Recently, Ronveaux [11] has shown how to use a combination of a Riccati transformation and a homographie transformation to estimate both from below and above the distance between a zero and the succeeding or preceding extremum (zero of y' ) of solutions of

1.1

In this paper, we show how such transformations can be used to derive an equation from which the distance between successive zeros of a solution y of (1.1) can be estimated directly.

Throughout this paper we consider associative rings with unity elements. In §1 various results on the representation of rings by rings of sections of special rings are compared. In particular, it is shown that results enunciated by Dauns and Hofmann, Koh, and the present author may all be deduced from one statement, the proof of which appears in §3.

Let (G, ρ) be a finite connected (undirected) graph without loops and multiple edges. So x, y being two elements of G (vertices of the graph (G, ρ)), 〈x, y〉 ∊ ρ means that x and y are connected by an edge. Two vertices x, y ∊ G have the distance μ(x, y) equal to n, if n is the smallest number with the following property: there exists a sequence x0, x1, …, xn of vertices such that x0 = x, xn = y and 〈xi-1, Xi〉 ∊ ρ for i = 1, …, n. If x ∊ G, we put μ(x, x) = 0.

In [2], A. Sade gives a construction for quasigroups which he calls the singular direct product. In this paper we generalize Sades' construction. As an application we obtain a recursive construction for quasigroups orthogonal to their transposes. All quasigroups considered in this paper will be finite.

In [ 2 ] P. Fletcher proved that a finite topological space has a unique compatible quasi-uniformity; C. Barnhill and P. Fletcher showed in [1] that a topological space (X, ), with finite, has a unique compatible quasiuniformity. In this note we give some necessary conditions for unique quasiuniformizability.

For a sequence σ = (x1, …, xn) of real numbers, let σi and respectively denote the cyclic permutation (xi, xi+1, …, xi-1) and the reverse cyclic permutation (xj, xj-1, …, xj+1), and let . Also denote by Mrj(σ) and mrj(σ) the rth largest and the rth smallest numbers respectively, among the first j partial sums s1, s2, …, Sj for 1≤r≤j≤n. As usual, let the superscripts + and — respectively mean maximize and minimize with zero.

In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.

The purpose of this note is to provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]). This latter result we state as

Theorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by

1

then

2

unless f≡0. The constant is the best possible.

In [1] Anderson uses a simple random sample of p-component vectors from N(μ, Σ) to test the hypothesis H, Σ = σ2I, where I is the identity matrix and σ2 is unknown. Let W= λ2/N, where λ is the likelihood ratio criterion for testing H.

A function f(x) is said to be completely monotonic on (0, ∞) if

1

Familiar examples of such functions are given by f(x) = exp(—αx) and f(x) = (x+β)-α, where α ≥ 0, β ≥ 0. A discussion of completely monotonic functions is given in [5, Ch. IV].

Dans cette note on donne une démonstration, fondée sur des propriétés élémentaires des opérateurs linéaires dans des espaces munis de deux normes, d'un théorème d'interpolation établi par J. Lions et J. Peetre [2] dans le cadre de leur théorie des espaces de moyenne.

In [1] we constructed a (G, σ)-space and determined a condition which is both necessary and sufficient for the (G, σ)-product of the vectors v1, v2, …, vn to be zero. The purpose of the present paper is to give a criterion for v1 Δ v2 Δ … Δ vn to be zero, in the particular case when V is a unitary space and the group G belongs to a special class of groups which we shall define below.

We follow [6] and [7] for all terminologies. The purpose of this note is to prove

Theorem 1. Let X and Y be any two integer-compact spaces. The following are equivalent:

1. (1) X is homeomorphic to Y.

2. (2) C(X, Z) and C(Y, Z) are isomorphic as rings.

3. (3) C(X, Z) and C(F, Z) are isomorphic as lattices.

4. (4) C(X, Z) and C(Y, Z) are isomorphic as p.o. groups.

5. (5) C(X, Z) and C(Y, Z) are isomorphic as multiplicative semigroups.

In [1] Whitehead defined a CW-complex as a closure finite cell complex with the weak topology (i.e. a topology coherent with the family of closed cells, in Spanier's terminology). The purpose of this note is to show that these two conditions imposed on cell complexes can be replaced by a single one.

Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.

Raghavarao and Chandrasekhararao [3] introduced a family of PBIB designs having three associate classes known as cubic designs. In this paper a detailed analysis of the case of PBIB designs having four associate classes, which are called quartic designs, is given. Results are obtained pertaining to construction and existence of quartic designs. Moreover, using methods similar to those used by Shah [5], [6], [7], the block structure of certain quartic designs is studied.

Stone lattices can be considered as forming a category of abstract algebras and thus there is a forgetful functor from this category to the category of distributive lattices with zero and unit. In this note we consider Stone lattices in this light (cf. [3], [4]) and describe an adjoint to the forgetful functor. The Stone extension of a distributive lattice with zero unit which we obtain differs markedly from the one given in [1].

Since 1963 several authors ([13], [2], [6], [14], [10], [11], [12], [9]) have considered integral equations each one of which is contained as a special case in one of the two equations

1.1

1.2

for Re b > 0 and x ∊ [α, β].

This note originated in an attempt to generalize the assertion of Problem 164 which was proposed by Moser [2].

Let us first state our generalization.

It is known (cf. M. A. McKiernan [6]) that the only measurably bounded solutions ƒ of the equations

1

where x ∊ Rn, t ∊ R, αi(i= 1, …, m) span the space , and Σi ∊ 1 μi ≠0 for any I ⊂ {1, …, m}, are polynomials. The degree of these polynomials and the dimension of the solution space can be estimated by numbers depending on m and n. (For estimates and other details concerning equations (1) see see [1], [2], [3], [4], [5], [6].)