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.     

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.     

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.     

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).     

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.     

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     

Identities for Combinatorial Extremal Theory

Let be the set {1, 2, ..., n}, and let Ø be the emptyset. Let G be the family of all non-empty sets of subsets of. For A G and X , put An important discovery is the following. 1991 Mathematics SubjectClassification 05D05.     

Finite groups in Stone-Cech compactifications

Let be an infinite cardinal and let G = ₂. Now let β Gbe the Stone–ech compactification of G as a discrete semigroup,and let =_<c_{β G} {xG\{0}:minsupp (x)}. We show that thesemigroup contains no nontrivial finite group.     

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.     

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.     

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.     

Spaces Universal under Closed Embeddings for Finite-Dimensional Complete Metric Spaces

We shall prove that for every natural number n and every cardinalnumber there exists an n-dimensional complete metric spaceX_n, of weight such that every n-dimensional complete metricspace of weight is embeddable in X_n, as a closed subset.     

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.     

A Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S-Adic Von Koch Snowflake

Let D be an open set in Euclidean space R^m with boundary D,and let :D[0, ) be a bounded, measurable function. Let u:DDx[0,)[0, ) be the unique weak solution of the heat equation [formula] with initial condition [formula] and with inhomogeneous Dirichlet boundary condition [formula] Then u(x; t) represents the temperature at a point xD at timet if D has initial temperature 0, while the temperature at apoint xD is kept fixed at (x) for all t>0. We define thetotal heat content (or energy) in D at time t by [formula] In this paper we wish to examine the effect of imposing additionalcooling on some subset C on both u and E_D. 1991 MathematicsSubject Classification 35K05, 60J65, 28A80.     

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Let = 2cos (/5) and let []. Denote the normaliser ofG₀() of the Hecke group G₅ in PSL₂() by N(G₀()). Then N(G₀())= G₀(/h), where h is the largest divisor of 4 such that h² divides. Further, N(G₀())/G₀() is either 1 (if h = 1), ₂ x ₂ (if h= 2) or ₄ x ₄ (if h = 4).     

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.     

An Example on Sobolev Space Approximation

For each d2 we construct a connected open set R^d such that = int (clos()), and for each k 1 and each p [1, ), the subsetW^k, () fails to be dense in the Sobolev space W^{k, p}(), in thenorm of W^{k, p}(). 1991 Mathematics Subject Classification 46E35,46F05.     

On the Distribution of Denominators in Sylvester Expansions

For any x (0, 1], let the series be the Sylvester expansion of x. Galambos has shown that theLebesgue measure of the set [formula] is 1 when = e, the base of the natural logarithm. This paperprovides a proof that for any 1, A() has Hausdorff dimension1 when e. 2000 Mathematics Subject Classification 11K55 (primary),28A80 (secondary).     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.