Extension of Fourier $L\sp{p}---L\sp{q}$ multipliers
Michael G.
Cowling
1-33
Abstract: By $M_p^q(\Gamma )$ we denote the space of Fourier ${L^p} - {L^q}$ multipliers on the LCA group $ \Gamma$. K. de Leeuw [4] (for $\Gamma = {R^a}$), N. Lohoué [16] and S. Saeki [19] have shown that if ${\Gamma _0}$ is a closed subgroup of $\Gamma$, and $\phi$ is a continuous function in $M_p^p(\Gamma )$, then the restriction $ {\phi _0}$ of $ \phi$ to ${\Gamma _0}$ is in $M_p^p({\Gamma _0})$, and ${\left\Vert {{\phi _0}} \right\Vert _{M_p^p}} \leqslant {\left\Vert \phi \right\Vert _{M_p^p}}$. We answer here a natural question arising from this result: we show that every continuous function $ \psi$ in $M_p^p(\Gamma )$ is the restriction to ${\Gamma _0}$ of a continuous $M_p^p(\Gamma )$ function whose norm is the same as that of $\psi$. A Figà-Talamanca and G. I. Gaudry [8] proved this with the extra condition that ${\Gamma _0}$ be discrete: our technique develops their ideas. An extension theorem for $M_p^q({\Gamma _0})$ is obtained: this complements work of Gaudry [11] on restrictions of $M_p^q(\Gamma )$-functions to ${\Gamma _0}$.
Functions which are restrictions of $L\sp{p}$-multipliers
Michael G.
Cowling
35-51
Abstract: Raouf Doss has given a sufficient condition for a measurable function $ \phi$ on a measurable subset $\Lambda$ of an LCA group $\Gamma$ to be the restriction (l.a.e.) to $ \Lambda$ of the Fourier transform of a bounded measure, i.e., a Fourier multiplier of type (1, 1). We generalise Doss' theorem, and prove that, if the measurable function $ \phi$ on $\Lambda$ is approximable on finite subsets of $\Lambda$ by trigonometric polynomials which are Fourier multipliers of type (p, p) on $ \Gamma$ of norms no greater than C, then $\phi$ is equal locally almost everywhere to the restriction to $\Lambda$ of a Fourier multiplier of type (p, p) and norm no greater than C.
On a varietal structure of algebras
R. D.
Giri
53-60
Abstract: Shafaat introduced two successive generalisations of the variety of algebras: namely the semivariety and the quasivariety. We study a slightly more generalised concept which we call a pseudovariety.
An improved version of the noncompact weak canonical Schoenflies theorem
W. R.
Brakes
61-69
Abstract: The main result of this paper is that any proper collared embedding of ${R^{n - 1}}$ in ${R^n}$ can be extended to a homeomorphism of $ {R^n}$ such that the extension depends continuously on the original embedding in a stronger sense than previously known. Analogous results are proved for proper embeddings of $ {R^k}$ in ${R^n}$ (with the usual homotopy conditions when $k = n - 2$). An alternative proof of the usual compact form of the weak canonical Schoenflies theorem is also obtained.
Binary digit distribution over naturally defined sequences
D. J.
Newman;
Morton
Slater
71-78
Abstract: In a previous paper the first author showed that multiples of 3 prefer to have an even number of ones in their binary digit expansion. In this paper it is shown that in some general classes of naturally defined sequences, the probability that a member of a particular sequence has an even number of ones in its binary expansion is $ 1/2$.
Monads defined by involution-preserving adjunctions
Paul H.
Palmquist
79-87
Abstract: Consider categories with involutions which fix objects, functors which preserve involution, and natural transformations. In this setting certain natural adjunctions become universal and, thereby, become constructible from abstract data. Although the formal theory of monads fails to apply and the Eilenberg-Moore category fails to fit, both are successfully adapted to this setting, which is a 2-category. In this 2-category, each monad (= triple = standard construction) defined by an adjunction is characterized by a pair of special equations. Special monads have universal adjunctions which realize them and have both underlying Frobenius monads and adjoint monads. Examples of monads which do (respectively, do not) satisfy the special equations arise from finite monoids (= semigroups with unit) which are (respectively, are not) groups acting on the category of linear transformations between finite dimensional Euclidean (= positive definite inner product) spaces over the real numbers. More general situations are exposed.
