Rough Shapley functions for games with a priori unions
Abstract
A family of allocation rules for cooperative games with a priori unions is introduced in this paper. These allocation rules, which will be called rough Shapley values, are extensions of the well-known Shapley value for classical cooperative games. The family of rough Shapley values, which is constructed by using rough sets to describe different cooperative options, includes several of the extensions of the Shapley value found in the literature. We prove that the rough Shapley values are the allocation rules for games with a priori unions that satisfy some reasonable properties.
Full text
ARTICLE TEMPLATE Rough Shapley functions for games with a priori unions M. Basallotea, J.M. Gallardoa, C. Hern´andez-Manceraaand A. Jim´enez-Losadaa aDepartment of Applied Mathematics II, ETSI, University of Seville, Spain ARTICLE HISTORY Compiled January 12, 2024 ABSTRACT A family of allocation rules for cooperative games with a priori unions is introduced in this paper. These allocation rules, which will be called rough Shapley values, are extensions of the well-known Shapley value for classical cooperative games. The family of rough Shapley values, which is constructed by using rough sets to describe different cooperative options, includes several of the extensions of the Shapley value found in the literature. We prove that the rough Shapley values are the allocation rules for games with a priori unions that satisfy some reasonable properties. KEYWORDS game theory; cooperative games; rough sets; Shapley value; a priori unions 1. Introduction The theory of cooperative games with transferable utility studies situations in which a group of agents cooperate to obtain some profit. One of the main goals of this theory is to provide fair allocation rules for distributing the profit among the agents. The Shapley value [17] is the most commonly used allocation rule. But the use of the Shapley value requires symmetry in the relationships between the agents. However, there are often special alliances or incompatibilities which condition the way in which the agents interact. These situations require specific models. The model on which this article is based is that of cooperative games with a priori unions, introduced by Owen [13] in 1977, which is focused on the study of cooperative situations where there are prior group alliances between agents. Such alliances are described by partitions in the agent set. Owen’s model has been generalized using other types of structures, such as coalition configurations [1], proximity relations [5] or colored graphs [12]. Owen obtained and characterized a value for games with a priori unions. This value extends the Shapley value in the sense that in the particular case that there are no previous alliances, it is equal to the Shapley value. Since 1977, multiple extensions of the Shapley value have been obtained for games with a priori unions (see, for instance, [2,3,11]). Besides, modifications of the Shapley value for a priori unions were given by Brink and Dietz [4] and by Gon¸calves-Dosantos and Alonso-Meijide [8]. Regarding Owen’s model for cooperative games with a priori unions, the reasons for the formation of prior groups can be various, such as interests related to the game itself, external interests, family or friendship relationships or geographical proximity. It is at CONTACT A. Jim´enez-Losada. Email: a[email protected]
this point that the motivation for the present paper arises. It seems reasonable that the nature of the previous alliances will influence how players from different coalitions cooperate. Thus, a group may act coercively (this is the case of the situations studied in [9]), preventing its members from cooperating unless the entire group participates, or it may be indifferent, or it could even encourage its members to cooperate with other agents, as long as the whole group benefits from such cooperation. These different behaviors should affect the allocation rule. In order to formally describe the different possible behaviors of the prior groups, we propose to use rough sets, introduced by Pawlak [14] in 1982. In each set of agents (coalition) we will consider the partition induced by the prior groups, and then we will use some coefficients to describe the behavior of the sets in this partition. In game theory, rough sets have been used mostly in non-cooperative games (see, for instance, [10]). In cooperative games, Polkowski [15] used them to estimate the worths of certain coalitions in games with incomplete information. In this paper we use rough sets to obtain a family of values for cooperative games with a priori unions. These values are extensions of the Shapley value. The method that we use is constructive and it allows a parametric description of the values in the family, similarly to how Radzik and Driessen [18] describe a family of solutions for standard cooperative games. Besides, we introduce a new approach to obtain solutions for cooperative games with a priori unions, in the sense that we do not use a model based on a two-step negotiation. This paper is organized as follows. In Section 2 we recall some preliminaries concerning cooperative games and games with a priori unions. In Section 3 we present an axiomatic description of a family of extensions of the Shapley value for games with a priori unions. In Section 4, another family of extensions of the Shapley value is described. In this case the description is not axiomatic. Instead, this family is obtained by applying the Shapley value to a parametric modification of the underlying game taking into account the a priori unions. Finally in Section 5 we show the equivalence between the families obtained in Sections 3 and 4. 2. Games with a priori unions Let Nbe a finite set with |N|=n. A cooperative game on Nis defined by a characteristic function v: 2N→Rwhere 2Nis the set of all subsets of Nand v(∅) = 0. The elements of Nare called players, the subsets S⊆Ncoalitions and v(S) is the worth of S, that is, the collective payment that the players in Swould obtain if the cooperate. We denote by GNthe family of cooperative games with set of players N. The set of games GNis a real vector space with dimension 2n−1 if we consider the operations v+wand av with v, w ∈GNand a∈Rgiven by (v+w)(S) = v(S)+w(S) and (av)(S) = av(S) for every S⊆N. If T∈2N\{∅} the unanimity game of Tis the game uT∈GNdefined by uT(S) = 1 if T⊆Sand uT(S) = 0 otherwise. Unanimity games form a basis of GN, that is, every game v∈GNcan be written as v=X {T⊆N:T6=∅} ∆v TuT,(1) where the numbers ∆v Tare called the Harsanyi coefficients of v. Player i∈Nis said to be null in v∈GNif v(S) = v(S\ {i}) for every S⊆N. If θis a permutation of Nthen we denote θi =θ(i) for each i∈Nand θS ={θi :i∈S}for each S⊆N. If v∈GNthen the permuted game is the game θv ∈GNwith θv(θS) = v(S) 2
for every S⊆N. A payoff vector for a game v∈GNis any vector x∈RN. The component xiis interpreted as the payment of player i∈N. A value on GNis a function fN:GN→RN, which assigns a payoff vector fN(v) to each game v∈GN. The best-known value for cooperative games is the Shapley value [17], φN. For each game v∈GNthe payoff of a player i∈Nis defined by φN i(v) = X {S⊆N:i∈S} γn s[v(S)−v(S\{i})] ,with γn s=(s−1)!(n−s)! n!.(2) where s=|S|. The Shapley value satisfies the following properties: (S1) efficiency: if v∈GN, then Pi∈NφN i(v) = v(N); (S2) linearity: if v, w ∈GNand a, b ∈R, then φN(av +bw) = aφN(v) + bφN(w); (S3) null player property: if i∈Nis a null player for v∈GN, then φN i(v) = 0; (S4) symmetry: if θis a permutation of Nand v∈GN, then φN θi(θv) = φN i(v) for every i∈N. Notice that symmetry implies equal treatment of symmetric players, that is, if i, j ∈N,v∈GNand v(S∪{i}) = v(S∪{j}) for all S⊆N\{i, j}, then φi(v) = φj(v). These properties characterize the Shapley value, that is, φis the unique value on GNthat satisfies linearity, efficiency, null player property and symmetry. A solution function for cooperative games is a mapping f that assigns a value fNon GNfor each finite set N. The Shapley function is the solution function φthat assigns to each finite set of players Nthe Shapley value on N. If v∈GNand N0⊂N, then the game v∈GN0is the characteristic function v restricted to 2N0. The Shapley function satisfies this property: (S5) if Nis a finite set of players and i∈Nis a null player in v∈GN, then φN j(v) = φN\{i} j(v) for all j∈N\{i}. Owen [13] introduced games with a priori unions. Starting from the idea that in a cooperative situation some players are more likely to act together than others, he considered prior alliances in the set of players. An a priori unions structure or partition is a family Pof disjoint nonempty sets of players, namely a partition of a certain set N(P) = SA∈P A. In our case N=N(P). Each element A∈ P is called a union or group and it is interpreted as a set of players with common interests. A partition P is individualist if |A|= 1 for all A∈ P, and it is globalist if |P| = 1. A coalitional solution function is a mapping fthat assigns a value fPon GNto each partition P. In particular, all solution functions are coalitional solution functions which do not take into account the partition, that is, fP(v) = fP0(v) if N(P) = N(P0) = N. Owen [13] described a two-step method to obtain a coalitional solution function, which turned out to be an extension of the Shapley function. Let P={A1, ..., Am}be a partition and v∈GN. Let us denote M={1, ..., m}. The quotient game vP∈GMfor vis given by vP(Q) = vN(PQ)for all Q⊆Mwhere PQ={Aq:q∈Q}. For each S⊆A∈ P we define the restricted partition PS= (P \{A})∪{S}. A game wq∈GAqfor each q∈Mis now introduced as wq(S) = φM q(vPS) for all S⊆Aq. Finally, the Owen function is defined as the coalitional solution function ψwhere for every partition P, v∈GNand i∈N, ψP i(v) = φAq i(wq),(3) where Aqis the group in Psuch that i∈Aq. In both globalist and individualist partitions the Owen function and the Shapley function coincide, that is, ψP(v) = φN(v) for all v∈GN. The Owen value ψPsatisfies (S1-S3) but not (S4). The Owen function has two main criticisms: it satisfies the null player (thus denying a reasonable 3
