Suppose that f(z) is a transcendental entire function and that the Fatou set F(f)≠∅. Setandwhere the supremum supU is taken over all components of F(f). If B1(f)<∞ or B2(f)<∞, then we say F(f) is strongly uniformly bounded or uniformly bounded respectively. We show that, under some conditions, F(f) is (strongly) uniformly bounded.

We provide a new self contained proof of the following result of H. Schubert: If K is a (p,q)-torus knot, then the bridge number of K is min{p, q}.

We give sufficient conditions, as well as some necessary conditions, for the Orlicz–Lorentz space Λϕ,ω to have the weak-star uniform Kadec–Klee property. These results generalize the characterization of the weak-star uniform Kadec–Klee property in the Lorentz space Λω = Lω,1 due to Dilworth and Hsu.

We prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A neutral GL-projection on a pre-symmetric space represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system. Two neutral GL-projections S and T on the pre-symmetric space A* are said to be L-orthogonal if for all elements x in SA* and y in TA*,By studying the algebraic properties of the dual space A of A*, which is a JBW*-triple, it is shown that, provided that the orthogonal neutral GL-projections S and T satisfy a certain geometrical condition, there exists a smallest neutral GL-projection S∨T majorizing both S and T, and that S, T and S∨T form a compatible family.

We characterize projective and injective Banach modules in approximate terms, generalizing thereby a characterization of contractible Banach algebras given by F. Ghahramani and R. J. Loy. As a corollary, we show that each uniformly approximately amenable Banach algebra is amenable. Some applications to homological dimensions of Banach modules and algebras are also given.

The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. In terms of the representation theory of the symplectic group, we give a complete description of cup products of two classes in the first rational cohomology of the Johnson kernel obtained by the rational dual of the second Johnson homomorphism.

We consider random functions formed as sums of pulseswhere Xn are independent random vectors, 0<α<1, and G is an elementary “pulse” or “bump”. Typically such functions have fractal graphs and we find the Hausdorff dimension of these graphs using a novel variant on the potential theoretic method.

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n, 1), n > 0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under free group endomorphisms.

Ghenciu and Lewis introduced the notion of a strong Dunford–Pettis set and used this notion to study the presence or absence of isomorphic copies of c0 in Banach spaces. The authors asserted that they could obtain a fundamental result of J. Elton without resorting to Ramsey theory. While the stated theorems are correct, unfortunately there is a flaw in the proof of the first theorem in the paper which also affects subsequent corollaries and theorems. The difficulty is discussed, and Elton's results are employed to establish a Schauder basis proposition which leads to a quick proof of the theorem in question. Additional results where questions arise are discussed on an individual basis.

If two permutations are strongly conjugate, then their corresponding positive permutation braids (also called simple braids) are conjugate. In this paper we exhibit two conjugate simple braids whose associated permutations are not strongly conjugate. In terms of the grey and black graphs with vertices in the ultra summit set defined in [1], this result can be reformulated by saying that there are ultra summit sets with simple braids (hence with canonical length k = 1) in which the grey graph is not connected. Birman, Gebhardt and González Meneses have given similar examples, but with k ≥ 2 [1]. Recall that the set of simple braids on n strings is a basis of the Hecke algebra Hn. If two simple braids on n strings are conjugate, the associated permutations are centrally conjugate, which means that the coefficients of any central element of Hn corresponding to these simple braids are equal. Working on topological dynamics, Hall and Carvalho [8] have discovered two braids on 12 strings (one and its reverse) which are not conjugate. Since one braid is the reverse of the other, their corresponding permutations are centrally conjugate. For n ≤ 6 we checked that central conjugacy implies conjugacy of the corresponding simple braids.

For a polyhedral domain , and a Hilbert metric nonexpansive map T:Σ→Σ which does not have a fixed point in Σ, we prove that the omega limit set ω(x;T) of any point x ∈ Σ is contained in a convex subset of the boundary ∂Σ. We also identify a class of order-preserving homogeneous of degree one maps on the interior of the standard cone which demonstrate that there are Hilbert metric nonexpansive maps on an open simplex with omega limit sets that can contain any convex subset of the boundary.

