Overgroups of irreducible linear groups, II
Ben
Ford
3869-3913
Abstract: Determining the subgroup structure of algebraic groups (over an algebraically closed field $K$ of arbitrary characteristic) often requires an understanding of those instances when a group $Y$ and a closed subgroup $G$ both act irreducibly on some module $V$, which is rational for $G$ and $Y$. In this paper and in Overgroups of irreducible linear groups, I (J. Algebra 181 (1996), 26-69), we give a classification of all such triples $(G,Y,V)$ when $G$ is a non-connected algebraic group with simple identity component $X$, $V$ is an irreducible $G$-module with restricted $X$-high weight(s), and $Y$ is a simple algebraic group of classical type over $K$ sitting strictly between $X$ and $% \operatorname{SL}(V)$.
Rates of convergence of diffusions with drifted Brownian potentials
Yueyun
Hu;
Zhan
Shi;
Marc
Yor
3915-3934
Abstract: We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.
The space of complete minimal surfaces with finite total curvature as lagrangian submanifold
Joaquín
Pérez;
Antonio
Ros
3935-3952
Abstract: The space ${\cal M}$ of nondegenerate, properly embedded minimal surfaces in ${\mathbb R}^3$ with finite total curvature and fixed topology is an analytic lagrangian submanifold of ${\mathbb C}^n$, where $n$ is the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for the second fundamental form of this submanifold. We also consider the compact boundary case, and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane, having a horizontal end of finite total curvature, is a locally convex curve in the plane ${\mathbb C}$.
Morse homology for generating functions of Lagrangian submanifolds
Darko
Milinkovic
3953-3974
Abstract: The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the ``finite dimensional'' symplectic invariants constructed via generating functions to the ``infinite dimensional'' ones constructed via Floer theory in Y.-G. Oh, Symplectic topology as the geometry of action functional. I, J. Diff. Geom. 46 (1997), 499-577.
Extendability of Large-Scale Lipschitz Maps
Urs
Lang
3975-3988
Abstract: Let $X,Y$ be metric spaces, $S$ a subset of $X$, and $f \colon S \to Y$ a large-scale lipschitz map. It is shown that $f$ possesses a large-scale lipschitz extension $\bar f \colon X \to Y$ (with possibly larger constants) if $Y$ is a Gromov hyperbolic geodesic space or the cartesian product of finitely many such spaces. No extension exists, in general, if $Y$ is an infinite-dimensional Hilbert space. A necessary and sufficient condition for the extendability of a lipschitz map $f \colon S \to Y$ is given in the case when $X$ is separable and $Y$ is a proper, convex geodesic space.
Chern classes for singular hypersurfaces
Paolo
Aluffi
3989-4026
Abstract: We prove a formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety as a variation on another definition of the homology Chern class of singular varieties, introduced by W. Fulton; and we discuss the relation between these classes and others, such as Mather's Chern class and the $\mu$-class we introduced in previous work.
On the depth of the tangent cone and the growth of the Hilbert function
Juan
Elias
4027-4042
Abstract: For a $d-$dimensional Cohen-Macaulay local ring $(R, \mathbf{m})$ we study the depth of the associated graded ring of $R$ with respect to an $ \textbf{ m}$-primary ideal $I$ in terms of the Vallabrega-Valla conditions and the length of $I^{t+1}/JI^{t}$, where $J$ is a $J$ minimal reduction of $I$ and $t\ge 1$. As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to $\mathbf{m}$-primary ideals. We also study the growth of the Hilbert function.
Alexander invariants of complex hyperplane arrangements
Daniel
C.
Cohen;
Alexander
I.
Suciu
4043-4067
Abstract: Let $\mathcal{A}$ be an arrangement of $n$ complex hyperplanes. The fundamental group of the complement of $\mathcal{A}$ is determined by a braid monodromy homomorphism, $\alpha:F_{s}\to P_{n}$. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of $\mathcal{A}$. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of $\mathcal{A}$. We also provide a combinatorial criterion for when these lower bounds are attained.
