Median algebra
John R.
Isbell
319-362
Abstract: A study of algebras with a ternary operation $ (x,\,y,\,z)$ satisfying some identities, equivalent to embeddability in a lattice with $(x,\,y,\,z)$ realized as, simultaneously, $(x\, \wedge \,(y\, \vee \,z))\, \vee \,(y\, \wedge \,z)$ and $(x\, \vee \,(y\, \wedge \,z))\, \wedge \,(y\, \vee \,z)$. This is weaker than embeddability in a modular lattice, where those expressions coincide for all x, y, and z, but much of the theory survives the extension. For actual embedding in a modular lattice, some necessary conditions are found, and the investigation is carried much further in a special, geometrically described class of examples ("2-cells"). In distributive lattices $(x,\,y,\,z)$ reduces to the median $(x\, \wedge \,y)\, \vee \,(x\, \wedge \,z)\, \vee \,(y\, \wedge \,z)$, previously studied by G. Birkhoff and S. Kiss. It is shown that Birkhoff and Kiss found a basis for the laws; indeed, their algebras are embeddable in distributive lattices, i.e. in powers of the 2-element lattice. Their theory is much further developed and is connected into an explicit Pontrjagin-type duality.
A separation theorem for $\Sigma \sp{1}\sb{1}$ sets
Alain
Louveau
363-378
Abstract: In this paper, we show that the notion of Borel class is, roughly speaking, an effective notion. We prove that if a set A is both $\prod _\xi ^0$ and $\Delta _1^1$, it possesses a $\Pi _\xi ^0$-code which is also $\Delta _1^1$. As a by-product of the induction used to prove this result, we also obtain a separation result for $\Sigma _1^1$ sets: If two $\Sigma _1^1$ sets can be separated by a $\Pi _\xi ^0$ set, they can also be separated by a set which is both $ \Delta _1^1$ and $\Pi _\xi ^0$. Applications of these results include a study of the effective theory of Borel classes, containing separation and reduction principles, and an effective analog of the Lebesgue-Hausdorff theorem on analytically representable functions. We also give applications to the study of Borel sets and functions with sections of fixed Borel class in product spaces, including a result on the conservation of the Borel class under integration.
Every contractible fan is locally connected at its vertex
Lex G.
Oversteegen
379-402
Abstract: We prove that each contractible fan is locally connected at its vertex. It follows that every contractible fan is embeddable in the plane. This gives a solution to a problem raised by J. J. Charatonik and C. A. Eberhart.
Nonexistence of continuous selections of the metric projection for a class of weak Chebyshev spaces
Manfred
Sommer
403-409
Abstract: The class $\mathfrak{B}$ of all those n-dimensional weak Chebyshev subspaces of $ C\,[a,\,b]$ whose elements have no zero intervals is considered. It is shown that there does not exist any continuous selection of the metric projection for G if there is a nonzero g in G having at least $ n\, + \,1$ distinct zeros. Together with a recent result of Nürnberger-Sommer, the following characterization of continuous selections for $ \mathfrak{B}$ is valid: There exists a continuous selection of the metric projection for G in $ \mathfrak{B}$ if and only if each nonzero g in G has at most n distinct zeros.
Inner product spaces associated with Poincar\'e complexes
Seiya
Sasao;
Hideo
Takahashi
411-419
Abstract: We consider the homotopy type classification of a certain kind of Poincaré complex. First we define an inner product space associated with such a Poincaré complex and we investigate the relation between the inner product space and the homotopy type of the Poincaré complex. As an application, some results for manifolds are proved.
Homotopy operations for simplicial commutative algebras
W. G.
Dwyer
421-435
Abstract: The indicated operation algebra is studied by methods dual to the usual ones for studying the Steenrod algebra. In particular, the operations are constructed using higher symmetries of the shuffle map and their ``Adem relations'' are computed using the transfer map in the cohomology of symmetric groups.
Higher divided squares in second-quadrant spectral sequences
W. G.
Dwyer
437-447
Abstract: The geometric action of the Steenrod algebra on many mod 2 cohomology spectral sequences is complemented by the action of a completely different algebra.
$q$-extension of the $p$-adic gamma function
Neal
Koblitz
449-457
Abstract: p-adic functions depending on a parameter q, $0\, < \,\vert q\, - \,1{\vert _p}\, < \,1$, are defined which extend Y. Morita's p-adic gamma function and the derivative of J. Diamond's p-adic log-gamma function in the same way as the classical q-gamma function ${\Gamma _q}(x)$ extends $ \Gamma (x)$. Properties of these functions which are analogous to the basic identities satisfied by $ {\Gamma _q}(x)$ are developed.
Univalence criteria in multiply-connected domains
Brad G.
