Spherical functions on Cartan motion groups
Cary
Rader
1-45
Abstract: This paper gives a reasonably complete treatment of harmonic analysis on Cartan motion groups. Included is an explicit parameterization of irreducible spherical functions of general $K$-type, and of the nonunitary dual (and its topology). Also included is the explicit Plancherel measure, the Paley Wiener theorem, and an asymptotic expansion of general matrix entries. (These are generalized Bessel functions.) However the main result is Theorem 19, a technical result which measures the size of the centralizer of $K$ in the universal enveloping algebra of the corresponding reductive group.
Holomorphic maps from ${\bf C}\sp n$ to ${\bf C}\sp n$
Jean-Pierre
Rosay;
Walter
Rudin
47-86
Abstract: We study holomorphic mappings from $ {{\mathbf{C}}^n}$ to ${{\mathbf{C}}^n}$, and especially their action on countable sets. Several classes of countable sets are considered. Some new examples of Fatou-Bieberbach maps are given, and a nondegenerate map is constructed so that the volume of the image of ${{\mathbf{C}}^n}$ is finite. An Appendix is devoted to the question of linearization of contractions.
The existence of least area surfaces in $3$-manifolds
Joel
Hass;
Peter
Scott
87-114
Abstract: This paper presents a new and unified approach to the existence theorems for least area surfaces in $3$-manifolds.
Decay rates of Fourier transforms of curves
B. P.
Marshall
115-126
Abstract: Let $d\mu$ be a smooth measure on a nondegenerate curve in $ {{\mathbf{R}}^n}$. This paper examines the decay rate of spherical averages of its Fourier transform $ \widehat{d\mu }$. Thus estimates of the following form are considered: $\displaystyle {\left( {\int_{{\sum _r}} {\vert\widehat{d\mu }(\xi {\vert^p}d\xi } } \right)^{1/p}} \leqslant C{r^{ - \sigma }}\vert\vert f\vert\vert$ where ${\sum _r} = \{ \xi \in {{\mathbf{R}}^n}:\vert\xi \vert = r\}$.
Iterating the basic construction
Mihai
Pimsner;
Sorin
Popa
127-133
Abstract: Let $N \subset M$ be a pair of type II$_{1}$ factors with finite Jones' index and $N \subset M \subset {M_1} \subset {M_2} \subset \cdots \subset {M_n} \subset \cdots \subset {M_{2n + 1}}$ be the associated tower of type II$_{1}$ factors obtained by iterating Jones' basic construction. We give an explicit formula of a projection in ${M_{2n + 1}}$ which implements the conditional expectation of ${M_n}$ onto $N$, thus showing that $ {M_{2n + 1}}$ comes naturally from the basic construction associated to the pair $N \subset {M_n}$. From this we deduce several properties of the relative commutant
The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set
Murray
Gerstenhaber;
Samuel D.
Schack
135-165
Abstract: To each presheaf (over a poset) of associative algebras $\mathbb{A}$ we associate an algebra $\mathbb{A}!$. We define a full exact embedding of the category of (presheaf) $ \mathbb{A}$-bimodules in that of $ \mathbb{A}!$-bimodules. We show that this embedding preserves neither enough (relative) injectives nor enough (relative) projectives, but nonetheless preserves (relative) Yoneda cohomology. The cohomology isomorphism links the deformations of manifolds, algebraic presheaves, and algebras. It also implies that the cohomology of any triangulable space is isomorphic to the Hochschild cohomology of an associative algebra. (The latter isomorphism preserves all known cohomology operations.) We conclude the paper by exhibiting for each associative algebra and triangulable space a "product" which is again an associative algebra.
Stable extensions of homeomorphisms on the pseudo-arc
Judy
Kennedy
167-178
Abstract: We prove the following: Theorem. If $P'$ is a proper subcontinuum of the pseudoarc $P,\,h'$ is a homeomorphism from $ P'$ onto itself, and $\Theta$ is an open set in $P$ that contains $P'$, then there is a homeomorphism $ h$ from $ P$ onto itself such that $h\vert P' = h'$ and $h(x) = x$ for $x \notin \Theta$.
A construction of pseudo-Anosov homeomorphisms
Robert C.
Penner
179-197
Abstract: We describe a generalization of Thurston's original construction of pseudo-Anosov maps on a surface $F$ of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map $f:F \to F$ arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by $f$. To finally prove that $f$ is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov.
