Proceedings of the Edinburgh Mathematical Society

Volume 38 (Series II) Part 1 February 1995

Page 133—Paper by Daqing Wan, Gary L. Mullen and Peter Jau-Shyong Shiue

Due to a printer's error a mathematical sign was missed in the article heading which should now read:

The Number of Permutation Polynomials of the Form f(x) + cx Over a Finite Field

Unlike ring modules certain faithful N-groups are unique. The main theorem is that if N is a 2-tame ring-free near-ring where N/J(N) has DCCR, then all faithful 2-tame N-groups are finite and N-isomorphic. The finiteness of such an N-group follows easily from the fact it has a composition series. It is then shown that the length of a composition series depends only on N. This fact is used at key points in the proof. The situations where the N-group has or has not a minimal submodule require different analysis. The first case makes use of other interesting results and the second makes strong use of the inductive assumption.

A star operation is defined and studied for the Steenrod algebra. Numerous product formulas of Steenrod operations are presented.

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.

Let X = (X0, X1) and Y = (Y0, Y1) be Banach couples and suppose T:X→Y is a linear operator such that T:X0→Y0 is compact. We consider the question whether the operator T:[X0, X1]θ→[Y0, Y1]θ is compact and show a positive answer under a variety of conditions. For example it suffices that X0 be a UMD-space or that X0 is reflexive and there is a Banach space so that X0 = [W, X1]α, for some 0<α<1.

In response to a letter from Goldbach, Euler considered sums of the form

where s and t are positive integers.

As Euler discovered by a process of extrapolation (from s + t ≦ 13), σh(s, t) can be evaluated in terms of Riemann ζ-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we give a formula for the series

This evaluation involves ζ-functions and σh(2, m).

In view of understanding the Hopf algebra structure of the loop space homology in terms of H*(ΩE) and the map f, we consider the homotopy fibre F of the inclusion map In [15], the case when H*(Ωω) is surjective (the “inert” case) was studied, and in [11] a weaker condition, called “lazy”, was considered. Here we give several new characterizations of inert and lazy cell attachments in terms of properties of F. We also show how these results extend to the case of the mapping cone of an arbitrary map f: W→E.

For a pair of continuous linear operators T and S on complex Banach spaces X and Y, respectively, this paper studies the local spectral properties of the commutator C(S, T) given by C(S, T)(A): = SA−AT for all A∈L(X, Y). Under suitable conditions on T and S, the main results provide the single valued extension property, a description of the local spectrum, and a characterization of the spectral subspaces of C(S, T), which encompasses the closedness of these subspaces. The strongest results are obtained for quotients and restrictions of decomposable operators. The theory is based on the recent characterization of such operators by Albrecht and Eschmeier and extends the classical results for decomposable operators due to Colojoară, Foiaş, and Vasilescu to considerably larger classes of operators. Counterexamples from the theory of semishifts are included to illustrate that the assumptions are appropriate. Finally, it is shown that the commutator of two super-decomposable operators is decomposable.

We construct examples of unit-regular rings R for which K0(R) has torsion, thus answering a longstanding open question in the negative. In fact, arbitrary countable torsion abelian groups are embedded in K0 of simple unit-regular algebras over arbitrary countable fields. In contrast, in all these examples K0(R) is strictly unperforated.

We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials (where Pj(x) is a polynomial of degree j) under the restriction that the sup-norm of is bounded on the interval [ −b, b] (b>0). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval [ −b, b].