the process of flame propagation presents an extremely complicated mathematical problem .the governing equations include the nonlinear flow equations for the fuel and the products of combustion , as well as the transfer equations governing the heat conduction and species diffusion inside the flame front .fortunately , in practice , an inner flame scale determined by the latter processes is large compared to the flame front thickness , implying that the flame can be considered as a gasdynamic discontinuity .the initial problem is reduced thereby to a purely hydrodynamic problem of determining the propagation of a surface of discontinuity in an incompressible fluid , the laws of this propagation given by the usual navier - stokes and continuity equations complemented by the jump conditions at the surface , expressing the mass and momentum conservation across the flame front .the asymptotic methods allow one to express these conditions in the form of a power series with respect to the small flame front thickness .  despite this considerable progress, however , a closed theoretical description of the flame propagation is still lacking .what is meant by the term `` closed description '' here is the description of flame dynamics as dynamics of the flow variables _ on _ the flame front surface .reduction of the system of bulk equations and jump conditions , mentioned above , to this `` surface dynamics '' implies solving the flow equations for the fuel and the combustion products , satisfying given boundary conditions and the jump conditions at the flame front , and has only been carried out asymptotically for the case where is the gas expansion coefficient defined as the ratio of the fuel density and the density of burnt matter  difficulties encountered in trying to obtain a closed description of flames are conditioned by the following two crucial aspects :  \(1 ) characterized by the flow velocities which are typically well below the speed of sound , deflagration represents an essentially nonlocal process , in the sense that the flame - induced gas flows , both up- and downstream , strongly affect the flame front structure itself .a seeding role in this reciprocal process is played by the landau - darrieus ( ld ) instability of zero thickness flames a very important factor of non - locality of the flame propagation is the vorticity production in the flame , which highly complicates the flow structure downstream . in particular , the local relation between pressure and velocity fields upstream , expressed by the bernoulli equation , no longer holds for the flow variables downstream .\(2 ) deflagration is a highly nonlinear process which requires an adequate non - perturba- + tive description of flames with arbitrary values of the flame front slope . as a result of development of the ld - instability , exponentially growing perturbations with arbitrary wavelengths make any initially smooth flame front configuration corrugated .characteristic size of the resulting `` cellular '' structure is of the order of the cutoff wavelength given by the linear theory of the ld - instability is the flame front thickness .the exponential growth of unstable modes is ultimately suppressed by the nonlinear effects . since for arbitrary governing equations do not contain small parameters , it is clear that the ld - instability can only be suppressed by the nonlinear effects if the latter are not small , and therefore so is the flame front slope .the stabilizing role of the nonlinear effects is best illustrated in the case of stationary flame propagation .numerical experiments on 2d flames with show that even in very narrow tubes ( tube width of the order ) , typical values for the flame front slope are about nonlinearity can be considered small only in the case of small gas expansion , where one has the estimate for the slope , so that it is possible to derive an equation for the flame front position in the framework of the perturbation expansion in powers of   this perturbative method gives results in a reasonable agreement with the experiment only for flames with propagating in very narrow tubes ( tube width of the order ) , so that the front slope does not exceed unity .flames of practical importance , however , have up to 10 , and propagate in tubes much wider than as a result of development of the ld - instability , such flames turn out to be highly curved , which leads to a noticeable increase of the flame velocity . in connection with this , a natural question arises whether it is possible to develop a non - perturbative approach to the flame dynamics , closed in the sense mentioned above , which would be applicable to flames with arbitrary gas expansion .a deeper root of this problem is the following _ dilemma : _ as was mentioned above , the flame propagation is an essentially non - local process ; on the other hand , this non - locality itself is determined by the flame front configuration and the structure of gas flows near the front , so the question is whether an explicit bulk structure of the flow downstream is necessary in deriving an equation for the flame front position . in other words ,we look for an approach that would provide a closed description of flames more directly , without the need to solve the flow equations explicitly .the purpose of this paper is to develop such approach in the stationary case .the paper is organized as follows .the flow equations and related results needed in our investigation are displayed in sec .[ flowequations ] .a formal integral of the flow equations is derived in sec .[ integral ] on the basis of the thomson circulation theorem . with the help of this integral, an expression for the vortex mode of the burnt gas flow near the flame front is obtained in sec .[ structure ] . to close the system of equations relating the flow variables on the flame front , potentiality of the flow upstream , and of the burnt gas flow after extracting the vortex mode are to be expressed through the values of the fuel velocity at the front .this is done in sec .[ dispersion ] in the form of dispersion relations , using which we obtain an equation relating the fuel velocity at the front and the front configuration in sec .[ mainrel ] .complemented by the relation defining the local burning rate ( the evolution equation ) , the found equation provides description of stationary flames , which is closed in the above - mentioned sense .the developed approach is applied to the particular case of zero - thickness flames in sec .furthermore , it is shown in sec .[ ltransport ] how effects related to the finite flame front thickness can be taken into account in the obtained equation .finally , the case of weak gas expansion is considered in sec .[ smallt ] , where a single equation for the flame front position is obtained in the third and the fourth orders in analytical solutions of the third- and fourth - order equations are found in secs .[ ansol ] and [ ansol4 ] .the results obtained are discussed in sec .[ discussion ] .appendix [ appa ] contains a consistency check for the expression of the vortex mode , derived in sec .[ structure ] .some auxiliary mathematical results used in the text are summarized in appendix [ appb ] .as was mentioned in the point ( 1 ) of introduction , an important factor of the flow non - locality downstream is the vorticity production in curved flames , which highly complicates relations between the flow variables . in the presence of vorticity, pressure is expressed through the velocity field by an integral relation , its kernel being the green function of the laplace operator .it should be noted , however , that the jump condition for the pressure across the flame front only serves as the boundary condition for determining an appropriate green function , being useless in other respects .thus , it is convenient to exclude pressure from our consideration from the very beginning .the basis for this is provided by the well - known thomson circulation theorem .thus , we begin in sec .[ flowequations ] with the standard formulation of the problem of flame propagation , and then construct a formal implicit solution of the flow equations with the help of this theorem in sec .[ integral ] .let us consider a 2d stationary flame propagating in an initially uniform premixed ideal fluid .let the cartesian coordinates be chosen so that the -axis is in the direction of flame propagation , being in the fresh fuel .it will be convenient to introduce the following dimensionless variables  where is the velocity of a plane flame front , is the initial pressure in the fuel far ahead of the flame front , and is some characteristic length of the problem ( e.g. , the tube width ) . the fluid density will be normalized on the fuel density   as numerical experiments show , stationary flames exist only in sufficiently narrow tubes .hence , assuming the tube walls ideal , and denoting its width by we will deal below with spatially -periodic flames .more precisely , given a flame configuration described by the functions , ] according to and then periodically continue it to the whole -axis .  in connection with this procedure the following circumstance should be emphasized .all subsequent analysis is carried out under assumption that there exists a short wavelength cut - off for the flame perturbations . in other words ,we consider flames with small but non - zero front thickness existence of such a cut - off ensures smoothness of the functions under consideration . in particular , it prevents development of singularities such as the edge points occurring at the front of a zero - thickness flame , which lead to discontinuities in the values of the flow variables or their derivatives .the end points of the flame front , however , still represent a potential source of such discontinuities even in the case of flames with non - zero thickness because of the possibility of stream line refraction at these points , resulting in a formation of stagnation zones in the flow of burnt matter ( see ref . 11 for more detail ) .having imposed the boundary condition we thereby exclude this possibility . in view of what has just been said ,it is natural to assume further that considered as functions of the complex argument , together with the `` on - shell '' values of the flow velocity ,  are analytical functions of in a vicinity of the real axis .this assumption , simplifying subsequent analysis , is only technical , and can be weakened if necessary .as always , we assume that the process of flame propagation is nearly isobaric . then the velocity and pressure fields obey the following equations in the bulk where and summation over repeated indices is implied .acting on eq .( [ flow2 ] ) by the operator where and using eq .( [ flow1 ] ) , one obtains a 2d version of the thomson circulation theorem where according to eq .( [ thomson ] ) , the vorticity is conserved along the stream lines . as a simple consequence of this theorem , one can find the general solution of the flow equations upstream .namely , since the flow is potential at (  where is the velocity of the flame in the rest frame of reference of the fuel ) , it is potential for every therefore , where the linear hilbert operator is defined by and in the coordinate representation , acts on a function such that for according to   it will be shown in the next section how the thomson theorem can be used to obtain a formal integral of the flow equations downstream .consider the quantity where is the distance from an infinitesimal fluid element to the point of observation  and integration is carried over taking into account eq .( [ flow1 ] ) , one has for the divergence of  where  boundary of and its element .next , let us calculate using eq .( [ diva ] ) , we find analogously , together , these two equations can be written as substituting the definition of into the latter equation , and integrating by parts gives the first two terms on the right of eq .