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Intertwining Operators of Double Affine Hecke Algebras

a rXiv:q -alg/96514v511J u l1996June (1996),1–36Dedicated to Yuri I.Manin on the occasion of his 60birthdayIntertwining operatorsof double aﬃne Hecke algebras By Ivan Cherednik*Continuing [C3,C4],we study the intertwining operators of double aﬃne Hecke algebras H .They appeared in several papers (especially in [C2,C4,C6]).However for the ﬁrst time here we apply them systematically to create the non-symmetric [M3,C4]and symmetric [M2]Macdonald polynomials for arbitrary root systems and to start the theory of induced and co-spherical H -modules.The importance of this technique was clearly demonstrated in recent pa-pers by F.Knop and S.Sahi [Kn],[KS],[S].Using the intertwiners of the double aﬃne Hecke algebras in the case of GL (dual to those considered in [C1,C2])they proved the q,t -integrality conjecture by I.Macdonald [M1]and managed to establish the positivity of the coeﬃcients of the Macdonald polynomials in the diﬀerential case.As to the integrality,we mention another approach based on the so-called Vinet operators (see [LV]and a recent work by Kirillov,Noumi),and the results by Garsia,Remmel,and Tesler.We do not try in this paper to get the best possible estimates for the denominators of the Macdonald polynomials (generally speaking,the problem looks more complicated than in the stable GL -case).However even rather straightforward analysis of the intertwiners gives a lot.For instance,it is enough to ensure the existence of the restricted Macdonald polynomials at roots of unity from [C3,C4],where we used less convenient methods based directly on the deﬁnition or on the recurrence relations.The technique of intertwiners combined with the (projective)action of GL (2,Z )from [C3]gives another proof of the norm and the evaluation formulas (see [C4]).Here the H -embedding of the space of nonsymmetricpolynomialsinto the space of functions on the aﬃne Weyl group ˜W ([C4],Proposition 5.2)plays a key role.The latter representation when restricted to the aﬃne Hecke subalgebra turns into the classical one from [IM]as t is a power of p and q →0(˜W is identiﬁed with the set of double cosets of the corresponding p -adic group with respect to the Iwahori subroup).2IV AN CHEREDNIKAnother important application is a calculation of the Fourier transforms of the Macdonald polynomials in the sense of[C3,C4].For instance,it gives a canonical identiﬁcation of the polynomial representation of the aﬃne Hecke algebra with the representation in functions on the weight lattice(which col-lapses in the p-adic limit).We introduce a proper discretization of theµ-function(the truncated theta-function making Macdonald’s polynomials pairwise orthogonal)and the corresponding discrete inner product on Funct(˜W).It readily gives the pro-portionality of the norms of the Macdonald polynomials[M2,C2,M3,C4]and those deﬁned for the Jackson integral taken instead of the constant term in the inner product.The coeﬃcient of proportionality is described by the Aomoto conjecture(see[A,Ito])recently proved by Macdonald(to calculate it one can also follow[C2],replacing the shift operators by their discretizations).We note that the Macdonald polynomials considered as functions on˜W are square integrable forﬁnitely many weights only.Here|q|=1and the real partℜ(k)for t=q k is to be negative(otherwise we have none).The program is to describe all integrable and non-integrable eigenfunctions of the discrete Dunkl operators in this representation and to study the corresponding Fourier transform.In contrast to the classical p-adic harmonic analysis(see e.g.[HO])the Plancherel measure coincides with the discretization ofµ(the Fourier transform is self-dual).More generally,we consider the action of the double aﬃne Hecke algebra in the same space Funct(˜W)depending on an arbitrary given weight.Its sub-module generated by the delta-functions is induced(from a character of the standard polynomial subalgebra)and co-spherical.Mainly following[C5],we ﬁnd out when arbitrary induced representations(in the same sense)are irre-ducible and co-spherical using the technique of intertwiners.The answer is a natural”aﬃnization”of the well-known statements in the p-adic case(see e.g.[KL],[C5]).The classiﬁcation of co-spherical representations is impor-tant for the harmonic analysis and plays the key role in the theory of aﬃne Knizhnik-Zamolodchikov equations(see[C6,C7,C8]).We also induce up irre-ducible representations of aﬃne Hecke subalgebras([C6]is devoted to appli-cations of such representations).If q is suﬃciently general the H-modules we get are irreducible,so one can use the classiﬁcation of[KL].Thus in this paper we begin a systematic study of the representations of double aﬃne Hecke algebras and related harmonic analysis.The polyno-mial representation considered in the series of papers[C2-4]devoted to the Macdonald conjectures is remarkable,but still just an example.The paper was started during my stay at RIMS(Kyoto University),con-tinued at CRM in Montreal,and completed at the University of Nijmegen.I am grateful to T.Miwa,L.Vinet,G.Heckman and my colleagues at theseINTERTWINERS OF DOUBLE HECKE ALGEBRAS3 institutes for the kind invitations and the hospitality.The author thanks E. Frenkel,G.Heckman,D.Kazhdan,I.Macdonald,and E.Opdam for useful discussions.1.Aﬃne Weyl groupsLet R={α}⊂R n be a root system of type A,B,...,F,G with respect to a euclidean form(z,z′)on R n∋z,z′,normalized by the standard condition that(α,α)=2for longα.Let usﬁx the set R+of positive roots(R−=−R+),the corresponding simple rootsα1,...,αn,and their dual counterparts a1,...,a n,a i=α∨i,whereα∨=2α/(α,α).The dual fundamental weights b1,...,b n are determined from the relations(b i,αj)=δj i for the Kronecker delta.We will also use the dual root system R∨={α∨,α∈R},R∨+,and the latticesA=⊕n i=1Z a i⊂B=⊕n i=1Z b i,A±,B±for Z±={m∈Z,±m≥0}instead of Z.(In the standard notations, A=Q∨,B=P∨-see[B].)Later on,να=να∨=(α,α),νi=ναi,νR={να,α∈R}⊂{2,1,2/3}.(1.1)ρν=(1/2) να=να=(ν/2)νi=νb i,forα∈R+.The vectors˜α=[α,k]∈R n×R⊂R n+1forα∈R,k∈Z form the aﬃne root system R a⊃R(z∈R n are identiﬁed with[z,0]).We addα0def=[−θ,1] to the simple roots for the maximal rootθ∈R.The corresponding set R a+of positive roots coincides with R+∪{[α,k],α∈R,k>0}.We denote the Dynkin diagram and its aﬃne completion with{αj,0≤j≤n}as the vertices byΓandΓa(m ij=2,3,4,6ifαi andαj are joined by0,1,2,3laces respectively).The set of the indices of the images ofα0by all the automorphisms ofΓa will be denoted by O(O={0}for E8,F4,G2). Let O∗=r∈O,r=0.The elements b r for r∈O∗are the so-called minuscule weights((b r,α)≤1forα∈R+).Given˜α=[α,k]∈R a,b∈B,lets˜α(˜z)=˜z−(z,α∨)˜α,b′(˜z)=[z,ζ−(z,b)](1.2)for˜z=[z,ζ]∈R n+1.The aﬃne Weyl group W a is generated by all s˜α(simple reﬂections s j= sαjfor0≤j≤n are enough).It is the semi-direct product W⋉A,where the non-aﬃne Weyl group W is the span of sα,α∈R+.Here and futher we identify b∈B with the corresponding translations.For instance,a=sαs[α,1]=s[−α,1]sαfor a=α∨,α∈R.(1.3)4IV AN CHEREDNIKThe extended Weyl group W b generated by W and B is isomorphic to W⋉B:(wb)([z,ζ])=[w(z),ζ−(z,b)]for w∈W,b∈B.(1.4)Given b+∈B+,letωb+=w0w+0∈W,πb+=b+(ωb+)−1∈W b,ωi=ωbi,πi=πbi,(1.5)where w0(respectively,w+0)is the longest element in W(respectively,in W b+ generated by s i preserving b+)relative to the set of generators{s i}for i>0.The elementsπr def=πbr,r∈O leaveΓa invariant and form a group denoted byΠ,which is isomorphic to B/A by the natural projection{b r→πr}.As to{ωr},they preserve the set{−θ,αi,i>0}.The relationsπr(α0)=αr= (ωr)−1(−θ)distinguish the indices r∈O∗.Moreover(see e.g.[C2]): W b=Π⋉W a,whereπr s iπ−1r=s j ifπr(αi)=αj,0≤j≤n. (1.6)Givenν∈νR,r∈O∗,˜w∈W a,and a reduced decomposition˜w=s jl ...s j2s j1with respect to{s j,0≤j≤n},we call l=l(ˆw)the length ofˆw=πr˜w∈W b.Setting(1.7)λ(ˆw)={˜α1=αj1,˜α2=s j1(αj2),˜α3=s j1s j2(αj3),......,˜αl=˜w−1s jl(αjl)},one can represent(1.8)l=|λ(ˆw)|= νlν,for lν=lν(ˆw)=|λν(ˆw)|,λν(ˆw)={˜αm,ν(˜αm)=ν(˜αjm)=ν},1≤m≤l, where||denotes the number of elements,ν([α,k])def=να.Let us introduce the following aﬃne action of W b on z∈R n:(1.9)(wb) z =w(b+z),w∈W,b∈B,s˜α z =z−((z,α)+k)α∨,˜α=[α,k]∈R a,and the pairing([z,ζ],z′+d)def=(z,z′)+ζ,where we treat d formally(see e.g. [K]).The connection with(1.2,1.3)is as follows:(1.10)(ˆw([z,ζ]),ˆw z′ +d)=([z,ζ],z′+d)forˆw∈W b.Using the aﬃne Weyl chamberC a=nj=0Lαj,L˜α={z∈R n,(z,α)+k>0},INTERTWINERS OF DOUBLE HECKE ALGEBRAS5(1.11)λν(ˆw)={˜α∈R a+, C a ⊂ˆw L˜α ,ν(˜α)=ν} ={˜α∈R a+,lν(ˆws˜α)<lν(ˆw)}.It coincides with(1.8)due to the relations(1.12)λν(ˆwˆu)=λν(ˆu)∪ˆu−1(λν(ˆw)),λν(ˆw−1)=−ˆw(λν(ˆw)) if lν(ˆwˆu)=lν(ˆw)+lν(ˆu).The following proposition is from[C4].Proposition1.1.Given b∈B,the decomposition b=πbωb,ωb∈W can be uniquely determined from the following equivalent conditionsi)l(πb)+l(ωb)=l(b)and l(ωb)is the biggest possible,ii)ωb(b)=b−∈B−and l(ωb)is the smallest possible,iii)πb 0 =b andλ(πb)∩R=∅.We will also use thatλ(b)={˜α,(b,α)>k≥0ifα∈R+,(1.13)(b,α)≥k>0ifα∈R−},λ(πb)={˜α,α∈R−,(b−,α)>k>0if(α,b)<0,(1.14)(b−,α)≥k>0if(α,b)>0},andλ(π−1b)={˜α,−(b,α)>k≥0}for˜α=[α,k]∈R a+.(1.15)Convexity.Let us introduce two orderings on B.Here and further b±are the unique elements from B±which belong to the orbit W(b).Namely, b−=ωbπb=ωb(b),b+=w0(b−)=ω−b(b).So the equality c−=b−(or c+=b+)means that b,c belong to the same orbit.Setb≤c,c≥b for b,c∈B if c−b∈A+,(1.16)b c,c b if b−<c−or b−=c−and b≤c.(1.17)We use<,>,≺,≻respectively if b=c.For instance,c≻b+⇔b+>W(c)>b−,c b−⇔c∈W(b−)or c≻b+.The following sets(1.18)σ(b)def={c∈B,c b},σ∗(b)def={c∈B,c≻b},σ+(b)def={c∈B,c−>b−}=σ∗(b+).are convex.Moreoverσ+is W-invariant.By convex,we mean that if c,d= c+rα∨∈σforα∈R+,r∈Z+,then{c,c+α∨,...,c+(r−1)α∨,d}⊂σ.(1.19)6IV AN CHEREDNIKThe elements fromσ(b)strictly between c and d(i.e.c+qα,0<q<r) belong toσ+(b).πb,where i p are from any se-Proposition 1.2.a)Letˆu=s˜αi m...s˜αi1quence1≤i1<i2<...<i m≤l=l(b)in a reduced decomposition of ˆw=π−1b(see(1.7)).In other words,ˆu is obtained by crossing out any number of{s j}from a reduced decomposition ofπb.Then c def=ˆu 0 ∈σ∗(b).More-over,c∈σ+(b)if and only if at least one of˜αi p=[α,k]for1≤p≤m has k>0.b)If c,b belong to the same W-orbit then the converse is ly, settingωbc def=πbπ−1c,the following relations are equivalent:(i)c≻b(which means that c>b),(ii)(α,c)>0for allα∈λ(ωbc),(iii)l(πb)=l(ωbc)+l(πc),It also results from(i)thatωbc is the smallest possible element w∈W such that b=w(c).Proof.Assertion a)is a variant of Proposition1.2from[C4].For the sake of completeness we will outline the proof of b).Taking u(c)≤b<c,we will check(ii),(iii)by induction supposing that{u′(c)≤b′<c}⇒{(ii),(iii)}for all b′,u′such that l(u′)<l(u),which is obvious when l(u′)=0.Settingβ=u(α)forα∈λ(u),u(sα(c))=u(c)−(α,c)β∨andβ∈R−(see the deﬁnition ofλ(α)).One can assume that(α,c)>0for all suchα. Otherwise usα(c)≤u(c)≤c and we can argue by induction.Applying(1.12) and(1.13),we see that l(uc)=l(u)+l(c).Indeed,the intersection ofλ(c)andc−1(λ(u))={[α,(c,α)],α∈λ(u)}is empty.Hence the product uπc is reduced(i.e.l(uπc)=l(u)+l(πc))and λ(uπc)=ωc c−1(λ(u))∪λ(πc)contains no roots from R+.Finally,Proposition 1.1leads to(iii)(and the uniqueness of u of the minimal possible length).This reasoning gives the equivalence of(ii)and(iii)as well.Assertion(i)readily results from(ii).We will also use(cf.Proposition5.2,[C4])the relationsπb=πrπc for b=πr c and any c∈B,r∈O and the equivalence of the following three conditions:(1.20)(αj,c+d)>0⇔αj∈λ(π−1c)⇔{s jπc=πb,c≻b}for0≤j≤n.When j>0it is a particular case of Proposition1.2b). Assuming that(α0,c+d)=1−(θ,c)>0,b=s0 c =c+(α0,c+d)θ>c>c−θ=sθ(b).INTERTWINERS OF DOUBLE HECKE ALGEBRAS7 Hence c∈σ+(b).If the product s0πc is reducible then we can apply statement a)to come to a contradiction.Therefore s0πc=πb,since s0is simple.The remaining implications are obvious.2.Intertwining operatorsWe put m=2for D2k and C2k+1,m=1for C2k,B k,otherwise m=|Π|.Let us sett˜α=tν(˜α),t j=tαj,where˜α∈R a,0≤j≤n,X˜b =ni=1X k i i q k if˜b=[b,k],(2.1)for b=ni=1k i b i∈B,k∈18IV AN CHEREDNIKdoes not depend on the choice of the reduced decomposition(because{T} satisfy the same“braid”relations as{s}do).Moreover,Tˆv Tˆw=Tˆvˆw whenever l(ˆvˆw)=l(ˆv)+l(ˆw)forˆv,ˆw∈W b.(2.3)In particular,we arrive at the pairwise commutative elementsY b=ni=1Y k i i if b=ni=1k i b i∈B,where Y i def=T bi,(2.4)satisfying the relations(2.5)T−1iY b T−1i=Y b Y−1a iif(b,αi)=1,T i Y b=Y b T i if(b,αi)=0,1≤i≤n.The following maps can be extended to involutions of H(see[C1,C3]):ε:X i→Y i,Y i→X i,T i→T−1i ,(2.6)tν→t−1ν,q→q−1,τ:X b→X b,Y r→X r Y r q−(b r,b r)/2,Yθ→X−10T−20Yθ,(2.7)T i→T i,tν→tν,q→q,1≤i≤n,r∈O∗,X0=qX−1θ.Let us give some explicit formulas:(2.8)ε(T0)=XθT−1Yθ=XθT sθ,ε(πr)=X r Tω−1r,τ(T0)=X−10T−1,τ(πr)=q−(b r,b r)/2X rπr=q(b r,b r)/2πr X−1r∗,πr X r∗π−1r=q(b r,b r)X−1r,X r∗TωrX r=T−1ωr∗.Theorem2.3from[C3]says that the map(2.9) 0−1−10 →ε, 1101→τcan be extended to a homomorphism of GL2(Z)up to conjugations by the central elements from the group generated by T1,...,T n.The involutionη=τ−1ετcorresponding to the matrix −1011 will play an important role in the paper:η:X r→q(b r,b r)/2X−1r Y r=πr X r∗Tωr ,(2.10)Y r→q(b r,b r)/2X−1r Y r X r=πr T−1ωr∗,Yθ→T−10T−1sθ,T j→T−1j(0≤j≤n),πr→πr(r∈O∗),tν→t−1ν,q→q−1.INTERTWINERS OF DOUBLE HECKE ALGEBRAS9 We note thatεandηcommute with the main anti-involution∗from[C2]:(2.11)X∗i=X−1i,Y∗i=Y−1i,T∗i=T−1i,tν→t−1ν,q→q−1,0≤i≤n,(AB)∗=B∗A∗.The X-intertwiners(see e.g.[C2,C5,C6])are introduced as follows:Φj=T j+(t1/2j −t−1/2j)(X aj−1)−1,G j=Φj(φj)−1,˜G j=(φj)−1Φj,(2.12)φj=t1/2j +(t1/2j−t−1/2j)(X aj−1)−1,for0≤j≤n.They belong to the extension of H by theﬁeld C q,t(X)of rational functions in{X}.The elements G j and G′j satisfy the same relations as{s j,πr}do,{Φj}satisfy the relations for{T j}(i.e.the homogeneous Coxeter relations and those withπr).Hence the elements(2.13)Gˆw=πr G jl ···G j1,whereˆw=πr s jl···s j1∈W b,are well-deﬁned and G is a homomorphism of W b.The same holds for˜G.AstoΦ,the decomposition ofˆw should be reduced.The simplest way to see this is to use the following property of{Φ}which ﬁxes them uniquely up to left or right multiplications by functions of X:Φˆw X b=Xˆw(b)Φˆw,ˆw∈W b.(2.14)Oneﬁrst checks(2.14)for s j andπr,then observes thatΦfrom(2.13)satisfy (2.14)for any choice of the reduced decomposition,and uses the normalizing conditions to see that they are uniquely determined from the intertwining relations(2.14).We note thatΦj,φj are self-adjoint with respect to the anti-involution (2.11).HenceΦ∗ˆw=Φˆw−1,G∗ˆw=˜Gˆw−1,ˆw∈W b.(2.15)It follows from the quadratic relations for T.To deﬁne the Y-intertwiners we apply the involutionεtoΦˆw and to G,˜G. The formulas can be easily calculated using(2.8).In the case of GL n one getsthe intertwiners from[Kn].For w∈W,we just need to replace X b by Y−1b and conjugate q,t(cf.[C4]).However it will be more convenient to consider η(Φ)instead ofε(Φ)to create the Macdonald polynomials.Both constructions gives the intertwiners satisfying the∗-relations from(2.15).3.Standard representationsIt was observed in[C4],Section5that there is a natural passage from the representation of H in polynomials to a representation in functions on10IV AN CHEREDNIKW b.We will continue this line,beginning with the construction of the basic representaions of level0,1.Settingx˜b =ni=1x k i i q k if˜b=[b,k],b=ni=1k i b i∈B,k∈1representation V0is induced from the character{T j→t j,πr→1}.Namely, the imageˆH is uniquely determined from the following condition:(3.7)ˆH(f(x))=g(x)for H∈H if Hf(X)−g(X)∈ni=0H(T i−t i)+ r∈O∗H(πr−1).To make the statement about V1quite obvious let us introduce the Gauss-ianγ=Const qΣn i=1z i zαi/2,where formallyx b=q z b,z a+b=z a+z b,z i=z bi,(wa)(z b)=z w(b)−(a,b),a,b∈R n. More exactly,it is a W-invariant solution of the following diﬀerence equations:(3.8)b j(γ)=Const q(1/2)Σn i=1(z i−(b j,b i))(zαi−δj i)=q−z j+(b j,b j)/2γ=q(b j,b j)/2x−1jγfor1≤j≤n.The Gaussian commutes with T j for1≤j≤n because it is W-invariant.A straightforward calculation gives that(3.9)γ(X)T0γ(X)−1=X−10T−1=τ(T0),γ(X)Y rγ(X)−1=q−(b r,b r)/2X r Y r=τ(Y r),r∈O.Hence the conjugation byγinducesτ.We can put in the following way.There is a formal H-homomorphism:(3.10)V0∋v→ˆv def=vγ−1∈V1.One has to complete V0,1to make this map well-deﬁned(see the discrete rep-resentations below).We will later need an extended version of Proposition3.6from[C2].Proposition3.2.a)The operators{Y i,1≤i≤n}acting in V0preserve Σ(b)def=⊕c∈σ(b)C q,t x c andΣ∗(b)(deﬁned forσ∗(b))for arbitrary b∈B.b)The operators{T j,0≤j≤n}acting in V0preserveΣ+(b)=Σ∗(b+):(3.11)ˆTj(x b)modΣ+(b)=t1/2jx b if(b,αj)=0,=t1/2js j(x b)+(t1/2j−t−1/2j)x b if(b,αj)<0,=t−1/2js j(x b)if(b,αj)>0.c)Coming to V1,if(αj,b+d)>0(0≤j≤n)then(3.12)ˆT j x b modΣ+(s j b )=t−1/2js j x b .Otherwise,(αj ,b +d )≤0and(3.13)ˆTj x b ∈Σ(b )for (αj ,b +d )≤0,ˆT j x b =t 1/2jx b if (αj ,b +d )=0.Proof.Due to Proposition 3.3from [C4]it suﬃces to check c)for j =0.The ﬁrst inequality,the deﬁnition of ˆT 0 x b = c ∈B u bc x c ,and (1.20)readily give that (for nonzero u )c =b +rθ(r ∈Z )and(3.14)s θ(b ′)=b −θ<b <c ≤b +(α0,b +d )θ=s 0 b def =b ′.Hence c ∈σ+(b ′)if c =b ′.The coeﬃcient u bb ′equals t −1/20.Let (α0,b +d )≤0.Then(3.15)s θ(b )=b −(b,θ)θ<c ≤b and c ∈{Σ+(b )∪b }∈Σ(b )(cf.Proposition 1.2,a)).Discretization.We go to the lattice version of the functions and opera-tors.Let ξbe a ”generic”character of C [x ]:x a (ξ)def =n i =1ξk i i ,a =n i =1k i b i ∈B,for independent parameters ξi .The discretizations of functions g (x )in x ∈C n and the operators from the algebra A def =⊕ˆu ∈W b C q,t (X )ˆu ,are described by the formulas:δx a (bw )=x a (q b w (ξ))=q (a,b )x w −1(a )(ξ),(δˆu (δg ))(bw )=δg (ˆu −1bw ).(3.16)For instance,(δX a (δg ))(bw )=x a (bw )g (bw )(we will sometimes omitδand put g (ˆw )instead of δg (ˆw )).The image of g ∈C q,t (x )belongs to the space F ξdef=Funct(W b ,C ξ)of C ξ-valued functions on W b ,where C ξdef =C q,t (ξ1,...,ξn ).Considering the discretizations of operators ˆHfor H ∈H we come to the functional representation of H in F ξ.Similarly,introducing the group algebra C ξ[W b ]=⊕ˆw ∈W b C ξδˆwfor (for-mal)delta-functions ,we can consider the dual anti-action on the indices:(3.17)δ(g (x )ˆu )( ˆw ∈W b c ˆw δˆw )= ˆw ∈W bc ˆw g (ˆw )δˆu −1ˆw ,c ˆw ∈C ξ.Composing it with the anti-involution of H(3.18)T ⋄j =T j (0≤j ≤n ),π⋄r =π−1r (r ∈O ),X ⋄i =X i (0≤i ≤n ),sending q,t to q,t(and AB to B⋄A⋄),we get the delta-representation∆ξof H in Cξ[W b]:(3.19)H→δ(ˆH⋄)def=δ(H)for H∈H.Explicitly,δπr=πr=δ(πr),r∈O,and forˆw=bwδ(Ti(g))(ˆw))=t1/2ix ai(w(ξ))q(a i,b)−t−1/2ix ai(w(ξ))q(a i,b)−1g(ˆw)for0≤i≤n,(3.20)δ(T i)(δˆw)=t1/2ix ai(w(ξ))q(a i,b)−t−1/2x ai(w(ξ))q(a i,b)−1δˆw for0≤i≤n.(3.21)There is a natural Cξ-linear pairing between Fξand∆ξ.Given g∈Funct(W b,Cξ),ˆw∈W b,{g,δˆw}def=g(ˆw),{H(g),δˆw}={g,H⋄(δˆw)},H∈H.(3.22)It also gives a nondegenerate pairing between V0and∆ξ.For arbitrary ope-rators A∈A,the relation is as follows:{δA(g),δˆw}={g,δA(δˆw)}.Let us extend the discretization map and the pairing with∆ξto V1.We use the map from(3.10)for theδ-Gaussian:(3.23)δγ(bw)def=q(b,b)/2xb(w(ξ)),which satisﬁes(3.8)and is a discretization ofγfor a proper constant(cf.[C4], (6.20)).The representations Fξand∆ξcan be introduced when q,t,{ξi}are con-sidered as complex numbers ensuring that x˜a(ξ)=1for all˜a∈(R a)∨.Follow-ing Proposition5.2from[C4],let us specialize the deﬁnition of∆forξ=t−ρ. In this casex a(bw)=x a(q b t−w(ρ))=q(a,b) νt−(w(ρν),a)ν.(3.24)Proposition3.3.The H-module∆(−ρ)def=∆t−ρcontains the H-sub-module∆#def=⊕b∈B Cδπb.This also holds for any q∈C∗and generic t. Moreover,∆#is irreducible if and only if q is not a root of unity.When q→0and t is a power of prime p the action of the algebra H a generated by{T j,0≤j≤n}in∆(−ρ)coincides with the standard action ofthe p-adic Hecke algebra H(G//I)∼=H a on the(linear span of)delta-functions on I\G/I∼=W a.Here I is the Iwahori subgroup of the split semisimple p-adic group G(see[IM]).However∆#does not remain a submodule in this limit.Multiplying the delta-functions on the right by the operator of t-symmet-rization we can get an H a-submodule isomorphic to∆#(upon the restriction to H a).Its limit readily exists and coincides with the space of delta-functions on I\G/K for the maximal parahoric subgroup K.However the latter space can be identiﬁed with neither spaces of delta-functions for smaller subsets of W a(as in Proposition3.3).It is possible only for the q-deformation under consideration.Practically,when calculating with right K-invariant functions in the p-adic case one needs to consider their values on the whole W a(that is an obviousﬂaw since much fewer number of points is enough to reconstruct them uniquely).4.OrthogonalityThe coeﬃcient of x0=1(the constant term)of a polynomilal f∈C q,t[x] will be denoted by f 0.Let(4.1)µ= a∈R∨+∞ i=0(1−x a q i a)(1−x−1a q i+1a).(1−x a(tρ)t a q i a)(1−x a(tρ)t−1a q i a)Here x b(t±ρq c)=q(b,c) νt±(b,ρν)ν.We note thatµ∗0=µ0with respect to the involutionx∗b=x−b,t∗=t−1,q∗=q−1.Setting(4.3)f,g 0= µ0f g∗ 0= g,f ∗0for f,g∈C(q,t)[x],we introduce the non-symmetric Macdonald polynomials e b(x),b∈B−,by means of the conditions(4.4)e b−x b∈Σ∗(b), e b,x c =0for c∈σ∗={c∈B,c≻b}in the setup of Section1.They can be determined by the Gram-Schmidt process because the pairing is non-degenerate and form a basis in C(q,t)[x].This deﬁnition is due to Macdonald[M3](for tν=q k,k∈Z+)who extended Opdam’s nonsymmetric polynomials introduced in the degenerate (diﬀerential)case in[O2].He also established the connection with the Y-operators.The general case was considered in[C4].The notations are from Proposition1.1and(1.1).We use the involution ¯x a=x−1a,¯q=q,¯t=t,a∈B.Proposition4.1.a)For any H∈H and the anti-involution∗from (2.11), ˆH(f),g 0= f,ˆH∗(g) 0.Here f,g are either from V0or from V1.All products of{X b,Y b,T j,πr,q,tν}are unitary operators.b)The polynomials{e b,b∈B}are eigenvectors of the operators{L f def= f(Y1,···,Y n),f∈C[x]}:L¯f(e b)=f(#b)e b,where#b def=πb=bω−1b ,(4.5)x a(#b)def=x a(q b t−ω−1b(ρ))=q(a,b) νt−(ω−1b(ρν),a)ν,w∈W.(4.6)Proof.Assertion a)for V0is from[C2].Using(3.10)we come to V1(a formal proof is equally simple).Since operators{Y b}are unitary relative to , 0and leave allΣ(a),Σ∗(a)invariant(Proposition3.2),their eigenvectors in C q,t[x]are exactly{e}.See[M3,C4].The theorem results immediately in the orthogonality of{e b}for pairwise distinct b.Macdonald also gives the formula for the squares of e b(for tν= q k,k∈Z+)and writes that he deduced it from the corresponding formula in the W-symmetric case(proved in[C2]).The general case was considered in [C4]where we used the recurrence relations.A direct simple proof(based on the intertwiners)will be given below.The symmetric Macdonald polynomials form a basis in the space C q,t[x]W of all W-invariant polynomials and can be expressed as follows:(4.7)p b=P t b e b=P1b e b,b=b+∈B+,P t def= c∈W(b) νt lν(w c)/2νˆTw c,w cdef=ω−1c w0,P1b=P t=1b.This presentation is from[M3,C4](from[O2]in the diﬀerential case). Here one can take the complete symmetrizations(with proper coeﬃcients) since e b is W b-invariant for the stabilizer W b of b.Macdonald introduced these polynomials in[M1,M2]by the conditionsp b−m b∈Σ+(b), p b,m c 0=0,b∈B+,c≻b,(4.8)for the monomial symmetric functions m b= c∈W(b)x c.One can also deﬁne {p}as eigenvectors for the(W-invariant)operators L f,f∈C q,t[x]W:L f(p b)=f(q b o tρ)p b,b∈B+,b o=−w0(b),(4.9)normalized as above.Applying any elements from H Y=<T j,Y b>to e c(c∈W(b+))we get solutions of(4.9),because symmetric Y-polynomials are central in H Y(due to I.Bernstein).It readily gives the coincidence of(4.7)and(4.9).Functional representations.The representations Fξ,∆ξalso have in-variant skew-symmetric forms.Letµ1(bw)=µ(bw)/µ(1)def=(4.10)a∈R∨+∞i=0(1−x a(bw)q i a)(1−x−1a(bw)q i+1a)(1−x a(1)t a q i a)(1−x−1a(1)t a q i+1a)t1/2α−qjαt−1/2αx a(ξ),(4.11)where a=α∨,and we extend the conjugation∗from C q,t to Cξsettingξ∗i=ξ−1i.b)The following Cξ-valued scalar product is well-deﬁned for f,g from the H-submodule ofﬁnitely supported functions Fξ⊂Fξ=Funct(W b,Cξ):f,g 1= ˆw∈W bµ1(ˆw)f(ˆw)g(ˆw)∗= g,f ∗1.(4.12)c)Assertion a)from Proposition4.1holds for Fξand∆ξ,where the latter module is endowed with the scalar productf,g −1= ˆw∈W b(µ1(ˆw))−1uˆw v∗ˆw,f= uˆwδˆw,g= vˆwδˆw.(4.13)Namely, H(f),g ±1= f,H∗(g) ±1.Proof.Since x˜a(ˆw)=x˜a′(1)for˜a=˜α∨∈(R a+)∨,where˜a′def=ˆw−1(˜a), one has forˆw=bw:(4.14)µ1(ˆw)= ˜α∈R a+(1−x˜a(ˆw))(1−t a x˜a(1))(1−x˜a(1))(1−t a x˜a′(1)) = ˜α∈λ(ˆw)(1−x−1˜a(1))(1−t a x˜a(1))Here we use thatˆw−1(R a+)={−λ(ˆw)}∪{R a+\λ(ˆw)}.The invariance of µ1(ˆw)∈Cξwith respect to the conjugation∗is obvious.Other statements are completely analogous to those forµ0(and follow from them).The key relation(4.15)ˆHµ(X)=µ(X)(ˆH∗)+,H∈H,readily holds after the discretization.Here by+we mean the anti-involutionˆw+=ˆw−1∈W b,x+b=x−1b,b∈B,q,t→q−1,t−1.Its discretization conjugates the values of functions from Fξand the coeﬃcients ofδˆw in∆ξ(ﬁxingδˆw).The characteristic functions fˆw∈Fξ(ˆw∈W b)are deﬁned from the rela-tions fˆw(ˆu)=δˆw,ˆu for the Kronecker delta.The action of the operators X b on them is the same as for{δˆw}:X b(fˆw)=x b(ˆw)fˆw,X b(δˆw)=x b(ˆw)δˆw,b∈B,ˆw∈W b.Moreover the mapfˆw→µ1(ˆw)δˆw,ˆw∈W b,(4.16)establishes an H-isomorphism between Fξand∆ξ,taking , 1to , −1.It readily results from the formulas:δTi(fˆw)=t1/2ix−1ai(w(ξ))q−(a i,b)−t−1/2x ai(w(ξ))q(a i,b)−1fˆw for0≤i≤n,(4.17)and the formulas for the action of{πr}.Let us consider the special caseξ=t−ρ(see(3.24)).Using the pairing(3.22),we see that the subspace(4.18)F#=⊕ˆw∈#B C q,t fˆw⊂F(−ρ)=F t−ρ,where#B={#b=πb∈W b,b∈B},is an H-submodule.It is exactly the radical of the form , 1,which is well-deﬁned for suchξ.Indeed,anyˆw can be uniquely represented in the form(see[C2])ˆw=πb w,where b=ˆw b ,w∈W,l(ˆw)=l(πb)+l(w). Hence,{ˆw∈#B}⇒{αi∈λ(ˆw)for some i>0}⇒{µ1(ˆw)=0}.On the other hand,µ1(#b)= a∈R∨+ t−1/2α−q jαt1/2αx a(tρ)。

