Filters and the weak almost periodic compactification of a discrete semigroup
John F.
Berglund;
Neil
Hindman
1-38
Abstract: The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers.
Sieved ultraspherical polynomials
Waleed
Al-Salam;
W. R.
Allaway;
Richard
Askey
39-55
Abstract: The continuous $ q$-ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allaway's Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allaway's and showed that these polynomials arose from the $ q$-ultraspherical polynomials when $q$ approached a root of unity. A second class of such polynomials is found, and the recurrence relation and orthogonality relation are found for each of these polynomials. The orthogonality is interesting because the weight function has a finite number of zeros in $ (-1, 1)$. Generating functions and other formulas are also found.
Stable rank $2$ reflexive sheaves on ${\bf P}\sp{3}$ with small $c\sb{2}$ and applications
Mei-Chu
Chang
57-89
Abstract: We investigate the moduli spaces of stable rank two reflexive sheaves on $ {{\mathbf{P}}^3}$ with small Chern classes. As an application to curves of low degree in $ {{\mathbf{P}}^3}$, we prove the curve has maximal rank and that the corresponding Hilbert scheme is irreducible and unirational.
Localization of equivariant cohomology rings
J.
Duflot
91-105
Abstract: The main result of this paper is the "calculation" of the Borel equivariant cohomology ring ${H^{\ast} }(EG \times_G\,X,{\mathbf{Z}}/p{\mathbf{Z}})$ localized at one of its minimal prime ideals. In case $X$ is a point, the work of Quillen shows that the minimal primes $ {\mathfrak{P}_A}$ are parameterized by the maximal elementary abelian $ p$-subgroups $ A$ of $G$ and the result is $\displaystyle {H^{\ast} }{(BG,{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}} \... ...{H^{\ast} }(B{C_G}(A),{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}^{{W_G}(A)}$ . Here, $ {C_G}(A)$ is the centralizer of $A$ in $G$, and ${W_G}(A) = {N_G}(A)/{C_G}(A)$, where $ {N_G}(A)$ is the normalizer of $A$ in $G$. An example is included.
The law of exponential decay for expanding transformations of the unit interval into itself
M.
Jabłoński
107-119
Abstract: Let $T:[0,1] \to [0,1]$ be an expanding map of the unit interval and let $ {\xi _\varepsilon }(x)$ be the smallest integer $n$ for which ${T^n}(x) \in [0,\varepsilon ]$; that is, it is the random variable given by the formula $\displaystyle {\xi _\varepsilon }(x) = \min \{ n:{T^n}\;(x) \leqslant \varepsilon \}.$ It is shown that for any $z \geqslant 0$ and for any integrable function $f:[0,1] \to {R^ + }$ the measure $ {\mu _f}$ (where $ \mu$ is Lebesgue measure and ${\mu _f}$ is defined by $d{\mu _f} = fd\mu$) of the set of points $ x$ for which ${\xi _\varepsilon }(x) \leqslant z/\varepsilon$ tends to an exponential function of $z$ as $ \varepsilon$ tends to zero.
The heat equation with a singular potential
Pierre
Baras;
Jerome A.
Goldstein
121-139
Abstract: Of concern is the singular problem $\partial u/\partial t = \Delta u + (c/\vert x{\vert^2})\,u + f(t,x), u(x,0) = u_{0}(x)$, and its generalizations. Here $c \geqslant 0,x \in {{\mathbf{R}}^N},t > 0$, and $f$ and ${u_0}$ are nonnegative and not both identically zero. There is a dimension dependent constant ${C_{\ast} }(N)$ such that the problem has no solution for $c > {C_{\ast} }(N)$. For $c \leqslant {C_{\ast} }(N)$ necessary and sufficient conditions are found for $f$ and ${u_0}$ so that a nonnegative solution exists.
Actions of finite groups on homotopy $3$-spheres
M. E.
Feighn
141-151
Abstract: It is conjectured that the action of a finite group of diffeomorphisms of the $3$-sphere is equivariantly diffeomorphic to a linear action. This conjecture is verified if both of the following conditions hold: (i) Each isotropy subgroup is dihedral or cyclic. (ii) There is a point with cyclic isotropy subgroup of order not $ 1,2,3$ or $5$.
Operators with $C\sp{\ast} $-algebra generated by a unilateral shift
John B.
Conway;
Paul
McGuire
153-161
Abstract: If $T$ is an operator on a Hilbert space $\mathcal{H}$, this paper gives necessary and sufficient conditions on $T$ such that $ {C^{\ast} }(T)$, the ${C^{\ast} }$-algebra generated by $T$, is generated by a unilateral shift of some multiplicity. This result is then specialized to the cases in which $T$ is a hyponormal or subnormal operator. In particular, it is shown how to prove a recent conjecture of C. R. Putnam as a consequence of our result.