( [ vint ] ) represent the potential component of the fluid velocity while the third corresponds to the vortex component. our aim below will be to transform the latter to an integral over the flame front surface .to this end , we decompose into elementary as follows .let us take a couple of stream lines crossing the flame front at points and ( see fig .[ fig1 ] ) . consider the gas elements moving between these lines , which cross the front between the time instants and during this time interval , these elements fill a space element adjacent to the flame front .for sufficiently small the volume of  where the subscript`` '' means that the corresponding quantity is evaluated just behind the flame front , _i.e. , _ for and is the normal velocity of the burnt gas , being the unit vector normal to the flame front ( pointing to the burnt matter ) .after another time interval of the same duration the elements move to a space element adjacent to since the flow is incompressible , is of the same volume as continuing this , the space between the two stream lines turns out to be divided into an infinite sequence of s of the same volume , adjacent to each other .thus , summing over all the third term in eq .( [ vint ] ) can be written as where denotes the flame front surface ( the front line in our 2d case ) , and trajectory of a particle crossing the point at   substituting expression ( [ vint1 ] ) into eq .( [ vint ] ) gives this representation of the flow velocity downstream will be used in the next section to determine the structure of the vortex mode near the flame front .to determine the flame front dynamics , it is sufficient to know the flow structure near the flame front . as to the flow upstream, it is described by eqs .( [ solup1 ] ) , ( [ solup2 ] ) for all given the solution upstream , velocity components of the burnt gas at the flame front can be found from the jump conditions which express the mass and momentum conservation across the front . on the other hand ,these components are required to be the boundary values ( for ) of the velocity field satisfying the flow equations . as was shown in the preceding section, the latter can be represented in the integral form , eq .( [ vint2 ] ) .any velocity field can be arbitrarily decomposed into a potential ( ) and vortex ( ) modes : our strategy below will be to use the integral representation to determine the near - the - front structure of the vortex mode described by the last term in eq .( [ vint2 ] ) . asto the condition of its potentiality , expressed in the form of a `` dispersion relation '' at the flame front , will eventually close the system of integro - differential equations at the front .equation ( [ vint2 ] ) reveals the following important fact . up to a potential ,the value of the vortex mode at a given point of the flow downstream is determined by only one point in the range of integration over namely that satisfying this is , of course , a simple consequence of the thomson theorem underlying the above derivation of eq .( [ vint2 ] ) .it can be verified directly by calculating the curl of the right hand side of eq .( [ vint2 ] ) : contracting this equation with using and taking into account the relation one finds since for the product of -functions picks the point ( [ p ] ) out of the whole range of integration in the right hand side of eq .( [ twod ] ) .now , let us take the observation point sufficiently close to the flame front , _i.e. , _ [ . in view of what has just been said ,the vortex component for such points is determined by a contribution coming from the integration over near the flame front , which corresponds to small values of integration over all other gives rise to a potential contribution .  the small contribution to the integral kernel be calculated exactly .for such s , one can write let the equality of two fields up to a potential field be denoted as then , substituting eq .( [ stream ] ) into eq .( [ kernel ] ) , and integrating gives {-\frac{\bm{r v_+}}{v_+}}^{v_+ t_0 - \frac{\bm{r v_+}}{v_+}}\,.\end{aligned}\ ] ] here denotes the absolute value of the velocity field at the flame front , and is assumed small enough to justify the approximate equations ( [ stream ] ) .  as we know , the only point in the range of integration over that contributes to the vortex mode is the one satisfying eq .( [ p ] ) or , after integrating over  u_+(\tilde{\eta } ) - [ \xi - f(\tilde{\eta } ) ] w_+(\tilde{\eta } ) = 0\,.\end{aligned}\ ] ] the distance between this point and the point of observation tends to zero as the latter approaches to the flame front surface .thus , taking  ] because of the reasons that will be clear in sec .[ dispersion ] .after we have determined the near - the - front structure of the vortex component of the gas velocity downstream , we can write down a closed system of equations governing the stationary flame propagation . as was explained in the introduction , the term `` closed '' means that these equations relate only quantities defined on the flame front surface , without any reference to the flow dynamics in the bulk .this system consists of the jump conditions for the velocity components at the front , and the so - called evolution equation that gives the local normal fuel velocity at the front as a function of the front curvature . these equations ( except for the evolution equation )are consequences of the mass and momentum conservation across the flame front . in sec .[ gen ] , we obtain a closed system in the most general form , without specifying the form of the jump conditions , and then apply it to the case of zero thickness flames in sec . [ zero ] .   first of all , we need to find the `` on - shell '' expression for the vortex component , i.e. its limiting form for for this purpose we note that in this limit , therefore , eq . ( [ vintf ] ) gives let us denote the jump  ] here then we can write   = v^p_{i+ } + \int\limits_{f } d l~\chi(\eta - \tilde{\eta})\sigma_{+ } \frac{(v^n_- + [ v^n ] ) ( v_{i- } + [ v_i])}{2 ( v_-^2 + [ v^2])}\,.\end{aligned}\ ] ]  in every finite order of the asymptotic expansion with respect to the flame front thickness , the jumps  ] and taking into account periodicity of the function shows that because in view of the definition ( [ antisym ] ) .next , multiplying eq .( [ w ] ) by subtracting it from eq .( [ u ] ) , and using the evolution equation ( [ evolution1 ] ) , we arrive at the single equation for the flame front position the constant can be expressed through the flame velocity using the normalization condition ( [ lvcondition ] ) .namely , integrating eq .( [ 3ordereq ] ) over , ] which is small for all rather than in powers of for this purpose we first rewrite it as an integral over the contour [ cf .( [ apb2 ] ) ] integrating by parts then yields where denotes a contour in the complex -plane , which is run by when runs next , let us make the following change of the integration variable in eq .( [ apb6 ] ) this gives where the contour of -integration is obtained by a uniform vertical shift of by an amount note that = \tilde{\eta } + o(\alpha).$ ] hence , within the framework of the asymptotic expansion in  is to be considered as a small deformation of the initial contour taking small enough we can always secure from crossing singularities of the integrand ( including the pole ) under this deformation . assuming this , we deform back to the real axis , and obtain where is a function of the real variable defined by .\ ] ] resolving this relation with respect to in the form of a series \nonumber\\ & = & \bar{\eta } - i [ f(\bar{\eta } ) - f(\eta ) ] - f'(\bar{\eta})[f(\tilde{\eta } ) - f(\eta ) ] + \frac{i}{2 } f''(\bar{\eta})[f(\tilde{\eta } ) - f(\eta)]^2 + \dots \nonumber\\ & = & \bar{\eta } - i [ f(\bar{\eta } ) - f(\eta ) ] - f'(\bar{\eta})[f(\bar{\eta } ) - f(\eta ) ] + i \left[f'(\bar{\eta})\right]^2 [ f(\bar{\eta } ) - f(\eta ) ] \nonumber\\ & & + \frac{i}{2}f''(\bar{\eta})[f(\bar{\eta } ) - f(\eta)]^2 + \dots\,,\end{aligned}\ ] ] and substituting it into eq . ( [ apb8 ] ), we finally arrive at the following expansion of  - a'(\bar{\eta})f'(\bar{\eta})[f(\bar{\eta } ) - f(\eta ) ] \nonumber\\ & & + i a'(\bar{\eta})\left[f'(\bar{\eta})\right]^2 [ f(\bar{\eta } ) - f(\eta ) ] - \frac{1}{2}a''(\bar{\eta})[f(\bar{\eta } ) - f(\eta)]^2 + \dots\bigr\}\,.\end{aligned}\ ] ] since each term in the curly brackets in the integrand on the right of eq .( [ apb10 ] ) is periodic , the corresponding improper integrals are well - defined by the rule ( [ rulec ] ) .now it is straightforward to write down the result of asymptotic action of on a general namely , using in the first term in eq .( [ apb10 ] ) , and making the substitution throughout this equation yields - a(\bar{\eta})f'(\bar{\eta})[f(\bar{\eta } ) - f(\eta ) ] \nonumber\\ & & + i a(\bar{\eta})\left[f'(\bar{\eta})\right]^2 [ f(\bar{\eta } ) - f(\eta ) ] - \frac{1}{2}a'(\bar{\eta})[f(\bar{\eta } ) - f(\eta)]^2 + \dots\bigr\}\,.\end{aligned}\ ] ]  in the course of derivation of the third order equation in sec .[ smallt ] , we found it necessary to determine the asymptotic action of on the curly bracket in the left hand side of eq .( [ main1 ] ) taking into account terms . in this case , and therefore , it is sufficient to keep only the first term in the integrand of eq .( [ apb10 ] ) , which gives immediately               to dispersion relation for the gas velocity upstream , it can be written directly in terms of itself , since the analog of eq .( [ cauchy ] ) in this case is free of any divergences _ a priori _ , and gives in the limit     this equation one should remember that the evolution equation still has the form ( [ evolution1 ] ) [ all -contributions are already collected in the right hand side of eq .( [ main1 ] ) ] .    way to show that is to recall that no terms of the form appear in the linear equation for the flame front position . however , we prefer the reasoning given in the text , because it avoids referring back to the linear theory . .vaynblat and m. matalon , `` stability of pole solutions for planar propagating flames : i. exact eigenvalues and eigenfunctions & ii .properties of eigenvalues and eigenfunctions with implication to flame stability , '' siam j. applied math . * 60 * , 679 , 703 ( 2000 ) .fig.2 : dependence of the maximal flame velocity increase on the gas expansion coefficient , given by the sivashinsky equation ( dotted line ) , and its corrections eq . ( [ 3ordereqf ] ) ( dashed line ) , and eq . ( [ 4ordereqff ] ) ( full line ) .the marks represent the results of numerical experiment ( accuracy of the experimental data is about ) .fig.3 : near - the - front structure of the flow downstream . is the normal component of the velocity .since the observation point is close to the flame front , the stream line and the part of the front near this point can be considered straight .

Abstract: the problem of non - perturbative description of stationary flames with arbitrary gas expansion is considered . on the basis of the thomson circulation theorem an implicit integral of the flow equations is constructed . with the help of this integral , a simple explicit expression for the vortex mode of the burnt gas flow near the flame front is obtained . furthermore , a dispersion relation for the potential mode at the flame front is written down , thus reducing the initial system of bulk equations and jump conditions for the flow variables to a set of integro - differential equations for the flame front position and the flow velocity at the front . the developed approach is applied to the case of thin flames . finally , an asymptotic expansion of the derived equations is carried out in the case where is the gas expansion coefficient , and a single equation for the front position is obtained in the second post - sivashinsky approximation . it is demonstrated , in particular , how the well - known problem of correct normalization of the front velocity is resolved in the new approach . it is verified also that in the first post - sivashinsky approximation , the new equation reduces to the sivashinsky - clavin equation corrected according to cambray and joulin . analytical solutions of the derived equations are found , and compared with the results of numerical simulations .


