The Schwartz space of a general semisimple Lie group. II. Wave packets associated to Schwartz functions
Rebecca A.
Herb
1-69
Abstract: Let $G$ be a connected semisimple Lie group. If $G$ has finite center, Harish-Chandra used Eisenstein integrals to construct Schwartz class wave packets of matrix coefficients and showed that every $ K$-finite function in the Schwartz space is a finite sum of such wave packets. This paper is the second in a series which generalizes these results of Harish-Chandra to include the case that $ G$ has infinite center. In this paper, the Plancherel theorem is used to decompose $ K$-compact Schwartz class functions (those with $K$-types in a compact set) as finite sums of wave packets. A new feature of the infinite center case is that the individual wave packets occurring in the decomposition of a Schwartz class function need not be Schwartz class. These wave packets are studied to obtain necessary conditions for a wave packet of Eisenstein integrals to occur in the decomposition of a Schwartz class function. Applied to the case that $f$ itself is a single wave packet, the results of this paper yield a complete characterization of Schwartz class wave packets.
The construction of analytic diffeomorphisms for exact robot navigation on star worlds
Elon
Rimon;
Daniel E.
Koditschek
71-116
Abstract: A Euclidean Sphere World is a compact connected submanifold of Euclidean $n$-space whose boundary is the disjoint union of a finite number of $(n - 1)$ dimensional Euclidean spheres. A Star World is a homeomorph of a Euclidean Sphere World, each of whose boundary components forms the boundary of a star shaped set. We construct a family of analytic diffeomorphisms from any analytic Star World to an appropriate Euclidean Sphere World "model." Since our construction is expressed in closed form using elementary algebraic operations, the family is effectively computable. The need for such a family of diffeomorphisms arises in the setting of robot navigation and control. We conclude by mentioning a topological classification problem whose resolution is critical to the eventual practicability of these results.
Linear series with an $N$-fold point on a general curve
David
Schubert
117-124
Abstract: A linear series $(V,\mathcal{L})$ on a curve $X$ has an $N$-fold point along a divisor $D$ of degree $N$ if $\dim (V \cap {H^0}\;(X,\mathcal{L}\,(- D))) \geq \dim \;V - 1$. The dimensions of the families of linear series with an $N$-fold point are determined for general curves.
Weighted Sobolev-Poincar\'e inequalities and pointwise estimates for a class of degenerate elliptic equations
Bruno
Franchi
125-158
Abstract: In this paper we prove a Sobolev-Poincaré inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Hölder regularity of the weak solutions follows in a standard way.
A dense set of operators with tiny commutants
Domingo A.
Herrero
159-183
Abstract: For a (bounded linear) operator $T$ on a complex, separable, infinite-dimensional Hilbert space $ \mathcal{H}$, let $\mathcal{A}\,(T)$ and ${\mathcal{A}^a}(T)$ denote the weak closure of the polynomials in $T$ and, respectively, the weak closure of the rational functions with poles outside the spectrum of $ T$. Let $\mathcal{A}^{\prime}(T)$ and $\mathcal{A}''(T)$ denote the commutant and, respectively, the double commutant of $T$. We say that $T$ has a tiny commutant if $ \mathcal{A}^{\prime}(T)= {\mathcal{A}^a}(T)$. By constructing a