Questions concerning extensions of polynomials or analytic functions from a Banach space E to its bidual E", and others about reflexivity and related properties on spaces of homogeneous polynomials Pk(E), have recently spurred interest regarding the structure of the bidual of a space of polynomials [4, 14, 22].

In this paper we study the relationship between the bidual of Pk(E) and the space of polynomials over E". We define a map through which elements of the bidual of Pk(E) may be viewed as polynomials over E"

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and study this map to obtain information about Pk(E)". We have tried as far as possible to avoid imposing restricting conditions on E.

We give a hyperkähler analogue of the classification of compact homogeneous Kähler manifolds. Namely we show that, for a compact semisimple Lie group G, the complete G-invariant hyperkähler manifolds which are also locally Gc-homogeneous with respect to one of the complex structures are precisely the hyperkähler metrics of Kronheimer, Biquard and Kovalev on coadjoint semisimple orbits of Gc. As a consequence the only complete hyperkähler metrics which are of cohomogeneity one with respect to a triholomorphic action of compact semisimple Lie group are the flat metric on ℍn and the Calabi metric on T*[Copf]Pn.

For knots in S3 we obtain new criteria for periodicity. We show that the Casson–Gordon invariants of a periodic knot are preserved under the periodic action lifted to the cyclic covers. As an application, we consider a family of knots with the Seifert form of a period 3 knot, and using Casson–Gordon invariants show that knots in this family do not have period 3. We also obtain periodicity criteria in terms of the homology groups of cyclic branched covers of S3.

It is shown that the analogue of the Helson–Reiter theorem for ideals in group algebras of locally compact abelian groups holds for a class of groups including all divisible nilpotent discrete groups.

Let μ and ν be Radon probability measures on ℝ with compact supports. We will consider natural intersections of μ and the image of ν under an isometry f[ratio ]ℝn→ℝn. Our main result gives a lower bound for the packing dimensions of these intersection measures, if μ and ν satisfy some regularity assumptions. Our method, which is originally from [1], is the same as used in [3] where similarities were considered instead of isometries.

The present paper is a continuation of [Ga], [GL1] and [GO]. Using a key lemma we compare two currently existing definitions of finite type invariants of oriented integral homology spheres and show that type 3m invariants in the sense of Ohtsuki [Oh] are included in type m invariants in the sense of the first author [Ga]. This partially answers question 1 of [Ga]. We show that type 3m invariants of integral homology spheres in the sense of Ohtsuki map to type 2m invariants of knots in S3, thus answering question 2 from [Ga].

Given a locally compact group G and a probability measure μ on G it is of interest to know, in various situations, whether there exist divergent sequences {gn} such that {gn μg−1n is relatively compact (see for example [DM3] and [DS]); this phenomenon may be viewed as ‘collapsing’ of the measure. It is the purpose of this note to prove Theorem 1 below and give certain applications to the asymptotic behaviour of concentration functions.

We completely classify all the twistor harmonic (non-minimal) Lagrangian immersions of compact surfaces in the complex projective plane [Copf]ℙ2, i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over [Copf]ℙ2 are harmonic maps.

Examples are given of legendrian links in the manifold of cooriented contact elements of the plane, or equivalently, in the 1-jet space of the circle which are not equivalent via an isotopy of contact diffeomorphisms. These examples have generalizations to linked legendrian spheres in contact manifolds diffeomorphic to ℝn×Sn−1. These links are distinguished by applying the theory of generating functions to contact manifolds.

Let f[ratio ]X→[Copf]ℙp be a finite complex analytic map into complex projective space, with dimX<p. We obtain a result on the equivalence of low homotopy groups between the image of f and [Copf]ℙp, the level of comparison is a function of p, the maximal number of preimages of f and how bad the singularities of X are. This global result is deduced from a generalisation of a theorem of H. Hamm on the local structure of singularities, see [7].

