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

Up to a duality, the known finite projective planes which are not translation planes all are equivalent to affine planes which contain the type of structure defined below to be a "derivable net". (Insofar as the known finite planes are concerned, this means that the intimate connection between projective geometry and linear algebra still holds for non-Desarguesian planes.)

This paper proves that if р < s, then 0 is the only function that multiplies a Bergman Lр space into a Bergman Ls space.

J. Marica and J. Schönhein [4], using a theorem of M. Hall, Jr. [3], see below, proved that if any n − 1 arbitrarily chosen elements of the diagonal of an n × n array are prescribed, it is possible to complete the array to form an n × n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2], to the effect that an n × n incomplete latin square with at most n − 1 places occupied can be completed to an n × n latin square. When the complete diagonal is prescribed, it is easy to see that a counterexample is provided by the case that one letter appears n − 1 times on the diagonal and a second letter appears once. In the present paper, we prove that except in this case the completion to a full latin square is always possible. Completion to a symmetric latin square is also discussed.

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 $A\,:=\,-\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,.\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,+\,V$ be a magnetic Schrödinger operator on ${{\mathbb{R}}^{n}}$, where

$$\vec{a}:=\left( {{a}_{1}},...,{{a}_{n}} \right)\in L_{\text{loc}}^{2}\left( {{\mathbb{R}}^{n}},{{\mathbb{R}}^{n}} \right)\operatorname{and}0\le V\in L_{\text{loc}}^{1}\left( {{\mathbb{R}}^{n}} \right)$$

satisfy some reverse Hölder conditions. Let $\phi :\,{{\mathbb{R}}^{n}}\,\times \,[0,\,\infty )\,\to \,[0,\,\infty )$ be such that $\phi \left( x,\,. \right)$ for any given $x\,\in \,{{\mathbb{R}}^{n}}$ is an Orlicz function, $\phi \left( ^{.}\,,\,t \right)\,\in \,{{\mathbb{A}}_{\infty }}\left( {{\mathbb{R}}^{n}} \right)$ for all $t\,\in \,\left( 0,\,\infty \right)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I\left( \phi \right)\,\in \,(0,\,1]$. In this article, the authors prove that second-order Riesz transforms $V{{A}^{-1}}$ and ${{\left( \nabla \,-\,i\overrightarrow{a} \right)}^{2}}{{A}^{-1}}$ are bounded from the Musielak–Orlicz–Hardy space ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$, associated with $A$, to theMusielak–Orlicz space ${{L}^{\phi }}\left( {{\mathbb{R}}^{n}} \right)$. Moreover, we establish the boundedness of $V{{A}^{-1}}$ on ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$. As applications, some maximal inequalities associated with $A$ in the scale of ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$ are obtained.

The concepts of repleteness and more generally measure repleteness are investigated for set-theoretic lattices on specific subspaces of a lattice space. These general results are then applied to specific topological spaces, and we obtain as special cases some known theorems as well as some new results concerning for example, ∝-completeness, realcompactness, measure compactness and Borel-measure compactness.

It is shown that it follows from PFA that there is no compact scattered space of height greater than $\omega $ in which the sequential order and the scattering heights coincide.

We call a ring R (associative and with identity) for which every quasi-injective right R-module is injective a QII-ring. Similarly R is called an SSI-ring when every semisimple right R-module is injective. Clearly every semisimple, artinian ring is a QII-ring and every QII-ring is an SSI-ring. One then asks whether these inclusions among classes of rings are proper. The purpose of this note is to point out an instance when SSI implies QII It is then easy to see that an example of Cozzens shows that the class of QII-rings properly contains the class of semisimple, artinian rings.

Let f(x) be a polynomial of degree d ≥ 2 defined over the finite field kq with q = pn elements. Let

If f*(x, y) has no irreducible factor over kq which is absolutely irreducible, f is called an exceptional polynomial [1]. Davenport and Lewis have noted that when d is small compared with p, a permutation (substitution) polynomial is necessarily an exceptional polynomial. It is the purpose of this paper to prove the converse; that is, we will show the existence of a constant a(d), depending only on d, such that if f(x.) is an exceptional polynomial over kq, where p ≥ a(d), then f(x) is a permutation polynomial.

Let pij be the number of rooted convex polyhedra with i + 1 vertices and j + 1 faces. We express pij as a singly indexed summation whose terms decrease geometrically. From this we deduce that

uniformly as max(i, j) → ∞.

Let $G=\left( \mathbb{Z}/a\rtimes \mathbb{Z}/b \right)\times \text{S}{{\text{L}}_{2}}\left( {{\mathbb{F}}_{p}} \right)$, and let $X\left( n \right)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\text{Aut}\left( G \right)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X\left( 2dn-1 \right)$, where $2d$ is the period of $G$. The groups $\varepsilon \left( X\left( 2dn-1 \right)/\mu \right)$ of self homotopy equivalences of space forms $X\left( 2dn-1 \right)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X\left( 2dn-1 \right)$ are determined as well.

Answering a question of H. Petersson, we provide a class of examples of a pair of octonion algebras over a ring having isometric norms.

We show that any differentiable involution on a closed manifold whose fixed point set is a disjoint union of odd-dimensional real projective spaces must be a bounding involution.

Let be a representation of the finite group G over the field . If the order |G| of G is relatively prime to the characteristic of or n = 1 or 2, then it is known that the ring of invariants is Cohen-Macaulay. There are examples to show that need not be Cohen-Macaulay when |G| is divisible by the characteristic of . In all such examples is at least 4. In this note we fill the gap between these results and show that rings of invariants in three variables are always Cohen-Macaulay.

In [2], Herstein proves the following result:

Theorem. Let R be a prime ring, d≠0 a derivation of R such that d(x) d(y) = d(y) d(x) for all x, y ∈ R. Then, if char r≠2, R is commutative, and if char R = 2, R is commutative or an order in a simple algebra which is 4-dimensional over its center.

Let A be the local ring at a singular point p of an algebraic reduced curve. Let M (resp. Ml,..., Mh) be the maximal ideal of A (resp. of Ā). In this paper we want to classify ordinary singularities p with reduced tangent cone: Spec(G(A)). We prove that G(A) is reduced if and only if: p is an ordinary singularity, and the vector spaces span the vector space . If the points of the projectivized tangent cone Proj(G(A)) are in generic position then p is an ordinary singularity if and only if G(A) is reduced. We give an example which shows that the preceding equivalence is not true in general.

Two sequences of natural numbers are said to be complementary if they contain all the positive integers without repetition or omission. S. Beatty [l] observed that the sequences

(1)

(2)

(where square brackets denote the integral part function) are complementary if and only if α > 0 and α is irrational. We call the pair (1),(2) Beatty sequences of argument α.

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.

Knopp's inequality for convex functions ϕ on an interval I = [m,M] states that

for an explicit functional H, and all integrable g: [0, 1] → I. In this paper we give results of this kind in which the integral operator, ∫, is replaced by a general isotonic linear functional.