works on coding for gaussian multiple access channel with finite complex alphabets have been reported in - .constellation constrained ( cc ) capacity regions of a two - user gmac ( shown in fig .[ gmac_model ] ) have been computed in wherein the unique decodability ( ud ) property at the destination is achieved by appropriate rotation between the alphabets of the two users .the angles of rotation between the alphabets which enlarge the cc capacity regions have also been computed .+ for a fixed equal average power constraint for the two users , the gaussian capacity region of two - user gmac is shown in fig .[ cc_region_equal ] along with the cc capacity regions of two pairs of finite alphabets . in fig .[ cc_region_equal ] , the average power constraints for all the three cases are equal and , hence , the gaussian capacity region encloses the other two regions . for the channel model in ,code pairs based on trellis coded modulation ( tcm ) are proposed in such that sum - rates close to the sum capacity can be achieved . in such a setup , the number of information bits transmitted by the two users are equal . in other words ,coding schemes in can potentially approach any rate pair only on the degrees line within the capacity region shown in fig .[ cc_region_equal ] ( for example , the point q ) .note that , using the set up in , rate pairs other than those on the degrees line in fig .[ cc_region_equal ] can not be achieved ( for example , the point r or the point p shown in fig .[ cc_region_equal ] can not be achieved ) . +      in this paper , we propose tcm based trellis codes to approach _ any rate pair _( for example , the points r , p and p shown in fig .[ cc_region_equal ] ) within the capacity region of a two - user gmac with equal power constraint for the two users .in other words , unlike , the setup here permits the two users to transmit unequal number of information bits to the destination .the proposed codes are applicable when the two users have psk signal sets . since the signal sets can have unequal size and equal average power constraint, the corresponding cc capacity region can be asymmetric .for example , see fig .[ cc_region_equal ] which shows the cc capacity region of unequal sized signal sets with ( where and are the sizes of signal sets of user-1 and user-2 respectively ) . + throughout the paper , the terms alphabet and signal set are used interchangeably .all the rates mentioned in this paper are in bits per channel use . unless specified , whenever and are unequal , we assume throughout the paper .the contributions of the paper may be summarized as below :  * for a two - user gmac with psk signal sets and equal average power constraints for the two users , we use a generalised version of the metric in , to compute relative angle(s ) of rotation , between the signal sets to provide maximum enlargement of the cc capacity region .in particular , when user- and user- are equipped with -psk and -psk respectively , it is shown that , between the alphabets maximally enlarges the cc capacity region ( section [ sec3 ] ) .+ * for each , if user- employs the trellis labelled with the symbols of -psk , it is clear that the destination views the sum trellis , labelled with the symbols of the sum alphabet , of -psk and -psk signal sets in an equivalent siso ( single - input single - output ) awgn channel .when the angle , is , it is shown that , ungerboeck labelling on the trellis of each user induces a labelling on which maximizes the guaranteed minimum squared euclidean distance , of ( see theorem [ ungerboeck_theorem ] in section [ sec3_subsec3 ] ) . hence , for a given power constraint and a given pair ( ) , the proposed labelling scheme can be used systematically to construct trellis code pairs to approach any rate pair on the line inclined at degrees with the x - axis , within cc capacity region . as an example , for the cc capacity region shown in fig .[ cc_region_equal ] , any rate pair on the line joining the origin and the point r can be achieved .+ * we focus on approaching rate pairs on the line segment a - b shown in fig .[ cc_region_equal ] since such rate pairs maximize the sum capacity . for a given rate pair , say point p within the sector o - a - b shown in fig .[ cc_region_equal ] , we propose a method to choose a sequence of ( , ) pairs which when employed in our coding scheme can approach any rate pair on the line segment p-p ( section [ sec3_subsec4 ] ) .the above contributions along with  and establish that , whenever the input alphabets are psk signal sets and the average power constraints for both the users are equal , explicit practical tcm schemes approaching any rate pair within the capacity region can be obtained for two - user gaussian multiple access channels ._ notations : _ cardinality of the set is denoted by .absolute value of a complex number is denoted by and $ ] denotes the expectation of the random variable . a circularly symmetric complex gaussian random vector , with mean and covariance matrix is denoted by .also , the set of all real and complex numbers are denoted by and respectively . for two points , denotes the euclidean distance between and .the remaining content of the paper is organized as follows : in section [ sec2 ] , the two - user gmac model along with the description of the coding problem is presented . in section [ sec3 ] , details on designing tcm based code pairs with a pair of unequal sized psk signal sets are presented .section [ sec4 ] constitutes conclusion and some directions for possible further work .the model of a two - user gaussian mac as shown in fig .[ gmac_model ] consists of two users that need to convey information to a single destination .it is assumed that user-1 and user-2 communicate to the destination at the same time and in the same frequency band .symbol level synchronization is assumed at the destination .the two users are equipped with alphabets and of size and respectively .when user-1 and user-2 transmit symbols and respectively , the destination receives a symbol given by , where is the power used by each user for every channel use . henceforth , unless specified , we assume for simplicity . for the above channel , in ,cc capacity regions have been computed for some well known alphabets such as -psk , -qam etc and the angles of rotation between the alphabets that maximally enlarges the cc capacity region have also been presented . in this paper, we design code pairs over psk signal sets to approach rate pairs , ( ) ( for user-1 and for user-2 such that and can be unequal ) within the gaussian capacity region .+ for each , let user- be equipped with a convolutional encoder with input bits and output bits as shown in fig .[ scheme_1 ] . throughout the paper ,we consider convolutional encoders which add 1 bit of redundancy only .let the output bits of take values from a -dimensional signal set such that .henceforth , the codes ( set of codewords generated from ) and ( set of codewords generated from ) are represented by trellises and respectively .the definition of sum trellis , for the trellis pair and sum alphabet , of and have been introduced in .+ we assume that the destination performs joint decoding of the symbols of user-1 and user-2 by decoding for a sequence over on . for the trellis pair and the alphabet pair , the destination views an equivalent siso awgn channel with a virtual source equipped with the trellis , labelled with the elements of .due to the existence of an equivalent awgn channel in the two - user gmac set - up , has to be labelled with the elements of satisfying the ungerboeck design rules .however , such a labelling rule can be obtained on only through the pairs and .hence , in this paper , we propose labelling rules on and using and respectively such that is labelled with the elements of as per ungerboeck rules .    