Higher-order Sugawara operators for affine Lie algebras
Roe
Goodman;
Nolan R.
Wallach
1-55
Abstract: Let $\hat{\mathfrak{g}}$ be the affine Lie algebra associated to a simple Lie algebra $\mathfrak{g}$. Representations of $\hat{\mathfrak{g}}$ are described by current fields $ X(\zeta)$ on the circle $ {\mathbf{T}}\;(X \in \mathfrak{g}$ and $\zeta \in {\mathbf{T}})$. In this paper a linear map $ \sigma$ from the symmetric algebra $ S(\mathfrak{g})$ to (formal) operator fields on a suitable category of $\hat{\mathfrak{g}}$ modules is constructed. The operator fields corresponding to $\mathfrak{g}$-invariant elements of $S(\mathfrak{g})$ are called Sugawara fields. It is proved that they satisfy commutation relations of the form $(\ast)$ $\displaystyle [\sigma (u)(\zeta),X(\eta)] = {c_\infty }D\delta (\zeta /\eta)\sigma ({\nabla _X}u)(\zeta) + {\text{higher-order}}\;{\text{terms}}$ with the current fields, where $ {c_\infty }$ is a renormalization of the central element in $\hat{\mathfrak{g}}$ and $D\delta$ is the derivative of the Dirac delta function. The higher-order terms in $(\ast)$ are studied using results from invariant theory and finite-dimensional representation theory of $ \mathfrak{g}$. For suitably normalized invariants $u$ of degree $4$ or less, these terms are shown to be zero. This vanishing is also proved for $\mathfrak{g} = {\text{sl}}(n,{\mathbf{C}})$ and $u$ running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever $ {c_\infty }$ is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for $\hat{\mathfrak{g}}$.
Functional equations, tempered distributions and Fourier transforms
John A.
Baker
57-68
Abstract: This paper introduces a method for solving functional equations based on the Fourier transform of tempered distributions.
Multiresolution approximations and wavelet orthonormal bases of $L\sp 2({\bf R})$
Stephane G.
Mallat
69-87
Abstract: A multiresolution approximation is a sequence of embedded vector spaces ${({{\mathbf{V}}_j})_{j \in {\text{z}}}}$ for approximating $ {{\mathbf{L}}^2}({\mathbf{R}})$ functions. We study the properties of a multiresolution approximation and prove that it is characterized by a $2\pi$-periodic function which is further described. From any multiresolution approximation, we can derive a function $\psi (x)$ called a wavelet such that ${(\sqrt {{2^j}} \psi ({2^j}x - k))_{(k,j) \in {{\text{z}}^2}}}$ is an orthonormal basis of $ {{\mathbf{L}}^2}({\mathbf{R}})$. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space $ {{\mathbf{H}}^s}$.
Injectivity of operator spaces
Zhong-Jin
Ruan
89-104
Abstract: We study the structure of injective operator spaces and the existence and uniqueness of the injective envelopes of operator spaces. We give an easy example of an injective operator space which is not completely isometric to any $ {C^\ast }$-algebra. This answers a question of Wittstock [23]. Furthermore, we show that an operator space $E$ is injective if and only if there exists an injective ${C^\ast }$-algebra $A$ and two projections $p$ and $q$ in $A$ such that $E$ is completely isometric to $pAq$.
Invariant measures and equilibrium states for piecewise $C\sp {1+\alpha}$ endomorphisms of the unit interval
Christopher J.
Bose
105-125
Abstract: A differentiable function is said to be $ {C^{1 + \alpha }}$ if its derivative is a Hölder continuous function with exponent $\alpha > 0$. We show that three well-known results about invariant measures for piecewise monotonic and ${C^2}$ endomorphisms of the unit interval are in fact true for piecewise monotonic and ${C^{1 + \alpha }}$ maps. We show the existence of unique, ergodic measures equivalent to Lebesgue measure for $ {C^{1 + \alpha }}$ Markov maps, extending a result of Bowen and Series for the $ {C^2}$ case. We present a generalization of Adler's Folklore Theorem for maps which satisfy a restricted mixing condition, and we show that these $ {C^{1 + \alpha }}$ mixing endomorphisms possess unique equilibrium states, a result which was shown for the ${C^2}$ case by P. Walters.
On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semi-continuous payoffs
M. Ali
Khan
127-146
Abstract: We report results on the existence of a Cournot-Nash equilibrium distribution for games in which the action space is not necessarily metrizable and separable and the payoff functions are not necessarily continuous. Our work relies on the theory of Radon measures as developed by Schwartz-Topsoe and on the epitopology as developed by Dolecki-Salinetti-Wets
The structure of quasimultipliers of $C\sp *$-algebras
Hua Xin
Lin
147-172
Abstract: Let $A$ be a ${C^\ast }$-algebra and ${A^{\ast\ast}}$ its enveloping ${W^\ast }$-algebra. Let ${\text{LM}}(A)$ be the left multipliers of $ A$, ${\text{RM}}(A)$ the right multipliers of $ A$ and ${\text{QM}}(A)$ the quasi-multipliers of $ A$. A question was raised by Akemann and Pedersen [1] whether $ {\text{QM}}(A) = {\text{LM}}(A) + {\text{RM}}(A)$. McKennon [20] gave a nonseparable counterexample. L. Brown [6] shows the answer is negative for stable (separable) ${C^\ast }$-algebras also. In this paper, we mainly consider $\sigma $-unitial ${C^\ast }$-algebras. We give a criterion for ${\text{QM}}(A) = {\text{LM}}(A) + {\text{RM}}(A)$. In the case that $A$ is stable, we give a necessary and sufficient condition for ${\text{QM}}(A) = {\text{LM}}(A) + {\text{RM}}(A)$. We also give answers for other ${C^\ast }$-algebras.
Sarkovski\u\i's theorem for hereditarily decomposable chainable continua
Piotr
Minc;
W. R. R.
Transue
173-188
Abstract: Sarkovskii's theorem, which fails to hold for chainable continua, is shown to hold for hereditarily decomposable chainable continua.
Hamburger moment problems and orthogonal polynomials
T. S.
Chihara
189-203
Abstract: We consider a sequence of orthogonal polynomials given by the classical three term recurrence relation. We address the problem of deciding the determinacy or indeterminacy of the associated Hamburger moment problem on the basis of the behavior of the coefficients in the three term recurrence relation. Comparisons are made with other criteria in the literature. The efficacy of the criteria obtained is illustrated by application to many specific examples of orthogonal polynomials.
A generalization of the Levine-Tristram link invariant
Lawrence
Smolinsky
205-217
Abstract: Invariants to $ m$-component links are defined and are shown to be link cobordism invariants under certain conditions. Examples are given.
A new $3$-dimensional shrinking criterion
Robert J.
Daverman;
Dušan
Repovš
219-230
Abstract: We introduce a new shrinking criterion for cell-like upper semicontinuous decompositions $G$ of topological $3$-manifolds, such that the embedding dimension (in the sense of Štan'ko) of the nondegeneracy set of $ G$ is at most one. As an immediate application, we prove a recognition theorem for $3$-manifolds based on a new disjoint disks property.
The structure of some equivariant Thom spectra
Steven R.
Costenoble
231-254
Abstract: We show that the equivariant Thom spectra $M{O_{{{\text{Z}}_2}}}$ and $m{O_{{{\text{Z}}_2}}}$ do not split as wedges of equivariant Eilenberg-Mac Lane spectra, as they do nonequivariantly. This is done by finding two-stage Postnikov towers giving these spectra, and determining the nontrivial $k$-invariants. We also consider the question: In what sense is the spectrum $m{O_{{{\text{Z}}_2}}}$ representing unoriented bordism unique?
Compact abelian prime actions on von Neumann algebras
Klaus
Thomsen
255-272
Abstract: We classify the compact abelian actions on semifinite injective von Neumann algebras with factor fixed point algebra. The method uses that the (nonzero) eigenspaces of such an action contain unitaries which give rise to a classifying invariant.
Optimal $L\sp p$ and H\"older estimates for the Kohn solution of the $\overline\partial$-equation on strongly pseudoconvex domains
Der-Chen E.
Chang
273-304
Abstract: Let $\Omega$ be an open, relatively compact subset in $ {{\mathbf{C}}^{n + 1}}$, and assume the boundary of $\Omega$, $ \partial \Omega$, is smooth and strongly pseudoconvex. Let $\operatorname{Op}(K)$ be an integral operator with mixed type homogeneities defined on $\overline \Omega$: i.e., $K$ has the form as follows: $\displaystyle \sum\limits_{k,l \geq 0} {{E_k}{H_l},}$ where ${E_k}$ is a homogeneous kernel of degree $ - k$ in the Euclidean sense and ${H_l}$ is homogeneous of degree $- l$ in the Heisenberg sense. In this paper, we study the optimal ${L^p}$ and Hölder estimates for the kernel $ K$. We also use Lieb-Range's method to construct the integral kernel for the Kohn solution $\overline {{\partial^\ast}} {\mathbf{N}}$ of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to $ \overline {{\partial^\ast}} {\mathbf{N}}$. On the other hand, we prove Lieb-Range's kernel gains $1$ in "good" directions (hence gains $1/2$ in all directions) via Phong-Stein's theory. We also discuss the transferred kernel from the Siegel upper-half space to $\Omega$.
Nonlinearly equivalent representations of quaternionic $2$-groups
Washington
Mio
305-321
Abstract: We construct new examples of nonlinearly equivalent finite-dimensional real linear representations of quaternionic $2$-groups, which cannot be obtained from equivalent representations of cyclic groups by induction and composition techniques.
Coupled points in the calculus of variations and applications to periodic problems
Vera
Zeidan;
Pier Luigi
Zezza
323-335
Abstract: The aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions, and to develop second order necessary conditions for optimality. When one of the end points is fixed, our necessary conditions reduce to the known ones involving conjugate points. We also apply our results to the periodic problems of the calculus of variations.
Cauchy integral equalities and applications
Boo Rim
Choe
337-352
Abstract: We study bounded holomorphic functions $\pi$ on the unit ball ${B_n}$ of $ {\mathbb{C}^n}$ satisfying the following so-called Cauchy integral equalities: \begin{displaymath}\begin{array}{*{20}{c}} {C[{\pi ^{m + 1}}\bar \pi ] = {\gamma _m}{\pi ^m}} & {(m = 0,1,2, \ldots)} \end{array} \end{displaymath} for some sequence ${\gamma _m}$ depending on $\pi$. Among the applications are the Ahern-Rudin problem concerning the composition property of holomorphic functions on ${B_n}$, a projection theorem about the orthogonal projection of $ {H^2}({B_n})$ onto the closed subspace generated by holomorphic polynomials in $ \pi$, and some new information about the inner functions. In particular, it is shown that if we interpret ${\text{BMOA}}({B_n})$ as the dual of ${H^1}({B_n})$, then the map $g \to g \circ \pi$ is a linear isometry of ${\text{BMOA}}({B_1})$ into ${\text{BMOA}}({B_n})$ for every inner function $ \pi$ on ${B_n}$ such that $\pi (0) = 0$.
Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities
Alfonso
Castro;
Alexandra
Kurepa
353-372
Abstract: Let $p,\varphi :[0,T] \to R$ be bounded functions with $\varphi > 0$. Let $g:{\mathbf{R}} \to {\mathbf{R}}$ be a locally Lipschitzian function satisfying the superlinear jumping condition: (i) $ {\lim _{u \to - \infty }}(g(u)/u) \in {\mathbf{R}}$ (ii) $ {\lim _{u \to \infty }}(g(u)/{u^{1 + \rho }}) = \infty $ for some $\rho > 0$, and (iii) $ {\lim _{u \to \infty }}{(u/g(u))^{N/2}}(NG(\kappa u) - ((N - 2)/2)u \cdot g(u)) = \infty$ for some $ \kappa \in (0,1]$ where $ G$ is the primitive of $ g$. Here we prove that the number of solutions of the boundary value problem $\Delta u + g(u) = p(\left\Vert x\right\Vert) + c\varphi (\left\Vert x\right\Vert)$ for $x \in {{\mathbf{R}}^N}$ with $ \left\Vert x\right\Vert < T,u(x) = 0$ for $ \left\Vert x\right\Vert = T$ tends to $+ \infty$ when $c$ tends to $+ \infty$. The proofs are based on the "energy" and "phase plane" analysis.
A higher order invariant of differential manifolds
Gregory A.
Fredricks;
Peter B.
Gilkey;
Phillip E.
Parker
373-388
Abstract: We discuss conditions under which a lens space is $s$th order flat.
$\overline\partial\sb b$-equations on certain unbounded weakly pseudo-convex domains
Hyeonbae
Kang
389-413
Abstract: We found an explicit closed formula for the relative fundamental solution of $ {\bar \partial _b}$ on the surface ${H_k} = \{ ({z_1},{z_2}) \in {\mathbb{C}^2}:\operatorname{Im} {z_2} = \vert{z_1}{\vert^{2k}}\}$ . We then make estimates of the relative fundamental solution in terms of the nonisotropic metric associated with the surface. The estimates lead us to the regularity results. We also study the problem of finding weights $\omega$ so that ${\bar \partial _b}$ as an operator from $L_\omega ^2$ to ${L^2}$ has a closed range. We find the best possible weight among radial weights.
The transfer ideal of quadratic forms and a Hasse norm theorem mod squares
David B.
Leep;
Adrian R.
Wadsworth
415-432
Abstract: Any finite degree field extension $K/F$ determines an ideal ${\mathcal{T}_{K/F}}$ of the Witt ring $WF$ of $F$, called the transfer ideal, which is the image of any nonzero transfer map $WK \to WF$. The ideal ${\mathcal{T}_{K/F}}$ is computed for certain field extensions, concentrating on the case where $ K$ has the form $F\left({\sqrt {{a_1}} , \ldots ,\sqrt {{a_n}} } \right)$, ${a_i} \in F$. When $F$ and $K$ are global fields, we investigate whether there is a local global principle for membership in ${\mathcal{T}_{K/F}}$. This is shown to be equivalent to the existence of a "Hasse norm theorem mod squares," i.e., a local global principle for the image of the norm map $ {N_{K/F}}: {K^\ast}/{K^{\ast2}} \to {F^\ast}/{F^{\ast2}}$. It is shown that such a Hasse norm theorem holds whenever $K = F(\sqrt{a_1},\ldots,\sqrt{a_n})$, although it does not always hold for more general extensions of global fields, even some Galois extensions with group $ \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$.