The space wp of strongly Cesàro summable sequences of index p > 0 has been investigated by several authors. In [2], Kuttner proved that no Toeplitz matrix could sum all sequences in wP, a result which was extended to coregular matrices by Maddox [5]. In [1], Borwein considered the continuous dual space of wp. The more general space w(p) has also been considered [3, 4], where p = (pk) is a strictly positive sequence. The r-convexity of the spaces w∞(p) and w0(p) was dealt with in a partial way in [8]. In the present note we establish criteria for the rconvexity of some general classes of [A, p]0 and [A, p]∞ spaces (see [6] and [7] for definitions), and in particular we give the necessary and sufficient conditions for the r-convexity of w∞(p) and w0(p). For most of the relevant definitions and notation we refer to [8].

We are well-acquainted with the theorem about sequences which states that, the existence of

(1)

is sufficient to imply limk→∞ak = 0. Partially out of a growing interest in the theory of regularly varying sequences ([1]), and probably as an interesting problem, in and of itself, some mathematicians have tried to find conditions weaker than (1) that would guarantee limk→∞ak = 0.

The maximal number of pairwise edge disjoint 4-cycles in the complete graph Kn and the minimal number of 4-cycles whose union is Kn are determined.

This note presents characterizations of the injective and of the equationally compact ℵ0-semilattices, which are analogous, respectively, to the characterizations of the injective semilattices given by Bruns and Lakser [2] and of the equationally compact semilattices given by Grätzer and Lakser [3].

For a simple Lie algebra L over the complex numbers ℂ all irreducible representations admitting a highest weight have been constructed and characterized for example in [3, 6]. In [1] Bouwer considered the family of all irreducible representations of L admitting at least one one-dimensional weight space (this includes, of course, all those having a highest weight space) and showed, by construction, that this is a strictly larger class of representations.

Let B be a bounded linear operator having domain and range in a Banach space. If the second-order differential operator d2/dt2–B has a Bohr-Neugebauer type property for Bochner almost periodic functions, then any Stepanov-bounded solution of the differential equation (d2/dt2)u(t) – Bu(t) = g(t) is Bochner almost periodic, with g(t) being a Stepanov-almost periodic continuous function.

Sandomierski (Proc. A.M.S. 21 (1969), 205–207) has proved that a ring is semiperfect if and only if every simple module has a projective cover. This is generalized to semiperfect modules as follows: If P is a projective module then P is semiperfect if and only if every simple homomorphic image of P has a projective cover and every proper submodule of P is contained in a maximal submodule.

Let R be an associative ring and denote by the injective hull of the right module RR. If can be endowed with a ring multiplication which extends the existing module multiplication, we say that is a ring and the statement that R is a ring will always mean in this sense.

It is known that is a regular ring (in the sense of von Neumann) if and only if the singular ideal of R is zero.

Let (X, d) be a (nonempty) metric space. bc(X) will denote the family of all nonempty bounded closed subsets of X endowed with the Hausdorff metric D induced by d [2, pp. 205]. Let/be a map of X into bc(X). f is nonexpansive at a point x in X if for all y in X.

In measure theory and probability it is often useful to be able to extend a set function g to a measure μ. One situation in which such an extension arises is that of obtaining limit measures for inverse (or projective) systems of measure spaces ([1], [5]).

In this note first the weak almost-periodicity of the action of a weakly almost-periodic family of linear operators on an almost-periodic function is established. Then an application of this result is given.

Let {Xi:i=1, 2,…} be a real strictly stationary process (defined on a probability space (Ω, A, P)) which has absolutely continuous finite dimensional distributions (with respect to Lebesgue measure) and satisfies the ϕ-mixing condition: Let and denote the sub-cr-fields generated, respectively, by {Xi:i≤k} and {Xi:i≥k+n}; then, for each k≥1 and n≥l, and together imply.

W-groups are treated in the more general context of near-ring modules. It is shown that they share properties of d.g. modules with respect to central series. In particular, results of Magomaev are obtained and used to show that the group-theoretic and module-theoretic central series coincide for W-groups.

Let (Mn, F) be an n-dimensional Finsler space where Mn is the underlying n-dimensional manifold and is the Finsler fundamental function. F being a differentiable function of the point and element of support where T∞(Mn) is the tangent space of Mn at x and is positively homogeneous of degree one with respect to X.

The exact probability of making an error at the first stage of a stepwise regression is found, both when the independent variables are fixed and when they are random. The probabilities differ in the two cases because the random variables involved are pairwise independent but not jointly independent.

Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ = x+iy, y > 0), such that f(τ + λ)= f(τ) where λ is a positive real number and f(—1/τ) = γ(—iτ)kf(τ), k being a complex number. The function (—iτ)k is defined as exp(k log(—iτ) where log(—iτ) has the real value when —iτ is positive. Every such function is said to have signature (λ, k, γ) in the sense of E. Hecke [1] and has a Fourier expansion of the type f(τ) = a0 + σ an exp(2πin/λ), (n = 1,2,…), if we further assume that f(τ) = O(|y|-c) as y tends to zero uniformly for all x, c being a positive number.

All rings are associative but do not necessarily have identities. Definitions and basic results about radical classes can be found in [2]. A radical class is strong [3] if for every ring A, (A) contains all left and right -ideals of A.

Let R and S be rings with 1, G a group and RG and SG the corresponding group rings. In this paper, we study the problem of when RG≃SG implies R≃S. This problem was previously investigated in [8] for the case where G is assumed to be infinite cyclic. The corresponding question for polynomial rings, namely, when does R[x]≃S[x] imply R≃S, has been considered by several authors, particularly Coleman and Enochs [3].

Serfling (1968) has considered a central limit theorem in which assumptions are made concerning the expectation of variables conditioned on their distant predecessors. Dvoretsky (1972, theorem 5.3) has continued this investigation. Serfling showed that both martingales and φ-mixing sequences satisfied his conditions, and Dvoretsky extended this to Strong mixing sequences of random variables.

This paper generalizes some results on pivotal functions of maximum likelihood estimators of location and scale parameters and the related ancillary statistics obtained by Antle and Bain, and Fisher. It shows that the maximum likelihood estimator of the parameter in an invariant statistical model is an essentially equivariant estimator or a transformation variable in a structural model. In the latter case, ancillary statistics in the sense of Fisher used in conjunction with the maximum likelihood estimators can be easily recognized. It is also remarked that the values of maximum likelihood estimators from samples having the same “complexion” are simply related to those of other, perhaps simpler, transformation variables. In the development it also points out the importance of using the correct definition of the likelihood function originally proposed by Fisher.