Osgood
459-473
Abstract: Theorems due to Nehari and Ahlfors give sufficient conditions for the univalence of an analytic function in relation to the growth of its Schwarzian derivative. Nehari's theorem is for the unit disc and was generalized by Ahlfors to any simply-connected domain bounded by a quasiconformal circle. In both cases the growth is measured in terms of the hyperbolic metric of the domain. In this paper we prove a corresponding theorem for finitely-connected domains bounded by points and quasiconformal circles. Metrics other than the hyperbolic metric are also considered and similar results are obtained.
Products in the Atiyah-Hirzebruch spectral sequence and the calculation of $M{\rm SO}\sb\ast $
Brayton
Gray
475-483
Abstract: It is possible to put a multiplicative structure in the Atiyah-Hirzebruch spectral sequence in certain cases even though the spectra involved are not both ring spectra. As a special case, an easy calculation of the homotopy of MSO is obtained.
On the computational complexity of determining the solvability or unsolvability of the equation $X\sp{2}-DY\sp{2}=-1$
J. C.
Lagarias
485-508
Abstract: The problem of characterizing those D for which the Diophantine equation $ {X^2}\, - \,D{Y^2}\, = \, - \,1$ is solvable has been studied for two hundred years. This paper considers this problem from the viewpoint of determining the computational complexity of recognizing such D. For a given D, one can decide the solvability or unsolvability of ${X^2}\, - \,D{Y^2}\, = \, - \,1$ using the ordinary continued fraction expansion of $ \sqrt D$, but for certain D this requires more than $ \tfrac{1}{3}\,\sqrt D \,{(\log \,D)^{ - \,1}}$ computational operations. This paper presents a new algorithm for answering this question and proves that this algorithm always runs to completion in $O({D^{1/4\, + \,\varepsilon }})$ bit operations. If the input to this algorithm includes a complete prime factorization of D and a quadratic nonresidue $ {n_i}$ for each prime $ {p_i}$ dividing D, then this algorithm is guaranteed to run to completion in $O({(\log \,D)^5}\,(\log \,\log \,D)(\log \,\log \,\log \,D))$ bit operations. This algorithm is based on an algorithm that finds a basis of forms for the 2-Sylow subgroup of the class group of binary quadratic forms of determinant D.
Minimal skew products
S.
Glasner
509-514
Abstract: Let $(\sigma ,\,Z)$ be a metric minimal flow. Let Y be a compact metric space and let $\mathcal{g}$ be a pathwise connected group of homeomorphisms of Y. We consider a family of skew product flows on $Z\, \times \,Y\, = \,X$ and show that when $(\mathcal{g},\,Y)$ is minimal most members of this family have the property of being disjoint from every minimal flow which is disjoint from $(\sigma ,\,Z)$. From this and some further results about skew product flows we deduce the existence of a minimal metric flow which is disjoint from every weakly mixing minimal flow but is not PI.
A ${\bf Z}\times {\bf Z}$ structurally stable action
P. R.
Grossi Sad
515-525
Abstract: We consider in the product of spheres $ {S^m}\, \times \,{S^n}$ the $Z\, \times \,Z$-action generated by two simple Morse-Smale diffeomorphisms; if they have some kind of general position, the action is shown to be stable. An application is made to foliations.
An asymptotic theory for a class of nonlinear Robin problems. II
F. A.
Howes
527-552
Abstract: Various asymptotic phenomena exhibited by solutions of singularly perturbed Robin boundary value problems are studied in the case when the right-hand side grows faster than the square of the derivative.
Periodic orbits of continuous mappings of the circle
Louis
Block
553-562
Abstract: Let f be a continuous map of the circle into itself and let $ P(f)$ denote the set of positive integers n such that f has a periodic point of period n. It is shown that if $1\, \in \,P(f)$ and $n\, \in \,P(f)$ for some odd positive integer n then for every integer $m\, > \,n$, $ m\, \in \,P(f)$. Furthermore, if $P(f)$ is finite then there are integers m and n (with $ m\, \geqslant \,1$ and $n\, \geqslant \,0$) such that $P(f)\, = \,\{ m,\,2\,m,\,4\,m,\,8\,m,\,\ldots,\,{2^n}\,m\}$.
Composition series for analytic continuations of holomorphic discrete series representations of ${\rm SU}(n,\,n)$
Bent
Ørsted
563-573
Abstract: We study a certain family of holomorphic discrete series representations of the semisimple Lie group $G\, = \,SU(n,\,n)$ and the corresponding analytic continuation in the inducing parameter $\lambda$. At the values of $\lambda$ where the representations become reducible, we compute the composition series in terms of a Peter-Weyl basis on the Shilov boundary of the Hermitian symmetric space for G.
A Radon transform on spheres through the origin in ${\bf R}\sp{n}$ and applications to the Darboux equation
A. M.
Cormack;
E. T.
Quinto
575-581
Abstract: On domain ${C^\infty }\,({R^n})$ we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if $ f\, \in \,{C^\infty }\,({R^n})$ and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions $f\, \notin \,{C^\infty }\,({R^n})$ whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists.
