1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficients

Parseval's formula gives

If M is a mathematical system and End M is the set of singular endomorphisms of M, then End M forms a semigroup under composition of mappings. A number of papers have been written to determine the subsemigroup SM of EndM generated by the idempotents EM of End M for different systems M. The first of these was by J. M. Howie [4]; here the case of M being an unstructured set X was considered. Howie showed that if X is finite, then End X = Sx.

In [4] we have shown that any two semi-simple weighted convolution algebras L1(ω1) and L1(ω2) are isomorphic. In this paper, given any two radical weighted convolution algebras L1(ω1) and L1(ω2) we find necessary and sufficient conditions, in terms of ω1 and ω2, for L1(ω1) and L1(ω2) to be isomorphic.

In this paper, I introduce a class of arrangements called arrays of strength d and discuss methods of constructing them with the help of finite geometrical configurations and algebraic groups involving elements of a Galois field. The definitions of arrays of strength d and other configurations that are used are given below.

These equations are solved by the ordinary methods applicable to linear equations with constant coefficients.

There is a group of Tauberian theorems of which the simplest isone due to K. Ananda Rau [Theorem 2 of the paper numbered 1 inthe list of references at the end of the note]. More complicatedtheorems of the same group are discussed in a paper by S.Minakshisundaram and myself to be published by the LondonMathematical Society [4].

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

In previous papers [3, 4] the author has discussed the symmetric generalised Erdélyi–Kober operators of fractional integration defined by

where α>0, γ≧0 and the operators ℑiγ(η,α) and defined as in equations (1) and (2) respectively but with Jα−1, the Bessel function of the first kind replaced by Iα−1, the modified Bessel function of the first kind.

Let H be any group. We call a cardinal number r the rank r(H) of H if H can be generated by a generating system X with cardinal number r but not by a generating system Y with cardinal number s less than r. Let r(H) be the rank of H.

We call a generating system X of H a minimal generating system (M.G.S.) of H if X has the cardinal number r(H).

In this note we prove the following.

Given finite sets of cyclic words $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ in a finitely generated free group $F$ and two finite groups $A$ and $B$ of outer automorphisms of $F$, we produce an algorithm to decide whether there is an automorphism which conjugates $A$ to $B$ and takes $u_i$ to $v_i$ for each $i$. If $A$ and $B$ are trivial, this is the classic algorithm due to Whitehead. We use this algorithm together with Cohen and Lustig’s solution to the conjugacy problem for Dehn twist automorphisms of $F$ to solve the conjugacy problem for outer automorphisms which have a power which is a Dehn twist. This settles the conjugacy problem for all automorphisms of $F$ which have linear growth.

AMS 2000 Mathematics subject classification: Primary 20F32. Secondary 57M07

I. Soit ABC un triangle inscrit dans une circonférence donnée, qui a pour centre le point O, et pour rayon R la quantité OA. Si d'un point M pris sur la circonférence on abaisse les perpendiculaires ML, MN, MR sur les côtés BC, CA, AB du triangle, les pieds L, N, R de ces perpendiculaires sont situés sur une même droite RN, à laquelle on a donné le nom de pédale du point M; le point M est le point directeur de la pédale RN.

When the function f(u) is of “bistable type’, i.e. has two zeros h̲ and h+ at which f' is negative and (for simplicity) has only one other zero between them, then the constant functions u = h± are L∞-stable solutions of the nonlinear diffusion equation

In addition, there are travelling wave solutions u+(x, t) and u̲(x, t) which, if

connect h+ to h̲ in the sense that

the convergence being uniform on bounded x-intervals. These solutions are of the form

where U(z) is a monotone function (the wave's profile), U(±∞) = h±, and the velocity c is a specific positive number depending on the function f.

We have previously studied in some detail the multiplicative properties of a given arithmetic function f with respect to a fixed basic sequence (see, for example, (1), (2)). We investigate here the structure of M(f), the collection of all basic sequences such that f is multiplicative with respect to , and in particular we focus our attention on the maximal members of M(f). Our principal result will be a proof that each maximal member of M(f) contains the same set of type II primitive pairs. Moreover, we will give a simple criterion for determining, in terms of the behaviour of f, whether or not a particular primitive pair (p, p) is in any (and therefore every) maximal member of M(f).

Let f(x) be integrable L(0, 2π) and periodic with period 2π, and let ψ(t) be the conjugate function of with respect to the variable t, where x is onsidered as an arbitrary constant. The following theorems are due to K. K. Chen (1), (2), pp. 111–124.

According to Luther's translation 1 Kings vi. 31 should read— “At the entrance of the choir he made two doors with pentagonal door-posts.” This is probably a wrong translation, for nowhere on Asiatic monuments of this time has a pentagon been found. Prof. A. Merx in Cantor's Vorlesungen über Geschichte der Mathemalik, Vol. i., p. 91.

It is probable that the construction of the regular pentagon is due to the Pythagorean school as a consequence of I. 47. (Ibid, p. 151.)