The following type of argument is rendered almost believable by its frequent occurrence in elementary courses in statistics. Let ξi be a sequence of independent identically distributed random variables with means μ variances σ2.

Conditions are given under which the ring of quotients defined by an ideal is semisimple Artinian modulo its Jacobson radical.

Let us consider the retarded differential equation

1

for which the following assumptions are made:

(i) p: [t0, ∞) → [0, ∞) is continuous and not identically zero for all large t.

(ii) σ [t0, ∞)→ ℝ is continuous, strictly increasing,

(iii) φ: ℝ → ℝ is continuous, non-decreasing and

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

The following theorem is proved: Suppose R is a ring with identity which satisfies the identities xkyk = ykxk and xlyl = ylxl , where k and l are positive relatively prime integers. Then R is commutative. This theorem also holds for a group G. Furthermore, examples are given which show that neither R nor G need be commutative if either of the above identities is dropped. The proof of the commutativity of R uses the fact that G is commutative, where G is taken to be the group R* of units in R.

We consider the second-order operator

1

where the coefficients are real continuous functions on an interval ℐ with w and p positive. The operator is assumed singular at only one endpoint which we take to be either 0 (finite singularity) or ∞ (infinite singularity). Let be the Hilbert space of all complex-valued, measurable functions f satisfying

In classical information theory, the amount of information provided by an experiment is measured by a function of the probability distribution of the outcomes of the experiment. In this paper, information measures are functions of sequences of elements of a monoid (S, ∘) with identity e. It is assumed that the measures {μn: Sn → ℝ} of information are branching.

Recently H. Marubayashi [1,2] and S. Singh [10,11,12] generalized some results of torsion abelian groups for modules over some restricted rings, like bounded Dedekind prime rings, bounded hereditary Noetherian prime rings. Singh [12] introduced the concept of h-purity for a module MR satisfying the following conditions:

(I) Every finitely generated submodule of every homomorphic image of M is a direct sum of uniserial modules.

Every Banach space with a non-shrinking (unconditional) basis (Xi ) can be renormed so that the biorthogonal sequence has a much smaller (unconditional) basis constant than (xi ). On the other hand, if the unconditional constant of is C < 2 then the unconditional constant of (xi ) is at most C/(2—C). This estimate is sharp.

Let (X, p) and (Y, d) be metric spaces with at least two points. It is usual for introductory courses in topology to study the set Yx of all functions mapping X to Y with the pointwise, compact-open, uniform convergence, and uniform convergence on compacta topologies. Some care is taken to show sufficient conditions for these topologies to be equivalent [1, 2]. However, the question of necessary conditions are dismissed with examples showing that the topologies are not in general equivalent.

Let S denote the semigroup of all rectifiable, piecewise continuously difïerentiable paths in ℝn under concatenation. We prove a theorem to the effect that every finite collection of paths is contained in a subsemigroup of S which has the unique factorization property with respect to certain primes and straight lines. We also determine an abstract necessary sufficient condition for a subsemigroup of S to have this unique factorization property.

J. Marica and J. Schönhein [4], using a theorem of M. Hall, Jr. [3], see below, proved that if any n − 1 arbitrarily chosen elements of the diagonal of an n × n array are prescribed, it is possible to complete the array to form an n × n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2], to the effect that an n × n incomplete latin square with at most n − 1 places occupied can be completed to an n × n latin square. When the complete diagonal is prescribed, it is easy to see that a counterexample is provided by the case that one letter appears n − 1 times on the diagonal and a second letter appears once. In the present paper, we prove that except in this case the completion to a full latin square is always possible. Completion to a symmetric latin square is also discussed.

In this paper, we are concerned with the functional inequality

1

where 0 < Pi < l, 0 < qi < l, fi(p)≠0, for 0 < P < 1, (i = 1, 2,..., n) and n is a fixed positive integer, n ≥ 2.

Inequality (1) was studied by Rényi and Fischer, (see [1], [3]) in the special case

2

and this provided a characterization of Rényi's entropy.

This paper contains several characterizations of regular-closed and minimal regular topological spaces along with some related properties.

Certain operators essentially defined by convolution are considered. Their possible domain and range spaces are determined; then conditions are given under which the construction of the optimal continuous partner may be carried out for a suitable domain or range. Special cases of operators of convolution-type are useful in studying the boundedness properties of conjugate function operators and, more generally, classes of operators satisfying restricted weak-type conditions.

In a previous note on derivations [1] we determined the structure of a prime ring R which has a derivation d≠0 such that the values of d commute, that is, for which d(x) d(y) = d(y) d(x) for all x, y∈R. Perhaps even more natural might be the question: what elements in a prime ring commute with all the values of a non-zero derivation? We address ourselves to this question here, and settle it.

Fan ([2, Theorem 2]) has proved the following theorem:

Let K be a nonempty compact convex set in a normed linear space X. For any continuous map f from K into X, there exists a point u∈K such that

In this note, we prove that the above theorem is true for a continuous condensing map defined on a closed ball in a Banach space. We also prove that it is true for a continuous condensing map defined on a closed convex bounded subset of a Hilbert space.

The following result is due to Wielandt [1, Lemma 2.9]: Let A, B, K be N-submodules of some N-module, where N is a zero symmetric near-ring. Then the N-module, Γ: = (A + K) ∩ (B + K) | (A ∩ B) + K is commutative. Using this result Wielandt obtained density theorem for 2-primitive near-rings with identity. Betsch [1] used Wielandt's result to obtain the density theorem for O-primitive near-rings. The purpose of this paper is to extend this result for loops.