Poisson transforms on vector bundles
An
Yang
857-887
Abstract: Let $G$ be a connected real semisimple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $(\tau,V)$ be an irreducible unitary representation of $K$, and $G\times _K\,V$ the associated vector bundle. In the algebra of invariant differential operators on $G\times _K\,V$ the center of the universal enveloping algebra of $\operatorname{Lie}(G)$ induces a certain commutative subalgebra $Z_\tau$. We are able to determine the characters of $Z_\tau$. Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to $\tau$ the trivial representation.
Algebraic transition matrices in the Conley index theory
Robert
Franzosa;
Konstantin
Mischaikow
889-912
Abstract: We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.
Metric completions of ordered groups and $K_0$ of exchange rings
E.
Pardo
913-933
Abstract: We give a description of the closure of the natural affine continuous function representation of $K_0(R)$ for any exchange ring $R$. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes $K_0$ of exchange rings. As a consequence, the results about $K_0^+$ of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes $C^*$-algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications.
Sobolev estimates for operators given by averages over cones
Scipio
Cuccagna
935-946
Abstract: We prove a result related to work by A. Greenleaf and G. Uhlmann concerning Sobolev estimates for operators given by averages over cones. This is done using the almost orthogonality lemma of Cotlar and Stein, and the van der Corput lemma on oscillatory integrals.
Equations in a free $\mathbf Q$-group
O.
Kharlampovich;
A.
Myasnikov
947-974
Abstract: An algorithm is constructed that decides if a given finite system of equations over a free $\mathbf{Q}$-group has a solution, and if it does, finds a solution.
Euler products associated to metaplectic automorphic forms on the 3-fold cover of ${GSp}(4)$
Thomas
Goetze
975-1011
Abstract: If $\phi$ is a generic cubic metaplectic form on GSp(4), that is also an eigenfunction for all the Hecke operators, then corresponding to $\phi$ is an Euler product of degree 4 that has a functional equation and meromorphic continuation to the whole complex plane. This correspondence is obtained by convolving $\phi$ with the cubic $\theta$-function on GL(3) in a Shimura type Rankin-Selberg integral.
Tight contact structures on solid tori
Sergei
Makar-Limanov
1013-1044
Abstract: In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori. We prove the existence of a $\mathbb{Z}$-valued and a $\mathbb{R}/2\pi\mathbb{Z}$-valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic. At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori. At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.
On the elliptic equation $\Delta u+ku-Ku^p=0$ on complete Riemannian manifolds and their geometric applications
Peter
Li;
Luen-fai
Tam;
DaGang
Yang
1045-1078
Abstract: We study the elliptic equation $\Delta u + ku - Ku^{p} = 0$ on complete noncompact Riemannian manifolds with $K$ nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space ${\mathbf{R}}^{n}$ and the hyperbolic space ${\mathbf{H}}^{n}$ are carried out when $k$ is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.
Galois rigidity of pro-$l$ pure braid groups of algebraic curves
Hiroaki
Nakamura;
Naotake
Takao
1079-1102
Abstract: In this paper, Grothendieck's anabelian conjecture on the pro-$l$ fundamental groups of configuration spaces of hyperbolic curves is reduced to the conjecture on those of single hyperbolic curves. This is done by estimating effectively the Galois equivariant automorphism group of the pro-$l$ braid group on the curve. The process of the proof involves the complete determination of the groups of graded automorphisms of the graded Lie algebras associated to the weight filtration of the braid groups on Riemann surfaces.
Green's function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature
C.
Bandle;
A.
Brillard;
M.
Flucher
1103-1128
Abstract: We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.
On the equivariant Morse complex of the free loop space of a surface
Nancy
Hingston
1129-1141
Abstract: We prove two theorems about the equivariant topology of the free loop space of a surface. The first deals with the nondegenerate case and says that the ``ordinary'' Morse complex can be given an $O(2)$-action in such a way that it carries the $O(2)$-homotopy type of the free loop space. The second says that, in terms of topology, the iterates of an isolated degenerate closed geodesic ``look like'' the continuous limit of the iterates of a finite, fixed number of nondegenerate closed geodesics.
Dyadic equivalence to completely positive entropy
Adam
Fieldsteel;
J.
Roberto
Hasfura-Buenaga
1143-1166
Abstract: We show that every free ergodic action of $\bigoplus _1^\infty {\mathbb Z}_2$ of positive entropy is dyadically equivalent to an action with completely positive entropy.
The homological degree of a module
Wolmer
V.