payoff to any null player that is part of a coalition with at least one non-null player) and the equal treatment to the individualist partition and the globalist partition (it does not differentiate between these two extreme cases). 3. Coalitional Shapley functions Several extensions of the Shapley value for games with a priori unions have been studied in the literature [2–4,8,11]. Not all of them retain the essence of the Shapley value. For example, extensions have been defined that are not linear or that provide only group payoffs (not individual ones). With the idea of describing a family of coalitional solution functions, we propose an axiomatic definition that ensures that we maintain fundamental properties of the Shapley value. In addition, we have searched for axioms that are satisfied by the majority of extensions of the Shapley value for games with a priori unions introduced in the literature. We propose the following axioms for a coalitional solution function f. Efficiency. If Pis a partition and v∈GN, then X i∈N fP i(v) = v(N). Linearity. If Pis a partition, a, b ∈Rand v, w ∈GN, then fP(av +bw) = afP(v) + bfP(w). Let Pbe a partition. A permutation θover Nis P-compatible if θA ∈ P for all A∈ P. Compatible symmetry. If θis a permutation P-compatible over Nfor a partition P, then fP θi (θv) = fP i(v) for all v∈GN. Observe that compatible symmetry implies equal treatment of symmetric players i, j ∈Nin vif i, j are in the same group, in the sense that fP i(v) = fP j(v) (this is called intracoalitional symmetry in [7]). Let Pbe a partition. The set A∈ P is said to be a null group for v∈GNif any i∈Ais a null player for game v. The following axiom is a slight modification of the property of independence of null coalitions introduced in [7]. Null group property. Let Pbe a partition. If A∈ P is a null group for v∈GN then fP i(v) = 0,if i∈A, fP\A i(v),otherwise. If we take an individualistic partition then the axioms above correspond to the classical axioms (S1-S4) of the Shapley value plus condition (S5). Definition 3.1. A coalitional solution function is said to be a coalitional Shapley function if it satisfies efficiency, linearity, compatible symmetry and null group property. The Shapley function and the Owen function are examples of coalitional Shapley functions. Other examples are the values introduced in [11] and [3]. The proportional Shapley value introduced in [2] is not a coalitional Shapley function since it is not linear. Later we will show that there are more coalitional Shapley functions. In the following section we present a constructive method that allows to obtain a family of coalitional Shapley functions. With this method we will achieve the following: 4
•A parametric description of the family, similar to the description obtained by Radzik and Driessen [18] for a family of solutions for standard cooperative games. •A new approach to obtain solutions for games with a priori unions, which is not based on a two-step negotiation. •A way of modeling the attitude of the groups in the negotiation. 4. Rough Shapley functions In the approach proposed by Owen [13] to obtain a solution for games with a priori unions, two assumptions stand out: 1) in order to determine the collective payoff of a prior coalition, all the prior coalitions must act en bloc, and 2) in order to determine the payoff of each player in a prior coalition, the rest of the prior coalitions must act en bloc. Therefore, if the partition of Nis P={A1, ..., Am}, the only coalitions that matter in Owen’s model are those of the form Ai1∪. . . ∪Aik−1∪Ewhere E⊆Ak. The rest of the coalitions are totally ignored. This, in addition to entailing loss of information, seems somewhat arbitrary, since it presupposes certain attitudes of the players towards coalition formation. Our goal is to introduce a broader model that allows to consider a range of attitudes of the players, and in which Owen’s model is a particular case. Thus, in this paper, in order to obtain a coalitional Shapley value, we will have to somehow describe how partitioning affects coalition gains. 4.1. Rough versions of a game with a priori unions We propose to use rough set theory [14] in the context of cooperative games with a priori unions. Let Pbe a partition and N=N(P). If S⊆Nthen the interior and the closure of Sin Pare, respectively, ˚ S=[ {A∈P:A⊆S} Aand ¯ S=[ {A∈P:A∩S6=∅} A. (4) Coalition Sis said to be crisp if ˚ S=¯ S, that is, if it is the union of groups in the partition. Any non-crisp coalition is called a rough coalition. Suppose that the players in a coalition Sintend to cooperate. If Sis crisp, then it is clear all the players in Scan cooperate without any restrictions and the players in N\Swill play no role. If Sis not crisp, then the situation is different, since the players in Sthat are not in ˚ Sdo not have complete freedom to cooperate within S, since they are in prior groups which are not contained in S, and, therefore, the players in these groups could impede their cooperation. In this case, following rough set theory, we will use the lower approximation (i.e., the interior) and the upper approximation (i.e., the closure) of S, and we will use some coefficients to express the payment achievable by coalition S(taking into account the a priori union structure) as a linear combination of the payments (ignoring the a priori union structure) of the coalitions in the interval [˚ S, ¯ S] = {R:˚ S⊆R⊆¯ S}. The coefficients of this linear combination will be denoted αS Rfor each R∈[˚ S, ¯ S]. The set of all coefficients will be α=nαS RR∈[˚ S, ¯ S]:S⊆No. In this way we can consider a new game vαas vα(S) = X R∈[˚ S, ¯ S] αS Rv(R) (5) 5
for each S⊆N. It is clear that this construction is too general, and that we need to require conditions on the coefficients αS R. Definition 4.1. Let Pbe a partition of a finite set N. A rough coefficient for Pis a set of real numbers α=nαS RR∈[˚ S, ¯ S]:S⊆Nothat, for any A∈ P, satisfies the following three conditions: 1) Sharpness: αA A= 1. 2) Invariance: αS∪A R∪A=αS Rfor all S⊆N\Aand R∈[˚ S, ¯ S]. 3) Completeness: if S⊆N\A,R∈[˚ S, ¯ S] and S(T(S∪A, then X L∈[∅,A] αT R∪L=αS R. The family of rough coefficients for Pis denoted by RCP. For each α∈RCPand v∈GN, the game vαdefined by (5) will be called α-rough version of v. For any partition, the family of rough coefficients is nonempty, since we can take αS S= 1 for all S⊆Nand the rest of numbers equal to zero. This rough coefficient is called the crisp coefficient and it is denoted by δ. It is clear that vδ=vfor any cooperative game v∈GN. Moreover if Pis individualistic then the only rough coefficient is the crisp coefficient and hence the only rough version of the game is itself. Our aim now is to justify the conditions in Definition 4.1. The following proposition shows that the α-rough versions of cooperative games are the games vαwhich do not modify either the payments of crisp coalitions or the character of null players which are contained in null groups. Proposition 4.2. Let Pbe a partition of a finite set N. The numbers α= nαS RR∈[˚ S, ¯ S]:S⊆Noform a rough coefficient if and only if, for every cooperative game v∈GN, the following conditions hold: 1) vα(S) = v(S)for every crisp coalition S⊆N, 2) if i∈A∈ P and all players in Aare null players in v, then iis null in vα. Proof. Firstly we will take a rough coefficient αand prove that the conditions above are satisfied. 1) If Sis crisp then Sis a union of groups, that is, S=A1∪. . . ∪Akwith A1, . . . , Ak∈ P. By sharpness, αA1 A1= 1. By invariance, it follows that αA1∪A2 A1∪A2= 1. Successively applying the invariance property we obtain that αS S= 1. Since [˚ S, ¯ S] = {S}, we have vα(S) = αS Sv(S) = v(S). 2) Suppose that A∈ P and all players in Aare null players in v. If T={i1, ..., ip} ⊆ A, then, for all R⊆N, v(R) = v(R∪{i1}) = v(R∪{i1, i2}) = ··· =v(R∪{i1, ..., ip−1}) = v(R∪T).(6) Let S, T ⊆Nbe such that S⊆N\Aand T⊆A. We will prove that vα(S∪T) = vα(S).(7) 6
We can suppose that Tis nonempty. If T=Athen A⊆˚ S∪Aand hence we have [˚ S∪A, ¯ S∪A] = {R∪A:R∈[˚ S, ¯ S]}. For each R∈[˚ S, ¯ S] we have αS∪A R∪A=αS R by the invariance property of α, and also v(R∪A) = v(R) by (6). Therefore, vα(S∪A) = X R∈[˚ S, ¯ S] αS∪A R∪Av(R∪A) = X R∈[˚ S, ¯ S] αS Rv(R) = vα(S). Now we consider T6=A. In this case, we obtain the interval [˚ S∪T, ¯ S∪T] = {R∪L:R∈[˚ S, ¯ S], L ∈[∅, A]}. By completeness, each R∈[˚ S, ¯ S] satisfies PL∈[∅,A]αS∪T R∪L=αS R. We also have that v(R∪L) = v(R) by (6). Therefore, vα(S∪T) = X R∈[˚ S, ¯ S]X L∈[∅,A] αS∪T R∪Lv(R∪L) = X R∈[˚ S, ¯ S] v(R)X L∈[∅,A] αS∪T R∪L =X R∈[˚ S, ¯ S] v(R)αS R=vα(S). Finally we take i∈Aand we prove that iis a null player in vα. Let S⊆N\{i}. If we apply (7) to the pair (S\A, (S∩A)∪{i}) and also to the pair (S\A, S ∩A), we obtain vα(S∪{i}) = vα(S\A) = vα(S). Now let α=nαS RR∈[˚ S, ¯ S]:S⊆Nobe such that conditions 1 and 2 described in the proposition are satisfied. Let us prove that αis a rough coefficient. Take A∈ P. Since Ais crisp, we have, by condition 1, that uα A(A) = uA(A) = 1. Besides, by definition, uα A(A) = αA AuA(A) = αA A. We conclude that αsatisfies sharpness. Now consider S⊆N\Aand ∅ 6=S0⊆A. We aim to check that if R∈[˚ S, ¯ S], then X L∈[˚ S0,¯ S0] αS∪S0 R∪L=αS R.(8) We will proceed by induction on d(R) = |¯ S|− |R|. We will use the unanimity game uR. Notice that all players in Aare null players in uR. Base case. If d(R) = 0 (i.e., R=¯ S) we take the game u¯ S. By condition 2 all players in S0are null players in uα ¯ S. From (6) and definition (5) we obtain X L∈[˚ S0,¯ S0] αS∪S0 ¯ S∪L=uα ¯ S(S∪S0) = uα ¯ S(S) = αS ¯ S. Inductive step. Suppose that (8) is true for d(R)< k. Let us check that then it is true for d(R) = k. If we calculate uα R(S∪S0) and uα R(S) and take into account (6) and 7
condition 2, we obtain X R0∈[R, ¯ S]X L∈[˚ S0,¯ S0] αS∪S0 R0∪L=uα R(S∪S0) = uα R(S) = X R0∈[R, ¯ S] αS R0.(9) For each R0∈[R, ¯ S]\ {R}we have that d(R0)< k and we can apply the induction hypothesis. Therefore, PL∈[˚ S0,¯ S0]αS∪S0 R0∪L=αS R0. From this and (9) we conclude that X L∈[˚ S0,¯ S0] αS∪S0 R∪L=αS R. We have proved (8). Now, if we apply this equality to S0=Athen [˚ S0,¯ S0] = {A}and we obtain invariance. And if we apply (8) to S0∈[∅, A]\{∅, A}then [˚ S0,¯ S0] = [∅, A] and, by taking T=S∪S0, we obtain completeness. 4.2. The magic cube of a rough coefficient In this section we will identify a rough coefficient with a magic cube. Then we will solve the system of equations that determines a rough coefficient. This will give us information about the dimension of the family of rough coefficients. Let N={1, ..., n}and P={A1, ..., Am}. We assume that we have a coalition Sas S=S1∪ ··· ∪ Smwhere Sk(Akis a nonempty subset in each group1. The completeness condition says X R⊆Ak αS Sm p=1,p6=kTp∪R=αS\Sk Sm p=1,p6=kTp for any k= 1, ..., m and Tp⊆Apwith p= 1, ..., m,p6=k. In a similar way, using again the completeness property, X R⊆Ak αS\Sq∈QSq Sq∈QTq∪R=αS\Sq∈Q∪kSq Sq∈QTq for all nonempty Q⊆ {1, ..., m}, with k /∈Qand with cardinality |Q|< m −1. If |Q|=m−1, namely Q={1, .., m}\{k}, then we actually have X R⊆Ak αSk ∅∪R=α∅ ∅= 1. Thus we obtain a magic cube structure. Figure 1 shows this idea with three groups. We use the notation αS R1R2R3to represent αS R1∪R2∪R3with Rk⊆Akfor k= 1,2,3. We aim to calculate the dimension of the set of rough coefficients. We will solve the linear system formed by the conditions of the definition of rough coefficient for a given coalition S. Invariance implies again that we can assume S=S1∪···∪Smwith ∅ 6=Sk(Akfor every k= 1, ..., m. Let |Ak|=akand consider a1≤ ··· ≤ am. Our unknowns are the coefficients αS Rwith R⊆N. For this goal, we will describe the system 1If Sk=Akfor some kthen the invariance condition implies that this group can be removed. 8