for each , since the number of input bits to is ,there are edges originating from ( or converging to ; henceforth , we only refer to originating edges ) each state of .also , as there is only one bit redundancy to be added by the encoder and , the edges originating from each state have to be labelled with the elements of a proper subset of of size . therefore , for each , has to be partitioned into two sets namely , and and the diverging edges from each state of have to be labelled with the elements of either or . from the definition of a sum trellis, there are edges originating from each state of which gets labelled with the elements of one of the following sets , as per the ungerboeck design rules , the transitions originating from the same state of must be assigned symbols that are separated by largest minimum euclidean distance .hence , an optimal partitioning of into two sets and of equal cardinality has to be proposed such that the minimum euclidean distance , of each one of the sets in is maximized .however , since of the sets in can potentially be different , we try to find an optimal partitioning such that the minimum of the values of the sets in is maximized .the minimum of the values of the sets in is referred as the guaranteed minimum squared euclidean distance , of . .\ ] ]in this section , we propose a solution to the problem of designing labelling rules when and are symmetric psk signal sets of cardinality and respectively where and for some . without loss of generality , we assume that .let denote the ratio ( note that ) . to obtain the unique decodability property at the receiver , we employ a -rotated version of .when the two users are equipped with identical signal sets , a metric has been proposed in theorem 1 of , to compute relative angle(s ) of rotation between the alphabets such that the corresponding cc capacity region is maximally enlarged . however , the proof of theorem 1 in is not applicable when the cardinality of the signal sets are unequal .hence , proceeding on the similar lines of the results in theorem 1 of , a new metric shown in can be obtained to compute relative angle(s ) of rotation between two _ unequal _ sized alphabets . using the metric in , the angle, can be shown to maximally enlarge the cc capacity region of -psk and -psk signal sets .henceforth , we use throughout the paper .let and represent two symmetric psk signal sets of cardinality and respectively .the signal set is rotated by an angle .let and denote the points and of and respectively for and .the sum alphabet , of and can be written as given in ( at the top of the next page )    ' '' ''  where and such that for any .the phase components of the points and are given by and respectively .for a fixed , the set of points of the form and lie on a circle of radius and let that circle be denoted by . then , takes the structure of concentric _ asymmetric _ psk signal sets .it is to be noted that takes the structure of concentric _ symmetric _ psk signal sets when  .hence , the results of this paper is a non - trivial generalisation of the one in . as an example , see fig .[ sum_alphabet_pi_m ] which shows the when is a qpsk signal set and is a -psk signal set .the set containing the radii of the circles is given by   when = qpsk and = -psk when ,width=240 ]  henceforth , throughout the section , denotes the radius of the circle .since the radius of each circle is a cosine function , the elements of satisfies the relation , for the elements of , the following two propositions can be proved .[ prop3 ] the sequence from to is an increasing sequence .  using standard trigonometric identities , the term is given by . since , the sequence is an increasing sequence as increases from to .[ prop5 ] using the phase information of each point in , the following observations can be made :  1 .for a fixed , the angular separation between the two points and on is for all .similarly , for a fixed , the angular separation between the two points and on is for all .2 .  for a fixed , the angular separation between the point , on and the point on is for all .3 .  for a fixed ,the angular separation between the point on and the point on is for all .4 .  for a fixed ,the angular separation between the point on and the point on is for all .  in the next subsection , first , we partition both and into two groups ( due to one bit redundancy added by the two encoders ) using ungerboeck rules and then , exploiting the structure of , we compute the minimum euclidean distance , of each one of the sets in .   for each , let be partitioned into two sets of equal size using ungerboeck rules which results in two sets denoted by and such that of and is maximized .since the number of sets resulting from the partition is two , the minimum angular separation , between the points in each set is .the two groups of are of the form ,        similarly , the two groups of are of the form ,  it is clear that the four sets , , and form a partition of . throughout this paper , the set and its minimum euclidean distanceare denoted by and respectively and . in the next subsection ,only the structure of is studied and value is calculated .the values of the rest of the sets in can be calculated on the similar lines .the partition induced on due to the partition of and has been depicted in fig .[ partition ] .the elements of ( as given in at the top of the next page ) are of the form and where takes even value while and take odd values .when is odd , note that is odd and is even and hence will have points of the form and no points of the form on .similarly , when is even , will have will have points of the form and no points of the form on .  since takes only even values ,using observation 1 ) of proposition [ prop5 ] , between the points of on any circle is .hence the points of are maximally separated on every circle .the following two propositions are important in finding the minimum euclidean distance of the set . +    ' '' ''  [ prop6 ] and satisfy the following inequality for all to ,   for , let denote the line segment joining and . note that the three complex points and form the three vertices of an isosceles triangle in .since we have .therefore , the four points and form the vertices of an isosceles trapezoid such that is parallel to .also , note that is the length of the diagonal of the trapezoid . since the angle between the line segments and is obtuse , .[ prop7 ] for , satisfy the following inequality ,   here , we prove the inequality .the two terms across the inequality can be written as a ratio as shown below , since , we have to prove that .note that note that whenever .the inequality holds when .this completes the proof .  using the above two propositions , the value of the set is presented in the following lemma .+  [ lemma_1 ] the minimum euclidean distance of the set is given by   since the points of are maximally separated on every circle ( with minimum angular separation of ) and is the innermost circle , is a contender for .for this to be true , all other intra - distances in the set must be larger than or equal to .in particular , we have to show that the distances between the points on any two consecutive circles must be larger than .in that direction , the first observation is the equality , .from proposition [ prop3 ] , for all .hence , the points on and ( irrespective of their angular separation ) are separated by a distance larger than for all to .