The main result of this paper is little more than a juxtaposition of a remark in [Leroux] (§5, part A), with a theorem of [Banaschewski]. However, the results are individually little known and their juxtaposition not at all. More explicitly, Leroux remarks, in somewhat different form,

Let the category X be complete and cocomplete, and have a set of cogenerators. Then a necessary and sufficient condition that there exist infective effacements (with respect to the class of regular monomorphisms) is that X be coregular and satisfy, in addition, condition COEX 4*.

A subset consisting of k elements chosen, from n distinct ordered elements, with given restrictions is called are stricted choice. For example, one restriction on the k elements may be that no two consecutive elements appear, while another may be that no two alternate elements appear. Certain restricted choices may be used to obtain solutions to permutation problems ([1, p. 349]; [4]). Each restricted choice corresponds to a "restricted sequence of Bernoulli trials" as described in [1], In this paper an elementary method of obtaining more general types of restricted choices is given. Some special cases of the restricted choices and restricted Bernoulli trials are presented in the form of examples.

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.

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.

Let B be a ring with unity, A a imitai subring of the centre Cof B. Suppose further that B is A-integral. (That is, every element of B satisfies a monic polynomial with coefficients in A.) Under these assumptions, Hoechsmann [2] showed that "contraction to A" is a mapping from:

1. (1) The prime ideals of B onto the prime ideals of A,

2. (2) The maximal ideals of B onto the maximal ideals of A.

In this note we show that, under additional assumptions, a noncommutative version of the rest of the Cohen-Seidenberg "going up theorem" can be established.

An estimate for Kloosterman sums with oscillating coefficients is presented. Precisely we show: for any ϵ > 0 and a, b positive integers with (a, b) — 1 we have,

Similar techniques may be used to estimate other Kloosterman sums with oscillating coefficients which are not smooth.

H. T. Croft [1] has conjectured that among all tetrahedra with fixed total edge length the regular one has the greatest surface area. In this note we prove the following result, which includes this conjecture as a special case

For most classical and similitude groups, we show that each element can be written as a product of two transformations that preserve or almost preserve the underlying form and whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras, and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual.

One of the fundamental tools in the theory of totally ordered groups is Hahn’s Theorem (a detailed discussion may be found in Fuchs [3]), which asserts, roughly, that every abelian totally ordered group can be embedded in a lexicographically ordered (unrestricted) direct sum of copies of the ordered group of real numbers. Almost any general question regarding the structure of abelian totally ordered groups can be answered by reference to Hahn’s theorem. For the class of nonabelian totally ordered groups, a theorem which parallels Hahn’s Theorem is given in [5], and states that each totally ordered group can be o-embedded in an ordered wreath product of subgroups of the real numbers. In order to extend this theorem to include an “if and only if” statement, one must consider lattice ordered groups, as an ordered wreath product of subgroups of the real numbers is, in general, not totally-ordered, but lattice ordered.

A ring or near-ring R is called periodic if for each xϵR, there exist distinct positive integers n, m for which xn = xm. A well-known theorem of Herstein states that a periodic ring is commutative if its nilpotent elements are central [5], and Ligh [6] has asked whether a similar result holds for distributively-generated (d-g) near-rings. It is the purpose of this note to provide an affirmative answer.

We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

In a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, forma dense ${{G}_{\delta }}$. A similar theorem for bounded self-adjoint operators with a given normbound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon’s result, with simplicity of the spectrum added.

It is shown that techniques of Browder and Ishikawa for approximating fixed points of nonexpansive mappings extend in a more restricted sense to the locally nonexpansive case.

Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.

Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in ${{\mathbb{R}}^{3}}$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the Bäcklund transformation for constant torsion curves in ${{\mathbb{R}}^{3}}$, previously derived fromthe well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in ${{H}^{3}}$ or ${{S}^{3}}$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.

A topological space X is called a direct limit of a family (Xα) of subspaces of X if and only if

(1)

(2)

If X is a direct limit of an increasing sequence (Xn) of closed subspaces then it is well known and easy to prove that X is a T1-space resp. a T4-space provided all Xn are T1-spaces resp. T4-spaces.

Let $G$ be an infinite discrete group and let $\beta G$ be the Stone-Čech compactification of $G$. We take the points of $\beta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U\left( G \right)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\,\in \,U\left( G \right)$, define ${{I}_{p}}\,\subseteq \,\beta G$ by ${{I}_{p}}\,=\,{{\bigcap }_{A\in p}}\text{cl}\left( GU\left( A \right) \right)$, where $U\left( A \right)\,=\,\left\{ p\,\in \,U\left( G \right)\,:\,A\in \,p \right\}$. We show that if $\left| G \right|$ is a regular cardinal, then $\left\{ {{I}_{p}}\,:\,p\,\in \,U\left( G \right) \right\}$ is the finest decomposition of $U\left( G \right)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U\left( G \right)$ is Hausdorff.

Let R be an associative ring (not necessarily with identity).

R is a left П-ring if it has the following property: Let M be a finitely generated left R-module, N a submodule of M and ϕ:N→M an epimorphism. Then ϕ is an isomorphism.

Let $C$ be a hyperelliptic curve given by the equation ${{y}^{2}}\,=\,f(x)$ for $f\,\in \,\mathbb{Z}[x]$ without multiple roots. We say that points ${{P}_{i}}\,=\,({{x}_{i}},\,{{y}_{i}})\,\in \,C(\mathbb{Q})$ for $i\,=\,1,\,2,\,\ldots ,\,m$ are in arithmetic progression if the numbers ${{x}_{i}}$ for $i\,=\,1,\,2,\,\ldots ,\,m$ are in arithmetic progression.

In this paper we show that there exists a polynomial $k\,\in \,\mathbb{Z}[t]$ with the property that on the elliptic curve $\varepsilon \prime :{{y}^{2}}={{x}^{3}}+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four points in arithmetic progression that are independent in the group of all $\mathbb{Q}(t)$-rational points on the curve $\varepsilon \prime $. In particular this result generalizes earlier results of Lee and Vélez. We also show that if $n\,\in \,\mathbb{N}$ is odd, then there are infinitely many $k$'s with the property that on curves ${{y}^{2}}\,=\,{{x}^{n}}\,+\,k$ there are four rational points in arithmetic progressions. In the case when $n$ is even we can find infinitely many $k$'s such that on curves ${{y}^{2}}\,=\,{{x}^{n}}\,+\,k$ there are six rational points in arithmetic progression.

For relatively prime positive integers ${{u}_{0}}$ and $r$, we consider the least common multiple ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ of the finite arithmetic progression $\left\{ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$. We derive new lower bounds on ${{L}_{n}}$ that improve upon those obtained previously when either ${{u}_{0}}$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n\,+\,1$ for ${{u}_{0}}$ properly chosen, and is also nearly sharp as $n\,\to \,\infty$.