α∅∅∅ #$$$$ α∅%&∅ #$$$$$$ α∅%&%' #$$$$$$$ α%(%&∅ #$$$$$$$ α%(∅∅ #$$$$$$$ α%(%&%' #$$$$$$$$$ α%(∅%' #$$$$$$$$$ α∅∅%' #$$$$$$$ α%(∅ #\#& α∅∅ #\#& α%(%' #\#& α∅%' #\#& α ∅∅ # \ # ' α ∅%& # \ # ' α%(%& #\#' α%(∅ #\#' α∅∅ #\#(α%&∅ #\#( α∅%' #\#(α%&%' #\#( 1 α ∅ #$ α % $ #$ α∅ #& α%& #& α∅∅ #\#( α∅%& #\#( Figure 1. 3D rough magic cube. as a matrix using the following orders for unknowns and equations. We consider each family 2Akordered by the size and the labels. For example, if A1={1,2,3}then we take the order in 2A1:{∅,1,2,3,12,13,23, A1}. The unknowns are ordered as follows. First we set a subset of A1, then one of A2, and so on until we reach Amwhere we traverse all subsets of it using the above internal order in 2Am. We repeat the process by changing the element fixed in Am−1, later with Am−2an so on until A1. Let’s look at the idea in the three-group example. In this case, we go through the faces in the cube that face us (those of the direction A1) from the bottom towards us; in each face, from the left column to the right; and, in each column, from top to bottom: αS ∅∅∅, ..., αS ∅∅A3, ..., αS ∅A2∅, ..., αS ∅A2A3, ..., αS A1∅∅, ..., αS A1∅A3, ..., αS A1A2∅, ..., αS A1A2A3. The number of unknowns is 2Pm k=1 ak. We order the equations transversely to the unknowns. First we put the equations running through A1with the other groups with fixed coalitions (again taking into account the internal order of the groups). Then the ones that run through A2and so on. For example, in the figure the first equations are 9
by the reduction property of α, that vα(P)(S) = X T∈[˚ S, ¯ S] αS T(P)v(T) = X T∈[˚ S, ¯ S] αS T(P \A)v(T) = vα(P\A)(S) where [˚ S, ¯ S] is the same in Pand P \Abecause S⊆N\A. Let A={i1, ..., ip}. We proved before that i1, ..., ipare null players in vα(P). By (S5), φP,α i(v) = φN i(vα(P)) = φN\{i1} i(vα(P)) = ··· =φN\A i(vα(P)) =φN\A i(vα(P\A)) = φP\A,α i(v). In the following example we will show the construction of a rough Shapley function, which will be a coalitional Shapley function different from the Owen value and from the classic Shapley function. Example. We aim to define a particular family of rough coefficients α. To this end, we need a previous construction. Suppose that Ais a finite set and Sa nonempty set with S(A. For all T⊆Awe define hA(S, T) = |S∩T| |S|2|A|−1, and hA(A, A) = hA(∅,∅) = 1. If we randomly (and with equal probability) choose an element in Sand then randomly (and with equal probability) choose an subset of A that contains the element chosen, the number hA(S, T) is equal to the probability that the subset of Achosen is equal to T. The following equality holds: X T∈[∅,A] hA(S, T)=1.(11) If |S|=s,|A|=aand p=|S∩T|then X T∈[∅,A] hA(S, T) = X T∈[∅,A] |S∩T| |S|2|A|−1=1 |S|2|A|−1X T∈[∅,A]|S∩T| =1 s2a−1 s X p=0 s p2a−sp=1 s2s−1 s X p=1 s pp =1 s2s−1 s X p=1 s−1 p−1s=1 2s−1 s−1 X p=0 s−1 p= 1 where we have used two well-known combinatorial identities2. Now consider the partition P={A1, ..., Am}. For each S⊆Nand T∈[˚ S, ¯ S], we define αS T=Y k∈HP(S) hAk(Sk, Tk), 2Pr k=0 r k= 2r,kr k=rr−1 k−1 16
where Sk=S∩Akand HP(S) = {k:Sk6=∅}. Let us prove that αis a rough coefficient. Sharpness: since HP(Ap) = {p}we have that αAp Ap=hAp(Ap, Ap) = 1. Invariance: suppose that S⊆N\Ap. Clearly, HP(S∪Ap) = HP(S)∪{p}, (S∪Ap)p= Apand (S∪A)k=Skwith k6=p. Moreover, if T∈[˚ S, ¯ S] then T∪Apsatisfies the same properties. We have αS∪Ap T∪Ap=hAp(Ap, Ap)Y k∈HP(S) hAk(Sk, Tk) = αS T. Completeness: let S⊆N\Ap,R∈[˚ S, ¯ S] and T∈(S, S ∪Ap). Let T=S∪K with K∈(∅, Ap). Since HP(S∪K) = HP(S)∪ {p}we have that, if k6=p, then (S∪K)k=Skand (R∪L)k=Rk. Besides, (S∪K)p=Kand (R∪L)p=L. We have X L∈[∅,Ap] αS∪K R∪L=X L∈[∅,Ap]Y k∈HP(S∪K) hAk((S∪K)k,(R∪L)k) =X L∈[∅,Ap] hAp(K, L)Y k∈HP(S) hAk(Sk, Rk) =Y k∈HP(S) hAk(Sk, Rk)X L∈[∅,Ap] hAp(K, L) =Y k∈HP(S) hAk(Sk, Rk) = αS R=αS∪Ap R∪Ap, where we have used (11) and invariance. If we take the coefficient above α(P) for each partition Pwe obtain a family of rough coefficients, because, by construction, this family satisfies symmetry and reduction. Therefore, φαis a rough Shapley function, and, by Theorem 5.1, a coalitional Shapley function. Let us calculate φα(v) for a particular partition Pand a particular game v∈G{1,2,3}. Take P={{1,2},{3}} and the unanimity game u{13}∈G{1,2,3}. We have that h{1,2}({1},∅) = h{1,2}({2},∅) = h{1,2}({1},{2}) = h{1,2}({2},{1}) = 0, h{1,2}({1},{1}) = h{1,2}({2},{2}) = h{1,2}({1},{1,2}) = h{12}({2},{1,2})=1/2. Besides, by definition, h{1,2}(∅,∅) = h{1,2}({1,2},{1,2}) = h{3}(∅,∅) = h{3}({3},{3})=1. The rough version of a game v∈GNis vα({1}) = 1 2v({1}) + 1 2v({1,2}), vα({2}) = 1 2v({2}) + 1 2v({1,2}), vα({3}) = v({3}), vα({1,2}) = v({1,2}), vα({1,3}) = 1 2v({1,3})+1 2v(N), vα({2,3}) = 1 2v({2,3})+1 2v(N), 17
vα(N) = v(N). In particular, uα {1,3}({1,3}) = 1, uα {1,3}({2,3})=1/2, uα {1,3}(N) = 1 and uα {1,3}(S)=0 otherwise. The classical Shapley value and the Owen value coincide for this game, returning the payoff vector φN(u{1,3}) = (1/2,0,1/2). It is easy to check that φα(u{1,3}) = (4/12,1/12,7/12). Notice that φαdoes not satisfy the null player property. Indeed, φα 2(u{1,3})>0, although player 2 is null in u{1,3}. This player receives a payoff for being part of a coalition, {1,2}, that is not irrelevant. The following equality is a classic exercise of linear algebra that we will use in the proof of Theorem 5.2. Let x1, x2..., xn, y1, y2, ..., yn∈Rand n∈N. Then, Dn= x1+y1x1··· x1 x2x2+y2··· x2 . . .. . .··· . . . xnxn··· xn+yn = n Y k=1 yk"1 + n X k=1 xk yk#.(12) Moreover, Dn=xk n Y {l=1,l6=k} yl+ykDk n−1, where Dk n−1is the determinant obtained when we delete row kand column kin the expression of Dn The last result of this paper is devoted to prove that the rough Shapley functions “almost” complete the family of coalitional Shapley functions. A partition Pis said to be a partition into large groups if |A|>2 for every A∈ P. Games with a priori large unions model situations which are interesting to analyze. For instance, the Owen value was applied to these games to study the power in the European Parliament [6] and in the International Monetary Fund [16]. Theorem 5.2. If fis a coalitional Shapley function then it coincides with a rough Shapley function on partitions into large groups. Proof. Let fbe a coalitional Shapley function. Let Pbe a partition into large groups, that is, |A|>2 for all A∈ P. We will prove the result by induction on the cardinality |P| of the partition. Base case, |P| = 1.Consider frestricted to the games over the globalist partition P={N}. Notice that in this case compatible symmetry means symmetry because all the permutations over Nare compatible with {N}. Hence, fis a value that satisfies efficiency, linearity and symmetry (an ELS-value3following [18]). Proposition 2 in [18] says that there is a coefficient bsfor each cardinality s= 0, ..., n with b0= 14,bn= 1 3Moreover, the null group property only says f{N}(0) = 0. But combining efficiency with symmetry the same equality is achieved. Thus both families of values can be identified. 4Actually, in [18] took b0= 0 but this number is not significant. 18
and f{N} i(v) = X {S⊆N:i∈S} γn s[bsv(S)−bs−1v(S\{i})] . We take, for each non-crisp coalition S,αS T= 0 if ∅ 6=T6=S,αS S=b|S|and αS ∅= 1 − b|S|. Observe that we have constructed a rough coefficient for partition {N}. Sharpness: the only crisp coalitions in this partition are Nand ∅, and bn=b0= 1. Invariance: it is true because bn=b0. Completeness: let S=∅,A=Nand T∈2N\{∅, N}. We have X L∈[∅,N] αT L=αT ∅+b|T|= 1 = bn. Therefore, using this rough coefficient, we obtain φ{N},α =f{N}. Induction step. Suppose that fcoincides with a rough Shapley value for partitions with cardinality less than m, that is, there exists a family of rough coefficients αsuch that φP,α =fP(13) for any partition Pwith |P| < m. Take a partition P={A1, ..., Am}. Let N=SA∈P A={1, ..., n}. We look for a rough coefficient α(P) = αto define fP=φP,α. Reduction condition implies that all the numbers αS Twith S∩Ak=∅(or S∩Ak=Ak by invariance) for k= 1, ..., m are determined by (13). Following Section 3, coefficient αmust satisfy the rough magic cube. By (10) all the numbers αS Twith T∩Ak=∅for some group k= 1, ..., m are determined. Hence, only the numbers αS Twith T∩Ak6=∅ and S∩Ak6=Akfor all k= 1, ..., m are not determined. The rough coefficient must satisfy the equations φP,α i(v) = fP i(v) for each i∈Nand game von N. If we use the linearity axiom of fwe can reduce the system to the equations φi(uα H) = φP,α i(uH) = fP i(uH) (14) for all i∈Nand H⊆Nnonempty. Moreover, we can assume H∩Ak6=∅for all k= 1, ..., m. Indeed, if H∩Ak=∅for some k= 1, ..., m then, by the null group property, fP i(uH) = fP\Ak i(uH) for any i∈N\Akand fP i(uH) = 0 if i∈Ak. Therefore the payoffs in the games are determined. The same happens with any feasible φP,α. Suppose H=T1∪T2∪ ···Tmwith Tk⊆Akand Tk6=∅for any k= 1, ..., m. We consider the sets Tkin H(and also in P) such that: •Tk=Akfor k= 1, ..., p (p∈ {0, ..., m}), •Tk6=Akand |Tk|>1 for k=p+ 1, ..., p +q(q∈ {0, ..., m −p}), and • |Tk|= 1 for k=p+q+ 1, ..., m. Now we use the axiom of compatible symmetry for fPand φP,α to reduce the number of equalities in (14). So, we obtain the following cases: •For k= 1, ..., p : there is only one different payoff in Ak,φik(uα H) = fP ik(uH), because all the players are internal symmetric players in uH. 19
