Equivariant vector fields on spheres
Unni
Namboodiri
431-460
Abstract: We address the following question: If $G$ is a compact Lie group and $S(M)$ is the unit sphere of an $ R[G]$-module $ M$, then how many orthonormal $G$-invariant vector fields can be found on $ S(M)$? We call this number the $G$-field number of $M$. Under reasonable hypotheses on $M$, we reduce this question to determining when the difference of two $G$-vector bundles vanishes in a certain subquotient of the $K{O_G}$-theory of a real projective space. In this general setting, we solve the problem for $2$-groups, for odd-order groups, and for abelian groups. If $M$ also has "enough" orbit types (for example, all of them), then we solve the problem for arbitrary finite groups. We also show that under mild hypotheses on $M$, the $G$-field number depends only on the dimensions of the fixed point sets of $M$.
$F$-purity and rational singularity
Richard
Fedder
461-480
Abstract: We investigate singularities which are $F$-pure (respectively $F$-pure type). A ring $R$ of characteristic $p$ is $F$-pure if for every $R$-module $M,0 \to M \otimes R \to M \otimes\, ^1R$ is exact where $^1R$ denotes the $R$-algebra structure induced on $R$ via the Frobenius map (if $ r \in R$ and $s \in \, ^{1}R$, then $r \cdot s = {r^p}s$ in $^1R$). $F$-pure type is defined in characteristic 0 by reducing to characteristic $p$. It is proven that when $R = S/I$ is the quotient of a regular local ring $S$, $R$ is $F$-pure at the prime ideal $Q$ if and only if $({I^{[p]}}:I) \not\subset {Q^{[p]}}$. Here, ${J^{[p]}}$ denotes the ideal $\{ {a^p}\vert a \in J\} $. Several theorems result from this criterion. If $f$ is a quasihomogeneous hypersurface having weights $ ({r_1},\ldots,{r_n})$ and an isolated singularity at the origin: (1) $ \sum\nolimits_{i = 1}^n {{r_i} > 1}$ implies $ K[{X_1},\ldots,{X_n}]/(f)$ has $F$-pure type at $m = ({X_1},\ldots,{X_n})$. (2) $\sum\nolimits_{i = 1}^n {{r_i} < 1}$ implies $ K[{X_1},\ldots,{X_n}]/(f)$ does not have $F$-pure type at $m$. (3) $\sum\nolimits_{i = 1}^n {{r_i} = 1}$ remains unsolved, but does connect with a problem that number theorists have studied for many years. This theorem parallels known results about rational singularities. It is also proven that classifying $F$-pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the $ F$-pure locus in the maximal spectrum of $ K[{X_1},\ldots,{X_n}]/I$, where $K$ is a perfect field of characteristic $ P$, is Zariski open. An important conjecture is that $R/fR$ is $F$-pure (type) should imply $R$ is $F$-pure (type) whenever $R$ is a Cohen-Macauley, normal local ring. It is proven that $ \operatorname{Ext}^1{(^1}R,R) = 0$ is a sufficient, though not necessary, condition. A local ring $(R,m)$ of characteristic $p$ is $F$-injective if the Frobenius map induces an injection on the local cohomology modules $H_m^i(R) \to H_m^i{(^1}R)$. An example is constructed which is $F$-injective but not $F$-pure. From this a counterexample to the conjecture that $R/fR$ is $F$-pure implies $R$ is $F$-pure is constructed. However, it is not a domain, much less normal. Moreover, it does not lead to a counterexample to the characteristic 0 version of the conjecture.
Poles of a two-variable $P$-adic complex power
Leon
Strauss
481-493
Abstract: For almost all $ P$-adic completions of an algebraic number field, if $s \in {\mathbf{C}}$ is a pole of ${f^s} = \int_{}^{} {\int_{}^{} {\vert f(x,y){\vert^s}\vert dx{\vert _{{K_p}}}\vert dy{\vert _{{K_p}}}} }$ , where $f$ is a polynomial whose only singular point is the origin, $f(0,0) = 0$, and $f$ is irreducible in $\overline K [[x,y]]$, then $\operatorname{Re} (s)$ is $- 1$ or one of an explicitly given set of rational numbers, whose cardinality is the number of characteristic exponents of $f = 0$.
Measure and category approximations for $C$-sets
V. V.
Srivatsa
495-505
Abstract: The class of $ C$-sets in a Polish space is the smallest $\sigma$-field containing the Borel sets and closed under operation $ (\mathcal{A})$. In this article we show that any $C$-set in the product of two Polish spaces can be approximated (in measure and category), uniformly over all sections, by sets generated by rectangles with one side a $C$-set and the other a Borel set. Such a formulation unifies many results in the literature. In particular, our methods yield a simpler proof of a selection theorem for $C$-sets with $ {G_\delta }$-sections due to Burgess [4].
