In this article it is shown that for any Banach space E,L (l1,E) always contains uncountably many distinct A/-ideals that are closed subspaces of K(l1,E) and which are not complemented in L (l1,E) . Using standard duality arguments one obtains the result that infinitely many distinct subspaces of K(E, c0) are M-ideals in L(E, c0). In particular, for the case E = c0, this shows that the uniqueness conditions enjoyed by K(lp), p > 1, is not valid for E = c0. The results are obtained by utilizing the identification of L (l1,E) with the vector-valued sequence space lx(E) and to exploit natural decompositions of lx(E)’ afforded by a class of Lprojections on lx(E)’ induced by certain E'-valued vector measures.

This paper gives necessary conditions for mixed norm estimates from Lp to Lr (Lq) for solutions of the Klein-Gordon equation

These conditions are best possible if p = 2 or

The existence of metastable immersion or span for space forms and homogeneous spaces is shown to depend only on the two-type of the space.

Let R be an integral domain. It is proved that if a nonzero ideal I of R can be generated by n < ∞ elements, then I is invertible (i.e., flat) if and only if I(∩ Rai) = ∩ Iai for all { a1, . . ., a n﹜ ⊂ I. The article's main focus is on torsion-free R-modules E which are LCM-stable in the sense that E(Ra ∩ Rb) = Ea ∩ Eb for all a, b ∈ R. By means of linear relations, LCM-stableness is shown to be equivalent to a weak aspect of flatness. Consequently, if each finitely generated ideal of R may be 2-generated, then each LCM-stable R-module is flat. Finally, LCM-stableness of maximal ideals serves to characterize Prüfer domains, Dedekind domains, principal ideal domains, and Bézout domains amongst suitably larger classes of integral domains.

A composition operator CT on L2(X, Σ,m) is a bounded linear transformation induced by a mapping T : X → X via CTf = f∘ T.

If m has no atoms then the co-rank of CT (i.e., dim is either zero or infinite. As a corollary, when m has no atoms, CT is a Fredholm operator iff it is invertible.

The concept of the M-radical of a submodule B of an R-module A is discussed (R is a commutative ring with identity and A is a unitary fl-module). The M-radical of B is defined as the intersection of all prime submodules of A containing B. The main result of the paper is that if denotes the ideal radical of (B:A), then M-rad B = provided that A is a finitely generated multiplication module. Additionally, it is shown that if A is an arbitrary module, where for some

We show that a topological space X admits a unique quasiuniformity if and only if every interior-preserving open collection of X is finite.

It is a question of Arhangel'skiĭ [1] (Problem 2) whether the identity ψ(G) = X(G) holds for every minimal Hausdorff topological group G = 〈G,u〉). (Here, as usual, ψ(G), the pseudocharacter of G, is the least cardinal number K for which there is such that and and x(G), the character of G,is the least cardinality of a local base at e for (〈G,u〉.) That 〈G, u〉 is minimal means that, if v is a Hausdorff topological group topology for G and v ⊂ u, then v = u.

In this paper, we give some conditions on G sufficient to ensure a positive response to Arhangel'skiï's question, and we offer an example which responds negatively to a question on minimal groups posed some years ago (cf. [6] (p. 107) and [4] (p. 259)).

E. Pirtle has conjectured that if D is an almost Krull domain in which the height one prime ideals are divisorial then D is a Krull domain. An example is given to show that this is not the case. Further, let U = and let denote the set of prime ideals of D which are minimal over some ideal (a):(b), where a, b ∈ D. If Dp is a valuation ring for each let then Huckaba and Papick have asked whether D[x]U must be a Prufer domain. The given example shows that it need not be.

Let A and A denote the classes of ordinal spaces with the order topology and Σ-product spaces of the two point discrete space respectively. Characterizations are given in terms of ultrafiIters of clopen sets of those O-dimensional Hausdorff topological spaces that can be embedded homeomorphically as a closed subspace of a topological product of either spaces from the class Λ or the class Δ. Both classes consist of spaces that are ω0-bounded. An example is given of a O-dimensional Hausdorff ω0-bounded space that cannot be homeomorphically embedded as a closed subset of a product of spaces from either Λ or Δ, answering a question of R. G. Woods.

A subset of the line is called non-averaging if, whenever two points belong to the set, their average does not. This paper provides an example of a closed set which is small in the sense that it is non-averaging and has porosity 1 at each of its points and yet large in the sense that its Hausdorff dimension is 1.

A corollary of the main result is that if L is a finite-dimensional Galois extension of a field K and if w is a valuation of L extending a valuation v of K, then K is closed in L if and only if all valuations of L extending v are dependent. A further consequence is a generalization of Ostrowski's criterion for a real-valued valuation to be henselian.

S. Golomb noticed that Riemann's zeta function ζ induces a probability distribution on the positive integers, for any s > 1, and studied some of its properties connected to divisibility. The object of this paper is to show that the probability distribution mentioned above is the unique solution of an entropy-maximization problem.

Le but de cette note est de proposer une caractérisation des espaces projectifs complexes, des hyperquadriques et des hypersurfaces du troisième degré dans Pnc à l'aide de leurs points d'intersection avec l'ensemble des zéros d'une section d'un fibre positif donné sur la variété ambiante. Ceci généralise et complète ainsi certains résultats présentés par Badescu et Itoh.

If A is an elementary abelian group, let (A) denote the group of units, modulo torsion, of the group ring Z[A]. We study (A) by means of the composite

where C and B run over all cyclic subgroups and factor-groups, respectively. This map, γ, is known to be injective; its index, too, is known. In this paper, we determine the rank of γ tensored (over Z);with various fields. Our main result depends only on the functoriality of

Let Un be an n-dimensional subspace of C[0, 1]. We prove that if n ≥ 2, and Un contains a function which is strictly positive on (0, 1), then there exists an f ∈ C[0, 1] which has more than one best one-sided L '-approximation from Un. We also characterize those Un with the property that each f ∈ C[0, 1] has a unique best one-sided L1(w)-approximation from Un with respect to every strictly positive continuous weight function w.

Suppose f is a continuous endomorphism of an interval which has a periodic orbit, p0 < P1 < … < pn, that defines a permutation a by f(pi) = pσ(i). If σ is irreducible the topological entropy of f is bounded below by the logarithm of the spectral radius of an n x n matrix which is induced by σ.

In this paper we extend a result of Oberlin and Stein on Radon Transforms to /c-plane transforms for Specifically let

where the supremum is taken over all affine k-planes ∏ parallel to the vector k-plane π. We show that F is in Ln of the Grassmann manifold Gn,k whenever f is in the Lorentz space L(n/k, 1) of ℝn. The proof relies very heavily on the ideas of M. Christ.

If R is an uncountable commutative ring, it is shown that there exists a proper subring of R having the same cardinality as R. It is also shown that if |R| = ω is an uncountable regular cardinal, and if R1 is a subring of R containing an identity of R and such that |R1| < ω, then there exists a proper R1-subalgebra S of R such that |S| = |R|.

We shall evaluate the determinants of n x n matrices of the form [f(i,j)], where f (w, r) is an even function of m (mod r). Among the examples of determinants of this kind are H. J. S. Smith's determinant det [(i,j)], where (m, r) is the greatest common divisor of m and r, and a generalization of Smith's determinant due to T. M. Apostol.