The paper considers second-order, strongly elliptic, operators H with complex almost-periodic coefficients in divergence form on Rd. First, it is proved that the corresponding heat kernel is Hölder continuous and Gaussian bounds are derived with the correct small and large time asymptotic behaviour on the kernel and its Hölder derivatives. Secondly, it is established that the kernel has a variety of properties of almost-periodicity. Thirdly, it is demonstrated that the kernel of the homogenization Ĥ of H is the leading term in the asymptotic expansion of t [map ] Kt.

We consider the Dipper–James q-Schur algebra [Sscr ]q(n, r)k, defined over a field k and with parameter q ≠ 0. An understanding of the representation theory of this algebra is of considerable interest in the representation theory of finite groups of Lie type and quantum groups; see, for example, [6] and [11]. It is known that [Sscr ]q(n, r)k is a semisimple algebra if q is not a root of unity. Much more interesting is the case when [Sscr ]q(n, r)k is not semisimple. Then we have a corresponding decomposition matrix which records the multiplicities of the simple modules in certain ‘standard modules’ (or ‘Weyl modules’). A major unsolved problem is the explicit determination of these decomposition matrices.

Let Σg be a compact Riemann surface of genus g, and G = SU(n). The central element c = diag(e2πid/n, …, e2πid/n) for d coprime to n is introduced. The Verlinde formula is proved for the Riemann–Roch number of a line bundle over the moduli space [Mscr ]g, 1(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component, for which the loop around the boundary is constrained to lie in the conjugacy class of cexp(Λ) (for Λ ∈ t+), and also for the moduli space [Mscr ]g, b(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with s + 1 boundary components for which the loop around the 0th boundary component is sent to the central element c and the loop around the jth boundary component is constrained to lie in the conjugacy class of exp(Λ(j)) for Λ(j) ∈ t+. The proof is valid for Λ(j) in suitable neighbourhoods of 0.

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map

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is continuous, and a linear map D[ratio ]A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous.

The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.

Symplectic groups are well known as the groups of isometries of a vector space with a non-singular bilinear alternating form. These notions can be extended by replacing the vector space by a module over a ring R, but if R is non-commutative, it will also have to have an involution. We shall here be concerned with symplectic groups over free associative algebras (with a suitably defined involution). It is known that the general linear group GLn over the free algebra is generated by the set of all elementary and diagonal matrices (see [1, Proposition 2.8.2, p. 124]). Our object here is to prove that the symplectic group over the free algebra is generated by the set of all elementary symplectic matrices. For the lowest order this result was obtained in [4]; the general case is rather more involved. It makes use of the notion of transduction (see [1, 2.4, p. 105]). When there is only a single variable over a field, the free algebra reduces to the polynomial ring and the weak algorithm becomes the familiar division algorithm. In that case the result has been proved in [3, Anhang 5].

Symplectic groups are well known as the groups of isometries of a vector space with a non-singular bilinear alternating form. These notions can be extended by replacing the vector space by a module over a ring R, but if R is non-commutative, it will also have to have an involution. We shall here be concerned with symplectic groups over free associative algebras (with a suitably defined involution). It is known that the general linear group GLn over the free algebra is generated by the set of all elementary and diagonal matrices (see [1, Proposition 2.8.2, p. 124]). Our object here is to prove that the symplectic group over the free algebra is generated by the set of all elementary symplectic matrices. For the lowest order this result was obtained in [4]; the general case is rather more involved. It makes use of the notion of transduction (see [1, 2.4, p. 105]). When there is only a single variable over a field, the free algebra reduces to the polynomial ring and the weak algorithm becomes the familiar division algorithm. In that case the result has been proved in [3, Anhang 5].

The classification of finite groups acting freely on a homology 3-sphere is still incomplete. However it is known that these groups have periodic cohomology and groups with periodic cohomology are classified by the Suzuki–Zassenhaus theorem. The paper considers, more generally, finite groups which may act on a homology 3-sphere possibly with fixed points and the natural generalization of the Suzuki–Zassenhaus theorem to this case is found. According to the Suzuki–Zassenhaus theorem a group with periodic cohomology is either solvable or has a unique composition factor which is not cyclic, isomorphic to PSL(2, q) for q an odd prime. The main point of the work is that this fact is still true in the more general situation.

Let F be a non-Archimedean local field and G be the group of F-points of a connected reductive group defined over F. Let M be an F-Levi subgroup of G and P = MN be a parabolic subgroup with Levi decomposition P = MN. Jacquet, or truncated, restriction gives a functor from the category of smooth representations of G to that of M. The main result describes this functor in terms of homomorphisms and localizations of Hecke algebras attached to certain compact open subgroups of G and M. This leads to new and straightforward proofs of some fundamental results. The first computes the smooth dual of a Jacquet module of a smooth representation of G, generalizing the corresponding result for admissible representations due to Harish-Chandra and Casselman. The second identifies the co-adjoint of the Jacquet functor relative to P as the induction functor relative to the M-opposite of P, an unpublished result of J.-N. Bernstein.

The paper studies the connective complex K-theoretic Euler class of a certain bundle associated to a Euclidean immersion of the lens space L(2k)2n+1> to show that this manifold cannot be immersed in ℝ4n−2α(n) if k [ges ] α(n). The non-immersion is best possible in many cases. This suggests a close relationship between the immersion problem for complex projective spaces and that for ‘high’ 2-torsion lens spaces.

