The boundary value problem

is studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.

This boundary value problem arises in the search of positive radially symmetric solutions to

where Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

We show that there is a pair of handlebodies H1 and H2 with common boundary F with the properties:

(a) There is no essential simple closed curve in F bounding a disc in both H1 and H2.

(b) Given any positive number, there are essential simple curves Cii = 1, 2 on F, bounding discs in Hi whose distance apart in the Hausdorff topology on F is less than this positive number.

Such an example has consequences for Heegaard splittings and recognising the 3-sphere.

The effect of imposing a certain finiteness condition on the group of central automorphisms of a finite-by-nilpotent group is investigated. In particular it is shown that, if each central automorphism of a finite-by-nilpotent group G has finite order, then the factor group G/Z(G) has finite exponent.

It is shown that the lower radical construction of Tangeman and Kreiling need not terminate at any ordinal.

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.

Among the tricyclic Hamiltonian graphs with a prescribed number of vertices, the unique graph with maximal index is determined. Some subsidiary results are also included.

Under the assumption of the Selberg conjecture I establish by means of the Selberg trace formula the following:

Theorem. Let Γ denote Γ(q) or Γ0(q), q square-free. Let Δq denote the Laplace operator on L2(Γ\H), and let Σq denote its discrete spectrum. Then there exists an absolute positive constant A such that for q≧A

We consider certain exponential box splines E on a three-direction mesh whose exponents satisfy a symmetry condition. It is shown, in particular, that given bounded data on the integer lattice in R2, there is a unique bounded combination of integer translates of E that interpolates the data. When all exponents are zero, this reduces to a result of de Boor, Höllig and Riemenschneider in [2]. Unlike the proof in [2] we use only elementary analysis and do not employ any computer calculations.

We determine all functions f(z) meromorphic in the plane such that f′(z)/f(z) has finite order and f(z) and F(z) have only finitely many zeros, where F(z) = f″(z) + Af(z) for some constant A.

If R is a ring with identity and M is a left R-module then it is well known that the following statements are equivalent:

(1) M is flat.

(2) The functor −⊗M preserves embeddings of right ideals into R.

This paper investigates situations in which the analogous statements are equivalent in the context of S-sets over a monoid S.

Theorem. There exist non-Abelian finitely presented lattice-ordered groups which are totally ordered. This disproves a previous conjecture of the author [5]

A new interpretation of the Baker–Pym theorem is given in terms of operators and applies to a characterization of multipliers on a Banach algebra.

We investigate the equivalence classes of normal subdirect products of a product of free groups Fn1 × … × Fnk under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when k≧3, and list the finite number of classes when k = 2.

In the quotient ring of differential polynomials modulo cubic terms the usual odd order hierarchy of Korteweg–de Vries equations can be supplemented by an even order hierarchy. The full (even and odd) sequence is generated by an Olver recursion operator of order one and any pair has zero bracket in the quotient ring. The even order equations do not possess a Hamiltonian structure and thus their related Rosencrans densities are considered.

In this paper conditions are given that imply that a convolution operator has the same spectrum on all of the spaces Lp(G), 1≦p≦∞.