information theoretic security was first introduced by shannon in , which studied the problem of transmitting confidential information in a communication system in the presence of an eavesdropper with unbounded computational power . since then, an extensive body of work has been devoted to studying this problem for different network models by deriving fundamental transmission rate limits and designing low - complexity schemes to approach these limits in practice .secure communication using multiple antennas was extensively studied as well , see e.g. , .these works investigated efficient signaling mechanisms using the spatial degrees of freedom provided by multiple antennas to limit an eavesdropper s ability to decode information .the underlying information theoretic problem , the multi - antenna wiretap channel , was studied and the associated secrecy capacity was identified .we note that these works assumed that the eavesdropper s channel state information is available either completely or partially , although such an assumption may not be justified in practice .  as a more pessimistic but stronger assumption , references study secrecy capacity when the eavesdropper channel is arbitrarily varying and its channel states are known to the eavesdropper only .reference studies the single - user gaussian multi - input - multi - output ( mimo ) wiretap channel and characterizes the secrecy degrees of freedom ( s.d.o.f . ) . the same paper extended the single user analysis to the two user gaussian mimo multiple access ( mimo - mac ) channel .this was possible only when all the legitimate terminals have equal number of antennas , leaving the mimo - mac with arbitrary number of antennas at the terminals an open problem .our main contribution is to fully characterize the s.d.o.f .region of the two - transmitter mimo mac channel when the eavesdropper channel is arbitrarily varying .we show that the s.d.o.f .region can be achieved by a scheme that orthogonalizes the transmit signals of the two users at the intended receiver .moreover , it suffices to use a single - user wiretap channel code  and no coordination between the users is necessary except for synchronization and sharing the transmit dimensions . to establish the optimality of this scheme , our converse proof decomposes the mimo mac channel into a set of parallel and independent channels using the generalized singular value decomposition ( gsvd ) .a set of eavesdroppers , each monitoring a subset of links , is selected using an induction procedure and the resulting secrecy constraints are combined to obtain an upper bound on a weighted sum - rate expression .the outer bound matches the achievable rate in terms of the s.d.o.f .region , thus settling the open problem raised in for the case of two transmitters .interestingly , the s.d.o.f .region remains open for this model when the eavesdropper channel is perfectly known to all terminals .a significant body of literature already exists on this problem , see e.g. ,  .if the channel model has real inputs and outputs , gaussian signaling is in general suboptimal and user cooperating strategies as well as signal alignment techniques are necessary  . in  itis established that s.d.o.f . of achievable using real interference alignment for almost all configurations of channel gains .if the channel model has complex inputs and outputs , it is shown in ( * ? ? ?* section 5.16 ) that in general s.d.o.f . of is achievable using asymmetric gaussian signaling .in contrast , the best known upper bound on the s.d.o.f . of individual ratesis for both cases , established in ( * ? ? ?* section 5.5 ) .the remainder of this paper is organized as follows . in section [ sec :model ] , we describe the system model . the main result is stated as theorem [ thm : mainresult ] in section [ sec : mainresult ] .the proof of the theorem is divided into two parts .first , we establish the result for the case of parallel channels in section [ sec : parallel ] .subsequently , in section [ sec : generalmimo ] we establish the result for the general case by decomposing the mimo - mac channel into a set of independent parallel channels .such a reduction is used both in the proof of the converse as well as the coding scheme .section [ sec : conclusion ] concludes the paper .we use the following notation throughout the paper : for a set , and denote the set of random variables and respectively . denotes a non - negative sequence of that converges to when goes to .we use bold upper - case font for matrices and vectors and lower - case font for scalars .the distinction between matrices and vectors will be clear from the context .for a set , denotes its cardinality and a short hand notation is used for the sequence . denotes the empty set ., , . ]  as shown in figure [ fig:2usermac ] , we consider a discrete - time channel model where two transmitters communicate with one receiver in the presence of an eavesdropper .we assume transmitter has antennas , , the legitimate receiver has antennas whereas the eavesdropper has antennas .the channel model is given by  & ( i ) = _ k = 1 ^ 2 _ k _ k ( i ) + ( i ) [ eq : mimomac ] + & ( i ) = _ k = 1 ^ 2 _ k ( i ) _ k ( i ) [ eq : eavesdropperchannel ]  where denotes the time - index , are channel matrices and is the additive gaussian noise observed by the intended receiver , which is composed of independent rotationally invariant complex gaussian random variables with zero mean and unit variance . the sequence of eavesdropper channel matrices is an arbitrary sequence of length and only revealed to the eavesdropper . in contrast , are revealed to both the legitimate parties _ and _ the eavesdropper(s ) .we assume , the number of eavesdropper antennas , is known to the legitimate parties and the eavesdropper .user , , wishes to transmit a confidential message , , to the receiver over channel uses , while both messages , and , must be kept confidential from the eavesdropper .we use to index a specific sequence of over channel uses and use to represent the corresponding channel outputs for .the _ strong _ secrecy constraint is :  _ n i ( w_1 , w_2 ; _ ^n ) = 0 , [ eq : secrecy2 ]  where the convergence must be uniform over .the average power constraints for the two users are given by  _n _ i=1^n |_k(i)|^2 |p_k , k=1,2.[eq : power2 ]  the secrecy rate for user , , is defined as  r_s , k=_n h(w_k),k=1,2 . [ eq : def_r_e_k ]  such that can be reliably decoded by the receiver , and and are satisfied .we define the secrecy degrees of freedom as :  \{(d_1,d_2 ) : d_k=_|p_1=|p_2=|p , k=1,2}[eq : def_sdof_region ], , , ( b ) comparison between achievable s.d.o.f . region and a simple outer bound derived by considering one eavesdropper at a time . ]  before stating the main result , we illustrate the main difficulty in characterizing the s.d.o.f .region through a simple example .as illustrated in figure [ fig : motivatingexample](a ) , in this example , each transmitter has 2 antennas and the intended receiver has 3 antennas , while the eavesdropper has only 1 antenna .let denote the transmitted signals from the two users and denote the signals observed by the intended the receiver . and the main channel is given by  & y_1=x_1+z_1 , y_3=x_4+z_3 + & y_2=x_2+x_3+z_2  where denote additive channel noise .as shown in , a secrecy degree of freedom is achievable for a user if the other user remains silent .time sharing between these two users lead to the following achievable s.d.o.f .region :  d_1+d_2 1 , d_k 0 , k=1,2 [ eq : motivatingexampleachievableregion ]  for the converse , we begin by considering a simple upper bound , which reduces each channel to a single - user mimo wiretap channel . first , by revealing the signals transmitted by user to the intended receiver and assuming that the eavesdropper monitors either or we have that .similarly we argue that .to obtain an upper bound on the sum - rate we let the two transmitters to cooperate and reduce the system to a mimo link . the s.d.o.f . of this channel yields .this outer bound , illustrated in figure [ fig : motivatingexample](b ) , does not match with the achievable region given by .as we shall show in theorem [ thm : mainresult ] , is indeed the s.d.o.f .capacity region and hence a new converse is necessary to prove this result .our key observation is that the above upper bound only considers one eavesdropper at a time in deriving each of the three bounds .for example , when deriving , we assume there is only one eavesdropper which is monitoring either or . when deriving , we assume there is only one eavesdropper which is monitoring either or .similarly when deriving we again assume that there is one eavesdropper on either of the links . as we shall discuss below, a tighter upper bound is possible to find if we consider the simultaneous effect of two eavesdroppers .  in our system model , there are infinitely many possible eavesdroppers , each corresponding to a different channel state sequence .the challenge is to find out a finite number of eavesdroppers , whose joint effect leads to a tight converse .our choice of eavesdroppers is based on the following intuition : when an eavesdropper chooses which links to monitor , it should give precedence to those links over which only one user can transmit .this is because these links are the major contributor to the sum s.d.o.f . since they are dedicated links to a certain user . based on this intuition , we consider the following two eavesdroppers : one monitors for and the other monitors for .as we shall show later in lemma [ lemma : case1_r1r2_bound ] , the first eavesdropper implies the following upper bound on :  n(r_1 - _n ) i ( x_2^n ; y_2^n |y_1^n , x_\ { 3,4 } ^n ) [ eq : r1_motivatingexample ]  and the second eavesdropper implies the following upper bound on :  n(r_2 - _n ) i ( y_1^n , x_\ { 3,4 } ^n ; y_2^n ) [ eq : r2_motivatingexample ]  their joint effect can be captured by adding and , which lead to :  n(r_1 + r_2 - 2_n ) i ( x_2^n , y_1^n , x_\ { 3,4 } ^n ; y_2^n )  since there is only one term , which is , at the right side of the mutual information , we observe the sum s.d.o.f. can not exceed , thereby justifying that is indeed the largest possible s.d.o.f .region for figure [ fig : motivatingexample](a ) .  