Normally flat deformations
Bruce
Bennett
1-57
Abstract: We study flat families $Z/T$, together with a section $\sigma :T \to Z$ such that the normal cone to the image of $\sigma$ in Z is flat over T. Such a family is called a ``normally flat deformation (along $ \sigma$)"; it corresponds intuitively to a deformation of a singularity which preserves the Hilbert-Samuel function. We construct the versal normally flat deformation of an isolated singularity (X,x) in terms of the flat strata of the relative jets of the ``usual'' versal deformation of X. We give explicit criteria, in terms of equations, for a flat family to be normally flat along a given section. These criteria are applied to demonstrate the smoothness of normally flat deformation theoryand of the canonical map from it to the cone deformation theory of the tangent cone-in the case of strict complete intersections. Finally we study the tangent space to the normally flat deformation theory, expressing it as the sum of two spaces: The first is a piece of a certain filtration of the tangent space to the usual deformation theory of X; the second is the tangent space to the special fibre of the canonical map $N \to S$, where N (resp. S) is the parameter space for the versal normally flat deformation of (X, x) (resp. for the versal deformation of X). We discuss the relation of this second space to infinitesimal properties of sections.
The classification of stunted projective spaces by stable homotopy type
S.
Feder;
S.
Gitler
59-81
Abstract: A complete classification of stable homotopy types of complex and quaternionic stunted projective spaces, denoted by ${\mathbf{C}}P_n^k$ and ${\mathbf{Q}}P_n^k$ respectively, is obtained. The necessary conditions for such equivalences are found using K-theory and various characteristic classes introduced originally by J. F. Adams. As a by-product one finds the J-orders of the Hopf bundles over $ {\mathbf{C}}{P^n}$ and $ {\mathbf{Q}}{P^n}$ respectively. The algebraic part is rather involved. Finally a homotopy theoretical argument yields the constructions of such homotopy equivalences as are allowed by the fulfillment of the necessary conditions.
On the bordism of almost free $Z\sb{2k}$ actions
R. Paul
Beem
83-105
Abstract: An ``almost free'' ${Z_{{2^k}}}$ action on a manifold is one in which only the included ${Z_2}$ may possibly fix points of the manifold. For k = 2, these are the stationary-point free actions. It is shown that almost free ${Z_{{2^k}}}$ bordism is generated by three subalgebras: the extension from ${Z_2}$ actions, a coset of ${Z_2}$ extensions being the restrictions of circle actions and a certain ideal of elements which annihilate the whole ring. The additive structure is determined. Free $ {Z_{{2^k}}}$ bordism is shown to split as an algebra. It is shown that the kernel of the extension homomorphism from $ {Z_2}$ to ${Z_{{2^k}}}$ bordism is equal to the image of the corresponding restriction homomorphism.
Free $S\sp{3}$-actions on $2$-connected nine-manifolds
Richard I.
Resch
107-112
Abstract: In this paper a classification of free ${S^3}$-actions on 2-connected 9-manifolds is obtained by examining the corresponding principal $ {S^3}$-bundles. The orbit spaces that may occur are determined and it is proved that there are exactly two homotopy classes of maps from each of these spaces into the classifying space for principal ${S^3}$-bundles. It is shown that the total spaces of the corresponding bundles are distinct, yielding the main result that for each nonnegative integer k, there exist exactly two 2-connected 9-manifolds which admit free ${S^3}$-actions and, furthermore, the actions on each of these manifolds are unique.
Strong differentiability properties of Bessel potentials
Daniel J.
Deignan;
William P.
Ziemer
113-122
Abstract: This paper is concerned with the ``strong'' ${L_p}$ differentiability properties of Bessel potentials of order $ \alpha > 0$ of $ {L_p}$ functions. Thus, for such a function f, we investigate the size (in the sense of an appropriate capacity) of the set of points x for which there is a polynomial $ {P_x}(y)$ of degree $k \leqslant \alpha$ such that $\displaystyle \mathop {\lim \sup }\limits_{{\text{diam}}(S) \to 0} \;{({\text{d... ...rt S{\vert^{ - 1}}\int {\vert f(y) - {P_x}(y){\vert^p}dy} } \right\}^{1/p}} = 0$ where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.
On bounded univalent functions whose ranges contain a fixed disk
Roger W.
