Vortex rings: existence and asymptotic estimates
Avner
Friedman;
Bruce
Turkington
1-37
Abstract: The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter $\lambda$; as $ \lambda \to \infty$ the vortex ring tends to a torus whose cross-section is an infinitesimal disc.
The stable geometric dimension of vector bundles over real projective spaces
Donald M.
Davis;
Sam
Gitler;
Mark
Mahowald
39-61
Abstract: An elementary argument shows that the geometric dimension of any vector bundle of order ${2^e}$ over $R{P^n}$ depends only on $e$ and the residue of $ n\,\bmod \,8$ for $ n$ sufficiently large. In this paper we calculate this geometric dimension, which is approximately $2e$. The nonlifting results are easily obtained using the spectrum $bJ$. The lifting results require $ bo$-resolutions. Half of the paper is devoted to proving Mahowald's theorem that beginning with the second stage $ bo$-resolutions act almost like $ K({Z_2})$-resolutions.
An effective version of Dilworth's theorem
Henry A.
Kierstead
63-77
Abstract: We prove that if $(P,\,{ < ^P})$ is a recursive partial order with finite width $w$, then $P$ can be covered by $ ({5^w} - 1)/4$ recursive chains. For each $w$ we show that there is a recursive partial ordering of width $w$ that cannot be covered by $4(w - 1)$ recursive chains.
Geometric properties of homogeneous vector fields of degree two in ${\bf R}\sp{3}$
M. Izabel T.
Camacho
79-101
Abstract: In the space of homogeneous polynomial vector fields of degree two, those that project on Morse-Smale vector fields on $ {S^2}$ by the Poincaré central projection form a generic subset. The classification of those vector fields on ${S^2}$ without periodic orbits is given and applications to the study of local actions of the affine group of the line are derived.
Brouwerian semilattices
Peter
Köhler
103-126
Abstract: Let ${\mathbf{P}}$ be the category whose objects are posets and whose morphisms are partial mappings $\alpha :P \to Q$ satisfying (i) $ \forall p,\,q \in \operatorname{dom} \alpha [p < q \Rightarrow \alpha (p) < \alpha (q)]$ and (ii) $\forall p \in \operatorname{dom} \alpha \forall q \in Q[q < \alpha (p) \Rightarrow \exists r \in \operatorname{dom} \alpha [r < p\& \alpha (r) = q]]$. The full subcategory ${{\mathbf{P}}_f}$ of $ {\mathbf{P}}$ consisting of all finite posets is shown to be dually equivalent to the category of finite Brouwerian semilattices and homomorphisms. Under this duality a finite Brouwerian semilattice $ \underline A$ corresponds with $ M(\underline A )$, the poset of all meet-irreducible elements of $\underline A$. The product (in ${{\mathbf{P}}_f}$) of $n$ copies $ (n \in \mathbb{N})$ of a one-element poset is constructed; in view of the duality this product is isomorphic to the poset of meet-irreducible elements of the free Brouwerian semilattice on $n$ generators. If $ {\mathbf{V}}$ is a variety of Brouwerian semilattices and if $\underline A$ is a Brouwerian semilattice, then $\underline A $ is ${\mathbf{V}}$-critical if all proper subalgebras of $\underline A $ belong to ${\mathbf{V}}$ but not $ \underline A$. It is shown that a variety $ {\mathbf{V}}$ of Brouwerian semilattices has a finite equational base if and only if there are up to isomorphism only finitely many $ {\mathbf{V}}$-critical Brouwerian semilattices. This is used to show that a variety generated by a finite Brouwerian semilattice as well as the join of two finitely based varieties is finitely based. A new example of a variety without a finite equational base is exhibited.
Preservation of convergence of convex sets and functions in finite dimensions
L.
McLinden;
Roy C.
Bergstrom
127-142
Abstract: We study a convergence notion which has particular relevance for convex analysis and lends itself quite naturally to successive approximation schemes in a variety of areas. Motivated particularly by problems in optimization subject to constraints, we develop technical tools necessary for systematic use of this convergence in finite-dimensional settings. Simple conditions are established under which this convergence for sequences of sets, functions and subdifferentials is preserved under various basic operations, including, for example, those of addition and infimal convolution in the case of functions.
A partition relation for Souslin trees
Attila
Máté
143-149
Abstract: The aim of these notes is to give a direct proof of the partition relation Souslin tree $ \to (\alpha )_k^2$, valid for any integer $k$ and any ordinal $\alpha < {\omega _1}$. This relation was established by J. E. Baumgartner, who noticed that it follows by a simple forcing and absoluteness argument from the relation ${\omega _1} \to (\alpha )_k^2$, which is a special case of a theorem of Baumgartner and A. Hajnal.
