Real analytic desingularization and subanalytic sets: an elementary approach
H. J.
Sussmann
417-461
Abstract: We give a proof of a theorem on desingularization of real-analytic functions which is a weaker version of H. Hironaka's result, but has the advantage of being completely self-contained and elementary, and not involving any machinery from algebraic geometry. We show that the basic facts about subanalytic sets can be proved from this result.
On linear topological properties of $H\sp 1$ on spaces of homogeneous type
Paul F. X.
Müller
463-484
Abstract: Let $(X,d,\mu )$ be a space of homogeneous type. Let $B = \{ x \in X:\mu \{ x\} = 0\}$, then $\mu (B) > 0$ implies that ${H^1}(X,d,\mu )$ contains a complemented copy of ${H^1}(\delta )$. This applies to Hardy spaces $ {H^1}(\partial \Omega ,d,\omega )$ associated to weak solutions of uniformly elliptic operators in divergence form. Under smoothness assumptions of the coefficients of the elliptic operators, we obtain that ${H^1}(\partial \Omega ,d,\omega )$ is isomorphic to ${H^1}(\delta )$.
Equivariant BP-cohomology for finite groups
N.
Yagita
485-499
Abstract: The Brown-Peterson cohomology rings of classifying spaces of finite groups are studied, considering relations to the other generalized cohomology theories. In particular, ${\operatorname{BP} ^{\ast}}(M)$ are computed for minimal nonabelian $p$-groups $M$. As an application, we give a necessary condition for the existence of nonabelian $p$-subgroups of compact Lie groups.
The enumerative geometry of plane cubics. I. Smooth cubics
Paolo
Aluffi
501-539
Abstract: We construct a variety of complete plane cubics by a sequence of five blow-ups over $ {\mathbb{P}^9}$. This enables us to translate the problem of computing characteristic numbers for a family of plane cubics into one of computing five Segre classes, and to recover classic enumerative results of Zeuthen and Maillard.
Geometrical implications of certain infinite-dimensional decompositions
N.
Ghoussoub;
B.
Maurey;
W.
Schachermayer
541-584
Abstract: We investigate the connections between the "global" structure of a Banach space (i.e. the existence of certain finite and infinite dimensional decompositions) and the geometrical properties of the closed convex bounded subsets of such a space (i.e. the existence of extremal and other topologically distinguished points). The global structures of various--supposedly pathological-- examples of Banach spaces constructed by R. C. James turn out to be more "universal" than expected. For instance James-tree-type (resp. James-matrix-type) decompositions characterize Banach spaces with the Point of Continuity Property (resp. the Radon-Nikodým Property). Moreover, the Convex Point of Continuity Property is stable under the formation of James-infinitely branching tree-type "sums" of infinite dimensional factors. We also give several counterexamples to various questions relating some topological and geometrical concepts in Banach spaces.
Semisimple representations of quivers
Lieven
Le Bruyn;
Claudio
Procesi
585-598
Abstract: We discuss the invariant theory of the variety of representations of a quiver and present generators and relations. We connect this theory of algebras with a trace satisfying a formal Cayley-Hamilton identity
The domain spaces of quasilogarithmic operators
M.
Cwikel;
B.
Jawerth;
M.
Milman
599-609
Abstract: The construction of intermediate Banach spaces in interpolation theory and the study of commutator inequalities in this context are closely related to certain nonlinear operators $ \Omega$. Here an explicit characterization of the domain spaces of these operators $\Omega$ is obtained, and the characterization is related to logarithmic Sobolev inequalities.
The extreme projections of the regular simplex
P.
Filliman
611-629
Abstract: The largest and smallest projections of the regular $n$-dimensional simplex into a $ k$-dimensional subspace are determined for certain values of $n$ and $k$. These results suggest that the smallest $ k$-dimensional projection and the largest $(n - k)$-dimensional projection occur in orthogonal subspaces of ${R^n}$.
The spectra and commutants of some weighted composition operators
James W.
