Let be a Lie algebra, a complemented ideal of , and W an -module. Hochschild and Mostow have described the construction of a -module “induced” from W, which is finite-dimensional provided W is finite-dimensional and satisfies a nilpotent action condition. This note describes a modification of their construction which is functorial and a weak adjoint to the restriction functor from –modules to -modules.

We study Ohno–Zagier type relations for multiple t-values and multiple t-star values. We represent the generating function of sums of multiple t-(star) values with fixed weight, depth and height in terms of the generalised hypergeometric function $\,_3F_2$ . As applications, we get a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula for sums of multiple t-(star) values with fixed weight and depth.

For a positive integer n, let $\mathcal T(n)$ denote the set of all integers greater than or equal to n. A sum of generalised m-gonal numbers g is called tight $\mathcal T(n)$ -universal if the set of all nonzero integers represented by g is equal to $\mathcal T(n)$ . We prove the existence of a minimal tight $\mathcal T(n)$ -universality criterion set for a sum of generalised m-gonal numbers for any pair $(m,n)$ . To achieve this, we introduce an algorithm giving all candidates for tight $\mathcal T(n)$ -universal sums of generalised m-gonal numbers. Furthermore, we provide some experimental results on the classification of tight $\mathcal T(n)$ -universal sums of generalised m-gonal numbers.

Let G be a compact abelian group with dual group Γ. For 1 ≤ p < ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.

The aim of this paper is to study the relationship between the nature of an infinite series of real terms in which the general term tends to zero and the derived set (set of limit points) of the aggregate of the fractional parts of its partial sums. For all types of series (in which the nth term tends to zero) we determine the derived set.

We denote by F the set of the fractional parts of the partial sums of the series, and by F′ the derived set of F. The principal results of the paper can be stated as follows:

1. the series is convergent if and only if F has at most one limit point in [0, 1] or F′ ⊆ {0, 1};

2. the series, if non-oscillatory, is divergent if and only if F′ = [0, 1];

3. if the series is oscillatory, F′ is a closed sub-interval of [0, 1] or [0, 1] - (a, b), where (a, b) ⊊ (0, 1).

Integral comparison theorems of Hille-Wintner type of second order linear equations are shown to be valid for the third order linear equation y‴ + q(t)y = 0.

For a graph G, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of G. Given a positive integer m and a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as G ranges over all graphs on m edges that contain no copy of H. We prove bounds on $f(m,H)$ for some bipartite graphs H and give a bound for a conjecture of Alon et al. [‘MaxCut in H-free graphs’, Combin. Probab. Comput. 14 (2005), 629–647] concerning $f(m,K_{4,s})$.

Let D be a domain bounded by a Jordan curve. For 1 ≤ p ≤ ∞, let Lp(D) be the class of all functions f holomorphic in D such that where A is the area of D. For f ∈Lp(D), set

πn consists of all polynomials of degree at most n. Recently, Andre Giroux (J. Approx. Theory 28 (1980), 45–53) has obtained necessary and sufficient conditions, in terms of the rate of decrease of the approximation error , such that has an analytic continuation as an entire function having finite order and finite type. In the present paper we have considered the approximation error (*) on a Carathéodory domain and have extended the results of Giroux for the case 1 ≤ p < 2.

We give sufficient conditions for a homogeneous superposition operator to be a continuous mapping between Orlicz-Sobolev spaces. This extends a result of Marcus and Mizel concerning mappings between Sobolev spaces.

We study integral transforms of functions belonging to the Jakubowski class S(m, M) and determine the range of values of the exponent for which the integral is a convex or a close to convex function.

Fibonacci varieties were introduced by one of us in 1978 and a natural generalisation was studied shortly afterwards. We carry this investigation one stage further by giving a description of the free objects in these varieties. This is done in terms of the n-abelian groups of Levi.

Convergence notions are used extensively in the areas of probability and statistics. Many times proofs can be simplified by considering an appropriate convergence structure on the space and using well-known results from the theory of convergence spaces; for example, compactness arguments are sometimes simplified by using a generalized Ascoli theorem in the convergence space setting. The theory of convergence spaces is also used to generalize some results in probability and statistics.

We derive explicit formulae for the height of the Leech lattice and give evidence supporting the conjecture that the Leech lattice has minimum height and minimum values of the Epstein zeta function among all 24 dimensional lattices of determinant one.

The conditions in the strong law of large numbers given by Li et al. [‘A strong law for B-valued arrays’, Proc. Amer. Math. Soc.123 (1995), 3205–3212] for B-valued arrays are relaxed. Further, the compact logarithm rate law and the bounded logarithm rate law are discussed for the moving average process based on an array of random elements.

The asymptotic behavior of nonoscillatory solutions of nth order nonlinear functional differential equations

is investigated. Sufficient conditions are provided which ensure that all nonoscillatory solutions approach zero as t → ∞.