A convex polytope D (α) was defined in Barnes (1978) as the set of Minkowski-reduced forms with prescribed diagonal coefficients α1, α2,…αn. A local minimum of the determinant D(f) over D(α) must occur at a vertex of D(α). Here a criterion is obtained for a given vertex to provide a local minimum, completely analogous to Voronoï's criterion for a perfect form to be extreme.

In conjunction with an earlier work by Leong (1974a), this paper completes the solution of the isomorphism problem for finite nilpotent groups of class two with cyclic centre. A canonical decomposition for 2-groups of such type is obtained and proved.

The work establishes the estimates of linear independence, transcendence, and algebraic independence measures of the values at algebraic points of a set of E-functions, satisfying linear differential equations with coefficients from C(z), rather close to their natural bounds.

Bijective correspondences are established between the radical classes R (in a variety W of rings) with the property that a ring A is in R exactly when its finitely generated subrings are all in R, and certain filters of ideals in a free W-ring. It follows that such classes are determined by the polynomial identities satisfied by the finite subsets of their members. Analogous considerations are applied to radical classes R which, for a fixed integer n, have the property that a ring is in R if and only if its subrings generated by at most n elements are in R.

Here we consider a problem on weighted (0, 2) interpolation. We choose the interpolatory conditions in such a way that we get the polynomial of degree ≤2n−1, satisfying those conditions. Moreover we prove that the sequence of these interpolatory polynomials under certain conditions converges uniformly to a function belonging to the Zygmund class.

A radical class R of rings (not necessarily associtative) is called an n-radical class if it has the property that a ring is in R if and only if every subring generated by ≤n elements is in R. A transfer theorem is proved, relating n-radical classes in two universal varieties which share the same ≤n-generator rings. Partially through the use of this result, we obtain information about extension closed subvarieties of various universal varieties of power-associative rings.

In this note we give several new characterizations of arbitrary pseudocompact spaces, that is spaces characterized by the property that all continuous real-valued functions on the space are bounded.

Optimality conditions without constraint qualifications are given for the convex programming problem: Maximize f(x) such that g(x) ∈ B, where f maps X into R and is concave, g maps X into Rm and is B-concave, X is a locally convex topological vector space and B is a closed convex cone containing no line. In the case when B is the nonnegative orthant, the results reduce to some of those obtained recently by Ben-Israel, Ben-Tal and Zlobec.

The main result of this paper shows that the existence of commuting normal extension (c.n.e.) for an arbitrary family of commuting subnormal operators can be determined by considering appropriate families of multivariable weighted shifts. In proving this some known criteria for c.n.e. are generalized. It is also shown that a family of jointly quasi-normal operators has c.n.e.

The conjugacy of Cartan subalgebras of a Lie algebra L over an algebraically closed field under the connected automorphism group G of L is inherited by those G-stable ideals B for which B/Ci is restrictable for some hypercenter Ci of B. Concequently, if L is a restrictable Lie algebra such that L/Ci restrictable for some hypercenter Ci of L, and if the Lie algebra of Aut L contains ad L, then the Cartan subalgebras of L are conjugate under G. (The techniques here apply in particular to Lie algebras of characteristic 0 and classical Lie algebras, showing how the conjugacy of Cartan subgroups of algebraic groups leads quickly in these cases to the conjugacy of Cartan subalgebras.)

Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved.

Let N be a direct summand of a module which is a direct sum of modules of torsion-free rank one over a discrete valuation ring. Then there is a torsion module T such that N⊕T is also a direct sum of modules of torsion-free rank one.

Ordered vector spaces more general than vector lattices are represented as function spaces with ‘almost’ the pointwise order. The ‘pseudo-lattice’ property which is necessary (and sufficient in the finite-dimensional case) for the representation is also studied in relation to the lattice property.

A flat spot in a Banach space X is an element x ∈ Sx = {x ∈ X: ‖x‖ = 1} with the property that the infimum m(x) of the lengths of all curves in Sx joining x to −x is 2. Flat spots occur in every non-superreflexive space when suitably renormed. A study is made of the geometric implications of the existence of flat spots. Connections with other notions such as differentiability, decomposition constants and Kadec-Klee norms are explored and some renorming results for non-superreflexive spaces are proved.

The isoperimetric problem in the Euclidean plane is completely solved for bounded, convex sets which are symmetric about the origin, and which contain no non-zero point of the integral lattice.

R. Paré and W. Schelter (1978) have extended the Cayley-Hamilton theorem by showing that for each n<1 there is an integer k such that all n x n matrices over any (possibly noncommutative) ring satisfy a monic polynomial of degree k. We give a lower bound for this degree, namely π(n), which is defined as the shortest possible length of a sequence with entries from {1, 2, …, n}.

In this paper a general concept of regularity for rings is defined. It is shown that every regularity determines in a natural way a subradical and a radical for rings. A wide class of regularities is constructed: the class of polynomial regularities. All well-known regularities, such as the Perlis-Jacobson regularity, the von Neumann regularity and many others, belong to this class. Special attention is paid to regularities which are elementary in the sense that the so-called unic and nullic polynomial regularities can be thought of as intersections of the elementary ones.

In this article, it is shown that the Volterra integral equation of convolution type w − w⊗g = f has a continuous solution w when f, g are continuous functions on Rx and ⊗ denotes a truncated convolution product. A similar result holds when f, g are entire functions of several complex variables. Also simple proofs are given to show when f, g are entire, f⊗g is entire, and, if f⊗g=0, then f = 0 or g = 0. Finally, the set of exponential polynomials and the set of all solutions to linear partial differential equations are considered in relation to this convolution product.

We establish a general algebraic independence theorem for the solutions of a certain kind of functional equations. As a particular application, we prove that for any real irrational ζ, the numbers are algebraically independent, for multiplicatively independent algebraic αi with 0<|<| <1.

Two artin algebras Λ and Λ′ are said to be stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. In this paper a characterization is given of the artin algebras stably equivalent to Nakayama algebras of Loewy length (at most) four. The proof is an illustration of the technique of using irreducible maps to study problems about stable equivlence.