as captured by and , a simultaneous selection of two different eavesdroppers for the two usersreduces the effective signal dimension at the receiver from three to one , thus leading to a tighter converse .as we shall show later in section [ subsec : converse - parallel - case-2 ] , in generalizing this example we are required to systematically select a sequence of eavesdroppers using an induction procedure .in this section , we state the main result of this work . to express our result , we define as the rank of and as the rank of ].[thm : mainresult ]  : ( a ) , ( b ) , ( c )   fig .[ fig : dofregion ] illustrates the structure of the s.d.o.f .region as a function of the number of eavesdropping antennas . in fig .[ fig : dofregion ] ( a ) we have . in this casethe s.d.o.f .region is a polymatroid ( see e.g. , ( * ? ? ?* definition 3.1 ) ) described by and .[ fig : dofregion ] ( b ) illustrates the shape of the s.d.o.f.region when . in fig .[ fig : dofregion ] ( b ) , without loss of generality , we assume and the s.d.o.f .region is bounded by the lines , and  & ( r_1 + r_2 - r_0)d_1 + ( r_1-n_e)d_2 + & ( r_1-n_e ) ( r_2-n_e).[eq : dof_region_b ]  when , the s.d.o.f.region , as illustrated in fig .[ fig : dofregion ] ( c ) is bounded by and the line   the s.d.o.f .region in theorem [ thm : mainresult ] allows the following simple interpretation : the region can be expressed as a convex hull of a set of rectangles shown by figure [ fig : interpretation ] ( illustrated for figure [ fig : dofregion ] ( a ) ) .each rectangle is parameterized by the dimensions of the subspace occupied by the transmission signals from the two users , denoted by , where indicates the dimension of user , . then in order for the signals from both transmitters to be received reliably by the receiver , we must have  & t_1 + t_2 r_0 [ eq : t_i_constraint ] + & 0 t_i r_i , i = 1,2 [ eq : sum_t_i_constraint ]  each user then transmits confidential messages with ^+ ] , where is the number of degrees of freedom occupied by user .to achieve reliable transmission , we must have and . ], where . ]  in this section , we establish theorem [ thm : mainresult ] for the case of parallel channels . as illustrated in fig .[ fig : seta , b , c ] , the receiver observes the noise random variables across the sub - channels are independent and each is distributed according to and and denote the transmit symbols of user 1 and user 2 respectively .the parallel channel model is a special case of  with \!,~ { { \mathbf h}}_2 \!= \ ! \left [ \begin{array}{*{20}c }   { { { \mathbf o}}}_{|{\mathcal a}| }  & { } & { }  \\   { } & { { { \mathbf i } } _ { |{\mathcal b}| } } & { } \\    { } & { } & { { { \mathbf i}}_{|{\mathcal c}| } }  \end{array } \right]\ ! ,  \label{eq : h1h2}\end{aligned}\ ] ] where , and denote the identity matrices of size , and respectively , and and denote the matrices , all of whose entries are zeros . note that we do not make any assumption on the eavesdropper s channel model .it suffices to establish the achievability of points and in  and  respectively .the rest of the region follows through time - sharing between these points .note that for the proposed parallel channel model ^+ , \left[|{\mathcal c}|- n_e \right]^+\right)\label{eq : p3_parallel}\\ p_4 & = \left ( \left[|{\mathcal a}| - n_e\right]^+ , \left[|{\mathcal b}| + |{\mathcal c}|- n_e \right]^+\right)\label{eq : p4_parallel}\end{aligned}\ ] ]  to prove the achievability of we restrict user to transmit only on the last components of in and allow user to transmit over all of the components of in and .note that in this case , the signals of these two users do not interfere with each other at the intended receiver . from , user can transmit such that ^+ ] and  _n i(w_2 ; _ 2^n _ 2^n)=0[eq : w2issecure ]  where we use to denote the sequence .furthermore since is independent of we have that  _n i(w_1 ; _ 1^n _ 1^n,_2^n _ 2^n)=0[eq : w1issecure_2 ] + _ n i(w_2 ; _ 1^n _ 1^n,_2^n _ 2^n)=0[eq : w2issecure_2 ]  which imply :  & i ( w_1 ; _ 1^n _ 1^n , _ 2^n _2^n|w_2 ) + & i ( w_1 ; w_2 , _ 1^n _ 1^n,_2^n _2^n ) + = & i ( w_1 ; _ 1^n _ 1^n,_2^n _ 2^n ) + i( w_1 ; w_2 |_1^n _ 1^n , _ 2^n _2^n ) + & i ( w_1 ; _ 1^n _ 1^n , _ 2^n _2^n ) + i ( w_1 , _ 1^n _ 1^n;w_2 , _ 2^n _2^n ) + = & i ( w_1 ; _ 1^n _ 1^n )  where the last step follows from the fact that is independent from . therefore implies  _ n i ( w_1 ; _ 1^n _ 1^n , _ 2^n _2^n|w_2 ) = 0.[eq : w1issecuregivenw2 ]  adding and , we obtain  _ n i ( w_1 , w_2 ; _ 1^n _ 1^n , _ 2^n _ 2^n ) = 0[eq : w1issecuregivenw2b ]  and the secrecy constraint  follows from the data - processing inequality . also , since the convergence in in and is uniform , the convergence in and hence in is uniform as well .hence we have proved the point is achievable .the achievability of is proved by repeating the argument above by exchanging user with user .[ remark : independentsecrecyguarantee ] as is evident from , the secrecy guarantee achieved by one user is not affected by the transmission strategy of the other user .   we need to show that the s.d.o.f .region is contained within since  and  directly follow from the single user case in , we only need to show .let be the set of links such that an eavesdropper is monitoring for , . , .we establish the following upper bound on the achievable rate pairs .  &n ( r_s,1-_n ) i ( x_1 , a\e_1^n ; y_a\e_1 ^n ) + i ( x_1,b^n ; y_b^n | m ) [ eq : case1_r1_bound ] + & n ( r_s,2-_n ) i(x_2 , c\e_2^n ; y_c\e_2^n ) + i ( m ; y_b^n ) [ eq : case1_r2_bound ]  where . [ lemma : case1_r1r2_bound ]  the proof is provided in appendix [ app : r1r2_case1 ] .  the proof is completed upon adding and so that  & n(r_s,1+r_s,2 - 2_n ) + & ( x_1,a\e_1^n ; y_1,a\e_1^n)+ i(x_2,c\e_2^n;y_2,c\e_2^n ) + & + i(m , x_1,b^n;y_b^n )  and using  & d ( i ( x_a\e_1^n ; y_a\e_1 ^n ) ) |a|-n_e + & d ( i(x_c\e_2^n ; y_c\e_2^n ) ) |c|-n_e + & d ( i(m , x_1 , b^n ; y_b^n ) ) |b|  where characterizes the pre - log scaling of with respect to .   , , and when , and .( a ) case i , , .( b ) case ii , , , , , , . ]  without loss of generality , we assume . let be the set of links such that an eavesdropper is monitoring for , .let , , and .define the set such that , . since .then theorem [ thm : mainresult ] reduces to and  & |g| d_1 + |f| d_2 |f| |g| [ eq : converse_case2_parallel ]  which we now show .we first introduce the following lemma :  for any choice of and with appropriate cardinalities the rates and are upper bounded by  & n ( r_s,1-_n ) i ( x_1,f^n ; y_f^n | m , x_1 , b\f^n ) [ eq : r1bnd_case2_alt ] + & n ( r_s,2-_n ) i ( m , x_1 , b\g^n ; y_g^n ) [ eq : r2bnd_case2_alt ]  where .[ lemma : r1r2_case2 ]  the proof is provided in appendix [ app : r1r2_case2 ] .  the above bounds can be interpreted as follows ..  suggests that to bound the receiver is revealed the entire user 2 signal as well as ..  suggests that to bound we can restrict the observation of the receiver to .  for the remainder of the proofwe assume without loss of generality that .we fix while choosing different sets of elements : , the sets and a sequence of in the following recursive manner .let , . for construct as follows .1 .  * case i : * + let , where denotes the smallest element in .let , and .this case is illustrated in figure [ fig : macfg](a ) for .case ii : * + let , and , and , where .this case is illustrated in figure [ fig : macfg](b ) for .[ defn : rec ]  to interpret the above construction , we note that the set is a row - vector with elements and let be obtained by concatenating identical copies of the vector i.e. , }_{|{\mathcal f}| \text { copies}}\ ] ] as shown in figure [ fig : macfg ] , by our construction , the vector spans the first elements of , the vector spans the next elements of etc .the constant denotes the index number of copies of the vector necessary to cover .when latexmath:[ ] is , there exists unitary matrices and nonsingular upper triangular matrix such that for , , ,  &_ 1^h ^h_1 = _ 1 ( n_t_1 r_0 ) _( r_0 n_r ) [ eq : h1_decomp ] + & _ 2^h ^h_2 = _ 2 ( n_t_2 r_0 ) _( r_0 n_r ) [ eq : h2_decomp ] + & _ 1 = [ eq : sigma_1 ] + & _ 2 = [ eq : sigma_2 ]  where are identity matrices , are zero matrices , and are diagonal matrices with positive real elements on the diagonal line that satisfy , and . for clarity , the dimension of each matrix is shown in the parenthesis in the subscript . has the same number of columns as . has the same number of columns .however , are not necessarily square matrices and can be empty , i.e. , having zero number of rows .  for convenience in notationwe define and observe that is a square and non - singular matrix . then from theorem [ thm : gsvd ] , we have :  ^h _ t _ t = _ t^h , t = 1,2 .  without loss of generality , we can cancel and and rewrite as : {n_r  \times r_0 } \mathbf{\sigma}_1^h \mathbf{x}_1  \\ + \left [ \begin{array}{l }  \mathbf{a}_{r_0  \times r_0 } ^h  \\  \mathbf{0}_{(n_r - r_0 )\times r_0 } \\\end{array } \right]_{n_r  \times r_0 } \mathbf{\sigma}_2^h \mathbf{x}_2  + \mathbf{z}.\end{gathered}\ ] ] since and are unitary matrices , the components of are independent from each other and the power constraints of each transmitter remains the same as . because the components of are independent, the intended receiver can discard the last components in without affecting the secrecy capacity region of this channel .this means that we only need to consider the case where and rewrite as :  = _r_0 r_0 ^h ( _ 1^h _ 1 + _2^h _ 2 ) + .[ eq : mimomac2 ]   for establishing the converse , we further enhance the channel model in to the following  = _ 1^h _ 1 + _2^h _ 2 + _+ [ eq : mimomac3 ]  where is any sufficiently small constant such that , times the maximal eigenvalue of , is smaller than and is a circularly symmetric unit - variance gaussian noise vector .  to establish , note that we can express where is a gaussian random vector , independent of and with a covariance matrix  _ r_0 r_0 - _ + ^2 _ r_0 r_0^h _ r_0 r_0 [ eq : noisetodiscard ]  which is guaranteed to be positive semi - definite by our choice of . upon substituting  into ,we have we consider an enhanced receiver that is revealed .clearly this additional knowledge can only increase the rate and serves as an upper bound .it is also clear that since is independent of , it suffices to use this information to cancel in  and then discard it .furthermore since the matrix is invertible , upon canceling it , we obtain .we further enhance the receiver by replacing and with and so that the model reduces to  = _ 1^h _ 1 + _ 2^h+ [ eq : mimomac3e ]  where  & _ 1^h = _r_0 n_t_1 [ eq : barsigma_1 ] + & _ 2^h = _ r_0 n_t_2 [ eq : barsigma_2 ]  are obtained by replacing each diagonal by the identity matrix .the model  can only have a higher capacity , since each diagonal entry in is between .we observe that in the resulting channel model is identical to -  & |a|= r_0 - r_2 [ eq : ca ] + & |b|= s = r_1+r_2-r_0 [ eq : cb ] + & |c|= r_0 - r_1 [ eq : cc ]  except that the noise variance is reduced by a factor of . since a fixed scaling in the noise power does not affect the secure - degrees of freedom , an outer bound on the s.d.o.f .for the parallel channel model - with , and defined via , continues to be an outer bound on the s.d.o.f .region for the general mimo - mac channel .  