Barnard
123-144
Abstract: Let $\mathcal{S}$ denote the standard normalized class of regular, univalent functions in $K = {K_1} = \{ z:\vert z\vert < 1\}$. Let $\mathcal{F}$ be a given compact subclass of $\mathcal{S}$. We consider the following two problems. Problem 1. Find $ \max \vert{a_2}\vert$ for $f \in \mathcal{F}$. Problem 2. For $\vert z\vert = r < 1$, find the $\max \;(\min )\vert f(z)\vert$ for $f \in \mathcal{F}$. In this paper we are concerned with the subclass $\mathcal{S}_d^\ast(M) = \{ f \in \mathcal{S}:{K_d} \subset f(K) \subset {K_M}\} $. Through the use of the Julia variational formula and the Loewner theory we determine the extremal functions for Problems 1 and 2 for the class $ \mathcal{S}_d^\ast(M)$, for all d, M such that $\tfrac{1}{4} \leqslant d \leqslant 1 \leqslant M \leqslant \infty$.
A singular semilinear equation in $L\sp{1}({\bf R})$
Michael G.
Crandall;
Lawrence C.
Evans
145-153
Abstract: Let $\beta$ be a positive and nondecreasing function on R. The boundary-value problem $f \in {L^1}({\mathbf{R}})$. It is shown that this problem can have a solution only if $ \beta$ is integrable near $- \infty$, and that if this is the case, then the problem has a solution exactly when $\smallint _{ - \infty }^\infty f(x)dx > 0$.
Critical groups having central monoliths of a nilpotent by abelian product variety of groups
James J.
Woeppel
155-161
Abstract: Let $\mathfrak{N}$ be a variety of groups which has nilpotency class two and finite odd exponent. Let $\mathfrak{A}$ be an abelian variety of groups with finite exponent relatively prime to the exponent of $ \mathfrak{N}$. The existence in the product variety $\mathfrak{N}\mathfrak{A}$ of nonnilpotent critical groups having central monoliths is established. The structure of these critical groups is studied. This structure is shown to depend on an invariant, k. The join-irreducible subvariety of $\mathfrak{N}\mathfrak{A}$ generated by the nonnilpotent critical groups of $\mathfrak{N}\mathfrak{A}$ having central monoliths is determined, in particular, for k odd.
Homomorphisms of integral domains of characteristic zero
E.
Fried;
J.
Sichler
163-182
Abstract: Every category of universal algebras is isomorphic to a full subcategory of the category of all integral domains of characteristic zero and all their 1-preserving homomorphisms. Consequently, every monoid is isomorphic to the monoid of all 1-preserving endomorphisms of an integral domain of characteristic zero.
Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies
George S.
Shapiro
183-198
Abstract: Let $\Lambda$ be a discrete subset of an LCA group and E a compact subset of the dual group. Balayage is said to be possible for $(\Lambda ,E)$ if the Fourier transform of each measure on G is equal on E to the Fourier transform of some measure supported by $\Lambda$. Following Beurling, we show that this condition is equivalent to the possibility of bounding certain functions with spectra in E by their bounds on $\Lambda$. We derive consequences of this equivalence, among them a necessary condition on $\Lambda$ for balayage when E is compact and open (a condition analogous to a density condition Beurling and Landau gave for balayage in Euclidean spaces). We show that if balayage is possible for $(\Lambda ,E)$ and if $\Lambda '$ is close to $\Lambda$, then balayage is possible for $E \subset R$ with the property that there are ``arbitrarily sparse'' sets $\Lambda$ with balayage possible for $(\Lambda ,E)$.
Lebesgue summability of double trigonometric series
M. J.
Kohn
199-209
Abstract: We formulate a definition of symmetric derivatives of odd order for functions of two variables. Our definition is based on expanding in a Taylor's series a weighted average of the function about circles. The definition is applied to derive results on Lebesgue summability for spherically convergent double trigonometric series.
Completely unstable flows on $2$-manifolds
Dean A.
Neumann
211-226
Abstract: Completely unstable flows on 2-manifolds are classified under both topological and ${C^r}$-equivalence $(1 \leqslant r \leqslant \infty )$, in terms of the corresponding orbit spaces.
Stability in Witt rings
Thomas C.
Craven
227-242
Abstract: An abstract Witt ring R is defined to be a certain quotient of an integral group ring for a group of exponent 2. The ring R has a unique maximal ideal M containing 2. A variety of results are obtained concerning n-stability, the condition that ${M^{n + 1}} = 2{M^n}$, especially its relationship to the ring of continuous functions from the space of minimal prime ideals of R to the integers. For finite groups, a characterization of integral group rings is obtained in terms of n-stability. For Witt rings of formally real fields, conditions equivalent to n-stability are given in terms of the real places defined on the field.
