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.)

Let

be an entire function, where (αn) is a strictly increasing sequence of non-negativeintegers. The maximum modulus, M(r), the minimum modulus, m(r), and the maxi-mum term, μ(r), of f defined by

The paper is concerned with a curve F, the complete intersection of a quadric with a quartic surface, that admits a group of self-projectivities isomorphic to the symmetric group of degree 5. Every generator of the quadric is, as shown at the end of the paper, cut by F equianharmonically. F has 80 stalls, points where its osculating plane is stationary; they are of two kinds, 60 to be labelled ∑, the other 20 Ω. F also has inflections at 24 points which compose a figure encountered on earlier occasions. A search is made for tritangent planes of F of which, when reckoned according to proper multiplicity, there must be 2048. Among them are 60 all of whose three contacts are ∑ while a further 120 each involve a single ∑ among their contacts and 420 each involve a single Ω.

The proofs usually given that Sn = αn + βn + … can be expressed in terms of the elementary symmetric functions ∑α, ∑αβ, etc., though simple, are not in general elementary. The following demonstration will, I think, be found to combine these two qualities.