Operator methods and Lagrange inversion: a unified approach to Lagrange formulas
Ch.
Krattenthaler
431-465
Abstract: We present a general method of proving Lagrange inversion formulas and give new proofs of the $s$-variable Lagrange-Good formula [13] and the $q$-Lagrange formulas of Garsia [7], Gessel [10], Gessel and Stanton [11, 12] and the author [18]. We also give some $q$-analogues of the Lagrange formula in several variables.
Paracommutators---boundedness and Schatten-von Neumann properties
Svante
Janson;
Jaak
Peetre
467-504
Abstract: A very general class of operators, acting on functions in $ {L^2}({{\mathbf{R}}^d})$, is introduced. The name "paracommutator" has been chosen because of the similarity with the paramultiplication of Bony and also because paracommutators comprise as a special case commutators of Calderón-Zygmund operators, as well as many other interesting examples (Hankel and Toeplitz operators etc.). The main results, extending previous results by Peller and others, express boundedness and Schatten-von Neumann properties of a paracommutator in terms of its symbol.
On the mean value property of harmonic functions and best harmonic $L\sp 1$-approximation
Myron
Goldstein;
Werner
Haussmann;
Lothar
Rogge
505-515
Abstract: The present paper deals with the inverse mean value property of harmonic functions and with the existence, uniqueness, and characterization of a best harmonic $ {L^1}$-approximant to strictly subharmonic functions. The main theorem concerning the inverse mean value property of harmonic functions is based on a generalization of a theorem due to Ü. Kuran as well as on an approximation theorem proved by J. C. Polking and also by L. I. Hedberg. The inverse mean value property will be applied in order to prove necessary and sufficient conditions for the existence of a best harmonic $ {L^1}$-approximant to a subharmonic function $s$ satisfying $ \Delta s > 0$ a.e. in the unit ball.
Minimal $K$-types for $G\sb 2$ over a $p$-adic field
Allen
Moy
517-529
Abstract: We single out certain representations of compact open subgroups of $ {G_2}$ over a $ p$-adic field and show they play a role in the representation theory of $ {G_2}$ similar to minimal $ K$-types in the theory of real groups.
Measured laminations in $3$-manifolds
Ulrich
Oertel
531-573
Abstract: An essential measured lamination embedded in an irreducible, orientable $ 3$-manifold $M$ is a codimension $1$ lamination with a transverse measure, carried by an incompressible branched surface satisfying further technical conditions. Weighted incompressible surfaces are examples of essential measured laminations, and the inclusion of a leaf of an essential measured lamination into $M$ is injective on ${\pi _1}$. There is a space $ \mathcal{P}\mathcal{L}(M)$ whose points are projective classes of essential measured laminations. Projective classes of weighted incompressible surfaces are dense in $ \mathcal{P}\mathcal{L}(M)$. The space $ \mathcal{P}\mathcal{L}(M)$ is contained in a finite union of cells (of different dimensions) embedded in an infinite-dimensional projective space, and contains the interiors of these cells. Most of the properties of the incompressible branched surfaces carrying measured laminations are preserved under the operations of splitting or passing to sub-branched surfaces.
The space of incompressible surfaces in a $2$-bridge link complement
W.
Floyd;
A.
Hatcher
575-599
Abstract: Projective lamination spaces for $2$-bridge link complements are computed explicitly.
Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups
L.
Corwin;
F. P.
Greenleaf
601-622
Abstract: Let $G$ be a connected, simply connected nilpotent Lie group, $H$ a Lie subgroup, and $\sigma$ an irreducible unitary representation of $ H$. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of $ \operatorname{Ind} (H \uparrow G,\,\sigma )$. One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is algebraic-geometric in nature and amounts to an extension of the familiar fact that for almost all polynomials over $ R$ of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if $G$ is a complex nilpotent Lie group and $ H$ a complex Lie subgroup, then the multiplicity is a.e. constant.
Arens regularity of the algebra $A\hat\otimes B$
A.
Ülger
623-639
Abstract: Let $A$ and $B$ be two Banach algebras. On the projective tensor product $ A\hat \otimes \,B$ of $ A$ and $B$ there exists a natural algebra structure. In this note we study Arens regularity of the Banach algebra $ A\hat \otimes \,B$.
On Aitchison's construction by isotopy
Daniel
Silver
641-652
Abstract: We describe a method introduced by I. Aitchison for constructing doubly slice fibered $n$-knots. We prove that all high-dimensional simple doubly slice fibered $n$-knots can be obtained by this construction. (Even-dimensional $n$-knots are required to be $Z$-torsion-free.) We also show that any possible rational Seifert form can be realized by a doubly slice fibered classical knot.
The continuous $(\alpha, \beta)$-Jacobi transform and its inverse when $\alpha+\beta+1$ is a positive integer
G. G.
Walter;
A. I.
Zayed
653-664
Abstract: The continuous $(\alpha ,\,\beta )$-Jacobi transform is introduced as an extension of the discrete Jacobi transform by replacing the polynomial kernel by a continuous one. An inverse transform is found for both the standard and a modified normalization and applied to a version of the sampling theorem. An orthogonal system forming a basis for the range is shown to have some unusual properties, and is used to obtain the inverse.