Circle actions on homotopy spheres not bounding spin manifolds
Reinhard
Schultz
89-98
Abstract: Smooth circle actions are constructed on odd-dimensional homotopy spheres that do not bound spin manifolds. Examples are given in every dimension for which exotic spheres of the described type exist.
On $\pi \sb{3}$ of a finite $H$-space
J. R.
Hubbuck;
R.
Kane
99-105
Abstract: The third homotopy group of a finite H-space is shown to have no torsion.
On stable noetherian rings
Zoltán
Papp
107-114
Abstract: A ring R is called stable if every localizing subcategory of $_R{\text{M}}$ is closed under taking injective envelopes. In this paper the stable noetherian rings are characterized in terms of the idempotent kernel functors of $ _R{\text{M}}$ (O. Goldman [5]). The stable noetherian rings, the classical rings (Riley [11]) and the noetherian rings ``with sufficiently many two-sided ideals'' (Gabriel [4]) are compared and their relationships are studied. The close similarity between the commutative noetherian rings and the stable noetherian rings is also pointed out in the results.
M\"untz-Sz\'asz theorem with integral coefficients. II
Le Baron O.
Ferguson;
Manfred
von Golitschek
115-126
Abstract: The classical Müntz-Szász theorem concerns uniform approximation on [0, 1] by polynomials whose exponents are taken from a sequence of real numbers. Under mild restrictions on the exponents or the interval, the theorem remains valid when the coefficients of the polynomials are taken from the integers.
The residue calculus in several complex variables
Gerald Leonard
Gordon
127-176
Abstract: Let W be a complex manifold and V an analytic variety. Then homology classes in $W - V$ which bound in V, called the geometric residues, are studied. In fact, a long exact sequence analogous to the Thom-Gysin sequence for nonsingular V is formed by a geometric construction. A geometric interpretation of the Leray spectral sequence of the inclusion of $W - V \subset V$ is also given. If the complex codimension of V is one, then one shows that each cohomology class of $W - V$ can be represented by a differential form of the type $\theta \wedge \lambda + \eta$ where $ \lambda$ is the kernel associated to V and $\theta \vert V$ is the Poincaré residue of this class.
A decomposition for certain real semisimple Lie groups
H. Lee
Michelson
177-193
Abstract: For a class of real semisimple Lie groups, including those for which G and K have the same rank, Kostant introduced the decomposition $G = K{N_0}K$, where $ {N_0}$ is a certain abelian subgroup of N, and conjectured that the Jacobian of the decomposition with respect to Haar measure, as well as the spherical functions, would be polynomial in the canonical coordinates of $ {N_0}$. We compute here the Jacobian, which turns out to be polynomial precisely when the equality of ranks is satisfied. We also compute those spherical functions which restrict to polynomials on ${N_0}$.
A Galois theory for a class of inseparable field extensions
R. L.
Davis
195-203
Abstract: The structure of the group of rank n higher derivations in a field K is discussed and a characterization of its Galois subgroups is given. This yields a Galois type correspondence between these subgroups and the subfields over which K is purely inseparable, finite dimensional and modular.
$Z$-sets in ANR's
David W.
Henderson
205-216
Abstract: (1) Let A be a closed Z-set in an ANR X. Let $\mathcal{F}$ be an open cover of X. Then there is a homotopy inverse $f:X \to X - A$ to the inclusion $X - A \to X$ such that f and both homotopies are limited by $ \mathcal{F}$. (2) If, in addition, X is a manifold modeled on a metrizable locally convex TVS, F, such that F is homeomorphic to ${F^\omega }$, then there is a homotopy $j:X \times I \to X$ limited by $\mathcal{F}$ such that the closure (in X) of $ j(X \times \{ 1\} )$ is contained in $X - A$.