•For k=p+ 1, ..., m : there are two different payoffs in Ak,φik(uα H) = fP ik(uH) and φjk(uα H) = fP jk(uH) with ik∈Tkand jk∈Ak\Tk, because all the players in Tkare internal symmetric players in uHand also all the players in Ak\Tk. So, we have five types of payoffs (one in the first kind of equations, two in the second one, and two in the third one). We reduce (14) to p+ 2(m−p) equations with the structure X {S⊆N:ik∈S} γn |S|[uα H(S)−uα H(S\{ik})] = fP ik(uH), k = 1, ..., m (15) X {S⊆N:jk∈S} γn |S|[uα H(S)−uα H(S\{jk})] = fP jk(uH), k =p+ 1, ..., m (16) where uα H(S) = X R∈[˚ S∪H, ¯ S] αS R. We suppose them in the following order: for each k= 1, ..., p the equation in (15) and for every k=p+ 1, ..., m first the equation in (15) and second the equation in (16). We choose one player representing each group, that is, dH k∈Akfor any k= 1, ..., m, such that dH k∈Tkfor all k= 1, ..., p +qand dH k∈Ak\Tkfor all k=p+q+ 1, ..., m. This election is feasible because the groups are large. Let DH={dH 1, ..., dH m}. We select p+ 2(m−p) unknown numbers to solve (15,16): •for k= 1, ..., m :αDH∪{iH k} H, with iH k∈Tk\{dH k}for equation (15), •for k=p+ 1, ..., m :αDH∪{jH k} H, with jH k∈Ak\(Tk∪{dH k}) for equation (16). We consider these unknowns ordered like their associated equations. Let ak=|Ak| and tk=|Tk|for k= 1, ..., m. We will take into account the symmetry condition of αin order to join equal numbers and get the structure of the matrix in Figure 2, where we focus on the coefficients of our unknown numbers in order to study their determinant. The explanation of how we get the different coefficients is in the Annex of the paper. We note that some numbers Xl bwith l= 1,2,3,4,5 appear in Figure 2, whose formulation can be seen in the Annex, which are independent of the unknowns and which are repeated in all the equations of each row of blocks. In a second step we take out Xq bwith q= 1, ..., 5 for the different rows, observe that Xq b6= 0 for all k= 1, ..., 5. We also take out the values ac−1 2,tc−1 2, ac−tc,ac−2 2 from the columns of kinds 2,3,4 and 6 respectively. They are not zero because the groups are large. The reader must take into account the element of the main diagonal in each case. The result is Figure 3. We transform the main diagonal as 20
<latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="lfLj5ayaNSocNtcomVVIL7kzeBE=">AAAB6HicdZDLSgMxFIbPeK31VnXpJliELsowo8XaXcGNywr2Am0pmfRMG5u5kGSEMvQddCXqzufxBXwb02kFFf1XX87/B85/vFhwpR3nw1pZXVvf2Mxt5bd3dvf2CweHLRUlkmGTRSKSHY8qFDzEpuZaYCeWSANPYNubXM399j1KxaPwVk9j7Ad0FHKfM6rNqO2Wbdsux4NC0bFrmcgCqpUl1Fzi2k6mIizVGBTee8OIJQGGmgmqVNd1Yt1PqdScCZzle4nCmLIJHWHXYEgDVP00W3dGTv1IEj1Gkr2/Z1MaKDUNPJMJqB6r3958+JfXTbR/2U95GCcaQ2YixvMTQXRE5q3JkEtkWkwNUCa52ZKwMZWUaXObvKn/1ZH8D60z272wz28qxXppeYgcHMMJlMCFKtThGhrQBAYTeIQXeLXurAfryXpeRFes5Z8j+CHr7RMXgIxZ</latexit> 1,...,p <latexit sha1_base64="nmhA55uQ6b0yLqaOCKf/43IzGW0=">AAAB5HicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEqMICUxIp9yBhs5P2o4JIbyBroy684l8Ad/GMoyJGj2rr/ecJvdcPxFcacf5sAorq2vrG8XN0tb2zu5eef/gTsWpZNhmsYhl16cKBY+wrbkW2E0k0tAX2PEnlwu/c49S8Ti61dMEvZCOIh5wRrUZ3fCBPyhXHLuRiSyhXsuh4RLXdjJVIFdrUH7vD2OWhhhpJqhSPddJtDejUnMmcF7qpwoTyiZ0hD2DEQ1RebNs1Tk5CWJJ9BhJ9v6endFQqWnom0xI9Vj99hbDv7xeqoMLb8ajJNUYMRMxXpAKomOyaEyGXCLTYmqAMsnNloSNqaRMm7uUTP2vjuR/uDu13XP77LpWaVbzQxThCI6hCi7UoQlX0II2MBjBI7zAqxVYD9aT9byMFqz8zyH8kPX2Cfxli9g=</latexit> ib <latexit sha1_base64="5Fkg35Z2FUnas1FHsXr3ltfmo68=">AAAB5HicdZDLTgIxFIbP4A3xhrp000hMWJEZJSI7EjcuMcolgQnplDNQ6VzSdkwI4Q10ZdSdT+QL+DaWYUzU6L/6ev6/yfmPFwuutG1/WLmV1bX1jfxmYWt7Z3evuH/QVlEiGbZYJCLZ9ahCwUNsaa4FdmOJNPAEdrzJ5cLv3KNUPApv9TRGN6CjkPucUW1GN3cDb1As2ZV6KrKEWjWDukOcip2qBJmag+J7fxixJMBQM0GV6jl2rN0ZlZozgfNCP1EYUzahI+wZDGmAyp2lq87JiR9JosdI0vf37IwGSk0Dz2QCqsfqt7cY/uX1Eu1fuDMexonGkJmI8fxEEB2RRWMy5BKZFlMDlElutiRsTCVl2tylYOp/dST/Q/u04pxXzq6rpUY5O0QejuAYyuBADRpwBU1oAYMRPMILvFq+9WA9Wc/LaM7K/hzCD1lvn/3li9k=</latexit> jb <latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="nmhA55uQ6b0yLqaOCKf/43IzGW0=">AAAB5HicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEqMICUxIp9yBhs5P2o4JIbyBroy684l8Ad/GMoyJGj2rr/ecJvdcPxFcacf5sAorq2vrG8XN0tb2zu5eef/gTsWpZNhmsYhl16cKBY+wrbkW2E0k0tAX2PEnlwu/c49S8Ti61dMEvZCOIh5wRrUZ3fCBPyhXHLuRiSyhXsuh4RLXdjJVIFdrUH7vD2OWhhhpJqhSPddJtDejUnMmcF7qpwoTyiZ0hD2DEQ1RebNs1Tk5CWJJ9BhJ9v6endFQqWnom0xI9Vj99hbDv7xeqoMLb8ajJNUYMRMxXpAKomOyaEyGXCLTYmqAMsnNloSNqaRMm7uUTP2vjuR/uDu13XP77LpWaVbzQxThCI6hCi7UoQlX0II2MBjBI7zAqxVYD9aT9byMFqz8zyH8kPX2Cfxli9g=</latexit> ib <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="5Fkg35Z2FUnas1FHsXr3ltfmo68=">AAAB5HicdZDLTgIxFIbP4A3xhrp000hMWJEZJSI7EjcuMcolgQnplDNQ6VzSdkwI4Q10ZdSdT+QL+DaWYUzU6L/6ev6/yfmPFwuutG1/WLmV1bX1jfxmYWt7Z3evuH/QVlEiGbZYJCLZ9ahCwUNsaa4FdmOJNPAEdrzJ5cLv3KNUPApv9TRGN6CjkPucUW1GN3cDb1As2ZV6KrKEWjWDukOcip2qBJmag+J7fxixJMBQM0GV6jl2rN0ZlZozgfNCP1EYUzahI+wZDGmAyp2lq87JiR9JosdI0vf37IwGSk0Dz2QCqsfqt7cY/uX1Eu1fuDMexonGkJmI8fxEEB2RRWMy5BKZFlMDlElutiRsTCVl2tylYOp/dST/Q/u04pxXzq6rpUY5O0QejuAYyuBADRpwBU1oAYMRPMILvFq+9WA9Wc/LaM7K/hzCD1lvn/3li9k=</latexit> jb <latexit sha1_base64="4nlCEpLs9lt42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEhIBE5iQTrkDDZ2ftB0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65fiK40o7zYRXW1jc2t4rbpZ3dvf2D8uFRV8WpZNhhsYjlnU8VCh5hR3Mt8C6RSENfYM+fXi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUsvw9GMUtDjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Retlx0Ts6CWBI9QbJ8f89mNFRqFvomE1I9Ub+9xfAvr5/q4MrLeJSkGiNmIsYLUkF0TBZ9yYhLZFrMDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9tC/atUqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+t5FS1Y+Z9j+CHr7ROG94r8</latexit> b <latexit sha1_base64="ihMRwXZ4jfcYe29bFDJV6p2VUNM=">AAAB7nicdZDNSsNAFIUn9a/Wv6pLN4NF6CokVazdFdy4ESrYH2hDmUxv2qGTSZyZCKX0NXQl6s6H8QV8GydpBBU9qzP3nIH7XT/mTGnH+bAKK6tr6xvFzdLW9s7uXnn/oKOiRFJo04hHsucTBZwJaGumOfRiCST0OXT96WWad+9BKhaJWz2LwQvJWLCAUaLNyLsmTOARI+NIED4sVxy7kQkvTf0sNw0Xu7aTqYJytYbl98EookkIQlNOlOq7Tqy9OZGaUQ6L0iBREBM6JWPoGytICMqbZ0sv8EkQSawngLP39+6chErNQt90QqIn6neWDv/K+okOLrw5E3GiQVBTMVmQcKwjnLIbUAlU81lKTCUzW2I6IZJQbS5UMvhfjPh/06nZ7rl9elOrNKv5IYroCB2jKnJRHTXRFWqhNqLoDj2iF/RqxdaD9WQ9L6sFK/9ziH7IevsEF8KP9A==</latexit> Main diagonal <latexit sha1_base64="4nlCEpLs9lt42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEhIBE5iQTrkDDZ2ftB0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65fiK40o7zYRXW1jc2t4rbpZ3dvf2D8uFRV8WpZNhhsYjlnU8VCh5hR3Mt8C6RSENfYM+fXi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUsvw9GMUtDjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Retlx0Ts6CWBI9QbJ8f89mNFRqFvomE1I9Ub+9xfAvr5/q4MrLeJSkGiNmIsYLUkF0TBZ9yYhLZFrMDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9tC/atUqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+t5FS1Y+Z9j+CHr7ROG94r8</latexit> b <latexit sha1_base64="fJDXT3vCNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoKthSYNk+mkHTq5MDMRSgjoy+hK1J0P4Qv4Nk7TCCp6Vt/85wz85/diRoU0zQ+ttLK6tr5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uFv7NLeGCRuG1nMfECdAkpD7FSKqRq1dtxOIpGqWXo46Nk9hOqYsVZpnbcfWaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkxtMxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5fsKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKqv/VEf4P/RPDOjNOrxq1dr04RBlUwRGoAws0QRt0QBf0AAb34BG8gFftTnvQnrTnZbSkFX8OwQ9pb59HcJY9</latexit> ↵DH[{iH c} H <latexit sha1_base64="lfLj5ayaNSocNtcomVVIL7kzeBE=">AAAB6HicdZDLSgMxFIbPeK31VnXpJliELsowo8XaXcGNywr2Am0pmfRMG5u5kGSEMvQddCXqzufxBXwb02kFFf1XX87/B85/vFhwpR3nw1pZXVvf2Mxt5bd3dvf2CweHLRUlkmGTRSKSHY8qFDzEpuZaYCeWSANPYNubXM399j1KxaPwVk9j7Ad0FHKfM6rNqO2Wbdsux4NC0bFrmcgCqpUl1Fzi2k6mIizVGBTee8OIJQGGmgmqVNd1Yt1PqdScCZzle4nCmLIJHWHXYEgDVP00W3dGTv1IEj1Gkr2/Z1MaKDUNPJMJqB6r3958+JfXTbR/2U95GCcaQ2YixvMTQXRE5q3JkEtkWkwNUCa52ZKwMZWUaXObvKn/1ZH8D60z272wz28qxXppeYgcHMMJlMCFKtThGhrQBAYTeIQXeLXurAfryXpeRFes5Z8j+CHr7RMXgIxZ</latexit> 1,...,p <latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="fJDXT3vCNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoKthSYNk+mkHTq5MDMRSgjoy+hK1J0P4Qv4Nk7TCCp6Vt/85wz85/diRoU0zQ+ttLK6tr5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uFv7NLeGCRuG1nMfECdAkpD7FSKqRq1dtxOIpGqWXo46Nk9hOqYsVZpnbcfWaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkxtMxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5fsKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKqv/VEf4P/RPDOjNOrxq1dr04RBlUwRGoAws0QRt0QBf0AAb34BG8gFftTnvQnrTnZbSkFX8OwQ9pb59HcJY9</latexit> ↵DH[{iH c} H <latexit sha1_base64="fJDXT3vCNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoKthSYNk+mkHTq5MDMRSgjoy+hK1J0P4Qv4Nk7TCCp6Vt/85wz85/diRoU0zQ+ttLK6tr5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uFv7NLeGCRuG1nMfECdAkpD7FSKqRq1dtxOIpGqWXo46Nk9hOqYsVZpnbcfWaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkxtMxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5fsKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKqv/VEf4P/RPDOjNOrxq1dr04RBlUwRGoAws0QRt0QBf0AAb34BG8gFftTnvQnrTnZbSkFX8OwQ9pb59HcJY9</latexit> ↵DH[{iH c} H <latexit sha1_base64="2vt4dygS7EDahRMOKvCxEREiuqo=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoK9QJOGyXTSjp1cmJkIJQT0ZXQl6s6H8AV8G6dpBRU9q2/+cwb+83sxo0Ka5odWWFldW98obpa2tnd29/T9g66IEo5JB0cs4n0PCcJoSDqSSkb6MSco8BjpedOLud+7JVzQKLyWs5g4ARqH1KcYSTVy9bKNWDxBw/Ry2LJxEtvpjYsVZpnbcvWKaTRywQXUa0toWNAyzFwVsFTb1d/tUYSTgIQSMyTEwDJj6aSIS4oZyUp2IkiM8BSNyUBhiAIinDQvkcFjP+JQTgjM39+zKQqEmAWeygRITsRvbz78yxsk0j93UhrGiSQhVhHl+QmDMoLzW8AR5QRLNlOAMKdqS4gniCMs1cVKqv5XR/g/dE8M68w4vapVmtXlIYqgDI5AFVigDpqgBdqgAzC4B4/gBbxqd9qD9qQ9L6IFbfnnEPyQ9vYJSPeWPg==</latexit> ↵DH[{jH c} H <latexit sha1_base64="2vt4dygS7EDahRMOKvCxEREiuqo=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoK9QJOGyXTSjp1cmJkIJQT0ZXQl6s6H8AV8G6dpBRU9q2/+cwb+83sxo0Ka5odWWFldW98obpa2tnd29/T9g66IEo5JB0cs4n0PCcJoSDqSSkb6MSco8BjpedOLud+7JVzQKLyWs5g4ARqH1KcYSTVy9bKNWDxBw/Ry2LJxEtvpjYsVZpnbcvWKaTRywQXUa0toWNAyzFwVsFTb1d/tUYSTgIQSMyTEwDJj6aSIS4oZyUp2IkiM8BSNyUBhiAIinDQvkcFjP+JQTgjM39+zKQqEmAWeygRITsRvbz78yxsk0j93UhrGiSQhVhHl+QmDMoLzW8AR5QRLNlOAMKdqS4gniCMs1cVKqv5XR/g/dE8M68w4vapVmtXlIYqgDI5AFVigDpqgBdqgAzC4B4/gBbxqd9qD9qQ9L6IFbfnnEPyQ9vYJSPeWPg==</latexit> ↵DH[{jH c} H <latexit sha1_base64="ET2jitQP5EHo+rFRXeT1v16h8Ko=">AAAB93icbZDLSsNAFIZPvNZ6i5edm2ARurEkVdRlwY3LCvYCbQyT6Uk7dHJhZiLUkGfRlag738MX8G2c1iy09V99c/5/4JzfTziTyra/jKXlldW19dJGeXNre2fX3NtvyzgVFFs05rHo+kQiZxG2FFMcu4lAEvocO/74eup3HlBIFkd3apKgG5JhxAJGidIjzzzsev690w8EoRnx/FMnz+q5Z1bsmj2TtQhOARUo1PTMz/4gpmmIkaKcSNlz7ES5GRGKUY55uZ9KTAgdkyH2NEYkROlms+1z6ySIhaVGaM3ev7MZCaWchL7OhESN5Lw3Hf7n9VIVXLkZi5JUYUR1RHtByi0VW9MSrAETSBWfaCBUML2lRUdEd6B0VWV9vjN/7CK06zXnonZ2e15pVIsiSnAEx1AFBy6hATfQhBZQeIRneIN3Y2I8GS/G6090ySj+HMAfGR/fcjmSOA==</latexit> X1 b ab1 2 <latexit sha1_base64="oWnImmMjVj3ALKw/FQW5HpDcQ5k=">AAACCnicbVDNSsNAGNzUv1r/qh69BItYkZYkinosePFYwf5AGsOX7bZdupuE3Y1QQt9AX0ZPot48+gK+jdsaUFvnNDszC99MEDMqlWV9GrmFxaXllfxqYW19Y3OruL3TlFEiMGngiEWiHYAkjIakoahipB0LAjxgpBUMLyd+644ISaPwRo1i4nHoh7RHMSgt+cVD1+n0gXPwU35sj2/DShn8oOIc/aiOVj2/WLKq1hTmPLEzUkIZ6n7xo9ONcMJJqDADKV3bipWXglAUMzIudBJJYsBD6BNX0xA4kV46LTQ2D3qRMNWAmNP372wKXMoRD3SGgxrIWW8i/ue5iepdeCkN40SREOuI9noJM1VkTnYxu1QQrNhIE8CC6itNPAABWOn1Crq+PVt2njSdqn1WPbk+LdXK2RB5tIf2URnZ6BzV0BWqowbC6AE9oVf0Ztwbj8az8fIdzRnZn130B8b7F0DOmTQ=</latexit> [2n m+1 (ab2)n m+2] <latexit sha1_base64="7nywyKpFB9nmWZESxKxexZB0pKM=">AAAB93icbZDLSsNAFIZPvNZ6i5edm2ARurEkVdRlwY3LCvYCbQyT6Uk7dHJhZiLUkGfRlag738MX8G2c1iy09V99c/5/4JzfTziTyra/jKXlldW19dJGeXNre2fX3NtvyzgVFFs05rHo+kQiZxG2FFMcu4lAEvocO/74eup3HlBIFkd3apKgG5JhxAJGidIjzzzsev690w8EoRnx6KmTZ/XcMyt2zZ7JWgSngAoUanrmZ38Q0zTESFFOpOw5dqLcjAjFKMe83E8lJoSOyRB7GiMSonSz2fa5dRLEwlIjtGbv39mMhFJOQl9nQqJGct6bDv/zeqkKrtyMRUmqMKI6or0g5ZaKrWkJ1oAJpIpPNBAqmN7SoiOiO1C6qrI+35k/dhHa9ZpzUTu7Pa80qkURJTiCY6iCA5fQgBtoQgsoPMIzvMG7MTGejBfj9Se6ZBR/DuCPjI9vc72SOQ==</latexit> X1 