We study the asymptotic properties of positive solutions to the semilinear equation — Δu = f(x, u). Existence and asymptotic estimates are obtained for solutions in exterior domains, as well as entire solutions, for n ≧ 2. The study uses integral operator equations in Rn, and convergence theorems for solutions of Poisson's equation in bounded domains. A consequence of the method is that more precise estimates can be obtained for the growth of solutions at infinity, than have been obtained by other methods. As a special case the results are applied to the generalized Emden-Fowler equation — Δu = p(x)uγ, for γ > 0

Let us suppose one of the following conditions: (a) p ≧ 2 and F is a closed subspace of a projective limit (b) p = 1 and F is a complemented subspace of an echelon Köthe space of order 1, Λ(X,β,μ,gk); and (c) 1 > p > 2 and F is a quotient of a countable product of Lp(μn) spaces. Then, F is Montel if and only if no infinite dimensional subspace of F is normable.

Recent work by Krystock, Porter, and Vermeer has emphasized the importance of the concepts of Katětov spaces and H-sets in the theory of H-closed spaces. These properties are closely related to being the θ-closure of some set and being the adherence of an open filter. This relationship is developed by establishing, among other facts, that an H-closed space in which every closed set is the θ-closure of some set is compact and the θ-closure of a subset of an H-closed space is Katětov and characterizing the open filter adhérences of a space as precisely those sets which are the image of a closed set of the absolute of the space. Also, examples are given of a countable, scattered space which is not Katětov and an H-closed space with an H-closed subspace which is not the θ-closure of any subset of the given space.

It is shown that if V is a congruence distributive variety whose members have one element subalgebras, then the class of absolute retracts of V is closed under direct products. If V is residually small, then a characterisation of the amalgamation class of V is given.

Suppose X and Y are closed subspaces of (ΣXn)p and (ΣYn)q (1 < p ≦ q < ∞, dim Xn < ∞, dimYn < ∞), respectively. If K(X, Y), the space of the compact linear operators from X to Y, is dense in L(X, Y), the space of the bounded linear operators from X to Y, in the strong operator topology, then K(X, Y) is an M-ideal in L(X, Y).

We examine the convergence and analytic properties of a continued fraction of Ramanujan and its connection to the orthogonal polynomials of Meixner-Pollaczek.

We prove that if A is the infinitesimal generator of a bounded analytic semigroup in a sector {z ∊ C : |arg z| ≦ (απ)/2} of bounded linear operators on a Banach space, then the following inequalities hold:

for any x ∊ D(An) and for any 0 < β < α. This result helps us to answer in affirmative a question raised by M. W. Certain and T. G. Kurtz [3]. Similar inequalities are proved for cosine operator funtions.

Differentially simple local noetherian Q -algebras are shown to be always (a certain type of) subrings of formal power series rings. The result is established as an illustration of a general theory of differential filtrations and differential completions.

A bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

If G is any group then g ∊ G is called an involution if g ≠ 1 and g o g = 1. A group G is called bireflectional if every element in G is a product of two involutions. It is known that 2- dimensional, 3- dimensional, and some types of n-dimensional (n > 3) absolute geometries (in the sense of H. Kinder) are bireflectional. In this article the author proves the general result that every n-dimensional absolute geometry is bireflectional.

Recently we have introduced an enriched cohomology theory for categories that are tripleable (algebraic) over a category of modules. The cohomology admits a circle product, related to the obstruction problem for algebraic deformations, making the total complex a graded ring. We here offer similar constructions in two other situations - coalgebraic and bialgebraic categories. Examples include categories of bialgebras, sheaves of modules, and sheaves of algebras over a sheaf of rings.

Let f be an additive arithmetical function having a distribution F. For any sequence let

In this note, we determine the slowest growing function b so that Qn{b, f) tends weakly to F, for various f.

An easy proof will be given to show that for finite dimensional Hopf-algebras with nilpotent augmentation ideal over the field of p elements, the upper and lower Loewy-series coincide. In particular, this holds for the restricted universal envelope of nilpotent Lie-p-algebras with nilpotent p-map.

Using a technique of Agarwal and Andrews (1987), bijective proofs of some n-color partition identities discovered recently by the author, are given.

Let P be a Markov operator on L∞(X, Σ, m) which does not disappear (i.e., P1A ≡ 0 => 1A ≡ 0 ) . We study the relationship between the σ-algebras

(the deterministicσ-algebra), and the asymptoticσ-algebra

When m is a σ-finite invariant measure, measurable iff p*npnf = f, and also iff Pnf has the same distribution as f . The case of a convolution operator on a locally compact group is considered.

In [2], Erdös showed that the counting function A(n) of a B2-sequence satisfies Here it is shown that A(n) satisfies an analogous relationship for B4-sequences!

The definition of strong Cesaro summability with respect to a modulus is extended to a definition of strong A -summability with respect to a modulus when A is a nonnegative regular matrix summability method. It is shown that if a sequence is strongly A-summable with respect to an arbitrary modulus then it is A-statistically convergent and that Astatistical convergence and strong A-summability with respect to a modulus are equivalent on the bounded sequences.

A ring R is said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

In this paper we show that for a Banach space X having the Rn,k property and such that X is constrained in a dual space having the Radon-Nikodým property, the space of X-valued Bochner integrable functions has the Rn,k property.

Let K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:

(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.

(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.