+ secondly , we must prove that a point on and a point on are separated by a distance larger than for all to .in that direction , it can be shown that between a point on and a point on is one of the values from the set , depending on the value of for all to . using proposition [ prop6 ] ,we have furthermore , using the inequality in proposition [ prop7 ] , we have note that the above inequality holds only when . however , when , we have .therefore , will be the radius of the innermost circle wherein the minimum angular separation is .hence proposition [ prop7 ] is not applicable when .since , the inequality in can be extended to hence , . with this , we have proved that a point on and a point on are separated by a distance larger than for all to .this completes the proof .the values of the rest of the sets in can be calculated on the similar lines .the following lemma provides the values of , and .+  [ lemma_2 ] the minimum euclidean distances of the sets , and are given by   the proof is on the similar lines of the proof for lemma [ lemma_1 ] .+   in the preceding subsection , values of each one of the sets of induced by ungerboeck partition on and have been computed in and ( we refer = = = = ) . in this subsection , using these values , it is shown that a non - ungerboeck partition on either or results in a set such that the of at least one of the sets in is lesser than .[ ungerboeck_theorem ] for , ungerboeck partitioning on and into two sets is optimal in maximizing the minimum of the values of the sets in .let and be the two sets ( of equal cardinality ) resulting from a partition of for . if either or is not ungerboeck partitioned , then it is to be shown that , of at least one of the sets in the set is lesser than . here , we prove the above result when is not ungerboeck partitioned . on the similar lines ,the above result can be proved when is not ungerboeck partitioned as well .we show that between any two points in one of sets in is smaller than on the circle .it is assumed that there are exactly points on in each set of .otherwise , at least one set contains more than points on and hence between a pair of points in that set will be lesser than .therefore , the sub - optimality of the partition can be proved .hence , without loss of generality , we assume that for some such that .note that the elements of are of the form or . since is odd and is an even and odd pair , can be of the form and can be of the form or vice - verse . without loss of generality , we assume that is of the form and is of the form for some , .note that , the points and belong to one of the sets in and have an angular separation of .this implies that there exists a pair of points on such that between them is lesser than .this completes the proof .  from the above theorem, it is clear that ungerboeck labelling on the trellis of each user can be used systematically to construct trellis code pairs to approach _ any rate pair _ within the cc capacity region .using the results presented in the preceding subsection , we illustrate how to choose the cardinality of psk signal sets to achieve any rate pair ( assuming ) within the sector o - a - b shown in fig .[ cc_region_equal ] .we do not consider achieving rate pairs outside the sector o - a - b since such points can be moved either horizontally or vertically ( or both ) into the sector o - a - b which in - turn either increases the rate for both users or increases the rate for one of the users by keeping the rate for the other as it is . for example , achieving the rate pair denoted by p shown in fig .[ cc_region_equal ] is better than achieving the rate pair denoted by p since the latter has higher sum - rate by retaining the same rate for user-2 and increasing the rate for user-1 .+ for a given equal power constraint , to approach a rate pair , we choose sufficiently large values of and such that  1 . and satisfies the following approximation , 2 .the cc capacity region with -psk and -psk signal sets encloses the point .  for the above choice of ( ) ,if the number of input bits for user- is , and ungerboeck labelling is employed on the trellis ( with larger number of states ) of each user , then the rate pair can be approached . to approach larger rates ( close to the line segment a - b ) on the same line segment , larger values of and to be chosen satisfying according to points 1 ) and 2 ) mentioned above .therefore , with sufficiently large values of and satisfying the conditions 1 ) and 2 ) , any pair , in the gaussian capacity region can be approached .in this paper , we have designed the optimal labelling rule on the trellises of the two users ( with equal average power constraints on the two users ) to approach any rate pair within the capacity region .some possible directions for further work are as follows :  * in this paper , we have assumed equal average power constraint for both the users . if unequal power constraints are considered , then it is interesting to see if rotation between the alphabets provides enlargement of the cc capacity region .* we have designed tcm schemes with unequal rates for the two users even though the two users have equal average power constraints .however , when rates proportional to the power constraint of each user has to be transmitted ( with unequal average power constraints on the two users ) , tcm schemes have to be designed with pairs of signal sets which have unequal power constraints and unequal size .this work was partly supported by the drdo - iisc program on advanced research in mathematical engineering to b. s. rajan .1 f. n. brannstrom , t. m. aulin and l. k. rasmussen , `` constellation - constrained capacity for trellis code multiple access systems '' in the proceedings of _ ieee globecom 2001 _ , vol .2 , san antonio , texas , nov , 2001 , pp . 11 - 15 .t. aulin and r. espineira , `` trellis coded multiple access ( tcma ) '' in the proceedings of _ icc 99 _ , vancouver , bc , canada , june 1999 , pp .1177 - 1181 .fredrik n brannstrom , tor m. aulin and lars k. rasmussen `` iterative multi - user detection of trellis code multiple access using a posteriori probabilities '' in the proceedings of _ ieee icc 2001 _ , finland , june 2001 , pp .wei zhang , c. damours and a. yongacoglu , `` trellis coded modulation design form multi - user systems on awgn channels '' in the proceedings of _ ieee vehicular technology conference 2004 _ , pp .1722 - 1726 .ezio biglieri , `` coding for wireless channels '' , _springer - verlag new york , inc _ , 2005 .j. harshan and b. sundar rajan , `` finite signal - set capacity of two - user gaussian multiple access channel '' in the proceedings of _ ieee international symposium on information theory , ( isit 2008 ) _ , toronto , canada , july 06 - 11 , 2008 .1203 - 1207 .g. ungerboeck , `` channel coding with multilevel / phase signals , '' _ ieee trans .inform . theory _01 , 1982 , pp .j. harshan and b. sundar rajan , `` coding for two - user gaussian mac with psk and pam signal sets '' in the proceedings of _ ieee international symposium on information theory , ( isit 2009 ) _ ,seoul , south korea , june 28- july 03 , 2009 , pp . 1859 - 1863 .

Abstract: in this paper , code pairs based on trellis coded modulation are proposed over psk signal sets for a two - user gaussian multiple access channel . in order to provide unique decodability property to the receiver and to maximally enlarge the constellation constrained ( cc ) capacity region , a relative angle of rotation is introduced between the signal sets . subsequently , the structure of the _ sum alphabet _ of two psk signal sets is exploited to prove that ungerboeck labelling on the trellis of each user maximizes the guaranteed minimum squared euclidean distance , in the _ sum trellis_. hence , such a labelling scheme can be used systematically to construct trellis code pairs for a two - user gmac to approach _ any rate pair _ within the capacity region .   multiple access channels , ungerboeck partitioning , constellation constrained capacity , trellis coded modulation .


