We describe Jordan homomorphisms and Jordan triple homomorphisms onto 2-torsion free semiprime rings in which the annihilator of any ideal is a direct summand.

Let D be a domain in the complex ω-plane and let x: D → R3 be a regular minimal surface. Let M(K) be the set of points ω0 ∈ D where the Gauss curvature K attains local minima: K(ω0) ≤ K(ω) for |ω – ω0| < δ(ω0), δ(ω0) < 0. The components of M(K) are of three types: isolated points; simple analytic arcs terminating nowhere in D; analytic Jordan curves in D. Components of the third type are related to the Gauss map.

The l-adic representations associated to prime dimensional type IV absolutely simple abelian varieties over number fields are studied. The image of such a representation was computed. The results coincide with the well-known conjectures of Mumford and Tate.

The space l∞ is known to have no Schauder decomposition. It is proved here that l∞ does not even possess any strong Markuševič decomposition.

In this paper we discuss existence of positive solutions to a general nonlinear elliptic system of reaction-diffusion equations representing a predator-prey or competition model of interaction between two species, in a heterogeneous environment. We also consider homogeneous Dirichlet and/or Robin boundary conditions. In the predator-prey case we give necessary and sufficient conditions for the existence of positive solutions, while in the competition case we give sufficient conditions. We use index theory in a positive cone to attack our problem and characterise our results by the sign of the first eigenvalues of certain Schrodinger type operators.

Let the group H have presentation where m ≥ 3, pi ≥ 2 and (pi, pj) = 1 if i ≠ j. We show that H is a one-relator group precisely if H can be obtained from a suitable group 〈a, b; ap = bp〉 by repeated applications of a (two-stage) procedure consisting of applying central Nielsen transformations followed by adjoining a root of a generator. We conjecture that any one-relator group G with non-trivial centre and G/G′ not free abelian of rank two can be obtained in the same way from a suitable group 〈a, b; ap = bp〉.

Let A be a finite set. A partial clone on A is a composition closed set of operations containing all projections. It is well known that the partial clones on A, ordered by inclusion, form a lattice. We show that the minimal partial clones on A are:

(a) minimal clones of full operations or

(b) generated by partial projections defined on a totally reflexive and totally symmetric domain.

S(R) is the semigroup, under composition, of all continuous selfmaps of the space R of real numbers. We show that the primes of S(R) are precisely those continuous selfmaps which are surjective and have exactly two local extrema. Additional results are then derived from this. For example, if f is any surjective continuous selfmap of R with n ≥ 2 local extrema, then there exist homeomorphisms from R onto R such that m ≤ 1 + n/2 and

where P is the polynomial defined by P(x) = x3 − x. It follows from this that the homeomorphisms together with the polynomial P generate a dense subsemigroup of S(R) where the topology on S(R) is the compact-open topology.

Let P1 be the class of holomorphic functions on the unit disc U = {z: |z| < 1} for which f(0) = 1 and Re f > 0. Let also Pn be the corresponding class on the unit disc Un. The inequality |ak| ≤ 2 is known for the Taylor coefficients in the class P1. In this paper, it is generalised for the class Pn. If ρ = (ρ1, ρ2, …, ρn), with ρ1, ρ2, …, ρn nonegative integers whose greatest common divisor is equal to 1, we describe the form of the functions f ∈ Pn under the restriction |aρ| = 2. Under the same restriction, we give conditions for a function to be an extreme point of the class Pn.

The notion of relative normal-convexity is introduced and its connection to equations over groups is studied. We study the effect on relative normal-convexity by free products, amalgamations, and HNN extensions.

