Orbital parameters for induced and restricted representations
Ronald L.
Lipsman
433-473
Abstract: General formulas for the spectral decomposition of both induced and restricted representations are laid out for the case of connected Lie groups $H \subset G$. The formulas--which detail the actual spectrum, the multiplicits, and the spectral measure--are in terms of the usual parameters in the so-called orbit method. A proof of these formulas is given in the nilpotent situation. The proof is much simpler than a previously obtained proof using nilpotent algebraic geometry. It is also capable of generalization to nonnilpotent groups. With that in mind, many new examples are presented for semisimple and symmetric homogeneous spaces. Also, a start is made in the case of exponential solvable homogeneous spaces with the treatment of both normal and conormal subgroups.
Periodic orbits of maps of $Y$
Lluís
Alsedà;
Jaume
Llibre;
Michał
Misiurewicz
475-538
Abstract: We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of $Y = \{ z \in {\mathbf{C}}:{z^3} \in [0,1]\}$ into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into itself.
Remarks on classical invariant theory
Roger
Howe
539-570
Abstract: A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.
Projections onto translation-invariant subspaces of $H\sp 1({\bf R})$
Dale E.
Alspach;
David C.
Ullrich
571-588
Abstract: Recently I. Klemes has characterized the complemented translation-invariant subspaces of $ {H^1}(\mathbb{T})$. In this paper we investigate the case of ${H^1}(\mathbb{R})$. The main results are that the hull of a complemented translation-invariant subspace is $ \varepsilon$-separated for some $ \varepsilon > 0$, and that an $ \varepsilon$-separated subset of $ {\mathbb{R}^ + }$ which is in the ring generated by cosets of closed subgroups of $\mathbb{R}$ (intersected with ${\mathbb{R}^ + }$) and lacunary sequences is the hull of a complemented ideal.
Accessory parameters for punctured spheres
Irwin
Kra
589-617
Abstract: This paper contains some qualitative results about the accessory parameters for punctured spheres with signature. We show that the Fuchsian uniformizing connection, and hence also the accessory parameters, for the surface depends real analytically on moduli. We also show that the important invariants of a uniformization of a punctured sphere such as the accessory parameters, Fuchsian groups, Poincaré metrics, and covering maps vary continuously under degenerations such as coalescing of punctures.
A calculus approach to hyperfunctions. II
Tadato
Matsuzawa
619-654
Abstract: We consider any hyperfunctions with the compact support as initial values of the solutions of the heat equation. The main aim of this paper is to unify the theory of distributions and hyperfunctions as well as simplify proofs of some important results via heat kernel.
Band-limited functions: $L\sp p$-convergence
Juan A.
Barceló;
Antonio
Córdoba
655-669
Abstract: We consider the set ${B_p}(\Omega)$ (functions of ${L^p}({\mathbf{R}})$ whose Fourier spectrum lies in $[ - \Omega , + \Omega ]$). We prove that the prolate spheroidal wave functions constitute a basis of this space if and only if $4/3 < p < 4$. The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hilbert transform.
Small zeros of quadratic forms over number fields. II
Jeffrey D.
Vaaler
671-686
Abstract: Let $F$ be a nontrivial quadratic form in $ N$ variables with coefficients in a number field $k$ and let $ \mathcal{Z}$ be a subspace of ${k^N}$ of dimension $M,1 \leq M \leq N$. If $F$ restricted to $ \mathcal{Z}$ vanishes on a subspace of dimension $ L,1 \leq L < M$, and if the rank of $F$ restricted to $ \mathcal{Z}$ is greater than $M - L$, then we show that $F$ must vanish on $M - L + 1$ distinct subspaces $ {\mathcal{X}_0},{\mathcal{X}_1}, \ldots ,{\mathcal{X}_{M - L}}$ in $\mathcal{Z}$ each of which has dimension $ L$. Moreover, we show that for each pair $ {\mathcal{X}_0},{\mathcal{X}_1},1 \leq l \leq M - L$, the product of their heights $ H({\mathcal{X}_0})H({\mathcal{X}_1})$ is relatively small. Our results generalize recent work of Schlickewei and Schmidt.
A characterization of nonchaotic continuous maps of the interval stable under small perturbations
D.
Preiss;
J.
Smítal
687-696
Abstract: Recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles (sometimes this is possible even in the case when the trajectory is not asymptotically periodic) or is $ \varepsilon$-chaotic for some $\varepsilon > 0$. In certain cases, the first property is stable under small perturbations. This means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small. In other words, there are nonchaotic maps without "chaos explosions". In the paper we give a characterization of these maps along with some consequences. Using the main result it is possible to prove that generically the nonchaotic maps are stable.
Optimal stopping of two-parameter processes on nonstandard probability spaces
Robert C.
Dalang
697-719
Abstract: We prove the existence of optimal stopping points for upper semicontinuous two-parameter processes defined on filtered nonstandard (Loeb) probability spaces that satisfy a classical conditional independence hypothesis. The proof is obtained via a lifting theorem for elements of the convex set of randomized stopping points, which shows in particular that extremal elements of this set are ordinary stopping points.