Fronts propagating with curvature dependent speed Algorithms Based on Hamilton-Jacobi Formulations

reaching out into the unburnt gas somehow propagate slower than do concave regions which are hot gases surrounding a small unburnt pocket. At the same time, particles along the ﬂame front undergo an increase in volume as they burn, creating a jump in velocity across the ﬂame front. This discontinuity in the velocity ﬁeld creates vorticity along the burning ﬂame, which can be related to the local curvature, and this new vorticity ﬁeld contributes to the advection of the propagating ﬂame. Thus, there are at least two distinct ways in which the speed of the moving ﬂame depends on the local curvature. Typically, there have been two types of numerical algorithms employed in the solution of such problems. The ﬁrst parameterizes the moving front by some variable and discretizes this parameterization into a set of marker points [39]. The positions of the marker points are updated in time according to approximations to the equations of motion. Such techniques can be extremely accurate in the attempt to follow the motions of small perturbations. However, for large, complex motion, several problems soon occur. First, marker particles come together in regions where the curvature of the propagating front builds, causing numerical instability unless a regridding technique is employed. The regridding mechanism usually contains a error term which resembles diffusion and dominates the real effects of curvature under analysis. Secondly, such methods suffer from topological problems; when two regions "burn" together to form a single one, ad-hoc techniques to eliminate parts of the boundary are required to make the algorithm work. Other algorithms commonly employed fall under the category of "volume of ﬂuid " techniques, which, rather than track the boundary of the propagating front, track the motion of the interior region. An example of this type of algorithm is SLIC [26]. In these algorithms, the interior is discretized, usually by employing a grid on the domain and assigning to each cell a "volume fraction" corresponding to the amount of interior ﬂuid currently located in that cell. An advantage of such techniques is that no new computational elements are required as the calculation progresses (unlike the parameterization methods), and complicated topological boundaries are easily handled, see [4,32]. Unfortunately, it is difﬁcult to calculate the curvature of the front from such a representa-