The coordinatization of Arguesian lattices
Alan
Day;
Douglas
Pickering
507-522
Abstract: We show that the auxiliary planar ternary ring of an $n$-frame in an Arguesian lattice, $n \geqslant 3$, is indeed an associative ring with unit. The addition of two weak necessary conditions allows us to coordinatize a hyperplane of this $ n$-frame. This generalizes the classical work of von Neumann, Baer-Inaba, Jónsson and Jónsson-Monk.
Support points of families of analytic functions described by subordination
D. J.
Hallenbeck;
T. H.
MacGregor
523-546
Abstract: We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the typically-real functions, and the functions which are subordinate to a given majorant. If the majorant $F$ is univalent then each support point has the form $F \circ \;\phi $, where $\phi$ is a finite Blaschke product and $\phi (0) = 0$. This completely characterizes the set of support points when $F$ is convex. The set of support points is found for some specific majorants, including $F(z) = {((1 + z)/(1 - z))^p}$ where $p > 1$. Let $ K$ and ${\text{St}}$ denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families $ {K^{\ast} }$ and ${\text{St}}^{\ast}$ defined by the property of being subordinate to some member of $K$ or $ {\text{St}}$, respectively.
$I\sp{X}$, the hyperspace of fuzzy sets, a natural nontopological fuzzy topological space
R.
Lowen
547-564
Abstract: Let $X$ be a uniform topological space, then on the family ${I^X}$ (resp. $\Phi (X)$) of all nonzero functions (resp. nonzero uppersemicontinuous functions) from $X$ to the unit interval $I$, a fuzzy uniform topology is constructed such that ${2^X}$ (resp. $ \mathcal{F}(X)$), the family of all nonvoid (resp. nonvoid closed) subsets of $ X$ equipped with the Hausdorff-Bourbaki structure is isomorphically injected in $ {I^X}$ (resp. $ \Phi (X)$). The main result of this paper is a complete description of convergence in ${I^X}$, by means of a notion of degree of incidence of members of ${I^X}$. Immediate consequences are that first it can be shown that this notion of convergence refines some particular useful notions of convergence of fuzzy sets used in applications, and that second it follows from its construction and properties that for each ordinary uniform topological space $ X$ there exists a natural nontopological fuzzy uniform topology on ${I^X}$.
Cocycles and local product decomposition
Jun-ichi
Tanaka
565-572
Abstract: As an application of cocycles, we establish a relation between the classical Hardy spaces on the real line $R$ and simply invariant subspaces on a quotient of the Bohr group. When this result is specialized suitably, it yields the well-known results concerning the elements of invariant subspaces. We also study, by using Gamelin's representation theorem, unitary functions which are the values of cocycles.
Conformally flat manifolds with nilpotent holonomy and the uniformization problem for $3$-manifolds
William M.
Goldman
573-583
Abstract: A conformally flat manifold is a manifold with a conformal class of Riemannian metrics containing, for each point $x$, a metric which is flat in a neighborhood of $x$. In this paper we classify closed conformally flat manifolds whose fundamental group (more generally, holonomy group) is nilpotent or polycyclic of rank $3$. Specifically, we show that such conformally flat manifolds are covered by either the sphere, a flat torus, or a Hopf manifold--in particular, their fundamental groups contain abelian subgroups of finite index. These results are applied to show that certain $ {T^2}$-bundles over $ {S^1}$ (namely, those whose attaching map has infinite order) do not have conformally flat structures. Apparently these are the first examples of $3$-manifolds known not to admit conformally flat structures.
Separatrix and limit cycles of quadratic systems and Dulac's theorem
Carmen
Chicone;
Douglas S.
Shafer
585-612
Abstract: Separatrix cycles for a planar quadratic vector field are studied. The results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles.
The divisor classes of the hypersurface $z\sp{p\sp{m}}=G(x\sb{1},\cdots ,x\sb{n})$ in characteristic $p>0$
Jeffrey
Lang
613-634
Abstract: In this article we use P. Samuel's purely inseparable descent techniques to study the divisor class groups of normal affine hypersurfaces of the form ${z^p} = G({x_1},\ldots,{x_n})$ and develop an inductive procedure for studying those of the form $ {z^{p^m}} = G$. We obtain results concerning the order and type of these groups and apply this theory to some specific examples.
Diophantine approximation properties of certain infinite sets
Wolfgang M.
Schmidt
635-645
Abstract: We exhibit various infinite sets of reals whose finite subsets do not have good simultaneous rational approximations. In particular there is an infinite set such that each finite subset is "badly approximable" in the sense that Dirichlet's theorem is best possible up to a multiplicative constant.
