61,638 research outputs found
Herding as a Learning System with Edge-of-Chaos Dynamics
Herding defines a deterministic dynamical system at the edge of chaos. It
generates a sequence of model states and parameters by alternating parameter
perturbations with state maximizations, where the sequence of states can be
interpreted as "samples" from an associated MRF model. Herding differs from
maximum likelihood estimation in that the sequence of parameters does not
converge to a fixed point and differs from an MCMC posterior sampling approach
in that the sequence of states is generated deterministically. Herding may be
interpreted as a"perturb and map" method where the parameter perturbations are
generated using a deterministic nonlinear dynamical system rather than randomly
from a Gumbel distribution. This chapter studies the distinct statistical
characteristics of the herding algorithm and shows that the fast convergence
rate of the controlled moments may be attributed to edge of chaos dynamics. The
herding algorithm can also be generalized to models with latent variables and
to a discriminative learning setting. The perceptron cycling theorem ensures
that the fast moment matching property is preserved in the more general
framework
A new numerical method for constructing quasi-equilibrium sequences of irrotational binary neutron stars in general relativity
We propose a new numerical method to compute quasi-equilibrium sequences of
general relativistic irrotational binary neutron star systems. It is a good
approximation to assume that (1) the binary star system is irrotational, i.e.
the vorticity of the flow field inside component stars vanishes everywhere
(irrotational flow), and (2) the binary star system is in quasi-equilibrium,
for an inspiraling binary neutron star system just before the coalescence as a
result of gravitational wave emission. We can introduce the velocity potential
for such an irrotational flow field, which satisfies an elliptic partial
differential equation (PDE) with a Neumann type boundary condition at the
stellar surface. For a treatment of general relativistic gravity, we use the
Wilson--Mathews formulation, which assumes conformal flatness for spatial
components of metric. In this formulation, the basic equations are expressed by
a system of elliptic PDEs. We have developed a method to solve these PDEs with
appropriate boundary conditions. The method is based on the established
prescription for computing equilibrium states of rapidly rotating axisymmetric
neutron stars or Newtonian binary systems. We have checked the reliability of
our new code by comparing our results with those of other computations
available. We have also performed several convergence tests. By using this
code, we have obtained quasi-equilibrium sequences of irrotational binary star
systems with strong gravity as models for final states of real evolution of
binary neutron star systems just before coalescence. Analysis of our
quasi-equilibrium sequences of binary star systems shows that the systems may
not suffer from dynamical instability of the orbital motion and that the
maximum density does not increase as the binary separation decreases.Comment: 20 pages, 18 figures, more results of convergence tests are added,
revised version accepted for publication in PR
Dynamical transition in the TASEP with Langmuir kinetics: mean-field theory
We develop a mean-field theory for the totally asymmetric simple exclusion
process (TASEP) with open boundaries, in order to investigate the so-called
dynamical transition. The latter phenomenon appears as a singularity in the
relaxation rate of the system toward its non-equilibrium steady state. In the
high-density (low-density) phase, the relaxation rate becomes independent of
the injection (extraction) rate, at a certain critical value of the parameter
itself, and this transition is not accompanied by any qualitative change in the
steady-state behavior. We characterize the relaxation rate by providing
rigorous bounds, which become tight in the thermodynamic limit. These results
are generalized to the TASEP with Langmuir kinetics, where particles can also
bind to empty sites or unbind from occupied ones, in the symmetric case of
equal binding/unbinding rates. The theory predicts a dynamical transition to
occur in this case as well.Comment: 37 pages (including 16 appendix pages), 6 figures. Submitted to
Journal of Physics
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
Selection theorem for systems with inheritance
The problem of finite-dimensional asymptotics of infinite-dimensional dynamic
systems is studied. A non-linear kinetic system with conservation of supports
for distributions has generically finite-dimensional asymptotics. Such systems
are apparent in many areas of biology, physics (the theory of parametric wave
interaction), chemistry and economics. This conservation of support has a
biological interpretation: inheritance. The finite-dimensional asymptotics
demonstrates effects of "natural" selection. Estimations of the asymptotic
dimension are presented. After some initial time, solution of a kinetic
equation with conservation of support becomes a finite set of narrow peaks that
become increasingly narrow over time and move increasingly slowly. It is
possible that these peaks do not tend to fixed positions, and the path covered
tends to infinity as t goes to infinity. The drift equations for peak motion
are obtained. Various types of distribution stability are studied: internal
stability (stability with respect to perturbations that do not extend the
support), external stability or uninvadability (stability with respect to
strongly small perturbations that extend the support), and stable realizability
(stability with respect to small shifts and extensions of the density peaks).
Models of self-synchronization of cell division are studied, as an example of
selection in systems with additional symmetry. Appropriate construction of the
notion of typicalness in infinite-dimensional space is discussed, and the
notion of "completely thin" sets is introduced.
Key words: Dynamics; Attractor; Evolution; Entropy; Natural selectionComment: 46 pages, the final journal versio
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