a - posteriori analysis of deterministic pde ( partial differential equations ) is a well developed tool .see for example the book or the results for allen - cahn and related equations .the strength of the method is usually the derivation of error indicators for the refinement of meshes in adaptive schemes .see for an example in a stochastic setting .also for spdes ( stochastic pdes ) there are recent results on a - posteriori analysis .the results of use a - posteriori estimates in polynomial or wiener - chaos expansion , and the results of show a - posteriori mean square error estimates , where the whole numerical approximation is known ( at least several moments of it ) .in our work we follow a different more path - wise approach .we measure the error in mean square conditioned on the calculated numerical data .given a single realization of the numerical approximation , without using a - priori information on the solution we show analytic bounds , that can be calculated numerically , and guarantee a - posteriori that the true solution is close to the one realization of the numerical approximation , which we calculated . in the numerical example we even get a meaningful result for a relatively poor approximation .the general philosophy of a - priori error analysis is to use the true solution , which is plugged into the numerical scheme to calculate the residual . then using the discrete in time equation given by the numerical scheme, we can derive a discrete equation for the error , which has coefficients depending on the true solution . using a - priori information of the solution , asymptotic bounds for the errorare derived . in a - posteriorianalysis we use a time - continuous interpolation of the numerical data , which is plugged into the spde , in order to derive the residual . for the errorwe obtain a pde which is continuous in time and has coefficients depending on the numerical data . herewe can use now standard a - priori spde - type methods to derive error bounds , that depend only on numerical data and the residual .although for simplicity of presentation , we use a much simpler equation of allen - cahn - type , our result is motivated by equations where the global existence of solutions is not known , and thus global a - priori estimates are not available .typical examples are the three - dimensional navier - stokes equation or a somewhat simpler equation from surface growth . for the latter in - posteriori analysis was used for the deterministic pdes to prove numerically the regularity of solutions and thus the global existence and uniqueness .  herewe focus as a starting point for simplicity on an one - dimensional equation of allen - cahn type . hereeven the asymptotic convergence results of numerical schemes are well known see for example or for a truncated scheme .moreover , there is no problem with existence and uniqueness of solutions .see for example .  for the spatial discretization we use the spectral galerkin - scheme , which simplifies the analysis .moreover , for the time - discretization we use a variant of the exponential scheme introduced by .asymptotically , both results should be equivalent , but the variant we use is slightly easier in the analysis .  the precise functional analytic set - up and the equation itselfis presented in section [ sec : sett ] . in section [ sec : sb ]we present analytic results for stochastic terms we can not evaluate numerically .one is the infinite - dimensional reminder of the stochastic convolution at discretization times .the second one bounds fluctuations in between discretization times .here we need to analyze an ornstein - uhlenbeck bridge - process , as we now the stochastic convolution at all discretization times , both in the future and the past .  in the main result we present in section [ sec : res ]analytic error estimates for the residual that depend only on the numerically calculated data , the initial condition , and the stochastic terms already bounded in section [ sec : sb ] .we provide a bound in moments of the -norm , which is conditioned on the given numerical data . in section [ sec : app ] we study the conditional mean square error of the approximation in -norm , given the numerical data .nevertheless , this is a property of the equation and not of the data .we need to quantify the continuous dependence of solutions on additive perturbations like the stochastic convolution . due to the relatively simple structure of the equation with a stable nonlinearity and a stable linear part ,this is relatively straightforward .  in the final section ,we give numerical examples to illustrate the result . herewe use a quite poor discretization given that the solution is very rough and still obtain meaningful error bounds .the main source of error comes from bounds on term that appear between the discretization points and not at the discrete times where the approximation is calculated .consider the following spde on the hilbert - space ) ] , where is the laplacian , some cylindrical -wiener process .finally , is the locally - lipschitz nonlinearity .the dirichlet laplacian is diagonal w.r.t . , and generates an analytic semigroup on .moreover , it is a contraction semigroup on any .this follows in as the largest eigenvalue of is and thus . in is true by the maximum principle .then by the riesz - thorin theorem for any -space we have for  let us remark that by geometric series is an invertible operator in with bounded inverse .moreover , for simplicity we assume that the covariance operator is also diagonal in the fourier basis , and denote the eigenvalues by , i.e.  . in the numerical examples we consider space - time - white noise of order one that corresponds to for all , which is a case of solutions of quite poor regularity with strong fluctuations .nevertheless we could allow for even rougher noise .we assume that for some , which guarantees that the the stochastic convolution ( or ornstein - uhlenbeck process )  is continuous both in space and time ( cf . ) .the mild formulation of ( [ spde ] ) is defined by the fixed point equation here the existence and uniqueness of solutions is standard .see for example .here we just need for some that , l^p) ] by linear interpolation with .the aim of this paper is to bound the conditional mean - square error given the numerical data , i.e , we want to obtain = \mathbb{e } [ \|u-\varphi\|^2\ | \ \ { z_k \}_{k\in{\mathbb{n } } } ] = \text { small}\;,\ ] ] which is not an asymptotic result , but one that holds for the given approximation . in theorem [ thm : main ] we present the main analytic result for this statement .the term `` small '' depends on one hand on the the numerical data , and we will evaluate this part only numerically . on the other hand , we have infinite - dimensional parts and random fluctuations between discretization points , which we have to bound analytically , as there is no data available .the general philosophy is to evaluate as much as possible of the error bounds using the numerical data , and only rely on analytic estimates if no numerical evaluation is possible .here we present analytic results for stochastic terms we can not evaluate numerically .one is the infinite - dimensional reminder of the stochastic convolution at discretization times .the other one arises from fluctuations in between discretization times , where we need to analyze an ornstein - uhlenbeck bridge - process .   