In recent years there has been considerable attention paid to the behavior of solutions of elliptic boundary value problems in domains with piecewise smooth boundary. In two dimensions the study concerns the behavior of a solution near a corner, and in three (or more) dimensions two cases have been given considerable attention: a conical vertex on the boundary, or an edge.

The solution of such a problem may be singular at the nonsmooth boundary points. The standard example in two dimensions is a solution in polar coordinates of the Dirichlet problem near a corner of interior angle πα;u = r1/α sin θ/α is a function which is harmonic in the sector 0 < θ < πα, has zero boundary values near the corner, and yet at the origin has unbounded derivatives of order > 1/α unless 1/α is an integer.

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.

In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the type

with r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

Throughout this paper when we refer to a semilattice S we shall mean that S is a meet semilattice. We shall denote the infimum of two elements a, b of S by a ∧ b, and the supremum, if it exists, by a ∨ b. A prime semilattice is a meet semilattice such that the infimum distributes over all existing finite suprema, in the sense that if x1 ∨ x2 … ∨ xn exists then (x ∧ x1) ∨ (x ∧ x2) … ∨ (x ∧ xn) exists for any x and equals x ∧ (x1 ∨ x2 … ∨ xn). Such semilattices were first studied by Balbes [1] and we use his terminology.

A non-empty subset F of S is a filter provided that x ∧ y ∊ F if and only if x ∊ F and y ∊ F.

In the study of optimization problems it is necessary to consider functions that are not differentiable. This has led to the consideration of generalized gradients and a corresponding calculus for certain classes of functions. Rockafellar [16] and others have developed a very strong and elegant theory of subgradients for convex functions. This convex theory gives point-wise criteria for the existence of extrema in optimization problems.

There are however many optimization problems that involve functions which are neither differentiable nor convex. Such functions arise in many settings including optimal value functions [15]. In order to deal with such problems Clarke [3] defined a type of subgradient for nonconvex functions. This definition was initially for Lipschitz functions on R”. Clarke extended this definition to include lower semicontinuous (l.s.c.) functions on Banach spaces through the use of a directional derivative, the distance function from a closed set and tangent and normal cones to closed sets.

Let Λ = {λ0,…, λn} denote a set of distinct integers and let Π(Λ) denote the set of all generalized polynomials of the form

For any given ζ on the unit circle U with

we consider the set Zk of points 1, ζ, ζ2, …, ζk−1 where

We shall denote by or the class of Λ-splines S(z) which satisfy the following conditions:

• (i) S(z) ∊ Cn−1(U)

• (ii) S(z)|Av ∊ Π(Λ) where

  

The strong ϕ topology. Let S be a linear space of real sequences written in functional notation

There is a natural duality between S and the space ϕ of sequences which are eventually ϕ given by the equation

The series has only a finite number of nonzero terms since t is in ϕ.

A subset B of ϕ is called S-bounded if

for each s in S.

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.

Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .

If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.

Let R = k[X1, …, Xn] with k a field, and let I ⊂ R be a homogeneous ideal. The algebra R/I is said to have a pure resolution if its homogeneous minimal resolution has the form

Some of the known examples of pure resolutions include the coordinate rings of: the tangent cone of a minimally elliptic singularity or a rational surface singularity [15], a variety defined by generic maximal Pfaffians [2], a variety defined by maximal minors of a generic matrix [3], a variety defined by the submaximal minors of a generic square matrix [6], and certain of the Segre-Veronese varieties [1].

If I is in addition Cohen-Macaulay, then Herzog and Kühl have shown that the betti numbers bi are completely determined by the twists di.

In [2] by means of a fairly lengthy argument involving Hermitian varieties it was shown that PG(2n, q2) can be partitioned into (q2n++ 1 + 1)/(q + l)-caps. Moreover, these caps were shown to constitute the “large points” of a PG(2n, q) in a natural way. In [3] a similar argument was used to show that once two disjoint (n – l)-subspaces are removed from PG(2n, q2), the remaining points can be partitioned into (q2n – 1)/(q2 – l)-caps.

The purpose of this paper is to give a short proof of the results found in [2], and then use the technique developed to partition PG(2n, q) into (qn + l)-caps for n even and q any prime-power. Moreover, these caps can be treated in a natural way as the “large points” of a PG(n – 1, q).

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the form

For example, the solution of the initial value problem

is given by u(x, t) = Ttf(x) where

Peral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only if

From the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

Let |aij|n×n denote the n × n determinant with (i, j)-entry aij, and hk = hk(x1, …, xn) denote the kth-homogeneous symmetric function of x1, …, xn defined by

where the summation is over all m1, …, mn ≧ 0 such that m1 + … + mn = k. We adopt the convention that hk = 0 for k < 0. For integers α1 ≧ α2 … ≧ αn ≧ 0, the Jacobi-Trudi identity (see [6], [7]) states that

In this paper we give a combinatorial proof of an equivalent identity, Theorem 1.1, obtained by moving the denominator on the RHS to the numerator on the LHS.

Let A be an element of a complex Banach algebra with identitI. The ordinary spectrum of A, sp(A), consists of those points z in the complex plane such that A — zI has no inverse in . If Q is any other element of , we define spQ(A), the spectrum of A relative to Q, or Q-spectrum of A, as those points z such that has no inverse in . Thus if Q = 0 the Q-spectrum of A is the same as the ordinary spectrum of A.

The generalized notion of spectrum, spQ(A), retains many of the properties of the ordinary spectrum, particularly when A and Q commute and the ordinary spectrum of Q does not meet the unit circle. Under these conditions the Q-spectrum of A is a nonempty compact subset of the plane, and if both sp(A) and sp(Q) are finite (or countable), so is spQ(A).

Suppose that G is a reductive algebraic group defined over Q. There occurs in the trace formula a remarkable distribution on G(A)1 which is supported on the unipotent set. It is defined quite concretely in terms of a certain integral over G(Q)\G(A)1. Despite its explicit description, however, this distribution is not easily expressed locally, in terms of integrals on the groups G(QV). For many applications of the trace formula, it will be essential to do this. In the present paper we shall solve the problem up to some undetermined constants.

The distribution, which we shall denote by Junip, was defined in [1] and [3] as one of a family {Jo} of distributions. It is the value at T = T0 of a certain polynomial . We shall recall the precise definition in Section 1.