Many-one reducibility within the Turing degrees of the hyperarithmetic sets $H\sb{a}(x)$
G. C.
Nelson
1-44
Abstract: Spector [13] has proven that the hyperarithmetic sets $ {H_a}(x)$ and $ {H_b}(x)$ have the same Turing degree iff $\vert a\vert = \vert b\vert$. Y. Moschovakis has proven that the sets ${H_a}(x)$ under many-one reducibility for $\vert a\vert = \gamma$ and $a \in \mathcal{O}$ have nontrivial reducibility properties if $\gamma$ is not of the form $\alpha + 1$ or $ \alpha + \omega$ for any ordinal a. In particular, he proves that there are chains of order type ${\omega _1}$ and incomparable many-one degrees within these Turing degrees. In Chapter II, we extend this result to show that any countable partially ordered set can be embedded in the many-one degrees within these Turing degrees. In Chapter III, we prove that if $\gamma$ is also not of the form $\alpha + {\omega ^2}$ for some ordinal a, then there is no minimal many-one degree of the form $ {H_a}(x)$ in this Turing degree, answering a question of Y. Moschovakis posed in [8]. In fact, we prove that given ${H_a}(x)$ there are ${H_b}(x)$ and ${H_c}(x)$ both many-one reducible to ${H_a}(x)$ with incomparable many-one degrees, $ \vert a\vert = \vert b\vert = \vert c\vert = \gamma $.
On Fourier transforms
C.
Nasim
45-51
Abstract: If $f(x)$ and $g(x)$ satisfy the equations $\displaystyle g(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}g(t){k_1}(xt)dt,$ then we call f and g a pair of ${k_1}$-transforms, where $\displaystyle {k_1} = \frac{1}{{2\pi i}}\int _{{\raise0.5ex\hbox{$\scriptstyle ... ...r0.25ex\hbox{$\scriptstyle 2$}} + i\infty }\frac{{K(s)}}{{1 - s}}{x^{1 - s}}ds.$ In this paper alternative sets of conditions are established for f and g to be ${k_1}$-transform provided $K(s)$ is decomposable in a special way. These conditions involve simpler functions, which replace the kernel ${k_1}(x)$. Results are proved for the function spaces ${L^2}$. The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.
Splitting the tangent bundle
Wolf
Iberkleid
53-59
Abstract: We determine those unoriented cobordism classes which can be realized by a manifold whose tangent bundle splits into a sum of real (complex) line bundles.
Free products of topological groups which are $k\sb{\omega }$-spaces
Edward T.
Ordman
61-73
Abstract: Let G and H be topological groups and $G \ast H$ their free product topologized in the manner due to Graev. The topological space $G \ast H$ is studied, largely by means of its compact subsets. It is established that if G and H are ${k_\omega }$-spaces (respectively: countable CW-complexes) then so is $G \ast H$. These results extend to countably infinite free products. If G and H are ${k_\omega }$-spaces, $G \ast H$ is neither locally compact nor metrizable, provided G is nondiscrete and H is nontrivial. Incomplete results are obtained about the fundamental group $ \pi (G \ast H)$. If $ {G_1}$ and ${H_1}$ are quotients (continuous open homomorphic images) of G and H, then ${G_1} \ast {H_1}$ is a quotient of $ G \ast H$.
Interpolation between $H\sp{p}$ spaces: the real method
C.
Fefferman;
N. M.
Rivière;
Y.
Sagher
75-81
Abstract: The interpolation spaces in the Lions-Peetre method between $ {H^p}$ spaces, $0 < p < \infty$, are calculated.
Analytic centers and analytic diameters of planar continua
Steven
Minsker
83-93
Abstract: This paper contains some basic results about analytic centers and analytic diameters, concepts which arise in Vitushkin's work on rational approximation. We use Carathéodory's theorem to calculate $ \beta (K,z)$ in the case in which K is a continuum in the complex plane. This leads to the result that, if $g:\Omega (K) \to \Delta (0;1)$ is the normalized Riemann map, then $\beta (g,0)/\gamma (K)$ is the unique analytic center of K and $\beta (K) = \gamma (K)$. We also give two proofs of the fact that $\beta (g,0)/\gamma (K) \in {\text{co}}\;(K)$. We use Bieberbach's and Pick's theorems to obtain more information about the geometric location of the analytic center. Finally, we obtain inequalities for $\beta (E,z)$ for arbitrary bounded planar sets E.
Norm inequalities for the Littlewood-Paley function $g\sp{\ast} \sb{\lambda }$
Benjamin
Muckenhoupt;
Richard L.
