Unitary representations of Lie groups with cocompact radical and applications
L.
Pukanszky
1-49
Abstract: The paper gives a necessary and sufficient condition in order that a connected and simply connnected Lie group with cocompact radical be of type I. This result is then applied to a characterization of Lie groups, all irreducible unitary representations of which are completely continuous.
$L\sb{\infty }{}\sb{\lambda }$-equivalence, isomorphism and potential isomorphism
Mark
Nadel;
Jonathan
Stavi
51-74
Abstract: It is well known that two structures are ${L_{\infty \omega }}$-equivalent iff they are potentially isomorphic [that is, isomorphic in some (Cohen) extension of the universe]. We prove that no characterization of ${L_{\infty \lambda }}$-equivalence along these lines is possible (at least for successor cardinals $\lambda$) and the potential-isomorphism relation that naturally comes to mind in connection with ${L_{\infty \lambda }}$ is often not even transitive and never characterizes ${ \equiv _{\infty \lambda }}$ for $\lambda > \omega$. A major part of the work is the construction of ${\kappa ^ + }$-like linear orderings (also Boolean algebras) A, B such that ${N_{{\kappa ^ + }}}({\mathbf{A}},{\mathbf{B}})$, where ${N_\lambda }({\mathbf{A}},{\mathbf{B}})$ means: A and B are nonisomorphic ${L_{\infty \lambda }}$-equivalent structures of cardinality $\lambda$.
Fr\'echet differentiable functionals and support points for families of analytic functions
Paul
Cochrane;
Thomas H.
MacGregor
75-92
Abstract: Given a closed subset of the family $ {S^\ast}(\alpha )$ of functions starlike of order $\alpha$ of a particular form, a continuous Fréchet differentiable functional, J, is constructed with this collection as the solution set to the extremal problem $\max \operatorname{Re} J(f)$ over ${S^\ast}(\alpha )$. Similar results are proved for families which can be put into one-to-one correspondence with $ {S^\ast}(\alpha )$. The support points of $ {S^\ast}(\alpha )$ and $K(\alpha )$, the functions convex of order $ \alpha$, are completely characterized and shown to coincide with the extreme points of their respective convex hulls. Given any finite collection of support points of ${S^\ast}(\alpha )$ (or $ K(\alpha )$), a continuous linear functional, J, is constructed with this collection as the solution set to the extremal problem $ \max \operatorname{Re} J(f)$ over $ {S^\ast}(\alpha )$ (or $K(\alpha )$).
The metabelian $p$-groups of maximal class
R. J.
Miech
93-119
Abstract: This paper presents a solution to the isomorphism problem for the set of metabelian p-groups of maximal class.
Invariant measures and equilibrium states for some mappings which expand distances
Peter
Walters
121-153
Abstract: For a certain collection of transformations T we define a Perron-Frobenius operator and prove a convergence theorem for the powers of the operator along the lines of the theorem D. Ruelle proved in his investigation of the equilibrium states of one-dimensional lattice systems. We use the convergence theorem to study the existence and ergodic properties of equilibrium states for T and also to study the problem of invariant measures for T. Examples of the transformations T considered are expanding maps, transformations arising from f-expansions and shift systems.
Global structural stability of a saddle node bifurcation
Clark
Robinson
155-171
Abstract: S. Newhouse, J. Palis, and F. Takens have recently proved the global structural stability of a one parameter unfolding of a saddle node when the nonwandering set is finite and transversality conditions are satisfied. (The diffeomorphism is Morse-Smale except for the saddle node.) Using their local unfolding of a saddle node and our method of compatible families of unstable disks (instead of the more restrictive method of compatible systems of unstable tubular families), we are able to extend one of their results to the case where the nonwandering set is infinite. We assume that a saddle node is introduced away from the rest of the nonwandering set which is hyperbolic (Axiom A), and that a (strong) transversality condition is satisfied.
A nonlinear semigroup for a functional differential equation
Dennis W.
Brewer
173-191
Abstract: A representation theorem is obtained for solutions of the nonlinear functional differential equation
Nearnesses, proximities, and $T\sb{1}$-compactifications
Ellen E.
