The Norton algebra of a Q -polynomial distance-regular graph
aa r X i v : . [ m a t h . C O ] J un The Norton algebra of a Q -polynomialdistance-regular graph Paul Terwilliger
Abstract
We consider the Norton algebra associated with a Q -polynomial primitive idempo-tent of the adjacency matrix for a distance-regular graph. We obtain a formula for theNorton algebra product that we find attractive. Keywords . Bose-Mesner algebra; Krein parameter; Q -polynomial; Leonard system. . Primary: 05C50; Secondary: 05E30. There is a family of highly regular graphs said to be distance-regular [1, 2, 4]. Examplesinclude the Johnson graphs [2, Section 9.1], the Hamming graphs [2, Section 9.2], the Grass-mann graphs [2, Section 9.3], and the dual polar graphs [2, Section 9.4]. The graphs in thesefour families are particularly attractive for several reasons: (i) they have a Q -polynomialstructure, according to which their Krein parameters vanish in a certain attractive pattern;(ii) these graphs come with a ranked partially ordered set that can be used to analyze thegraph.In the analysis of any distance-regular graph Γ, one often considers the eigenspaces for theadjacency matrix A of Γ. By [3] these eigenspaces possess an algebra structure, called theNorton algebra, that is commutative but not necessarily associative. The Norton product ⋆ is described as follows. Let X denote the vertex set of Γ. The rows and columns of A areindexed by X . The matrix A acts on a vector space V over R , consisting of column vectorswhose coordinates are indexed by X . For x ∈ X let ˆ x denote the vector in V that has x -coordinate 1 and all other coordinates zero. So { ˆ x | x ∈ X } is a basis for V . The entry-wiseproduct ◦ : V × V → V satisfies ˆ x ◦ ˆ y = δ x,y ˆ x for all x, y ∈ X . The matrix A is diagonalizablesince it is symmetric, so V is a direct sum of the eigenspaces of A . For an eigenspace of A ,the corresponding primitive idempotent E acts as the identity on the eigenspace, and zeroon the other eigenspaces of A . Thus E is the projection from V onto the eigenspace. Theeigenspace is EV . For u, v ∈ EV we have u ⋆ v = E ( u ◦ v ).Earlier we mentioned Q -polynomial structures. For a given Q -polynomial structure on Γthe adjacency matrix A has a distinguished primitive idempotent, said to be Q -polynomial.Recently, several authors have considered the Norton algebra EV for a Q -polynomial prim-itive idempotent E of A . This was done by C. Maldonado and D. Penazzi in [8], under the1ssumption that Γ is a Johnson graph, Hamming graph, or Grassmann graph. It was doneby F. Levstein, C. Maldonado, and D. Penazzi in [7], under the assumption that Γ is a dualpolar graph. In both articles the authors compute ˇ u ⋆ ˇ v for all u, v ∈ L , where { ˇ u | u ∈ L } is a certain spanning set for EV indexed by the set L of rank 1 elements in the associatedposet. The results of [7, 8] are used by J. Huang in [5] to investigate the extent to which theNorton product is nonassociative.In the present paper we consider the Norton algebra EV , where E is a Q -polynomial primitiveidempotent of the adjacency matrix A for any distance-regular graph Γ with diameter d ≥ x, y of Γ we give an explicit formula for the Norton product E ˆ x ⋆ E ˆ y , interms of a few eigenvalues θ , θ , θ of A and a sequence of scalars { θ ∗ i } di =0 called the dualeigenvalues of Γ associated with E . We give two versions of our formula. The first version ismore straightforward. To obtain the second version, we use the balanced set condition [9,10]to make the symmetry E ˆ x ⋆ E ˆ y = E ˆ y ⋆ E ˆ x explicit. Our main results are Theorems 3.7 and4.4.The paper is organized as follows. In Section 2 we review some basic concepts concerningdistance-regular