On the universal theory of classes of finite models
S.
Tulipani
163-170
Abstract: First order theories for which the truth of a universal sentence on their finite models implies the truth on all models are investigated. It is proved that an equational theory has such a property if and only if every finitely presented model is residually finite. The most common classes of algebraic structures are discussed.
The theory of ordered abelian groups does not have the independence property
Y.
Gurevich;
P. H.
Schmitt
171-182
Abstract: We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmitt's Elementary properties of ordered abelian groups, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every $n$-type in the theory of coloured chains has at most ${2^n}$ coheirs, thereby strengthening a result by B. Poizat.
Points of continuity for semigroup actions
Jimmie D.
Lawson
183-202
Abstract: The purpose of this paper is to provide a more unified approach to questions involving the existence of points of joint continuity in separately continuous semigroup actions by deriving a small number of general principles which suffice to deduce previously derived results and generalizations thereof. The first major result gives sufficient conditions for a point to be a point of joint continuity in a general setting of "migrants", a useful symmetric generalization of semigroup actions. Results concerning actions of semigroups with group-like properties follow. In the latter part of the paper the notion of a subordinate point is introduced and joint continuity at subordinate points for various settings is proved. Finally, these results are applied to linear actions on locally convex spaces.
Complexity of subcases of Presburger arithmetic
Bruno
Scarpellini
203-218
Abstract: We consider formula subclasses of Presburger arithmetic which have a simple structure in one sense or the other and investigate their computational complexity. We also prove some results on the lower bounds of lengths of formulas which are related to questions on quantifier elimination.
Mean value properties of the Laplacian via spectral theory
Robert S.
Strichartz
219-228
Abstract: Let $\phi ({z^2})$ be an even entire function of temperate exponential type, $L$ a selfadjoint realization of $- \Delta + c\,(x)$, where $\Delta$ is the Laplace-Beltrami operator on a Riemannian manifold, and $ \phi \,(L)$ the operator given by spectral theory. A Paley-Wiener theorem on the support of $\phi \,(L)$ is proved, and is used to show that $Lu = \lambda u$ on a suitable domain implies $\phi \,(L)\,u = \phi \,(\lambda)\,u$, as well as a generalization of Àsgeirsson's theorem. A concrete realization of the operators $\phi \,(L)$ is given in the case of a compact Lie group or a noncompact symmetric space with complex isometry group.
Unicellular operators
José
Barría;
Kenneth R.
Davidson
229-246
Abstract: An operator is unicellular if its lattice of invariant subspaces is totally ordered by inclusion. The list of nests which are known to be the set of invariant subspaces of a unicellular operator is surprisingly short. We construct unicellular operators on ${l^p},1 \leqslant p < \infty$, and on $ {c_0}$ with lattices isomorphic to $ \alpha + X + {\beta ^{\ast}}$ where $\alpha$ and $\beta$ are countable (finite or zero) ordinals, and $X$ is in this short list. Certain other nests are attained as well.
Positive-definiteness and its applications to interpolation problems for holomorphic functions
Frank
Beatrous;
Jacob
Burbea
247-270
Abstract: Holomorphic interpolation problems of the Pick-Nevanlinna and Loewner types as well as abstract interpolation theorems on functional Hilbert spaces are considered. Various characterizations are presented for restrictions of bounded holomorphic functions. In addition, certain norm estimates for restrictions and extensions of holomorphic functions are obtained.
Moments of balanced measures on Julia sets
M. F.
Barnsley;
A. N.
Harrington
271-280
Abstract: By a theorem of S. Demko there exists a balanced measure on the Julia set of an arbitrary nonlinear rational transformation on the Riemann sphere. It is proved here that if the transformation admits an attractive or indifferent cycle, then there is a point with respect to which all the moments of a balanced measure exist; moreover, these moments can be calculated exactly. An explicit balanced measure is exhibited in an example where the Julia set is the whole sphere and for which the moments, with respect to any point, do not all exist.
Singular Vietoris-Begle theorems for relations
D. G.
Bourgin;
Robert M.
Nehs
281-318
Abstract: The Vietoris-Begle theorem with singularities, for three spaces $ X$, $Y$, $T$, is extended to the case that a closed relation replaces a continuous map and more generally to set valued maps. The developments are carried out based on modification of the topology of $T$ so that in general it is no longer even Hausdorff. This entails interpretation of dimension of singulars sets in terms of considertions in $Y$ rather than $T$. The techniques are those of sheaf and spectral sequence theory.