On the Complexity of the Interlace Polynomial

Although the interlace polynomial q(G; x, y) is a diﬀerent object from the celebrated Tutte polynomial (also known as dichromatic polynomial, see, for instance, [Tut84]), they are also similar to each other. While the Tutte polynomial can be deﬁned recursively by a deletion-contraction identity on edges, the interlace polynomial satisﬁes recurrence relations involving several operations on vertices (deletion, pivotization, complementation).
∗A preliminary version of this work has appeared in the proceedings of STACS 2008.
1
polynomial”, qR(G; x) = q(G; x, 2) is the new “vertex-rank interlace polynomial” and I(G; x) = q(G; 1, 1 + x) is the independent set polynomial1 [ABS04b].
arXiv:0707.4565v3 [] 16 Aexity of the Interlace Polynomial∗
Markus Bl¨aser, Christian Hoﬀmann

Hadamard ideals and Hadamard matrices with two circulant cores

b Department c Centre
for Computer Security Research, School of Information Technology and Computer Science, University of Wollongong, Wollongong, NSW 2522, Australia
2
GL-pairs
Georgiou, Koukouvinos and Seberry [?] point out that GL-pairs, which can be used to construct Hadamard matrices of order 2 + 2 with two circulant cores, exist for (i) is a prime (see for example [23]). (ii) 2 + 1 is a prime power (these arise from Szekeres diﬀerence sets, see for example [23] or [25]). (iii) = 2k − 1, k ≥ 2 (two Galois sequences are a GL-pair, see for example [27]). (iv) = p(p + 2) where p and p + 2 are both primes (two such sequences are a GL-pair,see for example, [28,29]). (v) = 49, 57 (these have been found by a non-exhaustive computer search that uses generalized cyclotomy and master-switch techniques, see [25,26]). (vi) = 3, 5, . . . , 45 (these have been found and classiﬁed by exhaustive computer searches, see [23]). (vii) = 47, 49, 51, 53 and 55 (these have been found and classiﬁed by partial computer searches, see [23]). (viii) = 143 (also veriﬁed the results for = 3, 5, 7, 11, 13, 15, 17, 19, 23, 25, 31, 35, 37, 41, 43, 53, 59, 61, 63 see [24]). GL-pairs do not exist for even lengths. It is indicated in [23] that the following lengths ≤ 200 are unresolved: 77, 85, 87, 91, 93, 115, 117, 121, 123, 129, 133, 145, 147, 159, 161, 169, 171, 175, 177, 185, 187 and 195. We note here that a GL-pair for length = 143 is constructed easily since 143 = 11 · 13 is a product of twin primes. 2

Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators

1
Introduction
Although being central to topics such as the derivation of complete sets of mutually unbiased bases [1, 2], or to an in depth understanding of quantum entanglement [3, 4], the commutation relations between the generalized Pauli operators of ﬁnite-dimensional quantum systems are still not well understood. Recently, considerable progress has been made in this respect by employing ﬁnite geometries such as ﬁnite projective lines [4, 5], generalized quadrangles [6, 7, 8] and polar spaces [9, 10] to treat dimensions d = 2N and, most recently [11, 7], also the case of d = 9. In this paper, after introducing the basic notation about generalized Pauli operators and Pauli graphs and brief recalling the established results, we will ﬁrst have a look at the smallest composite dimension, d = 6, as this is the ﬁrst case where we expect to ﬁnd serious departures from what is known about Hilbert spaces whose dimension is a power of a prime. We shall, indeed, ﬁnd that the ﬁnite geometry here is more intricate, exhibiting more than one line sharing two distinct points. In light of this ﬁnding, we then revisit the d = 32 case and, ﬁnally, brieﬂy address the case of d = 23 . A complete orthonormal set of operators in a p-dimensional Hilbert space (p a prime number) is of cardinality of p2 − 1. These operators can be derived from the shift and clock operators X and Z

lecture_09

Binomial equations
We introduce in this section a particular kind of polynomial equations, that have nice computational properties. A binomial system of polynomial equations is one where each equation has only two terms. We also assume that the system has only a ﬁnite number of solutions, i.e., the solution set is a ﬁnite set of points in Cn . We are interested in determining the exact number of solutions, and in eﬃcient computational procedures for solving the system. Let’s start with an example. Consider the binomial system given by 8x2 y 3 − 1 = 0 2x3 y 2 − yx = 0. (1)
2
Newton polytopes
Many of the polynomial systems that appear in practice are far from being “generic,” but rather present a number of structural features that, when properly exploited, allow for much more eﬃcient computational techniques. This is quite similar to the situation in numerical linear algebra, where there is a big diﬀerence in performance between algorithms that take into account the sparsity structure of a matrix and those that do not. For matrices, the standard notion of sparsity is relatively straightforward, and relates mostly to the number of nonzero coeﬃcients. In computational algebra, however, there exists a much more reﬁned notion of sparsity that refers not only to the number of zero coeﬃcients of a polynomial, but also to the underlying combinatorial structure. This notion of sparsity for multivariate polynomials is usually presented in terms of the Newton polytope of a polynomial, deﬁned below. � Deﬁnition 2. Consider a multivariate polynomial p(x1 , . . . , xn ) = α cα xα . The Newton polytope of p, denoted by New(f ), is deﬁned as the convex hull of the set of exponents α, considered as vectors in Rn . Thus, the Newton polytope of a polynomial always has integer extreme points, given by a subset of the exponents of the polynomial. Example 3. Consider the polynomial p(x, y ) = 5 − xy − x2 y 2 + 3y 2 + x4 . Its Newton polytope New(f ), displayed in Figure 1, is the convex hull of the points (0, 0), (1, 1), (2, 2), (0, 2), (4, 0). Example 4. Consider the polynomial p(x, y ) = 1 − x2 + xy + 4y 4 . Its Newton polytope New (p) is the triangle in R2 with vertices {(0, 0), (2, 0), (0, 4)}. Newton polytopes are an essential tool when considering polynomial arithmetic because of the fol lowing fundamental identity: New(g · h) = New(g ) + New(h), where + denotes the Minkowski addition of polytopes. 92

A New Algebraization of the Lame Equation

uction
ψ ′′ (x) + E − m ℓ(ℓ + 1) sn2 x ψ (x) = 0 , (1.1)
The Lam´ e equation,
where ℓ is a real parameter1 , and sn x ≡ sn(x|m) is the usual Jacobian elliptic function of modulus m, occupies a central position in the theory of diﬀerential equations with periodic coeﬃcients. The study of its properties has attracted the attention of many illustrious mathematicians over the last century; classical references are [1–4]. Basic properties of the Lam´ e equation are as follows. First, it arises by separation of variables in the Laplace equation in ellipsoidal coordinates. Secondly, it possesses two linearly independent 2K (k ) or 4K (k )-periodic solutions (for characteristic √ values of E ) if and only if ℓ is a nonnegative integer. Here k = m, and K (k ) (denoted by K from now on) is the complete elliptic integral of the ﬁrst kind with parameter k :

