Borel classes and closed games: Wadge-type and Hurewicz-type results
A.
Louveau;
J.
Saint-Raymond
431-467
Abstract: For each countable ordinal $\xi$ and pair $ ({A_0},\,{A_1})$ of disjoint analytic subsets of $ {2^\omega }$, we define a closed game ${J_\xi }({A_0},\,{A_1})$ and a complete $\Pi _\xi ^0$ subset ${H_\xi }$ of $ {2^\omega }$ such that (i) a winning strategy for player I constructs a $\sum _\xi ^0$ set separating ${A_0}$ from ${A_1}$; and (ii) a winning strategy for player II constructs a continuous map $\varphi :{2^\omega } \to {A_0} \cup {A_1}$ with ${\varphi ^{ - 1}}({A_0}) = {H_\xi }$. Applications of this construction include: A proof in second order arithmetics of the statement "every $\Pi _\xi ^0$ non $ \sum _\xi ^0$ set is $\Pi _\xi ^0$-complete"; an extension to all levels of a theorem of Hurewicz about $\sum _2^0$ sets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieff's and Wadge's hierarchies.
Longer than average intervals containing no primes
A. Y.
Cheer;
D. A.
Goldston
469-486
Abstract: We present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes. The first method is based on an argument of Erdös which uses a sieve upper bound for prime twins to bound the density function for gaps between primes. The second method uses known results about the first three moments for the distribution of intervals with a given number of primes. Better results are obtained by assuming that the first $n$ moments are Poisson. The related problem of longer than average gaps between primes is also considered.
Manifolds on which only tori can act
Kyung Bai
Lee;
Frank
Raymond
487-499
Abstract: A list of various types of connected, closed oriented manifolds are given. Each of the manifolds support some of the well-known compact transformation group properties enjoyed by aspherical manifolds. We list and describe these classes and their transformation group properties in increasing generality. We show by various examples that these implications can never be reversed. This establishes a hierarchy in terms of spaces in one direction and the properties they enjoy in the opposite direction.
Comparison between analytic capacity and the Buffon needle probability
Takafumi
Murai
501-514
Abstract: We show that analytic capacity and the Buffon needle probability are not comparable.
Weak limits of projections and compactness of subspace lattices
Bruce H.
Wagner
515-535
Abstract: A strongly closed lattice of projections on a Hilbert space is compact if the associated algebra of operators has a weakly dense subset of compact operators. If the lattice is commutative, there are necessary and sufficient conditions for compactness, one in terms of the structure of the lattice, and the other in terms of a measure on the lattice. There are many examples of compact lattices, and two main types of examples of noncompact lattices. Compactness is also related to the study of weak limits of certain projections.
Dense morphisms in commutative Banach algebras
Gustavo
Corach;
Fernando Daniel
Suárez
537-547
Abstract: Using a new notion of stability we compute exactly the stable rank of the polydisc algebra, extend Oka's extension theorem to $ n$-tuples of functions without common zeros and give an estimation for a question raised by Swan concerning the stable rank of a dense subalgebra of a given Banach algebra.
Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups
L.
Corwin;
F. P.
Greenleaf;
G.
Grélaud
549-583
Abstract: Let $K$ be a Lie subgroup of the connected, simply connected nilpotent Lie group $G$, and let $ \mathfrak{k}$, $\mathfrak{g}$ be the corresponding Lie algebras. Suppose that $\sigma$ is an irreducible unitary representation of $K$. We give an explicit direct integral decomposition of ${\operatorname{Ind} _{k \to G}}\sigma$ into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between $G^\wedge$ and the coadjoint orbits in $ {\mathfrak{g}^{\ast}}$ (and similarly for $ K^\wedge,\,{\mathfrak{k}^{\ast}}$). Let $P:{\mathfrak{k}^{\ast}} \to {\mathfrak{g}^{\ast}}$ be the canonical projection, let $ {\mathcal{O}_\sigma } \subset {\mathfrak{k}^{\ast}}$ be the orbit corresponding to $\sigma$, and, for $ \pi \in G^\wedge$, let ${\mathcal{O}_\pi } \subset {\mathfrak{g}^{\ast}}$ be the corresponding orbit. The main result of the paper says essentially that $ \pi \in G^\wedge$ appears in the direct integral iff ${P^{ - 1}}({\mathcal{O}_\sigma })$ meets $ {\mathcal{O}_\pi }$; the multiplicity of $\pi$ is the number of ${\operatorname{Ad} ^{\ast}}(K)$-orbits in ${\mathcal{O}_\pi } \cap {P^{ - 1}}({\mathcal{O}_\sigma })$. There is also a natural description of the measure class in the integral.