On the Weil-Petersson metric on Teichm\"uller space
A. E.
Fischer;
A. J.
Tromba
319-335
Abstract: Teichmüller space for a compact oriented surface $M$ without boundary is described as the quotient $ \mathcal{A}/{\mathcal{D}_0}$, where $ \mathcal{A}$ is the space of almost complex structures on $M$ (compatible with a given orientation) and ${\mathcal{D}_0}$ are those ${C^\infty }$ diffeomorphisms homotopic to the identity. There is a natural ${\mathcal{D}_0}$ invariant ${L_2}$ Riemannian structure on $\mathcal{A}$ which induces a Riemannian structure on $ \mathcal{A}/{\mathcal{D}_0}$. Infinitesimally this is the bilinear pairing suggested by Andre Weil--the Weil-Petersson Riemannian structure. The structure is shown to be Kähler with respect to a naturally induced complex structure on $ \mathcal{A}/{\mathcal{D}_0}$.
Fixed point sets of metric and nonmetric spaces
John R.
Martin;
William
Weiss
337-353
Abstract: A space $ X$ is said to have the complete invariance property $ ($CIP$)$ if every nonempty closed subset of $ X$ is the fixed point set of some self-mapping of $X$. It is shown that connected subgroups of the plane and compact groups need not have CIP, and CIP need not be preserved by self-products of Peano continua, nonmetric manifolds or 0-dimensional spaces. Sufficient conditions are given for an infinite product of spaces to have CIP. In particular, an uncountable product of real lines, circles or two-point spaces has CIP. Examples are given which contrast the behavior of CIP in the nonmetric and metric cases, and examples of spaces are given where the existence of CIP is neither provable nor refutable with the usual axioms of set theory.
The family approach to total cocompleteness and toposes
Ross
Street
355-369
Abstract: A category with small homsets is called total when its Yoneda embedding has a left adjoint; when the left adjoint preserves pullbacks, the category is called lex total. Total categories are characterized in this paper in terms of special limits and colimits which exist therein, and lex-total categories are distinguished as those which satisfy further exactness conditions. The limits involved are finite limits and intersections of all families of subobjects. The colimits are quotients of certain relations (called congruences) on families of objects (not just single objects). Just as an arrow leads to an equivalence relation on its source, a family of arrows into a given object leads to a congruence on the family of sources; in the lex-total case all congruences arise in this way and their quotients are stable under pullback. The connection with toposes is examined.
Liouville theorems, partial regularity and H\"older continuity of weak solutions to quasilinear elliptic systems
Michael
Meier
371-387
Abstract: This paper describes the connections between Liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth. It is shown that equivalence does in general hold only in some restricted sense. A complete correspondence can be established in certain cases, e.g. for small solutions and for minima of quadratic variational integrals.
Universal families for conull FK spaces
A. K.
Snyder
389-399
Abstract: This paper considers the problem of determining a useful family of sequence spaces which is universal for conull FK spaces in the following sense: An FK space is conull if and only if it contains a member of the family. In the equivalent context of weak wedge spaces, an appropriate family of subspaces of boundedness domains ${m_A}$ of matrices is shown to be universal. Most useful is the fact that the members of this family exhibit unconditional sectional convergence. The latter phenomenon is known for wedge spaces. Another family of spaces which is universal for conull spaces among semiconservative spaces is provided. The spaces are designed to simplify gliding humps arguments. Improvements are thereby obtained for some pseudoconull type theorems of Bennett and Kalton. Finally, it is shown that conull spaces must contain pseudoconull BK algebras.
Milnor's invariants and the completions of link modules
Lorenzo
Traldi
401-424
Abstract: Let $L$ be a tame link of $\mu \geqslant 2$ components in ${S^3}$, $H$ the abelianization of its group ${\pi _1}({S^3} - L)$, and $IH$ the augmentation ideal of the integral group ring $ {\mathbf{Z}}H$. The $ IH$-adic completions of the Alexander module and Alexander invariant of $ L$ are shown to possess presentation matrices whose entries are given in terms of certain integers $\mu ({i_1}, \ldots ,{i_q})$ introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of $L$ with each other to all be zero. In the special case $\mu = 2$, it is shown that the various Milnor invariants $ \bar \mu ([r + 1,s + 1])$ are determined (up to sign) by the Alexander polynomial of $L$, and that this Alexander polynomial is 0 iff $ \bar \mu ([r + 1,s + 1]) = 0$ for all $ r,s \geqslant 0$ with $ r + s$ even; also, the Chen groups of $L$ are determined (up to isomorphism) by those nonzero $ \bar \mu ([r + 1,s + 1])$ with $r + s$ minimal. In contrast, it is shown by example that for $ \mu \geqslant 3$ the Alexander polynomials of a link and its sublinks do not determine its Chen groups.
Proper holomorphic mappings that must be rational
Steven
Bell
425-429
Abstract: Suppose $f:{D_1} \to {D_2}$ is a proper holomorphic mapping between bounded domains in ${{\mathbf{C}}^n}$. We shall prove that under certain circumstances $f$ must be a rational mapping, i.e., that the $ n$ component functions $ {f_i}$ of $f$ are rational functions.