CONFLUENCE OF SWALLOWTAIL SINGULARITES OF THE HYPERBOLIC SCHWARZ MAP DEFINED BY THE HYPERGE

CONFLUENCE OF SW ALLOWTAIL SINGULARITES OF THE HYPERBOLIC SCHW ARZ MAP DEFINED BY THEHYPERGEOMETRIC DIFFERENTIAL EQUATION MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDAAbstract.The papers[G´a lvez et al.2000,Kokubu et al.2003,Kokubu et al.2005]gave a method of constructingﬂat surfaces in the three-dimesnional hyperbolicspace.Such surfaces have generically singularities,since any closed nonsigularﬂat surface is isometric to a horosphere or a hyperbolic cylinder.In the pa-per[Sasaki et al.2006],we deﬁned a map,called the hyperbolic Schwarz map,from the one-dimensional projective space to the three-dimensional hyperbolicspace by use of solutions of the hypergeometric diﬀerential equation.Its imageis aﬂat front and its generic singularities are cuspidal edges and swallowtailsingularities.In this paper we study the curves consisting of cuspidal edges andcreation/elimination of swallowtail singularities depending on the parametersof the hypergeometric equation.Contents1.Introduction12.Singularities of the image surface23.Conﬂuence of swallowtail singularities9References111.IntroductionWe consider the hypergeometric diﬀerential equationx(1−x)u +{c−(a+b+1)x}u −abu=0,where(a,b,c)are complex parameters.By a change of the unknown u by multi-plying a non-zero function,we deform the equation into the SL-form:u −q(x)u=0,whereq=−141−µ20x+1−µ21(1−x)+1+µ2∞−µ20−µ21x(1−x)=−14(1−µ2∞)x2+(µ2∞+µ20−µ21−1)x+1−µ20x2(1−x)2,andµ0=1−c,µ1=c−a−b,µ∞=b−a.Date:February20,2007.2000Mathematics Subject Classiﬁcation.33C05,53C42.Key words and phrases.hypergeometric diﬀerential equation,hyperbolic Schwarz map,ﬂat front,swallowtail singularity.12MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDA For two linearly independent solutions u0and u1to this equation,we deﬁne the (multi-valued)hyperbolic Schwarz map(HS)HS:X=C−{0,1} x−→H(x)=U(x)t¯U(x)∈H3,whereU=u0u 0u1u 1;The image lies in the three-dimensional hyperbolic space H3identiﬁed with the space of positive2×2-hermitian matrices modulo diagonal ones.We remark that the(multi-valued)Schwarz map(S)S:X x−→u0(x):u1(x)∈P1,and the(multi-valued)derived Schwarz map(DS)DS:X x−→u 0(x):u 1(x)∈P1,are regarded as the maps with images in the ideal boundary of H3,which is iden-tiﬁed with the complex projective line.The maps S and D S are connected by a one-parameter family ofﬂat fronts in H3and the map HS is one member of this family.We refer to[G´a lvez et al.2000,Kokubu et al.2003,Sasaki et al.2007]for these maps.The image surface of X under HS is one ofﬂat fronts studied in[Kokubu et al.2003]. The points0and1are singularities of the diﬀerential equation and they may deﬁne ends generally.On the other hand,it is well-known that generic singularities of fronts are cuspidal eges and swallowtails.In[Sasaki et al.2006],we drew picturesof surfaces either when the monodromy group of the equation isﬁnite or when itis the elliptic modular group,especially paying attention to the curves of cuspidal edges and to the swallowtail singularities.In this paper,we study the motion of such singularities depending on the pa-rameters a,b and c.Actually,we treat the case where the parameters take special values:a=1/2,b=1/2and c=1−p,where p is a real parameter.The reason why we treat this case is that the hypergeometric diﬀerential equation admits a rich symmetry in this case so that computational arguments work fairly well.Moreover, when p=0,we succeeded to draw nice pictures as in[Sasaki et al.2006].Whenp takes a general value,the number of points in the plane X where the map HS has swallowtail singularities is counted and when p takes some special values,we encounter conﬂuences of swallowtail singularities.Referring to the general theoryof such conﬂuences given in[Arnold1976,Langevin et al.1995],we study the hap-penings in our case and show that the three among theﬁve types of conﬂuencesin[Langevin et al.1995]really happen and one more type of conﬂuence appears. From a computational point of view,we relied on the primary-decomposition algo-rithm of related ideals to get the special values of p and on computing the Sturm sequence associated with polynomials in order to count the number of swallowtail singularities.2.Singularities of the image surfaceRelative to the diﬀerential equation in an SL-form above,the conditions on the coeﬃcient q so that the surface has cuspidal edges and swallowtails are given as follows;we refer to[Sasaki et al.2006].Lemma2.1.(1)A point p∈X is a singular point of the hyperbolic Schwarz map HS if and only if|q(p)|=1,CONFLUENCE OF SW ALLOWTAIL SINGULARITIES 3(2)a singular point x ∈X of HS is equivalent to the cuspidal edge if and onlyifq (x )=0and q 3(x )¯q (x )−q (x )=0,(3)and a singular point x ∈X of HS is equivalent to the swallowtail if andonly ifq (x )=0,q 3(x )¯q (x )−q (x )=0,and 1q q (x )q (x ) −12 q (x )q (x )2 =0.We apply the lemma to the hypergeometric equation.The set {|q |=1}is givenas the curveC ={x ;P 1(x,¯x )=0},whereP 1(x,¯x )=|Q |2−16|x 2(1−x )2|2andQ (x )=(1−µ2∞)x 2+(µ2∞+µ20−µ21−1)x +1−µ20.The topological type of the curve C heavily depends on the number of real roots of the equation P 1=0.We next deﬁne R byq=−Q x (1−x )−2Q (1−2x )4x 3(1−x )3=−R (x )4x 3(1−x )3,S byq 3(x )¯q (x )−q (x )=S (x,¯x )4x (1−x )¯x (1−¯x ),and T by 1q q (x )q (x ) −12 q (x )q (x ) 2 =T (x,¯x )Q (x )Q (¯x ).We thus have three polynomials in x ,¯x ,a ,b and c :R (x )=x (1−x )Q −2(1−2x )Q,S (x,¯x )=Q 3(x )R (¯x )+64x 3(1−x )3¯x 3(1−¯x )3R (x ),T (x,¯x )=x (1−x )Q (¯x )(−16Q 2−8(1−2x )QQ +6x (1−x )Q 2−4x (1−x )QQ ).We setP 2= (S ),P 3= (S ),R 1= (R ),R 2= (R )and R 3= (T ).Then the singularity is a cuspidal edge if and only ifP 1=0,{P 2=0or P 3=0},{R 1=0or R 2=0}and is a swallowtail if and only if P 1=0,P 2=0,P 3=0,{R 1=0or R 2=0},R 3=0.2.1.Shape of the curve C .We restrict our attention to the case (a,b,c )=(1/2,1/2,c ),and set x =(1/2+s )+it and c =1−p .The polynomials P i and R i are polynomials of p ,s and t .They are symmetric relative to the reﬂections:s ↔−s ,t ↔−t and p ↔−p .Hence,we pay attention only to the case p ≥0in the following.Since x =0,1are singularities of the coeﬃcient q ,the points (s,t )=(±1/2,0)are out of consideration.Since Q =2c −c 2−x +x 2and D =−4x 2(1−x )2,P 1is a polynomial of totaldegree eight given as follows:P 1=1/2+5/2s 2−2p 2s 2+16t 2s 4−16t 4s 2−64t 2s 6−96t 4s 4−64t 6s 2+6t 2s 2+2t 2p 2−3/2p 2−16t 6−16t 8−5t 4−5/2t 2−5s 4+16s 6−16s 8+p 44MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDAThe curve C ={(s,t );P 1(p,s,t )=0}for each p chages its shape as in Figures 3and 4,where the value of constants p i will be given in the next subsection.2.2.Swallowtail points.We study the set Z :={(p,s,t );P 1=P 2=P 3=0}.It is done by use of the primary-decomposition-algorithm that ﬁnds a set of generators of every minimal associated prime of the ideal I := P 1,P 2,P 3 in the ring Q [p,s,t ].The result is the following.Lemma 2.2.The set Z is the union of the sets deﬁned by the following ideals.1.I 1:= p −1,(2s −1)2+4t 2 2.I 2:= p −1,(2s +1)2+4t 2 3.I 3:= p +1,(2s −1)2+4t 2 4.I 4:= p +1,(2s +1)2+4t 2 5.I 5:= t,4s 4−s 2−p 2+1 6.I 6:= s,8t 4+6t 2+2p 2−1 7.I 7:= H 0(p,t ),H 1(p,s,t ),H 2(p,s,t ),H 3(p,s,t ) .The polynomials H 0(p,t ),H 1(p,s,t ),H 2(p,s,t ),H 3(p,s,t )are given in the ap-pendix.Although the result was obtained by a “black box”algorithm,it can be veriﬁedif we are allowed to use the method of ﬁnding Groebner basis.Let G be a Groebner basis of an ideal J with respect to a term order.We denote the remainder of a polynomial f by G by NF G (f ).Then f ∈J if and only if NF G (f )=0;in this way,we can verify an ideal inclusion.Lemma 2.2is shown as follows.We ﬁrst see I ⊂I i ,which implies V (I i )⊂Z .The converse inclusion Z ⊂7 i =1V (I i )follows from 7 i =1V (I i )=V (I 1I 2···I 7)andI 1I 2···I 7⊂√I .In fact,I 1I 2···I 7is generated by P :={g 1g 2···g 7;g i ∈I i (i =1,...,7)}and we can verify g 2∈I for each g ∈P by using a Groebner basis of I .Lemma 2.2implies that the set Z consists ofZ 1:={(p,s,0);4s 4−s 2−p 2+1=0},Z 2:={(p,0,t );8t 4+6t 2+2p 2−1=0},Z 3:={(p,s,t );H 0(p,t )=H 1(p,s,t )=H 2(p,s,t )=H 3(p,s,t )=0}.Figure 1draws the set Z 1projected to the space (s,p ).The explicit representa-tion is s 2= 1± 16p −15 /8,p =± 4s 4−s 2+1.The right ﬁgure enlarges the upper part of the left ﬁgure.As we see later in the next subsection,the circles in the right ﬁgure denote those points that are worse than swallowtail singularity:the p -coordinate of the top one is 1,that of middle ones is p 5,and that of the bottom ones is p 3,where p 3:=√15/4∼0.9682458365,p 5∼0.9713175204.Thus,for any value of p ∈(p 3,1),p =p 5,we have four swallowtail points,and for any value p >1two swallowtail points,both on the axis {t =0}.For the set Z 2,to have real points,it is necessary and suﬃcient that −p 1≤p ≤p 1,where p 1:=1/√2∼0.7071067810.The explicit relation is t 2= −3+ 17−16p 2 /8,p =± 1/2−3t 2−4t 4.For each value p ∈[0,p 1)we have two swallowtail points lying on the line {s =0}.In the case p =p 1,the point (s,t )=(0,0)is not a swallowtail.CONFLUENCE OF SW ALLOWTAIL SINGULARITIES5Figure1.The set Z1projected to the space(s,p) Summarizing the argument above,the number of the swallowtail points belong-ing to Z1∪Z2is given as follows:p0∗p1∗p3∗p5∗1∗Z10000042402Z22200000000Here∗stands for any value between the values of the both sides.To study the set Z3,weﬁrst deal with the polynomials H2and H3.We can writeH1(p,s,t)=c1(p)s2+c0(p,t),wherec1=−8192000p18+71680000p16−267673600p14+565094400p12−749209600p10 +651990400p8−375410912p6+138995680p4−30234388p2+2960012. Then we can see that•c1H2and(c1)2H3belong to the ideal H0,H1 ,which is veriﬁed by showing c1H2 mod H0and(c1)2H3mod H0are divisible by H1.Hence,for each value where c1(p)=0,the set Z3is the same as the set{H0= H1=0}.Moreover,we can see that•c20belongs to the ideal c1,H0 because c1divides the remainder c20mod H0with respect to t.Hence,the set{(p,s,t);p≥0,c1=H0=H1=0}consists of lines in(p,s,t)-space deﬁned by c1(p)=H0(p,t)=0,which are given as(p,t)=(p4,±0.011811323560992964937),(p7,±0.000192205787502698965),(p10,±0.022606558445778182272),wherep4∼0.97127920368420120746,p7∼1.00370488167353310415,p10∼1.03276891081183183482.The set{(p,s,t);p≥0,c1=H0=H1=H2=H3=0}consists of the points(p,s,t)=(p4,±0.4448235948,±0.01181132356),(p4,±0.2944179698,±0.01181132356),(p7,±0.5074847467,±0.00019220578).6MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDA Taking care of these exceptions,it is enough to study the set{(p,s,t);H0=H1= 0}.We next deal with the curve H0(t,p)=0in the tp-plane where p≥0;look at Figure2.In order to know the precise shape of the curve,we use the Sturm sequence.For a polynomial f(t),the Sturm sequence{f0,f1,...,f l}is deﬁned by the following recurrence:(2.1)f0=f,f1=d fdt,f i=−(f i−2mod f i−1)(i=2,...,l),f l−1mod f l=0,where(f i mod f i−1)is the polynomial remainder.We deﬁneσ(a)to be the num-ber of sign changes in the sequence{f0(a),f1(a),...,f l(a)},where zeros are not counted.Theorem2.3.(Sturm)[Jacobson1975]For a,b∈R such that a<b and f(a),f(b)= 0,the number of roots of f(t)in the interval(a,b)isσ(a)−σ(b).In particular,the number of all real roots of f(x)is determined by the signs of the leading coeﬃcients and the degrees of f i’s.Let ={f0(t),...,f l(t)}(f i∈Q(p)[t])be a sequence of polynomials obtained by applying the recurrence(2.1)to f0=H0with respect to t.Let M⊂R be the zeros of the numerators and denominators of the leading coeﬃcients of f i’s. R\M is a disjoint union of open intervals I k.Then the sequence gives the correct Sturm sequence at each p∈R\M and Theorem2.3ensures that the number of roots of H0is constant for all p∈I k.It is clear that each branch t=t(p)such that H0(p,t(p))=0over I k is a continuous function of p.If p∈M,p is a root of an irreducible polynomial over Q and we can compute the Sturm sequence over an algebraic numberﬁeld Q(p).The Sturm sequence at each p tells the number of roots t(p)within any interval,and we can draw the curve with desired precision.The set M contains the zeros of the discriminant of the equation H0=0,and we get the coordinates(p,t)of several extreme points as in theﬁgure:X=(p10,0.022********),A=(p9,0.020********),B=(p8,0),C=(p6,0),D=(p5,0),E=(p4,0.01181132356),F=(p2,0.08654627008),wherep2∼0.94237741898935061,p6=1,p8∼1.03077640441513745,p9∼1.03230371163023017.(Since the point X is very near to A,the point(p10,−0.022*******)is drawn in theﬁgure.)For each point(p,t)on the curve H0(p,t)=0,we solve the equation H1(p,s,t)= 0.Then,since H1(p,s,t)=c1(p)s2+c0(p,t),the number of real solutions depends on the signs of c1(p)and c0(p,t).To determine the sign of c0(p,t)/c1(p),we enlarge the set M by adjoining the values of p satisfying c0=H0=0for some t,and the zeros of c1,and then recompute I k.Let t=t(p)be the continuous function over I k discussed above.Then c0(p,t(p))/c1(p)is also continuous and its sign is constant over I k because the numerator does not vanish over I k.Thus we can determine the the number of solutions by evaluating c0(p,t(p))/c1(p)at a point p∈I k.For p∈M,we have to deal with an algebraic numberﬁeld again;we omit the details.2.3.Non-swallowtail points.If for some p the point(s,t)is a swallowtail,then (p,s,t)∈Z.However,not all points in Z are swallowtails.We need to check the condition(R1=0or R2=0)and the condition R3=0,namely,the conditionCONFLUENCE OF SW ALLOWTAIL SINGULARITIES 7Figure 2.The curve c 1(t,p )=0q =0and the condition (T )=0,respectively.This check is done by studying the setsE 1:={(p,s,t );p ≥0,P 1=P 2=P 3=R 1=R 2=0}andE 2:={(p,s,t );p ≥0,P 1=P 2=P 3=R 3=0},by relying on the primary decomposition of the corresponding ideals P 1,P 2,P 3,R 1,R 2 and P 1,P 2,P 3,R 3 .Lemma 2.4.The set deﬁned by the ideal P 1,P 2,P 3,R 1,R 2 is the union of the sets deﬁned by the ideals.1.[p −1,t,s ],2.[p −1,t,2s −1],3.[p −1,t,2s +1],4.[p −1,4s −1,16t 2+1],5.[p −1,4s +1,16t 2+1],6.[p −1,s,4t 2+1],7.[p +1,t,s ],8.[p +1,t,2s −1],9.[p +1,t,2s +1],10.[p +1,4s −1,16t 2+1],11.[p +1,4s +1,16t 2+1],12.[p +1,s,4t 2+1],13.[2p 2−1,t,s ],14.[16p 2−17,t,8s 2−3],15.[16p 2−15,t,8s 2−1],16.[16p 2−17,s,8t 2+3],17.[16p 2−15,s,8t 2+1],18.[20p 2−19,80t 2+3,80s 2−3].By this lemma we can see the following.The ideals 4,5,6,10,11,12,16,17and 18have no real points.The ideals 1,2,3,7,8and 9yield three points(1,0,0),(1,±1/2,0),and the ideals 13,14and 15yield 5points(p 1,0,0),(p 8,±s 1,0),(p 3,±s 2,0),in E 1(recall we assumed p ≥0),where p 8=√17/4∼1.03077640,s 1= 3/8∼0.61237,s 2=1/√8∼0.35355.8MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDA Lemma2.5.The set deﬁned by the ideal P1,P2,P3,R3 is the union of the sets deﬁned by the ideals.19.[p−1,4s2−4s+4t2+1],20.[p−1,4s2+4s+4t2+1],21.[p+1,4s2−4s+4t2+1],22.[p+1,4s2+4s+4t2+1],23.[4p2−3,4s−1,16t2+1],24.[4p2−3,4s+1,16t2+1],25.[r1(p),s,r2(p,t)],26.[r3(p),t,r4(p,s)],27.[r5(p),r6(p,s),r7(p,t)].wherer1(p)=256p6−464p4+224p2−19,r2(p,t)=24t2+64p4−84p2+29,r3(p)=256p6−560p4+416p2−109,r4(p,s)=24s2−64p4+108p2−47,r5(p)=262144p14−1458176p12+3352128p10−4064896p8+2723312p6−934456p4+111981p2+7964,r6(p,s)=105137152s2+805737070592p12−3734052323328p10+6842337426624p8−6160590983296p6+2674420712784p4−400807788840p2−27034171281, r7(p,t)=105137152t2+1275266859008p12−6003856457728p10+11187800841024p8−10256897785728p6+4544374234288p4−701552747480p2−45094112399.In this lemma,the cases19,20,21and22yield two points(1,±1/2,0)in E2,and the cases23,24and25yield no real points.For the ideal26,by solving r3(p)=0,we get a real positive solution p=p5.The corresponding values of s are determined by the equation r4(p,s)=0.We thus have two points(p5,±0.2939504177,0)in E2.For the last ideal27,by solving r5(p)=0,we get real positive solutions p= p2and p9.The corresponding values of(s,t)are determined by solving r6(p,s)= r7(p,t)=0.The result is(p2,±0.3834951026,±0.08654627008),(p9,±0.5955418899,±0.020********).Summarizing the argument above,the number of the swallowtail points belong-ing to Z3where p≥0is given as follows:p p2∗p5∗p6=1∗p8∗p9∗Z308440448002.4.Number of swallowtail singularities.As we mentioned earlier,we restrict our consideration to the range of p to the interval[0,∞).Combining the consid-eration in the previous two subsections,the exceptional values of p for which the point(p,s,t)in Z is not a swallowtail are(2.2)p1,p2,p3,p5,p6=1,p8,p9.We remark that the values p4and p7that appeared in the course of study of the set Z turn out to be not exceptional and that the exceptional values of p are classiﬁed into the three cases:q =0and (T)=0:p1,p3,p6at(0,0),p8,q =0and (T)=0:p2,p5,p9,q =0and (T)=0:p6at(±1/2,0).Now we sum up the above data and get the number N of swallowtail singularities: p0∗p1∗p2∗p3∗p5∗p6=1∗p8∗p9∗N220008812680661022 In Figures3and4,the swallowtail singularities are represented by circles;other symbols represent worse singularities explained in the next section.CONFLUENCE OF SW ALLOWTAIL SINGULARITIES9 Remark2.6.The number N counts the swallowtail singularites in the plane x= 1/2+s+it,not on the image surface;recall that the hyperbolic Schwarz map HS is multi-valued.3.Confluence of swallowtail singularitiesIn the previous section,we studied the variation of the number of swallowtail singularities when p runs from0to∞.On the other hand,the conﬂuence of swallowtail singularities was studied by Arnold[Arnold1976].Figure6cites the Figure3of[Langevin et al.1995](p.547),which showsﬁve types(1,...,5)of conﬂuence(bifurcation)of swallowtail singularities.From the study in the last section,we observe that the types2,3and5actually occur in our move,and that another type of conﬂuence also occurs.In Figures3 and4,they are denoted by•(Type2), (Type3), (Type5), ,respectively.In this section,the(local)image surface is denoted by S,and the(local)image of the curve C under the hyperbolic Schwarz map HS is denoted by C(⊂S).3.1.•Around p=p1,p=p3,p=p6and p=p8.When p<p1,there isa pair of swallowtail singularities on S carried by a pair of cuspidal components of C.When p tends to p1,two singularities come together and kiss,and when p1<p,C becomes a pair of nonsigular curves.We observe that this move is of type2in Figure6.See Figure7of the image surface S under HS of the square {(s,t);−0.5<s<0.5,−0.5<t<0.5}.Similar happens when p p3,p p8and p p6around(s,t)=(0,0).3.2. Around p=p2and p=p9.When p≤p2,there are no swallowtail singularity.However,when p>p2,there are four pairs of swallowtail singularities, eight in all.The move around p2is observed to be of type5in Figure6.A picture of a pair is drawn in Figure8(left)when p=0.945.The lower picture is the curve C carrying two swallowtails,which reside at the two cusps of this curve.The upper picture is a tubular neighborhood in the surface S of this curve C.Similar happens when p p9.3.3. Around p=p5.When p p5,three swallowtail singularities shrink to one point that is not a swallowtail singularity and,instantly after passing p5one swallowtail singularity reappears.This move is not in theﬁve moves of Arnold in Figure6.Figure8(right)shows the curve C with three cusps,and the surface S around this curve.3.4. Around p=p6.When p=p6=1,there are no swallowtail.If p goes apart from1,then three swallowtail singularities appear in both directions;this move is observed to be of type3in Figure6.In Figure9,the left picture is the image of a small square situated right of the point(s,t)=(−0.5,0)(singular point of the diﬀerential equation)including three swallowtails when p=0.975.They shrink to one point as p tends to1.The right picture is the image of a small square situated left of the point(−0.5,0)including three swallowtails when p=1.028.3.5.Drawing Figures.We give some assignments on theﬁgures.Figures3–5 draw the curve C.Eachﬁrst row gives a global view and the second row for p≥0.78gives aﬁner view of a part of C.The marks•, , and are drawn in a somewhat emphasized manner.In the third row(in the second row when p≤p1),the images curves C are given;the range of drawing is indicated by a dotted quadrangle in theﬁgures of theﬁrst rows.When p<1,the map HS deﬁned10MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDAin the upper half plane is continued analytically through the interval{0<x<1},and,when p>1,through the negative real axis{x<0}.Figure7draws the imagesS of the square{(s,t);s∈(−0.5,0.5),t∈(−0.5,0.5)},where the image curves Care drawn more in detail than in Figure3.Figures8and Figure9draw S and Cfor distinct values of p.Appendix:List of polynomialsH0=(262144p8−524288p6+262144p4)t8+(524288p10−1572864p8+1687552p6−735232p4+79872p2+16384)t6+(393216p12−1654784p10+2817024p8−2459136p6+1142464p4−260480p2+21700)t4+(131072p14−737280p12+1755392p10−2289728p8+1761920p6−794968p4+192308p2−18715)t2+16384p16−118784p14+376000p12−679104p10+766068p8−553296p6+250232p4−64912p2+7412,H1=(−8192000p18+71680000p16−267673600p14+565094400p12−749209600p10+651990400p8−375410912p6+138995680p4−30234388p2+2960012)s2+(−83886080p16+461373440p14−1033895936p12+1196949504p10−740818944p8+222822400p6−22544384p4)t6+(−125829120p18+786432000p16−2102657024p14+3123380224p12−2789900288p10+1502928896p8−452747264p6+55562240p4+4239360p2−1409024)t4+(−62914560p20+458424320p18−1490173952p16+2838306816p14−3490809856p12+2877347840p10−1598347904p8+588634656p6−138705184p4+19590612p2−1352780)t2−10485760p22+94371840p20−391053312p18+980131840p16−1640966144p14+1914822656p12−1580043904p10+917850432p8−366915360p6+96240468p4−15040195p2+1087441.H2=(−172748519424t2+342090457088000p16−2647506597888000p14+8502392953446400p12−15007782165760000p10+16125961477836800p8−10936659362160000p6+4626862256233568p4−1127440491359040p2+122038029361684)s2+(3503006280581120p14−15725697150484480p12+27286047559778304p10−22422671555297280p8+8286510388346880p6−927195522924544p4)t6+(5254509420871680p16−27529427791380480p14+59988842462314496p12−69830531227779072p10+45967246621245440p8−16318586888937472p6+2406740047872000p4+118811578937344p2−58122468702208)t4+(2627254710435840p18−16487753934110720p16+45567759200370688p14−72489552047398912p12+72546907718641664p10−46869348690018304p8+19388801939481216p6−4981168098598560p4+752756242889920p2−55743159881812)t2+437875785072640p20−3498277391564800p18+12794850594193408p16−28001940223336448p14+40236544719603712p12−39314960801359872p10+26264510377602688p8−11791213573216192p6+3403925190427360p4−576100608409972p2+44796791787623,H3=−43273504115712s4+(−174591366701056000p16+1354597407135744000p14−4361070217826508800p12 +7714396815022745600p10−8303627443711283200p8+5639582815676545920p6−2388886969950578656p4+582795932673655872p2−63153571220692772)s2+(−1787815595018813440p14+8060676765258874880p12−14035315945780543488p10+11564439040729546752p8−4282247261844406272p6+480262996655341568p4)t6+(−2681723392528220160p16+14102307692284477440p14−30819600746391273472p12+35956859092128694272p10−23711874589423796224p8+8431178804896194560p6−1246058944783564800p4−60931259457007616p2+30059710795074560)t4+(−1340861696264110080p18+8440901125082644480p16−23384966557443178496p14+37273429941273878528p12−37362998720665520128p10+24170842385410605056p8−10009973619071190144p6+2574206274932313888p4−389407674475545536p2+28850050377704036)t2−223476949377351680p20+1789753918478090240p18−6558969382275055616p16+14379092033155440640p14−20693107121308362752p12+20246624022842148864p10−13542078892737183872p8+6086106587227785920p6−1758722499457332320p4+297968123692665860p2−23197985921481043.CONFLUENCE OF SW ALLOWTAIL SINGULARITIES 111tststs–0.20.20.40.6–0.50.5–0.4–0.2–0.20.20.40.6–0.4–0.20.20.4–0.3–0.2–0.12–0.200.2t–0.500.5s–0.200.2t–0.500.5stst s0.20.4–0.20.2–0.1–0.050.4650.470.4750.48–0.050.050.660.680.70.720.740.460.47–0.050.050.70.720.740.76Figure 3.The curve C and the curve C(1)References[Arnold 1976]V.I.Arnol’d ,Wave front evolution and equivariant Morse lemma ,Comm.pureappl.Math.,29(1976),557–582.[G´a lvez et al.2000]J.A.G ´a lvez,A.Mart ´ınez and ´an ,Flat surfaces in hyperbolic 3-space ,Math.Annalen,316(2000),419–435.[Iwasaki et al.1991]K.Iwasaki,H.Kimura,S.Shimomura and M.Yoshida ,From Gauss toPainlev ´e–A modern theory of special functions ,Vieweg Verlag,Wiesbaden,1991.12MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDA3tststststst s0.280.2820.284–0.0020.0020.9180.920.9220.2680.270.272–0.0020.0020.9240.9260.9280.2540.2560.258–0.0020.0020.9320.9340.936p =p 5∼0.9713175p =0.975p =p 6=1tstststst s0.2420.2440.246–0.0020.0020.9380.940.9420.210.2120.2140.216–0.0020.0020.9540.9560.958Figure 4.The curve C and the curve C (2)CONFLUENCE OF SW ALLOWTAIL SINGULARITIES 138tsts tststst s–0.1866–0.1864–0.1862–0.1860.0140.0160.0180.020.96150.96160.9617–0.201–0.20050.0180.020.0220.95480.955–0.2045–0.2040.0180.020.0220.0240.9530.95329tsttststs–0.2085–0.2080.020.0250.95080.9510.9512–0.2435–0.243–0.24250.030.0350.930.93050.931Figure 5.The curve C and the curve C (3)14MASAYUKI NORO,TAKESHI SASAKI,KOTARO YAMADA,AND MASAAKI YOSHIDAFigure 6.Conﬂuence of swallowtail singularities from[Langevin et al.1995],p.547[Jacobson1975]N.Jacobson,Lectures in abstract algebra3:Theory ofﬁelds and Galois theory, Graduate Texts in Math.32,1975,Springer.CONFLUENCE OF SW ALLOWTAIL SINGULARITIES15Figure8.The images under HS(2)[Kokubu et al.2003]M.Kokubu,M.Umehara and K.Yamada,An elementary proof of Small’s formula for null curves in P SL(2,C)and an analogue for Legendreian curves in P SL(2,C),Osaka J.Math.40(2003),697–715.[Kokubu et al.2005]M.Kokubu,W.Rossman,K.Saji,M.Umehara and K.Yamada,Singu-larities ofﬂat fronts in hyperbolic space,Paciﬁc J.Math.221(2005),303–351. [Langevin et al.1995]ngevin,G.Levitt and H.Rosenberg,Classes d’homotopie de sur-faces avec rebroussements et queues d’aronde dans R3,Can.J.Math.47(1995),544–572. [Sasaki et al.2006]T.Sasaki K.Yamada and M.Yoshida,Hyperbolic Schwarz map for the hy-pergeometric equation,preprint,math.CA/0609196;revised2007.[Sasaki et al.2007]T.Sasaki K.Yamada and M.Yoshida,Derived Schwarz map of the hyper-geometric diﬀerential equation and a parallel family ofﬂat fronts,preprint,2007. [Yoshida1997]M.Yoshida,Hypergeometric Functions,My Love,Vieweg Verlag,Wiesbaden, 1997.(Noro)Department of Mathematics,Kobe University,Kobe657-8501,JapanE-mail address:noro@math.kobe-u.ac.jp(Sasaki)Department of Mathematics,Kobe University,Kobe657-8501,JapanE-mail address:sasaki@math.kobe-u.ac.jp(Yamada)Faculty of Mathematics,Kyushu University,Fukuoka812-8581,Japan E-mail address:kotaro@math.kyushu-u.ac.jp(Yoshida)Faculty of Mathematics,Kyushu University,Fukuoka810-8560,Japan E-mail address:myoshida@math.kyushu-u.ac.jp。

