Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm p_{n}=O(n^{-\alpha})$ with some $\alpha>2/3$, then the existence of bounded solutions of $Ju=\lambda u$ is proved for almost every $\lambda\in(-2,2)$ with the WKB-type asymptotic behavior.

In this paper we characterise the kernel of a little Hankel operator on the Bergman space L2a(D) in terms of the inner divisors, and obtain a characterisation for finite rank little Hankel operators using the invariant subspace theory technique.

Let $\Cal F$ be a holomorphic foliation (possibly with singularities) on a non-singular manifold $M$, and let $V$ be a complex analytic subset of $M$. Usual residue theorems along $V$ in the theory of complex foliations require that $V$ be tangent to the foliation (that is, a union of leaves and singular points of $V$ and $\Cal F$); this is the case for instance for the blow-up of a non-dicritical isolated singularity. In this paper, residue theorems are introduced along subvarieties that are not necessarily tangent to the foliation, including the blow-up of the dicritical situation.

Higher algebraic K-theoretic analogues of Hasse's norm theorem in class field theory are proved. The result gives a partial affirmative answer to a question posed by Bak and Rehmann.

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on $L^{2}(\mathbb{S}^{n-1})\times \ldots\times L^{2}(\mathbb{S}^{n-1})$.

The notions of controlled truncations for operators in the Roe algebras $C^*(X)$ of a coarse space $(X, \cal{E})$ with uniformly locally finite coarse structure, and rank distributions on $(X, \cal{E})$ are introduced. It is shown that the controlled propagation operators in an ideal $I$ of $C^*(X)$ are exactly the controlled truncations of elements in $I$. It follows that the lattice of the ideals of $C^*(X)$ in which controlled propagation operators are dense is isomorphic to the lattice of all rank distributions on $(X, \cal{E})$. If $X$ is a discrete metric space with Yu's property A, then the ideal structure of the Roe algebra $C^*(X)$ is completely determined by the rank distributions on $X$.

The following result is established.

THEOREM. Let G be a periodic, residually finite group with all subgroups sub-normal. Then G is nilpotent.

The well-known groups of Heineken and Mohamed [1] show that the hypothesis of residual finiteness cannot be omitted here, while examples in [5] show that a residually finite group with all subgroups subnormal need not be nilpotent. The proof of the Theorem will use the results of Möhres that a group with all subgroups subnormal is soluble [3] and that a periodic hypercentral group with all subgroups subnormal is nilpotent [4]. Borrowing an idea from [2], the plan is to construct certain subgroups H and K that intersect trivially, and to show that the subnormality of both leads to a contradiction.

Let $H$ be a not necessarily separable Hilbert space, and let $\mathcal{B}(H)$ denote the space of all bounded linear operators on $H$. It is proved that a commutative lattice $\mathcal{D}$ of self-adjoint projections in $H$ that contains $0$ and $I$ is spatially complete if and only if it is a closed subset of $\mathcal{B}(H)$ in the strong operator topology. Some related results are obtained concerning commutative lattice-ordered cones of self-adjoint operators that contain $\mathcal{D}$.

The Fredholm properties of Toeplitz operators $T_a$ on Hardy spaces $H^p$ ($1<p<\infty$) with continuous symbols $a$ are well understood. We consider $T_a$ acting on $H^1$, where the operator is bounded provided that $a$ belongs to the class of symbols given by Janson and Stegenga's result on the pointwise multipliers on $H^1$. A necessary and sufficient condition for $T_a$ to be a Fredholm operator is given when $a$ is continuous and satisfies a mild additional condition (much weaker than Hölder continuity). A formula for the index of $T_a$ is also derived. In addition, we study the case of matrix-valued symbols and Toeplitz operators on $\rm{BMO}_A$.

I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.

Tao has shown that hard summation (summing Fourier-type series by using terms in order of decreasing size of the Fourier coefficients) works for wavelets, but the present author has shown that it fails for classical Fourier series. This paper, which is intended for a general audience, exhibits the underlying ideas in the context of Haar and Walsh series, where many of the proofs simplify.

The topological disc (De Paepe's) \[ P=\{(z^2,\bar{z}^2+\bar{z}^3):|z|\le 1\}\subset {\bb C}^2 \]

is shown here to have non-trivial polynomially convex hull. In fact, the authors show that this holds for all discs of the form $X=\{(z^2,f(\bar{z})):|z|\le r\}$ , where $f$ is holomorphic on $|z|\le r$ , and $f(z)=z^2+a_3z^3+\ldots$ , with all coefficients $a_n$ real, and at least one $a_{2n+1}\ne 0$ .

In this paper the domain in which any two points are joined by finite numbers of arbitrarily small homeomorphisms is studied, and two examples are given.

A theorem of Beurling and Nevanlinna gives a lower bound for the harmonic measure of a compact set in the unit disk in terms of the harmonic measure of a projection of the set. In this paper, a generalization of that result is proved, involving the sum of harmonic measures evaluated at two points.

In this paper, the behaviour of the positive eigenfunction $\phi$ of $\Lp u=\la |u|^{p-2}u$ in $\Om$, $u_{|\p \Om} =0$, $p>1$, is studied near its critical points. Under some convexity and symmetry assumptions on $\Om$, $\phi$ is seen to have a unique critical point at $x=0$; also, the behaviour of both $\phi$ and $\nabla\phi$ is determined nearby. Positive solutions $u$ to some general problems $\Lp u=f(u)$ in $\Om$, $u_{|\p \Om} =0$, are also considered, with some convexity restrictions on $u$.

Sufficient conditions are given for the global existence of solutions to the Rayleigh equation.

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\backslash G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion that involves root systems.

This paper proves that a difference field (E, σ) admits quantifier elimination if and only if E is an algebraically closed field, and σ is an integer power of the Frobenius automorphism.

The distribution on the torus ℝ/ℤ of a set of fractions of the form

(formula here)

is investigated, where q is a large integer, m is the inverse of m modulo q, R(x) is a rational function defined modulo q, and [Uscr ], [Mscr ], [Nscr ] are subsets of {1,…,q}. Under some natural assumptions, it is shown that the set [Rscr ] is uniformly distributed on ℝ/ℤ.

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in ${\rm SL}(2,{\mathbb Z})$. It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example of a holomorphic almost modular form is the logarithm of $\prod_{n=1}^\infty (1-\exp(2\pi\i n^2 z))$. The paper is motivated by the author's previous studies [Int. Math. Res. Not. 39 (2003) 2131–2151] on the connection between almost modular functions and the distribution of the sequence $n^2x$ modulo one.