmath4=3 k),@xcite laonias ( @xmath4=2.75 k),@xcite bani@xmath0p@xmath0 ( @xmath4=2.4 k),@xcite bani@xmath0as@xmath0 ( @xmath4=0.7 k),@xcite and srni@xmath0as@xmath0 ( @xmath4=0.62 k).@xcite . it is intriguing to understand why @xmath4 is low in the ni - based compounds , which may facilitate understanding the high-@xmath4 in iron - based ones . bafe@xmath0as@xmath0 , a typical parent compound of iron - based superconductor with the thcr@xmath0si@xmath0 structure , exhibits a structural transition from tetragonal to orthorhombic , concomitant with a spin - density - wave ( sdw ) transition at 140 k.@xcite the structural transition was suggested to be driven by the magnetic degree of freedom.@xcite bani@xmath0as@xmath0 with the same structure , displays a structural transition around 131 k , however , from a tetragonal phase to a lower symmetry triclinic phase.@xcite no evidence of sdw is reported in bani@xmath0as@xmath0 so far . on the other hand , it was pointed out that the transition in bani@xmath0as@xmath0 is a first - order one , while that is more second - order - like in bafe@xmath0as@xmath0.@xcite moreover , the c - axis resistivity drops by two orders of magnitude and a thermal hysteresis of in - plane resistivity is present at the transition of bani@xmath0as@xmath0.@xcite although there are intriguing resemblance as well as differences between bafe@xmath0as@xmath0 and bani@xmath0as@xmath0 , the electronic structure of the latter is still not exposed . here we report the angle - resolved photoemission spectroscopy ( arpes ) study of the electronic structure of bani@xmath0as@xmath0 . our data are compared to the band structure calculation of bani@xmath0as@xmath0 and the results of iron pnictides reported before , revealing their similarities and differences . particularly , no band folding is found in the electronic structure of bani@xmath0as@xmath0 , confirming that there is no collinear sdw type of magnetic ordering . because of the intimate relation between superconductivity and magnetism,@xcite the absence of magnetic ordering is possibly related to the low-@xmath4 of bani@xmath0as@xmath0 . furthermore , a hysteresis is observed for the band shift , resembling the hysteresis in the resistivity data . the band shift can be accounted for by the significant lattice distortion in bani@xmath0as@xmath0 , in contrast to iron pnictides , where the band shift is largely caused by the magnetic ordering . as@xmath0 in the triclinic phase taken with 100 ev incident electrons . , width=94 ] bani@xmath0as@xmath0 single crystals were synthesized by self - flux method , and a similar synthesis procedure has been described in ref . . its stoichiometry was confirmed by energy dispersive x - ray ( edx ) analysis . arpes measurements were performed ( 1 ) with circularly - polarized synchrotron light and randomly - polarized 8.4 ev photons from a xenon discharge lamp at beamline 9 of hiroshima synchrotron radiation center ( hsrc ) , ( 2 ) with linearly polarized synchrotron light at the surface and interface spectroscopy ( sis ) beamline of swiss light source ( sls ) , and ( 3 ) with randomly - polarized 21.2 ev photons from a helium discharge lamp . scienta r4000 electron analyzers are equipped in all setups . the typical energy resolution is 15 mev , and angular resolution is 0.3@xmath8 . the samples were cleaved _ in situ _ , and measured in ultrahigh vacuum better than 5@xmath9 mbar . the high quality sample surface was confirmed by the clear pattern of low - energy electron diffraction ( leed ) , where no sign of surface reconstruction is observed ( fig . [ leed ] ) . -@xmath3 in the triclinic phase . ( a)-(e ) photoemission intensity plots of cuts 1 - 5 as indicated in panel g , taken in hsrc with circularly polarized 21.2 ev photons at 20 k. ( f ) photoemission intensity plot of cut 6 as indicated in panel g , taken with randomly polarized 21.2 ev photons from a helium lamp at 25 k. ( g ) the mdcs corresponding to panel f. ( h ) cuts 1 - 6 are indicated in the projected 2d brillouin zone . the dashed curves and markers trace the band dispersions.,width=321 ] -@xmath1 measured at 10 k in the triclinic phase with 100 ev linearly polarized light at sls . ( a ) experimental setup for the s and p polarization geometries , and the indication of the @xmath2-@xmath1 cut in the projected 2d brillouin zone . ( b ) and ( c ) photoemission intensity plots measured in the s and p polarization geometries respectively . the image contrast in the rectangular region as enclosed by dash - dotted lines in panel c is adjusted to reveal the bands in this region . ( d ) stack of mdcs in the s and p polarization geometries . each mdc is normalized by its integrated weight . dashed curves and markers trace the band dispersions . labels are explained in the text.,width=321 ] figures [ gm](a ) and [ gm](c ) show the fermi surface maps measured with circularly polarized 22.5 ev photons in the tetragonal and triclinic phases , respectively . there are four patches near the @xmath2 point as indicated by the arrows , and two electron pockets around the @xmath1 point . sixteen cuts from @xmath2 to @xmath1 are presented in fig . [ gm](b ) to illustrate the electronic structure evolution in the tetragonal phase , and the corresponding data in the triclinic phase are shown in fig . [ gm](d ) . in the tetragonal phase , the parabolic - shaped band in cuts 1 - 2 around @xmath2 is referred as @xmath10 . the @xmath10 band appears to be @xmath11-shaped in cuts 3 - 4 . from cut 1 to cut 4 , the parabolic part shrinks continuously , and eventually only the inverted parabolic part is observable in cuts 5 - 7 . the evolution of the electronlike bands around @xmath1 are shown in cuts 8 - 16 , many of which are complex due to the rapid change of dispersions . nonetheless , two bands ( @xmath12 and @xmath13 ) can be resolved as indicated by the dashed curves in cut 16 and will be further elaborated in fig . [ aroundm ] . the data in the triclinic phase [ fig . [ gm](d ) ] are generally similar to those in the tetragonal phase . nonetheless , we note that @xmath10 already shows some bending near the fermi energy ( @xmath14 ) in the tetragonal phase as indicated by the arrows on the data taken along cuts 1 - 2 . on the other hand , @xmath10 just passes through @xmath14 without bending in the triclinic phase . moreover , the band top of @xmath10 is below @xmath14 in cuts 6 - 7 of fig . [ gm](b ) as indicated by the arrows , but barely touches @xmath14 in fig . [ gm](d ) , where cuts 6 - 7 pass through one of the four patches around @xmath2 . therefore , the four spectral weight patches around @xmath2 [ as marked by four arrows in fig . [ gm](a ) ] are due to the residual spectral weight of the @xmath10 band in the tetragonal phase . however , the @xmath10 band shifts up and they evolve into small holelike fermi surfaces in the triclinic phase . this will be further illustrated in fig . [ gx ] . of note , from this complete set of data , we do not observe any sign of band folding or splitting like that in the iron pnictides.@xcite to further illustrate the electronic structure of bani@xmath0as@xmath0 , figs . [ gx](a)-[gx](e ) present photoemission intensities along five cuts parallel to the @xmath2-@xmath3 direction in the triclinic phase . the @xmath11-shaped feature originated from @xmath10 simply moves towards @xmath14 from cut 1 to cut 5 , touching @xmath14 at cuts 4 - 5 , which confirms that the four small fermi surfaces around @xmath2 are holelike . note that the downward part of @xmath10 [ indicated by the arrow in fig . [ gx](e ) ] is clearly resolved here , while it is barely observable in the same momentum region when the cuts are along the @xmath2-@xmath1 direction , as shown in cuts 1 - 2 of figs . [ gm](b ) and [ gm](d ) . it highlights the matrix element effects since the 3@xmath15 orbitals have specific orientations . figure [ gx](f ) shows the photoemission intensity along @xmath2-@xmath3 , taken with randomly polarized 21.2 ev photons from a helium lamp in the triclinic phase . the determined band structure is traced by dashed curves , where the broad spectral weight around @xmath3 are attributed to an electronlike band @xmath16 , which is further shown by markers in the corresponding momentum distribution curves ( mdcs ) [ fig . [ gx](g ) ] . therefore , there is an electron fermi pocket around @xmath3 . similar to iron pnictides , the bands near @xmath14 are quite complicated and mainly contributed to by the ni 3@xmath15 electrons in bani@xmath0as@xmath0 . to resolve the complex bands around @xmath1 , we utilize the linearly polarized light , which could only detect bands with certain symmetry , so that the measured partial electronic structure helps reducing the complexity in analysis.@xcite figure [ aroundm ] presents data along @xmath2-@xmath1 , taken with linearly polarized 100 ev photons in sls in the triclinic phase . two polarization geometries ( s and p ) are illustrated in fig . [ aroundm](a ) . in the s polarization geometry , we resolve two bands , whose dispersions are depicted by dashed curves in the photoemission intensity plot [ fig . [ aroundm](b ) ] . while in the p polarization geometry , one intense parabolic electronlike band around @xmath1 is resolved with the band bottom at about -0.57 ev . the dispersions in both geometries are marked in the corresponding mdcs [ fig . [ aroundm](d ) ] . the asymmetry of the dispersion indicated by triangles may be due to the slight sample misalignment . the image contrast in the dash - dotted region in fig . [ aroundm](c ) is adjusted to highlight the @xmath10 feature . the observed @xmath10 feature is consistent with the data in figs . [ gm ] and [ gx ] . by comparing with the fermi crossings observed in cuts 1 and 16 of fig . [ gm](d ) , we attribute the three bands to @xmath10 , @xmath12 , and @xmath13 as shown in fig . [ aroundm](d ) , where @xmath12 and @xmath13 are two electronlike bands around @xmath1 . moreover , since the experimental setup under the s ( p ) polarization geometry detects states with odd ( even ) symmetry with respect to the mirror plane , the @xmath12 band is of mainly odd symmetry while @xmath13 is of even symmetry . we note that @xmath10 is observed in both geometries [ fig . [ aroundm](b ) and [ aroundm](c ) ] , suggesting that the @xmath10 band has mixed symmetries . = 0 for data shown in panels b and c. data were taken in hsrc with circularly polarized 22.5 ev photons for panels b - d . ( e ) edcs at @xmath17=0 for a cooling - warming - cooling cycle , measured with randomly polarized 8.4 ev photons from a xenon discharge lamp . the short bars indicate the peak positions . ( f ) summary of peak positions obtained from panel e , which exhibits a hysteresis.,width=321 ] orbitals . ( b)-(f ) contributions of the @xmath18 , @xmath19 , @xmath20 , @xmath21 , and @xmath22 orbitals to the calculated band structure of bani@xmath0as@xmath0 respectively . the contribution is represented by both the size of the symbols and the color scale.,width=321 ] to study the first order transition of bani@xmath0as@xmath0 , the temperature dependence is presented in fig . the photoemission intensity plots along @xmath2-@xmath3 are shown in figs . [ loop](b ) and [ loop](c ) for the tetragonal and the triclinic phases respectively . the corresponding energy distribution curves ( edcs ) at @xmath17=0 are stacked in fig . [ loop](d ) . interestingly , the band bottom of the @xmath11-shaped feature is moved from -200 mev in the tetragonal phase to -170 mev in the triclinic phase . in other words , the @xmath11-shaped band moves towards @xmath14 and its electronic energy is raised up . however , another feature at higher binding energies shifts away from @xmath14 . its band top is moved from -350 mev at 145 k to -390 mev at 12 k , which partially saves the electronic energy . since the resistivity shows a hysteresis loop,@xcite it is intriguing to investigate whether a similar hysteresis could be observed for the electronic structure . data in figs . [ loop](e)-[loop](f ) are taken with randomly polarized 8.4 ev photons from a xenon discharge lamp , in a cooling - warming - cooling cycle . the edcs at @xmath17=0 across the transition are stacked in fig . [ loop](e ) , where the peak positions are indicated by short bars . the temperature dependence of peak positions is summarized in fig . [ loop](f ) , showing a clear hysteresis with the band shift as much as 25 mev . such electronic structure demonstration of a hysteresis of 3 k is so far the most obvious . a hysteresis in the electronic structure has been observed in fete , but with a loop width of only 0.5 k.@xcite our observation here is consistent with the bulk transport properties , which indicates that the measured electronic structure reflects the bulk properties . the measured band structure and fermi surface are summarized in figs . [ calc](a ) , [ calc](d ) and [ calc](e ) . for comparison , local density approximation calculations which have been reported before in ref . are reproduced in figs . [ calc](b ) , [ calc](f ) and [ calc](g ) . the notations for bands near @xmath14 are labeled in figs . [ calc](a ) and [ calc](b ) . qualitatively , although not all calculated bands were observed , the main features of the experiments are captured by the calculation , such as the dispersion nature of the bands . the @xmath10 , @xmath12 , and @xmath13 bands of the experimental results in the @xmath2-@xmath1 direction are similar to the numerical results in the @xmath23-@xmath24 direction . as shown in fig . [ calc](c ) , the measured @xmath10 band along @xmath2-@xmath1 matches the calculation well after the calculated bands are renormalized by a factor of 1.66 and shifted down by 0.08 ev . this renormalization factor is consistent with the results of optical measurements.@xcite although not all bands could match , it may suggest that the correlation in bani@xmath0as@xmath0 is weaker than that in iron pnictides.@xcite along @xmath2-@xmath3 , the observed @xmath10 and @xmath16 bands partially resemble the calculated dispersions along both @xmath23-@xmath5 and @xmath25-@xmath26 , as highlighted by the shaded regions [ fig . [ calc](b ) ] , but the energy positions do not match . our data along in this direction might correspond to a @xmath27 between @xmath25 and @xmath23 . as expected from the differences in the experimentally determined and calculated band structures , the fermi surface topologies are quite different in both the experiments and the calculations . in our data [ figs . [ calc](d ) and [ calc](e ) ] , we observe four small fermi pockets around @xmath2 only in the triclinic phase . around @xmath1 , two electronlike fermi pockets are resolved in both phases . around @xmath3 , the observed fermi crossings are from an electronlike pocket . as a comparison , in the calculated fermi surface of high - t tetragonal phase [ figs . [ calc](f ) ] , there are two large warped cylinders of electron pockets around the zone corner , a pocket interconnected from the zone center to a large deformed cylinder around the zone corner , a 3d electron pocket around @xmath23 , and 3d pockets located between @xmath26 and @xmath5 . in the low - t triclinic phase [ figs . [ calc](g ) ] , 3d pockets around @xmath23 and between @xmath26 and @xmath5 are gapped out . the large electron fermi pockets around @xmath1 observed in our data are generally consistent with that in the calculation , which is a direct consequence of two more electrons from ni than fe . note that the @xmath27-dispersion is significant in the calculation . however , we have measured with four different photon energies , including the more bulk - sensitive 8.4 ev photons , and no obvious differences in dispersion have been observed . therefore , the @xmath27-dispersions in bani@xmath0as@xmath0 may be weaker than calculated . for a multiband and multiorbital superconductor , it is crucial to understand the orbital characters of the band structure . because of the symmetry of 3@xmath15 orbitals with respect to the mirror plane , the s polarization geometry in photoemission can only detect the @xmath18 and @xmath19 orbitals while the p polarization geometry can only detect the @xmath20 , @xmath21 , and @xmath22 orbitals [ fig . [ orbital](a)].@xcite the contributions of the five 3@xmath15 orbitals to the calculated band structure are presented in fig . [ orbital](b)-[orbital](f ) , which therefore can be compared to our polarization dependent data . along @xmath23-@xmath24 , the @xmath10 band is consisted of mainly the odd @xmath18 orbital and some contributions of odd @xmath19 and even @xmath22 , thus can be observed in both the s and p polarization geometries ; while the @xmath13 band is consisted of the even @xmath20 and @xmath21 orbitals , thus can only be observed in the p polarization geometry . they are in good agreement with our observation . the @xmath12 band is consisted of the odd @xmath18 , @xmath19 and even @xmath22 orbitals in the calculation , thus should be observed in both the s and p polarization geometries . however , @xmath12 is mainly detected in the s polarization geometry , possibly because in the p polarization geometry it is buried in the intense peak of @xmath13 . the consistency between our data and the calculated orbital characters confirms that our data along @xmath2-@xmath1 match the band structure calculation along @xmath23-@xmath24 . it was observed in the optical data that the phase transition leads to a reduction of conducting carriers , consistent with the removal of small fermi surfaces shown by the calculation.@xcite however , we do not observe such behavior by arpes . on the contrary , instead of the disappearance of small fermi surfaces in the triclinic phase , we observe that bands shift up in energy , leading to additional four fermi surfaces . the inconsistency between the optical data and our photoemission data suggests that the changes in optical data across the phase transition are possibly an integrated effect of band structure reorganization over the entire brillouin zone , instead of the disappearance of certain fermi surface sheets ; but it is also possible that only limited @xmath28-space has been probed in the current photoemission study . as a sibling compound of iron pnictides , bani@xmath0as@xmath0 exhibits quite different properties and electronic structure . the parent compounds of iron pnictides show a second - order - like transition that is the sdw transition concomitant with a structural transition . however , bani@xmath0as@xmath0 shows a strong first - order - like structural transition , without magnetic ordering reported to date . from the aspect of electronic structure , iron pnictides possess several hole pockets around @xmath2 , and several electron pockets around @xmath1 , but have no pockets around @xmath3 , while the band structure of bani@xmath0as@xmath0 is dramatically different from that of the iron pnictides . moreover , no signature of folding could be found in our data , confirming that no collinear magnetic ordering exists in bani@xmath0as@xmath0 . because of the intimate relation between the magnetism and superconductivity,@xcite the absence of magnetic ordering might be related to the low-@xmath4 in bani@xmath0as@xmath0 . across the structural transition in bani@xmath0as@xmath0 , the ni - ni distance changes from 2.93 @xmath29 to 2.8 @xmath29 ( or 3.1 @xmath29 ) , corresponding to a lattice distortion as much as @xmath305% in average.@xcite a rough estimation can be made for the hopping parameter @xmath31 between certain @xmath15-@xmath15 orbitals after the lattice distortion according to ref . , @xmath32 where @xmath16 is the relative lattice distortion ; @xmath33 is the hopping parameter before the distortion ; @xmath34 is the induced hopping parameter change . therefore , the 5% lattice distortion would cause @xmath3017.5% of change to @xmath33 . since the measured bandwidth of @xmath10 is at least 200 mev , the induced band shift would be larger than 35 mev , more than enough to account for the measured band shift of 25 mev . note that the differences between the calculated band structures of tetragonal and triclinic phases in fig . [ calc](f ) are solely induced by considering the different lattice parameters . for instance , the band along @xmath23-@xmath5 near @xmath14 has a shift of 25% of the bandwidth , generally consistent with our observation . therefore , our bani@xmath0as@xmath0 data provide a prototypical experimental showcase of band shift due to significant lattice distortion . as a comparison , the lattice distortion in nafeas is 0.36% , which would induce only 1 mev of band shift , much smaller than the observed 16 mev by arpes.@xcite similar results can be found in other iron pnictides.@xcite the minor lattice distortion can not account for the large band shift observed in iron pnictides , therefore it has been concluded that the only promising explanation left is that the band shift is related to the magnetism.@xcite to summarize , we report the first electronic structure study of bani@xmath0as@xmath0 by arpes . in comparison with the band calculation of bani@xmath0as@xmath0 and reports of iron pnictides , we conclude several points as following : 1 . we observe four small fermi pockets around @xmath2 only in the triclinic phase , an electronlike pocket around @xmath3 and two electronlike pockets around @xmath1 in both tetragonal and triclinic phases . the main features of the measured band structure along @xmath2-@xmath1 is qualitatively captured by the band calculations , however differences exist along @xmath2-@xmath3 . the electronic structure of bani@xmath0as@xmath0 is also distinct from the that of iron pnictides . moreover , the correlation effects in bani@xmath0as@xmath0 seems to be weaker than that in iron pnictides , as the band renormalization factor is smaller for bani@xmath0as@xmath0 . 2 . unlike iron pnictides , we do not observe any sign of band folding in bani@xmath0as@xmath0 , confirming no collinear sdw related magnetic ordering . since the magnetism intimately relates to the superconductivity , possibly this is why the @xmath4 is much lower in bani@xmath0as@xmath0 than in iron pnictides . the sdw / structural transition in iron pnictides is second - order - like , while the structural transition in bani@xmath0as@xmath0 is first - order and a thermal hysteresis is observed for its band shift . the band shift in bani@xmath0as@xmath0 is caused by the significant lattice distortion . on the other hand , the band shifts in the iron pnictides can not be accounted for by the minor lattice distortion there , but are related to the magnetic ordering . part of this work was performed at the surface and interface spectroscopy beamline , swiss light source , paul scherrer institute , villigen , switzerland . we thank c. hess and f. dubi for technical support . this work was supported by the nsfc , moe , most ( national basic research program no . 2006cb921300 and 2006cb601002 ) , stcsm of china . l. x. yang , b. p. xie , y. zhang , c. he , q. q. ge , x. f. wang , x. h. chen , m. arita , j. jiang , k. shimada , m. taniguchi , i. vobornik , g. rossi , j. p. hu , d. h. lu , z. x. shen , z. y. lu , and d. l. feng , phys . b * 82 * , 104519 ( 2010 ) . a. j. drew , ch . niedermayer , p. j. baker , f. l. pratt , s. j. blundell , t. lancaster , r. h. liu , g. wu , x. h. chen , i. watanabe , v. k. malik , a. dubroka , m. rssle , k. w. kim , c. baines , and c. bernhard , nat . mater . * 8 * , 310 ( 2009 ) . a. d. christianson , e. a. goremychkin , r. osborn , s. rosenkranz , m. d. lumsden , c. d. malliakas , i. s. todorov , h. claus , d. y. chung , m. g. kanatzidis , r. i. bewley , and t. guidi , nature * 456 * , 930 ( 2008 ) . y. zhang , j. wei , h. w. ou , j. f. zhao , b. zhou , f. chen , m. xu , c. he , g. wu , h. chen , m. arita , k. shimada , h. namatame , m. taniguchi , x. h. chen , and d. l. feng , phys . lett . * 102 * , 127003 ( 2009 ) . l. x. yang , y. zhang , h. w. ou , j. f. zhao , d. w. shen , b. zhou , j. wei , f. chen , m. xu , c. he , y. chen , z. d. wang , x. f. wang , t. wu , g. wu , x. h. chen , m. arita , k. shimada , m. taniguchi , z. y. lu , t. xiang , and d. l. feng , phys . lett . * 102 * , 107002 ( 2009 ) . bo zhou , yan zhang , le - xian yang , min xu , cheng he , fei chen , jia - feng zhao , hong - wei ou , jia wei , bin - ping xie , tao wu , gang wu , masashi arita , kenya shimada , hirofumi namatame , masaki taniguchi , x. h. chen , and d. l. feng , phys . b * 81 * , 155124 ( 2010 ) . liu , h .- y . liu , l. zhao , w .- t . zhang , x .- w . jia , j .- q . meng , x .- dong , j. zhang , g. f. chen , g .- wang , y. zhou , y. zhu , x .- y . wang , z .- y . xu , c .- t . chen , and x. j. zhou , phys . b * 80 * , 134519 ( 2009 ) . y. zhang , b. zhou , f. chen , j. wei , m. xu , l. x. yang , c. fang , w. f. tsai , g. h. cao , z. a. xu , m. arita , h. hayashi , j. jiang , h. iwasawa , c. h. hong , k. shimada , h. namatame , m. taniguchi , j. p. hu , d. l. feng , arxiv:0904.4022 ( unpublished ) . fei chen , bo zhou , yan zhang , jia wei , hong - wei ou , jia - feng zhao , cheng he , qing - qin ge , masashi arita , kenya shimada , hirofumi namatame , masaki taniguchi , zhong - yi lu , jiangping hu , xiao - yu cui , and d. l. feng , phys . b , * 81 * , 014526 ( 2010 ) .

bani@xmath0as@xmath0 , with a first order phase transition around 131 k , is studied by the angle - resolved photoemission spectroscopy . the measured electronic structure is compared to the local density approximation calculations , revealing similar large electronlike bands around @xmath1 and differences along @xmath2-@xmath3 . we further show that the electronic structure of bani@xmath0as@xmath0 is distinct from that of the sibling iron pnictides . particularly , there is no signature of band folding , indicating no collinear sdw related magnetic ordering . moreover , across the strong first order phase transition , the band shift exhibits a hysteresis , which is directly related to the significant lattice distortion in bani@xmath0as@xmath0 .

transitional disks have a small or no excess from @xmath0@xmath1 m to @xmath0@xmath2 m relative to their full disk cousins , but a significant excess at longer wavelength @xcite , suggesting cleared out inner disk . this interpretation dates back to the era of the _ infrared astronomical satellite _ ( @xcite , @xcite ) , and was later developed with the help of detailed near - infrared ( nir ) to mid - infrared ( mir ) spectra provided by the infrared spectrograph ( irs ) on - board _ spitzer space telescope_. detailed radiative transfer modeling suggests that this kind of sed is consistent with disk models which harbor a central ( partially ) depleted region ( i.e. a cavity or a gap ) , while a `` wall - like '' structure at the outer edge of this region can be responsible for the abrupt rise of the sed at mir @xcite . the disk+cavity model based on sed - only fitting usually contains large uncertainties , because the sed samples the emission from the whole disk ; by tuning the ingredients in the fitting , one could fit the irs sed with different models ( see the example of ux tau a , @xcite and ( * ? ? ? * hereafter a11 ) ) . better constraints on the disk structure can be obtained from resolved images of the transitional disks . using the _ submillimeter array _ ( sma ) interferometer @xcite , resolved images of transitional disks at sub - mm wavelength have provided direct detections of these cavities @xcite , and measurements of their properties . recently , a11 observed a sample of 12 nearby transitional disks ( at a typical distance of @xmath0@xmath3 pc ) . combining both the sma results and the sed , they fit detailed disk+cavity model for each object , and the cavity size ( @xmath0@xmath4 au ) is determined with @xmath0@xmath5 uncertainty . they concluded that large grains ( up to @xmath0mm - sized ) inside the cavity are depleted by at least a factor of 10 to 100 ( the `` depletion '' in this work is relative to a `` background '' value extrapolated from the outer disk ) . under the assumption that the surface density of the disk is described by their model , the infrared spectral fitting demands that the small grains ( micron - sized and smaller ) inside the cavity to be heavily depleted by a factor of @xmath0@xmath6 . recently , most objects in this sample have been observed by the subaru high - contrast coronographic imager for adaptive optics ( hiciao ) at nir bands , as part of the the strategic explorations of exoplanets and disks with subaru project , seeds , @xcite . seeds is capable of producing polarized intensity ( pi ) images of disks , which greatly enhances our ability to probe disk structure ( especially at the inner part ) by utilizing the fact that the central source is usually not polarized , so that the stellar residual in pi images is much smaller than in full intensity ( fi ) images @xcite . the seeds results turned out to be a big surprise in many cases the polarized nir images do not show an inner cavity , despite the fact that the inner working angle of the images ( the saturation radius or the coronagraph mask size , @xmath7 , or @xmath0@xmath8 au at the distance to taurus @xmath0@xmath3 pc , see section [ sec : psf ] ) is significantly smaller than the cavity sizes inferred from sub - mm observations . high contrast features such as surface brightness excesses or deficits exist in some systems , but they are localized and do not appear to be central cavities . instead , the image is smooth on large scales , and the azimuthally averaged surface brightness radial profile ( or the profile along the major axis ) increases inward smoothly until @xmath9 , without any abrupt break or jump at the cavity edge ( the slope may change with radius in some systems ) . examples include rox 44 ( m. kuzuhara et al . 2012 , in prep . ) , sr 21 ( k. follette et al . 2012 , in prep . ) , gm aur ( j. hashimoto et al . 2012a , in prep . ) , and sao 206462 @xcite ; see also the sample statistics ( j. hashimoto et al . 2012b , in prep . ) . some objects such as ux tau a also do not show a cavity ( r. tanii et al . 2012 , in prep . ) , however the inner working angle of their seeds images is too close to the cavity size , so the status of the cavity is less certain . we note that lkca 15 also does not exhibit a clear cavity in its pi imagery ( j. wisniewski et al . 2012 , in prep . ) , but does exhibit evidence of the wall of a cavity in its fi imagery @xcite . this apparent inconsistency between observations at different wavelengths reveals something fundamental in the transitional disk structure , as these datasets probe different components of protoplanetary disks . at short wavelengths ( i.e. nir ) where the disk is optically thick , the flux is dominated by the small dust ( micron - sized or so ) at the surface of the disk ( where the stellar photons get absorbed or scattered ) , and is sensitive to the shape of the surface ; at long wavelength ( i.e. sub - mm ) , disks are generally optically thin , so the flux essentially probes the disk surface density in big grains ( mm - sized or so ) , due to their large opacity at these wavelengths @xcite . combining all the three pieces of the puzzle together ( sed , sub - mm observation , and nir imaging ) , we propose a disk model that explains the signatures in all three observations simultaneously : the key point is that the spatial distributions of small and big dust are decoupled inside the cavity . in this model , a well defined cavity ( several tens of au in radius ) with a sharp edge exists only in spatial distribution of the big dust and reproduces the central void in the sub - mm images , while no discontinuity is found for the spatial distribution of the small dust at the cavity edge . inside the cavity , the surface density of the small dust does not increase inwardly as steeply as it does in the outer disk ; instead it is roughly constant or declines closer to the star ( while maintaining an overall smooth profile ) . in this way , the inner region ( sub - au to a few au ) is heavily depleted in small dust , so that the model reproduces the nir flux deficit in the sed ( but still enough small dust surface density to efficiently scatter near - ir radiation ) . modeling results show that the scattered light images for this continuous spatial distribution of the small dust appear smooth as well , with surface brightness steadily increasing inwardly , as seen in many of the seeds observations . the structure of this paper is as follows . in section [ sec : modeling ] we introduce the method that we use for the radiative transfer modeling . in section [ sec : image ] we give the main results on the scattered light images : first a general interpretation of the big picture through a theoretical perspective , followed by the modeling results of various disk+cavity models . we investigate the sub - mm properties of these models in section [ sec : submm ] , and explore the degeneracy in the disk parameter space on their model sed in section [ sec : sed ] . we summarize the direct constraints put by the three observations on this transitional disk sample in section [ sec : discussion ] , as well as the implications of our disk models . our generic solution , which qualitatively explains the signatures in all the three observations , is summarized in section [ sec : summary ] . in this section , we introduce the model setup in our radiative transfer calculations , and the post processing of the raw nir polarized scattered light images which we perform in order to mimic the observations . the purpose of this modeling exercise is to `` translate '' various physical disk models to their corresponding nir polarized scattered light images , sub - mm emission images , and sed , for comparison with observations . we use a modified version of the monte carlo radiative transfer code developed by @xcite , @xcite , and b. whitney et al . 2012 , in prep . ; for the disk structure , we use a11 and @xcite for references . the nir images ( this section ) and sed ( section [ sec : sed ] ) are produced from simulations with @xmath10 photon packets , and for the sub - mm images ( section [ sec : submm ] ) we use @xmath11 photon packets . by varying the random seeds in the monte carlo simulations , we find the noise levels in both the radial profile of the convolved images ( section [ sec : psf ] ) and the sed to be @xmath12 in the range of interest . in our models , we construct an axisymmetric disk ( assumed to be at @xmath0140 pc ) 200 au in radius on a @xmath13 grid in spherical coordinates ( @xmath14 ) , where @xmath15 is in the radial direction and @xmath16 is in the poloidal direction ( @xmath17 is the disk mid - plane ) . we include accretion energy in the disk using the shakura & sunyaev @xmath18 disk prescription @xcite . disk accretion under the accretion rate assumed in our models below ( several @xmath19 yr@xmath20 ) does not have a significant effect on the sed or the images ( for simplicity , accretion energy from the inner gas disk is assumed to be emitted with the stellar spectrum , but see also the treatment in @xcite ) . we model the entire disk with two components : a thick disk with small grains ( @xmath0@xmath21m - sized and smaller , more of less pristine ) , and a thin disk with large ( grown and settled ) grains ( up to @xmath0mm - sized ) . figure [ fig : sigma ] shows the schematic surface density profile for both dust population . the parametrized vertical density profiles for both dust populations are taken to be gaussian ( i.e. @xmath22 , isothermal in the vertical direction @xmath23 ) , with scale heights @xmath24 and @xmath25 being simple power laws @xmath26 ( we use subscripts `` s '' and `` b '' to indicate the small and big dust throughout the paper , while quantities without subscripts `` s '' and `` b '' are for both dust populations ) . following a11 , to qualitatively account for the possibility of settling of big grains , we fix @xmath27 in most cases to simplify the models , unless indicated otherwise . radially the disk is divided into two regions : an outer full disk from a cavity edge @xmath28 to 200 au , and an inner cavity from the dust sublimation radius @xmath29 to @xmath28 ( @xmath29 is determined self - consistently as where the temperature reaches the sublimation temperature @xmath301600 k , @xcite , usually around 0.1@xmath310.2 au ) . at places in the disk where a large surface area of material is directly exposed to starlight , a thin layer of material is superheated , and the local disk `` puffs '' up vertically @xcite . to study this effect at the inner rim ( @xmath29 ) or at the cavity wall , we adopt a treatment similar to a11 . in some models below we manually raise the scale height @xmath32 at @xmath29 or @xmath28 by a certain factor from its `` original '' value , and let the puffed up @xmath32 fall back to the underlying power law profile of @xmath32 within @xmath00.1 au as @xmath33 . we note that these puffed up walls are vertical , which may not be realistic @xcite . for the surface density profile in the outer disk , we assume @xmath34 where @xmath35 is the surface density at the cavity edge ( normalized by the total disk mass ) , @xmath36 is a characteristic scaling length , and the gas - to - dust ratio is fixed at 100 . following a11 , we take @xmath37 of the dust mass to be in large grains at @xmath38 . for the inner disk ( i.e. @xmath39 ) three surface density profiles have been explored : & & _ i(r)=(_cav_cav)e^(r_cav - r)/r_c ( rising _ i(r ) ) , [ eq : sigmai - andrews ] + & & _ i(r)=_cav_cav ( flat _ i(r ) ) , and [ eq : sigmai - flat ] + & & _ i(r)=(_cav_cav ) ( declining _ i(r ) ) , [ eq : sigmai - linearneg ] and their names are based on their behavior when moving inward inside the cavity . we note that equation and together form a single @xmath40 scaling relation for the entire disk ( with different normalization for the inner and outer parts ) , as in a11 . we define the depletion factor of the total dust inside the cavity as @xmath41 where @xmath42 is found by extrapolating @xmath43 from the outer disk , i.e. evaluating equation at @xmath39 ( or equation with @xmath44 ) . in addition , we define @xmath45 and @xmath46 as the cavity depletion factors for the small and big dust respectively as @xmath47 where 0.15 and 0.85 are the mass fractions of the small and big dust in the outer disk . we note that unlike previous models such as a11 , our cavity depletion factors are radius dependent ( a constant @xmath45 or @xmath46 means a uniform depletion at all radii inside the cavity ) . specifically , we define the depletion factor right inside the cavity edge as @xmath48 with the same @xmath49 , different models with different @xmath50 profiles ( equations - ) have similar @xmath50 ( and @xmath51 ) in the outer part of the cavity , but very different @xmath50 ( and @xmath51 ) at the innermost part . lastly , the mass averaged cavity depletion factor @xmath52 is defined as @xmath53 sma observations have placed strong constraints on the spatial distribution of the big dust , while the constraints on the small grains from the sed are less certain , especially beyond @xmath54 au . based on this , we adopt the spatial distribution of big grains in a11 ( i.e. equation , and no big grains inside the cavity ) , and focus on the effect of the distribution of small grains inside the cavity . therefore , the sub - mm properties of our models are similar to those of the models in a11 ( section [ sec : submm ] ) , since large grains dominate the sub - mm emission . we tested models with non - zero depletion for the big grains , and found that they make no significant difference as long as their surface density is below the sma upper limit . from now on we drop the explicit radius dependence indicator @xmath55 from various quantities in most cases for simplicity . for the small grains we try two models : the standard interstellar medium ( ism ) grains ( @xcite , @xmath0micron - sized and smaller ) , and the model that @xcite employed to reproduce the hh 30 nir scattered light images , which are somewhat larger than the ism grains ( maxim size @xmath020 @xmath21 m ) . these grains contain silicate , graphite , and amorphous carbon , and their properties are plotted in figure [ fig : dust ] . the two grain models are similar to each other , and both are similar to the small grains model which a11 used in the outer disk and the cavity grains which a11 used inside the cavity and on the cavity wall . we note that for detailed modeling which aims at fitting specific objects , the model for the small grains needs to be turned for each individual object . for example , the strength and shape of the silicate features indicate different conditions for the small grains in the inner disk ( @xcite , and @xcite , who also pointed out that the silicate features in transitional disks typically show that the grains in the inner disk are dominated by small amorphous silicate grains similar to ism grains ) . however , since we do not aim at fitting specific objects , we avoid tuning the small dust properties and assume ism grains @xcite for the models shown below , to keep our models generalized and simple . for the large grains we try three different models , namely models 1 , 2 , and 3 from @xcite . the properties of these models are plotted in figure [ fig : dust ] . they adopt a power - law size distribution ( i.e. as in @xcite ) with an exponential cutoff at large size , and the maxim size is @xmath01 mm . these grains are made of amorphous carbon and astronomical silicates , with solar abundances of carbon and silicon . these models cover a large parameter space , however we find that they hardly make any difference in the scattered light image and the irs sed , due to their small scale height and their absence inside the cavity . for this reason we fix our big grains as described by model 2 in @xcite ( which is similar to the model of the big grains in a11 ) . we note that small grains have much larger opacity than big grains at nir , and it is the other way around at sub - mm ( figure [ fig : dust ] ) . to obtain realistic images which can be directly compared to seeds observations , the raw nir images of the entire disk+star system from the radiative transfer simulations need to be convolved with the point spread function ( psf ) of the instrument . seeds can obtain both the fi and the pi images for any object , either with or without a coronagraph mask . the observation could be conducted in several different observational modes , including angular differential imaging ( adi , @xcite ) , polarization differential imaging ( pdi , @xcite ) , and spectral differential imaging ( sdi , @xcite ) . for a description of the instrument see @xcite and @xcite . in this work , we produce both the narrow band 880 @xmath21 m images and @xmath56 band nir images . while at 880 @xmath21 m we produce the full intensity images , for the nir scattered light images we focus on the pi images ( produced in the pdi mode , both with and without a coronagraph mask ) . this is because ( 1 ) pdi is the dominate mode for this sample in seeds , and ( 2 ) it is more difficult to interpret fi ( adi ) images since its reduction process partially or completely subtracts azimuthally symmetric structure . other authors had to synthesize and reduce model data in order to test for the existence of features like cavities @xcite or spatially extended emission @xcite . for examples of pdi data reduction and analysis , see @xcite . when observing with a mask , @xmath9 in the pi images is the mask size ( typically @xmath57 in radius ) , and when observing without a mask , @xmath9 is determined by the saturation radius , which typically is @xmath0@xmath58 . to produce an image corresponding to observations made without a mask , we convolve the raw pi image of the entire system with an observed unsaturated hiciao @xmath56-band psf . the resolution of the psf is @xmath0@xmath59 ( @xmath01.2@xmath60 for an 8-m telescope ) and the strehl ratio is @xmath040% @xcite . the integrated flux within a circle of radius @xmath61 is @xmath080% of the total flux ( @xmath090% for a circle of radius @xmath62 ) . we then carve out a circle at the center with @xmath58 in radius to mimic the effect of saturation . we call this product the convolved unmasked pi image . to produce an image corresponding to observations with a mask , we first convolve the part of the raw pi image which is not blocked by the mask with the above psf . we then convolve the central source by an observed pi coronagraph stellar residual map ( the psf under the coronagraph ) , and add this stellar residual to the disk images ( the flux from the inner part of the disk which is blocked by the mask , @xmath020 au at @xmath0140 pc , is added to the star ) . lastly , we carve out a circle @xmath57 in radius from the center from the combined image to indicate the mask . we call this product the convolved masked pi image . we note that the stellar residual is needed to fully reproduce the observations , but in our sample the surface brightness of the stellar residual is generally well below the surface brightness of the disk at the radius of interest , so it does nt affect the properties of the images much . in this study , the disk is assumed to be face - on in order to minimize the effect of the phase function in the scattering , so that we can focus on the effect of the disk structure . this is a good approximation since most objects in this irs / sma / subaru sample have inclinations around @xmath025@xmath63 ( i.e. minor to major axis ratio @xmath00.9 . an observational bias towards face - on objects may exist , since they are better at revealing the cavity ) . additional information about the scattering properties of the dust could be gained from analyzing the detailed azimuthal profile of the scattered light in each individual system , which we defer to the future studies . to calculate the azimuthally averaged surface brightness profiles , we bin the convolved images into a series of annuli @xmath59 in width ( the typical spatial resolution ) , and measure the mean flux within each annulus . with the tools described above , we investigate what kinds of disk structure could simultaneously reproduce the gross properties of all three kinds of observations described in section [ sec : introduction ] . in this section , we first investigate the properties of the scattered light images from a semi - analytical theoretical point of view ( section [ sec : imagetheory ] ) , then we present the model results from the monte carlo simulations ( section [ sec : imageresult ] ) . in a single nir band , when the ( inner ) disk is optically thick ( i.e. not heavily depleted of the small grains ) , the scattering of the starlight can be approximated as happening on a scattering surface @xmath64 where the optical depth between the star and surface is unity ( the single scattering approximation ) . this surface is determined by both the disk scale height ( particularly @xmath65 in the simple vertically - isothermal models ) , and the radial profile of the surface density of the grains . the surface brightness of the scattered light @xmath66 scales with radius as @xcite @xmath67 where @xmath68 is the stellar luminosity at this wavelength , @xmath69 is a geometrical scattering factor , @xmath70 is the polarization coefficient for pi ( @xmath71 for fi ) , and @xmath72 is the grazing angle ( the angle between the impinging stellar radiation and the tangent of the scattering surface ) . we note that both @xmath69 and @xmath70 depend on azimuthal angle , inclination of the disk , and the scattering properties of the specific dust population responsible for scattering at the particular wavelength . however , if the disk is relatively face - on and not too flared , they are nearly position independent , because the scattering angle is nearly a constant throughout the disk and the dust properties of the specific dust population do not change much with radius . the grazing angle is determined by the curvature of the scattering surface . in axisymmetric disks , assuming the surface density and scale height of the small dust to be smooth functions of radius , the grazing angle is also smooth with radius , and two extreme conditions can be constrained as follows : 1 . for disks whose scattering surface is defined by a constant poloidal angle @xmath16 ( such as a constant opening angle disk ) , @xmath73 ( @xcite , where @xmath74 is the radius of the star ) , so the brightness of the scattered light scales with @xmath15 as @xmath75 2 . for flared disks ( but not too flared , @xmath76 ) , the grazing angle can be well approximated as @xmath77 in this case , @xcite explicitly calculated the position of the scattering surface @xmath64 at various radii ( see also @xcite ) , and found @xmath78 where the coefficient @xmath79 a few and is nearly a constant . combined with the fact that @xmath80 ( @xmath81 as typical values in irradiated disks , @xcite ) , we have the intensity of the scattered light scales with radius as @xmath82 although the above calculations are under two extreme conditions ( for complete flat or flared disks ) , and they are based on certain assumptions and the observed images have been smeared out by the instrument psf , the radial profiles ( azimuthally averaged , or along the major axis in inclined systems ) of seeds scattered light images for many objects in this sample lie between equations ( [ eq : ir-3 ] ) and ( [ eq : ir-2 ] ) in the radius range of interest . in order to guide the eye and ease the comparison between modeling results and observations , we use the scaling relation @xmath83 to represent typical observational results , and plot it on top of the radiative transfer results , which will be presented in section [ sec : imageresult ] ( with arbitrary normalization ) . on the other hand , an abrupt jump in the surface density or scale height profile of the small dust in the disk produces a jump in @xmath72 at the corresponding position . the effect of this jump will be explored in section [ sec : imageresult ] . first , we present the simulated @xmath56 band pi images for a face - on transitional disk with a uniformly heavily depleted cavity with @xmath84 as equation ( figure [ fig : image - andrews ] ) , and the associated surface brightness radial profiles ( the thick solid curves in figure [ fig : rp - andrews ] ) . in each figure , the three panels show the raw image , the convolved unmasked image , and the convolved masked image , respectively . this model is motivated by the disk+cavity models in a11 ; we therefore use parameters typical of those models . experiments show that the peculiarities in each individual a11 disk model hardly affect the qualitative properties of the images and their radial profiles , as long as the cavity is large enough ( @xmath85 ) and the disk is relatively face - on . this disk harbors a giant cavity at its center with @xmath86 au ( @xmath0@xmath87 at 140 pc ) . the disk has the same inwardly rising @xmath88 scaling both inside and outside the cavity as equations ( [ eq : sigmao ] ) and ( [ eq : sigmai - andrews ] ) . outside the cavity both dust populations exist , with the big / small ratio as @xmath89 . inside the cavity there is no big dust ( @xmath90 ) , and the small dust is uniformly heavily depleted to @xmath91 ( @xmath92 ) . the surface density profiles for both dust populations can be found in figure [ fig : sigma ] . the disk has total mass of @xmath93 ( gas - to - dust mass ratio 100 ) , @xmath94 at 100 au with @xmath95 , @xmath96 au , and accretion rate @xmath97 yr@xmath20 . the central source is a 3 @xmath98 , @xmath99 , 5750 k g3 pre - main sequence star . we puff up the inner rim and the cavity wall by 100% and 200% , respectively . the most prominent features of this model in both the unmasked and masked pi images are the bright ring at @xmath28 , and the surface brightness deficit inside the ring ( i.e. the cavity ) . correspondingly , the surface brightness profile increases inwardly in the outer disk , peaks around @xmath28 , and then decreases sharply . this is very different from many seeds results ( such as the examples mentioned in section [ sec : introduction ] ) , in which both the bright ring and the inner deficit are absent , and the surface brightness radial profile keeps increasing smoothly all the way from the outer disk to the inner working angle , as illustrated by the scaling relation ( [ eq : seeds ] ) in figure [ fig : rp - andrews ] . this striking difference between models and observations suggests that the small dust can not have such a large depletion at the cavity edge . figure [ fig : rp - andrews ] shows the effect of uniformly filling the cavity with small dust on the radial profile for unmasked images ( left ) and masked images ( right ) , leaving the other model parameters fixed . as @xmath100 gradually increases from @xmath0@xmath101 to 1 , the deviation in the general shape between models and observations decreases . the @xmath102 model corresponds to no depletion for small grains inside the cavity ( i.e. a full small dust disk ) ; this model agrees much better with the scattered light observations , despite a bump around the cavity edge produced by its puffed up wall ( section [ sec : image - puffingup ] ) , although this model fails to reproduce the transitional - disk - like sed ( section [ sec : sed ] ) . we note that this inconsistency between uniformly heavily depleted cavity models and observations is intrinsic and probably can not be solved simply by assigning a high polarization to the dust inside the cavity . the polarization fraction ( pi / fi ) in the convolved disk images of our models ranges from @xmath00.3 to @xmath00.5 , comparable to observations ( for example @xcite ) . even if we artificially increase the polarization fraction inside the cavity by a factor of 10 , by scaling up the cavity surface brightness in the raw pi images ( which results in a ratio pi / fi greater than unity ) while maintaining the outer disk unchanged , the convolved pi images produced by these uniformly heavily depleted cavity models still have a prominent cavity at their centers . we also note that the contrast of the cavity ( the flux deficit inside @xmath28 ) and the strength of its edge ( the brightness of the ring at @xmath28 ) are partially reduced in the convolved images compared with the raw images . this is due both to the convolution of the disk image with the telescope psf , which naturally smooths out any sharp features in the raw images , and to the superimposed seeing halo from the bright innermost disk ( especially for the unmasked images ) . in the rest of section [ sec : image ] we focus on the small grains inside the cavity while keeping the big grains absent , and study the effect of the parameters @xmath84 , @xmath103 , and the puffing up of the inner rim and cavity wall on the scattered light images . the convolved images for three representative models are shown in figure [ fig : image - variation ] , and the radial profiles for all models are shown in figure [ fig : rp - variation ] . each model below is varied from one standard model , which is shown as the top panel in figure [ fig : image - variation ] and represented by the thick solid curve in all panels in figure [ fig : rp - variation ] . this fiducial model has a flat @xmath84 with no discontinuity at the cavity edge for the small dust ( i.e. @xmath104 ) , and no puffed up rim or wall , but otherwise identical parameters to the models above . first we study the effect of three surface density profiles inside the cavity , namely rising ( equation ) , flat ( equation ) , and declining @xmath84 ( equation ) . the surface densities for these models are illustrated in figure [ fig : sigma ] . panel ( a ) in figure [ fig : rp - variation ] shows the effect of varying the surface density on the image radial profiles . except for shifting the entire curve up and down , different @xmath84 produce qualitatively very similar images and radial profiles , and all contain the gross features in many seeds observations ( illustrated by the scaling relation ( [ eq : seeds ] ) ) . this is due to both the fact that smooth surface density and scale height profiles yield a smooth scattering surface , and the effect of the psf . we note that there is some coronagraph edge effect at the inner working angle in the masked images , which is caused by that the part of the disk just outside ( but not inside ) the mask is convolved with the psf . this results in a narrow ring of flux deficit just outside the mask . in general , the flux is trustable beyond about one fwhm of the psf from @xmath9 ( @xmath105 ) ( * ? ? ? * some instrumental effects in observations may also affect the image quality within one fwhm from the mask edge as well ) . we investigate the effect of different @xmath103 with flat @xmath84 ( [ eq : sigmai - flat ] ) . panel ( b ) in figures [ fig : rp - variation ] shows the effect on the radial profile of the convolved images . when deviating from the fiducial model with @xmath104 ( i.e. a continuous small dust disk at @xmath28 ) , a bump around the cavity edge and a surface brightness deficit inside the cavity gradually emerge . we quantify this effect by measuring the relative flux deficit at @xmath106 ( @xmath107 ) as a function of @xmath103 , the small - dust discontinuity at the cavity edge ( subpanel in each plot ) . for each model we calculate the ratio of the flux at @xmath106 , @xmath108 , to @xmath109 , the flux at @xmath110 ( @xmath111 ) , normalized by @xmath112 in the fiducial , undepleted model . for the model with a @xmath113 discontinuity in @xmath114 at @xmath28 ( @xmath115 ) , @xmath112 is @xmath015% lower than in the fiducial model in the convolved unmasked images , and @xmath020% lower in the convolved masked images . the latter is larger because the mask suppresses the halo of the innermost disk , thus the relative flux deficit in the masked image is closer to its intrinsic value , i.e. the deficit in the raw images . in this sense the masked images are better at constraining the discontinuity of the small dust than the unmasked images . the middle row in figure [ fig : image - variation ] shows the model images for @xmath116 ( illustrated by the thin solid curve in figure [ fig : sigma ] ) . the edge of the cavity is quite prominent in the raw image , while it is somewhat smeared out but still visible in the convolved images . for many objects in this irs / sma / subaru cross sample , the seeds images are grossly consistent with a continuous small dust disk at the @xmath28 , while in some cases a small discontinuity may be tolerated . for the purpose of comparison with the observations , we now discuss the detectability of a finite surface density discontinuity at the cavity edge , given the sensitivity and noise level of the seeds data ( the numerical noise level in our simulations is well below the noise level in the observations , see section [ sec : setup ] ) . in typical seeds observations with an integration time of several hundred seconds , the intrinsic poisson noise of the surface brightness radial profile due to finite photon counts is usually a few tenth of one percent at radius of interests , smaller than the error introduced by the instrument and the data reduction process . @xcite estimated the local noise level of the surface brightness to be @xmath010% at @xmath117 for the seeds sao 206462 pi images , which should be an upper limit for the noise level in the azimuthally averaged surface brightness ( more pixels ) and in the inner region of the images ( brighter ) . if this is the typical value for the instrument , then our modeling results indicate that seeds surface brightness measurements should be able to put relatively tight constraints on the surface density discontinuity for the small dust at @xmath28 . for example , seeds should be able to distinguish a disk of small dust continuous at @xmath28 from a disk of small dust with a @xmath113 density drop at @xmath28 . thus , a lower limit on the small dust depletion factor at the cavity edge can be deduced from detailed modeling for each individual object . this lower limit is likely to be higher than the upper limit from sma on the depletion factor of the big dust ( @xmath118 ) , for objects with relatively smooth radial profiles . if this is confirmed for some objects in which the two limits are both well determined , it means that the density distribution of the small dust needs to somehow _ decouple _ from the big dust at the cavity edge . we will come back to this point in section [ sec : discussion ] . lastly , we explore the effects of the puffed up inner rim and the cavity wall on the images , by comparing the fiducial model ( no puffing up anywhere ) to a model with the inner rim puffed up by 100% , and another model with the cavity wall puffed up by 200% . panel ( c ) in figure [ fig : rp - variation ] shows the effects . puffing up the inner rim has little effect on the image , while the puffed up wall produces a bump at the cavity edge , similar to the effect of a gap edge . images for the model with the puffed up wall are shown in figure [ fig : image - variation ] ( bottom row ) , where the wall is prominent in the raw image while been somewhat smeared out but still visible in the convolved images . scattered light images should be able to constrain the wall for individual objects . without digging deeply into this issue , we simply note here that the seeds images for many objects in this sample are consistent with no or only a small puffed up wall . for all models shown in section [ sec : image ] , the disk is optically thin in the vertical direction at 880 @xmath21 m ( @xmath119 , where @xmath120 is the opacity per gram at @xmath121 ) . in this case , the intensity @xmath122 at the surface of the disk at a given radius may be expressed as : @xmath123 where @xmath124 ghz at 880 @xmath21 m , @xmath125 is the planck function at @xmath126 ( the temperature at @xmath23 ) , and @xmath127 is the vertical density distribution ( @xmath128 and @xmath129 depend on @xmath15 as well ) . outside the cavity , the big dust dominates the 880 @xmath21 m emission , since the big dust is much more efficient at emitting at @xmath0880 @xmath21 m than the small dust ( @xmath130 at these wavelengths ) . inside the cavity there is _ no _ big dust by our assumption , so the sub - mm emission comes only from the small dust . the thick curves in figure [ fig : intensity880 ] show the 880 @xmath21 m intensity as a function of radius for the models in section [ sec : image - sigmai ] ( i.e. continuous small dust disks with rising , flat , or declining @xmath84 ) . in other words , this is an 880 @xmath21 m version of the surface brightness radial profile for the `` raw '' image , without being processed by a synthesized beam dimension ( the sma version of the `` psf '' ) . the bottom thin dashed curve is for the uniformly heavily depleted cavity model as in figure [ fig : image - andrews ] ( @xmath91 , rising @xmath84 ) and no big dust inside the cavity , which represents the sub - mm behavior of the models for most systems in a11 . since the disk is roughly isothermal in the vertical direction near the mid - plane where most dust lies @xcite , @xmath126 may be approximated by the mid - plane temperature @xmath131 . the planck function can be approximated as @xmath132 at this wavelength due to @xmath133 ( 880 @xmath21 m @xmath016 k , marginally true in the very outer part of the disk ) . in addition , since the two dust populations have roughly the same mid - plane temperature , but very different opacities ( @xmath134 ) , equation can be simplified to @xmath135 for @xmath38 and @xmath136 for @xmath39 . for our continuous small disk models with a complete cavity for the big dust , the intensity at 880 @xmath21 m drops by @xmath02.5 orders of magnitude when moving from outside ( @xmath137 ) to inside ( @xmath138 ) the cavity edge , due to both the higher opacity of the big dust and the fact that big dust dominates the mass at @xmath38 . inside the cavity , the intensity ( now exclusively from the small dust ) is determined by the factor @xmath139 . for an irradiated disk @xmath131 increases inwardly , typically as @xmath140 @xcite , while @xmath84 in our models could have various radial dependencies ( equations - ) . in the flat @xmath84 models ( equation ) , the intensity inside the cavity roughly scales with @xmath15 as @xmath141 the same as \{@xmath142 } and is 1.5 orders of magnitude lower than @xmath137 in the innermost disk ( around the sublimation radius ) . on the other hand , if there is no depletion of the small dust anywhere inside the cavity ( i.e. the rising @xmath84 with @xmath102 case ) , the intensity at the center ( thick dashed curve ) can exceed @xmath137 by two orders of magnitude . in a11 , for 880 @xmath21 m images of models with no big dust inside the cavity , the residual emission near the disk center roughly traces the quantity @xmath143 . in most cases , a11 found those residuals to be below the noise floor . due to the sensitivity limit , the constraint on @xmath143 inside the cavity is relatively weak ; nevertheless , a11 were able to put an upper limit equivalent to @xmath144 for the mm - sized dust ( with exceptions such as lkca 15 ) . here we use a mock disk model to mimic this constraint . the top thin dashed curve in figure [ fig : intensity880 ] is from a model with uniform depletion factors @xmath145 for _ both _ dust populations inside the cavity ( so the entire disk has the same dust composition everywhere ) . the result shows that , qualitatively , various models with a continuous small dust disk and a complete cavity for the big dust are all formally below this mock sma limit , though a quantitative fitting of the visibility curve is needed to constrain @xmath100 and @xmath103 , in terms of upper limits , on an object by object basis . this may line up with another sed - based constraint on the amount of small dust in the innermost disk , as we will discuss in the section [ sec : sed ] . while the intensity discussion qualitatively demonstrates the sub - mm properties of the disks , figure [ fig:880um ] shows the narrow band images at 880 @xmath21 m for two disk models . the top row is from the model which produces figure [ fig : image - andrews ] ( also the bottom thin dashed curve in figure [ fig : intensity880 ] and the left panel in figure [ fig : sigma ] ) , which is an a11 style model with a uniformly heavily depleted cavity with rising @xmath84 , @xmath91 , and no big dust inside the cavity . the bottom row is from the fiducial model in section [ sec : imageresult ] ( which produces the top row in figure [ fig : image - variation ] , and the thick solid curve in figure [ fig : intensity880 ] and in the left panel in figure [ fig : sigma ] ) , which has a continuous distribution for the small dust with flat @xmath84 and @xmath104 , and no big dust at @xmath39 as well . the panels are the raw images from the radiative transfer simulations ( left ) , images convolved by a gaussian profile with resolution @xmath0@xmath87 ( middle , to mimic the sma observations , a11 ) and @xmath0@xmath58 ( right , to mimic future alma observations ( section [ sec : future ] ) . both models reproduce the characteristic features in the sma images of this transitional disk sample : a bright ring at the cavity edge and a flux deficit inside , agree with the semi - analytical analysis in section [ sec : submm - intensity ] , but _ very different _ nir scattered light images . the intrinsic reason for this apparent inconsistency is , as we discussed above , that big and small dust dominate the sub - mm and nir signals in our models , respectively . thus two disks can have similar images at one of the two wavelengths but very different images at the other , if they share similar spatial distributions for one of the dust populations but not the other . lastly , we comment on the effect of big to small dust ratio , which is fixed in this work as 0.85/0.15 to simplify the model ( see the discussion of depletion of the small dust in the surface layer of protoplanetary disks , @xcite ) . the scattering comes from the disk surface and is determined by the grazing angle , which only weakly depends on the small dust surface density , if it is continuous and smooth ( section [ sec : image - sigmai ] ) . changing the mass fraction of the big dust in the outer disk from 0.85 to 0.95 in our @xmath104 models ( effectively a factor of 3 drop in surface density of the small dust everywhere ) introduces a @xmath020% drop in the surface brightness of the scattered light images , but a factor of 3 drop in the cavity 880 @xmath21 m intensity ( @xmath146 at @xmath39 ) . lastly , we note that since the big - to - small dust sub - mm emission ratio is @xmath147 , the small grains must contain more than 90% of the total dust mass to dominate the sub - mm emission . in this section , we explore the parameter degeneracy in reproducing the transitional - disk - like sed with their distinctive nir - mir dips . sed fitting ( particularly of the irs spectrum ) can only provide constraints on the spatial distribution of the small dust within a few or a few tens au from the center , and it contains strong degeneracy in the parameter space ( a11 ) . below , we show that disk models with different cavity structures can produce roughly the same sed , containing the transitional disk signature , as long as their innermost parts are modestly depleted ( by a factor of @xmath01000 or so ) . except for the specifically mentioned parameters , the other parameters of these models are the same as for the fiducial model in section [ sec : imageresult ] ; in particular there is no big dust inside the cavity . figure [ fig : sed ] shows the sed for four disk models varied based on the fiducial model in section [ sec : imageresult ] . the model for the thick dashed curve has a uniformly heavily depleted cavity with rising @xmath84 , @xmath148 , and no big dust inside the cavity ( illustrated by the thin dashed curve in the left panel of figure [ fig : sigma ] ) . the scale height profile has @xmath149 and @xmath150 at 100 au . the inner rim is puffed up by 100% and the outer wall is puffed up by 200% . the inclination is assumed to be @xmath151 , @xmath152 au , and @xmath153 au . this model is motivated by the a11 disk+cavity structure . the full small dust disk model ( the thin dashed curve ) has otherwise identical properties but @xmath102 ( i.e. completely filled cavity for the small dust ) . the other two smooth small - dust disk models have much more massive inner disks with @xmath104 ( i.e. a continuous small dust disk ) and no puffed up inner rim or cavity wall . the solid curve model has flat @xmath84 ( equation , illustrated by the thick solid curve in the left panel of figure [ fig : sigma ] ) , @xmath154 and @xmath155 at 100 au . the dash - dotted curve model has declining @xmath84 ( equation , illustrated by the dash - dotted curve in the left panel of figure [ fig : sigma ] ) , @xmath156 , and @xmath157 at 100 au . the two smooth small dust disk models with flat or declining @xmath84 produce qualitatively similar sed as the uniformly heavily depleted model ( in particular , roughly diving to the same depth at nir , and coming back to the same level at mir , as the signature of transitional disks ) , despite the fact that they have very different structures inside the cavity . the minor differences in the strength of the silicate feature and the nir flux could be reduced by tuning the small dust model and using a specifically designed scale height profile at the innermost part ( around the sublimation radius or so ) . the main reasons for the similarity are : 1 . the depletion factor ( or the surface density ) at the innermost part ( from @xmath29 to @xmath01 au or so ) . while the two smooth small dust disk models differ by @xmath05 orders of magnitude on the depletion factor ( or the surface density ) at the cavity edge from the uniformly heavily depleted model , the difference is much smaller at the innermost disk , where most of the nir - mir flux is produced . at the innermost disk , the small dust is depleted by @xmath03 orders of magnitude in the flat @xmath84 model , @xmath05 orders of magnitude for the declining @xmath84 model , and @xmath05 orders of magnitude in the uniformly heavily depleted model ( with rising @xmath84 ) . on the other hand , the integrated depletion factor @xmath158 for the small dust is @xmath00.3 for the flat @xmath84 model , @xmath00.2 for the declining @xmath84 , and @xmath010@xmath159 for the uniformly heavily depleted model , more in line with @xmath103 , because most of the mass is at the outer part of the cavity . we note that the total amount of small dust is not as important as its spatial distribution inside the cavity , and the amount of dust in the innermost part , in determining the nir - mir sed . the scale height of small grains @xmath25 at the innermost part . the two smooth small dust disk models are more flared than the uniformly heavily depleted model . while the three have roughly the same scale height outside the cavity , the difference increases inward . at 1 au , @xmath25 for the uniformly heavily depleted model is 1.7@xmath160 that of the flat @xmath84 model and @xmath161 that of the declining @xmath84 model . 3 . the puffed up inner rim . the inner rim scale height is doubled in the heavily depleted model , which increases the nir flux and reduces the mir flux since the puffed up rim receives more stellar radiation and shadows the disk behind it . the puffed up inner rim is removed in the flat or declining @xmath84 models . the surface density ( or the depletion factor ) and the scale height at the innermost part are considerably degenerate in producing the nir to mir flux in the sed ( a11 ) . in general , a disk which has a higher surface density and scale height at the innermost part and a puffed up inner rim intercepts more stellar radiation at small radii , and has more dust exposed at a high temperature , so it produces more nir flux . on the other hand , the shadowing effect cast by the innermost disk on the outer disk causes less mir emission @xcite . in this way , changes in some of these parameters could be largely compensated by the others so that the resulting sed are qualitatively similar . however , in order to reproduce the characteristic transitional disk sed , the value of the depletion factor inside the cavity can not be too high . the increasing surface density at small radii would eventually wipe out the distinctive sed deficit , and the resulting sed evolves to a full - disk - like sed , as illustrated by the full small dust disk model in figure [ fig : sed ] . in our experiments with not too flared @xmath65 ( comparing with the canonical @xmath81 in irradiated disk , @xcite ) , we find an upper limit on the order of @xmath162 for the depletion factor in the innermost part in our smooth disk models . we note that this limit depends on the detailed choices of the disk and cavity geometry , such as @xmath36 and @xmath28 , and the big - to - small - dust ratio in the outer disk . the inner rim is puffed up in a11 to intercept the starlight and to shadow material at larger radii . at a given radius , the scale height @xmath32 of the gas disk scales as @xmath163 where @xmath164 is the orbital frequency and @xmath165 is the isothermal sound speed in the disk @xmath166 where @xmath128 is the ( mid - plane ) temperature and @xmath21 is the average weight of the particles . if the scale heights of the dust and the gas are well coupled ( e.g. for well - mixed - gas - dust models or a constant level of dust settling ) , tripling the rim scale height ( not unusual in a11 ) means an order of magnitude increase in its temperature . due to the sudden change of the radial optical depth from @xmath00 to unity in a narrow transition region directly illuminated by the star , some puffing up may be present , but probably not that significant . in addition , @xcite pointed out that a realistic puffed up rim has a curved edge ( away from the star ) instead of a straight vertical edge due to the dependence of @xmath167 on pressure , which further limits the ability of the rim to shadow the outer disk . based on these reasons , we choose not to have the rim puffed up in our models , though we note that a relatively weak puffing up , as in the uniformly heavily depleted model here , does not make a major difference in the results . similar idea applies to the puffed up wall as well . the typical @xmath65 value assumed in a11 in this sample ( @xmath168 ) is small compared with the canonical values for irradiated disk models ( @xmath81 , @xcite ) . this leads to that the temperature determined by the input scale height ( @xmath169 , equation and ) may increase inwardly too steeply compared with the output mid - plane temperature calculated in the code ( @xmath170 ) . @xmath169 at 100 au in the three models are close to each other due to their similar scale height there , and all agree with @xmath170 ( @xmath030 k ) within @xmath171 . however , at 1 au , while @xmath169 in our smooth small dust disk models is close to @xmath170 ( @xmath0220 k , within @xmath172 ) , the input temperature in the uniformly heavily depleted model appears to be too high by a factor of @xmath03 . first , we review the _ direct _ , model - independent constraints on the disk structure which the three observations the infrared sed , sma sub - mm observations , and seeds nir polarized scattered light imaging put on many transitional disks in this cross sample : 1 . irs reveals a distinctive dip in the spectra around 10 @xmath21 m , which indicates that the small dust ( @xmath0@xmath21m - sized or so ) in the inner part of the disk ( from @xmath29 to several au ) must be moderately depleted . however , due to degeneracy in parameter space , the detailed inner disk structure is model dependent . models with different cavity depletion factors , @xmath84 , and scale heights at the innermost disk could all reproduce the transitional disk signature . the irs spectra are not very sensitive to the distribution of big dust . the sma images show a sub - mm central cavity , which indicates that the big dust ( mm - sized or so , responsible for the sub - mm emission ) is heavily depleted inside the cavity . however , while the observations can effectively constrain the spatial distribution of the big dust outside the cavity , they can place only upper limits on its total amount inside the cavity . sma observations do not place strong constraints on the distribution of the small dust , though a weak upper limit for the amount of small dust inside the cavity may be determined based on the sma noise level . seeds nir polarized scattered light images are smooth on large scales , and have no clear signs of a central cavity . the radial profiles of many images increase inwardly all the way from the outer disk to the inner working angle without sudden jumps or changes of slope , indicating that the scattering surfaces and their shapes are smooth and continuous ( outside @xmath9 ) . on the other hand , scattered light images are not very sensitive to the detailed surface density profiles and the total amount of small dust inside the cavity . the nir images normally do not provide significant constraints on the distribution of the big dust . in this work , we propose a generic disk model which grossly explains all three observations simultaneously . previous models in the literature which assume a full outer disk and a uniformly heavily depleted inner cavity can reproduce ( 1 ) and ( 2 ) , but fail at ( 3 ) , because they also produce a cavity in the scattered light images , which contradicts the new seeds results . through radiative transfer modeling , we find that qualitatively ( 3 ) is consistent with a smooth disk of small dust with little discontinuity in both surface density and scale height profile . table [ tab : models ] summarizes the key points in various models and compares their performances in these three observations . since we focus on generic disk models only which reproduce the gross features in observations , and we do not try to match the details of specific objects , we more or less freeze many nonessential parameters in sections [ sec : image]-[sec : sed ] which do not qualitatively change the big picture for simplicity . the important ingredients include the dust properties ( mostly for the small dust , both the size distribution and the composition ) , @xmath114 , the big - to - small - dust ratio , and @xmath173 ( both the absolute scale and @xmath65 ) . nir scattered light images are able to provide constraints on some of these parameters ( particularly @xmath173 and @xmath65 ) , due to the dependence of the position and the shape of the disk surface on them . these parameters were not well constrained previously using sub - mm observations and sed due to strong degeneracies ( a11 ) . we note that alternative models for explaining the scattered light images exist , but generally they require additional complications . as one example , if the small dust is not depleted in the outer part of the sub - mm cavity , but is heavily depleted inside a radius smaller than @xmath9 ( @xmath174 au ) , then it is possible to fit all the three observations , in which case the _ small dust cavity _ does not reveal itself in the scattered light images due to its small size . however , in this case one needs to explain why different dust populations have different cavity sizes . future scattered light imaging with even smaller @xmath9 may test this hypothesis . as another example , while we achieve a smooth scattering surface by having continuous surface density and scale height profiles for the small dust , it is possible to have the same result with discontinuities in both , but with just the right amount such that the combination of the two yields a scattering surface inside the cavity smoothly joining the outer disk . this may work if the cavity is optically thick ( i.e. not heavily depleted ) , so that a well - defined scattering surface inside the cavity exists . experiments show that for the models in section [ sec : imageresult ] , uniformly depleting the small dust by a factor of @xmath01000 and tripling the scale height inside the cavity would roughly make a smooth scattering surface . however , fine tuning is needed to eliminate the visible edge from a small mismatch in the two profiles . also , the thicker cavity shadows the outer disk , and makes it much dimmer in scattered light ( by about one order of magnitude ) . lastly , without tuning on the scale height and/or surface density in the innermost part , this model produces too much nir - mir flux and too small flux at longer wavelengths in its sed , due to its big scale height at small @xmath15 and the subsequent shadowing effect . there are several important conclusions that can be drawn based on our modeling of the transitional disks at different wavelengths . first , as we discussed in section [ sec : image - deltacav015 ] , for some objects the _ lower _ limit for the depletion of the _ small _ dust at the cavity edge ( as constrained by the scattered light images ) is likely to be above the _ upper _ limit for the _ big _ dust constrained by the sma , based on the modeling results and the noise level in the two instruments . this essentially means that the small dust has to spatially _ decouple _ from the big dust at the cavity edge . this is the first time that this phenomenon has been associated with a uniform sample in a systematic manner . detailed modeling of both images for individual objects is needed , particularly in order to determine how sharp the big dust cavity edge is from the sub - mm observations , to pin down the two limits and check if they are really not overlapping . while we defer this to future work , we note that having the big dust the same surface density as the small dust inside the cavity probably can not reproduce the sub - mm images . experiments show that even with our declining @xmath84 ( [ eq : sigmai - linearneg ] ) , a fixed big / small dust ratio and a continuous surface density for both throughout the disk ( i.e. @xmath175 ) produce a sub - mm cavity with a substantially extended edge , and the central flux deficit disappears in the smeared out image . if this is confirmed , it further leads to two possibilities : ( a ) whatever mechanism responsible for clearing the cavity have different efficiency for the small and big dust , or ( b ) there are other additional mechanisms which differentiate the small and big dust after the cavity clearing process . at the moment it is not clear which one of the two possibilities is more likely , and both need more thorough investigations . second , as we argued in section [ sec : sed ] , in order to reproduce the distinctive nir deficit in the transitional disk sed , an effective `` upper limit '' of @xmath100 at the innermost region is required , which in experiments with our disk parameters is on the order of @xmath162 . this is far from the _ lower limit _ of @xmath100 at the cavity edge ( close to 1 ) , constrained by the scattered light images . together , the two limits indicate that the spatial distribution of the small grains is very different inside and outside the cavity specifically , @xmath114 tends to be flat or even decrease inwardly inside the cavity . in addition , this implies that the gas - to - dust ratio needs to increase inwardly , given that most of these objects have non - trivial accretion rates ( @xmath176yr , a11 ) . for a steady shakura & sunyaev disk , the accretion rate @xmath177 is related to the gas surface density @xmath178 as : @xmath179 where @xmath18 is the shakura - sunyaev viscosity parameter , @xmath164 is the angular velocity of the disk rotation , and @xmath165 is given by equation . at @xmath00.1 au , equation predicts @xmath180@xmath181 g @xmath182 , assuming a temperature @xmath183 k , @xmath184yr , @xmath185 , and @xmath186 as typical t tauri values . this is very different from our upper limit of @xmath1801 g @xmath182 in the innermost disk , obtained assuming a fixed gas - to - dust ratio of 100 ( the flat @xmath84 models in figure [ fig : sigma ] ) . to simultaneously have large @xmath178 but small @xmath187 in the innermost disk , the gas - to - dust ratio needs to increase substantially from the nominal value of 100 ( by a factor of @xmath188 in our models , echos with @xcite ) . this could put constraints on the cavity depletion mechanism or dust growth and settling theory . at the moment , the mechanism(s ) which are responsible for clearing these giant cavities inside transitional disks are not clear ( see summary of the current situation in a11 ) . regarding the applications of our model on this subject , we note two points here . the flat / declining surface density of the small grains inside the cavity in our models is consistent with the grain growth and settling argument , that the small grains in the inner disk are consumed at a faster rate due to higher growth rate there @xcite . also , the so - called dust filtration mechanism seems promising for explaining why small dust but not big dust is present inside the cavity @xcite , since it could effectively trap the big dust at a pressure maximum in the disk but filter through the small dust . particularly , combining the two ( dust growth and dust filtration ) , @xcite proposed a transitional disk formation model from a theoretical point of view to explain the observations , and their predicted spatial distribution of both dust populations in the entire disk is well consistent with the ones here . imaging is a very powerful tool for constraining the structure of protoplanetary disks and the spatial distribution of both the small and big dust , and there are many ongoing efforts aiming at improving our ability to resolve the disks . in the direction of optical - nir imaging , updating existing coronagraph and adaptive optics ( ao ) systems , such as the new coronagraphic extreme adaptive optics ( scexao ) system on subaru ( @xcite , which could raise the strehl ratio to @xmath00.9 ) , are expected to achieve better performance and smaller @xmath9 in the near future . to demonstrates the power of the optimal performances of these next generation instruments in the nir imaging , figure [ fig : nextgenerationpsf ] shows the surface brightness radial profile of several masked @xmath56-band disk images convolved from the _ same _ raw image by _ different _ psf . except the dotted curve , all the other curves are from the model corresponding to the middle row in figure [ fig : image - variation ] , which has a @xmath87 radius cavity , flat @xmath84 , and @xmath116 ( a @xmath189 drop in @xmath114 at @xmath28 ) . we use three psfs : the current hiciao psf in h band , which could be roughly approximated by a diffraction limited core of an 8-m telescope ( resolution @xmath0@xmath59 ) with a strehl ratio of @xmath00.4 plus an extended halo ; mock psf i ( to mimic scexao ) , which is composed of a diffraction limited core of an 8-m telescope with a strehl ratio of 0.9 , and an extended halo similar in shape ( but fainter ) as the current hiciao psf ; model psf ii ( to mimic the next generation thirty - to - forty meter class telescopes ) , which is composed of a diffraction limited core of a 30-m telescope ( resolution @xmath0@xmath190 ) with a strehl ratio of 0.7 , and a similar halo as the previous two . compared with the full small dust disk case , all the convolved images of the @xmath116 model shows a bump at @xmath28 and a relative flux deficit at @xmath39 . however , from the current hiciao psf to model psf i and ii , the contrast level of the cavity becomes higher and higher , and closer and closer to the raw image ( which essentially has an infinite spatial resolution ) . with these next generation instruments , the transition of the spatial distribution of the small dust at the cavity edge will be better revealed . on the other hand , in radio astronomy , the atacama large millimeter array ( alma ) is expected to revolutionize the field , with its much better sensitivity level and exceptional spatial resolution ( @xmath0@xmath58 or better ) . as examples , the right panels in figure [ fig:880um ] show images convolved by a gaussian profile with resolution @xmath0@xmath58 , which mimic the ability of alma and show two prominent improvements over the images under the current sma resolution ( @xmath0@xmath87 , middle panels ) . first , the edge of the cavity is much sharper in the mock alma images . this will make the constraint on the transition of the big dust distribution at the cavity edge much better . second , while the weak emission signal in the bottom model is overwhelmed by the halo of the outer disk in the mock sma image , resulting in that the bottom model is nearly indistinguishable from the top model ( which essentially produces zero cavity sub - mm emission ) , the mock alma image successfully resolves the signal as an independent component from the outer disk , and separates the two models . this weak emission signal traces the spatial distribution of the dust ( both populations ) inside the cavity , which is the key in understanding the transitional disk structure . at this stage , the total number of objects which have been observed by all three survey - scale projects ( using irs / sma / subaru ) is still small . increasing the number in this multi - instrument cross sample will help clear the picture . in addition , future observations which produce high spatial resolution images at other wavelengths , such as uv , optical , or other nir bands ( for example using hst , @xcite , or the future jwst ) , or using interferometer ( such as the astrometric and phase - referencing astronomy project on keck , @xcite , and amber system on very large telescope interferometer , @xcite ) should also be able to provide useful constraints on the disk properties . we summarize this paper by coming back to the question which we raised at the beginning : what kind of disk structure is consistent with and is able to reproduce the characteristic signatures in all three observations of transitional protoplanetary disks : a high contrast cavity in sub - mm images by sma , a nir deficit in sed by spitzer irs , and a smooth radial profile in nir polarized scattered light images by subaru hiciao . we propose one generic solution for this problem , which is feasible but by no means unique . the key points are : 1 . a cavity with a sharp edge in the density distribution of big grains ( up to @xmath0mm - sized ) and with a depletion factor of at least 0.1 - 0.01 inside is needed to reproduce the sma sub - mm images , as pointed out by a11 . 2 . right inside the cavity edge ( @xmath015 - 70 au ) , the surface density for the small dust ( @xmath0micron - sized and smaller ) does not have a big sudden ( downward ) jump ( a small discontinuity may exist ) . the seeds nir scattered light images , which typically detect the disk at @xmath19115 au ( the inner working angle in seeds ) , generally require continuous / smooth profiles for the surface density and scale height of the small dust . the small dust in the innermost region ( i.e. within a few au , on a scale smaller than measured by seeds ) has to be moderately depleted in order to produce the transitional - disk - like sed , assuming the disk is not too flared , but the exact depletion factor is uncertain and model dependent . as we discussed in section [ sec : ourfeature ] , combining all the above points , our model suggests that the spatial distributions of the big and small dust are _ decoupled _ inside the cavity ( particularly at the cavity edge ) . also , our model argues that the surface density of the small dust inside the cavity is flat or decreases with radius , consistent with the predictions in dust growth models . combined with the accretion rate measurement of these objects , it further implies that the gas - to - dust ratio increases inwardly inside the cavity of transitional disks . r.d . thanks sean andrews , nuria calvet , eugene chiang , bruce draine , catherine espaillat , elise furlan , and jim stone for useful conversations and help . this work is partially supported by nsf grant ast 0908269 ( r. d. , z. z. , and r. r. ) , ast 1008440 ( c. g. ) , ast 1009314 ( j. w. ) , ast 1009203 ( j. c. ) , nasa grant nnx22sk53 g ( l. h. ) , and sloan fellowship ( r. r. ) . we thank pascale garaud and doug lin for organizing the international summer institute for modeling in astrophysics ( isima ) at kavli institute for astronomy and astrophysics , beijing , which facilitated the discussion of radiative transfer modeling among r. d. , l. h. , t. m. , and z. z .. we would also like to thank the anonymous referee for suggestions that improved the quality of the draft .