Polyhedra and Polytopes

q
A=
j =1
Cj .
Consequently, we get
p q
A=A∩E =
i=1
((Ei )+ ∩ (Ei )− ) ∩
j =1
Cj ,
which proves that A is also an H-polyhedron in E . The following simple proposition shows that we may assume that E = En : Proposition 4.2 Given any two aﬃne Euclidean spaces, E and F , if h : E → F is any aﬃne map then: (1) If A is any V -polytope in E , then h(E ) is a V -polytope in F . (2) If h is bijective and A is any H-polyhedron in E , then h(E ) is an H-polyhedron in F .
p
A=A∩E =
i=1
(Ci ∩ E ),
where Ci ∩ E is one of the closed half-spaces determined by the hyperplane, Hi = Hi ∩ E , in E . Thus, A is also an H-polyhedron in E . Conversely, assume that A is an H-polyhedron in E and that d < n. As any hyperplane, H , in E can be written as the intersection, H = H− ∩ H+ , of the two closed half-spaces that it bounds, E itself can be written as the intersection,

On the absolute Mahler measure of polynomials having all zeros in a sector

1. Introduction Let P (z ) = z be a monic polynomial with integer coeﬃcients, irreducible over the rationals, of degree d ≥ 1, and having zeros α1 , · · · , αd . Its relative Mahler measure M (P ), given by
d
1≤
i=1
αa i R(αi ) =
|αi |≤1
|αa i R(αi )| ×
|αi >1
a+r 1 |αi R(α− i )| a+r α2 i |αi |>1
=
|αi |≤1
|αa i R(αi )| ×
|αi |>1 (2a+r )d
1 a −1 |(α− i ) R(αi )| × 2a+r
j
Received by the editor March 12, 2003 and, in revised form, August 10, 2003. 2000 Mathematics Subject Classiﬁcation. Primary 11R04, 12D10.
c 2004 American Mathematical Society
MAHLER MEASURE OF POLYNOMIALS
385
A, and in fact it has the smallest absolute measure among factors A of measure > 1. It follows that Ω(P∗ ) is the smallest value of the absolute measure for polynomials having all zeros in | arg z | ≤ θ for θ ∈ [θi , θi ]. Hence, c(θ) = Ω(P∗ ) for these θ. One of the main problems in the previous paper was to ﬁnd for each interval suitable polynomials to use to obtain a good auxiliary function. In fact they only usproduced a table of good Pj which were for almost all polynomials of one of the following six types: z n Q(z + z −1 − k ) z n S (z + z −1 − 2) z n (Q(z ) + Q(1/z )) (k = 3, 2, 1, 0) (types 1, 2, 3, 4),

Generalized WDVV equations for F4 pure N=2 Super-Yang-Mills theory

We start with the family of Riemann surfaces [11],[12] associated with pure F4 Seiberg-Witten theory 1/3 1/3 2 3 2 3 −q − q + 4p µ b1 (x) 1 −q + q + 4p + (2.1) z + = W (x, u1 , ..., u4 ) = − z 24 2 2 2
1
2
Associative algebra for F4
where p, q, b1 are polynomials in x, u1 , ..., u4 which can be found in Appendix A. The Seiberg-Witten diﬀerential on this curve is dz x (∂x W ) dx (2.2) λSW = x = z W 2 − 4µ and its derivatives with respect to the moduli parameters ui are holomorphic [9] ωi = ∂λSW ∼ ∂W =− ∂ui ∂ui dx W2 − 4µ = φi dx W 2 − 4µ (2.3)
2.1
The polynomial ring and ideals
−2 3
q 2 + 4 p3 Due to the cubic and square roots in (2.1), the φi = − ∂W ∂ui have terms containing −q + which are certainly not polynomial in x. For classical gauge groups this problem does not occur and the φi are basically in a polynomial ring. It is desirable to work with a polynomial ring because it will lead to associativity of the algebra structure (2.4). For this purpose we set c= a=p b=p q 2 + 4 p3 −q + c 2 −q − c 2

New relations for two-dimensional Hermite polynomials

∞ p
R12 a2 R22 a2 2 1+2 + 2 R11 a1 R11 a1
p
.
we can rewrite the square bracket in the right-hand side as 1 + 2rα + α2 = (α + r )2 − (r 2 − 1) = (1 − r 2 ) 1 − (ρ + γ )2 , where ρ= √ r r2 −1 , γ=√ r2 α . −1
Developing the function [1 − (ρ + γ ) ] with respect to variable γ we arrive at the expansion
2 R11 2 R22 r2 − 1 E= p p!k ! 2 p=0 k =0
2 p
k
∞
2p
p− k
p− k 2
1
Introduction
The Hermite polynomials of several variables arise quite naturally almost in all problems relating to quantum systems described by means of multidimensional quadratic Hamiltonians: see, e.g., [1-4] and references therein. However, until now they were not widely used by other authors in the papers on quantum mechanics and quantum optics (for a few of exceptions see, e.g., [5,6]), due to the absence of simple explicit formulas, which would be convenient for the calculations and for the analysis of the relations obtained. The aim of the present paper is to give the expressions for multidimensional Hermite polynomials in terms of the well known classical orthogonal polynomials. Besides, we shall give new sum rules and generating functions, as well as asymptotic formulas for various combinations of the parameters. Although some of the results could be found in odd form in other references [1-11], we believe that bringing them together will be useful for the further applications. Just having in mind these applications, we pay a special attention to the speciﬁc sets of parameters deﬁning the polynomials, which are typical for the quantum mechanical problems. We shall consider mainly the case of the Hermite polynomials of two variables, but some generalizations to higher dimensions will be also given. The structure of the paper is as follows. In the next section we derive the explicit expressions for the two-dimensional Hermite polynomials in terms of the Jacobi, Gegenbauer, Legendre, and usual Hermite polynomials. The asymptotics of the two-dimensional Hermite polynomials of zero arguments are considered in Section 3. New sum rules and generating functions for the “diagonal” two-dimensional Hermite polynomials are given in Section 4, and their generalizations to higher dimensions are considered in Section 5. Section 6 is devoted to the physical applications. In that section we discuss brieﬂy two problems: the photon statistics in squeezed mixed (thermal) quantum states, and the transition probabilities between the energy levels of a quantum oscillator with a time-dependent frequency.

Mesh generation-MarshallBern

Xerox Palo Alto Research Center, 3333 Coyote Hill Rd., Palo Alto, CA 94304; bern@ Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA 16802; plassman@ Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the O ce of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.
without specifying the exact form of this representation. Computational geometers typically assume exact, combinatorial data structures, such as linked lists for simple polygons and polygons with holes, doubly connected edge lists 103] or quad-edge structures 63] for planar multiple domains, and winged-edge structures 44, 62] for polyhedral domains. In practice, complicated domains are designed on computer aided design (CAD) systems. These systems use surface representations designed for visual rendering, and then translate the nal design to another format for input to the mesh generator. The Stereolithography Tessellation Language (STL) le format, originally developed for the rapid prototyping of solid models, speci es the boundary as a list of surface polygons (usually triangles) and surface normals. The advantages of the STL input format are that a \watertight" model can be ensured and model tolerance (deviation from the CAD model) can be speci ed by the user. The Initial Graphics Exchange Speci cation (IGES) format enables a variety of surface representations, including higher-order representations such as B-splines and NURBs. Perhaps due to its greater complexity or to sloppy CAD systems or users, IGES les often contain incorrect geometry (either gaps or extra material) at surface intersections. An alternative approach to format translation is to directly query the CAD system with, say, point-enclosure queries, and then construct a new representation based on the answers to those queries. This approach is most advantageous when the translation problem is di cult, as it may be in the case of implicit surfaces (level sets of complicated functions) or constructive solid geometry formulas. With either approach, translation or reconstruction by queries, the CAD model must be topologically correct and su ciently accurate to enable meshing. We expect to see greater integration between solid modeling and meshing in the future. A structured mesh is one in which all interior vertices are topologically alike. In graphtheoretic terms, a structured mesh is an induced subgraph of an in nite periodic graph such as a grid. An unstructured mesh is one in which vertices may have arbitrarily varying local neighborhoods. A block-structured or hybrid mesh is formed by a number of small structured meshes combined in an overall unstructured pattern. In general, structured meshes o er simplicity and easy data access, while unstructured meshes o er more convenient mesh adaptivity (re nement/dere nement based on an initial solution) and a better t to complicated domains. High-quality hybrid meshes enjoy the advantages of both approaches, but hybrid meshing is not yet fully automatic. We shall discuss unstructured mesh generation at greater length than structured or hybrid mesh 2