for the stochastic term we can not use any numerical data to evaluate it .moreover it is infinite dimensional .the main result here is lemma [ lembouq ] below .first we need estimates of a gaussian in the -norm using the expansion in fourier - series .let with a covariance - operator on such that .denote the eigenvalues and eigenfunctions by and suppose that for all we have , then   by assumption , we obtain that for all the real - valued random variable .thus by tonelli s theorem which implies the claim .note that we used that we can explicitly calculate the fourth moment of a centered scalar gaussian .recall that the fourier - basis is defined by here we have for where  thus we have for  now we can verify this yields the following lemma :  [ lem : oul4 ] let with a covariance - operator on such that .let be diagonal w.r.t .the fourier - basis , then   we furthermore obtain :  [ lembouq ] the sequence is independent of and bounded in -norm for all by   a stronger result as proven in , where we even could take the supremum in time over bounded intervals inside the expectation and use instead of -norms .the bound of lemma [ lembouq ] is still not numerically computable , but given a bounded sequence , it is usually straightforward to evaluate ( or bound ) the series explicitly .see remark [ rem : mainres ] .we start by using lemma [ lem : oul4 ] this easily implies the claim .   in order to treat random fluctuations between discretization points ,we define for  first the processes are independent and identically distributed .  moreover ,it is easy to see that } ] the law of given is a gaussian with mean and covariance with ^{-1 } a z\ ] ] and   we follow the result of , but our setting is much simpler .first all operators involved are diagonal and thus symmetric .furthermore , they all commute .we can also treat degenerate noise by restricting the results of to the hilbert - space given by the range of , which is in general only a subset of .but then both and are invertible on that space .first , = \lambda(t , z)+\hat{z}_t\ ] ] where by ( * ? ? ?2.11 ) is a centered gaussian with and by ( * ? ? ?2.13 ) with the following operators , this implies ^{-1 } z\;.\ ] ] and   the following bound is surely not optimal , but a slight simplification of the exact bound .[ lem : boundn ] we bound where for any fixed we define {5 } }  \|[i - e^{2ha}]^{-1 } a z\|_{l^4 } + \big(\frac{3}{2\pi}\big)^{1/4 } \left ( h^{-1/2 } \sigma_r(h ) + \sum_{k = r+1}^\infty  \frac{\alpha_k^2 } { 2k^2 }  \right)^{1/2 }  \big]^4\ ] ] where   we can explicitly calculate an upper bound for by using numerical data for and for the infinite series we can used for large   first using lemmas [ lem : oub ] and [ lem : oul4 ] now and thus in order to obtain a numerically computable bound , we fix large , and bound the -norm directly , using that all factors are bounded by .thus for the mean value , we obtain from lemma [ lem : oub ] using ^{-1 } a z\|_{l^4 } \;.\ ] ] thus ^{-1 } a z\|^4_{l^4 } \;.\ ] ]this section is devoted bounds on the residual .first we consider the discretization points , and later we focus on the points , , which are in between .  the residual measures the quality of an arbitrary numerical approximation . for is defined as    at first for the points we have   in the following lemma we estimate the residual at the discretization points using the numerical data .[ lem : resnh ] the residual at discrete times is given as where given by the recursive schema , ds\ ] ] and .\ ] ] moreover , is bounded by assumptions on the regularity of and is random and independent of the numerical data .the value of just depends on the numerical data and .note that the cubic terms depending on and are all in and thus computable .the integrals are all diagonal matrices that can be calculated explicitly in the numerical evaluation .we have \tau  \\ & -  q_nz(nh )  \\ = :  & -e^{nha}q_n u_0 + i_2(n ) + i_3(n ) -  q_n z(nh)\;.  \end{split}\ ] ] for we can not use any numerical data .we need assumptions on the regularity of the initial condition to bound it .for the two integrals we use for  ] we have with defined in and bounded in and was defined in .now we are going to bound the residual for intermediate times .let us fix and  ] and obtain from ( [ resnhtau ] ) by triangle inequality \big)^{1/4 }  \\&=  \big ( \mathbb{e}[\int _ { 0}^{t } \|\text{res}(n(s)h+\tau(s))\|_{l_4}^4 ds|(z_k)_{k \in \mathbb{n } } ] \big)^{1/4 }  \\ &  \leq  \big(\mathbb{e}[\int_{0}^{t }  \|\text{res } ( n(s)h)\|^4_{l^4 } ds |(z_k)_{k \in \mathbb{n } } ] \big)^{1/4 }  +  \big(\int_{0}^{t}\tau(s)^4 \|au_{n(s)}\|_{l^4}^4 ds  \big)^{1/4 }  \\ &   + \big(\int_{0}^{t}\frac{\tau(s)^4}{h^4}\|d_{n(s)+1}\|^4_{l^4}ds \big)^{1/4 }  +  \big(\int_{0}^{t}\|i(\tau(s))\|^4_{l^4}ds \big)^{1/4 }  \\ &  + \big(\mathbb{e}[\int_{0}^{t}\|z_{n(s)}(\tau(s))\|^4_{l^4 } ds |(z_k)_{k \in \mathbb{n}}\big)^{1/4 }  .\end{split}\ ] ] we now bound all the terms above separately .let be the largest integer , such that . from lemma [ lem : resnh ] again by triangle inequality , we get \big)^{1/4 }\leq  \big(\int_{0}^{t } \|\text{res}^{\text{dat}}_{n(s)}\|_{l_4}^4 ds  \big)^{1/4 } \\&+ \big ( \mathbb{e}\int_{0}^{t } \|\text{res}^{\text{stoch}}_{n(s)}\|_{l_4}^4 ds  \big)^{1/4 } + \big ( \int_{0}^{t } \|\text{res}^{\text{ic}}_{n(s)}\|_{l_4}^4 ds  \big)^{1/4 } \\ & \leq  \big(h\sum_{n=1}^{m(t ) } \|\text{res}^{\text{dat}}_n\|_{l_4}^4  \big)^{1/4 } + \big(h \sum_{n=1}^{m(t ) } \mathbb{e}\|\text{res}^{\text{stoch}}_n\|_{l_4}^4  \big)^{1/4 } + \big(h  \sum_{n=1}^{m(t ) } \|\text{res}^{\text{ic}}_n\|_{l_4}^4  \big)^{1/4 } \\ &\leq  \big(h\sum_{n=1}^{m(t ) } \|\text{res}^{\text{dat}}_n\|_{l_4}^4  \big)^{1/4 } +   \big(\frac{3hm(t)}{8\pi}\big)^{1/4 } \big ( \sum_{k > n } \frac{\alpha_k^2}{k^2 }  \big)^{1/2 } + \big(h  \sum_{n=1}^{m(t ) } \|e^{nha}q_nu_0\|_{l_4}^4  \big)^{1/4 } \end{split}\ ] ] where we used that , so that all sums start at .  for the next term and similarly for the integral - term by   \big)^{1/4 } \;.\end{split}\ ] ] finally for the ornstein - uhlenbeck bridge , we have from lemma [ lem : boundn ] } \big)^{1/4 } \leq  \big ( h \sum_{n=0}^{m(t ) } \mathcal{s}_h(z_{n+1}-e^{ha}z_n ) ) \big)^{1/4}\;.