Let ξ be an irrational number with simple continued fraction expansion ξ= [a0;a1,a2,…], Pn/qn be its nth convergent, . The following two theorems were proved by Müller [9] and rediscovered by Bagemihl and McLaughlin [1]:

Theorem 1.For n>1,

Let ρA + σB =[ραμν + σbμν] be a pencil of type m × m′, i.e. with m rows and m′ columns, where A and B are matrices with constant elements which are not mere scalar multiples of each other; and ρ and σ are homogeneous parameters.

Two commutative Banach algebras A and B are said to be similar if there exists a Banach algebra D such that [xD]− = D for some x in D, and two one-to-one continuous homomorphisms φ:D→A and ψ:D→B such that φ(D) is a dense ideal of A and ψ(D) a dense ideal of B.

We prove in this paper that the Volterra algebra is similar to A0/e-z A0 where A0 is the commutative uniform, separable Banach algebra of all continuous functions on the closed right-hand half plane , analytic on H and vanishing at infinity. We deduce from this result that multiplication by an element of A0/e-z A0 is a compact mapping.

Let Γ be a finite graph together with a group Gv at each vertex v. The graph productG(Γ) is obtained from the free product of all Gv by factoring out by the normal subgroup generated by for all adjacent v, w.

In this note we construct a projective resolution for G(Γ) given projective resolutions for each Gv, and obtain some applications.

In response to a letter from Goldbach, Euler considered sums of the form

where s and t are positive integers.

As Euler discovered by a process of extrapolation (from s + t ≦ 13), σh(s, t) can be evaluated in terms of Riemann ζ-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we give a formula for the series

This evaluation involves ζ-functions and σh(2, m).

A real Jordan algebra which is also a Banach space with a norm which satisfies

for each pair a, b of elements, is said to be a JB-algebra. A JB-algebra which is also a Banach dual space is said to be a JBW-algebra.

Let A = [aij] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n>l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).

We prove that any infinite Coxeter group has a finite index subgroup which surjects ℤ.

1. In the first part of this paper we have considered merely the contours of curves, that is, contour points, and the method of obtaining the various physical diagrams. In this part we shall consider chiefly the contours of surfaces; that is, contour lines.

If any curve be cut by planes parallel to that of (x, y) and if the various points of intersection be projected on any one of these planes, say z = 0, the contour points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break in continuity by projecting every point of the curve upon the plane z = 0. But such a line may be regarded as the intersection, by the plane z = 0, (see fig. 48) of a cylindrical surface whose generating lines are parallel to the z-axis and are drawn from the given curve to meet that plane. We have here then the intersection of a given surface by a surface over which z is constant. But this satisfies our definition of a contour line. This case of a cylindrical surface supplies the simplest system of contour lines by giving z different values. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is that in which the same values of the x and y co-ordinates of a point on the curve correspond to different values of the z-co-ordinate.

We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials (where Pj(x) is a polynomial of degree j) under the restriction that the sup-norm of is bounded on the interval [ −b, b] (b>0). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval [ −b, b].

When Etherington (2) introduced linear commutative non-associative algebras in connection with problems in theoretical genetics, he pointed out that various sequences of elements in these algebras represented different mating systems. In all such systems it was however assumed that the generations did not overlap, and this restriction has been kept in later work in this field. In this paper we treat sequences which make it possible to find the probability distribution in successive generations in a discrete time model where the generations may be overlapping. We also consider idempotents in genetic algebras and outline how the method used on the overlapping generation sequence may be applied to other sequences.