Multi-scale analysis implies strong dynamical localization

a r X i v :mat h-ph/99122v12Dec1999MULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION DAVID DAMANIK 1,2AND PETER STOLLMANN 11Fachbereich Mathematik,Johann Wolfgang Goethe-Universit¨a t,60054Frankfurt,Germany 2Department of Mathematics 253–37,California Institute of Technology,Pasadena,CA 91125,U.S.A.E-mail:damanik@,stollman@math.uni-frankfurt.de Abstract.We prove that a strong form of dynamical localization follows from a variable energy multi-scale analysis.This abstract result is applied to a number of models for wave propagation in disordered media.1.Introduction In the present paper we prove that a variable energy multi-scale analysis implies dynamical localization in a strong (expectation)form.Thus we accomplish a goal of a long line of research.Ever since Anderson’s paper [5],the dynamics of waves in random media has been a subject of intensive research in mathematical physics.The breakthrough as far as mathematically rigorous results are concerned came with the paper [19]by Fr¨o hlich and Spencer in which absence of diﬀusion is proven.They also introduced a technique of central importance to the topic:multi-scale analysis.The next step was a proof of exponential localization,by which one understands pure point spectrum with exponentially decaying eigenfunctions;see the bibliogra-phy for a list of results in diﬀerent generality.However,from the point of view oftransport properties,exponential localization does not yield too much information.We refer to [13,14]where a strengthening of exponential decay is introduced,prop-erty (SULE),which in fact allows one to prove dynamical localization.In [20]it was shown that a variable energy multi-scale analysis implies (SULE)almost surely,so thatsup t>0 |X |p e −itH (ω)P I (H (ω))χK <∞P -a.s.,where H (ω)is a random Hamiltonian which admits multi-scale analysis in the interval I ,P I is the spectral projector onto that interval,and K is compact.We will strengthenthe last statement to E sup t>0 |X |p e −itH (ω)P I (H (ω))χK <∞.Here,as in [20],the p which is admissible depends on the characteristic parameters of multi-scale analysis.In order to explain this we will sketch in the next section an abstract form of multi-scale analysis and introduce the necessary set-up.In12 D.DAMANIK AND P.STOLLMANNSection3,we show that multi-scale analysis implies dynamical localization in the expectation.We do so by showing that(more or less)forη∈L∞,suppη⊂I(the localized region),E χΛ1η(H(ω))χΛ2 ≤ η ∞·dist(Λ1,Λ2)−2ξ,whereξis one of the characteristic exponents of multi-scale analysis.We should note here that the main progress concerns continuum models,since for discrete models the Aizenman technique[1,3]is available,which gives even exponential decay of the expectation above(see[2]for an exposition in which a number of applications is presented and the very recent[4]which shows that the Aizenman technique is applicable in the energy region in which multi-scale analysis works).However,our results clearly apply to discrete models with singular single-site distribution,most notably the one-dimensional Bernoulli-Anderson model.Moreover,we refer to[7] where a study of time means instead of the sup is undertaken.However,the latter paper does not contain too much about continuum models,and the results we present contain the estimates given there.In Section4we present our applications to a number of models for wave propagation in disordered media,including band edge dynamical localization for Schr¨o dinger and divergence form operators as well as Landau Hamiltonians.2.The multi-scale scenarioIn this section we present the abstract framework for multi-scale analysis de-veloped in[29].We start with a number of properties which are easily veriﬁed for the applications we shall discuss,where H(ω)is a random operator in L2(R d) and HΛ(ω)denotes its restriction to an open cubeΛ⊂R d with suitable boundary conditions.We call a cubeΛ=ΛL(x)of sidelength L centered at x suitable if x∈Z d and L∈3N\6N.In this caseΛL/3(x)are unions of closed unit cubes centered on the lattice.DenoteΛint:=ΛL/3(x),Λout:=ΛL(x)\ΛL−2(x),and denote the respective characteristic functions byχint=χintΛ=χint L,x:=χΛintandχout=χoutΛ=χoutL,x:=χΛout.Theﬁrst condition concerns measurability and independence:(INDY)(Ω,F,P)is a probability space;for every cubeΛ,HΛ(ω)is a self-adjointoperator in L2(Λ),measurable inω,such that HΛL (x)(ω)is stationary in x∈Z dand HΛand HΛ′are independent for disjoint cubesΛandΛ′.So far,HΛand HΛ′are not related ifΛ⊂Λ′.The next condition supplies a relation.In concrete examples it is the so-called geometric resolvent inequality which follows from commutator estimates and the resolvent identity.For E∈ρ(HΛ(ω)),we denoteRΛ(E)=RΛ(ω,E)=(HΛ(ω)−E)−1.(GRI)For given bounded I0⊂R,there is a constant C geom such that for all suitable cubesΛ,Λ′withΛ⊂Λ′,A⊂Λint,B⊂Λ′\Λ,E∈I0andω∈Ω,theMULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION3following inequality holds:χB RΛ′(E)χA ≤C geom· χB RΛ′(E)χoutΛ · χoutΛRΛ(E)χA .Finally,we need an upper bound for the trace of the local Hamiltonians HΛin a given bounded energy region I0,which follows from Weyl’s law in concrete cases at hand.(WEYL)For each interval J⊂I0,there is a constant C such thattr(P J(HΛ(ω))≤C·|Λ|for allω∈Ω.Here P J(·)denotes the spectral projection of the operator in question.Given this basic set-up,multi-scale analysis deals with an inductive proof of resolvent decay estimates.This resolvent decay is measured in terms of the following concept:Deﬁnition2.1.LetΛ=ΛL(x),x∈Z d,L∈2N+1.Λis called(γ,E)-good for ω∈Ωifχout RΛ(E)χint ≤exp(−γ·L).Λis called(γ,E)-bad forω∈Ωif it is not(γ,E)-good forω.We can now deﬁne the property on which we base our induction:G(I,L,γ,ξ)∀x,y∈Z d,d(x,y)≥L the following estimate holds:P{∀E∈I:ΛL(x)orΛL(y)is(γ,E)-good forω}≥1−L−2ξ.The basic idea of the multi-scale induction is that we consider some larger cube Λ′with sidelength L′=Lα.With high probability there are not too many disjoint bad cubes of sidelength L inΛ′.Of course,since the number of cubes inΛ′is governed byα,this will only hold ifαis not too large,depending onξ.By virtue of the geometric resolvent inequality(GRI),each of the good cubes of sidelength L inΛ′will add to exponential decay on the big cube.In order to make this work,we will additionally need a“worst case estimate.”This is given by the following weak form of a Wegner estimate:W(I,L,Θ,q)For all E∈I andΛ=ΛL(x),x∈Z d,the following estimate holds: P{dist(σ(HΛ(ω)),E)≤exp(−LΘ)}≤L−q.We have the following theorem:Theorem2.2.Let I0⊂R be a bounded open set and assume that HΛ(ω)satisﬁes (INDY),(GRI)and(WEYL)for I0.Assume that there are L0∈2N+1,q>d,Θ∈(0,12−α≤ξ0∧1I⊂I0,L≥L∗,L∈3N\6N,andγL≥Lβ−1,the estimate G(I,L,γL,ξ0) is satisﬁed,then G(I,L′,γL′,ξ)also holds,where(i)L′∈3N\6N,Lα≤L′≤Lα+6,(ii)ξ≥ξ0∧[14 D.DAMANIK AND P.STOLLMANN(iii)γL′≥γL(1−8L1−α)−C1·L−1−6Lα(Θ−1)≥(L′)1−β.For a proof of the result in this form we refer to[29].It is modelled after the variable multi-scale analysis by von Dreifus-Klein[15].See also[17]and[20]for continuum versions.Let us now formulate an immediate consequence of the preceding theorem. Corollary2.3.Let I0,(HΛ(ω)),ξ0,β,q,Θ,α∈(1,2)be as in Theorem2.2.There exists L(ξ0,β,Θ,q,C geom,α)such that the following holds.If I⊂I0and G(I,L,γL,ξ0)is satisﬁed for someγL≥Lβ−1and some L≥4(q−d).(ii)Lαk≤L k+1≤Lαk+6.3.Multi-scale estimates imply strong dynamical localizationWe keep the framework introduced in the preceding section.Thus we start out with a family HΛof random local Hamiltonians whereΛruns through the suitable cubes.Now we introduce a link to a Hamiltonian on the whole space R d.Consider the statement(EDI)Assume that H(ω)is a self-adjoint operator in L2(R d),measurable with respect toω,and suppose that there is a measurable setΩ1with P(Ω1)=1and a constant C EDI such that for everyω∈Ω1,the spectrum of H(ω)in I0is pure point and every eigenfunction u of H(ω)corresponding to E∈I0satisﬁesχintΛu ≤C EDI· χoutΛ(HΛ(ω)−E)−1χintΛu · χoutΛu .(EDI)For the operators H(ω)we shall consider in Section4and HΛ(ω)the restric-tion toΛwith respect to suitable boundary conditions,the eigenfunction decay inequality(EDI)readily follows.Moreover,in this case,we can use the multi-scale machinery to prove pure point spectrum almost surely.Therefore,the condition above seems to be a natural abstract condition.We can now state the main result of the present paper:Theorem3.1.Assume that H(ω)and HΛ(ω)satisfy(INDY),(GRI),(WEYL) and(EDI)above for a given bounded open set I0⊂R.Moreover,assume(i)χΛP I(H(ω))χΛis trace class for every suitable cubeΛandtr(χΛP I(H(ω))≤C tr·|Λ|κfor someﬁxedκ.(ii)There exist L0∈N,q>d andΘ∈(0,14(q−d).Then there existsL,there is an open interval I=∅,I⊂I0,such that G(I,L,Lβ−1,ξ0)holds,MULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION5then for everyη∈L∞with suppη⊂I,it follows thatE{ |X|pη(H(ω))χK }<∞for every compact set K⊂R d.Let usﬁrst sketch the idea of the proof which is quite simple.Of course,by|X|p we denote the operator of multiplication with|x|p.We writeχΛ1η(H(ω))χΛ2≤ E n∈I0 χΛ1φn(ω) · χΛ2φn(ω) ·|η(E n(ω))|,(3.1)where E n(ω),φn(ω)denote the eigenvalues and eigenfunctions of H(ω)in I.The probability that bothΛ1andΛ2are bad for the same E n(ω)is small,roughly poly-nomially in the distance betweenΛ1,Λ2.If one of them is good,the eigenfunction decay inequality(EDI)says that one of the norms appearing in the rhs of(3.1)is exponentially small.This leads to a polynomial decay of E{ χΛ1η(H(ω))χΛ2}once the interval I is suitably chosen to guarantee the necessary probabilistic esti-mates.The assumption in(iii)of the theorem ensures that the polynomial growth of|X|p is killed by this polynomial decay.To make all of this work we have to over-come the diﬃculty that in the sum in(3.1)we have inﬁnitely many terms.This is taken care of by analyzing the centers of localization x n(ω)ofφn(ω).All this will be done relatively to a certain length scale L k.We proceed in several steps.Theﬁrst steps will be used to choose an appropriate αand set up a multi-scale scenario.Then we take care of thoseφn(ω)whose centers are far away from K.To this end,we employ the Weyl-type trace condition(i). Proof.Step1.Chooseα∈(1,2)such that4d α−14(q−d)∧ξ0=:ξand3d(α−1)+αp<2ξ.Note that the latter condition can be achieved forα>1small enough,since p<2ξ. For this choice ofα,let3Z d andE j={ω∈Ω;for some E∈I there exist y,z∈Γj∩Λ3Lj+1such thatΛLj(y)andΛLj(z)are disjoint and both not(γ,E)-good}.SinceΓj∩Λ3Lj+1≤ 9L j+16 D.DAMANIK AND P.STOLLMANNClaim.For every k∈N,P(Ωk2bad )≤c(α,d,ξ)·L2d(α−1)−2ξk.(3.2)Proof.We haveP(Ωk2bad )≤c d· j≥k L2d(α−1)−2ξj≤c d·L2d(α−1)−2ξk· 1+ j≥k+1 L j L k≥Lαj−kk4,whereχL is shorthand forχΛL (0) .Proof.We divideΛc3L j+2into angular regions M i,M i=Λ3Li+1\MULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION7SinceΛLi (˜x n)is(γ,E n(ω))-bad andω∈Ωk2good,it follows thatΛLi(˜x)is(γ,E n(ω))-good so thatχint Li ,˜xφn 2≤(C EDI)2·e−2γL i.Since#M i∩Γi grows only polynomially in L i,the assertion follows.Step4:There exists C=C(γ,α,d,κ,C tr)such that forω∈Ωk2good,j≥k,#{n;x n(ω)∈ΛLj+1}≤C·Lακdj+1.Proof.Since#{...}is non-decreasing in j,and since j0from Step3only depends on(γ,α,d),we can restrict ourselves to the case j≥j0and adapt the constant C.We start by observingx n∈ΛLj+1(χ3Lj+2P I(H(ω))χ3Lj+2φn(ω)|φn(ω))≤tr(χ3Lj+2P I(H(ω))).We want to show that each of the terms in the sum is at least14=(φn|φn)−((1−χ3Lj+2)φn|φn)−12.Plugging this into the above estimate on the trace,we get the claimed bound for the number#{n;...}.Step5:There is k1=k1(C EDI,α,C tr,κ,L0,γ,d)such that for k≥k1,ω∈Ωk2good and x∈Γk∩ΛLk+1\ΛLk,χint Lk ,xη(H(ω))χint Lk,0≤exp(−γ2exp(−γ8 D.DAMANIK AND P.STOLLMANN Using Step4again,we see that∞j=k+1x n∈ΛLj+1\ΛLjχint Lk,xφn(ω) · χint Lk,0φn(ω) ≤C·C EDI·∞ j=k+1e−γL j Lακd j+1≤12L k)if k≥k1(C EDI,α,C tr,κ,L0,γ,d).The latter estimate,together with(3.3)and (3.4),gives the assertion.Step6:For k≥k1from Step5and x∈Γk∩ΛLk+1\ΛLk,we haveE{ χintL k,x η(H(ω))χint Lk,0}≤ η ∞· c(α,d,ξ)·L2d(α−1)−2ξk+exp(−γ2L k)P(Ωk2good)≤ η ∞· c(α,d,ξ)L2d(α−1)−2ξk+exp(−γ2L j) <∞,sinceαp+3d(α−1)−2ξ<0and the L j grow fast enough.Although we cannot apply the theorem directly,a look at the proof,particularly at Steps5to7,shows that we have the following:Corollary3.2.Let the assumptions of Theorem3.1be satisﬁed.Then we haveE sup t>0 |X|p e−itH(ω)P I(H(ω))χK <∞.4.ApplicationsIn this section we present a list of models for which the variable energy multi-scale analysis has been established and which therefore exhibit strong dynamical localization by the results of the preceding section.MULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION9 4.1.Periodic plus Anderson.Here we discuss band edge localization for alloy-type models which consist of a periodic background operator with impurities sitting on the periodicity lattice.We take Z d as this lattice simply for notational conve-nience;a reformulation for more general lattices presents no diﬃculties whatsoever. Note that compared with most results available in the literature,we assume mini-mal conditions on the single-site measure:1.Let p=2if d≤3and p>d/2if d>3.2.Let V0∈L ploc (R d),V0periodic w.r.t.Z d and H0=−∆+V0.3.Let f∈L p(Λ1(0)),f≥0and f≥σonΛs(0)for someσ>0,s>0;f iscalled the single-site potential.4.Letµbe a probability measure on R,with suppµ=[q−,q+],where q−<q+∈R;µis called the single-site measure.5.LetΩ=[q−,q+]Z d,P= Z dµonΩand q k:Ω→R,q k(ω)=ωk.6.LetVω(x):= k∈Z d q k(ω)f(x−k)andH A(ω)=−∆+V0+Vω.For an elementary discussion of this model and all the ingredients necessary to prove localization,we refer to[29];see also[22,23].Note that to conform with standard notation,we denote by p both the power of the moment operator in the dynamical bounds and the power deﬁning the appropriate L p space the potentials have to belong to.This,however,should not lead to any real confusion. Theorem4.1.Let H A(ω)be as above.Assume that the single-site measureµis H¨o lder continuous,that is,there exists a>0such that for every interval J of length small enough,µ(J)≤|J|a.DenoteΣ=σ(H A(ω))a.e.and E0=infΣ.Let p>0. Then there existsε0>0such that forη∈L∞(R)with suppη⊂[E0,E0+ε0]and compact K,we haveE{ |X|pη(H A(ω))χK }<∞.Moreover,for I⊂[E0,E0+ε0]and K compact:E sup t |X|p e−iH A(ω)t P I(H A(ω))χK <∞.Proof.It is well known that(INDY),(WEYL),(GRI)and(EDI)are satisﬁed if we take for H AΛthe operator H A restricted toΛwith periodic boundary conditions. Due to[22,28]we have a Wegner estimate of the formP{dist(σ(H AΛ(ω)),E0)≤exp(−LΘ)}≤C·L2·d·exp(−aLΘ),where L denotes the sidelength of the cubeΛ.In particular,W(I0,L,Θ,q)is satisﬁed for a neighborhood I0of E0,arbitrarily givenΘand q,and L large enough. For given p>0,we can start the multi-scale induction with2ξ>p by Lifshitz asymptotics.Note that the above theorem includes the case of single-site potentials with small support.Moreover,using Klopp’s analysis of internal Lifshitz tails[26],Veselic establishes the necessary initial length scale estimates at lower band edges in the10 D.DAMANIK AND P.STOLLMANNcase where H0exhibits a non-degenerate behavior at the corresponding edge[30], so the result above extends to this case.If one does not know that H0has a non-degenerate band edge,one can still derive an initial length scale estimate by requiring a disorder assumption.This,however,might put some restriction on the power p.Theorem4.2.Let H A(ω)be as above.Assume(i)The single-site measureµis H¨o lder continuous.(ii)There existsτ>d such that for small h>0,µ([q−,q−+h])≤hτandµ([q+−h,q+])≤hτ.DenoteΣ=σ(H(ω))a.e.and let E0∈∂Σ.Let p<2(2τ−d).Then there exists ε0>0such that forη∈L∞(R)with suppη⊂[E0−ε0,E0+ε0]and compact K, we haveE{ |X|pη(H A(ω))χK }<∞.Moreover,for I⊂[E0−ε0,E0+ε0]and K compact:E sup t |X|p e−iH A(ω)t P I(H A(ω))χK <∞.Proof.We have already checked everything except for the initial length scale es-timate G(I,L,γ,ξ),and in particular how largeξcan be taken.By an elementary argument,we can takeξsubject to the conditionξ<2τ−d(see[22]),which gives the claimed result.With a modiﬁcation of independent multi-scale analysis given in[23]we can also treat the correlated or long-range case,by which we understand that the single-site potential f is no longer assumed to have support in the unit cube;see[23]. Theorem4.3.Let H A(ω)be as above,with condition3replaced by,f≥0and f≥σonΛs(0)for someσ>0,s>0;3.Let f∈L plocf≤C|x|−m for|x|large.Then the conclusions of Theorems4.1and4.2hold true withp<2 m4−d ∧2(2τ−d),respectively.Remark.Although discrete models are not explicitly included in the above frame-work,our principal strategy pursued in Section3is clearly able to treat random operators inℓ2(Z d)for which a multi-scale analysis has been established.In partic-ular,building on results from[8]one may establish strong dynamical localization for the discrete Anderson model,where for d=1,even pure point single-site measures (e.g.,the Bernoulli case)are within the scope of this result.See[8]for explicit re-quirements to make the multi-scale machinery work.We thus obtain new results on strong dynamical localization also in the discrete case since the Aizenman method does not cover single-site distributions which are too singular(e.g.,the Bernoulli case).MULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION11 4.2.Random divergence form operators.The following type of model has been introduced in[17,27]in order to study classical waves(see[17]for a motiva-tion).These models are also intensively studied in[29].1.Let a0:R d→M(d×d)be measurable,Z d-periodic and such that for someη>0,M>0,η≤a0(x)≤M for all x∈R das matrices,that is,η ζ 2≤(a0(x)ζ|ζ)≤M ζ 2for everyζ∈C d.2.Let S=[0,λmax]d×O(d),whereλmax>0and O(d)denotes the orthogonalmatrices.3.Letνbe a probability measure on O(d)and letγi,i=1,...,d be probabilitymeasures on R with suppγi=[0,λmax].4.S is called the single-site space andµ=γ1⊗···⊗γd⊗νis called the single-sitemeasure.5.LetΩ=S Z d,P=µZ d,and forω(k)=(λ1(k),...,λd(k),u(k)),deﬁnea k(ω)=u(k)∗diag(λ1(k),...,λd(k))u(k),where diag(λ1(k),...,λd(k))denotes the diagonal matrix with the indicated diagonal elements.6.Deﬁneaω(x):= k∈Z dχΛ1(k)(x)a k(ω)andH DIV(ω)=−∇(a0+aω)∇.Although the formulas may seem intricate,it is easy to see what is happening. For site k,we choose a non-negative matrix a k(ω)at random by choosing its d eigenvalues and a unitary conjugation matrix.This is done independently at diﬀer-ent sites and we get an Anderson-like random matrix function aωwhich is used as a perturbation to the perfectly periodic medium a0.Note that a0+aωhave uniform upper and lower bounds(ηand M+λmax)so that the operators can be deﬁned via quadratic forms with the Sobolev space W1,2(R d)as common form domain.The initial value problem we are now interested in is governed by the wave equation∂2v|t=0=v1(WE)∂trather than the Schr¨o dinger equation.Solutions are given byv(t)=cos t H DIV(ω) w1, where v1=12 D.DAMANIK AND P.STOLLMANNDenote Σ=σ(H DIV (ω))a.e.and let E 0∈∂Σ\{0}.Then there exists ε0>0such that for η∈L ∞(R )with supp η⊂[E 0−ε0,E 0+ε0]and compact K ,we haveE { |X |p η(H DIV (ω))χK }<∞.Moreover,for I ⊂[E 0−ε0,E 0+ε0]and K compact:E sup t|X |p cos t H DIV (ω) P I (H DIV (ω))χK <∞.By the results from [17,27]the conditions for multi-scale analysis are satisﬁed.4.3.Random quantum waveguides.Quantum waveguides have been intro-duced for the investigation of two-or three-dimensional motion of electrons in small channels,tubes or layers of crystalline matter of high purity.Mathematically speaking,one considers the free Laplacian in a domain which should be thought of as a perturbation of a strip.The following random model is taken from [24],where all the necessary conditions for multi-scale analysis are veriﬁed:It consists of a collection of randomly dented versions of a parallel strip R ×(0,d max )=D max .More precisely,let d max >0,0<d <d max ,and consider Ω=[0,d ]Z .The i -th coordinate ω(i )of ω∈Ωgives the deviation of the width of the random strip from d max ,that is,d i (ω):=d max −ω(i ),which lies between d min =d max −d and d max .Deﬁne γ(ω):R →[d min ,d max ]as the polygon in R 2joining the points {(i,d i (ω))}i ∈Z andD (ω)={(x 1,x 2)∈R 2|0<x 2<γ(ω)(x 1)}.The following picture will help in visualizing this domain:6x 2...........d min ...........d max t t t t t tt t t ti −1i i +1t t t t t t t t t t ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................D (ω)We ﬁx a probability measure µon [0,d ]with 0∈supp µ={0}and introduce P =µZ ,a probability measure on Ω.Consider H W (ω)=−∆D (ω),the LaplacianMULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION13 on D(ω)with Dirichlet boundary conditions,which is a self-adjoint operator in L2(D(ω)).Note thatπ2infσ(H W(ω))=E0:=x2 2+ ∂2−B214 D.DAMANIK AND P.STOLLMANNMoreover,for I⊂[E n(B)+εn(B),E n+1(B)−εn(B)]and K compact:E sup t |X|p e−iH L(ω)t P I(H L(ω))χK <∞.In[31],a proof of exponential localization is given for a case which includes single-site potentials of changing sign.However,the use of microlocal techniques requires smoothness of the potential.Acknowledgements.The results presented here were obtained during a visit of P.S.to Caltech.Partialﬁnancial support by the DFG and the hospitality at Caltech and Santa Monica are gratefully acknowledged. D.D.receivedﬁnancial support from the German Academic Exchange Service through Hochschulsonder-programm III(Postdoktoranden).Moreover,we would like to thank F.Germinet, S.Jitomirskaya and A.Klein for helpful discussions.References[1]M.Aizenman:Localization at weak disorder:Some elementary bounds.Rev.Math.Phys.6,1163–1182(1994)[2]M.Aizenman and G.M.Graf:Localization bounds for an electron gas.J.Phys.A:Math.Gen.31,6783–6806(1998)[3]M.Aizenman and S.Molchanov:Localization at large disorder and at extreme energies:Anelementary mun.Math.Phys.157,245–278(1993)[4]M.Aizenman,J.H.Schenker,R.H.Friedrich and D.Hundertmark:Finite-volume criteria forAnderson localization.Preprint1999,math-ph/9910022v2[5]P.W.Anderson:Absence of diﬀusion in certain random lattices.Phys.Rev.109,1492–1505(1958)[6]J.M.Barbaroux,bes and P.D.Hislop:Localization near band edges for randomSchr¨o dinger operators.Helv.Phys.Acta70,16–43(1997)[7]J.M.Barbaroux,W.Fischer and P.M¨u ller:Dynamical properties of random Schr¨o dingeroperators.Preprint1999,math-ph/9907002[8]R.Carmona,A.Klein and F.Martinelli:Anderson localization for Bernoulli and other sin-gular mun.Math.Phys.108,41–66(1987)[9]bes and P.D.Hislop:Localization for some continuous,random Hamiltonians ind-dimensions.J.Funct.Anal.124,149–180(1994)[10]bes and P.D.Hislop:Landau Hamiltonians with random potentials:Localizationand density of mun.Math.Phys.177,603–630(1996)[11]T.C.Dorlas,N.Macris and J.V.Pul´e:Localization in a single-band approximation to randomSchr¨o dinger operators in a magneticﬁeld.Helv.Phys.Acta68,329–364(1995)[12]T.C.Dorlas,N.Macris and J.V.Pul´e:Localization in single Landau bands.J.Math.Phys.37(4),1574–1595(1996)[13]R.del Rio,S.Jitomirskaya,st and B.Simon:What is localization?Phys.Rev.Lett.75,117–119(1995)[14]R.del Rio,S.Jitomirskaya,st and B.Simon:Operators with singular continuousspectrum,IV.Hausdorﬀdimensions,rank one perturbations,and localization.J.d’Analyse Math.69,153–200(1996)[15]H.von Dreifus and A.Klein:A new proof of localization in the Anderson tight bindingmun.Math.Phys.124,285–299(1989)[16]A.Figotin and A.Klein:Localization phenomenon in gaps of the spectrum of random latticeoperators,J.Stat.Phys.75,997–1021(1994)[17]A.Figotin and A.Klein:Localization of classical waves,I:Acoustic mun.Math.Phys.180,439–482(1996)[18]J.Fr¨o hlich,F.Martinelli,E.Scoppola and T.Spencer:Constructive proof of localization inthe Anderson tight binding mun.Math.Phys.101,21–46(1985)[19]J.Fr¨o hlich and T.Spencer:Absence of diﬀusion in the Anderson tight binding model forlarge disorder or low mun.Math.Phys.88,151–184(1983)MULTI-SCALE ANALYSIS IMPLIES STRONG DYNAMICAL LOCALIZATION15 [20]F.Germinet and S.de Bi`e vre:Dynamical localization for discrete and continuous randomSchr¨o dinger mun.Math.Phys.194,323–341(1998)[21]H.Holden and F.Martinelli:On the absence of diﬀusion for a Schr¨o dinger operator onL2(Rν)with a random mun.Math.Phys.93,197–217(1984)[22]W.Kirsch,P.Stollmann and G.Stolz:Localization for random perturbations of periodicSchr¨o dinger operators.Random Oper.Stochastic Equations6,241–268(1998)[23]W.Kirsch,P.Stollmann and G.Stolz:Anderson localization for random Schr¨o dinger opera-tors with long range mun.Math.Phys.195,495–507(1998)[24]F.Kleespies and P.Stollmann:Localization and Lifshitz tails for random quantum wave-guides,Rev.Math.Phys.,to appear[25]F.Klopp:Localization for some continuous random Schr¨o dinger mun.Math.Phys.167,553–569(1995)[26]F.Klopp:Internal Lifshitz tails for random perturbations of periodic Schr¨o dinger operators.Duke Math.J.98(2),335–396(1999)[27]P.Stollmann:Localization for acoustic waves in random perturbations of periodic media.Israel J.Math.107,125–139(1998)[28]P.Stollmann:Wegner estimates and localization for continuum Anderson models with somesingular distributions.Arch.Math.,to appear[29]P.Stollmann:Caught by Disorder:Lectures on Bound States in Random Media.Book,inpreparation[30]I.Veselic:Localisation for random perturbations of periodic Schr¨o dinger operators withregular Floquet eigenvalues.Preprint1998,mparc/98-569[31]W.M.Wang:Microlocalization,percolation,and Anderson localization for the magneticSchr¨o dinger operator with a random potential.J.Funct.Anal.146,1–26(1997)。

