We solve two problems on convex bodies stated on p. 38 of S. M. Ulam's book, A collection of mathematical problems (New York, 1960).

Problem 1. This problem is due to Mazur. In three-dimensional Euclidean space there is given a convex surface W and a point O in its interior. Consider the set V of all points P defined by the requirement that the length of the interval OP is equal to the area of the plane section of W through O and perpendicular to OP. Is the (centrally symmetric) set V a convex surface?

Let Nn be the number of real roots of a random algebraic equation The coefficients ξν are independent random variables identically distributed with expectation zero; the variance and third absolute moment are finite and non-zero. It is proved that

where εν tends to zero, but εν log n tends to infinity. The measure of the exceptional set tends to zero as n tends to infinity.

Let a, b be relatively prime integers with |a| > |b| > 0. For any integer n > 0, let π n denote the nth cyclotomic polynomial, denned by

where ζn is a primitive nth root of unity.

It was proved by Davenport (2) that a cubic form in n variables with rational coefficients always represents zero if n > 32; and this condition was later (3) relaxed to n ≥ 29. The object of the present paper is to establish conditions under which a cubic form will represent every rational number other than zero, for rational values of the variables. The main result is as follows.

Let X be a compact polyhedron, Xn the topological product of n factors equal to X. The symmetric group Sn operates on Xn by permuting the factors, and hence if G is any subgroup of Sn we have an orbit space Xn/G obtained by identifying each point of Xn with its images under G. In particular Xn/Sn is the nth symmetric product of X, and if G is a cyclic subgroup of order n then Xn/G is the nth cyclic product of X.

In a recent paper, (2), Baumslag showed that a wreath product of A by B is nilpotent if and only if A and B are nilpotent p-groups for the same prime p, with A of finite exponent and B finite. We shall calculate the (nilpotency) class of such groups when A and B are Abelian. This provides a lower bound for the class in the general case. We give a simple construction for a set of non-nilpotent metabelian groups which satisfy a finite Engel condition. With Engel class defined by Definition 6·1, we show that there are nilpotent groups of arbitrarily large nilpotency class for which the nilpotency class is equal to the Engel class.

In this note we consider a particular type of covering problem. Let T be given a plane set and Tθ be the set obtained from T by a rotation about some point in the plane through an angle θ in the clockwise sense. If a set K is such that for every θ there is a translation which transforms Tθ into a subset of K then we say that K is a rotation cover of T. The problem considered here is to determine for fixed n, l the triangle of least area which is a rotation cover for an n-sided regular polygon of side length l. In each case the solution is an equilateral triangle the altitude of which is given by

This solution is unique.

A group G is called characteristically simple if it has no characteristic subgroups other than itself and the unit subgroup. For brevity, we call such groups -groups; we also use to denote the class of all characteristically simple groups.

We construct various pathological curves and sequences of curves to solve three problems of Steinhaus.

A group G is called locally soluble if every finitely generated subgroup of G is soluble. Terms like ‘locally nilpotent’ and ‘locally finite’ are defined similarly.

The study of the divisibility of polynomials in a single variable over a field is essentially based on the division algorithm, which may be stated as follows: (A) Given polynomials a, b such that there exist polynomials q and a1, such that† .

Interest in the theory of linear graphs has grown recently in several fields which at first sight seem rather far apart. Recent publications include an excellent review by Bryant(1) of electrical network theory, statistical mechanical work by Sykes(2) and in the field of linear programming a paper by Glicksman, Johnson and Eselson (3). It is in this latter field that the subject of this paper arose, but we shall confine attention here to the graph theoretical aspect alone.

The kernel of a non-trivial linear functional φ on a linear space E is a maximal proper linear subspace of E which determines φ up to a non-zero multiple. Does a similar result hold for homomorphisms of a group G into the additive group R of the real numbers?

Singular points on irreducible primals were investigated briefly by C. Segre(8), where the author classified multiple points by the nature of the nodal tangent cone. For surfaces the problem of classification was investigated by, amongst others, Du Val(1) and a complete classification of isolated double points of surfaces lying on non-singular threefolds was given by Kirby(5). In (3) we classified certain types of double points on algebraic primals in Sn. An isolated double point which after a finite number of resolutions gave rise to at most a finite number of isolated double points was called a double point of rank zero. We found that the only isolated double points of rank zero are those which are analogous to the binodes, unodes and exceptional unodes (2) of surfaces.

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).

In a recent paper (3) Mordell proposed for solution

PROBLEM 1. To find the constants a1, …, an so that

where xn+i = xi ≥ 0 for all i.

Let f(z) = u(z) + iv(z) be a function regular for |z| < 1 with real and imaginary parts u(z) and u(z) respectively. The following theorems have been proved by Miss M. L. Cartwright(1) and (2).

Theorem A. if v(0) = 0, 0 < α < ∞ and

for 0 ≤ r < 1, 0 ≤ θ < 2π then

for 0 ≤ r < 1, 0 ≤ θ < 2π

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).

The object of this paper is to obtain the original function of which the Laplace transform (l) is the product

where, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equation

In § 2 it will be shown that this original function is

where the symbol Δ(n; α) represents the set of parameters