transitional circumstellar disks around young stellar objects have a distinctive infrared deficit around 10 microns in their spectral energy distributions ( sed ) , recently measured by the _ spitzer infrared spectrograph _ ( irs ) , suggesting dust depletion in the inner regions . these disks have been confirmed to have giant central cavities by imaging of the submillimeter ( sub - mm ) continuum emission using the _ submillimeter array _ ( sma ) . however , the polarized near - infrared scattered light images for most objects in a systematic irs / sma cross sample , obtained by hiciao on the subaru telescope , show no evidence for the cavity , in clear contrast with sma and spitzer observations . radiative transfer modeling indicates that many of these scattered light images are consistent with a smooth spatial distribution for micron - sized grains , with little discontinuity in the surface density of the micron - sized grains at the cavity edge . here we present a generic disk model that can simultaneously account for the general features in irs , sma , and subaru observations . particularly , the scattered light images for this model are computed , which agree with the general trend seen in subaru data . decoupling between the spatial distributions of the micron - sized dust and mm - sized dust inside the cavity is suggested by the model , which , if confirmed , necessitates a mechanism , such as dust filtration , for differentiating the small and big dust in the cavity clearing process . our model also suggests an inwardly increasing gas - to - dust - ratio in the inner disk , and different spatial distributions for the small dust inside and outside the cavity , echoing the predictions in grain coagulation and growth models .

at specific molecular sites . the vertical position @xmath0 controls the tip - molecule coupling @xmath1 , while the molecule - substrate coupling is fixed . ( b ) planar approach : by embedding a bottom - up synthesized magnetic molecule into a solid - state device , one can control its energy levels through a gate - voltage @xmath2 . this _ scanning in energy space _ grants access to both regimes of `` real '' ( redox ) charging and `` virtual '' charging ( scattering ) and their nontrivial crossover . ( c ) conductance map showing a range of features in the regime ( center ) , regimes ( far left and right ) as well as the crossover regime . these are analyzed in detail in fig . [ fig:10 ] and fig . [ fig:15 ] . ] both the fundamental and applied studies on transport phenomena in electronic devices of molecular dimensions have bloomed over the past decade@xcite . an interesting aspect of this development is that it has increasingly hybridized the diverse fields of chemistry , nanofabrication and physics with the primary ambition of accessing properties like high spin and large exchange couplings , vibrational modes , large charging energies and long electronic / nuclear spin coherence times , subtle electronic orbital interplay , self - organisation @xcite and chirality @xcite . this is rendered possible by the higher energy scales of the molecular systems a direct consequence of their size and their complex , chemically tailorable , inner structures which have proven to be effective in addressing , for instance , the spin - phonon@xcite , shiba@xcite and kondo physics@xcite , quantum interference effects@xcite and nuclear spin manipulation@xcite . in most of the works , in particular those concerning molecular _ spin _ systems , two complementary approaches have contributed to explore these effects . on the one hand stands transport spectroscopy , which is the major tool of choice in the scanning - tunneling microscopy ( stm ) approach to nanoscale spin systems @xcite , depicted in fig . [ fig:1](a ) . is also dominant in the field of mechanically - controlled break junctions ( mcbj)@xcite to study vibrations @xcite and , less often , spin effects@xcite . on the other hand , transport spectroscopy , originating in the multi - terminal fabrication of quantum dots ( qds , fig . [ fig:1](b))@xcite , is a well - developed tool applied to a broad range of excitations in nanostructures , @xcite including spin.@xcite l l l l * regime * ' '' '' & * section * & * qd community * & * stm / mcbj community * + ' '' '' & sec . [ sec : qd ] & single - electron tunneling ( set ) & resonant tunneling + & & sequential / incoherent tunneling & + & sec . [ sec : stm ] & ( in)elastic co - tunneling ( cot ) & ( in)elastic electron tunneling spectroscopy ( eets / iets ) + & & coherent tunneling & + & & schrieffer - wolff ( transformation ) & appelbaum ( hamiltonian ) + & sec . [ sec : cot ] & & pump - probe ( co)tunneling spectroscopy + crossover & sec . [ sec : crossover ] & cotunneling - assisted single - electron & + & & tunneling ( coset , cast ) & + the key difference between and approaches is the former s reliance on energy - level control _ independent _ of the transport bias , i.e. , true _ gating _ of the molecular levels@xcite , which should be distinguished from the capacitive level shift in stm which is _ caused _ by the bias . in terms of physical processes , this difference corresponds to spectroscopy relying on `` real '' charging of the molecule and transport involving only `` virtual '' charging . in this contribution we discuss a comprehensive picture of transport applicable to a large family of nanoscale objects . this is motivated by the experimental spectrum of a molecular junction depicted in fig . [ fig:1](c ) . such a conductance map is so full of detail that it warrants a systematic joint experimental and theoretical study . in particular , we discuss several effects which are often overlooked despite their importance to electron transport spectroscopy and despite existing experimental @xcite and theoretical works @xcite . for instance , it turns out that `` inelastic '' or `` off - resonant '' transport is _ not _ simply equivalent to the statement that `` resonant processes play no role '' . in fact , we show that generally less than @xmath3 of the parameter regime of applied voltages that nominally qualified as `` off - resonant '' is actually described by the widely used inelastic ( co)tunneling ( or iets ) picture . although in many experiments to date this has not been so apparent , our experimental evidence suggests that this needs consideration . in theoretical considerations , and transport regimes are often taken as complementary . our measurements illustrate how this overlooks an important class of relaxation processes . the breakdown of the picture in the regime presents , in fact , new opportunities for studying the relaxation of molecular spin - excitations which are of importance for applications . interestingly , these resonances are qualitative indicators of a device of high quality , e.g. , for applications involving spin - pumping . we illustrate _ experimentally _ the ambiguities that the sole modeling of conductance curves can run into . for instance , we show that this may lead one to infer quantum states that do not correspond to real excitations , but are simply _ mirages _ of lower lying excitations , including their zeeman splittings . although elaborated here for a spin system , our conclusions apply generally , for example to electronic @xcite and vibrational excitations in nano electro - mechanical systems ( nems ) @xcite . the outline of the paper is as follows : in sec . [ sec : theory ] we review the physical picture of electron tunneling spectroscopy and outline how a given spectrum manifests itself in and transport spectra . in sec . [ sec : crossover ] we discuss how these two spectra continuously transform into each other as the energy levels are varied relative to the bias voltage . with this in hand , we put together a physical picture capturing all discussed effects which will be subsequently applied to describe the experiment in sec . [ sec : experiment ] . in sec . [ sec : experiment ] we follow the reverse path of experimental transport spectroscopy : we reconstruct the excitation spectrum of a high - spin molecular junction based on the feature - rich transport spectra as a function of bias voltage , magnetic field , and gate voltage . starting from the analysis , we use the boundary conditions imposed by the spectrum to resolve a number of ambiguities in the state - assignment . with the full model in hand we highlight two informative transport features : ( i ) _ nonequilibrium _ , i.e. , a pump - probe spectroscopy using the electronic analog of raman transitions and ( ii ) _ mirages _ of resonances that occur well inside the regime . we conclude with an outlook in sec . [ sec : discussion ] . since we aim to bring the insights from various communities together , we summarize in table [ tab : compare ] the different but equivalent terminology used . for clarity reasons we set @xmath4 for the rest of this discussion . with different charge @xmath5 and further quantum numbers denoted by @xmath6 , as sketched in the lower panel . due to the capacitive coupling to a gate electrode these energy differences can be tuned to be ( b ) on - resonance and ( c ) off - resonance with the electrode continuum . ( b ) `` real '' charging : absorption of an electron , reduces the molecule for real , @xmath7 , going from the ground state @xmath8 for charge @xmath5 to an excited state @xmath9 with charge @xmath10 . since this is a one - step process , the rate scales with @xmath11 , the strength of the tunnel coupling . ( c ) `` virtual charging '' : the `` scattering '' of an electron `` off '' or `` through '' the molecule proceeds via any virtual intermediate state , for example , starting from the ground state @xmath8 and ending in a final excited state @xmath9 , @xmath12 . the rate of such a two - step process scales as @xmath13 . in this case charging is considered only `` virtual '' , as no redox reaction takes place : although energy and angular momentum are transferred onto the molecule , the electron number remains fixed to @xmath10 . ] the two prevalent conceptual approaches to transport through molecular electronic devices are characterized by the simple physical distinction , sketched in fig . [ fig:2 ] , between `` real '' charging chemical reduction or oxidation and `` virtual '' charging electrons `` scattering '' between contacts `` through '' a molecular `` bridge '' . theoretically , the distinction rests on whether the physical processes appear in the leading or next - to - leading order in the tunnel coupling strength , @xmath11 , relative to the thermal fluctuation energy @xmath14 . experimentally , this translates into distinct applied voltages under which these processes turn on . these conditions are the primary spectroscopic indicators , allowing the distinction between `` real '' and `` virtual '' transport processes , and take precedence over line shape and lifetime broadening . for reviews on theoretical approaches to molecular transport see refs . . real charging forms the starting point of what we will call the picture of transport ( see table [ tab : compare ] for other nomenclature ) . its energy resolution is limited by the heisenberg lifetime set by the tunnel coupling @xmath15 @xmath16 allowing for sharp transport spectroscopy of weakly coupled systems . this relation has a prominent place in the field of qds which covers artificial structures as `` artificial atoms '' and `` artificial molecules '' with redox spectra @xcite very similar to real atoms @xcite and simple molecules @xcite . resonant transport also plays a role in stm although its energy resolution is often limited by the strong coupling typical of the asymmetric probe - substrate configuration . given sufficient weak coupling / energy resolution , much is gained when the energy - level dependence of these transport spectra , can be mapped out as function of _ gate - voltage_. this dependence allows a detailed model to be extracted involving just a few electronic orbitals @xcite , their coulomb interactions @xcite and their interaction with the most relevant degrees of freedom ( e.g. , isotropic @xcite and anisotropic spins @xcite , quantized vibrations @xcite , and nuclear spins @xcite ) . in particular electronic @xcite , spin - orbit @xcite structure as well as electro - mechanical coupling @xcite of cnts have been very accurately modeled this way . in molecular electronics transport spectroscopy takes a prominent role since imaging of the device is challenging . by moving to molecular - scale gated structures one often compromises real - space imaging . in this paper we highlight the advantages that such structures offer . nevertheless , electrical gates that work simultaneously with a scanning tip @xcite or a mcbj @xcite have been realized , but with rather low gate coupling . notably , mechanical gating @xcite by lifting a single molecule from the substrate has been demonstrated , resulting in stability diagrams where the role of @xmath17 taken over by the tip - height @xmath0 in fig . [ fig:1 ] . a scanning quantum - dot @xcite has also been realized using a single - molecule @xcite . in the transport regime one considers processes of the leading order in the tunnel coupling @xmath11 , cf . ( [ eq : lifetime - set ] ) . although most of this is in principle well - known , we review this approach @xcite since some of its basic consequences for the _ regime _ discussed below are often overlooked . typically , analysis of spectra requires a model hamiltonian @xmath18 that involves at most tens of states in the most complex situations @xcite . its energies @xmath19 are labeled by the charge number @xmath5 and a further quantum numbers ( orbital , spin , vibrational ) collected into an index @xmath6 . crucial for the following discussion is the voltage - dependence of this energy spectrum . we assume it is uniform , i.e. , @xmath20 , independent of further quantum numbers @xmath6 . this can be derived from a capacitive description of the coulomb interactions between system and electrodes referred to as the _ constant interaction model _ @xcite . in this case , @xmath21 where @xmath19 are constants and @xmath22 ( @xmath23 ) is the potential applied at source ( drain ) electrode . here , @xmath24 for @xmath25 , @xmath26,@xmath27 are capacitive parameters of which only two are independent since @xmath28 . in sec . [ sec : break ] we discuss corrections to this often good assumption@xcite . unless stated otherwise , we will set for simplicity @xmath29 , i.e. , the negative shift of the energy levels equals the gate voltage . the bias is applied to the electron source , @xmath30 , and the drain is grounded , @xmath31 , giving @xmath32 and @xmath33 with constant @xmath34 . unless stated otherwise , schematics are drawn assuming @xmath35 , corresponding to symmetric and dominant source - drain capacitances @xmath36 . the hamiltonian for the complete transport situation takes the generic form @xmath37 where @xmath38 is a sum of tunneling hamiltonians that each transfers a single electron across one of the junctions to either metal electrodes . the electrodes , labeled by @xmath39l(left ) , r(right ) , are described by @xmath40 essentially through their densities of states and by their electrochemical potentials @xmath41 and temperature @xmath14 . for the present purposes this level of detail suffices , e.g. , see ref . for details . for a tunneling process involving such a transfer of precisely one electron , one of the electrochemical potentials has to fulfill @xmath42 in order for the electron to be injected into an @xmath5-electron state @xmath6 , resulting in the final @xmath43-electron state @xmath44 . below this threshold the state @xmath45 is `` unstable '' , and decays back to @xmath46 by expelling the electron back into the electrode . the rate for the injection process , @xmath47 , is given by familiar `` golden rule '' expressions and depends on the difference of both sides of eq . ( [ eq : set - res ] ) relative to temperature @xmath14 . when the process `` turns on '' by changing @xmath48 , it gives rise to a peak in the differential conductance , , corresponding to a sharp step in current , of width @xmath14 and height @xmath49 ( in units of @xmath50 ) since we are assuming weak coupling and high temperature . if the total system conserves both the spin and its projection along some axis ( e.g. , the @xmath51-field axis ) , the rate involves a selection - rule - governed prefactor . this prefactor is zero unless the change of the molecular spin and its projection satisfy @xmath52 these conditions reflect the fact that only a single electron is available for transferring spin to the molecule . incidentally , we note that this picture is very useful even beyond the weak couplings and high temperatures assumed here . close to the resonance defined by condition ( [ eq : set - res ] ) the transport still shows a peak which is , however , modified by higher - order corrections . the width of the current step becomes broadened @xmath53 , giving a conductance peak @xmath54 in units of @xmath50 . its energy position may shift on the order of @xmath11 . ) ( shaded ) for two charge states @xmath5 and @xmath43 . the boundary lines ( bold ) , where @xmath55 for @xmath56 , have slopes @xmath57 and @xmath58 , respectively , allowing the capacitive parameters to be determined . the green lines , offset horizontally by @xmath59 , indicate the window of accessibility of the excited state @xmath60 and are defined by @xmath61 . ( b ) similar to figure ( a ) , for three charge states @xmath5 , @xmath43 and @xmath62 . this adds a copy of the bias window of ( a ) that is horizontally offset by the energy @xmath63 [ eq . ( [ eq : u ] ) ] with boundaries @xmath64 for @xmath65 . the excitation lines on the right ( green ) are mirrored horizontally , @xmath66 for @xmath56 , since electron processes relative to @xmath43 have become hole processes . ] it is now clear in which regime of applied voltages the above picture applies . in fig . [ fig : set ] this is sketched in the plane of applied bias ( @xmath48 ) and gate voltage ( @xmath17 ) . here , we call such a ( schematic ) intensity plot also known as `` stability diagram '' or `` coulomb - diamond'' a _ transport spectrum_. the indicated vertical line cuts through this diagram correspond to traces measured in stm or mcbj experiments . applied to the ground states of subsequent charge states labeled by @xmath8 , eq . ( [ eq : set - res ] ) gives the two inequalities @xmath67 these define the shaded bias window in fig . [ fig : set](a ) , delimited by the `` cross '' . here , a single electron entering from the left can exit to the right , resulting in a net directed current . it is now tempting to naively define the _ regime _ as the complement of the grey regime in fig . [ fig : set](a ) , i.e. , by moving across its boundaries by more than @xmath14 or @xmath11 . a key point of our paper is that this simple rationale is not correct already for a small finite bias matching some excitation at energy @xmath68 , indicated by green lines in . only in the linear - response regime@xcite around @xmath69 the regime can be defined as the complement of the resonant regime : @xmath70 in subsequent charge states analogous considerations apply : transitions between charge states @xmath43 and @xmath62 give rise to a shifted `` copy '' of the bias window as shown in fig . [ fig : set](b ) . the shift experimentally directly accessible is denoted by : @xmath71 this includes the charging energy of the molecule , but also the magnitude of orbital energy differences and the magnetic field . and magnetic field @xmath51 one finds @xmath72 due to the opposite spin - filling enforced by the pauli principle . ] the above rules are substantiated by a simple master equation for the stationary - state occupations @xmath73 of the states with energy @xmath19 that can be derived from the outlined model , see , e.g. , ref . . this approach is used in sec . [ sec : qd - exp ] to model part of our experiment . for the @xmath74 resonance regime the stationary - state equation reads ( for notational simplicity we here set @xmath75 ) @xmath76 here , @xmath77 is the matrix of transition rates @xmath78 between states @xmath79 and @xmath80 , and analogously for @xmath81 . for example , one of the equations , @xmath82 describes the balance between the gain in occupation probability due to all transitions @xmath83 , and the leakage @xmath84 from the state @xmath80 . the entries of the diagonal matrices @xmath85 and @xmath86 have negative values @xmath87 and @xmath88 , respectively , such that probability normalization @xmath89 is preserved in eq . ( [ eq : set - master ] ) . in the leading order in @xmath11 , the rate matrix has separate contributions from the left ( @xmath90 ) and right ( @xmath91 ) electrode : @xmath92 . these allow the stationary current to be computed by counting the electrons transferred by tunnel processes through the @xmath93-th junction , @xmath94_{f , i}^{n_f , n_i } p^{n_i}_{i } , \label{eq : set - current}\end{aligned}\ ] ] where stationarity guarantees @xmath95 . we note that , because we are considering only single - electron tunneling processes ( first order in @xmath11 ) , the primed sum is constrained to @xmath96 by charge conservation . we now take the opposite point of view and consider transport entirely due to `` virtual charging '' or `` scattering through '' the molecule . the resulting transport spectroscopy , alternatively called cotunneling ( ) or iets spectroscopy , dates back to lambe and jacklevic @xcite . the discussion of the precise conditions under which the picture applies is postponed to sec . [ sec : crossover ] . throughout we will denote by the label unless stated otherwise _ inelastic _ cotunneling . the attractive feature of relative to spectroscopy is the higher energy resolution as we explain below [ eq . ( [ eq : lifetime - cot ] ) ff . ] . exploiting this in combination with the stm s imaging capability has allowed chemical identification @xcite . this in turn has enabled atomistic modeling of the junction using _ ab - initio _ calculations @xcite , also including strong interaction effects @xcite , giving a detailed picture of transport on the atomic scale @xcite . in recent years , spectroscopy has been also intensively applied to spin systems @xcite in more symmetric @xcite stm configurations . however , it is sometimes not realized that the same spectroscopy also applies to gated molecular junction , and more generally to qds @xcite . in fact , motivated by the enhanced energy resolution , spectroscopy of discrete spin - states was introduced in gate - controlled semiconductor qds @xcite before it was introduced in stm as `` spin - flip '' spectroscopy @xcite , see also @xcite . spectroscopy is also used to study molecular properties other than spin , e.g. , vibrational states @xcite . ( b ) , now indicating the `` equilibrium '' resonance ( green horizontal line ) in which the excitation @xmath97 is reached from the ground state @xmath98 by . this horizontal line always connects to the resonances for @xmath99 and @xmath100 ( green tilted lines ) . ( b ) `` nonequilibrium '' resonances corresponding to the transition @xmath101 . case ( i ) and ( ii ) are discussed in the text . note that there is never a corresponding resonance ( crossed - out red dashed line ) connecting to a non - equilibrium excitation as is the case for `` equilibrium '' lines in ( a ) . ] in the picture , one considers transport due to next - to - leading order processes , i.e. of order @xmath13 in the tunnel rates . this involves elastic ( inelastic ) processes involving two electrons from the electrodes and a zero ( net ) energy transfer of energy . when the maximal energy supplied by the electrons one electron coming in from , say , @xmath90 at high energy @xmath102 , and the other outgoing to @xmath91 at low energy @xmath34 exceeds a discrete energy difference of the molecule , @xmath103 transport may be altered with @xmath48 . importantly , on the right hand side , all @xmath48 and @xmath17 dependences of the energies cancel out [ cf . [ sec : qd - gate ] ] since we assumed that the applied voltages _ uniformly _ shift the excitation spectrum for fixed charge.@xcite the occurrence of such a process depends on whether the initial state @xmath6 is occupied or not by another already active process . it thus depends on whether we are in the `` equilibrium '' or `` nonequilibrium '' regime , both of which are accessible in our experiment in sec . [ sec : experiment ] . the spectroscopy rules require the following separate discussion . [ [ equilibrium - inelastic ] ] `` equilibrium '' inelastic + + + + + + + + + + + + + + + + + + + + + + + + + + + already in the linear transport regime , @xmath104 and @xmath11 ) , ] there is scattering through the molecule in a fixed stable charge state in the form of _ elastic _ @xcite , see table [ tab : compare ] for the varied nomenclature . this gives rise to a small current scaling @xmath105 . with increasing bias @xmath48 , this mechanism yields a nonlinear background current which is , however , featureless . when the voltage provides enough energy to reach the lowest excitation @xmath9 of the @xmath43-electron ground state @xmath8 , the transition @xmath106 is enabled , cf . [ fig:2](c ) . this occurs when the _ gate - voltage independent _ criterion set by eq . ( [ eq : cot - res ] ) with @xmath107 and @xmath108 is satisfied : @xmath109 the above energy condition is the tell - tale sign of an process : as sketched in fig . [ fig : cot](a ) , this allows for a clear - cut distinction from processes with a gate dependent energy condition ( [ eq : set - res ] ) . importantly , such a feature always connects to the gate - dependent resonance corresponding to excitation @xmath110 . as in the regime , we stress that criterion ( [ eq : cot - res2 ] ) uses the peak position in the ( @xmath111 plane as a primary indicator . the line shape along a vertical cut in the figure , as measured in stm , may be less clear . in theoretical modeling the line shape is also not a unique indicator . the line shape is a good secondary indicator of the nature of a process . [ [ nonequilibrium - inelastic - electronic - pump - probe - spectroscopy . ] ] `` nonequilibrium '' inelastic : electronic pump - probe spectroscopy. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the above `` equilibrium '' picture of transport has been successfully applied in many instances . however , as the first excited state @xmath97 is accessed , the rules of the game change . if the relaxation induced by sources other than transport is weak enough @xcite , the occupation of the excited states can become non - negligible . in such a case , as illustrated in fig . [ fig : cot](b ) , a secondary inelastic process from the excited state @xmath9 to an even higher excited state @xmath112 should be considered . such secondary processes , with the generic condition : @xmath113 indicate a device with an intrinsic relaxation rate small compared to rates @xmath114 . as discussed in fig . [ fig : cot](b ) , such excitations _ never _ connect to a corresponding excitation in the transport spectrum . at this point , two cases have to be considered , both of which are relevant to the our experiment in sec . [ sec : pump - probe ] . \(i ) if @xmath115 i.e . , the gaps in the energy spectrum grow with energy an extra _ `` nonequilibrium '' inelastic _ resonance at bias @xmath116 appears , as illustrated in panel ( i ) of fig . [ fig : cot](b ) . this extra resonance is very useful since it provides a further consistency check on the excitations @xmath117 and @xmath118 observed independently in the . is not allowed by a selection rule , the secondary resonance may be the only evidence of this state . ) ] clearly , the intensity of such secondary `` nonequilibrium '' resonances is generally expected to be lower than the primary ones that start from the ground state . in sec . [ sec : pump - probe ] we will experimentally control this sequential `` electronic pump - probe '' excitations by tuning a magnetic field . \(ii ) in the opposite case , @xmath119 , no _ extra _ excitation related to @xmath112 appears : there is no change in the current at the lower voltage @xmath120 because the initial state @xmath97 only becomes occupied at the _ higher _ voltage @xmath117 . this is illustrated in panel ( ii ) of fig . [ fig : cot](b ) . examples of both these cases occur in the spectra of molecular magnets due to the interesting interplay of their easy - axis and transverse anisotropy , see the supplement of ref . . similar to the case , the conditions ( [ eq : cot - res])-([eq : cot - res - neq ] ) are incorporated in a simple stationary master equation for transport whose derivation we discuss further below . in particular , the occupation probabilities @xmath121 in the stationary transport state are determined by ( as previously , we put @xmath75 ) @xmath122 here , @xmath86 is a matrix of rates @xmath123 for transitions between states @xmath124 . since in the regime charging is only `` virtual '' , these transitions now occur for a fixed charge state . the matrix takes the form @xmath125 , including rate matrices @xmath126 for back - scattering from the molecule ( to the same electrode , @xmath127 ) and scattering through it ( between electrodes @xmath128 ) . the current is obtained by counting the net number of electrons transferred from one electrode to the other : @xmath129 the inclusion into this picture of the above discussed `` non - equilibrium '' effects depends whether one solves the master equation ( [ eq : cot ] ) or not . to obtain the simpler description of `` equilibrium '' inelastic one can insert _ by hand _ equilibrium populations @xmath130 directly into eq . ( [ eq : stm - current ] ) . solving , instead , eq . ( [ eq : cot ] ) without further assumptions gives the `` nonequilibrium '' inelastic case @xcite discussed above [ case ( ii ) ] . in practice , these two extreme limits both computable without explicit consideration of intrinsic relaxation are always useful to compare since any more detailed modeling of the intrinsic relaxation will lie somewhere in between. the electron tunneling rates in eq . ( [ eq : cot ] ) are made up entirely of contributions of order @xmath13 . there are two common ways of computing these rates , and we now present the underlying physics relevant for the discussion in sec . [ sec : crossover ] . [ [ appelbaum - schrieffer - wolff - hamiltonian . ] ] appelbaum - schrieffer - wolff hamiltonian. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + a conceptual connection between the `` virtual '' charging picture and the picture of `` real '' charging in sec . [ sec : qd ] emerges naturally when applying the unitary transformation @xcite due to appelbaum @xcite , schrieffer and wolff @xcite ( ) to the transport _ hamiltonian _ @xmath131 [ cf . [ sec : qd ] ] . the effective model obtained in this way allows one to easily see the key features of the spectroscopy . in this approach , the one - electron tunneling processes described by the hamiltonian @xmath15 , are transformed away and the charge state is fixed by hand to a definite integer . with that , also all the gate - voltage dependence of resonance _ positions _ [ eq . ( [ eq : cot - res2 ] ) ff . ] drops out . this new model is obtained by applying a specially chosen unitary transformation @xmath132 to the original hamiltonian such that : @xmath133 the term @xmath134 is effectively replaced by @xmath135 , which involves only @xmath13 processes and represents exclusively scattering of electrons `` off '' and `` through '' the molecule . in many cases of interest @xcite this coupling @xmath135 contains terms describing the potential ( scalar ) and exchange ( spin - spin ) scattering of electrons and holes with amplitudes @xmath136 and @xmath137 , respectively : @xmath138 in the equation above , the operators @xmath139 ( @xmath140 ) describe spin-(in)dependent intra- [ @xmath127 ] and inter - electrode [ @xmath128 ] scattering of electrons , see ref . for details . this scattering is coupled to the molecule through its charge and spin ( @xmath143 ) . _ selection rules . _ the coupling @xmath144 has selection rules that differ from the original single - electron tunnel coupling @xmath38 : @xmath145 these reflect that the two electrons involved in the scattering process have integer spin 0 or 1 available for exchange with the molecule . we will apply this in sec . [ sec : spectro ] . this is illustrated by the example model ( [ eq : ha ] ) where the spin - operator @xmath143 has matrix elements that obey eq . ( [ eq : rules - cot ] ) . _ lifetime . _ after transforming to this new effective picture , scattering becomes the leading order transport mechanism . the `` golden rule '' approach can be then applied analogously to the case of of the regime , but now with respect to the scattering @xmath144 . in this way eq . ( [ eq : cot ] ) is obtained together with an expression for the corresponding rate matrix @xmath86 . the given by eq . ( [ eq : stm - current ] ) shows gate - voltage - independent _ steps _ at energies set by eq . ( [ eq : cot - res ] ) . although at high temperatures these steps get thermally broadened @xcite , at low enough @xmath14 their broadening is smaller than that of the peaks . while calculation of this lineshape requires higher - order contributions to @xmath86 , the relevant energy scale ( inverse lifetime ) is given by the magnitude of the `` golden rule '' rates _ for the effective coupling @xmath144 _ scaling as @xmath146 this results in a much larger lifetime compared to the one from due to the role of the interactions on the molecule suppressing charge fluctuations . the smaller intrinsic broadening is a key advantage of vs. spectroscopy @xcite . _ line shape . _ due to nonequilibrium effects i.e . , the voltage - dependence of the occupations obtained by solving eq . ( [ eq : cot]) a small peak can develop on top of the step @xcite . moreover , processes beyond the leading - order in @xmath144 , which is all the approach accounts for , can have a similar effect . these turn the tunneling step into a peak and are in use for more precise modeling of experiments @xcite . spin - polarization @xcite and spin - orbit effects @xcite , however , also affect the peak shape and asymmetry . at low temperatures and sufficiently strong coupling a nonequilibrium kondo effect develops which has been studied in great detail @xcite . these works show that the peak amplitude is then enhanced nonperturbatively in the tunnel coupling , in particular for low lying excitations . this requires nonequilibrium renormalization group methods beyond the present scope and we refer to various reviews @xcite . in particular , it requires an account of the competition between the kondo effect and the current - induced decoherence @xcite in the ( generalized ) quantum master equation for the nonequilibrium density operator @xcite . from the present point of view of spectroscopy , the kondo effect can be considered as a limit of an inelastic feature at @xmath147 as @xmath148 , see fig . [ fig : cot](a ) . its position is simply @xmath149 at gate voltages sufficiently far between adjacent resonances by criterion ( [ eq : naive ] ) . in particular , for transport spectroscopy of atomic and molecular spin systems the kondo effect and its splitting into features @xcite is very important especially in combination with strong magnetic anisotropy @xcite . we refer to reviews on stm @xcite and qd @xcite studies . [ [ golden-rule-mathcalt-matrix-rates.sectmatrix ] ] `` golden rule '' @xmath150-matrix rates.[sec : tmatrix ] + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + a second way of arriving at the master equation ( [ eq : cot ] ) and the rates in @xmath86 is the so - called @xmath150-matrix approach . in essence , here is regarded as a scattering process : in the `` golden rule '' the next - to - leading order @xmath150-matrix @xcite , @xmath151 is used instead of the coupling @xmath15 , where @xmath152 is the scattering energy . the main shortcoming of this approach is that the @xmath150-matrix rates so obtained are infinite . the precise origin of the divergences was identified in ref . to the neglect of contributions that formally appear in _ first - order in @xmath11 _ but which effectively contribute only in second order to the stationary state @xcite . by taking these contributions consistently into account @xcite , finite effective rates@xcite for the master equation for the probabilities are obtained . in both the and approach these contributions are ignored and , instead , finite expressions for the rates are obtained only after _ ad - hoc _ infinite subtractions @xcite . this regularization `` by hand '' can and in practice does lead to rates _ different _ from the consistently - computed finite rates , see ref . for explicit comparisons . these problems have also been related @xcite to the fact that calculation of stationary transport using a density matrix ( occupations ) is _ not _ a scattering problem although it can be connected to it @xcite in the following sense : the coupling to the electrodes is never adiabatically turned off at large times ( i.e. , there is no _ free _ `` outgoing state '' ) . as we discuss next , such a consistent first _ plus _ second order approach is not only technically crucial but this also leads to additional physical effects that we measure in sec . [ sec : experiment ] . ( a ) , now indicating the regimes where the excited state @xmath97 relaxes by ( blue ) or by ( red ) . ( b ) relaxation mechanisms after excitation by : _ left panel _ : relaxation in two steps via `` real '' occupation of charge state @xmath5 . _ right panel _ : when the process `` set 1 '' is energetically not allowed excitation and relaxation proceeds by using charge state @xmath5 only `` virtually '' . ] having reviewed the two prominent , complementary pictures of transport due to `` real '' and `` virtual '' charging , we now turn to the crossover regime where these two pictures coexist . this has received relatively little attention , but our experiment in sec . [ sec : experiment ] highlights its importance . as we have seen , despite the fact that charging is only `` virtual '' , an energy exchange between molecule and scattering electrons can occur . depending on the energy - level positions , this `` virtual '' tunneling can `` heat '' the molecule so as to switch on `` real '' charging processes even well _ outside _ the regime . however , in contrast to real heating , which leads to smearing of transport features , this nonequilibrium effect actually results in sharp features in the transport as a function of bias voltage . it thus becomes a new tool for _ spectroscopy_. we first consider the simple case of a single excited state at energy @xmath153 for @xmath43 electrons . in fig . [ fig : crossover](a ) we see that the resulting resonance at @xmath147 ( red ) connects to the excited - state resonance @xmath154 ( blue ) , see also fig . [ fig : cot](a ) . the other resonance condition for the excited state , @xmath155 defines the green line dividing the inelastic regime @xmath156 into two regions shaded red and blue . in the one shaded blue , at the point marked with a circle , the excited state created by a process is stable , that is , it can not decay by a single - electron process since @xmath157 . as shown in the right panel of fig . [ fig : crossover](b ) , the relaxation of this stable state can then only proceed by another process via `` virtual '' charging and it is thus slow ( @xmath105 ) . essentially , this means that the molecule is not `` hot '' enough to lift the coulomb blockade of the _ excited _ state . in contrast , in the red shaded area , at the point marked with a star , this stability is lost as @xmath158 . now the relaxation proceeds much faster through a single - electron process ( order @xmath11 ) as sketched in the left panel of fig . [ fig : crossover](b ) . the molecule gets charged for `` real '' ( either @xmath5 or @xmath62 ) and quickly absorbs / emits an electron returning to the stable @xmath43 electron _ ground _ state , where the system idles waiting for the next excitation . notably , this quenching of the excited state takes place far away from the resonant transport regime in terms of the resonance width , i.e. , violating the linear - response criterion ( [ eq : naive ] ) for being `` off - resonance '' . the enhanced relaxation induced by first - order tunneling , occurring when moving from the circle to the star in fig . [ fig : crossover](a ) , leads to a change in current if no other processes ( e.g. , phonons , hyperfine coupling , etc . ) dominate this relaxation channel ( @xmath53 ) . as a results , the presence of such a resonance signals a `` good '' molecular device , i.e. , one in which the intrinsic relaxation is small compared to the `` transport - coupling '' @xmath11 . we refer to this resonance , first pointed out in refs . and studied further @xcite , as cotunneling - assisted or . the resonance has both and character . on the one hand , the geometric construction in fig . [ fig : crossover](a ) and fig . [ fig : mirage ] shows that it stems from _ the same _ excitation as the step at @xmath147 . however , its position @xmath159 has the same strongly gate - voltage dependence as a resonance , in contrast to the original resonance at @xmath147 . yet , the peak requires to appear and its amplitude is relatively weak , whereas the peak is strong and does not require . for this reason , the peak can be seen as a _ `` mirage '' _ of the excitation and a `` mirror image '' of the @xmath99 peak , as constructed in fig . [ fig : mirage](a ) . the resulting mirrored energy conditions can easily be checked in an experiment cf . [ fig:13] and impose constraints on spectroscopic analysis : if shows a resonance as a function of bias _ outside _ the regime , a resonance at the mirrored position _ inside _ the regime should be present . the vertical cut on the right shows a resonance at @xmath147 and its _ mirage _ at some bias @xmath160 . to identify the latter as such , a corresponding resonance must be present at the mirrored gate voltage @xmath161 , as in the vertical cut shown on the left . note that the indicated construction works for nonsymmetric capacitive coupling . for symmetric coupling , one can literally mirror the gate - voltage position relative to @xmath68 on the horizontal axis . ] besides the appearance of mirages , the crossover regime provides further important pieces of spectroscopic information by constraining how and spectra continuously connect as the gate voltage is varied . this is discussed in sec . [ sec : theory - qd ] , [ sec : stm ] and later on in sec . [ sec : qd - exp ] , but we summarize the rules here . first , only `` equilibrium '' transitions the only ones connecting to an resonance as we explained in fig . [ fig : crossover] can exhibit a mirage . second , excited - excited transitions ( i.e. , for the same charge state @xmath43 ) _ never _ connect to a corresponding feature , as we illustrated in panel ( i ) of fig . [ fig : cot](b ) . finally , transitions between excited states with different charge visible in the regime _ never _ connect to a feature as will be illustrated in fig . [ fig:11 ] . these are strict consistency requirements when analyzing the transport spectra in the -crossover regime . we are now in the position to determine the region in which the physical picture of scattering through the molecule of sec . [ sec : stm ] applies . this is illustrated in fig . [ fig : validity ] . the key necessary assumption of the approach often not stated precisely is that all excited states @xmath97 that are _ accessible _ from the ground state @xmath98 must be `` stable '' with respect to _ first - order _ relaxation processes : @xmath162 this is the case if the condition ( [ eq : set - res ] ) additionally holds for the _ excited _ states , i.e. , for @xmath163 in eq . ( [ eq : set - res ] ) : @xmath164 for both @xmath65 . we note that in theoretical considerations , it is easy to lose sight of condition ( [ eq : stableex ] ) when `` writing down '' an hamiltonian model ( or only @xmath150-matrix rates for ) [ sec . [ sec : theory - stm ] ] and assuming the couplings to be fitting _ parameters _ of the theory . ) that `` accessible '' means also accessible via nonequilibrium cascades of transitions ( `` nonequilibrium ' ' ) , but we will not discuss this further complication . ) ] in fig . [ fig : crossover](a ) we already shaded in light blue the region bounded by the first condition ( [ eq : stableex ] ) where the picture applies . in fig . [ fig : validity](a ) we now show that the full restrictions imposed by both `` virtual '' charge states @xmath5 and @xmath62 in ( [ eq : stableex ] ) strongly restrict the validity regime of the approach for states with `` real '' occupations and charge @xmath43 . in fig . [ fig : validity](b ) and its caption we explain that for any individual excitation @xmath165 the picture _ always _ breaks down in the sense that it works only for _ elastic _ , i.e. , for @xmath166 . this amounts to @xmath3 of the _ nominal _ regime . when accounting for several excited states below the threshold @xmath167 , a sizeable fraction of this region must be further excluded . in fig . [ fig : validity](c ) we construct the regime of validity ( blue ) for some example situations . the shape and size of this validity regime ( light blue ) depends on the details of the excitation spectrum . the center panel illustrates that for a harmonic spectrum the picture in fact applies in only @xmath168 of the nominal regime ( i.e. , obtained by taking the complement of the regime ) . the left and right panel in fig . [ fig : validity](c ) show how this changes for anharmonic spectra characteristic of quantum spins with positive and negative magnetic anisotropy , respectively . ( a ) for an excitation @xmath169 . regions where the approach is valid ( fails ) are colored blue ( red ) , as in fig . [ fig : crossover ] . in the light blue region @xmath166 there is only elastic ( dashed black construction lines are not resonances ) , but for @xmath170 inelastic does excite the molecule . however , the approach only applies in the darker blue triangle where both excitation _ and relaxation _ proceed by `` virtual '' charging . this regime is restricted from both sides [ eq . ( [ eq : stableex ] ) ] and shrinks as @xmath68 increases . ( b ) for @xmath171 the picture works only for _ elastic _ , i.e. , for @xmath166 , which amounts to @xmath172 of the _ nominal _ regime . the inset explains the threshold @xmath169 : for fixed gate voltage at the center ( @xmath173 ) the best - case scenario for the picture to work there is no relaxation by as long as the bias satisfies @xmath174 . requiring this to hold at the onset of excitation by , @xmath147 , gives the threshold . ( c ) several excitations from a superharmonic ( left ) , harmonic ( center ) and subharmonic ( right ) spectrum for charge @xmath43 . in the limit of vanishing harmonic energy spacing , the blue region where the picture works approaches @xmath175 of the nominal off - resonant regime . ] in summary , processes _ always _ dominate the relaxation of excitations at energy @xmath176 populated by excitation because they are `` too hot '' : for such excitations there is no `` deep '' or `` far off - resonant '' regime where considerations based on the picture alone are valid . for lower - energy excitations , @xmath177 , there is a triangular - shaped region in which one is still truly `` far off resonance '' and excitations are not quenched . the size of that region varies according to ( [ eq : stableex ] ) and is much smaller than naively expected by extending the linear - response criterion ( [ eq : naive ] ) . although theoretical @xcite and experimental @xcite studies on exist , this point seems to have been often overlooked and is worth emphasizing . experimentally , to be sure that the picture applies to unidentified excitation one must at least have an estimate of the gap @xmath63 and of the level position or , preferably , a map of the dependence of transport on the level position independent of the bias as in gated experiment discussed in sec . [ sec : experiment ] or stm situations allowing for mechanical gating @xcite . due to their hybrid character , mirages do not emerge in a picture of either `` real '' or `` virtual '' charging alone . in particular , processes are omitted when deriving the rates by means of the transformation [ sec . [ sec : stm ] ] , and , for this reason , that picture can not account for these phenomena . instead , a way to capture these effects is to extend eqs . ( [ eq : set - master ] ) and ( [ eq : cot ] ) to a master equation which simultaneously includes transition rates of leading ( @xmath11 ) and next - to - leading order ( @xmath13 ) . this has been done using the @xmath150-matrix approach @xcite , requiring the _ ad - hoc _ regularization by hand mentioned in sec . [ sec : tmatrix ] . a systematic expansion which avoids these problems is , however , well - known @xcite . and we refer to refs . for calculation of the rates . relevant to our experiment in sec . [ sec : experiment ] is that with the computed rates in hand , a stationary master equation needs to be solved to obtain the occupation of the states and from these the current . we stress that _ even when far off - resonance _ where naively speaking `` the charge is fixed '' to , say , @xmath43 a description of the transport requires a model which also includes _ both the @xmath5 and @xmath62 charge states _ , together with their relative excitations . this is essential to correctly account for the relaxation mechanisms that visit these states `` for real '' and not `` virtual . '' and @xmath137 in eq . ( [ eq : ha ] ) , ] the mirages are missed since @xmath144 only accounts for scattering processes . the minimal master equation required for transport thus takes then the form : @xmath178 where as before @xmath75 for simplicity . here the rates for the various processes change whenever one of the energetic conditions ( [ eq : set - res ] ) and ( [ eq : cot - res ] ) is satisfied . examination of the various contributions in the expression of the rate matrices @xcite reveals that the following effects are included : * @xmath77 is a matrix of rates that change when condition ( [ eq : set - res ] ) is met . it also includes @xmath13-corrections that _ shift and broaden _ the resonance . * @xmath86 is matrix of both and rates . the latter change when condition ( [ eq : cot - res ] ) is met . * @xmath179 and @xmath180 are matrices of _ pair - tunneling _ rates , e.g. , @xmath181 for transitions between states differing by _ two _ electrons , @xmath182 . these lead to special resonances discussed in sec . [ sec : break ] . the solution of the full stationary master equation ( [ eq : full ] ) requires some care @xcite due to the fact that it contains both small rates and large rates whose interplay produces the mirages . even though the _ ( first - order ) rates _ are large , they have a small albeit non - negligible effect since , in the stationary situation , the initial states for these transitions may have only small occupations . these occupations , in turn , depend on the competition between _ all _ processes / rates in the stationary limit . this is the principal reason why one can not avoid solving the master equation ( [ eq : full ] ) with both first and second order processes included . to conclude , eq . ( [ eq : full ] ) captures the delicate interplay of ( ) and ( ) processes leading to mirages ( ) . the appearance of such mirages indicates that _ intrinsic _ relaxation rates are smaller than transport rates ( @xmath53 ) . `` nonequilibrium '' is also included in this approach and the appearance of its additional features in our experiment signals a molecular device with even lower intrinsic relaxation rates , i.e. , smaller than the relaxation rates ( @xmath105 ) . -dependence . upon crossing the red line , the ground state changes from singlet to triplet , see main text . ( b - c ) electron - pair tunneling resonance ( red ) for ( b ) repulsive electron interaction @xmath183 and ( c ) effectively attractive interaction @xmath184 . ( d ) transport feature ( red ) due to coherent spin - dynamics on a single , interacting orbital coupled to nearly antiparallel ferromagnets . although it looks like a resonance with anomalous gate and bias dependence it does not correspond to any state on the system . it is instead a sharp amplitude modulation caused by the _ orientation _ of the accumulated nonequilibrium spin relative to the electrode polarization vectors . ] the above account of the basic rules of transport spectroscopy , although extensive , is by no means exhaustive . the key conditions are eq . ( [ eq : set - res ] ) and ( [ eq : cot - res ] ) , determining the _ resonance positions _ as a function of applied voltages . readers interested mostly in the application of these rules to a high - resolution transport experiment can skip the remainder of this section and proceed directly to sec . [ sec : experiment ] . here , we give an overview of a variety of additional effects that bend or break these rules , found in experimental and theoretical studies . in fig . [ fig : break ] we sketch a number of transport spectra that can not be understood from what we have learned in the previous discussion . [ [ nonuniform - level - shifts - due - to - voltages . ] ] nonuniform level shifts due to voltages. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the assumption made so far [ sec . [ sec : qd - gate ] ] that all energy levels are uniformly shifted by applied gate and bias voltages may not be valid in case of local electric field gradients . in fact , this was already seen in the first experiment on gated spectroscopy of a single - triplet semiconductor dot @xcite due to the change of the confining potential with gate voltage . in molecular junctions this has also been observed . figure [ fig : break](a ) schematizes how the transport spectrum in refs . displays such effects . in this case , the resonances can still be identified as weakly gate - dependent resonances , which is not a trivial issue as the experiments in ref . show . however , a qualitatively new and strongly gate - dependent resonance@xcite ( red line ) appears upon ground state change . piecing together all the evidence , it was shown that this effect originates from a change in amplitude of the background , without requiring the introduction of any additional states into the model . these effects are included in eq . ( [ eq : full ] ) , which was shown @xcite to reproduce the experimental data of ref . in detail . [ [ pair - tunneling . ] ] pair tunneling. + + + + + + + + + + + + + + + + in all the schematics so far we left out resonances that are caused by electron _ pair tunneling_. these are described @xcite by the rates @xmath179 and @xmath180 included in the master equation ( [ eq : full ] ) . in fig . [ fig : break](b ) we sketch where these pair - tunneling resonances ( red lines ) are expected to appear : their positions are obtained by taking the _ bias - averaged positions _ of the two _ subsequent resonances . _ this condition follows by requiring the maximal energy of an electron pair in the electrode @xmath93 to match a corresponding molecular energy change . for example , for a single orbital at energy @xmath185 one obtains @xmath186 where @xmath187 is the charging energy . this gives a bias window in which pair tunneling @xmath188 can contribute to transport , @xmath189 provided that the @xmath5 and / or the @xmath62 state is occupied . the effective _ charging energy _ for each electron is _ halved _ since the energy @xmath187 is available for both electrons together in a single process . although small ( comparable with ) its distinct resonance position and shape clearly distinguish the pair - tunneling current from current ref . that dominates in the regime where it occurs . [ [ electron - attraction . ] ] electron attraction. + + + + + + + + + + + + + + + + + + + + + clearly , pair tunneling effects are expected to become important if the effective interaction energy @xmath187 is attractive @xcite . such attraction in fact appears in various systems . in molecular systems this is known as electrochemical `` potential - inversion '' @xcite . in artificial qds a negative @xmath187 have been observed experimentally @xcite in transport spectra of the type sketched in fig . [ fig : break](c ) , see also ref . . interestingly , in this case the ground state has either @xmath5 or @xmath62 electrons and never @xmath43 since starting from @xmath190 the single - electron transition energies @xmath191 and @xmath192 are higher than electron - pair transition energy _ per electron _ @xmath193 . this is also included in the approach ( [ eq : full ] ) , see ref . . [ [ coherence - effects . ] ] `` coherence '' effects. + + + + + + + + + + + + + + + + + + + + + + + finally , we turn to the assumption used in sec . [ sec : theory - qd ] that the molecular state is described by `` classical '' occupation probabilities of the quantum states ( statistical mixture ) . for instance , each degenerate spin multiplet is treated as an `` incoherent '' mixture of different spin projections ( no quantum superpositions of spin - states states ) . equivalently , the spin has no average polarization in the direction transverse to the quantization axis . however , when in contact with , e.g. , spin - polarized electrodes , such polarization does arise already in order @xmath11 . in that case one must generalize eq . ( [ eq : full ] ) to include off - diagonal density - matrix in the energy eigenbasis . , see discussion in sec . [ sec : tmatrix ] . ) ] in physical terms , this means that one must account for the coupled dynamics of charge , spin - vector and higher - rank spin tensors @xcite . in the regime , such effects can lead to a nearly 100% modulation of the transport current @xcite due to quantum interference . this emphasises that @xcite the first order approximation in @xmath11 is not `` incoherent '' or `` classical '' as some of the nomenclature in table [ tab : compare ] seems to imply . similar coherence effects can arise from orbital polarization in qds @xcite and stm configurations @xcite , from an interplay between spin and orbital coherence @xcite , or from charge superpositions of electron pairs . finally , for high - spin systems coherence effects of tensorial character can arise . this leads to the striking effect that in contact with ferromagnets ( vector polarization ) they can produce a magnetic anisotropy ( tensor ) @xcite , see also related work @xcite . an extension of the approach ( [ eq : full ] ) also describes these effects @xcite . the perhaps most striking effect of spin - coherence is depicted in fig . [ fig : break](d ) : resonances can _ split _ for no apparent reason @xcite and wander off deep into the regime @xcite ( red line ) . depending on the junction asymmetry , this feature of coherent nonequilibrium spin dynamics can appear as a pronounced gate - voltage dependent current peak or as a feature close to the linear response regime , mimicking a kondo resonance , see also ref . . in the second part of this paper we present feature - rich experimental transport spectra as a function of gate - voltage and magnetic field . their analysis requires all the spectroscopic rules that we outlined in the first part of the paper . we show how the underlying hamiltonian model can be reconstructed from the transport data , revealing an interesting high - spin quantum system with low intrinsic relaxation . the molecule used to form the junction is a @xmath194 single - molecule magnet ( smm ) with formula [ fe@xmath195(l)2(dpm)6 ] @xmath196 et2o where hdpm is 2,2,6,6- tetramethyl - heptan-3,5-dione . here , h3l is the tripodal ligand 2-hydroxymethyl-2-phenylpropane-1,3-diol , carrying a phenyl ring @xcite . after molecular quantum - dot formation , the device showed interesting isotropic high - spin behavior and the clearest signatures of to date @xcite for any quantum - dot structure . before turning to the measurements and their analysis , we first discuss specific challenges one faces probing spin - systems using either or spectroscopy . isotropic , _ high - spin _ molecules have molecular states labeled by the spin length @xmath197 and spin - projection @xmath198 . to detect them two types of selection rules are frequently used in stm and qd studies . using these we construct the possible spectroscopic and fingerprints that we can expect to measure . spectroscopy using conductance as a function of magnetic field @xmath51 ( `` spin - flip spectroscopy''@xcite ) has been a key tool in both stm and break - junction studies . this approach assumes that `` virtual '' charging processes dominate . these processes involve two electrons for which the selection rules ( [ eq : rules - cot ] ) apply . however , for high - spin molecules considered here , there can be multiple spin - spectrum assignments that fit the same transport spectrum . an indication for this is that in the present experiment some of the spectra are very similar to those of entirely different nanostructures @xcite . to see how this comes about we construct in fig . [ fig : spectro](a)-(c ) the three possible different fingerprints that two spin - multiplets can leave in the transport spectrum based on selection rules ( [ eq : rules - cot ] ) alone . for simplicity , we assume that all processes start from the ground state @xmath199 , i.e. , in the `` equilibrium '' situation discussed in sec . [ sec : cot ] . this figure shows that one can determine only whether the spin value changes by 1 or remains the same upon excitation , but not on the _ absolute _ values of the spin lengths ( unless the ground state has spin zero ) . ) vs. a magnetic field @xmath51 . the right panels show the corresponding transport spectra , i.e. , the resonant bias positions in matching an energy difference ( @xmath200 ) . ( a ) if the spin increases upon excitation , @xmath201 , there is a _ three - fold _ splitting of the transport - spectrum ( blue ) starting at @xmath147 for @xmath202 due to the transitions to the excited multiplet . the ground multiplet gives a line ( green ) starting at @xmath149 and increasing with @xmath51 if @xmath203 . only for @xmath204 this green line is _ missing_. ( b ) if the spin length does not change upon excitation , @xmath205 , the excited multiplet appears in the transport spectrum through a _ double _ line starting at @xmath147 . the ground multiplet gives a line ( green ) starting at @xmath149 and increasing with @xmath51 if @xmath203 . clearly , for @xmath206 the @xmath51-dependent lines are _ missing_. ( c ) if the spin length decreases upon excitation , @xmath207 , the excited multiplet appears in the transport spectrum through a single line ( blue ) starting at @xmath147 , increasing with @xmath51 . since in this case the ground spin @xmath208 is always nonzero , there is an intra - multiplet line ( green ) starting at @xmath149 . ] a second key tool in the study of spin effects is the transport in the regime @xcite . this provides additional constraints that reduce the nonuniqueness in the spin - assignment . in the regime , the linear - transport part is governed by the transition between the two ground - state multiplets with different charge , @xmath190 and @xmath98 , for which selection rules ( [ eq : rules - set ] ) hold . as sketched in fig . [ fig : spin_blockade ] , if linear transport is observed , then the ground - state spin values are necessarily linked by @xmath209 this constraint , used in refs . , restricts the set of level assignments inferred through spectroscopy on each of the _ two _ subsequent charge states , by fixing the relative ground state spins @xmath210 and @xmath211 . their absolute values remain , however , undetermined , unless one of two happens to be zero . arguments based on the presence of the additional spin - multiplets can then be used to motivate a definite assignment of spin values . molecules for which eq . ( [ eq : nospinblock ] ) fails can be identified by a clear experimental signature : the transport is blocked up to a finite bias as explained in fig . [ fig : spin_blockade ] . such _ spin - blockade _ has been well - studied experimentally @xcite and theoretically @xcite and finds application in spin - qubits ( `` pauli - spin blockade '' ) . it has been reported also for a molecular junction @xcite . clearly , when several excited spin multiplets / charge states are involved , both the and spin - spectroscopy become more complex . however , selection rules similar to eq . ( [ eq : nospinblock ] ) also apply to _ excited _ states and thus `` lock '' the two spin spectra together . in addition , the nonequilibrium occupations of the states contribute to further restricts@xcite the set of possible spin - values as we will now illustrate in our experimental spectroscopic analysis . , then transport is suppressed ( red dashed cross ) . transport sets in only when a finite bias makes the lowest _ spin - compatible _ excitation energetically accessible . this can be either and @xmath43 state with @xmath212 ( shown ) or an @xmath5 electron state with @xmath213 ( not shown ) . ] molecular junctions are produced starting from a three - terminal solid - state device @xcite consisting of an oxide - coated metallic local gate electrode with a thin gold nanowire deposited on top . on such a device , a low - concentration solution of molecules ( @xmath214 mm ) is drop - casted . the nanowire is then electromigrated at room temperature and allowed to self - break @xcite so that a clean nanogap is formed , with a width of @xmath215 nm . the solution is evaporated and the electromigrated junctions are cooled down in a dilution fridge ( @xmath216 mk ) equipped with a vector magnet and low - noise electronics . all the measurements are performed in a two - probe scheme either by applying a dc bias @xmath48 and recording the current @xmath217 or by measuring @xmath218 with a standard lock - in ac modulation of the bias . a molecular junction as sketched in fig . [ fig:1](b ) is formed when a molecule physisorbs@xcite on the gold leads , and thus establishes a tunneling - mediated electrical contact . the presence of the molecule in the junction is signaled by large transport gaps @xmath63 exceeding @xmath219 mev and low - bias inelastic fingerprints . numerous molecular systems have been investigated in this configuration . as a side remark , the fact that we do not observe pronounced magnetic anisotropy effects is not unexpected : the formation of a molecular junction may involve surface interactions . in several cases previously studied clear spectroscopic signatures of the `` bare '' molecular structure ( before junction formation ) , such as the magnetic anisotropy @xcite , were observed also in junctions . however , depending on the mechanical and electrical robustness of the molecule , this and other spin - related parameters may undergo quantitative @xcite or qualitative changes @xcite and sometimes offer interesting opportunities for molecular spin control @xcite . image - charge stabilization effects , for example , can lead to entirely new spin structure such as a singlet - triplet pair @xcite on opposite sides of a molecular bridge . we now turn to the analysis of the feature - rich transport spectrum anticipated in fig . [ fig:1](c ) and reproduced in fig . [ fig:10 ] . it consists of two regimes on the left and right with fixed charge states provisionally labeled @xmath5 and @xmath43 and a regime in the center surrounded by a significant crossover regime . we first separately identify the electronic spectrum for each of the two accessible charge states @xmath5 and @xmath43 using the approach discussed in sec . [ sec : stm ] . in fig . [ fig:10 ] ( a ) we show the color map and the corresponding steps for fixed @xmath220 v as a function of magnetic field , @xmath51 . two steps ( peaks in ) starting from @xmath221 mev and @xmath222 mev at @xmath223 t shift upward in energy and parallel to each other as the magnetic field increases . in the standard picture each step signal to the opening of an inelastic transport channel through the molecule . transport takes place via virtual charging involving a real spin - flip excitation with selection rules on spin - length @xmath224 and magnetization @xmath225 . the charge of the molecule remains fixed , and is labeled @xmath43 . the shift in magnetic field of both steps indicates a nonzero spin ground state multiplet with spin @xmath226 . according to sec . [ sec : spectro ] , the presence of only one other finite - bias excitation shifting in magnetic field relates the spin - values as @xmath227 , but leaves their absolute values undetermined . as we will see later , other spectroscopic information constrains the ground spin to be a triplet @xmath14 , @xmath228 , with a singlet excited state labeled @xmath197 . from the excitation voltage , a ferromagnetic ( fm ) interaction energy @xmath229 mev can be extracted . such type of excitation has been seen in other molecular structures @xcite . spectra of this kind have also been obtained earlier in other quantum - dot heterostructures , such as few - electron single and double quantum dots , albeit typically characterized by smaller and antiferromagnetic couplings @xcite . we now change the gate voltage to more negative values so that the molecule is oxidized @xmath230 , i.e. , we extract exactly one electron from the molecule . this can be inferred from the transport regime that we traverse along the way . in this new charge state we perform an independent spectroscopy . in fig . [ fig:10](b ) we show the for @xmath231 v as a function of the magnetic field @xmath51 with corresponding line cuts . at @xmath223 t two sets of peaks in appear at @xmath232 mev and @xmath233 mev and split each in three peaks at higher magnetic fields . a weak excitation shifting upwards in @xmath51 from @xmath234 v is also present . with the help of fig . [ fig : spectro](a ) the weak excitation and the first set of peaks are associated to @xmath235 , while the second set , corresponding instead to the spectrum depicted in , fixes the spin to @xmath236 . the crucial information provided by the clear absence of spin blockade in the intermediate regime eventually constrains @xmath210 to @xmath237 or @xmath238 according to ( [ eq : nospinblock ] ) . the only two spin configurations compatible with the observations are therefore : a ground doublet @xmath239 , an excited quartet @xmath240 and a second doublet @xmath241 or , alternatively , a ground quartet , an excited sextuplet and a quartet . as we will see in the next section , the latter can be rigorously ruled out by analyzing the spectrum . the presence of the excited quartet state @xmath240 implies that the charge state @xmath5 is a _ three - spin _ system , @xmath242 , as sketched in the top panel of fig . [ fig:10](b ) . the system with one extra electron in fig . [ fig:10](a ) is thus actually a @xmath243 electron system with one closed shell , as sketched in the figure . upon extraction of an electron , the spectrum of the molecular device changes drastically , transforming from a ferromagnetic _ high - low _ spin spectrum for @xmath243 into a nonmonotonic _ low - high - low _ spin excitation sequence for @xmath242 . the spin - excitation energies extracted from the two independent analyses are : @xmath244 and @xmath245 these energy differences provide the starting point of a more atomistic modeling of the magnetic exchanges in the two charge states . we stress that for the transport spectroscopy this is not necessary and it goes beyond the present scope . we only note that while the @xmath243 state requires only one fixed ferromagnetic exchange coupling @xmath246 [ fig . [ fig:10](a ) ] together with the assumption that two other electrons occupy a closed shell ; the @xmath242 spectrum requires , in the most general case , three distinct exchange couplings between the three magnetic centers [ fig . [ fig:10](b ) ] . these relate to the two available energy differences through @xmath247 and @xmath248 to a complicated function @xmath249 . since this involves three unknowns for two splittings , only microscopic symmetry considerations or detailed consideration of the transport current magnitude are needed to uniquely determine the microscopic spin structure . this type of microscopic modeling has proven successful in many instances , see ref . and references therein . however , the underlying assumptions on localized spins and fixed charge occupations can only be made when sufficiently far away from resonance , i.e. , such that does not take place as expressed by conditions ( [ eq : stableex_rates ] ) and ( [ eq : stableex ] ) . using the ability to control the energy levels with the gate , the analysis can be complemented by a spectroscopy in the central part of fig . [ fig:10](c ) . here , `` real '' charging processes dominate . for example , starting from the ground state @xmath239 , addition of a single electron leads to occupation of the @xmath14 ground state . this is evidenced by the clear presence of a regime of transport down to the linear - response limit . inside the regime additional lines parallel to the edges of the cross appear as well . as we explained in fig . [ fig : set ] , these correspond to `` real '' charging processes where excess ( deficit ) energy is used to excite ( relax ) the molecule . these additional lines , schematized for our experiment in fig . [ fig:11](a ) , fall into two categories according to the criteria : 1 . lines terminating at the boundary of the regime correspond to the _ ground _ @xmath5 to excited @xmath250 transitions or _ vice versa_. 2 . lines that never reach the boundary , but terminate inside the regime at a line parallel to this boundary . these correspond to _ excited _ @xmath5 to excited @xmath250 transitions . their earlier termination indicates that that the initial excited state must become first occupied through another process . the line _ at which _ it terminates corresponds to the onset of this `` activating '' process . . transitions between a _ ground and excited _ state ( blue , red ) reach the boundary of the regime at the black circle from where they continue horizontally as a excitation . the inset depicts the chemical potential configuration at such a black circle where and connect . the transitions between _ two excited _ states ( orange , green ) do not connect to any excitation . ( b ) transport spectrum computed using the master equations ( [ eq : set - master])-([eq : set - current ] ) . the energies are extracted independently from the two spectra in fig . [ fig:10 ] and the capacitive parameters @xmath251 , @xmath252 @xmath253 are fixed by the observed slopes of the lines [ cf . [ fig : set](a ) ] , leaving the tunnel rates ( [ eq : rates_t])-([eq : rates_s ] ) as adjustable parameters . the broadening of the peaks in the experiment is due to tunneling , @xmath254 ( fwhm ) , rather than temperature , @xmath255 . ( [ eq : set - master])-([eq : set - current ] ) do not include this @xmath11-broadening and we crudely simulate it by an effective higher temperature @xmath256 . the master equations ( [ eq : set - master])-([eq : set - current ] ) are valid for small effective tunnel coupling @xmath257 , which only sets the overall scale of plotted current and not the relative intensities of interest . the caption to fig . [ fig:15 ] explains that @xmath258 should not be adjusted to match the larger experimental current magnitude . ] in fig . [ fig:10](c ) and fig . [ fig:11](a ) the transitions @xmath259 , @xmath260 and @xmath261 fall into category ( a ) , while the @xmath262 and @xmath263 transitions belong to ( b ) . due to the large difference in spin - length values of the spin - spectra the latter transition , marked in dashed - green , is actually forbidden by the selection rules ( [ eq : rules - set ] ) . following this line , we find that it terminates at a strong negative differential conductance ( ndc ) feature ( white in the stability diagram in fig . [ fig:10 ] ) marking the onset of the transition @xmath259 . to test our earlier level assignment based , we now compute the expected transport spectrum the first - order ( @xmath11 ) master equations ( [ eq : set - master])-([eq : set - current ] ) and by adjusting the result , we extract quantitative information about the tunnel coupling . the model hamiltonian is constructed from the energies ( [ eq : energies_3])-([eq : energies_4 ] ) and their observed spin - degeneracies . assuming that spin is conserved in the tunneling , the rates between magnetic sublevels are fixed by clebsch - gordan spin - coupling coefficients @xcite incorporating both the and selection rules eqs . ( [ eq : rules - set ] ) and ( [ eq : rules - cot ] ) . the tunnel parameters in units of an overall scale @xmath258 are adjusted to fit the relative experimental intensities : @xmath264 and @xmath265 their relative magnitudes provide further input the further microscopic modeling of the 3 - 4 spin system mentioned at the end of sec . [ sec : stm - exp ] . as shown in fig . [ fig:11](b ) , the _ ( ) part _ of the experimental conductance in fig . [ fig:10](c ) , as schematized in fig . [ fig:11](a ) is reproduced in detail . this includes transitions exciting the molecule from its ground states , but also a transition between excited states.@xcite the ndc effect is explained in more detail later on together with the full calculation in fig . [ fig:14 ] . as discussed in fig . [ fig : cot]-[fig : crossover ] and indicated in fig . [ fig:11](a ) the excitations corresponding to the ground @xmath5 to excited @xmath250 transitions connect continuously to the excitations . those corresponding to two excited states , each of a different charge state , has no corresponding excitation to connect to . in this sense , the spectrum effectively ties the two separately - obtained spin spectra and allows a consistency check on their respective level assignments , cf . [ sec : connecting ] . for instance , from the fact that the @xmath260 transition is clearly visible marked red in fig . [ fig:10](c) we conclude that the first excited multiplet of the @xmath5 charge state _ can not _ be a sextuplet ( @xmath266 ) since such transition would be spin - forbidden and thus weak . another example is given by the presence of the @xmath267 transition [ orange in fig . [ fig:10](c ) ] , which implies that the second excited multiplet of the @xmath5 charge state be a quartet . the fact that this transition does not continue into any of the ones is also consistent with its excited - to - excited character . these two exclusions considerations were anticipated in sec . [ sec : stm - exp ] and are crucial for our assignment in the three - electron state and has now allowed us to reverse - engineer the effective many - electron molecular hamiltonian . with this in hand , we turn to the main experimental findings and investigate the `` nonequilibrium '' through the molecule [ sec . [ sec : pump - probe ] ] and the crossover regime where `` real '' and `` virtual '' tunneling nontrivially compete in the relaxation of spin excitations [ sec . [ sec : coset ] ] . , we highlight here the transitions that are involved in the spin pumping process ( green dotted lines ) . ( a ) spectra measured as a function of @xmath51-field at @xmath268 v in the @xmath243 charge state . the @xmath269 nonequilibrium spin - excitation shows up as a weak , field - independent step vanishing at higher field . for @xmath270 the intra - triplet transition ( red arrow ) requires lower energy than the `` nonequilibrium '' @xmath269 transition . for @xmath271 , the intra - triplet is unlocked ( activated ) at an energy higher than the @xmath269 and only one transition of the cascade is visible . ( b ) spectra measured as a function of @xmath51-field at @xmath272 v in the @xmath242 charge state . here the nonequilibrium excitation has a negative slope . for @xmath270 the excited state of the ground - state doublet @xmath239 is populated enough to promote a second , nonequilibrium excitation to the excited doublet @xmath241 ( green dotted line ) . as @xmath271 the @xmath273 transition crosses over , lowering , in consequence , the population of the spin - up state . this results into a quench of the nonequilibrium excitation . due to the proximity to regime as compared to fig . [ fig:10](a ) , a feature ( orange dotted line ) appears as a mirage of a spin - excitation . ] we first investigate how spectrum evolves as we further _ approach _ the regime from either side . [ fig:12](a ) shows the analogous of fig . [ fig:10](a ) but closer to the regime , at @xmath268 v. a horizontal , @xmath51-field independent line appears ( dotted green line in the center - panel schematic ) that terminates at @xmath274 t , precisely upon crossing the intra - triplet excitation ( blue line ) . this indicates that the excited triplet ( spin @xmath275 perpendicular to the field , @xmath276 ) lives long enough for a secondary process to excite the system to the singlet state ( reducing the spin length to @xmath277 ) . strong evidence for this is the termination of this line : once the initial state ( @xmath276 excited triplet ) for this transition is no longer accessible for @xmath271 , the `` nonequilibrium '' cascade of transitions is interrupted . we consistently observe this effect , also when approaching the regime from the side of the other charge state ( @xmath242 ) with different spin . in fig . [ fig:12](b ) we show the magnetic field spectrum taken at @xmath278 v. here the lowest @xmath241 excitation gains strength@xcite relative to fig . [ fig:10](b ) . in this case , the excited @xmath239 state is the starting point of a `` nonequilibrium '' cascade . as for the previous case , it terminates when levels cross at @xmath279 t for similar reasons : once the @xmath240 state gains occupation for @xmath271 ( since the @xmath273 transition becomes energetically more favorable ) the excited @xmath198-substates of the @xmath239 multiplet are depleted causing the line to terminate . in both charge states , the observed current gives an estimate for the spin - relaxation time , @xmath280 s. nonequilibrium transitions can thus give rise to clear excitations at _ lower _ energy than expected from the simple selection - rule plus equilibrium arguments of sec . [ sec : spectro ] . in this type of processes , two event ( @xmath13 ) happen in sequence , so that a total of four electrons are involved.@xcite in this sense , the phenomena can be regarded as a single - molecule _ electronic pump - probe _ experiment , that is , the excess energy left behind by the first process ( pump ) allows the second process to reach states ( probe ) that would be otherwise inaccessible at the considered bias voltage . this has been successfully applied in stm studies @xcite for dynamical spin - control . [ [ mirages . ] ] mirages. + + + + + + + + + we now further reduce the distance to resonance , again coming from either side , and _ enter _ the crossover regime discussed in sec . [ sec : crossover ] . we are , however , still `` well away from resonance '' by the linear - response condition ( [ eq : naive ] ) . in the upper panel of fig . [ fig:13](a ) we show traces taken at various magnetic fields for a constant gate voltage . at high bias voltage the steeply rises due to the onset of the main resonance . below this onset , we note a step - like excitation at @xmath281 mev ( black arrow ) which shifts up in magnetic field with the same @xmath8-factor ( @xmath282 ) as the other lower - lying excitations.@xcite if one adopts the picture this excitation is attributed to the opening of an independent `` channel '' . this attribution proves to be erroneous : keeping @xmath283 t fixed and varying the gate voltage ( fig . [ fig:13](a ) , lower panel ) , we observe that the lower excitations are left unchanged , whereas the higher one under consideration _ shifts linearly _ with @xmath17 , revealing that it is _ not _ a excitation . this attribution to can be further ruled out by looking at the full gate - voltage dependence in the stability diagram shown in the left panel of fig . [ fig:13](b ) . the excitation ( red arrow ) has the same gate dependence as the resonances , even though it is definitely not in the regime by the linear - response criterion ( [ eq : naive ] ) . in fact , it is a mirage of the _ same _ lowest gate - voltage independent excitation as we explained in fig . [ fig : mirage ] . its bias ( energy ) position does not provide information about the excitation energy @xmath68 : depending on the energy level position the mirage s excitation voltage @xmath159 can lie anywhere above the threshold voltage @xmath147 , see sec . [ sec : crossover ] . in the stability diagram in the right panel of fig . [ fig:13](b ) , we connect by dashed lines all the resonances to their corresponding excitations according to the scheme in fig . [ fig : mirage ] . we find that mirages appear for virtually all spin - related excitations of the molecule . the stability diagram in fig . [ fig:13](c ) [ same color coding as in ( b ) ] shows that at high magnetic field @xmath284 t these mirages persist . the clearly visible resonances mark the lines where the relaxation mechanism changes from `` virtual '' ( ) to `` real '' ( ) charging . they indicate that any intrinsic relaxation is comparable or slower than . mirages are thus a signature of slow intramolecular relaxation , in particular they indicate that the intrinsic relaxation time is bounded from below by the magnitude of the observed currents @xmath285 s , consistent with the sharper lower bound we obtained above from nonequilibrium spectroscopy . [ [ spin - relaxation . ] ] spin relaxation. + + + + + + + + + + + + + + + + + to shed light on what the relaxation mechanism by transport entails in our device , we return to the stability diagram for @xmath202 t , which is shown as in the right panel of fig . [ fig:13](d ) . highlighted at negative bias are the two crossover - regime bands within which , rather than , dominates the relaxation . the left panel shows the different relaxation paths for these two bands . focusing on the orange band , we start out on the far left of fig . [ fig:13](d ) moving at fixed bias @xmath286 mev along the onset of inelastic . fig . [ fig:13](e ) depicts the corresponding energies ( left ) and energy differences ( right ) . here , the molecule is in the spin - doublet @xmath239 ground state and is occasionally excited to the high - spin quartet @xmath240 by from where it relaxes via path 1 ( @xmath287 s ) , again by . when reaching the circle ( @xmath173 ) in fig . [ fig:13](d ) the _ relaxation _ mechanism changes : path 1 is overridden by the faster relaxation path 2 ( @xmath288 s ) which becomes energetically allowed [ eq . ( [ eq : stableex ] ) ] . the top panel of fig . [ fig:13](e ) illustrates that although the ground state @xmath239 is off - resonant ( highlighted in red ) , after exciting it by to @xmath240 increasing the spin - length the system has enough _ spin - exchange _ energy ( green ) to expel a single electron in a `` _ _ real _ _ '' tunneling processes leaving a _ charged _ triplet state behind . at the star ( @xmath289 ) in fig . [ fig:13](d ) the _ excitation _ mechanism changes from to , leaving the relaxation path unaltered . now the ground state @xmath239 becomes unstable with respect to `` real '' charging : there is enough energy to expel an electron to the right electrode and sequentially accept another one from the left . we thus have an transport cycle , i.e. , the stationary state is a statistical mixture of the @xmath5 and @xmath43 ground states . ) and its corresponding current formula ( not shown , see refs . ) for the same parameters as in fig . [ fig:11](b ) . ( a ) transport spectrum for @xmath202 corresponding to fig . [ fig:10](c ) . ( b ) corresponding color plots of the occupation probabilities of the five spin multiplets ( probabilities summed of degenerate levels ) . the effective coupling @xmath258 merely an overall scale factor in fig . [ fig:11](b) now controls the magnitude of the and current corrections _ relative _ to the current . although elaborate , these corrections still neglect nonperturbative broadening effects and must kept small for consistency by explicitly setting @xmath290 . more advanced master equation approaches based on renormalization - group@xcite ( rg ) or hierarchical@xcite ( hqme ) methods can deal with both this broadening and the corresponding larger currents . ] the regime is delimited by mirage resonances and situated between the two positions @xmath173 and @xmath289 . failure to identify the difference between this `` band '' and the pure happening on the left of @xmath173 , besides yielding a wrong qualitative spin multiplet structure , leads to an overestimation of the relaxation time : in the regime the spin - excitations created by inelastic are _ quenched_. [ [ quenching - of - spin - excitations . ] ] quenching of spin - excitations. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we now assess this quenching in detail for the experimental situation by a calculation based on the master equation @xcite ( [ eq : full ] ) that includes all @xmath11 and @xmath13 processes using the model determined earlier [ eqs . ( [ eq : energies_3])-([eq : rates_s ] ) ] , simulating the broadening as before by an effective temperature [ fig . [ fig:11 ] ] . the computed conductance for @xmath202 is shown in fig . [ fig:14](a ) . besides the excitations including the ndc effect obtained earlier in fig . [ fig:11](c ) , we capture the main features of the experimental data in fig . [ fig:11](c ) and fig . [ fig:13](d ) : the three horizontal excitations and two prominent lines . we can now explore the nonequilibrium occupations of the five spin - multiplets as the transport spectrum is traversed . these are shown in fig . [ fig:14](b ) . the lowest panels show in the left ( right ) regime the ground multiplet @xmath239 with @xmath5 electrons ( @xmath14 with @xmath43 electrons ) is occupied with probability 1 at low bias voltage ( black regions ) . in contrast , in the regime these two ground states are both partially occupied due to processes . we compare the occupations along three different vertical line cuts in fig . [ fig:14](a ) . \(i ) increasing the bias voltage in the regime , starting from @xmath291 v , one first encounters in fig . [ fig:14](a ) a dip ( ndc , white ) . this is caused by the occupation of the @xmath197 state , as the @xmath197-panel in fig . [ fig:14](b ) shows . this drains so much probability from the @xmath14 multiplet [ with a higher transition rate to the @xmath239 multiplet , eq . ( [ eq : rates_t])-([eq : rates_s ] ) ] that the current goes down . increasing the bias further depopulates the @xmath197 state again , thereby restoring the current through a series of peaks . \(ii ) increasing the bias voltage starting from the right regime the excited @xmath197-state becomes populated by decreasing the average spin - length of the molecule . when crossing the resonance at higher bias this excitation is _ completely quenched _ ( white diagonal band ) well before reaching the regime , _ enhances _ the molecular spin , restoring the triplet . \(iii ) when starting from the left regime , the population of the excited @xmath240-state enhances the average spin - length of the molecule . as before , crossing the resonance at higher bias _ quenches _ this excitation . now this _ reduces _ molecular spin , restoring the doublet . along the way , the @xmath241 state also becomes occupied by and subsequently quenched by . because of its higher energy , the white band in the @xmath241-panel of fig . [ fig:14](b ) is much broader . ) characteristic of molecular qd devices . ( b ) strongly asymmetric couplings , ( @xmath292 ) . this is typical for molecular stm junctions , where the energy levels `` pin '' to one electrode ( substrate ) , leaving the tip electrode to act as a probe . in this case the energy @xmath293 represents the _ level alignment _ with the fermi - energy . ] the results show that the widths of the two bands where the excitations are quenched by are unrelated to the width of the resonances , set by the maximum of @xmath11 and @xmath14 . they are , instead , set by the _ excitation spectrum _ one wishes to probe . in fig . [ fig:15](a ) we quantify how far the energy level has to be detuned from resonance in order avoid this quenching in our molecular qd device structure . when this detuning lies in the window @xmath294 one is sure to run into the band with increasing bias . only for @xmath295 there is a finite window where the excitation is not quenched . for the excitations @xmath14 , @xmath240 , @xmath241 in our experiment , this amounts to @xmath296 , @xmath296 , and @xmath296 times the resonance width . in fig . [ fig:15](b ) we show the corresponding construction for strong capacitive asymmetry typical of stm setups . to avoid quenching for any bias polarity , one now needs to stay further away from resonance @xmath297 . interestingly , for @xmath298 excitation at forward @xmath147 is not quenched , whereas at reverse bias @xmath299 it is . for asymmetric junctions , the mechanism thus leads to a _ strong bias - polarity dependence _ of relaxation of excitations in the nominal regime . for @xmath300 one is sure to run into the band for forward bias . whereas in the present experiment we encountered relatively low - lying spin - excitations ( @xmath301 few mev ) atomic and molecular devices can boast such excitations up to tens of mev . to gauge the impact of mirage resonances , consider an excitation at @xmath302 mev that we wish to populate by , e.g. , for the purpose of spin - pumping @xcite . to avoid the quenching of this excitation @xmath147 the distance to the fermi - energy at @xmath149 ( level - alignment ) needs to exceed _ room temperature _ , even when operating the device at mk temperatures . for vibrational and electronic excitations on the 100 mev scale the implications are more severe . moreover , even for excitations that do satisfy these constraints , cascades of `` nonequilibrium '' excitation may if even higher excitations are available ( e.g. , vibrations) provide a path to excitations that do decay by processes . whereas all these effects can be phrased loosely as `` heating '' in this paper we demonstrated the discrete nature of these processes , their _ in - situ _ tuneability , and the role they play as a spectroscopic tool . we have used electron transport on a single - molecule system to comprehensively characterize the spin degree of freedom and its interaction with the tunneling electrons . three key points applicable to a large class of systems emerged with particular prominence : \(i ) combining and spectroscopy in a single stable device provides new tools for determining spin properties _ within _ and _ across _ molecular redox states . this is crucially relevant for the understanding of the different spin - relaxation mechanisms , even in a _ single _ redox state . \(ii ) nonequilibrium pump - probe electron excitation using two processes ( four electrons ) was demonstrated in our three - terminal molecular device and signals a substantial intrinsic spin relaxation time of about 1 ns , much larger than the transport times . \(iii ) mirages of resonances arise from the nontrivial interplay of and . these resonances signal a sharp increase of the relaxation rate and can occur far away from resonance ( many times the resonance width ) . this limits the regime where spin - pumping works by quenching nonequilibrium populations created by a current . the appearance of a mirage of a certain excitation indicates that the relaxation of the corresponding molecular degree of freedom dominates over all possible unwanted , intrinsic mechanism . thus , `` good devices show mirages '' and `` even better devices '' show nonequilibrium transitions . energy level control turns out to be essential for `` imaging '' in energy space , distinguishing mirages from real excitations . whereas real - space imaging seems to be of little help in this respect , the mechanical gating possible with scanning probes overcomes this problem . however , even when energy - level control is available , spectroscopy of molecular junctions still requires extreme care as we illustrated in sec . [ sec : break ] by several examples that break spectroscopic rules . moreover , our work underlines that level alignment has to be treated on a more similar footing as as coupling ( @xmath11 ) and temperature ( @xmath14 ) broadening in the engineering of molecular spin structures and their spin - relaxation rates @xcite . beyond electron charge transport , recent theoretical work @xcite has pointed out that importance of is amplified when moving to nanoscale transport of _ heat _ @xcite . whereas in charge transport all electrons carry the same charge , in energy transport electrons involved in processes effectively can carry a quite different energy from that acquired in a process only and therefore dominate energy currents @xcite . thus , the sensitivity to spin - relaxation processes is dramatically increased in heat transport , indicating an interesting avenue @xcite for a _ spin - caloritronics _ @xcite on the nanoscale . we thank a. cornia for the synthesis of the molecules , m. leijnse and m. josefsson for assistance with the calculations , and s. lounis , m. dos santos dias and t. esat for discussions . we acknowledge financial support by the dutch organization for fundamental research ( nwo / fom ) and an advanced erc grant ( mols@mols ) . m. m. acknowledges financial support from the polish ministry of science and higher education through a young scientist fellowship ( 0066/e-336/9/2014 ) , and from the polish ministry of science and education as iuventus plus project ( ip2014 030973 ) in years 2015 - 2017 . thanks funds from the eu fp7 program , project 618082 acmol through a nwo - veni fellowship . 265ifxundefined [ 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 , , and , eds . , @noop _ _ , ( , , ) @noop * * , ( ) , ed . , @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1021/nl2021637 [ * * , ( ) ] link:\doibase 10.1146/annurev - physchem-040214 - 121554 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase doi 10.1126/science.1202204 [ * * , ( ) ] @noop ( ) @noop * * , ( ) link:\doibase 10.1021/acs.nanolett.5b02188 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.77.125306 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , ed . 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( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.77.201406 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , http://stacks.iop.org/0957-4484/19/i=6/a=065401 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , oxford graduate texts ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.105.086103 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' @noop * * , ( ) @noop * * , ( ) and , eds . , @noop _ _ , , vol . 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molecular systems can exhibit a complex , chemically tailorable inner structure which allows for targeting of specific mechanical , electronic and optical properties . at the single - molecule level , two major complementary ways to explore these properties are molecular quantum - dot structures and scanning probes . this article outlines comprehensive principles of electron - transport spectroscopy relevant to both these approaches and presents a new , high - resolution experiment on a high - spin single - molecule junction exemplifying these principles . such spectroscopy plays a key role in further advancing our understanding of molecular and atomic systems , in particular the relaxation of their spin . in this joint experimental and theoretical analysis , particular focus is put on the crossover between _ resonant _ regime [ single - electron tunneling ( set ) ] and the _ off - resonant _ regime [ inelastic electron ( co)tunneling ( iets ) ] . we show that the interplay of these two processes leads to unexpected _ mirages _ of resonances not captured by either of the two pictures alone . although this turns out to be important in a large fraction of the possible regimes of level positions and bias voltages , it has been given little attention in molecular transport studies . combined with nonequilibrium iets four - electron pump - probe excitations these mirages provide crucial information on the relaxation of spin excitations . our encompassing physical picture is supported by a master - equation approach that goes beyond weak coupling . the present work encourages the development of a broader connection between the fields of molecular quantum - dot and scanning probe spectroscopy .

