Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).

It is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

This paper attempts to classify the least ordinal α0 for which E(α0) (the E closure of α0 ∪ {α0}) is inadmissible. Among the result proved are (i)Lα0 = ZFC-; (ii)α0 is very large in comparison with the least ordinal satifying (i); (iii) (α0, α] marks precisely an ω-Gap, where α¯ = E(α0) ∩ ON; (iv) the Kr-sequence of α0 has length ω.

Using the theory of the Satake diagrams associated with the non-compact simple Lie algebras over the real number field R, we shall construct a family of simple groups over a field K which are called the simple groups associated with the Satake diagrams. The list of these simple groups includes all Chevalley groups and twisted groups, and all simple algebraic groups of adjoint type defined over R if K is the complex number field C (except two types given by Table II′). Furthermore, the simple groups associated with the Satake diagrams of type AIII, BI, DI are identified with the simple groups obtained from the unitary or orthogonal groups of non-zero indices. The quasi-Bruhat decomposition of the “non-split” simple groups associated with the Satake diagrams which are not Chevalley groups or twisted groups will be given in this paper.

In this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

An eulerian chain in a directed graph is a continuous directed route which traces every arc of the digraph exactly once. Such a route may be finite or infinite, and may have 0, 1 or 2 end vertices. For each kind of eulerian chain, there is a characterization of those diagraphs possessing such a route. In this survey paper we strealine these characterizations, and then synthesize them into a single description of all digraphs having some eulerian chain. Similar work has been done for eulerian chains in undirected graphs, so we are able to compare corresponding results for graphs and digraphs.

If there is a W (n, p), then there is a symmetric W (n2, p2).

This paper calculates the central Borel 2 cocycles for certain 2-step nilpotent Lie groups G with values in the injectives A of the category of 2nd countable locally compact abelian groups. The G's include, among others, all groups locally isomorphic to a Heisenberg group. The A's are direct sums of vector groups and (possibly infinite dimensional) tori, and in particular include R, T, and Cx. The main results are as follows.

(4.1) Every symmetric central 2 cocycle is trivial.

(4.2) Every central 2 cocycle is cohomologous with a skew symmetric bimultiplicative one (which is necessarily jointly continuous by [7]).

(4.3) The corresponding cohomology group H2cent (G, A) is calculated as the skew symmetric jointly continuous bimultiplicative maps modulo Homcont ([G, G]–, A).

These results generalize the case when G is a connected abelian Lie group and A = T, due to Kleppner [3]. Using standard facts of the cohomology of groups they can be interpreted as classifying all continuous central extensions (1) → A → E → G → (1) of the group G by the abelian group A. Finally some counterexamples are given to extending these results.

The bidual of a unital infrabarrelled l.m.c. C* algebra E, equipped with the bidual topology and the regualr Arens product, is always an l.m.c. C*-algebra. On the other hand, a unital l.m.c. *-algebra E has the C*-property if and only if every self-adjoint element x of E is strongly hermitian (x has real numerical range), or the sets of normalized states and normalized continuous positive linear forms of E coincide. Finally, every unital cpmplete l.m.c. C* algebra satisfying, locally, the property ‘the extreme points are dense in that set of continuous positive linear forms” (antiliminal algebra) has the complexes as its only normal elements.

We prove the following statements about bounded linear operators on a complex separable infinite dimensional Hilbert space. (1) Let A and B* be subnormal operators. If A2X = XB2 and A3X = XB3 for some operator X, then AX = XB. (2) Let A and B* be subnormal operators. If A2X – XB2 ∈ Cp and A3X – XB3 ∈ Cp for some operator X, then AX − XB ∈ C8p. (3) Let T be an operator such that 1 − T*T ∈ Cp for some p ≥1. If T2X − XT2 ∈ Cp and T3X – XT3 ∈ Cp for some operator X, then TX − XT ∈ Cp. (4) Let T be a semi-Fredholm operator with ind T < 0. If T2X − XT2 ∈ C2 and T3X − XT3 ∈ C2 for some operator X, then TX − XT ∈ C2.

