Let a group G be covered by finitely many disjoint cosets of subgroups Gi. We study conditions which imply that the subgroups Gi are conjugate in G.

The cyclotomic identity

where and μ is the classical Möbius function, is shown to be a consequence of a natural isomorphism of species.

Let f(z) be an n–Valued algebroid function of finite lower order. In the present paper, we give a spread relation of f(z) and some applications of the spread relation.

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

We give sufficient conditions for C∞ vector field systems on Rn with genus g = 1 to be diffeomorphic to a contact structure. The diffeomorphism is explicitly constructed and used to give the most general integral submanifolds for the systems. Finally the implications of these results for integrable hyperbolic partial differential equations in the plane is discussed.

In this paper we study the isometric deformations of surfaces in E3, which preserve the lines of curvature. We call a surface M and LC-surface if it admits a non-trivial deformation of this type. We distinguish three types of LC-surfaces, and obtain some new results about these three types of surfaces.

Let G be a finite group. It is proved that the localised Buinside ring Ωp(G) is of finite representation type if and only if for each p–perfect subgroup H of G, , where means the conjugacy class of K.

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

We study the following question: given any local formation of finite groups, do there exist maximal local subformations? An answer is given by providing a local definition of the intersection of all maximal local subformations.

Let S(G) be a Segal algebra on an infinite compact Abelian group G. We study the existence of many discontinuous translation invariant linear functionals on S(G). It is shown that if G/CG contains no finitely generated dense subgroups, then the dimension of the linear space of all translation invariant linear functionals on S(G) is greater than or equal to 2C and there exist 2C discontinuous translation invariant linear functionals on S(G), where c and CG denote the cardinal number of the continuum and the connected component of the identity in G, respectively.

An orbital integral formula is proven for the direct integral decomposition of an induced representation of a connected nilpotent Lie group. Previous work required simple connectivity. An explicit description of the spectral measure and spectral multiplicity function is derived in terms of orbital parameters. It is also proven that connected (but not necessarily simply connected) exponential solvable symmetric spaces are multiplicity free. Finally, the qualitative properties of the spectral multiplicity function are examined via several illuminating examples.

In this paper, we study a nonlinear partial differential equation on a compact manifold;

where a > 1 is a constant, r is a positive constant, and H is a prescribed smooth function.

Kazdan and Warner showed that if λ1(g) < 0 and < 0, where is the mean of H, then there is a constant 0 < r0(H) ≤ ∞ such that one can solve this equation for 0 < r < r0(H), but not for r > r0(H). They also proved that if r0(H) = ∞, then H(x) ≤ 0 (≢0) for all x ∈ M. They conjectured that this necessary condition might be sufficient.

I show that this conjecture is right; that is, if H(x) ≤ 0 (≠ 0) for all x ∈ M, then r0(H) = ∞.

A generalisation of the notion of “sets of measure zero” for arbitrary Banach spaces is defined so that continuous convex functions are automatically Gateaux differentiable “almost everywhere”. It is then shown that this class of sets satisfies all the properties tht one expects of sets of measure zero. Moreover (in a certain large class of Banach spaces, at least) nonempty open sets are not of “measure zero”.

In this paper we show that the average distance constant of a general polygon which is a subset of an M-space with non-positive curvature can be expressed as the extreme value of either of two nonlinear programs and discuss the practical application of one of these nonlinear programs for the determination of the average distance constant for a polygon in general, and in particular for a planar triangle.

We discuss the structure of some inverse semigroups and the associated C* algebras. In particular, we study the bicyclic semigroup and the free monogenic inverse semigroup, following earlier work of Conway, Duncan and Paterson. We then associate to each zero-one matrix A an inverse semigroup CA, and show that the C*-algebra OA of Cuntz and Krieger is closely related to the semigroup algebra C*(CA).

This paper resulted from an attempt to answer questions like: does βP have a subspace homeomorphic to βQ and does βQ have a subspace homeomorphic to P, where P denotes the space of all irrational numbers. These questions are answered in the negative by providing the appropriate machinery which can also be applied to other examples. En route we prove that weakly Fréchet realcompact spaces have homeomorphic Stone-Čech compactifications if and only if they are homeomorphic.