A convergent series expansion for hyperbolic systems of conservation laws
Eduard
Harabetian
383-424
Abstract: We consider the discontinuous piecewise analytic initial value problem for a wide class of conservation laws that includes the full three-dimensional Euler equations. The initial interaction at an arbitrary curved surface is resolved in time by a convergent series. Among other features the solution exhibits shock, contact, and expansion waves as well as sound waves propagating on characteristic surfaces. The expansion waves correspond to the one-dimensional rarefactions but have a more complicated structure. The sound waves are generated in place of zero strength shocks, and they are caused by mismatches in derivatives.
A rigidity property for the set of all characters induced by valuations
Robert
Bieri;
John R. J.
Groves
425-434
Abstract: If $K$ is a field and $G$ a finitely generated multiplicative subgroup of $K$ then every real valuation on $K$ induces a character $G \to {\mathbf{R}}$. It is known that the set $ \Delta (G) \subseteq {{\mathbf{R}}^n}$ of all characters induced by valuations is polyhedral. We prove that $\Delta (G)$ satisfies a certain rigidity property and apply this to give a new and conceptual proof of the Brewster-Roseblade result [4] on the group of automorphisms of $K$ stabilizing $G$.
Spectral theory of the linearized Vlasov-Poisson equation
Pierre
Degond
435-453
Abstract: We study the spectral theory of the linearized Vlasov-Poisson equation, in order to prove that its solution behaves, for large times, like a sum of plane waves. To obtain such an expansion involving damped waves, we must find an analytical extension of the resolvent of the equation. Then, the poles of this extension are no longer eigenvalues and must be interpreted as eigenmodes, associated to ``generalized eigenfunctions'' which actually are linear functionals on a Banach space of analytic functions.
The Auslander-Reiten quiver of a simple curve singularity
Ernst
Dieterich;
Alfred
Wiedemann
455-475
Abstract: With any simple curve singularity (plane, complex, affine-algebraic) of Dynkin-type $\Delta$ we associate the category of all finitely generated torsionfree modules over its complete local ring. For each of these module categories we calculate the Auslander-Reiten quiver. We suggest the construction of the ``twisted quiver'' of a quiver with involution and valuation of arrows which gives rise to a (purely combinatorial) one-to-one correspondence between the Auslander-Reiten quiver and the Dynkin diagram $ \Delta$.
General convergence of continued fractions
Lisa
Jacobsen
477-485
Abstract: We introduce a new concept of convergence of continued fractions--general convergence. Moreover, we compare it to the ordinary convergence concept and to strong convergence. Finally, we prove some properties of general convergence.
Volume of mixed bodies
Erwin
Lutwak
487-500
Abstract: By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body. These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body. The inequality obtained for the volume of mixed bodies is also used to derive a form of the Brunn-Minkowski inequality involving mixed bodies. As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.
Martingale transforms and complex uniform convexity
J.
Bourgain;
W. J.
Davis
501-515
Abstract: Martingale transforms and Calderon-Zygmund singular integral operators are bounded as operators from ${L_2}({L_1})$ to $ {L_2}({L_q})$ when $0 < q < 1$. If $ Y$ is a reflexive subspace of ${L_1}$ then ${L_1}/Y$ can be renormed to be $2$-complex uniformly convex. A new proof of the cotype 2 property of ${L_1}/{H_1}$ is given.
On $K\sb 3$ of truncated polynomial rings
Janet
Aisbett
517-536
Abstract: Group homology spectral sequences are used to investigate ${K_3}$ of truncated polynomial rings. If $F$ is a finite field of odd characteristic, we show that relative ${K_2}$ of the pair $(F\left[ t \right]/({t^q}),\,({t^k}))$, which has been identified by van der Kallen and Stienstra, is isomorphic to ${K_3}(F\left[ t \right]/({t^k}),\,(t))$ when $q$ is sufficiently large. We also show that $ {H_3}({\text{SL}}\,{\mathbf{Z}}\left[ t \right]/({t^k});{\mathbf{Z}}) = {{\mathbf{Z}}^{k - 1}} \oplus {\mathbf{Z}}/24$ and is isomorphic to the associated ${K_3}$ group modulo an elementary abelian $ 2$-group.
