These are the differentials of order $n$
Dan
Laksov;
Anders
Thorup
1293-1353
Abstract: We answer P.-A. Meyer's question ``Qu'est ce qu'une différentielle d'ordre $n$?''. In fact, we present a general theory of higher order differentials based upon a construction of universal objects for higher order differentials. Applied to successive tangent spaces on a differentiable manifold, our theory gives the higher order differentials of Meyer as well as several new results on differentials on differentiable manifolds. In addition our approach gives a natural explanation of the quite mysterious multiplicative structure on higher order differentials observed by Meyer. Applied to iterations of the first order Kähler differentials our theory gives an algebra of higher order differentials for any smooth scheme. We also observe that much of the recent work on higher order osculation spaces of varieties fits well into the framework of our theory.
A counterexample concerning the relation between decoupling constants and UMD-constants
Stefan
Geiss
1355-1375
Abstract: For Banach spaces $X$ and $Y$ and a bounded linear operator $T:X \rightarrow Y$ we let $\rho(T):=\inf c$ such that \begin{displaymath}\left( AV_{\theta _l = \pm 1} \left\|\sum\limits _{l=1}^\infty \theta _l \left( \sum\limits _{k=\tau _{l-1}+1}^{\tau _l} h_k T x_k \right)\right\|_{L_2^Y}^2 \right)^{\frac{1}{2}} \le c \left\| \sum\limits _{k=1}^\infty h_k x_k \right\| _{L_2^X} \end{displaymath} for all finitely supported $(x_k)_{k=1}^\infty \subset X$ and all $0 = \tau _0 < \tau _1 < \cdots$, where $(h_k)_{k=1}^\infty \subset L_1[0,1)$ is the sequence of Haar functions. We construct an operator $T:X \rightarrow X$, where $X$ is superreflexive and of type 2, with $\rho(T)<\infty$ such that there is no constant $c>0$ with \begin{displaymath}\sup _{\theta _k = \pm 1} \left\| \sum\limits _{k=1}^\infty \theta _k h_k T x_k \right\| _{L_2^X} \le c \left\| \sum\limits _{k=1}^\infty h_k x_k \right\| _{L_2^X}. \end{displaymath} In particular it turns out that the decoupling constants $\rho(I_X)$, where $I_X$ is the identity of a Banach space $X$, fail to be equivalent up to absolute multiplicative constants to the usual $\operatorname{UMD}$-constants. As a by-product we extend the characterization of the non-superreflexive Banach spaces by the finite tree property using lower 2-estimates of sums of martingale differences.
Transition operators of diffusions reduce zero-crossing
Steven
N.
Evans;
Ruth
J.
Williams
1377-1389
Abstract: If $u(t,x)$ is a solution of a one-dimensional, parabolic, second-order, linear partial differential equation (PDE), then it is known that, under suitable conditions, the number of zero-crossings of the function $u(t,\cdot)$ decreases (that is, does not increase) as time $t$ increases. Such theorems have applications to the study of blow-up of solutions of semilinear PDE, time dependent Sturm Liouville theory, curve shrinking problems and control theory. We generalise the PDE results by showing that the transition operator of a (possibly time-inhomogenous) one-dimensional diffusion reduces the number of zero-crossings of a function or even, suitably interpreted, a signed measure. Our proof is completely probabilistic and depends in a transparent manner on little more than the sample-path continuity of diffusion processes.
Eigenvalue estimate on complete noncompact Riemannian manifolds and applications
Manfredo
P.
do Carmo;
Detang
Zhou
1391-1401
Abstract: We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.
Geometric groups. I
Valera
Berestovskii;
Conrad
Plaut;
Cornelius
Stallman
1403-1422
Abstract: We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.
Universal constraints on the range of eigenmaps and spherical minimal immersions
Gabor
Toth
1423-1443
Abstract: The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space.
$C^\ast$-algebras generated by a subnormal operator
Kit
C.
Chan;
Zeljko
Cuckovic
1445-1460
Abstract: Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if $T$ is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under $T$ cannot be orthogonal to each other. Then we show that the $C^*$-algebra generated by $T$ and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that $C^*$-algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy space and the Bergman space of the unit ball.