Carlson
631-654
Abstract: An operator $ {T_{ug}}$ on a Hilbert space $H$ of functions on a set $X$ defined by ${T_{ug}}(f) = u(f \circ g)$, where $f$ is in $H,\;u:X \to {\mathbf{C}}$ and $ g:X \to X$, is called a weighted composition operator. In this paper $X$ is the set of integers and $H = {L^2}({\mathbf{Z}},\mu )$, where $\mu$ is a measure whose sigma-algebra is the power set of $ {\mathbf{Z}}$. One distinguished space is ${l^2} = {L^2}({\mathbf{Z}},\mu )$, where $\mu$ is counting measure. The most important results given here are the determination of the spectrum of ${T_{ug}}$ on ${l^2}$ and a characterization of the commutant of ${T_g}$ on ${L^2}({\mathbf{Z}},\mu )$. To obtain many of the results it was necessary to assume the function $ g$ to be one-to-one except on a finite subset of the integers.
Rational approximations to L-S category and a conjecture of Ganea
Barry
Jessup
655-660
Abstract: The rational version of Ganea's conjecture for L-S category, namely that $\operatorname{cat} (S \times {\Sigma ^k}) = \operatorname{cat} (S) + 1$, if $S$ is a rational space and ${\Sigma ^k}$ denotes the $k$-sphere, is still open. Recently, a module type approximation to $ \operatorname{cat} (S)$, was introduced by Halperin and Lemaire. We have previously shown that $ M\operatorname{cat}$ satisfies Ganea's conjecture. Here we show that for $ (r - 1)$ connected $ S$, if $M\operatorname{cat} (S)$ is at least $\dim S/2r$, then $ M\operatorname{cat} (S) = \operatorname{cat} (S)$. This yields Ganea's conjecture for these spaces. We also extend other properties of $ M\operatorname{cat}$, previously unknown for cat, to these spaces.
Metric transforms and Euclidean embeddings
M.
Deza;
H.
Maehara
661-671
Abstract: It is proved that if $0 \leqslant c \leqslant 0.72/n$ then for any $n$-point metric space $(X,d)$, the metric space $(X,{d^c})$ is isometrically embeddable into a Euclidean space. For $6$-point metric space, $c = \tfrac{1} {2}{\log _2}\tfrac{3} {2}$ is the largest exponent that guarantees the existence of isometric embeddings into a Euclidean space. Such largest exponent is also determined for all $ n$-point graphs with "truncated distance".
Weak stability in the global $L\sp 1$-norm for systems of hyperbolic conservation laws
Blake
Temple
673-685
Abstract: We prove that solutions for systems of two conservation laws which are generated by Glimm's method are weakly stable in the global ${L^1}$-norm. The method relies on a previous decay result of the author, together with a new estimate for the ${L^1}$ Lipschitz constant that relates solutions at different times. The estimate shows that this constant can be bounded by the supnorm of the solution, and is proved for any number of equations. The techniques do not rely on the existence of a family of entropies, and moreover the results would generalize immediately to more than two equations if one were to establish the stability of solutions in the supnorm for more than two equations.
Enlacements du mouvement brownien autour des courbes de l'espace
Jean-François
Le Gall;
Marc
Yor
687-722
Abstract: Limit theorems are proved for the winding numbers of a three-dimensional Brownian motion around certain curves in space. In particular, the joint asymptotic distribution of the winding numbers around two curves is obtained. This joint distribution generalizes the asymptotic law of the winding numbers of a planar Brownian motion around two points, which has recently been given by Pitman and Yor. The limiting distributions are closely related to the time spent by a linear Brownian motion above and below a multiple of its maximum process. Proofs rely on stochastic calculus for continuous semi-martingales.
Perturbed dynamical systems with an attracting singularity and weak viscosity limits in Hamilton-Jacobi equations
B.
