Let D be the unit disc in the complex plane. It is shown that

(i) for 0 < p < 2, there exists an analytic function f ∊ BMOA for which

and

(ii) for 2 < p < ∞, there exists an analytic function f ∉ BMOA for which

This settles the question of Stroethoff [5] on BMOA.

We generalize the notion of t-orthogonality in p-adic Banach spaces by introducing t-frames (§2). This we use to prove that a Fréchet-Montel (FM-)space is of countable type (Theorem 3.1), the non-archimedeancounterpart of a well known theorem in functional analysis over ℝ or ℂ ([6], p. 231). We obtain several characterizations of FM-spaces (Theorem 3.3) and characterize the nuclear spaces among them (§4).

Suppose that A is either the group algebra L1 (G) of a locally compact group G, or the Volterra algebra or a weighted convolution algebra with a regulated weight. We characterize: a) Module homomorphisms of A*, when A* is regarded an A** left Banach module with the Arens product, b) all the weak*-weak* continuous left multipliers of A**.

In this paper it is shown that the computation of higher dimensional harmonic volume, defined in [1], can be reduced to Harris' computation in the onedimensional case (See [3]), so that higher dimensional harmonic volume may be computed essentially as an iterated integral. We then use this formula to produce a specific smooth curve , namely a specific double cover of the Fermat quartic, so that the image of the second symmetric product of in its Jacobian via the Abel-Jacobi map is algebraically inequivalent to the image of under the group involution on the Jacobian.

Suppose u is a harmonic function on a domain D and x, x' are in D. We estimate |u(x) — u(x')| using two Brownian motions started at x and x' and killed on exiting a cube Q ⊂ D. By selecting appropriate versions of the two Brownian motions, a classical gradient estimate for u is easily derived.

Dans cet article, nous considérons le problème d'existence d'une solution périodique du problème de la forme: y″ = f(t,y,y′), 0 ≦ t ≦ 1 où f: [0,1 ] X R x R —> R est une fonction continue qui n'est pas nécessairement périodique et sans condition de croissance sur la variable y′. Nous obtenons certaines extensions d'un théorème du type Nirenberg ainsi que de l'équation de Liénard.

We give an example of an elliptic second order pseudodifferential operator with a non-zero eta invariant. The operator is constructed on homogeneous bundles over compact Lie groups and is formed by composing differential operators and an operator of class In general it is not elliptic but in the special case of even dimensional bundles over SU(2) it is elliptic. The eta invariant is calculated in the special case and in the non elliptic case a difference eta invariant is obtained.

We consider the directed graph representing the obstruction relation between objects moving along the streamlines of a two-dimensional velocity field. A collection of objects is sequentially separable if and only if the corresponding graph has no directed cycles. A sufficient condition for this is the permeability of closed Jordan curves.

We present a new elementary proof of the fact that a norm one multiplier ϕ on LP(T) satisfying ϕ(0) = ϕ(k) = 1 is k-periodic, and extend this result, when possible, to multipliers on translation invariant subspaces of LP. A consequence of our work is that all such multipliers on HP(T) are the restriction of a norm one multiplier on LP(T).

In this paper, under certain hypothesis, we characterize generalized hyperbolic cylinders as the only null 2-type hypersurfaces in a Lorentz space.

There have been a number of papers which give necessary conditions for a ring to be finite, and a few, most notably H. E. Bell [1], which do the same for near-rings. We wish to make a contribution to this latter theme. Most of Bell's results concern distributive near-rings. Our main contribution is to extend a number of these results to weakly distributive near-rings.

Let E be a Banach ideal space and X be a Banach space. The Banach function space E(X) does not contain a copy of C0 if and only if neither E nor X contains a copy of c0. Some extensions of this result are also noted.

Suppose is a sequence of functions on ℝn with Δfk = fk+1 (where Δ is the Laplacian) that satisfies the growth condition: |fk(x)| ≤ Mk{1 + |x|)a where a ≥ 0 and the constants have sublinear growth Then Δf0 = —f0- This characterizes eigenfunctions f of Δ with polynomial growth in terms of the size of the powers Δkf, —∞ < k < ∞. It also generalizes results of Roe (where a = 0, Mk = M, and n = 1 ) and Strichartz (where a = 0, Mk = M for n). The analogue holds for formally self-adjoint constant coefficient linear partial differential operators on ℝn.

Let q be a positive power of an odd prime p, and let Fq(t) be the function field with coefficients in the finite field of q elements. Let denote the ideal class number of the real quadratic function field obtained by adjoining the square root of an even-degree monic . The following theorem is proved: Let n ≧ 1 be an integer not divisible by p. Then there exist infinitely many monic, squarefree polynomials, such that n divides the class number, . The proof constructs an element of order n in the ideal class group.

This presentation concerns the relation of chain conditions on a space X, with the weights of compact sets in Cp(X), generalizing up to the class of dσ-bounded spaces, or stable spaces. In the last case, stronger results are obtained for Corson compact subsets of CP(X).

We consider the self-adjoint second-order scalar difference equation (1) Δ(rnΔxn) +pnXn+1 = 0 and the matrix system (2) Δ(RnΔXn) + PnXn+1 = 0, where are seQuences of real numbers (d x d Hermitian matrices) with rn > 0(Rn > 0). The oscillation and nonoscillation criteria for solutions of (1) and (2), obtained in [3, 4, 10], are extended to a much wider class of equations by Riccati and averaging techniques.

We obtain inequalities relating the eigenvalues of the Dirichlet and the Neumann problems for the Laplacian in the unit ball in Rn.

Let X be a projective nonsingular variety over a real closed field R such that the set X(R) of R-rational points of X is nonempty. Let ClR(X) = Cl(X)/Γ(X), where Cl(X) is the group of classes of linearly equivalent divisors on X and Γ(X) is the subgroup of Cl(X) consisting of the classes of divisors whose restriction to some neighborhood of X(R) in X is linearly equivalent to 0. It is proved that the group ClR(X) is isomorphic to (Z/2)s for some non-negative integer s. Moreover, an upper bound on s is given in terms of the Z/2-dimension of the group cohomology modules of Gal(C/R), where , with values in the Néron-Severi group and the Picard variety of Xc = X xR C.

A method will be introduced to solve problems utt — uss = h(s, t), u(t,t) - u(1+t,1 - t), u(s,0) = g(s), u(1,1) = 0 and for (s, t) in the characteristic triangle R = {(s,t) : t ≤ s ≤ 2 — t, 0 ≤ t ≤ 1}. Here represent the directional derivatives of u in the characteristic directions e1 = (— 1, — 1) and e2 = (1, — 1), respectively. The method produces the symmetric Green's function of Kreith [1] in both cases.

We explicitly identify the possible probability entrance laws for a class of measure-valued processes that are constructed by taking a particular measure-valued Markov branching process and conditioning it to stay away from the zero measure trap. The set of extreme points of the entrance space is larger than the state space of the conditioned process, and contains elements which correspond to starting the conditioned process at the zero measure.