The hyperspace of the closed unit interval is a Hilbert cube
R. M.
Schori;
J. E.
West
217-235
Abstract: Let X be a compact metric space and let ${2^X}$ be the space of all nonvoid closed subsets of X topologized with the Hausdorff metric. For the closed unit interval I the authors prove that ${2^I}$ is homeomorphic to the Hilbert cube ${I^\infty }$, settling a conjecture of Wojdyslawski that was posed in 1938. The proof utilizes inverse limits and near-homeomorphisms, and uses (and developes) several techniques and theorems in infinite-dimensional topology.
Ergodic theorems for the asymmetric simple exclusion process
Thomas M.
Liggett
237-261
Abstract: Consider the infinite particle system on the integers with the simple exclusion interaction and one-particle motion determined by $p(x,x + 1) = p$ and $p(x,x - 1) = q$ for $x \in Z$, where $p + q = 1$ and $p > q$. If $\mu$ is the initial distribution of the system, let ${\mu _t}$ be the distribution at time t. The main results determine the limiting behavior of ${\mu _t}$ as $t \to \infty$ for simple choices of $\mu$. For example, it is shown that if $ \mu$ is the pointmass on the configuration in which all sites to the left of the origin are occupied, while those to the right are vacant, then the system converges as $t \to \infty$ to the product measure on ${\{ 0,1\} ^Z}$ with density $ {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. For the proof, an auxiliary process is introduced which is of interest in its own right. It is a process on the positive integers in which particles move according to the simple exclusion process, but with the additional feature that there can be creation and destruction of particles at one. Ergodic theorems are proved for this process also.
Embeddings and immersions of manifolds in Euclidean space
David R.
Bausum
263-303
Abstract: The problem of computing the number of embeddings or immersions of a manifold in Euclidean space is treated from a different point of view than is usually taken. Also, a theorem dealing with the existence of an embedding of $ {M^m}$ in ${R^{2m - 2}}$ is given.
A formula for the tangent bundle of flag manifolds and related manifolds
Kee Yuen
Lam
305-314
Abstract: A formula is given for the tangent bundle of a flag manifold G in terms of canonically defined vector bundles over G. The formula leads to a unified proof of some parallelizability theorems of Stiefel manifolds. It can also be used to deduce some immersion theorems for flag manifolds.
Heegaard splittings of branched coverings of $S\sp{3}$
Joan S.
Birman;
Hugh M.
Hilden
315-352
Abstract: This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegaard genus 2 is a 3-sphere. 2. We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3-manifolds are ``unique'' to the question of whether plat representations of knots and links are ``unique". 3. We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that ${S^3}$ could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for ${\pi _1}{S^3}$ are always ``nice". The final section of the paper studies a special class of genus 2 Heegaard splittings: the 2-fold covers of ${S^3}$ which are branched over closed 3-braids. It is established that no counterexamples to the ``genus 2 Poincaré conjecture'' occur in this class of 3-manifolds.
Picard's theorem and Brownian motion
Burgess
Davis
353-362
Abstract: Properties of the paths of two dimensional Brownian motion are used as the basis of a proof of the little Picard theorem and its analog for complex valued functions, defined on simply connected n dimensional manifolds, which map certain diffusions into Brownian motion.
The absolute continuity of phase operators
J.
Dombrowski;
G. H.
Fricke
363-372
Abstract: This paper studies the spectral properties of a class of operators known as phase operators which originated in the study of harmonic oscillator phase. Ifantis conjectured that such operators had no point spectrum. It was later shown that certain phase operators were, in fact, absolutely continuous and that all phase operators at least had an absolutely continuous part. The present work completes the discussion by showing that all phase operators are absolutely continuous.
On Dedekind's problem: the number of isotone Boolean functions. II
D.
Kleitman;
G.