b ac1 2 <latexit sha1_base64="2R6l73kkvjoi8MGeUTy3cBPMy3A=">AAAB93icbZDLSsNAFIYnXmu9xcvOTbAI3ViSKuqy4MZlBXuBNobJ9KQdOrkwcyLUkGfRlag738MX8G2c1iy09V99c/5/4JzfTwRXaNtfxtLyyuraemmjvLm1vbNr7u23VZxKBi0Wi1h2fapA8AhayFFAN5FAQ19Axx9fT/3OA0jF4+gOJwm4IR1GPOCMoh555mHX8++dfiApy9Bjp06e1XPPrNg1eyZrEZwCKqRQ0zM/+4OYpSFEyARVqufYCboZlciZgLzcTxUklI3pEHoaIxqCcrPZ9rl1EsTSwhFYs/fvbEZDpSahrzMhxZGa96bD/7xeisGVm/EoSREipiPaC1JhYWxNS7AGXAJDMdFAmeR6S4uNqO4AdVVlfb4zf+witOs156J2dnteaVSLIkrkiByTKnHIJWmQG9IkLcLII3kmb+TdmBhPxovx+hNdMoo/B+SPjI9vkK+STA==</latexit> X1 b tc1 2 <latexit sha1_base64="cECEHYMWfALr52aBM7nCwkjOkQM=">AAAB73icbZDLTsJAFIZPvSLeUJduJhITXEhaNeqSxI1LTOSSQGmmwxQmTC/OnJqQhufQlVF3vosv4Ns4YBcK/qtvzv9Pcv7jJ1JotO0va2l5ZXVtvbBR3Nza3tkt7e03dZwqxhsslrFq+1RzKSLeQIGStxPFaehL3vJHN1O/9ciVFnF0j+OEuyEdRCIQjKIZ9dqe33Mq1GOn6LETr1S2q/ZMZBGcHMqQq+6VPrv9mKUhj5BJqnXHsRN0M6pQMMknxW6qeULZiA54x2BEQ67dbLb1hBwHsSI45GT2/p3NaKj1OPRNJqQ41PPedPif10kxuHYzESUp8oiZiPGCVBKMybQ86QvFGcqxAcqUMFsSNqSKMjQnKpr6znzZRWieVZ3L6vndRblWyQ9RgEM4ggo4cAU1uIU6NICBgmd4g3frwXqyXqzXn+iSlf85gD+yPr4B1pePCA==</latexit> X1 b(actc) <latexit sha1_base64="/K+2LrCeTXysN2SXpVo30tauTtE=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaJeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/4AyrNbthL0RWwSmgBoXaw+rnYJSwLMJYM0GV6jt2qr2cSs2ZwFllkClMKZvQMfYNxjRC5eWLZWfkLEgk0SGSxft3NqeRUtPIN5mI6lAte/Phf14/08G1l/M4zTTGzESMF2SC6ITMO5MRl8i0mBqgTHKzJWEhlZRpc5mKqe8sl12FznnDuWxc3DVrrXpxiDKcwCnUwYEraMEttMEFBhye4Q3erdB6sl6s159oySr+HMMfWR/fMvOL1w==</latexit> X1 b <latexit sha1_base64="ba4qVAVS7hO4wgMO4Ze2ZY8GevM=">AAAB93icbZDLSsNAFIZPvNZ6i5edm2ARurEkVdRlwY3LCvYCbQyT6Uk7dHJhZiLUkGfRlag738MX8G2c1iy09V99c/5/4JzfTziTyra/jKXlldW19dJGeXNre2fX3NtvyzgVFFs05rHo+kQiZxG2FFMcu4lAEvocO/74eup3HlBIFkd3apKgG5JhxAJGidIjzzzsev59vR8IQjPl+adOntVzz6zYNXsmaxGcAipQqOmZn/1BTNMQI0U5kbLn2IlyMyIUoxzzcj+VmBA6JkPsaYxIiNLNZtvn1kkQC0uN0Jq9f2czEko5CX2dCYkayXlvOvzP66UquHIzFiWpwojqiPaClFsqtqYlWAMmkCo+0UCoYHpLi46I7kDpqsr6fGf+2EVo12vORe3s9rzSqBZFlOAIjqEKDlxCA26gCS2g8AjP8AbvxsR4Ml6M15/oklH8OYA/Mj6+AZC4kkw=</latexit> X2 b tb1 2 <latexit sha1_base64="16cBt3UgqnChdPs8wblrYl8yPJ8=">AAAB93icbZDLSsNAFIZPvNZ6i5edm2ARurEkVdRlwY3LCvYCbQyT6Uk7dHJhZiLUkGfRlag738MX8G2c1iy09V99c/5/4JzfTziTyra/jKXlldW19dJGeXNre2fX3NtvyzgVFFs05rHo+kQiZxG2FFMcu4lAEvocO/74eup3HlBIFkd3apKgG5JhxAJGidIjzzzsev59vR8IQjPi0VMnz+q5Z1bsmj2TtQhOARUo1PTMz/4gpmmIkaKcSNlz7ES5GRGKUY55uZ9KTAgdkyH2NEYkROlms+1z6ySIhaVGaM3ev7MZCaWchL7OhESN5Lw3Hf7n9VIVXLkZi5JUYUR1RHtByi0VW9MSrAETSBWfaCBUML2lRUdEd6B0VWV9vjN/7CK06zXnonZ2e15pVIsiSnAEx1AFBy6hATfQhBZQeIRneIN3Y2I8GS/G6090ySj+HMAfGR/fdUqSOg==</latexit> X2 b ac1 2 <latexit sha1_base64="j+SzuZQQh8y2aABfMZXqqoHaJJ0=">AAAB93icbZDLSsNAFIZPvNZ6i5edm2ARurEkVdRlwY3LCvYCbQyT6aQdOrkwcyLUkGfRlag738MX8G2c1iy09V99c/5/4JzfTwRXaNtfxtLyyuraemmjvLm1vbNr7u23VZxKylo0FrHs+kQxwSPWQo6CdRPJSOgL1vHH11O/88Ck4nF0h5OEuSEZRjzglKAeeeZh1/Pv6/1AEpqhR0+dPKvnnlmxa/ZM1iI4BVSgUNMzP/uDmKYhi5AKolTPsRN0MyKRU8Hycj9VLCF0TIaspzEiIVNuNts+t06CWFo4Ytbs/TubkVCpSejrTEhwpOa96fA/r5dicOVmPEpSZBHVEe0FqbAwtqYlWAMuGUUx0UCo5HpLi46I7gB1VWV9vjN/7CK06zXnonZ2e15pVIsiSnAEx1AFBy6hATfQhBZQeIRneIN3Y2I8GS/G6090ySj+HMAfGR/fkjySTQ==</latexit> X2 b tc1 2 <latexit sha1_base64="42G9NbBMzxEXl+Pp3iruJkRc9ic=">AAAB73icbZDLTgIxFIY7eEO8oS7dNBITXEhm0KhLEjcuMZFLwmXSKWegoXOxPWNCJjyHroy68118Ad/GgrNQ8F99Pf/f5PzHi6XQaNtfVm5ldW19I79Z2Nre2d0r7h80dZQoDg0eyUi1PaZBihAaKFBCO1bAAk9CyxvfzPzWIygtovAeJzH0AjYMhS84QzPqt12vXy0zl5+hy0/dYsmu2HPRZXAyKJFMdbf42R1EPAkgRC6Z1h3HjrGXMoWCS5gWuomGmPExG0LHYMgC0L10vvWUnviRojgCOn//zqYs0HoSeCYTMBzpRW82/M/rJOhf91IRxglCyE3EeH4iKUZ0Vp4OhAKOcmKAcSXMlpSPmGIczYkKpr6zWHYZmtWKc1k5v7so1crZIfLkiByTMnHIFamRW1InDcKJIs/kjbxbD9aT9WK9/kRzVvbnkPyR9fEN2B6PCQ==</latexit> X2 b(actc) <latexit sha1_base64="WDECJzj+yJE0AsucnWJQq96Q4EQ=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaNOqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnWbDuWyc313UWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATRxi9g=</latexit> X2 b <latexit sha1_base64="FIyLt+d0haFKnSce35coXvxhMHg=">AAAB93icbZDLSsNAFIZPvNZ6i5edm2ARurEkVdRlwY3LCvYCbQyT6Uk7dHJhZiLUkGfRlag738MX8G2c1iy09V99c/5/4JzfTziTyra/jKXlldW19dJGeXNre2fX3NtvyzgVFFs05rHo+kQiZxG2FFMcu4lAEvocO/74eup3HlBIFkd3apKgG5JhxAJGidIjzzzsev690w8EoRnx6Gk9z+q5Z1bsmj2TtQhOARUo1PTMz/4gpmmIkaKcSNlz7ES5GRGKUY55uZ9KTAgdkyH2NEYkROlms+1z6ySIhaVGaM3ev7MZCaWchL7OhESN5Lw3Hf7n9VIVXLkZi5JUYUR1RHtByi0VW9MSrAETSBWfaCBUML2lRUdEd6B0VWV9vjN/7CK06zXnonZ2e15pVIsiSnAEx1AFBy6hATfQhBZQeIRneIN3Y2I8GS/G6090ySj+HMAfGR/fdT+SOg==</latexit> X1 b ac2 2 <latexit sha1_base64="jffFyJNKWx+oJRJ7DBQ6vpbMcMY=">AAAB93icbZDJTsMwFEUdxlKmMOzYWFRI3VAlAQHLSmxYFokOUhsix31prTqDbAcpRPkWWCFgx3/wA/wNbskCWu7q+N1r6b3rJ5xJZVlfxtLyyuraemWjurm1vbNr7u13ZJwKCm0a81j0fCKBswjaiikOvUQACX0OXX9yPfW7DyAki6M7lSXghmQUsYBRovTIMw97nn/vDAJBaE48euoUuVN4Zs1qWDPhRbBLqKFSLc/8HAxjmoYQKcqJlH3bSpSbE6EY5VBUB6mEhNAJGUFfY0RCkG4+277AJ0EssBoDnr1/Z3MSSpmFvs6ERI3lvDcd/uf1UxVcuTmLklRBRHVEe0HKsYrxtAQ8ZAKo4pkGQgXTW2I6JroDpauq6vPt+WMXoeM07IvG2e15rVkvi6igI3SM6shGl6iJblALtRFFj+gZvaF3IzOejBfj9Se6ZJR/DtAfGR/fdsySOw==</latexit> X2 b ac2 2 <latexit sha1_base64="KxyhTJmOcvnkLgY5CHSt8h1g4eE=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaMeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9lo3l3UWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATXvi9k=</latexit> X3 b <latexit sha1_base64="KxyhTJmOcvnkLgY5CHSt8h1g4eE=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaMeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9lo3l3UWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATXvi9k=</latexit> X3 b <latexit sha1_base64="KxyhTJmOcvnkLgY5CHSt8h1g4eE=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaMeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9lo3l3UWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATXvi9k=</latexit> X3 b <latexit sha1_base64="KxyhTJmOcvnkLgY5CHSt8h1g4eE=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaMeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9lo3l3UWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATXvi9k=</latexit> X3 b <latexit sha1_base64="KxyhTJmOcvnkLgY5CHSt8h1g4eE=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaMeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9lo3l3UWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATXvi9k=</latexit> X3 b <latexit sha1_base64="KxyhTJmOcvnkLgY5CHSt8h1g4eE=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaMeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9lo3l3UWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATXvi9k=</latexit> X3 b <latexit sha1_base64="cAVFb4Uz6oXEk1Mdm6IwGl7p7XI=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaJeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9m4uGvWWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATdti9o=</latexit> X4 b <latexit sha1_base64="cAVFb4Uz6oXEk1Mdm6IwGl7p7XI=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaJeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9m4uGvWWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATdti9o=</latexit> X4 b <latexit sha1_base64="cAVFb4Uz6oXEk1Mdm6IwGl7p7XI=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaJeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9m4uGvWWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATdti9o=</latexit> X4 b <latexit sha1_base64="cAVFb4Uz6oXEk1Mdm6IwGl7p7XI=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaJeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9m4uGvWWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATdti9o=</latexit> X4 b <latexit sha1_base64="cAVFb4Uz6oXEk1Mdm6IwGl7p7XI=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaJeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9m4uGvWWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATdti9o=</latexit> X4 b <latexit sha1_base64="cAVFb4Uz6oXEk1Mdm6IwGl7p7XI=">AAAB5nicbZDNTsJAFIVv8Q/xD3XpZiIxYUVaJeqSxI1LTCyQQCXT4ZZOmP5kZmpCGl5BV0bd+UC+gG/jgF0oeFbf3HMmuef6qeBK2/aXVVpb39jcKm9Xdnb39g+qh0cdlWSSocsSkcieTxUKHqOruRbYSyXSyBfY9Sc3c7/7iFLxJL7X0xS9iI5jHnBGtRm5vaH/0BxWa3bDXoisglNADQq1h9XPwShhWYSxZoIq1XfsVHs5lZozgbPKIFOYUjahY+wbjGmEyssXy87IWZBIokMki/fvbE4jpaaRbzIR1aFa9ubD/7x+poNrL+dxmmmMmYkYL8gE0QmZdyYjLpFpMTVAmeRmS8JCKinT5jIVU99ZLrsKnfOGc9m4uGvWWvXiEGU4gVOogwNX0IJbaIMLDDg8wxu8W6H1ZL1Yrz/RklX8OYY/sj6+ATdti9o=</latexit> X4 b <latexit sha1_base64="aXTP5K66qEaq8fw/fSNtedw2+Tc=">AAAB5nicbZDLTsJAFIZP8YZ4Q126mUhMWJHW+5LEjUtMLJBAJdPhlE6YXjIzNSENr6Aro+58IF/At3HALhT8V9+c/5/k/MdPBVfatr+s0srq2vpGebOytb2zu1fdP2irJJMMXZaIRHZ9qlDwGF3NtcBuKpFGvsCOP76Z+Z1HlIon8b2epOhFdBTzgDOqzcjtDvyHi0G1ZjfsucgyOAXUoFBrUP3sDxOWRRhrJqhSPcdOtZdTqTkTOK30M4UpZWM6wp7BmEaovHy+7JScBIkkOkQyf//O5jRSahL5JhNRHapFbzb8z+tlOrj2ch6nmcaYmYjxgkwQnZBZZzLkEpkWEwOUSW62JCykkjJtLlMx9Z3FssvQPm04l42zu/Nas14cogxHcAx1cOAKmnALLXCBAYdneIN3K7SerBfr9Sdasoo/h/BH1sc3OOuL2w==</latexit> X5 b <latexit sha1_base64="aXTP5K66qEaq8fw/fSNtedw2+Tc=">AAAB5nicbZDLTsJAFIZP8YZ4Q126mUhMWJHW+5LEjUtMLJBAJdPhlE6YXjIzNSENr6Aro+58IF/At3HALhT8V9+c/5/k/MdPBVfatr+s0srq2vpGebOytb2zu1fdP2irJJMMXZaIRHZ9qlDwGF3NtcBuKpFGvsCOP76Z+Z1HlIon8b2epOhFdBTzgDOqzcjtDvyHi0G1ZjfsucgyOAXUoFBrUP3sDxOWRRhrJqhSPcdOtZdTqTkTOK30M4UpZWM6wp7BmEaovHy+7JScBIkkOkQyf//O5jRSahL5JhNRHapFbzb8z+tlOrj2ch6nmcaYmYjxgkwQnZBZZzLkEpkWEwOUSW62JCykkjJtLlMx9Z3FssvQPm04l42zu/Nas14cogxHcAx1cOAKmnALLXCBAYdneIN3K7SerBfr9Sdasoo/h/BH1sc3OOuL2w==</latexit> X5 b <latexit sha1_base64="aXTP5K66qEaq8fw/fSNtedw2+Tc=">AAAB5nicbZDLTsJAFIZP8YZ4Q126mUhMWJHW+5LEjUtMLJBAJdPhlE6YXjIzNSENr6Aro+58IF/At3HALhT8V9+c/5/k/MdPBVfatr+s0srq2vpGebOytb2zu1fdP2irJJMMXZaIRHZ9qlDwGF3NtcBuKpFGvsCOP76Z+Z1HlIon8b2epOhFdBTzgDOqzcjtDvyHi0G1ZjfsucgyOAXUoFBrUP3sDxOWRRhrJqhSPcdOtZdTqTkTOK30M4UpZWM6wp7BmEaovHy+7JScBIkkOkQyf//O5jRSahL5JhNRHapFbzb8z+tlOrj2ch6nmcaYmYjxgkwQnZBZZzLkEpkWEwOUSW62JCykkjJtLlMx9Z3FssvQPm04l42zu/Nas14cogxHcAx1cOAKmnALLXCBAYdneIN3K7SerBfr9Sdasoo/h/BH1sc3OOuL2w==</latexit> X5 b <latexit sha1_base64="aXTP5K66qEaq8fw/fSNtedw2+Tc=">AAAB5nicbZDLTsJAFIZP8YZ4Q126mUhMWJHW+5LEjUtMLJBAJdPhlE6YXjIzNSENr6Aro+58IF/At3HALhT8V9+c/5/k/MdPBVfatr+s0srq2vpGebOytb2zu1fdP2irJJMMXZaIRHZ9qlDwGF3NtcBuKpFGvsCOP76Z+Z1HlIon8b2epOhFdBTzgDOqzcjtDvyHi0G1ZjfsucgyOAXUoFBrUP3sDxOWRRhrJqhSPcdOtZdTqTkTOK30M4UpZWM6wp7BmEaovHy+7JScBIkkOkQyf//O5jRSahL5JhNRHapFbzb8z+tlOrj2ch6nmcaYmYjxgkwQnZBZZzLkEpkWEwOUSW62JCykkjJtLlMx9Z3FssvQPm04l42zu/Nas14cogxHcAx1cOAKmnALLXCBAYdneIN3K7SerBfr9Sdasoo/h/BH1sc3OOuL2w==</latexit> X5 b <latexit sha1_base64="aXTP5K66qEaq8fw/fSNtedw2+Tc=">AAAB5nicbZDLTsJAFIZP8YZ4Q126mUhMWJHW+5LEjUtMLJBAJdPhlE6YXjIzNSENr6Aro+58IF/At3HALhT8V9+c/5/k/MdPBVfatr+s0srq2vpGebOytb2zu1fdP2irJJMMXZaIRHZ9qlDwGF3NtcBuKpFGvsCOP76Z+Z1HlIon8b2epOhFdBTzgDOqzcjtDvyHi0G1ZjfsucgyOAXUoFBrUP3sDxOWRRhrJqhSPcdOtZdTqTkTOK30M4UpZWM6wp7BmEaovHy+7JScBIkkOkQyf//O5jRSahL5JhNRHapFbzb8z+tlOrj2ch6nmcaYmYjxgkwQnZBZZzLkEpkWEwOUSW62JCykkjJtLlMx9Z3FssvQPm04l42zu/Nas14cogxHcAx1cOAKmnALLXCBAYdneIN3K7SerBfr9Sdasoo/h/BH1sc3OOuL2w==</latexit> X5 