Showing papers in "Journal of The London Mathematical Society-second Series in 1999"

On Restrictions and Extensions of the Besov and Triebel–Lizorkin Spaces with Respect to Lipschitz Domains

Abstract: The restrictions B s pq (Ω) and F s pq (Ω) of the Besov and Triebel–Lizorkin spaces of tempered distributions B s pq (ℝ n ) and F s pq (ℝ n ) to Lipschitz domains Ω⊂ℝ n are studied. For general values of parameters ( s ∈ℝ, p >0, q >0) a ‘universal’ linear bounded extension operator from B s pq (Ω) and F s pq (Ω) into the corresponding spaces on ℝ n is constructed. The construction is based on a new variant of the Calderon reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of B s pq (Ω) and F s pq (Ω) in terms of their values in Ω are also obtained.

Definable Compactness and Definable Subgroups of o-Minimal Groups

Abstract: The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring

Abstract: Two rings A and B are said to be derived Morita equivalent if the derived categories Db(Mod A) and Db(Mod B) are equivalent. If A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T such that the functor T[otimes ]LA−[ratio ]Db(Mod A) → Db(Mod B) is an equivalence. The complex T is called a tilting complex.When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). This group acts naturally on the Grothendieck group Ko(A).It is proved that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables one to compute DPic(A) in these cases.Assume that A is noetherian. Dualizing complexes over A are complexes of bimodules which generalize the commutative definition. It is proved that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. This classification is used to deduce properties of rigid dualizing complexes.Finally finite k-algebras are considered. For the algebra A of upper triangular 2×2 matrices over k, it is proved that t3 = s, where t, s ∈ DPic(A) are the classes of A*:= Homk(A, k) and A[1] respectively. In the appendix, by Elena Kreines, this result is generalized to upper triangular n×n matrices, and it is shown that the relation tn+1 = sn−1 holds.

Strong and Weak Type Inequalities for Some Classical Operators in Orlicz Spaces

Abstract: Inequalities involving classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals of convolutive type have been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [4, 20, 24]. Generalizations of these results to Zygmund spaces are presented in [4]. An exhaustive treatment of the problem of boundedness of such operators in Lorentz and Lorentz–Zygmund spaces is given in [3]. See also [8, 9] for further extensions in the framework of generalized Lorentz–Zygmund spaces.As far as Orlicz spaces are concerned, a characterization of Young functions A having the property that the Hardy–Littlewood maximal operator or the Hilbert and Riesz transforms are of weak or strong type from the Orlicz space LA into itself is known (see for example [13]). In [17, 23] conditions on Young functions A and B are given for the fractional integral operator to be bounded from LA into LB under some restrictions involving the growths and certain monotonicity properties of A and B.The main purpose of this paper is to find necessary and sufficient conditions on general Young functions A and B ensuring that the above-mentioned operators are of weak or strong type from LA into LB. Our results for (fractional) maximal operators are presented in Section 2, while Section 3 deals with fractional and singular integrals. In particular, we re-cover a result concerning the standard Hardy–Littlewood maximal operator which has recently been proved in [2, 11, 12]. Finally, in Section 4, the resolvent operator of some differential problems is taken into account and a priori bounds for Orlicz norms of solutions to elliptic boundary value problems in terms of Orlicz norms of the data are established. Let us mention that part of the results of the present paper were announced in [6].

INVERTIBLE SPECTRA IN THE E ( n )-LOCAL STABLE HOMOTOPY CATEGORY

Abstract: Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with C. There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category S. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere S [HMS94, Str92]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE . The most well-known case is E = K(n), the nth Morava K-theory, considered in [HMS94]. In this paper we study the case E = E(n), where E(n) is the Johnson-Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by L. Our main theorem is the following result.

Cellular algebras: inflations and morita equivalences

Abstract: Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra.The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper.The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

Multi‐Peak Solutions for a Wide Class of Singular Perturbation Problems

Abstract: This paper concerns a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. The corresponding elliptic equations in a bounded domain without any symmetry assumptions are studied. It is assumed that the mean curvature of the boundary has M isolated, non-degenerate critical points. Then it is shown that for any positive integer M [les ] M there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. The method is based on Lyapunov–Schmidt reduction.

A minimax principle for the eigenvalues in spectral gaps

Abstract: A minimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded self-adjoint operator. It allows the nth eigenvalue of the Dirac operator with Coulomb potential from below to be bound by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application it is shown that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schrodinger operator with the same potential.

Acyclic Colourings of Planar Graphs with Large Girth

Abstract: A proper vertex-colouring of a graph is acyclic if there are no 2-coloured cycles. It is known that every planar graph is acyclically 5-colourable, and that there are planar graphs with acyclic chromatic number v a fl 5 and girth g fl 4. It is proved here that a planar graph satisfies v a % 4i fg & 5 and v a % 3i fg & 7.

ONE-DIMENSIONAL p -ADIC SUBANALYTIC SETS

Abstract: Hence the semialgebraic complexity of the fibers S x ̄2y `Z p : (x, y) `S ́ remains bounded as the parameter x ranges over Zm p . The proof of [2, 3.32] depends on the compactness of Z p in a way that makes it hard to see how it could be adapted to give the stronger Theorem A. Instead we take a model-theoretic approach and work in nonstandard models to prove the Pminimality of the analytic theory of Q p in the sense of [5], which implies Theorem A. To explain this we now introduce some notation used throughout the paper. Let K be a field complete with respect to a nontrivial non-Archimedean absolute value r r, for instance K ̄Q p ; let R ̄2x `K : rxr% 1 ́ denote its valuation ring. Let Y ̄ (Y \" ,...,Y n ) be a tuple of distinct indeterminates, and let K©Ya be the ring of power series f(Y ) ̄3ν aν Y ν `K[[Y ]] such that raνr! 0 as rνr!¢ (the Tate ring in Y over K ). Here ν ̄ (ν \" ,..., ν n ) ranges over multi-indices in Nn and rνr ̄ ν \" ­...­ν n . Given such a power series f(Y ) we define a function f :Kn MNK by

Characterisation of Graphs which Underlie Regular Maps on Closed Surfaces

Abstract: It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v . Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that H v is the cyclic subgroup of index 2 in G v . An analogous result is proved for orientably-regular embeddings.

