Johnstone and Paré have each given a way of constructing the largest topology allowing a given object to be a sheaf. In this paper we use the notion of a partial map to construct such a topology in a simple way.

In this paper we introduce the concept of almost nilpotence for Γ-rings, similar to the corresponding concept for rings, as defined by Van Leeuwen and Heyman. An almost mlpotent radical property Α0 is introduced for Γ-rings, and shown to be supernilpotent. If M is a Γ-ring with left and right operator rings L and R respectively, then Α(L)+ = Α0(M) = Α(R)*, where Α(−) denotes the almost nilpotent radical of a ring. If M is a Γ-ring and m, n are positive integers, then Α0(Mm, n) is the almost nilpotent radical of the Γn, m-ring Mm, n.

This paper considers a special class of stochastic optimal parameter selection problem subject to probability constraints on the state. The system dynamics are governed by a linear Ito stochastic differential equation with controllable parameters appearing nonlinearly in the dynamics. The problem seeks to optimise a cost functional which is quadratic in the state with weighting matrices being time invariant but depending nonlinearly on the parameters. Although the inclusion of the probability state constraints renders the problem insolvable by the conventional LQG theory, we show that the problem can in fact be transformed into an equivalent deterministic optimal parameter selection problem solvable by an existing software MISER. Numerical examples are presented to demonstrate the feasibility and efficiency of the proposed approach.

In terms of a new notion, called prebalancedness, those extensions of a Butler group by a Butler group which are again Butler groups are characterised. It is shown that the prebalanced extensions of torsion-free abelian groups form a divisible subgroup in the group of extensions.

Let g(n) denote the number of multiplicative partitions of the natural number n. We prove that

For a measurable function f on a probability space a basic inequality is ‖f‖p ≤ ‖f‖q where 1 ≤ p < q < ∞ and ‖f‖p denotes the Lp norm of f. The above inequality becomes an equality provided |f| is a constant almost everywhere. We obtain an improvement of the above inequality in all cases that |f| is not a constant almost everywhere.

A quadratic operator Q of Hilbert-Schmidt class on a real separable Hilbert space H is shown to be uniquely representable as a sequence of self-adjoint linear operators of Hilbert-Schmidt class on H, such that Q(x) = Σk〈Lkx, x〉uk with respect to a Hilbert basis . It is shown that with the norm | ‖Q‖ | = (Σk ‖Lk‖2)½ and inner-product 〈〈〈Q, P〉〉〉 = Σk 〈〈Lk, Mk〉〉, together with a multiplication defined componentwise through the composition of the linear components, the vector space of all Hilbert-Schmidt quadratic operators Q on H becomes a linear H*-algebra containing an ideal of nuclear (trace class) quadratic operators. In the finite dimensional case, each Q is also shown to have another representation as a block-diagonal matrix of Hilbert-Schmidt class which simplifies the practical computation and manipulation of quadratic operators.

The usual proof that dentability implies the Radon-Nikodým property involves a clever but rather baroque exhaustion argument. This note presents a very short and simple proof of this implication. The techniques in this new proof are then generalised to derive some direct proofs of recent results concerning strongly regular operators on L1.

Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,a2,…,an,…], let pn/qn be its nth convergent and let θn = qn|qnξ − pn|. In this paper a general method is introduced to deduce a series of inequalities involving the triple (θn−1, θn, θn+1).

A slant immersion is defined as an isometric immersion from a Riemannnian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this article we give some fundamental results concerning slant immersions. Several results on slant surfaces in ℂ2 are also proved.

In this paper relations between T(r, f) and T(r, f(k)) are established for a class of meromorphic functions f(z), where T(r, f) and T(r, f(k)) are the Nevanlinna characteristic functions f(z) and f(k)(z) respectively. An example is provided to show that a result of Singh is not true. The conclusions obtained here correct and generalise the result of Singh.

We generalise some existing results on strict radical classes, and introduce some new notions of hereditariness of radical classes. This is achieved by considering the classes of rings for which a radical class is hereditary and/or strict (in the general sense).

Let K be a function field in one variable over a finite field of characteristic three. If a ∈ K is not a cube, we show that the equation y2 = x3 + a has only finitely many solutions x, y ∈ K.

The main purpose of this work is to give a general and elementary treatment of the fixed point and the coincidence problems for compositions of set-valued maps with not necessarily locally convex domains and to display, once more, the central rôle played by the selection property.

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

The infinitary divisors of a natural number n are the products of its divisors of the , where py is an exact prime-power divisor of n and (where yα = 0 or 1) is the binary representation of y. Infinitary harmonic numbers are those for which the infinitary divisors have integer harmonic mean. One of the results in this paper is that the number of infinitary harmonic numbers not exceeding x is less than 2.2 x1/2 2(1+ε)log x/log log x for any ε > 0 and x > n0(ε). A corollary is that the set of infinitary perfect numbers (numbers n whose proper infinitary divisors sum to n) has density zero.

Let S be the family of functions f(z) = z + a2z2 + … which are analytic and univalent in |z| < 1. We find the value

as a function of r 0 < r < 1. The known lower estimate of

is improved. Relations with the growth theorem are considered and the radius of univalence of f(z)/z is discussed.

Corresponding to each ordered set there is a variety, determined up to equivalence, generated by an algebra whose term operations are all the monotone operations on the ordered set. We produce several characterisations of the finite bounded ordered sets for which the corresponding variety is congruence-distributive. In particular, we find that congruence-distributivity, congruence-modularity, and residual smallness are equivalent for these varieties.

The object of this paper is to prove a conjecture given recently by Nunokawa that if f(z) ∈ Α(p) satisfies Re{1 + zf″(z)/f′(z)} < p + 1/2 in the unit disk U, then f(z) is p-valently starlike in U.