We first define an inscribed center of a bounded convex body in a normed linear space as the center of a largest open ball contained in it (when such a ball exists). We then show that completeness is a necessary condition for a normed linear space to admit inscribed centers. We show that every weakly compact convex body in a Banach space has at least one inscribed center, and that admitting inscribed centers is a necessary and sufficient condition for reflexivity. We finally apply the concept of inscribed center to prove a type of fixed point theorem and also deduce a proposition concerning so-called Klee caverns in Hilbert spaces.

In this paper, linear embeddings of partial designs into designs are found where no repeated blocks are introduced in the embedding process. Triple systems, pure cyclic triple systems, cyclic and directed triple systems are considered. In particular, a partial triple system with no repeated triples is embedded linearly in a triple system with no repeated triples.

John W. Moon has discovered several computational errors in our article above (J. Austral. Math. Soc. (Series A) 20 (1975), 483–503). The five constants reported for identity trees below Table 1 at the bottom of page 502 are all wrong. The correct values are

The notion of quasi-regularity, defined for optimization problems in Rn, is extended to the Banach space setting. Examples are given to show that our definition of quasi-regularity is more natural than several other possibilities in the general situation. An infinite dimensional version of the Lagrange multiplier rule is established.

Convergent iterative sequences are constructed for the polynomials fm = z + zm, m ≧ 2, with initial point the lemniscate {z: |fm (z)| ≦1}. In the particular case m = 2 convergent iterative sequences are constructed also for f-1m, (z) with an arbitrary initial point. The method is based on a certain variational principle which allows reducing the problem to the well known situation of an analytic function mapping a simply connected domain into a proper subset of itself and possessing a fixed point in the domain.

The leading term approach to rates of convergence is employed to derive non-uniform and global descriptions of the rate of convergence in the central limit theorem. Both upper and lower bounds are obtained, being of the same order of magnitude, modulo terms of order n-r. We are able to derive general results by considering only those expansions with an odd number of terms.

A near 1-factor of a graph of order 2n ≧ 4 is a subgraph isomorphic to (n − 2) K2 ∪ P3 ∪ K1. Wallis determined, for each r ≥ 3, the order of a smallest r-regular graph of even order without a 1-factor; while for each r ≧ 3, Chartrand, Goldsmith and Schuster determined the order of a smallest r-regular, (r − 2)-edge-connected graph of even order without a 1-factor. These results are extended to graphs without near 1-factors. It is known that every connected, cubic graph with less than six bridges has a near 1-factor. The order of a smallest connected, cubic graph with exactly six bridges and no near 1-factor is determined.

Suppose H is a complex Hilbert space and K is a nonempty closed convex subset of H. Suppose T: K → H is a monotomc Lipschitzian mapping with constant L ≧ 1 such that, for x in K and h in H, the equation x + Tx Tx = h has a solution q in K. Given x0 in K, let {Cn}∞n=0 be a real sequence satisfying: (i) C0 = 1, (ii) 0 ≦ Cn < L-2 for all n ≧ 1, (iii) ΣnCn(1 − Cn) diverges. Then the sequence {Pn}∞n=0 in H defined by pn = (1 − Cn)xn + CnSxn, n ≧ 0, where {xn}∞n=0 in K is such that, for each n ≧ 1, ∥ xn – Pn−1 ∥ = infx ∈ k ∥ Pn−1 − x ∥, converges strongly to a solution q of x + Tx = h. Explicit error estimates are given. A similar result is also proved for the case when the operator T is locally Lipschitzian and monotone.

Lagrangian necessary conditions for optimality, of both Fritz John and Kuhn Tucker types, are obtained for a constrained minimization problem, where the functions are locally Lipschitz and have directional derivatives, but need not have linear Gâteaux derivatives; the variable may be constrained to lie in a nonconvex set. The directional derivatives are assumed to have some convexity properties as functions of direction; this generalizes the concept of quasidifferentiable function. The convexity is not required when directional derivatives are replaced by Clarke generalized derivatives. Sufficient Kuhn Tucker conditions, and a criterion for the locally solvable constraint qualification, are obtained for directionally differentiable functions.

The notion of strong lifting compactness is introduced for completely regular Hausdorff spaces, and its structural properties, as well as its relationship to the strong lifting, to measure compactness, and to lifting compactness, are discussed. For metrizable locally convex spaces under their weak topology, strong lifting compactness is characterized by a list of conditions which are either measure theoretical or topological in their nature, and from which it can be seen that strong lifting compactness is the strong counterpart of measure compactness in that case.