Wheeden
95-111
Abstract: Weighted norm inequalities for ${L^p}$ and ${H^p}$ are derived for the Littlewood-Paley function $g_\lambda ^ \ast $. New results concerning the boundedness of this function are obtained, by a different method of proof, even in the unweighted case. The proof exhibits a connection between $g_\lambda ^\ast$ and a maximal function for harmonic functions which was introduced by C. Fefferman and E. M. Stein. A new and simpler way to determine the behavior of this maximal function is given.
Free $S\sp{1}$ actions and the group of diffeomorphisms
Kai
Wang
113-127
Abstract: Let ${S^1}$ act linearly on ${S^{2p - 1}} \times {D^{2q}}$ and ${D^{2p}} \times {S^{2q - 1}}$ and let $f:{S^{2p - 1}} \times {S^{2q - 1}} \to {S^{2p - 1}} \times {S^{2q - 1}}$ be an equivariant diffeomorphism. Then there is a well-defined ${S^1}$ action on ${S^{2p - 1}} \times {D^{2q}}{ \cup _f}{D^{2p}} \times {S^{2q - 1}}$. An ${S^1}$ action on a homotopy sphere is decomposable if it can be obtained in this way. In this paper, we will apply surgery theory to study in detail the set of decomposable actions on homotopy spheres.
Pointwise differentiability and absolute continuity
Thomas
Bagby;
William P.
Ziemer
129-148
Abstract: This paper is concerned with the relationships between ${L_p}$ differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in ${L_p}$, and $0 \leq l \leq k$, then f has an ${L_p}$ derivative of order l everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an ${L_p}$ derivative everywhere except for a set small in capacity and if these derivatives are in $ {L_p}$, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.
Complex approximation for vector-valued functions with an application to boundary behaviour
Leon
Brown;
P. M.
Gauthier;
W.
Seidel
149-163
Abstract: This paper deals with the qualitative theory of uniform approximation by holomorphic functions. The first theorem is an extension to vector-valued mappings of N. U. Arakélian's theorem on uniform holomorphic approximation on closed sets. Our second theorem is on asymptotic approximation and yields, as in the scalar case, applications to cluster sets.
Extensions of the $v$-integral
J. R.
Edwards;
S. G.
Wayment
165-184
Abstract: In Representations for transformations continuous in the BV norm [J. R. Edwards and S. G. Wayment, Trans. Amer. Math. Soc. 154 (1971), 251-265] the $\nu$-integral is defined over intervals in $ {E^1}$ and is used to give a representation for transformations continuous in the BV norm. The functions f considered therein are real valued or have values in a linear normed space X, and the transformation $ T(f)$ is real or has values in a linear normed space Y. In this paper the $ \nu$-integral is extended in several directions: (1) The domain space to (a) $ {E^n}$, (b) an arbitrary space S, a field $\Sigma$ of subsets of S and a bounded positive finitely additive set function $\mu$ on $\Sigma$ (in this setting the function space is replaced by the space of finitely additive set functions which are absolutely continuous with respect to $ \mu$); (2) the function space to (a) bounded continuous, (b) $ {C_c}$, (c) ${C_0}$, (d) C with uniform convergence on compact sets; (3) range space X for the functions and Y for the transformation to topological vector spaces (not necessarily convex); (4) when X and Y are locally convex spaces, then a representation for transformations on a $ {C_1}$-type space of continuously differentiable functions with values in X is given.
The continuity of Arens' product on the Stone-\v Cech compactification of semigroups
Nicholas
Macri
185-193
Abstract: A discrete semigroup is said to have the compact semigroup property (c.s.p.) [the compact semi-semigroup property (c.s.s.p.)] if the multiplication Arens' product, on its Stone-Čech compactification, is jointly [separately] ${w^ \ast }$-continuous. We obtain an algebraic characterization of those semigroups which have c.s.p. by characterizing algebraically their almost periodic subsets. We show that a semigroup has c.s.p. if and only if each of its subsets is almost periodic. This characterization is employed to prove that for a cancellation semigroup to have c.s.p., it is necessary and sufficient that each of its countable subsets be almost periodic. We answer in the negative a heretofore open question--is c.s.p. equivalent to c.s.s.p.
Entire vectors and holomorphic extension of representations. II
Richard
Penney
195-207
Abstract: Let G be a connected, simply connected Lie group and let $ {G_c}$ be its complexification. Let U be a unitary representation of G. The space of vectors v at which U is holomorphically extendible to ${G_c}$ is denoted $\mathcal{H}_\infty ^\omega (U)$. In [9] we characterized those U for which $\mathcal{H}_\infty ^\omega $ is dense. In the present work we study $\mathcal{H}_\infty ^\omega $ as a topological vector space, proving e.g., that $\mathcal{H}_\infty ^\omega $ is a Montel space if U is irreducible and G is nilpotent. We prove a representation theorem for
An approach to fixed-point theorems on uniform spaces
E.
