Indecomposable representations of semisimple Lie groups
Birgit
Speh
1-34
Abstract: Let $G$ be a semisimple connected linear Lie group, ${\pi _1}$ a finite-dimensional irreducible representation of $G$, ${\pi _2}$ an infinite-dimensional irreducible representation of $G$ which has a nontrivial extension with $ {\pi _1}$. We study the category of all Harish-Chandra modules whose composition factors are equivalent to ${\pi _1}$ and ${\pi _2}$
Approximating topological surfaces in $4$-manifolds
Gerard A.
Venema
35-45
Abstract: Let ${M^2}$ be a compact, connected $ 2$-manifold with $ \partial {M^2} \ne \emptyset$ and let $ h:{M^2} \to {W^4}$ be a topological embedding of ${M^2}$ into a $4$-manifold. The main theorem of this paper asserts that if ${W^4}$ is a piecewise linear $4$-manifold, then $h$ can be arbitrarily closely approximated by locally flat PL embeddings. It is also shown that if the $4$-dimensional annulus conjecture is correct and if $ W$ is a topological $ 4$-manifold, then $ h$ can be arbitrarily closely approximated by locally flat embeddings. These results generalize the author's previous theorems about approximating disks in $4$-space.
On connectivity in matroids and graphs
James G.
Oxley
47-58
Abstract: In this paper we derive several results for connected matroids and use these to obtain new results for -connected graphs. In particular, we generalize work of Murty and Seymour on the number of two-element cocircuits in a minimally connected matroid, and work of Dirac, Plummer and Mader on the number of vertices of degree two in a minimally $ 2$-connected graph. We also solve a problem of Murty by giving a straightforward but useful characterization of minimally connected matroids. The final part of the paper gives a matroid generalization of Dirac and Plummer's result that every minimally $2$-connected graph is $3$-colourable.
Random ergodic sequences on LCA groups
Jakob I.
Reich
59-68
Abstract: Let $ {\{ X(t,\omega )\} _{t \in {{\mathbf{R}}^ + }}}$ be a stochastic process on a locally compact abelian group $G$, which has independent stationary increments. We show that under mild restrictions on $ G$ and $\{ X(t,\omega )\}$ the random families of probability measures $\displaystyle {\mu _T}( \cdot ,\omega ) = B_T^{ - 1}\int\limits_0^T {f(t){x_{( \cdot )}}} (X(t,\omega ))dt\quad {\text{for}}\;T > 0{\text{,}}$ where $ f(t)$ is a function from ${{\mathbf{R}}^ + }$ to ${{\mathbf{R}}^ + }$ of polynomial growth and ${B_T} = \int_0^T {f(t)} \;dt$, converge weakly to Haar measure of the Bohr compactification of $ G$. As a consequence we obtain mean and individual ergodic theorems and asymptotic occupancy times for these processes.
Iteration and the solution of functional equations for functions analytic in the unit disk
Carl C.
Cowen
69-95
Abstract: This paper considers the classical functional equations of Schroeder $f \circ \varphi = \lambda f$, and Abel $f \circ \varphi = f + 1$, and related problems of fractional iteration where $ \varphi$ is an analytic mapping of the open unit disk into itself. The main theorem states that under very general conditions there is a linear fractional transformation $ \Phi$ and a function $ \sigma$ analytic in the disk such that $\Phi \circ \sigma = \sigma \circ \varphi$ and that, with suitable normalization, $\Phi$ and $\sigma$ are unique. In particular, the hypotheses are satisfied if $\varphi$ is a probability generating function that does not have a double zero at 0. This intertwining relates solutions of functional equations for $ \varphi$ to solutions of the corresponding equations for $\Phi$. For example, it follows that if $ \varphi$ has no fixed points in the open disk, then the solution space of $ f \circ \varphi = \lambda f$ is infinite dimensional for every nonzero $ \lambda$. Although the discrete semigroup of iterates of $\varphi$ usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself, we find that for each $z$ in the disk, all sufficiently large fractional iterates of $\varphi$ can be defined at $z$. This enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of $ \varphi$. If the iterates of $\varphi$ can be embedded in a continuous semigroup, we show that the semigroup must come from the corresponding semigroup for $\Phi$, and thus be real analytic in $t$. The proof of the main theorem is not based on the well known limit technique introduced by Koenigs (1884) but rather on the construction of a Riemann surface on which an extension of $\varphi$ is a bijection. Much work is devoted to relating characteristics of $ \varphi$ to the particular linear fractional transformation constructed in the theorem.
Lewy's curves and chains on real hypersurfaces
James J.
Faran
97-109
Abstract: Lewy's curves on an analytic real hypersurface $M = \{ r(z,z) = 0\}$ in ${{\mathbf{C}}^2}$ are the intersections of $ M$ with any of the Segre hypersurfaces ${Q_w} = \{ z:r(z,w) = 0\} $. If $M$ is the standard unit sphere, these curves are chains in the sense of Chern and Moser. This paper shows the converse in the strictly pseudoconvex case: If all of Lewy's curves are chains, $ M$ is locally biholomorphically equivalent to the sphere. This is proven by analyzing the holomorphic structure of the space of chains. A similar statement is true about real hypersurfaces in $ {{\mathbf{C}}^n}$, $n > 2$, in which case the proof relies on a pseudoconformal analogue to the theorem in Riemannian geometry which states that a manifold having "sufficiently many" totally geodesic submanifolds is projectively flat.
Compactness properties of an operator which imply that it is an integral operator
A. R.
Schep
111-119
Abstract: In this paper we study necessary and (or) sufficient conditions on a given operator to be an integral operator. In particular we give another proof of a characterization of integral operators due to W. Schachermayer.
Tensor products of principal series for the De Sitter group
Robert P.