graphs. In Section 3 we recall the Norton algebra and obtain the first versionof our Norton product formula. In Section 4 we use the balanced set condition to obtain thesecond version of our Norton product formula. In Section 5 we remark how certain equationsin Sections 3, 4 can be obtained using the theory of Leonard systems. In this section we review some basic concepts concerning distance-regular graphs. For morebackground information we refer the reader to [1, 2, 4].Let R denote the real number field. Let X denote a nonempty finite set. Let Mat X ( R )denote the R -algebra consisting of the matrices that have rows and columns indexed by X and all entries in R . Let I denote the identity matrix in Mat X ( R ), and let J denote thematrix in Mat X ( R ) that has all entries 1. Let V = R X denote the vector space over R consisting of the column vectors that have coordinates indexed by X and all entries in R .The algebra Mat X ( R ) acts on V by left multiplication. For x ∈ X let ˆ x denote the vectorin V that has x -coordinate 1 and all other coordinates 0. The vectors { ˆ x | x ∈ X } form abasis for V . Let denote the vector in V that has all entries 1. So = P x ∈ X ˆ x . Note that J ˆ x = for all x ∈ X .Let Γ = ( X, R ) denote an undirected connected graph, without loops or multiple edges, withvertex set X , edge set R , and path-length distance function ∂ . Recall the diameter d =max { ∂ ( x, y ) | x, y ∈ X } . For x ∈ X and an integer i ≥ i ( x ) = { y ∈ X | ∂ ( x, y ) = i } .We abbreviate Γ( x ) = Γ ( x ). For an integer k ≥
0, Γ is said to be regular with valency k whenever k = | Γ( x ) | for x ∈ X . The graph Γ is said to be distance-regular wheneverfor all integers h, i, j (0 ≤ h, i, j ≤ d ) and all x, y ∈ X at distance ∂ ( x, y ) = h , the scalar p hi,j = | Γ i ( x ) ∩ Γ j ( y ) | is independent of x and y . The scalars p hi,j are called the intersectionnumbers of Γ. For the rest of this paper, we assume that Γ is distance-regular with diameter d ≥
2. By construction p hi,j = p hj,i for 0 ≤ h, i, j ≤ d . By the triangle inequality we find thatfor 0 ≤ h, i, j ≤ d , 2i) p hi,j = 0 if one of h, i, j is greater than the sum of the other two;(ii) p hi,j = 0 if one of h, i, j is equal to the sum of the other two.We abbreviate c i = p i ,i − (1 ≤ i ≤ d ), a i = p i ,i (0 ≤ i ≤ d ), b i = p i ,i +1 (0 ≤ i ≤ d − c i = 0 for 1 ≤ i ≤ d and b i = 0 for 0 ≤ i ≤ d −
1. The graph Γ is regular withvalency k = b . Moreover k = c i + a i + b i for 0 ≤ i ≤ d , where c = 0 and b d = 0.Next we recall the Bose-Mesner algebra of Γ. For 0 ≤ i ≤ d let A i denote the matrix inMat( R ) that has ( x, y )-entry( A i ) x,y = ( , if ∂ ( x, y ) = i ;0 , if ∂ ( x, y ) = i ( x, y ∈ X ) . We call A i the i th distance-matrix of Γ. Note that A = I . We abbreviate A = A and callthis the adjacency matrix of Γ. By the construction, A i A j = d X h =0 p hi,j A h (0 ≤ i, j ≤ d ) . By these comments the matrices { A i } di =0 form a basis for a commutative subalgebra ofMat X ( R ). This algebra is denoted by M and called the Bose-Mesner algebra of Γ. Thealgebra M is generated by A [1, p. 190].Next we recall the primitive idempotents and eigenvalues of Γ. By [2, p. 45] the vector space M has a basis { E i } di =0 such that (i) E = | X | − J ; (ii) I = P di =0 E i ; (iii) E i E j = δ i,j E i (0 ≤ i, j ≤ d ). This basis is unique up to permutation of { E i } di =1 . We call { E i } di =0 the primitiveidempotents of M (or Γ). The primitive idempotent E is called trivial . By construction,there exist real numbers { θ i } di =0 such that A = P di =0 θ i E i . The { θ i } di =0 are mutually distinctsince A generates M . Using E = | X | − J we obtain θ = k . The scalars { θ i } di =0 are calledthe