large family of "models" and by using standard techniques of approximation, it is shown that $T \in \mathcal{L}\,(\mathcal{H}):T$ has a tiny commutant is norm-dense in the algebra $ \mathcal{L}\,(\mathcal{H})$ of all operators acting on $ \mathcal{H}$. Other related results: Let $ \operatorname{Lat}\;\mathcal{B}$ denote the invariant subspace lattice of a subalgebra $\mathcal{B}$ of $ \mathcal{L}(\mathcal{H})$. For a Jordan curve $\gamma \subset {\mathbf{C}}$, let $\hat \gamma$ denote the union of $\gamma$ and its interior; for $T \in \mathcal{L}\;(\mathcal{H})$, let ${\rho _{s - F}}\,(T)= \{ \lambda \in {\mathbf{C}}:\lambda - T$ is a semi-Fredholm operator, and let $ \rho _{s - F}^ + (T)(\rho _{s - F}^ - (T))= \{ \lambda \in {\rho _{s - F}}(T):{\text{ind}}(\lambda - T) > 0\;(< 0,{\text{resp.)\} }}$. With this notation in mind, it is shown that ${\{ T \in \mathcal{L}(\mathcal{H}):\mathcal{A}(T)= {\mathcal{A}^a}(T)\} ^ - } ... ...peratorname{Lat}\;{\mathcal{A}^a}(T)\} ^ - }= \{ A \in \mathcal{L}(\mathcal{H})$ if $\gamma$ (Jordan curve) $\subset\rho _{s - F}^ \pm (A)$, then $\hat \gamma \subset \sigma (A)\}$; moreover, $\{ A \in \mathcal{L}(\mathcal{H})$: if $\gamma$ (Jordan curve) $\subset\rho _{s - F}^ \pm (A)$, then ${\text{ind}}(\lambda - A)$ is constant on $\hat \gamma \cap {\rho _{s - F}}(A)\} \subset {\{ T \in \mathcal{L}(\mathcal{H... ...orname{Lat}\;\mathcal{A}^{\prime}(T)\} \subset\{ A \in \mathcal{L}(\mathcal{H})$: if $\gamma$ (Jordan curve) $\subset\rho _{s - F}^ \pm (A)$, then $ \hat \gamma \cap {\rho _{s - F}}(A) \subset\rho _{s - F}^ \pm (A)\} \subset \{ T \in \mathcal{L}(\mathcal{H}):\mathcal{A}(T)= {\mathcal{A}^a}(T)\}$. (The first and the last inclusions are proper.) The results also include a partial analysis of $ \operatorname{Lat}\;\mathcal{A}''(T)$.
Automorphisms and twisted forms of generalized Witt Lie algebras
William C.
Waterhouse
185-200
Abstract: We prove that the automorphisms of the generalized Witt Lie algebras $ W(m,{\mathbf{n}})$ over arbitrary commutative rings of characteristic $p \geq 3$ all come from automorphisms of the algebras on which they are defined as derivations. By descent theory, this result then implies that if a Lie algebra over a field becomes isomorphic to $ W(m,{\mathbf{n}})$ over the algebraic closure, it is a derivation algebra of the type studied long ago by Ree. Furthermore, all isomorphisms of those derivation algebras are induced by isomorphisms of their underlying associative algebras.
Alexander duality and Hurewicz fibrations
Steven C.
Ferry
201-219
Abstract: We explore conditions under which the restriction of the projection map $ p:{S^n} \times B \to B$ to an open subset $ U \subset S^n \times B$ is a Hurewicz fibration. As a consequence, we exhibit Hurewicz fibrations $p:E \to I$ such that: (i) $p:E \to I$ is not a locally trivial bundle, (ii) $p^{ - 1}(t)$ is an open $n$-manifold for each $t$, and (iii) $p\; \circ \;{\text{proj:E}} \times {R^1} \to I$ is a locally trivial bundle. The fibers in our examples are distinguished by having nonisomorphic fundamental groups at infinity. We also show that when the fibers of a Hurewicz fibration with open $ n$-manifold fibers have finitely generated $ (n - 1){\text{st}}$ homology, then all fibers have the same finite number of ends. This last shows that the punctured torus and the thrice punctured two-sphere cannot both be fibers of a Hurewicz fibration $p:E \to I$ with open $2$-manifold fibers.