\ ] ]  summarizing , we have the following bound :  [ thm : mainres ] for and with we have \leq \mathcal{k}_m^4\ ] ] with   \big)^{1/4 }  \\ &  + \big(h \sum_{n=0}^{m } \mathcal{s}_h(z_{n+1}-e^{ha}z_n ) \big)^{1/4 } \end{split}\ ] ]  [ rem : mainres ] the quantity is almost numerically computable using numerical data .moreover , we can update the sums in the numerical computation , so that we do not need to calculate them in every step .the only two terms that are not yet computable are the following two :  * for the one depending on the infinite dimensional remainder of the initial condition , one can easily give an analytic bound that depends on the regularity of .for the numerical examples we just suppose that .for example if or .* for the sum depending on the for one can easily give an upper bound , once the are given .in this section , we are mainly based on the properties of the equation .the numerical data only comes into play via the residual . we need to quantify the continuous dependence of solutions on additive perturbations like the stochastic convolution or the residual .recall the mild solution of ( [ spde ] ) and the definition of the residual therefore by putting we have with .let therefore ,\ ] ] which means is the solution of the following differential equation recall that , so .now we use standard a - priori estimates for the equation for .this yields good estimates , as both the linear part and the nonlinear part are stable , which simplifies the error estimate significantly . the following lemma is necessary to bound the cubic .it is not optimal , but sufficient for our purposes .[ lem : cubic ] for all we have \leq r^4 + 3r^2\varphi^2\ ] ]  first = - 3 r \int_{\varphi}^{r+r+\varphi } \zeta^2 d\zeta = - 3 r \int_{0}^{r+r } ( \zeta+\varphi)^2 d\zeta \;.\ ] ] thus the term is non - positive if and have the same sign ( i.e. , in the case or  ]  \leq \|q_n u_0\|^2 + 2 ( \mathcal{k}_m)^4 + 6(\mathcal{k}_m)^2  \big(\int_0^t \|\varphi\|_{l^4}^4 ds \big)^{1/2}\ ] ] in order to have a fully numerically computable quantity , we need to take care of the integral .we proceed similarly to and use to obtain \\ \end{split}\ ] ]  [ thm : main ] for and with we have  \leq \|q_n u_0\|^2 + 2 ( \mathcal{k}_m)^4 + 6 ( \mathcal{k}_m)^2 ( \mathcal{i}_m ) ^{1/2}\ ] ] where was defined in theorem [ thm : mainres ] and \end{split}\ ] ]  as we expect to be small , and the solution of the numerical scheme not , the second term should dominate in the error estimate .this is also confirmed in the numerical example .for the numerical result we focus on space - time white noise of strength , which means that all . moreover , as both the linear part and the nonlinearity are stable , we expect the solution to be of order with rare events , where the solution is larger .nevertheless , we expect solutions to be quite rough .due to poor regularity properties , we do not expect the numerical approximation to be very accurate , but still we tried a relatively poor discretization with fourier - modes and relatively large time - steps with a terminal time . herewe intend to show the robustness of the a - posteriori error estimate , which even for this poor discretization yields an almost reasonable result .first in figure [ fig : km and error ] we plotted the residual for together with the final error .as expected is small and the error term from theorem [ thm : main ] is bounded by the error term involving and the numerical data .  in figure[ fig : comparison ] we plot two terms of the residual , for .one of the main terms in which depends on the numerical data is , therefore we plot in figure [ fig : residual ] to see impact of this terms on the residual - bound , which seems to be negligible .moreover in figure [ fig : ou - bridge ] we plot , i.e , the term in which arises from the ou - bridge . by comparing [ fig : km ] and [ fig : ou - bridge ] we can see the impact of the ou - bridge on .this gives a substantial , but not the most dominant term in .we can also see that this error term is almost growing linear .the reason for this is that the part in that depends on the numerical data is quite small and the deterministic part of the estimate bounding the fluctuations dominates the bound .the final bound for the error  ] are data controlled terms for error that arises between discretization points . as the step size is quite largethese terms seem to dominate although in they seem to be of smaller order .finally bounding the stochastic fluctuation in between is not that large as we would expected it to be . [cols="^,^,^,^,^",options="header " , ]     bartels .s , mller .r , christoph , o , _ robust a priori and a posteriori error analysis for the approximation of allen - cahn and ginzburg - landau equations past topological changes _, siam j. numer49 ( 2011 ) , no . 1 , 110 - 134 .     blmker .d , romito .m , _ stochastic pdes and lack of regularity ( a surface growth equation with noise : existence , uniqueness , and blow - up ) _ jahresbericht der deutschen mathematiker - vereinigung , 117(4):233 - 286 , 2015 .kloeden , p.e . ; lord , g.j .; neuenkirch , a. ; shardlow , t. _ the exponential integrator scheme for stochastic partial differential equations : pathwise error bounds . _ j. comput235 , no . 5 , 1245.1260 ( 2011 ) .moon , kyoung - sook ; von schwerin , erik ; szepessy , anders ; tempone , raul . _ an adaptive algorithm for ordinary , stochastic and partial differential equations ._ recent advances in adaptive computation , 325343 , contemp . math . , 383 , amer . math .soc . , providence , ri , 2005   yang , xiaoyuan ; duan , yuanyuan ; guo , yuhua ._ a posteriori error estimates for finite element approximation of unsteady incompressible stochastic navier - stokes equations , _siam j. numer .48 ( 2010 ) , no .4 , 15791600 .yang , xiaoyuan ; qi , ruisheng ; duan , yuanyuan . _ a posteriori analysis of finite element discretizations of stochastic partial differential delay equations ._ j. difference equ .appl . 18 ( 2012 ) , no . 10 , 16491663 .

Abstract: the aim of this paper is the derivation of an a - posteriori error estimate for a numerical method based on an exponential scheme and spectral galerkin methods . we obtain analytically a rigorous bound on the mean square error conditioned to the calculated data , which is numerically computable and relies on the given numerical data . thus one can check a - posteriori the error for a given numerical computation without relying on an asymptotic result .   all estimates are only based on the numerical data and the structure of the equation , but they do not use any a - priori information of the solution , which makes the approach applicable to equations where global existence of solutions is not known . for simplicity of presentation , we develop the method here in a relatively simple situation of a stable one - dimensional allen - cahn equation with additive forcing .