In this paper, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion B*-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real C*-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend the main results in [12] (namely, we will give the full versions of the Gelfand–Naimark theorem and the Gelfand theorem for quaternion B*-algebras). On our way to these results, we compare, clarify and unify the term ‘quaternion Hilbert spaces’ in the literatures.

We find two bases for the lattices of the SU(2)-TQFT-theory modules of the torus over given rings of integers. One basis is a variation on the bases defined in [GMW] for the lattices of the SO(3)-TQFT-theory modules of the torus. Moreover, we discuss the quantization functors (Vp, Zp) for p = 1, and p = 2. Then we give concrete bases for the lattices of the modules in the 2-theory. We use the above results to discuss the ideal invariant defined in [FK]. The ideal can be computed for all the 3-manifolds using the 2-theory, and for all 3-manifolds with torus boundary using the SU(2)-TQFT-theory. In fact, we show that this ideal using the SU(2)-TQFT-theory is contained in the product of the ideals using the 2-theory and the SO(3)-TQFT-theory under a certain change of coefficients, and with equality in the case of torus boundary.

We prove a Hurewicz-type theorem for generalized separation: we present a method which allows us to test if for a sequence of Borel sets (Ai)i > ω satisfying there is a sequence (Bi)i < ω of Π0ξ sets such that or not. We also prove an analogous result for generalized reduction. The results of the paper are motivated by a Hurewicz-type theorem of A. Louveau and J. Saint Raymond on ordinary separation of analytic sets.

A coupled cell system is a collection of dynamical systems, or ‘cells’, that are coupled together. The associated coupled cell network is a labelled directed graph that indicates how the cells are coupled, and which cells are equivalent. Golubitsky, Stewart, Pivato and Török have presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of the concept of a ‘balanced equivalence relation’, which depends solely on the network architecture. In their approach the network is assumed to be finite. We prove that the set of all balanced equivalence relations on a network forms a lattice, in the sense of a partially ordered set in which any two elements have a meet and a join. The partial order is defined by refinement. Some aspects of the theory make use of infinite networks, so we work in the category of networks of ‘finite type’, a class that includes all locally finite networks. This context requires some modifications to the standard framework. As partial compensation, the lattice of balanced equivalence relations can then be proved complete. However, the intersection of two balanced equivalence relations need not be balanced, as we show by a simple example, so this lattice is not a sublattice of the lattice of all equivalence relations with its usual operations of meet and join. We discuss the structure of this lattice and computational issues associated with it. In particular, we describe how to determine whether the lattice contains more than the equality relation. As an example, we derive the form of the lattice for a linear chain of identical cells with feedback.

We use the theory of sectorial sesquilinear forms to characterize the closure of the Creation Operator of Quantum Mechanics in the classical set-up. Further, we bring that theory to bear upon the class of unbounded cyclic subnormal operators that admit analytic models; in particular, we provide a sufficient condition for the existence of complete sets of eigenvectors for certain sectorial operators related to unbounded subnormals. The relevant theory is illustrated in the context of a class of analytic models of which the classical Segal–Bargmann space is a prototype. The framework of sectorial sesquilinear forms is also shown to be specially useful for treating questions related to the existence, uniqueness and stability of certain parabolic evolution equations naturally associated with such analytic models.

This paper is devoted to 3-manifolds which admit two distinct Dehn fillings producing a Klein bottle.

Let M be a compact, connected and orientable 3-manifold whose boundary contains a 2-torus T. If M is hyperbolic then only finitely many Dehn fillings along T yield non-hyperbolic manifolds. We consider the situation where two distinct slopes γ1, γ2 produce a Klein bottle. We give an upper bound for the distance Δ(γ1, γ2), between γ1 and γ2. We show that there are exactly four hyperbolic manifolds for which Δ(γ1, γ2) > 4.

We study the spectral gaps of the Schrödinger operatorwhere κ∈(0,2π) and are parameters. Let τ=2π−κ. Suppose that the ratio κ0:=τ/κ is irrational. We denote the jth gap of the spectrum of H by Gj, its length by |Gj|. We obtain a relationship between the asymptotic behaviour of |Gj| as j→∞ and the Diophantine properties of κ0. In particular, we show that if β1+β2=0, thenwhere M(κ0) stands for the Markov constant of κ0.

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so called null convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.