In this paper we have established some fixed point theorems in complete and compact metric spaces.

We present some results concerning the structure of polynomially normal operators. It is shown, among other things, that if Tn is normal for some n > 1, then T is quasi–similar to a direct sum of a normal operator and a compact operator and if p(T) is normal with T essentially normal, then T can be written as the sum of a normal operator and a compact operator. Utilizing the direct integral theory of operators we finally show that if p(T) is normal and T*T commutes with T + T*, then T must be normal.

In a large university with a wide range of student options it is often impossible to cater for all allowable student courses of study within the constraints of a practical timetable, and a decision must be made to exclude some options. The entropy of choice available to students is used to develop a measure of timetable efficiency which balances the desirability of having both popular and rarer options available against the need to deny some students of their chosen courses. This efficiency gives a way for ranking the merit of different timetable exclusions. A simple numerical example is used for illustrations.

Rao, Rodabaugh, Wilke and Zemmer [J. Combin. Theory Ser. A. 11 (1971), 72–92] constructed a number of new VW systems called C-systems from the exceptional near–fields and established that they coordinatize translation planes not isomorphic to generalized André planes. In this paper the translation complement of the plane coordinatized by the C-system I–1 has been found. This plane has the interesting property that its translation complement divides the ideal points into two orbits of lengths 10 and 16. Further, the translation complement contains a subgroup isomorphic to SL(2,5) and therefore one of the exceptional Walker's planes of order 25 [H. Luneberg, Translation Planes, Springer-Verlag (1980), pp.235–244] is indeed the C–plane corresponding to the C–system I–1, which was discovered in 1969.

Let (A, B) denote the class of certain p-valent starlike functions. Recently G. Lakshma Reddy and K.S. Padmanabhan [Bull. Austral. Math. Soc. 25 (1982), 387–396] have shown that the function g defined by

belongs to the class (A, B) if f ∈ (A, B). The technique used by them fails when c is any positive real number. In this paper, by employing a more powerful technique, we improve their result to the case when c is any real number such that c ≥ −p(1+A)/(1+B).

It is known that diagonal forms over a finite field have non-trivial zeros if the number of variables, or the size of the field, is large enough. We consider diagonal forms of degree up to five in cases where the size hypotheses are not satisfied. There are finitely many fields not covered by the known results, but a direct computational test of all possible equations is impractical. We describe means of cutting down considerably on the number of fields and the number of equations for which there exist diagonal forms, of degree up to 5 and in 3 variables, with no non-trivial zero.

A class of translation planes of order p2r, where r is an odd natural number and p is a prime, p ≥ 7, p ≢ ± (mod 10) is constructed. A salient feature shared by all these planes is that one ideal point is fixed by the translation complement and the remaining ideal points are divided into at least two orbits, one of which is of length pr.

We give examples of finite groups of odd prime power order in which the commutators lying in the centre do not generate the intersection of the centre and the commutator subgroup.

Given positive integers k, l, m, the (k, l, m) triangle group has presentation δ(k, l, m) = < X, Y, Z | Xk = Yl = Zm = XYZ = 1 >. This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.

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.

We prove the existence of generalized periodic solutions of the boundary value problem for the nonlinear heat equation. The proof is based on classical Leray-Schauder's techniques and coincidence degree.

Let α be an algebraic integer of degree n > 1 with conjugates α = α1, α2, …, αn. Let D(α) denote the diameter of {α1, …, αn} and diam . In 1929 Favard showed that diam and that, when n = 2 or 3, the minimal values of diam (α) are less than 2. The author shows that diam and He also finds all algebraic integers a with diam (α) ≤ 2 when n ≤ 5. Similar results are found for D(α) and, in particular, D(α) > 2 when n = 5.

We consider a certain real hypersurface M of a complex projective space. The purpose of this paper is to characterize M in terms of Ricci curvatures.

Let A ⊂ C(S) be a function algebra. In this paper we prove that is a weak peak set for A if and only if for any open neighbourhood U of S0 there is an f in A such that ∥f∥ ≤ 2, |f(x) − l| ≤ 1/3 on S0 and |f(x)| ≤ 1/3 on S\U.

Let R be a commutative subdirectly irreducible ring, with minimal ideal M. It is shown that either R is a field, or M2 = 0. A construction is given which yields commutative sub-directly irreducible rings possessing nonzero-divisors, and nonzero nilpotent elements either with a unity element, or without. Such a ring without unity has been constructed by Divinsky. The same technique enables the construction of subdirectly irreducible rings with mixed additive groups.

Suppose X is a 4-dimensional complete intersection in CPr+4 of multidegree d1, …, dr. We show that X supports infinitely many almost complex structures for exactly 8 possible multi-degrees. In particular, a hypersurface of degree d in CP5 admits infinitely many almost complex structures if and only if d = 2 or 6. This generalizes a result of E. Thomas [4] for CP4. We give also some tables of possible Todd genera and a result for complex surfaces.

Coproducts and tensor products of algebras for essentially algebraic theories are exhibited as Kan extensions and relationships with derivations on monoidal closed categories are described.