b <latexit sha1_base64="aXTP5K66qEaq8fw/fSNtedw2+Tc=">AAAB5nicbZDLTsJAFIZP8YZ4Q126mUhMWJHW+5LEjUtMLJBAJdPhlE6YXjIzNSENr6Aro+58IF/At3HALhT8V9+c/5/k/MdPBVfatr+s0srq2vpGebOytb2zu1fdP2irJJMMXZaIRHZ9qlDwGF3NtcBuKpFGvsCOP76Z+Z1HlIon8b2epOhFdBTzgDOqzcjtDvyHi0G1ZjfsucgyOAXUoFBrUP3sDxOWRRhrJqhSPcdOtZdTqTkTOK30M4UpZWM6wp7BmEaovHy+7JScBIkkOkQyf//O5jRSahL5JhNRHapFbzb8z+tlOrj2ch6nmcaYmYjxgkwQnZBZZzLkEpkWEwOUSW62JCykkjJtLlMx9Z3FssvQPm04l42zu/Nas14cogxHcAx1cOAKmnALLXCBAYdneIN3K7SerBfr9Sdasoo/h/BH1sc3OOuL2w==</latexit> X5 b <latexit sha1_base64="pnvN6Vwi32wmOF/gVPCN2Xy57Uc=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHWZhwuwjM70mZMOP6MmoNz/FH/BvHHAPCtapuqs66aoglUKj43xZpbX1jc2t8nZlZ3dvv2ofHHZ1kinGOyyRieoFVHMpYt5BgZL3UsVpFEj+EExu5vrDI1daJPE9TlPuRXQUi1Awimbl29VBqCjLqc/O3FnenPl2zWk4C5BV4hakBgXavv05GCYsi3iMTFKt+66TopdThYJJPqsMMs1TyiZ0xPuGxjTi2ssXj8/IaZgogmNOFvNvb04jradRYDwRxbFe1ubL/7R+huG1l4s4zZDHzFiMFmaSYELm+clQKM5QTg2hTAnzJWFjajpA01LFxHeXw66SbrPhXjbO7y5qrXpRRBmO4QTq4MIVtOAW2tABBhk8wxu8W2g9WS/W64+1ZBU3R/AH1sc3JNiQXw==</latexit> ac1 2 <latexit sha1_base64="pnvN6Vwi32wmOF/gVPCN2Xy57Uc=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHWZhwuwjM70mZMOP6MmoNz/FH/BvHHAPCtapuqs66aoglUKj43xZpbX1jc2t8nZlZ3dvv2ofHHZ1kinGOyyRieoFVHMpYt5BgZL3UsVpFEj+EExu5vrDI1daJPE9TlPuRXQUi1Awimbl29VBqCjLqc/O3FnenPl2zWk4C5BV4hakBgXavv05GCYsi3iMTFKt+66TopdThYJJPqsMMs1TyiZ0xPuGxjTi2ssXj8/IaZgogmNOFvNvb04jradRYDwRxbFe1ubL/7R+huG1l4s4zZDHzFiMFmaSYELm+clQKM5QTg2hTAnzJWFjajpA01LFxHeXw66SbrPhXjbO7y5qrXpRRBmO4QTq4MIVtOAW2tABBhk8wxu8W2g9WS/W64+1ZBU3R/AH1sc3JNiQXw==</latexit> ac1 2 <latexit sha1_base64="pnvN6Vwi32wmOF/gVPCN2Xy57Uc=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHWZhwuwjM70mZMOP6MmoNz/FH/BvHHAPCtapuqs66aoglUKj43xZpbX1jc2t8nZlZ3dvv2ofHHZ1kinGOyyRieoFVHMpYt5BgZL3UsVpFEj+EExu5vrDI1daJPE9TlPuRXQUi1Awimbl29VBqCjLqc/O3FnenPl2zWk4C5BV4hakBgXavv05GCYsi3iMTFKt+66TopdThYJJPqsMMs1TyiZ0xPuGxjTi2ssXj8/IaZgogmNOFvNvb04jradRYDwRxbFe1ubL/7R+huG1l4s4zZDHzFiMFmaSYELm+clQKM5QTg2hTAnzJWFjajpA01LFxHeXw66SbrPhXjbO7y5qrXpRRBmO4QTq4MIVtOAW2tABBhk8wxu8W2g9WS/W64+1ZBU3R/AH1sc3JNiQXw==</latexit> ac1 2 <latexit sha1_base64="CvSlFxZeN4CGw0+twJZmqBQACT8=">AAAB8nicbVDLTgJBEJzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHXphwuwjM70mZMOP6MmoNz/FH/BvHHAPCtapuqs66aoglUKj43xZpbX1jc2t8nZlZ3dvv2ofHHZ1kikOHZ7IRPUCpkGKGDooUEIvVcCiQMJDMLmZ6w+PoLRI4nucpuBFbBSLUHCGZuXb1UGoGM+ZH5w1Z3lz5ts1p+EsQFeJW5AaKdD27c/BMOFZBDFyybTuu06KXs4UCi5hVhlkGlLGJ2wEfUNjFoH28sXjM3oaJoriGOhi/u3NWaT1NAqMJ2I41svafPmf1s8wvPZyEacZQsyNxWhhJikmdJ6fDoUCjnJqCONKmC8pHzPTAZqWKia+uxx2lXSbDfeycX53UWvViyLK5JickDpxyRVpkVvSJh3CSUaeyRt5t9B6sl6s1x9rySpujsgfWB/fJNaQXw==</latexit> ab2 2 <latexit sha1_base64="LCgJ8YrgS5n3gylMYKUxJw9Z7SU=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHWZhwuwjM70mZMOP6MmoNz/FH/BvHHAPCtapuqs66aoglUKj43xZpbX1jc2t8nZlZ3dvv2ofHHZ1kinGOyyRieoFVHMpYt5BgZL3UsVpFEj+EExu5vrDI1daJPE9TlPuRXQUi1Awimbl29VBqCjL0Wdn7ixvzny75jScBcgqcQtSgwJt3/4cDBOWRTxGJqnWfddJ0cupQsEkn1UGmeYpZRM64n1DYxpx7eWLx2fkNEwUwTEni/m3N6eR1tMoMJ6I4lgva/Plf1o/w/Day0WcZshjZixGCzNJMCHz/GQoFGcop4ZQpoT5krAxNR2gaali4rvLYVdJt9lwLxvndxe1Vr0oogzHcAJ1cOEKWnALbegAgwye4Q3eLbSerBfr9cdasoqbI/gD6+MbQcqQcg==</latexit> tc1 2 <latexit sha1_base64="LCgJ8YrgS5n3gylMYKUxJw9Z7SU=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHWZhwuwjM70mZMOP6MmoNz/FH/BvHHAPCtapuqs66aoglUKj43xZpbX1jc2t8nZlZ3dvv2ofHHZ1kinGOyyRieoFVHMpYt5BgZL3UsVpFEj+EExu5vrDI1daJPE9TlPuRXQUi1Awimbl29VBqCjL0Wdn7ixvzny75jScBcgqcQtSgwJt3/4cDBOWRTxGJqnWfddJ0cupQsEkn1UGmeYpZRM64n1DYxpx7eWLx2fkNEwUwTEni/m3N6eR1tMoMJ6I4lgva/Plf1o/w/Day0WcZshjZixGCzNJMCHz/GQoFGcop4ZQpoT5krAxNR2gaali4rvLYVdJt9lwLxvndxe1Vr0oogzHcAJ1cOEKWnALbegAgwye4Q3eLbSerBfr9cdasoqbI/gD6+MbQcqQcg==</latexit> tc1 2 <latexit sha1_base64="LCgJ8YrgS5n3gylMYKUxJw9Z7SU=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHWZhwuwjM70mZMOP6MmoNz/FH/BvHHAPCtapuqs66aoglUKj43xZpbX1jc2t8nZlZ3dvv2ofHHZ1kinGOyyRieoFVHMpYt5BgZL3UsVpFEj+EExu5vrDI1daJPE9TlPuRXQUi1Awimbl29VBqCjL0Wdn7ixvzny75jScBcgqcQtSgwJt3/4cDBOWRTxGJqnWfddJ0cupQsEkn1UGmeYpZRM64n1DYxpx7eWLx2fkNEwUwTEni/m3N6eR1tMoMJ6I4lgva/Plf1o/w/Day0WcZshjZixGCzNJMCHz/GQoFGcop4ZQpoT5krAxNR2gaali4rvLYVdJt9lwLxvndxe1Vr0oogzHcAJ1cOEKWnALbegAgwye4Q3eLbSerBfr9cdasoqbI/gD6+MbQcqQcg==</latexit> tc1 2 <latexit sha1_base64="4EA/0fg7Ud9AYIrnQC+x4Djn2co=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHXphwuwjM7MmZMOP6MmoNz/FH/BvHHAPCtapuqs66aogFVxpx/mySmvrG5tb5e3Kzu7eftU+OOyqJJMMOywRiewFVKHgMXY01wJ7qUQaBQIfgsnNXH94RKl4Et/raYpeREcxDzmj2qx8uzoIJWU59dlZc5Y3Z75dcxrOAmSVuAWpQYG2b38OhgnLIow1E1Spvuuk2sup1JwJnFUGmcKUsgkdYd/QmEaovHzx+IychokkeoxkMf/25jRSahoFxhNRPVbL2nz5n9bPdHjt5TxOM40xMxajhZkgOiHz/GTIJTItpoZQJrn5krAxNR1o01LFxHeXw66SbrPhXjbO7y5qrXpRRBmO4QTq4MIVtOAW2tABBhk8wxu8W9p6sl6s1x9rySpujuAPrI9vJlqQYA==</latexit> ac2 2 <latexit sha1_base64="4EA/0fg7Ud9AYIrnQC+x4Djn2co=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHXphwuwjM7MmZMOP6MmoNz/FH/BvHHAPCtapuqs66aogFVxpx/mySmvrG5tb5e3Kzu7eftU+OOyqJJMMOywRiewFVKHgMXY01wJ7qUQaBQIfgsnNXH94RKl4Et/raYpeREcxDzmj2qx8uzoIJWU59dlZc5Y3Z75dcxrOAmSVuAWpQYG2b38OhgnLIow1E1Spvuuk2sup1JwJnFUGmcKUsgkdYd/QmEaovHzx+IychokkeoxkMf/25jRSahoFxhNRPVbL2nz5n9bPdHjt5TxOM40xMxajhZkgOiHz/GTIJTItpoZQJrn5krAxNR1o01LFxHeXw66SbrPhXjbO7y5qrXpRRBmO4QTq4MIVtOAW2tABBhk8wxu8W9p6sl6s1x9rySpujuAPrI9vJlqQYA==</latexit> ac2 2 <latexit sha1_base64="4EA/0fg7Ud9AYIrnQC+x4Djn2co=">AAAB8nicbVDLTgJBEOzFF+KDVY9eJhITLpJdNOqRxItHTARJYLOZHXphwuwjM7MmZMOP6MmoNz/FH/BvHHAPCtapuqs66aogFVxpx/mySmvrG5tb5e3Kzu7eftU+OOyqJJMMOywRiewFVKHgMXY01wJ7qUQaBQIfgsnNXH94RKl4Et/raYpeREcxDzmj2qx8uzoIJWU59dlZc5Y3Z75dcxrOAmSVuAWpQYG2b38OhgnLIow1E1Spvuuk2sup1JwJnFUGmcKUsgkdYd/QmEaovHzx+IychokkeoxkMf/25jRSahoFxhNRPVbL2nz5n9bPdHjt5TxOM40xMxajhZkgOiHz/GTIJTItpoZQJrn5krAxNR1o01LFxHeXw66SbrPhXjbO7y5qrXpRRBmO4QTq4MIVtOAW2tABBhk8wxu8W9p6sl6s1x9rySpujuAPrI9vJlqQYA==</latexit> ac2 2 <latexit sha1_base64="OQNxgDjd3+ahFYdQ1ZY+iAKZrg0=">AAAB6nicbZC9TsMwFIVv+C3lr8DIYlEhlYEqAQSMlVgYi0R/UBtVjnvTWrWTyHaQqqgvARMCNh6HF+BtcEsGaDnT53uOpXtukAiujet+OUvLK6tr64WN4ubW9s5uaW+/qeNUMWywWMSqHVCNgkfYMNwIbCcKqQwEtoLRzdRvPaLSPI7uzThBX9JBxEPOqLGjhwrtsVPTYye9UtmtujORRfByKEOueq/02e3HLJUYGSao1h3PTYyfUWU4EzgpdlONCWUjOsCOxYhK1H42W3hCjsNYETNEMnv/zmZUaj2Wgc1IaoZ63psO//M6qQmv/YxHSWowYjZivTAVxMRk2pv0uUJmxNgCZYrbLQkbUkWZsdcp2vrefNlFaJ5Vvcvq+d1FuVbJD1GAQziCCnhwBTW4hTo0gIGEZ3iDd0c4T86L8/oTXXLyPwfwR87HN5J5jS4=</latexit> (actc) <latexit sha1_base64="OQNxgDjd3+ahFYdQ1ZY+iAKZrg0=">AAAB6nicbZC9TsMwFIVv+C3lr8DIYlEhlYEqAQSMlVgYi0R/UBtVjnvTWrWTyHaQqqgvARMCNh6HF+BtcEsGaDnT53uOpXtukAiujet+OUvLK6tr64WN4ubW9s5uaW+/qeNUMWywWMSqHVCNgkfYMNwIbCcKqQwEtoLRzdRvPaLSPI7uzThBX9JBxEPOqLGjhwrtsVPTYye9UtmtujORRfByKEOueq/02e3HLJUYGSao1h3PTYyfUWU4EzgpdlONCWUjOsCOxYhK1H42W3hCjsNYETNEMnv/zmZUaj2Wgc1IaoZ63psO//M6qQmv/YxHSWowYjZivTAVxMRk2pv0uUJmxNgCZYrbLQkbUkWZsdcp2vrefNlFaJ5Vvcvq+d1FuVbJD1GAQziCCnhwBTW4hTo0gIGEZ3iDd0c4T86L8/oTXXLyPwfwR87HN5J5jS4=</latexit> (actc) <latexit sha1_base64="OQNxgDjd3+ahFYdQ1ZY+iAKZrg0=">AAAB6nicbZC9TsMwFIVv+C3lr8DIYlEhlYEqAQSMlVgYi0R/UBtVjnvTWrWTyHaQqqgvARMCNh6HF+BtcEsGaDnT53uOpXtukAiujet+OUvLK6tr64WN4ubW9s5uaW+/qeNUMWywWMSqHVCNgkfYMNwIbCcKqQwEtoLRzdRvPaLSPI7uzThBX9JBxEPOqLGjhwrtsVPTYye9UtmtujORRfByKEOueq/02e3HLJUYGSao1h3PTYyfUWU4EzgpdlONCWUjOsCOxYhK1H42W3hCjsNYETNEMnv/zmZUaj2Wgc1IaoZ63psO//M6qQmv/YxHSWowYjZivTAVxMRk2pv0uUJmxNgCZYrbLQkbUkWZsdcp2vrefNlFaJ5Vvcvq+d1FuVbJD1GAQziCCnhwBTW4hTo0gIGEZ3iDd0c4T86L8/oTXXLyPwfwR87HN5J5jS4=</latexit> (actc) <latexit sha1_base64="Qks+Fj9l0Nj56BfDyn7mdLogLE0=">AAACCnicbVDNSsNAGNzUv1r/oh69LBaxIi1JFPVY8OKxgv2BNIbNdtMu3U3C7kYooW+gL6MnUW8efQHfxm0NqK1zmp2ZhW8mSBiVyrI+jcLC4tLySnG1tLa+sbllbu+0ZJwKTJo4ZrHoBEgSRiPSVFQx0kkEQTxgpB0MLyd++44ISePoRo0S4nHUj2hIMVJa8s1D1+n2EefIz/ixPb6NqhXlB1Xn6Ed1tOr5ZtmqWVPAeWLnpAxyNHzzo9uLccpJpDBDUrq2lSgvQ0JRzMi41E0lSRAeoj5xNY0QJ9LLpoXG8CCMBVQDAqfv39kMcSlHPNAZjtRAznoT8T/PTVV44WU0SlJFIqwj2gtTBlUMJ7vAHhUEKzbSBGFB9ZUQD5BAWOn1Srq+PVt2nrScmn1WO7k+Ldcr+RBFsAf2QQXY4BzUwRVogCbA4AE8gVfwZtwbj8az8fIdLRj5n13wB8b7F16kmUc=</latexit> [2n m+1 (tb2)n m+2] <latexit sha1_base64="oAEos6TWFNW9EimHifPNRhuPnGU=">AAAB/nicbVDLSsNAFJ34rPVVddlNsAgFoSQq6rLgxmUF+4Cmlsn0Jhk6k4SZG6GEgv6MrkTd+RH+gH/jtHahrWd17j3nwj3HTwXX6Dhf1tLyyuraemGjuLm1vbNb2ttv6SRTDJosEYnq+FSD4DE0kaOATqqASl9A2x9eTfT2PSjNk/gWRyn0JA1jHnBG0az6pbInIMCuF1IpaT+Xx+74LvYUDyOs9UsVp+ZMYS8Sd0YqZIZGv/TpDRKWSYiRCap113VS7OVUIWcCxkUv05BSNqQhdA2NqQTdy6chxvZRkCgbI7Cn829vTqXWI+kbj6QY6XltsvxP62YYXPZyHqcZQsyMxWhBJmxM7EkX9oArYChGhlCmuPnSZhFVlKFprGjiu/NhF0nrpOae105vzir16qyIAimTQ1IlLrkgdXJNGqRJGHkkz+SNvFsP1pP1Yr3+WJes2c0B+QPr4xtQ+JWR</latexit> ⇥n m+1 <latexit sha1_base64="pa7fIYcBZNtR9xgYMV0huwvmcxU=">AAACCnicbVDLSsNAFJ34tr6qLt0Ei1gRS6KiLgU3LhWsCk0MN+NNOziThJkboYT+gf6MrkTdufQH/BuntQtfBy6ce88ZmHPiXApDnvfhjIyOjU9MTk1XZmbn5heqi0vnJis0xybPZKYvYzAoRYpNEiTxMtcIKpZ4Ed8c9fWLW9RGZOkZdXMMFbRTkQgOZE9RdT2QmFBjqw5RvEV2/I2gDUpBVKrN7d5VGmjR7lAYVWtewxvA/Uv8IamxIU6i6ntwnfFCYUpcgjEt38spLEGT4BJ7laAwmAO/gTa2LE1BoQnLQaCeu5Zk2qUOuoP9u7cEZUxXxdajgDrmt9Y//qe1CkoOwlKkeUGYcmuxWlJIlzK334t7LTRykl1LgGthf+nyDmjgZNur2Pj+77B/yfl2w99r7Jzu1g7rwyKm2ApbZXXms312yI7ZCWsyzu7ZI3thr86d8+A8Oc9f1hFn+GaZ/YDz9gmOrZls</latexit> (abtb1)n m+2⇤ <latexit sha1_base64="qWvnJlH+cjrgVOr6knB7ebv8HlA=">AAAB73icbZDLSgMxFIYz9VbrrerSzWARCkKZUVGXBTcuK9gL9MaZ9EwbmmTGJCOUoc+hK1F3vosv4NuY1llo67/6cv4/cP4TxJxp43lfTm5ldW19I79Z2Nre2d0r7h80dJQoinUa8Ui1AtDImcS6YYZjK1YIIuDYDMY3M7/5iEqzSN6bSYxdAUPJQkbB2FGvMwQhoJ+KU3/ak/1iyat4c7nL4GdQIplq/eJnZxDRRKA0lIPWbd+LTTcFZRjlOC10Eo0x0DEMsW1RgkDdTedbT92TMFKuGaE7f//OpiC0nojAZgSYkV70ZsP/vHZiwutuymScGJTURqwXJtw1kTsr7w6YQmr4xAJQxeyWLh2BAmrsiQq2vr9YdhkaZxX/snJ+d1GqlrND5MkROSZl4pMrUiW3pEbqhBJFnskbeXcenCfnxXn9ieac7M8h+SPn4xsMUo/U</latexit> n m+1 <latexit sha1_base64="ibUgfaD1vEWb628nJkJLDqkWyug=">AAACCnicbVDNSsNAGNzUv1r/oh69BItYkZakFfVY8OKxgv2BNoYv2027dDcJuxuhhL6BvoyeRL159AV8G7c1oFbnNDszC9+MHzMqlW1/GLmFxaXllfxqYW19Y3PL3N5pySgRmDRxxCLR8UESRkPSVFQx0okFAe4z0vZHF1O/fUuEpFF4rcYxcTkMQhpQDEpLnnnYrfYGwDl4KT92JjdhuQSeX64dfatVrbqeWbQr9gzWX+JkpIgyNDzzvdePcMJJqDADKbuOHSs3BaEoZmRS6CWSxIBHMCBdTUPgRLrprNDEOggiYakhsWbvn9kUuJRj7usMBzWU895U/M/rJio4d1MaxokiIdYR7QUJs1RkTXex+lQQrNhYE8CC6istPAQBWOn1Crq+M1/2L2lVK85ppXZ1UqyXsiHyaA/toxJy0Bmqo0vUQE2E0T16RC/o1bgzHown4/krmjOyP7voF4y3T0JcmTU=</latexit> [2n m+1 (ab3)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] . texto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] Figure 2. Kinds of rows and main diagonal. Step 1. <latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="lfLj5ayaNSocNtcomVVIL7kzeBE=">AAAB6HicdZDLSgMxFIbPeK31VnXpJliELsowo8XaXcGNywr2Am0pmfRMG5u5kGSEMvQddCXqzufxBXwb02kFFf1XX87/B85/vFhwpR3nw1pZXVvf2Mxt5bd3dvf2CweHLRUlkmGTRSKSHY8qFDzEpuZaYCeWSANPYNubXM399j1KxaPwVk9j7Ad0FHKfM6rNqO2Wbdsux4NC0bFrmcgCqpUl1Fzi2k6mIizVGBTee8OIJQGGmgmqVNd1Yt1PqdScCZzle4nCmLIJHWHXYEgDVP00W3dGTv1IEj1Gkr2/Z1MaKDUNPJMJqB6r3958+JfXTbR/2U95GCcaQ2YixvMTQXRE5q3JkEtkWkwNUCa52ZKwMZWUaXObvKn/1ZH8D60z272wz28qxXppeYgcHMMJlMCFKtThGhrQBAYTeIQXeLXurAfryXpeRFes5Z8j+CHr7RMXgIxZ</latexit> 