The calculus of partition sequences, changing cofinalities, and a question of Woodin
Arthur
W.
Apter;
James
M.
Henle;
Stephen
C.
Jackson
969-1003
Abstract: We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory ``ZF$+$DC$+\omega _1$ is the only regular, uncountable cardinal $\le\omega _{\omega _1+1}$.'' This gives a partial answer to a question of Woodin.
Decomposition theorems for groups of diffeomorphisms in the sphere
R.
de la Llave;
R.
Obaya
1005-1020
Abstract: We study the algebraic structure of several groups of differentiable diffeomorphisms in $\mathbf{S}^n$. We show that any given sufficiently smooth diffeomorphism can be written as the composition of a finite number of diffeomorphisms which are symmetric under reflection, essentially one-dimensional and about as differentiable as the given one.
Deforming a map into a harmonic map
Deane
Yang
1021-1038
Abstract: This paper presents some existence and uniqueness theorems for harmonic maps between complete noncompact Riemannian manifolds. In particular, we obtain as a corollary a recent result of Hardt-Wolf on the existence of harmonic quasi-isometries of the hyperbolic plane.
Sums of squares of regular functions on real algebraic varieties
Claus
Scheiderer
1039-1069
Abstract: Let $V$ be an affine algebraic variety over $\mathbb{R}$ (or any other real closed field $R$). We ask when it is true that every positive semidefinite (psd) polynomial function on $V$ is a sum of squares (sos). We show that for $\dim V\ge 3$ the answer is always negative if $V$ has a real point. Also, if $V$ is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if $V$ is a smooth surface with only real divisors at infinity. The ``compact'' case is harder. We completely settle the case of smooth curves of genus $\le 1$: If such a curve has a complex point at infinity, then every psd function is sos, provided the field $R$ is archimedean. If $R$ is not archimedean, there are counter-examples of genus $1$.
On the distribution of points in projective space of bounded height
Kwok-Kwong
Choi
1071-1111
Abstract: In this paper we consider the uniform distribution of points in compact metric spaces. We assume that there exists a probability measure on the Borel subsets of the space which is invariant under a suitable group of isometries. In this setting we prove the analogue of Weyl's criterion and the Erdös-Turán inequality by using orthogonal polynomials associated with the space and the measure. In particular, we discuss the special case of projective space over completions of number fields in some detail. An invariant measure in these projective spaces is introduced, and the explicit formulas for the orthogonal polynomials in this case are given. Finally, using the analogous Erdös-Turán inequality, we prove that the set of all projective points over the number field with bounded Arakelov height is uniformly distributed with respect to the invariant measure as the bound increases.
The second bounded cohomology of an amalgamated free product of groups
Koji
Fujiwara
1113-1129
Abstract: We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we show that a group with infinitely many ends has infinite dimensional second bounded cohomology.
Embeddings in generalized manifolds
J.
L.
Bryant;
W.
Mio
1131-1147
Abstract: We prove that a ($2m-n+1$)-connected map $f\colon M^m\to X^n$ from a compact PL $m$-manifold $M$ to a generalized $n$-manifold $X$ with the disjoint disks property, $3m\le 2n-2$, is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact $M$ and proper maps $f$ that are properly ($2m-n+1$)-connected. The techniques developed lead to a general position result for arbitrary maps $f\colon M\to X$, $3m\le 2n-2$, and a Whitney trick for separating $P\hspace*{-1pt}L$submanifolds of $X$ that have intersection number 0, analogous to the well-known results when $X$ is a manifold.
Odd primary $bo$ resolutions and classification of the stable summands of stunted lens spaces
Jesús
González
1149-1169
Abstract: The classification of the stable homotopy types of stunted lens spaces and their stable summands can be obtained by proving the triviality of certain stable classes in the homotopy groups of these spaces. This is achieved in the 2-primary case by Davis and Mahowald using classical Adams spectral sequence techniques. We obtain the odd primary analogue using the corresponding Adams spectral sequence based at the spectrum representing odd primary connective $K$-theory. The methods allow us to answer a stronger problem: the determination of the smallest stunted space where such stable classes remain null homotopic. A technical problem prevents us from giving an answer in all situations; however, in a quantitative way, the number of cases missed is very small.
Characterizations of spectra with $\mathcal{U}$-injective cohomology which satisfy the Brown-Gitler property
David
J.
Hunter;
Nicholas
J.