Stability of minimal orbits
John E.
Brothers
537-552
Abstract: Let $G$ be a compact Lie group of isometries of a riemannian manifold $M$. It is well known that the minimal principal orbits are those on which the volume function ${\mathbf{v}}$, which assigns to $p \in M$ the volume of the orbit of $ p$, is critical. It is shown that stability of a minimal orbit on which the hessian of $ {\mathbf{v}}$ is nonnegative is determined by the degree of involutivity of the distribution of normal planes to the orbits. Specifically, if the lengths of the tangential components of Lie brackets of pairs of orthonormal normal vector fields are sufficiently small relative to the hessian of ${\mathbf{v}}$, then the minimal orbit is stable, and conversely. Computable lower bounds are obtained for the values of these parameters at which stability turns to instability. These lower bounds are positive even in the case where $ {\mathbf{v}}$ is constant, and are finite unless the normal distribution is involutive. Several examples in which $M$ is a compact classical Lie group and $G$ is a subgroup of $M$ are discussed, showing in particular that the above estimates are sharp.
Unitary quasilifting: applications
Yuval Z.
Flicker
553-565
Abstract: Let $U(3)$ be the quasi-split unitary group in three variables defined using a quadratic extension $ E/F$ of number fields. Complete local and global results are obtained for the $ \sigma$-endo-(unstable) lifting from $U(2)$ to $ {\text{GL}}(3,\,E)$. This is used to establish quasi-(endo-)lifting for automorphic forms from $U(2)$ to $U(3)$ by means of base change from $U(3)$ to ${\text{GL}}(3,\,E)$. Base change quasi-lifting is also proven. Continuing the work of $\left[ {\mathbf{I}} \right]$, the exposition is elementary, and uses only a simple form of an identity of trace formulas, and base change transfer of orbital integrals of spherical functions.
Resonance and quasilinear ellipticity
Victor L.
Shapiro
567-584
Abstract: Two resonance-type existence theorems for periodic solutions of second order quasilinear elliptic partial differential equations are established. The first theorem is a best possible result, and the second theorem presents conditions which are both necessary and sufficient.
On the ampleness of homogeneous vector bundles
Dennis M.
Snow
585-594
Abstract: A formula is proved which expresses the ampleness of a homogeneous vector bundle over $G/P$ in terms of the distance of the weights of the representation of $P$ to certain dominant weights of $G$.
Spherical polynomials and the periods of a certain modular form
David
Kramer
595-605
Abstract: The space of cusp forms on $ {\text{S}}{{\text{L}}_2}({\mathbf{Z}})$ of weight $2k$ is spanned by certain modular forms with rational periods.
Conformally flat manifolds whose development maps are not surjective. I
Yoshinobu
Kamishima
607-623
Abstract: Let $M$ be an $n$-dimensional conformally flat manifold. A universal covering of $ M,\,\tilde M$ admits a conformal development map into ${S^n}$. When a development map is not surjective, we can relate the boundary of the development image with the limit set of the holonomy group of $ M$. In this paper, we study properties of closed conformally flat manifolds whose development maps are not surjective.
Fixed points of topologically stable flows
Mike
Hurley
625-633
Abstract: This paper concerns certain necessary conditions for a flow to be topologically stable (in the sense of P. Walters). In particular, it is shown that under fairly general conditions one can conclude that a topologically stable flow has a finite number of fixed points, and each of these is isolated in the chain recurrent set of the flow.
