Definition 1. The standard definition is that a mould is a function on “a variable number of variables”. To flesh out this definition, in the general case, let A, B be sets and K be an algebra. A mould, M • = (M •, A, K), is a map from the free monoid A∗ into K and a bimould is defined as a function on the free monoid of the Cartesian product of two sets, (A × B)∗: w = (w1, ., wr) → M w N • : (A × B)∗ → K
1 , · · · , r → Nw. Examples Ze•∗ is the bimould defined by
∗, Q/Z × N∗, C) := Ze∗n1>n2>···nr>0
with s1 ≥ 2. If we take i = 0 ∀i then we obtain the usual multiple zeta values. Sometimes people say thatelements in the image of this mould are “colored multiple zeta values”. r , r−1− r ,··· , 1− 2
∗, {e2πik; k ∈ Q} ∪ {0}, C) := W ae2πi 10s1−1···e2πi r 0sr−1
Hence we require that the first term be a root of unity and the last term be 0. Operations on Moulds
Given two moulds (resp. bimoulds) (M •, A(resp. × B), K) and (N •, A(resp. × B), K) addition and multi-plication are given by
M • + N • = C• : Cw = M w + N w M • × N• = mu(M •, N •) = C• : Cw = M w1 · Nw2. w=w1·w2 swap : (M •, A × B, K) → (M •, B × A, K)
nepar(M •)(w1,.,wr) = (−1)rM (−w1,.,−wr)
Flexions A-semi-group, B-abelian group w = .w1 · w2. w1 = u1 · · · ur , w2 = ur+1 · · · us Symmetries Definition 2. A mould/bimould A• is symmetral (resp. alternal) if ∀w1, w2, Aw = Aw1Aw2 (resp. = 0), w∈sha(w1,w2) where sha(w1, w2) denotes the shuffle product of sequences. We say such a mould is “as” (resp. “al”). W a•∗ is as. A mould/bimould A• is symmetrel (resp. alternel) if∀w1, w2, Aw = Aw1Aw2 (resp. = 0), w∈she(w1,w2) where she(w1, w2) denotes the “contracting shuffle” or “stuffle” product of sequences, which is given by the recursion relation, w1 = (a1, ., ar), w2 = (ar+1, ., ar+s) she(wi, ∅) = wi she(w1, w2) = a1 · she((a2, ., ar), w2) ar+1 · she(w1, (ar+2, ., ar+s))
(a1 + ar+1) · she((a2, ., ar), (ar+2, ., ar+s)).
Such a mould is called es (resp. el). Ze•∗ is es. More Examples
The following examples of moulds define two generating series for multiple zeta values, and provide a methodof regularization of multiple zeta values. Regularization
There exists a unique mould, Ze•, such that
· Ze• = Ze•∗ wherever Ze•∗ is defined,· Ze• is defined on all of (Q/Z × N∗)∗
Likewise, there exists a unique mould, W a•, such that
· W a• = W a•∗ wherever Wa•∗ is defined,· W a• is defined on all {e2πik; k ∈ Q} ∪ {0},· W a(1) = W a(0) = 0,· W a• is symmetral. Generating Series vs1−1 · · · vsr−1.W a(e2πi 10s1−1,··· ,e2πi r 0sr−1)us1−1(u
1 + u2)s2−1 · · · (u1 + u2 + · · · + ur)sr−1.Zag• is symmetral, whereas Zig• satifies another symmetry, symmetril, closely related to symmetrel. Conversion
(mono•, Q/Z × N∗, C) := 1 +
(−1)k−1ζ(k) ki] × Q/Z, C) := miniProposition 3. mini• × swap(Zag•) = Zig•
We say then that Zag• is as/is, since it’s symmetral and its swap is symmetril (up to multiplication by
Key Results ARI/GARI
In order to keep simplicity for this talk, we take the following definition, which is more restricted than theusual general definition. Definition 4. Let ARIal/il be the Lie algebra with the following definition. • As a vector space over Q,
(ARIal/il, Q[ui] × Q/Z, C[[ui]]) := M • : M ∅ = 0, M is al, swap(M •) is alternil∗∗ ,ari(M •, N •)w = Aw1Bw2 − Bw1Aw2 + M w3N w2 w4 − N w3M w2 w4 w=w2w3w4 M w1 w3N w2 − Nw1 w3M w2 w=w1w2w3 (** The alternil condition means “up to a multiplication by a commutative bimould”.)• The ari bracket is equal to the Lie-Poisson bracket {, }, up to a variable change, on the usual Lie algebra
of multizeta values (dm). However, the ari bracket can be defined on a more general set of vector spaces,which is a tool Ecalle uses in his proofs. • The flexions ( , , , ) in the definition of the “ari” bracket correspond to the derivations, Df(x) =
0, Df (y) = [y, f ], which give the definition of the above mentioned Poisson bracket. Definition 5. By taking the ari-exponential of the Lie algebra, ARIal/il, we obtain a Lie group, GARIas/is which has the following presentation.