Integrals for braided Hopf algebras

Abstract Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object Int H is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integration with cross-product and transmutation are studied. 1991 Mathematics Subject Classiﬁcation. Primary 16W30, 18D15, 17B37; Secondary 1troduction 2 Preliminaries 3 Integrals and the generalized Radford formula 4 Proofs of the main results 5 Examples of braided Hopf algebras 6 Integrals and related structures 7 Applications of integrals 1 5 10 14 30 40 45
1
Introduction

堆垛机英文资料翻译

堆垛机英文资料翻译翻译英文原文Realization of Neural Network Inverse System with PLC in Variable Frequency Speed-Regulating SystemAbstract. The variable frequency speed-regulating system which consists of an induction motor and a general inverter, and controlled by PLC is widely used in industrial field. .However, for the multivariable, nonlinear and strongly coupled induction motor, the control performance is not good enough to meet the needs of speed-regulating. The mathematic model of the variable frequency speed-regulating system in vector control mode is presented and its reversibility has been proved. By constructing a neural network inverse system and combining it with the variable frequency speed-regulating system, a pseudo-linear system is completed, and then a linear close-loop adjustor is designed to get high performance. Using PLC, a neural network inverse system can be realized in actural system. The results of experiments have shown that the performances of variable frequency speed-regulating system can be improved greatly and the practicability of neural network inverse control was testified.1.IntroductionIn recent years, with power electronic technology, microelectronic technology and modern control theory infiltrating into AC electric driving system, inverters have been widely used in speed-regulating of AC motor. The variable frequency speed-regulating system which consists of an induction motor and a general inverter is used to take the place of DC speed-regulating system. Because of terrible environmentand severe disturbance in industrial field, the choice of controller is an important problem. In reference [1][2][3], Neural network inverse control was realized by using industrial control computer and several data acquisition cards. The advantages of industrial control computer are high computation speed, great memory capacity and good compatibility withother software etc. But industrial control computer also has some disadvantages in industrial application such as instability and fallibility and worse communication ability. PLC control system is special designed for industrial environment application, and its stability and reliability are good. PLC control system can be easily integrated into field bus control system with the high ability of communication configuration, so it is wildly used in recent years, and deeply welcomed. Since the system composed of normal inverter and induction motor is a complicated nonlinear system, traditional PID control strategy could not meet the requirement for further control. Therefore, how to enhance control performance of this system is very urgent.The neural network inverse system [4][5] is a novel control method in recent years. The basic idea is that: for a given system, an inverse system of the original system is created by a dynamic neural network, and the combination system of inverse and object is transformed into a kind of decoupling standardized system with linear relationship. Subsequently, a linear close-loop regulator can be designed to achieve high control performance. The advantage of this method is easily to be realized in engineering. The linearization and decoupling control of normal nonlinear system can realize using this method.Combining the neural network inverse into PLC can easily make up the insufficiency of solving the problems of nonlinearand coupling in PLC control system. This combination can promote the application of neural network inverse into practice to achieve its full economic and social benefits.In this paper, firstly the neural network inverse system method is introduced, and mathematic model of the variable frequencyspeed-regulating system in vector control mode is presented. Then a reversible analysis of the system is performed, and the methods and steps are given in constructing NN-inverse system with PLC control system.Finally, the method is verified in experiments, and compared with traditional PI control and NN-inverse control.2.Neural Network Inverse System Control MethodThe basic idea of inverse control method [6] is that: for a given system, anα-th integral inverse system of the original system is created by feedback method, and combining the inverse system with original system, a kind of decoupling standardized system with linear relationship is obtained, which is named as a pseudo linear system as shown in Fig.1. Subsequently, a linear close-loop regulator will be designed to achieve high control performance.Inverse system control method with the features of direct, simple and easy to understand does not like differential geometry method [7], which is discusses the problems in "geometry domain". The main problem is the acquisition of the inverse model in the applications. Since non-linear system is a complex system, and desired strict analytical inverse is very difficult to obtain, even impossible. The engineering application of inverse system control doesn’t meet the expectations. As neural network has non-linear approximate ability, especially fornonlinear complexity system, it becomes the powerful tool to solve the problem.a ?th NN inverse system integrated inverse system with non-linear ability of the neural network can avoid the troubles of inverse system method. Then it is possible to apply inverse control method to a complicated non-linear system.a ?th NN inverse system method needs less system information such as the relative order of system, and it is easy to obtain the inverse model by neural network training. Cascading the NN inverse system with the original system, a pseudo-linear system is completed. Subsequently, a linear close-loop regulator will be designed.3. Mathematic Model of Induction Motor Variable FrequencySpeed-Regulating System and Its ReversibilityInduction motor variable frequency speed-regulating system supplied by the inverter of tracking current SPWM can be expressed by 5-th order nonlinear model in d-q two-phase rotating coordinate. The model was simplified as a 3-order nonlinear model. If the delay of inverter is neglected, the model is expressed as follows:(1) where denotes synchronous angle frequency, and is rotate speed.are stator’s current, and are rotor’s flux linkage in (d,q)axis. is number of poles. is mutual inductance, and is rotor’s inductance. J i s moment of inertia.is rotor’s time constant, and is load torque.In vector mode, thenSubstituted it into formula (1), then(2) Taking reversibility analyses of forum (2), thenThe state variables are chosen as followsInput variables areTaking the derivative on output in formula(4), then(5)(6) Then the Jacobi matrix is Realization of Neural Network Inverse System with PLC(7)(8)As so and system is reversible. Relative-order of system isWhen the inverter is running in vector mode, the variability of flux linkage can be neglected (considering the flux linkage to be invariableness and equal to the rating). The original system was simplified as an input and an output system concluded by forum (2).According to implicit function ontology theorem, inverse system of formula (3)can be expressed as(9)When the inverse system is connected to the original system in series, the pseudo linear compound system can be built as the type of4. Realization Steps of Neural Network Inverse System4.1 Acquisition of the Input and Output Training SamplesTraining samples are extremely important in the reconstruction of neural network inverse system. It is not only need to obtain the dynamic data of the original system, but also need to obtain the static date. Reference signal should include all the work region of original system, which can be ensure the approximate ability. Firstly the step of actuating signal is given corresponding every 10 HZ form 0HZ to 50HZ, and the responses of open loop are obtain. Secondly a random tangle signal is input, which is a random signal cascading on the step of actuatingsignal every 10 seconds, and the close loop responses is obtained. Based on these inputs, 1600 groupstraining samples are gotten.4.2 The Construction of Neural NetworkA static neural network and a dynamic neural network composed of integral is used to construct the inverse system. The structure of static neural network is 2 neurons in input layer, 3 neurons in output layer, and 12 neurons in hidden layer. The excitation function of hidden neuron is monotonic smooth hyperbolic tangent function. The output layer is composed of neuron with linear threshold excitation function. The training datum are the corresponding speed of open-loop, close-loop, first orderderivative of these speed, and setting reference speed. After 50 times。