our story begins with a theorem of gromov , proved in 1980 . [ theorem : polynomial : growth ] let @xmath0 be any finitely generated group . if @xmath0 has polynomial growth then @xmath0 is virtually nilpotent , i.e. @xmath0 has a finite index nilpotent subgroup . gromov s theorem inspired the more general problem ( see , e.g. @xcite ) of understanding to what extent the asymptotic geometry of a finitely generated solvable group determines its algebraic structure . one way in which to pose this question precisely is via the notion of quasi - isometry . a ( coarse ) _ quasi - isometry _ between metric spaces is a map @xmath1 such that , for some constants @xmath2 : 1 . @xmath3 for all @xmath4 . the @xmath5-neighborhood of @xmath6 is all of @xmath7 . @xmath8 and @xmath7 are _ quasi - isometric _ if there exists a quasi - isometry @xmath9 . note that the quasi - isometry type of a metric space @xmath8 is unchanged upon removal of any bounded subset of @xmath8 ; hence the term `` asymptotic '' . quasi - isometries are the natural maps to study when one is interested in the geometry of a group . in particular : the word metric on any f.g . group is unique up to quasi - isometry . any injective homomorphism with finite index image is a quasi - isometry , as is any surjective homomorphism with finite kernel . the equivalence relation generated by these two types of maps can be described more compactly : two groups @xmath10 are equivalent in this manner if and only if they are _ weakly commensurable _ , which means that there exists a group @xmath11 and homomorphisms @xmath12 , @xmath13 each having finite kernel and finite index image ( proof : show that `` weakly commensurable '' is in fact an equivalence relation ) . this weakens the usual notion of _ commensurability _ , i.e. when @xmath0 and @xmath14 have isomorphic finite index subgroups . weakly commensurable groups are clearly quasi - isometric . any two cocompact , discrete subgroups of a lie group are quasi - isometric . there are cocompact discrete subgroups of the same lie group which are not weakly commensurable , for example arithmetic lattices are not weakly commensurable to non - arithmetic ones . the polynomial growth theorem was an important motivation for gromov when he initiated in @xcite the problem of classifying finitely - generated groups up to quasi - isometry . theorem [ theorem : polynomial : growth ] , together with the fact that nilpotent groups have polynomial growth ( see [ section : nilpotent ] below ) , implies that the property of being nilpotent is actually an asymptotic property of groups . more precisely , the class of nilpotent groups is _ quasi - isometrically rigid _ : any finitely - generated group quasi - isometric to a nilpotent group is weakly commensurable to some nilpotent group . sometimes this is expressed by saying that the property of being nilpotent is a _ geometric property _ , i.e. it is a quasi - isometry invariant ( up to weak commensurability ) . the natural question then becomes : [ question : rigidity ] which subclasses of f.g . groups are quasi - isometrically rigid ? for example , are polycyclic groups quasi - isometrically rigid ? metabelian groups ? nilpotent - by - cyclic groups ? in other words , which of these algebraic properties of a group are actually geometric , and are determined by apparently cruder asymptotic information ? a. dioubina @xcite has recently found examples which show that the class of finitely generated solvable groups is _ not _ quasi - isometrically rigid ( see [ section : dioubina ] ) . on the other hand , at least some subclasses of solvable groups are indeed rigid ( see [ section : abc ] ) . along with question [ question : rigidity ] comes the finer classification problem : [ problem : classification ] classify f.g . solvable ( resp . nilpotent , polycyclic , metabelian , nilpotent - by - cyclic , etc . ) groups up to quasi - isometry . as we shall see , the classification problem is usually much more delicate than the rigidity problem ; indeed the quasi - isometry classification of finitely - generated nilpotent groups remains one of the major open problems in the field . we discuss this in greater detail in [ section : nilpotent ] . the corresponding rigidity and classification problems for irreducible lattices in semisimple lie groups have been completely solved . this is a vast result due to many people and , among other things , it generalizes and strengthens the mostow rigidity theorem . we refer the reader to @xcite for a survey of this work . in contrast , results for finitely generated solvable groups have remained more elusive . there are several reasons for this : finitely generated solvable groups are defined algebraically , and so they do not always come equipped with an obvious or well - studied geometric model ( see , e.g. , item 3 below ) . dioubina s examples show not only that the class of finitely - generated solvable groups is not quasi - isometrically rigid ; they also show ( see [ section : abc ] below ) that the answer to question [ question : rigidity ] for certain subclasses of solvable groups ( e.g. abelian - by - cyclic ) differs in the finitely presented and finitely generated cases . there exists a finitely presented solvable group @xmath15 of derived length 3 with the property that @xmath15 has unsolvable word problem ( see @xcite ) . solving the word problem for a group is equivalent to giving an algorithm to build the cayley graph of that group . in this sense there are finitely presented solvable groups whose geometry can not be understood , at least by a turing machine . solvable groups are much less rigid than irreducible lattices in semisimple lie groups . this phenomenon is exhibited concretely by the fact that many finitely generated solvable groups have infinite - dimensional groups of self quasi - isometries , with the operation of composition , becomes a group @xmath16 once one mods out by the relation @xmath17 if @xmath18 in the @xmath19 norm . ] ( see below ) . [ problem : flexible ] for which infinite , finitely generated solvable groups @xmath15 is @xmath20 infinite dimensional ? in contrast , all irreducible lattices in semisimple lie groups @xmath21 have countable or finite - dimensional quasi - isometry groups . at this point in time , our understanding of the geometry of finitely - generated solvable groups is quite limited . in [ section : nilpotent ] we discuss what is known about the quasi - isometry classification of nilpotent groups ( the rigidity being given by gromov s polynomial growth theorem ) . beyond nilpotent groups , the only detailed knowledge we have is for the finitely - presented , nonpolycyclic abelian - by - cyclic groups . we discuss this in depth in [ section : abc ] , and give a conjectural picture of the polycyclic case in [ section : abc2 ] . one of the interesting discoveries described in these sections is a connection between finitely presented solvable groups and the theory of dynamical systems . this connection is pursued very briefly in a more general context in [ section : final ] , together with some questions about issues beyond the limits of current knowledge . this article is meant only as a brief survey of problems , conjectures , and theorems . it therefore contains neither an exhaustive history nor detailed proofs ; for these the reader may consult the references . it is a pleasure to thank david fisher , pierre de la harpe , ashley reiter , jennifer taback , and the referee for their comments and corrections . recall that the _ wreath product _ of groups @xmath22 and @xmath23 , denoted @xmath24 , is the semidirect product @xmath25 , where @xmath26 is the direct sum of copies of b indexed by elements of a , and a acts via the `` shift '' , i.e. the left action of @xmath22 on the index set @xmath22 via left multiplication . note that if @xmath22 and @xmath23 are finitely - generated then so is @xmath24 . the main result of dioubina @xcite is that , if there is a bijective quasi - isometry between finitely - generated groups @xmath22 and @xmath23 , then for any finitely - generated group @xmath27 the groups @xmath28 and @xmath29 are quasi - isometric . dioubina then applies this theorem to the groups @xmath30 where @xmath31 is a finite nonsolvable group . it is easy to construct a one - to - one quasi - isometry between @xmath22 and @xmath23 . hence @xmath32 and @xmath33 are quasi - isometric . now @xmath0 is torsion - free solvable , in fact @xmath34 is an abelian - by - cyclic group of the form @xmath35$]-by-@xmath36 . on the other hand @xmath14 contains @xmath37 , and so is not virtually solvable , nor even weakly commensurable with a solvable group . hence the class of finitely - generated solvable groups is not quasi - isometrically rigid . dioubina s examples never have any finite presentation . in fact if @xmath24 is finitely presented then either @xmath22 or @xmath23 is finite ( see @xcite ) . this leads to the following question . is the class of finitely presented solvable groups quasi - isometrically rigid ? note that the property of being finitely presented is a quasi - isometry invariant ( see @xcite ) . while the polynomial growth theorem shows that the class of finitely generated nilpotent groups is quasi - isometrically rigid , the following remains an important open problem . classify finitely generated nilpotent groups up to quasi - isometry . the basic quasi - isometry invariants for a finitely - generated nilpotent group @xmath0 are most easily computed in terms of the set @xmath38 of ranks ( over @xmath39 ) of the quotients @xmath40 of the lower central series for @xmath0 , where @xmath41 is defined inductively by @xmath42 and @xmath43 $ ] . one of the first quasi - isometry invariants to be studied was the growth of a group , studied by dixmier , guivarch , milnor , wolf , and others ( see @xcite , chapters vi - vii for a nice discussion of this , and a careful account of the history ) . the _ growth _ of @xmath0 is the function of @xmath44 that counts the number of elements in a ball of radius @xmath44 in @xmath0 . there is an important dichotomy for solvable groups : let @xmath0 be a finitely generated solvable group . then either @xmath0 has polynomial growth and is virtually nilpotent , or @xmath0 has exponential growth and is not virtually nilpotent . when @xmath0 has polynomial growth , the degree @xmath45 of this polynomial is easily seen to be a quasi - isometry invariant . it is given by the following formula , discovered around the same time by guivarch @xcite and by bass @xcite : @xmath46 where @xmath47 is the degree of nilpotency of @xmath0 . another basic invariant is that of virtual cohomological dimension @xmath48 . for groups @xmath0 with finite classifying space ( which is not difficult to check for torsion - free nilpotent groups ) , this number was shown by gersten @xcite and block - weinberger @xcite to be a quasi - isometry invariant . on the other hand it is easy to check that @xmath49 where @xmath47 is the degree of nilpotency , also known as the hirsch length , of @xmath0 . as bridson and gersten have shown ( see @xcite ) , the above two formulas imply that any finitely generated group @xmath15 which is quasi - isometric to @xmath50 must have a finite index @xmath50 subgroup : by the polynomial growth theorem such a @xmath15 has a finite index nilpotent subgroup @xmath51 ; but @xmath52 and so @xmath53 which can only happen if @xmath54 for @xmath55 , in which case @xmath51 is abelian . give an elementary proof ( i.e. without using gromov s polynomial growth theorem ) that any finitely generated group quasi - isometric to @xmath50 has a finite index @xmath50 subgroup . as an exercise , the reader is invited to find nilpotent groups @xmath56 which are not quasi - isometric but which have the same degree of growth and the same @xmath57 . there are many other quasi - isometry invariants for finitely - generated nilpotent groups @xmath15 . all known invariants are special cases of the following theorem of pansu @xcite . to every nilpotent group @xmath15 one can associate a nilpotent lie group @xmath58 , called the _ malcev completion _ of @xmath15 ( see @xcite ) , as well as the associated graded lie group @xmath59 . [ theorem : pansu ] let @xmath60 be two finitely - generated nilpotent groups . if @xmath61 is quasi - isometric to @xmath62 then @xmath63 is isomorphic to @xmath64 . we remark that there are nilpotent groups with non - isomorphic malcev completions where the associated gradeds are isomorphic ; the examples are 7-dimensional and somewhat involved ( see @xcite , p.24 , example 2 ) . it is not known whether or not the malcev completion is a quasi - isometry invariant . theorem [ theorem : pansu ] immediately implies : the numbers @xmath65 are quasi - isometry invariants . in particular we recover ( as special cases ) that growth and cohomological dimension are quasi - isometry invariants of @xmath15 . to understand pansu s proof one must consider _ carnot groups_. these are graded nilpotent lie groups @xmath51 whose lie algebra @xmath66 is generated ( via bracket ) by elements of degree one . chow s theorem @xcite states that such lie groups @xmath51 have the property that the left - invariant distribution obtained from the degree one subspace @xmath67 of @xmath66 is a _ totally nonintegrable _ distribution : any two points @xmath68 can be connected by a piecewise smooth path @xmath69 in @xmath51 for which the vector @xmath70 lies in the distribution . infimizing the length of such paths between two given points gives a metric on @xmath51 , called the _ carnot carethodory _ metric @xmath71 . this metric is non - riemannian if @xmath72 . for example , when @xmath51 is the @xmath73-dimensional heisenberg group then the metric space @xmath74 has hausdorff dimension @xmath75 . one important property of carnot groups is that they come equipped with a @xmath76-parameter family of dilations @xmath77 , which gives a notion of _ ( carnot ) differentiability _ ( see @xcite ) . further , the differential @xmath78 of a map @xmath79 between carnot groups @xmath56 which is ( carnot ) differentiable at the point @xmath80 is actually a _ lie group homomorphism _ @xmath81 . * sketch of pansu s proof of theorem [ theorem : pansu ] . * if @xmath82 is a nilpotent group endowed with a word metric @xmath83 , the sequence of scaled metric spaces @xmath84 has a limit in the sense of gromov - hausdorff convergence : @xmath85 ( see @xcite , and @xcite for an introduction to gromov - hausdorff convergence ) . it was already known , using ultralimits , that some subsequence converges @xcite . pansu s proof not only gives convergence on the nose , but it yields some additional important features of the limit metric space @xmath86 : * ( identifying limit ) it is isometric to the carnot group @xmath87 endowed with the carnot metric @xmath71 . * ( functoriality ) any quasi - isometry @xmath88 bewteen finitely - generated nilpotent groups induces a _ bilipschitz homeomorphism _ note that functoriality follows immediately once we know the limit exists : the point is that if @xmath90 is a @xmath91 quasi - isometry of word metrics , then for each @xmath47 the map @xmath92 is a @xmath93 quasi - isometry , hence the induced map @xmath94 is a @xmath95 quasi - isometry , i.e. is a bilipschitz homeomorphism . given a quasi - isometry @xmath96 , we thus have an induced bilipschitz homeomorphism @xmath97 between carnot groups endowed with carnot - carethodory metrics . pansu then proves a regularity theorem , generalizing the rademacher - stepanov theorem for @xmath98 . this general regularity theorem states that a bilipschitz homeomorphism of carnot groups ( endowed with carnot - carethodory metrics ) is differentiable almost everywhere . since the differential @xmath99 is actually a group homomorphism , we know that for almost every point @xmath100 the differential @xmath101 is an isomorphism . the first progress on question [ question : rigidity ] and problem [ problem : classification ] in the non-(virtually)-nilpotent case was made in @xcite and @xcite . these papers proved classification and rigidity for the simplest class of non - nilpotent solvable groups : the _ solvable baumslag - solitar groups _ @xmath102 these groups are part of the much broader class of abelian - by - cyclic groups . a group @xmath15 is _ abelian - by - cyclic _ if there is an exact sequence @xmath103 where @xmath22 is an abelian group and @xmath104 is an infinite cyclic group . if @xmath15 is finitely generated , then @xmath22 is a finitely generated module over the group ring @xmath105 $ ] , although @xmath22 need not be finitely generated as a group . by a result of bieri and strebel @xcite , the class of finitely presented , torsion - free , abelian - by - cyclic groups may be described in another way . consider an @xmath106 matrix @xmath107 with integral entries and @xmath108 . let @xmath109 be the ascending hnn extension of @xmath50 given by the monomorphism @xmath110 with matrix @xmath107 . then @xmath109 has a finite presentation @xmath111=1 , ta_it^{-1}=\phi_m(a_i ) , i , j=1,\ldots , n\rangle\ ] ] where @xmath112 is the word @xmath113 and the vector @xmath114 is the @xmath115 column of the matrix @xmath107 . such groups @xmath109 are precisely the class of finitely presented , torsion - free , abelian - by - cyclic groups ( see @xcite for a proof involving a precursor of the bieri - neumann - strebel invariant , or @xcite for a proof using trees ) . the group @xmath109 is polycyclic if and only if @xmath116 ( see @xcite ) . the results of @xcite and @xcite are generalized in @xcite , which gives the complete classification of the finitely presented , nonpolycyclic abelian - by - cyclic groups among all f.g . groups , as given by the following two theorems . the first theorem in @xcite gives a classification of all finitely - presented , nonpolycyclic , abelian - by - cyclic groups up to quasi - isometry . it is easy to see that any such group has a torsion - free subgroup of finite index , so is commensurable ( hence quasi - isometric ) to some @xmath109 . the classification of these groups is actually quite delicate the standard quasi - isometry invariants ( ends , growth , isoperimetric inequalities , etc . ) do not distinguish any of these groups from each other , except that the size of the matrix @xmath107 can be detected by large scale cohomological invariants of @xmath109 . given @xmath117 , the _ absolute jordan form _ of @xmath107 is the matrix obtained from the jordan form for @xmath107 over @xmath118 by replacing each diagonal entry with its absolute value , and rearranging the jordan blocks in some canonical order . [ theorem : classification ] let @xmath119 and @xmath120 be integral matrices with @xmath121 for @xmath122 . then @xmath123 is quasi - isometric to @xmath124 if and only if there are positive integers @xmath125 such that @xmath126 and @xmath127 have the same absolute jordan form . theorem [ theorem : classification ] generalizes the main result of @xcite , which is the case when @xmath128 are positive @xmath129 matrices ; in that case the result of @xcite says even more , namely that @xmath123 and @xmath124 are quasi - isometric if and only if they are commensurable . when @xmath130 , however , it s not hard to find @xmath131 matrices @xmath128 such that @xmath132 are quasi - isometric but not commensurable . polycyclic examples are given in @xcite ; similar ideas can be used to produce nonpolycyclic examples . the following theorem shows that the algebraic property of being a finitely presented , nonpolycyclic , abelian - by - cyclic group is in fact a geometric property . [ theorem : rigidity ] let @xmath133 be a finitely presented abelian - by - cyclic group , determined by an @xmath106 integer matrix @xmath107 with @xmath134 . let @xmath0 be any finitely generated group quasi - isometric to @xmath15 . then there is a finite normal subgroup @xmath135 such that @xmath136 is commensurable to @xmath137 , for some @xmath106 integer matrix @xmath51 with @xmath138 . theorem [ theorem : rigidity ] generalizes the main result of @xcite , which covers the case when @xmath107 is a positive @xmath129 matrix . the @xmath129 case is given a new proof in @xcite , which is adapted in @xcite to prove theorem [ theorem : rigidity ] . the `` finitely presented '' hypothesis in theorem [ theorem : rigidity ] can not be weakened to `` finitely generated '' , since diuobina s example ( discussed in [ section : dioubina ] ) is abelian - by - cyclic , namely @xmath35$]-by-@xmath36 . one new discovery in @xcite is that there is a strong connection between the geometry of solvable groups and the theory of dynamical systems . assuming here for simplicity that the matrix @xmath107 lies on a @xmath76-parameter subgroup @xmath139 in @xmath140 , let @xmath141 be the semi - direct product @xmath142 , where @xmath143 acts on @xmath98 by the @xmath76-parameter subgroup @xmath139 . we endow the solvable lie group @xmath141 with a left - invariant metric . the group @xmath141 admits a _ vertical flow _ : @xmath144 there is a natural _ horizontal foliation _ of @xmath141 whose leaves are the level sets @xmath145 of time . a quasi - isometry @xmath146 is _ horizontal respecting _ if it coarsely permutes the leaves of this foliation ; that is , if there is a constant @xmath147 so that @xmath148 where @xmath149 denotes hausdorff distance and @xmath150 is some function , which we think of as a _ time change _ between the flows . a key technical result of @xcite is the phenomenon of _ time rigidity _ : the time change @xmath151 must actually be _ affine _ , so taking a real power of @xmath107 allows one to assume @xmath152 . it is then shown that `` quasi - isometries remember the dynamics '' . that is , @xmath153 coarsely respects several foliations arising from the partially hyperbolic dynamics of the flow @xmath154 , starting with the weak stable , weak unstable , and center - leaf foliations . by keeping track of different exponential and polynomial divergence properties of the action of @xmath154 on tangent vectors , the weak stable and weak unstable foliations are decomposed into flags of foliations . using time rigidity and an inductive argument it is shown that these flags are coarsely respected by @xmath153 as well . relating the flags of foliations to the jordan decomposition then completes the proof of : [ theorem : horizontal ] if there is a horizontal - respecting quasi - isometry @xmath146 then there exist nonzero @xmath155 so that @xmath156 and @xmath157 have the same absolute jordan form . the `` nonpolycyclic '' hypothesis ( i.e. @xmath134 ) in theorem [ theorem : classification ] is used in two ways . first , the group @xmath109 has a model space which is topologically a product of @xmath98 and a regular tree of valence @xmath158 , and when this valence is greater than @xmath159 we can use coarse algebraic topology ( as developed in @xcite , @xcite , and @xcite ) to show that any quasi - isometry @xmath160 induces a quasi - isometry @xmath161 satisfying the hypothesis of theorem [ theorem : horizontal ] . second , we are able to pick off _ integral _ @xmath162 by developing a `` boundary theory '' for @xmath109 ; in case @xmath134 this boundary is a self - similar cantor set whose bilipschitz geometry detects the primitive integral power of @xmath163 by cooper s theorem @xcite , finishing the proof of theorem [ theorem : classification ] . [ problem : extend ] extend theorem [ theorem : classification ] and theorem [ theorem : rigidity ] to the class of finitely - presented nilpotent - by - cyclic groups . of course , as the classification of finitely - generated nilpotent groups is still open , problem [ problem : extend ] is meant in the sense of reducing the nilpotent - by - cyclic case to the nilpotent case , together with another invariant . this second invariant for a nilpotent - by - cyclic group @xmath0 will perhaps be the absolute jordan form of the matrix which is given by the action of the generator of the cyclic quotient of @xmath0 on the nilpotent kernel of @xmath0 . the polycyclic , abelian - by - cyclic groups are those @xmath109 for which @xmath116 , so that @xmath109 is cocompact and discrete in @xmath141 , hence quasi - isometric to @xmath141 . in this case the proof of theorem [ theorem : classification ] outlined above breaks down , but this is so in part because the answer is quite different : the quasi - isometry classes of polycyclic @xmath109 are much coarser than in the nonpolycyclic case , as the former are ( conjecturally ) determined by the absolute jordan form up to _ real _ , as opposed to integral , powers . the key conjecture is : [ conjecture : poly1 ] suppose that @xmath164 , and that @xmath107 and @xmath51 have no eigenvalues on the unit circle . then every quasi - isometry of @xmath165 is horizontal - respecting . the general ( with arbitrary eigenvalues ) case of conjecture [ conjecture : poly1 ] , which is slightly more complicated to state , together with theorem [ theorem : horizontal ] easily implies : suppose that @xmath164 . then @xmath109 is quasi - isometric to @xmath137 if and only if there exist nonzero @xmath155 so that @xmath166 and @xmath167 have the same absolute jordan form . here by @xmath166 we mean @xmath168 , where @xmath169 is a @xmath76-parameter subgroup with @xmath170 ( we are assuming that @xmath107 lies on such a subgroup , which can be assumed after squaring @xmath107 ) . now let us concentrate on the simplest non - nilpotent example , which is also one of the central open problems in the field . the 3-dimensional geometry @xmath171 is the lie group @xmath141 where @xmath172 is any matrix with 2 distinct real eigenvalues ( up to scaling , it does nt matter which such @xmath107 is chosen ) . [ conjecture : solv ] the @xmath73-dimensional lie group @xmath171 is quasi - isometrically rigid : any f.g . group @xmath0 quasi - isometric to @xmath171 is weakly commensurable with a cocompact , discrete subgroup of @xmath171 . there is a natural boundary for @xmath171 which decomposes into two pieces @xmath173 and @xmath174 ; these are the leaf spaces of the weak stable and weak unstable foliations , respectively , of the vertical flow on @xmath175 , and are both homeomorphic to @xmath143 . the isometry group @xmath176 acts on the pair @xmath177 affinely and induces a faithful representation @xmath178 whose image consists of the pairs @xmath179 just as quasi - isometries of hyperbolic space @xmath180 are characterized by their quasiconformal action on @xmath181 ( a fact proved by mostow ) , giving the formula @xmath182 , we conjecture : [ conjecture : qigroup ] @xmath183 where @xmath184 denotes the group of bilipschitz homeomorphisms of @xmath143 , and @xmath185 acts by switching factors . there is evidence for conjecture [ conjecture : qigroup ] : the direction @xmath186 is not hard to check ( see @xcite ) , and the analogous theorem @xmath187 was proved in @xcite . by using convergence groups techniques and a theorem of hinkkanen on uniformly quasisymmetric groups ( see @xcite ) , we have been able to show : conjecture [ conjecture : poly1 ] ( in the @xmath188 case ) @xmath189 conjecture [ conjecture : qigroup ] @xmath189 conjecture [ conjecture : solv ] here is a restatement of conjecture [ conjecture : poly1 ] in the @xmath190 case : every quasi - isometry @xmath191 is horizontal respecting . [ conjecture : solvhorizontal ] here is one way _ not _ to prove conjecture [ conjecture : solvhorizontal ] . one of the major steps of @xcite in studying @xmath192 was to construct a model space @xmath193 for the group @xmath192 , study the collection of isometrically embedded hyperbolic planes in @xmath193 , and prove that for any quasi - isometric embedding of the hyperbolic plane into @xmath193 , the image has finite hausdorff distance from some isometrically embedded hyperbolic plane . however , @xmath171 has quasi - isometrically embedded hyperbolic planes which are _ not _ hausdorff close to isometrically embedded ones . the natural left invariant metric on @xmath171 has the form @xmath194 from which it follows that the @xmath195-planes and @xmath196-planes are the isometrically embedded hyperbolic planes . but none of these planes is hausdorff close to the set @xmath197 which is a quasi - isometrically embedded hyperbolic plane . an even stranger example is shown in figure [ figurewindvane ] . these strange quasi - isometric embeddings from @xmath198 to @xmath171 do share an interesting property with the standard isometric embeddings , which may point the way to understanding quasi - isometric rigidity of @xmath171 . we say that a quasi - isometric embedding @xmath199 is _ @xmath22-quasivertical _ if for each @xmath200 there exists a vertical line @xmath201 such that @xmath202 is contained in the @xmath22-neighborhood of @xmath203 , and @xmath203 is contained in the @xmath22-neighborhood of @xmath204 . in order to study @xmath171 , it therefore becomes important to understand whether every quasi - isometrically embedded hyperbolic plane is quasi - vertical . specifically : show that for all @xmath205 there exists @xmath22 such that each @xmath205-quasi - isometrically embedded hyperbolic plane in @xmath171 is @xmath22-quasivertical . [ problemquasivertical ] arguing by contradiction , if problem [ problemquasivertical ] were impossible , fixing @xmath205 and taking a sequence of examples whose quasi - vertical constant @xmath22 goes to infinity , one can pass to a subsequence and take a renormalized limit to produce a quasi - isometric embedding @xmath206 whose image is entirely contained in the upper half @xmath207 of @xmath171 . but we conjecture that this is impossible : there does not exist a quasi - isometric embedding @xmath208 whose image is entirely contained in the upper half space @xmath209 . while we have already seen that there is a somewhat fine classification of finitely presented , nonpolycyclic abelian - by - cyclic groups up to quasi - isometry , this class of groups is but a very special class of finitely generated solvable groups . we have only exposed the tip of a huge iceberg . an important next layer is : the first step in attacking this problem is to find a workable method of describing the geometry of the natural geometric model of such groups @xmath0 . such a model should fiber over @xmath98 , where @xmath47 is the rank of the maximal abelian quotient of @xmath0 ; inverse images under this projection of ( translates of ) the coordinate axes should be copies of the geometric models of abelian - by - cyclic groups . * polycyclic versus nonpolycyclic . * we ve seen the difference , at least in the abelian - by - cyclic case , between polycyclic and nonpolycyclic groups . geometrically these two classes can be distinguished by the trees on which they act : such trees are lines in the former case and infinite - ended in the latter . it is this branching behavior which should combine with coarse topology to make the nonpolycyclic groups more amenable to attack . note that a ( virtually ) polycyclic group is never quasi - isometric to a ( virtually ) nonpolycyclic solvable group . this follows from the theorem of bieri that polycylic groups are precisely those solvable groups satisfying poincare duality , together with the quasi - isometric invariance of the latter property ( proved by gersten @xcite and block - weinberger @xcite ) . * solvable groups as dynamical systems . * the connection of nilpotent groups with dynamical systems was made evident in @xcite , where gromov s polynomial growth theorem was the final ingredient , combining with earlier work of franks and shub @xcite , in the positive solution of the _ expanding maps conjecture _ : every locally distance expanding map on a closed manifold @xmath107 is topologically conjugate to an expanding algebraic endomorphism on an infranil manifold ( see @xcite ) . in [ section : abc ] and [ section : abc2 ] we saw in another way how invariants from dynamics give quasi - isometry invariants for abelian - by - cyclic groups . this should be no big surprise : after all , a finitely presented abelian - by - cyclic group is describable up to commensurability as an ascending hnn extension @xmath109 over a finitely - generated abelian group @xmath50 . the matrix @xmath107 defines an endomorphism of the @xmath47-dimensional torus . the mapping torus of this endomorphism has fundamental group @xmath109 , and is the phase space of the suspension semiflow of the endomorphism , a semiflow with partially hyperbolic dynamics ( when @xmath107 is an automorphism , and so @xmath109 is polycyclic , the suspension semiflow is actually a flow ) . here we see an example of how the geometric model of a solvable group is actually the phase space of a dynamical system . but bieri - strebel @xcite have shown that _ every _ finitely presented solvable group is , up to commensurability , an ascending hnn extension with base group a finitely generated solvable group . in this way every finitely presented solvable group is the phase space of a dynamical system , probably realizable geometrically as in the abelian - by - cyclic case . e. ghys and p. de la harpe , infinite groups as geometric objects ( after gromov ) , in _ ergodic theory , symbolic dynamics and hyperbolic spaces _ , ed . by t. bedford and m. keane and c. series , oxford univ . press , 1991 .

a survey of problems , conjectures , and theorems about quasi - isometric classification and rigidity for finitely generated solvable groups .

many simply connected , smoothable topological 4manifolds ( with @xmath3 odd ) are known to admit infintely many distinct smooth structures . the existence of such _ exotic _ structures are , however , less clear when the euler characteristic of the 4manifold is small , for example if the manifold is homeomorphic to the ( blow up of the ) complex projective plane . using the rational blow down construction @xcite together with knot surgery @xcite in double node neighborhoods @xcite , many new smooth 4manifolds have been discovered with @xmath4 and @xmath5 @xcite . similar ideas can be applied to get examples of irreducible exotic simply connected 4manifolds with @xmath1 and relatively small @xmath6 . the study of exotic structures on simply connected manifolds with @xmath1 has a rich history . recall that the @xmath7 surface @xmath8 is such an example with @xmath9 . applying appropriate logarithmic transformations on @xmath8 it was shown that @xmath10 admits infinitely many smooth structures @xcite . later works , relying on gompf s symplectic normal connected sum operation , together with donaldson theory showed that the topological manifolds @xmath11 with @xmath12 admit infinitely many smooth structures @xcite . more recent results of park @xcite proved the same statement with @xmath13 . our main result in this paper improves this bound : [ t : m1 ] the simply connected topological 4manifold @xmath0 admits infinitely many distinct smooth structures . by modifying the construction used in the proof of theorem [ t : m1 ] , we can define a set of 4manifolds which still have @xmath1 , but their euler characteristic is smaller than the above examples . in these cases , however , we were unable to show that the manifolds are simply connected . [ t : m2 ] there are infinitely many pairwise nondiffeomorphic smooth , closed 4manifolds with vanishing first homology , @xmath1 , @xmath2 and nontrivial seiberg witten invariants . the proof of the results will involve two steps . first we desribe how to construct the manifolds claimed and then we use seiberg witten theory in proving that they are nondiffeomorphic . in the construction we will apply mapping class group arguments and the theory of lefschetz fibrations . using the knot surgery construction then we can identify configurations of curves in the resulting 4manifolds which can be rationally blown down , leading us to the desired examples . recall that a genus1 lefschetz fibration @xmath14 can be described by the word in the mapping class group @xmath15 of the 2torus @xmath16 corresponding to the monodromy presentation of @xmath17 . more precisely , a point @xmath18 where @xmath19 is not onto ( called a singular point of the fibration ) gives rise to a singular fiber @xmath20 , and the monodromy of the fibration around such a fiber can be given by the composition of dehn twists along the circles corresponding to the vanishing cycles of the singular points of the fiber . by traversing through the singular fibers in a counterclockwise manner relative to a fixed base point @xmath21 , we get the above mentioned word describing the fibration . notice that we do not assume that @xmath17 is injective on the set of its singular points , that is , a singular fiber can contain more than one singular points . the assumption that the map @xmath17 is a lefschetz fibration implies that the vanishing cycles corresponding to the singular points in one fixed singular fiber can be chosen to be disjoint . it is known that the mapping class group @xmath15 can be generated by two elements @xmath22 which are subject to the two relations @xmath23 in fact , @xmath15 can be shown to be isomorphic to @xmath24 by mapping @xmath25 to @xmath26 and @xmath27 to @xmath28 . since the forgetful map from the mapping class group @xmath29 of the 2torus with one marked point to @xmath15 is an isomorphism , we get that any genus1 lefschetz fibration admits a section . genus1 lefschetz fibrations were classified by moishezon @xcite , who showed that after a possible perturbation such a fibration over @xmath30 is equivalent to one of the fibrations given by the words @xmath31 ( @xmath32 ) in @xmath15 . the resulting 4manifold is usually called @xmath33 ( the simply connected elliptic surface with section and of holomorphic euler characteristic @xmath34 ) , and @xmath8 is the famous @xmath7 surface . it can be shown that a section of @xmath35 has self intersection @xmath36 . following @xcite we call a fiber with monodromy conjugate to @xmath37 of _ type @xmath38 _ ( @xmath39 ) . when @xmath40 , the corresponding fiber is also called a _ fishtail _ fiber . it is easy to see that topologically a singular fiber of type @xmath41 ( @xmath42 ) is a plumbing of @xmath43 smooth 2spheres of self intersection @xmath44 plumbed along a circle ( see @xcite ) , while a fishtail fiber is an immersed 2sphere with one positive double point . since in @xmath16 nonisotopic simple closed curves necessarily intersect each other , it is easy to see that a genus1 lefschetz fibration can have only @xmath41fibers as singular fibers . the fibration @xmath17 can be perturbed near a singular fiber @xmath45 of type @xmath41 into a fibration @xmath46 which is the same as @xmath17 outside of @xmath45 but breaks @xmath45 into two singular fibers of types @xmath47 and @xmath48 with @xmath49 . this fact implies that a generic genus1 lefschetz fibration admits only fishtail fibers . in our study however , we will find it most helpful to understand what kind of other singular fibers an elliptic fibration can admit . we start with a proposition showing the existence of a particular genus1 fibration on @xmath8 . [ p : mcg ] there exists an elliptic lefschetz fibration on the @xmath7 surface @xmath8 with a section , a singular fiber @xmath45 of type @xmath51 , three singular fibers @xmath52 of type @xmath53 and two further fishtail fibers . it is not hard to see that the word @xmath54 ( defining a genus1 lefschetz fibration on the @xmath7 surface @xmath8 ) in the mapping class group @xmath15 of the torus is equivalent to @xmath55 ( alternatively , by substituting @xmath25 and @xmath27 with the matrices they correspond to under the map @xmath56 , we can check that the above product is equal to the identity matrix . ) by collecting the powers of @xmath25 in the front using conjugation , we get @xmath57 followed by the product of three squares of some conjugates of @xmath27 and two further conjugates of @xmath27 . since a conjugate of @xmath27 by a word @xmath58 corresponds to the dehn twist along the image under the diffeomorphism @xmath59 of the curve inducing @xmath27 , the proposition follows . as we already mentioned , genus1 lefschetz fibrations always admit sections . perturb first the above fibration near the @xmath53 fibers @xmath60 ( @xmath61 ) in a way that these give rise to fishtail fibers @xmath62 with isotopic vanishing cycles . let us denote the resulting lefschetz fibration by @xmath63 . let @xmath64 be three twist knots as depicted in @xcite . let @xmath65 denote the 4manifold we get after performing three knot surgeries with knots @xmath64 along three regular fibers in the fibration on the @xmath7 surface found above . because of the existence of fishtail fibers in the complement with nonisotopic vanishing cycles , we conclude that @xmath66 . if we perform the surgeries in the double node neighborhoods near the fishtail fibers @xmath67 , and @xmath68 and @xmath69 respectively , then we can find a pseudo section @xmath70 as in @xcite which is an immersed sphere of self intersection @xmath71 with three positive double points , intersecting the further fishtail fibers and the @xmath51 fiber @xmath45 transversally . next smooth the intersections of this pesudo section @xmath70 with two further fishtail fibers . the result is an immersed sphere of self intersection 2 having 5 positive double points . now blow up the 4manifold @xmath65 in the five double points of this sphere , and find an embedded sphere of self intersection @xmath72 in @xmath73 . let @xmath74 denote the tubular neighborhood of the linear plumbing of spheres given by this @xmath75sphere together with 14 of the @xmath71spheres in the @xmath51 fiber @xmath45 . it is not hard to see that @xmath74 is diffeomorphic to @xmath76 in the notation of @xcite , cf . also @xcite . it is then easy to show that @xmath77 can be given as the oriented boundary of the rational ball @xmath78 , see @xcite . define @xmath79 as the rational blow down of @xmath73 along @xmath74 , that is , @xmath80 notice that @xmath65 is simply connected , and the complement of @xmath74 in @xmath81 is simply connected since the @xmath71sphere in @xmath45 intersecting the last @xmath71sphere of the linear chain @xmath74 provides a hemisphere which contracts the generator of the fundamental group of @xmath82 . since @xmath83 surjects onto @xmath84 under the natural embedding , van kampen s theorem implies that @xmath79 is simply connected . now simple signature and euler characteristics computation together with freedman s theorem @xcite verifies the result . for short , let @xmath85 denote the 4manifold @xmath79 if @xmath86 the @xmath34twist knot @xmath87 . the following proposition is true in a wider generality , we restrict our attention to the special case @xmath88 in order to keep our discussion as simple as possible . using the results of @xcite the seiberg witten invariants of @xmath85 can be easily computed . this computation immediately shows by ( * ? ? ? * theorem 1.1 ) the seiberg witten invariant of @xmath93 can be computed to be equal to @xmath94 ( use the facts that the seiberg witten function of the @xmath7 surface is equal to 1 and the alexander polynomial of @xmath87 is @xmath95 ) . this result , together with the blow up formula shows that the five fold blow up @xmath96 has exactly two seiberg witten basic classes @xmath97 which evaluate on the @xmath75sphere of the configuration @xmath74 as @xmath98 . moreover , the value of the seiberg witten function on these basic classes is @xmath99 . now * theorem 8.5 ) implies that @xmath85 has two basic classes , on which the value of the seiberg witten function is equal to @xmath99 , verifying the result . the 4manifolds @xmath85 provide an infinite family of smooth 4manifolds all homeomorphic to @xmath0 according to proposition [ p : hom ] , and by corollary [ c : nondiffeo ] these manifolds are pairwise nondiffeomorphic . therefore the set @xmath100 provides an infinite collection of distinct smooth structures on @xmath0 , hence the proof is complete . using a variation of the above procedure , closed 4manifolds with @xmath1 and @xmath2 can be constructed as follows . consider the fibration @xmath63 found above , containing a singular fiber of type @xmath51 and eight fishtail fibers , out of which three pairs @xmath101 ( @xmath61 ) have isotopic vanishing cycles . proceed as before by doing three knot surgeries along three regular fibers with twist knots @xmath64 and using the double node neighborhoods provided by the fishtail fibers @xmath102 identify the pseudo section , which is again an immersed sphere with homological square @xmath44 and has three positive double points . as before , resolve the two positive intersections of this pseudo section with the remaining two fishtail fibers , and find the immersed sphere with 5 double point and homological square 2 . blow up the 4manifold at the double points of the pesudo section , and consider the resulting sphere of square @xmath72 in @xmath65 . this sphere intersects the @xmath51 fiber @xmath45 transversally in a unique point @xmath103 which is on the @xmath71sphere @xmath104 . now @xmath105 is intersected by two other spheres in @xmath45 , let @xmath106 be one of them and denote the intersection point of @xmath105 and @xmath106 by @xmath107 . apply 17 infinitely close blow ups at @xmath107 . the resulting plumbing manifold @xmath108 can be given by the linear plumbing @xmath109 @xmath110 as before , it is routine to see that the boundary @xmath111 can be given as the boundary of the rational ball @xmath112 , hence we can blow it down , resulting the 4manifold @xmath113 . since the normal circle of the pseudo section ( which circle generates the first homology of @xmath111 ) vanishes in the homology of the complement @xmath114 ( as it is shown by a regular fiber ) , the mayer vietoris sequence implies that @xmath115 vanishes . simple euler characteristic and signature computations now imply finally , by computing seiberg witten invariants of these manifolds in the fashion it was done in theorem [ t : sw ] , we get that the 4manifolds @xmath116 are pairwise nondiffeomorphic , all with nonvanishing seiberg witten invariants , verifying the claim of theorem [ t : m2 ] .

we construct an infinite family of simply connected , pairwise nondiffeomorphic 4manifolds , all homeomorphic to @xmath0 . similar ideas provide examples of 4manifolds with @xmath1 , @xmath2 , vanishing first homology and nontrivial seiberg witten invariants .

