In this article, we prove that for a finite field Fq with even q > 3, any complete mapping polynmial of Fq has reduced degree at most q − 3. This is a solution to a problem of Niederreiter and Robinson about finite fields.;

Generalised k-statistics associated with multi-indexed arrays of random variables satisfying a generalised form of exchangeability are studied. By showing that they form multi-indexed reversed martingales and that the associated family of σ-fields possesses certain conditional independence properties, conditions for the a.s. convergence of generalised k-statistics are obtained. When the arrays of random variables are sums of independent arrays of independent effects, as is the case with the standard random effects anova models, the limits are identified as the associated generalixed cumulants.

We say that a function has locally small Riemann sums on an interval if for each point x in the interval, and for each positive number ε, all sufficiently fine partitions of intervals lying in neighborhood of x but not containing x have Riemann sums of absolute value less than ε. The main result is then as the title states. We use the generalized Riemann approach to Perron integration, assuming that functions are measurable only to insure that conditions involving the positive and negative parts of the functions are satisfied.

A technique is described for calculating the number of block ideals of FG, where F is a algebraically closed field of characteristic p, and where G is a p-soluble finite group. Among its consequences are the following: if U is a G-invariant irreducible FOp′(G)-module, then there is a unique block ideal of FG whose restriction to Op′(G) has all its composition factors isomorphic to U; and if G has p′-length 1, the number of block ideals of FG is the number of G-conjugacy classes of Op′(G)

In this paper two theorems are proved that give a partial answer to a question posed by G. Behrendt and P. Neumann. Firstly, the existence of a group of cardinality ℵ1 with exactly ℵ1 normal subgroups, yet having a subgroup of index 2 with 2ℵ1 normal subgroups, is consistent with ZFC (the Zermelo-Fraenkel axioms for set theory together with the Axiom of Choice). Secondly, the statement “Every metabelian-by-finite group of cardinality ℵ1 has 2ℵ1 normal subgroups” is consistent with ZFC.

It is shown that for arbitrary ε > 0 there is a function x(t, x) defined on the square [0,1] × [0,1] such that x(t, s) represents an extremal point of the unit ball in the space of Lipschitz continuous functions, and the gradient of x(t, s) is equal to 0 except on a set of measure at most ε.

An infinite family of 2-groups is produced. These groups have no direct factors and have a non-abelian automorphism group in which all automorphisms are central.

It is shown that if G is a non-amenable group, then there are no non-zero translation invariant functionals on Lp(G) for 1 < p < ∞. Furthermore, if G contains a closed, non-abelian free subgroup, then there are no non-zero translation invariant functionals on C0(G). The latter is proved by showing that a certain non-invertible convolution operator on C0(G) is surjective.

An infinite collection of indecomposable isols such that no isol is comparable to certain infinitary combinations of the others is constructed, extending a result of Dekker and Myhill. This collection is then used to investigate differences between the arithmetic of classical RET's and that of RET's on recursive manifolds, a difference relevant to the manifold equivalent of the Schröder-Bernstein Theorem.

The paper is devoted to showing that if the factorized group G = AB is almost solvable, if A and B are π-subgroups with min-p for some prime p in π and also if the hypercenter factor group A/H(A) or B/H(B) has min p for the prime p. then G is a π-group with min-p for the prime p.

Given Hausdorff spaces X and Y and a perfect irreducible and θ-continuous map f from X onto Y, technique that carries open (ultra) filters on X to open (ultra) filters on Y back and forth in a natural way is introduced. It is proved that if f is a perfect irreducible and θ-continuous map from X onto Y, then X is almost realcompact if and only if Y is almost realcompact. Several commutativity relations between the ‘absolutes of almost realcompactifications’ and the ‘almost realcompactifications of absólutes’ of a space X are discussed.

We present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

The purpose of this paper is to construct a class of groups which properly contains the class of N-constrained groups, and which is such that all groups in this class have N-injectors.

For any group G, we introduce the subset S(G) of elements g which are conjugate to for some positive integer k. We show that, for any bounded representation π of G any g in S(G), either π(g) = 1 or the spectrum of π(g) is the full unit circle in C. As a corollary, S(G) is in the kernel of any homomorphism from G to the unitary group of a post-liminal C*-algebra with finite composition series.

Next, for a topological group G, we consider the subset of elements approximately conjugate to 1, and we prove that it is contained in the kernel of any uniformly continuous bounded representation of G, and of any strongly continuous unitary representation in a finite von Neumann algebra.

We apply these results to prove triviality for a number of representations of isotropic simple algebraic groups defined over various fields.

In this paper we define generalization of paracompactness for bitopological spaces. (X, τ1, τ2) is Δ-pairwise paracompact if and only if every τi open cover admits a τ1 ∨ τ2 open refinement which is τ1 ∨ τ2 locally finite. Every quasimetric space (X, τp, τq) is Δ-pairwise paracompact. An analogue of Michael's characterization of regular paracompact spaces is proved for Δ-pairwise paracompact spaces.

Every continuous Volterra right inverse to the derivative in the space of complex-valued infinitely differentiable functions has the form of an integral.

The notion of an almost Krull domain is extended to rings satisfying a polynomial identity. Some general structural results are obtained. We also prove that skew polynomial rings R [ X, σ] remain almost Krull if R is an almost Krull ring. Finally, we study when semigroup ring R[S] are almost Krull rings, in the case when the group of quotients of S has the ascending chain condition on cyclic subgroups. An example is included to show that the general (semi-) group ring case is much more difficult to deal with.

For an integrable function f on T, we introduce a modified partial sum and establish its L1-convergence property. The relation between the sum and L1-convergence classes is also established. As a corollary, a new L1-convergence class is obtained. It is shown that this class covers all known L1-convergence classes.

Reflective Subcategories, Localizations and Factorization Systems: Corrigenda C. Cassidy, M. Hébert and G. M. Kelly, 1980 Mathematics subject classification (Amer. Math. Soc.): 18 A 20.