A complex linear homogeneous differential equation of the nth order is called strong disconjugate in a domain G if, for every n points z1,…, zn in G and for every set of positive integers, k1…, kl, k1 + … + kl = n, the only solution y(z) of the equation which satisfies

is the trivial one y(z) = 0. The equation y(n)(z) = 0 is strong disconjugate in the whole plane and for every other set of conditions of the form y(mk(zk) = 0, k = 1 , . . . , n, m1 ≤ m2... mn, there exist, in any given domain, points z1 , . . . , zn and nontrivial polynomials of degree smaller than n, which satisfy these conditions. An analogous results holds also for real disconjugate differential equations.

Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.

B. Grünbaum and J. N. Lillington have considered inequalities defined by three lines meeting in a compact convex subset of the plane. We prove a conjecture of Lillington and propose some conjectures of our own.

If R is a commutative Noetherian ring and I is a nonzero ideal of R, it is known that R+I[x] is a Noetherian ring exactly when I is idempotent, and so, when R is a domain, I = R and R has identity. In this paper, the noncommutative analogues of these results, and the corresponding ones for power series rings, are proved. In the general case, the ideal I must satisfy the idempotent condition that TI = T for each right ideal T of R contained in I. It is also shown that when every ideal of R satisfies this condition, and when R satisfies the descending chain condition on right annihilators, R must be a finite direct sum of simple rings with identity.

Let R be a ring with 1 whose nil subrings are nilpotent modulo the sum of nilpotent ideals. It is proved that if G is a locally solvable group of unipotent elements in R, then the subring generated by {g −1 g ∈ G} is nil. This result implies a result of Sizer showing that a solvable group of unipotent matrices over a skew field can be simultaneously triangularized.

The strongest positive results in the stability theory of C*-algebras assert that if are sufficiently close C*-subalgebras of (H) of certain kinds, then there is a unitary operator U on H near I, such that . We give examples of C*-algebras , both isomorphic to the algebra of continuous functions from [0, 1] to the algebra of compact operators on Hilbert space, which can be as close as we like, yet for which there is no isomorphism α: → with . Thus the results mentioned do not extend to these C*-algebras.

In this paper we discuss the relationship between local properties such as freeness and projectivity of a group and the freeness or projectivity of its pro-C-completion. We show that for certain classes, C, of finite groups (e.g. p-groups, nilpotent groups, super-solvable groups) the pro-C-completion of a locally free pro-C-group is a free pro-C-group. We also show that under certain circumstances the converse is also true but we leave open the question, for example, of whether a locally free pro-p-group is free.

Best biased and one-sided Chebyshev approximation with respect to a varisolvent approximating function on an interval are considered. The uniform limit of best biased approximations is the (unique) best one-sided approximation if the best one-sided approximation is of maximum degree. Examples are given where the best one-sided approximation is not of maximum degree and failure of uniform convergence and of existence occurs.

The purpose of this note is to tie together some results concerning doubly stochastic matrices and their representations as convex combinations of permutation matrices.

New characterizations of weakly-continuous and θ-continuous functions are presented, and θ-continuity is applied to characterize H(i) spaces; a recent characterization of closed graph functions is utilized to characterize H-closed spaces. Noiri has shown that a function λ which is almost-continuous in the sense of Husain is weakly-continuous if cl(λ−1(W)) ⊂ λ−1(cl(W)) for all open W. It is established here that almost-continuity is superfluous in this statement.

Nӧrlund methods of summability are studied as mappings from I1 into I1. Conditions are given for an arbitrary l-l method to include a Nӧrlund method. In particular necessary and sufficient conditions are given for a row finite l-l method to include a Nӧrlund mean.

A counter-example is provided to the conjecture of Y. S. Pawar and N. K. Thakare that a semilattice S with 0 is 0-distributive if and only if for each filter F and each ideal I such that F ∩ I = Ø, there exists a prime filter containing F and disjoint from I. This shows that 0-distributivity is not equivalent to weak distributivity. A characterization is also given of finite P-uniform semilattices in terms of 0-distributivity.

We extend the comprehensive treatment of k-spaces and sequential spaces provided by Franklin's refined notion of a natural cover to kR-spaces and sR-spaces. For this purpose, an apparently unstudied class of maps of topological spaces, the class of R-quotient maps, is introduced.

The duals of the Camillo-Zelmanowitz formulas for Goldie dimension are shown to hold for Varadarajan's notion of corank, subject to the existence of certain cocomplements. In particular, the formulas hold for modules over perfect rings. Also, if R is semiperfect, then the vector space dimension formulas hold for all modules over R for Goldie dimension iff they hold for corank iff R is semisimple.

Let h(m) denote the class number of the quadratic field Q(√m). In this paper necessary and sufficient conditions for h (m) to be divisible by 16 are determined when m = −p, where p is a prime congruent to 1 modulo 8, and when m = −2p, where p is a prime congruent to ±1 modulo 8.

We show certain generic rings for stably free modules are smooth. Using the theory of smooth algebras we deduce that these rings are regular when the base ring is regular. Also this enables us to calculate the dimensions of these rings. Gabel used these generic rings to determine the freeness of certain stably free modules. Our results allow a strengthening of his results when restrictions are placed on the type of stably free module—for example orthogonal stably free modules.

Let X be a complemented subspace of a Banach lattice E. It is shown that if every Dunford-Pettis operator from L1[0,1] into X is Pettis-representable then X has the Radon-Nikodym property.

The paper is concerned with the Liouville-Riemann and Weyl fractional integrals. Necessary and sufficient conditions are obtained for a function to have a fractional integral which is equivalent to a constant.

We are concerned with the oscillatory behavior of the second order elliptic equation

1

where Δ is the Laplace operator in n-dimensional Euclidean space Rn, E is an exterior domain in Rn, and c:E × R → R and f:E → R are continuous functions.

A function v : E − R is called oscillatory in E if v(x) has arbitrarily large zeros, that is, the set {x ∈ E : v(x) = 0} is unbounded. For brevity, we say that equation (1) is oscillatory in E if every solution u ∈ C2(E) of (1) is oscillatory in E.

Embedding theorems are employed to show that many important categories do not possess non-trivial retracts or injectives. E.g., the categories of monoids, groups, rings, rings with unity, polynomial identity rings, nilpotent groups, solvable groups, and several varieties of groups.