oscillation experiments over vastly different baselines and a range of neutrino energies have filled up a vast portion of the mass and mixing jigsaw of the neutrino sector . yet , we still remain in the dark with regard to cp - violation in the lepton sector . neither do we know the mass ordering whether it is normal or inverted . further open issues are the absolute mass scale of neutrinos and whether they are of majorana or dirac nature . while we await experimental guidance for each of the above unknowns , there have been many attempts to build models of lepton mass which capture much of what is known . here we propose a neutrino mass model based on the direct product group @xmath1 . the elements of @xmath15 correspond to the permutations of three objects can be found in appendix [ apps3 ] . ] . needless to say , @xmath15-based models of neutrino mass have been considered earlier @xcite . a popular point of view @xcite has been to note that a permutation symmetry between the three neutrino states is consistent with ( see appendix [ apps3 ] ) . so these models entail fine tuning . ] ( a ) a democratic mass matrix , @xmath16 , all whose elements are equal , and ( b ) a mass matrix proportional to the identity matrix , @xmath17 . a general combination of these two forms , e.g. , @xmath18 , where @xmath19 are complex numbers , provides a natural starting point . one of the eigenstates , namely , an equal weighted combination of the three states , is one column of the popular tribimaximal mixing matrix . many models have been presented @xcite which add perturbations to this structure to accomplish realistic neutrino masses and mixing . variations on this theme @xcite explore mass matrices with such a form in the context of grand unified theories , in models of extra dimensions , and examine renormalisation group effects on such a pattern realised at a high energy . other variants of the @xmath15-based models , for example @xcite , rely on a 3 - 3 - 1 local gauge symmetry , tie it to a @xmath20-extended model , or realise specific forms of mass matrices through soft symmetry breaking , etc . as discussed later , the irreducible representations of @xmath15 are one and two - dimensional . the latter provides a natural mechanism to get maximal mixing in the @xmath21 sector @xcite . the present work , also based on @xmath15 symmetry , breaks new ground in the following directions . firstly , it involves an interplay of type - i and type - ii see - saw contributions . secondly , it presents a general framework encompassing many popular mixing patterns such as tribimaximal mixing . further , the model does not invoke any soft symmetry breaking terms . all the symmetries are broken spontaneously . we briefly outline here the strategy of this work . we use the standard notation for the leptonic mixing matrix the pontecorvo , maki , nakagawa , sakata ( pmns ) matrix @xmath22 . @xmath23 where @xmath24 and @xmath25 . the neutrino masses and mixings arise through a two - stage mechanism . in the first step , from the type - ii see - saw the larger atmospheric mass splitting , @xmath26 , is generated while the solar splitting , @xmath27 , is absent . also , @xmath28 , @xmath29 and the model parameters can be continuously varied to obtain any desired @xmath4 . of course , in reality @xmath30 @xcite , the solar splitting is non - zero , and there are indications that @xmath9 is large but non - maximal . experiments have also set limits on @xmath31 . the type - i see - saw addresses all the above issues and relates the masses and mixings to each other . the starting form incorporates several well - studied mixing patterns such as tribimaximal ( tbm ) , bimaximal ( bm ) , and golden ratio ( gr ) mixings within its fold . these alternatives all have @xmath32 and @xmath2 . they differ only in the value of the third mixing angle @xmath4 as displayed in table [ t1 ] . the fourth option in this table , no solar mixing ( nsm ) , exhibits the attractive feature symmetry @xcite which built on previous work along similar lines @xcite . ] that the mixing angles are either maximal , i.e. , @xmath33 ( @xmath9 ) or vanishing ( @xmath3 and @xmath4 ) . .the solar mixing angle , @xmath34 for this work , for the tbm , bm , and gr mixing patterns . nsm stands for the case where the solar mixing angle is initially vanishing . [ cols="^,^,^,^,^",options="header " , ] the group has two 1-dimensional representations denoted by @xmath35 and @xmath36 , and a @xmath37-dimensional representation . @xmath35 is inert under the group while @xmath38 changes sign under the action of @xmath39 . for the 2-dimensional representation a suitable choice of matrices with the specified properties can be readily obtained . we choose @xmath40 where @xmath41 is a cube root of unity , i.e. , @xmath42 . for this choice of @xmath39 and @xmath43 the remaining matrices of the representation are : @xmath44 the product rules for the different representations are : @xmath45 one can see that each of the @xmath46 matrices @xmath47 in eqs . ( [ s3_21 ] ) and ( [ s3_22 ] ) satisfies : @xmath48 where @xmath49 for @xmath50 and @xmath51 for @xmath52 . if @xmath53 and @xmath54 are two field multiplets transforming under @xmath15 as doublets then using eqs . ( [ s3_21 ] ) and ( [ s3inv ] ) : @xmath55 sometimes we have to deal with hermitian conjugate fields . noting the nature of the complex representation ( see , for example , @xmath43 in eq . ( [ s3_21 ] ) ) the conjugate @xmath15 doublet is @xmath56 . as a result , one has in place of ( [ s3_prod ] ) @xmath57 eqs . ( [ s3_prod ] ) and ( [ s3_prod2 ] ) are essential in writing down the fermion mass matrices in sec . [ model ] . as seen in table [ tab1s ] this model has a rich scalar field content . in this appendix we write down the scalar potential of the model keeping all these fields and derive conditions which must be met by the coefficients of the various terms so that the desired _ vev_s can be achieved . these conditions ensure that the potential is locally minimized by this choice . table [ tab1s ] displays the behaviour of the scalar fields under @xmath1 besides the gauged electroweak @xmath58 . the fields also carry a lepton number . the scalar potential is the most general polynomial in these fields with up to quartic terms . our first step will be to write down the explicit form of this potential . here we do not exclude any term permitted by the symmetries . @xmath59 invariance of the terms as well as the abelian lepton number and @xmath60 conservation are readily verified . it is only the @xmath15 behaviour which merits special attention . there are a variety of scalar fields in this model , e.g. , @xmath61 singlets , doublets , and triplets . therefore , the scalar potential has a large number of terms . for simplicity we choose all couplings in the potential to be real . in this appendix we list the potential in separate parts : ( a ) those belonging to any one @xmath61 sector , and ( b ) inter - sector couplings of scalars . the @xmath61 singlet _ vev_s , which are responsible for the right - handed neutrino mass , are significantly larger than those of other scalars . so , in the second category we retain only those terms which couple the singlet fields to either the doublet or the triplet sectors . the @xmath61 singlet sector comprises of two fields @xmath62 and @xmath63 transforming as @xmath64 and @xmath65 of @xmath66 respectively . they have @xmath67 . the scalar potential arising out of these is : @xmath68 ^ 2 + { \lambda_2^s \over 2}\left[\gamma^\dagger\gamma\right]^2 \nonumber \\ & + & { \lambda_3^s \over 2}(\chi^\dagger\chi)(\gamma^\dagger\gamma ) + \lambda_4^s\left\{(\gamma^\dagger \chi)(\gamma^\dagger\chi)+ h.c.\right \ } \;\ ; , \label{vs_s3}\end{aligned}\ ] ] where the coefficient of the cubic term , @xmath69 , carries the same dimension as mass while the @xmath70 are dimensionless . the @xmath61 doublet sector of the model has two fields @xmath71 that are doublets of @xmath15 , in addition to @xmath72 , @xmath73 , and @xmath74 which are @xmath15 singlets . among them , all fields except @xmath75 and @xmath72 ( @xmath76 @xmath77 ) are invariant under @xmath78 . @xmath79 leaving aside @xmath15 properties for the moment , to which we return below , out of any @xmath80 doublet @xmath81 one can construct two quartic invariants @xmath82 and @xmath83 . needless to say , this can be generalised to the case where several distinct @xmath80 doublets are involved . in order to avoid cluttering , in eq . ( [ vd_s3 ] ) we have displayed only the first combination for all quartic terms . the quartic terms involving @xmath84 to @xmath85 in eq . ( [ vd_s3 ] ) are combinations of two pairs of @xmath15 doublets . each pair can combine in accordance to @xmath86 resulting in three terms . the @xmath15 invariant in the potential arises from a combination of the @xmath87 from one pair with the corresponding term from the other pair . thus , for each such term of four @xmath15 doublets , three possible singlet combinations exist ( recall , eq . ( [ s3prodapp ] ) ) and we have to keep an account of all of them . we elaborate on this using as an example the @xmath88 term which actually stands for a set of terms : @xmath89 ^ 2 + \lambda^d_{1_{1'}}\left[(\phi_1^\dagger\phi_1)-(\phi_2^\dagger\phi_2)\right]^2 + \lambda^d_{1_{2}}\left[(\phi_1^\dagger\phi_2)(\phi_2^\dagger\phi_1 ) + ( \phi_2^\dagger\phi_1)(\phi_1^\dagger\phi_2)\right].\ ] ] substituting @xmath90 , @xmath91 and @xmath92 and defining @xmath93 and @xmath94 we get : @xmath95 + { \lambda^d_{a_2}\over2}(v_1^*v_1)(v_2^*v_2 ) . \label{t6}\ ] ] similarly , @xmath96 + { \lambda^d_{b_2}\over2}(v_3^*v_3)(v_4^*v_4 ) \label{t7}\ ] ] where , @xmath97 and @xmath98 . further , @xmath99 & \rightarrow & \lambda^d_{3_{1}}\left[(\phi_1^\dagger\phi_1+\phi_2^\dagger\phi_2)(\phi_3^\dagger\phi_3+\phi_4^\dagger\phi_4)\right ] + \lambda^d_{3_{1'}}\left[(\phi_1^\dagger\phi_1-\phi_2^\dagger\phi_2)(\phi_3^\dagger\phi_3-\phi_4^\dagger\phi_4)\right ] \nonumber\\ & + & \lambda^d_{3_{2}}\left[(\phi_1^\dagger\phi_2)(\phi_4^\dagger\phi_3 ) + ( \phi_2^\dagger\phi_1)(\phi_3^\dagger\phi_4)\right].\end{aligned}\ ] ] substituting the respective @xmath90 and defining @xmath100 , @xmath101 and @xmath102 we get ; @xmath99 & \longrightarrow & { \lambda^d_{{ab}_1}\over 2 } \left[(v_1^*v_1)(v_3^*v_3)+(v_2^*v_2)(v_4^*v_4)\right ] + { \lambda^d_{{ab}_2}\over 2 } \left[(v_1^*v_1)(v_4^*v_4)+(v_2^*v_2)(v_3^*v_3)\right ] \nonumber\\ & + & \lambda^d_{{ab}_3 } \left[(v_1^*v_2)(v_4^*v_3)+(v_2^*v_1)(v_3^*v_4)\right ] . \label{t8}\end{aligned}\ ] ] in a similar fashion the @xmath103 term when expanded will lead to @xmath104 & \longrightarrow & { \tilde{\lambda}^d_{{ab}_1}\over 2 } \left[(v_1^*v_3)(v_3^*v_1)+(v_2^*v_4)(v_4^*v_2)\right ] + { \tilde{\lambda}^d_{{ab}_2}\over 2 } \left[(v_1^*v_3)(v_4^*v_2)+(v_2^*v_4)(v_3^*v_1)\right ] \nonumber\\ & + & \tilde{\lambda}^d_{{ab}_3 } \left[(v_1^*v_4)(v_4^*v_1)+(v_2^*v_3)(v_3^*v_2)\right ] . \label{t9}\end{aligned}\ ] ] adding eqs.([t8 ] ) and eq.([t9 ] ) we get : @xmath99 + { \lambda_4^d\over 2}\left[(\phi_a^\dagger\phi_b)(\phi_b^\dagger\phi_a)\right ] & = & { \hat{\lambda}^d_{{ab}_1}\over 2 } \left[(v_1^*v_1)(v_3^*v_3)+(v_2^*v_2)(v_4^*v_4)\right ] \nonumber\\ & + & { \hat{\lambda}^d_{{ab}_2}\over 2 } \left[(v_1^*v_1)(v_4^*v_4)+(v_2^*v_2)(v_3^*v_3)\right ] \nonumber\\ & + & \hat{\lambda}^d_{{ab}_3 } \left[(v_1^*v_2)(v_4^*v_3)+(v_2^*v_1)(v_3^*v_4)\right ] ; \label{sum_t8nt9}\end{aligned}\ ] ] where , @xmath105 , @xmath106 and @xmath107 . also , summing up the @xmath108 and @xmath109 terms lead to @xmath110 , where @xmath111 . both the @xmath61 triplets present in our model ( @xmath112 , @xmath113 ) that are responsible for majorana mass generation of the left handed neutrinos happen to be @xmath15 invariants and differ only in their @xmath78 properties i.e. , @xmath114 and @xmath115 . @xmath116 ^ 2 + { \lambda_2^t \over 2}\left[\rho_l^\dagger\rho_l\right]^2 + { \lambda_3^t \over 2}(\delta_l^\dagger\delta_l)(\rho_l^\dagger\rho_l ) \nonumber \\ & + & { \lambda_4^t\over 2}(\delta_l^\dagger\rho_l)(\rho_l^\dagger\delta_l ) + { \lambda_5^t \over 2}(\delta_l\rho_l)(\delta_l\rho_l)^\dagger \;\;. \label{vt_s3}\end{aligned}\ ] ] it is noteworthy that when we write the minimized potential in terms of the vacuum expectation values , the @xmath117 , @xmath118 and @xmath119 terms will be providing the same contribution as far as potential minimization is concerned . thus we can club these couplings together as @xmath120 . so far we have listed those terms in the potential which arise from scalars of any specific @xmath61 behaviour singlets , doublets , or triplets . in addition , there can be terms which couple one of these sectors to another . since the vacuum expectation values of the singlet scalars are the largest we only consider here the couplings of this sector to the others . the @xmath61 triplet sector vev is very small and we drop the doublet - triplet cross - sector couplings . couplings between the @xmath61 singlet and doublet scalars in the potential give rise to the terms : @xmath121 + \lambda_2^{ds}\left[(\phi_b^\dagger\phi_a)_{1}\chi + h.c.\right ] + \lambda_3^{ds}\left[(\alpha^\dagger\eta)\chi + h.c.\right ] \nonumber \\ & + & { \lambda_1^{ds}\over 2}(\phi_a^\dagger\phi_a)(\chi^\dagger\chi ) + { \lambda_2^{ds}\over 2}(\phi_a^\dagger\phi_a)(\gamma^\dagger\gamma ) + { \lambda_3^{ds}\over 2}(\phi_b^\dagger\phi_b)(\chi^\dagger\chi ) + { \lambda_4^{ds}\over 2}(\phi_b^\dagger\phi_b)(\gamma^\dagger\gamma ) \nonumber \\ & + & { \lambda_5^{ds}\over 2}(\alpha^\dagger\alpha)(\chi^\dagger\chi ) + { \lambda_6^{ds}\over 2}(\alpha^\dagger\alpha)(\gamma^\dagger\gamma ) + { \lambda_7^{ds}\over 2}(\eta^\dagger\eta)(\chi^\dagger\chi ) + { \lambda_8^{ds}\over 2}(\eta^\dagger\eta)(\gamma^\dagger\gamma ) \nonumber \\ & + & \lambda_9^{ds}\left[(\phi_a^\dagger\phi_b)\chi^2 + h.c.\right ] + \lambda_{10}^{ds}\left[(\phi_a^\dagger\phi_b)\gamma^2 + h.c.\right ] + \lambda_{11}^{ds}\left[(\eta^\dagger\alpha)\chi^2 + h.c.\right ] + \lambda_{12}^{ds}\left[(\eta^\dagger\alpha)\gamma^2 + h.c.\right ] \nonumber \\ & + & \lambda_{13}^{ds}\left[(\phi_a^\dagger\phi_b)_{1'}(\chi\gamma)+ h.c.\right ] + { \lambda_{14}^{ds}\over 2}(\beta^\dagger\beta)(\chi^\dagger\chi ) + { \lambda_{15}^{ds}\over 2}(\beta^\dagger\beta)(\gamma^\dagger\gamma ) \;\;. \label{vds_s3}\end{aligned}\ ] ] the terms in the potential which arise from couplings between the @xmath61 singlet and triplet scalars are : @xmath122 + { \lambda_1^{ts } \over 2 } ( \delta_l^\dagger\delta_l)(\chi^\dagger\chi ) + { \lambda_2^{ts}\over 2}(\delta_l^\dagger\delta_l)(\gamma^\dagger\gamma ) + { \lambda_3^{ts } \over 2}(\rho_l^\dagger\rho_l)(\chi^\dagger\chi ) \nonumber \\ & + & { \lambda_4^{ts}\over 2}(\rho_l^\dagger\rho_l)(\gamma^\dagger\gamma ) + \lambda_5^{ts } \left \ { ( \delta_l^\dagger\rho_l)\chi^2 + h.c.\right \ } + \lambda_6^{ts } \left \ { ( \delta_l^\dagger\rho_l)\gamma^2 + h.c.\right \ } \;\;. \label{vts_s3}\end{aligned}\ ] ] @xmath61 doublets : @xmath125 , @xmath126 , @xmath127 , @xmath128 and @xmath129 . recall that from the structure of the charged lepton mass matrix eq . ( [ vevrat ] ) requires @xmath130 where the real quantity @xmath131 . we often also need @xmath132 . define @xmath137 . @xmath138 \nonumber\\ & + & v_\alpha\left [ { \lambda_5^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_6^{ds}\over2}(u_\gamma^*u_\gamma ) \right ] + v_\eta\left [ \lambda_{17}^d ( v_1^*v_3 ) b + \lambda_3^{ds}u_\chi + \lambda_{11}^{ds}(u_\chi^*)^2 + \lambda_{12}^{ds}(u_\gamma^*)^2 \right ] = 0.\nonumber\\ \label{mint_d1}\end{aligned}\ ] ] @xmath140 \nonumber\\ & + & v_\eta\left [ { \lambda_{7}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{8}^{ds}\over2}(u_\gamma^*u_\gamma ) \right ] + v_\alpha\left [ \lambda_{17}^d(v_3^*v_1)b + \lambda_3^{ds}u_\chi^*+\lambda_{11}^{ds}(u_\chi)^2 + \lambda_{12}^{ds}(u_\gamma)^2 \right ] \nonumber\\ & = & 0 . \label{mint_d3}\end{aligned}\ ] ] @xmath141 \nonumber\\ & + & v_1\left [ \left\{{\lambda_{5}^{d}\over2}(v_\eta^*v_\eta ) + { \lambda_{6}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{7}^{d}\over2}(v_\beta^*v_\beta ) \right \ } + \left\{{\lambda_{1}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{2}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & v_3\left [ \left\{\lambda_{17}^d(v_\alpha^*v_\eta ) \right \ } + \left\ { \lambda_1^{ds}u_\gamma^*+ \lambda_2^{ds}u_\chi^*+ \lambda_9^{ds}(u_\chi)^2 + \lambda_{10}^{ds}(u_\gamma)^2 + \lambda_{13}^{ds}(u_\chi u_\gamma ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d4}\end{aligned}\ ] ] @xmath142 \nonumber\\ & + & av_1\left [ \left\{{\lambda_{5}^{d}\over2}(v_\eta^*v_\eta ) + { \lambda_{6}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{7}^{d}\over2}(v_\beta^*v_\beta ) \right \ } + \left\{{\lambda_{1}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{2}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & av_3\left [ \left\{\lambda_{17}^d(v_\alpha^*v_\eta ) \right \ } + \left\ { -\lambda_1^{ds}u_\gamma^*+ \lambda_2^{ds}u_\chi^*+ \lambda_9^{ds}(u_\chi)^2 + \lambda_{10}^{ds}(u_\gamma)^2 -\lambda_{13}^{ds}(u_\chi u_\gamma ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d5}\end{aligned}\ ] ] @xmath143 \nonumber\\ & + & v_3\left [ \left\ { { \lambda_{8}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{9}^{d}\over2}(v_\beta^*v_\beta ) + { \lambda_{10}^{d}\over2}(v_\eta^*v_\eta ) \right \ } + \left\{{\lambda_{3}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{4}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & v_1\left [ \left\{\lambda_{17}^d(v_\eta^*v_\alpha ) \right \ } + \left\ { \lambda_1^{ds}u_\gamma+ \lambda_2^{ds}u_\chi+ \lambda_9^{ds}(u_\chi^*)^2 + \lambda_{10}^{ds}(u_\gamma^*)^2 + \lambda_{13}^{ds}(u_\chi^ * u_\gamma^ * ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d6}\end{aligned}\ ] ] @xmath144 \nonumber\\ & + & av_3\left [ \left\ { { \lambda_{8}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{9}^{d}\over2}(v_\beta^*v_\beta ) + { \lambda_{10}^{d}\over2}(v_\eta^*v_\eta ) \right \ } + \left\{{\lambda_{3}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{4}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & av_1\left [ \left\{\lambda_{17}^d(v_\eta^*v_\alpha ) \right \ } + \left\ { -\lambda_1^{ds}u_\gamma+ \lambda_2^{ds}u_\chi+ \lambda_9^{ds}(u_\chi^*)^2 + \lambda_{10}^{ds}(u_\gamma^*)^2 -\lambda_{13}^{ds}(u_\chi^ * u_\gamma^ * ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d7}\end{aligned}\ ] ] 100 see , for example , p. f. harrison and w. g. scott , phys . lett . b * 557 * , 76 ( 2003 ) [ hep - ph/0302025 ] . s. l. chen , m. frigerio and e. ma , phys . d * 70 * , 073008 ( 2004 ) erratum : [ phys . d * 70 * , 079905 ( 2004 ) ] [ hep - ph/0404084 ] ; 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( t2k collaboration ) , arxiv:1502.01550v2 [ hep - ex ] ( see fig .

_ we develop a see - saw model for neutrino masses and mixing with an @xmath0 symmetry . it involves an interplay of type - i and type - ii see - saw contributions of which the former is subdominant . the @xmath1 quantum numbers of the fermion and scalar fields are chosen such that the type - ii see - saw generates a mass matrix which incorporates the atmospheric mass splitting and sets @xmath2 . the solar splitting and @xmath3 are absent , while the third mixing angle can achieve any value , @xmath4 . specific choices of @xmath4 are of interest , e.g. , @xmath5 ( tribimaximal ) , @xmath6 ( bimaximal ) , @xmath7 ( golden ratio ) , and @xmath8 ( no solar mixing ) . the role of the type - i see - saw is to nudge all the above into the range indicated by the data . the model results in novel interrelationships between these quantities due to their common origin , making it readily falsifiable . for example , normal ( inverted ) ordering is associated with @xmath9 in the first ( second ) octant . cp - violation is controlled by phases in the right - handed neutrino majorana mass matrix , @xmath10 . in their absence , only normal ordering is admissible . when @xmath10 is complex the dirac cp - phase , @xmath11 , can be large , i.e. , @xmath12 , and inverted ordering is also allowed . the preliminary results from t2k and nova which favour normal ordering and @xmath13 are indicative , in this model , of a lightest neutrino mass of 0.05 ev or more . _ ` key words : neutrino mixing , \theta_{13 } , solar splitting , s3 , see - saw , leptonic cp - violation ` + soumita pramanick@xmath14 , amitava raychaudhuri@xmath14 +

the vast majority of the free electrons in the ism of the milky way reside in a thick ( @xmath9 900-pc scale height ) diffuse layer known as the reynolds layer or the warm ionized medium ( e.g. reynolds 1993 ) . this phase fills about 20% of the ism volume , with a local midplane density of about 0.1 @xmath10 . such a phase is now known to be a general feature of external star - forming galaxies , both spirals ( e.g. walterbos 1997 ; rand 1996 ) and irregulars ( e.g. hunter & gallagher 1990 ; martin 1997 , hereafter m97 ) , where it is commonly referred to as diffuse ionized gas ( dig ) . however , for edge - on spirals , only in the more actively star - forming galaxies does the gas manifest itself as a smooth , widespread layer of emission detectable _ above _ the hii region layer ( rand 1996 ) . one such galaxy , ngc 891 , is an attractive target for study , not only because of its prominent dig layer ( rand , kulkarni , & hester 1990 ; dettmar 1990 ) , but also its proximity ( @xmath11 mpc will be assumed here ) and nearly fully edge - on aspect ( @xmath12 ; swaters 1994 ) . one of the outstanding problems in the astrophysics of the ism is the ionization of these layers . for the reynolds layer , the local ionization requirement ( @xmath13 s@xmath14 per @xmath15 of galactic disk ; reynolds 1992 ) is comfortably exceeded ( by a factor of 6 or 7 ) only by the ionizing output of massive stars . alternatively , the ionization would require essentially all the power put out by supernovae ( reynolds 1984 ) hence , this energy source could contribute at some level but probably can not explain all of the diffuse emission . photo - ionization models , on the other hand , must explain how the ionizing photons can travel @xmath9 1 kpc or more from their origin in the thin disk of massive stars to maintain this distended layer . crucial information on both the ionization and thermal balance of dig comes from emission line ratios . in the reynolds layer , ratios of [ s@xmath2ii ] @xmath16 and [ n@xmath2ii ] @xmath17 to h@xmath1 are generally enhanced relative to their hii - region values , while [ o@xmath2iii ] @xmath18/h@xmath1 is much weaker . these contrasts are in accordance with models in which photons leak out of hii regions and ionize a larger volume , with the radiation field becoming increasingly diluted with distance from the hii region [ mathis 1986 ; domg@xmath19rgen , & mathis 1994 ; sokolowski 1994 ( hereafter s94 ; see also bland - hawthorn , freeman , & quinn 1997 ) ] . the effect of this dilution , measured by the ionization parameter , @xmath20 , is primarily to allow species such as s and o , which are predominantly doubly ionized in hii regions , to recombine into a singly ionized state . the effect may be less noticeable for n because it is mostly singly ionized in hii regions . the wisconsin h@xmath1 mapper ( wham ) has been used to determine [ o@xmath2i]/h@xmath1 in three low - latitude directions , resulting in values @xmath21 to 0.04 ( haffner & reynolds 1997 ) . such low values imply , since the ionization of o and h are strongly coupled by a charge exchange reaction , that the diffuse gas is nearly completely ionized ( reynolds 1989 ) . although weak , [ o@xmath2iii ] emission has been detected in two directions in the reynolds layer at @xmath22 ( reynolds 1985 ) , with the result [ o@xmath2iii]/h@xmath23 . reynolds postulated that the [ o@xmath2iii ] emission does not arise from diluted stellar ionization but from gas at about 10@xmath24 k , presumably the same gas as seen in c@xmath2 iv @xmath251550 and o@xmath2iii ] @xmath26 emission by martin & bowyer ( 1990 ) . the origin of this rapidly cooling gas is unclear . [ o@xmath2iii ] emission from the dig of ngc 891 and the implications for dig ionization is one of the main subjects of this paper . in external spiral galaxies , smooth increases in [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 vs. distance from hii regions have been observed in both the in - plane and vertical directions in accordance with photo - ionization models [ walterbos & braun 1994 ; dettmar & schulz 1992 ; rand 1997a ( hereafter r97 ) ; golla , dettmar , & domg@xmath27rgen 1996 ; greenawalt , walterbos , & braun 1997 ; wang , heckman , & lehnert 1997 ] . the same trends are seen in irregulars ( hunter & gallagher 1990 ; m97 ) . this behavior has been revealed in ngc 891 through spectra using long slits oriented vertically to the plane . dettmar & schulz ( 1992 ) placed a slit at @xmath28 ne of the nucleus , while r97 took a deeper spectrum at @xmath29 ne . r97 found that [ n@xmath2ii]/h@xmath1 rises to a value of 1.4 , implying a very hard ionizing spectrum . s94 , which pays particular attention to modeling the dig of ngc 891 , can predict such a high value only by assuming a stellar imf extending to 120 m@xmath30 , a reduction in cooling efficiency due to elemental depletions , and hardening of the radiation field by the intervening gas . [ o@xmath2i]/h@xmath1 was not detected by dettmar & schulz ( 1992 ) in the halo of ngc 891 at @xmath28 ne , with an upper limit of 0.05 . dettmar ( 1992 ) also reported an upper limit on [ o@xmath2iii]/h@xmath4 of 0.4 at the same location . a wealth of forbidden - line long - slit data on bright dig and hii regions in irregular galaxies has recently been published by m97 . through the use of line - diagnostic diagrams ( e.g. baldwin , phillips , & terlevich 1981 ; veilleux & osterbrock 1987 ) , she finds that while photo - ionization models can explain the line ratio behavior in many galaxies , the rather shallow fall - off of [ o@xmath2iii]/h@xmath4 with distance from hii regions and the sharp rise in [ o@xmath2i]/h@xmath1 seen in some galaxies imply a second source of ionization . shocks are favored as the most likely second source . the forbidden lines , though bright , are sensitive to metallicity and temperature and thus their interpretation in terms of ionization scenarios is complicated by uncertainties in abundances , degree of depletion , and sources of non - ionization heating . a more direct constraint on the ionizing spectrum has come from the very weak he@xmath2i @xmath255876 line . / h@xmath1 is relatively easy to interpret in terms of the ratio of helium- to hydrogen - ionizing photons , allowing the hardness of the ionizing spectrum , the mean spectral type of the responsible stars , and the upper imf cutoff to be inferred , assuming pure stellar photoionization . the results for the reynolds layer ( reynolds & tufte 1995 ) , for hi worms from equivalent radio recombination lines ( heiles et al . 1996 ) and for ngc 891 ( r97 ) all imply a much softer spectrum than do the forbidden lines . further consequences of this discrepancy are discussed in the above three references . the goal of this paper is to make further progress in understanding the ionization of dig in spirals . the motivations are two - fold . first , the dig halo of ngc 891 features several bright filaments and shells . it is likely that some of these are chimney walls ( norman & ikeuchi 1989 ) surrounding regions of space evacuated by many supernovae . in this case , radiation from any continuing star formation near the base of the chimney will have an unimpeded journey to the walls , and thus the filaments may be directly ionized and show a spectrum more like an hii region than diffuse gas , which receives a significant contribution from relatively soft diffuse re - radiation ( norman 1991 ) . if true , then the filaments should show lower [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 than the surrounding gas . to this end , a spectrum has been taken with a slit oriented parallel to the major axis , but offset into the halo gas , traversing several filaments . the second purpose is to study in more detail the dependence of line ratios on @xmath6 beyond the results reported in r97 for he@xmath2i / h@xmath1 and [ n@xmath2ii]/h@xmath1 . by adding measurements of [ s@xmath2ii ] , [ o@xmath2iii ] , [ o@xmath2i ] , and h@xmath4 , one can form diagnostic diagrams and thus constrain the source(s ) of ionization in the spirit of m97 . the spectra were obtained at the kpno 4-m telescope on 1996 december 1213 . the slit positions are shown in figure 1 ( plate 00 ) overlaid on the h@xmath1 image of rkh and on a version of the image in which a median filter has been applied and the resulting smooth image subtracted to reveal the filaments clearly . note also how the filaments connect onto the brightest hii regions in the disk , with few exceptions . for the first night , the slit was oriented parallel to the major axis and offset from it by 15 along the se side of the minor axis . this slit position will be referred to as the parallel slit . the slit position for the second night was the same as in r97 s observations : oriented perpendicular to the major axis and centered @xmath31 on the nw ( approaching ) side of ngc 891 the perpendicular slit . the slit length is 5 , and the spatial scale is 0.69 " per pixel . the kpc-24 grating was used with the t2 kb 2048x2048 ccd , providing a dispersion of 0.53@xmath32 per pixel , a resolution of 1.3@xmath32 , and a useful coverage of about 800 @xmath32 . for observations of red lines , the grating was tilted to give a central wavelength of about 6600@xmath32 , allowing h@xmath1 , [ n@xmath2ii ] @xmath17 , [ s@xmath2ii ] @xmath16 , and [ o@xmath2i ] @xmath33 to be observed . for the blue lines [ o@xmath2iii ] @xmath34 and h@xmath4 , the central wavelength was set to about 5000@xmath32 . for the parallel slit , seven half - hour spectra were taken , along with separate sky exposures because no part of the slit covered pure sky . for the perpendicular slit , seven half - hour spectra were taken covering the blue lines , and three covering the red lines . sky subtraction for these spectra was achieved using regions of pure sky at both ends of the slit . exact slit center positions were varied to allow removal of chip defects in the stacking process . the reduction was carried out with the iraf package . small - scale variations in response were removed using projector flats . the slit illumination correction was determined with sky flats , and the spectral response function with standard stars . arc lamp exposures were used to calibrate the wavelength scale as a function of location along the slit . the final calibrated , sky - subtracted spectra were spatially aligned and stacked . the noise in continuum - free regions of the stacked perpendicular slit exposures is @xmath35 and @xmath36 erg @xmath37 s@xmath14 @xmath38 arcsec@xmath39 in the blue and red spectra , respectively . for the stacked parallel slit exposures , continuum covers the entire slit length and consequently the pixel - to - pixel variations are higher : @xmath40 erg @xmath37 s@xmath14 @xmath38 arcsec@xmath39 . since the spectral lines appear well represented by gaussians , line properties were determined with gaussian fits . the continuum level was estimated from a linear fit to the continuum on each side of the line . error bars reflect the noise in the spectra and the uncertainty in the fit of the spectral response function . line ratios for the perpendicular slit are consistent with those in r97 , given the calibration and slit positioning uncertainties . however , there is a mistake in the velocity scale of figure 10 of r97 : all velocities are about 35 km s@xmath14 too low . the h@xmath1 , [ n@xmath2ii ] , and [ s@xmath2ii ] lines were detected along the entire useable slit length , which corresponds to about 15 kpc at the assumed distance of ngc 891 . reliable parameters could be measured over a 13 kpc region . [ o@xmath2i ] is detected from about 3 kpc sw of the peak in continuum emission to the ne edge of the slit . however , confusion with a sky line limits measurement of reliable parameters on the ne side to a maximum distance of about 5 kpc from the continuum peak . figure 2 shows the runs of [ s@xmath2ii ] @xmath41/h@xmath1 , [ n@xmath2ii ] @xmath256583/h@xmath1 , [ o@xmath2i ] @xmath42/h@xmath1 , [ s@xmath2ii ] @xmath41/[n@xmath2ii]@xmath256583 , and normalized h@xmath1 surface brightness with position along the slit . the data have been averaged over 10 pixels , or about 300 pc , except in the case of [ o@xmath2i]/h@xmath1 , where the averaging is over 20 pixels . figure 1b shows that the slit traverses four bright filaments . these are apparent as peaks in the h@xmath1 profile ( marked in figure 2a ) . the emission is also obviously much brighter on the ne side than on the sw side , confirming the result from the images of rkh and dettmar ( 1990 ) . the ratio of the two [ s@xmath2ii ] lines is consistent with the low - density limit of 1.5 ( osterbrock 1989 ) , even for the filaments . at first glance , it would seem that the data support the expectation outlined in i : there is a definite reduction in [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 at the positions of the four bright filaments . [ o@xmath2i ] @xmath42/h@xmath1 also tends to be lower at the filament crossings . however , this result reflects a more general trend of these line ratios with h@xmath1 surface brightness , as shown in figure 3 . the ratios at the positions of the three brightest filaments define the correlation at h@xmath1 surface brightnesses @xmath43 erg @xmath37 s@xmath14 arcsec@xmath39 . there is no evidence for a discontinuity or steepening of the correlation at these intensities . in fact , the slope becomes nearly flat here . what is the reason for these very good correlations ? very similar results are found for the perpendicular slit data ( figure 7 ) , suggesting , in a stellar photo - ionization scenario , that they reflect the well - established variation ( e.g. s94 ) of line ratios with ionization parameter , @xmath20 , which measures the diluteness of the radiation field ( see 1 ) . if this is the case , then it is implied that @xmath20 is lower along lines of sight with faint dig , even when comparing at the same @xmath6 . gas along lines of sight with faint h@xmath1 emission may be relatively remote from ionizing stars in the disk , so that large columns of intervening gas and geometric dilution result in a low @xmath20 . alternatively , it is possible that such gas is no more remote from ionizing stars , but that @xmath20 is low because the responsible clusters feature fewer such stars , leading to an intrinsically weak emergent ionizing radiation field . the fact that the lines of sight passing through the filaments feature the brightest dig while the filaments are clearly associated with bright visible hii regions in the disk ( figure 1b ) would tend to suggest that the former explanation is correct . however , the dust lane may be hiding numerous fainter hii regions whose stars may be the primary source of ionization for gas along lines of sight with fainter emission . on the other hand , such hii regions should contain fewer massive stars and have , if the imf varies little , a lower probability of containing the most massive stars , resulting in softer spectra on average . a softer spectrum leads to lower line ratios ( s94 ) and would offset the dilution effect to some degree . this question will probably be resolved from studies of dig in more face - on galaxies where the in - plane variations of h@xmath1 surface brightness and line ratios can be related to the distribution of ionizing stars . regardless of the explanation , since the filaments in ngc 891 follow the overall correlation in figure 3 , it can not be claimed that the line ratios are lower in the filaments because they surround _ evacuated _ regions and are directly ionized by hii regions below . this does not imply that the filaments are not chimney walls , but does point out the potential difficulties in deriving information on isolated structures in edge - on galaxies . it is interesting that [ s@xmath2ii]/[n@xmath2ii ] shows almost no spatial variation or dependence on h@xmath1 surface brightness compared to the other line ratios . this ratio will be discussed further in the next section , where it will become clear that there is little dependence on @xmath6 either . figure 4 shows heliocentric velocity centroids , formed from a weighted average of the h@xmath1 , [ s@xmath2ii ] , and [ n@xmath2ii ] line centroids , along with the emission profile . one of the filaments on the ne side and a broad region centered on a filament on the sw side show velocities further from the systemic velocity than expected from the smooth , nearly linear trend of velocity with position . again , this is not necessarily an indication that the filaments have peculiar velocities , but may simply indicate that they are located in the inner disk . if so , then these velocities are more heavily weighted by inner disk material than are adjacent ones . inner disk gas will show velocities further from the systemic velocity compared to outer disk gas because of the greater projection of the rotation velocity vector along the line of sight . hence , the velocity deviations may simply be due to geometrical effects . the h@xmath1 line , the [ n@xmath2ii ] @xmath44 line and the [ s@xmath2ii ] lines are detected up to about @xmath45 kpc on each side of the plane . [ o@xmath2i ] is detected to about half this height , while [ o@xmath2iii ] and h@xmath4 are detected up to about @xmath46 kpc . shown in figure 6 are the vertical runs of [ s@xmath2ii ] @xmath47/h@xmath1 , [ n@xmath2ii ] @xmath44/h@xmath1 , [ o@xmath2i]/h@xmath1 , [ s@xmath2ii ] @xmath41/[n@xmath2ii ] @xmath44 , and [ o@xmath2iii]/h@xmath4 . the h@xmath1 profile is also plotted the local minimum at @xmath7 kpc is due to the dust lane . the data are averaged over 10 pixels . [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 show a smooth and remarkably similar increase with @xmath6 , from about 0.35 in the midplane to over 1.0 at @xmath48 kpc . [ s@xmath2ii]/[n@xmath2ii ] is nearly constant at about 0.6 at all @xmath6 where the uncertainties are not too large . [ o@xmath2i]/h@xmath1 increases from about 0.03 at @xmath7 kpc to 0.08 at @xmath49 kpc . however , the most surprising result is that [ o@xmath2iii]/h@xmath4 _ rises _ from 0.3 in the midplane to 0.8 at @xmath50 kpc . figure 7 shows the correlation of these ratios with h@xmath1 surface brightness . these are very similar to the correlations in figure 3 . again , the ratio of the two [ s@xmath2ii ] lines is everywhere consistent with the low - density limit of 1.5 . again assuming @xmath51k , [ o@xmath2i]/h@xmath1 values imply h is essentially 100% ionized at @xmath7 kpc , decreasing to about 90% at @xmath8 kpc . if @xmath52k , then these ionization fractions are 90% and 80% . except for the discrepancy mentioned in 2 , the mean velocities of the lines are consistent with the results of r97 and give no additional information . therefore , we will not discuss the kinematics further . we now attempt to understand whether the above emission line properties can be understood by massive - star photo - ionization alone . in doing so , we temporarily ignore the problems posed by the low he@xmath2i / h@xmath1 but will return briefly to the reconciliation of this ratio with the forbidden line ratios in 4 . we use unpublished models from s94 since they are the only models which specifically attempt to reproduce the line ratios in the dig of ngc 891 . in these models an ionizing spectrum of radiation from a population of stars with an imf slope of 2.7 ( intermediate between salpeter and miller - scalo values ) and stellar atmospheres from kurucz ( 1979 ) is considered . this radiation field is allowed to propagate through a slab of gas ( representing a clump of halo gas ) in a one - dimensional calculation , the dilution being measured by the ionization parameter , @xmath20 , at the front of the slab . as discussed in 1 , as lower values of @xmath20 are considered , the predominant ionization state of s and o ( and n to a lesser extent ) changes from doubly to singly ionized . as a consequence , [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 rise while [ o@xmath2iii]/h@xmath4 falls . at the end of the model slab , an increasingly neutral zone appears as lower values of @xmath20 are considered , leading to a slow rise in [ o@xmath2i]/h@xmath1 with @xmath20 . in this one - dimensional calculation of pure photo - ionization , @xmath20 is expected to decline exponentially with @xmath6 . this dependence is probably more complicated in a real galaxy , although @xmath20 should generally fall with increasing height . thus , further free parameters are the value of @xmath20 at @xmath7 kpc , and the run of @xmath20 with @xmath6 . both radiation bounded models and matter bounded models with various terminating total atomic hydrogen columns for the clumps were considered by s94 . we will use only his models with the hardest stellar spectrum considered ( with an upper imf cutoff of 120 m@xmath30 ) , hardening of the radiation field as it propagates through the intervening gas before reaching the slab , and heavy element depletions . only these models are able to yield [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 in the range 11.5 . these models have also been recently published by bland - hawthorn et al . we examine as two extremes the matter - bounded model ( which will be referred to as pm ) with the lowest terminal hydrogen column considered for the individual clumps , 2@xmath53 @xmath37 , and the radiation - bounded model ( pr ) . figures 8 and 9 show these ratios in diagnostic diagrams of [ o@xmath2iii]/h@xmath4 and [ o@xmath2i]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 , and [ s@xmath2ii]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 . along the sequence of points , @xmath6 generally increases from 0 kpc at the left end to 2 kpc for plots of [ o@xmath2iii]/h@xmath4 , 1.3 kpc for [ o@xmath2i]/h@xmath1 , and 3 kpc for [ s@xmath2ii]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 at the right end . models pr and pm are shown in figures 8 and 9 as the small open circles joined by solid lines . values of log@xmath54 are marked as explained in the captions . it is immediately obvious that neither model is a good match to the data . most importantly , the flatness of [ o@xmath2iii]/h@xmath4 below @xmath8 kpc and its rise with [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 ( and @xmath6 ) above @xmath8 kpc is at complete odds with the models . also , the typical value of [ o@xmath2iii]/h@xmath4 is poorly predicted by a model chosen to match the observed [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 . in model pr , while [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 require log@xmath54 in the range 2.3 to 2.7 at @xmath7 kpc , and 3.3 to 3.7 at @xmath50 kpc , the predicted [ o@xmath2iii]/h@xmath4 is @xmath55 times that observed for most of this range of @xmath20 . in model pm , on the other hand , if we use [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 to set @xmath20 at @xmath7 kpc , we require log@xmath54 in the range 3.7 to 4.1 , and an extra source of [ o@xmath2iii ] emission at lower @xmath20 . while such a source can be identified and tested ( see below ) , it is not clear that such a low value of @xmath20 should apply to the midplane , given that the disk contains both hii regions and diffuse gas . however , at low @xmath6 the kinematics indicate that we receive emission preferentially from the outer disk because of the absorbing dust layer ( r97 ) . this outer disk gas may , like the high-@xmath6 gas , see a relatively dilute radiation field because star formation is concentrated in the inner disk ( rand 1997b ) . in that case the appropriate @xmath20 for @xmath7 may be quite low and the gradient of @xmath20 with @xmath6 rather shallower than expected in a galaxy without such a dust lane . both models are more successful at reproducing the runs of [ o@xmath2i]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 , [ o@xmath2i]/h@xmath1 vs. [ s@xmath2ii]/h@xmath1 and [ s@xmath2ii]/[n@xmath2ii ] . if [ n@xmath2ii]/h@xmath1 were lower by about 0.2 dex at low @xmath6 , pr would fit these data quite well , with log@xmath56 at @xmath7 kpc , @xmath57 at @xmath8 kpc , and @xmath58 at @xmath50 kpc . model pm would fit equally well for log@xmath59 at @xmath7 , @xmath60 at @xmath8 kpc , and @xmath61 at @xmath50 kpc . a similar discrepancy in the observed vs. modeled vertical run of [ s@xmath2ii]/[n@xmath2ii ] was noted by golla et al . ( 1996 ) in ngc 4631 . [ n@xmath2ii]/h@xmath1 is expected to show less disk - halo contrast than [ s@xmath2ii]/h@xmath1 because the change in predominant ionization state between hii regions and diffuse gas is smaller for n due to its higher ionization potential . this trend is reflected in the models of both s94 and domg@xmath19rgen , & mathis ( 1994 ) . in both ngc 891 and ngc 4631 , however , the disk - halo contrast in [ s@xmath2ii]/h@xmath1 is rather similar to that in [ n@xmath2ii]/h@xmath1 . one important factor in explaining the common behavior of [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 might be a _ radial _ abundance gradient in league with the dust absorption effect noted above . rubin , ford , & whitmore ( 1984 ) found that log ( [ s@xmath2ii]/[n@xmath2ii ] ) generally increases in hii regions in spirals by 0.3 from inner to outer hii regions . if such a gradient is present in ngc 891 , then the fact that we preferentially observe outer disk gas at low-@xmath6 means that [ s@xmath2ii]/[n@xmath2ii ] should be higher there for a given @xmath20 . the gradual inclusion of more inner disk gas with increasing @xmath6 will then tend to offset the dependence of [ s@xmath2ii]/[n@xmath2ii ] on @xmath20 in the models . however , the parallel slit data suggests little dependence of the ratio on distance along the major axis , although there is a slight trend in the right direction between 2 and 7 kpc from the center on both sides . hence , inasmuch as the major - axis dependence of [ s@xmath2ii]/[n@xmath2ii ] at @xmath0 pc reflects the radial dependence at @xmath7 pc , it would seem that an abundance gradient does not affect the line ratios significantly . other expected effects of an abundance gradient are also not seen in the parallel slit data . domg@xmath19rgen , & mathis ( 1994 ) find that [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 increase with abundance , while [ o@xmath2i]/h@xmath1 shows little dependence . in the presence of a radial gradient , these trends , if strong enough , might be revealed as a decline in [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 with distance along the major axis . the dominant trend there is with h@xmath1 brightness and thus it seems that variations in @xmath20 are the more important effect . again , though , it must be noted that the averaging along the line of sight in the parallel slit data will diminish the observable effects of an abundance gradient . so far , it has been difficult to find much variation in [ s@xmath2ii]/[n@xmath2ii ] in the dig of edge - on galaxies . this question remains open to further exploration . the most surprising result for the perpendicular slit is the behavior of [ o@xmath2iii]/h@xmath4 with @xmath6 . although such high ( and higher ) values of [ o@xmath2iii]/h@xmath4 are found in hii regions , they are a feature of high - excitation ( high @xmath20 ) conditions . if the level of excitation were increasing with @xmath6 , however , we would also expect to see [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 falling , contrary to what is observed . as discussed above , their rise is qualitatively consistent with a smooth transition from predominantly doubly - ionized to singly - ionized states , as expected in the dilute photo - ionization models . hence , it is very unlikely that the [ o@xmath2iii ] emission arises from the same dig component as the [ n@xmath2ii ] and [ s@xmath2ii ] . we therefore require a second source of diffuse h@xmath1 emission which features bright [ 0@xmath2iii ] . two plausible mechanisms for producing such emission , shocks and turbulent mixing layers are discussed in this and the next subsection . respectively . both can produce bright [ o@xmath2iii ] emission . these mechanisms were also considered for dig in irregular galaxies by m97 . it should be noted , though , that the dig in these irregulars is generally much brighter than that studied here . we first consider whether the line ratio data can be explained if some of the dig emission is produced by shock ionization . we consider only the pm model further because the pr model would require a second source of h@xmath1 emission with highly unusual properties : if the value of @xmath20 at the midplane is to be roughly 2.3 to 2.7 , then the second component must dominate the stellar - ionized component by a factor of 30 or more and have essentially no [ o@xmath2iii ] emission if the runs of [ o@xmath2iii]/h@xmath4 vs. [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 are to be explained . low - speed shocks do have this property , but also produce large amounts of [ o@xmath2i ] emission , further confouding the problem . alternatively , if log@xmath62 at @xmath7 kpc , then the second component must have essentially no [ n@xmath2ii ] or [ s@xmath2ii ] emission and account for about 75% of the h@xmath1 emission at @xmath7 kpc in order to reproduce the values of [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 . in either case , it is also difficult to see how the subsequent rise in all the line ratios with @xmath6 would be achieved without the model being highly contrived . with the pm model , there is some hope of matching the data by adding a strong source of [ o@xmath2iii ] emission as long as @xmath20 is low enough in the midplane and the other ratios can be matched . we employ the shock models of shull & mckee ( 1979 ) , as were also considered by m97 . the line ratios in shock models , especially [ o@xmath2iii]/h@xmath4 , are very sensitive to the shock velocity . other variables include the preshock gas density and ionization state , abundance , and transverse magnetic field strength . the gas is assumed to be initially neutral at @xmath63 @xmath10 and subsequently penetrated by a precursor ionization front . more appropriate to our case would be a lower initial density ( of order 0.1 @xmath10 ) and a high initial ionization fraction . we should also consider depleted abundances to be consistent with the stellar ionization models , but sm calculated only one such model to show the general effect of depletions . we do not perform an exhaustive search of parameter space or carry out a statistical test of the goodness of fit , firstly because an examination of figures such as figure 9 can quickly reveal which composite models are most successful , and secondly because some of the fixed parameters are probably inappropriate for the halo of ngc 891 in any case . in the composite models , we still consider that the stellar radiation field is characterized by a decrease of @xmath20 with @xmath6 , and that some fraction of h@xmath1 emission from shock ionization is added , with this fraction possibly changing with @xmath6 . although no model can reproduce the line ratio behavior to within the errors , we find that some of the main characteristics of the data can be reproduced . one of the most successful models is shown overlaid on the data in figure 9 as the dashed lines joining open circles , which mark values of log@xmath54 . the shock speed is 90 km s@xmath14 . at @xmath7 kpc , 7% of the h@xmath1 emission arises from a 90 km s@xmath14 shock , rising to 30% by @xmath50 kpc . the composite model is most successful if log@xmath59 at @xmath7 kpc , @xmath60 at @xmath8 kpc ( the limit of the [ o@xmath2i]/h@xmath1 data ) , and @xmath61 at @xmath50 kpc ( the limit of the [ o@xmath2iii]/h@xmath4 data ) . note that in the figure , circles indicate log@xmath64 4.0 , 4.3 , 4.7 and 5.0 , corresponding to @xmath65 kpc , in all the panels despite the fact that the [ s@xmath2ii]/[n@xmath2ii ] ratio includes data up to @xmath45 kpc and [ o@xmath2i]/h@xmath1 is only detected up to @xmath8 kpc . also shown in figure 9 are the line ratios for the shock models alone , namely , a 90 km s@xmath14 and a 100 km s@xmath14 shock with standard abundances , and a 100 km s@xmath14 shock with depleted abundances . the major shortcoming of this composite model is in reproducing [ s@xmath2ii]/[n@xmath2ii ] at low @xmath6 . for a given amount of [ s@xmath2ii ] emission , the model overpredicts [ n@xmath2ii ] . at high @xmath6 this ratio is somewhat more successfully modeled , but [ n@xmath2ii]/h@xmath1 simply does not show as much disk - halo contrast as observed , as was the case for the pure photo - ionization models . also , the predicted [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 reach a plateau at 1.0 at the lowest @xmath20 considered by s94 , and thus can not match the observed continuing rise beyond @xmath50 kpc ( figure 9e ) . in this regard radiation - bounded models are more successful ( s94 ) , at least for [ s@xmath2ii]/h@xmath1 . in that model , it is still rising at the lowest considered @xmath20 value , so that its observed continued rise may be explainable . finally , a low value of @xmath20 at @xmath7 pc is still required , but again this may not be unreasonable given the arguments mentioned above . there is some freedom for variation of the shock parameters . a 130 km s@xmath14 shock ( the highest speed considered by sm ) contributing 2% of the h@xmath1 emission at @xmath7 kpc , rising to 7% at @xmath50 kpc , produces almost identical curves in the diagnostic diagrams . since only the [ o@xmath2iii]/h@xmath4 ratio is large ( 7.35 ) for this model , adding such a small amount of this emission affects mainly this ratio . the composite model can be further explored by keeping the shock contribution constant with @xmath6 but varying the shock speed . a model with a shock giving rise to 25% of the h@xmath1 emission , with a speed of 60 km s@xmath14 at @xmath7 kpc , and 90 km s@xmath14 at @xmath50 kpc ( but with log@xmath66 now ) can reproduce the ranges of [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/[n@xmath2ii ] , and [ o@xmath2iii]/h@xmath4 as successfully as the above models , but low - speed shocks produce far too much [ o@xmath2i ] emission . sm find that the effect of introducing depleted abundances for a 100 km s@xmath14 shock is to raise [ s@xmath2ii]/[n@xmath2ii ] , while [ o@xmath2iii]/h@xmath4 drops and [ o@xmath2i]/h@xmath1 rises . however , assuming that the fractional changes also apply for a 90 km s@xmath14 shock , depleted abundances do not solve the [ s@xmath2ii]/[n@xmath2ii ] problem , even if the composite model is started at a different value of log@xmath54 . the effect on [ s@xmath2ii]/[n@xmath2ii ] is insufficient because of the small contribution from shocks needed to fit [ o@xmath2iii]/h@xmath4 . further latitude in the composite models may be gained by considering a larger range of values for some of the other variables . for instance , dopita & sutherland ( 1996 ) calculate properties of lower density , magnetized shocks with speeds from 150 to 300 km s@xmath14 . they consider densities and magnetic field strengths obeying the relation @xmath67 g @xmath68 . these shocks have strong radiative precursors which contribute a significant fraction of the emission . a model with shock speed 150 km s@xmath14 , @xmath69 @xmath10 and no magnetic field produces line ratios similar to the 90 km s@xmath14 model considered above , with the exception that [ o@xmath2i]/h@xmath1 is about 80% higher . the main conclusion from this subsection is that shocks are feasible as a secondary source of energy input into the dig of ngc 891 , but it is difficult to constrain their parameters with much confidence . we have not speculated on the origin of the putative shocks . but given that the observed filaments suggest the presence of supershells and chimneys , it is plausible that the shocks originate in such expanding structures . the shock speeds considered above are reasonable when compared to superbubble calculations ( e.g. maclow , mccray , & norman 1989 ) . in fact , the slit passes close to one of the most prominent filaments , which may still have an associated shock . this possibility highlights the importance of observing with many such slit positions . finally , we point out that m97 also found that if a constant shock speed model is used , then the fraction of the dig emission that must come from shocks increases with @xmath6 . turbulent mixing layers ( tmls ) are expected to occur at the interfaces of hot and cold ( or hot and warm ) gas in the ism of galaxies ( begelman & fabian 1990 ; slavin , shull , & begelman 1993 ) . superbubble walls are a likely location for such layers . shear flows are expected along the interface , leading to kelvin - helmholtz instabilities and subsequent mixing of the gas . the result is a layer of intermediate temperature gas , probably at @xmath70 . this gas may produce some fraction of the dig emission in galaxies . like typical dig , it features enhanced [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 relative to hii regions and low [ o@xmath2i]/h@xmath1 . however , unlike stellar - ionized gas , [ o@xmath2iii]/h@xmath4 should be of order 13 . hence , tmls provide a source of enhanced [ o@xmath2iii]/h@xmath4 and therefore may be relevant to the current problem . slavin , shull , & begelman ( 1993 ) calculate line ratios for tmls as a function of shear velocity , mixing layer temperature , initial cold layer temperature , and abundance ( solar vs. depleted ) . we will consider depleted abundances here . again , we will introduce a contribution to the dig emission from tmls at some @xmath6 or @xmath20 , allowing for an increase in this contribution with @xmath6 as @xmath20 continues to decline , and consider only the pm model for the photo - ionized component . the best match to the data features a shear velocity of 25 km s@xmath14 ( the lowest modeled ) , a mixing layer temperature of log@xmath71 , and an initially warm layer at 10@xmath72 k rather than a cold layer . this model is shown overlaid on the diagnostic diagrams in figure 10 . at @xmath7 kpc , 3% of the h@xmath1 emission arises from tmls , rising to 15% at @xmath50 kpc . the rough relation of log@xmath54 with @xmath6 is similar to the previous composite model : log@xmath73 at @xmath7 kpc , @xmath60 at @xmath8 kpc ( the limit of the [ o@xmath2i]/h@xmath1 data ) , and @xmath61 at @xmath50 kpc ( the limit of the [ o@xmath2iii]/h@xmath4 data ) . again , note that in the figure , circles indicate log@xmath64 4.1 , 4.3 , 4.7 and 5.0 , corresponding to @xmath65 kpc , in all the panels despite the varying maximum @xmath6 of the line ratio determinations . the model is reminiscent of the best shock models and appears to be somewhat more successful , but suffers from the same primary shortcoming : the underprediction of [ s@xmath2ii]/[n@xmath2ii ] at low @xmath6 . other tml models will not alleviate this problem because the model in figure 10 already features maximal [ s@xmath2ii]/[n@xmath2ii ] . as is the case for shocks , there is room for flexibility in the parameters . for instance , [ o@xmath2iii]/h@xmath4 is fairly constant in all the models with depleted abundances , and all models regardless of abundance feature very low [ o@xmath2i]/h@xmath1 . [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 show somewhat more variation with the input parameters , but as most of the [ s@xmath2ii ] and [ n@xmath2ii ] emission arises from the stellar - ionized gas , and the tml contribution is small , there is reasonable latitude for variation of the parameters without altering the resulting values of these ratios . again , the main point is to demonstrate the feasibility of tmls as a secondary source of energy input rather than to find the best fitting parameters , or indeed to show whether tmls or shocks are preferred as the second component . shapiro & benjamin ( 1993 ) consider cooling , falling galactic fountain gas initially raised from the midplane by supernovae . the calculation is followed from an initial temperature of 10@xmath74 k to a final value of 10@xmath72 k. while there are not yet predictions of optical emission line ratios from such gas , one can expect [ o@xmath2iii ] emission as the gas cools . the run of [ o@xmath2iii]/h@xmath4 with @xmath6 in a composite model with fountain gas would depend on the fraction of diffuse h@xmath1 emission produced by the latter ( the authors estimate that it could account for perhaps 40% in the milky way ) and the details of the cooling and dynamics , including the interaction with halo gas from other processes . another idea that has not been deeply explored is heating of the halo by microflares from magnetic reconnection events ( raymond 1992 ) . this process may have a role in producing ultraviolet emission and absorption lines , soft x - ray halo emission , and reynolds layer emission . as the theory stands now , a broad range of [ o@xmath2iii]/h@xmath1 values may result from this process , and there are enough uncertainties as not to warrant a detailed comparison with the data at this point . the relevance of microflares should be revealed with further refinements of the theory and additional observations . the problem of explaining low he@xmath2i / h@xmath1 in combination with high [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 has motivated work on other sources of heating for the dig . since the forbidden lines are highly temperature sensitive , small changes in temperature can be important . minter & balser ( 1997 ) find that the dissipation of turbulent energy in the ism could raise the temperature of the dig by about 2000 k without additional ionization of helium , thus providing a reasonable match to these line ratios in the reynolds layer . however , there is still insignificant [ o@xmath2iii ] emission . photo - electric heating from dust grains ( reynolds & cox 1992 ) should also have little effect on [ o@xmath2iii ] emission . finally , scattered light from hii regions could be a source of [ o@xmath2iii ] emission in the halo , but the rise in [ o@xmath2iii]/h@xmath4 with @xmath6 would not be expected . using a slit oriented parallel to , and offset 700 pc above , the major axis of ngc 891 , a spectrum of the dig has been taken which reveals a clear correlation of ratios of forbidden lines to h@xmath1 with the h@xmath1 surface brightness . the original motivation for this observation was to search for variations in line ratios on and off the filaments of dig as further evidence that they are walls around evacuated chimneys . but although the filaments do show reduced [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 relative to gas on adjacent lines of sight , the contrast merely reflects the overall correlation . the relationship probably indicates that regions of brighter h@xmath1 emission receive a radiation field with a higher ionization parameter . also , although some of the filaments show deviations from the observed smooth trend of mean velocity with position along the major axis , these departures could simply be due to geometric effects : if the filaments are inner galaxy features , they will bias the mean velocity for their line of sight away from the systemic velocity . [ s@xmath2ii]/[n@xmath2ii ] surprisingly shows no significant variation along the slit . finally , the h@xmath1-emitting halo gas at this height is about 8095% ionized , based on the observed range of [ o@xmath2i]/h@xmath1 and assuming @xmath3 k. the correlation of [ o@xmath2i]/h@xmath1 with surface brightness probably reflects a higher degree of ionization where the photon field is more intense . results from this observation emphasize the difficulty in interpreting dig observations of edge - on galaxies . confusion is caused by uncertainties in the location of a parcel of gas along the line of sight , its effective distance from a source of ionization and other unrelated gas in the same direction . it is difficult from such observations to draw conclusions about the environment of the filament , for example . spectra from a slit oriented perpendicular to the plane at @xmath29 along the major axis on the ne side show a rise of [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 with @xmath6 . at the midplane , [ o@xmath2i]/h@xmath1 values indicate that h is essentially 100% ionized , dropping to 90% at @xmath8 kpc , assuming @xmath75k . the @xmath6-dependence of these line ratios is expected if the gas is ionized by massive stars in the disk . however , it is unexpectedly found that [ o@xmath2iii]/h@xmath4 also rises with @xmath6 , whereas it should decline with @xmath6 in photo - ionization models . this result necessitates the consideration of secondary sources of ionization . [ s@xmath2ii]/[n@xmath2ii ] unexpectedly shows essentially no dependence on @xmath6 and h@xmath1 surface brightness . put another way , [ n@xmath2ii]/h@xmath1 shows the same disk - halo contrast as that of [ s@xmath2ii]/h@xmath1 , whereas a smaller contrast is expected . strong [ o@xmath2iii ] emission is expected from several energetic procesess . we considered shocks and turbulent mixing layers as sources of such gas . models in which a small fraction of the h@xmath1 emission comes from one of these mechanisms can be made to fit the data reasonably well , but most noticeably the remarkable constancy of [ s@xmath2ii]/[n@xmath2ii ] with @xmath6 is still difficult to reproduce . in the case of shocks , it is difficult to constrain the shock speed or the contributed fraction of the dig emission at this point . of course , the line of sight may sample shocks with a range of speeds and some mean value . there is also significant latitude in the parameters in the case of tmls . other sources of strong [ o@xmath2iii ] emission may include cooling galactic fountain gas and microflares from magnetic reconnection . these should be explored further in light of the current results . because of these facts and possibilities , the results are meant only to indicate the feasibility of such classes of models and the likelihood that one or more physical processes is producing intermediate temperature gas in the halo of ngc 891 . on the other hand , the finding that the second source of line emission becomes more important as @xmath6 increases may be reasonable . in the case of tmls , for example , shull & slavin ( 1994 ) point out that this process may indeed be more common at large @xmath6 , where superbubbles break out of the thin disk gas layers , producing rayleigh - taylor instabilities and shear flows that lead to the mixing . these authors were attempting to explain the larger scale - height of c@xmath2iv uv absorption line gas relative to n@xmath2v ( sembach & savage 1992 ) as an increasing predominance of tmls over sn bubbles with height off the plane the former producing higher c@xmath2iv / n@xmath2v . if the tml process begins only at the approximate height where breakout occurs , while the stellar radiation field is increasingly diluted with @xmath6 , then an increasing fraction of h@xmath1 emission from tmls may be quite reasonable . the rough fractions found in 3.2.3 are comparable to those expected by slavin et al . ( 1993 ) for the milky way diffuse h@xmath1 emission . if the photo - ionized and secondary components of the dig emission both arise in exponential layers with different scale - heights , then the composite models , although illustrative , can be used to estimate roughly the relative scale - heights . in the two shock models and one tml model considered , the fraction of emission arising from the second component is 35 times higher at @xmath50 kpc than at @xmath7 kpc . assuming exponential layers , the scale - height of the second component must be 34 times that of the photo - ionized component . this conclusion is very tentative , however , given the uncertainties in the modeling and the lack of information on [ o@xmath2iii]/h@xmath4 at higher @xmath6 . it it is tempting to identify the second component with the high-@xmath6 tail of h@xmath1 emission found in the deeper spectrum of rand ( 1997a ) . however , the exponential scale - height of this tail is 57 times that of the main component , and it contributes about 50% of the emission at @xmath50 kpc . while these numbers do not quite match those for the second component proposed here , there may yet prove to be a connection . for the photo - ionized component , we found the most success by using the model from s94 with the lowest terminal hydrogen column considered for individual clouds in the dig layer . using larger columns tends to push the model curves towards those of the radiation - bounded case , which is found to be very difficult to incorporate in a successful composite model . this constraint suggests that the dig consists of quite small clumps ( or filaments or sheets , since s94 s calculation is one - dimensional ) of several pc thickness , for a representative density of @xmath76 @xmath10 . if this conclusion is not borne out by future observations , the composite models presented here will need to be reconsidered . the s94 model used here also features the hardest emergent stellar spectrum considered ( in order to produce high [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 ) , but more modest spectra may be allowable if other sources of non - ionization heating are at work ( e.g. minter & balser 1997 ) . despite complications introduced by the second dig component , it should be noted that the observed properties of the three ratios [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 with @xmath6 are still reasonably explained _ to first order _ by photo - ionization models alone . their smooth increase with @xmath6 is as predicted as are their rough values . emission from the second component probably has only a secondary effect on these ratios . the @xmath6-independence of [ s@xmath2ii]/[n@xmath2ii ] is not understood in either a pure photo - ionization or composite model . an undesirable aspect of the composite models considered here is that emission from photo - ionized and ( for example ) shock ionized gas is simply added together with no unified physical picture in mind . it would be desirable eventually to have , say , a calculation of the evolution of a superbubble which included photo - ionization , shocks and tmls in a more self - consistent way . for instance , what is the effect of the radiative precursor of a shock which enters gas already ionized by dilute stellar radiation ? the emission line properties revealed by the perpendicular slit share some similarities with those of the halo of the starburst galaxy m82 . in the fabry - perot data of shopbell & bland - hawthorn ( 1997 ) , [ n@xmath2ii]/h@xmath1 shows a general tendency to rise with @xmath6 on the n side , up to the limit of measurability at about @xmath6=750 pc . on the s side , there is little dependence on z , with perhaps 0.6 typical . [ o@xmath2iii]/h@xmath1=0.03 at @xmath7 pc and 0.08 at @xmath77 pc ( values of [ o@xmath2iii]/h@xmath4 are about 3 times higher assuming little extinction ) . in a long - slit spectrum through the halo of m82 , m97 sees higher values of [ o@xmath2iii]/h@xmath4 , reaching 0.7 at @xmath8 kpc . the sequence of points in her diagnostic diagram of [ s@xmath2ii]/h@xmath1 vs. [ o@xmath2iii]/h@xmath4 has a rising slope , as in ngc 891 , although much steeper ( other irregulars show a falling or nearly flat slope ) . [ o@xmath2i]/h@xmath1 also rises with @xmath6 in m82 , and shows a range of values ( about 0.01 to 0.1 ) similar to those reported here . shopbell & bland - hawthorn ( 1997 ) point out that their ratios become more shock - like with distance from the starburst , as also noted by heckman , armus , & miley ( 1990 ) . this behavior is now seen in a spiral halo as well . other evidence for multi - phase halos is provided by the study of ngc 4631 by martin & kern ( 1998 ) . they detect an extensive halo of [ o@xmath2iii ] emission which spatially coexists with the observed soft x - ray and h@xmath1 emitting halos . within this halo are several bright [ o@xmath2iii ] condensations in which the measured [ o@xmath2iii]/h@xmath4 ratio is @xmath78 . on this basis , they argue that the h@xmath1 and [ o@xmath2iii ] emission is tracing distinct components in a multi - phase halo medium . it should be pointed out that [ o@xmath2iii]/h@xmath1 values of 0.120.21 ( comparable to values in ngc 891 ) are seen in the dig of m31 ( greenawalt et al . there is little correlation of this ratio with the brightness of the dig or its remoteness from visible hii regions . the high [ o@xmath2iii]/h@xmath1 in this case may be due to a radiation field not as dilute as in the halo of ngc 891 ( for instance , [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 are substantially lower than in the halo of ngc 891 at @xmath79 kpc ) . greenawalt et al . ( 1997 ) also find that tmls may contribute some fraction of the h@xmath1 emission in regions of very faint dig , but no more than about 20% and most likely only a few percent , similar to the findings for ngc 891 . more light has been shed on the question of [ o@xmath2iii]/h@xmath4 trends in the dig of face - ons by wang et al . ( 1997 ) . for three of their five galaxies with good [ o@xmath2iii ] detections in their dig spectra , they find that [ o@xmath2iii]/h@xmath4 is in the range @xmath80 to 2 and , for a given slit , is systematically higher than in the hii regions in that slit . they also find that the [ o@xmath2iii ] line widths are usually larger than those of [ n@xmath2ii ] , and thus refer to a quiescent photo - ionized dig component which accounts for the bulk of the h@xmath1 , [ n@xmath2ii ] and [ s@xmath2ii ] emission , and a disturbed component ( shocks and tmls are considered ) contributing a minority ( @xmath81 20% ) of the h@xmath1 emission but responsible for the [ o@xmath2iii ] emission . for vertical hydrostatic equilibrium , the contrast in line - widths indicates that the scale - height of the disturbed component is 1.52 times greater than that of the quiescent dig . in ngc 891 , the line - widths are dominated by galactic rotation and thus such an analysis can not be carried out . the existence of a second source of line emission may be relevant for the issue of the low he@xmath2i / h@xmath1 ratio . the value of 0.027 for the lower halo of ngc 891 implies a much softer spectrum than is required to explain the high [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 ( r97 ) . apart from the possibility that the forbidden line emission is complicated by additional sources of ionization and non - ionizing heating , he@xmath2i / h@xmath1 may also be affected by secondary ionization sources if they contribute sufficient h@xmath1 emission . for instance , the ratio is fairly sensitive to shock conditions . the sm 100 km s@xmath14 model gives a value of 0.027 . the ratio is 0.005 for an 80 km s@xmath14 shock , but this model predicts insignificant [ o@xmath2iii ] emission . the dopita & sutherland ( 1996 ) higher velocity , lower density ( n=1 ) , magnetized models give a much more significant ionized precursor . for the lowest velocity considered , 150 km s@xmath14 , the shock itself gives a low ratio of 0.019 , but the line fluxes for the precursor are not given . for a 200 km s@xmath14 shock , the ratio from combined shock and precursor is 0.046 . slavin , shull , & begelman ( 1993 ) do not predict he@xmath2i emission . regardless , if shocks or tmls can provide , say 25% of the h@xmath1 emission at @xmath50 kpc , then there may be a region of parameter space which can produce low enough he@xmath2i / h@xmath1 so that the composite line ratio is significantly reduced below that of the stellar - ionized gas alone . the contributions of these sources required in 3 may not be sufficiently large , but the effect is worth future consideration . the inferred stellar temperature , mean spectral type and upper imf cutoff would then all be underestimated . finally , it is worth re - emphasizing that all emission line fluxes and ratios presented here are averaged along a line of sight through the dig layer , and that local variations in , for example , the derived ionization fraction of h surely exist . also , although the vertical dependence of the line ratios has been very revealing , only one slit position has been observed , covering the halo above the most active region of star formation in the disk , and close to an h@xmath1 filament . a key question is how these halo properties vary with environment . does [ o@xmath2iii]/h@xmath4 show the same behavior above more quiescent parts of the disk ? is this behavior peculiar to the halo of ngc 891 only , or is it a general feature of dig halos ? these questions will be addressed by further observations . the author has benefited from many useful discussions about dig ionization from r. reynolds , r. walterbos ( whose comments as referee also improved the paper ) , j. slavin , r. benjamin , j. shields , and others . the help of the kpno staff is also greatly appreciated .