Tarafdar
209-225
Abstract: Diaz and Metcalf [2] have some interesting results on the set of successive approximations of a self mapping which is either a nonexpansion or a contraction on a metric space with respect to the set of fixed points of the mapping. We have extended most of these results to a Hausdorff uniform space. We have also proved a Banach's contraction mapping principle on a complete Hausdorff uniform space and indicated some applications in locally convex linear topological spaces.
Semicellularity, decompositions and mappings in manifolds
Donald
Coram
227-244
Abstract: If X is an arbitrary compact set in a manifold, we give algebraic criteria on X and on its embedding to determine that X has an arbitrarily small, closed neighborhood each component of which is a p-connected, piecewise linear manifold which collapses to a q-dimensional subpolyhedron from some p and q. This property generalizes cellularity. The criteria are in terms of UV properties and Alexander-Spanier cohomology. These criteria are then applied to decide when the components of a given compact set in a manifold are elements of a decomposition such that the quotient space is the n-sphere. Conversely, algebraic criteria are given for the point inverses of a map between manifolds to have arbitrarily small neighborhoods of the type mentioned above; these criteria are considerably weaker than for an arbitrary compact set.
On homeomorphisms of infinite-dimensional bundles. I
Raymond Y. T.
Wong
245-259
Abstract: In this paper we present several aspects of homeomorphism theory in the setting of fibre bundles modeled on separable infinite-dimensional Hilbert (Fréchet) spaces. We study (homotopic) negligibility of subsets, separation of sets, characterization of subsets of infinite-deficiency and extending homeomorphisms; in an essential way they generalize previously known results for manifolds. An important tool is a lemma concerning the lifting of a map to the total space of a bundle whose image misses a certain closed subset presented as obstruction; from this we are able to obtain a result characterizing all subsets of infinite deficiency (for bundles) by their restriction to each fibre. Other results then follow more or less routinely by employing the rather standard methods of infinite-dimensional topology.
On homeomorphisms of infinite dimensional bundles. II
T. A.
Chapman;
R. Y. T.
Wong
261-268
Abstract: This paper presents some aspects of homeomorphism theory in the setting of (fibre) bundles modeled on separable Hilbert manifolds and generalizes results previously established. The main result gives a characterization of subsets of infinite deficiency in a bundle by means of their restriction to the fibres, from which we are able to prove theorems of the following types: (a) mapping replacement, (b) separation of sets, (c) negligibility of subsets, and (d) extending homeomorphisms.
On homeomorphisms of infinite dimensional bundles. III
T. A.
Chapman;
R. Y. T.
Wong
269-276
Abstract: In this paper we continue the study of homeomorphisms and prove an analogue of the homeomorphism extension theorem for bundles modeled on Hilbert cube manifolds; thus we generalize previous results for Q-manifolds (Anderson-Chapman). This analogy, as in the case of manifolds, requires a consideration of proper maps and proper homotopies. The approach to the present problem is similar to that considered in our previous papers. Bear in mind several distinct difficulties occur in our setting.
$\lambda $ connected plane continua
Charles L.
Hagopian
277-287
Abstract: A continuum M is said to be $ {\mathbf{\lambda }}$ connected if any two distinct points of M can be joined by a hereditarily decomposable continuum in M. Recently this generalization of arcwise connectivity has been related to fixed point ptoblems in the plane. In particular, it is known that every ${\mathbf{\lambda }}$ connected nonseparating plane continuum has the fixed point property. The importance of arcwise connectivity is, to a considerable extent, due to the fact that it is a continuous invariant. To show that $ {\mathbf{\lambda }}$ connectivity has a similar feature is the primary purpose of this paper. Here it is proved that if M is a ${\mathbf{\lambda }}$ connected continuum and f is a continuous function of M into the plane, then $f(M)$ is $ {\mathbf{\lambda }}$ connected. It is also proved that every semiaposyndetic plane continuum is $ {\mathbf{\lambda }}$ connected.
Homeomorphisms of a certain cube with holes
Donald
Myers
289-299
Abstract: For some manifolds the group of isotopy classes of self-homeomorphisms is known. In this paper this group is computed for a well-known cube with two holes. Two related manifolds are defined and the groups of isotopy classes on these manifolds are given without proof. One of these cubes with holes is such that every homeomorphism is isotopic to the identity.
Rational approximation on product sets
Otto B.
Bekken
301-316
Abstract: Our object here is to study pointwise bounded limits, decomposition of orthogonal measures and distance estimates for $R({K_1} \times {K_2})$ where $ {K_1}$ and ${K_2}$ are compact sets in the complex plane.