eigenvalues of A (or Γ).Next we recall the Krein parameters of Γ. For 0 ≤ i, j ≤ d we have A i · A j = δ i,j A i , where · denotes the entry-wise product for Mat X ( R ). Therefore M is closed under the · product.Consequently there exist q hi,j ∈ R (0 ≤ h, i, j ≤ d ) such that E i · E j = | X | − d X h =0 q hi,j E h (0 ≤ i, j ≤ d ) . (1)By construction q hi,j = q hj,i for 0 ≤ h, i, j ≤ d . By [2, Proposition 4.1.5] we have q hi,j ≥ ≤ h, i, j ≤ d . The scalars q hi,j are called the Krein parameters of Γ.We describe one significance of the Krein parameters. In this description, we will use thefollowing notation. For u ∈ V and x ∈ X let u x denote the x -coordinate of u . So u = P x ∈ X u x ˆ x . For u, v ∈ V their entry-wise product u ◦ v is the vector in V that has x -coordinate u x v x for all x ∈ X . So u ◦ v = P x ∈ X u x v x ˆ x . For x, y ∈ X we haveˆ x ◦ ˆ y = ( ˆ x, if x = y ;0 , if x = y . (2)3or v ∈ V we have ◦ v = v . For subspaces Y, Z of V define Y ◦ Z = Span { y ◦ z | y ∈ Y, z ∈ Z } .By [3, Proposition 5.1] we have E i V ◦ E j V = X ≤ h ≤ dq hij =0 E h V (0 ≤ i, j ≤ d ) . (3)Next we recall the Q -polynomial property. The given ordering { E i } di =1 of the nontrivialprimitive idempotents of Γ is said to be Q -polynomial whenever for 0 ≤ h, i, j ≤ d ,(i) q hi,j = 0 if one of h, i, j is greater than the sum of the other two;(ii) q hi,j = 0 if one of h, i, j is equal to the sum of the other two.Let E denote a nontrivial primitive idempotent of Γ. We say that E is Q -polynomial wheneverthere exists a Q -polynomial ordering { E i } di =1 of the nontrivial primitive idempotents of Γsuch that E = E . For the rest of this paper we assume that E is Q -polynomial. Byconstruction, there exist real numbers { θ ∗ i } di =0 such that E = | X | − d X i =0 θ ∗ i A i . (4)By [1, p. 260] the scalars { θ ∗ i } di =0 are mutually distinct. The scalars { θ ∗ i } di =0 are called the dual eigenvalues of Γ associated with E . For notational convenience let θ ∗− and θ ∗ d +1 denoteindeterminates. Taking the trace of each side of (4) yields θ ∗ = rank( E ). Also, multiplyingboth sides of (4) by A and evaluating the result yields c i θ ∗ i − + a i θ ∗ i + b i θ ∗ i +1 = θ θ ∗ i (0 ≤ i ≤ d ) . (5) We continue to discuss the distance-regular graph Γ and its Q -polynomial primitive idem-potent E . In this section we turn the vector space EV into a commutative nonassociativealgebra called the Norton algebra. Definition 3.1. (See [3, Proposition 5.2].) The
Norton algebra of Γ consists of the vectorspace EV , together with the product u ⋆ v = E ( u ◦ v ) ( u, v ∈ EV ) . The Norton algebra is commutative, but not necessarily associative.The vector space EV is spanned by the vectors { E ˆ x | x ∈ X } . These vectors are nonzero,mutually distinct, and linearly dependent [9, Theorem 1.1]. As we investigate the Nortonproduct ⋆ it is natural to consider E ˆ x ⋆ E ˆ y for all x, y ∈ X . In the next two lemmas wediscuss some extremal cases. 4 emma 3.2. For x ∈ X , E ˆ x ⋆ E ˆ x = | X | − q , E ˆ x. (6) Proof.
By (1) we have E · E = | X | − P dh =0 q h , E h . For this equation, multiply each side by E to obtain E ( E · E ) = | X | − q , E . For this equation, compare column x of each side toobtain (6). Lemma 3.3.
The following are equivalent: (i) E ˆ x ⋆ E ˆ y = 0 for all x, y ∈ X ; (ii) u ⋆ v = 0 for all u, v ∈ EV ; (iii) The Krein parameter q , = 0 .Proof. By (3) and the construction.We have been discussing some extremal cases. Before we proceed to the general case, webring in some notation. To motivate things, observe that for x ∈ X and 0 ≤ i ≤ d , A i ˆ x = X z ∈ Γ i ( x ) ˆ z. (7) Lemma 3.4.
For x, y ∈ X and ≤ i, j ≤ d we have A i ˆ x ◦ A j ˆ y = X z ∈ Γ i ( x ) ∩ Γ j ( y ) ˆ z. Proof.
