Calculs De Facteurs Epsilon De Paires Pour GLn Sur Un Corps Local, I

Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GL_n(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GL_n(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.     

The Polynomials Associated with a Julia Set

We prove that, with two exceptions, the set of polynomials withJulia set J has the form {pⁿ:nN,} where p is one of these polynomialsand is the symmetry group of J. The exceptions occur when Jis a circle or a straight line segment.     

Metric Entropy of Convex Hulls in Hilbert Spaces

We show in this note the following statement which is an improvementover a result of R. M. Dudley and which is also of independentinterest. Let X be a set of a Hilbert space with the propertythat there are constants , >0, and for each n N, the setX can be covered by at most n balls of radius n^–. Then,for each n N, the convex hull of X can be covered by 2ⁿ ballsof radius . The estimate is best possible for all n N, apart from the value c=c(, , X).In other words, let N(, X), >0, be the minimal number ofballs of radius covering the set X. Then the above result isequivalent to saying that if N(, X)=O(^–1/) as 0, thenfor the convex hull conv (X) of X, N(, conv (X)) =O(exp(^–2/(12))). Moreover, we give an interplay between several coveringparameters based on coverings by balls (entropy numbers) andcoverings by cylindrical sets (Kolmogorov numbers). 1991 MathematicsSubject Classification 41A46.     

Sub- and Superadditive Properties of Fejer's sine Polynomial

Let be Fejér's sine polynomial. We prove the following statements.

1. The inequality holds for all x, y (0, ) with x + y < if and only if 0 and + rß 1.
2. The converse of the above inequality is valid for allx, y (0, ) with x + y < if and only if 0 and + rß 1.
3. For all n N and x, y [0, ] we have . Both bounds are best possible.

2000 Mathematics Subject Classification 42A05, 26D05 (primary),39B62 (secondary).     

A variant of the Hales-Jewett theorem

It was shown by Bergelson that any set B with positive uppermultiplicative density contains nicely intertwined arithmeticand geometric progressions: for each k there exist a, b, d such that {b(a+id)^j: i, j {1, 2, ..., k}}B. In particular,one cell of each finite partition of contains such configurations.We prove a Hales–Jewett-type extension of this partitiontheorem.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

Polynomial symplectomorphisms

Let be the field of real or complex numbers. Let (X ²ⁿ, )be a symplectic affine space. We study the group of polynomialsymplectomorphisms of X. We show that for an arbitrary k thegroup of polynomial symplectomorphisms acts k-transitively onX. Moreover, if 2 l 2n – 2 then elements of this groupcan be characterized by polynomial automorphisms which preservethe symplectic type of all algebraic l-dimensional subvarietiesof X.     

Invariant Subspaces of C1.-Contractions with Non-Reductive Unitary Extensions

Let T be a contraction acting on the Hilbert space H such thatlim_n||Tⁿh||0, for every nonzero h;H. It is proved that if theunitary operator attached to T in a canonic way contains thebilateral shift, then T has a non-trivial invariant subspace.Furthermore, if in addition lim_n||T^*nh||0 holds for every nonzeroh H, then T is shown to be reflexive.     

Reaping Numbers of Boolean Algebras

A subset A of a Boolean algebra B is said to be (n,m)-reapedif there is a partition of unity p B of size n such that |{b p:b a 0}| m for all a A. The reaping number r_n,m (B) ofa Boolean algebra B is the minimum cardinality of a set A B\{0}which cannot be (n,m)-reaped. It is shown that for each n, thereis a Boolean algebra B such that r_n+1,2(B) r_n,2(B). Also, {r_n,m(B):mn } consists of at most two consecutive cardinals. The existenceof a Boolean algebra B such that r_n,m (B) r_n',m' (B) is equivalentto a statement in finite combinatorics which is also discussed.     