two long - slit spectra of the diffuse ionized gas in ngc 891 are presented . the first reveals variations parallel to the major axis in emission line ratios in the halo gas at @xmath0 pc . it is found that filaments of h@xmath1 emission show lower values of [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 and [ o@xmath2i]/h@xmath1 . although this result is expected if the filaments represent the walls of evacuated chimneys , it merely reflects a more general correlation of these ratios with h@xmath1 surface brightness along the slit , and may simply arise from radiation dilution effects . halo regions showing low line ratios are probably relatively close to ionizing sources in the disk below . the results highlight difficulties inherent in observations of edge - on galaxies caused by lack of knowledge of structure in the in - plane directions . the [ s@xmath2ii]/[n@xmath2ii ] ratio shows almost no dependence on distance along the major axis or h@xmath1 surface brightness . values of [ o@xmath2i]/h@xmath1 indicate that h is 8095% ionized ( assuming @xmath3 k ) , with the higher ionization fractions correlating with higher surface brightness . much more interesting information on the nature of this gaseous halo comes from the second observation , which shows the vertical dependence of [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 , [ o@xmath2i]/h@xmath1 , and [ o@xmath2iii]/h@xmath4 through the brightest region of the dig halo . the most surprising result , in complete contradiction to models in which the dig is ionized by massive stars in the disk , is that [ o@xmath2iii]/h@xmath4 rises with height above the plane for @xmath5 kpc ( even as [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 are rising , in line with expectations from such models ) . the run of [ s@xmath2ii]/[n@xmath2ii ] is also problematic , showing essentially no contrast with @xmath6 . the [ o@xmath2iii ] emission probably arises from shocks , turbulent mixing layers , or some other secondary source of ionization . composite models in which the line emission comes from a mix of photo - ionized gas and shocks or turbulent mixing layers are considered in diagnostic diagrams , with the result that many aspects of the data can be explained . problems with the run of [ s@xmath2ii]/[n@xmath2ii ] still remain , however . there is a reasonably large parameter space allowed for the second component . for the photo - ionized component , only matter - bounded models succeed , putting a fairly strong restriction on the clumpiness of the halo gas . given the many uncertainties , the composite models can do little more than demonstrate the feasibility of these processes as secondary sources of energy input . a fairly robust result , however , is that the fraction of h@xmath1 emission arising from the second component probably increases with @xmath6 . from values of [ o@xmath2i]/h@xmath1 , h is essentially 100% ionized at @xmath7 kpc and 90% ionized at @xmath8 kpc ( again assuming @xmath3 k ) .

since its discovery in 2003 @xcite , the @xmath4 charmonium is subject of many experimental and theoretical efforts aimed at disclosing its nature for a recent review see @xcite . the most recent data on the mass of the @xmath3 is @xcite m_x=(3871.850.27(stat)0.19(syst ) ) , [ xmass ] with a width of @xmath5 . however , the problem of the quantum numbers for the @xmath3 is not fully resolved yet : while the analysis of the @xmath6 decay mode of the @xmath4 yields either @xmath2 or @xmath7 quantum numbers @xcite , the recent analysis of the @xmath8 mode seems to favour the @xmath7 assignment @xcite , though the @xmath2 option is not excluded . this question is clearly very central , for the most promising explanations for the @xmath3 in the @xmath9-wave @xmath10 molecule model @xcite as well as in the coupled - channel model @xcite require the quantum numbers @xmath2 . in addition , the @xmath3 can not be a naive @xmath11 @xmath7 state , for its large branching fraction for the @xmath12 mode @xcite is not compatible with the quark - model estimates for the @xmath7 charmonium @xcite . so , for the @xmath7 quantum numbers , very exotic explanations for the @xmath3 would have to be invoked . the aim of the present paper is to perform a combined analysis of the data on the @xmath13 and @xmath14 mass distribution in the @xmath6 and @xmath8 mode , respectively . we find that the @xmath9-wave amplitudes from the decay of a @xmath2 state provide a better overall description of the data than the @xmath15-wave ones from the @xmath7 , especially when the parameter range is restricted to realistic values . we conclude then that the existing data favour @xmath2 quantum numbers of the @xmath3 , however , improved data in the @xmath8 mode are necessary to allow for definite conclusions regarding the @xmath3 quantum numbers . recently belle announced @xcite the updated results of the measurements for the reaction @xmath16 : _ 2^+&=&[8.610.82(stat)0.52(syst)]10 ^ -6 , + b_2 ^ 0&=&[4.31.2(stat)0.4(syst)]10 ^ -6 , for the charged @xmath17 and neutral @xmath18 mode , respectively , with @xmath19 being the product branching fraction @xmath20 in the corresponding mode . the number of events in the background - subtracted combined distribution is @xmath21 for the decay @xmath22 babar reports @xcite _ 3^+&=&[0.60.2(stat)0.1(syst)]10 ^ -5 , + b_3 ^ 0&=&[0.60.3(stat)0.1(syst)]10 ^ -5 , for the charged mode @xmath23 and for the neutral mode @xmath24 , respectively . similarly to the two - pion case above , @xmath25 stands for the product branching fraction @xmath26 in the corresponding mode . the number of events in the combined distribution is n^sig+bg_3=34.06.6,n_3^bg=8.91.0 , and we assume a flat background . note , the spectrum reported in @xcite and used below appears not to be efficiency corrected . however , since only the shape of this spectrum plays a role for the analysis ( see number - of - event distributions ( [ n23pi ] ) below ) and we can reproduce the theoretical spectra of @xcite , which have the efficiency of the detector convoluted in via a monte carlo simulation , the invariant mass dependence of the efficiency corrections is expected to be mild and therefore should not affect our analysis significantly . thus the updated ratio of branchings reads @xcite = = 0.80.3 . [ brratio ] in our analysis we use the ratio ( [ brratio ] ) as well as n_2=196,n_3=25.1 , [ bandn ] and he corresponding bin sizes are @xmath27 mev and @xmath28 mev . as in previous analyses , we assume that the two - pion final state is mediated by the @xmath31 in the intermediate state , while the three - pion final state is mediated by the @xmath32 . it was shown in @xcite that the description of the @xmath6 spectrum with the @xmath7 assumption is improved drastically if the isospin - violating @xmath31-@xmath32 mixing is taken into account . theoretical issues of the @xmath31-@xmath32 mixing are discussed , for example , in @xcite . here we include this effect with the help of the prescription used in @xcite , where the transition amplitude is described by the real parameter @xmath33 . thus , the amplitudes for the decays @xmath34 and @xmath35 take the form & & a_2=a_xj / g_a_2 + & & + a_xj / g_g_a_2 , + & & a_3=a_xj / g_a_3 + & & + a_xj / g_g_a_3 , where the vector meson propagators are g_v^-1=m_v^2-m^2-im_v_v(m),v= , , with @xmath36 ( @xmath37 ) being the @xmath29 ( @xmath30 ) invariant mass in the @xmath6 ( @xmath8 ) final state . masses of the vector mesons used below are @xcite @xmath38 the complex mixing amplitude multiplying the @xmath32 propagator used , for example , in @xcite to analyse the two - pion spectrum , in our notation reads as @xmath39 ; in particular we reproduce naturally the phase of 95@xmath40 quoted in @xcite . note , as we shall only study the invariant mass distributions of the two final states , we do not need to keep explicitly the vector nature of the intermediate states . for the `` running '' @xmath31 meson width we use @xmath41 ^ 2,\ ] ] where @xmath42 , @xmath43 , with @xmath44 gev@xmath45 and with the nominal @xmath31 meson width @xmath46 mev . for the @xmath32 meson `` running '' width ( the nominal width being @xmath47 mev ) , the @xmath48 and @xmath49 decay modes are summed , with the branchings ( 3)=89.1%,()=8.28% . in particular , ( m)=_^(0)^3 , while , for the @xmath50 , we resort to the expressions derived in @xcite , with a reduced contact term which provides the correct nominal value of the @xmath51 decay width @xcite . the transition amplitudes for the decays @xmath52 are parameterised in the standard way , namely , a_xj / v = g_xvf_lx(p ) , with the blatt - weisskopf `` barrier factor '' f_0x(p)=1 , f_1x(p)=(1+r^2 p^2)^-1/2 , for the @xmath2 and @xmath7 assignment , respectively . here @xmath53 denotes the @xmath54 momentum in the @xmath3 rest frame . the `` radius '' @xmath55 is not well understood . if one associates it with the size of the @xmath56 vertex , it might be related to the range of force . in the quark model this radius is @xmath57 . this is also in line with the inverse mass of the lightest exchange particle allowed between @xmath54 and @xmath58 , namely , @xmath59 . on the other hand , a larger value @xmath60 gev@xmath45 is used in the experimental analysis of @xcite . therefore , in the analysis presented below we use both values @xmath61 gev@xmath45 as well as @xmath60 gev@xmath45 , keeping in mind that smaller values of @xmath55 are preferred by phenomenology . the theoretical invariant mass distributions for the @xmath29 and @xmath30 final state take the form : = bm__2 p^2l+1f_lx^2(p)|r_xg_+g_g_|^2 , + [ br23pi ] + = bm__3 p^2l+1f_lx^2(p)|g_+r_x g_g_|^2,where @xmath62 and the parameter @xmath63 absorbs the details of the short - ranged dynamics of the @xmath3 production . the theoretical number - of - event distributions read n_2(m)= , + [ n23pi ] + n_3(m)=. the @xmath31-@xmath32 mixing parameter @xmath64 is extracted from the @xmath65 decay width ( @xmath66 ) . the corresponding amplitude reads a_2=g_a_2 , and we find that 3.410 ^ -3 ^2 . [ cols="^,^,^,^,^,^,^,^",options="header " , ] the number - of - event distributions ( [ n23pi ] ) possess 3 free parameters : the `` barrier '' factor @xmath55 , the ratio of couplings @xmath67 and the overall normalisation parameter @xmath63 . as outlined above , we perform the analysis for two values of @xmath55 , namely , the preferred value of 1 gev@xmath45 and a significantly larger value of 5 gev@xmath45 used in earlier analyses . since the normalisation factor @xmath63 drops out from the ratio of the two branchings , we extract the ratio @xmath67 directly from the integrated data , that is from the relation ( dm)/(dm).=b_3/b_2 , where the value of the ratio on the right - hand side is fixed by eq . ( [ brratio ] ) , and in the integration above we have cut off the @xmath29 invariant mass at 400 mev , as in @xcite , and the @xmath30 invariant mass at 740 mev , as in @xcite . therefore the norm @xmath63 is our only fitting parameter which governs the overall strength of the signal in both channels simultaneously , while the shape of the curves is fully determined from other sources . in table [ t1 ] , we list the parameters of the 3 combined fits to the data , found for the 2 values of the blatt - weisskopf parameter @xmath55 . the corresponding line shapes and the result of the integration in bins are shown in fig . [ fig1 ] . one can see from table [ t1 ] and fig . [ fig1 ] that the best overall description of the data for the two channels under consideration is provided by the @xmath9-wave fit . the @xmath15-wave fit is capable to provide the description of the data of a comparable ( however somewhat lower ) quality , only for large values of the blatt - weisskopf parameter @xmath55 , @xmath60 gev@xmath45 . the @xmath15-wave fit becomes poorer when the blatt - weisskopf parameter is decreased , and for values of @xmath55 of order 1 gev@xmath45 , the quality of the @xmath15-wave fit is unsatisfactory , which is the result of a very poor description of the two - pion spectrum see the dashed ( green ) curve in fig . [ fig1 ] . varying the ratio of branchings @xmath68 around its central value within the experimental uncertainty interval [ see eq . ( [ brratio ] ) ] leads only to minor changes in the fits and does not affect the conclusions . since no charged partners of the @xmath4 are observed experimentally , it is supposed to be ( predominantly ) an isoscalar . then the ratio @xmath62 measures the strength of the isospin violation in the @xmath69 decay vertex . as discussed above , this ratio is extracted directly from the data on the ratio of the branchings ( [ brratio ] ) . an isospin - violating observable for a compact charmonium is the ratio of the branching fractions for the @xmath70 decays into @xmath71 and @xmath72 final states as r_(2s)== 0.03 , [ rcharm ] where @xmath73 and @xmath74 are the center - of - mass momenta of @xmath75 and @xmath76 , respectively , and the @xmath70 branching fractions are taken from @xcite . however , since here also the denominator violates a symmetry , namely , su(3 ) , and there might be significant meson - loop contributions @xcite , the estimate ( [ rcharm ] ) is to be regarded as a conservative upper bound for the isospin violation strength for compact charmonia . in contrast to this , in the @xmath9-wave molecular picture for the @xmath3 , isospin violation is enhanced significantly compared to the strength ( [ rcharm ] ) for it proceeds via intermediate @xmath77 states and is therefore driven by the mass difference @xmath78 mev of the neutral and charged @xmath77 threshold see , for example , @xcite . an order - of - magnitude estimate is provided by the expression r_x^mol~|| ~ ~0.13 , [ rmol ] where @xmath79 is the @xmath80 meson mass , while @xmath81 and @xmath82 denote the amplitudes corresponding to loop diagrams with neutral and charged @xmath10 intermediate states , respectively , evaluated at the @xmath3 mass . they are composed of two terms , the strongly channel - dependent analytic continuation of the unitarity cut , proportional to the typical momentum of the meson pair , and the weakly channel - dependent principle value term , whose size is identified with the inverse range of forces of order of 1 gev ( see above ) . this estimate is within a factor of 2 consistent with the value @xmath83 found from our @xmath9-wave fit see table [ t1 ] . on the other hand , if the @xmath3 has the quantum numbers @xmath7 , one should expect the isospin violation in the @xmath3 wave function to be of the natural charmonium size , and thus of the order of @xmath84see discussion below eq . ( [ rcharm ] ) , since the @xmath10 loop effects are suppressed by the additional centrifugal barrier : the estimate analogous to eq . ( [ rmol ] ) now reads @xmath85 . thus , for the state with the quantum numbers @xmath7 , one expects values of at most @xmath86 , that are significantly smaller than those following from the data ( see table [ t1 ] ) . one is led to conclude therefore that for @xmath87 , needed for the quantum numbers @xmath7 to be consistent with the data on the @xmath3 decays , a new , yet unknown , isospin violation mechanism would have to be invoked . we conclude therefore that , although the present quality of the data in the @xmath35 channel is not sufficient to draw a definite conclusion concerning the quantum numbers of the @xmath4 , the combined analysis of the existing two- and three - pion spectra favours the @xmath9-wave fit , related to the @xmath2 assignment for the @xmath4 , over the @xmath15-wave fit , related to the @xmath7 assignment . we notice that an acceptable @xmath15-wave fit calls for a large range parameter in the blatt - weisskopf form factor which meets certain difficulties with its phenomenological interpretation . in addition , while the value @xmath62 can be understood theoretically for the @xmath2 assignment , the value extracted for the @xmath7 assignment is too large to be explained from known mechanisms of the isospin violation . we acknowledge useful discussions with e. braaten , s. eidelman , f .- k . guo , and r. mizuk . the work was supported in parts by funds provided from the helmholtz association ( grant nos vh - ng-222 and vh - vi-231 ) , by the dfg ( grant nos sfb / tr 16 and 436 rus 113/991/0 - 1 ) , by the eu hadronphysics2 project , by the rffi ( grant nos rffi-09 - 02 - 91342-nnioa and rffi-09 - 02 - 00629a ) , and by the state corporation of russian federation `` rosatom . 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we re - analyse the two- and three - pion mass distributions in the decays @xmath0 and @xmath1 and argue that the present data favour the @xmath2 assignment for the quantum numbers of the @xmath3 .

the application of the dirac quantization programme to generally covariant systems triggers a ` conceptual problem ' since the quantization of the hamiltonian constraint yields a seemingly ` timeless ' evolution equation @xmath0 and thereby leads to the issue of ` frozen dynamics ' in the quantum theory . while classically the presence of a hamiltonian constraint does not lead to an impeding obstacle because we can always resort to a time coordinate along the flow of the hamiltonian with respect to which one can order physical relations , such a time coordinate is absent in the physical hilbert space . the standard ` conceptual solution ' to this problem is _ relational dynamics _ @xcite , according to which time evolution arises by relating dynamical quantities to other internal variables which we use as clocks to keep track of internal time ; such relational statements are invariant under the gauge flow of the hamiltonian constraint and could thus , in principle , be promoted into well defined operators on the physical hilbert space . nevertheless , even when adopting this ` conceptual solution ' one encounters a whole plethora of _ technical _ problems in the quantum theory @xcite . here we want to address four such problems : 1 . the _ hilbert space problem_. which hilbert space representation is one to use and , in particular , how is one to construct a positive - definite inner product on the space of solutions ? 2 . the _ multiple choice problem_. which relational clock variable should one employ ? ( different relational clocks may yield _ a priori _ different quantum theories . ) the _ global time problem_. good global clocks which parametrize the gauge orbits such that every trajectory intersects every constant clock time slice once and only once may not exist . the _ observable problem_. it is a notoriously difficult challenge to construct explicit observables in generally covariant theories , especially in the quantum theory . in the sequel we summarise the effective approach to the problem of time @xcite which at least in the semiclassical regime deals with some of these problems and circumvents others . rather than employing special clock choices and attempting to justify these , we follow the basic premise that there are to be no distinguished clocks and that , instead , we ought to treat all clocks on an equal footing . our goal consists in employing local internal times , possibly translating between different clocks and thereby evolving ( relational ) data along complete semiclassical trajectories . the central idea of the effective approach is to sidestep the _ hilbert space problem _ altogether ; instead of employing wave functions or density matrices to describe states in a fixed hilbert space , we regard states as ( _ a priori _ complex ) linear functionals on an algebra of kinematical variables , say polynomials in a canonial pair @xmath1 , and use expectation values @xmath2 and @xmath3 , and moments . ] @xmath4 to parametrize states @xcite . this construction immediately carries over to an arbitrary number of canonical pairs . this space of states can be given a ( quantum ) phase space structure via a poisson bracket defined as follows for any operators @xmath5 polynomial in the canonical variables [ poisson ] \{,}:= . this definition can be extended to the moments by linearity and the leibnitz rule and yields ` classical poisson brackets ' for the expectation values of basic canonical variables , i.e. @xmath6 , vanishing poisson brackets between expectation values and moments @xmath7 and more complicated brackets between the moments @xcite . in this summary we focus on finite dimensional systems with a single constraint @xmath8 playing the role of the hamiltonian constraint of general relativity . the dirac programme requires physical states to satisfy @xmath9 . the spectrum of @xmath8 is essential for the construction of the physical hilbert space : if zero lies in the discrete part of the spectrum , the physical hilbert space turns out to be a subspace of the kinematical hilbert space , while a new hilbert space with a new ( positive definite ) inner product must be constructed for solutions if zero lies in the continuous part of the spectrum . there exist techniques for constructing such physical hilbert spaces in the latter case @xcite , however , finding physical hilbert spaces in practice remains a difficult task . the effective techniques , on the other hand , work for both zero in the discrete or continuous part of the spectrum . at the effective level , physical states must clearly satisfy [ qcon ] ( , , ( qp),)=0 , however , this is not sufficient since the expectation value of @xmath8 may be zero even if @xmath10 . furthermore , when solving one ( first class ) constraint classically we can eliminate an entire canonical pair , while on the quantum phase space after solving ( [ qcon ] ) and factoring out its flow we would be left with an infinite tower of ( unconstrained ) moments of the eliminated canonical pair . clearly , we must impose a further set of constraints to account for this . it turns out that [ pol ] c_pol:= = 0 , where @xmath11 stands for all polynomials in the basic canonical operators , provides a correct independent set of constraint functions on quantum phase space which together with ( [ qcon ] ) is then equivalent to the dirac condition @xcite . as a result , we have infinitely many constraints for infinitely many variables . in order to reduce the system to a tractable finite size , we impose a very general semiclassical approximation : assume @xmath12 and truncate the system at @xmath13 by neglecting all terms of higher order . this is consistent with the quantum poisson bracket structure ( [ poisson ] ) which preserves orders in @xmath13 and generates the flows and dynamics on quantum phase space , once suitable initial data for the relevant expectation values and moments has been chosen . rather than defining a quantum theory , the effective approach in its present form provides an effective tool for evaluating the quantum dynamics of given ( finite dimensional ) systems . let us outline the effective approach to semiclassical relational dynamics with an explicit toy model ( for a more general discussion and details see @xcite ) . the model we are considering is the isotropic 2d harmonic oscillator with prescribed total energy , whose classical solutions are closed orbits in phase space such that no global clock exists @xcite . it is subject to the constraint [ quant - rov ] could construct clock functions are the @xmath14 . in order to obtain a notion of evolution from this _ a priori _ timeless system , we may choose a variable as a local clock , say @xmath15 , and deparametrize locally at the classical level by factorizing the constraint c=(p_1+h(q_1))(p_1-h(q_1 ) ) , h(q_1)=. quantization of the first factor ( in the region where it vanishes ) yields a local schrdinger regime [ schrod ] i(q_1,q_2)=(_2,_2 ; q_1)(q_1,q_2)=(q_1,q_2 ) with a non self adjoint @xmath16 ( defined by spectral decomposition ) due to non unitarity in @xmath15 evolution ( @xmath15 is not globally valid ) . given this construction , we can calculate expectation values and moments in @xmath15 time and compare their evolution to the effective treatment . at the effective level , on the other hand , we retain 14 kinematical degrees of freedom at order @xmath13 , namely four expectation values @xmath17 , four spreads @xmath18 , and six covariances @xmath19 , where the @xmath20 are any of the @xmath14 . at order @xmath13 the constraint ( [ quant - rov ] ) translates via ( [ qcon ] ) and ( [ pol])into the following five first class quantum constraint functions @xcite [ effrov ] c&= & p_1 ^ 2+p_2 ^ 2+q_1 ^ 2+q_2 ^ 2+(p_1)^2+(p_2)^2+(q_1)^2+(q_2)^2-m= 0 + c_q_1&= & 2p_1(q_1p_1)+2p_2(q_1p_2)+2q_1(q_1)^2 + 2q_2(q_1q_2)+ip_1= 0 + c_p_1&= & 2p_1(p_1)^2 + 2p_2(p_1p_2)+2q_1(p_1q_1)+2q_2(p_1q_2)-iq_1= 0 + c_q_2&=&2p_1(p_1q_2)+ 2p_2 ( q_2p_2)+2q_1(q_1q_2)+2q_2(q_2)^2+ip_2 = 0 + c_p_2&=&2p_1(p_1p_2)+ 2p_2(p_2)^2 + 2q_1(q_1p_2)+2q_2(q_2p_2)-iq_2 = 0.these five constraints generate only four independent flows since at order @xmath13 we have to deal with a degenerate poisson structure @xcite . it is convenient to fix three of these independent flows . selecting @xmath15 as a relational clock it should not be represented as an operator ; we therefore choose a gauge which ` projects this clock to a parameter ' by setting its fluctuations to zero ( fixing three @xmath21 flows and leaving us with one ` hamiltonian flow ' ) [ zeitgeist ] ( q_1)^2=(q_1q_2)=(q_1p_2)=0 . indeed , the choice of an internal time variable in the effective framework is best described and interpreted in a corresponding gauge @xcite : we refer to such a choice and gauge ( e.g. ( [ zeitgeist ] ) ) as a _ zeitgeist_. after choosing a local relational clock and corresponding zeitgeist , local relational observables at the effective level are given by correlations of expectation values and moments with the expectation value of the chosen clock ( here @xmath22 ) evaluated in its zeitgeist . we call such state dependent local observables _ fashionables _ because they comprise the complete physical information about the system ( at order @xmath13 ) as long as the zeitgeist is valid , but may fall out of fashion when a zeitgeist necessarily changes at turning points of local clocks . it turns out that the fashionables in @xmath15 time of the effective framework agree perfectly with those computed in the @xmath15 schrdinger regime ( [ schrod ] ) @xcite . in its turning region , @xmath15 becomes complex valued and its zeitgeist incompatible with the semiclassical expansion , such that evolution in @xmath15 breaks down _ before _ the turning point a signature of non unitarity . thus , a new local internal time ( here @xmath23 ) is needed and for a full evolution we must switch between @xmath15 and @xmath23 time . furthermore , since a given internal time is best described in a corresponding choice of gauge , we must switch also between the @xmath15 and @xmath23 zeitgeister . explicit gauge transformations between these zeitgeister can , indeed , be constructed and initial data can be evolved consistently around the entire closed semiclassical orbit through the turning points of various clock variables @xcite . it should be emphasized that in each gauge we evolve a _ different _ set of fashionables which highlights the local nature of the relational concept in the absence of global clock functions . the effective approach summarised here sidesteps the technical problems mentioned in the introduction . in particular , the _ hilbert space problem _ is avoided altogether since no use of any hilbert space representation has been made . at the effective level one can make sense of local time evolution and switching between different local clocks can essentially be achieved by an additional gauge transformation which enables one to handle the _ multiple choice _ and _ global time problem _ and implies the equivalence of different clock choices at semiclassical order . note , however , that if the relational clock is non global , it assumes complex values in its turning region @xcite . state dependent fashionables arise naturally in this framework which simplifies the _ observable problem _ due to the classical treatment of the effective system . in the example outlined here and other models , the effective evolution agrees with local deparametrizations at a hilbert space level @xcite . finally , the notion of relational evolution disappears in highly quantum states of systems without global clocks @xcite . the application of this effective framework to the more interesting closed frw model with a massive scalar field will appear elsewhere @xcite . * acknowledgments * the author would like to thank martin bojowald and artur tsobanjan for collaboration on this subject . 9 slides at http://loops11.iem.csic.es/loops11/index.php?option=com_content&view=article&id=101 c. rovelli , _ quantum gravity _ ( cup , cambridge , 2004 ) ; 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we provide a synopsis of an effective approach to the problem of time in the semiclassical regime . the essential features of this new approach to evaluating relational quantum dynamics in constrained systems are illustrated by means of a simple toy model .

the large values of the solar ( @xmath11 ) and atmospheric ( @xmath12 ) mixing angles may be telling us about some new symmetries of leptons not presented in the quark sector and may provide a clue to the nature of the quark - lepton physics beyond the standard model . if there exists such a flavor symmetry in nature , the tribimaximal ( tbm ) @xcite pattern for the neutrino mixing will be a good zeroth order approximation to reality : @xmath13 for example , in a well - motivated extension of the standard model through the inclusion of @xmath14 discrete symmetry , the tbm pattern comes out in a natural way in the work of @xcite . although such a flavor symmetry is realized in nature leading to exact tbm , in general there may be some deviations from tbm . recent data of the t2k @xcite and minos @xcite collaborations and the analysis based on global fits @xcite of neutrino oscillations enter into a new phase of precise measurements of the neutrino mixing angles and mass - squared differences , indicating that the tbm mixing for three flavors of leptons should be modified . in the weak eigenstate basis , the yukawa interactions in both neutrino and charged lepton sectors and the charged gauge interaction can be written as @xmath15 when diagonalizing the neutrino and charged lepton mass matrices @xmath16 , one can rotate the neutrino and charged lepton fields from the weak eigenstates to the mass eigenstates @xmath17 . then we obtain the leptonic @xmath18 unitary mixing matrix @xmath19 from the charged current term in eq . ( [ lagrangiana ] ) . in the standard parametrization of the leptonic mixing matrix @xmath20 , it is expressed in terms of three mixing angles and three _ cp_-odd phases ( one for the dirac neutrino and two for the majorana neutrino ) @xcite @xmath21 where @xmath22 and @xmath23 , and @xmath24 is a diagonal phase matrix which contains two _ cp_-violating majorana phases , one ( or a combination ) of which can be in principle explored through the neutrinoless double beta ( @xmath25 ) decay @xcite . for the global fits of the available data from neutrino oscillation experiments , we quote two recent analyses : one by gonzalez - garcia _ et al . _ @xcite @xmath26 in @xmath27 ( @xmath7 ) ranges , or equivalently @xmath28 and the other given by fogli _ et al . _ with new reactor neutrino fluxes @xcite : @xmath29 corresponding to @xmath30 the analysis by fogli _ et al . _ includes the t2k @xcite and minos @xcite results . the t2k collaboration @xcite has announced that the value of @xmath8 is non - zero at @xmath31 c.l . with the ranges @xmath32 or @xmath33 for @xmath34 , @xmath35 and the normal ( inverted ) neutrino mass hierarchy . the minos collaboration found @xmath36 with a best fit of @xmath37 for @xmath34 , @xmath35 and the normal ( inverted ) neutrino mass hierarchy . the experimental result of non - zero @xmath38 implies that the tbm pattern should be modified . however , properties related to the leptonic _ violation remain completely unknown yet . the trimaximal neutrino mixing was first proposed by cabibbo @xcite ] if one considers @xmath14 discrete symmetry , it will have two subgroups , namely , @xmath39 and @xmath40 . the trimaximal matrix given in eq . ( [ cabibbo ] ) is obtained under @xmath40 . ] ( see also @xcite ) @xmath41 with @xmath42 being a complex cube - root of unity . this mixing matrix has maximal _ cp _ violation with the jarlskog invariant @xmath43 . however , this trimaximal mixing pattern has been ruled out by current experimental data on neutrino oscillations . in their original work , harrison , perkins and scott ( hps ) @xcite proposed to consider the simple mass matrices @xmath44 that can lead to the tribimaximal mixing , where @xmath45 and @xmath46 are real parameters , in eq . ( [ mass1 ] ) can be in general introduced as complex : e.g. , @xmath47 and @xmath48 . this case has been considered by xing @xcite who pointed out that the off - diagonal terms in @xmath49 will acquire a phase from the complex @xmath50 . it has the interesting implication that a nonzero @xmath51 will result from the phase of @xmath50 . however , the corresponding jarlskog invariant is exactly zero and the absence of intrinsic _ cp _ violation makes this possibility less interesting . ] @xmath52 and @xmath53 . the mass matrices are diagonalized by the trimaximal matrix @xmath54 for charged lepton fields and the bimaximal matrix @xmath49 defined below for neutrino fields , that is , @xmath55 and @xmath56 . the combination of trimaximal and bimaximal matrices leads to the so - called tbm mixing matrix : @xmath57 it is clear by now that the tribimaximal mixing is not consistent with the recent experimental data on the reactor mixing angle @xmath8 because of the vanishing matrix element @xmath58 in @xmath0 . in this work we consider an extension of the tribimaximal mixing by considering small perturbations to the mass matrices @xmath59 and @xmath60 which we will call @xmath61 and @xmath62 , respectively ( see eq . ( [ mass2 ] ) below ) so that @xmath63 is no longer in the bimaximal form and @xmath64 deviates from the trimaximal structure , where @xmath2 is the unitary matrix needed to diagonalize the matrix @xmath65 . as a consequence , @xmath66 small perturbations . hence , the corrections to the tbm pattern arise from both charged lepton and neutrino sectors . inspired by the t2k and minos measurements of a sizable reactor angle @xmath8 , there exist in the literature intensive studies of possible deviations from the exact tbm pattern . however , most of these investigations were focused on the modification of tbm arising from either the neutrino sector @xcite or the charged lepton part @xcite , but not both simultaneously . the paper is organized as follows . in sec . ii , we set up the model by making a general extension to the charged lepton and neutrino mass matrices . then in sec . iii we study the phenomenological implications by considering two different scenarios for the charged lepton mixing matrix . our conclusions are summarized in sec . in order to discuss the deviation from the tbm mixing , let us consider a simple and general extension of the original proposal by hps given in eq . ( [ mass1 ] ) , by taking into account perturbative effects on the mass matrices @xmath67 and @xmath68 . the generalized mass matrices @xmath69 and @xmath70 can be introduced as , dihedral groups , @xmath71 , etc . by considering higher order and radiative effects , the matrices in eq . ( [ mass2 ] ) can be realized . for example , we have shown in ref . @xcite that these matrices can be obtained by introducing dimension-5 operators to the lagrangians . ] @xmath72 where @xmath69 and @xmath70 are defined as the hermitian square of the mass matrices @xmath73 and @xmath74 , respectively , with the subscript @xmath75 denoting charged fermion fields ( charged leptons or quarks ) . due to the hermiticity of @xmath69 and @xmath70 , the parameters @xmath76 are real , while @xmath77 and @xmath78 are complex . the parameters @xmath79 , @xmath78 and @xmath80 represent small perturbations . note that the ( 11 ) , ( 13 ) , ( 22 ) elements ( _ i.e. , _ @xmath81 , @xmath82 and @xmath83 ) in @xmath70 are assumed to contain any perturbative effects on the elements @xmath84 , @xmath50 , and @xmath46 in @xmath68 , respectively . for simplicity , it is assumed that @xmath85 is real just as the other elements in @xmath70 and the vanishing off - diagonal elements in @xmath68 remain zeros in @xmath70 . the parameters @xmath86 and @xmath77 are encoded in @xcite as @xmath87 where the subscript @xmath88 indicates a generation of charged fermion field , and @xmath89 represents a bare mass of @xmath88 , for example , @xmath90 for charged lepton fields . we first discuss the hermitian square of the neutrino mass matrix , @xmath70 , in eq . ( [ mass2 ] ) . it can be diagonalized by @xmath91 with @xmath92 and @xmath93 where the diagonal phase matrix @xmath94 contains two additional phases , which can be absorbed into the neutrino mass eigenstate fields . for a small perturbation @xmath95 , the mixing parameter @xmath96 can be expressed in terms of @xmath97 @xmath3 is then reduced to @xmath98 the neutrino mass eigenvalues are obtained as @xmath99 and their differences are given by @xmath100 from which we have a relation @xmath101 . it is well known that the sign of @xmath102 is positive due to the requirement of the mikheyev - smirnov - wolfenstein resonance for solar neutrinos . the sign of @xmath103 depends on that of @xmath104 : @xmath105 for the normal mass spectrum and @xmath106 for the inverted one . the quantities @xmath107 ( or @xmath108 ) are determined by the four parameters @xmath109 , while the majorana phases in eq . ( [ unu ] ) are hidden in the squared mass eigenvalues . we next turn to the hermitian square of the mass matrix for charged fermions in eq . ( [ mass2 ] ) . this modified charged fermion mass matrix is no longer diagonalized by @xmath54 @xmath110 where @xmath111 corresponding to @xmath112 , respectively , and @xmath113 is composed of the combinations of @xmath79 and @xmath78 . to diagonalize @xmath114 , we need an additional matrix @xmath115 which can be , in general , parametrized in terms of three mixing angles and six phases : @xmath116 where @xmath117 , @xmath118 and a diagonal phase matrix @xmath119 which can be rotated away by the phase redefinition of left - charged fermion fields . the charged fermion mixing matrix now reads @xmath120 . finally , we arrive at the general expression for the leptonic mixing matrix @xmath121 a simple and general extension of the mass matrices given in eq . ( [ mass2 ] ) thus leads to two possible sources of corrections to the tribimaximal mixing : @xmath2 measures the deviation of the charged lepton mixing matrix from the trimaximal form and @xmath3 characterizes the departure of the neutrino mixing from the bimaximal one . the charged lepton mass matrix in eq . ( [ mass2 ] ) or ( [ aa ] ) has 12 free parameters . three of them are replaced by the phases @xmath122 in eq . ( [ vl ] ) which can be eliminated by a redefinition of the physical charged lepton fields . the remaining 9 parameters can be expressed in terms of @xmath123 . > from eqs . ( [ aa ] ) and ( [ vl ] ) the mixing angles and phases can be expressed as @xmath124+\arg(\eta_{12 } ) \ , \end{aligned}\ ] ] with the condition @xmath125 . in the charged fermion sector , there is a qualitative feature that distinguishes the neutrino sector from the charged fermion one . the mass spectrum of the charged leptons exhibits a similar hierarchical pattern to that of the down - type quarks , unlike that of the up - type quarks which show a much stronger hierarchical pattern . for example , in terms of the cabbibo angle @xmath126 , the fermion masses scale as @xmath127 , @xmath128 and @xmath129 . this may lead to two implications : ( i ) the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite is mainly governed by the down - type quark mixing matrix , and ( ii ) the charged lepton mixing matrix is similar to that of the down - type quark one . therefore , we shall assume that ( i ) @xmath130 and @xmath131 , where @xmath132 is associated with the diagonalization of the down - type ( up - type ) quark mass matrix and @xmath133 is a @xmath18 unit matrix , and ( ii ) the charged lepton mixing matrix @xmath134 has the same structure as the ckm matrix , that is , @xmath135 or @xmath136 . recently , we have proposed a simple _ ansatz _ for the charged lepton mixing matrix @xmath2 , namely , it has the qin - ma - like parametrization in which the _ cp_-odd phase is approximately maximal @xcite . armed with this _ ansatz _ , we notice that the 6 parameters @xmath137 in @xmath2 are reduced to four independent ones @xmath138 . it has the advantage that the tbm predictions of @xmath139 and especially @xmath140 will not be spoiled and that a sizable reactor mixing angle @xmath8 and a large dirac _ cp_-odd phase are obtained in the mixing @xmath141 . the qin - ma ( qm ) parametrization of the quark ckm matrix is a wolfenstein - like parametrization and can be expanded in terms of the small parameter @xmath142 @xcite . however , unlike the original wolfenstein parametrization @xcite , the qm one has the advantage that its _ cp_-odd phase @xmath143 is manifested in the parametrization and is near maximal , _ i.e. , _ @xmath144 . this is crucial for a viable neutrino phenomenology . it should be stressed that one can also use any parametrization for the ckm matrix as a starting point . as shown in @xcite , one can adjust the phase differences in the diagonal phase matrix @xmath145 in eq . ( [ vl ] ) in such a way that the prediction of @xmath146 will not be considerably affected . for @xmath147 , the qm parametrization @xcite can be obtained from eq . ( [ vl ] ) by the replacements @xmath148 : @xmath149 on the other hand , for @xmath150 the qm parametrization is obtained by the replacements @xmath151 : @xmath152 where the superscript @xmath75 denotes @xmath153 ( down - type quarks ) or @xmath154 ( charged leptons ) . from the global fits to the quark mixing matrix given by @xcite we obtain @xmath155 because of the freedom of the phase redefinition for the quark fields , we have shown in @xcite that the qm parametrization is indeed equivalent to the wolfenstein one in the quark sector . finally , the leptonic mixing parameters ( @xmath156 ) except majorana phases can be expressed in terms of five parameters @xmath96 ( or @xmath108 ) , @xmath157 , the last four being the qm parameters in the lepton sector . if we further assume that all the qm parameters except @xmath143 have the same values in both the ckm and pmns matrices , then only two free parameters left in the lepton mixing matrix are @xmath108 and @xmath143 . if @xmath143 is fixed to be the same as the ckm one , then there will be only one free parameter @xmath108 in our calculation . in the next section , we shall study the dependence of the mixing angles @xmath158 and the jarlskog invariant @xmath159 on @xmath143 and @xmath108 . to make our point clearer , let us summarize the reduction of the number of independent parameters in this work . in the leptonic sector , we start with 16 free parameters ( 12 from the charged lepton mass matrix @xmath160 and 4 from the neutrino mass matrix @xmath161 ) as shown in eq . ( [ mass2 ] ) . among the 12 parameters from @xmath160 , three phases can be rotated away by the redefinition of the charged lepton fields . the remaining 9 parameters correspond to three charged lepton masses ( @xmath162 ) and six angles in the charged lepton mixing matrix @xmath163 as shown in eq . ( [ vl ] ) , while the 4 parameters from @xmath161 correspond to three neutrino masses ( @xmath164 ) plus one angle ( @xmath96 or @xmath108 ) in the neutrino mixing matrix @xmath165 as shown in eq . ( [ unu ] ) or ( [ epsilon ] ) . with our _ ansatz _ for @xmath163 discussed before , the 6 angles in @xmath163 are reduced to four qm parameters ( @xmath166 ) . thus , the number of parameters finally becomes five ( @xmath166 plus @xmath96 ( or @xmath108 ) ) , except for the six lepton masses . under the further assumption of the qm parameters @xmath167 having the same values in both the ckm and pmns matrices , these five parameters are reduced to only two ones @xmath143 and @xmath108 . we now proceed to discuss the low energy neutrino phenomenology with the neutrino mixing matrix @xmath168 ( see eq . ( [ unu ] ) ) characterized by the mixing angle @xmath96 or the small parameter @xmath108 and the charged lepton mixing matrix @xmath169 in which @xmath134 is assumed to have the similar expression as the qm parametrization @xcite given by @xmath170 or @xmath4 ( see eq . ( [ vl1 ] ) and eq . ( [ vl2 ] ) , respectively ) . the lepton mixing matrix thus has the form @xmath171 therefore , the corrections to the tbm matrix within our framework arise from the charged lepton mixing matrix @xmath2 characterized by the parameters @xmath138 and the matrix @xmath3 specified by the parameter @xmath108 whose size is strongly constrained by the recent t2k data . indeed , the parameters @xmath172 and @xmath143 in the lepton sector are _ a priori _ not necessarily the same as that in the quark sector . hereafter , we shall use the central values in eq . ( [ eq : qmfh ] ) of the parameters @xmath173 for our numerical calculations . in the following we consider both cases : 0.4 cm * ( i ) * @xmath174 0.5 cm with the help of eqs . ( [ hps ] ) and ( [ vl1 ] ) , the leptonic mixing matrix corrected by the replacements @xmath175 and @xmath176 , can be written , up to order of @xmath177 and @xmath178 , as @xmath179 note that @xmath180 here contains five independent parameters ( @xmath181 and @xmath108 ) . . ] by rephasing the lepton and neutrino fields @xmath182 , @xmath183 , @xmath184 and @xmath185 , the pmns matrix is recast to @xmath186 where @xmath187 is an element of the pmns matrix with @xmath188 corresponding to the lepton flavors and @xmath189 to the light neutrino mass eigenstates . in eq . ( [ pmns2 ] ) the phases defined as @xmath190 , @xmath191 , @xmath192 , @xmath193 and @xmath194 have the expressions : @xmath195 > from eq . ( [ pmns2 ] ) , the neutrino mixing parameters can be displayed as @xmath196 it follows from eqs . ( [ leptona ] ) and ( [ mixing1 ] ) that the solar neutrino mixing angle @xmath11 can be approximated , up to order @xmath197 and @xmath178 , as @xmath198 this indicates that the deviation from @xmath199 becomes small when @xmath200 approaches to zero and the magnitude of @xmath108 is less than @xmath142 . since it is the first column of @xmath2 that makes the major contribution to @xmath201 , this explains why we need a phase of order @xmath202 for the element @xmath203 : when @xmath204 , the present data of the solar mixing angle can be accommodated even for a large @xmath205 ( but less than @xmath142 ) . the behavior of @xmath201 as a function of @xmath143 is plotted in fig . [ fig1 ] where the horizontal dashed lines denote the upper and lower bounds of the experimental data in @xmath7 ranges . the allowed regions for @xmath143 ( in radian ) lie in the ranges of @xmath206 and @xmath207 , recalling that the qm phase is @xmath208 . likewise , the atmospheric neutrino mixing angle @xmath12 comes out as @xmath209\right ) \ . \label{atm } \end{aligned}\ ] ] fig . [ fig1 ] shows a small deviation from the tbm atmospheric mixing angle with @xmath210 for @xmath211 . owing to the absence of corrections to the first order of @xmath142 or @xmath108 in eq . ( [ atm ] ) , the deviation from the maximal mixing of @xmath12 comes mainly from the terms associated with @xmath212 or @xmath213 . especially , for @xmath214 we have the approximation @xmath215 , which implies @xmath216 for @xmath211 . we see from fig . [ fig1 ] that @xmath217 lies in the ranges @xmath218 for @xmath219 . the reactor mixing angle @xmath8 now reads @xmath220 evidently , @xmath51 depends considerably on the parameters @xmath142 and @xmath108 . thus , we have a non - vanishing @xmath8 with a central value of @xmath221 or @xmath222 for @xmath223 @xcite . note that the size of the unknown parameter @xmath108 is constrained by the plot of @xmath51 versus @xmath143 in fig . [ fig1 ] where the horizontal dot - dashed lines represent the present t2k data for the normal neutrino mass hierarchy . for a negative value of @xmath108 , the plot for @xmath51 versus @xmath143 is flipped upside - down . assuming @xmath224 , we see from eq . ( [ masssquare ] ) that a positive ( negative ) value of @xmath108 leads to a normal ( inverted ) neutrino mass spectrum . for example , we find @xmath225 ( @xmath226 ) for @xmath227 and @xmath228 ( @xmath229 ) . leptonic _ cp _ violation can be detected through the neutrino oscillations which are sensitive to the dirac _ cp_-phase @xmath230 , but insensitive to the majorana phases in @xmath20 @xcite . it follows from eqs . ( [ mixing elements2 ] ) and ( [ mixing1 ] ) that the dirac phase @xmath231 has the expression @xmath232 where terms of order @xmath233 have been neglected in both numerator and denominator . assuming @xmath224 , we show in table [ diraccp11 ] the predictions for @xmath234 and @xmath8 as a function of @xmath108 , where we have used the central values of eq . ( [ eq : qmfh ] ) . .[diraccp11 ] predictions of @xmath230 and @xmath8 as a function of @xmath108 in the case of @xmath174 . [ cols="^,^,^",options="header " , ] the strength of @xmath235 violation @xmath159 can be expressed in a similar way to eq . ( [ jcp1 ] ) @xmath236 which can be approximated as @xmath237 . when @xmath214 , it is further reduced to @xmath238 for @xmath239 . assuming @xmath224 , we see from fig . [ fig2 ] that @xmath240 ( @xmath241 ) for @xmath242 ( @xmath243 ) and @xmath227 . in their original work , harrison , perkins and scott proposed simple charged lepton and neutrino mass matrices that lead to the tribimaximal mixing @xmath0 . in this paper we considered a general extension of the mass matrices so that the lepton mixing matrix becomes @xmath1 . hence , corrections to the tribimaximal mixing arise from both charged lepton and neutrino sectors : the charged lepton mixing matrix @xmath2 measures the deviation of from the trimaximal form and the @xmath3 matrix characterizes the departure of the neutrino mixing from the bimaximal one . following our previous work to assume a qin - ma - like parametrization @xmath4 for @xmath2 in which the _ cp_-odd phase is approximately maximal , we study the phenomenological implications in two different scenarios : @xmath5 and @xmath6 . we found that both scenarios are consistent with the data within @xmath7 ranges . especially , the predicted central value of the reactor neutrino mixing angle @xmath222 is in good agreement with the recent t2k data . however , the data of @xmath146 can be easily accommodated in the second scenario but only marginally in the first one . hence , the precise measurements of the solar mixing angle in future experiments will test which scenario is more preferable . the leptonic _ violation characterized by the jarlskog invariant @xmath9 is generally of order @xmath10 . 99 # 1#2#3phys . * b#1 * , ( # 3 ) # 2 # 1#2#3nucl . * b#1 * , ( # 3 ) # 2 # 1#2#3phys . * d#1 * , ( # 3 ) # 2 # 1#2#3phys . # 1 * , ( # 3 ) # 2 # 1#2#3mod . * a#1 * , ( # 3 ) # 2 # 1#2#3phys . rep . * # 1 * , ( # 3 ) # 2 # 1#2#3science * # 1 * , ( # 3 ) # 2 # 1#2#3astrophys . j. * # 1 * , ( # 3 ) # 2 # 1#2#3eur . j. * c#1 * , ( # 3 ) # 2 # 1#2#3jhep * # 1 * , ( # 3 ) # 2 # 1#2#3j . * g#1 * , ( # 3 ) # 2 # 1#2#3int . j. mod . # 1 * , ( # 3 ) # 2 # 1#2#3prog . * # 1 * , ( # 3 ) # 2 p. f. harrison , d. h. perkins and w. g. scott , phys . lett . b * 530 * , 167 ( 2002 ) [ arxiv : hep - ph/0202074 ] ; p. f. harrison and w. g. scott , phys . b * 535 * , 163 ( 2002 ) [ arxiv : hep - ph/0203209 ] . x. g. he , y. y. keum and r. r. volkas , jhep * 0604 * , 039 ( 2006 ) [ arxiv : hep - ph/0601001 ] . m. c. gonzalez - garcia , m. maltoni and j. salvado , jhep * 1004 * , 056 ( 2010 ) [ arxiv:1001.4524v3 [ hep - ph ] ] . g. l. fogli , e. lisi , a. marrone , a. palazzo and a. m. rotunno , phys . d * 84 * , 053007 ( 2011 ) [ arxiv:1106.6028 [ hep - ph ] ] . k. nakamura _ et al_. 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harrison , perkins and scott have proposed simple charged lepton and neutrino mass matrices that lead to the tribimaximal mixing @xmath0 . we consider in this work an extension of the mass matrices so that the leptonic mixing matrix becomes @xmath1 , where @xmath2 is a unitary matrix needed to diagonalize the charged lepton mass matrix and @xmath3 measures the deviation of the neutrino mixing matrix from the bimaximal form . hence , corrections to @xmath0 arise from both charged lepton and neutrino sectors . following our previous work to assume a qin - ma - like parametrization @xmath4 for the charged lepton mixing matrix @xmath2 in which the _ cp_-odd phase is approximately maximal , we study the phenomenological implications in two different scenarios : @xmath5 and @xmath6 . we find that the latter is more preferable , though both scenarios are consistent with the data within @xmath7 ranges . the predicted reactor neutrino mixing angle @xmath8 in both scenarios is consistent with the recent t2k and minos data . the leptonic _ cp _ violation characterized by the jarlskog invariant @xmath9 is generally of order @xmath10 .

m. rybarsch would like to thank the studienstiftung des deutschen volkes for financial support of this work .

spin models of neural networks and genetic networks are considered elegant as they are accessible to statistical mechanics tools for spin glasses and magnetic systems . however , the conventional choice of variables in spin systems may cause problems in some models when parameter choices are unrealistic from the biological perspective . obviously , this may limit the role of a model as a template model for biological systems . perhaps less obviously , also ensembles of random networks are affected and may exhibit different critical properties . we here consider a prototypical network model that is biologically plausible in its local mechanisms . we study a discrete dynamical network with two characteristic properties : nodes with binary states 0 and 1 , and a modified threshold function with @xmath0 . we explore the critical properties of random networks of such nodes and find a critical connectivity @xmath1 with activity vanishing at the critical point . we currently experience a revived interest in dynamical networks of nodes with binary states , driven by two active fields of research : modeling of molecular information processing networks ( as , e.g. , genetic networks or protein networks ) @xcite , as well as modeling of adaptive networks @xcite . these network models with binary states are reminiscent of artificial neural networks as studied in the statistical mechanics community about two decades ago . an early motivation of networks with binary node states @xmath2 was given by mcculloch and pitts in 1943 @xcite as a model for neural information processing . a model for associative memory constructed from such nodes by hopfield in 1982 @xcite attracted considerable interest among physicists as it is conveniently accessible to equilibrium statistical mechanics methods @xcite . a simple redefinition of weights and thresholds maps the model onto the mathematical representation of a spin glass with states @xmath3 , which has become the usual form of the hopfield model in the physics literature . the corresponding redefinition of weights and thresholds does not affect the functioning of the model , as its mechanism of an associative memory works on the redefined weights as well . in some circumstances , however , when faithful representation of certain biological details is important , the exact definition matters . in the spin version of a neural network model , for example , a node with negative spin state @xmath4 will transmit non - zero signals through its outgoing weights @xmath5 , despite representing an inactive ( ! ) biological node . in the model , such signals arrive at target nodes @xmath6 , e.g. , as a sum of incoming signals @xmath7 . however , biological nodes , as genes or neurons , usually do not transmit signals when inactive . in biochemical network models each node represents whether a specific chemical component is present @xmath8 or absent @xmath9 . thus the network itself is mostly in a state of being partially absent as , e.g. , in a protein network where for every absent protein all of its outgoing links are absent as well @xcite . in the spin state convention , this fact is not faithfully represented . another example for an inaccurate detail is the common practice to use the standard convention of the heaviside step function as an activation function in discrete dynamical networks ( or the sign function in the spin model context ) . the convention @xmath10 is not a careful representation of biological circumstances . both , for genes and neurons , a silent input frequently maps to a silent output . therefore , we use a redefined threshold function defined as @xmath11 when studying statistical properties of ensembles of threshold networks with random links , these details have a considerable influence on the network s dynamics and critical properties . when simulating ensembles via networks of spins @xmath3 , care should be taken to properly renormalize weights and activation thresholds to ensure faithful implementation of the original model with states @xmath2 . however , this is frequently omitted , resulting in the statistics of a system of limited biological plausibility @xcite . another example where normalization and the definition of the nodes thresholds matters are adaptive networks , currently discussed in the context of neural networks @xcite . when defining local adaptive mechanisms , it is particularly important to base it on biologically plausible definitions of nodes and circuits . while these mechanisms work also for spin type networks @xcite , such an implementation is not realizable in a biological context , as it would require signals over links which are in fact silent , due to the inactivity of their source nodes . an adaptive algorithm based on such correlations of non - activity is therefore not plausible . in this paper , we first define a binary threshold network that does not include explicitly forbidden states in the context of biological examples . then we study its critical properties which we find to be distinctly different from those of random boolean networks @xcite and random threshold networks @xcite . in particular , activity of the network now influences criticality in a non - trivial way , as recently observed for random threshold networks with bistable nodes @xcite . let us consider randomly wired threshold networks of @xmath12 nodes @xmath13 . at each discrete time step , all nodes are updated in parallel according to @xmath14 using the input function @xmath15 in particular we choose @xmath16 for plausibility reasons ( zero input signal will produce zero output ) . while the weights take discrete values @xmath17 with equal probability for connected nodes , we select the thresholds @xmath18 for the following discussion . for any node @xmath6 , the number of incoming links @xmath19 is called the in - degree @xmath20 of that specific node . @xmath21 denotes the average connectivity of the whole network . with randomly placed links , the probability for each node to actually have @xmath22 incoming links follows a poissonian distribution : @xmath23 to analytically derive the critical connectivity of this type of network model , we first study damage spreading on a local basis and calculate the probability @xmath24 for a single node to propagate a small perturbation , i.e. to change its output from 0 to 1 or vice versa after changing a single input state . the calculation can be done closely following the derivation for spin - type threshold networks in ref . @xcite , but one has to account for the possible occurrence of ` 0 ' input signals also via non - zero links . the combinatorial approach yields a result that directly corresponds to the spin - type network calculation via @xmath25 . however , this approach does not hold true for our boolean model in combination with the defined theta function @xmath16 as it assumes a statistically equal distribution of all possible input configurations for a single node . in the boolean model , this would involve an average node activity of @xmath26 over the whole network . instead we find ( fig . [ fig : activity ] ) that the average activity on the network is significantly below @xmath27 . at @xmath28 ( which will turn out to be already far in the supercritical regime ) , less than 30 percent of all nodes are active on average . around @xmath29 ( where we usually expect the critical connectivity for such networks ) , the average activity is in fact below 10 percent . thus , random input configurations will more likely consist of a higher number of ` 0 ' signal contributions than of @xmath30 inputs . therefore , when counting input configurations for the combinatorial derivation of @xmath24 , we need to weight all relevant configurations according to their realization probability as given by the average activity @xmath31 . for the first @xmath32 this yields @xmath33 which generalizes to @xmath34 using @xmath35 and @xmath36 . as the in - degree @xmath37 is not equal for all nodes , the expectation value of @xmath38 is essential to determine the critical connectivity of the whole network . this will yield the average probability for damage spreading for a certain average connectivity @xmath21 . consider a network of size @xmath39 . for large @xmath12 , the problem studied in the above section is equivalent to connecting a new node @xmath40 with state @xmath41 to an arbitrary node @xmath6 , increasing @xmath20 to @xmath42 . if now @xmath41 is changed , this will result in a state change of node @xmath6 with a probability @xmath43 . as mentioned above , the link distribution follows a poissonian ; thus we can calculate the expectation value @xmath44 for the thermodynamical limit @xmath45 as @xmath46 which can be computed numerically using @xmath24 as given in eq . ( [ eq : ps_k ] ) . the actual calculation of the critical connectivity @xmath47 for the network model presented above is now done by applying the annealed approximation as introduced by derrida and pomeau @xcite . the application of this method on threshold networks is discussed in detail in @xcite , which can be directly transferred on the network model discussed in the present work . the critical connectivity can thus be obtained by solving @xmath48 however , @xmath47 now depends on the average network activity , which in turn is a function of the average connectivity @xmath21 itself as shown in fig . [ fig : activity ] . from the combined plot in fig . [ fig : activityvskc ] we find that both curves intersect at a point where the network dynamics due to the current connectivity @xmath49 exhibit an average activity which in turn yields a critical connectivity @xmath47 that exactly matches the given connectivity . this intersection thus corresponds to the critical connectivity of the present network model . however , the average activity still varies with different network sizes , which is obvious from figure [ fig : activityvskc ] . therefore , also the critical connectivity is a function of @xmath12 . table [ tab : kc_sizes ] lists results for different values of @xmath12 . .critical connectivity @xmath47 for different sizes as determined from curve intersections in figure [ fig : activityvskc ] . [ cols="^,^,^,^,^,^,^",options="header " , ] for an analytic approach to the infinite size limit , we can now calculate the probability for a node at given in - degree @xmath37 and average network activity @xmath50 at time @xmath51 to exhibit output state 1 . this probability equals the average activity for the next time step @xmath52 . by examining all relevant input configurations , we find that for given constant @xmath37 this generalizes to @xmath53 again , we have to account for the poissonian distribution of links in our network model , so the average evolution of network activity is obtained by @xmath54 it is now possible to distinguish between the different dynamical regimes by solving @xmath55 for the critical line . the solid line in figure [ fig : activityvskc ] depicts the evolved activity in the long time limit . we find that for infinite system size , the critical connectivity is at @xmath56 while up to this value all network activity vanishes in the long time limit ( @xmath57 ) . for any average connectivity @xmath58 , a certain fraction of nodes remains active . in finite size systems , both network activity evolution and damage propagation probabilities are subject to finite size effects , thus increasing @xmath47 to a higher value . as a numerical verification , we can also derive the critical connectivity for infinite system size @xmath59 using finite size scaling of the above simulation results ( table [ tab : kc_sizes ] ) . the optimum fit is shown in figure [ fig : kc - scaling ] and yields @xmath60 which perfectly supports the analytical calculation . the same consideration is also possible to obtain the average activity @xmath61 for increasing network size . this can be done both at constant values of @xmath49 as well as along the critical line in figure [ fig : activityvskc ] using the values of @xmath62 from table [ tab : kc_sizes ] . for the critical line , we indeed find vanishing network activity ( inset of figure [ fig : kc - scaling ] ) : @xmath63 this does also hold true for any connectivity below the critical line . for supercritical networks at @xmath64 , the numerical simulation yields a non - zero fraction of nodes which remains active on average in good agreement with the analytic computation ( see squares in figure [ fig : activityvskc ] ) . in additional numerical simulations using the standard step function , we obtain a critical connectivity of @xmath65 , which is analytically supported by a combinatorial calculation following ref . @xcite where we find @xmath25 . the networks exhibit significantly higher average activity , while most of the active nodes are frozen in the active state . on a side note , if we chose to calculate the critical connectivity based on an assumed average activity of @xmath27 , the activity - dependent calculation via ( [ eq : ps_k ] ) and ( [ eq : k_ccalculation ] ) would effectively reproduce the same result ( @xmath65 ) as the original combinatiorial approach from @xcite would yield for the new boolean model . as the assumption does not hold true here , this result can only be viewed as an additional plausibility check for the correspondence between both approaches . in passing we note that also for the spin type threshold network studied in @xcite we find activity levels different from @xmath27 , such that conditions for a valid combinatorial approach might not be met in that case , as well . finally , let us have a closer look on the average length of attractor cycles and transients . as shown in fig . [ fig : attractors ] , the behavior is strongly dependent of the dynamical regime of the network . as expected and in accordance with early works on random threshold networks @xcite as well as random boolean networks @xcite , we find an exponential increase of the average attractor lengths with network size @xmath12 in the chaotic regime ( @xmath66 ) , whereas we can observe a sub - linear increase in the frozen phase ( @xmath67 ) . we find similar behavior for the scaling of transient lengths ( inset of figure [ fig : attractors ] ) . in summary we studied threshold networks with boolean node states that are biologically more plausible than current boolean and threshold networks and which are simpler than the recently introduced networks with bistable threshold nodes @xcite . a major observation is that activity of the nodes depends on connectivity which also renders critical properties of the networks activity - dependent , as found earlier for random threshold networks with bistable nodes @xcite . we extend the annealed approximation to correct for these effects and find connectivity @xmath1 and vanishing activity at the critical point in the thermodynamic limit . going beyond the statistics of random network ensembles , also real biological circuits can be implemented with great ease using the threshold networks defined here . we successfully reproduced the dynamical trajectory of the budding yeast cell cycle network as implemented with bistable threshold functions in @xcite , as well as for the corresponding network in fission yeast @xcite . to conclude , let us remind ourselves of the original idea of using random boolean networks for characterizing typical properties of biological networks @xcite . in 1969 , using random boolean networks as a null model for genetic networks was a logical approach , given our complete ignorance of the circuitry of genetic networks at that time . thus the best guess was to treat all possible boolean rules as equally probable . today , however , we have much more details knowledge about certain properties of genetic networks and , therefore , about more realistic ensembles of random networks . a biologically motivated and carefully defined threshold network , as attempted in this paper , may provide a more suited null model for the particular properties of biological networks than random boolean networks with equally distributed boolean functions .

in highly anisotropic layered superconductors , tilting a magnetic field at a direction oblique to the cuo@xmath1 planes leads to novel states of vortex matter . bitter decoration@xcite , hall probe microscopy and lorentz microscopy have confirmed that the attraction between josephson vortex ( jv ) stacks and pancake vortices ( pv ) in the vortex solid state of bi@xmath1sr@xmath1cacu@xmath1o@xmath2 leads to the so - called crossing - lattice@xcite . this state is only one of many constituting a particularly rich phase diagram . the occurence of other structural phases of vortex matter , _ e.g. _ , the lattice of tilted pv stacks , the combined perpendicular and tilted pv stacks@xcite , or the pv - soliton lattice , depends on the interplay between the magnetic and josephson coupling contributions to the pv lattice tilt modulus @xcite . the josephson plasma resonance ( jpr ) frequency @xmath4 is sensitive to the superconducting phase difference , @xmath5 , between cuo@xmath1 layers @xmath6 , through @xmath7 where @xmath8 stands for thermal and disorder average . since @xmath5 intimately depends on the alignment of pv stacks , jpr can in principle be used to detect and identify different vortex phases . the bi@xmath1sr@xmath1cacu@xmath1o@xmath2 single crystals ( @xmath3 k ) were cut from a larger underdoped bi@xmath1sr@xmath1cacu@xmath1o@xmath2 crystal , grown by the travelling solvent floating zone technique . the jpr was measured using the cavity perturbation technique in the tm@xmath9 modes , with the microwave electrical field aligned along the sample @xmath0-axis . two orthogonal coils were used to apply field components parallel ( @xmath10 ) and perpendicular ( @xmath11 ) to the cuo@xmath1 layers . varying the mode @xmath12 , allowed us to change the jpr frequency and to probe both the vortex solid and the liquid state . magnetic - field dependance of the microwave absorption ( arbitrary units ) of a bi@xmath1sr@xmath1cacu@xmath1o@xmath2 single crystal obtained at different frequencies . ] figure [ fig : dissipation ] shows the microwave absorption obtained by sweeping @xmath11 at constant @xmath10 and temperature . when @xmath13 , the jpr is identified as the maximum of the microwave absorption , for the field @xmath14 ( arrows in figure [ fig : dissipation ] ) . following the evolution of the absorption , the lineshape is modified by @xmath10 in two ways . first , the intensity of the microwave response decreases for all frequencies . it has been pointed out that the presence of a josephson vortex lattice ( jvl ) strongly modulates the _ c_-axis critical current@xcite . thus , the jpr can not develop in the stacks of jvs . however , for low values of the in - plane field ( @xmath15 , where @xmath16 is the anisotropy ratio and @xmath17 the interlayer distance ) , the distance between two stacks of jvs is sufficiently high so that the phase remains unaffected far from the cores . the remaining microwave absorption mainly arises from jv - free regions , the extent of which decreases linearly with @xmath10 . second , @xmath10 changes the field @xmath14 at which the maximum absorption occurs : the decrease of @xmath14 observed in the vortex solid at 31.2 and 39.4 ghz implies the decrease of @xmath18 and is consistent with the addition of a jvl . however , for 19.2 and 22.9 ghz , @xmath14 , measured in the vortex liquid , _ increases_. the @xmath14-loci for different temperatures are shown in fig . [ fig : result ] together with the pv vortex lattice melting line in @xmath13 . the increase of @xmath14 is only observed in the vortex liquid state and for sufficiently low pv densities . in the solid phase , it has been shown@xcite that the addition of pvs on a dense lattice of jvs ( @xmath19 ) increases the @xmath0-axis critical current : the pvs adjust their positions to the jvl so as to maximize @xmath18@xcite . the same effect could be present in the vortex liquid phase , giving rise to an effective attraction between pvs and jvs and to a _ correlated vortex liquid _ in which the density of pvs is smaller in jv - free regions . since the microwave absorption due to the jpr mainly comes from those regions , increasing @xmath10 also increases @xmath14 . however , even though the presence of well - defined josephson vortices in the presence of a pv liquid is still controversial , we believe that it can be the case for low densities of pvs . the correlated vortex liquid is therefore stable close to the pv melting line , for low values of the perpendicular magnetic field .

