Bundle theories for topological manifolds
C. B.
Hughes;
L. R.
Taylor;
E. B.
Williams
1-65
Abstract: Manifold approximate fibrations arise in the geometric topology of manifolds and group actions on topological manifolds. The primary purpose of this paper is to classify manifold approximate fibrations in terms of the lifting problem for a certain bundle. Our classification meshes well with the classical classifications of fibrations and bundles and, hence, we are able to attack questions such as the following. When is a fibration controlled homotopy equivalent to a manifold approximate fibration? When is a manifold approximate fibration controlled homeomorphic to a bundle?
H\"older domains and Poincar\'e domains
Wayne
Smith;
David A.
Stegenga
67-100
Abstract: A domain $D \subset {R^d}$ of finite volume is said to be a $p$-Poincaré domain if there is a constant $ {M_p}(D)$ so that $\displaystyle {\int\limits_D {\vert u - {u_D}\vert} ^p}dx \leq M_p^p(D){\int\limits_D {\vert\nabla u\vert} ^p}dx$ for all functions $u \in {C^1}(D)$. Here ${u_D}$ denotes the mean value of $u$ over $D$. Techniques involving the quasi-hyperbolic metric on $D$ are used to establish that various geometric conditions on $D$ are sufficient for $D$ to be a $p$-Poincaré domain. Domains considered include starshaped domains, generalizations of John domains and Hàlder domains. $D$ is a Hàlder domain provided that the quasi-hyperbolic distance from a fixed point ${x_0} \in D$ to $x$ is bounded by a constant multiple of the logarithm of the euclidean distance of $x$ to the boundary of $D$. The terminology is derived from the fact that in the plane, a simply connected Hàlder domain has a Hàlder continuous Riemann mapping function from the unit disk onto $D$. We prove that if $D$ is a Hàlder domain and $ p \ge d$, then $ D$ is a $ p$-Poincaré domain. This answers a question of Axler and Shields regarding the image of the unit disk under a Hàlder continuous conformal mapping. We also consider geometric conditions which imply that the imbedding of the Sobolev space $ {W^{1,p}}(D) \to {L^p}(D)$ is compact, and prove that this is the case for a Hàlder domain $D$.
Upper bounds for ergodic sums of infinite measure preserving transformations
Jon
Aaronson;
Manfred
Denker
101-138
Abstract: For certain conservative, ergodic, infinite measure preserving transformations $T$ we identify increasing functions $A$, for which $\displaystyle \limsup \limits_{n \to \infty } \frac{1} {{A(n)}}\sum\limits_{k = 1}^n {f \circ } {T^k} = \int_X {fd\mu } \quad {\text{a}}{\text{.e}}{\text{.}}$ holds for any nonnegative integrable function $f$. In particular the results apply to some Markov shifts and number-theoretic transformations, and include the other law of the iterated logarithm.
Complete localization of domains with noncompact automorphism groups
Kang-Tae
Kim
139-153
Abstract: We prove a characterization of the domains in ${{\mathbf{C}}^n}$ with an automorphism orbit accumulating at a boundary point at which the boundary is real analytic and convex up to a biholomorphic change of local coordinates. This result generalizes the well-known Wong-Rosay theorem on strongly pseudoconvex domains to the case of locally convex domains with real analytic boundaries.
A complete classification of the piecewise monotone functions on the interval
Stewart
Baldwin
155-178
Abstract: We define two functions $f$ and $g$ on the unit interval $[0,1]$ to be strongly conjugate iff there is an order-preserving homeomorphism $h$ of $[0,1]$ such that $g = {h^{ - 1}}fh$ (a minor variation of the more common term "conjugate", in which $ h$ need not be order-preserving). We provide a complete set of invariants for each continuous (strictly) piecewise monotone function such that two such functions have the same invariants if and only if they are strongly conjugate, thus providing a complete classification of all such strong conjugacy classes. In addition, we provide a criterion which decides whether or not a potential invariant is actually realized by some piecewise monotone continuous function.
Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in ${\bf R}\sp 2$
Shigeru
Sakaguchi
179-190
Abstract: We consider some two-dimensional semilinear elliptic boundary value problems over a bounded convex domain in ${{\mathbf{R}}^2}$ and show the uniqueness of the critical point of the solutions.
Power series space representations of nuclear Fr\'echet spaces
Dietmar
Vogt
191-208
Abstract: Let $E$ be a nuclear graded Fréchet space such that the norms satisfy inequalities $\vert\vert\vert\vert _k^2 \leq {C_k}\vert\vert\vert{\vert _{k - 1}}\vert\vert\vert{\vert _{k - 1}}$ for all $k$, let $F$ be a graded Fréchet space such that the dual (extended real valued) norms satisfy inequalities $\vert\vert\vert\vert _k^{*2} \leq {D_k}\vert\vert\vert\vert _{k - 1}^*\vert\vert\vert\vert _{k + 1}^*$ for all $k$, and let $A$ be a tame (resp. linearly tame) linear map from $F$ to $E$. Then there exists a tame (resp. linearly tame) factorization of $A$ through a power series space $\Lambda _\infty ^2(\alpha )$. In the case of a tame quotient map, $E$ is tamely equivalent to a power series space of infinite type. This applies in particular to the range of a tame (resp. linearly tame) projection in a power series space $\Lambda _\infty ^2(\alpha )$. In this case one does not need nuclearity. It also applies to the tame spaces in the sense of the various implicit function theorems. If they are nuclear, they are tamely equivalent to power series spaces ${\Lambda _\infty }(\alpha )$.
