On a relation between $\widetilde {\rm SL}\sb{2}$ cusp forms and cusp forms on tube domains associated to orthogonal groups
S.
Rallis;
G.
Schiffmann
1-58
Abstract: We use the decomposition of the discrete spectrum of the Weil representation of the dual reductive pair $({\tilde{SL}_2},\;O(Q))$ to construct a generalized Shimura correspondence between automorphic forms on $ O(Q)$ and $\widetilde{S{L_2}}$. We prove a generalized Zagier identity which gives the relation between Fourier coefficients of modular forms on $\widetilde{S{L_2}}$ and $O(Q)$. We give an explicit form of the lifting between $ \widetilde{S{L_2}}$ and $ O(n,2)$ in terms of Dirichlet series associated to modular forms. For the special case $n = 3$, we construct certain Euler products associated to the lifting between $S{L_2}$ and ${\text{S}}{{\text{p}}_2} \cong O(3,2)$ (locally).
Families of real and symmetric analytic functions
Yusuf
Abu-Muhanna;
Thomas H.
MacGregor
59-74
Abstract: We introduce families of functions analytic in the unit disk and having rotational symmetries. The families include the $ k$-fold symmetric univalent functions which have real coefficients. We relate the families to special classes of functions with a positive real part and then determine their extreme points. The case $k = 2$ corresponds to the odd functions which "preserve quadrants" and the extreme points of this set are characterized by having a radial limit which is real or imaginary almost everywhere. We also find estimates on the initial coefficients of functions in the families.
On the prevalence of horseshoes
Lai Sang
Young
75-88
Abstract: In this paper the symbolic dynamics of several differentiable systems are investigated. It is shown that many well-known dynamical systems, including Axiom $ {\text{A}}$ systems, piecewise monotonic maps of the interval, the Lorenz attractor and Abraham-Smale examples, have inside them subsystems conjugate to subshifts of finite type. These subsystems have hyperbolic structures and hence are stable. They can also be chosen to have entropy arbitrarily close to that of the ambient system.
Nonstandard analysis and lattice statistical mechanics: a variational principle
A. E.
Hurd
89-110
Abstract: Using nonstandard methods we construct a configuration space appropriate for the statistical mechanics of lattice systems with infinitely many particles and infinite volumes. Nonstandard representations of generalized equilibrium measures are obtained, yielding as a consequence a simple proof of the existence of standard equilibrium measures. As another application we establish an extension for generalized equilibrium measures of the basic variational principle of Landord and Ruelle. The same methods are applicable to continuous systems, and will be presented in a later paper.
Stationary logic and ordinals
D. G.
Seese
111-124
Abstract: The $L({\mathbf{aa}})$-theory of ordinals is investigated. It is proved that this theory is decidable and that each ordinal is finitely determinate.
Subnormal operators, Toeplitz operators and spectral inclusion
Gerard E.
Keough
125-135
Abstract: Let $S$ be a subnormal operator on the Hilbert space $H$, and let $N = \int z \;dE(z)$ be its minimal normal extension on $K$. Let $\mu$ be a scalar spectral measure for $ N$. If $f \in {L^\infty }(\mu )$, define ${T_f} = Pf(N){\vert _H}:\;H \to H$, where $ P:K \to H$ denotes orthogonal projection. $S$ has the $ {C^ \ast }$-Spectral Inclusion Property ( $ {C^ \ast }$-SIP) if $ \sigma (f(N)) \subseteq \sigma ({T_f})$, for all $f \in C(\sigma (N))$, and $S$ has the ${W^\ast}$-Spectral Inclusion Property ($ {W^\ast}$-SIP) if $ \sigma (f(N)) \subseteq \sigma ({T_f})$, for all $f \in {L^\infty }(\mu )$. It is shown that $ S$ has the ${C^\ast}$-SIP if and only if $\sigma (N) = \Pi (S)$, the approximate point spectrum of $S$. This is equivalent to requiring that $E(\Delta )K$ have angle 0 with $H$, for all nonempty, relatively open $ \Delta \subseteq \sigma (N)$. $S$ has the ${W^\ast}$-SIP if this angle condition holds for all proper Borel subsets of $ \sigma (N)$. If $ S$ is pure and has the $ {C^\ast}$ or $ {W^\ast}$-SIP, then it is shown that $\sigma (f(N)) \subseteq {\sigma _e}({T_f})$, for all appropriate $f$.
A partition theorem for the infinite subtrees of a tree
Keith R.
Milliken
137-148
Abstract: We prove a generalization for infinite trees of Silver's partition theorem. This theorem implies a version for trees of the Nash-Williams partition theorem.
Multivariate rearrangements and Banach function spaces with mixed norms
A. P.
Blozinski
149-167
Abstract: Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the $ L(P,Q;\ast)$ spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz $L(p,g)$ spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the $ L(P,Q;\ast)$ spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Young's inequality) Theorem.