by measuring the josephson plasma resonance , we have probed the influence of an in - plane magnetic field on the pancake vortex correlations along the @xmath0-axis in heavily underdoped bi@xmath1sr@xmath1cacu@xmath1o@xmath2 ( @xmath3 k ) single crystals both in the vortex liquid and in the vortex solid phase . whereas the in - plane field enhances the interlayer phase coherence in the liquid state close to the melting line , it slightly depresses it in the solid state . this is interpreted as the result of an attractive force between pancake vortices and josephson vortices , apparently also present in the vortex liquid state . the results unveil a boundary between a correlated vortex liquid in which pancakes adapt to josephson vortices , and the usual homogeneous liquid . josephson plasma resonance , bi:2212 , josephson vortex 74.25.qt , 74.50.+r

low - ionization nuclear emission - line regions ( liners ; heckman 1980 ) are found in many nearby bright galaxies ( e.g. , ho , filippenko , & sargent 1997a ) . extensive studies at various wavelengths have shown that type 1 liners ( liner 1s , i.e. , those galaxies having broad h@xmath3 and possibly other broad balmer lines in their nuclear optical spectra ) are powered by a low - luminosity agn ( llagn ) with a bolometric luminosity less than @xmath14 ergs s@xmath2 ( ho et al . 2001 ; terashima , ho , & ptak 2000a ; ho et al . 1997b ) . on the other hand , the energy source of liner 2s is likely to be heterogeneous . some liner 2s show clear signatures of the presence of an agn , while others are most probably powered by stellar processes , and the luminosity ratio @xmath6/@xmath15 can be used to discriminate between these different power sources ( e.g. , prez - olea & colina 1996 ; maoz et al . 1998 ; terashima et al . it is interesting to note that currently there are only a few liner 2s known to host an obscured agn ( e.g. , turner et al . this paucity of obscured agn in liners may indicate that liner 2s are not simply a low - luminosity extension of luminous seyfert 2s , which generally show heavy obscuration with a column density averaging @xmath10 @xmath16 @xmath11 @xmath12 ( e.g. , turner et al . 1997 ) . alternatively , biases against finding heavily obscured llagns may be important . for example , objects selected through optical emission lines or x - ray fluxes are probably biased in favor of less absorbed ones , even if one uses the x - ray band above 2 kev . in contrast , radio observations , particularly at high frequency , are much less affected by absorption . although an optical spectroscopic survey must first be done to find the emission lines characteristic of a llagn , follow up radio observations can clarify the nature of the activity . for example , vlbi observations of some llagns have revealed a compact nuclear radio source with @xmath17 k , which is an unambiguous indicator of the presence of an active nucleus and can not be produced by starburst activity ( e.g. , falcke et al . 2000 ; ulvestad & ho 2001 ) . a number of surveys of seyfert galaxies at sub arcsecond resolution have been made with the vla ( ulvestad & wilson 1989 and references therein ; kukula et al . 1995 ; nagar et al . 1999 ; thean et al . 2000 ; schmitt et al . 2001 ; ho & ulvestad 2001 ) and other interferometers ( roy et al . 1994 ; morganti et al . 1999 ) , but much less work has been done on the nuclear radio emission of liners . nagar et al . ( 2002 ) have reported a vla 2 cm radio survey of all 96 llagns within a distance of 19 mpc . these llagns come from the palomar spectroscopic survey of bright galaxies ( ho et al . 1997a ) . as a pilot study of the x - ray properties of llagns , we report here a _ chandra _ survey of a subset , comprising 15 galaxies , of nagar et al s ( 2002 ) sample . fourteen of these galaxies have a compact nuclear radio core with a flat or inverted radio spectrum ( nagar et al . 2000 ) . we have detected 13 of the galactic nuclei with _ chandra_. we also examine the `` radio loudness '' of our sample and compare it with other classes of agn . a new measure of `` radio loudness '' is developed , in which the 5 ghz radio luminosity is compared with the 210 kev x - ray luminosity ( @xmath9=@xmath18 ( 5 ghz)/@xmath19(210 kev ) ) rather than with the b - band optical luminosity ( @xmath20(5 ghz)/@xmath8(b ) ) , as is usually done . @xmath9 has the advantage that it can be measured for highly absorbed nuclei ( @xmath10 up to several times @xmath11 @xmath12 ) which would be totally obscured ( @xmath21 up to a few hundred mag for the galactic gas to dust ratio ) at optical wavelengths , and that the compact , hard x - ray source in a llagn is less likely to be confused with emission from stellar - powered processes than is an optical nucleus . this paper is organized as follows . the sample , observations , and data reduction are described in section 2 . imaging results and x - ray source detections are given in section 3 . section 4 presents spectral results . the power source , obscuration in llagns , and radio loudness of llagns are discussed in section 5 . section 6 summarizes the findings . we use a hubble constant of @xmath22 km s@xmath2 mpc@xmath2 and a deceleration parameter of @xmath23 throughout this paper . our sample is based on the vla observations by nagar et al . their sample of 48 objects consists of 22 liners , 18 transition objects , which show optical spectra intermediate between liners and nuclei , and eight low - luminosity seyferts selected from the optical spectroscopic survey of ho et al . ( 1997a ) . the sample is the first half of a distance - limited sample of llagns ( nagar et al . 2002 ) , as described in section 1 . we selected 14 objects showing a flat to inverted spectrum radio core ( @xmath24 , @xmath25 ) according to nagar et al.s ( 2000 ) comparison with longer wavelength radio data published in the literature . one object ( the liner 2 ngc 4550 ) has a flat spectrum radio source at a position significantly offset from the optical nucleus . this object was added as an example of a liner without a detected radio core . the target list and some basic data for the final sample are summarized in table 1 . the distances are taken from tully ( 1988 ) in which @xmath22 km s@xmath2 mpc@xmath2 is assumed . the sample consists of seven liner 1s , four liner 2s , two seyfert 1s , one seyfert 2 , and one transition 2 object . 12 out of these 15 objects have been observed with the vlba and high brightness temperature ( @xmath26 k ) radio cores were detected in all of them ( falcke et al . 2000 ; ulvestad & ho 2001 ; nagar et al . therefore , these objects are strong candidates for agns . results of archival / scheduled _ chandra _ observations of more liners with a compact flat / inverted spectrum radio core found by nagar et al . ( 2002 ) will be presented in a future paper . a log of _ chandra _ observations is shown in table 1 . the exposure time was typically two ksec each . all the objects were observed at or near the aim point of the acis - s3 back - illuminated ccd chip . eight objects were observed in our guaranteed time observation program and the rest of the objects were taken from the _ chandra _ archives . the eight objects were observed in 1/8 sub - frame mode ( frame time 0.4 s ) to minimize effects of pileup ( e.g. , davis 2001 ) . 1/2 sub - frame modes were used for three objects . ciao 2.2.1 and caldb 2.7 were used to reduce the data . in the following analysis , only events with _ asca _ grades 0 , 2 , 3 , 4 , 6 ( `` good grades '' ) were used . for spectral fitting , xspec version 11.2.0 was employed . an x - ray nucleus is seen in all the galaxies except for ngc 4550 and ngc 5866 . some off - nuclear sources are also seen in some fields . the source detection algorithm `` wavdetect '' in the ciao package was applied to detect these nuclear and off - nuclear sources , where a detection threshold of @xmath27 and wavelet scales of 1 , @xmath28 , 2 , @xmath29 , 4 , @xmath30 , 8 , @xmath31 , and 16 pixels were used . source detections were performed in the three energy bands 0.58 kev ( full band ) , 0.52 kev ( soft band ) , and 28 kev ( hard band ) . the resulting source lists and raw images were examined by eye to exclude spurious detections . the source and background counts were also determined by manual photometry and compared with the results of wavdetect . in the few cases that the two methods gave discrepant results , we decided to use the results of the manual photometry after inspection of the raw images . some sources were detected in only one or two energy bands . in such cases , we calculated the upper limits on the source counts in the undetected band(s ) at the 95% confidence level by interpolating the values in table 2 of kraft , burrows , & nousek ( 1991 ) . table 2 shows the positions , detected counts , band ratios ( hard / soft counts ) , fluxes and luminosities in the 210 kev band of the nuclear sources . the same parameters for off - nuclear sources with signal - to - noise ratios greater than three are summarized in table 3 . for bright objects ( @xmath32 40 counts ) , the fluxes were measured by spectral fits presented in the next section . for faint objects , fluxes were determined by assuming the galactic absorption column density and power law spectra , with photon indices determined from the band ratios . when only lower or upper limits on the band ratio were available , a photon index of 2 was assumed if the limit is consistent with @xmath33 . when the band ratio is inconsistent with @xmath33 , the upper or lower limit is used to determine the photon index . luminosities were calculated only if the source is spatially inside the optical host galaxy as indicated by comparing the position with the optical image of the digitized sky survey . possible identifications for off - nuclear sources are also given in the last column of table 3 . the positions of the x - ray nuclei coincide with the radio core positions to within the positional accuracy of _ chandra_. the nominal separations between the x - ray and radio nuclei are in the range 0.05 0.95@xmath34 . inspection of the images shows that the nucleus in most objects appears to be unresolved , while some objects show faint extended emission . the soft and hard band images of the nuclear regions of ngc 3169 and ngc 4278 are shown in fig 1 as examples of extended emission . in the soft band image of ngc 3169 , emission in the nuclear region extending @xmath35 arcsec in diameter is clearly visible , while the nucleus itself is not detected ( table 2 ) . about 25 counts were detected within a 10 pixel ( 4.9 arcsec ) radius in the @xmath36 kev band . this extended emission is not seen in the hard band . the soft band image of ngc 4278 consists of a bright nucleus and a faint elongated feature with a length of @xmath37 and a position angle of @xmath38 . the hard band image is unresolved . the nuclear regions ( 10 arcsec scale ) of the other galaxies look compact to within the current photon statistics . more extended diffuse emission at larger scales ( @xmath39 ) is seen in a few objects . the nuclei of ngc 2787 and ngc 4203 are embedded in soft diffuse emission with diameters of @xmath40 and @xmath41 , respectively . ngc 4579 shows soft diffuse emission with a similar morphology to the circumnuclear star forming ring , in addition to a very bright nucleus ( see also eracleous et al . ngc 4565 shows extended emission along the galactic plane . ngc 5866 has soft extended emission @xmath42 ( 2 kpc ) in diameter and no x - ray nucleus is detected ( table 2 ) . any diffuse emission associated with the other galaxies is much fainter . spectral fits were performed for the relatively bright objects those with @xmath32 40 detected counts in the 0.58 kev band . the spectrum of one fainter object ( ngc 4548 ) showing a large ( = hard ) hardness ratio was also fitted . some objects are so bright that pileup effects are significant . column 7 of table 1 gives the count rates per ccd read - out frame time and can be used to estimate the significance of pileup . in the two objects ngc 3147 and ngc 4278 , the pileup is mild and we corrected for the effect by applying the pileup model implemented in xspec , where the grade morphing parameter @xmath3 was fixed at 0.5 ( after initially treating it as a free parameter [ davis 2001 ] since @xmath3 is not well constrained ) . the pileup effect for the three objects with the largest count rate per frame ( ngc 4203 , ngc 4579 , and ngc 5033 ) is serious and we did not attempt detailed spectral fits . instead , we use the spectra and fluxes measured with _ asca _ for these three objects ( terashima et al . 2002b and references therein ) in the following discussions . we confirmed that the nuclear x - ray source dominates the hard x - ray emission within the beam size of _ asca _ ( see appendix ) . the other objects in the sample are faint enough to ignore the effects of pileup . x - ray spectra were extracted from a circular region with a radius between 4 pixels ( 2.0@xmath34 ; for faint sources ) and 10 pixels ( 4.9@xmath34 ; for bright sources ) depending on source brightness . background was estimated using an annular region centered on the target . a maximum - likelihood method using the c - statistic ( cash 1979 ) was employed in the spectral fits . in the fit with the c - statistic , background can not be subtracted , so we added a background model ( measured from the background region ) with fixed parameters to the spectral models , after normalizing by the ratio of the geometrical areas of the source and background regions . the errors quoted represent the 90% confidence level for one parameter of interest ( @xmath43=2.7 ) . a power - law model modified by absorption was applied and acceptable fits were obtained in all cases ( fig . the best - fit parameters for the nuclear sources are given in table 4 . the observed fluxes and luminosities ( the latter corrected for absorption ) in the 210 kev band are shown in table 2 . the results for a few bright off - nuclear sources are presented in the appendix . the photon indices of the nuclear sources are generally consistent with the typical values observed in llagns ( photon index @xmath44 , e.g. , terashima et al . 2002a , 2002b ) , although errors are quite large due to the limited photon statistics . the spectral slope of ngc 6500 ( @xmath45 ) is somewhat steeper than is typical of llagns . this may indicate that there is soft x - ray emission from a source other than the agn and/or the intrinsic slope of the agn is steep . one galaxy ( ngc 3169 , a liner 2 ) has a large absorption column ( @xmath10=@xmath46 @xmath12 ) , while two galaxies ( ngc 4548 , a liner 2 , and ngc 3226 , a liner 1.9 ) show substantial absorption ( @xmath47 @xmath12 , and @xmath48 @xmath12 , respectively ) . others have small column densities which are consistent with ` type 1 ' agns . no meaningful limit on the equivalent width of an fe k@xmath3 line was obtained for any of the objects because of limited photon statistics in the hard x - ray band . one object ngc 2787 has only 8 detected photons in the 0.58 kev band and is too faint to obtain spectral information . a photon index of 2.0 and the galactic absorption of @xmath49 @xmath12 were assumed to calculate the flux and luminosity which are shown in table 2 . an x - ray nucleus is detected in all the objects except for ngc 4550 and ngc 5866 . we test whether the detected x - ray sources are the high energy extension of the continuum source which powers the optical emission lines by examining the luminosity ratio @xmath6/@xmath15 . the h@xmath3 luminosities ( @xmath15 ) were taken from ho et al . ( 1997a ) and the reddening was estimated from the balmer decrement for narrow lines and corrected using the reddening curve of cardelli , clayton , & mathis ( 1989 ) , assuming the intrinsic h@xmath3/h@xmath50 flux ratio = 3.1 . the x - ray luminosities ( corrected for absorption ) in the @xmath51 kev band are used . the h@xmath3 luminosities and logarithm of the luminosity ratios @xmath6/@xmath15 are shown in table 5 . the @xmath6/@xmath15 ratios of most objects are in the range of agns ( @xmath52 @xmath6/@xmath15 @xmath16 12 ) and in good agreement with the strong correlation between @xmath6 and @xmath15 for llagns , luminous seyferts , and quasars presented in terashima et al . ( 2000a ) and ho et al . this indicates that their optical emission lines are predominantly powered by a llagn . note that this correlation is not an artifact of distance effects , as shown in terashima et al . ( 2000a ) . the four objects ngc 2787 , ngc 4550 , ngc 5866 , and ngc 6500 , however , have much lower @xmath6/@xmath15 ratios ( @xmath52 @xmath6/@xmath15 .5ex0 ) than expected from the correlation ( @xmath52 @xmath6/@xmath15 @xmath53 ) , and their x - ray luminosities are insufficient to power the h@xmath3 emission ( terashima et al . this x - ray faintness could indicate one or more of several possibilities such as ( 1 ) an agn is the power source , but is heavily absorbed at energies above 2 kev , ( 2 ) an agn is the power source , but is currently switched - off or in a faint state , and ( 3 ) the optical narrow emission lines are powered by some source(s ) other than an agn . we briefly discuss these three possibilities in turn . if an agn is present in these x - ray faint objects and absorbed in the hard energy band above 2 kev , only scattered and/or highly absorbed x - rays would be observed , and then the intrinsic luminosity would be much higher than that observed . this can account for the low @xmath6/@xmath15 ratios and high radio to x - ray luminosity ratios ( @xmath54(5 ghz)/@xmath6 ; table 5 and section 5.3 ) . if the intrinsic x - ray luminosities are about one or two orders of magnitude higher than those observed , as is often inferred for seyfert 2 galaxies ( turner et al . 1997 , awaki et al . 2000 ) , @xmath6/@xmath15 and @xmath54(5 ghz)/@xmath6 become typical of llagns . alternatively , the agn might be turned off or in a faint state , with a higher activity in the past being inferred from the optical emission lines , whose emitting region is far from the nucleus ( e.g. , eracleous et al . also , the radio observations were made a few years before the _ chandra _ ones . this scenario might thus explain their relatively low @xmath6/@xmath15 ratios and their relatively high @xmath55/@xmath6 ratios . if this is the case , the size of the radio core can be used to constrain the era of the active phase in the recent past . the upper limits on the size of the core estimated from the beam size ( @xmath56 2.5 mas ) are 0.16 , 0.19 , and 0.48 pc for ngc 2787 , ngc 5866 , and ngc 6500 , respectively ( falcke et al . therefore , the agn must have been active until @xmath570.52 , @xmath570.60 , and @xmath571.6 years , respectively , before the vlba observations ( made in 1997 june ) and inactive at the epochs ( 2000 jan 2002 jan , see table 1 ) of the x - ray observations . this is an ad hoc proposal and such abrupt declines of activity are quite unusual , but it can not be completely excluded . it may also be possible that the ionized gas inferred from the optical emission lines is ionized by some sources other than an agn , such as hot stars . if the observed x - rays reflect the intrinsic luminosities of the agn , a problem with the agn scenario for the three objects ngc 2787 , ngc 5866 , and ngc 6500 is that these galaxies have very large @xmath18(5 ghz)/@xmath6 ratios , and would thus be among the radio loudest llagns . the presence of hot stars in the nuclear region of ngc 6500 is suggested by uv spectroscopy ( maoz et al . maoz et al . ( 1998 ) studied the energy budget for ngc 6500 by using the h@xmath3 and uv luminosity at 1300 a and showed that the observed uv luminosity is insufficient to power the h@xmath3 luminosity even if a stellar population with the salpeter initial mass function and a high mass cutoff of 120@xmath58 are assumed . this result indicates that a power source in addition to hot stars must contribute significantly , and supports the obscured agn interpretation discussed above . the first possibility , i.e. , an obscured low - luminosity agn as the source of the x - ray emission , seems preferable for ngc 2787 , ngc 5866 and ngc 6500 , although some other source(s ) may contribute to the optical emission lines . additional lines of evidence which support the presence of an agn include the fact that all three of these galaxies ( ngc 2787 , ngc 5866 , and ngc 6500 ) have vlbi - detected , sub - pc scale , nuclear radio core sources ( falcke et al . 2000 ) , a broad h@xmath3 component ( in ngc 2787 , and an ambiguous detection in ngc 5866 ; ho et al . 1997b ) , a variable radio core in ngc 2787 , and a jet - like linear structure in a high - resolution radio map of ngc 6500 with the vlba ( falcke et al . only an upper limit to the x - ray flux is obtained for ngc 5866 . if an x - ray nucleus is present in this galaxy and its luminosity is only slightly below the upper limit , this source could be an agn obscured by a column density @xmath10@xmath59 @xmath12 or larger . if the apparent x - ray luminosity of the nucleus of ngc 5866 is _ much _ lower than the observed upper limit , and the intrinsic x - ray luminosity conforms to the typical @xmath6/@xmath15 ratio for llagn ( @xmath52 @xmath6/@xmath15 @xmath60 ) , then the x - ray source must be almost completely obscured . the optical classification ( transition object ) suggests the presence of an ionizing source other than an agn , so the low observed @xmath6/@xmath15 ratio could alternatively be a result of enhanced h@xmath3 emission powered by this other ionizing source . the x - ray results presented above show that the presence of a flat ( or inverted ) spectrum compact radio core is a very good indicator of the presence of an agn even if its luminosity is very low . on the other hand , ngc 4550 , which does not possess a radio core , shows no evidence for the presence of an agn and all the three possibilities discussed above are viable . if the _ rosat _ detection is real ( halderson et al . 2001 ) , the time variability between the _ rosat _ and _ chandra _ fluxes may indicate the presence of an agn ( see appendix ) . it is notable that type 2 liners without a flat spectrum compact radio core may be heterogeneous in nature . for instance , some liner 2s without a compact radio core ( e.g. , ngc 404 and transition 2 object ngc 4569 ) are most probably driven by stellar processes ( maoz et al . 1998 ; terashima et al . 2000b ; eracleous et al . 2002 ) . in our sample , we found at least three highly absorbed llagns ( ngc 3169 , ngc 3226 , and ngc 4548 ) . in addition , if the x - ray faint objects discussed in section 5.1 are indeed agns , they are most probably highly absorbed with @xmath10@xmath61 @xmath12 . among these absorbed objects , ngc 2787 is classified as a liner 1.9 , ngc 3169 , ngc 4548 , and ngc 6500 as liner 2s , and ngc 5866 as a transition 2 object . thus , heavily absorbed liner 2s , of which few are known , are found in the present observations demonstrating that radio selection is a valuable technique for finding obscured agns . along with heavily obscured llagns known in low - luminosity seyfert 2s ( e.g. , ngc 2273 , ngc 2655 , ngc 3079 , ngc 4941 , and ngc 5194 ; terashima et al . 2002a ) , our observations show that at least some type 2 llagns are simply low - luminosity counterparts of luminous seyferts in which heavy absorption is often observed ( e.g. , risaliti , maiolino , & salvati 1999 ) . however , some liner 2s ( e.g. , ngc 4594 , terashima et al . 2002a ; ngc 4374 , finoguenov & jones 2001 ; ngc 4486 , wilson & yang 2002 ) and low - luminosity seyfert 2s ( ngc 3147 ; section 4 and appendix ) show no strong absorption . therefore , the orientation - dependent unified scheme ( e.g. , antonucci 1993 ) does not always apply to agns in the low - luminosity regime , as suggested by terashima et al . . combination of x - ray and radio observations is valuable for investigating a number of areas of agn physics , including the `` radio loudness '' , the origin of jets , and the structure of accretion disks . low - luminosity agns ( liners and low - luminosity seyfert galaxies ) are thought to be radiating at very low eddington ratios ( @xmath62/@xmath63 ) and may possess an advection - dominated accretion flow ( adaf ; see e.g. , quataert 2002 for a recent review ) . a study of radio loudness in llagns can constrain the jet production efficiency by an adaf - type disk . earlier studies have suggested that llagns tend to be radio loud compared to more luminous seyferts based on the spectral energy distributions of seven llagns ( ho 1999 ) and , for a larger sample , on the conventional definition of radio loudness @xmath64(5 ghz)/@xmath8(b ) ( the subscript `` o '' , which stands for optical , is usually omitted but we use it here to distinguish from @xmath9 see below ) , with @xmath65 being radio loud ( kellermann et al . 1989 , 1994 ; visnovsky et al . 1992 ; stocke et al . 1992 ; ho & peng 2001 ) . ho & peng ( 2001 ) measured the luminosities of the nuclei by spatial analysis of optical images obtained with _ hst _ to reduce the contribution from stellar light . a caveat in the use of optical measurements for the definition of radio loudness is extinction , which will lead to an overestimate of @xmath66 if not properly allowed for . although ho & peng ( 2001 ) used only type 11.9 objects , some objects of these types show high absorption columns in their x - ray spectra . in this subsection , we study radio loudness by comparing radio and hard x - ray luminosities . since the unabsorbed luminosity for objects with @xmath10 .5ex@xmath11 @xmath12 can be reliably measured in the 210 kev band , which is accessible to _ asca _ , _ xmm - newton _ , and _ chandra _ , and such columns correspond to @xmath21 .5ex50 mag , it is clear that replacement of optical by hard x - ray luminosity potentially yields considerable advantages . in addition , the high spatial resolutions of _ xmm - newton _ and especially _ chandra _ usually allow the nuclear x - ray emission to be identified unambiguously , while the optical emission of llagn can be confused by surrounding starlight . in the following analysis , radio data at 5 ghz taken from the literature are used since fluxes at this frequency are widely available for various classes of objects . we used primarily radio luminosities obtained with the vla at .5ex@xmath67 resolution for the present sample . high resolution vla data at 5 ghz are not available for several objects . for four such cases , vlba observations at 5 ghz with 150 mas resolution are published in the literature ( falcke et al . 2000 ) and are used here . for two objects , we estimated 5 ghz fluxes from 15 ghz data by assuming a spectral slope of @xmath68 ( cf . nagar et al . the radio luminosities used in the following analysis are summarized in table 5 . since our sample is selected based on the presence of a compact radio core , the sample could be biased to more radio loud objects . therefore , we constructed a larger sample by adding objects taken from the literature for which 5 ghz radio , 210 kev x - ray , and @xmath69 measurements are available . first , we introduce the ratio @xmath70(5 ghz)/@xmath6 as a measure of radio loudness and compare the ratio with the conventional @xmath69 parameter . the x - ray luminosity @xmath6 in the 210 kev band ( source rest frame ) , corrected for absorption , is used . , which utilizes monochromatic b - band luminosities . this alternative provides completely identical results if the x - ray spectral shape is known and the range of spectral slopes is not large . for example , the conversion factor @xmath8(2 kev)/@xmath6 is 0.31 , 0.26 , and 0.22 kev@xmath2 for photon indices of 2 , 1.8 , and 1.6 , respectively , and no absorption . ] we examine the behavior of @xmath9 using samples of agn over a wide range of luminosity , including llagn , the seyfert sample of ho & peng ( 2001 ) and pg quasars which are also used in their analysis . @xmath69 parameters and radio luminosities were taken from ho & peng ( 2001 ) for the seyferts and kellermann et al . ( 1989 ) for the pg sample . the values of @xmath69 in kellermann et al . ( 1989 ) have been recalculated by using only the core component of the radio luminosities . the optical and radio luminosities of the pg quasars were calculated assuming @xmath71 and @xmath72 ( @xmath73 ) . the x - ray luminosities ( mostly measured with _ asca _ ) were compiled from terashima et al . ( 2002b ) , weaver , gelbord , & yaqoob ( 2001 ) , george et al . ( 2000 ) , reeves & turner ( 2000 ) , iwasawa et al . ( 1997 , 2000 ) , sambruna , eracleous , & mushotzky ( 1999 ) , nandra et al . ( 1997 ) , smith & done ( 1996 ) , and cappi et al . note that only a few objects ( ngc 4565 , ngc 4579 , and ngc 5033 ) in our radio selected sample have reliable measurements of nuclear @xmath8(b ) . 3 . compares the parameters @xmath69 and @xmath9 for the seyferts and pg sample . these two parameters correlate well for most seyferts . some seyferts have higher @xmath69 values than indicated by most seyferts . this could be a result of extinction in the optical band . seyferts showing x - ray spectra absorbed by a column greater than @xmath74 @xmath12 ( ngc 2639 , 4151 , 4258 , 4388 , 4395 , 5252 , and 5674 ) are shown as open circles in fig . at least four of them have a value of @xmath69 larger than indicated by the correlation . the correlation between @xmath75 and @xmath76 for the less absorbed seyferts can be described as @xmath75 = 0.88 @xmath77 + 5.0 . according to this relation , the boundary between radio loud and radio quiet object ( @xmath75 = 1 ) corresponds to @xmath78 . the values of @xmath69 and @xmath9 for a few obscured seyferts are consistent with the correlation , indicating that optical extinction is not perfectly correlated with the absorption column density inferred from x - ray spectra . the pg quasars show systematically lower @xmath69 values than those of seyferts at a given @xmath76 . for the former objects , @xmath79 corresponds to @xmath80 . this apparently reflects a luminosity dependence of the shape of the sed : luminous objects have steeper optical - x - ray slopes @xmath81 ( @xmath82 ; e.g. , elvis et al . 1994 , brandt , laor , & wills 2000 ) , where @xmath83 is often measured as the spectral index between 2200 a and 2 kev , while less luminous agns have @xmath84 ( ho 1999 ) . this is related to the fact that luminous objects show a more prominent `` big blue bump '' in their spectra . fig . 8 of ho ( 1999 ) demonstrates that low - luminosity objects are typically 11.5 orders of magnitude fainter in the optical band than luminous quasars for an given x - ray luminosity . note that none of the pg quasars used here shows a high absorption column in its x - ray spectrum below 10 kev . the definition of radio loudness using the hard x - ray flux ( @xmath85 ) appears to be more robust than that using the optical flux because x - rays are less affected by both extinction at optical wavelengths and the detailed shape of the blue bump , as noted above . further , measurements of nuclear x - ray fluxes of seyferts and llagns with _ chandra _ are easier than measurements of nuclear optical fluxes , since in the latter case the nuclear light must be separated from the surrounding starlight , a difficult process for llagns . 4 shows the x - ray luminosity dependence of @xmath9 . in this plot , the llagn sample discussed in the present paper is shown in addition to the seyfert and pg samples used above . this is an `` x - ray version '' of the @xmath75-@xmath86 plot ( fig . 4 in ho & peng 2001 ) . radio galaxies taken from sambruna et al . ( 1999 ) are also added and we use radio luminosities from the core only . our plot shows that a large fraction ( @xmath87% ) of llagns ( @xmath6@xmath88 ergs s@xmath2 ) are `` radio loud '' . this is a confirmation of ho & peng s ( 2001 ) finding . note , however , that our sample is not complete in any sense , and this radio - loud fraction should be measured using a more complete sample . since radio emission in llagns is likely to be dominated by emission from jets ( nagar et al . 2001 ; ulvestad & ho 2001 ) , these results suggest that , in llagn , the fraction of the accretion energy that powers a jet , as opposed to electromagnetic radiation , is larger than in more luminous seyfert galaxies and quasars . since llagns are thought to have an adaf - type accretion flow , such might indicate that an adaf can produce jets more efficiently than the geometrically thin disk believed present in more luminous seyferts . the three llagns with the largest @xmath9 in fig . 4 are the three x - ray faint objects discussed in section 5.1 ( ngc 2787 , ngc 5866 , and ngc 6500 ) and which are most probably obscured agns . if their intrinsic x - ray luminosities are 12 orders of magnitude higher than those observed , their values of @xmath9 become smaller by this factor and are then in the range of other llagns . even if we exclude these three llagns , the radio loudness of llagns is distributed over a wide range : the radio - loudest llagns have @xmath9 values similar to radio galaxies and radio - loud quasars , while some llagns are as radio quiet as radio - quiet quasars . a comparison with blazars is of interest to compare our sample with objects for which the nuclear emission is known to be dominated by a relativistic jet and thus strongly beamed . the average @xmath89 for high - energy peaked bl lac objects ( hbls ) , low - energy peaked bl lac objects ( lbls ) , and flat spectrum radio quasars ( fsrqs ) are 3.10 , 1.27 , and 0.95 , respectively , where we used the average radio and x - ray luminosities for a large sample of blazars given in table 3 of donato et al . the average @xmath76 for hbls is similar to that for llagns in our sample , while the latter two classes are about two orders of magnitude more radio loud than llagns . although llagns and hbls have similar values of @xmath89 , the spectral slope in the x - ray band is different : llagns have a photon index in the range 1.72.0 ( see also terashima et al . 2002 ) , while hbls usually show steeper spectra ( photon index @xmath32 2 , e.g. , fig . 1 in donato et al . 2001 ) , and the x - ray emission is believed to be dominated by synchrotron radiation . furthermore , blazars with a lower bolometric luminosity tend to have a synchrotron peak at a higher frequency and a steeper x - ray spectral slope than higher bolometric luminosity blazars ( donato et al . 2001 ) . we also constructed an @xmath9-@xmath6 plot ( fig . 5 ) using the _ total _ radio luminosities of the radio source ( i.e. including the core , jets , lobes , and hot spots , if present ) . the radio data were compiled from vron - cetty & vron ( 2001 ) , kellermann et al . ( 1989 ) , and sambruna et al . the pg sample and other quasars are shown with different symbols . this plot appears similar to fig . 4 for llagns , seyferts , and radio - quiet quasars since these objects do not possess powerful jets or lobes and off - nuclear radio emission associated with the agn is generally of low luminosity ( ulvestad & wilson 1989 , nagar et al . 2001 , ho & ulvestad 2001 , kellermann et al . 1989 ) . on the other hand , radio galaxies have powerful extended radio emission and consequently the @xmath9 values calculated using the total radio luminosities become higher than if only nuclear luminosities are used . we used the same x - ray luminosities as in fig . 4 , because jets , lobes , and hot spots are almost always much weaker than the nucleus in x - rays . in fact , in our observations of llagns , we found no extended emission directly related to the agn . thus , the differences between fig . 4 and fig . 5 result from the extended radio emission . fourteen galaxies with a nuclear radio source having a flat or inverted spectrum have been observed with _ chandra _ with a typical exposure time of 2 ksec . an x - ray nucleus is detected in all but one object ( ngc 5866 ) . 11 galaxies have x - ray and h@xmath3 luminosities in good accord with the correlation known for agns over a wide range of luminosity , which indicates that these objects are agns and that the agn is the dominant power source of their optical emission lines . their x - ray luminosities are between @xmath0 and @xmath1 ergs s@xmath2 . the three objects ngc 2787 , ngc 5866 , and ngc 6500 have significantly lower x - ray luminosities than expected from the @xmath6-@xmath15 correlation . various observations suggest that these objects are most likely to be heavily obscured agns . these observational results show that radio and hard x - ray observations provide an efficient way to find llagn in nearby galaxies , even if the nuclei are heavily obscured . one object ( the liner 2 ngc 4550 ) , which does not show a radio core , was also observed for comparison . no x - ray nucleus is detected . if the x - ray source detected in this galaxy with _ rosat _ is indeed the nucleus , the nucleus must be variable in x - rays , which would indicate the presence of an agn . we have used the ratio @xmath90(5 ghz)/@xmath6 as a measure of radio loudness and found that a large fraction of llagns are radio loud . this confirms earlier results based on nuclear luminosities in the optical band , but our results based on hard x - ray measurements are much less affected by obscuration and the detailed shape of the `` big blue bump '' . we speculate that the increase in @xmath9 as @xmath6 decreases below @xmath91 ergs s@xmath2 may result from the presence of an advection - dominated accretion flow in the inner part of the accretion flow in low - luminosity objects . however , the steep x - ray spectra in our sample of llagns rule out high temperature thermal bremsstrahlung as the x - ray emission mechanism . is supported by the japan society for the promotion of science postdoctoral fellowship for young scientists . this research was supported by nasa through grants nag81027 and nag81755 to the university of maryland . in this appendix , we compare our results with previously published results particularly in the hard x - ray band obtained with _ asca _ and _ chandra_. the optical spectroscopic classification is given in parentheses after the object name . _ ngc 3147 ( s2)_. this object was observed with _ asca _ in 1993 september and the observed flux was @xmath92 ergs s@xmath2@xmath12 in the 210 kev band ( ptak et al . 1996 , 1999 ; terashima et al . our _ chandra _ image is dominated by the nucleus and shows that the off - nuclear source contribution within the _ asca _ beam is negligible . therefore , a comparison between the observed _ chandra _ flux ( @xmath93 ergs s@xmath2@xmath12 ) , which is 2.3 times larger than that of _ asca _ , implies time variability providing additional evidence for the presence of an agn . in the _ asca _ spectrum , a strong fe - k emission line is detected at @xmath94 kev ( source rest frame ) with an equivalent width of @xmath95 ev . one interpretation of this relatively large equivalent width is that the nucleus is obscured by a large column density and the observed x - rays are scattered emission ( ptak et al . 1996 ) . however , the luminosity ratios @xmath6/@xmath15 and @xmath6/@xmath96\lambda 5007}$ ] suggest small obscuration ( terashima et al . 2002b ) . the observed variability supports the interpretation that the x - ray emission is not scattered emission from a heavily obscured nucleus . this galaxy is an example of a seyfert 2 with only little absorption in the x - ray band . _ ngc 3226 ( l1.9)_. this galaxy was observed with the _ chandra _ hetg in 1999 december ( george et al . they obtained an intrinsic luminosity of @xmath97 ( @xmath98 ergs s@xmath2 , 68% confidence limit ) in the 210 kev band after conversion to a distance of 23.4 mpc . this luminosity is consistent with our value of @xmath99 ( @xmath100 ergs s@xmath2 , 90% confidence range ) after correction for absorption . _ ngc 4203 ( l1.9)_. a result on the same data set is presented in ho et al . the nucleus of this object has a large x - ray flux and pileup is severe in this observation . a bright source is seen 2@xmath101 se of the nucleus which was also separated from the nucleus with _ asca _ sis observations ( iyomoto et al . 1998 ; terashima et al . the _ chandra _ observations show that there is no source confusing the _ asca _ observation of the nucleus . therefore , we used an _ asca _ flux in the discussions . we analyzed archival _ data observed on 1998 may 24 . the effective exposure times after standard data screening were 19.6 ksec for each sis and 23.9 ksec for each gis . spectrum is well fitted with a power law with a photon index 1.85 ( 1.77@xmath1021.94 ) . the best - fit absorption column is @xmath10=0 , with an upper limit of @xmath103 @xmath12 . the observed flux in the 2@xmath10210 kev band is @xmath104 ergs s@xmath2@xmath12 . our _ chandra _ image in the hard energy band is dominated by the nucleus and no bright source is seen in the field . therefore , the hard x - ray measurement with _ asca _ seems reliable . the _ chandra _ flux in the 210 kev band ( @xmath105 ergs s@xmath2@xmath12 ) is about one - third of the _ asca _ flux indicating variability . _ ngc 4550 ( l2)_. this source is not detected with the present _ chandra _ observation . a detection with the _ rosat _ pspc is reported by halderson et al . the observed _ rosat _ flux in the 0.12.4 kev band is @xmath106 ergs s@xmath2 @xmath12 . rosat _ source is offset from the optical nucleus by 10@xmath34 . if this source is indeed the nucleus , our non detection by _ ( @xmath107 ergs s@xmath2 @xmath12 ) indicates time variability . _ ngc 4565 ( s1.9)_. the nuclear region is dominated by two sources : the nucleus and an off - nuclear source which is brighter than the nucleus . the observed _ chandra _ fluxes of these two source in the 210 kev band ( @xmath108 and @xmath109 ergs s@xmath2@xmath12 ) are slightly lower than those obtained with _ asca _ ( @xmath110 and @xmath111 ergs s@xmath2@xmath12 ; mizuno et al . 1999 ; terashima et al . 2002b ) , respectively . these differences appear not to be significant given the statistical , calibration , and spectral - modeling uncertainties . ( the uncertainties on the _ chandra _ fluxes are dominated by the statistical errors , which are @xmath112 % for the off - nuclear source and @xmath1650% for the nucleus , while the error in the _ asca _ fluxes is dominated by calibration uncertainties of @xmath35 % . ) the _ chandra _ spectrum of the off - nuclear source can be fitted by an absorbed power law model with a photon index of @xmath113 and @xmath10 = @xmath114 @xmath12 . a multicolor disk blackbody model also provides a good fit with best - fit parameters @xmath115 kev and @xmath10 = @xmath116 @xmath12 . _ ngc 4579 ( l1.9/s1.9)_. a result on the same data set is presented by ho et al . the nucleus is significantly piled up in the _ chandra _ observation . hard band image is dominated by the nucleus and no bright source is seen in the field . therefore , we used _ fluxes observed in 1995 and 1998 . detailed _ asca _ results are published in terashima et al . ( 1998 , 2000c ) . a long ( 33.9 ksec exposure ) _ chandra _ observation performed in 2000 may is presented in eracleous et al . the 210 kev flux reported is @xmath117 ergs s@xmath2@xmath12 which is similar to that of the second _ asca _ observation in 1998 ( @xmath118 ergs s@xmath2@xmath12 ) . _ ngc 5033 ( s1.5)_. a result on the same data set is presented in ho et al . the nucleus is significantly piled up in the _ chandra _ observation . the five off - nuclear sources shown in table 3 are located within the _ asca _ beam . the sum of the counts from these sources is less than 44 counts in the 28 kev band , while 380 counts are detected from the nucleus before correction for pileup . therefore , the _ asca _ flux ( @xmath119 ergs s@xmath2@xmath12 ; terashima et al . 1999 , 2002b ) is probably larger than the true nuclear flux by @xmath120% or less , unless the off - nuclear sources show drastic time variability . we used the _ asca _ flux without any correction for the off - nuclear source contribution . the 10% uncertainty does not affect any of the conclusions . we performed a spectral fit to the brightest off - nuclear source ( cxou j131329.7 + 363523 ) . an absorbed power law model was applied , and @xmath10 = 0.40 ( @xmath571.2 ) @xmath121 @xmath12 and a photon index @xmath122 were obtained . _ ngc 5866 ( t2)_. extended emission of diameter @xmath56 30@xmath34 ( @xmath1232 kpc ) is seen . the spectrum of this emission may be represented by a mekal plasma model with @xmath124 1 kev and abundance of 0.15 solar . this component could be identified with a gaseous halo of this s0 galaxy . falcke , h. , lehr , j. , barvainis , r. , nagar , n. m. , & wilson , a. s. 2001 , probing the physics of active galactic nuclei by multiwavelength monitoring , eds . b. m. peterson , r. w. pogge , & r. s. polidan , asp conf . series 224 , p.265 , ( asp : san francisco ) ptak , a. , yaqoob , t. , serlemitsos , p.j . , kunieda , h. , & terashima , y. 1996 , , 459 , 542 quataert , e. 2002 , `` probing the physics of active galactic nuclei '' , eds . b. m. peterson , r. w. pogge , and r. s. polidan , ( san francisco : astronomical society of the pacific ) , asp conference proceedings , vol . 224 , p.71 terashima , y. , ho , l .c . , & ptak , a. f. 2000a , , 539 , 161 terashima , y. , ho , l .c . , ptak , a. f. , mushotzky , r. f. , serlemitsos , p. j. , yaqoob , t. , & kunieda , h. 2000b , , 533 , 729 terashima , y. , ho , l .c . , ptak , a. f. , yaqoob , t. , kunieda , h. , misaki , k. , & serlemitsos , p. j. 2000c , , 535 , l79 terashima , y. , iyomoto , n. , ho , l. c. , & ptak , a. f. 2002b , , 139 , 1 cccccccc name & @xmath125 & class & date & exposure & & notes + & ( mpc ) & & & ( s ) & ( s@xmath2 ) & ( frame@xmath2 ) & + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) + ngc 266 & 62.4 & l1.9 & 2001 jun 1 & 2033 & 0.020 & 0.0080 & a + ngc 2787 & 13.3 & l1.9 & 2000 jan 7 & 1050 & 0.0075 & 0.024 & c + ngc 3147 & 40.9 & s2 & 2001 sep 19 & 2202 & 0.54 & 0.21 & a + ngc 3169 & 19.7 & l2 & 2001 may 2 & 1953 & 0.081 & 0.033 & a + ngc 3226 & 23.4 & l1.9 & 2001 mar 23 & 2228 & 0.094 & 0.038 & a + ngc 4143 & 17.0 & l1.9 & 2001 mar 26 & 2514 & 0.063 & 0.025 & a + ngc 4203 & 9.7 & l1.9 & 1999 nov 4 & 1754 & 0.17 & 0.54 & c + ngc 4278 & 9.7 & l1.9 & 2000 apr 20 & 1396 & 0.18 & 0.33 & b + ngc 4548 & 16.8 & l2 & 2001 mar 24 & 2746 & 0.0097 & 0.0039 & a + ngc 4550 & 16.8 & l2 & 2001 mar 24 & 1885 & ... & ... & a + ngc 4565 & 9.7 & s1.9 & 2000 jun 30 & 2828 & 0.045 & 0.081 & b + ngc 4579 & 16.8 & l1.9/s1.9 & 2000 feb 23 & 2672 & 1.1 & 0.47 & a + ngc 5033 & 18.7 & s1.5 & 2000 apr 28 & 2904 & 0.33 & 0.59 & b + ngc 5866 & 15.3 & t2 & 2002 jan 10 & 2247 & ... & ... & a + ngc 6500 & 39.7 & l2 & 2000 aug 1 & 2104 & 0.020 & 0.064 & c + cccccccccl name & ra & dec . & & counts & & hard / soft & flux & luminosity & notes + & ( j2000 ) & ( j2000 ) & ( 0.58 kev ) & ( 0.5 - 2 kev ) & ( 2 - 8 kev ) & band ratio & ( 210 kev ) & ( 210 kev ) + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) & ( 9 ) & ( 10 ) + ngc 266 & 0 49 47.81 & 32 16 40.0 & 40.7@xmath1266.4 & 29.8@xmath1265.5 & 10.9@xmath1263.3 & 0.37@xmath1260.13 & 1.6 & 7.5 & a + ngc 2787 & 9 19 18.70 & 69 12 11.3 & 7.9@xmath1262.8 & @xmath5712.0 & @xmath576.4 & ... & 0.25 & 0.053 & b + ngc 3147 & 10 16 53.75 & 73 24 02.8 & 1180.1@xmath12634.4 & 843.4@xmath12629.1 & 333.5@xmath12618.3 & 0.40@xmath1260.03 & 37 & 76 & a , c + ngc 3169 & 10 14 15.05 & 03 27 57.9 & 159.0@xmath12612.6 & @xmath5711.8 & 151.0@xmath12612.3 & @xmath3212.9 & 24 & 26 & a + & & & & & & & 26 & 22 & a , d + ngc 3226 & 10 23 27.01 & 19 53 55.0 & 209.3@xmath12614.5 & 125.2@xmath12611.2 & 80.5@xmath1269.0 & 0.64@xmath1260.09 & 7.6 & 5.5 & a + ngc 4143 & 12 9 36.07 & 42 32 03.0 & 157.4@xmath12612.6 & 121.3@xmath12611.1 & 32.6@xmath1265.7 & 0.27@xmath1260.05 & 3.1 & 1.1 & a + ngc 4203 & 12 15 05.02 & 33 11 49.9 & 294.3@xmath12617.2 & 198.7@xmath12614.1 & 91.7@xmath1269.6 & 0.46@xmath1260.06 & ... & ... & e + ngc 4278 & 12 20 06.80 & 29 16 51.6 & 255.6@xmath12616.4 & 209.6@xmath12614.9 & 52.0@xmath1267.4 & 0.25@xmath1260.04 & 8.1 & 0.91 & a , c + ngc 4548 & 12 35 26.46 & 14 29 46.7 & 26.6@xmath1265.2 & 8.7@xmath1263.0 & 17.7@xmath1264.2 & 2.02@xmath1260.85 & 1.6 & 0.61 & a + ngc 4550 & ... & ... & @xmath127 & @xmath128 & @xmath128 & ... & @xmath129 & @xmath130 & + ngc 4565 & 12 36 20.78 & 25 59 15.7 & 127.3@xmath12611.3 & 92.5@xmath1269.6 & 34.9@xmath1265.9 & 0.38@xmath1260.08 & 3.2 & 0.36 & a + ngc 4579 & 12 37 43.52 & 11 49 05.4 & 3067.9@xmath12655.6 & 2240.5@xmath12647.7 & 812.3@xmath12628.5 & 0.36@xmath1260.01 & ... & ... & e + ngc 5033 & 13 13 27.47 & 36 35 38.1 & 946.5@xmath12630.9 & 562.1@xmath12623.8 & 380.2@xmath12619.5 & 0.68@xmath1260.05 & ... & ... & e + ngc 5866 & ... & ... & @xmath127 & @xmath131 & @xmath132 & ... & @xmath133 & @xmath134 & b + ngc 6500 & 17 55 59.78 & 18 20 18.0 & 42.4@xmath1266.6 & 41.5@xmath1266.5 & @xmath577.6 & @xmath570.18 & 0.28 & 0.55 & a + & & & & & & & 0.69 & 1.3 & a , d + ccccccccccl ngc 2787 & 9 19 23.05 & 69 14 24.4 & j091923.1 + 691424 & 21.0@xmath1264.6 & 17.9@xmath1264.2 & @xmath578.0 & @xmath570.44 & 0.68 & ... & a , b + ngc 3147 & 10 16 51.50 & 73 24 08.9 & j101651.5 + 732409 & 6.7@xmath1262.6 & @xmath576.8 & @xmath579.2 & ... & 0.10 & 0.20 & b + ngc 3169 & 10 14 14.35 & 03 28 10.8 & j101414.3 + 032811 & 6.9@xmath1262.6 & 6.9@xmath1262.6 & @xmath573.0 & @xmath570.43 & 0.11 & 0.051 & b + & 10 14 17.90 & 03 28 55.2 & j101417.9 + 032855 & 10.9@xmath1263.3 & 8.0@xmath1262.8 & @xmath579.4 & @xmath571.18 & 0.18 & 0.084 & b + ngc 3226 & 10 23 26.69 & 19 54 06.8 & j102326.7 + 195407 & 8.9@xmath1263.0 & 7.9@xmath1262.8 & @xmath574.7 & @xmath570.59 & 0.13 & 0.085 & b + ngc 4203 & 12 15 09.20 & 33 09 54.7 & j121509.2 + 330955 & 240.1@xmath12615.5 & 196.4@xmath12614.0 & 40.8@xmath1266.4 & 0.21@xmath1260.04 & ... & ... & a , c , ton 1480 + & 12 15 14.33 & 33 11 04.7 & j121514.3 + 331105 & 11.9@xmath1263.5 & @xmath576.3 & 9.9@xmath1264.3 & @xmath321.57 & 2.0 & ... & a , d j121514.3 + 331105 , g + & 12 15 15.34 & 33 13 54.0 & j121515.3 + 331354 & 6.0@xmath1262.4 & 6.0@xmath1262.4 & @xmath573.0 & @xmath570.50 & 0.10 & ... & a , b + & 12 15 15.64 & 33 10 12.3 & j121515.6 + 331012 & 16.9@xmath1265.2 & 14.9@xmath1265.0 & @xmath576.3 & @xmath570.42 & 0.30 & ... & a , b , star + & 12 15 19.84 & 33 10 12.2 & j121519.8 + 331012 & 15.9@xmath1264.0 & 10.9@xmath1263.3 & @xmath574.7 & @xmath570.43 & 0.29 & ... & a , b + ngc 4550 & 12 35 21.30 & 12 14 04.5 & j123521.3 + 121405 & 6.0@xmath1262.4 & 5.9@xmath1262.4 & @xmath573.0 & @xmath570.50 & 0.10 & ... & a , b + & 12 35 27.76 & 12 13 38.9 & j123527.8 + 121339 & 35.8@xmath1266.0 & 26.6@xmath1265.2 & 8.0@xmath1262.8 & 0.30@xmath1350.12 & 1.1&11000 & a , d , qso 1232 + 125 , h + ngc 4565 & 12 36 14.65 & 26 00 52.5 & j123614.7 + 260052 & 14.9@xmath1265.0 & 8.9@xmath1264.1 & 6.0@xmath1263.6 & 0.67@xmath1260.51 & 0.76 & 0.086 & d , a30 , i + & 12 36 17.40 & 25 58 55.5 & j123617.4 + 255856 & 269.5@xmath12616.4 & 209.7@xmath12614.5 & 59.9@xmath1267.7 & 0.29@xmath1260.04 & 5.8 & 0.67 & e , f , a32 , i + & 12 36 18.64 & 25 59 34.6 & j123618.6 + 255935 & 8.8@xmath1263.0 & @xmath5713.5 & @xmath576.2 & ... & 0.098 & 0.011 & b + & 12 36 19.02 & 25 59 31.5 & j123619.0 + 255932 & 6.9@xmath1262.6 & @xmath5710.7 & @xmath576.2 & ... & 0.077 & 0.009 & b + & 12 36 19.03 & 26 00 27.0 & j123619.0 + 260027 & 19.9@xmath1264.5 & 18.0@xmath1264.2 & @xmath576.4 & @xmath570.36 & 0.22 & 0.025 & b , a33 , i + & 12 36 20.92 & 25 59 26.7 & j123620.9 + 255927 & 5.9@xmath1262.4 & @xmath5710.6 & @xmath574.7 & ... & 0.065 & 0.007 & b + & 12 36 27.39 & 25 57 32.7 & j123627.4 + 255733 & 15.9@xmath1264.0 & 14.9@xmath1263.9 & @xmath574.8 & @xmath570.32 & 0.18 & 0.020 & b , a37 ? , i + & 12 36 28.12 & 26 00 00.9 & j123628.1 + 260001 & 12.9@xmath1263.6 & 11.0@xmath1263.3 & @xmath576.4 & @xmath570.58 & 0.14 & ... & a , b + & 12 36 31.28 & 25 59 36.9 & j123631.3 + 255937 & 12.0@xmath1264.6 & @xmath5718.9 & @xmath574.8 & ... & 0.13 & ... & a , b , a43 ? , i + ngc 5033 & 13 13 24.78 & 36 35 03.7 & j131324.8 + 363504 & 13.9@xmath1263.7 & 10.0@xmath1263.2 & @xmath579.4 & @xmath570.94 & 0.15 & 0.063 & b + & 13 13 28.88 & 36 35 41.0 & j131328.9 + 363541 & 6.7@xmath1262.6 & @xmath5710.2 & @xmath576.3 & ... & 0.072 & 0.030 & b + & 13 13 29.46 & 36 35 17.3 & j131329.5 + 363517 & 34.6@xmath1265.9 & 31.7@xmath1265.7 & @xmath577.9 & @xmath570.25 & 0.37 & 0.16 & b + & 13 13 29.66 & 36 35 23.1 & j131329.7 + 363523 & 47.7@xmath1266.9 & 31.8@xmath1265.7 & 15.9@xmath1265.1 & 0.50@xmath1260.18 & 2.0 & 0.89 & f + & 13 13 35.56 & 36 34 04.4 & j131335.6 + 363404 & 7.0@xmath1262.6 & 6.0@xmath1262.4 & @xmath574.8 & @xmath570.80 & 0.075 & 0.031 & b + ngc 6500 & 17 56 01.59 & 18 20 22.6 & j175601.6 + 182023 & 7.0@xmath1262.6 & @xmath5710.4 & @xmath576.2 & ... & 0.12 & 0.23 & b + ngc 266 & @xmath138 & @xmath139 & 10.6 ( 6 ) + ngc 3147 & @xmath140 & @xmath141 & 45.4 ( 41 ) & a + ngc 3169 & @xmath142 & @xmath143 & 21.3 ( 26 ) + ngc 3226 & @xmath144 & @xmath145 & 8.1 ( 12 ) + ngc 4143 & @xmath146 & @xmath147 & 12.2 ( 11 ) + ngc 4278 & @xmath148 & @xmath149 & 16.9 ( 18 ) & a + ngc 4548 & @xmath150 & @xmath151 & 3.7 ( 5 ) + ngc 4565 & @xmath152 & @xmath153 & 1.6 ( 4 ) + ngc 6500 & @xmath154 & @xmath155 & 8.5 ( 8) + cccccc name & @xmath52@xmath15 & @xmath52@xmath6/@xmath15 & @xmath156(5 ghz ) & @xmath156(5 ghz)/@xmath6 & notes + & ( erg s@xmath2 ) & & ( erg s@xmath2 ) & & + & ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) + ngc 266 & 39.36 & 1.52 & 37.87 & @xmath1023.00 & a + ngc 2787 & 38.56 & 0.16 & 37.22 & @xmath1021.50 & b + ngc 3147 & 40.02 & 1.86 & 38.01 & @xmath1023.87 & c + ngc 3169 & 39.52 & 1.82 & 37.19 & @xmath1024.16 & a + ngc 3226 & 38.93 & 1.81 & 37.20 & @xmath1023.54 & a + ngc 4143 & 38.69 & 1.34 & 37.16 & @xmath1022.87 & b + ngc 4203 & 38.35 & 2.02 & 36.79 & @xmath1023.59 & b , e + ngc 4278 & 39.20 & 0.76 & 37.91 & @xmath1022.05 & b + ngc 4548 & 38.48 & 1.31 & 36.31 & @xmath1023.48 & d + ngc 4550 & 38.50 & @xmath1570.09 & 36.07 & @xmath1582.34 & d + ngc 4565 & 38.46 & 1.10 & 36.15 & @xmath1023.41 & b + ngc 4579 & 39.48 & 1.82 & 37.65 & @xmath1023.59 & b , e + ngc 5033 & 39.70 & 1.67 & 36.79 & @xmath1024.57 & c , e + ngc 5866 & 38.82 & @xmath1570.56 & 36.89 & @xmath1581.18 & b + ngc 6500 & 40.48 & @xmath1020.37 & 38.90 & @xmath1021.21 & a +

the results of _ chandra _ snapshot observations of 11 liners ( low - ionization nuclear emission - line regions ) , three low - luminosity seyfert galaxies , and one -liner transition object are presented . our sample consists of all the objects with a flat or inverted spectrum compact radio core in the vla survey of 48 low - luminosity agns ( llagns ) by nagar et al . ( 2000 ) . an x - ray nucleus is detected in all galaxies except one and their x - ray luminosities are in the range @xmath0 to @xmath1 ergs s@xmath2 . the x - ray spectra are generally steeper than expected from thermal bremsstrahlung emission from an advection - dominated accretion flow ( adaf ) . the x - ray to h@xmath3 luminosity ratios for 11 out of 14 objects are in good agreement with the value characteristic of llagns and more luminous agns , and indicate that their optical emission lines are predominantly powered by a llagn . for three objects , this ratio is less than expected . comparing with properties in other wavelengths , we find that these three galaxies are most likely to be heavily obscured agn . we use the ratio @xmath4(5 ghz)/@xmath5 , where @xmath6 is the luminosity in the 210 kev band , as a measure of radio loudness . in contrast to the usual definition of radio loudness ( @xmath7(5 ghz)/@xmath8(b ) ) , @xmath9 can be used for heavily obscured ( @xmath10 .5ex@xmath11 @xmath12 , @xmath13 mag ) nuclei . further , with the high spatial resolution of _ chandra _ , the nuclear x - ray emission of llagns is often easier to measure than the nuclear optical emission . we investigate the values of @xmath9 for llagns , luminous seyfert galaxies , quasars and radio galaxies and confirm the suggestion that a large fraction of llagns are radio loud .

