An upper bound for P[Σ Xi≥tσ, Σ Yi ≥ tσ], where (Xi, Yi), i = 1, 2, …, n are bounded independent random variables, was given by Mullen (1973). An improvement to the bound is possible without further assumptions.

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.

Let U be a bounded open subset of the complex plane. By a well known result of A. M. Davie, C(bU) is the uniformly-closed linear span of A(U) and the powers (z-zi)-n, n = 1, 2, 3, … with zi a point in each component of U. We show that if A(U) is a Dirichlet algebra and bU is of infinite length, then one power of (z - zi) is superfluous.

In this note we present an array of results which deals with the question "When is the product of two [a, b]-compact spaces an [a, b]-compact space".

In section 1, we give some essential terminology. In section 2, we define some new classes of functions and then obtain some product theorems. In section 3, we give some applications of the product theorems obtained in section 2.

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.

We extend the following result to several variables: For any sequence {Nj}, we have C{Nj} = C{Mj}, with {Mj} logarithmically convex i.e.

In [4], Niven proved that the set A of integers for all s ≥ l and all n ≥ 1 has density zero, being the sum of the sth powers of all positive divisors of n. However his argument contains a mistake (see Remark 1). In this paper we give a proof of Niven's result and establish several related results, one of which generalizes a result of Dressier (See Theorem 3 and Remark 2).

A series ∑ an is said to be summable (I) to the limit A if

(*)

Clearly the limit is the same whether x→∞ through all real values or only positive integer values, and the expression whose limit is being taken can also be expressed in the two equivalent forms

where [x] is the greatest integer ≤x.