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.

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.

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

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.

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.

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.

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.

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 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.

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.

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