Use (2) and (7).
Definition 3.5.
Pick x, y ∈ X and write i = ∂ ( x, y ). Define x + y = A ˆ x ◦ A i +1 ˆ y = X z ∈ Γ( x ) ∩ Γ i +1 ( y ) ˆ z, (8) x y = A ˆ x ◦ A i ˆ y = X z ∈ Γ( x ) ∩ Γ i ( y ) ˆ z, (9) x − y = A ˆ x ◦ A i − ˆ y = X z ∈ Γ( x ) ∩ Γ i − ( y ) ˆ z, (10)where we understand A − = 0, Γ − ( x ) = ∅ and A d +1 = 0, Γ d +1 ( x ) = ∅ .We clarify the notation (8)–(10). Pick x, y ∈ X . If ∂ ( x, y ) = d then x + y = 0. If ∂ ( x, y ) = 1then x − y = ˆ y . If x = y then x y = 0 and x − y = 0. Lemma 3.6.
For x, y ∈ X we have x + y + x y + x − y = A ˆ x, (11) Ex + y + Ex y + Ex − y = θ E ˆ x. (12)5 roof. To verify (11), note that each side is equal to P z ∈ Γ( x ) ˆ z . To get (12), apply E to eachside of (11), and use EA = θ E .The following is our first main result. Theorem 3.7.
Assume that Γ is Q -polynomial with respect to E . Then for all x, y ∈ X wehave E ˆ x ⋆ E ˆ y = ( θ ∗ i − − θ ∗ i ) Ex − y + ( θ ∗ i +1 − θ ∗ i ) Ex + y + ( θ − θ ) θ ∗ i E ˆ x + ( θ − θ ) E ˆ y | X | ( θ − θ ) (13) where i = ∂ ( x, y ) . Here θ ∗− and θ ∗ d +1 denote indeterminates.Proof. We consider the vector E ( A ˆ x ◦ E ˆ y ) − θ E (ˆ x ◦ E ˆ y ) . We evaluate this vector in two ways. For the first evaluation, use A − θ I = P dh =0 ( θ h − θ ) E h to obtain E ( A ˆ x ◦ E ˆ y ) − θ E (ˆ x ◦ E ˆ y ) = d X h =0 ( θ h − θ ) E ( E h ˆ x ◦ E ˆ y ) . For the above sum, we examine the h -summand for 0 ≤ h ≤ d . For h = 0 the summand is( θ − θ ) | X | − E ˆ y because E ˆ x ◦ E ˆ y = | X | − J ˆ x ◦ E ˆ y = | X | − ◦ E ˆ y = | X | − E ˆ y. For h = 1 the summand is ( θ − θ ) E ˆ x ⋆ E ˆ y by Definition 3.1. For h = 2 the summandis zero by construction. For 3 ≤ h ≤ d the summand is zero by (3) and the definition of Q -polynomial. By these comments, E ( A ˆ x ◦ E ˆ y ) − θ E (ˆ x ◦ E ˆ y ) = ( θ − θ ) | X | − E ˆ y + ( θ − θ ) E ˆ x ⋆ E ˆ y. (14)For the second evaluation, use E = | X | − P dℓ =0 θ ∗ ℓ A ℓ to obtain( A − θ I )ˆ x ◦ E ˆ y = | X | − d X ℓ =0 θ ∗ ℓ ( A − θ I )ˆ x ◦ A ℓ ˆ y. For the above sum, we examine the ℓ -summand for 0 ≤ ℓ ≤ d . The term A ˆ x ◦ A ℓ ˆ y is equalto x − y (if ℓ = i −
1) and x y (if ℓ = i ) and x + y (if ℓ = i + 1) and zero (if | ℓ − i | > x ◦ A ℓ ˆ y is equal to ˆ x (if ℓ = i ) and zero (if ℓ = i ). By these comments,( A − θ I )ˆ x ◦ E ˆ y = | X | − ( θ ∗ i − x − y + θ ∗ i x y + θ ∗ i +1 x + y − θ θ ∗ i ˆ x ) . Therefore E ( A ˆ x ◦ E ˆ y ) − θ E (ˆ x ◦ E ˆ y ) = | X | − ( θ ∗ i − Ex − y + θ ∗ i Ex y + θ ∗ i +1 Ex + y − θ θ ∗ i E ˆ x ) . (15)Comparing (14), (15) we obtain | X | ( θ − θ ) E ˆ x ⋆ E ˆ y = θ ∗ i − Ex − y + θ ∗ i Ex y + θ ∗ i +1 Ex + y − θ θ ∗ i E ˆ x + ( θ − θ ) E ˆ y. (16)In (16), eliminate the term Ex y using (12). In the resulting equation, solve for E ˆ x ⋆ E ˆ y andwe are done. 6eferring to Theorem 3.7, the formula for E ˆ x ⋆ E ˆ y can be simplified if i ∈ { , , d } . Thissimplification is discussed next. Corollary 3.8.
