We give the generating function of split $(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split $n$-colour partitions. We derive a combinatorial relation between the number of restricted split $n$-colour partitions and the function $\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with $d(a)$ copies of each part $a$ and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’, Indian J. Pure Appl. Math 22(9) (1991), 737–743].

In this note we consider Hermite-Fejér interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejér type operators and special values of α, β it was proved by the first author in recent papers that one has uniform convergence on the whole interval [−1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejér interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejér type interpolation polynomials , converge pointwise to f for arbitrary α, β > −1. The convergence is uniform on [−1 + δ,1 − δ].

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

The isodiametric problem in the Euclidean plane is solved for bounded convex sets, which are symmetric about the origin, and which contain no interior non-zero point of an arbitrary lattice L.

A relationship is pointed out between the results in a recent paper of Shin's and those in a previously published paper by M.J.C. Gover.

For three possible types of tide-well systems the non-dimensional head response, Y(τ), to a sinusoidal fluctuation of the sea-level is given by the differential equation Estimates of the non-dimensional well response Z(τ) = sinτ - Y(τ) are found by considering the steady state solutions of the above equation. With n = 2 the equation is linear and an exact solution can be found; for n ≠ 2 the equation is non-linear and several methods which give approximate solutions are described. The methods used can be extended to cover other values of n; for example, with n = 4 the equation corresponds to one governing oscillations near resonance in open pipes.

It is shown that a saturated formation F has the property that in each group G each F-projector of G is complemented if and only if F is the formation of finite soluble π-groups for some set π of primes.

Müntz's theorems give necessary and sufficient conditions for a sequence of powers in one variable to be complete in the spaces of all real-valued continuous functions or square integrable functions with the usual norms.

A consideration of the separation properties of pre-neighbourhood lattices, leads to the definition and characterisation of T1-neighbourhood lattices in terms of the properties of the neighbourhood mapping, independently of points. It is then shown that if net convergence is defined in neighbourhood lattices as a consequence of replacing ‘point’ by ‘set’ in topological convergence, then the limits of convergent nets are unique. The relationship between continuity and convergence is established with the proof of the statement that a residuated function between conditionally complete T1-neighbourhood lattices is continuous if and only if it preserves the limit of convergent nets. If P(X) denotes the power set of X, then the observation that a filter in a topological space (X, T) is a net in P(X) leads to a discussion of the net convergence of a filter as a special case of net convergence. Particular attention is paid to maximal filters, Fréchet filters and to the filter generated by the limit elements of a net. Further, if the ‘filter’ convergence of a filter F in a topological space (X, T) is given by , if η(x) ⊆ F, then the relationship between ‘filter’ convergence and the net convergence of a filter in P(X) is established. Finally, it is proved that, in the neighbourhood system ‘lifted’ from a topological space to P(P(X)), the continuous image of a filter which converges to a singleton set is a convergent filter with the appropriate image set as the limit.

According to results of Heineken and Stadelmann, a non-torsion group is a 2-Baer group if and only if it is 2-Engel, and it has all subgroups 2-subnormal if and only if it is nilpotent of class 2. We extend some of these results to values of n greater than 2. Any non-torsion group which is an n-Baer group is an n-Engel group. The converse holds for n = 3, and for all n in the case of metabelian groups. A non-torsion group without involutions having all subgroups 3-subnormal has nilpotency class 4, and this bound is sharp.

Every subgroup of finite index of the modular group PSL (2, ℤ) has a signature consisting of conjugacy-invariant integer parameters satisfying certain conditions. In the case of genus zero, these parameters also constitute a prescription for the degree and the orders of the poles of a rational function F with the property:

Functions correspond to subgroups, and we use this to establish necessary and sufficient conditions for existence of subgroups with a certain subclass of allowable signatures.

We consider Fitting classes for which the injectors in any finite solvable group are modular subgroups. It is shown that only normal Fitting classes have this property. In fact, we prove two more general results demonstrating that modular Fitting functors and submodular Fitting classes are normal.

The author has noticed a faulty argument in his paper [1]; it occurs on p. 85 and requires the following changes:

At the end of line 9, for (Qi)i=1,2,… read (Qi)i ∈ I; in line 10, the equation should read ; lines 15 to 20 should be replaced by the following: then there exists some n–tuple δ = (δ1, …, δn) such that for every io there exists i ≥ io with εi = δ. Then δ1a1, …, δnan are positively independent over Q; for if rj ∈ PR and then there exists io such that , and for appropriate i ≥ io it follows that rj = 0 for j = 1, …, n.

Willis has established a structure theory of locally compact totally disconnected groups. An important feature of this theory is the notion of a tidy subgroup. In this paper we provide several results regarding these subgroups.

In this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

It is shown that the problem of deciding whether a polygonal curve c in a finite surface complex K is knotted in K is complete recursively enumerable, and hence unsolvable.

We study the following Dirichlet problem for the degenerate elliptic Monge-Ampère equation: Given , f ≥ 0 and , find a solution , t ≥ 2, satisfying in Ω and u = g on ∂Ω. Since f is nonnegative, we cannot apply any standard elliptic methods. In this paper, we use an iteration scheme of Nash-Moser type and a priori estimates for degenerate elliptic operators, and solve the Dirichlet problem for a certain class of f and g.

Sufficient conditions are derived for the solutions of a system of linear delay-differential equations to behave asymptotically as those of the same system without the argument delays. A generalisation of the result to integrodifferential systems is indicated.

In this paper we modify the classical Gröbner basis technique and prove the existence of a characteristic polynomial in two variables associated with a finitely generated module over a Weyl algebra. We determine invariants of such a polynomial and show that some of the invariants are not carried by the Bernstein dimension polynomial of the module.