Sufficient conditions are obtained for all solutions of a general scalar linear functional differential equation to be oscillatory. Our main theorem concerns some particular cases of a conjecture of Hunt and Yorke.

We prove in this paper that every finitely generated critical module over A[X] is compressible where A is a Noetherian ring integral over a subring of its centre. Here A[X] denotes the polynomial ring over A in a commuting indeterminate X.

Whereas a locally convex Hausdorff space X inherits any completeness properties that the space of continuous linear operators, L (X), in X, may have (for the topology of pointwise convergence in X), this is not so in the converse situation and is the problem discussed here. The barrelledness of X in its Mackey topology plays an important role: if L (X) is quasicomplete, then X is barrelled for its Mackey topology. Consequently, for Mackey spaces X is turns out that L (X) is quasicomplete if and only if X is quasicomplete and barrelled: this is false if sequential completeness is substituted for quasicompleteness. Furthermore, there exist non-barrelled spaces X for which X and L (X) are quasicomplete (sequentially complete). Hence, although barrelledness is a sufficient condition for completeness of L (X) in various senses, it is certainly not necessary.

The planar Navier-Stokes equations with periodic boundary conditions are shown to have a nearby asymptotically stable attracting set whenever a Galerkin approximation involving the eigenfunctions of the Stokes operator has such an attracting set, provided the approximation has sufficiently many terms and its attracting set is sufficiently strongly stable. Lyapunov functions are used to characterize the stability of these attracting sets, which are compact sets of arbitrary geometric shape. This generalizes earlier results of Constantin, Foias and Temam and of the author for asymptotically stable steady solutions in the Navier-Stokes equations and such Galerkin approximations.

We prove a Korovkin-type theorem on approximation of bivariate functions in the space of B-continuous functions (introduced by K. Bögel in 1934). As consequences, some sequences of uniformly approximating pseudopolynomials are obtained.

Let w be a nonnegative weight function in L1 = L1 (dθ/2π). Let Q and P denote the orthogonal projections to the closed linear spans in L2(wdθ/2π) of {einθ: n ≤ 0} and {einθ: n > 0}, respectively. The commutator of Q and P is studied. This has applications for prediction problems when such a weight arises as the spectral density of a discrete weakly stationary Gaussian stochastic process.

The purpose of this note is to record a few formulas relating the indices of various lattices and sublattices all arising from the decomposition of a euclidean space E into three mutually orthogonal subspaces E = E0 ⊥ E1 ⊥ E2 which are rational with respect to a given lattice Λ ⊂ E. In the case that Λ is unimodular these formulas simplify to give very simple identities between various intersection lattices.

The relation between the subclass X of balanced one-sided ideals, which contains all two-sided ideals, and weakly prime one-sided ideals is considered. If a member P of X is prime, then it is prime in its idealizer I (P). Furthermore, if the ring is left and right noetherain, then I (P) modulo P is a prime Goldie ring.

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

In this paper we extend Kurose's structure theorem to characterize a closed derivation in the algebra of continuous functions on the unit square, under the conditions that the range is the whole algebra and the kernel is the set of all functions depend only on the second variable, as a partial derivative with respect to signed measures on the unit square.

Recently, Ky Fan extended his will known lemma (which is an extension of the classical theorem of Knaster, Kuratowski and Mazurkiewica) to the noncompact case. Using this result, another interesting lemma of Fan is generalized in this paper. As applications of our theorem, we obtain a generalizationof Browder's variational inequality and derive Fan's other recent results directly from our theorem. Also, in this paper, we give a slight extension recent results of K. K. Tan, which themselves are generalizations of many well-known results on minimax and variational inequalities.

We present some results concerning the trace of certain trace class commutators of operators acting on a separable, complex Hilbert space. It is shown, among other things, that if X is a Hilbert-Schmidt operator and A is an operator such that AX − XA is a trace class operator, then tr (AX − XA) = 0 provided one of the following conditions holds: (a) A is subnormal and A*A − AA* is a trace class operator, (b) A is a hyponormal contraction and 1 − AA* is a trace class operator, (c) A2 is normal and A*A − AA* is a trace class operator, (d) A2 and A3 are normal. It is also shown that if A is a self - adjoint operator, if f is a function that is analytic on some neighbourhood of the closed disc{z: |z| ≥ ||A||}, and if X is a compact operator such that f (A) X − Xf (A) is a trace class operator, then tr (f (A) X − Xf (A))=0.

Necessary and sufficient conditions are given for certain classes of eventually regular semigroups to the group-bound or even periodic.

In this paper the Perron integrability of the limit of a sequence of functions is proved under certain convergene conditions.

The object of this paper is to generalize the classic theorems of Eilenberg and Debreu on the existence of continuous order-preserving transformations on ordered topological spaces and to prove them in a different way. The proof of the theorems is based on Nachbin's generalization to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.

We give a new proof of the hypercentre theorem of Herstein.

In [1], Herstein has defined the hypercentre of a ring R as follows:

Herstein has proved:

THEOREM. If R is a ring without non-zero nil ideals, thenT (R) = Z (R).