2,314 research outputs found
Noise-Induced Stabilization of Planar Flows I
We show that the complex-valued ODE
\begin{equation*}
\dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0,
\end{equation*} which necessarily has trajectories along which the dynamics
blows up in finite time, can be stabilized by the addition of an arbitrarily
small elliptic, additive Brownian stochastic term. We also show that the
stochastic perturbation has a unique invariant measure which is heavy-tailed
yet is uniformly, exponentially attracting. The methods turn on the
construction of Lyapunov functions. The techniques used in the construction are
general and can likely be used in other settings where a Lyapunov function is
needed. This is a two-part paper. This paper, Part I, focuses on general
Lyapunov methods as applied to a special, simplified version of the problem.
Part II of this paper extends the main results to the general setting.Comment: Part one of a two part pape
Towards a first observation of magneto-electric directional anisotropy and linear birefringence in gases
In this contribution to PSAS'2010 we report on recent progress on an
experiment aimed at measuring small optical directional anisotropies by
frequency metrology in a high finesse ring cavity. We focus on our first
experimental goal, the measurement of magneto-electric effects in gases. After
a review of the expected effects in our set-up, we present the apparatus and
the measurement procedure, showing that we already have the necessary
sensitivity to start novel experiments.Comment: Proceedings of PSAS'2010, to be published in Canadian Journal of
Physics, 2011 Ja
A stochastic-Lagrangian particle system for the Navier-Stokes equations
This paper is based on a formulation of the Navier-Stokes equations developed
by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to
appear), where the velocity field of a viscous incompressible fluid is written
as the expected value of a stochastic process. In this paper, we take
copies of the above process (each based on independent Wiener processes), and
replace the expected value with times the sum over these
copies. (We remark that our formulation requires one to keep track of
stochastic flows of diffeomorphisms, and not just the motion of particles.)
We prove that in two dimensions, this system of interacting diffeomorphisms
has (time) global solutions with initial data in the space
\holderspace{1}{\alpha} which consists of differentiable functions whose
first derivative is H\"older continuous (see Section \ref{sGexist} for
the precise definition). Further, we show that as the system
converges to the solution of Navier-Stokes equations on any finite interval
. However for fixed , we prove that this system retains roughly
times its original energy as . Hence the limit
and do not commute. For general flows, we only
provide a lower bound to this effect. In the special case of shear flows, we
compute the behaviour as explicitly.Comment: v3: Typo fixes, and a few stylistic changes. 17 pages, 2 figure
Simple Systems with Anomalous Dissipation and Energy Cascade
We analyze a class of linear shell models subject to stochastic forcing in
finitely many degrees of freedom. The unforced systems considered formally
conserve energy. Despite being formally conservative, we show that these
dynamical systems support dissipative solutions (suitably defined) and, as a
result, may admit unique (statistical) steady states when the forcing term is
nonzero. This claim is demonstrated via the complete characterization of the
solutions of the system above for specific choices of the coupling
coefficients. The mechanism of anomalous dissipations is shown to arise via a
cascade of the energy towards the modes () with higher ; this is
responsible for solutions with interesting energy spectra, namely \EE |a_n|^2
scales as as . Here the exponents depend on
the coupling coefficients and \EE denotes expectation with respect to
the equilibrium measure. This is reminiscent of the conjectured properties of
the solutions of the Navier-Stokes equations in the inviscid limit and their
accepted relationship with fully developed turbulence. Hence, these simple
models illustrate some of the heuristic ideas that have been advanced to
characterize turbulence, similar in that respect to the random passive scalar
or random Burgers equation, but even simpler and fully solvable.Comment: 32 Page
Self-expansion within sexual minority relationships
According to the self-expansion model, people increase their positive self-concept content when they form and maintain romantic relationships, and self-expansion is an important predictor of relationship outcomes. Although thought to be universal, no prior research has examined self-expansion among sexual minority individuals. In the current study, sexual minority (N = 226) and heterosexual (N = 104) participants completed measures of self-expansion and relationship outcomes, and sexual minority participants completed measures of sexual minority stress. Overall, sexual minorities reported similar levels of self-expansion as heterosexuals, and sexual minority status did not moderate the association between self-expansion and relationship satisfaction, investments, or quality of alternatives. However, sexual minority status moderated the association between self-expansion and commitment. For sexual minority participants, self-expansion negatively correlated with sexual minority stressors (i.e., internalized homonegativity, concealment, inauthenticity) and moderated the association between internalized homonegativity and relationship satisfaction and commitment, as well as concealment and relationship satisfaction and commitment, such that the negative association between sexual minority stressors and relationship outcomes was weaker in relationships characterized by high (vs. low) levels of self-expansion
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