3,429 research outputs found

### A Model of Heat Conduction

We define a deterministic ``scattering'' model for heat conduction which is
continuous in space, and which has a Boltzmann type flavor, obtained by a
closure based on memory loss between collisions. We prove that this model has,
for stochastic driving forces at the boundary, close to Maxwellians, a unique
non-equilibrium steady state

### Spectral Properties of Hypoelliptic Operators

We study hypoelliptic operators with polynomially bounded coefficients that
are of the form K = sum_{i=1}^m X_i^T X_i + X_0 + f, where the X_j denote first
order differential operators, f is a function with at most polynomial growth,
and X_i^T denotes the formal adjoint of X_i in L^2. For any e > 0 we show that
an inequality of the form |u|_{delta,delta} <= C(|u|_{0,eps} + |(K+iy)u|_{0,0})
holds for suitable delta and C which are independent of y in R, in weighted
Sobolev spaces (the first index is the derivative, and the second the growth).
We apply this result to the Fokker-Planck operator for an anharmonic chain of
oscillators coupled to two heat baths. Using a method of Herau and Nier [HN02],
we conclude that its spectrum lies in a cusp {x+iy|x >= |y|^tau-c, tau in
(0,1], c in R}.Comment: 3 figure

### Curvature of Co-Links Uncovers Hidden Thematic Layers in the World Wide Web

Beyond the information stored in pages of the World Wide Web, novel types of
``meta-information'' are created when they connect to each other. This
information is a collective effect of independent users writing and linking
pages, hidden from the casual user. Accessing it and understanding the
inter-relation of connectivity and content in the WWW is a challenging problem.
We demonstrate here how thematic relationships can be located precisely by
looking only at the graph of hyperlinks, gleaning content and context from the
Web without having to read what is in the pages. We begin by noting that
reciprocal links (co-links) between pages signal a mutual recognition of
authors, and then focus on triangles containing such links, since triangles
indicate a transitive relation. The importance of triangles is quantified by
the clustering coefficient (Watts) which we interpret as a curvature
(Gromov,Bridson-Haefliger). This defines a Web-landscape whose connected
regions of high curvature characterize a common topic. We show experimentally
that reciprocity and curvature, when combined, accurately capture this
meta-information for a wide variety of topics. As an example of future
directions we analyze the neural network of C. elegans (White, Wood), using the
same methods.Comment: 8 pages, 5 figures, expanded version of earlier submission with more
example

### Non-Linear Stability Analysis of Higher Order Dissipative Partial Differential Equations

We extend the invariant manifold method for analyzing the asymptotics of
dissipative partial differential equations on unbounded spatial domains to
treat equations in which the linear part has order greater than two. One
important example of this type of equation which we analyze in some detail is
the Cahn-Hilliard equation. We analyze the marginally stable solutions of this
equation in some detail. A second context in which such equations arise is in
the Ginzburg-Landau equation, or other pattern forming equations, near a
codimension-two bifurcation

### Scattering Phases and Density of States for Exterior Domain

For a bounded open domain $\Omega\in \real^2$ with connected complement and
piecewise smooth boundary, we consider the Dirichlet Laplacian -\DO on
$\Omega$ and the S-matrix on the complement $\Omega^c$. Using the restriction
$A_E$ of $(-\Delta-E)^{-1}$ to the boundary of $\Omega$, we establish that
$A_{E_0}^{-1/2}A_EA_{E_0}^{-1/2}-1$ is trace class when $E_0$ is negative and
give bounds on the energy dependence of this difference. This allows for
precise bounds on the total scattering phase, the definition of a
$\zeta$-function, and a Krein spectral formula, which improve similar results
found in the literature.Comment: 15 pages, Postscript, A

### Transport Properties of the Lorentz Gas in Terms of Periodic Orbits

We establish a formula relating global diffusion in a space periodic
dynamical system to cycles in the elementary cell which tiles the space under
translations.Comment: 8 pages, Postscript, A

### Temperature Profiles in Hamiltonian Heat Conduction

We study heat transport in the context of Hamiltonian and related stochastic
models with nearest-neighbor coupling, and derive a universal law for the
temperature profiles of a large class of such models. This law contains a
parameter $\alpha$, and is linear only when $\alpha=1$. The value of $\alpha$
depends on energy-exchange mechanisms, including the range of motion of tracer
particles and their times of flight.Comment: Revised text, same results Second revisio

### Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

In this paper we describe invariant geometrical ~structures in the phase
space of the Swift-Hohenberg equation in a neighborhood of its periodic
stationary states. We show that in spite of the fact that these states are only
marginally stable (i.e., the linearized problem about these states has
continuous spectrum extending all the way up to zero), there exist finite
dimensional invariant manifolds in the phase space of this equation which
determine the long-time behavior of solutions near these stationary solutions.
In particular, using this point of view, we obtain a new demonstration of
Schneider's recent proof that these states are nonlinearly stable.Comment: 44 pages, plain tex, 0 figure

- â€¦