Throughout this paper, all rings have the identity 1 and ring homomorphisms are assumed to preserve 1. We use p to denote a prime integer and F to denote a field of characteristic p. For an element α in F, we set

A = F[ϰ]/(ϰp - α)F[ϰ].

Moreover, by D and R, we denote the derivation of A induced by the ordinary derivation of F[ϰ] and the skew polynomial ring A[X,D] where aX = Xa+D(a) (a ∈ A), respectively (cf. [2]).

In [3], R. W. Gilmer determined all the B-automorphisms of B[X] for any commutative ring B. Since then, some extensions or generalizations of his results have been obtained ([1], [2] and [5]). As to the characterization of automorphisms of skew polynomial rings, M. Rimmer [5] established a thorough result in case of automorphism type, while M. Ferrero and K. Kishimoto [2], among others, have made some progress in case of derivation type.

This paper is a continuation of [3] which initiated a systematic study of sufficient conditions for the weighted interpolation inequality of sum form, 1.1 to hold. Here ϕ, θ are non-negative functions of m, j, p, q, r, Ω is a bounded or unbounded domain in Rn, ∊ belongs to an interval Γ=(0, ∊0), u is in a certain Banach space E(Ω), and N, W, P are measurable real functions satisfying N≧ 0, W, P > 0, as well as additional conditions stated below. Finally the constant K does not depend on u although it may depend on the other parameters.

Given a Riemannian manifold M, the Riemann tensor R induces the curvature operator on the exterior power of the tangent space, defined by the formula where the inner product is defined by From the symmetries of R, it follows that ρ is self-adjoint and so has only real eigenvalues. R also induces the sectional curvature function K on 2-planes in is an orthonormal basis of the 2-plane π.

Let X be a space with two metrics d1 and d2. Let S :(X, d1) → (X, d2) be continuous. We say S has the generalized pseudoorbit shadowing property with respect to the metrics d1 and d2 if for every every δ-pseudo-orbit in d1 can be ∊-shadowed by a true orbit in d2, i.e., if {x0, x1,…} satisfies for all i ≧ 0, then for all i ≧ 0. The main result of this note shows that certain Markov operators P : L1→ L1 have the generalized shadowing property on weakly compact subsets of the space of probability density functions, where d1 is the metric of norm convergence and d2 is the metric of weak convergence. An important class of such operators are the Frobenius-Perron operators induced by certain expanding and nonexpanding maps on the interval. When there is exponential convergence of the iterates to the density, we can express δ in terms of ∊. We also show that, unlike the situation in the space X itself, the generalized shadowing property is valid for all parameters in families of maps and that there is stability of the shadowing property.

Let E be a Banach Lattice. We will consider E to be weakly sequentially complete and to have a weak unit u. Thus we may represent E as a lattice of real valued functions defined on a measure space (χ, , μ). There is a set R ⊂ χ such that R supports a maximal invariant function Φ for a postive contraction T on E [5]. Let N = χ — R be the complement of R. Akcoglu and Sucheston showed that where E+ is the positive cone of E. If in addition a monotone condition (UMB) is satisfied, then the same authors showed [4] that converges in norm.

Let be a compact Riemann surface, be the complement of a nonvoid finite subset of and A() be the ring of finite sums of meromorphic functions in with finite divisor. In this paper it is proved that every nonzero f ∈ A() can be decomposed as a product αβ, where α is either a unit or a product of powers of irreducible elements of A(), uniquely determined by f up to multiplication by units, and β is a product of functions of the type eφ – 1, with φ holomorphic and nonconstant in . Furthermore, a similar result is obtained for a certain class of subrings of A().

Let G be a group and n(≧ 2) an integer. We say that G belongs to the class of groups Pn if every product of n elements can be reordered, i.e. for all n-tuples , there exists a non-trivial element σ in the symmetric group Σn such that Let P denote the union of the classes Pn, n ≧ 2. Clearly every finite group belongs to P and each class Pn is closed with respect to forming subgroups and factor groups.

We construct a family of unitals in the Hughes plane. We prove that they are not isomorphic with the classical unitals, and in so doing we exhibit a configuration that exists in the latter, but not in the former. This new configurational property of the classical unitals might serve in the future again as an isomorphism test.

A particular instance of our construction has appeared in [11]. But it only concerns itself with the case where the matrix involved is the identity, whereas the present article treats the general case of symmetric matrices over a suitable field. Furthermore, [11] does not answer the isomorphism question. It states that (the English translation is ours) “It remains to be seen whether the unitary designs constructed in this note are isomorphic or not with known designs”.

In this paper it is shown how the graphical methods developed by Stephen for analyzing inverse semigroup presentations may be used to study varieties of inverse semigroups. In particular, these methods may be used to solve the word problem for the free objects in the variety of inverse semigroups generated by the five-element combinatorial Brandt semigroup and in the variety of inverse semigroups determined by laws of the form xn = xn + 1. Covering space methods are used to study the free objects in a variety of the form ∨ where is a variety of inverse semigroups and is the variety of groups.

The object of this paper is to prove certain p-adic orbital integral identities needed in order to accomplish the symmetric square transfer via the twisted Arthur trace formula. Only §5 of this article contains original material, the rest of it is due to R. Langlands. Very briefly, we reduce the problem of proving certain orbital integral identities for “matching” functions in the respective Hecke algebras to two counting problems on the buildings. We give Langlands’ solution of one of these problems in the case of the unit elements of the respective Hecke algebras and §5 provides the solution to the other one, again, in the unit element case. The main results assume p ≠ 2.