(GARIas/is, Q[ui] × Q/Z, C[[ui]]) := {M • : M ∅ = 1, M is as, swap(M •) is symmetril∗∗},gari(A•, B•)w = A b1 · · · A bs Ba1 · · · Bas+1 (B−1) c1 · · · (B−1) cs, w=a1b1c1···bscsas+1 where s ≥ 0, bi = ∅ (∀1 ≤ i ≤ s), ci · ai+1 = ∅ (∀1 ≤ i ≤ s − 1) and (B−1) denotes the inverse for standard mould multiplication. • invgari(M•) is inverse of a mould M• for the gari product,gari(invgari(A•), A•) = gari(A•, invgari(A)•) = 1•, where 1∅ = 1, 1w = 0. IMPORTANT FACT
The mould Zag• is an element of the Lie group, GARI. Canonical Decomposition into Irreductibles Theorem 6. The mould Zag• may be decomposed into three factors, Zag• = gari(Zag•, Zag• , Zag• )
• The even/odd length components of Zag•are even/odd functions of w, while the even/odd legth are odd/even functions of w; • Each component is decomposed as a series in a basis of ARIal/il, which when evaluated at i = 0 areirreducible elements of the Q algebra of multiple zeta values, Zeta;• The irreducibles appearing as coefficients in the factors give us a factorization for the multiple zeta value
Zeta := ZetaI ⊗ ZetaII ⊗ ZetaIII.· The factor Zag• is the most simple to express explicitly,
gari(Zag• , Zag• ) = gari(nepar(invgari(Zag•)), Zag•).
By linearizing, you can see that indeed this provides an odd/even function on components of even/oddlength. · The length 1 component is given by
· The associated factor in the multiple zeta value algebra, ZetaIII, is generated by irreducibles of odd depth,
i.e. linear combinations of ζ(s1, ., sr) where r is odd. The mould of such irreducibles is denoted byIrr• . · We get an explicit expression for the set of irreducible multiple zeta values in factor Zag• in terms of a
mould loma•, which is a generating mould which (vaguely speaking) forms a basis of rational polynomialsfor ARIal/il (the explicit construction is out of the scope of this talk). We have
Irrs1,.,sr loma• · · · loma• ,
where loma• is the restriction of loma• to the weight s· Zagw ∈ Q[[u
1 = 0), which in the language of zetas, means that the corresponding factor in
the Zeta algebra, ZetaI, is generated by 6ζ(2) = π2. · The explicit factorization of Zag• from Zag• is a very costly analytic contruction, whose difficulty comes
from getting rid of unwanted singularities. The formula is the following:
Zag• = gari(tal•, invgari(pal•), expari(roma•)),
and again the definition of pal and roma go out of the scope, since they are very long. · Zag• is explicitly calculated by factoring Zag• by Zag• and Zag• . · Zag• may be factored as a generating series for the irreducible multiple zeta values of even depth in the
same manner as Zag• , providing a set of “canonical” irreducibles for ZetaIrrs1,.,sr loma• · · · loma• .
sional time. In her 100 breaststroke she is only INSIDE THE ATHLETE three seconds away and if she accomplished this Sydney Foster time this winter, she would be ahead of where she Sydney Foster started out as just an ordinary kid was as a nine year old trying to accomplish the with ordinary dreams and an ordinary life. She same standard. Since there
Revision Date 07/12/2012Revision 2Supersedes date 07/12/2012 SAFETY DATA SHEET SODIUM HYDROXIDE PEARL LRG According to Regulation (EC) No 1907/2006 SECTION 1: IDENTIFICATION OF THE SUBSTANCE/MIXTURE AND OF THE COMPANY/UNDERTAKING 1.1. Product identifier 1.2. Relevant identified uses of the substance or mixture and uses advised against Processes involving incompatible materials. P