Reed
193-207
Abstract: Gagrat, Naimpally, and Thron together have shown that separated Lodato proximities yield ${T_1}$-compactifications, and conversely. This correspondence is not $1 - 1$, since nonequivalent compactifications can induce the same proximity. Herrlich has shown that if the concept of proximity is replaced by that of nearness then all principal (or strict) ${T_1}$-extensions can be accounted for. (In general there are many nearnesses compatible with a given proximity.) In this paper we obtain a 1-1 correspondence between principal ${T_1}$-extensions and cluster-generated nearnesses. This specializes to a 1-1 match between principal $ {T_1}$-compactifications and contigual nearnesses. These results are utilized to obtain a 1-1 correspondence between Lodato proximities and a subclass of $ {T_1}$-compactifications. Each proximity has a largest compatible nearness, which is contigual. The extension induced by this nearness is the construction of Gagrat and Naimpally and is characterized by the property that the dual of each clan converges. Hence we obtain a 1-1 match between Lodato proximities and clan-complete principal ${T_1}$-compactifications. When restricted to EF-proximities, this correspondence yields the usual map between ${T_2}$-compactifications and EF-proximities.
The Riemann hypothesis for Selberg's zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator
Burton
Randol
209-223
Abstract: Much of that part of the theory of the Riemann zeta-function based on the Riemann hypothesis carries over to zeta-functions of Selberg's type, and in this way one can get asymptotic information about various eigenvalue problems. The methods are illustrated in the case of a compact Riemann surface.
Relativized weak disjointness and relatively invariant measures
Douglas C.
McMahon
225-237
Abstract: In this paper we study the relativized weak disjointness and the relativized regionally proximal relation for homomorphisms of point-transitive transformation groups, under the assumption of a relativized invariant measure. We also include a proof of a Folner-type result for syndetic subsets of an amenable group.
Leaf prescriptions for closed $3$-manifolds
John
Cantwell;
Lawrence
Conlon
239-261
Abstract: Our basic question is: What open, orientable surfaces of finite type occur as leaves with polynomial growth in what closed 3-manifolds? This question is motivated by other work of the authors. It is proven that every such surface so occurs for suitable $ {C^\infty }$ foliations of suitable closed 3-manifolds and for suitable $ {C^1}$ foliations of all closed 3-manifolds. If the surface has no isolated nonplanar ends it also occurs for suitable ${C^\infty }$ foliations of all closed 3-manifolds. Finally, a large class of surfaces with isolated nonplanar ends occurs in suitable $ {C^\infty }$ foliations of all closed, orientable 3-manifolds that are not rational homology spheres.
Approximation theorems for uniformly continuous functions
Anthony W.
Hager
263-273
Abstract: Let X be a set, A a family of real-valued functions on X which contains the constants, ${\mu _A}$ the weak uniformity generated by A, and $ U({\mu _A}X)$ the collection of uniformly continuous functions to the real line R. The problem is how to construct $U({\mu _A}X)$ from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely A-equiuniform, order-one subsets of ${A^ + }$ is uniformly dense in $U({\mu _A}X)$. Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly locally-A (resp., uniformly locally piecewise-A) is uniformly dense in $ U({\mu _A}X)$. Further, for any A, a finitely A-uniform function is just a composition $F \circ ({a_1}, \ldots ,{a_p})$ for some $ {a_1}, \ldots ,{a_p} \in A$ and F uniformly continuous on the range of $({a_1}, \ldots ,{a_p})$ in $ {R^p}$. Thus, such compositions are dense in $ U({\mu _A}X)$. For $BU({\mu _A}X)$, the compositions with $F \in BU({R^p})$ are dense (B denoting bounded functions). So, in a sense, to know $U({\mu _A}X)$ it suffices to know A and subspaces of the spaces ${R^p}$, and to know $ BU({\mu _A}X)$, A and the spaces ${R^p}$ suffice.
The homological dimensions of symmetric algebras
James E.
Carrig
275-285
Abstract: Let D be a Dedekind domain and M a rank-one torsion-free D-module. An analysis of $A = {S_D}(M)$, the symmetric algebra of M, yields the following information: Theorem. (1) Tor-dim $A \leqslant 2\;and\; = 1\;iff\;M = K$, the quotient field of D; (2) A is coherent; (3) Global $\dim A = 2$. For higher rank modules coherence is not assured and only rough estimates of the dimensions are found. On the other hand, if $ {S_D}(M)$ is a domain of global dimension two, then M has rank one but the dimension of D may be two. If D is local of dimension two then $M = K$.