Contributions from conjugacy classes of regular elliptic elements in Hermitian modular groups to the dimension formula of Hermitian modular cusp forms
Min King
Eie
635-645
Abstract: The dimension of the vector space of hermitian modular cusp forms on the hermitian upper half plane can be obtained from the Selberg trace formula; in this paper we shall compute the contributions from conjugacy classes of regular elliptic elements in hermitian modular groups by constructing an orthonomal basis in a certain Hilbert space of holomorphic functions. A generalization of the main Theorem can be applied to the dimension formula of cusp forms of $SU(p,\,q)$. A similar theorem was given for the case of regular elliptic elements of $ {\text{Sp}}(n,\,{\mathbf{Z}})$ in [5] via a different method.
Functional equations for character series associated with $n\times n$ matrices
Edward
Formanek
647-663
Abstract: Let $A$ be either the ring of invariants or the trace ring of $r$ generic $n \times n$ matrices. Then $ A$ has a character series $ \chi (A)$ which is a symmetric rational function of commuting variables ${x_1}, \ldots ,{x_r}$. The main result is that if $r \geq {n^2}$, then $\chi (A)$ satisfies the functional equation $\displaystyle \chi (A)(x_1^{ - 1}, \ldots ,x_r^{ - 1}) = {( - 1)^d}{({x_1} \cdots {x_r})^{{n^2}}}\chi (A)({x_1}, \ldots ,{x_r})$ , where $d$ is the Krull dimension of $A$.
Left separated spaces with point-countable bases
William G.
Fleissner
665-677
Abstract: Theorem 2.2 lists properties equivalent to left separated spaces in the class of ${T_1}$ with point-countable bases, with examples preventing plausible additions to this list. For example, $X$ is left iff $X$ is $\sigma$-weakly separated or $X$ has a closure preserving cover by countable closed sets, but $X$ is left separated does not imply that $ X$ is $\sigma $-discrete. Theorem 2.2 is used to show that the following reflection property holds after properly collapsing a supercompact cardinal to ${\omega _2}$: If $X$ is a not $\sigma$-discrete metric space, then $X$ has a not $\sigma$-discrete subspace of cardinality less than ${\omega _2}$. Similar reflection properties are shown true in some models and false in others.
On the generalized Nakayama conjecture and the Cartan determinant problem
K. R.
Fuller;
B.
Zimmermann-Huisgen
679-691
Abstract: For Artin algebras allowing certain filtered module categories, the Generalized Nakayama Conjecture is shown to be true; our result covers all positively graded Artin algebras and those whose radical cube is zero. For the corresponding class of left artinian rings we prove that finite global dimension forces the determinant of the Cartan matrix to be 1.
Weak type estimates for Bochner-Riesz spherical summation multipliers
Sagun
Chanillo;
Benjamin
Muckenhoupt
693-703
Abstract: We consider the Bochner-Riesz multiplier $\displaystyle \widehat{{T_\delta }f}(\xi ) = {(1 - {\left\vert \xi \right\vert^2})^\delta } + \hat f(\xi ),\qquad \delta > 0,$ where $\widehat{}$ denotes the Fourier transform. It is shown that the multiplier operator ${T_\delta }$ is weak type $({p_0},\,{p_0})$ acting on $ {L^{p0}}({{\mathbf{R}}^n})$ radial functions, where ${p_0}$ is the critical value $2n/(n + 1 + 2\delta )$.
The complex equilibrium measure of a symmetric convex set in ${\bf R}\sp n$
Eric
Bedford;
B. A.
Taylor
705-717
Abstract: We give a formula for the measure on a convex symmetric set $K$ in $ {{\mathbf{R}}^n}$ which is the Monge-Ampere operator applied to the extremal plurisubharmonic function ${L_K}$ for the convex set. The measure is concentrated on the set $K$ and is absolutely continuous with respect to Lebesgue measure with a density which behaves at the boundary like the reciprocal of the square root of the distance to the boundary. The precise asymptotic formula for $x \in K$ near a boundary point ${x_0}$ of $K$ is shown to be of the form $ c({x_0})/{[{\operatorname{dist}}(x,\,\partial K)]^{ - 1/2}}$, where the constant $ c({x_0})$ depends both on the curvature of $K$ at ${x_0}$ and on the global structure of $K$.
