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NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let {T[subscript n]} be a sequence of continuous linear transformations on L[superscript p](X) for finite measure space X and 1 0. The first result of this paper is to prove (1) from convergence under the weaker assumption that the sequence {T[subscript n]} commutes with each member of a family of measure-preserving transformations on X, a family which is large enough to have only trivial fixed sets. This result contains Stein's theorem, concludes maximal ergodic theorems from individual ergodic theorems, and applies in situations arising in probability theory. The conditions above are then weakened so that the domain of {T[subscript n]} becomes an F-space of functions satisfying a certain concordance condition on its topology, and the operators {T[subscript n]} become continuous in measure with range in the space of measurable functions on X. Then, under the assumption that {T[subscript n]} commutes with enough measure-preserving transformations as above, a slightly weaker version of (1) is concluded. Now, assume that {T[subscript n]} is a sequence of continuous-in-measure linear transformations of an abstract F-space [...] into measurable functions on finite measure space X, and that [...] for every f in a dense subset of E. A decomposition of the measure space X is then obtained, such that [...] on one of the sets for all f in E, and such that for all f in the complement of a set of the first category in E, [...] a.e. on the other set of the decomposition. A, theorem of Banach then applies on the first set to give a result which can be viewed as similar to (1). The decomposition is then applied to the preceeding results to prove new theorems

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this paper the relationship between the 2-length [...] and the 2-exponent [...] of a finite solvable group G is studied. It is shown that [...] - 1 provided that [...]. The special case of groups satisfying [...] = 2, i.e., groups whose Sylow 2-groups are of exponent 4, is investigated to determine whether [...] in this case. This question is not answered but it is shown that a certain normal subgroup (which may be the whole group) satisfies [...]. In addition if all the elements of order 4 are contained in this subgroup, then [...] for the whole group as well. As an application of this last result, it is proved that [...] in a group of exponent 12

The theory of optimal algorithmic processes is part of computational complexity. This paper deals with analytic computational complexity. The relation between the goodness of an iteration algorithm and its new function evaluation and memory requirements are analyzed. A new conjecture is stated.

In a recent paper R. Dudley gave a characterization of those sequences of independent and identically distributed random variables which are l p {l_p} -compatible for p ≧ 1 p \geqq 1 . In the present note we extend his result into p ∈ ( 0 , 1 ] p \in (0,1] and provide some conditions (necessary or sufficient) for l φ {l_\varphi } -compatibility of a sequence of independent random variables not necessarily identically distributed.

This paper is concerned with linear partial difference operators L having constant coefficients. The functions considered are defined only on the lattice points of the complex plane. It is shown that with any two solutions of Lu(z) = 0 there is associated a new solution which is represented as a convolution product. This product may be considered as a type of line integral and is based upon a discrete analogue of Green's formula. This development may be regarded as an anology to the pioneering work of H. Lewy concerning the composition of solutions of partial differential equations. It may also be considered a continuation of the investigation pursued by Duffin and Duris in the introduction of a convolution product for discrete analytic functions. HUNT LIBRARY CARNEGIE-MELLON UNIVERSITY A Convolution Product for the Solutions of Partial Difference Equations* by R. J. Duffin and Joan Rohrer CARNEGIE INSTITUTE OF TECHNOLOGY

The present paper is motivated by the observation that Weyl's equidistribution theorem for real sequences on a bounded interval can be formulated in a way which is also meaningful for sequences of selfadjoint operators on a Hilbert space. We shall provide general results on weak convergence of operator measures which yield this version of Weyl's theorem as a corollary. Further, by combining the above results with the von Neumann ergodic theorem, we will obtain a Cesaro convergence property, equivalently, an "ergodic theorem", which is valid for all (projectionvalued) spectral measures whose support is in a bounded interval, as well as for the more general class of positive operator-valued measures. Within the same circle of ideas we deduce a convergence property which completely characterizes those spectral measures associated with "strongly mixing" unitary transformations. The final sections are devoted to applications of the preceding results in the study of complexvalued Borel measures as well as to an extension of our results to summability methods other than CesAro convergence. In particular, we obtain a complete characterization, in purely measure theoretic terms, of those complex measures on a bounded interval whose Fourier-Stieltjes coefficients converge to zero.

or one of its rotations. The conjecture was proved for n = 2, 3, and 4 by Bieberbach [1], Loewner [2] and Garabedian and Schiffer [3], respectively. In recent papers, it has been proved to be true if f is sufficiently close to K in one or more of the topologies defined by n (l) ~~~~~E I ak- k | k=2 (2) Re(2 - a2), (3) Re(3 - a3).