Kuhn
1171-1190
Abstract: We work in the stable homotopy category of $p$-complete connective spectra having mod $p$ homology of finite type. $H^*(X)$ means cohomology with $\mathbf{Z}/p$ coefficients, and is a left module over the Steenrod algebra $\mathcal{A}$. A spectrum $Z$ is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum $X$ satisfies the Brown-Gitler property if the natural map $[X,Z] \rightarrow \operatorname{Hom}_{\mathcal{A}}(H^*(Z),H^*(X))$ is onto, for all spacelike $Z$. It is known that there exist spectra $T(n)$ satisfying the Brown-Gitler property, and with $H^*(T(n))$ isomorphic to the injective envelope of $H^*(S^n)$ in the category $\mathcal{U}$ of unstable $\mathcal{A}$-modules. Call a spectrum $X$ standard if it is a wedge of spectra of the form $L \wedge T(n)$, where $L$ is a stable wedge summand of the classifying space of some elementary abelian $p$-group. Such spectra have $\mathcal{U}$-injective cohomology, and all $\mathcal{U}$-injectives appear in this way. Working directly with the two properties of $T(n)$ stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if $X$ is a spectrum with $\mathcal{U}$-injective cohomology, the following conditions are equivalent: (A) there exist a spectrum $Y$ whose cohomology is a reduced $\mathcal{U}$-injective and a map $X \rightarrow Y$ that is epic in cohomology, (B) there exist a spacelike spectrum $Z$ and a map $X \rightarrow Z$ that is epic in cohomology, (C) $\epsilon:\Sigma^{\infty}\Omega^{\infty}X \rightarrow X$ is monic in cohomology, (D) $X$ satisfies the Brown-Gitler property, (E) $X$ is spacelike, (F) $X$ is standard. ($M \in \mathcal{U}$ is reduced if it has no nontrivial submodule which is a suspension.) As an application, we prove that the Snaith summands of $\Omega^2S^3$ are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture. Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an $n$-connected space $X$ admits a map to an $n$-fold suspension that is monic in mod $p$ homology, then $\epsilon: \Sigma^n\Omega^n X \rightarrow X$ is onto in mod $p$ homology.
The $KO$-theory of toric manifolds
Anthony
Bahri;
Martin
Bendersky
1191-1202
Abstract: Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an $n$-dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they generate the complex cobordism ring. We use the Adams spectral sequence to compute the $KO$-theory of all toric manifolds and certain singular toric varieties.
Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions
Masataka
Nishikawa;
Kenji
Nishihara
1203-1215
Abstract: This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave $r(\frac{x}{t})$ connecting $u_{+}$ and $u_{-}$ for the scalar viscous conservation law in two space dimensions. We assume that the initial data $u_{0}(x,y)$ tends to constant states $u_{\pm }$ as $x \rightarrow \pm \infty$, respectively. Then, the convergence rate to $r(\frac{x}{t})$ of the solution $u(t,x,y)$ is investigated without the smallness conditions of $|u_{+}-u_{-}|$ and the initial disturbance. The proof is given by elementary $L^{2}$-energy method.
Symmetry of ground states for a semilinear elliptic system
Henghui
Zou
1217-1245
Abstract: Let $n\ge 3$ and consider the following system \begin{equation*}\Delta \mathbf{u}+\mathbf{f}(\mathbf{u})=0,\quad \mathbf{u}>0,\qquad x\in\mathbf{R}^n.\end{equation*} By using the Alexandrov-Serrin moving plane method, we show that under suitable assumptions every slow decay solution of (I) must be radially symmetric.
Gauge Invariant Eigenvalue Problems in $\mathbb{R}^n$ and in $\mathbb{R}^n_+$
Kening
Lu;
Xing-Bin
Pan
1247-1276
Abstract: This paper is devoted to the study of the eigenvalue problems for the Ginzburg-Landau operator in the entire plane ${\mathbb{R}}^{2}$ and in the half plane ${\mathbb{R}}^{2}_{+}$. The estimates for the eigenvalues are obtained and the existence of the associate eigenfunctions is proved when $curl A$ is a non-zero constant. These results are very useful for estimating the first eigenvalue of the Ginzburg-Landau operator with a gauge-invariant boundary condition in a bounded domain, which is closely related to estimates of the upper critical field in the theory of superconductivity.
Natural extensions for the Rosen fractions
Robert
M.
Burton;
Cornelis
Kraaikamp;
Thomas
A.