1,...,p <latexit sha1_base64="nmhA55uQ6b0yLqaOCKf/43IzGW0=">AAAB5HicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEqMICUxIp9yBhs5P2o4JIbyBroy684l8Ad/GMoyJGj2rr/ecJvdcPxFcacf5sAorq2vrG8XN0tb2zu5eef/gTsWpZNhmsYhl16cKBY+wrbkW2E0k0tAX2PEnlwu/c49S8Ti61dMEvZCOIh5wRrUZ3fCBPyhXHLuRiSyhXsuh4RLXdjJVIFdrUH7vD2OWhhhpJqhSPddJtDejUnMmcF7qpwoTyiZ0hD2DEQ1RebNs1Tk5CWJJ9BhJ9v6endFQqWnom0xI9Vj99hbDv7xeqoMLb8ajJNUYMRMxXpAKomOyaEyGXCLTYmqAMsnNloSNqaRMm7uUTP2vjuR/uDu13XP77LpWaVbzQxThCI6hCi7UoQlX0II2MBjBI7zAqxVYD9aT9byMFqz8zyH8kPX2Cfxli9g=</latexit> ib <latexit sha1_base64="5Fkg35Z2FUnas1FHsXr3ltfmo68=">AAAB5HicdZDLTgIxFIbP4A3xhrp000hMWJEZJSI7EjcuMcolgQnplDNQ6VzSdkwI4Q10ZdSdT+QL+DaWYUzU6L/6ev6/yfmPFwuutG1/WLmV1bX1jfxmYWt7Z3evuH/QVlEiGbZYJCLZ9ahCwUNsaa4FdmOJNPAEdrzJ5cLv3KNUPApv9TRGN6CjkPucUW1GN3cDb1As2ZV6KrKEWjWDukOcip2qBJmag+J7fxixJMBQM0GV6jl2rN0ZlZozgfNCP1EYUzahI+wZDGmAyp2lq87JiR9JosdI0vf37IwGSk0Dz2QCqsfqt7cY/uX1Eu1fuDMexonGkJmI8fxEEB2RRWMy5BKZFlMDlElutiRsTCVl2tylYOp/dST/Q/u04pxXzq6rpUY5O0QejuAYyuBADRpwBU1oAYMRPMILvFq+9WA9Wc/LaM7K/hzCD1lvn/3li9k=</latexit> jb <latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="nmhA55uQ6b0yLqaOCKf/43IzGW0=">AAAB5HicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEqMICUxIp9yBhs5P2o4JIbyBroy684l8Ad/GMoyJGj2rr/ecJvdcPxFcacf5sAorq2vrG8XN0tb2zu5eef/gTsWpZNhmsYhl16cKBY+wrbkW2E0k0tAX2PEnlwu/c49S8Ti61dMEvZCOIh5wRrUZ3fCBPyhXHLuRiSyhXsuh4RLXdjJVIFdrUH7vD2OWhhhpJqhSPddJtDejUnMmcF7qpwoTyiZ0hD2DEQ1RebNs1Tk5CWJJ9BhJ9v6endFQqWnom0xI9Vj99hbDv7xeqoMLb8ajJNUYMRMxXpAKomOyaEyGXCLTYmqAMsnNloSNqaRMm7uUTP2vjuR/uDu13XP77LpWaVbzQxThCI6hCi7UoQlX0II2MBjBI7zAqxVYD9aT9byMFqz8zyH8kPX2Cfxli9g=</latexit> ib <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="5Fkg35Z2FUnas1FHsXr3ltfmo68=">AAAB5HicdZDLTgIxFIbP4A3xhrp000hMWJEZJSI7EjcuMcolgQnplDNQ6VzSdkwI4Q10ZdSdT+QL+DaWYUzU6L/6ev6/yfmPFwuutG1/WLmV1bX1jfxmYWt7Z3evuH/QVlEiGbZYJCLZ9ahCwUNsaa4FdmOJNPAEdrzJ5cLv3KNUPApv9TRGN6CjkPucUW1GN3cDb1As2ZV6KrKEWjWDukOcip2qBJmag+J7fxixJMBQM0GV6jl2rN0ZlZozgfNCP1EYUzahI+wZDGmAyp2lq87JiR9JosdI0vf37IwGSk0Dz2QCqsfqt7cY/uX1Eu1fuDMexonGkJmI8fxEEB2RRWMy5BKZFlMDlElutiRsTCVl2tylYOp/dST/Q/u04pxXzq6rpUY5O0QejuAYyuBADRpwBU1oAYMRPMILvFq+9WA9Wc/LaM7K/hzCD1lvn/3li9k=</latexit> jb <latexit sha1_base64="4nlCEpLs9lt42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEhIBE5iQTrkDDZ2ftB0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65fiK40o7zYRXW1jc2t4rbpZ3dvf2D8uFRV8WpZNhhsYjlnU8VCh5hR3Mt8C6RSENfYM+fXi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUsvw9GMUtDjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Retlx0Ts6CWBI9QbJ8f89mNFRqFvomE1I9Ub+9xfAvr5/q4MrLeJSkGiNmIsYLUkF0TBZ9yYhLZFrMDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9tC/atUqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+t5FS1Y+Z9j+CHr7ROG94r8</latexit> b <latexit sha1_base64="ihMRwXZ4jfcYe29bFDJV6p2VUNM=">AAAB7nicdZDNSsNAFIUn9a/Wv6pLN4NF6CokVazdFdy4ESrYH2hDmUxv2qGTSZyZCKX0NXQl6s6H8QV8GydpBBU9qzP3nIH7XT/mTGnH+bAKK6tr6xvFzdLW9s7uXnn/oKOiRFJo04hHsucTBZwJaGumOfRiCST0OXT96WWad+9BKhaJWz2LwQvJWLCAUaLNyLsmTOARI+NIED4sVxy7kQkvTf0sNw0Xu7aTqYJytYbl98EookkIQlNOlOq7Tqy9OZGaUQ6L0iBREBM6JWPoGytICMqbZ0sv8EkQSawngLP39+6chErNQt90QqIn6neWDv/K+okOLrw5E3GiQVBTMVmQcKwjnLIbUAlU81lKTCUzW2I6IZJQbS5UMvhfjPh/06nZ7rl9elOrNKv5IYroCB2jKnJRHTXRFWqhNqLoDj2iF/RqxdaD9WQ9L6sFK/9ziH7IevsEF8KP9A==</latexit> Main diagonal <latexit sha1_base64="4nlCEpLs9lt42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIXv4B/iH+rSTSMxYTWZUSKyI3HjEhIBE5iQTrkDDZ2ftB0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65fiK40o7zYRXW1jc2t4rbpZ3dvf2D8uFRV8WpZNhhsYjlnU8VCh5hR3Mt8C6RSENfYM+fXi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUsvw9GMUtDjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Retlx0Ts6CWBI9QbJ8f89mNFRqFvomE1I9Ub+9xfAvr5/q4MrLeJSkGiNmIsYLUkF0TBZ9yYhLZFrMDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9tC/atUqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+t5FS1Y+Z9j+CHr7ROG94r8</latexit> b <latexit sha1_base64="fJDXT3vCNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoKthSYNk+mkHTq5MDMRSgjoy+hK1J0P4Qv4Nk7TCCp6Vt/85wz85/diRoU0zQ+ttLK6tr5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uFv7NLeGCRuG1nMfECdAkpD7FSKqRq1dtxOIpGqWXo46Nk9hOqYsVZpnbcfWaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkxtMxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5fsKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKqv/VEf4P/RPDOjNOrxq1dr04RBlUwRGoAws0QRt0QBf0AAb34BG8gFftTnvQnrTnZbSkFX8OwQ9pb59HcJY9</latexit> ↵DH[{iH c} H <latexit sha1_base64="lfLj5ayaNSocNtcomVVIL7kzeBE=">AAAB6HicdZDLSgMxFIbPeK31VnXpJliELsowo8XaXcGNywr2Am0pmfRMG5u5kGSEMvQddCXqzufxBXwb02kFFf1XX87/B85/vFhwpR3nw1pZXVvf2Mxt5bd3dvf2CweHLRUlkmGTRSKSHY8qFDzEpuZaYCeWSANPYNubXM399j1KxaPwVk9j7Ad0FHKfM6rNqO2Wbdsux4NC0bFrmcgCqpUl1Fzi2k6mIizVGBTee8OIJQGGmgmqVNd1Yt1PqdScCZzle4nCmLIJHWHXYEgDVP00W3dGTv1IEj1Gkr2/Z1MaKDUNPJMJqB6r3958+JfXTbR/2U95GCcaQ2YixvMTQXRE5q3JkEtkWkwNUCa52ZKwMZWUaXObvKn/1ZH8D60z272wz28qxXppeYgcHMMJlMCFKtThGhrQBAYTeIQXeLXurAfryXpeRFes5Z8j+CHr7RMXgIxZ</latexit> 1,...,p <latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="Rr+Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKGZSUwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXVtfaO4Wdra3tndK+8f9BRPJCZdzBmXtwFShNGYdDXVjNwKSVAUMNIPppcLv39PpKI8vtEzQfwIjWMaUoy0GXmi5tZt266L2t2wXHHsVia4hGYjh5YLXdvJVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyCfWEwOz9PZuiSKlZFJhMhPRE/fYWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAsw4OARvIBXK7YerCfreRktWPmfQ/BD1tsnhHiNuA==</latexit> p+1,...,p+q <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="Pd5UqyzbYAszv0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/Wv6pLN8EiFFqGGS3W7gpuXFawrTAdSibNtKHJZEwyQhn6FroSdefT+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyura+kZxs7S1vbO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQfTC8Xfv+eSEVFdKNnMfE5Gkc0pBhpM/Li2l3Nrdu2XefDcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpSfZivP4UkoJNQTArP392yKuFIzHpgMR3qifnuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qntntun103Ku1qfogiOALHoApc0ARtcAU6oAswEOARvIBXK7IerCfreRktWPmfQ/BD1tsngYKNtQ==</latexit> p+q+1,...,m <latexit sha1_base64="fJDXT3vCNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoKthSYNk+mkHTq5MDMRSgjoy+hK1J0P4Qv4Nk7TCCp6Vt/85wz85/diRoU0zQ+ttLK6tr5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uFv7NLeGCRuG1nMfECdAkpD7FSKqRq1dtxOIpGqWXo46Nk9hOqYsVZpnbcfWaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkxtMxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5fsKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKqv/VEf4P/RPDOjNOrxq1dr04RBlUwRGoAws0QRt0QBf0AAb34BG8gFftTnvQnrTnZbSkFX8OwQ9pb59HcJY9</latexit> ↵DH[{iH c} H <latexit sha1_base64="fJDXT3vCNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoKthSYNk+mkHTq5MDMRSgjoy+hK1J0P4Qv4Nk7TCCp6Vt/85wz85/diRoU0zQ+ttLK6tr5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uFv7NLeGCRuG1nMfECdAkpD7FSKqRq1dtxOIpGqWXo46Nk9hOqYsVZpnbcfWaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkxtMxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5fsKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKqv/VEf4P/RPDOjNOrxq1dr04RBlUwRGoAws0QRt0QBf0AAb34BG8gFftTnvQnrTnZbSkFX8OwQ9pb59HcJY9</latexit> ↵DH[{iH c} H <latexit sha1_base64="2vt4dygS7EDahRMOKvCxEREiuqo=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoK9QJOGyXTSjp1cmJkIJQT0ZXQl6s6H8AV8G6dpBRU9q2/+cwb+83sxo0Ka5odWWFldW98obpa2tnd29/T9g66IEo5JB0cs4n0PCcJoSDqSSkb6MSco8BjpedOLud+7JVzQKLyWs5g4ARqH1KcYSTVy9bKNWDxBw/Ry2LJxEtvpjYsVZpnbcvWKaTRywQXUa0toWNAyzFwVsFTb1d/tUYSTgIQSMyTEwDJj6aSIS4oZyUp2IkiM8BSNyUBhiAIinDQvkcFjP+JQTgjM39+zKQqEmAWeygRITsRvbz78yxsk0j93UhrGiSQhVhHl+QmDMoLzW8AR5QRLNlOAMKdqS4gniCMs1cVKqv5XR/g/dE8M68w4vapVmtXlIYqgDI5AFVigDpqgBdqgAzC4B4/gBbxqd9qD9qQ9L6IFbfnnEPyQ9vYJSPeWPg==</latexit> ↵DH[{jH c} H <latexit sha1_base64="2vt4dygS7EDahRMOKvCxEREiuqo=">AAAB/nicdZDLSsNAGIUn9VbrLeqym8EidBUSLdbuCrrosoK9QJOGyXTSjp1cmJkIJQT0ZXQl6s6H8AV8G6dpBRU9q2/+cwb+83sxo0Ka5odWWFldW98obpa2tnd29/T9g66IEo5JB0cs4n0PCcJoSDqSSkb6MSco8BjpedOLud+7JVzQKLyWs5g4ARqH1KcYSTVy9bKNWDxBw/Ry2LJxEtvpjYsVZpnbcvWKaTRywQXUa0toWNAyzFwVsFTb1d/tUYSTgIQSMyTEwDJj6aSIS4oZyUp2IkiM8BSNyUBhiAIinDQvkcFjP+JQTgjM39+zKQqEmAWeygRITsRvbz78yxsk0j93UhrGiSQhVhHl+QmDMoLzW8AR5QRLNlOAMKdqS4gniCMs1cVKqv5XR/g/dE8M68w4vapVmtXlIYqgDI5AFVigDpqgBdqgAzC4B4/gBbxqd9qD9qQ9L6IFbfnnEPyQ9vYJSPeWPg==</latexit> ↵DH[{jH c} H <latexit sha1_base64="oWnImmMjVj3ALKw/FQW5HpDcQ5k=">AAACCnicbVDNSsNAGNzUv1r/qh69BItYkZYkinosePFYwf5AGsOX7bZdupuE3Y1QQt9AX0ZPot48+gK+jdsaUFvnNDszC99MEDMqlWV9GrmFxaXllfxqYW19Y3OruL3TlFEiMGngiEWiHYAkjIakoahipB0LAjxgpBUMLyd+644ISaPwRo1i4nHoh7RHMSgt+cVD1+n0gXPwU35sj2/DShn8oOIc/aiOVj2/WLKq1hTmPLEzUkIZ6n7xo9ONcMJJqDADKV3bipWXglAUMzIudBJJYsBD6BNX0xA4kV46LTQ2D3qRMNWAmNP372wKXMoRD3SGgxrIWW8i/ue5iepdeCkN40SREOuI9noJM1VkTnYxu1QQrNhIE8CC6itNPAABWOn1Crq+PVt2njSdqn1WPbk+LdXK2RB5tIf2URnZ6BzV0BWqowbC6AE9oVf0Ztwbj8az8fIdzRnZn130B8b7F0DOmTQ=</latexit> [2n m+1 (ab2)n m+2] <latexit sha1_base64="Qks+Fj9l0Nj56BfDyn7mdLogLE0=">AAACCnicbVDNSsNAGNzUv1r/oh69LBaxIi1JFPVY8OKxgv2BNIbNdtMu3U3C7kYooW+gL6MnUW8efQHfxm0NqK1zmp2ZhW8mSBiVyrI+jcLC4tLySnG1tLa+sbllbu+0ZJwKTJo4ZrHoBEgSRiPSVFQx0kkEQTxgpB0MLyd++44ISePoRo0S4nHUj2hIMVJa8s1D1+n2EefIz/ixPb6NqhXlB1Xn6Ed1tOr5ZtmqWVPAeWLnpAxyNHzzo9uLccpJpDBDUrq2lSgvQ0JRzMi41E0lSRAeoj5xNY0QJ9LLpoXG8CCMBVQDAqfv39kMcSlHPNAZjtRAznoT8T/PTVV44WU0SlJFIqwj2gtTBlUMJ7vAHhUEKzbSBGFB9ZUQD5BAWOn1Srq+PVt2nrScmn1WO7k+Ldcr+RBFsAf2QQXY4BzUwRVogCbA4AE8gVfwZtwbj8az8fIdLRj5n13wB8b7F16kmUc=</latexit> [2n m+1 (tb2)n m+2] <latexit sha1_base64="oAEos6TWFNW9EimHifPNRhuPnGU=">AAAB/nicbVDLSsNAFJ34rPVVddlNsAgFoSQq6rLgxmUF+4Cmlsn0Jhk6k4SZG6GEgv6MrkTd+RH+gH/jtHahrWd17j3nwj3HTwXX6Dhf1tLyyuraemGjuLm1vbNb2ttv6SRTDJosEYnq+FSD4DE0kaOATqqASl9A2x9eTfT2PSjNk/gWRyn0JA1jHnBG0az6pbInIMCuF1IpaT+Xx+74LvYUDyOs9UsVp+ZMYS8Sd0YqZIZGv/TpDRKWSYiRCap113VS7OVUIWcCxkUv05BSNqQhdA2NqQTdy6chxvZRkCgbI7Cn829vTqXWI+kbj6QY6XltsvxP62YYXPZyHqcZQsyMxWhBJmxM7EkX9oArYChGhlCmuPnSZhFVlKFprGjiu/NhF0nrpOae105vzir16qyIAimTQ1IlLrkgdXJNGqRJGHkkz+SNvFsP1pP1Yr3+WJes2c0B+QPr4xtQ+JWR</latexit> ⇥n m+1 <latexit sha1_base64="pa7fIYcBZNtR9xgYMV0huwvmcxU=">AAACCnicbVDLSsNAFJ34tr6qLt0Ei1gRS6KiLgU3LhWsCk0MN+NNOziThJkboYT+gf6MrkTdufQH/BuntQtfBy6ce88ZmHPiXApDnvfhjIyOjU9MTk1XZmbn5heqi0vnJis0xybPZKYvYzAoRYpNEiTxMtcIKpZ4Ed8c9fWLW9RGZOkZdXMMFbRTkQgOZE9RdT2QmFBjqw5RvEV2/I2gDUpBVKrN7d5VGmjR7lAYVWtewxvA/Uv8IamxIU6i6ntwnfFCYUpcgjEt38spLEGT4BJ7laAwmAO/gTa2LE1BoQnLQaCeu5Zk2qUOuoP9u7cEZUxXxdajgDrmt9Y//qe1CkoOwlKkeUGYcmuxWlJIlzK334t7LTRykl1LgGthf+nyDmjgZNur2Pj+77B/yfl2w99r7Jzu1g7rwyKm2ApbZXXms312yI7ZCWsyzu7ZI3thr86d8+A8Oc9f1hFn+GaZ/YDz9gmOrZls</latexit> (abtb1)n m+2⇤ <latexit