The gluing of maximal ideals---spectrum of a Noetherian ring---going up and going down in polynomial rings
Ada Maria
de Souza Doering;
Yves
Lequain
583-593
Abstract: If ${M_1},\,...\,,\,{M_s}$ are maximal ideals of a ring R that have isomorphic residue fields, then they can be ``glued'' in the sense that a subring D of R with R is integral over D and ${M_1}\, \cap \,D\, = \,...\, = \,{M_s}\, \cap \,D$ can be constructed. We use this gluing process to prove the following result: Given any finite ordered set $ \mathcal{B}$, there exists a reduced Noetherian ring B and an embedding $\psi :\,\mathcal{B}\, \to \,Spec\,B$ such that $ \psi$ establishes a bijection between the maximal (respectively minimal) elements of $ \mathcal{B}$ and the maximal (respectively minimal) prime ideals of B and such that given any elements $\beta '$, $\beta ''$ of $ \mathcal{B}$, there exists a saturated chain of prime ideals of length r between $\psi (\beta '')$ if and only if there exists a saturated chain of length r between $ \beta '$ and $ \beta ''$. We also use the gluing process to construct a Noetherian domain A with quotient field L and a Noetherian domain B between A and L such that: $ A\,\hookrightarrow \,B$ possesses the Going Up and the Going Down properties, $ A[X]\,\hookrightarrow \,B[X]$ is unibranched and $A[X]\,\hookrightarrow \,B[X]$ possesses neither the Going Up nor the Going Down properties.
Factorization of curvature operators
Jaak
Vilms
595-605
Abstract: Let V be a real finite-dimensional vector space with inner product and let R be a curvature operator, i.e., a symmetric linear map of the bivector space $\Lambda {\,^2}V$ into itself. Necessary and sufficient conditions are given for R to admit factorization as $ R\, = \,\Lambda {\,^2}L$, with L a symmetric linear map of V into itself. This yields a new characterization of Riemannian manifolds that admit local isometric embedding as hypersurfaces of Euclidean space.
Fourier inversion on Borel subgroups of Chevalley groups: the symplectic group case
Ronald L.
Lipsman
607-622
Abstract: In recent papers, the author and J. A. Wolf have developed the Plancherel theory of parabolic subgroups of real reductive Lie groups. This includes describing the irreducible unitary representations, computing the Plancherel measure, and-since parabolic groups are nonunimodular-explicating the (unbounded) Dixmier-Pukanszky operator that appears in the Plancherel formula. The latter has been discovered to be a special kind of pseudodifferential operator. In this paper, the author considers the problem of extending this analysis to parabolic subgroups of semisimple algebraic groups over an arbitrary local field. Thus far he has restricted his attention to Borel subgroups (i.e. minimal parabolics) in Chevalley groups (i.e. split semisimple groups). The results he has obtained are described in this paper for the case of the symplectic group. The final result is (perhaps surprisingly), to a large extent, independent of the local field over which the group is defined. Another interesting feature of the work is the description of the ``pseudodifferential'' Dixmier-Pukanszky operator in the nonarchimedean situation.
On the ranges of analytic functions
J. S.
Hwang
623-629
Abstract: Following Doob, we say that a function $f(z)$ analytic in the unit disk U has the property $K(\rho )\,$ if $ f(0)\, = \,0$ and for some $\operatorname{arc} \,A\,$ on the unit circle whose measure $\left\vert A \right\vert\, \geqslant \,2\rho \, > \,0$, $\displaystyle \mathop {\lim \,\inf }\limits_{i \to \infty } \,\left\vert {f({P_... ... \,1\,{\text{where}}\,{P_i}\, \to \,P\, \in \,A\,{\text{and}}\,{P_i}\, \in \,U.$ We recently have solved a problem of Doob by showing that there is an integer $ N(\rho )$ such that no function with the property $K(\rho )$ can satisfy $\displaystyle {f_n}(z)\, = \,1\, + \,(1\, - \,{z^n})/{n^2},$ shows that the condition ${f_n}(0)\, = \,0$ is necessary and cannot be replaced by ${f_n}(0)\, = \,r{e^{i\alpha }}$, for $ r\, > \,1$. Naturally, we may ask whether this can be replaced by ${f_n}(0)\, = \,r{e^{i\alpha }}$, for $ r\, < \,1$? The answer turns out to be yes, when $n\, > \,N\,(r,\,\rho )$, where $\displaystyle N(r,\,\rho )\,\doteqdot\,(1/(1\, - \,r))\log (1/(1\, - \,\cos \rho )).$ .
Algebras of Fourier transforms with closed restrictions
Benjamin B.
Wells
631-636
Abstract: Let G denote a compact abelian group and let B denote a Banach subalgebra of A, the algebra of complex-valued functions on G whose Fourier series is absolutely convergent. If B contains the constant functions, separates the points of G, and if the restriction algebra, $B(E)$, is closed in $A(E)$ for every closed subset E of G, then $B = A$.