5,609 research outputs found

### Finding Cycles and Trees in Sublinear Time

We present sublinear-time (randomized) algorithms for finding simple cycles
of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The
complexity of these algorithms is related to the distance of the graph from
being $C_k$-minor-free (resp., free from having the corresponding tree-minor).
In particular, if the graph is far (i.e., $\Omega(1)$-far) {from} being
cycle-free, i.e. if one has to delete a constant fraction of edges to make it
cycle-free, then the algorithm finds a cycle of polylogarithmic length in time
\tildeO(\sqrt{N}), where $N$ denotes the number of vertices. This time
complexity is optimal up to polylogarithmic factors.
The foregoing results are the outcome of our study of the complexity of {\em
one-sided error} property testing algorithms in the bounded-degree graphs
model. For example, we show that cycle-freeness of $N$-vertex graphs can be
tested with one-sided error within time complexity
\tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known $\Omega(\sqrt{N})$
query lower bound, and contrasts with the fact that any minor-free property
admits a {\em two-sided error} tester of query complexity that only depends on
the proximity parameter \e. For any constant $k\geq3$, we extend this result
to testing whether the input graph has a simple cycle of length at least $k$.
On the other hand, for any fixed tree $T$, we show that $T$-minor-freeness has
a one-sided error tester of query complexity that only depends on the proximity
parameter \e.
Our algorithm for finding cycles in bounded-degree graphs extends to general
graphs, where distances are measured with respect to the actual number of
edges. Such an extension is not possible with respect to finding tree-minors in
$o(\sqrt{N})$ complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree
Graphs, One-Sided vs Two-Sided Error Probability Updated versio

### The Global Baroclinic Instability in Accretion Disks. II: Local Linear Analysis

This paper contains a local linear stability analysis for accretion disks
under the influence of a global radial entropy gradient beta = - d log T / d
log r for constant surface density. Numerical simulations suggested the
existence of an instability in two- and three-dimensional models of the solar
nebula. The present paper tries to clarify, quantify, and explain such a global
baroclinic instability for two-dimensional flat accretion disk models. As a
result linear theory predicts a transient linear instability that will amplify
perturbations only for a limited time or up to a certain finite amplification.
This can be understood as a result of the growth time of the instability being
longer than the shear time which destroys the modes which are able to grow. So
only non-linear effects can lead to a relevant amplification. Nevertheless, a
lower limit on the entropy gradient ~beta = 0.22 for the transient linear
instability is derived, which can be tested in future non-linear simulations.
This would help to explain the observed instability in numerical simulations as
an ultimate result of the transient linear instability, i.e. the Global
Baroclinic Instability.Comment: 35 pages, 11 figures; ApJ in pres

### Long-term Stable Equilibria for Synchronous Binary Asteroids

Synchronous binary asteroids may exist in a long-term stable equilibrium,
where the opposing torques from mutual body tides and the binary YORP (BYORP)
effect cancel. Interior of this equilibrium, mutual body tides are stronger
than the BYORP effect and the mutual orbit semi-major axis expands to the
equilibrium; outside of the equilibrium, the BYORP effect dominates the
evolution and the system semi-major axis will contract to the equilibrium. If
the observed population of small (0.1 - 10 km diameter) synchronous binaries
are in static configurations that are no longer evolving, then this would be
confirmed by a null result in the observational tests for the BYORP effect. The
confirmed existence of this equilibrium combined with a shape model of the
secondary of the system enables the direct study of asteroid geophysics through
the tidal theory. The observed synchronous asteroid population cannot exist in
this equilibrium if described by the canonical "monolithic" geophysical model.
The "rubble pile" geophysical model proposed by \citet{Goldreich2009} is
sufficient, however it predicts a tidal Love number directly proportional to
the radius of the asteroid, while the best fit to the data predicts a tidal
Love number inversely proportional to the radius. This deviation from the
canonical and \citet{Goldreich2009} models motivates future study of asteroid
geophysics. Ongoing BYORP detection campaigns will determine whether these
systems are in an equilibrium, and future determination of secondary shapes
will allow direct determination of asteroid geophysical parameters.Comment: 17 pages, 1 figur

