2,103 research outputs found
Lattice initial segments of the hyperdegrees
We affirm a conjecture of Sacks [1972] by showing that every countable
distributive lattice is isomorphic to an initial segment of the hyperdegrees,
. In fact, we prove that every sublattice of any
hyperarithmetic lattice (and so, in particular, every countable locally finite
lattice) is isomorphic to an initial segment of . Corollaries
include the decidability of the two quantifier theory of
and the undecidability of its three quantifier theory. The key tool in the
proof is a new lattice representation theorem that provides a notion of forcing
for which we can prove a version of the fusion lemma in the hyperarithmetic
setting and so the preservation of . Somewhat surprisingly,
the set theoretic analog of this forcing does not preserve . On
the other hand, we construct countable lattices that are not isomorphic to an
initial segment of
Integral Fluctuation Relations for Entropy Production at Stopping Times
A stopping time is the first time when a trajectory of a stochastic
process satisfies a specific criterion. In this paper, we use martingale theory
to derive the integral fluctuation relation for the stochastic entropy production in a
stationary physical system at stochastic stopping times . This fluctuation
relation implies the law , which states
that it is not possible to reduce entropy on average, even by stopping a
stochastic process at a stopping time, and which we call the second law of
thermodynamics at stopping times. This law implies bounds on the average amount
of heat and work a system can extract from its environment when stopped at a
random time. Furthermore, the integral fluctuation relation implies that
certain fluctuations of entropy production are universal or are bounded by
universal functions. These universal properties descend from the integral
fluctuation relation by selecting appropriate stopping times: for example, when
is a first-passage time for entropy production, then we obtain a bound on
the statistics of negative records of entropy production. We illustrate these
results on simple models of nonequilibrium systems described by Langevin
equations and reveal two interesting phenomena. First, we demonstrate that
isothermal mesoscopic systems can extract on average heat from their
environment when stopped at a cleverly chosen moment and the second law at
stopping times provides a bound on the average extracted heat. Second, we
demonstrate that the average efficiency at stopping times of an autonomous
stochastic heat engines, such as Feymann's ratchet, can be larger than the
Carnot efficiency and the second law of thermodynamics at stopping times
provides a bound on the average efficiency at stopping times.Comment: 37 pages, 6 figure
Homological norms on nonpositively curved manifolds
We relate the Gromov norm on homology classes to the harmonic norm on the
dual cohomology and obtain double sided bounds in terms of the volume and other
geometric quantities of the underlying manifold. Along the way, we provide
comparisons to other related norms and quantities as well.Comment: 19 pages, minor change
A new measure of instability and topological entropy of area-preserving twist diffeomorphisms
We introduce a new measure of instability of area-preserving twist
diffeomorphisms, which generalizes the notions of angle of splitting of
separatrices, and flux through a gap of a Cantori. As an example of
application, we establish a sharp >0 lower bound on the topological entropy in
a neighbourhood of a hyperbolic, unique action-minimizing fixed point, assuming
only no topological obstruction to diffusion, i.e. no homotopically non-trivial
invariant circle consisting of orbits with the rotation number 0. The proof is
based on a new method of precise construction of positive entropy invariant
measures, applicable to more general Lagrangian systems, also in higher degrees
of freedom
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