On the sequence space $l\sb{(p\sb{n})}$ and $\lambda \sb{(p\sb{n})},$ $0<p\sb{n}\leq 1$
S. A.
Schonefeld;
W. J.
Stiles
243-257
Abstract: Let $({p_n})$ and $({q_n})$ be sequences in the interval $(0,1]$, let ${l_{({p_n})}}$ be the set of all real sequences $ ({x_n})$ such that $ \sum {\vert{x_n}{\vert^{{p_n}}} < \infty }$, and let ${\lambda _{({q_n})}}$ be the set of all real sequences $({y_n})$ such that $ {\sup _\pi }\sum {\vert{y_n}{\vert^{{q_{\pi (n)}}}} < \infty }$ where the sup is taken over all permutations $\pi$ of the positive integers. The purpose of this paper is to investigate some of the properties of these spaces. Our results are primarily concerned with (1) conditions which are necessary and/or sufficient for $ {l_{({p_n})}}$ (resp., $ {\lambda _{({p_n})}}$) to equal $ {l_{({q_n})}}$ (resp., $ {\lambda _{({q_n})}}$), and (2) isomorphic and topological properties of the subspaces of these spaces. In connection with (1), we show that the following four conditions are equivalent for any sequence $({\varepsilon _n})$ which decreases to zero and has ${\varepsilon _1} < 1$. (a) There exists a number $K > 1$ such that the series $\sum {1/{K^{1/{\varepsilon _n}}}}$ converges; (b) the elements $ {\varepsilon _n}$ of the sequence satisfy the condition ${\varepsilon _n} = O(1/\ln n)$; (c) the sequence $((\ln n)((1/n)\sum\nolimits_1^n {{\varepsilon _j}} ))$ is bounded; and (d) ${l_{(1 - {\varepsilon _n})}}$ equals ${l_1}$. In connection with (2), we show that the following are true when $({p_n})$ increases to one. (a) ${\lambda _{({p_n})}}$ contains an infinite-dimensional closed subspace where the ${l_{({p_n})}}$-topology and the ${\lambda _{({p_n})}}$-topology agree; (b) $ {l_{({p_n})}}$ and ${\lambda _{({p_n})}}$ contain closed subspaces isomorphic to ${l_1}$; and (c) ${\lambda _{({p_n})}}$ contains no infinite-dimensional subspace where the ${\lambda _{({p_n})}}$-topology agrees with the ${l_1}$-topology if and only if $\displaystyle \lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .$
Weierstrass normal forms and invariants of elliptic surfaces
Arnold
Kas
259-266
Abstract: Let $\pi :S \to B$ be an elliptic surface with a section $\sigma :B \to S$. Let ${L^{ - 1}} \to B$ be the normal bundle of $\sigma (B)$ in S, and let $W = P({L^{ \otimes 2}} \oplus {L^{ \otimes 3}} \oplus 1)$ be a ${{\mathbf{P}}^2}$-bundle over B. Let $ {S^\ast}$ be the surface obtained from S by contracting those components of fibres of S which do not intersect $\sigma (B)$. Then ${S^\ast}$ may be imbedded in W and defined by a ``Weierstrass equation": $\displaystyle {y^2}z = {x^3} - {g_2}x{z^2} - {g_3}{z^3}$ where ${g_2} \in {H^0}(B,\mathcal{O}({L^{ \otimes 4}}))$ and ${g_3} \in {H^0}(B,\mathcal{O}({L^{ \otimes 6}}))$. The only singularities (if any) of $ {S^\ast}$ are rational double points. The triples $ (L,{g_2},{g_3})$ form a set of invariants for elliptic surfaces with sections, and a complete set of invariants is given by $\{ (L,{g_2},{g_3})\} /G$ where $G \cong {{\mathbf{C}}^\ast} \times {\operatorname{Aut}}\;(B)$.
On subcategories of TOP
S. P.
Franklin;
D. J.
Lutzer;
B. V. S.
Thomas
267-278
Abstract: A categorical characterization of a subcategory S of TOP (or $ {T_2}$) is one which enables the identification of S in TOP (or $ {T_2}$) without requiring the reconstruction of the topological structure of its objects. In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory.
Sums of solid $n$-spheres
Lois M.
Broussard
279-294
Abstract: We prove that the sum of two solid Antoine n-spheres $(n \geqslant 3)$ by the identity on the boundary is homeomorphic to the n-sphere $ {S^n}$.