Simple Lie algebras of toral rank one
Robert Lee
Wilson
287-295
Abstract: Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic $p > 7$. Let L have Cartan decomposition $L = H + {\sum _{\gamma \in \Gamma }}{L_\gamma }$. If $\Gamma$ generates a cyclic group then L is isomorphic to $ {\text{sl}}(2,F)$ or to one of the simple Lie algebras of generalized Cartan type $ W(1:{\mathbf{n}})$ or $H{(2:{\mathbf{n}}:\Phi )^{(2)}}$.
Frattini subalgebras of finitely generated soluble Lie algebras
Ralph K.
Amayo
297-306
Abstract: This paper is motivated by a recent one of Stewart and Towers [8] investigating Lie algebras with ``good Frattini structure'' (definition below). One consequence of our investigations is to prove that any finitely generated metanilpotent Lie algebra has good Frattini structure, thereby answering a question of Stewart and Towers and providing a complete Lie theoretic analogue of the corresponding group theoretic result of Phillip Hall. It will also be shown that in prime characteristic, finitely generated nilpotent-by-finite-dimensional Lie algebras have good Frattini structure.
A global theorem for singularities of maps between oriented $2$-manifolds
J. R.
Quine
307-314
Abstract: Let M and N be smooth compact oriented connected 2-mani-folds. Suppose $f:M \to N$ is smooth and every point $p \in M$ is either a fold point, cusp point, or regular point of f i.e., f is excellent in the sense of Whitney. Let ${M^ + }$ be the closure of the set of regular points at which f preserves orientation and M the closure of the set of regular points at which f reverses orientation. Let ${p_1}, \ldots ,{p_n}$ be the cusp points and $\mu ({p_k})$ the local degree at the cusp point $ {p_k}$. We prove the following: $\displaystyle \chi (M) - 2\chi ({M^ - }) + \sum \mu ({p_k}) = (\deg f)\chi (N)$ where $ \chi$ is the Euler characteristic and deg is the topological degree. We show that it is a generalization of the Riemann-Hurwitz formula of complex analysis and give some examples.
Invariant means on the continuous bounded functions
Joseph
Rosenblatt
315-324
Abstract: Let G be a noncompact nondiscrete $\sigma$-compact locally compact metric group. A Baire category argument gives measurable sets $ \{ {A_\gamma }:\gamma \in \Gamma \}$ of finite measure with card $(\Gamma ) = c$ which are independent on the open sets. One approximates $\{ {A_\gamma }:\gamma \in \Gamma \}$ by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions $ \{ {f_\gamma }:\gamma \in \Gamma \}$ in $ {\text{CB}}(G)$ with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on ${\text{CB}}(G)$ there are ${2^c}$ mutually singular elements of LIM each of which is singular to TLIM.
Perturbation of translation invariant positivity preserving semigroups on $L\sp{2}({\bf R}\sp{N})$
Ira W.
Herbst;
Alan D.
Sloan
325-360
Abstract: The theory of singular local perturbations of translation invariant positivity preserving semigroups on $ {L^2}({{\mathbf{R}}^N},{d^N}x)$ is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups, ${e^{ - tH}}$, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of --$\Delta$.
The immersion conjecture for $RP\sp{8l+7}$ is false
Donald M.
Davis;
Mark
Mahowald
361-383
Abstract: Let $\alpha (n)$ denote the number of l's in the binary expansion of n. It is proved that if $n \equiv 7$ (8), $ \alpha (n) = 6$, and $n \ne 63$, then ${\mathbf{R}}{P^n}$ can be immersed in ${{\mathbf{R}}^{2n - 14}}$. This provides the first counterexample to the well-known conjecture that the best immersion is in ${{\mathbf{R}}^{2n - 2\alpha (n) + 1}}$ (when $\alpha (n) \equiv 1$ or $ 2 \bmod 4$). The method of proof is obstruction theory.
Spectral theory for contraction semigroups on Hilbert space
Larry
Gearhart
385-394
Abstract: In this paper we determine the relationship between the spectra of a continuous contraction semigroup on Hilbert space and properties of the resolvent of its infinitesimal generator. The methods rely heavily on dilation theory. In particular, we reduce the general problem to the case that the cogenerator of the semigroup has a characteristic function with unitary boundary values. We then complete the analysis by generalizing the scalar result of J. W. Moeller on compressions of the translation semigroup to the case of infinite multiplicity.