We give explicit optimal curvature pinching constants for the Riemannian (p, q)- football orbifolds under the assumption that they are realised as surfaces of revolution in ℝ3. We show that sufficiently pinched sectional curvature assumptions imply that a (p, q)-football must be good.

A ring R is called right FSG if every finitely generated right R-subgenerator is a generator. In this note we consider the question of when a ring extension of a given right FSG ring is right FSG and the converse. As a consequence we obtain some results about right FSG group rings.

For an ideal I in a commutative ring R we consider the ideal In = ({in | i ∈I}). We show that if n! is a unit in R, then In = In. We give an example of a doubly generated ideal I with Is not finitely generated.

We prove a necessary and sufficient condition for a local homeomorphism defined on an open, connected subset of a Euclidean space to be globally one-to-one and, at the same time, for the image to be convex. Among the applications we give a practical sufficiency test for invertibility for twice differentiable local diffeomorphisms defined on a ball.

Let R be a right Noetherian ring with right global dimension bounded by 2, which is integral over its centre, and let a be a regular non-unit element in R. Then R/a; R is right hereditary if and only if a; is not in the square of any maximal ideal of R. More generally, we compare for a right Noetherian ring R which is integral over its center, the global dimension of R with the global dimension of R/(a1R + a2R + … + arR) for a regular R-sequence {ai}, which will allow us to give a considerable extension of a result of Hillman.

Mendelsohn triple system of order ν which can be extended to a tetrahedral quadruple system of order ν + 1 we call a derived Mendelsohn triple system. We consider some properties of derived Mendelsohn triple systems and give some results on their existence.

Suppose the qth derivative of a function f is Hölder continuous of index α, where 0 < α ≤ 1, on the interval [-1,1]. Suppose further that pn is any polynomial of degree at most n such that |τn(x)| = |f(x) - pn(x)| ≤ c {max ((1 - x2)½/n, 1/n2) } q+a [-1,1]. If

then it is shown that

‖τn‖β ≤ cn−q−α+β, 0 < β ≤ 1.

This paper deals with the equivalence between u−density and m−density for the subrings of C(X). It was proved by Kurzweil that such equivalence holds for those subrings that are closed under bounded inversion. Here an example is given in C(N) of a u−dense subring that is not m−dense. It is deduced that the two types of density coincide only in the trivial case where these topologies are the same, that is, if and only if X is a pseudocompact space.

Recently, Bhargava, Adiga and Somashekara made use of Ramanujan's 1Ψ1 summation formula to prove a convolution identity for certain coefficients generated by the quotient of two infinite products. As special cases of this identity, they deduced several results (connected especially with the generalised Frobenius partition functions) including, for example, the convolution identities given earlier by Kung-Wei Yang. In this sequel to the aforementioned works, we provide a complete answer to an interesting question raised by Bhargava, Adiga and Somashekara in connection with one class of their convolution identities.

We define the notion of Darboux integrability for linear second order partial differential operators,

.

We then build on certain geometric results of E. Vessiot related to the theory of Monge characteristics to show that the Darboux integrable operators L can be used to obtain a solution of the A2 Toda field theory. This solution is parametrised by four arbitrary functions. The approach presented in this paper thereby represents an alternative means of linearising the A2 Toda equations and may be contrasted with the known linearisation via the Lax pair.

Let M be a finite quasigroup of order n. For any integer k ≥ 2, let H(k, M) be the smallest positive integer h such that there exist h subsets Ai (i = 1, 2, …, h) such that Ai … Ah = M and |Ai| = k for every i = 1, 2, …, h. Define H(k, n) = max H(k, M). It is proved in this paper that

.

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

It is shown that if a partially-ordered topological space X admits a finite-point T2-ordered compactification, then it admits a countable T2-ordered compactification if and only if it admits n−point T2-ordered compactifications for all n beyond some integer m.

In this paper, we investigate a general class of variational inequalities. Existence and multiplicity results are obtained by using minimax principles for lower semicontinuous functions due to A. Szulkin.

The abstract triangle groups Δ(2, 4, r) can be defined for any positive integer r by Δ(2, 4, r) = 〈x, y | x2 = y4 = (xy)r = 1〉. In this paper we show that for every r ≥ 6, all but finitely many of the alternating groups An can be obtained as quotients of Δ(2, 4, r).

A ring R is called right principally injective (right P-injective) if every R-linear map from a principal right ideal of R can be extended to R. If every ring homomorphic image of R is right P-injective, R is called completely right P-injective (right CP-injective). In this paper we characterise completely quasi-Frobenius rings in terms of CP-injectivity.

Let P be a class of topological groups such that every topological group is isomorphic to a topological subgroup of the direct product (with Tychonoff topology) of a subfamily of P. Then every Tychonoff space is homeomorphic to a subspace of a group from P.

It has recently been shown that a Banach space enjoys the weak fixed point property if it is ε0-inquadrate for some ε0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘ε0-inquadrate in every direction’ for some ε0 < 2.