Lie groups that are closed at infinity
Harry F.
Hoke
721-735
Abstract: A noncompact Riemannian manifold $M$ is said to be closed at infinity if no bounded volume form which is also bounded away from zero can be written as the exterior derivative of a bounded form on $M$ . The isoperimetric constant of $M$ is defined by $ h(M) = \inf \{ {\text{vol}}(\partial S)/{\text{vol}}(S)\}$ where $S$ ranges over compact domains with boundary in $ M$. It is shown that a Lie group $G$ with left invariant metric is closed at infinity if and only if $h(G) = 0$ if and only if $G$ is amenable and unimodular. This result relates these geometric invariants of $G$ to the algebraic structure of $ G$ since the conditions amenable and unimodular have algebraic characterizations for Lie groups. $G$ is amenable if and only if $G$ is a compact extension of a solvable group and $G$ is unimodular if and only if $ \operatorname{Tr}({\text{ad}}\,X) = 0$ for all $X$ in the Lie algebra of $G$. An application is the clarification of relationships between several conditions for the existence of transversal invariant measures for a foliation of a compact manifold by the orbits of a Lie group action.
Characterizations of normal quintic $K$-$3$ surfaces
Jin Gen
Yang
737-751
Abstract: If a normal quintic surface is birational to a $K$-$3$ surface then it must contain from one to three triple points as its only essential singularities. A $ K$-$3$ surface is the minimal model of a normal quintic surface with only one triple point if and only if it contains a nonsingular curve of genus two and a nonsingular rational curve crossing each other transversally. The minimal models of normal quintic $K$-$3$ surfaces with several triple points can also be characterized by the existence of some special divisors.
Extending homeomorphisms and applications to metric linear spaces without completeness
Tadeusz
Dobrowolski
753-784
Abstract: A method of extending homeomorphisms between compacta metric spaces is presented. The main application is that homeomorphisms between compacta of an infinite-dimensional locally convex metric linear space extend to the whole space. A lemma used in the proof of this fact together with the method of absorbing sets is employed to show that every $\sigma$ -compact normed linear space is homeomorphic to a dense linear subspace of a Hilbert space. A discussion of the relative topological equivalence of absorbing sets in noncomplete spaces is included. The paper is concluded with some controlled versions of an isotopy extension theorem.
Conjugation and the prime decomposition of knots in closed, oriented $3$-manifolds
Katura
Miyazaki
785-804
Abstract: In this paper we consider the prime decomposition of knots in closed, oriented $3$-manifolds. (For classical knots one can easily prove the uniqueness of prime decomposition by using a standard innermost disk argument.) We define a new relation, conjugation, between oriented knots in closed, oriented $3$-manifolds and prove the following results. (1) The prime decomposition is, roughly speaking, uniquely determined up to conjugation, (2) there is a prime knot $\mathcal{R}$ in $ {S^1} \times {S^2}$ such that $\mathcal{R}\char93 {\mathcal{K}_1} = \mathcal{R}\char93 {\mathcal{K}_2}$ if ${\mathcal{K}_1}$ is a conjugation of ${\mathcal{K}_2}$, and (3) if a knot $\mathcal{K}$ has a prime decomposition which does not contain $ \mathcal{R}$, then it is the unique prime decomposition of $\mathcal{K}$ .
Stable processes with drift on the line
Sidney C.
Port
805-841
Abstract: The stable processes on the line having a drift are investigated. Except for the symmetric Cauchy processes with drift these are all transient and points are nonpolar sets. Explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets.
Cyclic extensions of $K(\sqrt{-1})/K$
Jón Kr.
Arason;
Burton
Fein;
Murray
Schacher;
Jack
Sonn
843-851
Abstract: In this paper the height $ {\text{ht}}(L/K)$ of a cyclic $2$-extension of a field $K$ of characteristic $\ne 2$ is studied. Here ${\text{ht}}(L/K) \geq n$ means that there is a cyclic extension $E$ of $ K,E \supset L$, with $[E:L] = {2^n}$. Necessary and sufficient conditions are given for ${\text{ht}}(L/K) \geq n$ provided $K(\sqrt { - 1})$ contains a primitive $ {2^n}$th root of unity. Primary emphasis is placed on the case $L = K(\sqrt { - 1})$. Suppose $ {\text{ht}}(K(\sqrt { - 1})/K) \geq 1$. It is shown that $ {\text{ht}}(K(\sqrt { - 1})/K) \geq 2$ and if $K$ is a number field then $ {\text{ht}}(K(\sqrt { - 1})/K) \geq n$ for all $n$. For each $n \geq 2$ an example is given of a field $ K$ such that $ {\text{ht}}(K(\sqrt { - 1})/K) \geq n$ but ${\text{ht}}(K(\sqrt { - 1})/K) \ngeq n + 1$.
On the Hopf index and the Conley index
Christopher K.
McCord
853-860
Abstract: The following generalization of the Poincaré-Hopf index theorem is proved: If $S$ is an isolated invariant set of a flow on a manifold $M$, then the sum of the Hopf indices on $ S$ is equal (up to a sign) to the Euler characteristic of the homology Conley index of $S$.