many biological systems exhibit complex oscillatory dynamics that evolve over multiple time - scales , such as the spiking and bursting activity of neurons , sinus rhythms in the beating of the heart , and intracellular calcium signalling . such rhythms are often described by singularly perturbed systems of ordinary differential equations @xmath2 where @xmath3 is the ratio of slow and fast time - scales , @xmath4 is fast , @xmath5 is slow , and @xmath6 and @xmath7 are smooth functions . a relatively new type of oscillatory dynamic feature discovered in slow / fast systems with @xmath8 is the so - called torus canard @xcite . torus canards are solutions of that closely follow a family of attracting limit cycles of the fast subsystem of , and then closely follow a family of repelling limit cycles of the fast subsystem of for substantial times before being repelled . this unusual behaviour in the phase space typically manifests in the time course evolution as amplitude modulation of the rapid spiking waveform , as shown in figure [ fig : amspiking ] . and @xmath8 fast variables , one of which is @xmath9 . ( a ) the time evolution of the torus canard in this case is an amplitude - modulated spiking rhythm , which consists of rapid spiking ( blue ) wherein the envelope of the waveform ( red ) also oscillates . ( b ) the projection of the torus canard into the slow / fast phase plane shows that the torus canard arises in the neighbourhood of where an attracting family of limit cycles of the fast subsystem ( green , solid ) and a repelling family of limit cycles of the fast subsystem ( green , dashed ) meet ( inset ) . the trajectory alternately spends long times following both the attracting and repelling branches of limit cycles.,title="fig:",width=480 ] ( -364,142)(a ) ( -178,142)(b ) first discovered in a model for the neuronal activity in cerebellar purkinje cells @xcite , torus canards were observed as quasi - periodic solutions that would appear during the transition between bursting and rapid spiking states of the system . further insight into the dynamics of the torus canards in this cell model was presented in @xcite , where a 2-fast/1-slow rotated van der pol - type equation with symmetry breaking was studied . since then , torus canards have been encountered in several other neural models @xcite , such as hindmarsh - rose ( subhopf / fold cycle bursting ) , morris - lecar - terman ( circle / fold cycle bursting ) , and wilson - cowan - izhikevich ( fold / fold cycle bursting ) , where they again appeared in the transition between spiking and bursting states . additional studies have identified torus canards in chemical oscillators @xcite , and have shown that torus canards are capable of interacting with other dynamic features to create even more complicated oscillatory rhythms @xcite . three common threads link all of the examples mentioned above . first and foremost , the torus canards occur in the neighbourhood of a fold bifurcation of limit cycles , also known as a saddle - node of periodics ( snpos ) , of the fast subsystem ( see figure [ fig : amspiking](b ) for instance ) . that is , the torus canards arise in the regions of phase space where an attracting set of limit cycles meets a repelling set of limit cycles . second , the torus canards occur for parameter sets which are @xmath10 close to a torus bifurcation of the full system . third , in these examples , there is only one slow variable , and the torus canards are restricted to exponentially thin parameter sets . in other words , the torus canards in these examples are degenerate . torus canards in @xmath0 require a one - parameter family of 2-fast/1-slow systems in order to be observed , and they undergo a very rapid transition from rapid spiking to bursting ( i.e. , torus canard explosion ) in an exponentially thin parameter window @xmath10 close to a torus bifurcation of the full system . in principle , the addition of a second slow variable unfolds the torus canard phenomenon , making the torus canards generic and robust . this is analogous to the unfolding of planar canard cycles via the addition of a second slow variable . that is , canard solutions in @xmath0 are generic and robust , and their properties are encoded in folded singularities of the reduced flow @xcite . so far , to our knowledge , the only case study of torus canards in systems with more than one slow variable is in a model for respiratory rhythm generation in the pre - btzinger complex @xcite , which is a 6-fast/2-slow system . there , the torus canards were studied numerically by averaging the slow motions over limit cycles of the fast subsystem and examining the averaged slow drift along the manifold of periodics . in particular , folded singularities of the averaged slow flow were numerically identified and the properties of the torus canards were inferred based on canard theory . from their observations , the authors in @xcite conjectured that the average of a torus canard is a folded singularity canard . this leads to the generic torus canard problem , which can be stated simply as follows . there is currently no analytic way to identify , classify , and analyze torus canards in the same way that canards in @xmath0 can be classified and analyzed based on their associated folded singularity . it has been suggested that averaging methods should be used @xcite to reduce the torus canard problem to a folded singularity problem in a related averaged system . however , this approach is not rigorously justified since the averaging method breaks down in a neighbourhood of a fold of limit cycles , which is precisely where the torus canards are located . our main goal then is to extend the averaging method to the torus canard regime and hence solve the generic torus canard problem in @xmath1 . there are three types of results in this article : theoretical , numerical , and phenomenological . the theoretical contribution is that we extend the averaging method to folded manifolds of limit cycles and hence to the torus canard regime . in so doing , we inherit fenichel theory @xcite for persistent manifolds of limit cycles and in particular , we are able to make use of the powerful theoretical framework of canard theory @xcite . we provide analytic criteria for the identification and characterization of torus canards based on an underlying class of novel singularities for differential equations , which we call _ toral folded singularities_. we illustrate our assertions by studying a spatially homogeneous model for intracellular calcium dynamics @xcite . in applying our results to this model , we discover a novel type of bursting rhythm , which we call _ amplitude - modulated bursting _ ( see figure [ fig : ambursting ] for an example ) . we show that these amplitude - modulated bursting solutions can be well - understood using our torus canard theory . in the process , we provide the first numerical computations of intersecting invariant manifolds of limit cycles . the new phenomenological result that stems from our analysis is that we construct the torus canard analogue of a canard - induced mixed - mode oscillation @xcite . ) . the amplitude - modulated bursts alternate between active phases where the trajectory ( blue ) rapidly oscillates , and silent phases where the trajectory remains quiescent . during the active phase , the envelope ( red ) of the rapidly oscillating waveform exhibits small - amplitude oscillations which extend the burst duration . these amplitude - modulated bursts are torus canard - induced mixed - mode oscillations ( section [ sec : tcimmo]).,width=480 ] earlier reports of torus canards have been seen in the literature , even though that terminology was not used . in @xcite , it was remarked that bifurcation delay may result when a trajectory crosses from a set of attracting states to a set of repelling states where the states may be either fixed points or limit cycles . in @xcite , a canonical form for subcritical elliptic bursting near a bautin bifurcation of the fast subsystem was studied . the canonical model consists of two fast ( polar ) variables @xmath11 and a single slow variable @xmath12 . in these polar coordinates , the oscillatory states of the fast subsystem may be identified as stationary radii . within this framework , torus canards occur as canard cycles of the planar @xmath13 subsystem , and in parameter space they arise in the rapid and continuous transition between the spiking and bursting regimes of the canonical model . the outline of the paper is as follows . in sections [ sec : averaging ] and [ sec : classification ] , we give the main theoretical results of the article . namely , we state the generic torus canard problem in @xmath1 in the case of two fast variables and two slow variables , and then combine techniques from floquet theory @xcite , averaging theory @xcite , and geometric singular perturbation theory @xcite to show that the average of a torus canard is a folded singularity canard . in so doing , we devise analytic criteria for the identification and topological classification of torus canards based on their underlying toral folded singularity . we examine the main topological types of toral folded singularities and show that they encode properties of the torus canards , such as the number of torus canards that persist for @xmath14 . we then discuss bifurcations of torus canards and make the connection between torus canards and the torus bifurcation that is often observed in the full system . we apply our results to the politi - hfer model for intracellular calcium dynamics @xcite in sections [ sec : ph ] , [ sec : phtc ] and [ sec : tcimmo ] . we examine the bifurcation structure of the model and identify characteristic features that signal the presence of torus canards . using our torus canard theory , we explain the dynamics underlying the novel class of amplitude - modulated bursting rhythms . we show that the amplitude modulation is organised locally in the phase space by twisted , intersecting invariant manifolds of limit cycles . these sections serve the dual purpose of illustrating the predictive power of our analysis , and also giving a representative example of how to implement those results in practice . in section [ sec : explosion ] , we make the connection between our current work on torus canards and prior work on torus canards in @xmath0 explicit . we show that the theoretical framework developed in sections [ sec : averaging ] and [ sec : classification ] can be used to compute the spiking / bursting boundary in the parameter spaces of 2-fast/1-slow systems by simply tracking the toral folded singularity . we illustrate these results in the morris - lecar - terman , hindmarsh - rose , and wilson - cowan - izhikevich models for neural bursting . in section [ sec : arbitrarydimensions ] , we extend our averaging method for folded manifolds of limit cycles to slow / fast systems with two fast variables and @xmath15 slow variables , where @xmath15 is any positive integer . moreover , we provide asymptotic error estimates for the averaging method on folded manifolds of limit cycles . we then conclude in section [ sec : discussion ] , where we summarize the main results of the article , discuss their implications , and highlight several interesting open problems . in this section , we study generic torus canards in @xmath1 in the case of two fast variables and two slow variables . in section [ subsec : assumptions ] , we state the assumptions of the generic torus canard problem in @xmath1 . within this framework , we develop an averaging method for folded manifolds of limit cycles in section [ subsec : theoretical ] and derive a canonical form for the dynamics around a torus canard . in section [ subsec : averagedcoefficients ] , we list ( algorithmically ) the averaged coefficients that appear in the canonical form . we consider four - dimensional singularly perturbed systems of ordinary differential equations of the form @xmath16 where @xmath3 measures the time - scale separation , @xmath17 is fast , @xmath18 is slow , @xmath6 and @xmath7 are sufficiently smooth functions , and @xmath19 and their derivatives are @xmath20 with respect to @xmath21 . [ ass : man ] the layer problem of system , given by @xmath22 possesses a manifold @xmath23 of limit cycles , parametrized by the slow variables . for each fixed @xmath24 , let @xmath25 denote the corresponding limit cycle and assume that @xmath25 has finite , non - zero period @xmath26 . that is , @xmath27 the floquet exponents of @xmath25 are given by @xmath28 where @xmath29 corresponds to a floquet multiplier equal to unity , which reflects the fact that @xmath25 is neutrally stable to shifts along the periodic orbit @xcite . the stability then , of the periodic orbit @xmath25 , is encoded in the floquet exponent @xmath30 . if @xmath31 , then @xmath25 is an asymptotically stable solution of and if @xmath32 , then @xmath25 is an unstable solution of . [ ass : fold ] the layer problem possesses a manifold @xmath33 of snpos given by @xmath34 moreover , we assume that the manifold of periodics is a non - degenerate folded manifold so that @xmath23 can be partitioned into attracting and repelling subsets , separated by the manifold of snpos . that is , @xmath35 where @xmath36 is the subset of @xmath23 along which @xmath31 , and @xmath37 is the subset of @xmath23 along which @xmath32 . we refer forward to section [ subsec : averagedcoefficients ] for a more precise formulation of the non - degeneracy condition that @xmath30 changes sign along the manifold of snpos . a schematic of our setup is shown in figure [ fig : setup ] . each point on the folded manifold @xmath23 corresponds to a limit cycle of the layer problem . ( b ) attracting ( blue ) and repelling ( red ) manifolds of limit cycles joined by the folded limit cycle @xmath38 ( which corresponds to the black marker in ( a ) ) shown in the cross - section @xmath39 . ( c ) the folded limit cycle @xmath38 , indicated by the black marker in ( a ) , shown in the cross - section @xmath40 , with unit tangent and normal vectors , @xmath41 and @xmath42 , respectively for some fixed @xmath43.,title="fig:",width=480 ] ( -360,340)(a ) ( -160,340)(b ) ( -284,182)(c ) [ ass : homoclinic ] if the layer problem has a critical manifold @xmath44 , then @xmath44 and @xmath33 are disjoint . assumption [ ass : homoclinic ] guarantees that the periodic orbits of the layer problem in a neighbourhood of the manifold of snpos have finite period . note that we are not eliminating the possibility of the manifold of limit cycles from intersecting the critical manifold @xmath44 , as would be the case near a set of hopf bifurcations of the layer problem . instead , we restrict the problem so that the snpos of stay a reasonable distance from the critical manifold . an important step in the analysis to follow is identifying unit tangent and unit normal vectors to the periodic orbit , @xmath25 , of the layer problem . one choice of unit tangent and normal vectors , @xmath45 and @xmath46 , to the periodic @xmath47 for fixed @xmath48 , is given by @xmath49 where @xmath50 denotes the standard euclidean norm and @xmath51 is the skew - symmetric matrix @xmath52 . the idea of averaging theory is to find a flow that approximates the slow flow on the family of periodic orbits of the layer problem @xcite . these averaging methods can be used to show that the effective slow dynamics on a family of asymptotically stable periodics are determined by an appropriately averaged system @xcite . that is , the slow drift on @xmath36 can be approximated by averaging out the fast oscillations , and the error in the approximation is @xmath10 . however , to our knowledge , there are currently no theoretical results about the slow drift near folded manifolds of periodics . the following theorem extends the averaging method from normally hyperbolic manifolds of limit cycles to folded manifolds of limit cycles . [ thm : averaging ] consider system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] , and let @xmath53 . then there exists a sequence of near - identity transformations such that the averaged dynamics of in a neighbourhood of @xmath54 are approximated by @xmath55 where an overbar denotes an average over one period of @xmath25 , and the coefficients in system can be computed explicitly ( see section [ subsec : averagedcoefficients ] ) . the fast variable @xmath56 in system can be thought of as the averaged radial perturbation to @xmath25 in the direction of @xmath46 , and the slow variables @xmath12 describe the averaged evolution of @xmath48 . we present the proof of theorem [ thm : averaging ] in section [ sec : arbitrarydimensions ] . the idea of the proof is to switch to a coordinate frame that moves with the limit cycles , apply a coordinate transformation that removes the linear radial perturbation , and then average out the rapid oscillations . the significance of theorem [ thm : averaging ] is that the averaged radial - slow dynamics described by system are autonomous , singularly perturbed , and occur in the neighbourhood of a folded critical manifold . as such , system falls under the framework of canard theory @xcite . theorem [ thm : averaging ] is a formal extension of the averaging method to folded manifolds of limit cycles . we defer the statement of asymptotic error estimates ( i.e. , the validity of this averaging method ) to section [ sec : arbitrarydimensions ] . here we list the averaged coefficients that appear in theorem [ thm : averaging ] ( and theorem [ thm : averagingarbitraryslow ] ) . we denote the period of @xmath25 by @xmath57 . the functions @xmath19 , and their derivatives are all evaluated at the limit cycle @xmath54 of the layer problem . recall that @xmath45 and @xmath46 denote unit tangent and unit normals to @xmath25 , respectively . we give the coefficients for the case of @xmath58-fast variables and @xmath15-slow variables , where @xmath59 . let @xmath60 be the fundamental solution defined by @xmath61 we will show in section [ sec : arbitrarydimensions ] that @xmath60 is @xmath57-periodic and bounded for all time ( lemma [ lemma : linear ] ) . the coefficients of the linear slow terms and the quadratic radial term in the radial equation are given ( component - wise ) by @xmath62 note that the coefficient @xmath63 of the quadratic @xmath56 term is a scalar . we compute the auxiliary quantities @xmath64 and @xmath65 as solutions of @xmath66 for @xmath67 . we refer forward to equation for the interpretation of @xmath64 and @xmath65 . using these auxiliary functions , we can compute the coefficients of the mixed terms in the radial equation according to @xmath68 where the @xmath69 matrix @xmath70 is given by @xmath71 and @xmath72 is the average of @xmath73 over one period of @xmath25 . the coefficients of the linear @xmath56-terms in the slow equations are @xmath74 finally , the @xmath75 matrix of coefficients of the linear @xmath12-terms in the slow equations is @xmath76 we can now simply list the averaged coefficients as @xmath77 for @xmath78 and @xmath67 , where @xmath79 , or @xmath80 . note that the non - degeneracy condition in assumption [ ass : fold ] is given by the requirement that the averaged coefficient of the quadratic radial term is non - zero ( i.e. , @xmath81 ) . we point out that the leading order terms in the averaged slow directions are simply given by the averages of the slow components of the vector field over one period of @xmath25 . we now study the dynamics of the averaged radial - slow system using geometric singular perturbation techniques . in section [ subsec : toralfoldedsing ] , we define the notion of a toral folded singularity a special limit cycle in the phase space from which torus canard dynamics can originate . we provide a topological classification of toral folded singularities and their associated torus canards in section [ subsec : toralclass ] . we then present the dynamics of the torus canards in the main cases , including those that exist near toral folded nodes ( section [ subsec : toralfn ] ) , toral folded saddles ( section [ subsec : toralfs ] ) , and toral folded saddle - nodes ( section [ subsec : toralfsn ] ) . we begin our geometric singular perturbation analysis of system by rewriting it in the more succinct form @xmath82 where @xmath83 correspond to the right - hand - sides of , and the prime denotes derivatives with respect to the fast time @xmath84 ( which is related to the slow time @xmath85 by @xmath86 ) . the idea is to decompose the dynamics of into its slow and fast motions by taking the singular limit on the slow and fast time - scales . the fast dynamics are approximated by solutions of the layer problem , @xmath87 where @xmath88 and @xmath89 are parameters , and was obtained by taking the singular limit @xmath90 in . the set of equilibria of , given by @xmath91 is called the critical manifold and is a key object in the geometric singular perturbations approach . assuming at least one of @xmath92 and @xmath93 is non - zero , the critical manifold has a local graph representation , @xmath94 , say . in the case of , the critical manifold is ( locally ) a parabolic cylinder in the @xmath95 phase space . linear stability analysis of the layer problem shows that the attracting and repelling sheets , @xmath96 and @xmath97 , of the critical manifold are separated by a curve of fold bifurcations , defined by @xmath98 note that @xmath96 and @xmath97 correspond to the attracting and repelling manifolds of limit cycles , @xmath36 and @xmath37 , respectively , introduced in . moreover , the fold curve @xmath99 corresponds to the manifold of snpos , @xmath33 . to describe the slow dynamics along the critical manifold @xmath44 , we switch to the slow time - scale ( @xmath100 ) in system and take the singular limit @xmath90 to obtain the reduced system @xmath101 where the overdot denotes derivatives with respect to @xmath85 . typically , to obtain a complete description of the flow on @xmath44 , we would need to compute the dynamics in an atlas of overlapping coordinate charts . in this case , and as is the case in many applications , we can use the graph representation of @xmath44 to project the dynamics of onto a single coordinate chart @xmath102 . the dynamics on @xmath44 are then given by @xmath103 where all functions and their derivatives are evaluated along @xmath44 . an important feature of the reduced flow highlighted by this projection is that the reduced flow is singular along the fold curve @xmath99 . that is , solutions of the reduced flow blow - up in finite time at the fold curve and are expected to fall off the critical manifold . to remove this finite - time blow - up of solutions , we introduce the phase space dependent time - transformation @xmath104 , which gives the _ desingularized system _ @xmath105 where we have recycled the overdot to denote derivatives with respect to @xmath106 , and @xmath94 . on the attracting sheets @xmath96 , the desingularized flow is ( topologically ) equivalent to the reduced flow . however , on the repelling sheets @xmath97 ( where @xmath107 ) , the time transformation reverses the orientation of trajectories , and the reduced flow is obtained by reversing the direction of the flow of the desingularized system . thus , the reduced flow can be understood by examining the desingularized flow and keeping track of the dynamics on both sheets of @xmath44 . the desingularized system possesses two types of equilibria : _ ordinary _ and _ folded_. the set of ordinary singularities @xmath108 consists of isolated points which are equilibria of both the reduced and desingularized flows and are @xmath10 close to equilibria of the fully perturbed problem provided they remain sufficiently far from the fold curve @xmath99 . the set of folded singularities @xmath109 consists of isolated points along the fold curve where the right - hand - side of the @xmath56-equation in ( and ) vanishes . whilst folded singularities are equilibria of the desingularized system , they are not equilibria of the reduced system . instead , folded singularities are points where the @xmath56-equation of the reduced system has a zero - over - zero type indeterminacy , which means trajectories may potentially pass through the folded singularity with finite speed and cross from one sheet of the critical manifold to another . that is , a folded singularity is a distinguished point on the fold curve where the reduced vector field may actually be regular . [ def : identification ] system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] , possesses a _ folded singularity of limit cycles _ , or a _ toral folded singularity _ for short , if it has a limit cycle @xmath53 , such that @xmath110 where @xmath111 for @xmath112 , are the averaged coefficients in theorem [ thm : averaging ] ( listed in section [ subsec : averagedcoefficients ] ) . in slow / fast systems with two slow variables and one fast variable , a folded singularity , @xmath45 , of the reduced flow on a folded critical manifold is a point on the fold of the critical manifold where there is a violation of transversality : @xmath113 geometrically , this corresponds to the scenario in which the projection of the reduced flow into the slow variable plane is tangent to the fold curve at @xmath45 . definition [ def : identification ] gives the averaged analogue for torus canards . more precisely , a toral folded singularity is a folded limit cycle @xmath114 such that the projection of the averaged slow drift along @xmath23 into the averaged slow variable plane is tangent to the projection of @xmath33 into the @xmath115-plane at the toral folded singularity ( see figure [ fig : phtoralfn ] for an example ) . we see that a toral folded singularity is a folded singularity of the @xmath95 system where solutions of the slow flow along the folded critical manifold can cross with finite speed from @xmath96 to @xmath97 ( or vice versa ) . that is , a toral folded singularity allows for singular canard solutions of the averaged radial - slow flow . a singular canard solution of the @xmath95 system corresponds , in turn , to a solution in the original @xmath116 system that slowly drifts along the manifold of periodics @xmath23 and crosses from @xmath36 to @xmath37 ( or vice versa ) via a toral folded singularity . based on this , we define singular torus canard solutions as follows . [ def : singtc ] suppose system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] has a toral folded singularity . a _ singular torus canard _ is a singular canard solution of the averaged radial - slow system . a _ singular faux torus canard _ is a singular faux canard solution of the averaged radial - slow system . canard theory classifies and characterizes singular canards based on their associated folded singularity . we have the following classification scheme for toral folded singularities . suppose system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] possesses a toral folded singularity @xmath117 . let @xmath118 denote the eigenvalues of the desingularized flow of , linearized about the origin ( i.e. , about @xmath117 ) . [ def : class ] the toral folded singularity is * a _ toral folded saddle _ if @xmath119 , * a _ toral folded saddle - node _ if @xmath120 and @xmath121 , * a _ toral folded node _ if @xmath122 , or a _ faux toral folded node _ if @xmath123 , * a _ degenerate toral folded node _ if @xmath124 , or * a _ toral folded focus _ if @xmath125 . thus , we may exploit the known results about the existence and dynamics of canards near classical folded singularities of the averaged radial - slow system in order to learn about the existence and dynamics of torus canards near toral folded singularities . folded nodes , folded saddles , and folded saddle - nodes are known to possess singular canard solutions . we examine their toral analogues in sections [ subsec : toralfn ] , [ subsec : toralfs ] , and [ subsec : toralfsn ] , respectively . folded foci possess no singular canards and so any trajectory of the slow drift along @xmath23 that reaches the neighbourhood of a toral folded focus simply falls off the manifold of periodics . in this section , we consider the toral folded node ( tfn ) and its unfolding in terms of the averaged radial - slow flow . fenichel theory guarantees that the normally hyperbolic segments , @xmath96 and @xmath97 , of @xmath44 persist as invariant slow manifolds , @xmath126 and @xmath127 , of respectively for sufficiently small @xmath21 . fenichel theory breaks down in neighbourhoods of the fold curve @xmath99 where normal hyperbolicity fails . the extension of @xmath126 and @xmath127 by the flow of into the neighbourhood of the tfn leads to a local twisting of the invariant slow manifolds around a common axis of rotation . this spiralling of @xmath126 and @xmath127 becomes more pronounced as the perturbation parameter @xmath21 is increased . moreover , there is a finite number of intersections between @xmath126 and @xmath127 . these intersections are known as ( non - singular ) maximal canards . the properties of the maximal canards associated to a tfn are encoded in the tfn itself in the following way . let @xmath128 be the eigenvalues of the tfn , where we treat the tfn as an equilibrium of , and let @xmath129 be the eigenvalue ratio . then , provided @xmath130 is bounded away from zero , the total number of maximal canards of that persist for sufficiently small @xmath21 is @xmath131 , where @xmath132 and @xmath133 is the floor function @xcite . this follows by direct application of canard theory @xcite to the averaged radial - slow system in the case of a folded node . the first or outermost intersection of @xmath126 and @xmath127 is called the primary strong maximal canard , @xmath134 , and corresponds to the strong stable manifold of the tfn . the strong canard is also the local separatrix that separates the solutions which exhibit local small - amplitude oscillatory behaviour from monotone escape . that is , trajectories on @xmath126 on one side of @xmath134 execute a finite number of small oscillations , whilst trajectories on the other side of @xmath134 simply jump away . the innermost intersection of @xmath126 and @xmath127 is the primary weak canard , @xmath135 , and corresponds to the weak eigendirection of the tfn . the weak canard plays the role of the axis of rotation for the invariant slow manifolds , which again follows directly from the results of @xcite applied to system . the remaining @xmath136 secondary canards , @xmath137 , @xmath138 , further partition @xmath126 and @xmath127 into sectors based on the rotational properties . that is , for @xmath139 , the segments of @xmath126 and @xmath127 between @xmath140 and @xmath141 consist of orbit segments that exhibit @xmath15 small - amplitude oscillations in an @xmath142 neighbourhood of the tfn , where @xmath143 . bifurcations of maximal canards occur at odd integer resonances in the eigenvalue ratio @xmath130 . when @xmath144 is an odd integer , there is a tangency between the invariant slow manifolds @xmath126 and @xmath127 . increasing @xmath144 through this odd integer value breaks the tangency between the slow manifolds resulting in two transverse intersections , i.e. , two maximal canards , one of which is the weak canard . thus , increasing @xmath144 through an odd integer results in a branch of secondary canards that bifurcates from the axis of rotation @xcite . thus , in the case of a tfn , the averaged radial - slow dynamics are capable of generating singular canards , which perturb to maximal canards that twist around a common axis of rotation . since a maximal canard of , by definition , lives at the intersection of ( the extensions of ) @xmath126 and @xmath127 in a neighbourhood of a folded node of , the corresponding trajectory in the original variables lives at the intersection of the extensions of @xmath145 and @xmath146 in the neighbourhood of the tfn ( see figure [ fig : phinvariantmanifolds ] ) . as such , we define a non - singular maximal torus canard as follows . [ def : maximaltc ] suppose system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] has a tfn singularity . a _ maximal torus canard _ of is a trajectory corresponding to the intersection of the attracting and repelling invariant manifolds of limit cycles , @xmath145 and @xmath146 , respectively , in a neighbourhood of the tfn . the implication in moving from maximal canards of the averaged radial - slow system to maximal torus canards of the original system is that a torus canard associated to a tfn consists of three different types of motion working in concert . first , when the orbit is on @xmath145 , we have rapid oscillations due to limit cycles of the layer problem . the slow drift along @xmath145 moves the rapidly oscillating orbit towards the manifold of snpos and in particular , towards the tfn . in a neighbourhood of the tfn , we have canard dynamics occurring within the envelope of the waveform . combined , these motions ( rapid oscillations due to limit cycles of the layer problem , slow drift along the manifold of limit cycles , and canard dynamics on the radial envelope ) manifest as amplitude - modulated spiking rhythms . the maximal number of oscillations that the envelope of the waveform can execute is dictated by . there are two ways in which oscillations can be added to or removed from the envelope . first is the creation of an additional secondary torus canard via odd integer resonances in @xmath144 . we conjecture that this bifurcation of torus canards will correspond to a torus doubling bifurcation . the other method of creating / destroying oscillations in the envelope is by keeping @xmath21 and @xmath130 fixed , and varying the position of the trajectory relative to the maximal torus canards . when the trajectory crosses a maximal torus canard , it moves into a different rotational sector , resulting in a change in the number of oscillations in the envelope ( see figure [ fig : phsectors ] ) . in the case of a toral folded saddle , system possesses exactly one singular canard and one singular faux canard , which correspond to the stable and unstable manifolds of the toral folded saddle , respectively , when considered as an equilibrium of the desingularized flow . the singular canard in this case plays the role of a separatrix , which divides the phase space @xmath44 between those trajectories that encounter the fold curve @xmath99 , and those that turn away from it . the unfolding in @xmath21 of the toral folded saddle of shows that only the singular canard persists as a transverse intersection of the invariant slow manifolds . there is no oscillatory behaviour associated with this maximal canard , and only those trajectories that are exponentially close to the maximal canard can follow the repelling slow manifold . returning to the original @xmath116 variables , the toral folded saddle has precisely one maximal torus canard solution , which plays the role of a separatrix between those solutions that fall off the manifold of limit cycles at the manifold of snpos , and those that turn away from @xmath33 and stay on @xmath23 . we remark that the toral faux canard of a toral folded saddle plays the role of an axis of rotation for local oscillatory solutions of , analogous to the toral weak canard in the tfn case . however , these oscillatory solutions of the averaged radial - slow system start on @xmath127 and move to @xmath126 . that is , there is a family of faux torus canard solutions associated to the toral folded saddle that start on @xmath146 and move to @xmath145 . since these solutions are inherently unstable , we leave further investigation of their dynamics to future work . in canard theory , the special case in which one of the eigenvalues of the folded singularity is zero is called a folded saddle - node ( fsn ) . the fsn comes in a variety of flavours , each corresponding to a different codimension-1 bifurcation of the desingularized reduced flow . the most common types seen in applications are the fsn i @xcite and the fsn ii @xcite . the fsn i occurs when a folded node and a folded saddle collide and annihilate each other in a saddle - node bifurcation of folded singularities . geometrically , the center manifold of the fsn i is tangent to the fold curve . it has been shown that @xmath147 canards persist near the fsn i limit for sufficiently small @xmath21 . the fsn ii occurs when a folded singularity and an ordinary singularity coalesce and swap stability in a transcritical bifurcation of the desingularized reduced flow . in this case , the center manifold of the fsn ii is transverse to the fold curve and it has been shown that @xmath148 canards persist for sufficiently small @xmath21 @xcite . here we show that toral folded singularities may also be of the fsn types . we focus on the toral fsn ii and its implications for torus canards . in analogy with the classic fsn ii points , we define a _ toral fsn ii _ to be a fsn ii of the averaged radial - slow system . these occur when an ordinary singularity of crosses the fold curve , i.e. , under the conditions @xmath149 in which case the folded singularity automatically has a zero eigenvalue . in the classic fsn ii , there is always a hopf bifurcation at an @xmath10-distance from the fold curve @xcite . since the toral fsn ii is ( by definition ) a fsn ii of system , we have that system will possess a hopf bifurcation located at an @xmath10-distance from the fold curve . in terms of the original , non - averaged @xmath116 coordinates , the toral fsn ii is detected as a limit cycle , @xmath47 , of the layer problem such that @xmath150 along @xmath47 . that is , the toral fsn ii occurs when the averaged slow nullclines intersect the manifold of snpos . moreover , the averaged radial - slow dynamics undergo a singular hopf bifurcation @xcite , from which a family of small - amplitude limit cycles of the averaged radial - slow system emanate . this creation of limit cycles in the radial envelope corresponds to the birth of an invariant phase space torus in the non - averaged , fully perturbed problem . as such , we conjecture that the toral fsn ii unfolds in @xmath21 to a singular torus bifurcation of the fully perturbed problem . we provide numerical evidence to support this conjecture in sections [ subsec : phsingvsnonsing ] , [ subsubsec : mlttoralfs ] , [ subsec : tchr ] , and [ subsec : tcwci ] . we now demonstrate ( in sections [ sec : ph ] [ sec : tcimmo ] ) the predictive power of our analytic framework for generic torus canards , developed in sections [ sec : averaging ] and [ sec : classification ] , in the politi - hfer ( ph ) model @xcite . this model describes the interaction between calcium transport processes and the metabolism of inositol ( 1,4,5)-trisphosphate ( 3 ) , which is a calcium - releasing messenger . in section [ subsec : phmodel ] , we describe the ph model for intracellular calcium dynamics . in section [ subsec : phbifn ] , we investigate the bifurcation structure of the ph model , and report on a novel class of amplitude - modulated subcritical elliptic bursting rhythms . in the parameter space , these exist between the tonic spiking and bursting regimes . in section [ subsec : phlayer ] we formally show that the ph model is a 2-fast/2-slow system and follow in section [ subsec : phgeometry ] by showing that it falls under the framework of our torus canard theory . changes in the concentration of free intracellular calcium play a crucial role in the biological function of most cell types @xcite . in many of these cells , the calcium concentration is seen to oscillate , and an understanding of how these oscillations arise and determining the mechanisms that generate them is a significant mathematical and biological pursuit . the biological process in which calcium is able to activate calcium release from internal stores is known as calcium - induced calcium release . the sequence of events leading to calcium - induced calcium release is as follows . an agonist binds to a receptor in the external plasma membrane of a cell , which initiates a chain of reactions that lead to the release of 3 inside that cell . the 3 binds to 3 receptors on the endoplasmic reticulum , which leads to the release of calcium from the internal store through the 3 receptors . that there are calcium oscillations is indicative of the fact that there are feedback mechanisms from calcium to the metabolism of 3 at work . mathematical modelling of these feedback mechanisms is broadly split into three classes . class i models assume that the 3 receptors are quickly activated by the binding of calcium , and then slowly inactivated by slow binding of the calcium to separate binding sites . that is , class i models feature sequential ( fast ) positive and then ( slow ) negative feedback on the 3 receptor . class ii models assume that the calcium itself regulates the production and degradation rates of 3 , which provides an alternative mechanism for negative and positive feedback . the most biologically realistic scenario incorporates both mechanisms ( i.e. , calcium feedback on 3 receptor dynamics as well as on 3 dynamics ) , and such models are known as hybrid models . one hybrid model for calcium oscillations is the ph model @xcite , which includes four variables ; the calcium concentration @xmath151 in the cytoplasm , the calcium concentration @xmath152 in the endoplasmic reticulum stores , the fraction @xmath153 of 3 receptors that have not been inactivated by calcium , and the concentration @xmath45 of 3 in the cytoplasm . the calcium flux through the 3 receptors is given by @xmath154 where @xmath155 is the ratio of the cytosolic volume to the endoplasmic reticulum volume . the active transport of calcium across the endoplasmic reticulum ( via sarco / endoplasmic reticulum atp - ase or serca pumps ) and plasma membrane are given respectively by the hill functions @xmath156 the calcium flux into the cell via the plasma membrane is given by @xmath157 where @xmath158 is the leak into the cell , and @xmath159 is the steady - state concentration of 3 in the absence of any feedback effects of calcium on 3 concentration . the ph model equations are then given by @xmath160 here the @xmath151 and @xmath152 equations describe the balance of calcium flux across the plasma membrane ( @xmath161 ) of the cell and the endoplasmic reticulum ( @xmath162 ) . the parameter @xmath163 measures the relative strength of the plasma membrane flux to the flux across the endoplasmic reticulum . note that if @xmath164 , then this creates a closed cell model wherein the total calcium in the cell is conserved . the @xmath153-equation describes the inactivation of 3 receptors by calcium whilst the @xmath45-equation describes the balance of ( calcium - independent ) 3 production and calcium - activated 3 degradation . the kinetic parameters , their standard values , and their biological significance are detailed in table [ tab : phparams ] ( appendix [ app : ph ] ) . unless stated otherwise , all parameters will be fixed at these standard values . following @xcite , we take @xmath159 to be the principal bifurcation parameter , since it is relatively easy to manipulate in an experimental setting . the parameter @xmath159 represents the steady - state 3 concentration in the absence of calcium feedback . note that the maximal rate of 3 formation is given by @xmath165 . variations in @xmath159 can generate a wide array of different behaviours in the ph model . representative traces are shown in figure [ fig : phbifn ] . ( i.e. , @xmath166 ) , and ( a ) @xmath167 m , ( b ) @xmath168 m , ( c ) @xmath169 m , and ( d ) @xmath170 m . ( a ) the attractor is a stable equilibrium . ( b ) the system exhibits subcritical elliptic bursting . the inset shows the small oscillations due to slow passage through a delayed hopf bifurcation . ( c ) the model can also generate rapid spiking solutions . ( d ) the subcritical elliptic bursts here feature amplitude - modulation during the active burst phase . inset : magnified view of the oscillations in the envelope of the waveform . ( e ) bifurcation structure of with respect to @xmath159 . the equilibria ( black ) change stability at hopf bifurcations ( hb ) . emanating from the hopf bifurcations are families of limit cycles , which change stability at torus bifurcations ( tr ) . the torus bifurcations act as the boundaries between spiking ( red ) and bursting ( blue ) regions.,title="fig:",width=480 ] ( -364,372)(a ) ( -181,372)(b ) ( -364,262)(c ) ( -181,262)(d ) ( -364,150)(e ) for small 3 production rates ( i.e. , small @xmath159 ) , the negative feedback of calcium on 3 metabolism overwhelms the production of 3 . there are no calcium oscillations , and the system settles to a stable equilibrium ( figure [ fig : phbifn](a ) ) . with increased 3 production rate , the system undergoes a supercritical hopf bifurcation ( labelled hb@xmath171 ) at @xmath172 m , from which stable periodic orbits emanate . this family of periodics becomes unstable at a torus bifurcation ( tr@xmath171 ) at @xmath173 m and the stable spiking solutions give way to bursting trajectories ( figure [ fig : phbifn](b ) ) . these bursting solutions are in fact subcritical elliptic ( or subhopf / fold - cycle ) bursts @xcite . a subcritical elliptic burster has two primary bifurcations that determine its outcome . the active phase of the burst is initiated when the trajectory passes through a subcritical hopf bifurcation of the layer problem , and terminates when the trajectory reaches a snpo and falls off the manifold of limit cycles of the layer flow . these subcritical elliptic bursts persist in @xmath159 until there is another torus bifurcation ( tr@xmath174 ) at @xmath175 m , after which the system exhibits rapid spiking ( figure [ fig : phbifn](c ) ) . the branch of spiking solutions remains stable until another torus bifurcation ( tr@xmath176 ) at @xmath177 m is encountered . initially , for @xmath159 values @xmath10 close to tr@xmath176 , the system exhibits amplitude - modulated spiking ( not shown ) . the amplitude modulated spiking only exists on a very thin @xmath159 interval . moreover , the amplitude modulation becomes more dramatic as @xmath159 increases until @xmath178 m , after which the trajectory is a novel type of solution that combines features of amplitude - modulated spiking and bursting . these hybrid _ amplitude - modulated bursting _ ( amb ) solutions appear to be subcritical elliptic bursts with the added twist that there is amplitude modulation in the envelope of the waveform during the active burst phase ( compare figures [ fig : phbifn](b ) and ( d ) ) . we will carefully examine these amb rhythms in section [ sec : tcimmo ] . for sufficiently large @xmath159 , we recover subcritical elliptic bursting solutions like those shown in figure [ fig : phbifn](b ) , but with increasingly long silent phases . eventually these subcritical elliptic bursting solutions disappear in the torus bifurcation tr@xmath179 at @xmath180 m and the attractor of the system is a spiking solution , which disappears in a supercritical hopf bifurcation ( hb@xmath174 ) at @xmath181 m . the bifurcation structure of with respect to @xmath159 described above was computed using auto @xcite and is shown in figure [ fig : phbifn](e ) . two primary features of our bifurcation analysis here signal the presence of multiple time - scale dynamics in system . firstly , the presence of trajectories that have epochs of rapid spiking interspersed with silent phases ( i.e. , the bursting solutions ) indicates that there is an intrinsic slow / fast structure . second , the transition from rapid spiking to bursting via a torus bifurcation , together with the appearance of amplitude - modulated waveforms suggests the presence of torus canards , which naturally arise in slow / fast systems . motivated by this , we now turn our attention to the problem of understanding the underlying mechanisms that generate these novel bursting rhythms . we will use the analytic framework developed in sections [ sec : averaging ] and [ sec : classification ] as the basis of our understanding . to demonstrate the existence of a separation of time - scales in system , we first perform a dimensional analysis , following a procedure similar to that of @xcite . we define new dimensionless variables @xmath182 , and @xmath183 via @xmath184 where @xmath185 and @xmath186 are reference calcium and 3 concentrations , respectively , and @xmath187 is a reference time - scale . details of the non - dimensionalization are given in appendix [ app : ph ] . for the parameter set in table [ tab : phparams ] , natural choices for @xmath185 and @xmath186 are @xmath188 m and @xmath189 m . with these choices , a typical time scale for the dynamics of the calcium concentration @xmath151 is given by @xmath190 s. the @xmath152 dynamics evolve much more slowly with a typical time scale @xmath191 s. the @xmath153 dynamics have a typical time - scale @xmath192 s. the time - scale @xmath193 for the @xmath45 dynamics depends on @xmath159 and can range from @xmath194 s ( for @xmath195 m ) to @xmath196 s ( for @xmath197 m ) . setting the reference time - scale to be the slow time - scale ( i.e. , @xmath198 ) , and defining the dimensionless parameters @xmath199 leads to the dimensionless version of the ph model @xmath200 where the overdot denotes derivatives with respect to @xmath183 , and the functions @xmath201 , and @xmath202 are given in appendix [ app : ph ] . for the parameter set in appendix [ app : ph ] , we have that @xmath203 is small . thus , for a large regime of parameter space , the ph model is singularly perturbed , with two fast variables ( @xmath204 ) and two slow variables ( @xmath205 ) . we now show that the ph model satsfies assumptions [ ass : man ] [ ass : homoclinic ] . the first step is to examine the bifurcation structure of the layer problem @xmath206 where the prime denotes derivatives with respect to the ( dimensionless ) fast time @xmath207 , which is related ( for non - zero perturbations ) to the dimensionless slow time @xmath183 by @xmath208 . the geometric configuration of the layer problem is illustrated in figure [ fig : phlayerbifn ] . . the critical manifold ( red surface ) possesses a curve of subcritical hopf bifurcations @xmath209 ( red curve ) that separates the attracting and repelling sheets , @xmath96 and @xmath97 . also shown is the maximum @xmath151-value for the manifold of limit cycles ( blue surface ) emanating from @xmath209 . the manifold of limit cycles consists of attracting and repelling subsets , @xmath36 and @xmath37 , which meet in a manifold of snpos , @xmath33.,width=312 ] system has a critical manifold , @xmath44 , with a curve of subcritical hopf bifurcations , @xmath209 , that divide @xmath44 between its attracting and repelling sheets , @xmath96 and @xmath97 , respectively . the manifold , @xmath37 , of limit cycles that emerges from @xmath209 is repelling . the repelling family of limit cycles meets an attracting family of limit cycles , @xmath36 , at a manifold of snpos , @xmath33 . thus , the ph model has precisely the geometric configuration described in section [ subsec : assumptions ] . the bifurcation structure of contains many other features outside of the region shown here . the full critical manifold is cubic - shaped . only part of that lies in the region shown in figure [ fig : phlayerbifn ] , and outside this region , there are additional curves of fold and hopf bifurcations , cusp bifurcations , and bogdanov - takens bifurcations . in this work , we are only concerned with the region of phase space presented in figure [ fig : phlayerbifn ] . we now apply the results of section [ sec : classification ] to the ph model . in section [ subsec : phtoralfs ] , we show that the ph model possesses toral folded singularities . we carefully examine the geometry and maximal torus canards in the case of a tfn in section [ subsec : phmanifolds ] . in order to do so , we must compute the invariant manifolds of limit cycles , the numerical method for which is outlined in section [ subsec : phnumerical ] . we then show in section [ subsec : phtoralfsclass ] that the torus canards are generic and robust phenomena , and occur on open parameter sets . in this manner , we demonstrate the practical utility of our torus canard theory . we now proceed to locate and classify toral folded singularities of the ph model . for each limit cycle in @xmath33 , we numerically check the condition for toral folded singularities given in equation , @xmath210 where the overlined quantities are the averaged coefficients that appear in theorem [ thm : averaging ] . recall , that the condition @xmath211 corresponds geometrically to the scenario in which the projection of the averaged slow drift into the slow variable plane is tangent to @xmath33 ( figure [ fig : phtoralfn ] ) . for our computations , we take this condition to be satisfied if @xmath212 . for the representative parameter set given in table [ tab : phparams ] ( appendix [ app : ph ] ) , we find that the ph model possesses a toral folded singularity for @xmath213 m at @xmath214 where @xmath215 . m at @xmath216 . ( a ) projection of @xmath23 in a neighbourhood of the tfn into the @xmath217 phase space . the blue curves show the envelopes of the slow drift along @xmath23 for different initial conditions . ( b ) projection into the slow variable plane . the singular slow drift along @xmath23 is tangent to @xmath33 at the tfn.,title="fig:",width=480 ] ( -364,146)(a ) ( -160,146)(b ) once the toral folded singularity has been located , we simply compute the remaining averaged coefficients from theorem [ thm : averaging ] , which allows us to compute the eigenvalues of the toral folded singularity and hence classify it according to the scheme in definition [ def : class ] . for the toral folded singularity at @xmath216 , the eigenvalues are @xmath218 and @xmath219 , so that we have a tfn . the associated eigenvalue ratio of the tfn is @xmath220 , and so by , the maximal number of oscillations that the envelope of the waveform can execute for sufficiently small @xmath21 is @xmath221 ( compare with figure [ fig : phbifn](d ) where the envelope only oscillates 4 times ) . we point out that the type of toral folded singularity can change with the parameters ( see section [ subsec : phtoralfsclass ] ) . all other points on @xmath33 are regular folded limit cycles . that is , @xmath222 at all other points on @xmath33 ; and so , for @xmath223 m , there is only a single tfn . note that the tfn identified here is a simple zero of @xmath224 , i.e. , @xmath224 has opposite sign for points on @xmath33 on either side of the tfn . this tfn is the natural candidate mechanism for generating torus canard dynamics . we now examine the geometry of the ph model in a neighbourhood of the tfn away from the singular limit . recall that averaging theory @xcite together with fenichel theory @xcite guarantees that normally hyperbolic manifolds of limit cycles , @xmath36 and @xmath37 , persist as invariant manifolds of limit cycles , @xmath145 and @xmath146 , for sufficiently small @xmath21 . we showed in section [ subsec : toralfn ] that the extensions of @xmath145 and @xmath146 into a neighbourhood of a tfn results in a local twisting of these manifolds of limit cycles . and @xmath146 into a neighbourhood of the tfn projected into the @xmath217 phase space for @xmath223 m and @xmath225 . the attracting invariant manifold of limit cycles ( blue ) is computed until it intersects the hyperplane @xmath226 . similarly , the repelling invariant manifold of limit cycles ( red ) is computed up to its intersection with @xmath227 . the inset shows @xmath228 ( blue ) and @xmath229 ( red ) . their singular limit counterparts , @xmath230 and @xmath231 , are also shown for comparison . there are 13 intersections , @xmath232 , @xmath233 , of the invariant manifolds , each corresponding to a maximal torus canard ( with @xmath234).,width=480 ] figure [ fig : phinvariantmanifolds ] demonstrates that the attracting and repelling manifolds of limit cycles twist in a neighbourhood of the tfn , and intersect a countable number of times . for @xmath223 m , we find that there are 13 intersections , consistent with the prediction from section [ subsec : phtoralfs ] . these intersections of @xmath145 and @xmath146 are the maximal torus canards ( by definition [ def : maximaltc ] ) . the outermost intersection of @xmath145 and @xmath146 , denoted @xmath235 , is the maximal strong torus canard . the intersections , @xmath232 , @xmath236 , are the maximal secondary torus canards . the innermost intersection is the maximal weak torus canard , @xmath237 . the maximal strong torus canard , @xmath235 , is the local phase space separatrix that divides between rapidly oscillating solutions that exhibit amplitude modulation and those that do not . the maximal weak torus canard , @xmath237 , plays the role of a local axis of rotation . that is , the invariant manifolds twist around @xmath237 . the maximal secondary torus canards partition @xmath145 and @xmath146 into rotational sectors . every orbit segment on @xmath145 between @xmath238 and @xmath239 for @xmath240 , is an amplitude - modulated waveform where the envelope executes @xmath241 oscillations in a neighbourhood of the tfn . m and @xmath225 . left column : projection of @xmath145 and @xmath146 , onto the @xmath242 plane along with the maximal torus canards @xmath243 . also shown is the envelope of the transient solution , @xmath47 , of in the ( a ) sector bounded by @xmath235 and @xmath244 , ( b ) sector bounded by @xmath244 and @xmath245 , and ( c ) sector bounded by @xmath245 and @xmath246 . right column : corresponding time traces of the transient solution @xmath47 . note that time is given in seconds.,title="fig:",width=480 ] ( -364,452)(a ) ( -364,296)(b ) ( -364,140)(c ) figure [ fig : phsectors ] illustrates the sectors of amplitude - modulation formed by the maximal torus canards . for fixed parameters , it is possible to change the number of oscillations in the envelope of the rapidly oscillating waveform by adjusting the initial condition . more specifically , the trajectory of for an initial condition on @xmath145 between @xmath235 and @xmath244 is an amb with one oscillation in the envelope ( figure [ fig : phsectors](a ) ) . by changing the initial condition to lie in the rotational sector bounded by @xmath244 and @xmath245 ( figure [ fig : phsectors](b ) ) , the amplitude - modulated waveform exhibits two oscillations in its envelope . the deeper into the funnel of the tfn , the smaller and more numerous the oscillations in the envelope of the rapidly oscillating waveform ( figure [ fig : phsectors](c ) ) . moreover , each oscillation significantly extends the burst duration . figure [ fig : phinvariantmanifolds ] is the first instance of the numerical computation of twisted , intersecting , invariant manifolds of limit cycles of a slow / fast system with at least two fast and two slow variables . here , we outline the numerical method ( inspired by the homotopic continuation algorithms for maximal canards of folded singularities @xcite ) used to generate figure [ fig : phinvariantmanifolds ] . the idea of the computation of @xmath145 is to take a set of initial conditions on @xmath36 sufficiently far from both the tfn and manifold of snpos , and flow it forward until the trajectories reach the hyperplane @xmath247 where @xmath248 is the @xmath152 coordinate of the tfn identified in section [ subsec : phtoralfs ] . this generates a family of rapidly oscillating solutions that form a mesh of the manifold @xmath145 . the envelope of each of those rapidly oscillating solutions is then used to form a mesh of the projection of @xmath145 . and @xmath146 for the ph model for the same parameter set as figure [ fig : phinvariantmanifolds ] . the attracting invariant manifold of limit cycles , @xmath145 , is computed by taking the set @xmath249 and flowing it forward until it intersects the hyperplane @xmath227 . the blue curves illustrate the behaviour of the envelopes of the rapidly oscillating orbit segments that comprise @xmath145 . similarly , the repelling invariant manifold of limit cycles , @xmath146 , is computed by flowing @xmath250 backwards in time up to the hyperplane @xmath227 . the red curves illustrate the envelopes of these rapidly oscillating orbit segments that comprise @xmath146.,width=312 ] to initialise the computation , a suitable set of initial conditions must be chosen . note that the projection of @xmath33 into the slow variable plane is a curve , @xmath251 , say ( see figure [ fig : phtoralfn](b ) ) . we choose our initial conditions to be a manifold of attracting limit cycles , @xmath249 , such that the projection of @xmath252 into the @xmath253 plane is approximately parallel to @xmath251 , and is sufficiently far from @xmath251 ( figure [ fig : phnumericalmethod ] ) . similarly , to compute @xmath146 , we initialize the computation by choosing a set of repelling limit cycles , @xmath250 , where the projection of @xmath254 into the @xmath253 plane is approximately parallel to , and sufficiently distant from , the curve @xmath251 . we then flow that set of initial conditions @xmath254 backwards in time until the trajectory hits the hyperplane @xmath227 . the envelopes of these rapidly oscillating trajectories are then used to visualize @xmath146 . to locate the maximal torus canard @xmath255 , we locate the initial condition within @xmath252 that forms the boundary between those orbit segments with @xmath241 oscillations in their envelope and orbit segments with @xmath256 oscillations in their envelope . we point out that whilst numerical methods exist for the computation and continuation of maximal canards of folded singularities @xcite , these methods will not work for maximal torus canards . there are currently no existing methods to numerically continue @xmath145 and @xmath146 . consequently , there are no numerical methods that will allow for the numerical continuation of maximal torus canards in parameters , which is essential for detecting bifurcations of torus canards . we have now carefully examined the torus canards associated to a tfn for a single parameter set . however , the ph model has two slow variables and so it supports tfns on open parameter sets . the theoretical framework developed in sections [ sec : averaging ] and [ sec : classification ] allows to determine how those tfns and their associated torus canards depend on parameters . figure [ fig : pheigenvalues](a ) shows the eigenvalue ratio , @xmath130 , of the toral folded singularity as a function of @xmath159 , for instance . . ( a ) the eigenvalue ratio , @xmath130 , of the toral folded singularity as a function of @xmath159 . the black markers indicate odd integer resonances in the eigenvalue ratio , where secondary torus canards bifurcate from the weak torus canard . there is a toral fsn ii at @xmath257 m . for @xmath258 m , the toral folded singularity is a toral folded saddle . bottom row : @xmath145 and @xmath146 , in a hyperplane passing through the tfn for @xmath259 and ( b ) @xmath260 m where @xmath261 and ( c ) @xmath262 m where @xmath263 . in both cases , the invariant manifolds are shown in an @xmath142 neighbourhood of the tfn . also shown are the attracting and repelling manifolds of limit cycles , @xmath36 and @xmath37 , of the layer problem.,title="fig:",width=480 ] ( -360,274)(a ) ( -366,138)(b ) ( -180,138)(c ) we find that the ph model has tfns and hence torus canard dynamics for @xmath264 m @xmath265 m . the black markers in figure [ fig : pheigenvalues](a ) indicate odd integer resonances in the eigenvalue ratio , @xmath130 , of the tfn . these resonances signal the creation of new secondary canards in the averaged radial - slow system . as such , when @xmath144 increases through an odd integer , we expect additional torus canards to appear . figures [ fig : pheigenvalues](b ) and ( c ) illustrate the mechanism by which these additional torus canards appear . namely , as @xmath159 decreases and @xmath144 increases , the invariant manifolds of limit cycles become more and more twisted , resulting in additional intersections . thus , figure [ fig : pheigenvalues](a ) essentially determines the number of torus canards that exist for a given parameter value . an alternative viewpoint is that figure [ fig : pheigenvalues](a ) determines the maximal number of oscillations that the envelope of the rapidly oscillating waveform can execute . for example , for @xmath159 on the interval between @xmath266 and @xmath267 , the amplitude - modulated waveform can have , at most , one oscillation in the envelope . for @xmath159 on the interval between @xmath268 and @xmath269 , the amplitude - modulated waveform can have , at most , two oscillations in the envelope , and so on . the ph model supports other types of toral folded singularities . for @xmath258 m , system has toral folded saddles . the toral folded saddle has precisely one torus canard associated to it . this torus canard , however , has no rotational behaviour , and instead acts as a local phase space separatrix between trajectories that fall off the manifold of periodics at @xmath33 and those that turn away from @xmath33 and stay on @xmath36 . the other main type of toral folded singularity that can occur is the toral folded focus . in the ph model , we find a set of toral folded foci for @xmath270 m . as stated in section [ subsec : toralclass ] , toral folded foci have no torus canard dynamics . the transition between tfn and toral folded saddle occurs at @xmath257 m in a toral fsn of type ii ( corresponding to @xmath271 ) , in which an ordinary singularity of the averaged radial - slow system coincides with the toral folded singularity . having carefully examined the local oscillatory behaviour of the ph model due to tfns and their associated torus canards , we proceed in this section to identify the local and global dynamic mechanisms responsible for the amb rhythms . in section [ subsec : amb ] , we study the effects of parameter variations on the amb solutions . we then show in section [ subsec : phsingtc ] that the ambs are torus canard - induced mixed - mode oscillations . in section [ subsec : phsingvsnonsing ] , we examine where the spiking , bursting , and amb rhythms exist in parameter space . in so doing , we demonstrate the origin of the amb rhythm and show how it varies in parameters . in section [ subsec : phbifn ] , we reported on the existence of amb solutions in the ph model ( see figure [ fig : phbifn](d ) ) . the novel features of these ambs are the oscillations in the envelope of the rapidly oscillating waveform during the active phase , which significantly extend the burst duration . changes in the parameter @xmath159 have a measurable effect on the amplitude modulation in these amb rhythms . increasing @xmath159 causes a decrease in the number of oscillations that the profile of the waveform exhibits . that is , as @xmath159 increases , the envelope of the bursting waveform gradually loses oscillations and the burst duration decreases . this progressive loss of oscillations in the envelope continues until @xmath159 has been increased sufficiently that all of the small oscillations disappear . for instance , for @xmath272 m , we observe 5 oscillations in the envelope ( figure [ fig : phampmod](a ) ) . this decreases to 4 oscillations for @xmath223 m ( figure [ fig : phbifn](d ) ) , down to 3 for @xmath273 m ( figure [ fig : phampmod](b ) ) , and then to 2 for @xmath274 m ( figure [ fig : phampmod](c ) ) . further increases in @xmath159 result in just 1 oscillation in the envelope ( not shown ) until , for sufficiently large @xmath159 , the oscillations disappear . once the amplitude modulation disappears ( figure [ fig : phampmod](d ) ) , the trajectories resemble the elliptic bursting rhythms discussed previously . m , ( b ) @xmath273 m , ( c ) @xmath274 m , and ( d ) @xmath275 m . increasing @xmath159 decreases the number of oscillations that the envelope of the waveform executes , and consequently decreases the burst duration.,title="fig:",width=480 ] ( -365,252)(a ) ( -180,252)(b ) ( -364,116)(c ) ( -180,116)(d ) it is currently unknown what kinds of bifurcations , if any , occur in the transitions between amb waveforms with different numbers of oscillations in the envelope . we conjecture that these transitions occur via torus doubling bifurcations ( since they are associated with tfns ; see section [ subsec : phsingtc ] ) . further investigation of the bifurcations that organise these transitions is beyond the scope of the current article . we first concentrate on understanding the mechanisms that generate the amplitude - modualted bursting rhythm seen in figure [ fig : phbifn](d ) , corresponding to @xmath276 m . to do this , we construct the singular attractor of the ph model for @xmath276 m ( figures [ fig : phtoralmmo](a ) and ( b ) ) . the singular attractor is the concatenation of four orbit segments . starting in the silent phase of the burst , there is a slow drift ( black , single arrow ) along the critical manifold @xmath96 that takes the orbit up to the curve @xmath209 of hopf bifurcations , where the stability of @xmath44 changes . this initiates a fast upward transition ( black , double arrows ) away from @xmath209 towards the attracting manifold of limit cycles , @xmath36 . once the trajectory reaches @xmath36 , there is a net slow drift ( black , single arrow ) that moves the orbit segment along @xmath36 towards @xmath33 . this net slow drift along @xmath36 can be described by an appropriate averaged system ( theorem [ thm : averaging ] ) . we find that for @xmath223 m , the fast up - jump from @xmath209 to @xmath36 projects the trajectory into the funnel of the tfn . as such , the slow drift brings the trajectory to the tfn itself ( green marker ) . at the tfn , there is a fast downward transition ( black , double arrows ) that projects the trajectory down to the attracting sheet of the critical manifold , thus completing one cycle . m and @xmath277 ( black ) , @xmath278 ( purple ) , @xmath279 ( orange ) , and @xmath280 ( olive ) . ( a ) trajectories superimposed on the critical manifold , @xmath281 , and manifold of limit cycles , @xmath282 . the singular attractor alternates between slow epochs ( single arrows ) on @xmath96 and @xmath36 , with fast jumps ( double arrows ) between them . the @xmath21-unfoldings of the singular attractor ( coloured trajectories ) spend long times near the tfn ( green marker ) . inset : projection into the slow variable plane in an @xmath142 neighbourhood of the tfn . the associated time traces of are shown in ( b ) for @xmath277 , ( c ) for @xmath278 , ( d ) for @xmath279 , and ( e ) for @xmath280 . the coloured envelopes in ( b)(e ) correspond to the coloured trajectories in ( a).,title="fig:",width=480 ] ( -340,406)(a ) ( -364,180)(b ) ( -180,180)(c ) ( -364,81)(d ) ( -180,81)(e ) figure [ fig : phtoralmmo ] shows that the singular attractor perturbs to the amb rhythm for sufficiently small @xmath21 ( purple , orange , and olive trajectories ) . that is , for small non - zero perturbations , the silent phase of the orbit is a small @xmath10-perturbation of the slow drift on the critical manifold . note that the trajectory does not immediately leave the silent phase when it reaches the hopf curve . dynamic bifurcation theory shows that the initial exponential contraction along @xmath96 allows trajectories to follow the repelling slow manifold for @xmath20 times on the slow time - scale @xcite . however , there eventually comes a moment where the repulsion on @xmath97 overwhelms the accumulative contraction on @xmath96 and the trajectory jumps away to the invariant manifold of limit cycles @xmath145 . we have established that for the segments of @xmath36 that are an @xmath20-distance from @xmath33 , the slow drift along @xmath145 is a smooth @xmath10 perturbation of the averaged slow flow along @xmath36 . in a neighbourhood of the manifold of snpos , and the tfn in particular , we have shown in section [ subsec : phmanifolds ] that torus canards are the local phase space mechanisms responsible for oscillations in the envelope of the rapidly oscillating waveform . these oscillations are restricted to an @xmath142-neighbourhood of the tfn ( figure [ fig : phtoralmmo](a ) , inset ) . note that as @xmath21 increases the position of the amb trajectory changes relative to the maximal torus canards . for instance , the purple and orange trajectories in figure [ fig : phtoralmmo ] are closer to one of the maximal torus canards than the olive trajectory . in fact , the olive solution lies in a different rotational sector than the purple and orange solutions and hence has fewer oscillations . thus , the amb consists of a local mechanism ( torus canard dynamics due to the tfn ) and a global mechanism ( the slow passage of the trajectory through a delayed hopf bifurcation , which re - injects the orbit into the funnel of the tfn ) . consequently , the amb can be regarded as a _ torus canard - induced mixed - mode oscillation_. the ph model supports canard - induced mixed - mode dynamics @xcite , since it has a cubic - shaped critical manifold with folded singularities . in fact , careful analyses of the canard - induced mixed - mode oscillations in were performed in @xcite . the difference between our work and @xcite is that we are concentrating on the amb behaviour near the torus bifurcation tr@xmath176 ( see figure [ fig : phbifn](e ) ) , whereas @xcite focuses on the mixed - mode oscillations near hb@xmath171 . we have demonstrated the origin of the amb rhythm for the specific parameter value @xmath276 m . the other amb rhythms observed in the ph model ( such as in figure [ fig : phampmod ] ) can also be shown to be torus canard - induced mixed - mode oscillations . the number of oscillations that the envelopes of the ambs exhibit is determined by two key diagnostics : the eigenvalue ratio of the tfn which determines how many maximal torus canards exist , and the global return mechanism ( slow passage through the delayed hopf ) which determines how many oscillations are actually observed . whilst we have carefully studied the local mechanism in sections [ subsec : phtoralfs][subsec : phtoralfsclass ] and identified the global return mechanism , we have not performed any careful analysis of the global return or its dependence on parameters . in particular , the boundary @xmath283 , corresponding to the special scenario in which the singular trajectory is re - injected exactly on the singular strong torus canard marks the boundary between those trajectories that reach the tfn ( and exhibit torus canard dynamics ) and those that simply reach the manifold of snpos and fall off without any oscillations in the envelope . furthermore , the global return is able to generate or lose oscillations in the envelope by re - injecting orbits into the different rotational sectors formed by the maximal torus canards . we leave the investigation of the global return for these ambs to future work . the bursting and spiking rhythms shown in figures [ fig : phbifn](b ) and ( c ) , respectively , can also be understood in terms of the bifurcation structure of the layer problem . in the bursting case , the trajectory can be decomposed into four distinct segments , analogous to the amb rhythm . the only difference is that the bursting orbit encounters @xmath33 at a regular folded limit cycle instead of a tfn , and so it simply falls off @xmath23 without exhibiting torus canard dynamics . such a bursting solution , with active phase initiated by slow passage through a fast subsystem subcritical hopf bifurcation , and active phase terminated at an snpo , is known as a subcritical elliptic burster @xcite . note that in the subcritical elliptic bursting ( figure [ fig : phbifn](b ) ) and amb ( figure [ fig : phbifn](d ) ) cases , the averaged radial - slow flow possesses a repelling ordinary singularity ( i.e. , unstable spiking solutions ) . in the tonic spiking case , the trajectory of can be understood by locating ordinary singularities of the averaged radial - slow system . we find that the averaged radial - slow system of the ph model has an attracting ordinary singularity , which corresponds to a stable limit cycle @xmath47 of the layer problem . since @xmath47 is hyperbolic , the full ph model exhibits periodic solutions which are @xmath10 perturbations of the normally hyperbolic limit cycle @xmath47 for sufficiently small @xmath21 . in this spiking regime , the system possesses a toral folded saddle ( corresponding to the region of negative @xmath130 in figure [ fig : pheigenvalues](a ) ) . we are interested in the transitions between the different dynamic regimes ( spiking , bursting , and amb ) of . figure [ fig : phtr](a ) shows the two - parameter bifurcation structure of in the @xmath284 plane . continuation of the torus bifurcations tr@xmath176 and tr@xmath179 ( from figure [ fig : phbifn](e ) ) generates a single curve , which separates the spiking and bursting regimes . the region enclosed by the tr@xmath176/tr@xmath179 curve consists of subcritical elliptic bursting solutions ( including the amb ) . by similarly continuing the hopf bifurcation hb@xmath174 , we find that the spiking regime is the region bounded by the hb@xmath174 curve and the curve of torus bifurcations . note that the branch tr@xmath176 , which separates the rapid spiking and amb waveforms , converges to the toral fsn ii at @xmath257 m in the singular limit @xmath90 . this supports our conjecture from section [ subsec : toralfsn ] that the @xmath21-unfolding of the toral fsn ii is a singular torus bifurcation . and tr@xmath179 , and the hopf bifurcation , hb@xmath174 , from figure [ fig : phbifn](e ) in ( a ) the @xmath284 plane , and ( b ) the @xmath285 plane . ( a ) the torus bifurcation curve encloses the bursting region . in the limit as @xmath90 , tr@xmath176 converges to the toral fsn ii at @xmath286 m . the region between the hopf curve and the torus curve is the spiking region . ( b ) in the singular limit @xmath90 , the bursting region is enclosed by the toral fsn ii and tr@xmath179 curves . the tr@xmath176 curve unfolds ( in @xmath21 ) from the toral fsn ii curve . thus , for @xmath287 , the amb solutions exhibit more oscillations in their envelopes the closer the parameters are chosen to the tr@xmath176 boundary.,title="fig:",width=480 ] ( -364,140)(a ) ( -184,140)(b ) we provide further numerical evidence to support this conjecture in figure [ fig : phtr](b ) , where we compare the loci of the toral fsns of type ii and torus bifurcation tr@xmath176 in the @xmath285 parameter plane for various @xmath21 . the coloured curves correspond to the torus bifurcation tr@xmath176 for @xmath288 ( blue ) , @xmath289 ( red ) , and @xmath290 ( green ; inset ) . as demonstrated in figure [ fig : phtr](b ) , the tr@xmath176 curve converges to the toral fsn ii curve in the singular limit . also shown are the tr@xmath179 and hb@xmath174 curves , which remain close to each other in the @xmath285 plane and enclose a very thin wedge of spiking solutions in the parameter space . thus , in the singular limit , the bursting region is enclosed by the toral fsn ii and tr@xmath179 curves ( figure [ fig : phtr](b ) ; shaded region ) . moreover , the curve tr@xmath176 of torus bifurcations that unfolds from the toral fsn ii curve forms the boundary between amb and amplitude - modulated spiking rhythms . that is , the closer the parameters are to the tr@xmath176 curve , the more oscillations in the envelope of the amb trajectories and hence the longer the burst duration . similarly , the spiking solutions that exist near the toral fsn ii curve exhibit amplitude modulation . the numerical computation and continuation of the curve of toral fsns of type ii requires careful numerics ; it requires solutions of a periodic boundary value problem subject to phase and integral conditions . we outline the procedure in appendix [ app : tfscurve ] . we saw from figure [ fig : pheigenvalues](a ) that the ph model supports tfn - type torus canards for @xmath264 m @xmath265 m . figure [ fig : phtr ] , however , shows that the amplitude modulated bursting exists on a more restricted interval of @xmath159 . this indicates the importance of the global return mechanism in shaping the outcome of the torus canard - induced mixed - mode dynamics . the bifurcations that separate the different amplitude - modulated waveforms are currently unknown and left to future work . having established the predictive power of our analysis of torus canards in @xmath1 , we now examine the connection between our analysis and prior work on torus canards in @xmath0 , namely in 2-fast/1-slow systems . in section [ subsec : tcexpmodels ] , we carefully study the transition from tonic spiking to bursting via amplitude - modulated spiking in the morris - lecar - terman system for neural bursting . we show that the boundary between spiking and bursting is given by the toral folded singularity of the system . we further demonstrate the power of our theoretical framework by tracking the toral folded singularities in the hindmarsh - rose ( section [ subsec : tchr ] ) and wilson - cowan - izhikevich ( section [ subsec : tcwci ] ) models . we note that the analysis in this section relies on the results in section [ sec : arbitrarydimensions ] , namely theorem [ thm : averagingarbitraryslow ] , which extends the averaging method for folded manifolds of limit cycles to slow / fast systems with two fast variables and an arbitrary number of slow variables . our theoretical framework also allows one to determine the parameter values for which a torus canard explosion occurs in 2-fast/1-slow systems ( theorems [ thm:3daveraging ] and [ cor : explosion ] ) . this predictive power is illustrated in appendix [ app : tcexplosion ] in the case of the forced van der pol equation . the morris - lecar - terman ( mlt ) model @xcite is an extension of the planar morris - lecar model for neural excitability in which the constant applied current is replaced with a linear feedback control , @xmath48 . the ( dimensionless ) model equations are @xmath291 where @xmath9 is the ( dimensionless ) voltage , @xmath292 is the recovery variable , and @xmath48 is the ( dimensionless ) applied current . the steady - state activation functions are given by @xmath293 and the voltage - dependent time - scale , @xmath294 , of the recovery variable @xmath292 is @xmath295 following @xcite , we treat @xmath15 and @xmath296 as the principal control parameters , and fix all other parameters at the standard values listed in table [ tab : mlt ] . .standard parameter set for the mlt model . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ] using table [ tab : fvdp ] , we find that the expression for @xmath297 simplifies greatly and the condition for the torus canard explosion reduces to @xmath298 recall that the actual analytic result is @xmath299 ( plus an exponentially small correction ) . thus , in the case of the fvdp oscillator , theorem [ thm:3daveraging ] and corollary [ cor : explosion ] give the location of the torus canard explosion ( for @xmath300 ) correct up to exponentially small error . this research was partially supported by nsf - dms 1109587 . i would like to thank tasso kaper , mark kramer , jonathan rubin , and martin wechselberger for helpful discussions . i am particularly grateful to tasso kaper and mark kramer for their careful and critical reading of the manuscript . i am especially indebted to tasso kaper for being an excellent sounding board for my ideas throughout the development of this project . j. burke , m. desroches , a. granados , t. j. kaper , m. krupa , and t. vo , _ from canards of folded singularities to torus canards in a forced van der pol equation _ , j. nonlinear sci . , * 26 * ( 2016 ) , pp . 405451 . e. j. doedel , a. r. champneys , t. f. fairgrieve , y. a. kuznetsov , k. e. oldeman , r. c. paffenroth , b. sanstede , x. j. wang and c. zhang , _ auto-07p : continuation and bifurcation software for ordinary differential equations _ , available from : http://cmvl.cs.concordia.ca/ a. politi , l. d. gaspers , a. p. thomas , and t. hfer , _ models of 3 and ca@xmath302 oscillations : frequency encoding and identification of underlying feedbacks _ , biophys . j. , * 90 * ( 2006 ) , pp . 31203133 . roberts , j. rubin , and m. wechselberger , _ averaging , foided singularities and torus canards : explaining transitions between bursting and spiking in a coupled neuron model _ , siam j. appl . * 14 * ( 2015 ) , pp . 18081844 . r. roussarie , _ techniques in the theory of local bifurcations : cyclicity and desingularization _ , in `` bifurcations and periodic orbits of vector fields '' ( ed . d. szlomiuk ) , kluwer academic , dordrecht ( 1993 ) , pp . 347382 .

[ sec : abstract ] torus canards are special solutions of slow / fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem . a relatively new dynamic phenomenon , torus canards have been found in neural applications to mediate the transition from tonic spiking to bursting via amplitude - modulated spiking . in @xmath0 , torus canards are degenerate : they require one - parameter families of 2-fast/1-slow systems in order to be observed and even then , they only occur on exponentially thin parameter intervals . the addition of a second slow variable unfolds the torus canard phenomenon , making them generic and robust . that is , torus canards in slow / fast systems with ( at least ) two slow variables occur on open parameter sets . so far , generic torus canards have only been studied numerically , and their behaviour has been inferred based on averaging and canard theory . this approach , however , has not been rigorously justified since the averaging method breaks down near a fold of periodics , which is exactly where torus canards originate . in this work , we combine techniques from floquet theory , averaging theory , and geometric singular perturbation theory to show that the average of a torus canard is a folded singularity canard . in so doing , we devise an analytic scheme for the identification and topological classification of torus canards in @xmath1 . we demonstrate the predictive power of our results in a model for intracellular calcium dynamics , where we explain the mechanisms underlying a novel class of elliptic bursting rhythms , called amplitude - modulated bursting , by constructing the torus canard analogues of mixed - mode oscillations . we also make explicit the connection between our results here with prior studies of torus canards and torus canard explosion in @xmath0 , and discuss how our methods can be extended to slow / fast systems of arbitrary ( finite ) dimension . * * keywords**torus canard , canard , geometric singular perturbation theory , folded singularity , averaging , bursting , spiking , amplitude - modulation , torus bifurcation * * ams subject classifications**34e17 , 34c29 , 34c15 , 37n25 , 34e15 , 37g15 , 34c20 , 34c45

the tucson - melbourne ( tm ) three - nucleon force due to two - pion exchange has a structure which , after an expansion of the invariant @xmath4n amplitudes in the inverse nucleon mass , was determined by the original implementation of chiral symmetry in the underlying @xmath4n scattering amplitude . given that structure , the strength constants ( the @xmath0 , @xmath1 , @xmath2 , and @xmath3 coefficients ) are then not free parameters but depend upon the @xmath4n scattering data base , which has improved greatly since the original determination of these coefficients . in this note , we review two recent developments in three - body force studies : i ) a critical analysis of the generic structure of a 2@xmath4 exchange three - body force ( tbf ) @xcite , and ii ) the new tm strength constants derived from invariant @xmath4n amplitudes @xcite corresponding to the contemporary data base which includes measurements taken at the meson factories since 1980 . we make updated tbf s of the tucson - melbourne type which reflect one or both developments , add them to a nn force , and calculate properties of the triton in order to see the effect of these developments in a simple nuclear system . to begin , we display the tucson - melbourne force ( leaving out an overall momentum conserving delta function ) : @xmath5 + ( i \vec { \tau}_3 \cdot \vec { \tau}_1 \times \vec { \tau}_2 ) d ( i \vec { \sigma}_3 \cdot { \vec q } \times { \vec { q ' } } ) \right\ } \ , , \label{eq : wpipi } \nonumber\\\end{aligned}\ ] ] where @xmath6 and @xmath7 and the pion rescatters from nucleon 3 . ( we refer the reader to refs . @xcite for diagrams , more extensive definitions , explanations of the other two cyclic terms , etc . needed for calculation but not directly relevant to the present discussion ) . now we review briefly the origin of this equation . the approach used in the tucson - melbourne ( tm ) family of forces is based upon applying the ward identities of current algebra to axial - vector nucleon scattering . the ward identities are saturated with nucleon and @xmath8 poles . then employing pcac ( partial conservation of the axial - vector current ) , one can derive expressions for the on - mass - shell pion - nucleon scattering amplitudes @xcite which map out satisfactorily the empirical coefficients of the hhler subthreshold crossing symmetric expansion based on dispersion relations @xcite and , after projection onto partial waves , describe the phase shifts reasonably well @xcite . the off - mass - shell extrapolation ( needed for the exchange of virtual , spacelike pions in a nuclear force diagram ) is trivial for the @xmath3 coefficient . it can be taken directly from the on - mass - shell theoretical or empirical amplitude @xmath9 since they coincide so closely ( see appendix a of ref . one can treat this coefficient more elaborately @xcite , but the result is the same . on the other hand , one really needs an off - shell @xmath10 amplitude for the important @xmath0 , @xmath1 , @xmath2 structure of eq . ( [ eq : wpipi ] ) . this structure relies on the fact that the off - pion - mass - shell amplitude @xmath11 can be written in a form which depends on measured on - shell amplitudes only . this rewriting of the pcac / current algebra amplitude exploits a convenient correspondence between the structure of the terms corresponding to spontaneously broken chiral symmetry and the structure of the model @xmath12 term . to see this , we note that the nonspin flip @xmath13-channel isospin even amplitude ( covariant nucleon pole term removed ) is @xmath14 where @xmath15 is the pion - nucleon @xmath15 term , @xmath16 mev , and the invariant amplitude @xmath17 is given in units of the charged pion mass ( 139.6 mev ) . the double divergence @xmath18 of the background axial vector amplitude denoted by @xmath19 contains the higher order @xmath12 isobar contribution . in general , @xmath20 must have the simple form @xcite @xmath21 on the other hand , the assumed form of the function @xmath22 , @xmath23 ( adapted @xcite for @xmath4n scattering from the @xmath24 generalization of the weinberg low energy expansion for @xmath25 scattering ) is such that @xmath26 satisfies soft pion theorems ( for a review see ref . @xcite ) , and ( with the aid of eq . ( [ eq : cexpa ] ) ) the constraint at the ( on - shell and measurable ) cheng - dashen point : @xmath27 the value of @xmath28 can be determined by taking the amplitude on - shell and comparing with on - shell data extrapolated into the subthreshold region @xcite , but it is not needed , as we will now demonstrate . neglecting the @xmath29 and @xmath30 terms in ( [ eq : fampli ] ) because they are of the order of @xmath31 or higher , the @xmath26 amplitude can be expanded in the three - vector pion momenta @xmath32 and @xmath33 as follows : @xmath34 the last equation explicitly exhibits the separation between the ( higher order in @xmath35 ) @xmath12 contribution contained in the @xmath36 term alone and the remaining chiral symmetry breaking terms . in ref . @xcite and subsequent discussions of the tm @xmath37 force , the @xmath36 and @xmath28 constants in the coefficient of the @xmath38 term were eliminated in favor of the on - shell ( measurable ) quantity @xmath39 @xmath40 from the expanded @xmath4n amplitude @xmath26 in conjunction with the @xmath4nn vertices @xmath41 and pion propagators , one constructs the three body force of eq . ( [ eq : wpipi ] ) . comparing eqs . ( [ eq : wpipi ] ) and ( [ eq : expan ] ) , ( @xmath42 and @xmath43 so that @xmath44 @xcite ) one sees that @xmath45 the @xmath12 constant @xmath36 contributes then to the overall coefficient b " that has been used in nuclear calculations ( @xmath46 ) @xmath47 \label{bcoef}\ ] ] finally the @xmath2-term of eq . ( [ eq : wpipi ] ) is given by @xmath48 the dominant part of @xmath2 comes from our ansatz eq.([eq : funf ] ) but a small part is due to the backward - propagating nucleon term @xmath49 ( z - graph " ) @xmath50 . this term ( which also appears in the @xmath3 coefficient ) is representation dependent and is the only local term of a consistent set of 15 terms derived some time ago @xcite . we note that the term proportional to @xmath51 did not appear before in eq . ( [ eq : expan ] ) . this term nevertheless is inserted in @xmath2 because both the backward - propagating part of the nucleon pole @xmath52 and the @xmath12 couple with the pion with a ( assumed the same ) form factor @xmath53 which is defined as @xmath54 . the chiral breaking @xmath15 term has no intrinsic @xmath55 dependence ( although it is multiplied by @xmath56 ) . it is convenient , if not necessary , however , since part of the amplitude is due to @xmath52 and @xmath19 , to multiply the final amplitude by form factors , dependent upon @xmath55 and @xmath57 . consequently , the constant term ( @xmath58 , labeled a " in the literature ) attains a spurious momentum dependence from the form factors . the term proportional to @xmath51 in eq . ( [ ccoef ] ) is inserted to correct for this spurious momentum dependence to the orders in @xmath55 and @xmath59 kept in the amplitude . the new development in the structure of a 2@xmath4 exchange tbf @xcite lies in another look at the decomposition of the @xmath2-term made originally @xcite to fourier transform eq . ( [ eq : wpipi ] ) , but true in general . begin with the schematic structure @xmath60 and rewrite it ( neglecting the isospin dependence in eq . ( [ wc ] ) ) as @xmath61 thus the @xmath2-term can be decomposed into a @xmath62-exchange term with the same operator structure as the @xmath0-term plus a short - range @xmath4-range term . without a form factor @xmath63 the short - range part would be a dirac delta function a zero - range or contact term . this operator structure is reflected in the coordinate space representations where one always finds the coefficient @xmath64 multiplying derivatives of two coordinate space yukawas " : see , for example , eqs . 3.9 - 3.11 of ref . @xcite or appendix a of ref . @xcite . without a form factor @xmath63 the short - range part would be a dirac delta function a zero - range or contact term . the tucson - melbourne force has an ( unadorned by @xmath65 ) @xmath2 coefficient multiplying a derivative of a product of a delta function and a coordinate space yukawa " as is easily seen in the same equations . it was the latter , rather singular , aspect of the tucson - melbourne force which made numerical work difficult in both coordinate space and momentum space ( the operator structure is the same ) . in addition , the recent trend toward a low mass cutoff @xmath66 in @xmath67 for pion exchange highlights the point already emphasized by the hokkaido group @xcite and , in the modern context , by the so paulo group @xcite . the contact terms ( those proportional to a coordinate - space @xmath68-function and its derivatives ) are spread out with increasing importance as @xmath66 becomes smaller and the ( strong interaction ) size of the nucleon grows . these groups contended that these contact terms , bringing the nucleon structure signature , should not be included in potential models . the subject of contact terms has been revived recently with the advent of effective field theories in which contact terms are used to emulate the short distance physics , and the long distance physics , including the physics of chiral symmetry , is retained explicitly . in these effective field theories ( chiral perturbation theory extended to two or more nucleons @xcite ) contact terms abound , both in the chosen chiral lagrangian and in the nucleon potentials . adapting a field redefinition technique first used in pion condensation @xcite , friar _ et al . _ @xcite were able to demonstrate , via a field theoretic calculation with an effective chiral lagrangian , why the contact term of eq . ( [ decomp ] ) does not appear in the @xmath62-three body force of chiral perturbation theory , even though that field theory can be transformed to emulate the soft pion theorems . in sum , although chiral symmetry in the form of pcac / current algebra motivated the ansatz eq . ( [ eq : funf ] ) which led to the operator structure of eq . ( [ decomp ] ) , chiral symmetry in the form of effective field theory dictates that only the @xmath62-exchange part ( @xmath69 for tm ) should be retained in a tbf from pion exchange . one moral which can be drawn from this new insight is that chiral constraints on the off - shell scattering amplitude are not enough to determine a three - nucleon force ; one must also satisfy chiral constraints on the on - shell three - nucleon @xmath70-matrix elements which are presumed to make up the force . this observation applies to other off - shell amplitudes embedded in nuclear force models @xcite . the removal of the spurious contact term from the tucson - melbourne force leaves a tbf with coefficients @xmath71 , @xmath1 , and @xmath3 which has been termed tm@xmath72 in ref . @xcite and subsequent works . in the following section we will examine the effects in the triton of the original tm tbf and the tm@xmath72 tbf . we consider tm and tm@xmath72 with the original strength constants and with strength constants from the current @xmath4n scattering data . we employ a variational monte carlo method developed for accurate numerical calculations of light nuclei @xcite . the urbana - type potentials " , suited to this variational approach , take the form of a sum of operators multiplied by functions of the interparticle distance . following our previous study of charge symmetry breaking in light hypernuclei @xcite , and in order to compare with other tbf studies @xcite , we use the reid soft core nucleon - nucleon potential in the form of the urbana - type reid @xmath73 potential @xcite . the reid @xmath73 is a simplified ( the sum of operators is truncated from a possible 18 @xcite to 8 operators ) @xmath74 force model which is equivalent to the original reid soft core nucleon - nucleon potential in the lower partial waves and can produce the dominant correlations in s - shell nuclei . to be specific , the reid @xmath73 is obtained from the reid soft core ( rsc ) potential in the singlet states @xmath75 and @xmath76 and the triplet states @xmath77 and @xmath78 . the binding energy of the triton , calculated with exact faddeev codes which include all partial waves @xmath79 ( 34 channels ) , is -7.59 mev for the reid @xmath73 ( as quoted in table iv of @xcite ) , to be compared with -7.35 mev obtained with the original rsc @xcite . this small discrepancy , presumably due to differences in the @xmath80-waves of the two potentials , should not affect our conclusions . the variational method we use , with monte carlo evaluations of the integrals , is described in ref . @xcite ( see also , ref . @xcite ) . here we specify only the _ differences _ from the equations in these references . in particular , the trial nuclear wave functions have the following structure : @xmath81 { \bf s}\left[\prod^{a}_{i < j}f_{ij}\right]\phi,\ ] ] where @xmath82 is an antisymmetric spin - isospin state , having appropriate values of total spin and isospin , with no spatial dependence , and @xmath83 is a symmetrization operator which makes 3 ! terms for the two - body correlation operator @xmath84 and one term for the three - body correlation operator @xmath85 . the nn correlation operator is @xmath86 and the triplet correlation induced by the three - body force has the usual linear form suggested by the first order perturbation theory @xcite @xmath87 where @xmath28 is a variational parameter . these pair correlations do not include the spin - orbit correlations described in ref . @xcite , nor do our triplet correlations include the more sophisticated three - body correlations introduced by arriaga _ et al . _ @xcite which reduce the difference between the variational upper bound and the faddeev binding energy of the triton to less than 2% . both improvements would be clearly desirable , but are beyond the scope of this preliminary investigation . we do , however , include the usual central three - body correlation @xmath88 multiplied by the correlation functions(@xmath89 , @xmath90 , @xmath91 , @xmath92 and @xmath93 ) : @xmath94\ ] ] with @xmath95 . with these correlations we get a binding energy of -7.28(3 ) mev with the reid @xmath73 alone , a number which compares favorably with variational results in table v of ref . @xcite , obtained with a slightly different trial wave function . we now demonstrate that our variational calculations track the faddeev results of ref . @xcite and suggest that the main outline of our results ( to be presented later ) will reflect the properties of the hamiltonians chosen , provided that the potentials are not too singular . the faddeev calculations we now examine used the rsc potential and the early parameters ( labeled tm(81 ) here ) of the tucson - melbourne tbf ( @xmath96a = + 1.13 , @xmath97b = -2.58 , @xmath97c = 1.00 , and @xmath97d = -0.753 in units of the charged pion mass : 139.6 mev ) obtained from an interior dispersion relation ( idr ) analysis of phase shifts circa 1973 @xcite . ten years ago there was little reason to look suspiciously at the @xmath2-term , and the goal of the exercise was to test the perturbative nature of the @xmath10 amplitude @xmath98-wave terms . to this end , a restricted model was chosen with @xmath99 , the faddeev eigenvalues calculated for a 34-channel solution for rsc / tm , and the solution tested by employing the resulting wave functions in a raleigh - ritz variational calculation . the variational result for this restricted hamiltonian coincided with the faddeev eigenvalue , indicating the high quality of the faddeev wave function . then the @xmath0 and @xmath2 terms were selectively set to their assigned values and the variational calculation was repeated . comparison of the results shows the non - perturbative role of the @xmath0 and the @xmath2 term on the triton wave function . to test our codes and to suggest that our methods can give insight into triton binding energy effects from the proposed redefinitions of the tm force , we made a parallel set of calculations with the reid @xmath73 and the old tm force , tm(81 ) , with the parameters given above . the results are shown in table 1 . we follow tradition and calculate the triton properties with the cutoff in the form factor @xmath100 . in the publications of the tucson - melbourne group @xmath101 has been recommended to match the goldberger - treiman discrepancy @xcite , another measure of chiral symmetry breaking @xcite . the value @xmath102 matches the goldberger - treiman discrepancy @xmath103 of the recent determinations of the @xmath4nn coupling constant @xmath104 @xcite . we do nt know the reason others have chosen @xmath105 as a test case but adopt it anyway . please notice from eq . ( [ ccoef ] ) that @xmath2 , and therefore @xmath106 , changes with different values of @xmath66 . from eq . ( [ ccoef ] ) , we see that @xmath107 , because the value of @xmath108 varies only between @xmath109 and @xmath110 as @xmath100 in eq . thus , the dependence of the value of @xmath106 with @xmath111 is slight , compared with the overall effect of the cutoff on the @xmath37 force . the results of our calculations are presented in figure 1 as the open circles and open squares . the plotted points include monte carlo error bars and the lines through the symbols are drawn to guide the eye . the open circles show the calculated triton binding energy with reid @xmath73/tm@xmath72(93 ) which has no short - range @xmath4-range term and the strength constants taken from twenty year old @xmath4n scattering data . the open squares indicate the results with the same nn potential and the updated strength constants of tm@xmath72(99 ) . each calculation was made variationally with the full hamiltonian with strength constants shown in table 2 . we indicate our calculated value of the binding energy of the triton with the reid @xmath73 alone ( @xmath112 mev ) by a horizontal ( sparse ) dotted line and the faddeev eigenvalue ( @xmath113 mev ) by the horizontal ( dense ) dotted line . our variational upper bounds are always above the corresponding faddeev eigenvalues . we compare our results with calculations in the literature with the old tbf tm(81 ) , where the lack of a prime means that the short - range @xmath4-range term is _ included_. we do not present our own variational estimates with this short - range @xmath4-range term included as they do not reflect the true situation ( see discussion of table 1 ) . the results of the combination rsc / tm(81 ) for the three cutoffs @xcite ( already quoted in table 1 for the cutoff @xmath114 ) are given by the points with an @xmath115 . another faddeev evaluation @xcite of the same hamiltonian ( rsc / tm(81 ) ) is shown as stars at the three values of @xmath116 and the short dashed line interpolates between the calculated values . the models tm@xmath72(93 ) and tm@xmath72(99 ) with the spurious short - range @xmath4-range tbf removed ( open circles and open squares ) give very similar binding energies in our calculation . the updating of the strength constants seems to have very little effect on the three nucleon bound state , once the spurious term is removed . it is difficult to estimate the effect of removing the short - range @xmath4-range force on the binding energy with the results available in figure 1 , because both the nn potential ( reid @xmath73 versus rsc ) and the tbf ( tm(81 ) and tm(93 ) ) are slightly different . however , once this spurious force is removed the two models tm@xmath72(93 ) and tm@xmath72(99 ) have a similar dependence upon @xmath66 ; those two curves are shifted vertically only slightly . it is noteworthy that the dependence upon @xmath66 is greater if the spurious short - range @xmath4-range term is included in the tbf @xcite ; and significantly greater for the momentum space calculations of ref . one would expect this as @xmath66 increases and the singular term ( in one nn separation ) becomes more like a delta function . it is a nice feature that removal of the spurious term makes the tucson - melbourne two - pion exchange force less sensitive to the cutoff .

