In a recent article, George E. Andrews considers a generalization of the Rogers-Ramanujan identities involving a pair of infinite sequences of positive integers, which he calls a Ramanujan pair. He lists the known Ramanujan pairs and conjectures that there are no more. The object of this note is to establish the existence of two further Ramanujan pairs.

A classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

For any class of rings it is shown that the class (M) of all rings each nonzero homomorphic image of which contains either a nonzero -ideal or an essential ideal is a radical class. If is a class of simple rings the upper radical generated by , (M), is shown to be equal to (M) where ' is the class of simple rings complementary to .

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

The aim of this paper is to show that some of the known properties of distributions in the domain of attraction of a stable law have counterparts for distributions which are stochastically compact in the sense of Feller. This enables us to unify the ideas of Feller and Doeblin, who first studied the concept of stochastic compactness, and give new characterizations of stochastic compactness and the domain of attraction of the normal distribution.

For totally real algebraic integers x of degree D(x), we examine the stucture of the set of values M(x)1D(x). where M(x) is the measure of x. We find a small limit point λ of this set, and show that the set is everywhere dense in (λ x).

This paper presents characterizations of optimality for the abstract convex program

when S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set and p and g are respectively convex and S-convex (on Ω). These characterizations, which include a Lagrange multiplier theorem and do not presume any a priori constraint qualification, subsume those presently in the literature.

The main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

We determine which varieties of commutative semigroups have the weak or strong amalgamation property. These are precisely the varieties of inflations of semilattices of abelian groups.

Reduced rings and lattice-ordered groups are examples of groups with Boolean orthogonalities. In this note we show that any group with a Boolean orthogonality satisfying a finiteness condition introduced by Stewart is isomorphic with a group of homeomorphisms of a topological space, in which two homeomorphisms are orthogonal if and only if they have disjoint supports.

The theme of the paper is a Möbius inversion principle for infinite sums. We deal with the origins and unprincipled use of this idea in the nineteenth century, its rigorous justification under minimal hypotheses and some applications to a problem in numerical integration.

Let R be a ring in which the multiplicative semigroup is completely semisimple. If R has the maximum (respectively, minimum) condition on principal multiplicative ideals. then R is semiprime artinian (respectively, a direct sum of dense rings of finite-rank linear transformations of vector spaces over division rings).

It is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y ∈ R, there exists a positive integer n depending on x and y such that (xy)k − xkyk is central for k = n,n+1, n+2, then R is commutative, thus generalizing a result of Kaya.

We are interested here in the Ramsey number r(T, C), where C is a complete k-uniform hypergraph and T is a “tree-like” k-graph. Upper and lower bounds are found for these numbers which lead, in some cases, to the exact value for r(T, C) and to a generalization of a theorem of Chváta1 on Ramsey numbers for graphs. In other cases we show that a determination of the exact values of r(T, C) would be equivalent to obtaining a complete solution to existence question for a certain class of Steiner systems.

An approach to p-adic interpolation via divided differences is used to give alternative proofs of results of van der Poorten on p-adic exponential polynomials and to derive a p-adic analogue of Turan's first main theorem on sums of powers.

It is shown that no proper ideal of a free inverse semigroup is free and that every isomorphism between ideals is induced by a unique automorphism of the whole semigroup. In addition, necessary and sufficient conditions are given for two principal ideals to be isomorhic.

It is shown that the simple groups G2(q), q = 3f, are characterized by their character table. This result completes characterization of the simple groups G2(q), q odd, by their character table.

The near-ring distributively generated by the semigroup of all endomorphisms of Sn, the symmetric group of degree n, for n ≥ 5, is close to being the near-ring of all mappings from Sn to itself respecting the identity. In this paper, the structure of these near-rings is studied in detail. In particular, addition and multiplication rules for the elements given in canonical form are determined. A complete list of all right ideals, left ideals, right invariant and left invariant subgroups is given.

Let G be a locally compact group G (which may be non-abelian) and Ap(G) the p-Fourier algebra of Herz (1971). This paper is concerned with the Fourier algebra Al, p(G) = Ap(G) ∩ L1(G) and various relations that exist between Al, p(G), Ap(G) and G.

Warren W. Wolfe obtained necessary conditions for the existence of orthogonal designs in terms of rational matrices. In this paper it is shown that these necessary conditions can be obtained in terms of integral matrices. In the integral form, Wolfe's theory is more useful in the construction of orthogonal designs.