In this paper we prove the following result: let $m,n\geq 1$ be distinct integers, let $R$ be an $mn(m+n)|m-n|$-torsion free semiprime ring and let $D:R\rightarrow R$ be an $(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation $(m+n)D(x^{2})=2mD(x)x+2nxD(x)$ for $x\in R$. Then $D$ is a derivation which maps $R$ into its centre.

We investigate images of higher-order differential operators of polynomial algebras over a field $k$. We show that, when $\operatorname{char}k>0$, the image of the set of differential operators $\{\unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D70F}_{i}\mid i=1,2,\ldots ,n\}$ of the polynomial algebra $k[\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{n},z_{1},\ldots ,z_{n}]$ is a Mathieu subspace, where $\unicode[STIX]{x1D70F}_{i}\in k[\unicode[STIX]{x2202}_{z_{1}},\ldots ,\unicode[STIX]{x2202}_{z_{n}}]$ for $i=1,2,\ldots ,n$. We also show that, when $\operatorname{char}k=0$, the same conclusion holds for $n=1$. The problem concerning images of differential operators arose from the study of the Jacobian conjecture.

The following theorem is proved. Let m, k and n be positive integers. There exists a number η=η(m,k,n) depending only on m, k and n such that if G is any residually finite group satisfying the condition that the product of any η commutators of the form [xm,y1,…,yk ] is of order dividing n, then the verbal subgroup of G corresponding to the word w=[xm,y1,…,yk ] is locally finite.

Starters with adders in abelian groups of odd order have been used extensively in the construction and study of Room squares. It is possible to define these concepts in non-abelian groups, with similar applicability to Room squares. Examples are given of adders for the “patterned starter” in the non-abelian groups of order pq, p and q primes larger than 3, p ≡ q ≡ 3 (mod 4), q ≡ 1 (mod p).

Let P(z) and Q(z) be two polynomials of the same degree n. If P(z) and Q(z) are apolar and if one of then has all its zeros in a circular region C, then according to a famous result known as Grace's Apolarity Theorem, the other will have at least one zero in C. In this paper we relax the condition that P(z) and Q(z) are of the same degree and present some generalizations of Grace's Apolarity theorem for the case when the circular region C is a closed half-plane. As an application of these results, we also generalize some results of Walsh and Szegö.

In this work we study the homogenisation problem for nonlinear elliptic equations involving $p$-Laplacian-type operators with sign-changing weights. We study the asymptotic behaviour of variational eigenvalues which consist of a double sequence of eigenvalues. We show that the $k$th positive eigenvalue goes to infinity when the average of the weights is nonpositive, and converges to the $k$th variational eigenvalue of the limit problem when the average is positive for any $k\geq 1$.

In 1919, Adolf Hurwitz formed the quaternion ring R composed of elements whose coordinates were either all integers or halves of odd integers. The objective of this paper is to examine the (two-sided) ideal structure in the hurwitzean polynomial ring R[x], formed by taking all polynomials with coefficients in R. The maximal and prime ideals of R[x] will be characterized with results surprisingly analogous to those in Z[x]. In addition, a canonical basis, of the type developed by G. Szekeres, 1952, for polynomial domains, will be developed for the ideals of R[x].

Two theorems for Lebesgue integrals, namely the Gauss-Green Theorem relating surface and volume integrals, and the integration-by-parts formula, are shown to possess generalizations where the integrands take values in a Banach space, the integrals are Bochner integrals, and derivatives are Fréchet derivatives. For integration-by-parts, the integrand consists of a continuous linear map applied to a vector-valued function. These results were required for a generalization of the calculus of variations, given in another paper.

We develop a generalised polynomial formalism which captures the concept of an algebra of piece-wise denned polynomials. The formalism is based on the Boolean power construction of universal algebra. A generalisation of the theory of substitution homomorphisms is developed. The abstract operation of composition of generalised polynomials in one variable is denned and shown to correspond to function composition.

For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,…,xk} such that for some positive integer d, xi + 1 − xi ∈ {d, 2d,…, md} for i = 1,…, k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2, …, n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r ≤ m, hm(m + r) ≤ 2c(m + r − 1) + 1, where c = ⌈m/(m − r)⌉. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

It is shown that any sheaf of R˜-modules, all of whose stalks are injective, is necessarily a flabby sheaf. This generalizes the result or Grothendieck that the sheaf M˜ determined by an injective module M over a commutative noetherian ring with 1 is flabby.

Aspects of duality relating to compact totally disconnected universal algebras are considered. It is shown that if P is a ““basic“ set of injectives in a variety of compact totally disconnected algebras then the category P of P-copresentable objects is in duality with the class of all G-copresentable algebras on P, where G: P → Ens is the forgetful functor and an algebra is taken to mean a finite-product-preserving functor from P to Ens.

We generalise the existence result for harmonic maps obtained by Hildebrandt-Kaul-Widman to the case where the domain manifold is complete noncompact.

Using generalized position vector fields we obtain new upper bound estimates of the first nonzero eigenvalue of a kind of elliptic operator on closed submanifolds isometrically immersed in Riemannian manifolds of bounded sectional curvature. Applying these Reilly inequalities we improve a series of recent upper bound estimates of the first nonzero eigenvalue of the Lr operator on closed hypersurfaces in space forms.

Namioka proved that the following condition (*) given below holds, if X is Čech-complete and Y is a locally compact, σ-compact space.

(*) Let X and Y be spaces, Z be a metric space and let f: X × Y → Z be separately continuous. Then there is a dense, Gδ set A in X such that A × Y ⊂ C(f).

Following Christensen a space X is called Namioka if (*) is true for any compact space Y. In this paper we introduce and study a new class of spaces which is closely related to Namioka spaces. Namely, we say that a space Y is co-Namioka if (*) holds for any Namioka space X.

In this paper we use the Dubovitski–Milyutin formalism to establish necessary and sufficient conditions for optimality in a nonlinear, distributed parameter control system, with convex cost criterion and initial condition not given a priori (that is it is not a known function but instead it belongs to a specified set). Our result extends a recent theorem of Lions. Finally a concrete example is worked out in detail.

An elementary proof is given of the fact that an n–tuple A = (A1, …, An) of self-adjoint matrices in a 2-dimensional Hilbert space consists of mutually commuting matrices Aj, 1 ≤ j ≤ n, if and only if γ(A) is non-empty. Here γ(A) ⊆ ℝn is the joint spectrum of A (in the sense of McIntosh and Pryde) consisting of those points β ∈ ℝn for which the matrix is not invertible.

In 1990 Allen Bell presented a sufficient condition for the primeness of the universal enveloping algebra of a Lie superalgebra. Let Q be a nonsingular bilinear form on a finite-dimensional vector space over a field of characteristic zero. In this paper we show that Bell's criterion applies to the Hamiltonian Cartan type superalgebras determined by Q, and hence that their enveloping algebras are semiprimitive.