On the structure of real transitive Lie algebras
Jack F.
Conn
1-71
Abstract: In this paper, we examine some of the ways in which abstract algebraic objects in a transitive Lie algebra $L$ are expressed geometrically in the action of each transitive Lie pseudogroup $ \Gamma$ associated to $ L$. We relate those chain decompositions of $\Gamma$ which result from considering $ \Gamma$-invariant foliations to Jordan-Hölder sequences (in the sense of Cartan and Guillemin) for $L$. Local coordinates are constructed which display the nature of the partial differential equations defining $\Gamma$; in particular, locally homogeneous pseudocomplex structures (also called ${\text{CR}}$-structures) are associated to the nonabelian quotients of complex type in a Jordan-Hölder sequence for $L$.
A method of lines for a nonlinear abstract functional evolution equation
A. G.
Kartsatos;
M. E.
Parrott
73-89
Abstract: Let $X$ be a real Banach space with $ {X^\ast}$ uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E) $\displaystyle u\prime(t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T].$ The operators $A(t):D \subset X \to X$ are $m$-accretive and $G(t,\phi )$ is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators $\hat A(t)$ and the corresponding use of nonlinear evolution operator theory.
Codimension $1$ orbits and semi-invariants for the representations of an equioriented graph of type $D\sb{n}$
S.
Abeasis
91-123
Abstract: We consider the Dynkin diagram ${D_n}$ equioriented and the variety $ \operatorname{Hom}({V_1},{V_3}) \times \Pi_{1 = 2}^n \operatorname{Hom} ({V_i},{V_{i + 1}})$, ${V_j}$ a vector space over $K$, on which the group $ G = \prod\nolimits_{i = 1}^n {{\text{GL}}} ({V_i})$ acts. We determine the maximal orbit and the codim. $1$ orbits of this action, giving their decomposition in terms of the irreducible representations of ${D_n}$. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.
$R$-sets and category
Rana
Barua
125-158
Abstract: We prove some category theoretic results for $R$-sets much in the spirit of Vaught and Burgess. Since the proofs entail many results on $ R$-sets and the $ R$-operator, we have studied them in some detail and have formulated many results appropriate for our purpose in, perhaps, a more unified manner than is available in the literature. Our main theorem is the following: Any $R$-set in the product of two Polish spaces can be approximated, in category, uniformly over all sections, by sets generated by rectangles with one side an $ R$-set and the other a Borel set. In fact, we prove a levelwise version of this result. For $C$-sets, this has been proved by V. V. Srivatsa.
Adapted probability distributions
Douglas N.
Hoover;
H. Jerome
Keisler
159-201
Abstract: We introduce a family of notions of equivalence for stochastic processes on spaces with an underlying filtration. These refine the notion of having the same distribution by taking account of the relation of the processes to their underlying filtrations. The weakest of these notions is the same as the notion of synonymity introduced by Aldous. Analysis of the strongest equivalence property leads to spaces with a strong universality property for adapted stochastic processes, which we call saturation. Spaces having this property contain 'strong' solutions to a large class of stochastic integral equations.
Interpolation and Gleason parts in $L$-domains
Michael Frederick
Behrens
203-225
Abstract: We describe the closure of $[ - 1/2,0)$ in the maximal ideal space $\mathcal{M}(\mathcal{D})$ of ${H^\infty }(\mathcal{D})$) for an arbitrary $L$-domain $ \mathcal{D}$. For $ L$-domains satisfying $ \sup ({c_{n + 1}}/{c_n}) < 1$ and $\Sigma {({r_n}/{c_n})^p} < \infty$, some $p \geqslant 1$, we describe all interpolation sequences for ${H^\infty }(\mathcal{D})$, we show that a homomorphism (except the distinguished homomorphism, when it exists) lies in a nontrivial Gleason part if and only if it is contained in the closure of an interpolating sequence, and we describe all the analytic structure occurring in $ \mathcal{M}(\mathcal{D})$.