substituting - in the upper bounds in section [ subsec : converse - parallel - case-1 ] , [ subsec : converse - parallel - case-2 ] and [ subsec : converse - parallel - case-3 ] we establish the converse in theorem [ thm : mainresult ] .   to establish the achievability for the general mimo case we further use a suitable degradation mechanism to reduce the model  to  = _ 1^h _ 1 + _ 2^h_ 2 + [ eq : mimomac5 ]  where is any sufficiently large constant such that , times the minimum eigenvalue of , is greater than and is a circularly symmetric unit - variance gaussian noise vector . since is non - singular we are guaranteed that all the singular values of are non - zero and hence a exists .  to establish ,let be a gaussian noise vector with covariance  ^2 _r_0 r_0^h _r_0 r_0 [ eq : noisetoadd ]  independent of and consider a degraded version of  which can be simulated at the receiver by adding additional noise to its output .since , we can express .substituting into  and canceling the non - singular matrix , we arrive at .let denote the minimum element on the diagonals of and in  and  respectively . by appropriately scaling down the transmit powers on each of the sub - channels we can further reduce  to where are defined in  and  respectively .the model  is identical to the parallel channel model - with the size of sets , and in - and with a noise power that is larger by a factor of .since a constant factor in the noise power does not affect the secrecy degrees of freedom , the coding schemes described in section [ subsec : achievable - parallel ] achieves the lower bound in theorem [ thm : mainresult ] .in this work we have studied the two - transmitter gaussian complex mimo - mac wiretap channel where the eavesdropper channel is arbitrarily varying and its state is known to the eavesdropper only , and the main channel is static and its state is known to all nodes .we have completely characterized the s.d.o.f .region for this channel for all possible antenna configurations .we have proved that this s.d.o.f .region can be achieved by a scheme that orthogonalizes the transmit signals of the two users at the intended receiver , in which each user achieves secrecy guarantee independently without cooperation from the other user .the converse was proved by carefully changing the set of signals available to the eavesdropper through an induction procedure in order to obtain an upper bound on a weighted - sum - rate expression .  as suggested by this work, the optimal strategy for a communication network where the eavesdropper channel is arbitrarily varying can potentially be very different from the case where the eavesdropper channel is fixed and its state is known to all terminals .this is also observed for example in the mimo broadcast channel and the two - way channel .characterizing secure transmission limits for a broader class of communication models with this assumption is hence important and is left as future work .for , from fano s inequality , we have  n(r_s,1-_n ) & i(w_1 ; y_abc^n)- i(w_1 ; y_e_1^n ) + & i ( w_1 ; y_abc^n |y_e_1 ^n ) + & i ( w_1 ; y_abc^n , x_2,bc^n |y_e_1 ^n ) + & = i ( w_1 ; y_ab^n , x_2,bc^n |y_e_1 ^n ) [ eq : beforemovex2c ]  where the last step  relies on the fact that the additive noise at each receiver end of each sub - channel in figure [ fig : seta , b , c ] is independent from each other and hence holds . since is independent from , and , can be written as :  & i ( w_1 ; y_ab^n |y_e_1 ^n , x_2 , bc^n ) + = & i ( w_1 ; y _( a\e_1 ) b^n |y_e_1 ^n , x_2,bc^n ) + = & i ( w_1 ; y_a\e_1 ^n |y_e_1 ^n , x_2,bc^n ) + i ( w_1 ; y_b^n |y_a^n , x_2,bc^n ) [ eq : r1_case1_split ]  where the last step  follows from the fact and hence .we separately bound each of the two terms above .& i ( w_1 ; y_a\e_1 ^n |y_e_1 ^n , x_2,bc^n ) + & i ( w_1 , y_e_1 ^n , x_2,bc^n ; y_a\e_1 ^n ) + & i ( w_1 , y_e_1 ^n , x_2,bc^n , x_1 , a\e_1^n ; y_a\e_1 ^n ) + & = i ( x_1 , a\e_1^n ; y_a\e_1 ^n ) [ eq : r1_case1_split_part1 ]  where the last step follows from the markov chain relation , we upper bound the second term in  as follows  & i ( w_1 ; y_b^n |y_a^n , x_2,bc^n ) + & i ( x^n_1,ab ; y_b^n |y_a^n , x_2,bc^n ) [ eq : r1_case1_w1_markov ] + & = i ( x^n_1,b ; y_b^n |y_a^n , x_2,bc^n ) + & + i ( x^n_1,a ; y_b^n |y_a^n , x_2,bc^n , x_1,b^n ) [ eq : r1_case1_y_markov ] + & = i ( x^n_1,b ; y_b^n |y_a^n , x_2,bc^n ) [ eq : r1_case1_split_part2 ]  where we use the markov relation in step and  follows from the fact markov relation note that  follows upon substituting  and  into .  for , from fano s inequality and the secrecy constraint , we have :  & n(r_s,2-_n ) i(w_2 ; y_abc^n)-i(w_2 ; x_2,e_2 ^n ) + & i ( w_2 ; y_abc^n |x_2,e_2 ^n ) + = & i ( w_2 ; y_bc^n |y_a^n , x_2,e_2 ^n ) [ eq : case1_r2_ya_markov ] + = & i ( w_2 ; y_(c\e_2 ) b^n |y_a^n , x_2,e_2 ^n ) [ eq : case1_r2_ye_markov ] + = & i ( w_2 ; y_c\e_2 ^n |y_a^n , x_2,e_2 ^n ) + i ( w_2 ; y_b^n |y_a^n , x_2,e_2 ^n , y_c\e_2 ^n ) [ eq : r2_case1_split ]  where  follows from the fact that is independent of and  follows from the fact that holds .we separately bound each term in .& i ( w_2 ; y_c\e_2 ^n |y_a^n , x_2,e_2 ^n ) i ( w_2 , y_a^n , x_2,e_2 ^n ; y_c\e_2 ^n ) + & i ( x_2,c\e_2^n , w_2 , y_a^n , x_2,e_2 ^n ; y_c\e_2 ^n ) + & = i ( x_2,c\e_2^n ; y_c\e_2 ^n ) , [ eq : r2_case1_split_part1 ]  where the justification for establishing  is identical to  and hence omitted .we finally bound the second term in .& i ( w_2 ; y_b^n |y_a^n , x_2,e_2 ^n , y_c\e_2 ^n ) + & i ( x_2,bc^n ; y_b^n |y_a^n , x_2,e_2 ^n , y_c\e_2 ^n ) + & i ( y_a^n , x_2,bc^n , x_2,e_2 ^n , y_c\e_2 ^n ; y_b^n ) + = & i ( y_a^n , x_2,bc^n , x_2,e_2 ^n ; y_b^n ) + & + i ( y_c\e_2 ^n ; y_b^n |y_a^n , x_2,bc^n , x_2,e_2 ^n ) + = & i ( y_a^n , x_2,bc^n ; y_b^n ) [ eq : r2_case1_split_part2 ]  where the justification for arriving at  is similar to  and hence omitted .  substituting  and  into  weestablish .assume the eavesdropper monitors and for .then for , from fano s inequality , we have :  & n(r_s,1-_n ) + & i ( w_1 ; y_abc^n ) - i(w_1 ; y_a^n , x_1,e_1 \a^n ) + & i ( w_1 ; y_abc^n |y_a^n , x_1,e_1 \a^n ) + = & i ( w_1 ; y _ bc^n |y_a^n , x_1,e_1 \a^n ) + & i ( w_1 ; y_bc^n , x_2,bc^n |y_a^n , x_1,e_1 \a^n ) + = & i ( w_1 ; y_bc^n |y_a^n , x_1,e_1 \a^n , x_2,bc^n ) [ eq : r1_case2_x2bc_indep ] + = & i ( w_1 ; y_f^n |y_a^n , x_1,e_1 \a^n , x_2,bc^n ) [ eq : r1_case2_f_markov ] + = & i ( w_1 ; y_f^n |y_a^n , x_1,b\f^n , x_2,bc^n ) [ eq : r1_case2_step1 ]  where  follows from the fact that is independent of . while  follows from the fact that since the noise across the channels is independent the markov condition holds and furthermore we have defined .& i ( w_1 ; y_f^n |y_a^n , x_1,b\f^n , x_2,bc^n ) + & i ( x_1 , ab^n ; y_f^n |y_a^n , x_1,b\f^n , x_2,bc^n ) + = & i ( x_1 , f^n ; y_f^n |y_a^n , x_1,b\f^n , x_2,bc^n ) + & + i ( x_1 , ab\f^n ; y_f^n |y_a^n , x_1,b^n , x_2,bc^n ) [ eq : r1_case2_yf_markov ] + = & i ( x_1 , f^n ; y_f^n |y_a^n , x_1,b\f^n , x_2,bc^n ) [ eq : r1_case2_step2 ]    & n(r_s,2-_n ) i ( w_2 ; y_abc^n)-i(w_2 ; x_2,e_2^n ) + & i ( w_2 ; y_abc^n |x_2,e_2^n ) + & i ( w_2 ; y_abc^n , x_1 , e_2 b^n |x_2,e_2^n ) + = & i ( w_2 ; y_bc^n |x_2,e_2^n , y_a^n , x_1 , e_2 b^n ) [ eq : r2_case2_tx1_indep ] + & i ( x_2,bc^n ; y _ bc^n |x_2,e_2^n , y_a^n , x_1 , e_2 b^n ) + = & i ( x_2,bc^n ; y _ ge_2^n |x_2,e_2^n , y_a^n , x_1 , e_2 b^n ) [ eq : r2_case2_eg_union ] + = & i ( x_2,bc^n ; y _ g^n |x_2,e_2^n , y_a^n , x_1 , e_2 b^n ) + & + i ( x_2,bc^n ; y _e_2 ^n |x_2,e_2^n ,y_a g^n , x_1 , e_2 b^n ) [ eq : r2_case2_tx1_ye_markov_split ] + = & i ( x_2,bc^n ; y _ g^n |x_2,e_2^n , y_a^n , x_1 , e_2 b^n ) [ eq : r2_case2_tx1_ye_markov ] + & i ( x_2,bc^n , y_a^n , x_1 , e_2 b^n ; y _g^n ) + & i ( m , x_1,b\g^n ; y _g^n )  where  follows from the fact that are the transmitted signals from user 1 and independent of and  follows from the fact that and and hence holds . eq.  follows from the fact that since the noise on each channel is markov , we have and hence the second term in is zero .a. thangaraj , s. dihidar , a. r. calderbank , s. mclaughlin , and j. merolla , `` on the application of ldpc codes to a novel wiretap channel inspired by quantum key distribution , '' _ ieee transactions on information theory _ ,53 , no . 8 , pp . 29332945 , august 2005 .r. liu , t. liu , and h. v. poor , `` multiple - input multiple - output gaussian broadcast channels with confidential messages , '' _ ieee transactions on information theory _ , vol .56 , no . 9 , pp .42154227 , september 2010 .e. molavianjazi , `` secure communication over arbitrarily varying wiretap channels , '' _ master thesis _ ,december 2009 , available online at http://etd.nd.edu/etd-db/theses/available/etd-12112009-112419/unrestricted/molavianjazie122009.pdf .x. he and a. yener , `` mimo wiretap channels with arbitrarily varying eavesdropper channel states , '' submitted to the ieee transactions on information theory , july , 2010 , in revision , available online at http://arxiv.org/abs/1007.4801 .e. tekin and a. yener , `` the general gaussian multiple access and two - way wire - tap channels : achievable rates and cooperative jamming , '' _ ieee transactions on information theory _ , vol .54 , no . 6 , pp . 27352751 , june 2008 .     x. he and a. yener , `` providing secrecy with structured codes : tools and applications to gaussian two - user channels , '' submitted to ieee transactions on information theory , july , 2009 , in revision , available online at http://arxiv.org/abs/0907.5388 .x. he , `` cooperation and information theoretic security in wireless networks , '' _ phd dissertation _ , august 2009 , available online at http://etda.libraries.psu.edu/theses/approved/worldwideindex/etd-5342/index.html .d. n. c. tse and s. v. hanly , `` multiaccess fading channels .i. polymatroid structure , optimal resource allocation and throughput capacities , '' _ ieee transactions on information theory _44 , no . 7 , pp .27962815 , 1998 .

