Consider the Fredholm equation of the second kind

where

and Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

Bonsall and Tomiuk have shown, in (3), the connection between the local compactness of a monothetic semi-algebra and the spectral properties of a generating element. This theme was developed, in (4), to give a complete characterisation of prime, strict locally compact monothetic semi-algebras in terms of the spectrum of a generator (Theorem A). Here we extend this result to the case of a semi-simple locally compact monothetic semi-algebra (Theorem B).

We consider a subclass of the Dirichlet series studied by Chandrasekharan and Narasimhan in (1). Our objective is to generalize some identities due to Landau (3) concerning r2(n), the number of representations of the positive integer n as the sum of 2 squares. We shall also give a slight extension of Theorem III in (1).

In (4) Vala proves a generalization of Schauder's theorem (3) on the compactness of the adjoint of a compact linear operator. The particular case of Vala's result that we shall be concerned with is as follows. Let t1 and t2 be non-zero bounded linear operators on the Banach spaces Y and X respectively, and denote by 1T2 the operator on B(X, Y) defined by

Let v = (a1 …, an) be a real n-tuple and be the numbers a1 …, an arranged in decreasing order. Let denote the sum of m greatest components of v and the sum of m smallest components of v, i.e.,

The following theorems in ring theory are well-known:

1. Let R be a ring. If e is a unique left identity, then e is also a right identity.

2. If R is a ring with more than one element such that aR = R for every nonzero element a ε R, then R is a division ring.

3. A ring R with identity e ≠ 0 is a division ring if and only if it has no proper right ideals.

Let B be a complex Banach algebra with an identity 1 and an involution x→x*. Kadison (1) has shown that, if B is a B*-algebra, [the set of extreme points of its unit ball coincides with the set of elements x of B for which

An n × n matrix An = (aij) is tri-diagonal if aij = 0 for |i−j|≧2. The latent roots of such matrices may be conveniently studied by forming the sequence of polynomials Ψk(λ)=|λI−Ak|, where Ak is the principal submatrix of Ak+1 obtained by deleting the last row and column of Ak+1, and then observing that these polynomials satisfy the following recurrence relation:

When studying the solutions of elliptic boundary value problems in a bounded, smoothly bounded domain D⊂Rn we often encounter the formula

where u(x)∈C2(D)∩C′() is a solution of the second order self-adjoint elliptic equation

and denotes differentiation along the inward normal to ∂D at x∈∂D.

Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra of projections on X such that has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in is EX the direct sum of the cyclic subspaces clm {Ex1:E∈} and clm {Ex2:E∈}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant ′ of is equal to A(), the algebra of operators generated by in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that has the first property.

In a paper (1) by Harding there is a tacit invitation to seek the connection between the following two problems:

(i) Find the number, ηk(N), of regions into which a k-dimensional space is partitioned by a set of N (k- l)-dimensional hyperplanes.

(ii) Find the number, vk(N), of distinct partitions of a given set of N points in a k-dimensional space E that can be induced by (k- 1)-dimensional hyperplanes.