Invariant subspaces in Banach spaces of analytic functions
Stefan
Richter
585-616
Abstract: We study the invariant subspace structure of the operator of multiplication by $z$, ${M_z}$, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class ${\mathcal{B}_1}(\overline \Omega )$. We say that an invariant subspace $ \mathcal{M}$ satisfies $ \operatorname{cod} \mathcal{M} = 1$ if $ z\mathcal{M}$ has codimension one in $ \mathcal{M}$. We give various conditions on invariant subspaces which imply that $ \operatorname{cod} \mathcal{M} = 1$. In particular, we give a necessary and sufficient condition on two invariant subspaces $\mathcal{M}$, $ \mathcal{N}$ with $\operatorname{cod} \mathcal{M} = \operatorname{cod} \mathcal{N} = 1$ so that their span again satisfies $ \operatorname{cod} (\mathcal{M} \vee \mathcal{N}) = 1$. This result will be used to show that any invariant subspace of the Bergman space $ L_a^p,\,p \geqslant 1$, which is generated by functions in $L_a^{2p}$, must satisfy $ \operatorname{cod} \mathcal{M} = 1$. For an invariant subspace $\mathcal{M}$ we then consider the operator $S = M_z^{\ast}\vert{\mathcal{M}^ \bot }$. Under some extra assumption on the domain of holomorphy we show that the spectrum of $S$ coincides with the approximate point spectrum iff $\operatorname{cod} \mathcal{M} = 1$. Finally, in the last section we obtain a structure theorem for invariant subspaces with $\operatorname{cod} \mathcal{M} = 1$. This theorem applies to Dirichlet-type spaces.
Nonlinear stability of vortex patches
Yun
Tang
617-638
Abstract: To establish the nonlinear (Liapunov) stability of both circular and elliptical vortex patches in the plane for the nonlinear dynamical system generated by the two-dimensional Euler equations of incompressible, inviscid hydrodynamics. This is accomplished by using a relative variational principle in terms of energy function. A counterexample shows that our result in the case of an elliptical vortex patch is the best one that can be attained by applying the energy estimate.
On the elliptic equations $\Delta u=K(x)u\sp \sigma$ and $\Delta u=K(x)e\sp {2u}$
Kuo-Shung
Cheng;
Jenn-Tsann
Lin
639-668
Abstract: We give some nonexistence results for the equations $\Delta u = K(x){u^\sigma }$ and $\Delta u = K(x){e^{2u}}$ for $K(x) \geqslant 0$.
Weighted norm estimates for Sobolev spaces
Martin
Schechter
669-687
Abstract: We give sufficient conditions for estimates of the form $\displaystyle {\int {\left\vert {u(x)} \right\vert} ^q}d\mu (x) \leqslant C\left\Vert u \right\Vert _{s,p}^1,\qquad u \in {H^{s,p}},$ to hold, where $ \mu (x)$ is a measure and $ {\left\Vert u \right\Vert _{s,p}}$ is the norm of the Sobolev space ${H^{s,p}}$. If $d\mu = dx$, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.
Linear series with cusps and $n$-fold points
David
Schubert
689-703
Abstract: A linear series $ (V,\,\mathcal{L})$ on a curve $X$ has an $n$-fold point along a divisor $D$ of degree $n$ if $\dim (V \cap {H^0}(X,\,\mathcal{L}( - D))) \geqslant \dim (V) - 1$. The linear series has a cusp of order $e$ at a point $P$ if $\dim (V \cap {H^0}(X,\,\mathcal{L}( - (e + 1)P))) \geqslant \dim (V) - 1$. Linear series with cusps and $n$-fold points are shown to exist if certain inequalities are satisfied. The dimensions of the families of linear series with cusps are determined for general curves.
Chaotic maps with rational zeta function
H. E.
Nusse
705-719
Abstract: Fix a nontrivial interval $X \subset {\mathbf{R}}$ and let $f \in {C^1}(X,\,X)$ be a chaotic mapping. We denote by ${A_\infty }(f)$ the set of points whose orbits do not converge to a (one-sided) asymptotically stable periodic orbit of $f$ or to a subset of the absorbing boundary of $ X$ for $f$. A. We assume that $f$ satisfies the following conditions: (1) the set of asymptotically stable periodic points for $ f$ is compact (an empty set is allowed), and (2) $A{\,_\infty }(f)\,$ is compact, $f$ is expanding on ${A_\infty }(f)$. Then we can associate a matrix $ {A_f}$ with entries either zero or one to the mapping $f$ such that the number of periodic points for $ f$ with period $ n$ is equal to the trace of the matrix ${\left[ {{A_f}} \right]^n}$; furthermore the zeta function of $f$ is rational having the eigenvalues of $ {A_f}$ as poles. B. We assume that $ f \in {C^3}(X,\,X)$ such that: (1) the Schwarzian derivative of $f$ is negative, and (2) the closure of ${A_\infty }(f)$ is compact and ${A_\infty }(f)$. Then we obtain the same result as in A.
