For i = 1,…, n let ai be a homogeneous polynomial of degree ri(>0) in the graded polynomial ring R[x1, …, xm], or R[x] for short, where R is a commutative ring with unity and x1, …, xm are indeterminates of degree 1. Let of degree - 1 be a formal inverse of xj and let U denote the graded R[x]-module In [2, §2] we introduced a graded complex of r-modules.

Let M be a compact flat Riemannian manifold of dimension n, and Γ its fundamental group. Then we have the following exact sequence (see [1])

where Zn is a maximal abelian subgroup of Γ and G is a finite group isomorphic to the holonomy group of M. We shall call Γ a Bieberbach group. Let T be a flat torus, and let Ggr act via isometries on T; then ┌ acts isometrically on × T where is the universal covering of M and yields a flat Riemannian structure on ( × T)/Γ. A flat-toral extension (see [9, p. 371]) of the Riemannian manifold M is any Riemannian manifold isometric to ( × T)/Γ where T is a flat torus on which Γ acts via isometries. It is convenient to adopt the convention that a single point is a 0-dimensional flat torus. If this is done, M is itself among the flat toral extensions of M. Roughly speaking, this is a way of putting together a compact flat manifold and a flat torus to make a new flat manifold the dimension of which is the sum of the dimensions of its constituents. It is, more precisely, a fibre bundle over the flat manifold with a flat torus as fibre.

A long time ago Ju. E. Vapne ([2], [3]) and, independently, the author ([4], [5]) classified those standard and complete wreath products that have faithful representations of finite degree over (commutative) fields. See [6] pages 37 40 & 150–154 for an account of this. Recently, in connection with finitary linear groups, I needed a more general wreath product. Somewhat to my surprise neither the classification nor the proof for these generalized wreath products was a straightforward translation from the standard case. The situation is intrinsically more complex and it seems worthwhile recording it separately.

Let X be a Banach space and Y its closed subspace having property U in X. We use a net (Aα) of continuous linear operators on X such that ‖ Aα ‖ ≤ 1, Aα (X) ⊂ Y for all α, and limαg(Aαy) = g(y), y ∈ Y, g∈Y* to obtain equivalent conditions for Y to be an HB-subspace, u-ideal or h-ideal of X. Some equivalent renormings of c0 and l2 are shown to provide examples of spaces X for which K(X) has property U in L(X) without being an HB-subspace. Considering a generalization of the Godun set [3], we establish some relations between Godun sets of Banach spaces and related operator spaces. This enables us to prove e.g., that if K(X) is an HB-subspace of L(X), then X is an HB-subspace of X**—the result conjectured to be true by Å. Lima [9].

We introduce a class of “differential operators” on graphs and we prove an energy estimate and a Liouville type theorem depending on some structural properties of the operators considered.

Given a p-subgroup P of a finite group G we express the number of p-blocks of G with defect group P as the p-rank of a symmetric integer matrix indexed by the N(P)/P-conjugacy classes in PC(P)/P. We obtain a combinatorial criterion for P to be a defect group in G.

We consider the spectral function, ρα(μ), for –∞<μ<∞ associated with the Sturm-Liouville equation

and the boundary condition

We suppose that q is a real-valued member of L1[0, ∞) and λ is a real parameter.

We investigate the distribution of the hitting time T defined by the first visit of the Brownian motion on the Sierpiński gasket at geodesic distance r from the origin. For this purpose we perform a precise analysis of the moment generating function of the random variable T. The key result is an explicit description of the analytic behaviour of the Laplace- Stieltjes transform of the distribution function of T. This yields a series expansion for the distribution function and the asymptotics for t →0.

In this paper the method of inner and outer sums [5], together with the computational power of computer symbolic manipulation, are used to extend to high order the asymptotic expansions in an appropriate limit of some infinite series arising in low Reynolds-number fluid mechanics. The enhanced applicability of the expansions is demonstrated, and the method is extended to treat alternating series.

Recently Edelman and Reiner suggested two poset structures, (n, d) and (n, d) on the set of all triangulations of the cyclic d-polytope C(n, d) with n vertices. Both posets are generalizations of the well-studied Tamari lattice. While (n, d) is bounded by definition, the same is not obvious for (n, d). In the paper by Edelman and Reiner the bounds of (n, d) were also confirmed for (n, d) whenever d≤5, leaving the general case as a conjecture.

Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum (i.e., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenvalue (or characteristic value) of odd multiplicity of the linearized problem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper considers a class of such nonlinear eigenvalue problems having simple eigenvalues and a “weak” nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further differentiability conditions it is shown that the continua are smooth, 1-dimensional curves and that there are no non-trivial solutions of the equation other than those lying on the bifurcating continua.

Given a convex function u, defined in an open bounded convex subset Ω of ℝn, we consider the set

where η is a Borel subset of Ω,ρ is nonnegative, and ∂u(x) denotes the subgradient (or subdifferential) of u at x. We prove that Pp(u; η) is a Borel set and its n-dimensional measure is a polynomial of degree n with respect to ρ. The coefficients of this polynomial are nonnegative measures defined on the Borel subsets of Ω. We find an upper bound for the values attained by these measures on the sublevel sets of u. Such a bound depends on the quermassintegrals of the sublevel set and on the Lipschitz constant of u. Finally we prove that one of these measures coincides with the Lebesgue measure of the image under the subgradient map of u.

A number of known estimates of the number of translates, or lattice translates, of a convex body H required to cover a convex body K are obtained as consequences of two simple results.