On the relative reflexivity of finitely generated modules of operators
Bojan
Magajna
221-249
Abstract: Let $\mathcal{R}$ be a von Neumann algebra on a Hilbert space $ \mathcal{H}$ with commutant $ \mathcal{R}^{\prime}$ and centre $ \mathcal{C}$. For each subspace $ \mathcal{S}$ of $\mathcal{R}$ let $ \operatorname{ref}_\mathcal{R}\,(\mathcal{S})$ be the space of all $B \in \mathcal{R}$ such that $XBY= 0$ for all $X,Y \in \mathcal{R}$ satisfying $X\,\mathcal{S}\,Y = 0$. If $ \operatorname{ref}_\mathcal{R}\,(\mathcal{S})= \mathcal{S}$, the space $\mathcal{S}$ is called $ \mathcal{R}$-reflexive. (If $\mathcal{R}= \mathcal{B}(\mathcal{H})$ and $\mathcal{S}$ is an algebra containing the identity operator, $ \mathcal{R}$-reflexivity reduces to the usual reflexivity in operator theory.) The main result of the paper is the following: if $\mathcal{S}$ is one-dimensional, or if $\mathcal{S}$ is arbitrary finite-dimensional but $\mathcal{R}$ has no central portions of type $ {{\text{I}}_n}$ for $ n > 1$, then the space $\overline {\mathcal{C}\mathcal{S}}$ is $ \mathcal{R}$-reflexive and the space $\overline {\mathcal{R}^{\prime}\,\mathcal{S}}$ is $ \mathcal{B}(\mathcal{H})$-reflexive, where the bar denotes the closure in the ultraweak operator topology. If $\mathcal{R}$ is a factor, then $ \mathcal{R}^{\prime}\,\mathcal{S}$ is closed in the weak operator topology for each finite-dimensional subspace $\mathcal{S}$ of $ \mathcal{R}$.
$q$-tensor space and $q$-Weyl modules
Richard
Dipper;
Gordon
James
251-282
Abstract: We obtain the irreducible representations of the $q$-Schur algebra, motivated by the fact that these representations give all the irreducible representations of $G{L_n}(q)$ in the nondescribing characteristic. The irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction.
A modified Schur algorithm and an extended Hamburger moment problem
Olav
Njåstad
283-311
Abstract: An algorithm for a Pick-Nevanlinna problem where the interpolation points coalesce into a finite set of points on the real line is introduced, its connection with certain multipoint Padé approximation problems is discussed, and the results are used to obtain the solutions of an extended Hamburger moment problem.
Almost tangent and cotangent structures in the large
G.
Thompson;
U.
Schwardmann
313-328
Abstract: We examine some global properties of integrable almost tangent and cotangent manifolds. In particular, we extend several results which essentially characterize tangent and cotangent bundles as, respectively, regular almost tangent and cotangent structures.
Solving ordinary differential equations in terms of series with real exponents
D. Yu.
GrigorЬev;
M. F.
Singer
329-351
Abstract: We generalize the Newton polygon procedure for algebraic equations to generate solutions of polynomial differential equations of the form $\sum\nolimits_{i = 0}^\infty {{\alpha _i}{x^{{\beta _i}}}}$ where the $ {\alpha _i}$ are complex numbers and the $ {\beta _i}$ are real numbers with ${\beta _0} > {\beta _1} > \cdots$. Using the differential version of the Newton polygon process, we show that any such a series solution is finitely determined and show how one can enumerate all such solutions of a given polynomial differential equation. We also show that the question of deciding if a system of polynomial differential equations has such a power series solution is undecidable.
First steps in descriptive theory of locales
John
Isbell
353-371
Abstract: F. Hausdorff and D. Montgomery showed that a subspace of a completely metrizable space is developable if and only if it is ${F_\sigma }$ and $ {G_\delta }$. This extends to arbitrary metrizable locales when " ${F_\sigma }$" and " $ {G_\delta }$" are taken in the localic sense (countable join of closed, resp. meet of open, sublocales). In any locale, the developable sublocales are exactly the complemented elements of the lattice of sublocales. The main further results of this paper concern the strictly pointless relative theory, which exists because--always in metrizable locales-- there exist nonzero pointless-absolute $ {G_\delta}^{\prime}{\text{s}}$, ${G_\delta }$ in every pointless extension. For instance, the pointless part $ {\text{pl}}({\mathbf{R}})$ of the real line is characterized as the only nonzero zero-dimensional separable metrizable pointless-absolute $ {G_\delta }$. There is no nonzero pointless-absolute $ {F_\sigma }$. The pointless part of any metrizable space is, if not zero, second category, i.e. not a countable join of nowhere dense sublocales.