Weighted join semilattices and transversal matroids
Richard A.
Brualdi
317-328
Abstract: We investigate join-semilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A join-subsemilattice of a Boolean lattice is weighted by cardinality, and we give a characterization of these in terms of the notion of a spread. The collection of flats with no coloops (isthmuses) of a matroid or pregeometry, partially ordered by set-theoretic inclusion, forms a join-semilattice which is weighted by rank. For transversal matroids these join-semilattices are isomorphic to join-subsemilattices of Boolean lattices. Using a previously obtained characterization of transversal matroids and results on weighted join-semilattices, we obtain another characterization of transversal matroids. The problem of constructing a transversal matroid whose join-semilattice of flats is isomorphic to a given join-subsemilattice of a Boolean lattice is then investigated.
Generalized almost periodicity in groups
Henry W.
Davis
329-352
Abstract: A module of almost periodic functions on a group is closed with respect to a quite general seminorm. The new space of functions is characterized in terms of the internal properties of its members. This yields new characterizations of Besicovitch and Weyl almost periodic functions in a variety of group-theoretic settings. Eberlein's theorem that weakly almost periodic functions on the real line are Weyl almost periodic is extended to locally compact groups.
On deformations of homomorphisms of locally compact groups
Dong Hoon
Lee
353-361
Abstract: The rigidity of homomorphisms of compactly generated locally compact groups into Lie groups is investigated.
A proof that $\mathcal{H}^2$ and $\mathcal{T}^2$ are distinct measures
Lawrence R.
Ernst
363-372
Abstract: It is proven that there exists a subset E of ${{\mathbf{R}}^3}$ such that the two-dimensional $\mathcal{J}$ measure of E is less than its two-dimensional Hausdorff measure. E is the image under the usual isomorphism of $ {\mathbf{R}} \times {{\mathbf{R}}^2}$ onto $ {{\mathbf{R}}^3}$ of the Cartesian product of $\{ x: - 4 \leq x \leq 4\} $ and a Cantor type subset of $ {{\mathbf{R}}^2}$; the latter term in this product is the intersection of a decreasing sequence, every member of which is the union of certain closed circular disks.
Construction of automorphic forms and integrals
Douglas
Niebur
373-385
Abstract: It is well known that modular forms of positive dimension have Fourier coefficients given by certain infinite series involving Kloostermann sums and the modified Bessel function of the first kind. In this paper a functional equation which characterizes all such Fourier series is found. It is also shown that these Fourier series have a construction similar to that of Poincaré series of negative dimension.
The Fredholm spectrum of the sum and product of two operators
Jack
Shapiro;
Morris
Snow
387-393
Abstract: Let $C(X)$ denote the set of closed operators with dense domain on a Banach space X, and $ L(X)$ the set of all bounded linear operators on X. Let ${\mathbf{\Phi }}(X)$ denote the set of all Fredholm operators on X, and ${\sigma _{\mathbf{\Phi }}}(A)$ the set of all complex numbers $ {\mathbf{\lambda }}$ such that $ ({\mathbf{\lambda }} - A) \notin {\mathbf{\Phi }}(X)$. In this paper we establish conditions under which $ {\sigma _{\mathbf{\Phi }}}(A + B) \subseteq {\sigma _{\mathbf{\Phi }}}(A) + {\... ...} ) \subseteq {\sigma _{\mathbf{\Phi }}}(A) \cdot {\sigma _{\mathbf{\Phi }}}(B)$, and ${\sigma _\Phi }(AB) \subseteq {\sigma _\Phi }(A){\sigma _\Phi }(B)$.
Spectral orders, uniform integrability and Lebesgue's dominated convergence theorem
Kong Ming
Chong
395-404
Abstract: Using the 'spectral' order relations $\prec$ and $ \prec\prec$ introduced by Hardy, Littlewood and Pólya, we characterize the uniform integrability of a family of integrable functions. We also prove an extension and a 'converse' of the classical Lebesgue's dominated convergence theorem in terms of the 'spectral' orders $\prec$ and $ \prec\prec$.
Limit theorems for variational sums
William N.
Hudson;
Howard G.
Tucker
405-426
Abstract: Limit theorems in the sense of a.s. convergence, convergence in $ {L_1}$-norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if $\{ X(t),0 \leq t \leq T\} $ is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form $\Sigma _{k = 1}^ng(X({t_{nk}}) - X({t_{n,k - 1}}))$ is shown to converge with probability one and in ${L_1}$-norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.
Correction to: ``The separable closure of some commutative rings'' (Trans. Amer. Math. Soc. {\bf 170} (1972), 109--124)
Andy R.
Magid
427-430