Sous-espaces bien dispos\'es de $L\sp{1}$-applications
Gilles
Godefroy
227-249
Abstract: RÉsumÉ. On montre que le quotient d'un espace ${L^1}$ par un sous-espace fermé dont la boule unité est fermée dans $ {L^0}$ est faiblement séquentiellement complet; cette situation se présente dans de nombreux cas concrets, tels que le quotient ${L^1}/{H^1}$. On applique le résultat général dans diverses situations: duaux de certaines algères uniformes, analyse harmonique, fonctions de plusieurs variables complexes. On montre ensuite comment peuvent s'appliquer les métheodes de $M$-structure; on considère aussi de nouvelles classes d'uniques préduaux. A titre d'exemples, on montre: (1) Le caractère f.s.c. d'espaces $ {\mathcal{C}_E}{(G)^\ast}$, pour de "gros" sous-ensembles $ E$ du groupe dual $\Gamma = \hat G$. (2) Le caractère f.s.c. d'espaces $ {L^1}/{H^1}$ mutli-dimensionnels, tels que $ {L^1}/{H^1}({D^n})$ et $ {L^1}/{H^1}({B^n})$. (3) L'unicité du prédual pour certaines sous-algèbres ultrafaiblement fermées non-autoadjointes de $ \mathcal{L}(H)$. One shows that the quotient of an ${L^1}$-space by a closed subspace, whose unit ball is closed in ${L^0}$, is weakly sequentially complete. This situation occurs in many natural cases, like ${L^1}/{H^1}$. This result is applied in several situations: uniform algebras, harmonic analysis, functions of several complex variables. One shows how to apply $M$-structure theory; several new classes of unique preduals are also obtained. As an example, one shows: (1) If $E$ is a "big" subset of the dual group $\Gamma = \hat G$, then $ {\mathcal{C}_E}{(G)^\ast}$ is w.s.c. (2) The spaces ${L^1}/{H^1}({D^n})$ and ${L^1}/{H^1}({B^n})$ are w.s.c. (3) Several classes of ${\omega ^\ast}$-closed non-self-adjoint subalgebras of $ \mathcal{L}(H)$ have unique preduals.
The amalgamation property for varieties of lattices
Alan
Day;
Jaroslav
Ježek
251-256
Abstract: There are precisely three varieties of lattices that satisfy the amalgamation property: trivial lattices, distributive lattices, and all lattices.
The evolution of random graphs
Béla
Bollobás
257-274
Abstract: According to a fundamental result of Erdös and Rényi, the structure of a random graph ${G_M}$ changes suddenly when $M \sim n/2$: if $M = \left\lfloor {cn} \right\rfloor$ and $c < \frac{1}{2}$ then a.e. random graph of order $ n$ and since $ M$ is such that its largest component has $O(\log n)$ vertices, but for $c > \frac{1}{2}$ a.e. ${G_M}$ has a giant component: a component of order $ (1-{\alpha _c}+o(1))n$ where ${\alpha _c} < 1$. The aim of this paper is to examine in detail the structure of a random graph ${G_M}$ when $M$ is close to $n/2$. Among others it is proved that if $M = n/2 + s$, $s = o(n)$ and $s \geq {(\log n)^{1/2}}{n^{2/3}}$ then the giant component has $ (4 + o(1))s$ vertices. Furthermore, rather precise estimates are given for the order of the $r$th largest component for every fixed $ r$.
The formation of the dead core in parabolic reaction-diffusion problems
Catherine
Bandle;
Ivar
Stakgold
275-293
Abstract: For some nonlinear parabolic problems of reaction-diffusion, a region of zero reactant concentration may be formed in finite time. Conditions are formulated for the existence of such a dead core and estimates for its time of onset are also given. These results complement previous ones that dealt with the stationary (elliptic) problem.