The sufficiency of the Matkowsky condition in the problem of resonance
Ching Her
Lin
647-670
Abstract: We consider the sufficiency of the Matkowsky condition concerning the differential equation $f(0,\varepsilon ) = 0$ identically in $\varepsilon ,{f_x}(0,\varepsilon ) \ne 0$ with $f > 0$ for $x < 0$ and $f < 0$ for $x > 0$. Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if $ f$ and $g$ are convergent power series for $\vert\varepsilon \vert < \rho \;(\rho > 0),f(x,0)=-2x$ and the interval $[ - a,b]$ is contained in a disc $D$ with center at 0. The main problem in this work is to remove from Sibuya's result the assumption that $D$ is a disc.
Stable complete constant mean curvature surfaces in ${\bf R}\sp{3}$ and $H\sp{3}$
Hiroshi
Mori
671-687
Abstract: We construct some $ 1$-parameter families of complete rotation surfaces with constant mean curvature in the hyperbolic $3$-space ${H^3}$ of constant sectional curvature $ -1$, and show that some of them are stable for the variational problem of area together with oriented volume, and that a complete connected, oriented surface with constant mean curvature in the Euclidean $3$-space ${R^3}$ which is stable for the variational problem is a plane.
Embedding $L\sp{1}$ in $L\sp{1}/H\sp{1}$
J.
Bourgain
689-702
Abstract: It is proved that $ {L^1}$ is isomorphic to a subspace of $ {L^1}/{H^1}$. More precisely, there exists a diffuse $\sigma$-algebra $ \mathfrak{S}$ on the circle such that the corresponding expectation $ {\mathbf{E}}:{H^\infty } \to {L^\infty }({\mathbf{C}})$ is onto. The method consists in studying certain martingales on the product $ {\prod ^{\mathbf{N}}}$.
Almost everywhere summability on nilmanifolds
Andrzej
Hulanicki;
Joe W.
Jenkins
703-715
Abstract: Let $G$ be a stratified, nilpotent Lie group and let $L$ be a homogeneous sublaplacian on $ G$. Let $E(\lambda )$ denote the spectral resolution of $ L$ on ${L^2}(G)$. Given a function $ K$ on $\mathbf{R}^+$, define the operator ${T_K}$ on ${L^2}(G)$ by ${T_k}f = \int_0^\infty \, {K(\lambda )\;dE(\lambda )\,f}$. Sufficient conditions on $ K$ to imply that $ {T_K}$ is bounded on $ {L^1}(G)$ and the maximal operator $K^{\ast} \varphi (x) = \sup_{t > 0}\vert{T_{K_t}}\varphi (x)\vert$ (where ${K_t}(\lambda ) = K(t\lambda )$) is of weak type $(1,1)$ are given. Picking a basis ${e_0},{e_1},\ldots$ of ${L^2}(G/\Gamma )$ ($\Gamma$ being a discrete cocompact subgroup of $G$) consisting of eigenfunctions of $ L$, we obtain almost everywhere and norm convergence of various summability methods of $ \Sigma (\varphi ,{e_j}){e_j},\varphi \in {L^p}(G/\Gamma ), 1 \leqslant p < \infty$.
Subcontinua with degenerate tranches in hereditarily decomposable continua
Lex G.
Oversteegen;
E. D.
Tymchatyn
717-724
Abstract: A hereditarily decomposable, irreducible, metric continuum $ M$ admits a mapping $ f$ onto $[0,1]$ such that each ${f^{ - 1}}(t)$ is a nowhere dense subcontinuum. The sets ${f^{ - 1}}(t)$ are the tranches of $ M$ and ${f^{ - 1}}(t)$ is a tranche of cohesion if $t \in \{ 0,1\}$ or ${f^{ - 1}}(t) = {\text{C1}}({f^{ - 1}}([0,t))) \cap {\text{C1}}\,({f^{ - 1}}((t,1]))$. The following answer a question of Mahavier and of E. S. Thomas, Jr. Theorem. Every hereditarily decomposable continuum contains a subcontinuum with a degenerate tranche. Corollary. If in an irreducible hereditarily decomposable continuum each tranche is nondegenerate then some tranche is not a tranche of cohesion. The theorem answers a question of Nadler concerning arcwise accessibility in hyperspaces.
Hereditarily additive families in descriptive set theory and Borel measurable multimaps
Roger W.
