Stochastic averaging with a flattened Hamiltonian: A Markov process on a stratified space (a whiskered sphere)
Richard
B.
Sowers
853-900
Abstract: We consider a random perturbation of a 2-dimensional Hamiltonian ODE. Under an appropriate change of time, we identify a reduced model, which in some aspects is similar to a stochastically averaged model. The novelty of our problem is that the set of critical points of the Hamiltonian has an interior. Thus we can stochastically average outside this set of critical points, but inside we can make no model reduction. The result is a Markov process on a stratified space which looks like a whiskered sphere (i.e, a 2-dimensional sphere with a line attached). At the junction of the sphere and the line, glueing conditions identify the behavior of the Markov process.
Finely $\mu$-harmonic functions of a Markov process
R.
K.
Getoor
901-924
Abstract: Let $X$ be a Borel right process and $m$ a fixed excessive measure. Given a finely open nearly Borel set $G$ we define an operator $\Lambda_G$ which we regard as an extension of the restriction to $G$ of the generator of $X$. It maps functions on $E$ to (locally) signed measures on $G$ not charging $m$-semipolars. Given a locally smooth signed measure $\mu$ we define $h$ to be (finely) $\mu$-harmonic on $G$ provided $(\Lambda_G + \mu) h = 0$ on $G$ and denote the class of such $h$ by $\mathcal H^\mu_f (G)$. Under mild conditions on $X$ we show that $h \in \mathcal H^\mu_f (G)$ is equivalent to a local ``Poisson'' representation of $h$. We characterize $\mathcal H^\mu_f (G)$ by an analog of the mean value property under secondary assumptions. We obtain global Poisson type representations and study the Dirichlet problem for elements of $\mathcal H^\mu_f (G)$ under suitable finiteness hypotheses. The results take their nicest form when specialized to Hunt processes.
Small profinite structures
Ludomir
Newelski
925-943
Abstract: We propose a model-theoretic framework for investigating profinite structures. We prove that in many cases small profinite structures interpret infinite groups. This corresponds to results of Hrushovski and Peterzil on interpreting groups in locally modular stable and o-minimal structures.
Groups definable in separably closed fields
E.
Bouscaren;
F.
Delon
945-966
Abstract: We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field $L$ is definably isomorphic to the group of $L$-rational points of an algebraic group defined over $L$.
Coloring ${\mathbb R}^n$
James
H.
Schmerl
967-974
Abstract: If $1 \leq m \leq n$ and $A \subseteq {\mathbb R}$, then define the graph $G(A,m,n)$ to be the graph whose vertex set is ${\mathbb R}^n$ with two vertices $x,y \in {\mathbb R}^n$ being adjacent iff there are distinct $u,v \in A^m$ such that $\Vert x-y\Vert = \Vert u-v\Vert$. For various $m$ and $n$ and various $A$, typically $A = {\mathbb Q}$ or $A = {\mathbb Z}$, the graph $G(A,m,n)$ can be properly colored with $\omega$ colors. It is shown that in some cases such a coloring $\varphi : {\mathbb R}^n \longrightarrow\omega$ can also have the additional property that if $\alpha : {\mathbb R}^m \longrightarrow{\mathbb R}^n$ is an isometric embedding, then the restriction of $\varphi$ to $\alpha(A^m)$ is a bijection onto $\omega$.
Some combinatorics of binomial coefficients and the Bloch-Gieseker property for some homogeneous bundles
Mei-Chu
Chang
975-992
Abstract: A vector bundle has the Bloch-Gieseker property if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle $\Omega ^{p}_{\mathbb{P}_{n}}(p+1)$ has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of $t^{k}$ in the rational function $\frac{(1+t)^{\binom n p} (1+3t)^{\binom {n}{p-2}} \cdots (1+(p-1)t)^{\binom n2}... ...1+2t)^{\binom {n}{p-1}} (1+4t)^{\binom {n}{p-3}} \cdots (1+pt)^{\binom {n}{1}}}$ (for $p$ even).
A dimension inequality for Cohen-Macaulay rings
Sean
Sather-Wagstaff
993-1005
Abstract: The recent work of Kurano and Roberts on Serre's positivity conjecture suggests the following dimension inequality: for prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ in a local, Cohen-Macaulay ring $(A,\mathfrak{n})$ such that $e(A_{\mathfrak{p}})=e(A)$ we have $\dim(A/\mathfrak{p})+\dim(A/\mathfrak{q})\leq\dim(A)$. We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when $R/\mathfrak{p}$ is regular.
The structure of linear codes of constant weight
Jay
A.
Wood
1007-1026
Abstract: In this paper we determine completely the structure of linear codes over $\mathbb Z/N\mathbb Z$ of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.
Symplectic $2$-handles and transverse links
David
T.
Gay
1027-1047
Abstract: A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching $2$-handles to convex boundaries of symplectic $4$-manifolds along links transverse to the induced contact structures. One method results in concave boundaries and depends on a fibration of the link complement over $S^1$; in this case the handles can be attached with any framing larger than a lower bound determined by the fibration. The other method results in a weaker convexity condition on the new boundary (sufficient to imply tightness of the new contact structure), and in this case the handles can be attached with any framing less than a certain upper bound. These methods supplement methods developed by Weinstein and Eliashberg for attaching symplectic $2$-handles along Legendrian knots.
Splittings of finitely generated groups over two-ended subgroups
Brian
H.
Bowditch
1049-1078
Abstract: We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of the JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated group which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.
Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications
A.
Fursikov;
M.
Gunzburger;
L.
Hou
1079-1116
Abstract: We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.
On a stochastic nonlinear equation in one-dimensional viscoelasticity
Jong
Uhn
Kim
1117-1135
Abstract: In this paper we discuss an initial-boundary value problem for a stochastic nonlinear equation arising in one-dimensional viscoelasticity. We propose to use a new direct method to obtain a solution. This method is expected to be applicable to a broad class of nonlinear stochastic partial differential equations.
Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations
Koichiro
Naito
1137-1151
Abstract: In this paper we introduce recurrent dimensions of discrete dynamical systems and we give upper and lower bounds of the recurrent dimensions of the quasi-periodic orbits. We show that these bounds have different values according to the algebraic properties of the frequency and we investigate these dimensions of quasi-periodic trajectories given by solutions of a nonlinear PDE.
Discrete decompositions for bilinear operators and almost diagonal conditions
Loukas
Grafakos;
Rodolfo
H.
Torres
1153-1176
Abstract: Using discrete decomposition techniques, bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct analogue of an almost diagonal condition for linear operators of Calderón-Zygmund type. Applications include a reduced $T1$ theorem for bilinear pseudodifferential operators and the extension of an $L^p$ multiplier result of Coifman and Meyer to the full range of $H^p$ spaces. The results of this article rely on decomposition techniques developed by Frazier and Jawerth and on the vector valued maximal function estimate of Fefferman and Stein.
Composite Bank-Laine functions and a question of Rubel
J.
K.
Langley
1177-1191
Abstract: A Bank-Laine function is an entire function $E$ satisfying $E = f \circ g$, with $f, g$ entire. Further, we prove that if $h$ is a transcendental entire function of finite order, then there exists a path tending to infinity on which $h$ and all its derivatives tend to infinity, thus establishing for finite order a conjecture of Rubel.
Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds
Emmanuel
Hebey
1193-1213
Abstract: Given $(M,g)$ a smooth compact Riemannian $n$-manifold, $n \ge 3$, we return in this article to the study of the sharp Sobolev-Poincaré type inequality \begin{displaymath}\Vert u\Vert_{2^\star}^2 \le K_n^2\Vert\nabla u\Vert_2^2 + B\Vert u\Vert_1^2\tag*{(0.1)}\end{displaymath} where $2^\star = 2n/(n-2)$ is the critical Sobolev exponent, and $K_n$ is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that $(0.1)$ is true if $n = 3$, that $(0.1)$is true if $n \ge 4$ and the sectional curvature of $g$ is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension $n$ is true and the sectional curvature of $g$ is nonpositive, but that $(0.1)$ is false if $n \ge 4$ and the scalar curvature of $g$ is positive somewhere. When $(0.1)$ is true, we define $B(g)$ as the smallest $B$ in $(0.1)$. The saturated form of $(0.1)$ reads as \begin{displaymath}\Vert u\Vert_{2^\star}^2 \le K_n^2\Vert\nabla u\Vert_2^2+B(g)\Vert u\Vert_1^2. \tag*{(0.2)}\end{displaymath} We assume in this article that $n \ge 4$, and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality $(0.1)$. We prove that $(0.1)$ is true, and that $(0.2)$ possesses extremal functions when the scalar curvature of $g$ is negative. A fairly complete answer to the question of the validity of $(0.1)$ under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given.
Scattering poles for asymptotically hyperbolic manifolds
David
Borthwick;
Peter
Perry
1215-1231
Abstract: For a class of manifolds $X$ that includes quotients of real hyperbolic $(n+1)$-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on $X$ coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for $X$. In order to carry out the proof, we use Shmuel Agmon's perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.
Spherical unitary highest weight representations
Bernhard
Krötz;
Karl-Hermann
Neeb
1233-1264
Abstract: In this paper we give an almost complete classification of the $H$-spherical unitary highest weight representations of a hermitian Lie group $G$, where $G/H$ is a symmetric space of Cayley type.
Hamburger and Stieltjes moment problems in several variables
F.-H.
Vasilescu
1265-1278
Abstract: In this paper we give solutions to the Hamburger and Stieltjes moment problems in several variables, in algebraic terms, via extended sequences. Some characterizations of the uniqueness of the solutions are also presented.