Stochastic representation and singularities of solutions of second order equations with semidefinite characteristic form
Kazuo
Amano
295-312
Abstract: In the theory of partial differential equations, there is no explicit representation of solutions for general degenerate elliptic-parabolic equations. However, Stroock and Varadhan [15] have obtained a stochastic representation for such a wider class of equations in $ {L^\infty }$ space. In this paper we establish, by using Stroock and Varadhan's stochastic representation, a method which enables us to construct solutions with singularities of second order equations with semidefinite characteristic form. Our theorems are not probabilistic paraphrases of the results obtained in the theory of partial differential equations. In fact, each assumption of the theorems is much weaker than any assumption of corresponding known results.
Bilinear forms on $H\sp{\infty }$ and bounded bianalytic functions
J.
Bourgain
313-337
Abstract: Given an arbitrary Radon probability measure on the circle $\pi$, a generlization of the classical Cauchy transform is obtained. These projections are used to prove that each bounded linear operator from a reflexive subspace of ${L^1}$ or $ {L^1}(\pi )/{H^1}$ into ${H^\infty }(D)$ admits a bounded extension. These facts lead to different variants of the cotype-$ 2$ inequality for ${L^1}(\pi )/{H^1}$. Applications are given to absolutely summing operators and the existence of certain bounded bianalytic functions. For instance, we derive the Hilbert space factorization of arbitrary bounded linear operators from ${H^\infty }(D)$ into its dual without an a priori approximation hypothesis, thus completing some of the work in [1]. Our methods give new information about the Fourier coefficients of ${H^\infty }(D \times D)$-functions, thus improving a theorem in [6].
On Skolem's exponential functions below $2\sp{2\sp{X}}$
Lou
van den Dries;
Hilbert
Levitz
339-349
Abstract: A result of Ehrenfeucht implies that the smallest class of number-theoretic functions $f:{\mathbf{N}} \to {\mathbf{N}}$ containing the constants $ 0,1,2, \ldots$, the identity function $X$, and closed under addition, multiplication and $f \to {f^X}$, is well-ordered by the relation of eventual dominance. We show that its order type is $ {\omega ^{{\omega ^\omega }}}$, and that for any two nonzero functions $ f,g$ in the class the quotient $f(n)/g(n)$ tends to a limit in ${E^ + } \cup \{ 0,\infty \}$ as $n \to \infty$, where ${E^ + }$ is the smallest set of positive real numbers containing $1$ and closed under addition, multiplication and under the operations $x \to {x^{ - 1}},x \to {e^x}$.
The Selberg trace formula. V. Questions of trace class
M. Scott
Osborne;
Garth
Warner
351-376
Abstract: The purpose of this paper is to develop criteria which will ensure that the $K$-finite elements of $C_c^\infty (G)$ are represented on $ L_{{\text{dis}}}^2(G/\Gamma )$ by trace class operators.
A general approach to the optimality of minimum distance estimators
P. W.
Millar
377-418
Abstract: Let $\Theta$ be an open subset of a separable Hilbert space, and $ {\xi _n}(\theta )$, $\theta \in \Theta$, a sequence of stochastic processes with values in a (different) Hilbert space $ B$. This paper develops an asymptotic expansion and an asymptotic minimax result for "estimates" $ {\hat \theta _n}$ defined by ${\inf _\theta }\vert{\xi _n}(\theta )\vert = \vert{\xi _n}({\hat \theta _n})\vert$, where $ \vert \cdot \vert$ is the norm of $B$. The abstract results are applied to study optimality and asymptotic normality of procedures in a number of important practical problems, including simple regression, spectral function estimation, quantile function methods, min-chi-square methods, min-Hellinger methods, minimum distance methods based on $ M$-functionals, and so forth. The results unify several studies in the literature, but most of the $ {\text{LAM}}$ results are new. From the point of view of applications, the entire paper is a sustained essay concerning the problem of fitting data with a reasonable, but relatively simple, model that everyone knows cannot be exact.
On hypersingular integrals and anisotropic Bessel potential spaces
H.
Dappa;
W.
Trebels
419-429
Abstract: In this paper we characterize anisotropic potential spaces in terms of hypersingular integrals of mixed homogeneity with respect to a general dilation matrix.