Abstract: a two - transmitter gaussian multiple access wiretap channel with multiple antennas at each of the nodes is investigated . the channel matrices at the legitimate terminals are fixed and revealed to all the terminals , whereas the channel matrix of the eavesdropper is arbitrarily varying and only known to the eavesdropper . the secrecy degrees of freedom ( s.d.o.f . ) region under a strong secrecy constraint is characterized . a transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver and uses a single - user wiretap code is shown to be sufficient to achieve the s.d.o.f .  region . the converse involves establishing an upper bound on a weighted - sum - rate expression . this is accomplished by using induction , where at each step one combines the secrecy and multiple - access constraints associated with an adversary eavesdropping a carefully selected group of sub - channels .


_ bayesian inference _ can be seen both as a sub - field of information theory and of general statistical inference .a typical problem in this field is : given observed noisy data and known statistical model of a noisy communication channel ( transition probability ) , as well as a _prior distribution _ for the input ( a pre - image ) , find the most likely pre - image , or compute the _ a posteriori _ marginal probability for some part of the pre - image .this field is also deeply related to _combinatorial optimization _ , which is a branch of optimization in computer science , related to operations research , algorithm theory and complexity theory .a typical problem in _ combinatorial optimization _ is : solve , approximate or count ( exactly or approximately ) instances of problems by exploring the exponentially large space of solutions . in many emerging applications ( in magnetic and optical recording , micro - fabrication , chip design , computer vision , network routing and logistics ) ,the data are structured in a two - dimensional grid ( array ) .moreover , data associated with an element of the grid are often binary and correlations imposed by the problem are local , so that only nearest neighbors on the grid are correlated .such problems are typically stated in terms of binary statistical models on planar graphs .  in this paper , we discuss a generic problem of bayesian inference defined on a planar graph .we focus on the problem of weighted counting , or ( from the perspective of statistical physics ) we aim to calculate the partition function of an underlying statistical model .as the seminal work of onsager on the two - dimensional ising model and its combinatorial interpretation by kac and ward have shown , the planarity constraint dramatically simplifies statistical calculations .by contrast , three - dimensional statistical models are much more challenging , and no exact results are known .building on the work of physicists , specifically on results of fisher and kasteleyn , barahona has shown that calculating the partition function of the spin glass ising model on an arbitrary planar graph is _ easy _ , as the number of operations required to evaluate the partition function scales algebraically , , with the size of the system . to prove this ,the partition function of the spin - glass ising model was reduced to a dimer model on an auxiliary graph , and the partition function was expressed as the pfaffian of a skew - symmetric matrix defined on the graph .the polynomial algorithm was later used in simulations of spin glasses .however , barahona also added a grain of salt to the exciting positive result , showing that generic planar binary problem is difficult . specifically , evaluating two - dimensional spin glass ising model in a magnetic field is np - hard , i.e. it is a task of likely exponential complexity .  when an exact computational algorithm of polynomial complexity is not available , efficient approximations become relevant .typically , the approximation is built around a tractable case .one such approximate algorithm built around the fisher - kasteleyn pfaffian formula was recently suggested by globerson and jaakkola in .although this approximation ( coined planar - graph decomposition " ) gives a provable upper bound for the partition function for some special graphical models , it constitutes just heuristics , i.e. it suffers from lack of error - control and the inability of gradual error - reduction .controlling errors in approximate evaluations of the partition function of a graphical model is generally difficult .however , one recent approach , developed by two of us and called loop calculus , offers a new method . loop calculus allows to express explicitly the partition function of a general statistical inference problem via an expansion ( the loop series ) , where each term is explicitly expressed via a solution of the belief propagation , or bethe - peierls ( bp ) equations .this brought new significance to the bp concept , which previously was seen as just heuristics .the bp equations are tractable for any graph ; generally , the number of terms in the loop series is exponentially large , so direct re - summation is not feasible .however , since any individual term in the series can be evaluated explicitly ( once the bp solution is known ) , the loop series representation offers a possibility for correcting the bare bp approximation perturbatively , accounting for loop contributions one after another sequentially .this scheme was shown to work well in improving bp decoding of low - density parity check codes in the error - floor regime , where the number of important loop contributions to the loop series is ( experimentally ) small , and the most important loop contributions ( comparable by absolute value to the bare bp one ) have a simple , single - connected structure . in spite of this progress ,the question remained : what to do with other truly difficult cases when the number of important loop corrections is not small , and when the important corrections are not necessarily single - connected ? in general, we still do not know how to answer these questions , while a partial answer for the important class of planar models is provided in this paper .   in this manuscriptwe show that , for any graph ( planar or not ) , the partial sum of the loop series over single - connected loops reduces to evaluation of the full partition function of an auxiliary dimer - matching model on an extended , regular degree-3 graph .weights of dimers calculated on the extended graph are expressed explicitly via solution of the respective bp equations .the dimer weights can be positive or negative .in general , summing the single - connected partition is not tractable .however , in the planar case , it reduces ( through manipulations reminiscent of the fisher - kasteleyn transformations ) to a pfaffian defined on the extended graph , which is also planar by construction .thus , we find a big class of planar graphical models which are computationally tractable by reduction ( via a bp / gauge transformation ) to a loop series including only single - connected loops , and summable into a pfaffian .moreover , we find that the partition function of the entire loop series is generally reducible to a weighted pfaffians series , where each higher - order pfaffian is associated with a sum of dimer configurations on a modified subgraph of the original graph .each term in the pfaffian series is computationally tractable via the belief propagation solution on the original graph .the material in the manuscript is organized as follows .a formal definition of the model is given in section [ subsec : model ] and a brief description of loop calculus forms section [ subsec : loopcalc ] .some introductory material on the graphical transformations is also given in [ sec : graph ] .section [ sec : singleloops ] is devoted to re - summation of the single - connected loops in the loop series ( we called it single connected partition ) . section [ subsec : todimer ] introduces graphical transformation from the original graph to the extended graph , reminiscent the fisher transformation .this allows to restate the single - connected loop partition of the loop series on the original graph in terms of a sum over dimer configurations on the extended graph .subsection [ subsec : pfaff ] adapts the kasteleyn transformation to our case , thus expressing the partition function of the single - connected series as a pfaffian of a matrix defined on the extended graph .section [ sec : easy ] describes a set of graphical models reducible under belief propagation gauge ( transformation ) to a loop series which is computationally tractable .section [ sec : full ] describes the representation of the loop series for planar graphs in terms of the pfaffian series , where each pfaffian sums dimer matchings on a graph extended from a subgraph of , with the later correspondent to exclusion of an even set of vertices from .grassmann representations , as well as fermionic models are discussed in section [ sec : grass ] : a general set of grassmann models on super - spaces is given in section [ subsec : grassgen ] , while section [ subsec : comments - fermions ] addresses the relation between binary models and integrable hierarchies .a brief list of future research topics is given in section [ sec : con ] .we introduce an undirected graph consisting of vertices and edges .this study focuses mainly on planar graphs , like those emerging in communication or logistics networks connecting or relating nearest neighbors on a 2d mesh or terrain . however , the material discussed in the present and the following subsections is general , and applies to any graph , planar or not .a binary variable , , which we will also be calling a spin , is associated with any edge .the graphical model is defined in terms of the probability function for a spin configuration . in ( [ p_sigma ] ) , is the vector built from all edge variables associated with the given vertex . s are positive and otherwise we will assume no restrictions on the factor functions . is the normalization factor , the so - called partition function of the graphical model .we refer to ( [ p_sigma ] ) as vertex - function " models , according to statistical physics notation . in the information theory ,they are known as forney - style graphical models .we will assume in the following that the degree of connectivity of any vertex in the graph is three .note that this is not a restrictive condition , as the -th order vertices , correspondent to -spin interactions with , can always be represented in terms of a product of triplet terms .then the -th degree vertex can be transformed into a planar graph consisting of degree three vertices .we discuss transformations to the triplets , in general but also on some examples ( ising model and parity check decoding of a linear code ) , in [ sec : graph ] .loop calculus gives an explicit expression for through the loop series : where can be any allowed generalized loop on the graph , i.e. is a subgraph of which does not contain any vertices of degree one ; is a set of vertices of graph which are also contained in the generalized loop ( by construction consists of two or three elements ) ; and and are beliefs associated with vertex and edge .the beliefs are defined via message variables  solving the following system of the belief propagation ( bp ) equations the bare ( bp ) partition function in eq .( [ zseries ] ) has the following expression in terms of the message variables : }.\label{z0}\ ] ]  bp equations ( [ bb ] ) are interpreted as conditions on the gauge transformations , leaving the partition function of the model invariant .these equations may allow multiple solutions , related to each other via respective gauge transformations .the multiple solutions correspond to multiple extrema of the bethe free energy and loop series can be constructed around any of the bp solutions .in the following we will show how to re - sum a part of the loop series accounting for all the single - connected loops , i.e. subgraphs of with all vertices of degree two where stands for the number of neighbors of within .the evaluation will consist of the following two steps :  * show that is equal to the partition function of the dimer - matching model on an auxiliary graph , .the graph will be constructed from the original by a transformation reminiscent of the fisher s trick , introduced in to streamline reduction of ising model to the dimer - matching model ; * use the pfaffian formula of kasteleyn to reduce to a pfaffian of a skew - symmetric matrix defined on .note that complexity of the pfaffian evaluation is , where is the size of .note : while a ) is valid for any graphical model , b ) applies only to the planar case .    