Assume that Γ is Q -polynomial with respect to E . Then (i)–(iii) hold below. (i) For x ∈ X , E ˆ x ⋆ E ˆ x = θ θ ∗ − θ θ ∗ + θ − θ | X | ( θ − θ ) E ˆ x. (ii) For x, y ∈ X at distance ∂ ( x, y ) = 1 , E ˆ x ⋆ E ˆ y = ( θ ∗ − θ ∗ ) Ex + y + ( θ − θ ) θ ∗ E ˆ x + ( θ − θ + θ ∗ − θ ∗ ) E ˆ y | X | ( θ − θ ) . (iii) For x, y ∈ X at distance ∂ ( x, y ) = d , E ˆ x ⋆ E ˆ y = ( θ ∗ d − − θ ∗ d ) Ex − y + ( θ − θ ) θ ∗ d E ˆ x + ( θ − θ ) E ˆ y | X | ( θ − θ ) . Proof. (i) We evaluate (13) with y = x and i = 0. We have x − y = 0 and x y = 0, so Ex + y = θ E ˆ x by Lemma 3.6.(ii) Set i = 1 in (13) and use x − y = ˆ y .(iii) Set i = d in (13) and use x + y = 0. Corollary 3.9.
Assume that Γ is Q -polynomial with respect to E . Then the Krein parameter q , satisfies q , = θ θ ∗ − θ θ ∗ + θ − θ θ − θ . Proof.
Compare Lemma 3.2 and Corollary 3.8(i).The eigenvalue θ appears in the above results. By [12, Lemma 19.21] we find that 1 + θ ,1 + θ ∗ are nonzero and 1 + θ θ − θ = 1 + θ ∗ θ ∗ − θ ∗ . (17) We continue to discuss the distance-regular graph Γ and its Q -polynomial primitive idem-potent E . Pick distinct x, y ∈ X . In the formula (13) we computed E ˆ x ⋆ E ˆ y . We have E ˆ x ⋆ E ˆ y = E ˆ y ⋆ E ˆ x , so the right-hand side of (13) must be invariant if we interchange x, y .In this section, we express the right-hand side of (13) in a form that makes this invarianceexplicit. We will use a result known as the balanced set condition.7 emma 4.1. (See [9, Theorem 1.1], [10, Theorem 3.3].) For distinct x, y ∈ X we have Ex − y − Ey − x = c i θ ∗ − θ ∗ i − θ ∗ − θ ∗ i ( E ˆ x − E ˆ y ) , (18) Ex + y − Ey + x = b i θ ∗ − θ ∗ i +1 θ ∗ − θ ∗ i ( E ˆ x − E ˆ y ) , (19) where i = ∂ ( x, y ) . Corollary 4.2.
For distinct x, y ∈ X we have C ( x, y ) = C ( y, x ) , B ( x, y ) = B ( y, x ) where C ( x, y ) = Ex − y − c i θ ∗ − θ ∗ i − θ ∗ − θ ∗ i E ˆ x, (20) B ( x, y ) = Ex + y − b i θ ∗ − θ ∗ i +1 θ ∗ − θ ∗ i E ˆ x (21) and i = ∂ ( x, y ) .Proof. Rearrange the terms in (18), (19).We clarify the meaning of (20) and (21). For i = 1 we have C ( x, y ) = E ˆ x + E ˆ y . For i = d we have B ( x, y ) = 0.For distinct x, y ∈ X we are going to express E ˆ x ⋆ E ˆ y in terms of C ( x, y ) and B ( x, y ). Thefollowing equation will be useful. Lemma 4.3.
