517,597 research outputs found
Integrability and chaos: the classical uncertainty
In recent years there has been a considerable increase in the publishing of
textbooks and monographs covering what was formerly known as random or
irregular deterministic motion, now named by the more fashionable term of
deterministic chaos. There is still substantial interest in a matter that is
included in many graduate and even undergraduate courses on classical
mechanics. Based on the Hamiltonian formalism, the main objective of this
article is to provide, from the physicist's point of view, an overall and
intuitive review of this broad subject (with some emphasis on the KAM theorem
and the stability of planetary motions) which may be useful to both students
and instructors.Comment: 24 pages, 10 figure
Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs
In this paper, we consider lower bounds on the query complexity for testing
CSPs in the bounded-degree model.
First, for any ``symmetric'' predicate except \equ
where , we show that every (randomized) algorithm that distinguishes
satisfiable instances of CSP(P) from instances -far
from satisfiability requires queries where is the
number of variables and is a constant that depends on and
. This breaks a natural lower bound , which is
obtained by the birthday paradox. We also show that every one-sided error
tester requires queries for such . These results are hereditary
in the sense that the same results hold for any predicate such that
. For EQU, we give a one-sided error tester
whose query complexity is . Also, for 2-XOR (or,
equivalently E2LIN2), we show an lower bound for
distinguishing instances between -close to and -far
from satisfiability.
Next, for the general k-CSP over the binary domain, we show that every
algorithm that distinguishes satisfiable instances from instances
-far from satisfiability requires queries. The
matching NP-hardness is not known, even assuming the Unique Games Conjecture or
the -to- Conjecture. As a corollary, for Maximum Independent Set on
graphs with vertices and a degree bound , we show that every
approximation algorithm within a factor d/\poly\log d and an additive error
of requires queries. Previously, only super-constant
lower bounds were known
Effective Potential for Complex Langevin Equations
We construct an effective potential for the complex Langevin equation on a
lattice. We show that the minimum of this effective potential gives the
space-time and Langevin time average of the complex Langevin field. The loop
expansion of the effective potential is matched with the derivative expansion
of the associated Schwinger-Dyson equation to predict the stationary
distribution to which the complex Langevin equation converges.Comment: 23 pages, 2 figure
Characteristics of Cosmic Time
The nature of cosmic time is illuminated using Hamilton-Jacobi theory for
general relativity. For problems of interest to cosmology, one may solve for
the phase of the wavefunctional by using a line integral in superspace. Each
contour of integration corresponds to a particular choice of time hypersurface,
and each yields the same answer. In this way, one can construct a covariant
formalism where all time hypersurfaces are treated on an equal footing. Using
the method of characteristics, explicit solutions for an inflationary epoch
with several scalar fields are given. The theoretical predictions of double
inflation are compared with recent galaxy data and large angle microwave
background anisotropies.Comment: 20 pages, RevTex using Latex 2.09, Submitted to Physical Review D Two
figures included in fil
Efficient size estimation and impossibility of termination in uniform dense population protocols
We study uniform population protocols: networks of anonymous agents whose
pairwise interactions are chosen at random, where each agent uses an identical
transition algorithm that does not depend on the population size . Many
existing polylog time protocols for leader election and majority
computation are nonuniform: to operate correctly, they require all agents to be
initialized with an approximate estimate of (specifically, the exact value
). Our first main result is a uniform protocol for
calculating with high probability in time and
states ( bits of memory). The protocol is
converging but not terminating: it does not signal when the estimate is close
to the true value of . If it could be made terminating, this would
allow composition with protocols, such as those for leader election or
majority, that require a size estimate initially, to make them uniform (though
with a small probability of failure). We do show how our main protocol can be
indirectly composed with others in a simple and elegant way, based on the
leaderless phase clock, demonstrating that those protocols can in fact be made
uniform. However, our second main result implies that the protocol cannot be
made terminating, a consequence of a much stronger result: a uniform protocol
for any task requiring more than constant time cannot be terminating even with
probability bounded above 0, if infinitely many initial configurations are
dense: any state present initially occupies agents. (In particular,
no leader is allowed.) Crucially, the result holds no matter the memory or time
permitted. Finally, we show that with an initial leader, our size-estimation
protocol can be made terminating with high probability, with the same
asymptotic time and space bounds.Comment: Using leaderless phase cloc
Convergence of large deviation estimators
We study the convergence of statistical estimators used in the estimation of
large deviation functions describing the fluctuations of equilibrium,
nonequilibrium, and manmade stochastic systems. We give conditions for the
convergence of these estimators with sample size, based on the boundedness or
unboundedness of the quantity sampled, and discuss how statistical errors
should be defined in different parts of the convergence region. Our results
shed light on previous reports of 'phase transitions' in the statistics of free
energy estimators and establish a general framework for reliably estimating
large deviation functions from simulation and experimental data and identifying
parameter regions where this estimation converges.Comment: 13 pages, 6 figures. v2: corrections focusing the paper on large
deviations; v3: minor corrections, close to published versio
Polynomial Hamiltonian form of General Relativity
Phase space of General Relativity is extended to a Poisson manifold by
inclusion of the determinant of the metric and conjugate momentum as additional
independent variables. As a result, the action and the constraints take a
polynomial form. New expression for the generating functional for the Green
functions is proposed. We show that the Dirac bracket defines degenerate
Poisson structure on a manifold, and a second class constraints are the Casimir
functions with respect to this structure. As an application of the new
variables, we consider the Friedmann universe.Comment: 33 pages, 1 figure, corrected reference
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