Berezin Quantization and Reproducing Kernels on Complex Domains
Miroslav
Englis
411-479
Abstract: Let $\Omega$ be a non-compact complex manifold of dimension $n$, $\omega =\partial \overline{\partial }\Psi$ a Kähler form on $\Omega$, and $K_\alpha ( x,\overline{y})$ the reproducing kernel for the Bergman space $A^2_\alpha$ of all analytic functions on $\Omega$ square-integrable against the measure $e^{-\alpha \Psi } |\omega ^n|$. Under the condition \begin{equation*}K_\alpha ( x,\overline{x})= \lambda _\alpha e^{\alpha \Psi (x)} \end{equation*} F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109--1163] was able to establish a quantization procedure on $(\Omega ,\omega )$ which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just $\Omega = \mathbf{C} ^n$ and $\Omega$ a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as $\alpha \to +\infty$. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in $\mathbf{C}^n$. Along the way, we also fix two gaps in Berezin's original paper, and discuss, for $\Omega$ a domain in $\mathbf{C}^n$, a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure $|\omega ^n|$.
A Construction of the Level 3 Modules for the Affine Lie Algebra $A_2^{(2)}$ and a New Combinatorial Identity of the Rogers-Ramanujan Type Amer. Math. Soc. 348 (1996), pp. 481-501.
Stefano
Capparelli
481-501
Abstract: We obtain a vertex operator construction of level 3 standard representations for the affine Lie algebra $A_2^{(2)}$. As a corollary, we also get new conbinatorial identities.
A Multivariate Faa di Bruno Formula with Applications
G.
M.
Constantine;
T.
H.
Savits
503-520
Abstract: A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of an infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.
Isomorphisms of adjoint Chevalley groups over integral domains
Yu
Chen
521-541
Abstract: It is shown that every automorphism of an adjoint Chevalley group over an integral domain containing the rational number field is uniquely expressible as the product of a ring automorphism, a graph automorphism and an inner automorphism while every isomorphism between simple adjoint Chevalley groups can be expressed uniquely as the product of a ring isomorphism, a graph isomorphism and an inner automorphism. The isomorphisms between the elementary subgroups are also found having analogous expressions.
Regularity and Algebras of Analytic Functions in Infinite Dimensions
R.
M.
Aron;
P.
Galindo;
D.
García;
M.
Maestre
543-559
Abstract: A Banach space $E$ is known to be Arens regular if every continuous linear mapping from $E$ to $E^{\prime}$ is weakly compact. Let $U$ be an open subset of $E$, and let $H_b(U)$ denote the algebra of analytic functions on $U$ which are bounded on bounded subsets of $U$ lying at a positive distance from the boundary of $U.$ We endow $H_b(U)$ with the usual Fréchet topology. $M_b(U)$ denotes the set of continuous homomorphisms $\phi:H_b(U) \to {\mathbb{C}}$. We study the relation between the Arens regularity of the space $E$ and the structure of $M_b(U)$.
An Index Theory For Quantum Dynamical Semigroups
B.
V. Rajarama
Bhat
561-583
Abstract: W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.
A cascade decomposition theory with applications to Markov and exchangeable cascades
Edward
C.
Waymire;
Stanley
C.
Williams
585-632
Abstract: A multiplicative random cascade refers to a positive $T$-martingale in the sense of Kahane on the ultrametric space $T = { \{ 0,1,\dots ,b-1 \} }^{\mathbf{N}}.$ A new approach to the study of multiplicative cascades is introduced. The methods apply broadly to the problems of: (i) non-degeneracy criterion, (ii) dimension spectra of carrying sets, and (iii) divergence of moments criterion. Specific applications are given to cascades generated by Markov and exchangeable processes, as well as to homogeneous independent cascades.
Harmonic Bergman Functions on Half-Spaces
Wade
C.
Ramey;
HeungSu
Yi
633-660
Abstract: We study harmonic Bergman functions on the upper half-space of $\bold{R}^n$. Among our main results are: The Bergman projection is bounded for the range $1< p < \infty$; certain nonorthogonal projections are bounded for the range $1\leq p < \infty$; the dual space of the Bergman $L^1$-space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range $1\leq p < \infty$; the Bergman norm is equivalent to a ``normal derivative norm'' as well as to a ``tangential derivative norm''.