Hansell
725-749
Abstract: A family $\mathcal{B}$ of Borel subsets of a space $ X$ is (boundedly) Borel additive if, for some countable ordinal $ \alpha$, the union of every subfamily of $ \mathcal{B}$ is a Borel set of class $\alpha$ in $X$. A problem which arises frequently in nonseparable descriptive set theory is to find conditions under which this property is "hereditary" in the sense that any selection of a Borel subset from each member of $\mathcal{B}$ (of uniform bounded class) will again be a Borel additive family. Similar problems arise for other classes of projective sets; in particular, for Souslin sets and their complements. Positive solutions to the problem have previously been obtained by the author and others when $X$ is a complete metric space or under additional set-theoretic axioms. We give here a fairly general solution to the problem, without any additional axioms or completeness assumptions, for an abstract "descriptive class" in the setting of generalized metric spaces (e.g., spaces with a $\sigma$-point-finite open base). A typical corollary states that any point-finite (co-) Souslin additive family in (say) a metrizable space is hereditarily (co-) Souslin additive. (There exists a point-countable ${F_\sigma }$ additive family of subsets of the real line which has a point selection which is not even Souslin additive.) Two structure theorems for "hereditarily additive" families are proven, and these are used to obtain a nonseparable extension of the fundamental measurable selection theorem of Kuratowski and Ryll-Nardzewski, and a complete solution to the problem of Kuratowski on the Borel measurability of complex and product mappings for nonseparable metric spaces.
Fully nonlinear, uniformly elliptic equations under natural structure conditions
Neil S.
Trudinger
751-769
Abstract: We derive first and second derivative estimates for classical solutions of fully nonlinear, uniformly elliptic equations which are subject to natural structure conditions analogous to those proposed and treated by Ladyzhenskaya and Ural'tseva for quasilinear equations. As an application we extend recent work of Evans and Lions on the Bellman equation for families of linear operators to families of quasilinear operators.
On some cheap control problems for diffusion processes
José-Luis
Menaldi;
Maurice
Robin
771-802
Abstract: We consider several cases of control problems for diffusion processes when the payoff functional does not depend explicitly on the control. We prove the continuity of the optimal cost function and give a characterization of this cost with a quasi-variational inequality interpreting the problem as limit of an impulse control problem when the cost of impulse tends to zero. Moreover, we show the existence of an optimal control for some particular situations.
Uniformly exhaustive submeasures and nearly additive set functions
N. J.
Kalton;
James W.
Roberts
803-816
Abstract: Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an $F$-space has a control measure. We also show that $ {c_0}$ (or any ${\mathcal{L}_\infty }$-space) is a $\mathcal{K}$-space, i.e. cannot be realized as the quotient of a nonlocally convex $F$-space by a one-dimensional subspace.
An application of asymptotic techniques to certain problems of spectral and scattering theory of Stark-like Hamiltonians
Matania
Ben-Artzi
817-839
Abstract: Let $ {L_0} = - \Delta + V({x_1}),L = {L_0} + {V_p}(x)$ be selfadjoint in ${L^2}({R^n})$. Here $V,{V_p}$ are real functions, $V({x_1})$ depends only on the first coordinate. Existence of the wave-operators $ {W_ \pm }\,(L,{L_0}) = s$ -${\lim_{t \to \pm \infty }}\,\exp (itL)\exp ( - it{L_0})$ is proved, using the stationary phase method. For this, an asymptotic technique is applied to the study of $ -{d^2}/d{t^2} + V(t)$ in $ {L^2}(R)$. Its absolute continuity is proved as well as a suitable eigenfunction expansion. $V$ is a "Stark-like" potential. In particular, the cases $ V({x_1}) = ( - \operatorname{sgn}{x_1})\vert{x_1}\,{\vert^\alpha },0 < \alpha \leqslant 2$, are included. ${V_p}$ may be taken as the sum of an $ {L^2}$-function and a function satisfying growth conditions in the $ + {x_1}$ direction. ${V_p}(x) = \vert x\vert^{ - 1}$ is included.
Group actions on aspherical $A\sb{k}(N)$-manifolds
Hsü Tung
Ku;
Mei Chin
Ku
841-859
Abstract: By an aspherical $ {A_k}(N)$-manifold, we mean a compact connected manifold $M$ together with a map $f$ from $M$ into an aspherical complex $N$ such that ${f^{\ast}}: H^k(N;Q)\to H^k(M;Q)$ is nontrivial. In this paper we shall show that if ${S^1}$ acts effectively and smoothly on a smooth aspherical ${A_k}(N)$-manifold, $k > 1$, $N$ a closed oriented Riemannian $k$-manifold, with strictly negative curvature, and the $K$-degree $ K(f) \ne 0$, then the fixed point set $F$ is not empty, and at least one component of $F = { \cup_{j}}{F_j}$ is an aspherical ${A_k}(N)$-manifold. Moreover, ${\operatorname{Sign}}(f) = {\Sigma_j}\,{\operatorname{Sign}}(f\vert{F_j})$. We also study the degree of symmetry and semisimple degree of symmetry of aspherical $ {A_k}(N)$-manifolds.