Coefficient ideals
Kishor
Shah
373-384
Abstract: Let $R$ be a $d$-dimensional Noetherian quasi-unmixed local ring with maximal ideal $M$ and an $M$-primary ideal $I$ with integral closure $ \overline I$. We prove that there exist unique largest ideals $ {I_k}$ for $1 \leq k \leq d$ lying between $I$ and $ \overline I$ such that the first $k + 1$ Hilbert coefficients of $ I$ and ${I_k}$ coincide. These coefficient ideals clarify some classical results related to $\overline I$. We determine their structure and immediately apply the structure theorem to study the associated primes of the associated graded ring of $ I$.
On complete congruence lattices of complete lattices
G.
Grätzer;
H.
Lakser
385-405
Abstract: The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In this paper, we characterize this lattice as a complete lattice. In other words, for a complete lattice $L$, we construct a complete lattice $ K$ such that $ L$ is isomorphic to the lattice of complete congruence relations of $K$. Regarding $K$ as an infinitary algebra, this result strengthens the characterization theorem of congruence lattices of infinitary algebras of G. Grätzer and W. A. Lampe. In addition, we show how to construct $ K$ so that it will also have a prescribed automorphism group.
Definable singularity
William J.
Mitchell
407-426
Abstract: The main result of this paper is a characterization of singular cardinals in terms of the core model, assuming that there is no model of $\exists \kappa \,o(\kappa)= {\kappa ^{ + + }}$. This characterization is used to prove a result in infinitary Ramsey theory. In the course of the proof we develop a simplified statement of the covering lemma for sequences of measures which avoids the use of mice. We believe that this development will be capable of isolating almost all applications of the covering lemma from the detailed structure of the core model.
Control of degenerate diffusions in ${\bf R}\sp d$
Omar
Hijab
427-448
Abstract: An optimal regularity result is established for the viscosity solution of the degenerate elliptic equation $\displaystyle - Av + F(x,\upsilon ,D\upsilon )= 0,$ $A= \frac{1}{2}\sum {{a_{ij}}(x){\partial ^2}/\partial \,x{_i}\,\partial \,{x_j}}, x \in {{\mathbf{R}}^d}$. We assume the equation is of Bellman type, i.e. $F(x,\upsilon ,p)= {\sup _{u \in U}}[b(x,u) \cdot p + c(x,u)\upsilon - f(x,u)]$, $U \subset{{\mathbf{R}}^d}$. If we set $\lambda \equiv {\inf _{x,u}}c(x,u)$, then there exists $ {\lambda _0} \geq 0$ such that $0 < \lambda < {\lambda _0}$ implies $\upsilon$ is Hölder, while $\lambda > {\lambda _0}$ implies $ \upsilon$ is Lipschitz. The following is established: Suppose the equation is also of Lipschitz type, i.e. suppose there is a Lipschitz function $ u(x,\upsilon ,p)$ such that the supremum in $ F\,(x,\upsilon ,p)$ is uniquely attained at $ u= u\,(x,\upsilon ,p)$; then there exists ${\lambda _1} > {\lambda _0}$ such that $\lambda > {\lambda _1}$ implies $ \upsilon$ is ${C^{1,1}},$ i.e. $D\upsilon$ exists and is Lipschitz.
Flowbox manifolds
J. M.
Aarts;
L. G.
Oversteegen
449-463
Abstract: A separable and metrizable space $X$ is called a flowbox manifold if there exists a base for the open sets each of whose elements has a product structure with the reals $\operatorname{Re}$ as a factor such that a natural consistency condition is met. We show how flowbox manifolds can be divided into orientable and nonorientable ones. We prove that a space $X$ is an orientable flowbox manifold if and only if $X$ can be endowed with the structure of a flow without restpoints. In this way we generalize Whitney's theory of regular families of curves so as to include self-entwined curves in general separable metric spaces. All spaces under consideration are separable and metrizable.