Hypoellipticity on the Heisenberg group-representation-theoretic criteria
Charles
Rockland
1-52
Abstract: A representation-theoretic characterization is given for hypoellipticity of homogeneous (with respect to dilations) left-invariant differential operators P on the Heisenberg group ${H_n}$; it is the precise analogue for ${H_n}$ of the statement for ${{\mathbf{R}}^n}$ that a homogeneous constant-coefficient differential operator is hypoelliptic if and only if it is elliptic. Under these representation-theoretic conditions a parametrix is constructed for P by means of the Plancherel formula. However, these conditions involve all the irreducible representations of ${H_n}$, whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely $P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m}$, where ${X_i},{Y_i},i = 1, \ldots ,n$, and Z generate the Lie algebra of ${H_n}$ via the commutation relations $[{X_i},{Y_j}] = {\delta _{ij}}Z$, and where m is a positive integer. In the course of the proof a connection is made between homogeneous left-invariant operators on ${H_n}$ and a class of degenerate-elliptic operators on $ {{\mathbf{R}}^{n + 1}}$ studied by Grušin. This connection is examined in the context of localization in enveloping algebras.
The conjugacy problem for boundary loops in $3$-manifolds
Benny D.
Evans
53-64
Abstract: A geometric solution of the word problem for fundamental groups of compact, orientable, irreducible, sufficiently large 3-manifolds has been given by F. Waldhausen. We present here a solution of a restricted version of the conjugacy problem for this same class of 3-manifolds; however, the conjugacy problem for 3-manifolds remains in general unsolved. The main results is that there is an algorithm that will determine for any two loops ${L_1},{L_2}$ in the boundary of a compact, orientable, irreducible sufficiently large 3-manifold M if ${L_1}$, is freely homotopic in M to $ {L_2}$.
Erickson's conjecture on the rate of escape of $d$-dimensional random walk
Harry
Kesten
65-113
Abstract: We prove a strengthened form of a conjecture of Erickson to the effect that any genuinely d-dimensional random walk $ {S_n},d \geqslant 3$, goes to infinity at least as fast as a simple random walk or Brownian motion in dimension d. More precisely, if $S_n^\ast$ is a simple random walk and $ {B_t}$, a Brownian motion in dimension d, and $\psi :[1,\infty ) \to (0,\infty )$ a function for which ${t^{ - 1/2}}\psi (t) \downarrow 0$, then $\psi {(n)^{ - 1}}\vert S_n^\ast\vert \to \infty$ w.p.l, or equivalently, $\psi {(t)^{ - 1}}\vert{B_t}\vert \to \infty$ w.p.l, iff $\smallint _1^\infty \psi {(t)^{d - 2}}{t^{ - d/2}} < \infty $; if this is the case, then also $\psi {(n)^{ - 1}}\vert{S_n}\vert \to \infty$ w.p.l for any random walk Sn of dimension d.
The cohomology of semisimple Lie algebras with coefficients in a Verma module
Floyd L.
Williams
115-127
Abstract: The structure of the cohomology of a complex semisimple Lie algebra with coefficients in an arbitrary Verma module is completely determined. Because the Verma modules are infinite-dimensional, the cohomology need not vanish (as it does for nontrivial finite-dimensional modules). The methods presented exploit the homological machinery of Cartan-Eilenberg [3]. The results of [3], when applied to the universal enveloping algebra of a semisimple Lie algebra and when coupled with key results of Kostant [12], Hochschild-Serre [9], yield the basic structure theorem-Theorem 4.19. Our results show, incidently, that an assertion of H. Kimura, Theorem 2 of [13] is false. A counterexample is presented in §6.
Existence, stability, and compactness in the $\alpha $-norm for partial functional differential equations
C. C.
Travis;
G. F.
Webb
129-143
Abstract: The abstract ordinary functional differential equation $ (a/dt)u(t) = - Au(t) + F({u_t}),{u_0} = \phi$, is studied, where $- A$ is the infinitesimal generator of an analytic semigroup of linear operators and F is continuous with respect to a fractional power of A.
Some exponential moments of sums of independent random variables
J.
Kuelbs
145-162
Abstract: If $\{ {X_n}\}$ is a sequence of vector valued random variables, $ \{ {a_n}\}$ a sequence of positive constants, and $M = {\sup _{n \geqslant 1}}\left\Vert {({X_1} + \cdots + {X_n})/{a_n}} \right\Vert$, we examine when $E(\Phi (M)) < \infty$ under various conditions on $ \Phi ,\{ {X_n}\}$, and $\{ {a_n}\}$. These integrability results easily apply to empirical distribution functions.
Variations, characteristic classes, and the obstruction to mapping smooth to continuous cohomology
Mark A.
Mostow
163-182
Abstract: In a recent paper, the author gave an example of a singular foliation on $ {{\mathbf{R}}^2}$ for which it is impossible to map the de Rham cohomology ${T_{{\text{DR}}}}$ to the continuous singular cohomology $ {T_{\text{c}}}$ (in the sense of Bott and Haefliger's continuous cohomology of spaces with two topologies) compatibly with evaluation of cohomology classes on homology classes. In this paper the obstruction to mapping ${T_{{\text{DR}}}}$ to ${T_{\text{c}}}$ is pinpointed by defining a whole family of cohomology theories ${T_{k,m,n}}$, based on cochains which vary in a $ {C^k}$ manner, which mediate between the two. It is shown that the obstruction vanishes on nonsingularly foliated manifolds. The cohomology theories are extended to Haefliger's classifying space $ (B{\Gamma _q} \to B{J_q})$, with its germ and jet topologies, by using a notion of differentiable space similar to those of J. W. Smith and K. T. Chen. The author proposes that certain of the ${T_{kmn}}$ be used instead of ${T_{\text{c}}}$ to study Bott and Haefliger's conjecture that the continuous cohomology of $(B{\Gamma _q} \to B{J_q})$ equals the relative Gel'fand-Fuks cohomology $ {H^\ast}({\mathfrak{a}_q},{O_q})$. It is shown that ${T_{kmn}}(B{\Gamma _q} \to B{J_q})$ may contain new characteristic classes for foliations which vary only in a ${C^k}$ manner when a foliation is varied smoothly.
Operators with small self-commutators
J. W.
Del Valle
183-194
Abstract: Let A be a bounded operator on a Hilbert space H. The self-commutator of A, denoted [A], is ${A^\ast}A - A{A^\ast}$. An operator is of commutator rank n if the rank of [A] is n. In this paper operators of commutator rank one are studied. Two particular subclasses are investigated in detail. First, completely nonnormal operators of commutator rank one for which ${A^\ast}A$ and $A{A^\ast}$ commute are completely characterized. They are shown to be special types of simple weighted shifts. Next, operators of commutator rank one for which $ \{ {A^n}e\} _{n = 0}^\infty$ is an orthogonal sequence (where e is a generator of the range of [A]) are characterized as a type of weighted operator shift.
On the growth of solutions of algebraic differential equations
Steven B.
Bank
195-212
Abstract: In this paper we determine estimates for the growth of both real-valued and complex-valued solutions of algebraic differential equations on an interval $({x_0}, + \infty )$. One of the main results of the paper (Theorem 3) confirms E. Borel's conjecture on the growth of real-valued solutions for a broad class of solutions of second-order algebraic differential equations. The conjecture had previously been shown to be false for third-order equations.
Classifying open principal fibrations
David A.
Edwards;
Harold M.
Hastings
213-220
Abstract: Let G be a compact metric group. We shall construct classifying spaces for open principal G-fibrations over compact metric spaces.
Continuous maps of the interval with finite nonwandering set
Louis
Block
221-230
Abstract: Let f be a continuous map of a closed interval into itself, and let $\Omega (f)$ denote the nonwandering set of f. It is shown that if $ \Omega (f)$ is finite, then $\Omega (f)$ is the set of periodic points of f. Also, an example is given of a continuous map g, of a compact, connected, metrizable, one-dimensional space, for which $ \Omega (g)$ consists of exactly two points, one of which is not periodic.
Ultrapowers and local properties of Banach spaces
Jacques
Stern
231-252
Abstract: The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the ${\mathcal{L}_p}$-spaces). Many examples are given; also, as an application, it is proved that any ${\mathcal{L}_p}$-space, $1 < p < \infty$, has an ultrapower which is isomorphic to an ${L_p}$-space.
The signature of symplectic manifolds
Leslie P.
Jones
253-262
Abstract: The motivation for this work was a calculation by Oshanin of the image of the signature homomorphism from the special unitary cobordism ring into the integers. Here we compute this image for symplectic cobordism. This is accomplished by proving two divisibility theorems and then giving examples to show the theorems are the best possible.
Conditionally compact semitopological one-parameter inverse semigroups of partial isometries
M. O.
Bertman
263-275
Abstract: The algebraic structure of one-parameter inverse semigroups has been completely described. Furthermore, if B is the bicyclic semigroup and if B is contained in any semitopological semigroup, the relative topology on B is discrete. We show that if F is an inverse semigroup generated by an element and its inverse, and F is contained in a compact semitopological semigroup, then the relative topology is discrete; in fact, if F is any one-parameter inverse semigroup contained in a compact semitopological semigroup, then the multiplication on F is jointly continuous if and only if the inversion is continuous on F, and we describe $\bar F$ in that case. We also show that if $\{ {J_t}\}$ is a one-parameter semigroup of bounded linear operators on a (separable) Hilbert space, then $\{ {J_t}\} \cup \{ J_t^\ast\}$ generates a one-parameter inverse semigroup T with $J_t^{ - 1} = J_t^\ast$ if and only if $\{ {J_t}\}$ is a one-parameter semigroup of partial isometries, and we describe the weak operator closure of T in that case.
$(E\sp{3}/X)\times E\sp{1}\approx E\sp{4}$ ($X$, a cell-like set): an alternative proof
J. W.
Cannon
277-285
Abstract: The author gives an alternative proof that a cell-like closed-0dimensional decomposition of ${E^3}$ is an ${E^4}$ factor. The argument is essentially 2-dimensional. The 3- and 4-dimensional topology employed is truly minimal.
On the Seifert manifold of a $2$-knot
M. A.
Gutierrez
287-294
Abstract: From geometric facts about embeddings $ {S^2} \to {S^4}$ we study the relationship between the smallest number of normal generators (weight) of a group and its preabelian presentations.
Strong differentiability of Lipschitz functions
C. J.
Neugebauer
295-306
Abstract: Let F be a differentiation basis in ${R^n}$, i.e., a family of measurable sets S contracting to 0 such that ${\left\Vert {{M_F}f} \right\Vert _p} \leqslant {A_p}{\left\Vert f \right\Vert _p}$, where ${M_F}$ is the Hardy-Littlewood maximal operator. For $f \in \Lambda _\alpha ^{pq}$, we let ${E_F}(f)$ be the complement of the Lebesgue set of f relative to F, and we show that $ {E_F}$ has $L_\alpha ^{pq}$-capacity 0, where $L_\alpha ^{pq}$ is a capacity associated with $\Lambda _\alpha ^{pq}$ in much the same way as the Bessel capacity ${B_{\alpha p}}$ is associated with $L_\alpha ^p$.
Paracompactness of box products of compact spaces
Kenneth
Kunen
307-316
Abstract: We consider box products of countably many compact Hausdorff spaces. Under the continuum hypothesis, the product is paracompact iff its Lindelöf degree is no more than the continuum; in particular, the product is paracompact if each space has weight continuum or less, or if each space is dispersed. Some partial results are proved under Martin's axiom.
Projective modules over subrings of $k[X, Y]$
David F.
Anderson
317-328
Abstract: In this paper we study projective modules over subrings of $k[X,Y]$. Conditions are given for projective modules to decompose into free $\oplus$ rank 1 modules. Our main result is that if k is an algebraically closed field and A a subring of $ B = k[X,Y]$ with $A \subset B$ integral and ${\text{sing}}(A)$ finite, then all f.g. projective A-modules have the form free $\oplus$ rank 1. We also give several examples of subrings of $k[X,Y]$ which have indecomposable projective modules of rank 2.
Projective varieties of low codimension in characteristic $p>0$
Robert
Speiser
329-343
Abstract: Let X be an s-dimensional closed Cohen-Macaulay subvariety of projective n-space, over an algebraically closed field of characteristic $p > 0$. Assume $s \geqslant \tfrac{1}{2}(n + 1)$. Then (1) every stratified vector bundle on X is trivial; (2) X is simply connected. Assertion (1) generalizes Gieseker's result for projective space, while (2) is a strengthened analogue of results of Barth and Ogus in characteristic zero.
Reductions of residuals are finite
R.
Hindley
345-361
Abstract: An important theorem of the $\lambda \beta K$-calculus which has not been fully appreciated up to now is D. E. Schroer's finiteness theorem (1963), which states that all reductions of residuals are finite. The present paper gives a new proof of this theorem and extends it from $\lambda \beta$-reduction to $\lambda \beta \eta $-reduction and reductions with certain extra operators added, for example the pairing, iteration and recursion operators. Combinatory weak reduction, with or without extra operators, is also included.
Infinitesimal calculus on locally convex spaces. I. Fundamentals
K. D.
Stroyan
363-383
Abstract: Differential calculus on nonnormed locally convex spaces suffers from technical difficulties (and the subsequent plethora of different definitions) partly because the families of multilinear maps over the spaces do not inherit a suitable topology. In this note we give the elementary ingredients of a strong differentiation based on Abraham Robinson's theory of infinitesimals. Though nontopologizable, our theory is a natural generalization of standard infinitesimal calculus (finite dimensional or Banach space), see Robinson [1966], Keisler [1976], or Stroyan and Luxemburg [1976]. It is simpler than many recent developments, e.g., Yamamuro [1974] and Keller [1974]. The technical improvement of our approach should lead to advances in a variety of subjects.
Sets of divergence on the group $2\sp{\omega }$
David C.
Harris;
William R.
Wade
385-392
Abstract: We show that there exist uncountable sets of divergence for $C({2^\omega })$. We also show that a necessary and sufficient condition that a set E be a set of divergence for ${L^p}({2^\omega }),1 < p < \infty$, is that E be of Haar measure zero.
Erratum: ``A partial surface variation for extremal schlicht functions'' (Trans. Amer. Math. Soc. {\bf 234} (1977), no. 1, 119--138)
T. L.
McCoy
393