## Mathematik.uni-muenchen.de

**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 **= (

*w*1

*, ., wr*)

*→ M ***w**
*N • *: (

*A × B*)

*∗ → K*
1

*, · · · , r → N***w***.*
**Examples**
*Ze•∗ *is the bimould defined by

*∗, *Q

*/*Z

*× *N

*∗, *C) :=

*Ze∗*
*n*1

*>n*2

*>···nr>*0
with

*s*1

*≥ *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

*∗, {e*2

*πik*;

*k ∈ *Q

*} ∪ {*0

*}, *C) :=

*W ae*2

*πi *10

*s*1

*−*1

*···e*2

*πi r *0

*sr−*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• *:

*C***w **=

*M ***w **+

*N ***w**
*M • × N• *=

*mu*(

*M •, N •*) =

*C• *:

*C***w **=

*M ***w**1

*· N***w**2

*.*
**w**=

**w**1

*·***w**2

*swap *: (

*M •, A × B, K*)

*→ *(

*M •, B × A, K*)

*nepar*(

*M •*)(

*w*1

*,.,wr*) = (

*−*1)

*rM *(

*−w*1

*,.,−wr*)

*Flexions*

A-semi-group,

*B*-abelian group

**w **=

*.***w**1

*· ***w**2

*.*

**w**1 =

*u*1

*· · · ur , ***w**2 =

*ur*+1

*· · · us*
**Symmetries**
**Definition 2. ***A mould/bimould A• is symmetral (resp. alternal) if*
*∀***w**1

*, ***w**2

*,*
*A***w **=

*A***w**1

*A***w**2

*(resp. *= 0)

*,*
**w***∈sha*(

**w**1

*,***w**2)

*where sha*(

**w**1

*, ***w**2)

*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*
*∀***w**1

*, ***w**2

*,*
*A***w **=

*A***w**1

*A***w**2

*(resp. *= 0)

*,*
**w***∈she*(

**w**1

*,***w**2)

*where she*(

**w**1

*, ***w**2)

*denotes the “contracting shuffle” or “stuffle” product of sequences, which is given by the*

recursion relation,
**w**1 = (

*a*1

*, ., ar*)

*, ***w**2 = (

*ar*+1

*, ., ar*+

*s*)

*she*(

**w***i, ∅*) =

**w***i*

she(

**w**1

*, ***w**2) =

*a*1

*· she*((

*a*2

*, ., ar*)

*, ***w**2)

*ar*+1

*· she*(

**w**1

*, *(

*ar*+2

*, ., ar*+

*s*))

(

*a*1 +

*ar*+1)

*· she*((

*a*2

*, ., 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

*{e*2

*πik*;

*k ∈ *Q

*} ∪ {*0

*}*,

*· W a*(1) =

*W a*(0) = 0,

*· W a• *is symmetral.

**Generating Series**
*vs*1

*−*1

*· · · vsr−*1

*.*
*W a*(

*e*2

*πi *10

*s*1

*−*1

*,··· ,e*2

*πi r *0

*sr−*1)

*us*1

*−*1(

*u*
1 +

*u*2)

*s*2

*−*1

*· · · *(

*u*1 +

*u*2 +

*· · · *+

*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−*1

*ζ*(

*k*)

*k*
*i*]

*× *Q

*/*Z

*, *C) :=

*mini*
**Proposition 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 **=

*A***w**1

*B***w**2

*− B***w**1

*A***w**2 +

*M ***w**3

*N ***w**2

**w**4

*− N ***w**3

*M ***w**2

**w**4

**w**=

**w**2

**w**3

**w**4

*M ***w**1

**w**3

*N ***w**2

*− N***w**1

**w**3

*M ***w**2

**w**=

**w**1

**w**2

**w**3

*(** 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 ***B***a1 ***· · · B***as**+

**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

*are*
*irreducible elements of the *Q

*algebra of multiple zeta values, *Z

*eta;*
*• The irreducibles appearing as coefficients in the factors give us a factorization for the multiple zeta value*
Z

*eta *:= Z

*etaI ⊗ *Z

*etaII ⊗ *Z

*etaIII.*
*· *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, Z

*etaIII*, is generated by irreducibles of odd depth,
i.e. linear combinations of

*ζ*(

*s*1

*, ., sr*) where

*r *is odd. The mould of such irreducibles is denoted by

*Irr• *.

*· *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

*Irrs*1

*,.,sr loma• · · · loma• ,*
where

*loma• *is the restriction of

*loma• *to the weight

*s*
*· Zag***w ***∈ *Q[[

*u*
1 = 0), which in the language of zetas, means that the corresponding factor in
the Z

*eta *algebra, Z

*etaI*, is generated by 6

*ζ*(2) =

*π*2.

*· *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 Z

*eta*
*Irrs*1

*,.,sr loma• · · · loma• .*
Source: http://www.mathematik.uni-muenchen.de/~carr/Kyoto-Moules.pdf

The Woman with Dysuria Finch University of Health Sciences/Chicago Medical School North Chicago, Illinois Bacterial cystitis is the most common bacterial infection occurring in women. Thirty percent of women will experience at least one episode of cystitis during their lifetime. About one third of patients presenting with symptoms of cystitis have upper urinary tract infections. A careful h

Acordo colectivo de trabalho n.º 9/2011 Acordo colectivo de entidade empregadora pública celebrado entre o Instituto da Segurança Social, I. P., e a Federação Nacional dos Sindicatos da Função Pública 1 - O presente Acordo Colectivo de Entidade Empregadora Pública, doravante designado por Acordo, aplica-se aos trabalhadores do Instituto da Segurança Social, I. P., adiante designado