Markowsky
373-390
Abstract: It is shown that $ \psi (n)$, the size of the free distributive lattice on n generators (which is the number of isotone Boolean functions on subsets of an n element set), satisfies $\displaystyle \psi (n) \leqslant {2^{(1 + O(\log \;n/n))\left( {\begin{array}{*{20}{c}} n {[n/2]} \end{array} } \right)}}.$ This result is an improvement by a factor $ \sqrt n$ in the 0 term of a previous result of Kleitman. In the course of deriving the main result, we analyze thoroughly the techniques used here and earlier by Kleitman, and show that the result in this paper is ``best possible'' (up to constant) using these techniques.
The predual theorem to the Jacobson-Bourbaki theorem
Moss
Sweedler
391-406
Abstract: Suppose $ R\xrightarrow{\varphi }S$ is a map of rings. S need not be an R algebra since R may not be commutative. Even if R is commutative it may not have central image in S. Nevertheless the ring structure on S can be expressed in terms of two maps $\displaystyle S{ \otimes _R}S\xrightarrow{{({s_1} \otimes {s_2} \to {s_1}{s_2})}}S,\quad R\xrightarrow{\varphi }S,$ which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an R-coring. Suppose R is an overing of B. Let $ {C_B} = R{ \otimes _B}R$. There are maps \begin{displaymath}\begin{array}{*{20}{c}} {{C_B} = R{ \otimes _B}R\xrightarrow{... ...{r_1} \otimes {r_2} \to {r_1}{r_2})}}R.} \end{array} \end{displaymath} These maps give ${C_B}$ an R-coring structure. The dual $^\ast{C_B}$ is naturally isomorphic to the ring ${\text{End}_{{B^ - }}}R$ of B-linear endomorphisms of R considered as a left B-module. In case B happens to be the subring of R generated by 1, we write ${C_{\text{Z}}}$. Then $^\ast{C_{\text{Z}}}$ is $ {\text{End}_{\text{Z}}}R$, the endomorphism ring of R considered as an additive group. This gives a clue how certain R-corings correspond to subrings of R and subrings of $ {\text{End}_{\text{Z}}}R$, both major ingredients of the Jacobson-Bourbaki theorem. $1 \otimes 1$ is a ``grouplike'' element in the R-coring $ {C_{\text{Z}}}$ (and should be thought of as a generic automorphism of R). Suppose R is a division ring and B a subring which is a division ring. The natural map ${C_{\text{Z}}} \to {C_B}$ is a surjective coring map. Conversely if $ {C_{\text{Z}}}\xrightarrow{\pi }D$ is a (surjective) coring map then $\pi (1 \otimes 1)$ is a grouplike in D and $\{ r \in R\vert r\pi (1 \otimes 1) = \pi (1 \otimes 1)r\}$ is a subring of R which is a division ring. This gives a bijective correspondence between the quotient corings of $C{ \otimes _{\text{Z}}}C$ and the subrings of R which are division rings. We show how the Jacobson-Bourbaki correspondence is dual to the above correspondence.
Homology with multiple-valued functions applied to fixed points
Richard
Jerrard
407-427
Abstract: Certain multiple-valued functions (m-functions) are defined and a homology theory based upon them is developed. In this theory a singular simplex is an m-function from a standard simplex to a space and an m-function from one space to another induces a homomorphism of homology modules. In a family of functions ${f_x}:Y \to Y$ indexed by $x \in X$ the fixed points of ${f_x}$ are taken to be the images at x of a multiple-valued function $\phi :X \to Y$. In certain circumstances $ \phi$ is an m-function, giving information about the behavior of the fixed points of ${f_x}$ as x varies over X. These facts are applied to self-maps of products of compact polyhedra and the question of whether such a product has the fixed point property for continuous functions is essentially reduced to the question of whether one of its factors has the fixed point property for m-functions. Some light is thrown on the latter problem by using the homology theory to prove a Lefschetz fixed point theorem for m-functions.
Erratum to: ``Regular overrings of regular local rings'' (Trans. Amer. Math. Soc. {\bf 171} (1972), 291--300)
Judith
Sally
429