as a valuable tool for various scientific disciplines , fractals introduced by mandelbrot  have received considerable attention in many fields in the past few decades  .the so - called regular fractals constitute an integral family of fractals , including the sierpinski triangle  , hierarchical lattice fractal  , vicsek fractals  , fractal  , peano basin fractal  , to name a few .these simple fractal structures have been a focus of extensive research  .a great advantage of regular fractals is that many problems about them can be treated analytically , presenting a first step in understanding the structural and dynamical proprieties of general fractals .a fundamental problem in the study of fractals is to find the relations between structural properties of fractals and their dynamics , which remains one of the great challenges in the realm of nonlinear science . as a paradigmatic dynamical process ,random walks on fractals play a crucial role in a wide range of applications  , therefore have been intensely investigated  .these investigations have uncovered some nontrivial effects of topological structures of different fractals on the behavior of random walks . among various random - walk issues , the trapping problem on fractals are becoming increasingly important since many phenomena built on fractals can be described by random walks  .particularly , the mean first - passage time ( mfpt ) to the trap node in a network has been widely studied  due to its role as an indicator of the transport efficiency .  in view of the importance of mfpt , a huge amount of effort has been devoted to developing techniques to determine mfpt to a given node , and numerous methods have been proposed , including the generating function  , renormalization group  , back - forward equation  , and so on .most of existing methods are presented for particular networks and thus are applicable only to some specific structures .a universal technique for general networks is still unavailable , with the exception of the spectral approach , through which the mfpt from a node to another can be expressed in terms of the eigenvalues and their corresponding eigenvectors of a related transition matrix  . for the trapping problem , the average trapping time ( att ) , i.e., the mfpt to the trap , can be approximated by the inverse of the smallest eigenvalue of an associated matrix  .irrespectively of the generality of the spectral approach , the computational complexity for determining the spectra of generic networks is very high  , which makes the task very difficult and even impossible for large networks . in this context , it is of interest to seek new techniques for evaluating the spectrum of a relevant matrix for some particular networks .  in this paper, we study a random walk problem on a family of treelike regular fractals  with a perfect trap fixed at a central node .the fractals under investigation include both the fractal  and the peano basin fractal  as their particular limiting cases . using the real - space decimation procedure , we derive an explicit recursive relation for the eigenvalues of a matrix associated with the trapping issue .we then continue to determine the full spectrum of eigenvalues and their multiplicities of the relevant matrix . in the end, we derive a recursion relation for the smallest eigenvalues at two consecutive iterations of the fractals , based on which we obtain an approximate value of the final smallest eigenvalue and then further evaluate the att .the fractals under consideration are constructed in the following iterative way  .let ( ) denote the fractals after iterations . initially ( ) , consists of an edge linking two nodes .in each subsequent iteration , is obtained from by performing the following operations on each existing edge as illustrated by fig .[ cons ] : replace the edge by a path two - links long , with the two end points of the path being the same end points of the original edge ; then , create ( an natural number ) new nodes and attach them individually to the middle node of the path . figure [ network ] describes the first three steps of the iterative construction process for a particular fractal corresponding to .the family of fractals , controlled by parameter , includes several well - known fractals as its limiting cases : when , it reduces to the fractal  ; when , it recovers the peano basin fractal  .  .]  the studied fractals have an obvious self - similar structure , which can be seen from their alternative construction approach .if we call the central node ( node in fig .[ network ] , for example ) the innermost node and those nodes farthest from the central node the outermost nodes , then the fractals can also be built as follows ( see fig . [ const2 ] ) : given the generation , we can obtain by merging copies of with the outermost nodes in separate replicas being coalesced into a single new node , i.e. , the innermost node in .   by joining copies of , represented by , , , , and . is the innermost node , while is an outermost node . ]it is easy to derive that the total numbers of edges and nodes in are and , respectively .some other properties of the fractals can also be determined  .their fractal dimension is , while their random - walk dimension is /\ln2 ] .in this section , we study the discrete - time unbiased random walks taking place on the fractals . in the process of the classical random walks , at each time step , the particle ( walker ) starts from its current location and moves to any of its nearest - neighboring nodes with the same probability equal to the reciprocal of the degree of the current node .now , consider the probability distribution function , which is the probability for the walker being on node at time .if the starting point for the walker is , then the stochastic master equation governing the evolution of is  with the initial condition .  in eq .( [ sme ] ) , if is not equal to , is the jumping probability from node to node , but otherwise ( ) .note that we can consider as an entry of matrix , called the transition matrix , which suggests that the powerful methods of the spectral theory for matrices  can be applied .we focus on a trapping problem which is a particular case of random walks on with a trap fixed on the central node of the fractals . for this case ,the transition matrix is a submatrix of , which is obtained from by deleting the row and column corresponding to the trap node . in the case without any confusion , we also use to represent the transition matrix of the trapping problem . belowwe will use the decimation method  to determine the complete spectrum of eigenvalues and their degeneracies , and demonstrate that the reciprocal of the smallest eigenvalue has the same scaling as that of the att .notice that previous works have studied analytically the laplacian spectra for different treelike graphs , e.g. , dendrimers  , vicsek fractals  , and related structures , say husimi - cacti  . herewe concentrate on the normalized laplacian matrix , since for an irregular graph , its laplacian matrix and normalized laplacian matrix behave quite differently , and the latter seems to be more natural  .the decimation method has an advantage that one can solve the eigenvalue equation based on a similar problem from the previous iteration .let denote the set of nodes belonging to , and the set of nodes generated at iteration .then , one can write the eigenvalue problem for the transition matrix in the following block form : \left[\begin{array}{cccc }  u_{\alpha } \\  u_{\beta } \end{array } \right]={\lambda}_{g+1 } \left[\begin{array}{cccc }  u_{\alpha } \\u_{\beta } \end{array } \right],\ ] ] where is the identity matrix since there are no transitions between the original nodes , is block diagonal with each block being the same matrix of the form ,\ ] ] and is an eigenvalue of .  equation ( [ t1 ] ) can be rewritten as two equations :  from eq .( [ t4 ] ) , one obtains provided that the concerned matrix is invertible . inserting eq .( [ t5 ] ) into eq .( [ t3 ] ) leads to {\alpha}={\lambda}_{g+1}u_{\alpha}.\ ] ]  thus , the problem of determining the eigenvalue for matrix of order is reduced to computing the eigenvalue problem of a matrix with a smaller order of .let $ ] .next , we will show that matrix can be related to matrix , and thus the eigenvalue for matrix can be expressed in terms of eigenvalue for matrix . to this end , we define where is the identity matrix of the same order as , and equation ( [ t7 ] ) indicates that the eigenvalues of are related to those of . in the appendix , we prove that , thus the eigenvalues of can be obtained through determining the spectrum of .  