The problem of establishing oscillation and non-oscillation criteria for elliptic equations has recently been considered by several authors. Extensive bibliographies may be found in the books by C. A. Swanson, [7], and by K. Kreith, [3].

Most of the interest has so far centered on equations of second order with some results also established for fourth order equations. Non-oscillation theorems for higher order equations have recently been established by the author, [1], and by Noussair and Yoshida, [5]. In particular, both in [1] and [5], Kneser-type theorems were established for classes of higher order elliptic equations.

Let Z(Z+) denote the set of all (positive) integers and let T denote the discrete Hilbert transform defined for suitable sequences a = {ak}kϵz by

where as usual the prime denotes omission of the term k = n.

Associated with any near-ring R, and any set S of distributive elements of R, one has a natural notion of S-distributive R-modules, analogous to that of modules over rings and including the latter as special case (Frohlich [6]; we recall the details in Section 1). Since near-rings can be viewed as slightly deficient rings, it makes sense to enquire whether such near-ring modules share with modules over rings the familiar and important property of having injective hulls.

A ring or near-ring R is called periodic if for each xϵR, there exist distinct positive integers n, m for which xn = xm. A well-known theorem of Herstein states that a periodic ring is commutative if its nilpotent elements are central [5], and Ligh [6] has asked whether a similar result holds for distributively-generated (d-g) near-rings. It is the purpose of this note to provide an affirmative answer.

Let be the density of thinnest k-fold covering of the plane by equal circles (of radius 1, say). Let Dk be the corresponding density when the centres of the circles are at the points of a lattice Λ. It is clear that

Our purpose is to deduce from a theorem of P. Hall the following observation.

Let G be a finitely generated group and F a free normal subgroup of G with G/F nilpotent. Then G is hopfian.

Here a group G is hopfian if every epiendomorphism G→G is an automorphism.

We show that Krasnoselski's Theorem, which is usually derived from Helly's Theorem, is in fact equivalent to it.

In [1] and [2], there was given a characterization for linear Jordan matrix algebras whose coordinatizing ring is *-prime Goldie or a Cayley-Dickson ring (C-D ring). If one considers the corresponding question in the more general setting of quadratic Jordan algebra as defined by McCrimmon in [11], then the result is similar. In this latter case the ample quadratic Jordan algebras, as studied by Montgomery in [12] and [13], are brought into play.

All topological spaces are assumed to be completely regular. C(X) (resp. C*(X)) will denote the ring of all (resp. all bounded) continuous real-valued functions on X. βX is the Stone-Cech compactification of X. A real number t is said to be an accumulation point of a function f ∊ C(X) if and only if f-1[[t-ε, t + ε]] is not compact for every ε > 0.

Donné (n + 2) points dans l'intervalle [-1,1],

1.1

L. Fejér a montré ([3]) qu'il existe toujours un polynôme de degré inférieur ou égal à n + 1 tel que

1.2

et

1.3

The permanent of an n ✕ n matrix A = (aij) is the matrix function

1

where the summation is over all permutations in the symmetric group, Sn. An n ✕ n matrix A is a circulant if there are scalars a1 …, an such that

2

where P is the n ✕ n permutation matrix corresponding to the cycle (12 … n) in Sn.

Let Sn denote the permutation group on {1, 2, 3, …, n}. Let k ≥ 1 and n ≥ 1 be integers. For σ ∊ Sn we define ([1]) mk, n: Sn → Z by

Let pn ∊ Sn be the reverse permutation: that is,

Dans un récent article [1], T. M. Apostol obtient une identité liant et , où f est une fonction complètement multiplicative, |f(n)| ≤ l, et p est un nombre premier. Il utilise cette identité pour prouver que

pour tout nombre premier p. Par la suite, T. M. Apostol obtenait d'autres identités [2].

Let be a finite measure space, and let T be a contraction in real Lp(X). (i.e. T is linear and ||T||≤1). It is said that the Dominated Ergodic Theorem holds for T, if there exists a constant cp such that, if M(T)f(x) = supn 1/n then ||M(T)f||p ≤ cp ||f||p for every f in Lp.

One of the nicest theorems of elementary number theory is the two-square theorem of Fermat.

A prime p is expressible as the sum of 2 squares if and only if -1 is a quadratic residue mod p.

For the autonomous system x' = F(x,∊), the case where the second Hopf-Friedrichs condition fails is analyzed in that sufficient conditions for bifurcation of non-trivial periodic solutions to occur are given. An application to mathematical ecology is also discussed.

Let γ be an increasing function on the real numbers such that γ(0) = 0 (which, by translation of axes, is no restriction) and suppose that γ(n) is a positive integer if n is a positive integer. Let γ- denote the inverse function of γ. Furthermore, let L(x) be the least integer ≥ x; let [x] be the greatest integer ≤x, and suppose that c0, c1 … is an arbitrary sequence of numbers.

Let X be an infinite connected CW-complex which is the union of an increasing sequence of subcomplexes Xr. Let cat X denote the Lusternik-Schnirelmann category of X, normalized to take the value 0 on contractible spaces. Suppose that cat Xr ≤ K (r ≥ 1). In his problem list [1], T. Ganea proved that cat Xr ≤ 2k +1 and asked (Problem 5) whether this is the best possible upper bound. The purpose of this note is to prove that cat X ≤ 2k.

We derive here simple asymptotic estimates for both the Euler and Bernoulli numbers. The derivations follow easily from known results, but I am unable to find them elsewhere in the literature. C. Jordan [1, p. 245 and p. 303] gives some related inequalities. Other properties of these two classical sets of numbers may be found in [1], [3] and [4].

Let (R, *) be a ring R with an involution *, i.e., * is a map RϵR such that for all a, bϵR

The trace and norm of an element a in (R, *) are respectively

(R, *) is said to be a normal ring if for all aϵR

or equivalently,