Let X be a smooth projective manifold over the complex number field [Copf] and L a Cartier divisor on X. Then (X, L) is called a polarized (resp. quasi-polarized) manifold if L is ample (resp. nef and big). For a polarized manifold (X, L), Takao Fujita introduced the notion of the Kodaira energy κε(X, L). The Kodaira energy of (X, L) is thought to have some interesting phenomena if κ(X)=−∞ (for example, Spectrum Conjecture which was proposed by T. Fujita). In order to study the Kodaira energy of (X, L), we consider the case in which X has a fibre space, that is, there exist a smooth projective manifold Y with dimX>dimY[ges ]1 and a surjective morphism f[ratio ]X→Y with connected fibres. In this case we introduce the notion of relative Kodaira energy and study some property of it. By using some results of relative Kodaira energy, we study some properties of the Kodaira energy.

We measure the distance of a general shape from the set of shapes having a certain sequence of their vertices collinear and also describe the behaviour of shapes as they move along a geodesic towards this set. This models problems which arise in computer vision and other applications of shape theory.

Some usual and important operations: change of variables, application of Fubini's theorem and derivation with respect to the isolated singularity (in the present work with respect to the origin of the spherical coordinates (r, θ)) are studied for the following singular integral

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where α∈C, Re(α)[les ]−n, j∈ℕ, α∈L1([sum ]n, C) and the symbol fp∫Ω, U means an integration on the set Ω in the finite part sense of Hadamard with respect to the domain configuration U. Moreover, applications to integral operators are outlined.

Let f∈L1(ℝd), and let fˆ be its Fourier integral. We study summability of the l-1 partial integral S(1)R, d(f; x)=∫[mid ]v[mid ][les ]R eiv·x fˆ(v)dv, x∈ℝd; note that the integral ranges over the l1-ball in ℝd centred at the origin with radius R>0. As a central result we prove that for δ[ges ]2d−1 the l-1 Riesz (R, δ) means of the inverse Fourier integral are positive, the lower bound being best possible. Moreover, we will give an l-1 analogue of Schoenberg's modification of Bochner's theorem on positive definite functions on ℝd as well as an extention of Polya's sufficiency condition.

This paper investigates the qualitative behaviour of solutions of the second order equation x¨+f(x, x¯) x¯+g(x)=0. The sufficient conditions for all solutions to be oscillatory are established. The theorems on the existence of a nontrivial periodic solution are proved and the criteria for the origin to be a global centre and globally asymptotically stable are given.

In a recent paper, K. B. Lee introduced the notion of an infra-solvmanifold of type (R). These manifolds are completely determined by their fundamental group Π. Such a Π is a finite extension of a lattice Γ of a solvable Lie group of type (R) and this lattice Γ is called the translational part of Π.

Having fixed an abstract group Π occurring as the fundamental group of an infra-solvmanifold of type (R), it seems to be hard to describe, in a formal algebraic language, which subgroup of Π is the translational part. In his paper Lee formulated a conjecture which would solve this problem, however, we show that this conjecture fails. Nevertheless, by defining a concept of eigenvalues for automorphisms of certain solvable groups (both Lie groups and discrete groups), we are able to prove a new theorem, characterizing completely the translational part of the fundamental group of an infra-solvmanifold of type (R).

The Fourier binest algebra is defined as the intersection of the Volterra nest algebra on L2(ℝ) with its conjugate by the Fourier transform. Despite the absence of nonzero finite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert–Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(−isx2/2) for s>0. This multinest is the reflexive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identification the unitary automorphism group of the algebra is determined as the semi-direct product ℝ2×κℝ for the action κt(λ, μ)=(etλ, e−t μ).

Let X be a compact totally ordered space made into a semigroup by the multiplication xy=max{x, y}. Suppose that there is a continuous regular Borel measure μ on X with supp μ=X. Then the space L1(μ) of μ-integrable functions becomes a Banach algebra when provided with convolution as multiplication. The second dual L1(μ)** therefore has two Arens multiplications, each making it a Banach algebra. We shall always consider L1(μ)** to have the first of these: if F, G∈L1(μ)** and F=w*−limi ϕi, G=w*−limj ψj, where (ϕi), (ψj) are bounded nets in L1(μ), then

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We consider the transmission problem for second-order differential equations with unbounded interfaces. The spaces B and B* are used to prove the limiting absorption principle for the steady-state problem and to establish the resolvent estimate at low frequencies for the steady-state operator. These are then applied in showing the validity of the limiting amplitude principle for such an operator.

A complete system (CS) [Jscr]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr] is a CS of surfaces for M, then any CS equivalent to [Jscr] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds: