The commutant of an analytic Toeplitz operator
Carl C.
Cowen
1-31
Abstract: For a function f in ${H^\infty }$ of the unit disk, the operator on ${H^2}$ of multiplication by f will be denoted by ${T_f}$ and its commutant by ${B^{ - 1}} \circ B$ motivates work on more general functions. A theorem is proved which gives conditions on a family $ \mathcal{F}$ of ${H^\infty }$ functions which imply that there is a function h such that $ \{ {T_h}\} ' = { \cap _{f \in \mathcal{F}}}\{ {T_f}\} '$. As a special case of this theorem, we find that if the inner factor of $ f - f(c)$ is a finite Blaschke product for some c in the disk, then there is a finite Blaschke product B with $ \{ {T_f}\} ' = \{ {T_B}\} '$. Necessary and sufficient conditions are given for an operator to commute with ${T_f}$ when f is a covering map (in the sense of Riemann surfaces). If f and g are in ${H^\infty }$ and $ f = h \circ g$, then $ \{ {T_f}\} ' \supset \{ {T_g}\} '$. This paper introduces a class of functions, the ${H^2}$-ancestral functions, for which the converse is true. If f and g are $ {H^2}$-ancestral functions, then $ \{ {T_f}\} ' \ne \{ {T_g}\} '$ unless $f = h \circ g$ where h is univalent. It is shown that inner functions and covering maps are ${H^2}$-ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function f which imply that ${T_f}$ does not commute with nonzero compact operators. It follows from one of these results that if f is an ${H^2}$-ancestral function, then $ {T_f}$ does not commute with any nonzero compact operators.
Asymptotic formulas for Toeplitz determinants
Estelle
Basor
33-65
Abstract: The object of this paper is to find an asymptotic formula for determinants of finite dimensional Toeplitz operators generated by a class of functions with singularities. The formula is a generalization of the Strong Szegö Limit Theorem.
Isotopy groups
Lawrence L.
Larmore
67-97
Abstract: For any mapping $ f:V \to M$ (not necessarily an embedding), where V and M are differentiable manifolds without boundary of dimensions k and n, respectively, V compact, let ${[V \subset M]_f} = {\pi _1}({M^V},E,f)$, i.e., the set of isotopy classes of embeddings with a specific homotopy to f (E = space of embeddings). The purpose of this paper is to enumerate ${[V \subset M]_f}$. For example, if $k \geqslant 3,n = 2k$, and M is simply connected, $ {[{S^k} \subset M]_f}$ corresponds to ${\pi _2}M$ or ${\pi _2}M \otimes {Z_2}$, depending on whether k is odd or even. In the metastable range, i.e., $3(k + 1) > 2n$, a natural Abelian affine structure on $ {[V \subset M]_f}$ is defined: if, further, f is an embedding ${[V \subset M]_f}$ is then an Abelian group. The set of isotopy classes of embeddings homotopic to f is the set of orbits of the obvious left action of ${\pi _1}({M^V},f)$ on ${[V \subset M]_f}$. A spectral sequence is constructed converging to a theory ${H^\ast}(f)$. If $3(k + 1) < 2n, {H^0}(f) \cong {[V \subset M]_f}$ provided the latter is nonempty. A single obstruction $\Gamma (f) \in {H^1}(f)$ is also defined, which must be zero if f is homotopic to an embedding; this condition is also sufficient if $3(k + 1) \leqslant 2n$. The ${E_2}$ terms are cohomology groups of the reduced deleted product of V with coefficients in sheaves which are not even locally trivial. ${[{S^k} \subset M]_f}$ is specifically computed in terms of generators and relations if $n = 2k, k \geqslant 3$ (Theorem 6.0.2).
Hyperarithmetically encodable sets
Robert M.
Solovay
99-122
Abstract: We say that a set of integers, A, is hyperarithmetically (recursively) encodable, if every infinite set of integers X contains an infinite subset Y in which A is hyperarithmetical (recursive). We show that the recursively encodable sets are precisely the hyperarithmetic sets. Let $\sigma$ be the closure ordinal of a universal $\Sigma _1^1$ inductive definition. Then A is hyperarithmetically encodable iff it is constructible before stage $\sigma$. We also prove an effective version of the Galvin-Prikry results that open sets, and more generally Borel sets, are Ramsey, and in the case of open sets prove that our improvement is optimal.
The theorem of Torelli for singular curves
Thomas
Jambois
123-146
Abstract: Let C be a compact (singular) curve embedded in a surface. Then C carries a canonical sheaf $\Omega$ which is locally free of rank 1. Moreover, C has a generalized Jacobian J which fits in an exact sequence $\displaystyle 0 \to F \to J \to A \to 0$ ($ \ast$) of algebraic groups such that A is an abelian variety and $F = {({{\mathbf{C}}^\ast})^r} \times {{\mathbf{C}}^s}$. Let $\underline{C}$ be the set of nonsingular points of C and let $\theta$ = Zariski-closure of the image of $(\underline{C})^{(g - 1)}$ in J. Then: Theorem. If C is irreducible and sections of $\Omega$ map C onto X in ${P^{g - 1}}$ then the isomorphism class of J together with the translation class of the divisor $ \theta$ on J determine the isomorphism class of X. As a corollary, if $\psi :C \to X$ is an isomorphism (in which case we call C nonhyperelliptic) the above data determine the isomorphism class of C. I do not know if this remains true when C is hyperelliptic. It should be noted that the linear equivalence class of $\theta$ is not enough to determine X. The principal idea of the proof is that of Andreotti, that is, to recover the curve as the dual of the branch locus of the Gauss map from $\theta$ to $ {P^{g - 1}}$; however our arguments are usually analytic. The organization of this paper is as follows: In §1 we prove a stronger than usual version of Abel's theorem for Riemann surfaces and in §2 we extend this theorem to apply to singular curves. In succeeding sections we construct the generalized Jacobian as a complex Lie group J and embed J in an analytic fibre bundle over A with projective spaces as fibre. This we use to endow J with the structure of an algebraic group. §7 contains a miscellany of facts about branch loci and dual varieties, and in §8 the main theorems are stated and proved. We should mention here that the variations on Abel's theorem proved in this paper (1.2.4 and 3.0.1) were proved by Severi, at least in the special case corresponding to ordinary double points [12].
Mesures invariantes sur les hypergroupes
R.
Spector
147-165
Abstract: The notion of an invariant or relatively invariant measure on a hypergroup is studied. The main result is that every commutative hypergroup carries an invariant measure.
A chain functor for bordism
Stanley O.
Kochman
167-196
Abstract: Chains of differential graded abelian monoids are defined for bordism and cobordism theories. These chains are used to define matric Massey products and can be filtered so as to define the Adams spectral sequence. From this point of view, we prove three basic theorems which show how Massey products behave in the Adams spectral sequence.
Tensor products for ${\rm SL}(2,\,k)$
Robert P.
Martin
197-211
Abstract: Let G be $ {\text{SL}}(2,k)$ where k is a locally compact, nondiscrete, totally disconnected topological field whose residual characteristic is not 2, $ {\pi _\sigma }$, be a principal series representation of G, and $\pi \in \hat G$ be arbitrary. We determine the decomposition of ${\pi _\sigma } \otimes \pi $ into irreducibles by reducing this problem to decomposing the restriction of each $T \in \hat G$ to a minimal parabolic subgroup B of G and decomposing certain tensor products of irreducibles of B.
Complete universal locally finite groups
Ken
Hickin
213-227
Abstract: This paper will partly strengthen a recent application of model theory to the construction of sets of pairwise nonembeddable universal locally finite groups [8]. Our result is Theorem. There is a set $\mathcal{U}$ of ${2^{{\aleph _1}}}$ universal locally finite groups of order $ {\aleph _1}$ with the following properties: 0.1. If $U \ne V \in \mathcal{U}$ and A and B are uncountable sugroups of U and V, then A and B are not isomorphic. Let A be an uncountable subgroup of $U \in \mathcal{U}$. 0.2. A does not belong to any proper variety of groups, and 0.3. A is not isomorphic to any of its proper subgroups. 0.4. Every $U \in \mathcal{U}$ is a complete group (every automorphism of U is inner).
Twist spinning revisited
Deborah L.
Goldsmith;
Louis H.
Kauffman
229-251
Abstract: This paper contains several applications of the following theorem: The 1-twist spin ${L_1}(k)$ of any knot $k \subset {S^{n - 1}}$ is interchangeable with the standard unknotted $(n - 2)$-sphere K in ${S^n}$ by means of a homeomorphism of triples $ h:({S^n},K,{L_1}(k)) \to ({S^n},{L_1}(k),K)$ which reverses the orientation of $ {S^n}$, and preserves the orientations of K and ${L_1}(k)$. One of these applications is Zeeman's Theorem about twist spun knots; another is a proof of a conjecture of R. H. Fox about certain manifolds which have the same fundamental group. We also prove that the iterated twist spun knot ${L_{a,b}}(k) \subset {S^{n + 1}}$ is fiber equivalent to one of $ {L_{0,g}}(k)$ or ${L_{g,g}}(k)$ where $g = {\text{g.c.d.}}(a,b)$.
Completeness theorems, incompleteness theorems and models of arithmetic
Kenneth
McAloon
253-277
Abstract: Let $\mathcal{A}$ be a consistent extension of Peano arithmetic and let $ \mathcal{A}_n^0$ denote the set of $\Pi _n^0$ consequences of $ \mathcal{A}$. Employing incompleteness theorems to generate independent formulas and completeness theorems to construct models, we build nonstandard models of $\mathcal{A}_{n + 2}^0$ in which the standard integers are $\Delta _{n + 1}^0$-definable. We thus pinpoint induction axioms which are not provable in $\mathcal{A}_{n + 2}^0$; in particular, we show that (parameter free) $ \Delta _1^0$-induction is not provable in Primitive Recursive Arithmetic. Also, we give a solution of a problem of Gaifman on the existence of roots of diophantine equations in end extensions and answer questions about existentially complete models of $ \mathcal{A}_2^0$. Furthermore, it is shown that the proof of the Gödel Completeness Theorem cannot be formalized in $\mathcal{A}_2^0$ and that the MacDowell-Specker Theorem fails for all truncated theories $\mathcal{A}_n^0$.
Diffeomorphisms almost regularly homotopic to the identity
Robert
Wells
279-292
Abstract: Let $f:M \to M$ be a self-map of a closed smooth n-manifold. Does there exist a diffeomorphism $\varphi :M \to M$ homotopic to f? Define $\varphi$ to be almost regularly homotopic to the identity if $\varphi \vert M - {\text{pt}}$. is regularly homotopic to the inclusion $M - {\text{pt}}. \subset M$. Let $\psi :M \to M \vee M$ be the result of collapsing the boundary of a smooth n-cell in M, and let $ \xi \in {\pi _n}(M)$ define $\tau (\xi )$ to be the composition $ n = 2l > 5$ with $ l\;{\nequiv}\;0 \bmod (4)$, then $ \tau (\xi )$ contains a diffeomorphism almost regularly homotopic to the identity iff $\xi$ is in the kernel of the stabilization map $ {\pi _n}(M) \to \pi _n^s(M)$.
Heisenberg manifolds and theta functions
R.
Tolimieri
293-319
Abstract: The algebraic structure of the $(2n + 1)$-dimensional Heisenberg group naturally induces a special class of differential operators whose solutions $(Df = 0)$ are related to classical theta function theory.
The classification of flat solvmanifolds
Alexander
Morgan
321-351
Abstract: This paper contains a complete algebraic characterization of the fundamental groups of flat solvmanifolds. This characterization is in terms of finite integral representations of free abelian groups and the associated cohomology. A classification of compact flat solvmanifolds follows, and a list of all compact flat solvmanifolds of dimensions 3, 4, and 5 (except the 5-dimensional with first betti number 1) is given. Some theorems on the classification of noncompact flat solvmanifolds have also been obtained. These give full results in some cases, partial results in others. For example, the odd order holonomy group case is completely settled.
Limiting distributions for branching random fields
Joseph
Fleischman
353-389
Abstract: In this paper we derive limiting distributions for branching Brownian motion. The cases considered are where the state space is (1) the line and (2) the plane where (a) initially there's but one particle and (b) where there's initially a random number of independent particles. In all cases, the branching process is critical and we obtain results for the growth of selectively neutral mutant types. We use moment generating functions to derive these results.
The existence of combinatorial formulae for characteristic classes
Norman
Levitt;
Colin
Rourke
391-397
Abstract: Given a characteristic class on a locally ordered combinatorial manifold M there exists a cocycle which represents the class on M and is locally defined, i.e. its value on $ \sigma \in M$ depends only on the ordered star ${\text{st}}(\sigma ,M)$. For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class.
On the rationality of divisors and meromorphic functions
Chia Chi
Tung
399-406
Abstract: Let E be a holomorphic vector bundle over a connected complex manifold X and D a divisor on E. Let $ A(D)$ be the set of all $x \in X$ for which $ ({\text{supp}}\;D) \cap {E_x}$ is a proper algebraic set in ${E_x}$. The purpose of this paper is to prove that the following conditions are equivalent: (i) $A(D)$ has positive measure in X; (ii) D extends to a unique divisor on the projective completion Ē of E; (iii) D is locally given by the divisors of rational meromorphic functions defined over open sets in X. Similar results for meromorphic functions are derived. The proof requires an extension theorem for analytic set: Assume E is a holomorphic vector bundle over a pure p-dimensional complex space X and S an analytic set in E of pure codimension 1. Then the closure S of S in E is analytic if and only if $S \cap {E_x}$ is a proper algebraic set for all x in a set of positive 2p-measure in every branch of X.