A Tranversality Theorem for Holomorphic Mappings and Stability of Eisenman-Kobayashi Measures
Sh.
Kaliman;
M.
Zaidenberg
661-672
Abstract: We show that Thom's Transversality Theorem is valid for holomorphic mappings from Stein manifolds. More precisely, given such a mapping $f:S\rightarrow M$ from a Stein manifold $S$ to a complex manifold $M$ and given an analytic subset $A$ of the jet space $J^{k} (S, M), \; f$ can be approximated in neighborhoods of compacts by holomorphic mappings whose $k$-jet extensions are transversal to $A$. As an application the stability of Eisenman-Kobayshi intrinsic $k$-measures with respect to deleting analytic subsets of codimension $>k$ is proven. This is a generalization of the Campbell-Howard-Ochiai-Ogawa theorem on stability of Kobayashi pseudodistances.
An existence result for linear partial differential equations with $C^\infty$ coefficients in an algebra of generalized functions
Todor
Todorov
673-689
Abstract: We prove the existence of solutions for essentially all linear partial differential equations with $C^\infty$-coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy's equation has solutions whenever its right-hand side is a classical $C^\infty$-function.
Characterizations of generalized Hermite and sieved ultraspherical polynomials
Holger
Dette
691-711
Abstract: A new characterization of the generalized Hermite polyno- mials and of the orthogonal polynomials with respect to the measure $|x|^\gamma (1-x^2)^{1/2}dx$ is derived which is based on a ``reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.
Asymptotic Expansion for Layer Solutions of a Singularly Perturbed Reaction-Diffusion System
Xiao-Biao
Lin
713-753
Abstract: For a singularly perturbed $n$-dimensional system of reaction-- diffusion equations, assuming that the 0th order solutions possess boundary and internal layers and are stable in each regular and singular region, we construct matched asymptotic expansions for formal solutions in all the regular, boundary, internal and initial layers to any desired order in $\epsilon$. The formal solution shows that there is an invariant manifold of wave-front-like solutions that attracts other nearby solutions. We also give conditions for the wave-front-like solutions to converge slowly to stationary solutions on that manifold.
Regularity theory and traces of $\mathcal{A}$-harmonic functions
Pekka
Koskela;
Juan
J.
Manfredi;
Enrique
Villamor
755-766
Abstract: In this paper we discuss two different topics concerning $\mathcal{A}$-harmonic functions. These are weak solutions of the partial differential equation \begin{equation*}\text{div}(\mathcal{A}(x,\nabla u))=0,\end{equation*} where $\alpha (x)|\xi |^{p-1}\le \langle \mathcal{A}(x,\xi ),\xi \rangle \le \beta (x) |\xi |^{p-1}$ for some fixed $p\in (1,\infty )$, the function $\beta$ is bounded and $\alpha (x)>0$ for a.e. $x$. First, we present a new approach to the regularity of $\mathcal{A}$-harmonic functions for $p>n-1$. Secondly, we establish results on the existence of nontangential limits for $\mathcal{A}$-harmonic functions in the Sobolev space $W^{1,q}(\mathbb{B})$, for some $q>1$, where $\mathbb{B}$ is the unit ball in $\mathbb{R}^n$. Here $q$ is allowed to be different from $p$.
On $CR$-mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions
Ruslan
Sharipov;
Alexander
Sukhov
767-780
Abstract: We prove the algebraicity of smooth $CR$-mappings between algebraic Cauchy-Riemann manifolds. A generalization of separate algebraicity principle is established.
Radial Solutions to a Dirichlet Problem Involving Critical Exponents when $N=6$
Alfonso
Castro;
Alexandra
Kurepa
781-798
Abstract: In this paper we show that, for each $\lambda > 0$, the set of radially symmetric solutions to the boundary value problem \begin{equation*}\begin{split} -\Delta u(x) & = % \lambda u(x) + u(x)\vert u(x)\vert,\quad x\in B := \{x\in R^6\colon\Vert x\Vert < 1\}, u(x) & = % 0, \quad x\in\partial B, \end{split} \end{equation*} is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
Factorizations of simple algebraic groups
Martin
W.
Liebeck;
Jan
Saxl;
Gary
M.
Seitz
799-822
Abstract: We determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups. Additional results are established which drop the maximality assumption, and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension.