We prove the existence of unbounded open subsets S of the complex plane with the following property. If f is a function transcendental and meromorphic in the plane, the poles of which have positive Nevanlinna deficiency, then f takes every finite value, with at most one exception, infinitely often in the complement of S.

We prove the existence of a (unique) S1-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kähler manifold X, extending the metric on X.

Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ = sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write l∈([Ufr ]).

In this short note, we exhibit examples of amenable (resp. weakly amenable) Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable), thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to [4, 9, 10] for background on amenability and weak amenability. For basic information about the Arens product in the second dual of a Banach algebra the reader can consult [5, 6].

In this paper finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of each type for all pure braid groups.

In this paper we prove that R7 admits smooth periodic maps with no fixed points for every period that is not a prime power. Results of P. A. Smith show that such examples do not exist in any lower dimensions.

In [4], it is proved that there exists a ‘unique’ adapted Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c) of constant sectional curvature 4c associated with each twisted product decomposition of a real-space-form if its twistor form is twisted closed. Conversely, if L: Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the immersion L is determined by the corresponding adapted Lagrangian isometric immersion of the twisted product decomposition. It is natural to ask the explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M˜n(4c) for each case: c = 0, c > 0 and c < 0.

Some aspects of the geometry and the dynamics of generalized Chaplygin systems are investigated. First, two different but complementary approaches to the construction of the reduced dynamics are reviewed: a symplectic approach and an approach based on the theory of affine connections. Both are mutually compared and further completed. Next, a necessary and sufficient condition is derived for the existence of an invariant measure for the reduced dynamics of generalized Chaplygin systems of mechanical type. A simple example is then constructed of a generalized Chaplygin system which does not verify this condition, thereby answering in the negative a question raised by Koiller.

In this paper, we will define the reduced cross-sectional C*-algebras of C*-algebraic bundles over locally compact groups and show that if a C*-algebraic bundle has the approximation property (defined similarly as in the discrete case), then the full cross-sectional C*-algebra and the reduced one coincide. Moreover, if a semi-direct product bundle has the approximation property and the underlying C*-algebra is nuclear, then the cross-sectional C*-algebra is also nuclear. We will also compare the approximation property with the amenability of Anantharaman-Delaroche in the case of discrete groups.

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.

Solutions to the Schrödinger, heat and stochastic Schrödinger equation with rather general potentials are represented, both in x- and p-representations, as integrals over the path space with respect to σ-finite measures. In the case of x-representation, the corresponding measure is concentrated on the Cameron–Martin Hilbert space of curves with L2-integrable derivatives. The case of the Schrödinger equation is treated by means of a regularization based on the introduction of either complex times or continuous non-demolition observations.

With the cancellation property of the bounded kernel, we prove that the generalized Marcinkiewicz integral operator is bounded on L2 (ℝn×ℝm) for all dimensions n, m.

Refined expressions are given for the error terms in the asymptotic expansion formulas for double zeta and double gamma functions, proved in the author's former paper [2]. Some inaccurate claims in [2] are corrected.

The paper concerns the identification of projective closed ideals of C*-algebras. We prove that, if a C*-algebra has the property that every closed left ideal is projective, then the same is true for all its commutative C*-subalgebras. Further, we say a Banach algebra A is hereditarily projective if every closed left ideal of A is projective. As a corollary of the stated result we show that no infinite-dimensional AW*-algebra is hereditarily projective. We also prove that, for a commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space, the following conditions are equivalent: (i) A is separable; and (ii) the C*-tensor product A [otimes ]minA is hereditarily projective. Howerever, there is a non-separable, hereditarily projective, commutative C*-algebra A contained in [Bscr ](H), where H is a separable Hilbert space.

Suppose that Ω is a compact Hausdorff space with a preorder [les ] whose graph is closed, and let Ω∘ be an open subset of Ω. This paper provides conditions sufficient to allow every increasing bounded real continuous function on Ω∘ to be extended to an increasing real continuous function on Ω. These conditions are: (i) that Ω is a Stonian space, and (ii) that the set C↑(Ω, [les ]) of increasing real continuous functions on Ω is a regular Dedekind complete sublattice of C(Ω). Under these conditions it is also shown that C↑(Ω, [les ]) is generated by idempotents, and an extension theorem for idempotents is proved.

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

Noncommutative Radon–Nikodým theorems have attracted a great deal of attention in the theory of operator algebras. There has been considerable work on non-commutative Radon–Nikodým theorems not only for C*-algebras but also for algebras of unbounded operators [3, 5, 8, 9, 12, 13]. In this paper, we will develop a Radon–Nikodým type theorem for completely bounded and completely positive invariant multilinear maps.

The concept of matricial order has turned out to be very important to understand the infinite-dimensional non-commutative structure of operator algebras. As the natural ordering attached to this structure, completely positive maps and completely bounded maps have been studied extensively. Results concerning completely bounded maps have many applications: cohomology of operator algebras, multipliers on group algebras, dilation theory, similarity theory, free product representations, and abstract characterizations of operator algebras.

We use a Volterra integral equation to derive lower bounds for the local maxima of |un(θ)| = (sin θ)λ|P(λ)n(cos θ)|, where P(λ)n (·) is the nth ultraspherical polynomial with parameter 0 < λ < 1. Moreover, inequalities for the critical points and inequalities between the extrema of un(θ) and un−1(θ) are obtained. The results are applied to show that, for every λ, the maxima of (n+λ)1−λ|un(θ)| form a strictly increasing sequence. This establishes a conjecture of Lorch [12, 13].

We introduce the concepts of local spectral and walk dimension for fractals. For a class of finitely ramified fractals we show that, if the Laplace operator on the fractal is defined with respect to a multifractal measure, then both the local spectral and walk dimensions will have associated non-trivial multifractal spectra. The multifractal spectra for both dimensions can be calculated and are shown to be transformations of the original underlying multifractal spectrum for the measure, but with respect to the effective resistance metric.