sha1_base64="qWvnJlH+cjrgVOr6knB7ebv8HlA=">AAAB73icbZDLSgMxFIYz9VbrrerSzWARCkKZUVGXBTcuK9gL9MaZ9EwbmmTGJCOUoc+hK1F3vosv4NuY1llo67/6cv4/cP4TxJxp43lfTm5ldW19I79Z2Nre2d0r7h80dJQoinUa8Ui1AtDImcS6YYZjK1YIIuDYDMY3M7/5iEqzSN6bSYxdAUPJQkbB2FGvMwQhoJ+KU3/ak/1iyat4c7nL4GdQIplq/eJnZxDRRKA0lIPWbd+LTTcFZRjlOC10Eo0x0DEMsW1RgkDdTedbT92TMFKuGaE7f//OpiC0nojAZgSYkV70ZsP/vHZiwutuymScGJTURqwXJtw1kTsr7w6YQmr4xAJQxeyWLh2BAmrsiQq2vr9YdhkaZxX/snJ+d1GqlrND5MkROSZl4pMrUiW3pEbqhBJFnskbeXcenCfnxXn9ieac7M8h+SPn4xsMUo/U</latexit> n m+1 <latexit sha1_base64="ibUgfaD1vEWb628nJkJLDqkWyug=">AAACCnicbVDNSsNAGNzUv1r/oh69BItYkZakFfVY8OKxgv2BNoYv2027dDcJuxuhhL6BvoyeRL159AV8G7c1oFbnNDszC9+MHzMqlW1/GLmFxaXllfxqYW19Y3PL3N5pySgRmDRxxCLR8UESRkPSVFQx0okFAe4z0vZHF1O/fUuEpFF4rcYxcTkMQhpQDEpLnnnYrfYGwDl4KT92JjdhuQSeX64dfatVrbqeWbQr9gzWX+JkpIgyNDzzvdePcMJJqDADKbuOHSs3BaEoZmRS6CWSxIBHMCBdTUPgRLrprNDEOggiYakhsWbvn9kUuJRj7usMBzWU895U/M/rJio4d1MaxokiIdYR7QUJs1RkTXex+lQQrNhYE8CC6istPAQBWOn1Crq+M1/2L2lVK85ppXZ1UqyXsiHyaA/toxJy0Bmqo0vUQE2E0T16RC/o1bgzHown4/krmjOyP7voF4y3T0JcmTU=</latexit> [2n m+1 (ab3)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPUffOj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEVfTCDiRXjbpM7IOwlhYqk+syft3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAArPV5B13dmy86TZrXinFVOrk+LtdJ0iDzaQ/uohBx0jmroCtVRA2H0gJ7QK3oz7o1H49l4+Y7mjOmfXfQHxvsXx7mY9w==</latexit> [n m+1 (ab1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzWv1r/Vj16CRalIi27VdRjwYvHCvYHtuuSTbNtaJJdkqxQlj6BvoyeRL159QV8G9NaUFvnNJmZwDcTJowq7TifVm5hcWl5Jb9aWFvf2Nyyt3eaKk4lJg0cs1i2Q6QIo4I0NNWMtBNJEA8ZaYWDy7HfuiNS0Vjc6GFCfI56gkYUI22kwD70Oj3EOQoyfuyObkW5pIOw7B79qFWj+oFddCrOBHCeuFNSBFPUA/uj041xyonQmCGlPNdJtJ8hqSlmZFTopIokCA9Qj3iGCsSJ8rNJnxE8iGIJdZ/Ayft3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQltRcCXEfSYS1Ga9g6ruzZedJs1pxzyon16fFWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9fEdz1vTPLvgD6/0L5Y+ZCg==</latexit> [n m+1 (tb1)n m+2] <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="N/+1tM9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZMrO7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1v57Z3dvUJx/6BpokQz3mCRjHQ7AMOlCHkDBUrejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0EvVaXX6YJWSW3HnoMvEy0iJZKj3ip/dfsQSxUNkEozpeG6MfgoaBZN8mu8mhsfAxjDkHUtDUNz46fzwKT0ZRJriiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRnfWnfaE5QzmxBJgW9krKRqCBof1S3tb3Fssuk2a14l1Wzu8uSrVy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenBfn9Se64mQ7h+QPnI9v72KQPQ==</latexit> n m+2 <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="Db51iQ8zL//nttJs7fXn1X3T/ps=">AAACCXicbVDNSsNAGNzUv1r/oh69BItSkZakinosePFYwf5AGsOX7aZdupuE3Y1QQp9AX0ZPot68+gK+jdtaUFvnNDszC99MkDAqlW1/GrmFxaXllfxqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpLvnnodnrAOfgZP3ZGt1G5BH5Qrh79qFWter5ZtCv2BNY8caakiKao++ZHpxvjlJNIYQZSuo6dKC8DoShmZFTopJIkgAfQI66mEXAivWzSZ2QdhLGwVJ9Yk/fvbAZcyiEPdIaD6stZbyz+57mpCi+8jEZJqkiEdUR7YcosFVvjWawuFQQrNtQEsKD6Sgv3QQBWeryCru/Mlp0nzWrFOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPxrPx8h3NGdM/u+gPjPcvyUeY+A==</latexit> [n m+1 (ab2)n m+2] <latexit sha1_base64="xaFiCSazm9iv7wQCmgRnk4yOx2s=">AAAB9HicbZDLSsNAFIZP6q3WW6pLN8EidGNJVNRlwY3LCvYCbQiT6aQdOrkwc1IpIW+iK1F3Pokv4Ns4rVlo67/65vz/wDm/nwiu0La/jNLa+sbmVnm7srO7t39gVg87Kk4lZW0ai1j2fKKY4BFrI0fBeolkJPQF6/qT27nfnTKpeBw94CxhbkhGEQ84JahHnlkdBJLQzMkz4vln6Pm5Z9bshr2QtQpOATUo1PLMz8EwpmnIIqSCKNV37ATdjEjkVLC8MkgVSwidkBHra4xIyJSbLVbPrdMglhaOmbV4/85mJFRqFvo6ExIcq2VvPvzP66cY3LgZj5IUWUR1RHtBKiyMrXkD1pBLRlHMNBAqud7SomOiW0DdU0Wf7ywfuwqd84Zz1bi4v6w160URZTiGE6iDA9fQhDtoQRsoPMIzvMG7MTWejBfj9SdaMoo/R/BHxsc3/OiRdQ==</latexit> 1 abtb Figure 3. Kinds of rows and main diagonal. Step 2. 21
•For b= 1, ..., p, 2γn m+1 −γn m+2(ab−2) = [γn m+1 −(ab−1)γn m+2]+[γn m+1 +γn m+2]. •For b=p+ 1, ..., p +qand ib, 2γn m+1 −(tb−2)γn m+2 = [γn m+1 −(tb−1)γn m+2]+[γn m+1 +γn m+2]. •For b=p+ 1, ..., p +qand jb, 1 ab−tb [γn m+1 −(ab−tb−1)γn m+2] = [−γn m+2] + 1 ab−tb [γn m+1 +γn m+2]. •For b=p+q+ 1, ..., m and ib, γn m+1 = [−γn m+2]+[γn m+1 +γn m+2]. •For b=p+q+ 1, ..., m and jb, 2γn m+1 −γn m+2(ab−3) = [γn m+1 −(ab−2)γn m+2]+[γn m+1 +γn m+2]. Following equality (12), we obtain the determinant D2m−p=γn m+1 +γn m+22m−p Qp+q k=p+1(ak−tk) 1 + p X k=1 γn m+1 −(ak−1)γn m+2 γn m+1 +γn m+2 + p+q X k=p+1 γn m+1 −(tk−1)γn m+2 γn m+1 +γn m+2 + p+q X k=p+1 −γn m+2(ak−tk) γn m+1 +γn m+2 + m X k=p+q+1 −γn m+2 γn m+1 +γn m+2 + m X k=p+q+1 γn m+1 −(ak−2)γn m+2 γn m+1 +γn m+2 =γn m+1 +γn m+22m−p−1 Qp+q k=p+1(ak−tk)(m+ 1)γn m+1 + (m−n+ 1)γn m+2. It is easy to check that D2m−p= 0, because (m+ 1)γn m+1 + (m−n+ 1)γn m+2 = 0. But there is a condition on the payoffs of fP, efficiency, that we have not used. We only needed to verify 2m−p−1 equalities, that is, we can consider the same determinant with one less equation and one less unknown. If there exists kwith ak6=tkthen we delete row kand, by (12) (it can be multiplied by ak−tk), Dk 2m−p−1=γn m+2(ak−tk)(γn m+1 +γn m+2)2m−p−2 Qp+q b=p+1,b6=k(ab−tb)6= 0, where Dk 2m−p−1is the determinant obtained when we delete row and column kin D2m−p. Now we suppose ak=tkfor all k= 1, ..., m. In this case Dk 2m−p−1= (γn m+1 −(ak−1)γn m+2)(γn m+1 +γn m+2)2m−p−2. 22
If there is k∈ {1, ..., m}with γn m+1 −(ak−1)γn m+2 6= 0, then we delete row kand we get Dk 2m−p−16= 0. If γn m+1 −(ak−1)γn m+2 = 0 for all k= 1, ..., m, then all the groups have the same size ak=n m+1.But if all the groups have the same size athen am =n and this implies a=n m. Funding This research has been supported by the FQM237 grant of the Andalusian Government References [1] Albizuri M.J., Aurrekoetxea J. and Zarzuelo J.M. Configuration values: extensions of the coalitional Owen value. Games and Economic Behavior; 2006 57: 1–17. [2] Alonso-Meijide, J.M. and Carreras, F. The proportional coalitional Shapley value. Expert Systems with Applications; 2011 38: 6967-6979. [3] Alonso-Meijide, J.M., Casas-M´endez, B., Gonz´alez-Rueda, M. and Lorenzo-Freire, S. Axiomatic of the Shapley value of a game with a priori unions. Top; 2014, 22(2): 749–770. [4] Brink, R. van den and Dietz, C. Union values for games with coalition structure. Decision Support Systems; 2014, 66: 1-8. [5] Fern´andez, J.R. , Gallego, I., Jim´enez-Losada A. and Ord´o˜nez, M. Cooperation among agents with a proximity relation. European Journal of Operational Research; 2016, 250: 555-565. [6] Fern´andez, J.R., Gallego, I., Jim´enez-Losada, A. and Ord´o˜nez, M. A Banzhaf value for games with a proximity relation among the agents. International Journal of Approximate Reasoning; 2017, 88: 192-208. [7] G´omez-R´ua, M. and Vidal-Puga, J. The axiomatic approach to three values in games with coalition structure. European Journal of Operational Research; 2010, 207: 795-806. [8] Gon¸calves-Dosantos, J.C., Alonso-Meijide J.M. New results on egalitarian values for games with a priori unions. Optimization; 2021, 67. [9] Huettner, F. and Wiese, H. The need for permission, the power to enforce, and duality in cooperative games with a hierarchy. International Game Theory Review; 2016, 18(4). [10] Herbert, J.P. and Yao, J.T. Game-theoretic rough sets. Fundamenta Informaticae; 2011, 108(3–4):267–286. [11] Kamijo, Y. A two-step Shapley value for cooperative games with coalition structures. International Game Theory Review; 2009, 11(2):207-214. [12] Ord´o˜nez, M., Fern´andez, J.R., Gallego, I. and Jim´enez-Losada, A. A value for games on colored communication structures. Operational Research; 2021, 21: 2247-2260. [13] Owen, G. Values of Games with a Priori Unions. In: Henn R., Moeschlin O. (eds) Mathematical Economics and Game Theory. Lecture Notes in Economics and Mathematical Systems, vol 141. Springer, Berlin, Heidelberg. 1977. [14] Pawlak, Z. Rough sets. International Journal of Parallel Programming; 1982, 11(5): 341–356. [15] Polkowski, L. and Araszkiewicz, B. A rough set approach to estimating the game value and the Shapley value from data. Fundamenta Informaticae; 2002, 82(4): 219–227. [16] Saavedra-Nieves, A. and Fiestras-Janeiro, G.M. Sampling methods to estimate the Banzhaf–Owen value. Annals of Operations Research; 2021, 301(1-2): 199-223. [17] Shapley, L.S. A Value for n-person Games. In Contributions to the Theory of Games, volume II, by H.W. Kuhn and A.W. Tucker, editors. Annals of Mathematical Studies v. 28, Princeton University Press: 307–317, 1953. [18] Radzik T., Driessen T. On a family of values for TU-games generalizing the Shapley value. Mathematical Social Sciences; 2013, 65: 105–111. 23
Appendix A. Explication of Figure 2 We explain here the five different kinds of rows showed in Figure 2. 1) Consider an equation corresponding to b∈ {1, ..., p}in (15). Take DH∪{i}with i /∈Ab. Observe that if there are Ksymmetric (in P) coalitions Scontaining ib to it in the equation, then there are also (ab−1)Ksymmetric S\{ib}options to it in the equation. So, the coefficient of number αDH∪{i} His Kγn m+1 −γn m+2(ab−1)(A1) Let X1 b= p Y k=1 k6=b ak p+q Y k=p+1 tk m Y k=p+q+1 (ak−1). We look for the coefficient of the chosen numbers in this equation, analyzing the five kinds of numbers plus the number corresponding to the equation. 1.1 Number αDH∪{iH b} H(we cannot apply (A1) here). It appears for all the coalitions equivalent by symmetry to S=DH∪{iH b}, namely (ab−1) p Y k=1 k6=b ak p+q Y k=p+1 tk m Y k=p+q+1 (ak−1) times, and with coalitions equivalent by symmetry to S\{ib}=DH∪{iH b}, namely ab−1 2p Y k=1 k6=b ak p+q Y k=p+1 tk m Y k=p+q+1 (ak−1) times. So, the coefficient of this number is X1 b ab−1 22γn m+1 −(ab−2)γn m+2. 1.2 Number αDH∪{iH c} H, with c∈ {1, ..., p}\{b}. The quantity Kof symmetric coalitions (compatible in P)Scontaining ibto DH∪{iH c}is K=ac 2p Y k=1 k6=b,c ak p+q Y k=p+1 tk m Y k=p+q+1 (ak−1). Since (A1) we get the coefficient of this number in the equation X1 b ac−1 2γn m+1 −(ab−1)γn m+2. 24
1.3 Number αDH∪{iH c} H,c∈ {p+ 1, ..., p +q}. We obtain in this case K=tc 2p Y k=1 k6=b ak p+q Y k=p+1 k6=c tk m Y k=p+q+1 (ak−1). Since (A1) we get the coefficient of this number X1 b tc−1 2γn m+1 −(ab−1)γn m+2. 1.4 Number αDH∪{jH c} H,c∈ {p+ 1, ..., p +q}. We get K=tc(ac−tc) p Y k=1 k6=b ak p+q Y k=p+1 k6=c tk m Y k=p+q+1 (ak−1), and then, from (A1), the coefficient of the number is X1 b(ac−tc)γn m+1 −(ab−1)γn m+2. 1.5 Number αDH∪{iH c} H,c∈ {p+q+ 1, ..., m}. It holds K= p Y k=1 k6=b ak p+q Y k=p+1 tk m Y k=p+q+1 (ak−1). Since (A1) the coefficient of the number is X1 bγn m+1 −(ab−1)γn m+2. 1.6 Number αDH∪{jH c} H,c∈ {p+q+ 1, ..., m}. We get K=ac−1 2p Y k=1 k6=b ak p+q Y k=p+1 tk m Y k=p+q+1 k6=c (ak−1). Using (A1), the coefficient of the number is X1 b ac−2 2γn m+1 −(ab−1)γn m+2. 2) Consider an equation corresponding to b∈ {p+1, ..., p+q}in (15). Take DH∪{i} with i /∈Tb. Observe that if there are Ksymmetric (in P) coalitions Scontaining ibto it in the equation, then there are also (tb−1)Ksymmetric S\{ib}options to it in the equation. So, the coefficient of number αDH∪{i} His Kγn m+1 −γn m+2(tb−1)(A2) 25