we introduce new values of the strength constants ( i.e. , @xmath0 , @xmath1 , @xmath2 , and @xmath3 coefficients ) of the tucson - melbourne ( tm ) 2@xmath4 exchange three nucleon potential . the new values come from contemporary dispersion relation analyses of meson factory @xmath4n scattering data . we make variational monte carlo calculations of the triton with the original and updated three - body forces to study the effects of this update . we remove a short - range @xmath4-range part of the potential due to the @xmath2 coefficient and discuss the effect on the triton binding energy .

early - type stars are themselves modest x - ray emitters , with a mean output in the 0.110 kev band of only @xmath2 of their optical / uv flux . thus , even a population of a million ob stars , typical of that found in a galaxy undergoing a vigorous starburst , will only produce an x - ray luminosity of @xmath3 . a single young neutron star similar to that in the crab nebula or a single o - star binary with a neutron star or black hole companion will thus outshine the entire population of main sequence stars . in order to determine the expected contribution of a young stellar population to the x - ray luminosity of a galaxy , then , it is necessary to estimate accurately the specific x - ray luminosity per o star , most of which comes from the deceased segment of the population . such an exercise is of interest in light of the relatively high x - ray luminosities of starburst galaxies and the potential contribution of such objects to the cosmic x - ray background . we attempt here a systematic , empirical census of the direct contributions ob stars , their lighter siblings , and their stellar remnants make to the hard ( 210 kev ) x - ray luminosity of a starburst galaxy by calculating the specific x - ray luminosity per o - star in the solar neighborhood , the galaxy as a whole , and the other members of the local group . we begin ( 2 ) with a cautionary tale concerning the calculation of long - term , mean x - ray luminosities for high - mass x - ray binaries by surveying the literature on the most luminous such system in the local group , smc x-1 . in 3 , we compile a list of all high - mass , accretion - powered x - ray binaries that lie within 3 kpc of the sun and , using the rossi x - ray timing explorer ( _ rxte _ ) all - sky monitor database and other archival data , compute the total x - ray luminosity within this volume arising from such systems . using a recent census of ob stars in the solar vicinity , we then calculate the specific x - ray luminosity per o star from accreting systems . this local estimate is then compared to that for the galaxy as a whole . the following section ( 4 ) repeats this analysis for the other local group galaxies , concluding with a commentary on the reported dependence of this value on metallicity . in 5 , we consider the other contributions an ob star population makes to the integrated x - ray luminosity of a galaxy from associated t tauri stars , stellar winds , and supernovae . we then go on ( 6 ) to assess the fractional contribution these direct sources of x - ray emission make to the total hard x - ray luminosity of starburst galaxies . we conclude with a summary of our results , a brief discussion of additional possible contributions to the x - ray luminosity of starburst galaxies , and the implications of these results for the origin of the cosmic x - ray background . the high - mass x - ray binary ( hmxb ) smc x-1 , the only persistent , bright , accretion - powered source in the small magellanic cloud ( smc ) , consists of a b0 supergiant primary accompanied by a neutron star with a 0.7 s pulse period in a 3.9 d orbit . the system is the most luminous x - ray binary in the local group , and is frequently cited as `` superluminous , '' given that its nominal x - ray luminosity exceeds the eddington limit for a @xmath4 neutron star . but just what is the mean , integrated luminosity of smc x-1 as observed from earth ? the most complete recent catalog of x - ray binaries is that of van paradijs ( 1995 ) which lists , among other system parameters , a maximum and minimum ( if available ) reported flux density for each source . in the case of smc x-1 , these values are 57 and 0.5 @xmath5jy . the incautious reader might adopt either the maximum value ( especially for the majority of sources in the catalog for which only one value is given ) , or simply average the two numbers to estimate the mean x - ray luminosity of the source . in fact , to convert these values to a mean observed x - ray luminosity requires adoption of a distance to the smc , a mean spectral form for the x - ray emission , and a bandwidth over which the emission is integrated , as well as a description of the temporal behavior of the source . compiling the variety of assumptions actually adopted in the literature is instructive : 45 kpc @xmath6 70 kpc ( howarth 1982 ; seward & mitchell 1981 ) ; @xmath7 ; ( angelini , white , & stella 1991 ; coe et al . 1981 ) ; @xmath8 @xmath9 ( kahabka & pietsch 1996 ; davison 1977 ) ; and 0.22.4 kev to 2100 kev ( kahabka & pietsch 1996 ; coe et al . 1981 ) . as a consequence , quoted luminosities range from @xmath10 erg @xmath11 ( seward & mitchell 1981 ) to @xmath12 ergs @xmath11 ( price et al . 1971 ) ; far from all of this uncertainty results from the source s intrinsic variability . furthermore , some reports undertake systematic data editing that bias the flux estimates upward leaving out data during the 16% of the time the x - ray source is in eclipse , ignoring periods when the source is at an undetectable level for a given instrument , etc.which , while usually well - documented and appropriate for the task at hand , require reversal when attempting to define the source s mean contribution to its galaxy s x - ray luminosity . to continue with this example , we adopt a distance to the small cloud of 65 kpc and , for smc x-1 itself , we employ spectral parameters @xmath13 , @xmath14 ( angelini et al . 1991 ) and a cutoff energy of 6.5 kev ; while these parameters ignore a reported soft ( _ kt _ @xmath15 kev ) component , they suffice to illustrate our point . correcting the mean fluxes for smc x-1 reported over monitoring times of weeks to a decade by angelini et al . ( 1991 ) , wojdowski et al . ( 1998 ) , whitlock & lochner ( 1994 ) , gruber & rothschild ( 1984 ) , and levine et al . ( 1996 ) to the 210 kev band , we find _ all _ are consistent within two sigma with the value of @xmath16 ergs @xmath11 , or 17 @xmath17 , a factor of 5 below the highest value in the literature , and a factor of 3.5 below the cataloged flux . this value is below the eddington limit for a neutron star mass of @xmath18 even before correcting for the lower metallicity of the accreting material in the smc ; more importantly , however , it represents the appropriate value to adopt in any summation of the integrated x - ray luminosity of the smc ob - star population ( see 4.2 ) and illustrates the need for caution when undertaking such a task . we begin by examining in detail the populations of hmxbs and ob stars where our information is most complete within 3 kpc of the sun . in table 1 , we list all 57 accretion - powered x - ray binaries with high - mass stellar primaries ever reported in the literature as lying within this distance . much of the data are taken from the catalog of van paradijs ( 1995 ) ; a survey of the literature in the intervening five years has been used to bring the list up to date . the first column contains the original source name , followed by a vernacular name ( if any ) . columns 24 list the optical counterpart name and j2000 coordinates . these have been taken from the hipparcos / tycho catalogs ( perryman et al . 1997 ; hg et al . 1997 ) if available ( see also chevalier & ilovaisky 1998 ) , and elsewise from the guide star catalog , or from the highest precision position reported in the literature . for sources without confirmed optical identifications , the best available x - ray coordinates are given , and the star listed is the brightest object in the error circle . the source of the position is given in column 5 ; if the positional uncertainty is greater than @xmath19 and/or the identification is uncertain , its value is given in parentheses . columns 6 and 7 give the star s visual magnitude and spectral type ; these are taken from the hipparcos or tycho catalogs when available , and otherwise from the literature . the source distances follow ( cols . 810 ) ; they include the smallest distance reported in the literature , a best estimate ( based in a few cases on hipparcos parallaxes , but mostly on a qualitative assessment of the literature ) , and the maximum plausible distance . for stars with hipparcos observations , the 1.5 @xmath20 lower limit is given . if a distance upper limit is greater than 3.0 kpc , it is quoted as `` @xmath21 kpc '' , and the source s contribution to the local x - ray binary luminosity is computed as if it were at 3 kpc for purposes of calculating an upper limit to this value . clearly , if the source lies at a greater distance , its inferred luminosity would be higher , but its contribution to the quantity of interest is zero , making this a conservative approach to calculating an upper limit to the x - ray luminosity of the population as a whole . the remaining columns of the table report source fluxes and luminosity measurements . when spectral parameters are provided in the literature , they have been used to correct the observed flux to the 210 kev band . when only an instrumental flux and bandwidth are quoted , we have adopted a power - law spectral form with a photon index of 1.0 and a plausible column density for the adopted distance(s ) ( using @xmath22 unless excess extinction is indicated ) . our results are not sensitive to this specific choice of parameters : varying the power law index over the range @xmath23 and the column density from @xmath24 @xmath9 changes the inferred luminosities by @xmath25 . since our goal is to calculate the best available long - term mean integrated flux for each source , we have utilized the _ rxte _ all - sky monitor ( asm ; levine et al . 1996 ) light curves when available ; more than half the sources have such light curves with largely continuous coverage ( 80% to 98% ) over more than 1650 days . for each of these , we include the number of days in the light curve for which no flux is available , the global mean flux , the mean flux plus 2 @xmath20 ( since many sources are not detected on most days , this value is useful as a 2 @xmath20 upper limit ) , the number of days the source flux exceeded 4 @xmath20 , the mean flux on those days , and the first and tenth brightest daily mean fluxes in the 4.5-year interval ; the latter values are included in order to ascertain whether or not one , or a few , large outbursts dominate the time - integrated luminosity . the 26 sources not included in the asm database have never been significantly above the asm threshold at any time throughout the last 4.5 years ( r. remillard , private communication ) . for these sources , we adopt an upper limit of 1.0 asm ct @xmath26 and calculate the luminosity limit for each source as described above . this limit is very conservative . as shown below , the mean asm flux value for a truly absent source ( smc x-3 , for example ) over the asm monitoring interval is @xmath27 ct s@xmath1 . the fact that all of the 31 regularly monitored sources have mean values a factor of five or more above this limit suggests that they are often present just below the detection threshold . but there is no such evidence for those sources which have never crossed the asm detection threshold , suggesting that a reasonable upper limit to their contribution could be at least an order of magnitude lower . in addition , for all sources , we have searched the high energy astrophysics science archive research center ( heasarc ) x - ray binary catalog which archives all observations of x - ray binaries in the center s large collection of databases . for each source , we list the number of detections ( col . 18 ) , the maximum count rate and , in column 21 , the catalog from which the count rate was taken . in no case does the integrated luminosity in a major outburst exceed the integrated luminosity over 30 years derived from the asm 4.5-year averages or our conservative upper limits thereto . as can be seen from the final line in table 1 , our best estimate for the integrated , mean 210 kev luminosity of the hmxb population within 3 kpc of the sun is @xmath28 . roughly one - third of this total comes from the black hole system cyg x-1 , one third from a handful of neutron star binaries such as vela x-1 and 4u170037 , and the final third from the upper limits adopted for the 26 sources not detected in the asm . the estimate is conservative , since it includes all the sources not detected in the asm as contributing at 1 ct s@xmath1 . furthermore , nine of the systems ( contributing 11% of the total flux ) have nominal distances beyond 3.0 kpc , but are included because their distance uncertainties allow membership in our volume - limited sample . if we take the 2 @xmath20 upper limits for _ all _ objects , the integrated value only increases by 40% . adopting the additional extreme assumption that all sources are at their maximum allowed distances still does not raise the conservative best estimate by a factor of two . while examination of the 30-year history of all sources does show much higher luminosities in some cases for brief intervals , there is no evidence to suggest that the last 4.5 years of asm data is in any way atypical . thus , we conclude that the hmxb population within the 38 kpc@xmath29 volume surrounding the sun produces a 210 kev x - ray luminosity of 23 @xmath30 . the final step in calculating the specific x - ray luminosity per o - star in the solar neighborhood is to find the number of o - stars within 3.0 kpc . we use the recent ( unpublished ) compilation of k. garmany ( private communication ) . she finds a total of 351 spectroscopically confirmed stars of types o3 through o9 out to 1.95 kpc from the sun ; the number in bins of constant projected area ( excepting a local minimum ) is roughly constant out to this distance , suggesting incompleteness is not a problem . in addition , there are 1915 b0b2 stars , plus a total of 772 stars with the colors of ob stars which lack spectroscopic types . adopting the same o / b ratio as for the classified stars ( 18% ) suggests as many as @xmath31 140 additional o stars should be added to the total . extrapolating with a constant surface density out to 3.0 kpc , then , yields a total of 1165 o stars within the volume . the specific accretion - powered x - ray luminosity per o star is @xmath32 , with a conservative upper limit ( 2 @xmath20 x - ray source upper limits , maximum distances , and no o stars in the unclassified portion of the stellar sample ) of @xmath33 . the census conducted here allows an estimate of the fraction of early type stars that eventually form x - ray binary systems . for example , there are 3038 stars of types o3 through b2 in the garmany compilation , implying a total of @xmath34 stars within the 3 kpc distance ( correcting for a modest incompleteness evident in the b0b2 star counts ) ; in this same region ( using best - estimate distances ) , we currently know of 24 hmxbs with primaries of these types , as well as 22 other accreting systems for which the spectral class is too poorly established to include them unambiguously . thus , @xmath35 of all the early - type stars are currently active accretion - powered x - ray sources . since the hmxb phase lasts for a few percent of an ob star s lifetime ( portegies - zwart & verbunt 1996 ) , @xmath36 of all ob stars must produce an hmxb . this is roughly consistent with population synthesis studies ( dewey & cordes 1987 ; meurs & van den heuvel 1989 ; dalton & sarazin 1995 ; lipunov , postnov , & prokhorov 1997 ; terman , taam , & savage 1998 ; portegies - zwart & van den heuvel 1999 ) , although the predictions of such calculations are quite sensitive to the assumed kick velocity imparted to neutron stars at birth . in our limited sample , at least , the fraction of x - ray active o stars is similar to that for b stars ; given their shorter lifetimes , the fraction of hmxbs produced must be larger , consistent with the notion that kick velocities become increasingly sucessful at unbinding binaries as the mass of the companion star decreases . note that these statistics include hmxbs with luminosities as low as @xmath37 erg s@xmath1 ; the fraction of systems with persistent luminosities @xmath38 erg s@xmath1 is an order of magnitude smaller . the integrated lyman continuum luminosity of the milky way is @xmath39 ( van den bergh & tammann 1991 ) . using table 5 of vacca ( 1994 ) for solar metallicity , a salpeter mass - function slope of 2.35 , and a mass upper limit of 80 @xmath40 implies a total galactic population of o stars of @xmath41 . this is @xmath42 greater than a straightforward extrapolation from the local population discussed above to the full galactic disk ( r = 12 kpc ) , consistent with the observed enhancement of star formation activity in the inner galaxy . it is also consistent with the claim of ratnatunga and van den bergh ( 1989 ) that the total pop i content of the galaxy is @xmath43 times that found in a 1 kpc@xmath44 area of the disk centered on the sun , and with an estimate ( van den bergh & tammann 1991 ) based on counts of embedded o stars from iras observations ( wood & churchwell 1989 ) . the uncertainty in the number of o stars is probably less than 50% . the total number of hmxbs in the galaxy is less well constrained . eight persistent sources are known with luminosities greater than @xmath45 ( see dalton & sarazin 1995 ) ; these produce a total _ peak _ x - ray luminosity ( see 2 ) of @xmath46 . other bright , unidentified x - ray sources in the galactic plane could add to this population ; dalton & sarazin s population synthesis model predicts 12 sources with @xmath47 and 43 sources with @xmath48 . tripling the four known sources with @xmath47 to match this prediction would yield a luminosity contribution of @xmath49 . integrating the dalton and sarazin predicted luminosity function down to @xmath50 and adding the be star population at a mean luminosity of @xmath51 yields a nominal hmxb luminosity for the galaxy of @xmath52 . this value is dominated by the luminous sources , as is observed to be the case in the solar neighorhood . adding the predicted flux from sources with @xmath53 to the observed @xmath54 of the eight bright sources yields a lower limit of @xmath55 for the galaxy s total @xmath54 . dividing the nominal value by the o star population derived above gives @xmath56 , the same as the upper limit on locally derived value and , again , uncertain by a factor @xmath57 . the specific luminosities derived above depend on a number of factors which could well be different in environments such as the nuclear starbursts to which we ultimately wish to apply our results . for example , metallicity can introduce a variety of effects : lower metallicity ( 1 ) increases the eddington luminosity of accreting sources by decreasing the x - ray scattering cross section of the infalling material , ( 2 ) lowers the mass of a star of a given spectral type ( and thus changes the conversion factor between the number of lyman continuum photons and the number of o stars ) , and , possibly , ( 3 ) changes the ratio of black holes to neutron stars formed in stellar collapse ( hutchings 1984 ; helfand 1984 ) . in order to explore the range of specific x - ray luminosities in different galactic environments , we have repeated the exercise of counting hmxbs and o stars in the four largest external members of the local group . the first x - ray sources discovered in the large magellanic cloud were detected with non - imaging rocket - borne instruments thirty years ago ( price et al . since then , systematic imaging surveys have been carried out by the _ einstein _ observatory ( long , helfand , & grabelsky 1981 ; wang et al . 1991 ) , _ exosat _ ( pakull et al . 1995 ; pietsch et al . 1989 ) , and _ rosat _ ( see haberl & pietsch 1999 , although the complete results have yet to be published ) . in addition , ten x - ray binary candidates have been monitored by the _ rxte _ asm , allowing us to calculate accurate mean fluxes on timescales of years . of the four bright , persistent accreting binaries in the lmc , one ( lmc x-2 ) is a low - mass system and does not concern us here . lmc x-1 and lmc x-3 are both strong black hole candidates ( hutchings et al . 1987 ; cowley et al . 1983 ) with steep x - ray spectra ( @xmath58 ; white & marshall 1984 ) ; lmc x-4 is a hmxb pulsar with a flat power - law index of @xmath59 ( kelley et al . 1983 ) . using these spectral parameters and the asm mean count rates calculated as above , the integrated 210 kev luminosity of these three sources is @xmath60 for an adopted distance of 50 kpc . in addition to these persistent sources , the original imaging surveys , high resolution images of 30 doradus ( wang & helfand 1991 ; wang 1995 ) , studies of variable sources in the _ rosat _ data ( haberl & pietsch 1999 ) , and monitoring observations by _ ariel v _ , _ rxte _ , _ cgro _ , etc . have led to the detection of an additional fourteen hmxb candidates in the large cloud . several of these have been confirmed as be - pulsar systems through the detection of x - ray pulses , although the majority have neither firm optical or x - ray confirmation of their identity . seven have been monitored by the asm for periods ranging from @xmath61 to @xmath62 days . while most of these have been detected on a few occasions , none has a mean flux in excess of 0.15 asm ct s@xmath1 ; for nominal spectral parameters of @xmath63 and @xmath64 , this corresponds to an upper limit of @xmath65 . of the remaining non - asm sources , none has ever been reported above a luminosity of @xmath66 for more than a single day in outburst . finally , there remain several dozen unidentified x - ray point sources from the _ einstein _ survey . although the majority of these are background interlopers , some could be additional hmxbs ; however , the integrated luminosity of the brightest ten sources is @xmath67 . an even larger number of ( mostly fainter ) point sources are to be found in the _ rosat _ survey , but , again , the majority will be interlopers , and the integrated luminosity of any lmc hmxbs will not affect our sums by more than a few percent . thus , we estimate the total accretion luminosity of the ob population in the lmc to be @xmath68 , or roughly half that of the milky way . kennicutt & hodge ( 1986 ) have derived the total lyman continuum flux from the integrated h@xmath69 luminosity of the lmc : @xmath70 . this value should be regarded as a lower limit owing to leakage of some lyman continuum photons from regions . oey & kennicutt ( 1998 ) estimate the leakage fraction ranges from @xmath71 to @xmath72 for a sample of 12 bright lmc regions ; we adopt their median value of 25% to correct the kennicutt & hodge estimate . from the integrated radio continuum flux , israel ( 1980 ) derives a value for @xmath73 of @xmath74 , which should be regarded as an upper limit owing to the nonthermal continuum radiation which has not been subtracted . we adopt @xmath75 as a conservative estimate . for a mean metallicity of one - third solar , the measured upper imf slope of @xmath76 ( massey et al . 1995 ) , and an upper mass cutoff of 80 @xmath40 ( both of which will be assumed throughout ) , vacca s ( 1994 ) tables provide an estimate for the total number of o stars in the large cloud of 5530 with an estimated uncertainty of @xmath72 . thus , the specific 210 kev x - ray luminosity is @xmath77 , a factor of 1.5 to 3 greater than those derived for the solar neighborhood and the galaxy as a whole . smc x-1 is the most luminous hmxb in the local group . as discussed in detail in 2 , however , its mean observed luminosity is not quite as extraordinary as is often implied . to complement all of the long - term studies cited above , we have used the _ rxte _ asm database to calculate its mean flux over the past 4.5 years as we have for the lmc and galactic binaries . using the spectral parameters quoted in 2 and a distance of 65 kpc , we find @xmath78 , completely consistent with the value found above from monitoring studies over the past three decades . early studies of smc x-1 with sas-3 ( clark et al . 1978 ) also led to the discovery of two other putative hmxbs in the smc at flux levels only a factor of @xmath79 lower . one of these , smc x-3 , has never been seen again , despite sensitive searches by imaging missions which reached flux levels nearly @xmath80 times lower . the other source , smc x-2 which had also disappeared a few months after its discovery ( clark , li , & van paradijs 1979 ) has been detected once more in the intervening 23 years by _ rosat _ at a level of @xmath81 ( adopting the spectral parameters of smc x-1 ) , although a subsequent observation with the same instrument failed to detect it at a level 650 times lower ( kahabka & pietsch 1996 ) . both sources have been monitored for the past 4.5 years by the _ rxte _ asm and have been detected with 4 @xmath20 significance on only one and three days , respectively ( roughly consistent with the number of such detections expected by chance , especially considering that the crowded region in which they reside raises the systematic uncertainties in daily flux determinations ) . their mean values are both consistent with zero , with @xmath82 upper limits of 0.05 asm ct @xmath26 or luminosities of @xmath83 . as with the lmc , a number of surveys and targeted observations with _ einstein _ and _ rosat _ , as well as monitoring observations by _ rxte _ and _ cgro _ have revealed several additional hmxbs and hmxb candidates in the small cloud . one new source , xte j0111.2 - 7317 has had a mean asm flux of 0.4 ct @xmath26 over the last 2 years , contributing a luminosity of @xmath84 during this interval . however , the source was not detected in either the _ einstein _ or _ _ surveys of the cloud at flux levels more than 100 times lower , so this is unlikely to represent an accurate estimate of its long - term mean luminosity . the other two hmxbs included in the _ rxte _ monitoring together contribute less than 20% of this luminosity , and the remaining thirteen candidates reported in the literature all have mean fluxes far below this level . finally , as with the lmc , the total number of remaining unidentified cloud members from the _ einstein _ ( seward & mitchell 1981 ; wang & wu 1992 ) and _ rosat _ ( kahabka & pietsch 1996 ; haberl et al . 2000 ) surveys would , if identified as hmxbs , increase the integrated luminosity of the population by only a few percent . thus , we estimate the total accretion luminosity of the ob population of the smc to be @xmath85 , or roughly equal to that for the lmc . it is important to note , however , that more than two - thirds of this luminosity arises in the singular system smc x-1 which , in addition to being the most luminous persistent hmxb in the local group , also contains the most rapidly rotating x - ray pulsar ( @xmath86 = 0.71 s ) , an object with a spin - up timescale of only 2000 years . in their detailed study of the spin and orbital evolution of smc x-1 , levine et al . ( 1993 ) estimate that the current high - luminosity phase of the binary s evolution will last at most a few times the pulsar spin - up time , or @xmath87 yr . compared with the @xmath88 yr main sequence lifetime of this 20 @xmath40 star , we have a chance of @xmath89 of seeing the system at this x - ray luminosity . since only @xmath90 of massive stars end up as short - period hmxbs ( portegies - zwart & van den heuvel 1999 ) , the number of expected systems in the smc is @xmath91 . thus , while it is not enormously improbable that we see smc x-1 at this luminosity , the long - term integrated x - ray luminosity of this galaxy is likely to be overestimated by a factor of several as a consequence of this one source s current strut upon the stage . we pursue this matter further below in discussing the putative dependence of a galaxy s x - ray luminosity on metallicity . we can estimate the total o star population for the smc in a manner exactly analogous to that used for the lmc . kennicutt & hodge ( 1986 ) report @xmath92 photons s@xmath1 from h@xmath69 data , while israel ( 1980 ) derives @xmath93 from the radio continuum emission ; using the same considerations cited in the previous section , we adopt @xmath94 photons s@xmath1 . for a metallicity of 0.1 solar , the tables in vacca yield an estimate of 1300 o stars . the resulting specific luminosity , then , is @xmath95 at the present time , although given the lifetime of smc x-1 and the arguments presented above , it is likely to be lower by a factor of @xmath96 on long timescales , making it more similar to , but still significantly in excess of , the values derived for the solar neighborhood , the galaxy , and the lmc . given its distance ( 720 kpc ) , individual x - ray binaries in m33 are not detectable by non - imaging or asm instruments , leaving the _ einstein _ ( long et al . 1981 ; markert & rallis 1983 ; trinchieri , fabbiano , & peres 1988 ) and _ rosat _ ( schulman & bregman 1995 ; long et al . 1996 ) surveys as our only views of its x - ray source population . the deepest image is that from the _ rosat _ pspc obtained by long et al . ( 1996 ) : 50 sources were detected within @xmath97 of the nucleus above a luminosity threshold of @xmath98 ergs s@xmath1 ( for our adopted spectral form of a power - law spectrum with @xmath99 and @xmath100but see below ) . five of the sources are identified with foreground stars , one is a background agn , and ten are positionally coincident with optically identified supernova remnants ; since the latter sources have mainly soft x - ray spectra , these associations are thought to be mostly correct . over 60% of the soft x - ray luminosity of the galaxy comes from a nuclear source which is unresolved with the _ rosat _ hri ( fwhm = @xmath101 ; schulman & bregman 1995 ) . the origin of this emission is unknown . there was evidence from the _ einstein _ data that the source is variable on timescales of days to months ( markert & rallis 1983 ; peres et al . 1989 ) ; more recently , dubus et al . ( 1997 ) claim evidence for a 20% modulation with a 106 d periodicity , although their result is not significant at the 3 @xmath20 level and the periodicity is apparently inconsistent with the _ einstein _ measurement ( see their figure 3 ) . the object s high x - ray luminosity in the soft band of the imaging experiments is in part a consequence of the source s soft spectrum . the asca observations of makishima et al . ( 2000 ) provide the most detailed spectral data in the harder x - ray band , and produce a luminosity estimate of @xmath102 . the unusual stellar content of the m33 nucleus ( oconnell 1983 ) and the absence of obvious signs of an agn at other wavelengths , has led to a variety of speculative notions concerning the nature of this source : an anomalous agn , a single black - hole hmxb , a cluster of hmxbs , intermediate - mass ( her x-1type ) binaries , lmxbs , and ( predictably ) a `` new '' type of x - ray source . while _ chandra _ observations will soon eliminate many of these options , it is at present unclear whether some or all of this source s luminosity should be charged to the ob population s accretion account . we calculate the specific luminosity for m33 both including and excluding this contribution . as for the remaining 33 x - ray sources with @xmath103 , one ( the third brightest ) is known to be an eclipsing binary pulsar ( dubus et al . 1999 ) . in the somewhat unlikely event that all 32 remaining sources also are hmxbs , we can estimate the integrated 210 kev luminosity from the _ exosat _ observations reported in gottwald et al . the non - imaging me detector s field of view includes all the x - ray emission from m33 . the me count rate was @xmath104 ct s@xmath1 in the 16 kev band and , while not a good fit , the spectrum can be characterized for our purposes of estimating a 210 kev luminosity by their best - fit power law parameters of @xmath105 @xmath9 . we find a total x - ray luminosity of @xmath106 . some small fraction of this emission will be contributed by the soft foreground stars and m33 snrs , so we adopt a 210 kev luminosity of @xmath107 including the nuclear source , and @xmath108 if it is excluded . there is substantial disagreement between the estimated thermal radio continuum fluxes of m33 between israel ( 1980 ) and berkuijsen ( 1983 ) . however , more recent radio results from buczilowski ( 1988 ) and the h@xmath69 measurements of devereux , duric , and scowen ( 1997 ) agree quite closely with berkuijsen s estimate which we adopt here . the implied lyman continuum flux is , then , @xmath109 ph s@xmath1 ; for a metallicity of 1/3 solar , we derive a total o star population of 3460 for m33 . this yields a range for the specific x - ray luminosity of @xmath110 ergs s@xmath1 star@xmath1 , a value comparable to that for the smc if the nuclear emission is included . as the largest member of the local group , m31 has been studied by all the major x - ray satellite missions . the _ einstein _ survey ( van speybroeck et al . 1979 ) revealed a luminous population of lmxbs both in globular clusters and in the galactic bulge , plus a disk population presumably consisting of hmxbs and supernova remnants . the recent _ rosat _ pspc survey ( supper et al . 1997 ) brought the number of detected sources in the vicinity of the galaxy to nearly 300 , lowered the luminosity threshold to @xmath111 ergs s@xmath1 , and confirmed the general picture of two source populations outlined above . in addition to these soft x - ray images , _ ginga _ carried out a long pointing at the galaxy in the 220 kev band of direct interest here . makishima et al . ( 1989 ) , fitted the high signal - to - noise integrated spectrum with a composite model to represent the dominant lmxb and hmxb populations ; the derived fluxes were carefully corrected for collimator response using the distribution of resolved sources in the _ einstein _ images . they find an upper limit to the hmxb contribution , translated to the 210 kev band using their spectral assumptions ( the galactic foreground absorption of @xmath112 @xmath9 and a cutoff power law with @xmath113 and @xmath114 kev ) of @xmath115 ergs s@xmath1 ; they demonstrate that this is consistent with a value derived by summing the _ einstein _ sources in its softer bandpass . radio - continuum , far - infrared , and h@xmath69 images indicate that the bulk of the star - formation in m31 occurs in a thin ring in the galactic disk @xmath116 kpc from the nucleus ( e.g. , beck & grve 1982 ; devereux et al . 1994 ; walterbos & braun 1994 ; xu & helou 1996 ) . this star - forming ring is the region in m31 where high - mass binaries might be expected to reside . after eliminating x - ray sources identified with foreground stars , background agn , and globular clusters , a comparison of the pspc source catalog of supper et al . ( 1997 ) and the 60 @xmath5 m image of xu & helou ( 1996 ) reveals that 20 _ rosat _ sources are positionally coincident with the star - forming ring ; four additional sources are located close to the ring , four more are coincident with 60 @xmath5m bright features outside the ring ( excluding the bulge , which is not a site of massive star formation ; see devereux et al . 1994 ) , and six others are found in an outer spiral arm northeast of the ring where there is some low - surface brightness ir emission . only two of these 34 sources ( both of which are weak and in the ring ) are identified with supernova remnants , so the remainder could all be hmxbs . the pspc count rates of the hmxb candidates sum to 0.36 ct s@xmath1 , roughly 30% of the 0.12.4 kev flux associated with the galaxy . applying the makishima et al . spectral parameters to this count rate and extrapolating to the 210 kev band suggests a maximum hmxb luminosity of @xmath117 in m31 . assuming a somewhat softer spectrum with @xmath118 or assigning only half the sources to the hmxb population yields a result consistent with the _ ginga _ analysis : @xmath119 erg s@xmath1 . as with the other local group galaxies , the ionizing photon luminosity of m31 can be inferred from the thermal fraction of its radio continuum emission and its h@xmath69 luminosity . beck & grve ( 1982 ) estimate that within the central 20 kpc , the thermal radio flux density at 2.7 ghz is @xmath120 jy , which suggests @xmath121 photons s@xmath1 . the extinction - corrected h@xmath69 luminosity of @xmath122 ergs s@xmath1 measured by walterbos & braun ( 1994 ) gives , assuming case b recombination , @xmath123 @xmath124 , and @xmath125 k , a nearly identical value for @xmath73 . thus , adopting solar metallicity , we estimate that there are 3660 o stars in m31 . an estimated luminosity of @xmath126 ergs s@xmath1 for the hmxb population yields a specific luminosity of @xmath127 ergs s@xmath1 star@xmath1 , very similar to the value we obtained for the smc and nearly a factor of ten larger than that for the galaxy . while surprising , we can think of no plausible loopholes in our argument to eliminate this difference ( see 7 ) . in table 2 , we summarize data relevant to the o - star populations and x - ray emissivity of the local group galaxies . for each object , we give our adopted distance ( uncertain by less than 10% ) , the blue luminosity @xmath128 and the adopted metallicity . the next column lists the quantity @xmath129 from vacca ( 1994 ) , the ratio of the number of equivalent o7 stars needed to produce the observed lyman continuum flux to the total number of actual o stars in the galaxy . this quantity depends on metallicity , and on the assumed slope and mass cutoff of the upper part of the imf ; we have adopted the salpeter @xmath130 for the milky way and @xmath131 for the other galaxies . varying the imf slope from 2.0 to 3.0 ( e.g. , hill , madore , & freedman 1994 ) changes the o - star counts by 22% to + 60% for solar metallicity , and 22% to + 36% for a metallicity of 0.1 solar . likewise , changing @xmath132 from @xmath133 to @xmath134 produces changes in the estimated o - star population of roughly @xmath135 . thus , the imf parameters are not a major source of uncertainty in our estimates . the number of lyman continuum photons inferred from the observations described in the text , and the resulting number of o stars are found in columns 6 and 7 . we then include several quantities which depend on the massive star population : star formation rate , core - collapse supernova rate , lyman continuum flux and number of o - stars , all normalized to the value for the milky way . clearly these quantities are not all independent , but they are listed to demonstrate that , within a factor of two , these four quantities are consistent for each galaxy , giving us some confidence that the o - star numbers by which we normalize our specific x - ray luminosities are not in error by more than a factor of two . the final two columns contain our estimates for the 210 kev @xmath54 discussed above and the x - ray luminosities per o star which constitute our principal result . the range of specific luminosities spans an order of magnitude . in two cases , we list a range of values based on differing assumptions about the assignment of x - ray flux to pop i binaries : for m33 , we quote the value including and excluding the nuclear source , and for the smc , we include the current value , as well as one - third of that value based on our arguments about the lifetime of smc x-1 . in both cases , however , the entire range of allowed values falls within the extremes defined by the milky way and m31 . while it is somewhat curious that the galaxy for which we have the best information the milky way has the lowest value , the consistency of the results we obtain for the solar neighborhood and the galaxy as a whole adds to the robustness of this conclusion . bringing the value for the earlier type galaxy m31 down by a factor of @xmath116 to agree with m0 appears to lie outside of the range of the uncertainties involved . the implications of this result for the putative dependence of pop i x - ray luminosity on metallicity is discussed below . shortly after the discovery of highly luminous pop i x - ray binaries in the magellanic clouds , clark et al . ( 1978 ) discussed the apparent shift in the mean x - ray luminosity of hmxbs in the clouds with respect to that in the milky way , and attributed the higher luminosities of the cloud binaries to the lower metallicity of the accreting gas . the discovery that the metal - poor extragalactic region ngc 5408 has a very high x - ray luminosity ( stewart et al . 1982 ) reinforced the notion that the pop i x - ray luminosity of a galaxy and its metallicity are inversely correlated . alcock & paczynski ( 1978 ) calculated evolutionary tracks for low - metallicity massive stars , and pointed out that such stars spend more time in evolutionary phases with massive stellar winds that power much hmxb emission , offering a possible explanation for this trend . hutchings ( 1984 ) offered an alternative explanation , postulating that the fraction of compact objects in hmxbs that are black holes may be higher in late - type ( lower metallicity ) galaxies ; indeed , two of the three persistently bright lmc pop i binaries are among the best black hole candidates . without any quantitative analysis of its significance or cause , numerous studies on the contribution of starbursts to the x - ray background ( xrb ; e.g. , bookbinder et al . 1980 ; griffiths & padovani 1990 ) have adopted this @xmath136 relation . the results presented here , however , suggest caution . while our value for the specific x - ray luminosity per o star in the solar neighborhood is similar to the number of @xmath137 ergs s@xmath1 per o star quoted by stewart et al . ( 1982 ) , our values for the lmc and smc disagree by factors of 4 to 8 ; a similar table from bookbinder et al . ( 1980 ) contains values higher by yet another factor of 4 . since no details on the derivation of these numbers are given in this earlier work , it is difficult to pinpoint the causes of these discrepancies , although the common overestimation of the x - ray luminosities of specific sources , exemplified by our discussion in 2 , is a likely culprit . our use of the asm data to obtain long - term mean @xmath54 values and our detailed analysis of the imaging data for each galaxy ( as well as modern estimates for o star counts ) has , we hope , reduced the uncertainties in these estimates . our conclusion that m31 , the most metal - rich member of the local group , has a specific x - ray luminosity per o star very similar to that of the smc ( the lowest metallicity galaxy ) casts serious doubt on the widely adopted notion that these two quantities are anticorrelated . the recognition that smc x-1 , which dominates the value for the small cloud , may be sufficiently short - lived that the current luminosity of that galaxy is several times greater than the long - term average would actually reverse the trend . the detailed census of 210 kev point sources in m31 , the resolution of the nature of the m33 nuclear source , and the resolution of the point source populations in more distant galaxies with _ chandra _ should help to constrain further the values derived here and to clarify the dependence , if any , of x - ray luminosity on metallicity . while hmxbs are the most luminous individual x - ray sources arising from star formation , several other high energy phenomena associated with massive stars also produce hard x - rays . for completeness , we evaluate their contributions to the specific x - ray luminosity per o star here . since most of these phenomena are , like the hmxbs , associated with all stars down to 8 @xmath40 ( the approximate dividing line between stars which end their lives in core - collapse supernovae and those which end as white dwarfs ) , we include the integrated contributions from stars down to this mass cut . furthermore , since these phenomena are mostly short - lived compared to the main sequence lifetimes of ob stars , we calculate the expected contribution to the instantaneous x - ray luminosity of the population by dividing total x - ray luminosity produced by the mean main - sequence lifetime of the population , weighted by the initial mass function : @xmath138 where @xmath139 is the initial mass function ( and we adopt the salpeter slope of @xmath140 ) , and @xmath141 ( stothers 1972 ) . the result , adopting lower- and upper - mass limits of 8 @xmath40 and 80 @xmath40 , respectively , is @xmath142 myr . in the steady state , such as obtains today in the milky way , this provides the appropriate comparison to our empirical specific luminosity per o star from the hmxbs . in a galaxy undergoing a starburst with a duration comparable to this timescale , the relative contributions of these various additional sources of x - ray emission will be a function of the starburst age . however , for a population of such starbursting systems , the steady state value provides a valid approximation . as noted in the introduction , ob stars on the main sequence produce x - rays which are thought to originate from shocks that develop in unsteady wind outflows ( lucy & white 1980 ; cooper & owocki 1994 ; feldmeier et al . the typical ratio of @xmath143 ( pallavicini et al . the characteristic temperature of the emission is @xmath144 kev ( chlebowski , harnden , & sciortino 1989 ) , implying a 210 kev luminosity of @xmath145 ergs s@xmath1 for all spectral classes . the total is , then , @xmath146 of the binary contribution and can be safely ignored . harder emission , both thermal and nonthermal , can arise when winds from neighboring stars collide ( cooke , fabian , & pringle 1978 ; chen & white 1991 ; wills , schild , & stevens 1995 ) . in the orion trapezium region , the total 210 kev x - ray luminosity , not all of which can reasonably be associated with this phenomenon is @xmath147 ergs s@xmath1 in the 210 kev band ( yamauchi & koyama 1993 ; yamauchi et al . with at least several o stars participating , this yields a specific luminosity of @xmath148 that of binary systems . while it is possible that in the massive ob associations found in starburst nuclei wind collisions could be significantly enhanced , it seems highly improbable that they will compete with binaries as a significant source of an ob star population s hard x - ray luminosity . lower mass stars , formed in association with massive stars , undergo a t tauri phase prior to descending onto the main sequence during which significant hard x - ray emission is produced ( e.g. , koyama et al . 1996 ) . again , using the local example of orion as a template , the x - ray luminosity associated with t tauri stars in the 210 kev band is @xmath149 ergs s@xmath1 ( yamauchi & koyama 1993 ) . since there are @xmath150 o stars in the orion complex , this yields a specific luminosity of @xmath151 ergs s@xmath1 . if there is a discrepancy in the ratio of high- to low - mass stars in starburst galaxies versus the local sites of star formation , it is likely to be in the direction of a deficit of lower mass stars , reducing this contribution to an even smaller value . in any event , it appears unlikely that the contribution from pre - main sequence low - mass stars will exceed 1% that of the hmxbs . the violent deaths of massive stars in core - collapse supernovae provide several means of producing x - ray emission : thermal emission from shock - heated gas left by the passage of the sn blast wave , nonthermal emission from particles accelerated at the shock front , nonthermal emission from a synchrotron nebula generated by a young , rapidly rotating neutron star , and emission from a hot young neutron star s surface and magnetosphere . we examine each of these in turn , taking the galaxy s supernova remnant ( snr ) population as exemplary . the hot gas generated by the outward moving shock wave from the sn explosion , along with the stellar ejecta heated by the reverse shock , produce thermal x - ray emission with a temperature characteristic of the shock velocities ; for most of a remnant s life these range from 300 to 3000 km s@xmath1 , yielding nominal temperatures from 0.2 to 20 kev , although delayed equilibration between the protons and electrons , nonequilibrium ionization , and inhomogeneities in the ambient and ejected material conspire to produce observed temperatures for the bulk of the emitting material of @xmath152 kev . while more sophisticated models of remnant x - ray emission have been constructed over the past few decades , it suffices for our purposes of estimating the total 210 kev energy radiated to use the simple sedov equations ( e.g. , gorenstein & tucker 1976 ) . for typical snr parameters ( explosion energy @xmath153 ergs , ambient density @xmath154 @xmath124 ) , we have calculated the temperature , shock velocity , and radius , as well as the fraction of the radiated flux emitted in the 210 kev band , as a function of time . as the swept - up material decelerates the shock , the temperature falls and the x - ray luminosity rises . however , the fraction of the emission in the 210 kev band also falls once @xmath155 km s@xmath1 ( @xmath156 yr ) , such that , for @xmath157 yr , the 210 kev band luminosity is constant to within a factor of two , with an average value of @xmath158 erg s@xmath1 ; for later times , the emission in this band rapidily declines into insignificance . thus , the integrated contribution from thermal remnant emission is @xmath159 erg s@xmath160 erg ; dividing by our mean o - star lifetime @xmath161 gives @xmath162 erg s@xmath1 per o star or roughly 1 - 2% of the hmxb contribution . in addition to heating ambient gas and supernova ejecta , the shock wave sweeps up magnetic fields and accelerates particles to relativistic energies . the primary consequence of this is the bright radio emission associated with snrs . however , for young remnants at least , the particle spectrum extends to very high energies , producing detectable synchrotron radiation in the x - ray band . the 210 kev x - ray luminosity of the historical remnant sn1006 is dominated by such synchrotron emission ( koyama et al . 1995 ) , and evidence for such nonthermal radiation has recently been detected in several other young remnants ( petre et al . 1999 and references therein ) . indeed , petre et al . claim that there is evidence that _ all _ young remnants have an x - ray synchrotron component , and that we only see this as a dominant contributor to the remnant s x - ray emission when the sn takes place in a very low density region of the interstellar medium and thus can form no significant reverse shock to illuminate the ejecta . the synchrotron luminosities of these sources are typically @xmath163 of the thermal @xmath54 , and the timescale over which this component is significant is less than that for the thermal emission . thus , its overall contribution to the hard x - ray luminosity is almost certainly @xmath164 that of the hmxb contribution . one of the most luminous hard x - ray sources in the galaxy is the crab nebula , a remnant of the supernova of 1054 ad powered by rotational kinetic energy loss from the young neutron star created in the explosion ; in the 210 kev band , @xmath165 ergs s@xmath1 ( harnden & seward 1984 ) . while often characterized as the prototypical young neutron star , the crab is , in fact , not typical . for example , the sn of 1181 ad also produced a pulsar - powered synchrotron nebula ( 3c 58 ) , but its 210 kev x - ray luminosity is @xmath166 times lower at only @xmath167 ergs s@xmath1 , despite its slightly younger age ( helfand , becker , & white 1995 ) . furthermore , evidence for young pulsars in the remnants of other core collapse supernovae has been notoriously difficult to find , and while more than three dozen such cases of snr / neutron star associations have now been suggested , none produce x - ray luminosities within a factor of five of the crab pulsar ( see helfand 1998 for a review ) . broad distributions of initial spin period and magnetic field strength for newly born neutron stars are likely to be responsible for the wide range of properties observed . a firm upper limit on the contribution such objects can make to the hard x - ray luminosity of a young stellar population can be derived by assuming that all neutron stars are born with @xmath168 msec , yielding a total rotational kinetic energy of @xmath169 erg , where @xmath170 g cm@xmath44 ) is the star s moment of inertia and @xmath172 is the rotational frequency . for the crab , the fraction of the rotational kinetic energy loss rate @xmath173 emerging in the 210 kev band is @xmath174 ; other young crab - like remnants such as 0540 - 693 in the lmc and 1509 - 58 show similar ratios of @xmath175 . thus , the upper limit to the contribution of young pulsar nebulae to the 210 kev luminosity of an ob population is @xmath176 erg @xmath177 , where @xmath178 is the fraction of supernovae that produce neutron stars , and @xmath179 is the fraction of the crab spin - down luminosity of the average young neutron star . although the mass cut dividing black hole and neutron star remnants of core collapse events is unknown , @xmath178 is likely to be of order unity . the quantity @xmath179 is less well - determined , but is clearly much less than unity : for a core - collapse sn rate of one per century , there should be ten sources with @xmath180 in the galaxy . in fact , there is only one source at 0.2 @xmath181 ( g29.7 - 0.3 ; helfand & blanton 1996 ) and no other sources within an order of magnitude . we adopt @xmath182 , although we regard this as a conservative upper limit . thus , the x - ray luminosity contribution from pulsar synchrotron nebulae could be as high as @xmath183 ergs s@xmath1 per o star or roughly 10% of the hmxb contribution ; if , as seems to be the case in the galaxy , the median x - ray luminosity of young neutron stars is at least a factor of ten less than that of the crab , the contribution of synchrotron nebulae will be @xmath184 of the hmxb value . the final source of x - ray emission resulting from a sn explosion is the thermal emission from the hot surface of the young neutron star and the nonthermal emission from its magnetosphere . since rapid neutrino cooling reduces the surface temperature to under @xmath185 k within a few decades , the contribution of thermal emission in the 210 kev band is completely negligible . nonthermal pulsed emission in the crab accounts for only @xmath186 of the total nonthermal emission produced by the pulsar nebula . becker and trumper ( 1997 ) have shown that @xmath187 for a wide range of pulsar ages and magnetic field strengths ; thus , magnetospheric x - ray emission from rotation - powered pulsars is negligible compared to the hmxb contribution . the large mechanical energy input to the interstellar medium of a starburst galaxy from stellar winds and supernovae results in a pressure - driven wind of hot plasma . such superwinds " ( e.g. , heckman , armus , & miley 1990 ) have been observed to be characteristic of galaxies with high star formation rates , and diffuse x - ray emission associated with them has been detected in a number of galaxies ( e.g. , m82 , ngc 253 [ fabbiano 1988 ] ; ngc 3256 [ moran , lehnert , and helfand 1999 ] , etc . ) . the characteristic temperature of these winds , however is @xmath188 to @xmath189 kev , and their contribution to the galaxies emission above 2 kev is negligible . having characterized the dominant role of hmxbs in the production of hard x - rays in the milky way and other local group galaxies , we can now discuss the direct contribution of ob stars and their remnants to the total hard x - ray luminosities of galaxies undergoing bursts of star formation . to do this , we need to relate the specific x - ray luminosity per o star to observable quantities for nearby starbursts total hard x - ray flux and infrared luminosity . the integrated 210 kev x - ray luminosity of high - mass binaries in a star - forming galaxy can be expressed as @xmath190_{_{\rm hmxb } } \times n({\rm o})$ ] , where @xmath191_{_{\rm hmxb}}$ ] is an adopted value of the specific x - ray luminosity per o star for hmxbs , and @xmath192 is the actual number of o stars present . assuming an imf slope of 2.35 , an upper mass cutoff of 100 @xmath40 , and solar metallicity , the models of leitherer & heckman ( 1995 ) predict that a region producing stars at a constant rate of 1 @xmath40 yr@xmath1 for at least @xmath193 yr will have @xmath194 o stars and an associated bolometric luminosity of @xmath195 ergs s@xmath1 . provided that the young stellar population dominates the host galaxy s bolometric luminosity which is approximately equal to its total infrared luminosity @xmath196the number of o stars can be scaled for a system of arbitrary star - formation rate : @xmath197 . the binary luminosity expression then becomes @xmath198_{_{\rm hmxb}}\ > l_{_{\rm ir}}$ ] , or in terms of fluxes , @xmath199_{_{\rm hmxb}}\ > f_{_{\rm ir}}$ ] . using the latter equation , we have computed the range of hmxb x - ray fluxes expected at a given ir flux for the range of local group values of @xmath191_{_{\rm hmxb}}$ ] . these are represented by the shaded region in figure 1 . this region is bounded on the lower - right by the specific x - ray luminosity per o star derived from direct counts of hmxbs and o stars in the solar neighborhood ( @xmath200 ergs s@xmath1 star@xmath1 ) , and on the upper - left by the value of @xmath0 ergs s@xmath1 star@xmath1 obtained for the smc and m31 . the dashed line represents the upper limit derived for the solar neighborhood which is roughly equal to the global milky way value . also plotted in figure 1 are the locations in the @xmath201 plane of several nearby starburst galaxies that have been studied with _ asca_. the ir fluxes of these objects have been calculated from the highest reported _ iras _ flux densities using the @xmath202 prescription of sanders & mirabel ( 1996 ) . their 210 kev x - ray fluxes have been collected from published _ asca _ results ( references are provided in the figure caption ) . note that the x - ray luminosities of the starbursts span several orders of magnitude , from @xmath203 ergs s@xmath1 ( ngc 1569 and ngc 4449 ) to @xmath204 ergs s@xmath1 ( ngc 253 and ngc 2146 ) to @xmath205 ergs s@xmath1 ( ngc 3256 and ngc 3690 ) . several important conclusions can be drawn from figure 1 . first , there is a clear tendency for the 210 kev x - ray fluxes of starburst galaxies to increase with @xmath202 , indicating that their hard x - ray luminosities are largely governed by sources whose contributions are proportional to the star - formation rate . as discussed in the previous section , hmxbs are expected to dominate over all other such contributors . however , in order for hmxbs to account for _ all _ of the hard x - rays produced in starbursts , their typical output per o star must be significantly greater than that observed in the milky way or the lmc . even in the starburst galaxies with the lowest @xmath206 ratios ( ngc 1569 , ngc 3256 , m83 , and ngc 253 ) , hmxbs would have to exhibit @xmath191_{_{\rm hmxb}}$ ] values that are 5 times higher than the milky way s . both direct observations and population syntheses ( e.g. , dalton & sarazin 1995 ) indicate that the bulk of the x - ray emission of a binary population arises from the small fraction of objects with the highest individual luminosities . thus , if hmxbs produce most of the hard x - ray flux of starburst galaxies , we would expect such systems to have many more high - luminosity objects ( per o star ) than the milky way and the lmc . for the nearest starbursts ( including several of the objects in fig . 1 ) , this hypothesis is testable with high - resolution _ chandra _ observations . the good correlation between @xmath207 and @xmath208 stands in contrast to the large scatter in a plot of @xmath209 vs. @xmath210 , the blue optical flux , from these same galaxies . thus , unlike normal galaxies which show a tight correlation between @xmath209 and @xmath211 ( fabbiano 1989)i.e . , the x - ray luminosity is proportional to the light from the whole stellar population and is dominated by long - lived , low - mass x - ray binary emission in starbursts , the dominant x - ray production is associated with the young stellar population . two objects , m82 and ngc 3310 , deviate significantly from the @xmath212 trend exhibited by the other starburst galaxies in figure 1 . the x - ray fluxes of these two objects are clearly inconsistent with the level of emission expected from an hmxb population , even one similar to that of the smc , suggesting that each galaxy possesses an extra component of hard x - ray luminosity that is weak ( or absent ) in the other starbursts . images of ngc 253 ( strickland et al . 2000 ) and m82 ( griffiths et al . 1999 ) , which have similar ir luminosities but 210 kev x - ray luminosities that differ by at least a factor of 5 , reveal strikingly different hard x - ray morphologies . the hard x - ray flux of ngc 253 is produced almost entirely by discrete sources , whereas in the more luminous m82 , about half of the hard x - ray emission arises from a diffuse component coincident with the most active region of star formation . we have suggested previously ( moran & lehnert 1997 ; moran , lehnert , & helfand 1999 ) that inverse - compton scattered emission , resulting from the interaction of ir photons with supernova - generated relativistic electrons , may in some circumstances contribute appreciably to the hard x - ray fluxes of starburst galaxies . m82 and ngc 3310 thus represent the best sites for the investigation of this possibility . we have demonstrated that the hard ( 210 kev ) x - rays produced directly by a population of ob stars , their remnants , and their accompanying lower - mass brethren are dominated by the small fraction of massive stars that form x - ray binaries . despite our efforts to assess with care the long - term mean luminosities of such systems through our use of the asm database and our consistent methods for deducing o - star number counts , we find a range of an order of magnitude in the specific x - ray luminosity per o star among the galaxies of the local group . while the specific luminosities of m33 and the smc are each dominated by a single source , and , as we have argued , could plausibly have a long - term value within a factor of 2 of the milky way , m31 remains an outlier . it is possible that only a small fraction of the 34 luminous x - ray sources coincident with star - forming regions in that galaxy are hmxbs , although no other local group galaxy shows a similar population of bright , non - hmxb objects . alternatively , the o - star population of m31 could have been severely underestimated if a large fraction ( @xmath213 ) of its lyman continuum photons escape from the galaxy without ionizing a hydrogen atom . were either ( or both ) of these scenarios to hold , and were we to ignore the bright single sources in m33 and the smc ( a somewhat uncomfortable chain of assumptions ) , the specific x - ray luminosity per o star in local group members could all fall within a factor of two of @xmath214 erg s@xmath1 per o star . in this case , the starburst galaxies would all require a source of x - ray luminosity in addition to the direct contributions of the ob star population . even if we allow the full observed range of specific luminosities and brand the milky way as atypical , however , some starbursts still require an additional hard x - ray component . while a buried active nucleus is a plausible candidate , _ chandra _ observations have ruled this out in the case of m82 . the diffuse nature of a significant fraction of the hard x - ray flux from m82 is consistent with our suggestion of ic emission as the origin of this additional component . the fact that the less intense and more diffuse starburst in ngc 253 ( 1 ) shows no significant diffuse hard emission , and ( 2 ) falls within the @xmath215 band predicted from binaries alone , is also consistent with this picture , since the predicted ic luminosity from ngc 253 would be negligible . natarajan and almaini ( 2000 ) have recently used global energetics arguments to conclude that ob stars and their products ( hmxbs and snrs ) contribute at most @xmath216 of the x - ray background at energies above 2 kev . they ( reasonably ) assume the hmxb population tracks the global star formation rate , although their normalization assumes a milky way hard x - ray luminosity a factor of 2.5 lower than we derive in 3.2 , and , thus , a factor of @xmath116 below the mean value for the local group . in addition , as they note , extra contributions such as ic emission are not included in their calculation . thus , we conclude it remains plausible that a significant contribution to the hard x - ray background arises from starburst galaxies . it should be emphasized that such a conclusion is not inconsistent with existing deep - field x - ray source counts or faint x - ray source identifications . given the steep redshift dependence of the star formation rate , the vast majority of the xrb contribution from starbursts will arise at redshifts greater than 1 . to illustrate , we use the results of moran et al . ( 1999 ) in which we showed that , owing to the tight correlation between far ir and centimetric radio emission for starburst galaxies ( and the correlation shown in figure 1 between far - ir and x - ray luminosity in these same galaxies ) , faint radio source counts can be used to constrain the surface density of starburst xrb contributors . for a 75% starburst fraction ( richards 1998 ) in the @xmath217 radio flux density range , @xmath218 arcmin@xmath219 ( fomalont et al . the _ rosat _ deep survey in the lockman hole ( hasinger et al . 1998 ) had a 0.52.0 kev limit of @xmath220 erg @xmath9 s@xmath1 for their complete sample of 50 sources over @xmath221 deg@xmath44 . using the ratio of 5 kev to 5 ghz flux density for starbursts found in moran et al . ( 1999 ) , @xmath222 ergs @xmath9 s@xmath1 kev@xmath223 , this implies an equivalent radio flux density limit of @xmath224 mjy ; and @xmath225 sources in the _ rosat _ and _ chandra _ surveys , respectively . ] assuming an x - ray spectral index of @xmath226 as observed in ngc 3256 , we should then expect @xmath227 starbursts deg@xmath219 or 0.1 such sources in the survey . for the largest _ chandra _ deep survey published to date ( giacconi et al . 2000 ) , the 210 kev limit is @xmath228 erg @xmath9 s@xmath1 which corresponds to a 5.7 mjy radio flux density and an expected surface density of 2.2 sources deg@xmath219 , or @xmath229 sources in the 0.096 deg@xmath44 survey area . in summary , the deepest surveys yet performed have not gone deep enough to reveal the population of starbursts at their expected luminosities . the continued flattening of the agn - dominated x - ray log@xmath230-log@xmath231 seen by _ observations at 210 kev flux levels above @xmath232 erg @xmath9 s@xmath1 strengthens the requirement for a new population of objects at fainter fluxes in order to account for the remaining 20 - 25% of the x - ray background . starburst galaxies remain an attractive candidate , and deeper _ chandra _ surveys should begin to find them at a surface density of 30 deg@xmath219 ( @xmath96 per _ chandra _ field ) when a flux threshold of @xmath233 erg @xmath9 s@xmath1 is reached . djh is grateful for the support of the raymond and beverly sackler fund , and joins ecm in thanking the institute of astronomy of the university of cambridge for hospitality during much of this work . this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center . the work of ecm is supported by nasa through _ chandra _ fellowship pf8 - 10004 awarded by the _ chandra _ x - ray center , which is operated by the smithsonian astrophysical observatory for nasa under contract nas8 - 39073 . djh acknowledges support from nasa grant nag 5 - 6035 . this paper is contribution number 696 of the columbia astrophysics laboratory . alcock , c. , & paczyinski , b. 1978 , apj , 223 , 244 angelini , l. , white , n.e . , & stella , l. 1991 , apj , 371 , 332 awaki , h. , ueno , s. , koyama , k. , tsuru , t. , & iwasawa , k. 1996 , pasj , 48 , 409 beck , r. , & grve , r. 1982 , a&a 105 , 192 becker , r.h . , helfand , d.j . , & szymkowiak , a.e . 1982 , apj , 255 , 557 becker , w. , & trumper , j. 1997 , a&a , 326 , 682 bookbinder , j. , cowie , l.l . , ostriker , j.p . , krolik , j.h . , & rees , m.r . 1980 , apj , 237 , 647 chen , w. , & white , r.l . 1991 , apj , 366 , 512 chevalier , c. , & ilovaisky , s.a . 1998 , a&a , 330 , 201 chlebowski , t. , harnden , f.r . jr . , & sciortino , s. 1989 , apj , 341 , 427 clark , g. , doxsey , r. , li , f. , jernigan , j.g . , & van paradijs , j. 1978 , apj , 221 , l37 clark , g. , li , f. , & van paradijs , j. 1979 , apj , 227 , 54 coe , m.j . , burnell , s.j.b . , engel , a.r . , evans , a.j . , & quenby , j.j . 1981 , mnras , 197 , 247 cooke , b.a . , fabian , a.c . , & pringle , j.e . 1978 , nature , 273 , 645 cooper , r.g . , & owocki , s.p . 1994 , ap&ss , 221 , 427 cowley , a.p . , crampton , d. , hutchings , j.b . , remillard , r. , & penfold , j.e . 1983 , apj , 272 , 118 dalton , w.w . , & sarazin , c.l . 1995 , apj , 448 , 369 davison , p.j.n . 1977 , mnras , 179 , 15p della ceca , r. , griffiths , r.e . , heckman , t.m , & mackenty , j.w . 1996 , apj , 469 , 662 della ceca , r. , griffiths , r.e . , & heckman , t.m . 1997 , apj , 485 , 581 della ceca , r. , griffiths , r.e . , heckman , t.m , lehnert , m.d . , & weaver , k.a . 1999 , apj , 514 , 772 devereux , n.a . , price , r. , wells , r.a . , & duric , n. 1994 , aj , 108 , 1667 dewey , r.j . , & cordes , j.m . 1987 , apj , 321 , 780 dubus , g. , charles , p.a . , long , k.s . , & hakala , p.j . 1997 , apj , 490 , l50 dubus , g. , charles , p.a . , long , k.s . , hakala , p. , & kuulkers , e. 1999 , mnras , 302 , 731 fabbiano , g. 1988 , apj , 330 , 672 fabbiano , g. 1989 , araa , 27 , 87 feldmeier , a. , kudritzki , r .- , palsa , r. , pauldrach , a.w.a . , & puls , j. 1997 , a&a , 320 , 899 fomalont , e.b . , windhorst , r.a . , kristian , j.a . , & kellerman , k.i . 1991 , aj , 102 , 1258 giacconi , r. , et al . 2000 , preprint ( astro - 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we present an empirical analysis of the integrated x - ray luminosity arising from populations of ob stars . in particular , we utilize results from the all - sky monitor on _ rxte _ , along with archival data from previous missions , to assess the mean integrated output of x - rays in the 210 kev band from accreting early - type binaries within 3 kpc of the sun . using a recent ob star census of the solar neighborhood , we then calculate the specific x - ray luminosity per o star from accretion - powered systems . we also assess the contribution to the total x - ray luminosity of an ob population from associated t tauri stars , stellar winds , and supernovae . we repeat this exercise for the major local group galaxies , concluding that the total x - ray luminosity per o star spans a broad range from 2 to @xmath0 erg s@xmath1 . contrary to previous results , we do not find a consistent trend with metallicity ; in fact , the specific luminosities for m31 and the smc are equal , despite having metallicities which differ by an order of magnitude . in light of these results , we assess the fraction of the observed 210 kev emission from starburst galaxies that arises directly from their ob star populations , concluding that , while binaries can explain most of the hard x - ray emission in many local starbursts , a significant additional component or components must be present in some systems . a discussion of the nature of this additional emission , along with its implications for the contribution of starbursts to the cosmic x - ray background , concludes our report .