Let Un be a U-statistic whose kernel depends on the size n of the sample under consideration. It is shown that when Un is suitably normalised its distribution function differs in Lp norm from the distribution function of a standard normal variable by a term of O(n-½).

The rational fraction

a, c, p, q positive integers, reduces to a polynomial under conditions specified in a result of Grosswald who also stated necessary and sufficient conditions for all the coefficients to tie nonnegative.

This last result is given a different proof using lemmas interesting in themselves.

The method of proof is used in order to give necessary and sufficient conditions for the positive coefficients to be equal to one. For a < 2pq, a = αp + βq, α, β nonnegative integers, c > 1, the exact positions of the nonzero coefficients are established. Also a necessary and sufficient condition for the number of vanishing coefficients to be minimal is given.

In this note we show that a paranormal operator T on a Banach space satisfies Weyl's theorem. This is accomplished by showing that

(i) every isolated point of its spectrum is an eigenvalue and the corresponding eigenspace has invariant complement,

(ii) for α ≠ 0, Ker(T-α) ⊥ Ker (T-β) (in the sense of Birkhoff) whenever β ≠ α.

It is shown that, for a semi-topological functor T: A → X, the category A is total, that is, the Yoneda embedding of A has a left adjoint, if X is total. In particular, monadic categories over Set (possibly without rank) are total, and full reflective subcategories of total categories are total.

Let K be a bounded closed convex set in the plane containing no points of the integral lattice in its interior and having width w, area A, perimeter p and circumradius R. The following best possible inequalities are established:

Recent developments on the isomorphism and other group ring problems are amply reviewed in Sehgal's book, Topics in group rings. The aim of this expository paper is to complement the content of Sehgal's book. Our main emphasis is the presentation of some results due to Saksonov which are published in Russian and do not seem well-known to the English reader. We also draw the reader's attention to some unpublished results of Higman.

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.

In this paper some nonlinear analogues of Gronwall's integral inequality are established and an application to differential equations is given.

Existence and uniqueness theorems for the nonlinear complementarity problem over closed convex cones in a reflexive real Banach space are established.

Spaces with homogeneous norms are closely related to the Beppo Levi spaces of Deny and Lions, to spaces of Riesz potentials, and to Sobolev spaces. In this paper we survey the literature on them, give a broad extension of their definition, and present their basic theory. Many of the properties of Sobolev spaces have their analogues. In fact, the two families are locally equivalent. Spaces with homogeneous norms are especially suited to the study of boundary value problems on for homogeneous elliptic operators with constant coefficients. We will use them extensively in a forthcoming paper to study elliptic partial differential equations with mixed boundary conditions on a smoothly bounded domain.

This paper considers the question of when a space with the property that each discrete subspace is countable is hereditarily Lindelbölf. The question is answered affirmatively for the class of ROP spaces and for the class of hereditarily meta-Lindelöf spaces. A characterization of hereditarily Lindelöf spaces in terms of countable subspaces is given.

In this note it is shown that any ribbon link is a sublink of a ribbon link for which surgery on the longitudes gives a connected sum of copies of S1 x S2. In particular there are many links for which the analogue of the knot theoretic Property R fails, and sublinks of homology boundary links need not be homology boundary links. Higher dimensional analogues of these results are also given and it is shown that if n ≥ 2 the group of a ν-component ribbon n-link has a presentation of deficiency ν. Hence there are high dimensional slice knots which are not ribbon knots.

This paper establishes the radii of convexity of the following two classes of regular functions,

where

.

We discuss a computationally stable numerical method for the solution of linear programs and games. The method is useful in obtaining approximate solutions to large numerically unstable linear programs.

Let F be a family of k-element subsets of an n-set, n > n0(k). Suppose any two members of F have non-empty intersection. Let τ(F) denote min|T|, T meets every member of F. Erdös, Ko and Rado proved and that if equality holds then τ(F) = 1. Hilton and Milner determined max|F| for τ(F) = 2. In this paper we solve the problem for τ(F) = 3.

The extremal families look quite complicated which shows the power of the methods used for their determination.

This paper gives a very general sufficient condition for the existence of constants B(n), C(n) for which either almost surely or almost surely, where Sn = X1 + X2 + … + Xn and Xi are independent and identically distributed random variables. The theorem is closely connected with results of Klass and Teicher on the one-sided boundedness of Sn, with the relative stability of Sn, and with a generalised law of the iterated logarithm due to Kesten. For non negative Xi the sufficient condition is shown to be necessary, and the results are partially generalised to the case when Xi form a stationary m-dependent sequence. Some connections with a generalised type of regular variation and with domains of partial attraction are also noted.

The following theorem is proved: Let D be a division ring such that for all x, y in D there exists a positive integer n = n(x, y) for which (xy)n − (yx)n is in the center of D. Then D is commutative. This theorem also holds for semisimple rings.

The main theorem of this paper shows that a complex p-normed algebra which is a pre-Bezout domain is isomorphic to the field of complex numbers, if it is a generalized unique factorization domain. This theorem generalizes the previous result of the authors proved by them in their paper Bull. Austral. Math. Soc. 20 (1979), 247–252. Some applications are then given.

Let Xi be independent and identically distributed random variables with Sn = X1 + X2 + … + Xn. We extend a classic result of Kesten, by showing that if Xiare in the domain of partial attraction of the normal distribution, there are sequences αn and B(n) for which

almost surely, and the almost sure limit points of (sn−αn)/b(n) coincide with the interval [−1, l]. The norming sequence B(n) is slightly different to that used by Kesten, and has properties that are less desirable. The converse to the above result is known to be true by results of Heyde and Rogozin.

In this paper we investigate the spectrum of a variety of quasigroups satisfying the 2-variable identity x(xy) = yx, called Stein quasigroups. Stein quasigroups are known to be self-orthogonal and have been given a considerable amount of attention because of this property. It is known that there are no Stein quasigroups of order 2, 3, 6, 7, 8, 10, 12, 14. The object of this paper is to show that for all but 36 values of n ≥ 15 there exists a Stein quasigroup of order n. In particular, the spectrum of Stein quasigroups contains all n ≥ 191.