Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions:

1. (1) Lim (axn + byn) = a Lim xn + b Lim yn

2. (2) If xn ≥ 0 for every n, then Lim xn ≥ 0

3. (3) Lim xn+1 = Lim xn

4. (4) If xn = 1 for every n, then Lim xn = 1

The existence of such limits is proved by Banach [1].

Let J(R) and G(R) respectively denote the Jacobson and Brown-McCoy radicals of the ring R and recall that R = G(R) if and only if R can not be homomorphically mapped onto a simple ring with unity [1, p. 120].

In general one knows that J(R) ⊆ G(R) [1, p. 118], while there do exist rings R for which J(R) ≠ G(R) (see [1, p. 120]).

In [S], H. Sharp characterizes each topology on a finite set S = {s1, s2,…sn} with a n×n zero-one matrix T = (tij) where tij=1 if and only if . In this paper we seek matrix characterizations of certain topological properties of finite spaces. Such characterizations will provide purely mechanical ways of determining if a space has a certain topological property.

Certain theorems of recent interest [1, 2] concerning sequential spaces may be deduced from the fact that the category of sequential spaces, is a coreflective subcategory of the category of topological spaces, J. A space is said to be sequential if it has the finest topology that permits the convergence of its convergent sequences.

We suppose that {x1, x2, …}, are increasing sequences of consecutive zeros of non-trivial solutions y(x), , of the differential equation

(1)

on an open interval I, and define, for any fixed λ > - 1,

In a recent note of Kolodner [2], the Cantor Intersection Theorem is used to give an alternative proof of the well known Contraction Mapping Principle. Kolodner applied Cantor's theorem first to a bounded metric space and then reduced the general case to this special case. Sometime ago, we found a somewhat different proof of the Contraction Mapping Principle using Cantor's theorem. Since our proof seems somewhat more direct we propose to present it here.

Consider a two associate class partially balanced incomplete block (PBIB) design [2] with parameters of the first kind t, b, r, λ,1 λ,2 n1, n2, and parameters of the second kind . Let the letters m, p, l, s represent treatments and define associates of treatment m, λmps = number of times the treatments m, p and s all occur together in the same block, λmpls = number of times the treatments m, p, l and s all occur together in the same block.

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.

Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].

This paper gives alternative straightforward and simpler proofs of some of the results of Laurent's [10], and Likes' [11], [13]. The derivation of the results is simplified by using the theory of Dirichlet's multiple integral and the transformation used to derive this multiple integral. Some applications of Dirichlet's transformation to order statistic theory from gamma, and normal populations, have been already given by Kabe [7].

Maitland's generalised Bessel function [4] is defined by the equation

where u is real and positive and v is any number real or complex. If u = 1, then (1.1) reduces to the form

In [1] Demidovic considered a system of linear differential equations

with A(t) continuous, T-periodic, odd, and skew symmetric. He proved that all solutions of (1) are either T-periodic or 2T-periodic0 In [2] Epstein used Floquet theory to prove that all solutions of (1) are T-periodic without the skew symmetric hypothesis. Epstein's results were then generalized by Muldowney in [7] using Floquet theory. Much of the above work can also be interpreted as being part of the general framework of autosynartetic systems discussed by Lewis in [5] and [6]. According to private correspondence with Lewis it seems that he was aware of these results well before they were published. However, it appears that these theorems were neither stated nor suggested in the papers by Lewis.

It is known that if K is a field, then the ring of formal power series in one or more variables, with coefficients in K, is Henselian at its maximal ideal. In this note we show that if R is a ring (commutative and with identity element) which is Henselian at the maximal ideals M1, M2, … then R[[x]] - the ring of formal power series in x with coefficients from R - is also Henselian at the maximal ideals M1 ⋅ R[[x]] + x⋅ R[[x]], etc.

Some of the most important G-structures of the first kind (1) are those defined by linear operators satisfying algebraic relations. Let J be a linear operator acting on the complexified space of a differentiable manifold V, and satisfying a relation of the form

where λ is a complex constant and I is the identity operator. In the case λ ≠ 0 the manifold has an almost product structure (2) which in the case λ = i reduces to an almost complex structure (3). In the remaining case, λ = 0, the manifold has an almost tangent structure (4).

Let (E, E') be a dual pair and G be a collection of σ(E, E') bounded sets in E with the following properties:

1. i) G contains all singletons.

2. ii) If A, B are in G there is a C in G with A ∪B ⊆ C.

3. iii) If A is in G then λ A ε G for any scaler λ.

Then sets of the form the polar of Ai, form base of neighbourhoods in E' for the polar topology TG of G convergence.

Let be the nth ultraspherical polynomial. Also let . The following generating relation is well known (3, p.98).

It can also be written as

1.1

This suggests the consideration of the class of polynomial sets {Qn(x), n = 0, 1, 2,…}, Qn(x) is of exact degree n and

1.2

The space Hm of homogeneous polynomials in n real variables x1, x2,…, xn of degree m may be considered as an inner product space with inner product ; where ds is the rotation-invariant measure on Sn-1 = {x ε Rn: |x| = 1}, . The problem solved in this paper is the following: given n-1 a linear functional ϕ on Hm, find Pϕ ε Hm so that ϕ(p) = (p, Pϕ) for all p ε Hm.

Let f(x) be a polynomial of degree d ≥ 2 defined over the finite field kq with q = pn elements. Let

If f*(x, y) has no irreducible factor over kq which is absolutely irreducible, f is called an exceptional polynomial [1]. Davenport and Lewis have noted that when d is small compared with p, a permutation (substitution) polynomial is necessarily an exceptional polynomial. It is the purpose of this paper to prove the converse; that is, we will show the existence of a constant a(d), depending only on d, such that if f(x.) is an exceptional polynomial over kq, where p ≥ a(d), then f(x) is a permutation polynomial.

Let X be a random variable whose density (or distribution if discrete) f(x; θ) depends on an unknown parameter θ, real or vector-valued. By making observations on X we want to know whether there exist estimates of prescribed accuracy for the real-valued parametric function g(θ). By an estimate of prescribed accuracy for g(θ) we mean a confidence interval of prescribed length and confidence coefficient or a point estimate with prescribed expected loss W. In the following our loss functions W will always satisfy the requirement that W(δ, θ) = V(|δ - θ|), where V is a strictly increasing function of its argument. The class of such loss functions includes among others the squared error loss.

The aim of this note is to point out a mistake in the proof of Theorem 2 of Wong's paper [1], We first give an example to show that the theore as stated is not true.

Example 1. Consider the cartesian plane and the graph H with equation .