By M. Aizenman (Chief Editor)

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**Example text**

We denote the canonical section on D → Dc as ψD . Now we discuss an example of a partial nonassociative dioperad which is important for us. Let U = ⊕n∈J U(n) be a graded vector space and J an index set. We denote the projection U → U(n) as Pn . Now we consider a family of spaces of multi-linear maps EU = {EU (m, n)}m,n∈N , where EU (m, n) := HomC (U ⊗m , U ⊗n ). 16) ( j) For f ∈ EU (m, n), g j ∈ EU (k j , l j ) and u p j ∈ U , 1 ≤ p j ≤ l j , j = 1, . . ,i n ) ( f ; g1 , . . ,sn ∈J (1) (n) ⊗ · · · ⊗ u ln ) (1) (n) (n) Ps1 g1 (u 1 ⊗ · · · ⊗ u l1 ) ⊗ · · · ⊗ Psn gn (u 1 ⊗ · · · ⊗ u ln ) is well-defined if the multiple sum converges absolutely.

Jn − w1 , . . , wn − +n + ; z −1 , z¯ −1 , . . , z −m − , z¯ −m − , z 1 , z¯ 1 , . . , z m + , z¯ m + ; s1 , . . 31) for p = −1, . . , −n − , q = 1, . . , n + , k = −1, . . , −m − and l = 1, . . , m + . 2. When rk = ∞ for some k = −m − , . . , −1, 1, . . , m + . 30) by exchanging 1op with wα(k) . 48 L. 16. νcl−op is S L(2, R)-invariant. Proof. The S L(2, R) is generated by the following three transformations 1. w → aw, ∀a ∈ R+ ; 2. w → w − b, ∀b ∈ R; 3. w → −1 w . That νcl−op is invariant under the first two transformations simply follows from the L(0)- and L(−1)-properties of m cl−op .

10. Let Pi ∈ S(m − , m + |n − , n + ), i = 1, 2 and Q ∈ K(m − , m + ). If I Q exists for 1 ≤ i ≤ m (1) , 1 ≤ j ≤ m (2) and 1 ≤ k ≤ n (1) , P1 i ∞−B j P2 and P1 k ∞−l + + − 1 ≤ l ≤ m − , then we have δ(P1 i ∞−B j P2 ) = δ(P1 ) (1) 2n + +i ∞−(2n(2) + j) δ(P2 ), − I δ(P1 k ∞−l Q) = (δ(P1 ) k ∞−l Q ) (1) n + +m + −1+k ¯ ∞−l Q. 11). The C-extension S˜ c (m − , m + |n − , n + ) of S(m − , m + |n − , n + ) is defined to be the pull˜ c (2n − + m − , 2n + + m + ). We denote the canonical section on S˜ c , which back bundle of K ˜ c , as ψS .

### Communications In Mathematical Physics - Volume 283 by M. Aizenman (Chief Editor)

by Anthony

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