Complete coinductive theories. I
A. H.
Lachlan
209-241
Abstract: Let $T$ be a complete theory over a relational language which has an axiomatization by $\exists \forall $-sentences. The properties of models of $T$ are studied. It is shown that quantifier-free formulas are stable. This limited stability is used to show that in $ \exists \forall$-saturated models the elementary types of tuples are determined by their $\exists$-types and algebraicity is determined by existential formulas. As an application, under the additional assumption that no quantifier-free formula has the FCP, the models $ \mathcal{M}$ of $ T$ are completely characterized in terms of certain 0-definable equivalence relations on cartesian powers of $M$. This characterization yields a result similar to that of Schmerl for the case in which $T$ is ${\aleph _0}$-categorical.
A Plancherel formula for parabolic subgroups
Mie
Nakata
243-256
Abstract: We obtain explicit Plancherel formulas for the parabolic subgroups $ P$ of $p$-adic unitary groups which fix one dimensional isotropic subspaces. By means of certain limits of difference operators (called strong derivatives), we construct a Dixmier-Pukanszky operator which compensates for the nonunimodularity of the group $P$. Then, we compute the Plancherel formula of $N \cdot A$, where $ N$ is the nilradical of $ P$ and
Intrinsic formality and certain types of algebras
Gregory
Lupton
257-283
Abstract: In this paper, a type of algebra is introduced and studied from a rational homotopy point of view, using differential graded Lie algebras. The main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces. That is, in the language of rational homotopy, whether or not such an algebra is intrinsically formal. Examples are given which show that, in general, this is not so--7.8 and 7.9. However, whilst it is true that not all such algebras are intrinsically formal, some of them are. The main results of this paper show a certain class of these algebras to be intrinsically formal--Theorem $2$ (6.1); and a second, different type of algebra also to be intrinsically formal--Theorem $ 1$ (5.2), which type of algebra overlaps with the first type in many examples of interest. Examples are given in $\S7$.
Almost split sequences and Zariski differentials
Alex
Martsinkovsky
285-307
Abstract: Let $R$ be a complete two-dimensional integrally closed analytic $k$-algebra. Associated with $R$ is the Auslander module $ A$ from the fundamental sequence $0 \to {\omega _R} \to A \to R \to k \to 0$ and the module of Zariski differentials ${D_k}{(R)^{ * * }}$. We conjecture that these modules are isomorphic if and only if $R$ is graded. We prove this conjecture for (a) hypersurfaces $f = X_3^n + {\text{g}}({X_1},{X_2})$, (b) quotient singularities, and (c) $R$ graded Gorenstein.
Isomorphism universal varieties of Heyting algebras
M. E.
Adams;
V.
Koubek;
J.
Sichler
309-328
Abstract: A variety $\mathbf{V}$ is group universal if every group $ G$ is isomorphic to the automorphism group ${\operatorname{Aut}}(A)$ of an algebra $A \in \mathbf{V}$; if, in addition, all finite groups are thus representable by finite algebras from $\mathbf{V}$, the variety $ \mathbf{V}$ is said to be finitely group universal. We show that finitely group universal varieties of Heyting algebras are exactly the varieties which are not generated by chains, and that a chain-generated variety $\mathbf{V}$ is group universal just when it contains a four-element chain. Furthermore, we show that a variety $ \mathbf{V}$ of Heyting algebras is group universal whenever the cyclic group of order three occurs as the automorphism group of some $A \in \mathbf{V}$. The results are sharp in the sense that, for every group universal variety and for every group $G$, there is a proper class of pairwise nonisomorphic Heyting algebras $ A \in \mathbf{V}$ for which ${\operatorname{Aut}}(A) \cong G$.
Surfaces of $E\sp 4$ satisfying certain restrictions on their normal bundle
Th.
Hasanis;
D.
Koutroufiotis;
P.
Pamfilos
329-347
Abstract: We consider smooth surfaces in ${E^4}$ whose normal bundles satisfy certain geometric conditions that entail the vanishing of the normal curvature, and prove that their Gauss curvatures cannot be bounded from above by a negative number. We also give some results towards a classification of flat surfaces with flat normal bundle in ${E^4}$.
A notion of rank for unitary representations of general linear groups
Roberto
Scaramuzzi
349-379
Abstract: A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the unipotent radical of a maximal parabolic subgroup. Irreducible representations of small rank are classified. It is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations.
Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems
N.
Dyn;
A.
Ron
381-403
Abstract: Local approximation order to smooth complex valued functions by a finite dimensional space $ \mathcal{H}$, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space $\mathcal{H}$ is based here on the identification of its dual with a certain space $\mathcal{P}$ of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from $\mathcal{P}$, with interpolation data characterized by the structure of $ \mathcal{H}$, and to construct bases of $ \mathcal{P}$ corresponding to the interpolation problem.
Curvatures and similarity of operators with holomorphic eigenvectors
Mitsuru
Uchiyama
405-415
Abstract: The curvature of the holomorphic vector bundle generated by eigenvectors of operators is estimated, and the necessary and sufficient conditions for contractions to be similar or quasi-similar with unilateral shifts are given.