Martin
121-135
Abstract: The decomposition of the tensor product of two principal series representations is determined for the simply connected double covering, $G = {\text{Spin}}(4,1)$, of the DeSitter group. The main result is that this decomposition consists of two pieces, ${T_c}$ and ${T_d}$, where ${T_c}$ is a continuous direct sum with respect to Plancherel measure on $\hat G$ of representations from the principal series only and ${T_d}$ is a discrete sum of representations from the discrete series of $G$. The multiplicities of representations occurring in ${T_c}$ and ${T_d}$ are all finite.
Genealogy of periodic points of maps of the interval
Robert L.
Devaney
137-146
Abstract: We describe the behavior of families of periodic points in one parameter families of maps of the interval which feature a transition from simple dynamics with finitely many periodic points to chaotic mappings. In particular, we give topological criteria for the appearance and disappearance of these families. Our results apply specifically to quadratic maps of the form ${F_\mu }(x) = \mu x(1 - x)$.
An approximation to $\Omega \sp{n}\Sigma \sp{n}X$
J.
Caruso;
S.
Waner
147-162
Abstract: For an arbitrary (nonconnected) based space $X$, a geometrical construction ${\tilde C_n}X$ is given, such that ${\tilde C_n}X$ is weakly homotopy-equivalent to ${\Omega ^n}{\Sigma ^n}X$ as a $ {\mathcal{C}_n}$-space.
A simpler approximation to $QX$
Jeffrey L.
Caruso
163-167
Abstract: McDuff's construction ${C^ \pm }(M)$ of a space of positive and negative particles is modified to a space ${C^ \pm }({R^\infty },X)$, which is weakly homotopy equivalent to ${\Omega ^\infty }{\Sigma ^\infty }X$, for a locally equi-connected, nondegenerately based space $ X$.
Higher derivation Galois theory of fields
Nickolas
Heerema
169-179
Abstract: A Galois correspondence for finitely generated field extensions $ k/h$ is presented in the case characteristic $h = p \ne 0$. A field extension $k/h$ is Galois if it is modular and $h$ is separably algebraically closed in $k$. Galois groups are the direct limit of groups of higher derivations having rank a power of $p$. Galois groups are characterized in terms of abelian iterative generating sets in a manner which reflects the similarity between the finite rank and infinite rank theories of Heerema and Deveney [9] and gives rise to a theory which encompasses both. Certain intermediate field theorems obtained by Deveney in the finite rank case are extended to the general theory.
Poincar\'e-Bendixson theory for leaves of codimension one
John
Cantwell;
Lawrence
Conlon
181-209
Abstract: The level of a local minimal set of a ${C^2}$ codimension-one foliation of a compact manifold is a nonnegative integer defined inductively, level zero corresponding to the minimal sets in the usual sense. Each leaf of a local minimal set at level $ k$ is at level $ k$. The authors develop a theory of local minimal sets, level, and how leaves at level $k$ asymptotically approach leaves at lower level. This last generalizes the classical Poincaré-Bendixson theorem and provides information relating growth, topological type, and level, e.g. if $ L$ is a totally proper leaf at level $k$ then $L$ has exactly polynomial growth of degree $ k$ and topological type $ k - 1$.
The arithmetic perfection of Buchsbaum-Eisenbud varieties and generic modules of projective dimension two
Craig
Huneke
211-233
Abstract: We prove the ideals associated with the construction of generic complexes are prime and arithmetically perfect. This is used to construct the generic resolution for modules of projective dimension two.
Unique minimality of Fourier projections
S. D.
Fisher;
P. D.
Morris;
D. E.
Wulbert
235-246
Abstract: The question of when the Fourier projection is the only one of least norm from a space of continuous functions on the circle onto spaces spanned by trigonometric polynomials is studied in two settings. In the first the domain space is the disc algebra and the range is finite-dimensional. In the second the domain space consists of all real continuous functions and the range has finite codimension.
Finitely additive Markov chains
S.
Ramakrishnan
247-272
Abstract: In this paper we develop the theory of Markov chains with stationary transition probabilities, where the transition probabilities and the initial distribution are assumed only to be finitely additive. We prove a strong law of large numbers for recurrent chains. The problem of existence and uniqueness of finitely additive stationary initial distributions is studied and the ergodicity of recurrent chains under a stationary initial distribution is proved.
On spaces of maps of $n$-manifolds into the $n$-sphere
Vagn Lundsgaard
Hansen
273-281
Abstract: The space of (continuous) maps of a closed, oriented manifold $ {C^n}$ into the $ n$-sphere ${S^n}$ has a countable number of (path-) components. In this paper we make a general study of the homotopy classification problem for such a set of components. For $ {C^n} = {S^n}$, the problem was solved in [4], and for an arbitrary closed, oriented surface ${C^2}$, it was solved in [5]. We get a complete solution for manifolds ${C^n}$ of even dimension $n \geqslant 4$ with vanishing first Betti number. For odd dimensional manifolds ${C^n}$ we show that there are at most two different homotopy types among the components. Finally, for a class of manifolds introduced by Puppe [8] under the name spherelike manifolds, we get a complete analogue to the main theorem in [4] concerning the class of spheres.
A class of extremal functions for the Fourier transform
S. W.
Graham;
Jeffrey D.
Vaaler
283-302
Abstract: We determine a class of real valued, integrable functions $f(x)$ and corresponding functions $ {M_f}(x)$ such that $ f(x) \leqslant {M_f}(x)$ for all $x$, the Fourier transform ${\hat M_f}(t)$ is zero when $\left\vert t \right\vert \geqslant 1$, and the value of ${\hat M_f}(0)$ is minimized. Several applications of these functions to number theory and analysis are given.