substituting for , eq .( [ t6 ] ) becomes which means that is , if denotes the eigenvalue of corresponding to the eigenvector , eq .( [ t12 ] ) implies plugging eqs .( [ t8 ] ) and ( [ t9 ] ) in eq .( [ t13 ] ) , one can see that the eigenvalue for transition matrix is related to that of via which can be recast as solving eq .( [ t15 ] ) gives its two roots which provide two eigenvalues of .by calculating the eigenvalues numerically , one can find some important properties of the spectra .first , all eigenvalues appearing at a given generation always exist at the next generation .second , all new eigenvalues at generation are just those generated via eq .( [ t16 ] ) by substituting with that are newly added to generation .that is , all eigenvalues can be obtained by the recurrence relation eq .( [ t16 ] ) .then , the next step is to determine the degeneracies of the eigenvalues , based on the two characteristics .let denote the multiplicity of eigenvalue of the transition matrix .since each eigenvalue in the fractals can be derived from eigenvalue 1 , we first determine the number of eigenvalue 1 of . for this purpose ,let be the rank of matrix .then , the number of eigenvalue of is   for the two limiting cases of and , we can easily obtain and .when , is the identity matrix of order , so . when , is block diagonal , with each of its blocks being the same matrix in the form of .\ ] ] obviously , the rank of each block is 2 .thus , we have .in order to determine for the case of , we define as the transition matrix of with the trap located at an outmost node , e.g. , node in fig .[ const2 ] , and define .we next show that it is possible to determine exactly the multiplicity of eigenvalue 1 for the fractals at every iteration .note that for , has the following form : ,\ ] ] where  satisfies ,\ ] ] in which each ( ) is a matrix of order , which has elements 0 and only one nonzero element describing the transition probability from one node in a replica of to the innermost node being joined together to form ( see fig . [ const2 ] ) ; and the superscript of a matrix represents its transpose .  by construction , for any node in ( ) that is directly connected to the center node , it must have a neighbor with degree 1 .thus , in matrix , there is only one nonzero element for row and for column , respectively , namely , and .hence , we can use elementary operations of a matrix to eliminate all nonzero element on the last row and the last column of .then , we have , which implies that .this , together with , leads to . combining the above - obtained results , the degeneracy of eigenvalue 1 is   we proceed to determine the multiplicities of other eigenvalues .for the convenience of description , on the basis of eq .( [ t14 ] ) , we define a set it is easy to verify that the cordiality ( number of elements of a set ) of is . on the other hand , from the above discussion, we can conclude that all the eigenvalues of belong to the set .moreover , extensive numerical computation on the eigenvalues of the studied fractals shows that the multiplicities of eigenvalues for the transmission matrix of the fractals satisfy . using this property ,we can obtain the degeneracy of each eigenvalue in as taking into account the degeneracies so obtained , the number of eigenvalues of is found to be which indicates that we have found all the eigenvalues of .   the probability distribution function considered above is very important since it describes the evolution of the system .many quantities of interest are encoded in this underlying probability distribution function .for example , in the limit of large system sizes , the att denoted by for the trapping problem on is asymptotically equal to the inverse of the smallest nonzero eigenvalue of , denoted by , that is , .here , is defined as the average of trapping time of node over all nodes except the innermost trap node .conceptually , is the expected time for a walker starting from node to reach the trap for the first time .  according to the recurrence relation given by eq .( [ t16 ] ) that dominates the eigenvalues of the transmission matrix for the fractals at two consecutive iterations , we can evaluate the smallest nonzero eigenvalue of . from eq .( [ t16 ] ) , we obtain the following obvious recursive relation : using taylor s formula , we have considering the initial condition , eq .( [ s2 ] ) can be solved by induction to yield ^{g-1}}\,.\ ] ] then , for large networks ,   we continue to represent as a function of the network size , in order to derive the scaling between these two quantities . from , we have .hence , for large systems , can be expressed in terms of as thus , for the entire family of the studied fractals , in the limit of large system sizes , the att grows as a power - law function of the network size .we note that the obtained exponent perfectly agree with the previously obtained result  .the so - called discrete - time random - walk problem on a graph can be described by a matrix , therefore the distribution of its eigenvalue spectrum is much of interest . in this paper , we have studied the trapping problem on a class of regular treelike fractals with a deep trap located at the central node .the self - similar structure of these fractals can be characterized analytically in terms of their eigenvalues .we have derived a recursive relationship between eigenvalues of a relevant matrix for random walks on the fractals at different iterations by making use of renormalization , for which the computational demands are very low . on the basis of the obtained recursion relation ,we have characterized the complete set of eigenvalues and further determined their degeneracies .moreover , we have provided a recursion relation between the smallest nonzero eigenvalues of the fractals at two successive iterations and obtained an approximate expression for the smallest nonzero eigenvalue .finally , we have evaluated the mfpt to the trap , showing that it is approximately equal to the inverse of the smallest eigenvalue .this work was supported by the national natural science foundation of china under grant no .61074119 and the hong kong research grants council under the grf grant cityu 1114/11e .in order to prove that , it suffices to show that their corresponding entries , denoted by and , are equal to each other .since both and are related to ( or ) , we first determine the entry of before evaluating and .  by the construction of the fractals , the element of the block matrix ( see eq .( [ t1 ] ) ) can be easily found . according to the properties of fractals , we have , where is the delta function defined as if , and otherwise . for and , we distinguish two cases : if the nodes and are directly connected by an edge in , then , where is the degree of node in ; otherwise .finally , for the diagonal block matrix , each entry is given in eq .( [ t2 ] ) .notice that is relevant to the inverse of the matrix .so we should determine .recall that is block diagonal , then is also block diagonal with each block being . since our goal is to calculate , as will be shown , it suffices to find the entry of at its first line and first column , which is denoted by . by definition , we have where is the complex adjugate matrix of .it is easy to verify that which yields                                            l. lovz , _ random walks on graphs : a survey _ , in combinatorics , paul erds is eighty vol .2 , edited by d. mikls , v. t. s , and t. sznyi ( jos bolyai mathematical society , budapest , 1996 ) , pp . 353 - 398 ; http://research.microsoft.com/ users / lovasz / erdos.ps .

Abstract: we study random walks on a family of treelike regular fractals with a trap fixed on a central node . we obtain all the eigenvalues and their corresponding multiplicities for the associated stochastic master equation , with the eigenvalues being provided through an explicit recursive relation . we also evaluate the smallest eigenvalue and show that its reciprocal is approximately equal to the mean trapping time . we expect that our technique can also be adapted to other regular fractals with treelike structures .