Let B0 ⊂ B1 ⊂ Bn ⊂ … ⊂: Bw be all the non-trivial varieties of distributive pseudocomplemented lattices (L; ∨, ∧, *, 0, 1) considered as algebras of type (2, 2, 1, 0, 0). A subset J of such an algebra L is a congruence-kernel if and only if it is a lattice-ideal and x** ∈ J for each x ∈ J. The smallest congruence having J as its kernel is Θ(J), where a ≡ b (Θ(J)), (a, b ∈ L) if and only if a ∧ c* = b ∧ c* for some c ∈ J. For given 0 ≤ n ≤ w, let Σn (J) be the smallest congruence having J as its kernel and such that the associated quotient algebra is in Bn. Of course, Σw (J) = Θ(J) and the main result of this paper shows that for 1 ≤ n < w, Σn(J) = ∩{Θ(P1 ∩ P2 ∩ … ∩ Pn): J ⊆ P1, P2, …, Pn ⊂ L are minimal prime ideals}.

Outcalt and Yaqub have extended the Wedderburn Theorem which states that a finite division ring is a field to the case where R is a ring with identity in which every element is either nilpotent or a unit. In this paper we generalize their result to the case where R has a left identity and the set of nilpotent elements is an ideal. We also construct a class of non-commutative rings showing that our generalization of Outcalt and Yaqub's result is real.

It is shown that the Petrov-Plebański classification of the trace-free Ricci tensors of some spherically symmetric metrics is invariant, contrary to an assertion by Takeno and Kitamura concerning these metrics.

Little seems to be known about the digits or sequences of digits in the decimal representation of a given irrational number like √2 or π. There is no difficulty in constructing an irrational number such that in its decimal representation certain digits or sequences of digits do not occur. On the other hand, well known theorems by Tchebychef, Kronecker, and Weyl imply that some integral multiple of the given irrational number always has any given finite sequence of digits occuring at least once in its decimal representation: for the fractional parts of the multiplies of the number lie dense in the interval (0, 1).

For a commutative l–group G with carrier lattice , G is weakly projectable iff the proper prime filters of are all maximal filters.

A constructive proof is given of Gelbaum's result that the maximal ideal space of the tensor product of commutative Banach algebras is homeomorphic to the cartesian product of the maximal ideal spaces.

Necessary conditions of the Fritz John type are given for a class of nonlinear programming problems over polyhedral cones in finite dimensional complex space.

Consider the problem to

where S is a polyhedral cone in, and Cm and f: C2n → C, g : C2n → Cm are differentiable functions. A necessary condition for a feasible point z0 to be optimal is that there exist τ≥0, ν ∈ S*, (τ, ν) ≠ 0, such that

and

In constructive mathematics the Dedekind cut definition of real number is not equivalent to the definition of real number by Cauchy sequences, and the Dedekind real numbers do not satisfy Heyting's axioms for constructive fields. A more general notion of constructive field is proposed which includes the Dedekind real numbers; some linear algebra is given which applies to such fields.

Chase has given several characterizations of a right coherent ring, among which are: every direct product of copies of the ring is left-flat; and every finitely generated submodule of a free right module is finitely related. We extend his results to obtain conditions for the ring of quotients of a ring with respect to a torsion theory to be coherent.

The existence of a certain type of Rudin-Shapiro sequence of functions is shown for all infinite compact groups which are not Lie, thus extending a recent result of the authors to all infinite compact groups.

As a special case of more general results, it is proved in this note that, if α is any real number and δ any positive number, then there exists a positive integer X such that the inequality

has infinitely many solutions in positive integers h and Yh.

The method depends on the study of infinite sequences of real linear forms in a fixed number of variables. It has relations to that used by Kronecker in the proof of his classical theorem and can be generalised.

This paper shows the existence of a one-to-one correspondence between a certain class of square matrices of arbitrary order and a related extension field. The elements of these matrices are obtained from certain basic linear recursive sequences by means of a generalization of the euclidean algorithm.

In this paper we present some approximation theorems for the eigenvalue problem of a compact linear operator defined on a Banach space. In particular we examine: criteria for the existence and convergence of approximate eigenvectors and generalized eigenvectors; relations between the dimensions of the eigenmanifolds and generalized eigenmanifolds of the operator and those of the approximate operators.

Carleson has proved that the Fourier series of functions belonging to the class L2 converge almost everywhere.

Improving his method, Hunt proved that the Fourier series of functions belonging to the class Lp (p > 1) converge almost everywhere. On the other hand, Kolmogoroff proved that there is an integrable function whose Fourier series diverges almost everywhere. We shall generalise Kolmogoroff's Theorem as follows: There is a function belonging to the class L(logL)p (p > 0) whose Fourier series diverges almost everywhere. The following problem is still open: whether “almost everywhere” in the last theorem can be replaced by “everywhere” or not. This problem is affirmatively answered for the class L by Kolmogoroff and for the class L(log logL)p (0 < p < 1) by Tandori.

If the finite soluble group G admits the dihedral group of order eight as a fixed-point-free group of automorphisms then the nilpotent length of G is at most three.

There are errors in my paper [1], Lemma 1 and from the end of Lemma 4 to the end of the paper.

LEMMA 1. If L is any finite lattice and L has a point which is both join and meet reducible then L is not weakly transferable.

We replace Lemma 5 by the following sequence.

Professor William F. Reynolds has discovered a mistake in the main theorem of [1]. The problem occurs with the first display on page 339 in that is only congruent modulo radZ(G) to . The theorem still holds if θ(radZ(G)) ⊆ radZ(H), but this is not the case in general.