Uniform partitions of an interval
Vladimir
Drobot
151-160
Abstract: Let $\{ {x_n}\}$ be a sequence of numbers in $ [0,\,1]$; for each $ n$ let ${u_0}(n), \ldots ,\,{u_n}(n)$ be the lengths of the intervals resulting from partitioning of $ [0,\,1]$ by $ \{ {x_1},\,{x_2}, \ldots ,\,{x_n}\}$. For $p > 1$ put $ {A^{(p)}}(n) = {(n + 1)^{p - 1}}\sum\nolimits_0^n {{{[{u_j}(n)]}^p}}$; the paper investigates the behavior of ${A^{(p)}}(n)$ as $ n \to \infty$ for various sequences $\{ {x_n}\}$. Theorem 1. If ${x_n} = n\theta \,(\bmod \,1)$ for an irrational $ \theta > 0$, then $\lim \,\inf \,{A^{(p)}}(n) < \infty$. However $ \lim \,\sup \,{A^{(p)}} < \infty$ if and only if the partial quotients of $\theta$ are bounded (in the continued fraction expansion of $\theta$). Theorem 2 gives the exact values for $\lim \,\inf$ and $\lim \,\sup$ when $\theta = \tfrac{1} {2}(1 + \sqrt 5 )$. Theorem 3. If $ \lim \,{A^{(p)}}(n) = \Gamma (p + 1)$ almost surely.
Singular sets and remainders
George L.
Cain;
Richard E.
Chandler;
Gary D.
Faulkner
161-171
Abstract: This paper characterizes the singular sets of several traditional classes of continuous mappings associated with compactifications. By relating remainders of compactifications to singular sets of mappings with compact range, new results are obtained about each.
Facial characterizations of complex Lindenstrauss spaces
A. J.
Ellis;
T. S. S. R. K.
Rao;
A. K.
Roy;
U.
Uttersrud
173-186
Abstract: We characterize complex Banach spaces $A$ whose Banach dual spaces are ${L^1}(\mu )$ spaces in terms of $L$-ideals generated by certain extremal subsets of the closed unit ball $K$ of $ {A^{\ast}}$. Our treatment covers the case of spaces $A$ containing constant functions and also spaces not containing constants. Separable spaces are characterized in terms of $ {w^{\ast}}$-compact sets of extreme points of $K$, whereas the nonseparable spaces necessitate usage of the $ {w^{\ast}}$-closed faces of $K$. Our results represent natural extensions of known characterizations of Choquet simplexes. We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of $ {C_{\mathbf{C}}}(X)$ which are complex Lindenstrauss spaces.
Measures with bounded powers on locally compact abelian groups
G. V.
Wood
187-210
Abstract: If $\mu$ is a measure on a locally compact abelian group with its positive and negative convolution powers bounded in norm by $K < \tfrac{1} {3}(4\cos (\pi /9) + 1) \sim 1.58626$ , then $\mu$ has the form $ \mu = \lambda (\cos \theta {\delta _x} + i\,\sin \theta {\delta _{xu}})$ where $\vert\lambda \vert = 1$ and ${u^2} = e$. Applications to isomorphism theorems are given. In particular, if $ {G_1}$ and ${G_2}$ are l.c.a. groups and $ T$ is an isomorphism of ${L^1}({G_1})$ onto $ {L^1}({G_2})$ with $\left\Vert T \right\Vert < \tfrac{1} {3}(4\,\cos (\pi /9) + 1)$, then either ${G_1}$ and ${G_2}$ are isomorphic, or they both have subgroups of order $2$ with isomorphic quotients.
On the $q$-analogues of some transformations of nearly-poised hypergeometric series
B.
Nassrallah;
Mizan
Rahman
211-229
Abstract: A number of transformation formulas for very well-poised basic hypergeometric series have been obtained which, in the limit $q \to 1 -$, approach the known transformation formulas for nearly-poised ordinary hypergeometric series.
Codazzi tensors and reducible submanifolds
Irl
Bivens
231-246
Abstract: An integral formula is derived for Codazzi tensors of type $ (k,\,k)$. Many of the classical Minkowski type integral formulas then become special cases of this one. If $M$ is a submanifold of Euclidean space and $ \pi$ is a parallel distribution on $M$ then each leaf of $\pi$ is a submanifold of Euclidean space with mean curvature normal vector field $\eta$. Using the above integral formula we show that the integral of ${\left\vert \eta \right\vert^2}$ over $ M$ is bounded below by an intrinsic constant and we give necessary and sufficient conditions for equality to hold. The reducible surfaces for which equality holds are characterized and related results concerned with Riemannian product manifolds are proved. Parallel tensors of type $ (1,\,1)$ are characterized in terms of the de Rham decomposition. It is shown that if $M$ is irreducible and $A$ is a parallel tensor of type $(1,\,1)$ on $M$ which is not multiplication by a constant then $M$ is a Kaehler manifold. Some further results are derived for manifolds whose simply connected cover is Kaehler.
Generic cohomology for twisted groups
George S.
Avrunin
247-253
Abstract: Let $G$ be a simple algebraic group defined and split over $ {k_0} = {{\mathbf{F}}_p}$, and let $\sigma$ be a surjective endomorphism of $ G$ with finite fixed-point set ${G_\sigma }$. We give conditions under which cohomology groups of $G$ are isomorphic to cohomology groups of ${G_\sigma }$.