Cluster values of bounded analytic functions
T. W.
Gamelin
295-306
Abstract: Let D be a bounded domain in the complex plane, and let $ \zeta$ belong to the topological boundary $ \partial D$ of D. We prove two theorems concerning the cluster set $ {\text{Cl}}(f,\zeta )$ of a bounded analytic function f on D. The first theorem asserts that values in ${\text{Cl}}(f,\zeta )\backslash f({\mathcyr{SH}_\zeta })$ are assumed infinitely often in every neighborhood of $\zeta$, with the exception of those lying in a set of zero analytic capacity. The second asserts that all values in $ {\text{Cl}}(f,\zeta )\backslash f({\mathfrak{M}_\zeta } \cap {\text{supp}}\;\lambda )$ are assumed infinitely often in every neighborhood of $\zeta$, with the exception of those lying in a set of zero logarithmic capacity. Here ${\mathfrak{M}_\zeta }$ is the fiber of the maximal ideal space $ \mathfrak{M}(D)$ of ${H^\infty }(D)$ lying over $\zeta$, ${\mathcyr{SH}_\zeta }$ is the Shilov boundary of the fiber algebra, and $\lambda$ is the harmonic measure on $\mathfrak{M}(D)$.
A general extremal problem for the class of close-to-convex functions
John G.
Milcetich
307-323
Abstract: For $\beta \geqslant 0,{K_\beta }$ denotes the set of functions $f(z) = z + {a_2}{z^2} + \cdots$ defined on the unit disc U with the representation $0 \leqslant \beta \leqslant 1$, and $\zeta \in U$, let $F(u,v)$ be analytic in a neighborhood of $ \{ (f(\zeta ),\zeta ):f \in {K_\beta }\}$. Then $ \max \{ \operatorname{Re} F(f(\zeta ),\zeta ):f \in {K_\beta }\}$ occurs for a function of the form $\displaystyle f(z) = {(\beta + 1)^{ - 1}}{(x - y)^{ - 1}}[{(1 + xz)^{\beta + 1}}{(1 + yz)^{ - \beta - 1}} - 1],$ where $\vert x\vert = \vert y\vert = 1$ and $ x \ne y$. If $0 < \beta < 1$ these are the only extremal functions. A consequence of this result is the determination of the value region $\{ f(\zeta )/\zeta :f \in {K_\beta }\}$ as $\{ {(\beta + 1)^{ - 1}}{(s - t)^{ - 1}}[{(1 + s)^{\beta + 1}}{(1 + t)^{ - \beta - 1}} - 1]:\vert s\vert,\vert t\vert \leqslant \vert\zeta \vert\}$.
End extensions, conservative extensions, and the Rudin-Frol\'\i k ordering
Andreas
Blass
325-340
Abstract: The ordering of ultrafilters on the natural numbers defined by ``E-prod N is an end extension of D-prod N,'' the ordering defined by ``E-prod N is a conservative extension of D-prod N,'' and the Rudin-Frolik ordering are proved to be distinct if the continuum hypothesis holds. These three orderings are also characterized in terms of (not necessarily internal) ultrafilters in the Boolean algebra of internal sets of natural numbers in a nonstandard universe.
Spectral geometry of symmetric spaces
Peter B.
Gilkey
341-353
Abstract: Let M be a compact Riemannian manifold without boundary. Let D be a differential operator on M. Let spec (D, M) denote the eigenvalues of D repeated according to multiplicity. Several authors have studied the extent to which the geometry of M is reflected by spec (D, M) for certain natural operators D. We consider operators D which are convex combinations of the ordinary Laplacian and the Bochner or reduced Laplacian acting on the space of smooth functions and the space of smooth one forms. We prove that is is possible to determine if M is a local symmetric space from its spectrum. If the Ricci tensor is parallel transported, the eigenvalues of the Ricci tensor are spectral invariants of M.
Limit theorems for convolution iterates of a probability measure on completely simple or compact semigroups
A.
Mukherjea
355-370
Abstract: This paper extends the study (initiated by M. Rosenblatt) of the asymptotic behavior of the convolution sequence of a probability measure on compact or completely simple semigroups. Let S be a locally compact second countable Hausdorff topological semigroup. Let $ \mu$ be a regular probability measure on the Borel subsets of S such that S does not have a proper closed subsemigroup containing the support F of $\mu$. It is shown in this paper that when S is completely simple with its usual product representation $X \times G \times Y$, then the convolution sequence ${\mu ^n}$ converges to zero vaguely if and only if the group factor G is noncompact. When the group factor G is compact, ${\mu ^n}$ converges weakly if and only if $ {\underline {\lim } _{n \to \infty }}{F^n}$ is nonempty. This last result remains true for an arbitrary compact semigroup S generated by F. Furthermore, we show that in this case there exist elements ${a_n} \in S$ such that ${\mu ^n} \ast {\delta _{{a_n}}}$ converges weakly, where $ {\delta _{{a_n}}}$ is the point mass at ${a_n}$. This result cannot be extended to the locally compact case, even when S is a group.