Let F be a non-Archimedean local field and G be the group of F-points of a connected reductive group defined over F. Let M be an F-Levi subgroup of G and P = MN be a parabolic subgroup with Levi decomposition P = MN. Jacquet, or truncated, restriction gives a functor from the category of smooth representations of G to that of M. The main result describes this functor in terms of homomorphisms and localizations of Hecke algebras attached to certain compact open subgroups of G and M. This leads to new and straightforward proofs of some fundamental results. The first computes the smooth dual of a Jacquet module of a smooth representation of G, generalizing the corresponding result for admissible representations due to Harish-Chandra and Casselman. The second identifies the co-adjoint of the Jacquet functor relative to P as the induction functor relative to the M-opposite of P, an unpublished result of J.-N. Bernstein.

The paper obtains the limiting behaviour of the expectations of the zeros of holomorphic sections of the canonical line bundles over a tower of covers of a compact complex manifold.

The paper obtains the limiting behaviour of the expectations of the zeros of holomorphic sections of the canonical line bundles over a tower of covers of a compact complex manifold.

The paper considers stationary critical points of the heat flow in sphere SN and in hyperbolic space ℍN, and proves several results corresponding to those in Euclidean space ℝN which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x0, is fixed and initial-boundary value problems are considered for the heat equation on bounded domains containing x0. It is shown that for any initial data satisfying the balance law with respect to x0 (or being centrosymmetric with respect to x0) the corresponding solution always has x0 as a stationary critical point, if and only if the domain is a geodesic ball centred at x0 (or is centrosymmetric with respect to x0, respectively).

The paper considers stationary critical points of the heat flow in sphere SN and in hyperbolic space ℍN, and proves several results corresponding to those in Euclidean space SN which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x0, is fixed and initial-boundary value problems are considered for the heat equation on bounded domains containing x0. It is shown that for any initial data satisfying the balance law with respect to x0 (or being centrosymmetric with respect to x0) the corresponding solution always has x0 as a stationary critical point, if and only if the domain is a geodesic ball centred at x0 (or is centrosymmetric with respect to x0, respectively).

The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system.

The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system.

Let E(ℤ) = {einx}n∈ℤ denote the trigonometrical exponential system. It is well known that E(ℤ) forms an orthogonal basis in the space L2(0, 2π). In 1964, H. Landau discovered that the trigonometrical system has the following property: certain small perturbations of E(ℤ) yield exponential systems which are complete in L2 on any finite union of 2π-periodic translations of any interval (ε, 2π−ε), 0 < ε < π.

Let E(ℤ) = {einx}n∈ℤ: denote the trigonometrical exponential system. It is well known that E(ℤ) forms an orthogonal basis in the space L2(0, 2π). In 1964, H. Landau discovered that the trigonometrical system has the following property: certain small perturbations of E(ℤ) yield exponential systems which are complete in L2 on any finite union of 2π-periodic translations of any interval (ε, 2π−ε), 0 < ε < π.

The derivation problem for a locally compact group G is to decide whether for each derivation D from L1(G) into L1(G) there is a bounded measure μ∈M(G) with D(a) = aμ−μa (a ∈ L1(G)). In this paper we obtain an affirmative answer for the case of connected groups. To explain the contents of this paper we give an equivalent formulation of the problem. Suppose that the group G acts as a group of homeomorphisms of the locally compact space X. Related to this there is an action of G on M(X). A bounded crossed homomorphism from G to M(X) is a map Φ with bounded range and satisfying Φ(gh) = gΦ(h)+Φ(g) (g, h ∈ G). The problem for bounded crossed homomorphisms is to decide if for each such Φ there is an element μ of M(X) with Φ(g) = gμ− μ (g ∈ G). The derivation problem is equivalent to this bounded crossed homomorphism problem for the special case X = G where G acts on X by conjugation (together with some mild continuity hypotheses about the map Φ:G→M(X) which are often automatically satisfied). The bounded crossed homomorphism problem always has a positive solution if G is amenable and a closely related calculation shows that in solving the bounded crossed homomorphism problem we need only solve it for functions Φ which are zero on H where H is a given amenable subgroup of G. It can happen that this condition of being zero on H forces Φ to be zero even when H is a comparatively small subgroup of G. If h is an element of G such that ‘hnx → ∞’ as n → ∞ for all x ∈ X then for any two measures μ and ν, for large values of n, μ and hnν have little overlap so ∥μ + hNν∥ → ∥μ∥ + ∥ν∥. Thus if H is the subgroup generated by h, for any g ∈ G

The derivation problem for a locally compact group G is to decide whether for each derivation D from L1(G) into L1(G) there is a bounded measure μ∈M(G) with D(a) = aμ−μa (a ∈ L1(G)). In this paper we obtain an affirmative answer for the case of connected groups. To explain the contents of this paper we give an equivalent formulation of the problem. Suppose that the group G acts as a group of homeomorphisms of the locally compact space X. Related to this there is an action of G on M(X). A bounded crossed homomorphism from G to M(X) is a map Φ with bounded range and satisfying Φ(gh) = gΦ(h)+Φ(g) (g, h ∈ G). The problem for bounded crossed homomorphisms is to decide if for each such Φ there is an element μ of M(X) with Φ(g) = gμ− μ (g ∈ G). The derivation problem is equivalent to this bounded crossed homomorphism problem for the special case X = G where G acts on X by conjugation (together with some mild continuity hypotheses about the map Φ:G→M(X) which are often automatically satisfied). The bounded crossed homomorphism problem always has a positive solution if G is amenable and a closely related calculation shows that in solving the bounded crossed homomorphism problem we need only solve it for functions Φ which are zero on H where H is a given amenable subgroup of G. It can happen that this condition of being zero on H forces Φ to be zero even when H is a comparatively small subgroup of G. If h is an element of G such that ‘hnx → ∞’ as n → ∞ for all x ∈ X then for any two measures μ and ν, for large values of n, μ and hnν have little overlap so ∥μ + hnν∥ → ∥μ∥ + ∥ν∥. Thus if H is the subgroup generated by h, for any g ∈ G

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