We have c i ( θ ∗ − θ ∗ i − )( θ ∗ i − − θ ∗ i ) θ ∗ − θ ∗ i + b i ( θ ∗ − θ ∗ i +1 )( θ ∗ i +1 − θ ∗ i ) θ ∗ − θ ∗ i = ( θ − θ ) θ ∗ i + θ − θ for ≤ i ≤ d − and c d ( θ ∗ − θ ∗ d − )( θ ∗ d − − θ ∗ d ) θ ∗ − θ ∗ d = ( θ − θ ) θ ∗ d + θ − θ . Proof.
In the equation 0 = E ˆ x ⋆ E ˆ y − E ˆ y ⋆ E ˆ x , expand the right-hand side using Theorem3.7 and evaluate the result using Lemma 4.1. Examine the outcome using the fact that E ˆ x = E ˆ y .The following is our second main result. Theorem 4.4.
Assume that Γ is Q -polynomial with respect to E . Then for distinct x, y ∈ X we have E ˆ x ⋆ E ˆ y = ( θ ∗ i − − θ ∗ i ) C ( x, y ) + ( θ ∗ i +1 − θ ∗ i ) B ( x, y ) + ( θ − θ )( E ˆ x + E ˆ y ) | X | ( θ − θ ) (22) where i = ∂ ( x, y ) . Here θ ∗ d +1 denotes an indeterminate. roof. To verify (22), expand the right-hand side using (20), (21) and evaluate the resultusing Theorem 3.7 along with Lemma 4.3.Referring to Theorem 4.4, the formula for E ˆ x ⋆ E ˆ y can be simplified if i ∈ { , d } . Thissimplification is discussed next. Corollary 4.5.
Assume that Γ is Q -polynomial with respect to E . Then (i), (ii) hold below. (i) For x, y ∈ X at distance ∂ ( x, y ) = 1 , E ˆ x ⋆ E ˆ y = ( θ ∗ − θ ∗ ) B ( x, y ) + ( θ − θ + θ ∗ − θ ∗ )( E ˆ x + E ˆ y ) | X | ( θ − θ ) . (ii) For x, y ∈ X at distance ∂ ( x, y ) = d , E ˆ x ⋆ E ˆ y = ( θ ∗ d − − θ ∗ d ) C ( x, y ) + ( θ − θ )( E ˆ x + E ˆ y ) | X | ( θ − θ ) . Proof. (i) Set i = 1 in (22) and use C ( x, y ) = E ˆ x + E ˆ y .(iii) Set i = d in (22) and use B ( x, y ) = 0. The algebraic structure of a Q -polynomial distance-regular graph can be described using theconcept of a Leonard system [11, Definition 1.4]; this concept was motivated by a theoremof D. A. Leonard [1, p. 260], [6]. We refer the reader to [11, 12] for the standard notationand basic results about Leonard systems. The equations below are routinely obtained usingthe formulas in [12, Sections 19, 20]. Let Φ denote any Leonard system with diameter d ≥ a ∗ = ( θ − a ) θ ∗ − ( θ − a ) θ ∗ + ( θ − θ ) c ∗ θ − θ . If we set a ∗ = q , and a = 0 and c ∗ = 1 then we recover the formula in Corollary 3.9.For Φ we also have c − a + θ θ − θ = c ∗ − a ∗ + θ ∗ θ ∗ − θ ∗ . If we set c = 1, a = 0 and c ∗ = 1, a ∗ = 0 then we recover (17).For Φ we also have c i ( θ ∗ − θ ∗ i − )( θ ∗ i − − θ ∗ i ) θ ∗ − θ ∗ i + b i ( θ ∗ − θ ∗ i +1 )( θ ∗ i +1 − θ ∗ i ) θ ∗ − θ ∗ i = ( θ − θ )( θ ∗ i − a ∗ ) + ( θ − θ ) c ∗ for 1 ≤ i ≤ d − c d ( θ ∗ − θ ∗ d − )( θ ∗ d − − θ ∗ d ) θ ∗ − θ ∗ d = ( θ − θ )( θ ∗ d − a ∗ ) + ( θ − θ ) c ∗ . If we set a ∗ = 0 and c ∗ = 1 then we recover the formulas in Lemma 4.3.9 Acknowledgement
The author thanks Jia Huang and Kazumasa Nomura for giving this paper a close readingand offering valuable comments.
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