to respective three - vertex of the extended graph .right panel : maps from the colorings of a vertex of to coloring of the respective -vertex of .notice , that the coloring of the external edges of are reversed in comparison with the coloring of original edges on .,title="fig:",width=226 ]  to respective three - vertex of the extended graph .right panel : maps from the colorings of a vertex of to coloring of the respective -vertex of .notice , that the coloring of the external edges of are reversed in comparison with the coloring of original edges on .,title="fig:",width=226 ]  following the construction of fisher , we expand each vertex of into a three - vertex of the extended graph , according to the scheme shown in the left panel of figure 1 . consider a vertex of and assume that are three neighbors of on . for each vertex , there are three contributions of degree two within a generalized loop , i.e. with , which can possibly contribute to the single - connected partition : .we associate the three weights with internal edges of the respective three - vertex of , while the weights of all the external edges of the three - vertex are equal to unity .then any coloring of the original graph , marking a single connected loop of , is in the one - to - one correspondence to a dimer - matching ( which we also call coloring ) of .the weights and coloring assignments are illustrated on an example at the left panel of figure 1 .an example of transformation mapping a single - connected - loop on respective dimer on is shown in figure [ trans_example ] .this map from the single - connected loops to dimers leads to the following representation for the single - connected partition  where the dimer - weights on are defined according to the simple rules explained in the previous paragraph .one finds that the right hand side of ( [ zsd ] ) is nothing but the partition function of a dimer - matching problem on .( upper left ) to ( lower right ) map .single connected loop of ( shown in red ) is in one - to - one correspondence with a valid dimer matching of , where dimers are also shown in red.,width=226 ]    orientation ( left panel ) and respective dimer ( matching ) configurations ( right panel ) correspondent to example of described by eq .( [ pfaf - eq]).,title="fig:",width=226 ]  orientation ( left panel ) and respective dimer ( matching ) configurations ( right panel ) correspondent to example of described by eq .( [ pfaf - eq]).,title="fig:",width=226 ]  kasteleyn has shown in ( see also ) that is equal to a pfaffian ( the square root of determinant ) of a skew - symmetric matrix of size , where is the number of vertices in .each element of the matrix with ( ordering is arbitrary , but it is fixed once and forever ) is , where .there are many possible choices of which guarantee the pfaffian relation : .a simple constructive way of choosing such a valid is to relate it to orientation of edges in a directed version of , built according to the following odd - face " rule : number of clockwise - oriented segments of any internal face of should be negative .example of a valid orientation is shown in figure [ trans_example ] and the respective expressions are   since calculating the determinant requires operations , one finds that re - summation of all the single - connected loops in the loop series expression for the partition function of a planar graphical model can be done efficiently in steps .in the case of a general vertex - function graphical model , the bp - gauge transformations , described by the set of bp equations ( [ bb ] ) , result in exact cancelation in the loop series of all the subgraphs containing at least one vertex of degree one within the subgraph .thus , for the graph with all vertices of degree three , any vertex contributing a generalized loop ( subgraph ) should be of degree two or three within the subgraph . as shown in the previous section ,if one ignores generalized loops with vertices of degree three and the original graph is planar , the resulting sub - series ( single - connected partition ) is computationally tractable , i.e. the number of operations required to evaluate the single - connected partition is cubic in the system size ( not exponential ! ) .  in this sectionwe discuss the class of planar models whose loop series do not contain any generalized loops with vertices of degree three . according to section [ sec : singleloops ] ,these models are tractable .indeed , it is known that bp eqs .( [ bb ] ) have at least one solution for the set of messages on any graph and for any factor functions defined on the vertices of the graph .the aforementioned requirement for the generalized loop not to contain any vertex of degree three translates into the following set of additional equations considered together , the set of eqs .( [ bb],[b3 ] ) is overdefined , i.e. it can not be solved in terms of variables for any values of the factor functions . however , if one allows flexibility in the factor functions , and , in fact , considers eqs .( [ bb],[b3 ] ) as a set of conditions on both the messages and the factor functions , one arrives at a big set of possible solutions.  therefore , eqs .( [ bb],[b3 ] ) define a big set of models reducible via bp transformations to a tractable loop series consisting only of single connected loops .moreover , the relations we established may be reversed .one may start from an arbitrary loop series consisting of only single connected loops , apply an arbitrary gauge transformation leaving the loop series invariant ( these transformations are not necessarily of bp type ) , and arrive at a graphical model with some set of factor functions . at first sight, the resulting graphical model might not look tractable , but it actually is , by construction .let us notice that the general planar problem ( e.g. spin glass in a magnetic field ) is np - hard , and it is thus not surprising that full re - summation does not allow expression in terms of a single pfaffian ( or a determinant ) .on the other hand , we already found that a part of the loop series , specifically its single - connected partition , reduces to a computationally tractable pfaffian .this suggests to represent the full loop series as a sum over terms , each representing a set of triplets ( fully colored vertices of degree tree on ) : where is either the empty set or any set of even nodes on ; are the weights from eq .( [ zseries ] ) associated with the triplet , such that ; and is the sum over all generalized loops ( proper loop series colorings , i.e. subgraphs ) of such that all nodes of are fully colored ( all edges adjusted to the nodes belong to the generalized loop ) , while any other vertices of are not colored or only partially colored .thus , the first term in eq .( [ ser_trip ] ) , where is the empty set , represents the single - connected partition , .we show here that not only the first term in eq .( [ ser_trip ] ) , associated with , but any term in eq .( [ ser_trip ] ) is computationally tractable , being equal to a pfaffian of a matrix defined on .indeed , it is straightforward to verify that the generalized loops associated with the given set of triplets ( fully colored vertices ) from the set are in one - to - one correspondence with the set of dimer matchings on , which is a subgraph of with all -vertices correspondent to , and external edges connected to the vertices , completely removed .notice that some vertices of are of degree two .( these are vertices neighboring the removed triplets of . )  an example of a construction is given in figure [ fig : psi ] .one associates weights to the edges of in exactly the same way as for the single - connected partition : the weights of all the external edges of -vertices of are equal to unity , while the internal edges are associated with the respective values , defined in eq .( [ mu_ab_a ] ) .correspondent to the same configuration of triplets , , and their respective dimer configurations on ( shown on the bottom).,title="fig:",width=226 ]  correspondent to the same configuration of triplets , , and their respective dimer configurations on ( shown on the bottom).,title="fig:",width=226 ]  for any of one constructs the skew - symmetric matrix according to the kasteleyn rule for the dimer - matching model described in section [ subsec : todimer ] . as before ,the dimensionality of the matrix is and each element of the matrix is the product of the respective dimer weight and orientation sign .notice that the choice of signs for the elements of depends on the set of excluded " triplets , and thus is not simply a minor of the original matrix , the one corresponding to the single - connected partition ( without exclusion ) .thus , eqs .( [ ser_trip],[zdm2 ] ) describe the pfaffian series representation for the loop series of the planar problem .any pfaffian in eq.([zdm2 ] ) allows a compact representation in terms of grassmann variables .indeed , let us associate a grassmann ( anti - commuting or fermionic ) variable with each vertex of .the grassmann variables satisfy and commute with ordinary -numbers .one also introduces the berezin integration rules over the grassmann variables this translates into the following rule of gaussian integration over the grassmann variables : where is the vector of the grassmann variables over the entire graph , and is an arbitrary skew - symmetric matrix on the graph .for example , applying this formula to the first term of the pfaffian series ( [ ser_trip ] ) one derives in general , any term in the pfaffian series of eq .( [ ser_trip ] ) can be represented as a gaussian grassmann integrable , however with different gaussian kernels , not reducible simply to minors of .   in this subsectionwe first consider graphical models on spaces generalizing the -point ( binary ) set to super - spaces containing commuting and anti - commuting parts .the models will be defined on arbitrary ( non necessarily planar ) graphs .then , we return to the simple example ( [ grassmann0 ] ) of pure dimer model with the grassmann ( anticommuting ) variables defined on vertices of , to see that the model can be restated as the vertex - function grassmann model on the original graph .  the general class of vertex - function models can be introduced as follows . for our graph , , consider a set of spaces , i.e. , we associate a space with any edge , , together with a vertex , , that belongs to the edge . for simplicitywe assume the spaces to be identical , i.e. , for all .the basic variables are .we also introduce the notation ( all products below are cartesian ) note that any is a two - component cartesian product .the vertex - function model is determined by a set of vertex functions defined on and a set of integration measures on .the model partition function is for the particular case when measures have supports restricted to the diagonals , i.e. , we can consider the basic variables that belong to the diagonals .this corresponds to a more conventional formulation of the vertex - function models with the variables residing on edges .note that the models introduced allow for loop - tower calculus , formulated in terms of fixing a proper gauge . the bp gauge fixing for a general vertex - function model described by eq .( [ define - z - continuous ] ) is nothing more than choosing basis sets in the vector spaces ( maybe infinite - dimensional ) of functions in . a standard binary model , defined in eq .( [ p_sigma ] ) , corresponds to the choice of the basic space to be a -point set .vertex models with -ary alphabet , e.g. discussed in , are described by .continuous models are obtained if is chosen to be a manifold of dimension .the continuous case can be extended to the choice of to be a supermanifold of dimension that contains grassmann ( anticommuting ) coordinates and whose substrate is an -dimensional manifold .note that a manifold can be considered as a supermanifold with zero odd dimension . in the remainder of this subsection we will be dealing with an opposite case of the zero even dimension , specifically with the purely grassmann case of the supermanifold .( [ grassmann0 ] ) is the partition function of a model stated in terms of grassmann variables defined on the vertices of .the extended graph is constructed from the original graph so that a vertex of extends into a triangle with three vertices of degree three ( see the left panel of figure [ trans ] ) .therefore , the three grassmann variables in ( [ grassmann0 ] ) are associated with a vertex of . then, eq . ( [ grassmann0 ] ) defined on allows an obvious reformulation in the vertex - function form ( [ define - z - continuous ] ) on , where represents the three grassmann variables that reside on the vertices of , obtained by expanding the vertex of the original graph .the dimer weights for the three edges of associated with the extended vertex of are encoded in the gaussian function .the dimer weight associated with an edge of that represents and edge of the original graph is encoded in the integration measure .also notice that the vertex - function grassmann model on a planar graph can be restated as a model on the triangulated graph , dual to , with complex fermion ( grassmann ) variables associated with the edges of the dual graph and functions associated with a face ( elementary triangle ) of the dual graph ( figure [ fig : triang ] illustrates the duality transformation ) .one interesting conclusion here is that the sequence of transformations discussed above leads us from a special binary model on a planar graph to a gaussian fermion ( grassmann ) model on the dual graph , thus representing an instance of the disorder operator approach of kadanoff - ceva developed originally for the ising model on a square lattice .   