We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

Given f ∈ C [−1, 1], let Hn, 3(f, x) denote the (0,1,2) Hermite-Fejér interpolation polynomial of f based on the Chebyshev nodes. In this paper we develop a precise estimate for the magnitude of the approximation error |Hn, 3(f, x) − f(x)|. Further, we demonstrate a method of combining the divergent Lagrange and (0,1,2) interpolation methods on the Chebyshev nodes to obtain a convergent rational interpolatory process.

The concept of a convex space is extended to an H-space; that is, a space having certain family of contractible subsets. For such spaces the KKM type theorems, the Fan-Browder fixed point theorem, the Ky Fan type matching theorem, and minimax inequalities are given. Moreover, applications to a von Neumann-Sion type minimax theorem, a saddle point theorem, a quasi-variational inequality, and a Kakutani type fixed point theorem are obtained.

We study the sets of periods of triangular maps on a cartesian product of arbitrary spaces. As a consequence we extend Kloeden's Theorem (in a 1979 paper) to a class of triangular maps on cartesian products of intervals and circles. We also show that, in some sense, this is the more general situation in which the Sharkovskiĭ ordering gives the periodic structure of triangular maps.

In this note, we prove a signature product formula for generalised Seifert fibrations. We also discuss how this result can be viewed using the theory of minimal models.

If (X′, τ′, ≤′) is an ordered compactification of the partially ordered topological space (X, τ, ≤) such that ≤′ is the smallest order that renders (X′, τ′, ≤′) a T2-ordered compactification of X, then X′ is called a Nachbin (or order-strict) compactification of (X, τ, ≤). If (X′, τ′, ≤*) is a finite-point ordered compactification of (X, τ, ≤), the Nachbin order ≤′ for (X′, τ′) is described in terms of (X, τ, ≤) and X′. When given the usual order relation between compactifications (ordered compactifications), posets of finite-point Nachbin compactifications are shown to have the same order structure as the poset of underlying topological compactifications. Though posets of arbitrary finite-point ordered compactifications are shown to be less well behaved, conditions for their good behavior are studied. These results are used to examine the lattice structure of the set of all ordered compactifications of the ordered topological space (X, τ, ≤).

This paper indicates how to apply the ellipsoid method directly to Linear Programming problems and proves that this kind of version of the ellipsoid method is almost as good as Karmarkar's type method in the theoretical sense.

With every poset we associate a class of coloured posets called zigzags. By means of zigzags we show that, if we delete a convex set from a finite lattice ordered set then the resulting poset has the strong selection property. We give the complete list of finite bounded irreducible posets admitting an n-ary near unanimity function, provided n ≤ 6. We present some examples and classes of posets with full descriptions of their zigzags.

The concepts of closed submodule, left and right annihilators are generalised and a necessary and sufficient condition on any Morita context is given so that the duality and projectivity between these sets exist. As a special case, we get Hutchinson's necessary and sufficient condition under a weaker hypothesis on the context.

This paper is a sequel to a recent paper by the author in this journal. We prove some -variants of the min-max type critical point theorems with relaxed boundary condition and then apply the abstract results to a semilinear elliptic boundary value problem.

If (Ω, Σ, μ) is a finite measure space and X is a normed space such that X* has the Radon-Nikodym property with respect to μ, we show first that each space Lp(μ, x), 1 < p < ∞, is ultrabornological whenever μ is atomless. When μ is arbitrary, we prove later on that the space Lp(μ, X) is ultrabornological if X* has the Radon-Nikodym property with respect to μ and X is itself an ultrabornological space.

This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others.

We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates.

We also establish a version of the Lévy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.

We give 8 short proof of the classical Stallings theorem that every finite n-dimensional cellular complex embeds up to homotopy in the 2n-dimensional Euclidean space. As an application we solve a problem of M. Kreck.

For a complete local ring R with maximal ideal m, we define a linear topology on a Matlis reflexive R-module M which coincides with the m-adic topology on M in case M is finitely generated. We show that a Matlis reflexive module is complete in this topology.

In this note we obtain a convolution identity for the coefficients Bn(α, θ, q) defined by

using Ramanujan's 1Ψ1 summation. The identity contains as special cases convolution identities of Kung-Wei Yang and a few more interesting analogue.

Sufficient conditions are given for the existence of an mth power root of one polynomial modulo another, over the complexes or the reals. Examples show the non-necessity of the conditions. In particular cases there can exist infinitely many square roots.

Let G be a separable locally compact group which admits a non-trivial compact normal subgroup. It is shown that the group C*-algebra C*(G) of G may be decomposed as a direct sum of ideals whose structure is determined up to *-isomorphism. Applications are given to Type 1, [FD]− groups; in particular it is shown that the group C*-algebra of such a group is a direct sum of homogeneous C*-algebras.