Perthame
723-748
Abstract: We give a new PDE proof of the Wentzell-Freidlin theorem concerning small perturbations of a dynamical system \begin{displaymath}\begin{gathered}{L_\varepsilon }{u_\varepsilon } = - \tfrac{\... ... \quad {\text{on}}\;\partial \Omega . \end{gathered} \end{displaymath} We prove that, if $b$ has a single attractive singular point, ${u_\varepsilon }$ converges uniformly on compact subsets of $\Omega$, and with an exponential decay, to a constant $\mu$, and we determine $\mu$. We also treat the case of Neumann boundary condition. In order to do so, we perform the asymptotic analysis for some ergodic measure which leads to a study of the viscosity limit of a Hamilton-Jacobi equation. This is achieved under very general assumptions by using a weak formulation of the viscosity limits of these equations. Résumé. Nous donnons une nouvelle preuve, par des méthodes EDP, du théorème de Wentzell-Freidlin concernant les petites perturbations d'un système dynamique: \begin{displaymath}\begin{gathered}{L_\varepsilon }{u_\varepsilon } = - \tfrac{\... ...\quad {\text{sur}}\;\partial \Omega . \end{gathered} \end{displaymath} Nous prouvons que, si $b$ a un seul point singulier attractif, alors ${u_\varepsilon }$ converge vers une constant $\mu$, uniformément sur tout compact, et avec une vitesse exponentielle. Nous déterminons $ \mu$. Nous traitons aussi le cas de conditions aux limites de Neuman. Pour cela, nous faisons l'analyse asymptotique d'une mesure ergodique intervenant naturellement dans le problème, ce qui revient à étudier la limite par viscosité évanescente dans une équation de Hamilton-Jacobi. Ceci est réalisé sous des hypothèses très générales gâce à un passage à la limite faible dans cette équation.
Prime ideals in differential operator rings. Catenarity
K. A.
Brown;
K. R.
Goodearl;
T. H.
Lenagan
749-772
Abstract: Let $R$ be a commutative algebra over the commutative ring $k$, and let $\Delta = \{ {\delta _1}, \ldots ,{\delta _n}\}$ be a finite set of commuting $k$-linear derivations from $R$ to $R$. Let $T = R[{\theta _1}, \ldots ,{\theta _n};{\delta _1}, \ldots ,{\delta _n}]$ be the corresponding ring of differential operators. We define and study an isomorphism of left $ R$-modules between $ T$ and its associated graded ring $ R[{x_1}, \ldots ,{x_n}]$, a polynomial ring over $R$. This isomorphism is used to study the prime ideals of $T$, with emphasis on the question of catenarity. We prove that $T$ is catenary when $R$ is a commutative noetherian universally catenary $k$-algebra and one of the following cases occurs: (A) $k$ is a field of characteristic zero and $ \Delta$ acts locally finitely; (B) $k$ is a field of positive characteristic; (C) $ k$ is the ring of integers, $R$ is affine over $k$, and $\Delta$ acts locally finitely.
The Mourre estimate for dispersive $N$-body Schr\"odinger operators
Jan
Dereziński
773-798
Abstract: We prove the Mourre estimate for a certain class of dispersive $ N$-body Schrödinger operators. Using this estimate we derive some properties of those operators such as the local finiteness of the finite spectrum and the absence of the singular continuous spectrum.
The Schubert calculus, braid relations, and generalized cohomology
Paul
Bressler;
Sam
Evens
799-811
Abstract: Let $X$ be the flag variety of a compact Lie group and let $ {h^{\ast}}$ be a complex-oriented generalized cohomology theory. We introduce operators on $ {h^{\ast}}(X)$ which generalize operators introduced by Bernstein, Gel'fand, and Gel'fand for rational cohomology and by Demazure for $K$-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin.
Construction by isotopy. II
Daniel S.
Silver
813-823
Abstract: Construction by isotopy is a technique introduced by Iain R. Aitchison for obtaining doubly slice fibered knots in any dimension. We show that if $k$ is any doubly slice fibered $(n - 2)$-knot, $n \geqslant 5$, such that ${\pi _1}({S^n} - k) \cong Z$, then $k$ is constructible by isotopy. We also prove that the $m$-twist-spin of any doubly slice knot is constructible by isotopy. Consequently, there exists a double slice knot constructible by isotopy that is not the double of any disk knot. We also give an example of a doubly slice fibered $6$-knot that is not constructible by isotopy.