Stable maps into free $G$-spaces
J. P. C.
Greenlees
199-215
Abstract: In this paper we introduce a systematic method for calculating the group of stable equivariant maps $ {[X,\,Y]^G}$ [3, 18] into a $G$-free space or spectrum $Y$. In fact the method applies without restriction on $X$ whenever $G$ is a $p$-group and $Y$ is $p$-complete and satisfies standard finiteness assumptions. The method is an Adams spectral sequence based on a new equivariant cohomology theory ${c^{\ast}}(X)$ which we introduce in $\S1$. This spectral sequence is quite calculable and provides a natural generalisation of the classical Adams spectral sequence based on ordinary $ \bmod p$ cohomology. It also geometrically realises certain inverse limits of nonequivariant Adams spectral sequences which have been useful in the study of the Segal conjecture [19, 5, 21, 9].
Moduli spaces of Riemann surfaces of genus two with level structures. I
Ronnie
Lee;
Steven H.
Weintraub
217-237
Abstract: The cohomology of modular varieties defined by congruence subgroups of $ {\operatorname{Sp} _4}({\mathbf{Z}})$ whose levels lie between $2$ and $4$ is studied. Using a counting argument and the techniques of zeta functions, the authors completely determine the cohomology of a particular variety of this type.
On the Wiener criterion and quasilinear obstacle problems
Juha
Heinonen;
Tero
Kilpeläinen
239-255
Abstract: We study the Wiener criterion and variational inequalities with irregular obstacles for quasilinear elliptic operators $ A$, $ A(x,\,\nabla u) \cdot \nabla u \approx \vert\nabla u{\vert^p}$, in ${{\mathbf{R}}^n}$. Local solutions are continuous at Wiener points of the obstacle function; if $ p > n - 1$, the converse is also shown to be true. If $p > n - 1$, then a characterization of the thinness of a set at a point is given in terms of $A$-superharmonic functions.
An Erd\H os-Wintner theorem for differences of additive functions
Adolf
Hildebrand
257-276
Abstract: An Erdös-Wintner type criterion is given for the convergence of the distributions ${D_x}(z) = {[x]^{ - 1}}\char93 \{ 1 \leqslant n \leqslant x:\,f(n + 1) - f(n) \leqslant z\}$, where $f$ is a real-valued additive function. A corollary of this result is that an additive function $ f$, for which $f(n + 1) - f(n)$ tends to zero on a set of density one, must be of the form $f = \lambda \log$ for some constant $\lambda$. This had been conjectured by Erdős.
Decompositions of continua over the hyperbolic plane
James T.
Rogers
277-291
Abstract: The following theorem is proved. Theorem. Let $X$ be a homogeneous continuum such that ${H^1}(X) \ne 0$. If $\mathcal{G}$ is the collection of maximal terminal proper subcontinua of $X$, then (1) The collection $\mathcal{G}$ is a monotone, continuous, terminal decomposition of $X$, (2) The nondegenerate elements of $\mathcal{G}$ are mutually homeomorphic, indecomposable, cell-like, terminal, homogeneous continua of the same dimension as $X$, (3) The quotient space is a homogeneous continuum, and (4) The quotient space does not contain any proper, nondegenerate, terminal subcontinuum. This theorem is related to the Jones' Aposyndetic Decomposition Theorem. The proof involves the hyperbolic plane and a subset of the circle at $\infty$, called the set of ends of a component of the universal cover of $X$.
Borel orderings
Leo
Harrington;
David
Marker;
Saharon
Shelah
293-302
Abstract: We show that any Borel linear order can be embedded in an order preserving way into $ {2^\alpha }$ for some countable ordinal $\alpha$ and that any thin Borel partial order can be written as a union of countably many Borel chains.
On a class of functionals invariant under a ${\bf Z}\sp n$ action
Paul H.
Rabinowitz
303-311
Abstract: Consider a system of ordinary differential equations of the form $ ({\ast})$ $\displaystyle \ddot q + {V_q}(t,\,q) = f(t)$ where $f$ and $V$ are periodic in $t$, $V$ is periodic in the components of $q = ({q_1}, \ldots ,{q_n})$, and the mean value of $f$ vanishes. By showing that a corresponding functional is invariant under a natural ${{\mathbf{Z}}^n}$ action, a simple variational argument yields at least $n + 1$ distinct periodic solutions of (*). More general versions of (*) are also treated as is a class of Neumann problems for semilinear elliptic partial differential equations.
Amalgamation for inverse and generalized inverse semigroups
T. E.
Hall
313-323
Abstract: For any amalgam $(S,\,T;\,U)$ of inverse semigroups, it is shown that the natural partial order on $S{{\ast}_U}T$, the (inverse semigroup) free product of $S$ and $T$ amalgamating $U$, has a simple form on $S \cup T$. In particular, it follows that the semilattice of $ S{{\ast}_U}T$ is a bundled semilattice of the corresponding semilattice amalgam $ (E(S),\,E(T);\,E(U))$; taken jointly with a result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered.