Decomposing Euclidean space with a small number of smooth sets
Juris
Steprans
1461-1480
Abstract: Let the cardinal invariant ${\mathfrak s}_{n}$ denote the least number of continuously smooth $n$-dimensional surfaces into which $(n+1)$-dimensional Euclidean space can be decomposed. It will be shown to be consistent that ${\mathfrak s}_{n}$ is greater than ${\mathfrak s}_{n+1}$. These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.
Intersections of $\mathbb{Q}$-divisors on Kontsevich's moduli space $\overline{M}_{0,n}(\mathbb{P}^r,d)$ and enumerative geometry
Rahul
Pandharipande
1481-1505
Abstract: The theory of $\mathbb Q$-Cartier divisors on the space of $n$-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of $\mathbb Q$-divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in $\mathbb P^r$ is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree $d$ plane curves is explicitly evaluated.
Limit sets of discrete groups of isometries of exotic hyperbolic spaces
Kevin
Corlette;
Alessandra
Iozzi
1507-1530
Abstract: Let $\Gamma$ be a geometrically finite discrete group of isometries of hyperbolic space $\mathcal{H}_{\mathbb{F}}^n$, where $\mathbb{F}= \mathbb{R}, \mathbb{C}, \mathbb{H}$ or $\mathbb{O}$ (in which case $n=2$). We prove that the critical exponent of $\Gamma$ equals the Hausdorff dimension of the limit sets $\Lambda(\Gamma)$ and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.
Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups
George
M.
Bergman
1531-1550
Abstract: For $F$ a free group of finite rank, it is shown that the fixed subgroup of any set $B$ of endomorphisms of $F$ has rank $\leq \operatorname {rank}(F)$, generalizing a recent result of Dicks and Ventura. The proof involves the combinatorics of derivations of groups. Some related questions are examined.
Entropy and periodic points for transitive maps
Ll.
Alsedà;
S.
Kolyada;
J.
Llibre;
L.
Snoha
1551-1573
Abstract: The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the $n$-star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.
Steady-state bifurcation with Euclidean symmetry
Ian
Melbourne
1575-1603
Abstract: We consider systems of partial differential equations equivariant under the Euclidean group $\mathbf{E}(n)$ and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when $n=1$ and $n=2$ and for reaction-diffusion equations with general $n$, reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of $\mathbf{E}(n)$. The representation theory of $\mathbf{E}(n)$ is driven by the irreducible representations of $\mathbf{O}(n-1)$. For $n=1$, this constitutes a mathematical statement of the `universality' of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When $n=2$, there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of $\mathbf{O}(1)$. There are infinitely many possibilities for each $n\ge 3$.
Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in $\mathbf R^N$
Der-Chen
Chang;
Galia
Dafni;
Elias
M.
Stein
1605-1661
Abstract: We study two different local $H^p$ spaces, $0 < p \leq 1$, on a smooth domain in $\mathbf{R}^n$, by means of maximal functions and atomic decomposition. We prove the regularity in these spaces, as well as in the corresponding dual BMO spaces, of the Dirichlet and Neumann problems for the Laplacian.
Vertex operators for twisted quantum affine algebras
Naihuan
Jing;
Kailash
C.
Misra
1663-1690
Abstract: We construct explicitly the $q$-vertex operators (intertwining operators) for the level one modules $V(\Lambda _i)$ of the classical quantum affine algebras of twisted types using interacting bosons, where $i=0, 1$ for $A_{2n-1}^{(2)}$ ($n\geq 3$), $i=0$ for $D_4^{(3)}$, $i=0, n$ for $D_{n+1}^{(2)}$ ($n\geq 2$), and $i=n$ for $A_{2n}^{(2)}$ ($n\geq 1$). A perfect crystal graph for $D_4^{(3)}$ is constructed as a by-product.
Haar Measure and the Artin Conductor
Benedict
H.
Gross;
Wee
Teck
Gan
1691-1704
Abstract: Let $G$ be a connected reductive group, defined over a local, non-archimedean field $k$. The group $G(k)$ is locally compact and unimodular. In On the motive of a reductive group, Invent. Math. 130 (1997), by B. H. Gross, a Haar measure $|\omega _G|$ was defined on $G(k)$, using the theory of Bruhat and Tits. In this note, we give another construction of the measure $|\omega _G|$, using the Artin conductor of the motive $M$ of $G$ over $k$. The equivalence of the two constructions is deduced from a result of G. Prasad.