Spaces of geodesic triangulations of the sphere
Marwan
Awartani;
David W.
Henderson
721-732
Abstract: We study questions concerning the homotopy-type of the space $\operatorname{GT} (K)$ of geodesic triangulations of the standard $n$-sphere which are (orientation-preserving) isomorphic to $K$. We find conditions which reduce this question to analogous questions concerning spaces of simplexwise linear embeddings of triangulated $n$-cells into $n$-space. These conditions are then applied to the $2$-sphere. We show that, for each triangulation $ K$ of the $2$-sphere, certain large subspaces of $ \operatorname{GT} (K)$ are deformable (in $ \operatorname{GT} (K)$) into a subsapce homeomorphic to $\operatorname{SO} (3)$. It is conjectured that (for $ n = 2$) $\operatorname{GT} (K)$ has the homotopy of $ \operatorname{SO} (3)$. In a later paper the authors hope to use these same conditions to study the homotopy type of spaces of geodesic triangulations of the $n$-sphere, $n > 2$.
On maximal functions associated to hypersurfaces and the Cauchy problem for strictly hyperbolic operators
Christopher D.
Sogge
733-749
Abstract: In this paper we prove a maximal Fourier integral theorem for the types of operators which arise in the study of maximal functions associated to averaging over hypersurfaces and also the Cauchy problem for hyperbolic operators. We apply the Fourier integral theorem to generalize Stein's spherical maximal theorem (see [8]) and also to prove a sharp theorem for the almost everywhere convergence to ${L^p}$ initial data of solutions to the Cauchy problem for second order strictly hyperbolic operators. Our results improve those of Greenleaf [3] and Ruiz [6]. We also can prove almost everywhere convergence to ${L^2}$ initial data for operators of order $m \geqslant 3$.
On the initial-boundary value problem for a Bingham fluid in a three-dimensional domain
Jong Uhn
Kim
751-770
Abstract: The initial-boundary value problem associated with the motion of a Bingham fluid is considered. The existence and uniqueness of strong solution is proved under a certain assumption on the data. It is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time-periodic.
Decompositions of Banach lattices into direct sums
P. G.
Casazza;
N. J.
Kalton;
L.
Tzafriri
771-800
Abstract: We consider the problem of decomposing a Banach lattice $Z$ as a direct sum $Z = X \oplus Y$ where $X$ and $Y$ are complemented subspaces satisfying a condition of incomparability (e.g. every operator from $ Y$ to $X$ is strictly singular). We treat both the atomic and nonatomic cases. In particular we answer a question of Wojtaszczyk by showing that ${L_1} \oplus {L_2}$ has unique structure as a nonatomic Banach lattice.
The equivariant Conner-Floyd isomorphism
Steven R.
Costenoble
801-818
Abstract: This paper proves two equivariant generalizations of the Conner-Floyd isomorphism relating unitary cobordism and $ K$-theory. It extends a previous result of Okonek for abelian groups to all compact Lie groups. We also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism.
Spectral measures, boundedly $\sigma$-complete Boolean algebras and applications to operator theory
Werner J.
Ricker
819-838
Abstract: A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise convergence topology it turns out, somewhat surprisingly, that the corresponding ${L^1}$-spaces for the spectral measure are isomorphic as vector spaces. This fact, together with I. Kluvanek's notion of closed vector measure (suitably developed in our particular setting) makes it possible to extend to the setting of locally convex spaces a classical result of W. Bade. Namely, it is shown that if $ B$ is a Boolean algebra which is complete (with respect to the bounded convergence topology) in Bade's sense, then the closed operator algebras generated by $B$ with respect to the bounded convergence topology and the pointwise convergence topology coincide.
Symmetry breaking for a class of semilinear elliptic problems
Mythily
Ramaswamy;
P. N.
Srikanth
839-845
Abstract: We study positive solutions of the Dirichlet problem for $- \Delta u = {u^p} - \lambda$, $p > 1$, $ \lambda > 0$, on the unit ball $\Omega$. We show that there exists a positive solution $ ({u_0},\,{\lambda _0})$ of this problem which satisfies in addition $\partial {u_0}/\partial n = 0$ on $\partial \Omega $. We prove also that at $ ({u_0},\,{\lambda _0})$, the symmetry breaks, i.e. asymmetric solutions bifurcate from the positive radial solutions.
Correction to: ``Meromorphic functions that share four values'' [Trans. Amer. Math. Soc. {\bf 277} (1983), no. 2, 545--567; MR0694375 (84g:30028)]
Gary G.
Gundersen
847-850