Fine convergence and admissible convergence for symmetric spaces of rank one
Adam
Korányi;
J. C.
Taylor
169-181
Abstract: The connections between fine convergence in the sense of potential theory and admissible convergence to the boundary for quotients of eigenfunctions of the Laplace-Beltrami operator are investigated. This leads to a version of the local Fatou theorem on symmetric spaces of rank one which is considerably stronger than previously known results. The appendix establishes the relationship between harmonic measures on the intersection of the Martin boundaries of a domain and a subdomain.
On the Picard group of a continuous trace $C\sp{\ast} $-algebra
Iain
Raeburn
183-205
Abstract: Let $A$ be a continuous trace $ {C^\ast}$-algebra with paracompact spectrum $T$, and let $C(T)$ be the algebra of bounded continuous functions on $T$, so that $C(T)$ acts on $A$ in a natural way. An $A - A$ bimodule $X$ is an $ A{ - _{C(T)}}A$ imprimitivity bimodule if it is an $A - A$ imprimitivity bimodule in the sense of Rieffel and the induced actions of $C(T)$ on the left and right of $X$ agree. We denote by $ {\text{Pi}}{{\text{c}}_{C(T)}}A$ the group of isomorphism classes of $A{ - _{C(T)}}A$ imprimitivity bimodules under ${ \otimes _A}$. Our main theorem asserts that $ {\text{Pi}}{{\text{c}}_{C(T)}}A \cong {\text{Pi}}{{\text{c}}_{C(T)}}{C_0}(T)$. This result is well known to algebraists if $A$ is an $n$-homogeneous ${C^\ast}$-algebra with identity, and if $ A$ is separable it can be deduced from two recent descriptions of the automorphism group $ {\text{Au}}{{\text{t}}_{C(T)}}A$ due to Brown, Green and Rieffel on the one hand and Phillips and Raeburn on the other. Our main motivation was to provide a direct link between these two characterisations of $ {\text{Au}}{{\text{t}}_{C(T)}}A$.
Holomorphic actions of ${\rm Sp}(n,\,{\bf R})$ with noncompact isotropy groups
Hugo
Rossi
207-230
Abstract: $U(p,q)$ is a subgroup of ${S_p}(n,R)$, for $p + q = n$. ${B_q} = {S_p}(n,r)/U(p,q)$ is realized as an open subset of the manifold of Lagrangian subspaces of ${{\mathbf{C}}^n} \times {{\mathbf{C}}^n}$. It is shown that ${B_q}$ carries a $(pq)$-pseudoconvex exhaustion function. ${B_{pq}} = {S_p}(n,r)/U(p) \times U(q)$ carries two distinct holomorphic structures making the projection to ${B_q}$, ${B_0}$ holomorphic respectively. The geometry of the correspondence between ${B_q}$ and ${B_0}$ via ${B_{pq}}$ is investigated.
Robinson's consistency theorem in soft model theory
Daniele
Mundici
231-241
Abstract: In a soft model-theoretical context, we investigate the properties of logics satisfying the Robinson consistency theorem; the latter is for many purposes the same as the Craig interpolation theorem together with compactness. Applications are given to H. Friedman's third and fourth problem.
Plane models for Riemann surfaces admitting certain half-canonical linear series. II
Robert D. M.
Accola
243-259
Abstract: For $r \geqslant 2$, closed Riemann surfaces of genus $ 3r + 2$ admitting two simple half-canonical linear series $g_{3r + 1}^r,h_{3r + 1}^r$ are characterized by the existence of certain plane models of degree $ 2r + 3$ where the linear series are apparent. The plane curves have $r - 2$ $3$-fold singularities, one $ (2r - 1)$-fold singularity $ Q$, and two other double points (typically tacnodes) whose tangents pass through $ Q$. The lines through $ Q$ cut out a $ g_4^1$ which is unique. The case where the $g_4^1$ is the set of orbits of a noncyclic group of automorphisms of order four is characterized by the existence of $3r + 3$ pairs of half-canonical linear series of dimension $r - 1$, where the sum of the two linear series in any pair is linearly equivalent to $g_{3r + 1}^r + h_{3r + 1}^r$.
Monotone decompositions of $\theta \sb{n}$-continua
E. E.
Grace;
Eldon J.
Vought
261-270
Abstract: We prove the following theorem for a compact, metric ${\theta _n}$-continuum (i.e., a compact, connected, metric space that is not separated into more than $ n$ components by any subcontinuum). The continuum $X$ admits a monotone, upper semicontinuous decomposition $ \mathfrak{D}$ such that the elements of $ \mathfrak{D}$ have void interiors and the quotient space $X/\mathfrak{D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $H$ of $X$, the continuum $ T(H) = \{ x\vert$ if $ K$ is a subcontinuum of $ X$ and $x \in {K^ \circ }$, then $K \cap H \ne \emptyset \} $ is nowhere dense. The elements of the decomposition are characterized in terms of the set function $T$. An example is given showing that the condition that requires $T(x)$ to have void interior for all $ x \in X$ is not strong enough to guarantee the decomposition.