Trellises formed by stable and unstable manifolds in the plane
Robert W.
Easton
719-732
Abstract: A trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diffeomorphism of a $2$-manifold. This paper describes and classifies some trellises. The set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold. Each homoclinic point is assigned a type number which is constant on its orbit. Combinatorial properties of trellises are studied using type numbers and the pair of linear orderings. Trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones.
Algebraic meridians of knot groups
Chichen M.
Tsau
733-747
Abstract: We propose the conjecture that every automorphism of a knot group preserves the meridian up to inverse and conjugation. We establish the conjecture for all composite knots, all torus knots, most cable knots, and at most one exception for hyperbolic knots; moreover we prove that the Property P Conjecture implies our conjecture. We also investigate hyperbolic knots in more detail, and give an example of figure-eight knot group and its automorphisms.
Vector bundles and projective modules
Leonid N.
Vaserstein
749-755
Abstract: Serre and Swan showed that the category of vector bundles over a compact space $X$ is equivalent to the category of finitely generated projective modules over the ring of continuous functions on $X$. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space $X$. Also the well-known homotopy classification of the vector bundles over compact $X$ up to isomorphism is extended to arbitrary $X$. It is shown that the ${K_0}$-functor and the Witt group of the ring of continuous functions on $X$ coincide, and they are homotopy-type invariants of $ X$.
Brownian motions of ellipsoids
J. R.
Norris;
L. C. G.
Rogers;
David
Williams
757-765
Abstract: The object of this paper is to provide an elementary treatment (involving no differential geometry) of Brownian motions of ellipsoids, and, in particular, of some remarkable results first obtained by Dynkin. The canonical right-invariant Brownian motion $G = \{ G(t)\}$ on $ {\text{GL}}(n)$ induces processes $X = G{G^T}$ and $Y = {G^T}G$ on the space of positive-definite symmetric matrices. The motion of the common eigenvalues of $ X$ and $Y$ is analysed. It is further shown that the orthonormal frame of eigenvectors of $ X$ ultimately behaves like Brownian motion on $ {\text{O}}(n)$, while that of $Y$ converges to a limiting value. The $ Y$ process is that studied by Dynkin and Orihara. From a naive standpoint, the $ X$ process would seem to provide a more natural model.
Global boundedness for a delay-differential equation
Stephan
Luckhaus
767-774
Abstract: The inequality $ ({\partial _t}u - \Delta u)(t,\,x)\qquad \leq \qquad u(t,\,x)(1 - u(t - \tau ,\,x))$ is investigated. It is shown that nonnegative solutions of the Dirichlet problem in a bounded interval remain bounded as time goes to infinity, whereas in a more dimensional domain, in general, this holds only if the delay is not too large.
$\Omega$-stable limit set explosions
S. E.
Patterson
775-798
Abstract: Certain diffeomorphisms of two-dimensional manifolds are considered. These diffeomorphisms have a finite hyperbolic limit set which contains a limit set cycle. The only nontransverse cycle connection in these cycles is a complete coincidence of one component of the unstable manifold of one periodic point with one component of the stable manifold of some other periodic point. A one-parameter family of diffeomorphisms containing the original diffeomorphism is described. It is shown that for parameter values arbitrarily near the parameter value corresponding to the original map these diffeomorphisms have a much enlarged limit set and are $\Omega$-stable.
$G$-deformations and some generalizations of H. Weyl's tube theorem
Oldřich
Kowalski;
Lieven
Vanhecke
799-811
Abstract: We prove an invariance theorem for the volumes of tubes about submanifolds in arbitrary analytic Riemannian manifolds under $ G$-deformations of the second order. For locally symmetric spaces or two-point homogeneous spaces we give stronger invariance theorems using only $G$-deformations of the first order. All these results can be viewed as generalizations of the result of H. Weyl about isometric deformations and the volumes of tubes in spaces of constant curvature. They are derived from a new formula for the volume of a tube about a submanifold.