Inverse eigenvalue problems on directed graphs
Robert
Carlson
4069-4088
Abstract: The differential operators $iD$ and $-D^2 - p$ are constructed on certain finite directed weighted graphs. Two types of inverse spectral problems are considered. First, information about the graph weights and boundary conditions is extracted from the spectrum of $-D^2$. Second, the compactness of isospectral sets for $-D^2 - p$ is established by computation of the residues of the zeta function.
Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces
Rong-Qing
Jia
4089-4112
Abstract: Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.
Remarks about global analytic hypoellipticity
Adalberto
P.
Bergamasco
4113-4126
Abstract: We present a characterization of the operators \begin{displaymath}L=\partial/\partial t+(a(t)+ib(t))\partial/\partial x\end{displaymath} which are globally analytic hypoelliptic on the torus. We give information about the global analytic hypoellipticity of certain overdetermined systems and of sums of squares.
Norm estimates and representations for Calderón-Zygmund operators using averages over starlike sets
David
K.
Watson;
Richard
L.
Wheeden
4127-4171
Abstract: We show that homogeneous singular integrals may be represented in terms of averages over starlike sets. This permits us to use the geometry of starlike sets to derive operator-specific weighted norm inequalities.
The $C^1$ closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups
Xiaofeng
Wang;
Duo
Wang
4173-4182
Abstract: In this paper a closing lemma for $C^1$ nonsingular endomorphisms equivariant under free actions of finite-groups is proved. Hence a recurrent trajectory, as well as all of its symmetric conjugates, of a $C^1$ nonsingular endomorphism equivariant under a free action of a finite group can be closed up simultaneously by an arbitrarily small $C^1$ equivariant perturbation.
Minimal lattice-subspaces
Ioannis
A.
Polyrakis
4183-4203
Abstract: In this paper the existence of minimal lattice-subspaces of a vector lattice $E$ containing a subset $B$ of $E_+$ is studied (a lattice-subspace of $E$ is a subspace of $E$ which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology $\tau$ on $E$ and $E_+$ is $\tau$-closed (especially if $E$ is a Banach lattice with order continuous norm), then minimal lattice-subspaces with $\tau$-closed positive cone exist (Theorem 2.5). In the sequel it is supposed that $B=\{x_1,x_2,\ldots,x_n\}$ is a finite subset of $C_+(\Omega)$, where $\Omega$ is a compact, Hausdorff topological space, the functions $x_i$ are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function $\beta(t) = \frac{r(t)}{\|r(t)\|_1}$ where $r(t) = \big(x_1(t),x_2(t),\ldots,x_n(t)\big)$. If $R(\beta)$ is the range of $\beta$ and $K$ the convex hull of the closure of $R(\beta)$, it is proved: (i) There exists an $m$-dimensional minimal lattice-subspace containing $B$ if and only if $K$ is a polytope of $\mathbb{R}^n$ with $m$ vertices (Theorem 3.20). (ii) The sublattice generated by $B$ is an $m$-dimensional subspace if and only if the set $R(\beta)$ contains exactly $m$ points (Theorem 3.7). This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.
Weighted Laplace transforms and Bessel functions on Hermitian symmetric spaces
Hongming
Ding
4205-4243
Abstract: This paper defines $\pi$-weighted Laplace transforms on the spaces of $\pi$-covariant functions. By the inverse Laplace transform we define operator-valued Bessel functions. We also study the holomorphic discrete series of the automorphism group of a Siegel domain of type II.
Classification of one $\textsf K$-type representations
Dan
Barbasch;
Allen
Moy
4245-4261
Abstract: Suppose $G$ is a simple reductive $p$-adic group with Weyl group $W$. We give a classification of the irreducible representations of $W$ which can be extended to real hermitian representations of the associated graded Hecke algebra $\mathbb{H}$. Such representations correspond to unitary representations of $G$ which have a small spectrum when restricted to an Iwahori subgroup.
A $K$ counterexample machine
Christopher
Hoffman
4263-4280
Abstract: We present a general method for constructing families of measure preserving transformations which are $K$ and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two $K$ transformations which are weakly isomorphic but not isomorphic, and a $K$ transformation with no roots. Ornstein's isomorphism theorem says families of Bernoulli shifts cannot have these properties. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations.