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.

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.

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 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.

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.

A ring R is called an l-ring (r-ring) in case R contains an indentity and every left (right) semigroup ideal is a left (right) ring ideal. A number of structure theorems are obtained for l-rings when R is left noetherian and left artinian. It is shown that left noetherian l-rings are local left principal ideal rings. When R is a finite dimensional algebra over a field, the property of being an l-ring is equivalent to being an r-ring. However, examples are given to show that these two concepts are in general not equivalent even in the artinian case.

For any group S let Ab(S) = {A∣A is an abelian subgroup of S of maximal order}. Let G be a Chevalley group of type An, Bn, Cn, or Dn over a finite field of characteristic p and let. In this paper Ab(U) is determined for all such groups.

A series of balanced incomplete block (BIB) designs with parameters , is constructed, where , w any integer.

Salem and Zygmund (1947, 1948), Baker (1972) and Dudley (1975) have shown that certain lacunary sets P of characters of a compact abelian group have sequences of the form where фk∈P converge to the normal distribution if suitably normalized. In this paper, a theorem of probability due to McLeish (1974) is applied to clarify and extend the previous results.

Properties of the funnel boundary are investigated for multivalued dynamical systems defined axiomatically in terms of attainability set mappings on complete, locally compact metric state spaces. The set of regular boundary events is shown to be dense in the funnel boundary and theorems of Fukuhara and Zaremba on peripheral attainability are generalized to the systems considered here.