a mapping of a classical inference problem onto finding an expectation value in a corresponding quantum model takes on a natural interpretation as a _quantum _ algorithm .this can be tried by using the theory of the infinite kadomtsev - petviashvilii ( kp ) hierarchy , specifically its fermionic formulation .consider lattice fermions with and introduce the population and shift operators .let denote the standard many - particle vacuum state where all single - fermion orbitals with are occupied , and is some uncorrelated ( i.e. represented by a single slater determinant ) many - particle state , which is sufficiently close to . introducing , , and we consider an expectation value the approach is based on mapping the partition function of a classical inference problem on a graph onto a calculation of an expectation value represented by eq .( [ define - z ] ) .we have established such a mapping for some simple grassmannian models on planar graphs , where all the details on the suggested approach will be presented .note that in the case and the expectation value is related to the -function of the kp integrable hierarchy .we conclude with a brief and incomplete discussion of future challenges and opportunities raised by this study .* we plan to extend the study looking at new approximate schemes for intractable planar problems . one new direction , suggested in section [ sec : easy ] , consists of exploring the vicinity of the computationally tractable models reducible via the bp - gauge transformation to the series of single - connected loops .it is also of great interest to explore the vicinity of integrable tractable models mentioned in [ subsec : comments - fermions ] . * perturbative exploration of a larger set of intractable non - planar problems which are close , in some sense , to planar problems , constitutes another interesting extension of the research .here , one would aim to blend the aforementioned planar techniques with planar ( or similar ) decomposition techniques , e.g. these of the type discussed in .* one important component of our analysis consisted in the pfaffian re - summation of the single - connected loop ( dimer ) contributions , which is a special feature of the graph planarity . on the other hand, it is known that the planarity is equivalent to the graph being minor - excluded with respect to and subgraphs .therefore , one wonders if there exists a generalization of the pfaffian reduction to partition functions of models from other and/or broader graph - minor classes defined within the graph - minor theory ?likewise , comparing with previous studies of the non - planar / non - spherical cases , based on the dimer approach . * extending the loop series analysis of the binary planar problem to the q - ary case seems feasible via the loop tower construction of .this research should be of a special interest in the context of recently proposed polynomial quantum algorithm for calculating partition function of the potts model .besides , recent progress shows that a kasteleyn - type approach is extendable to a -ary case , leading to the concept of heaps of dimers " , and ( in the continuum limit ) to fascinating connections with special , highly symmetric complex surfaces , known as harnack curves .* one would also be interested to study how ( and if ) phase transitions in the disorder - averaged planar ensembles , e.g. analyzed in , are related to distribution of parameters characterizing computational tractability ( complexity ) of the models . * in , the problem of finding all pseudo - codewords in a finite cycle code ( corresponding to the type of graphical model discussed in this paper ) , was addressed by constructing a generating function known as graph zeta function . the interesting fact discovered in that this generating function of pseudo - codewords has a determinant formulation , based on a discrete graph operator .hence , one may anticipate an existence of yet uncovered relation between the graph zeta function and a pfaffian - loop resummation of related graphical models .research of m.c . and r.t .was carried out under the auspices of the national nuclear security administration of the u.s .department of energy at los alamos national laboratory under contract no. de c52 - 06na25396 , and specifically the ldrd directed research grant on _ physics of algorithms_. m.c . also acknowledges support of the weston visiting professorship program at the weizmann institute of science , where he started to work on the manuscript .v.y.c . acknowledges support through the start - up funds from wayne state university .in this appendix we discuss graphical transformations reducing any binary problem to the vertex - function model described by eq .( [ p_sigma ] ) , where all vertices are of degree three .our main focus here is on the planar graphs , and on the graphical transformations preserving planarity .however some of the transformations and considerations discussed below apply to an arbitrary graph .  , to set of variables , on respective edges ., width=226 ]  often the original binary model is not represented in the vertex - function form .some or all binary variables describing a problem may actually be assigned to vertices of a graph , then respective functions are associated with edges and not vertices .obviously , one can also reformulate the model reducing it to the vertex ( canonical for our purposes ) form .the transformation is illustrated in figure [ fig : vtoe ] .algebraic form of the transformation shown in the figure reads , , where is the characteristic function equal to unity if all variables are equal each other and equal to zero otherwise .-vertex to two -vertices .it is assumed that ( 1 ) number of nodes in the gray area is not large , i.e. , ( 2 ) the new graph ( on the right ) is planar , ( 3 ) ordering ( say clockwise ) of the external nodes is preserved .the number of parameters characterizing the -vertex is or smaller , thus the number of parameters characterizing the two vertices and vertices from the gray area is sufficient , i.e. , to parameterize the original vertex.,width=226 ]  next , let us notice that , given a vertex - function model ( [ p_sigma ] ) with the degree of connectivity higher than three , one can always perform a sequence of transformations reducing the degree of connectivity of all the nodes in the resulting graphical model to three .an elementary graphical transformation of the kind is illustrated in figure [ fig : nn-1 ] .it is assumed that the transformation is applied sequentially to vertices of degree larger than three till none of these are left .the end result is that : ( a ) there are no vertices of degree larger than three left within the graph ; ( b ) the increase in the total number of vertices is polynomial ; ( c ) if the original graph is planar the resulting graph is also planar .the set of transformations just described is general , and thus often inefficient , in the sense that knowing specific form of the factor functions one can practically always do a more efficient , customized and simpler reduction .below we will illustrate this point on examples .     the spin glass ising model is usually defined in terms of variables associated with vertices of the graph where summation under the exponential on the r.h.s .goes over all edges of the graph , and associated with an edge can be positive or negative .obviously one can apply the vertex - to - edges transformation , explained in figure [ fig : vtoe ] , to restate the spin glass ising model as a vertex - function model .however , in this case one can also do a simpler transformation to the dual graph .let us consider a planar triangulated graph shown in black in figure [ fig : triang ] .all vertices of the respective dual graph , , shown in red in figure [ fig : triang ] , have degree of connectivity three .we assume that the spin glass ising model is defined on the planar triangulated graph . defining a new variable on an edge of as the product of two variables of the original graph connected by an edge of crossing the edge of , one finds that the sum on the r.h.s .( [ ising ] ) , rewritten in terms of the new variables , becomes , .however , the new variables , are not independent , but rather related to each other via a set of local constraints , : .then , eq . ( [ ising ] ) restated in terms of the new variables on the dual graph gets the following compact vertex - style form one interesting observation is that the allowed configurations of on the dual graph correspond exactly to the single - connected loops on , where the loops are built from the excited , , edges . therefore , andin accordance with discussion of section [ sec : singleloops ] , calculation of the partition function for the spin glass ising is reduced to evaluation of the respective pfaffian , which is the task of a polynomial complexity .notice also that adding a magnetic field ( linear in ) term in the expression under the exponent on the r.h.s . of eq .( [ isingg ] ) will raise the complexity level to exponential .     consider a linear code with the code - book defined in terms of the bi - partite tanner graph , consisting of bits and parity checks , and the set of edges relating bits to checks and checks to bits .then a message is a codeword of the code if it satisfies all the parity checks , i.e. .assuming that all the codewords are equally probable originally , and that the white channel transform a bit of the original codeword into the signal with the probability , one finds that the probability for to be a codeword resulted in the measurement is where , as usual , the partition function is fixed by the normalization condition , .( [ ldpc ] ) represents an example of a mixed graphical model , with variables defined on bit - vertices , the parity - check functions defined on check - vertices and the channel functions ( carrying the dependencies on the log - likelihoods ) also associated with the bit - vertices . in this case transformation to the vertex - style model is done by direct application of the vertex - to - edges procedure of figure [ fig : vtoe ] to all the bit - vertices of .then , the vertex - style version of eq .( [ ldpc ] ) becomes   in general , degree of connectivity of bit - vertices and check - vertices may be arbitrary .direct application of the general procedure explained above ( see figure [ fig : vtoe ] and discussion therein ) allows to reduce all the higher - degree nodes to a larger set of nodes of degree three .however , a simpler dendro - reduction is possible both for the bit - vertices and check - vertices . the dendro trick ( e.g. discussed in for complexity reduction of a linear programming decoding of ldpc codes )is schematically illustrated in the two right panels of figure [ fig : ldpc ] , where respective algebraic relations are and is equal to unity if all arguments are the same , and it is zero otherwise .99  chertkov m and chernyak v y , _ loop calculus in statistical physics and information science _ , 2006 phys .e * 73 * 065102(r ) [ cond - mat/0601487 ] chertkov m and chernyak v y , _ loop series for discrete statistical models on graphs _, 2006 j. stat .mech . p06009 [ cond - mat/0603189 ] fisher m e , _ statistical mechanics on a plane lattice _ , 1961 phys .rev * 124 * 1664                  peierls r , _ ising s model of ferromagnetism _ , 1936 proc .. soc . * 32 * 477 chertkov m and chernyak v y , _ loop calculus helps to improve belief propagation and linear programming decodings of low - density - parity - check codes _ , invited talk , 44th allerton conference 2006 [ arxiv : cs.it/0609154 ] chertkov m , _ reducing the error floor _ , invited talk at the information theory workshop on frontiers in coding " 2007 [ http://arxiv.org/abs/0706.2926 ]

Abstract: we discuss a generic model of bayesian inference with binary variables defined on edges of a planar graph . the loop calculus approach of is used to evaluate the resulting series expansion for the partition function . we show that , for planar graphs , truncating the series at single - connected loops reduces , via a map reminiscent of the fisher transformation , to evaluating the partition function of the dimer matching model on an auxiliary planar graph . thus , the truncated series can be easily re - summed , using the pfaffian formula of kasteleyn . this allows to identify a big class of computationally tractable planar models reducible to a dimer model via the belief propagation ( gauge ) transformation . the pfaffian representation can also be extended to the full loop series , in which case the expansion becomes a sum of pfaffian contributions , each associated with dimer matchings on an extension to a subgraph of the original graph . algorithmic consequences of the pfaffian representation , as well as relations to quantum and non - planar models , are discussed .