Schmidt
1277-1298
Abstract: The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. We find planar natural extensions for the associated interval maps. This allows us to easily prove that the interval maps are weak Bernoulli, as well as to unify and generalize results of Diophantine approximation from the literature.
Livsic theorems for hyperbolic flows
C.
P.
Walkden
1299-1313
Abstract: We consider Hölder cocycle equations with values in certain Lie groups over a hyperbolic flow. We extend Livsic's results that measurable solutions to such equations must, in fact, be Hölder continuous.
Random intersections of thick Cantor sets
Roger
L.
Kraft
1315-1328
Abstract: Let $C_{1}$, $C_{2}$ be Cantor sets embedded in the real line, and let $\tau _{1}$, $\tau _{2}$ be their respective thicknesses. If $\tau _{1}\tau _{2}>1$, then it is well known that the difference set $C_{1}-C_{2}$ is a disjoint union of closed intervals. B. Williams showed that for some $t\in \operatorname{int}(C_{1}-C_{2})$, it may be that $C_{1}\cap (C_{2}+t)$ is as small as a single point. However, the author previously showed that generically, the other extreme is true; $C_{1}\cap (C_{2}+t)$ contains a Cantor set for all $t$ in a generic subset of $C_{1}-C_{2}$. This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if $\tau _{1}\tau _{2}>1$, then $C_{1}\cap (C_{2}+t)$ contains a Cantor set for almost all $t$ in $C_{1}-C_{2}$.
Isometric Extensions of zero entropy {$\mathbb Z^{\lowercase{d}}$} Loosely Bernoulli Transformations
Aimee
S. A.
Johnson;
Ayse
A.
Sah. in
1329 - 1343
Positive definite spherical functions on Olshanskii domains
Joachim
Hilgert;
Karl-Hermann
Neeb
1345-1380
Abstract: Let $G$ be a simply connected complex Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{h}$ a real form of $\mathfrak{g}$, and $H$ the analytic subgroup of $G$ corresponding to $\mathfrak{h}$. The symmetric space ${\mathcal{M}}=H\backslash G$ together with a $G$-invariant partial order $\le$ is referred to as an Ol$'$shanskii space. In a previous paper we constructed a family of integral spherical functions $\phi _{\mu }$ on the positive domain ${\mathcal{M}}^{+} := \{Hx\colon Hx\ge H\}$ of ${\mathcal{M}}$. In this paper we determine all of those spherical functions on ${\mathcal{M}}^{+}$ which are positive definite in a certain sense.
The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras
K.
R.
Goodearl;
E.
S.
Letzter
1381-1403
Abstract: We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of $n\times n$ matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras $A$ over infinite fields $k$ of arbitrary characteristic. Our main result is the verification, for $A$, of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal $P$ of $A$ is primitive if and only if the center of the Goldie quotient ring of $A/P$ is algebraic over $k$, if and only if $P$ is a locally closed point - with respect to the Jacobson topology - in the prime spectrum of $A$. These equivalences are established with the aid of a suitable group $\mathcal{H}$ acting as automorphisms of $A$. The prime spectrum of $A$ is then partitioned into finitely many ``$\mathcal{H}$-strata'' (two prime ideals lie in the same $\mathcal{H}$-stratum if the intersections of their $\mathcal{H}$-orbits coincide), and we show that a prime ideal $P$ of $A$ is primitive exactly when $P$ is maximal within its $\mathcal{H}$-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which $\mathcal{H}$ acts transitively on the set of rational ideals within each $\mathcal{H}$-stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of $A$. For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.
Double coset density in classical algebraic groups
Jonathan
Brundan
1405-1436
Abstract: We classify all pairs of reductive maximal connected subgroups of a classical algebraic group $G$ that have a dense double coset in $G$. Using this, we show that for an arbitrary pair $(H, K)$ of reductive subgroups of a reductive group $G$ satisfying a certain mild technical condition, there is a dense $H, K$-double coset in $G$ precisely when $G = HK$ is a factorization.
Low-dimensional linear representations of ${Aut} F_n, n \geq 3$
A.
Potapchik;
A.
Rapinchuk
1437-1451
Abstract: We classify all complex representations of $\mathrm{Aut} \: F_n,$ the automorphism group of the free group $F_n$ $(n \geq 3),$ of dimension $\leq 2n - 2.$ Among those representations is a new representation of dimension $n + 1$ which does not vanish on the group of inner automorphisms.