In this paper we define the semigroup of an ordinary multiple point of an analytic plane curve f. This semigroup on tuples of integers is completely characterised in terms of the order of f, that is the number of distinct tangent directions to f at that point.

Let X and Y be Banach spaces and let C(X, Y) be the functions from X to Y continuous with respect to the weak topology on X and the strong topology on Y. By the Mosco topology τM on C(X, Y) we mean the supremum of the Fell topologies determined by the weak and strong topologies on X × Y, where functions are identified with their graphs. The function space is Hausdorff if and only if both X and Y are reflexive. Moreover, τM coincides with the stronger compact-open topology on C(X, Y) provided X is reflexive and Y is finite dimensional. We also show convergence in either sense is properly weaker than continuous convergence, even for continuous linear functionals, whenever X is infinite dimensional. For real-valued weakly continuous functions, τM is the supremum of the Mosco epitopology and the Mosco hypotopology if and only if X is reflexive.

Let E be a real Banach space with a uniformly convex dual, and let K be a nonempty closed convex and bounded subset of E. Suppose T: K → K is a continuous monotone map. Define S: K → K by Sx = f – Tx for each x in K and define the sequence iteratively by x0 ∈ K, xn+1 = (1 – Cn)xn + CnSxn, n ≥ 0, where is a real sequence satisfying appropriate conditions. Then, for any given f in K, the sequence converges strongly to a solution of x + Tx = f in K. Explicit error estimates are also computed. A related result deals with iterative solution of nonlinear equations of the dissipative type.

Let B be the Bloch space of all those functions f holomorphic in the open unit disc D of the complex plane satisfying . We establish sufficient conditions for the boundedness of functions f belonging to B satisfying a certain uniform radial boundedness condition, and, by introducing a wide class of subsets E of ∂D, which we call negligible sets for boundedness, we show that if f ∈ B and there is a constant K > 0 such that , then f is bounded in D. Hence a significant extension of a theorem of Goolsby is obtained.

Let F be any set of five points in R3 so situated that no four of the points are coplanar, and that the line xy through any two x and y of the points has a unique intersection point xy* with the plane determined by the other three. Let F^ denote the family of all such xy*. Let S(F) denote the set of all X ⊆ F^ which are maximal with respect to the property that X is a subset of a plane in R3. For k > 2 an integer, let S(k; F) denote the family of all k-membered elements in S(F).

A family 𝒟 of sets is said to be uniformly deep of depth d if and only if for every x ∈ ∪ 𝒟 there are exactly d distinct 𝒜 ∈ 𝒟 for which x ∈ 𝒜.

We establish the following result, and extend our ideas to general Euclidean spaces.

In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

This paper contains a precise description of the duality between the formal evolutions of flow charts and of circuits. In addition, it contains a new description of the free category-with-products on a multigraph as a familially representable construction.

A quantitative version of a classical result of S.N. Bernstein concerning the divergence of Lagrange interpolation polynomials based on equidistant nodes is presented. The proof is motivated by the results of numerical computations.

In 1969, Ligh proved that distributively generated (d-g) Boolean near-rings are rings, and hinted that some of the more complicated polynomial identities implying commutativity in rings may turn d-g near-rings into rings. In the present paper we investigate the following conditions: (1) xy = (xy)n(x, y); (2) xy = (yz)n (xy); (3) xy = ym (x, y)xn (x, y); (4) xy = xy n(x, y)x; (5) xy = xn(x, y)ym (x, y); finally prove that under appropriate additional hypotheses a d-g near-ring must be a commutative ring. Indeed the theorem proved here is a wide generalisation of many recently established results.

It is shown that weakly compact sets of James quasi reflexive space have normal structure.

We study a certain linear fourth order differential operator and show the existence of solutions to corresponding nonlinear problems. It will be shown that a maximum principle holds and that under certain conditions the linear operator has a positive principal eigenvalue with corresponding positive eigenfunction.

It is proved that the product of two non-zero derivations of a prime C*-algebra is bounded only if both of them are bounded.

A coefficient estimate for powers of a class of Bazilevic functions obtained by M.M. Elhosh, is disproved and some sharp bounds are given.

The Cauchy representation is not valid for every Schwartz distribution f ∈ (D(R))′ because (t – x)−l ∉ D(R). Since for all z ∈ {z: Im z ≠ 0} the kernel (t – z)−l belongs to (R) (1 > p > ∞), the Cauchy representation of the distributions in ((R))′ seems possible.

In this paper, we prove this fact. It is also proved that every probability density defines a generalised function on the space (R)(1 < p < ∞), of test functions. Applications of these results in probability theory are discussed.

The equivalence of a fixed point theorem and the Fan-Knaster-Kuratowski-Mazurkiewicz theorem in H-space has been established. The fixed point theorem is then applied to obtain a theorem on sets with H-convex sections, and also results on minimax inequalities.

We show that a compact convex body in En cannot be partitioned by n + 1 hyperplanes into 2n+l – 1 subsets of equal measure, thus generalising a result in the plane due to R.C. and E.F. Buck.

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

It has long been known that there is a close connection between stochastic independence of continuous functions on an interval and space-filling curves [9]. In fact any two nonconstant continuous functions on [0, 1] which are independent relative to Lebesgue measure are the coordinate functions of a space filling curve. (The results of Steinhaus [9] have apparently been overlooked in more recent work in this area [3, 5, 6].)

We give single laws for two subvarieties of Steiner quasigroups (squags); the subvariety of squags whose underlying triple systems are the affine spaces over GF[3], and the subvariety of squags whose underlying triple systems are the affine triple systems.