### Simulations of Incompressible MHD Turbulence

We simulate incompressible MHD turbulence in the presence of a strong
background magnetic field. Our major conclusions are: 1) MHD turbulence is most
conveniently described in terms of counter propagating shear Alfven and slow
waves. Shear Alfven waves control the cascade dynamics. Slow waves play a
passive role and adopt the spectrum set by the shear Alfven waves, as does a
passive scalar. 2) MHD turbulence is anisotropic with energy cascading more
rapidly along k_perp than along k_parallel, where k_perp and k_parallel refer
to wavevector components perpendicular and parallel to the local magnetic
field. Anisotropy increases with increasing k_perp. 3) MHD turbulence is
generically strong in the sense that the waves which comprise it suffer order
unity distortions on timescales comparable to their periods. Nevertheless,
turbulent fluctuations are small deep inside the inertial range compared to the
background field. 4) Decaying MHD turbulence is unstable to an increase of the
imbalance between the flux of waves propagating in opposite directions along
the magnetic field. 5) Items 1-4 lend support to the model of strong MHD
turbulence by Goldreich & Sridhar (GS). Results from our simulations are also
consistent with the GS prediction gamma=2/3. The sole notable discrepancy is
that 1D power law spectra, E(k_perp) ~ k_perp^{-alpha}, determined from our
simulations exhibit alpha ~ 3/2, whereas the GS model predicts alpha = 5/3.Comment: 56 pages, 30 figures, submitted to ApJ 59 pages, 31 figures, accepted
to Ap

### An Exact, Three-Dimensional, Time-Dependent Wave Solution in Local Keplerian Flow

We present an exact three-dimensional wave solution to the shearing sheet
equations of motion. The existence of this solution argues against transient
amplification as a route to turbulence in unmagnetized disks. Moreover, because
the solution covers an extensive dynamical range in wavenumber space, it is an
excellent test of the dissipative properties of numerical codes.Comment: 22 pages, 4 figures. To appear Apj Dec 1 200

### Gravity-Modes in ZZ Ceti Stars: IV. Amplitude Saturation by Parametric Instability

ZZ Ceti stars exhibit small amplitude photometric pulsations in multiple
gravity-modes. We demonstrate that parametric instability, a form of resonant
3-mode coupling, limits overstable modes to amplitudes similar to those
observed. In particular, it reproduces the observed trend that longer period
modes have larger amplitudes. Parametric instability involves the
destabilization of a pair of stable daughter modes by an overstable parent
mode. The 3-modes must satisfy exact angular selection rules and approximate
frequency resonance. The lowest instability threshold for each parent mode is
provided by the daughter pair that minimizes
$(\delta\omega^2+\gamma_d^2)/\kappa^2$, where $\kappa$ is the nonlinear
coupling constant, $\delta\omega$ is the frequency mismatch, and $\gamma_d$ is
the energy damping rate of the daughter modes. The overstable mode's amplitude
is maintained at close to the instability threshold value.
Although parametric instability defines an upper envelope for the amplitudes
of overstable modes in ZZ Ceti stars, other nonlinear mechanisms are required
to account for the irregular distribution of amplitudes of similar modes and
the non-detection of modes with periods longer than 1,200\s. Resonant 3-mode
interactions involving more than one excited mode may account for the former.
Our leading candidate for the latter is Kelvin-Helmholtz instability of the
mode-driven shear layer below the convection zone.Comment: 16 pages with 10 figures, abstract shortened, submitted to Ap

### Physical Constraints On Fast Radio Burst

Fast Radio Bursts (FRBs) are isolated, \ms radio pulses with dispersion
measure (DM) of order 10^3\DMunit. Galactic candidates for the DM of high
latitude bursts detected at \GHz frequencies are easily dismissed. DM from
bursts emitted in stellar coronas are limited by free-free absorption and those
from HII regions are bounded by the nondetection of associated free-free
emission at radio wavelengths. Thus, if astronomical, FRBs are probably
extra-galactic. FRB 110220 has a scattering tail of \sim 5.6\pm 0.1 \ms. If
the electron density fluctuations arise from a turbulent cascade, the
scattering is unlikely to be due to propagation through the diffuse
intergalactic plasma. A more plausible explanation is that this burst sits in
the central region of its host galaxy. Pulse durations of order \ms constrain
the sizes of FRB sources implying high brightness temperatures that indicates
coherent emission. Electric fields near FRBs at cosmological distances would be
so strong that they could accelerate free electrons from rest to relativistic
energies in a single wave period.Comment: 5 pages, accepted by ApJ

### Origin of chaos in the PrometheusâPandora system

We demonstrate that the chaotic orbits of Prometheus and Pandora are due to interactions associated with the 121:118 mean motion resonance. Differential precession splits this resonance into a quartet of components equally spaced in frequency. Libration widths of the individual components exceed the splitting resulting in resonance overlap which causes the chaos. A single degree of freedom model captures the essential features of the chaotic dynamics. Mean motions of Prometheus and Pandora wander chaotically in zones of width 1.8 deg yr^â1 and 3.1 deg yr^â1, respectively

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