Vasconcelos
1167-1179
Abstract: A homological degree of a graded module $M$ is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when $M$ is not Cohen-Macaulay. We construct a degree, $\operatorname{hdeg}(M)$, that behaves well under hyperplane sections and the modding out of elements of finite support. When carried out in a local algebra this degree gives a simulacrum of complexity à la Castelnuovo-Mumford's regularity. Several applications for estimating reduction numbers of ideals and predictions on the outcome of Noether normalizations are given.
Necessary conditions for optimal control problems with state constraints
R.
B.
Vinter;
H.
Zheng
1181-1204
Abstract: Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild `one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent `dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case. Proofs make use of recent `decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.
Values of Gaussian hypergeometric series
Ken
Ono
1205-1223
Abstract: Let $p$ be prime and let $GF(p)$ be the finite field with $p$ elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions \begin{equation*}_{2}F_{1}(x)=_{2} F_{1} \left ( \begin{matrix}\phi , & \phi & \epsilon \end{matrix} | x \right ) \ {\text{\rm and}} \ _{3}F_{2}(x)= _{3}F_{2} \left ( \begin{matrix}\phi , & \phi , & \phi & \epsilon , & \epsilon \end{matrix} | x \right ), \end{equation*} where $\phi$ and $\epsilon$ respectively are the quadratic and trivial characters of $GF(p).$ For all but finitely many rational numbers $x=\lambda ,$ there exist two elliptic curves $_{2}E_{1}(\lambda )$ and $_{3}E_{2}(\lambda )$ for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes $p$ for which $_{2}F_{1}(\lambda )$ is zero; however if $\lambda \neq -1,0, \frac{1}{2}$ or $2$, then the set of such primes has density zero. In contrast, if $\lambda \neq 0$ or $1$, then there are only finitely many primes $p$ for which $_{3}F_{2}(\lambda ) =0.$ Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that $_{3}E_{2}(8)$ is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating $_{3}F_{2}(4)$ over every $GF(p).$
The possible orders of solutions of linear differential equations with polynomial coefficients
Gary
G.
Gundersen;
Enid
M.
Steinbart;
Shupei
Wang
1225-1247
Abstract: We find specific information about the possible orders of transcendental solutions of equations of the form $f^{(n)}+p_{n-1}(z)f^{(n-1)}+\cdots +p_{0}(z)f=0$, where $p_0(z), p_1(z),\dots, p_{n-1}(z)$ are polynomials with $p_0(z) \not\equiv 0$. Several examples are given.
Bilinear operators on Herz-type Hardy spaces
Loukas
Grafakos;
Xinwei
Li;
Dachun
Yang
1249-1275
Abstract: The authors prove that bilinear operators given by finite sums of products of Calderón-Zygmund operators on $\mathbb{R}^{n}$ are bounded from $H\dot K_{q_{1}}^{\alpha _{1},p_{1}}\times H\dot K_{q_{2}}^{\alpha _{2},p_{2}}$ into $H\dot K_{q}^{\alpha ,p}$ if and only if they have vanishing moments up to a certain order dictated by the target space. Here $H\dot K_{q}^{\alpha ,p}$ are homogeneous Herz-type Hardy spaces with $1/p=1/p_{1}+1/p_{2},$ $0<p_{i}\le \infty ,$ $1/q=1/q_{1}+1/q_{2},$ $1<q_{1},q_{2}<\infty ,$ $1\le q<\infty ,$ $\alpha =\alpha _{1}+\alpha _{2}$ and $-n/q_{i}<\alpha _{i}<\infty$. As an application they obtain that the commutator of a Calderón-Zygmund operator with a BMO function maps a Herz space into itself.
A classification theorem for Albert algebras
R.
Parimala;
R.
Sridharan;
Maneesh
L.
Thakur
1277-1284
Abstract: Let $k$ be a field whose characteristic is different from 2 and 3 and let $L/k$ be a quadratic extension. In this paper we prove that for a fixed, degree 3 central simple algebra $B$ over $L$ with an involution $\sigma$ of the second kind over $k$, the Jordan algebra $J(B,\sigma,u,\mu)$, obtained through Tits' second construction is determined up to isomorphism by the class of $(u,\mu)$ in $H^1(k,SU(B,\sigma))$, thus settling a question raised by Petersson and Racine. As a consequence, we derive a ``Skolem Noether'' type theorem for Albert algebras. We also show that the cohomological invariants determine the isomorphism class of $J(B,\sigma,u,\mu)$, if $(B,\sigma)$ is fixed.