Homology of smooth splines: generic triangulations and a conjecture of Strang
Louis J.
Billera
325-340
Abstract: For $\Delta$ a triangulated $ d$-dimensional region in $ {{\mathbf{R}}^d}$, let $S_m^r(\Delta )$ denote the vector space of all $ {C^r}$ functions $ F$ on $\Delta$ that, restricted to any simplex in $\Delta$, are given by polynomials of degree at most $m$. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds $\Delta$ in the plane, getting lower bounds on the dimension of $ S{}_m^r(\Delta )$ for all $ r$. For $r = 1$, we prove a conjecture of Strang concerning the generic dimension of the space of $ {C^1}$ splines over a triangulated manifold in $ {{\mathbf{R}}^2}$. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.
Normal structure in dual Banach spaces associated with a locally compact group
Anthony To Ming
Lau;
Peter F.
Mah
341-353
Abstract: In this paper we investigated when the dual of a certain function space defined on a locally compact group has certain geometric properties. More particularly, we asked when weak$^{*}$ compact convex subsets in these spaces have normal structure, and when the norm of these spaces satisfies one of several types of Kadec-Klee property. As samples of the results we have obtained, we have proved, among other things, the following two results: (1) The measure algebra of a locally compact group has weak$^{*}$-normal structure iff it has property SUKK$ ^{*}$ iff it has property SKK$^{*}$ iff the group is discrete; (2) Among amenable locally compact groups, the Fourier-Stieltjes algebra has property SUKK$^{*}$ iff it has property SKK$^{*}$ iff the group is compact. Consequently the Fourier-Stieltjes algebra has weak$ ^{*}$-normal structure when the group is compact.
Extremal analytic discs with prescribed boundary data
Chin-Huei
Chang;
M. C.
Hu;
Hsuan-Pei
Lee
355-369
Abstract: This paper concerns the existence and uniqueness of extremal analytic discs with prescribed boundary data in a bounded strictly linearly convex domain $D$ in $ {{\mathbf{C}}^n}$. We prove that for any two distinct points $p$, $q$ in $\partial D$ (respectively, $p \in \partial D$ and a vector $v$ such that $\sqrt { - 1} v \in {T_p}(\partial D)$ and $\langle v,\,\overline \nu (p)\rangle = \sum\nolimits_1^n {{v_j}{{\overline \nu }_j}(p) > 0}$ where $ \nu (p)$ is the outward normal to $\partial D$ at $p$) there exists an extremal analytic disc $ f$ passing through $ p$, $q$ if $\partial D \in {C^k}$, $k \geqslant 3$ (respectively, $ f(1) = p$, $ \partial D \in {C^k}$, $k \geqslant 14$). Consequently, we can foliate $\overline D$ with these extremal analytic discs.
Banach spaces with separable duals
M.
Zippin
371-379
Abstract: It is proved that every Banach space with a separable dual embeds into a space with a shrinking basis. It follows that every separable reflexive space can be embedded in a reflexive space with a basis.
Finite covers of $3$-manifolds containing essential tori
John
Luecke
381-391
Abstract: It is shown in this paper that if a Haken $3$-manifold contains an incompressible torus that is not boundary-parallel then either it has a finite cover that is a torus-bundle over the circle or it has finite covers with arbitrarily large first Betti number.
Nonconvex variational problems with general singular perturbations
Nicholas C.
Owen
393-404
Abstract: We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of $\Gamma $-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in ${L^1}$ as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem.
A characterization of the weakly continuous polynomials in the method of compensated compactness
Robert C.
Rogers;
Blake
Temple
405-417
Abstract: We present a sufficient condition for weak continuity in the method of compensated compactness. The condition links weak continuity to the structure of the wave cone and the characteristic set for polynomials of degree greater than two. The condition applies to all the classical examples of weakly continuous functions and generalizes the Quadratic Theorem and the Wedge Product Theorem. In fact, the condition reduces to the Legendre-Hadamard Necessary Condition when the polynomial is quadratic, and also whenever a certain orthogonality condition is satisfied. The condition is derived by isolating conditions under which the quadratic theorem can be iterated.
Bifurcation phenomena associated to the $p$-Laplace operator
Mohammed
Guedda;
Laurent
Véron
419-431
Abstract: We determine the structure of the set of the solutions $u$ of $- {(\vert{u_x}{\vert^{p - 2}}{u_x})_x} + f(u) = \lambda \vert u{\vert^{p - 2}}u$ on $(0,\,1)$ such that $ u(0) = u(1) = 0$, where $p > 1$ and $\lambda \in {\mathbf{R}}$. We prove that the solutions with $k$ zeros are unique when $1 < p \leqslant 2$ but may not be so when $ p > 2$.