The object of this note is to exhibit a certain class of distribution spaces as being c-admissible in the sense of [1] and [2]. Throughout the terminology and notation are the same as in [1] and [2]. The only addition to this is that if x ∈ Rn then τx will denote the translation operator which carries each distribution u onto the distribution ux.

A normal matrix A = (aij) with complex elements is a matrix such that AACT = ACTA where ACT denotes the (complex) conjugate transpose of A. In an article by K. Morita [2] a quasi-normal matrix is defined to be a complex matrix A which is such that AACT = ATAC, where T denotes the transpose of A and AC the matrix in which each element is replaced by its conjugate, and certain basic properties of such a matrix are developed there. (Some doubt might exist concerning the use of ‘quasi’ since this class of matrices does not contain normal matrices as a sub-class; however, in deference to the original paper and the normal canonical form of Theorem 1 below, the terminology in [2] is used.)

For a ring R with unit, let RM denote the category of unitary left R-modules. Following S.E. Dickson [3], a(non-empty) class of R-modules is a torsion class in rM if is closed under factors, extensions, and direct sums. If, in addition, is closed under submodules, then is said to be hereditary.

Let k be a fixed integer ≧ 2. A positive integer m is called k-free if m is not divisible by the k'th power of any integer > 1. Let qk(m) be the characteristic function of the set of k-free integers; that is, qk(m) = 1 or 0 according as m is k-free or not. It can be easily shown that where μ(n) is the Mobius function. Let x ≧ 1 denote a real variable and n be a positive integer. Let Qk(x, n) and be the number and the sum of the reciprocals of the k-free integers ≦ x which are prime to n respectively.

In a recent paper Gregorac [1] has posed the following question: if P is a permutational product of the amalgam = A ∪ B|H and if P is residually finite are all the permutational products of residually finite?

The class s of functions f(z) which are regular and univalent in the open unit disk △ = {z: |z| < 1} each normalized by the conditionshas been studied intensively for over fifty years. A large and very successful portion of this work has dealt with subclasses of L characterized by some geometric property of f[Δ], the image of Δ under f(z), which is expressible in analytic terms. The class of starlike functions in L is one of these [3]; f(z) is starlike with respect to the origin if the segment [0,f(z)] is in f[Δ] for every z in Δ and this condition is equivalent to requiring that have a positive real part in Δ.

In the list of known finite non-abelian simple groups there are infinitely many pairs of non-isomorphic simple groups which have the same order. The smallest known example of two such groups are the simple groups A8 and PSL(3, 4), of order 20,160.

We shall be concerned with problem 7 of Hanna Neumann [7], which asks: Prove or disprove that if U and B are varieties, and neither of U and B is contained in the other, then U ∪ B and [U, B] are decomposable if and only if U and B have a common non-trivial right hand factor.

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.

Let R be a C-algebra over a commutative ring C of zero characteristic. An element a ∈ R will be called separable if there exists p ∈ C[x] for which p(a) = 0 and such that p′(a) is invertible, where p′ is the formal derivative of p. Call A the C-algebra generated by aR and aL the right and left multiplications by a, and write Da for the inner derivation defined by a. It will be shown that when a is separable there exists φ ∈ A such that [p′(a)]−1φ Da is idempotent. As a consequence it follows that the additive group of R may be decomposed into a direct sum of Ker Da and Im Da. Another result is that for an arbitrary C-derivation δ there exists d ∈ Im Da such that aδ = aDd. Thus Ker Da (and also Im Da) is a δ-subgroup of R+ if and only if aδ = 0.

In a previous communication [5] the author has dealt with the congruence properties of some restricted partition functions. The general category of such functions may be denoted.

Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.

Semirings of endomorphisms of semigroups form an important class of semirings. All examples which can be found in the literature concern semigroups with a subcommutative operation. We show that there exists a non-subcommutative semigroup whose endomorphisms form a semiring (this answers a question raised by Professor A. H. Clifford). We also give an example of a semigroup whose set of endomorphisms is not embeddable into a semiring; however it is the disjoint union of two semirings, but one of these semirings is not embeddable into a semiring with identity.

For a finite semigroup S and a field Φ, denote by Φ [S] the semigroup algebra of S over Φ and when S has a zero element, denote by Φ0[S] the contracted semigroup algebra of S over Φ (see § 5.2 [1]). Then theorem 5.31 [1], due to E.Hewitt and H. S. Zuckerman, gives a determination of the radical of Φ0[S] when S is commutative and the characteristic of Φ does not divide the order of any subgroup of S; and on page 168 [1] some results concerning the radical of Φ0[S] when S is completely 0-simple are given, the determination of the radical in the general case remaining open.

The methods developed up to now in the theory of plates do not lend themselves easily to the solution of problems in the case of plates of arbitrary shape. Such methods lend themselves only to the solution of problems for plates of certain special simple shapes. Moreover, because of the complicated mathematical tools involved in these methods, the methods are not suitable for engineers. Hence there arises the need for the formulation of an approximate theory which may appear quite suitable for attacking problems on plates of arbitrary shape. In the present paper the author develops a new method which aims at the solution of the problem quite independently of the shape of the plate.

In [2] Evans and Neumann raised the question of whether diassociativity in groupoids or loops is equivalent to a finite set of identities and in [3], Neumann still lists the problem as unsolved. It is the purpose of this paper to show that the answer to the question for groupoids is negative.