A group G is said to have the small squaring property on k-sets if |K2| < k2 for all k-element subsets K of G, and is said to have the small cubing property on k-sets if |K3| < k3 for all k-element subsets K. It is shown that a finite nonabelian group with the small squaring property on 3-sets is either a 2-group or is of the form TP with T a normal abelian odd order subgroup and P a nontrivial 2-group such that Q = Cp(T) has index 2 in P and P inverts T. Moreover either P is abelian and Q is elementary abelian, or Q is abelian and each element of P − Q inverts Q. Conversely each group of the form TP as above has the small squaring property on 3-sets. As for the nonabelian 2-groups with the small squaring property on 3-sets, those of exponent greater then 4 are classified and the examples are similar to dihedral or generalised quaternion groups. The remaining classification problem of exponent 4 nonabelian examples is not complete, but these examples are shown to have derived length 2, centre of exponent at most 4, and derived quotient of exponent at most 4. Further it is shown that a nonabelian group G satisfies |K2| < 7 for all 3-element subsets K if and only if G = S3. Also groups with the small cubing property on 2-sets are investigated.

Let f(z) be a meromorphic function; we shall investigate the asymptotic behaviour of the ratio T(r, f(z + α))/T(r, f(z)) and T(r, f(αz))/T(r, f(z)), and discuss the growth of the meromorphic solutions of some functional equations.

In 1867, Sylvester considered n × n matrices, (aij), with nonzero complex-valued entries, which satisfy (aij)(aij−1) = nI Such a matrix he called inverse orthogonal. If an inverse orthogonal matrix has all entries on the unit circle, it is a unit Hadamard matrix, and we have orthogonality in the usual sense. Any two inverse orthogonal (respectively, unit Hadamard) matrices are equivalent if one can be transformed into the other by a series of operations involving permutation of the rows and columns and multiplication of all the entries in any given row or column by a complex number (respectively a number on the unit circle). He stated without proof that there is exactly one equivalence class of inverse orthogonal matrices (and hence also of unit Hadamard matrices) in prime orders and that in general the number of equivalence classes is equal to the number of distinct factorisations of the order. In 1893 Hadamard showed this assertion to be false in the case of unit Hadamard matrices of non-prime order. We give the correct number of equivalence classes for each non-prime order, and orders ≤ 3, giving a complete, irredundant set of class representatives in each order ≤ 4 for both types of matrices.

Two sequences of rational functions are constructed from different expansions of (tqn − 1)−1 and extensions of certain known results in the theory of Walsh-type equiconvergence are sought.

Let x: M → Em be an immersion of finite type. In this paper we study the following two problems: (1) When is the spectral decomposition ofthe immersion x linearly independent? (2) When is the spectral decomposition orthogonal? Several results in this respect were obtained.

A space of periodic generalised functions, called boehmians, is investigated. The space of boehmians contains all periodic distributions. It is known that not every hyperfunction is a boehmian. We show that the converse is also true. We present some theorems which give sufficient conditions for a sequence of complex numbers to be the Fourier coefficients of a boehmian. Sufficient conditions (in terms of the Fourier coefficients) are obtained for a sequence of boehmians to converge. As an application, a Dirichlet problem is discussed.

We study the properties of the sup metric on infinite products Z = X.(If d is a bounded metric on X then ρ, defined by ρ((xα), (yα)) = , is the sup metric on Z.) In particular, we prove that if X is an AR(metric) or a topological group then so is Z.

It is shown that the best least-squares piecewise n degree polynomial approximation of xn+1 over [a, b] is obtained for a uniform partition. Moreover the approximation is continuous for n odd and discontinuous, with equal stepsizes at the nodes, for n even.

A semigroup T consisting of one-one mapping between certain principal left ideals in a type A semigroup S is constructed. T is shown to be a type A semigroup. A representation of S by T is then obtained which is analogous to Vagner-Preston's results on inverse semigroups.

We associate with an operator ideal 𝒜 (in the sense of Pietsch) a class of bounded sequences S𝒜 by using the 𝒜-variation of Astala. If 𝒜 and B are operator ideals, and we define (𝒜, B) as the class of operators which map a sequence of S𝒜 into a sequence of SB, we obtain the following:

Theorem. If Tn: X → Y is a sequence of operators and for every sequence (xn) ⊂ X in S𝒜 there exists p such that (Tpxn) belongs to SB, then Tm ∈ (𝒜, B) for some m.

The compact operators, weakly compact operators and some other operator ideals can be represented as (𝒜, B). Hence several results of Tacon and other authors are a consequence of this theorem.