Lacunary polynomials, multiple blocking sets and Baer subplanes

Abstract: New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Redei's results in the case when the derivative of the polynomial vanishes.

A Strong Law for the Largest Nearest‐Neighbour Link between Random Points

Abstract: Suppose that X1, X2, X3, … are independent random points in Rd with common density f, having compact support Ω with smooth boundary ∂Ω, with f[mid ]Ω continuous. Let Rni, k denote the distance from Xi to its kth nearest neighbour amongst the first n points, and let Mn, k = maxi[les ]nRni, k. Let θ denote the volume of the unit ball. Then as n → ∞,formula hereIf instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then nθMdn, k/ log n → (minKf)−1, almost surely.

Rationality of Moduli Spaces of Parabolic Bundles

Abstract: The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows that if rank and degree are coprime, the moduli space of vector bundles is stably rational, and the bound obtained on the level is strong enough to conclude rationality in many cases.

(2,3)-generation of exceptional groups

Abstract: We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)-generated, that is, a factor group of the modular group PSL 2 (ℤ). This completes the study of (2,3)-generation of groups of Lie type. Second, we complete the proof that groups of type E 7 and E 8 over fields of odd characteristic occur as Galois groups of geometric extensions of ℚ ab ( t ), where ℚ ab denotes the maximal Abelian extension field of ℚ. Finally, we show that all finite simple exceptional groups of Lie type have a pair of strongly orthogonal classes. The methods of proof in all three cases are very similar and require the Lusztig theory of characters of reductive groups over finite fields as well as the classification of finite simple groups.

On the affine Temperley-Lieb algebras.

Abstract: The paper describes the cell structure of the affine Temperley–Lieb algebra with respect to a monomial basis. A diagram calculus is constructed for this algebra.

Symmetric Polynomials on Rearrangement‐Invariant Function Spaces

Abstract: The exact representation of symmetric polynomials on Banach spaces with symmetric basis and also on separable rearrangement-invariant function spaces over [0, 1] and [0, infinity) is given. As a consequence of this representation it is obtained that, among these spaces, l(2n), L-2n[0, 1], L-2n[0, infinity) and L-2n[0, infinity) boolean AND L-2n[0, infinity) where n, nl are both integers are the only spaces that admit separating polynomials.

On Certain Exponential Sums and the Distribution of Diffie–Hellman Triples

Abstract: Let g be a primitive root modulo a prime p. It is proved that the triples (gx, gy, gxy), x, y = 1, …, p−1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let e>0 be fixed. Then uniformly for any integers a, b, c with gcd(a, b, c, p) = 1. Incomplete sums are estimated as well. The question is motivated by the assumption, often made in cryptography, that the triples (gx, gy, gxy) cannot be distinguished from totally random triples in feasible computation time. The results imply that this is in any case true for a constant fraction of the most significant bits, and for a constant fraction of the least significant bits.

The ubiquity of Thompson's group F in groups of piecewise linear homeomorphisms of the unit interval

Abstract: We show that Thompson's group F occurs with great frequency in the group of PL homeomorphisms of the unit interval.

On Cyclic Groups of Automorphisms of Riemann Surfaces

Abstract: The question of extendability of the action of a cyclic group of automorphisms of a compact Riemann surface is considered. Particular attention is paid to those cases corresponding to Singerman's list of Fuchsian groups which are not nitely-maximal, and more generally to cases involving a Fuchsian triangle group. The results provide partial answers to the question of which cyclic groups are the full automorphism group of some Riemann surface of given genus g > 1.

Dimension and Fundamental Groups of Asymptotic Cones

Abstract: Asymptotic cones were first used by Gromov in [ 6 ], where he constructed limit spaces of nilpotent groups in order to prove that groups with polynomial growth are virtually nilpotent. Gromov does not use the term asymptotic cone , which was introduced later by Van den Dries and Wilkie in [ 12 ], when they gave a nonstandard interpretation of Gromov's results. Ultrafilters appear in [ 12 ] for the first time in this context. Later, Gromov gave an extensive treatment of asymptotic cones in [ 7 ]. Since then several authors have used asymptotic cones to obtain interesting results, for instance, in identifying quasi-isometry classes of 3-manifolds [ 8 ] or relating asymptotic cones with Dehn functions of finitely presented groups (see [ 2 ] and [ 11 ], the results of which are stated in Section 2). The purpose of this paper is to develop some of the results stated in [ 7 ], in particular those describing the asymptotic cone of the Baumslag–Solitar groups and of Sol. According to [ 2 ], these spaces are not simply connected, since their Dehn functions are exponential, so our primary goal is to study their fundamental groups. It will be proved that these fundamental groups are uncountable and nonfree (Section 9), by constructing subgroups isomorphic to the fundamental group of the Hawaiian earring. These subgroups are constructed by finding subspaces in the asymptotic cones which are homotopically equivalent to the Hawaiian earring, and which induce injections in the fundamental group level (Section 8). Crucial to the proof of these facts is the computation of the covering dimension of these asymptotic cones (Section 7), which is done using a more general theorem on dimensions of spaces which admit certain maps into well-known spaces (Section 6).