Random perturbations of reaction-diffusion equations: the quasideterministic approximation
Mark I.
Freidlin
665-697
Abstract: Random fields ${u^\varepsilon }(t,\,x) = (u_1^\varepsilon (t,\,x), \ldots ,u_n^\varepsilon (t,\,x))$, defined as the solutions of a system of the PDE due. $\displaystyle \frac{{\partial u_k^\varepsilon }} {{\partial t}} = {L_k}u_k^\var... ...x;\,u_1^\varepsilon , \ldots ,u_n^\varepsilon ) + \varepsilon {\zeta _k}(t,\,x)$ are considered. Here ${L_k}$ are second-order linear elliptic operators, ${\zeta _k}$ are Gaussian white-noise fields, independent for different $k$, and $ \varepsilon$ is a small parameter. The most attention is given to the problem of determining the behavior of the invariant measure ${\mu ^\varepsilon }$ of the Markov process $u_t^\varepsilon = (u_1^\varepsilon (t,\, \cdot ), \ldots ,u_n^\varepsilon (t,\, \cdot ))$ in the space of continuous functions as $\varepsilon \to 0$, and also of describing transitions of $ u_t^\varepsilon$ between stable stationary solutions of nonperturbed systems of PDE. The behavior of ${\mu ^\varepsilon }$ and the transitions are defined by large deviations for the field ${u^\varepsilon }(t,\,x)$.
The density manifold and configuration space quantization
John D.
Lafferty
699-741
Abstract: The differential geometric structure of a Fréchet manifold of densities is developed, providing a geometrical framework for quantization related to Nelson's stochastic mechanics. The Riemannian and symplectic structures of the density manifold are studied, and the Schrödinger equation is derived from a variational principle. By a theorem of Moser, the density manifold is an infinite dimensional homogeneous space, being the quotient of the group of diffeomorphisms of the underlying base manifold modulo the group of diffeomorphisms which preserve the Riemannian volume. From this structure and symplectic reduction, the quantization procedure is equivalent to Lie-Poisson equations on the dual of a semidirect product Lie algebra. A Poisson map is obtained between the dual of this Lie algebra and the underlying projective Hilbert space.
A space-time property of a class of measure-valued branching diffusions
Edwin A.
Perkins
743-795
Abstract: If $d > \alpha$, it is shown that the $ d$-dimensional branching diffusion of index $\alpha$, studied by Dawson and others, distributes its mass over a random support in a uniform manner with respect to the Hausdorff ${\phi _\alpha }$-measure, where $ {\phi _\alpha }(x) = {x^\alpha }\log \log 1/x$. More surprisingly, it does so for all positive times simultaneously. Slightly less precise results are obtained in the critical case $d = \alpha$. In particular, the process is singular at all positive times a.s. for $d \geqslant \alpha$.
Some applications of tree-limits to groups. I
Kenneth
Hickin
797-839
Abstract: Sharper applications to group theory are given of an elegant construction -- the "tree-limit"--which S. Shelah circulated as a preprint in 1977 and used to obtain $\infty$-$\omega$-enlargements to power ${2^\omega }$ of certain countable homogeneous groups and skew fields. In this paper we enlarge the class of groups to which this construction can be interestingly applied and we obtain permutation representations of countable degree of the tree-limit groups; we obtain uncountable subgroup-incomparability for enlargements of countable existentially closed groups and even in nonhomogeneous cases we obtain the very strong "archetypal direct limit property" (which implies $ \infty$-$\omega $-equivalence (see (1.0)) of the permutation representations). We are able to control the permutation representations which get stretched by the tree-limit by varying the point-stabilizer subgroups (see (5.5)). In particular we can archetypally stretch in $ {2^\omega }$ subgroup-incomparable ways any homogeneous permutation representation of a countable locally finite group in which every finite subgroup has infinitely many regular orbits (Theorem 4). We discuss cases where tree-limits are subgroups of inverse limits.
Univalent functions in $H\cdot \overline H(D)$
Z.
Abdulhadi;
D.
Bshouty
841-849
Abstract: Functions in $H \cdot \overline H (D)$ are sense-preserving of the form $f = h \cdot \overline g$ where $h$ and $g$ are in $H(D)$. Such functions are solutions of an elliptic nonlinear P.D.E. that is studied in detail especially for its univalent solutions.
Correction to: ``Cartan subalgebras of simple Lie algebras'' [Trans. Amer. Math. Soc. {\bf 234} (1977), no. 2, 435--446; MR0480650 (58 \#806)]
Robert Lee
Wilson
851-855
Corrigendum to: ``Affine semigroups and Cohen-Macaulay rings generated by monomials'' [Trans. Amer. Math. Soc. {\bf 298} (1986), no. 1, 145--167; MR0857437 (87j:13032)]
Lê Tuân
Hoa;
Ngô Viêt
Trung
857