There is a skew-Hadamard matrix of order 92.

Previously the smallest order for which a skew-Hadamard matrix was not known was 92. We construct such a matrix below. The orders < 200 which are now undecided are 100, 116, 148, 156, 172, 188, 196; see [2], [3]. The existence of any Hadamard matrix of order 92 was unknown until 1962 [1].

Ralph P. Boas, Jr, proved the following theorem: Let g be an odd function, integrable on (0, π) and periodic with period 2π, and its Fourier series be Σ bn sinnt. If 0 < r < 1 and bn ≥ 0 b > for all n, then t−rg(t) ∈ L(0, π) if and only if the series Σ bn/n1−r converges. Philip Heywood asked whether the conditions g(t) ∈ L(0, π) and bn ≥ 0 can be replaced by tg(t) ∈ L(0, π) and bn ≥ −A/nr or not. We prove this problem affirmatively.

A calculation is made of the temperature distribution outside the laminar part of the plume produced by a small source of heat in an infinite fluid. It is shown in particular that the temperature variation is not exponentially small outside the plume, as might be expected in such a problem of boundary layer type.

In this paper we establish (c, 1) summability of the sequence {nBn (x)} and by using Tauber's Second Theorem, we deduce the convergence criterion of the conjugate series of a Fourier series.

Let A be a hermitian transformation on an n-dimensional unitary space En, with proper values a1 ≥ … ≥ an. Let M be a proper subspace of En. Suppose b1 ≥ … ≥ bh are the proper values of A/M and c1 ≥ … ≥ ck are the proper values of . Let i1. < … < ir and j1 < … < jr be sequences of positive integers, with ir ≤ k and jr ≤ h. then

.

This is a special case of one of the Thompson-Freede theorems which is proved by use of certain invariants.

Let for t ∈ [a, b] ⊂ [0, ∞)

where Ws is an n-dimensional Wiener process, f(s) an n-vector process and G(s) an n × m matrix process. f and G are nonanticipating and sample continuous. Then the set of limit points of the net

in Rn is equal, almost surely, to the random ellipsoid Et = G(t)Sm, Sm = {x ∈ Rm: |x| ≤ 1}. The analogue of Lévy's law is also given. The results apply to n-dimensional diffusion processes which are solutions of stochastic differential equations, thus extending the versions of Hinčin's and Lévy's laws proved by H.P. McKean, Jr, and W.J. Anderson.

There are only isolated instances of vectors of functions for which it is possible to obtain an explicit expression for the remainder functions obtained upon approximating by polynomials in the manner described by Mahler in his paper “Perfect systems”, Compositio Math. 19 (1968). We display appropriate identities and point to a pattern amongst these which suggests we should not expect convenient generalization to wider classes of functions. Proofs of perfectness do not require laborious computation but are immediate from the identities given.

It is the purpose of this paper to exhibit an infinite collection of prime, noninvertible links; each link in the collection is the union of two invertible knots. In fact, we present two such families of links. Our examples heuristically indicate that collections of links of this type exist in many varieties.

A power-associative ring A is called a p-ring provided there exists a prime p so that for every x in A, xp = x and px = 0. It is shown that if A is such a ring with p ≠ 2, then A is isomorphic to a subdirect sum of copies of GF(p), the Galois field with p elements.

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.

1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.

We prove that if C is an abelian category and M is the class of all coretractions, then the class of M-fibrations is the class of all retractions. As a corollary we prove that the class of all retractions is contained in the class of M-fibrations for any M.

An example is given of a finitely based variety of groups such that is not finitely based.

Let be the variety of groups determined by the laws (1) [[x1, x2], [x3, x4, [x5, x6]], (2) [[x1, x2, x3], [x4, x5]] [[x1x2], [x4, x5, x3]]−1, [[x1, x2, x3], [x1, x2]]. Then is not finitely based.

Let p denote a prime, G a. finite soluble p'-group and A an elementary abelian p-group of operators on G. Suppose that A has order p4 and that if ω ∈ A# then CG(ω) has nilpotent derived group. Then G has nilpotent derived group.

We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.

Torsion radicals in categories of modules have been studied extensively in connection with quotient categories and rings of quotients. (See [8], [12] and [13].) In this paper we consider the dual notion, which we have called a cotorsion radical. We show that the cotorsion radicals of the category RM correspond to the idempotent ideals of R. Thus they also correspond to TTF classes in the sense of Jans [9].

It is well-known that the trace ideal of a projective module is idempotent. We show that this is in fact a consequence of the natural way in which every projective module determines a cotorsion radical. As an application of these techniques we study a question raised by Endo [7], to characterize rings with the property that every finitely generated, projective and faithful left module is completely faithful. We prove that for a left perfect ring this is equivalent to being an S-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.

The two-dimensional flow around an infinite cylinder at low Reynolds number has interested fluid dynamicists for many years. In this paper it is shown that an infinite cylinder can propel itself through a viscous fluid (for example micro-organisms) if it has certain undulations on its surface.

There exists a non-solvable group which is third Engel. More generally, the existence of a non-solvable group in which every n-generator subgroup is nilpotent of class at most 2n - 1 is confirmed.

Let α be a generalized character of a (not necessarily finite) group G over an arbitrary field, and n a positive integer. It is shown that the function α(n) defined by α(n) (g) = α(gn) is also a generalized character of G. An application confirms a conjecture of Robert Higgins and David Ballew: if G is finite and k is also a positive integer, the k-th power of the number of n-th roots of g, summed over all g in G, is divisible by the order of G.