Feynman integrals and multiple polylogarithms

a r X i v :0705.0900v 2 [h e p -p h ] 10 M a y 2007MZ-TH/07-06Feynman integrals and multiple polylogarithms Stefan Weinzierl Institut für Physik,Universität Mainz,D -55099Mainz,Germany Abstract In this talk I review the connections between Feynman integrals and multiple polyloga-rithms.After an introductory section on loop integrals I discuss the Mellin-Barnes transfor-mation and shufﬂe algebras.In a subsequent section multiple polylogarithms are introduced.Finally,I discuss how certain Feynman integrals evaluate to multiple polylogarithms.1IntroductionIn this talk I will discuss techniques for the computation of loop integrals,which occur in pertur-bative calculations in quantumﬁeld theory.Particle physics has become aﬁeld where precisionmeasurements have become possible.Of course,the increase in experimental precision has tobe matched with more accurate calculations from the theoretical side.This is the“raison d’être”for loop calculations:A higher accuracy is reached by including more terms in the perturbativeexpansion.The complexity of a calculation increases obviously with the number of loops,butalso with the number of external particles or the number of non-zero internal masses associatedto propagators.To give an idea of the state of the art,speciﬁc quantities which are just purenumbers have been computed up to an impressive fourth or third order.Examples are the cal-culation of the4-loop contribution to the QCDβ-function[1],the calculation of the anomalousmagnetic moment of the electron up to three loops[2],and the calculation of the ratio of thetotal cross section for hadron production to the total cross section for the production of aµ+µ−pair in electron-positron annihilation to order O α3s [3].Quantities which depend on a single variable are known at the best to the third order.Outstanding examples are the computation ofthe three-loop Altarelli-Parisi splitting functions[4,5]or the calculation of the two-loop ampli-tudes for the most interesting2→2processes[6–16].For the calculation of these amplitudes,the knowledge of certain highly non-trivial two-loop integrals has been essential[17–19].Thecomplexity of a two-loop computation increases,if the result depends on more than one variable.An example for a two-loop calculation whose result depends on two variables is the computationof the two-loop amplitudes for e+e−→3jets[20–22].But in general,if more than one variable is involved,we have to content ourselves with next-to-leading order calculations.An example for the state of the art is here the computation of the electro-weak corrections to the process e+e−→4fermions[23,24].From a mathematical point of view loop calculations reveal interesting algebraic structures.Multiple polylogarithms play an important role to express the results of loop calculations.Themathematical aspects will be discussed in this talk.Additional material related to loop calcula-tions can found in the reviews[25–28]and the book[29].This paper is organised as follows:In the next section I review basic facts about Feynmanintegrals.Section3is devoted to the Mellin-Barnes transformation.In section4algebraic struc-tures like shufﬂe algebras are introduced.Section5deals with multiple polylogarithms.Section6combines the various aspects and shows,how certain Feynman integrals evaluate to multiplepolylogarithms.Finally,section7contains a summary.2Feynman integralsTo set the scene let us consider a scalar Feynman graph G.Fig.1shows an example.In thisexample there are three external lines and six internal lines.The momentaﬂowing in or outthrough the external lines are labelled p1,p2and p3and can be taken asﬁxed vectors.They areconstrained by momentum conservation:If all momenta are taken toﬂow outwards,momentum2p 1p 2p 3Figure 1:An example of a two-loop Feynman graph with three external legs.conservation requires thatp 1+p 2+p 3=0.(1)At each vertex of a graph we have again momentum conservation:The sum of all momenta ﬂowing into the vertex equals the sum of all momenta ﬂowing out of the vertex.A graph,where the external momenta determine uniquely all internal momenta is called a tree graph.It can be shown that such a graph does not contain any closed circuit.In contrast,graphs which do contain one or more closed circuits are called loop graphs.If we have to specify besides the external momenta in addition l internal momenta in order to determine uniquely all internal momenta we say that the graph contains l loops.In this sense,a tree graph is a graph with zero loops and the graph in ﬁg.1contains two loops.Let us agree that we label the l additional internal momenta by k 1to k l .Feynman rules allow us to translate a Feynman graph into a mathematical formula.For a scalar graph we have substitute for each internal line j a propagatorip 2j +m 2j .When integrating over E ,the integration contour has tobe deformed to avoid these two poles.Causality dictates into which directions the contour has to be deformed.The pole on the negative real axis is avoided by escaping into the lower complex half-plane,the pole at the positive real axis is avoided by a deformation into the upper complex half-plane.Feynman invented the trick to add a small imaginary part i δto the denominator,which keeps track of the directions into which the contour has to be deformed.In the following the i δ-term is omitted in order to keep the notation compact.The Feynman rules tell us also to integrate for each loop over the loop momentum:Z d 4k rHowever,there is a complication:If we proceed naively and write down for each loop an integral over four-dimensional Minkowski space,we end up with ill-deﬁned integrals,since these inte-grals may contain ultraviolet or infrared divergences!Therefore the ﬁrst step is to make these integrals well-deﬁned by introducing a regulator.There are several possibilities how this can be done,but the method of dimensional regularisation [30–32]has almost become a standard,as the calculations in this regularisation scheme turn out to be the simplest.Within dimensional regular-isation one replaces the four-dimensional integral over the loop momentum by an D -dimensional integral,where D is now an additional parameter,which can be a non-integer or even a complex number.We consider the result of the integration as a function of D and we are interested in the behaviour of this function as D approaches 4.It is common practice to parameterise the deviation of D from 4byD =4−2ε.(5)The divergences in loop integrals will manifest themselves in poles in 1/ε.In an l -loop integral ultraviolet divergences will lead to poles 1/εl at the worst,whereas infrared divergences can lead to poles up to 1/ε2l .We will also encounter integrals,where the dimension is shifted by units of two.In these cases we often writeD =2m −2ε,(6)where m is an integer,and we are again interested in the Laurent series in ε.Let us now consider a generic scalar l -loop integral I G in D =2m −2εdimensions with n propagators,corresponding to a graph G .Let us further make a slight generalisation:For each internal line j the corresponding propagator in the integrand can be raised to a power νj .Therefore the integral will depend also on the numbers ν1,...,νn .We deﬁne the Feynman integral byI G = e εγE µ2εl Z l ∏r =1d D k r 2n ∏j =11j −ln n =0.5772156649 (8)in the ﬁnal result.The integral measure is now d D k /(i πD /2)instead of d D k /(2π)D ,and each propagator is multiplied by i .The small imaginary parts i δin the propagators are not written explicitly.How to perform the D -dimensional loop integrals ?The ﬁrst step is to convert the products of propagators into a sum.This can be done with the Feynman parameter technique.In its full4generality it is also applicable to cases,where each factor in the denominator is raised to some powerν.The formula reads:n ∏i=11n∏i=1Γ(νi)1Zn∏i=1dx i xνi−1i δ 1−n∑i=1x in∏j=1Γ(νj)1Zn∏j=1dx j xνj−1j δ(1−n∑i=1x i)Uν−(l+1)D/2The function F 0is the sum over all such monomials times minus the corresponding invariant.The function F is then given by F0plus an additional piece involving the internal masses m j .Insummary,the functions U and F are obtained from thegraph as follows:U =∑T ∈T 1 ∏j ∈C (T ,G )x j,F 0=∑(T 1,T 2)∈T 2 ∏j∈C (T 1,G )x j (−s T 1),F =F 0+U n ∑j =1x j m 2j .(15)In general,U is a positive semi-deﬁnite function.Its vanishing is related to the UV sub-divergences of the graph.Overall UV divergences,if present,will always be contained in the prefactor Γ(ν−lD /2).In the Euclidean region,F is also a positive semi-deﬁnite function of the Feynman parameters x j .As an example we consider the graph in ﬁg.1.For simplicity we assume that all internal propagators are massless.Then the functions U and F read:U =x 15x 23+x 15x 46+x 23x 46,F =(x 1x 3x 4+x 5x 2x 6+x 1x 5x 2346)−p 21+(x 6x 3x 5+x 4x 1x 2+x 4x 6x 1235) −p 22+(x 2x 4x 5+x 3x 1x 6+x 2x 3x 1456)−p 23 .(16)Here we used the notation that x i j ...r =x i +x j +...+x r .Finally let us remark,that in eq.(7)we restricted ourselves to scalar integrals,where the numerator of the integrand is independent of the loop momentum.A priori more complicated cases,where the loop momentum appears in the numerator might occur.However,there is a general reduction algorithm,which reduces these tensor integrals to scalar integrals [34,35].The price we have to pay is that these scalar integrals involve higher powers of the propagators and/or have shifted dimensions.Therefore we considered in eq.(6)shifted dimensions and in eq.(7)arbitrary powers of the propagators.In conclusion,the integrals of the form as in eq.(7)are the most general loop integrals we have to solve.3The Mellin-Barnes transformationIn sect.2we saw that the Feynman parameter integrals depend on two graph polynomials U and F ,which are homogeneous functions of the Feynman parameters.In this section we will continue the discussion how these integrals can be performed and exchanged against a (multiple)sum over residues.The case,where the two polynomials are absent is particular simple:1Z 0 n ∏j =1dx j x νj −1j δ(1−n ∑i =1x i )=n∏j =1Γ(νj )With the help of the Mellin-Barnes transformation we now reduce the general case to eq.(17). The Mellin-Barnes transformation reads(A1+A2+...+A n)−c=1(2πi)n−1i∞Z−i∞dσ1...i∞Z−i∞dσn−1(18)×Γ(−σ1)...Γ(−σn−1)Γ(σ1+...+σn−1+c)Aσ11...Aσn−1n−1A−σ1−...−σn−1−c n.Each contour is such that the poles ofΓ(−σ)are to the right and the poles ofΓ(σ+c)are to the left.This transformation can be used to convert the sum of monomials of the polynomials U and F into a product,such that all Feynman parameter integrals are of the form of eq.(17).As this transformation converts sums into products it is the“inverse”of Feynman parametrisation. Eq.(18)is derived from the theory of Mellin transformations:Let h(x)be a function which is bounded by a power law for x→0and x→∞,e.g.|h(x)|≤Kx−c0for x→0,|h(x)|≤K′x c1for x→∞.(19)Then the Mellin transform is deﬁned for c0<Reσ<c1byh M(σ)=∞Zdx h(x)xσ−1.(20)The inverse Mellin transform is given byh(x)=1(1+x)c(22) with Mellin transform h M(σ)=Γ(−σ)Γ(σ+c)/Γ(c).For Re(−c)<Reγ<0we havex c2πiγ+i∞Zγ−i∞dσΓ(−σ)Γ(σ+c)2πiγ+i∞Zγ−i∞dσΓ(−σ)Γ(σ+c)Eq.(18)is then obtained by repeated use of eq.(24).With the help of eq.(17)and eq.(18)we may exchange the Feynman parameter integrals against multiple contour integrals.A single contour integral is of the formI=1Γ(σ+c2)...Γ(σ+c p)Γ(−σ+b1)...Γ(−σ+b n)2(m+n−p−q).(27)Then the integral eq.(25)converges absolutely forα>0[36]and deﬁnes an analytic function in|arg x|<min π,απn!,res(Γ(−σ+a),σ=a+n)=−(−1)nlemma states that1,Γ(a+b+c+d)(31) if none of the poles ofΓ(a+σ)Γ(b+σ)coincides with the ones fromΓ(c−σ)Γ(d−σ).Barnes second lemma reads1Γ(a+b+c+d+e+σ)Γ(a+d)Γ(b+d)Γ(c+d)Γ(a+e)Γ(b+e)Γ(c+e)=which are closely related:First note that the unit in an algebra can be viewed as a map from K to A and that the multiplication can be viewed as a map from the tensor product A⊗A to A(e.g. one takes two elements from A,multiplies them and gets one element out).A coalgebra has instead of multiplication and unit the dual structures:a comultiplication∆and a counit¯e.The counit is a map from A to K,whereas comultiplication is a map from A to A⊗A.Note that comultiplication and counit go in the reverse direction compared to multiplication and unit.We will always assume that the comultiplication is coassociative.The general form of the coproduct is∆(a)=∑ia(1)i⊗a(2)i,(37)where a(1)i denotes an element of A appearing in theﬁrst slot of A⊗A and a(2)i correspond-ingly denotes an element of A appearing in the second slot.Sweedler’s notation[51]consists in dropping the dummy index i and the summation symbol:∆(a)=a(1)⊗a(2)(38) The sum is implicitly understood.This is similar to Einstein’s summation convention,except that the dummy summation index i is also dropped.The superscripts(1)and(2)indicate that a sum is involved.A bialgebra is an algebra and a coalgebra at the same time,such that the two structures are compatible with each ing Sweedler’s notation,the compatibility between the multipli-cation and comultiplication is expressed as∆(a·b)= a(1)·b(1) ⊗ a(2)·b(2) .(39)A Hopf algebra is a bialgebra with an additional map from A to A,called the antipode S, which fulﬁlsa(1)·S a(2) =S a(1) ·a(2)=0for a=e.(40) With this background at hand we can now state the coproduct,the counit and the antipode for the shufﬂe algebra:The counit¯e is given by:¯e(e)=1,¯e(l1l2...l n)=0.(41) The coproduct∆is given by:∆(l1l2...l k)=k∑j=0 l j+1...l k⊗ l1...l j .(42)The antipode S is given by:S(l1l2...l k)=(−1)k l k l k−1...l2l1.(43)106t 1t 2=6t 1t 2+6t 1t 2Figure 2:Sketch of the proof for the shufﬂe product of two iterated integrals.The integral over the square is replaced by two integrals over the upper and lower triangle.The shufﬂe algebra is generated by the Lyndon words.If one introduces a lexicographic ordering on the letters of the alphabet A ,a Lyndon word is deﬁned by the propertyw <v(44)for any sub-words u and v such that w =uv .An important example for a shufﬂe algebra are iterated integrals.Let [a ,b ]be a segment of the real line and f 1,f 2,...functions on this interval.Let us deﬁne the following iterated integrals:I (f 1,f 2,...,f k ;a ,b )=bZ af 1(t 1)dt 1t 1Z af 2(t 2)dt 2...t k −1Z af k (t k )dt k(45)For ﬁxed a and b we have a shufﬂe algebra:I (f 1,f 2,...,f k ;a ,b )·I (f k +1,...,f r ;a ,b )=∑shufﬂesσI (f σ(1),f σ(2),...,f σ(r );a ,b ),(46)where the sum runs over all permutations σ,which preserve the relative order of 1,2,...,k and of k +1,...,r .The proof is sketched in ﬁg.2.The two outermost integrations are recursively replaced by integrations over the upper and lower triangle.We now consider generalisations of shufﬂe algebras.Assume that for the set of letters we have an additional operation(.,.):A ⊗A →A ,l 1⊗l 2→(l 1,l 2),(47)which is commutative and associative.Then we can deﬁne a new product of words recursively through(l 1l 2...l k )∗(l k +1...l r )=l 1[(l 2...l k )∗(l k +1...l r )]+l k +1[(l 1l 2...l k )∗(l k +2...l r )]+(l 1,l k +1)[(l 2...l k )∗(l k +2...l r )](48)This product is a generalisation of the shufﬂe product and differs from the recursive deﬁnition of the shufﬂe product in eq.(36)through the extra term in the last line.This modiﬁed product is known under the names quasi-shufﬂe product [52],mixable shufﬂe product [53]or stufﬂe6i 1j 1=6i 1j 1+6i 1j 1+6i 1j 1Figure 3:Sketch of the proof for the quasi-shufﬂe product of nested sums.The sum over the square is replaced by the sum over the three regions on the r.h.s.product [54].Quasi-shufﬂe algebras are Hopf ultiplication and counit are deﬁned as for the shufﬂe algebras.The counit ¯e is given by:¯e (e )=1,¯e (l 1l 2...l n )=0.(49)The coproduct ∆is given by:∆(l 1l 2...l k )=k∑j =0l j +1...l k ⊗ l 1...l j .(50)The antipode S is recursively deﬁned throughS (l 1l 2...l k )=−l 1l 2...l k −k −1∑j =1S l j +1...l k ∗ l 1...l j.(51)An example for a quasi-shufﬂe algebra are nested sums.Let n a and n b be integers with n a <n band let f 1,f 2,...be functions deﬁned on the integers.We consider the following nested sums:S (f 1,f 2,...,f k ;n a ,n b )=n b∑i 1=n af 1(i 1)i 1−1∑i 2=n af 2(i 2)...i k −1−1∑i k =n af k (i k )(52)For ﬁxed n a and n b we have a quasi-shufﬂe algebra:S (f 1,f 2,...,f k ;n a ,n b )∗S (f k +1,...,f r ;n a ,n b )=n b∑i 1=n af 1(i 1)S (f 2,...,f k ;n a ,i 1−1)∗S (f k +1,...,f r ;n a ,i 1−1)+n b ∑j 1=n a f k (j 1)S (f 1,f 2,...,f k ;n a ,j 1−1)∗S (f k +2,...,f r ;n a ,j 1−1)+n b ∑i =n af 1(i )f k (i )S (f 2,...,f k ;n a ,i −1)∗S (f k +2,...,f r ;n a ,i −1)(53)Note that the product of two letters corresponds to the point-wise product of the two functions:(f i ,f j )(n )=f i (n )f j (n ).(54)The proof that nested sums obey the quasi-shufﬂe algebra is sketched in Fig.3.The outermost sums of the nested sums on the l.h.s of (53)are split into the three regions indicated in Fig.3.5Multiple polylogarithmsIn the previous section we have seen that iterated integrals form a shufﬂe algebra,while nested sums form a quasi-shufﬂe algebra.In this context multiple polylogarithms form an interesting class of functions.They have a representation as iterated integrals as well as nested sums.There-fore multiple polylogarithms form a shufﬂe algebra as well as a quasi-shufﬂe algebra.The two algebra structures are independent.Let us start with the representation as nested sums.The multiple polylogarithms are deﬁned byLi m1,...,m k (x1,...,x k)=∑i1>i2>...>i k>0x i11i k m k.(55)The multiple polylogarithms are generalisations of the classical polylogarithms Li n(x)[55], whose most prominent examples areLi1(x)=∞∑i1=1x i1i21,(56)as well as Nielsen’s generalised polylogarithms[56]S n,p(x)=Li n+1,1,...,1(x,1,...,1p−1),(57) and the harmonic polylogarithms[57]H m1,...,m k (x)=Li m1,...,m k(x,1,...,1k−1).(58)Multiple polylogarithms have been studied extensively in the literature by physicists[57–70]and mathematicians[54,71–81].In addition,multiple polylogarithms have an integral representation.To discuss the integral representation it is convenient to introduce for z k=0the following functionsG(z1,...,z k;y)=yZdt1t2−z2...t k−1Zdt ky−z,(61)then one hasdk!(ln y)k.(64)This permits us to allow trailing zeros in the sequence(z1,...,z k)by deﬁning the function G with trailing zeros via(63)and(64).To relate the multiple polylogarithms to the functions G it is convenient to introduce the following short-hand notation:G m1,...,m k (z1,...,z k;y)=G(0,...,0m1−1,z1,...,z k−1,0...,0 m k−1,z k;y)(65)Here,all z j for j=1,...,k are assumed to be non-zero.One thenﬁndsLi m1,...,m k (x1,...,x k)=(−1)k G m1,...,m k 1x1x2,...,1z1,z1z k .(67)Eq.(66)together with(65)and(59)deﬁnes an integral representation for the multiple polyloga-rithms.To make this more explicit Iﬁrst introduce some notation for iterated integralsΛZ 0dtt−a1=ΛZdt nt n−1−a n−1×...×t2Zdt1t◦ m dt t◦...dt t−a.(69)The integral representation for Li m1,...,m k(x1,...,x k)reads thenLi m1,...,m k (x1,...,x k)=(−1)k1Zdt t−b1◦ dt t−b2◦...◦ dt t−b k,(70) where the b j’s are related to the x j’sb j=1t =∞∑n=1x nx −π22(ln(−x))2,Li2(x)=−Li2(1−x)+π2(i+1)!(−ln(1−x))i+1.(74)The generalisation to multiple polylogarithms proceeds along the same lines[67]:Using the integral representationG m1,...,m k (z1,z2,...,z k;y)=(75)yZdt t−z1 dt t−z2... dt t−z kone transforms all arguments into a region,where one has a converging power series expansion:G m1,...,m k (z1,...,z k;y)=∞∑j1=1...∞∑j k=11z1 j1×1z2 j2...1z kj k.(76)The multiple polylogarithms satisfy the Hölder convolution[54].For z1=1and z w=0this identity readsG(z1,...,z w;1)=(77) w∑j=0(−1)j G 1−z j,1−z j−1,...,1−z1;1−1p .The Hölder convolution can be used to accelerate the convergence for the series representation of the multiple polylogarithms.6Laurent expansion of Feynman integralsLet us return to the question on how to compute Feynman integrals.In section3we saw how to obtain from the Mellin-Barnes transformation(multiple)sums by closing the integration contours and summing up the residues.As a simple example let us consider that the sum of residues is equal to∞∑i=0Γ(i+a1+t1ε)Γ(i+a2+t2ε)i1m1...1i1m1...x i k kk is called the depth of the Z-sum and w=m1+...+m k is called the weight.If the sums go to inﬁnity(n=∞)the Z-sums are multiple polylogarithms:Z(∞;m1,...,m k;x1,...,x k)=Li m1,...,m k(x1,...,x k).(82) For x1=...=x k=1the deﬁnition reduces to the Euler-Zagier sums[83,84]:Z(n;m1,...,m k;1,...,1)=Z m1,...,m k(n).(83) For n=∞and x1=...=x k=1the sum is a multipleζ-value[54]:Z(∞;m1,...,m k;1,...,1)=ζm1,...,m k.(84) The usefulness of the Z-sums lies in the fact,that they interpolate between multiple polyloga-rithms and Euler-Zagier sums.The Z-sums form a quasi-shufﬂe algebra.UsingΓ(x+1)=xΓ(x),partial fractioning and an adjustment of the summation index one can transform eq.(78)into terms of the form∞∑i=1Γ(i+t1ε)Γ(i+t2ε)i m,(85)where m is an integer.Now using eq.(79)one obtainsΓ(1+ε)∞∑i=1(1+εt1Z1(i−1)+...)(1+εt2Z1(i−1)+...)i m.(86)Inverting the power series in the denominator and truncating inεone obtains in each order inεterms of the form∞∑i=1x ii m Z m1...m k(i−1),(88)which are special cases of multiple polylogarithms,called harmonic polylogarithms H m,m1,...,m k (x).This completes the algorithm for the expansion inεfor sums of the form as in eq.(78).The Hopf algebra of Z-sums has additional structures if we allow expressions of the formx n0i m Z(i−1;...)y n−ican again be expressed in terms of expressions of the form(89).In addition there is a conjugation,e.g.sums of the form−n∑i=1 n i(−1)i x ii mZ(i;...)y n−iΓ(i+a′1)...Γ(i+a k)Γ(j+b′1)...Γ(j+b l)Γ(i+j+c′1)...Γ(i+j+c m)Γ(i+a′1)...Γ(i+a k)Γ(j+b′1)...Γ(j+b l)Γ(i+j+c′1)...Γ(i+j+c 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