A Combinatorial Application of the Maximal Ergodic Theorem

Let X be a compact space,µ a Borel probability measureon X, T: X X a measure preserving continuous transformationand g: X R a continuous function. Then for some yX, This Lemma is used to give an alternative proof of a resultby Ruzsa [6], which implies the following extension of a resultof Bergelson [1]. If E N satisfies then there exists a set N such that n–1|[1,n]| (E) for all, n 1, and any finite subset{₁, ... _k} satisfies Ø. 7 Moria St., Ramat Hasharon, Israel     

The Adamjan-Arov-Krein Theorem on Hp-Spaces (2 <= p <= {infty}) and on the Disc Algebra

This paper is devoted to the following extension of the AAKtheorem. Let (p,q)[2, + ]². Let u: H^p x H^q C bea hankelianbilinear form and n N*. There is a hankelian bilinear formv: H^p x H^q C with rk(v) < n and ||u–v|| Ca_n(U) forsome constant C > 0 depending only on (p,q). Moreover, H^por H^q may be replaced by A in this statement.     

The Dirichlet problem by variational methods

Let ^N be a bounded open set and C( ). Assume that has an extensionC() such that H^–1().Then by the Riesz representation theorem there exists a unique

We show that u+ coincides with the Perron solutionof the Dirichlet problem

This extends recent results by Hildebrandt [Math. Nachr. 278(2005), 141–144] and Simader [Math. Nachr. 279 (2006),415–430], and also gives a possible answer to Hadamard'sobjection against Dirichlet's principle.     

Coincidence and The Colouring of Maps

In [8, 6] it was shown that for each k and n such that 2k >n, there exists a contractible k-dimensional complex Y and acontinuous map : Sⁿ Y without the antipodal coincidence property,that is, (x)(–x) for all x Sⁿ. In this paper it is shownthat for each k and n such that 2k > n, and for each fixed-pointfree homeomorphism f of an n-dimensional paracompact Hausdorffspace X onto itself, there is a contractible k-dimensional complexY and a continuous map :X Y such that (x)(f(x)) for all xX.Various results along these lines are obtained. 1991 MathematicsSubject Classication 55M10, 54C05.     

On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

A problem of Hirst on continued fractions with sequences of partial quotients

Let B denote an infinite sequence of positive integers b₁ <b₂ < ..., and let denote the exponent of convergence ofthe series _{n = 1} 1/b_n; that is, = inf {s 0 : _{n = 1} 1/b_n^s <}. Define E(B) = {x [0, 1]: a_n(x) B (n 1) and a_n(x) asn }. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221–227]proved the inequality dim_H E(B) /2 and conjectured (see ibid.,p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990)p. 278]) that equality holds. In this paper, we give a positiveanswer to this conjecture.     

An Upper Estimate for a Heat Kernel With Neumann Boundary Condition

Using an upper solution we obtain a bound from above for theheat kernel (x,y,t) for a region which is star-shaped withrespect to one of the points, say y. The estimate is for theNeumann problem and holds for short times. The form of the boundis moreover, for x\Y(y), Here Y(y) is a closed subset of R^Nwith measure zero, d(x,y) is the minimum distance between xand y via the boundary :d(x,y) = inf_Z(|x-z| + |y-z|), and f(.,y)is a positive function, continuous away from Y, and equal tounity on .     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

Trivial centralizers for codimension-one attractors

We show that if is a codimension-one hyperbolic attractor fora C^r diffeomorphism f, where 2 r , and f is not Anosov, thenthere is a neighborhood of f in Diff^r(M) and an open and denseset of such that any g has a trivial centralizer on thebasin of attraction for .     

Normal families and omitted functions II

Let k 2 be an integer and let be a family of functions meromorphicon a domain D in , all of whose poles are multiple and whosezeros all have multiplicity at least k + 1. Let h be a functionmeromorphic on D, h 0, . Suppose that for each f , f^(k)(z) h(z) for z D. Then is a normal family on D.     

On A Minimization Problem for Vector Fields in L1

This paper treats the problem of minimizing the norm of vectorfields in L¹ with prescribed divergence. The ridge of . playsan important role in the analysis, and in the case where R²is a polygonal domain, the ridge is thoroughly analysed andsome examples are presented. In the case where Rⁿ is a Lipschitzdomain and the divergence is a finite positive Borel measure,the infimum is calculated, and it is shown that if an extremalexists, then it is of the form ₁ = –Fd, where F is a nonnegativefunction and d(x) is the distance from x to the boundary .Finally, if R² is a polygonal domain and the measure is representedby a nonnegative continuous function, then an explicit expressionfor the extremal is given, and it is proven that this extremalis unique.