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## Closure testing the NNPDF3.0 methodology

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/09/2014
Português

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A thorough understanding of the issues surrounding the determination of
parton distributions is crucial due to their importance to calculations of LHC
observables. However, it is still not fully understood how much of an impact
methodological bias has on PDF fits. Closure tests, where a fit is performed to
pseudo-data generated using an existing PDF set, provide a way of directly
investigating whether current PDF fitting methodologies are successful. Here,
we present a sample of results from closure tests applying the NNPDF
methodology to data created using a variety of different PDF sets. The results
validate our methodology by showing that the initial PDFs can be reproduced
within uncertainties. We also briefly discuss our latest PDF determination,
NNPDF3.0, which has been developed making extensive use of the closure test
technique.; Comment: 4 pages, 4 figures, to appear in the proceedings of the 17th
High-Energy Physics International Conference in Quantum Chromodynamics
(QCD14), Montpellier, 30 June - 4 July, 2014

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## Understanding star formation in molecular clouds I. Effects of line-of-sight contamination on the column density structure

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Column-density maps of molecular clouds are one of the most important
observables in the context of molecular cloud- and star-formation (SF) studies.
With the Herschel satellite it is now possible to determine the column density
from dust emission. We use observations and simulations to demonstrate how LOS
contamination affects the column density probability distribution function
(PDF). We apply a first-order approximation (removing a constant level) to the
molecular clouds of Auriga, Maddalena, Carina and NGC3603. In perfect agreement
with the simulations, we find that the PDFs become broader, the peak shifts to
lower column densities, and the power-law tail of the PDF flattens after
correction. All PDFs have a lognormal part for low column densities with a peak
at Av~2, a deviation point (DP) from the lognormal at Av(DP)~4-5, and a
power-law tail for higher column densities. Assuming a density distribution
rho~r^-alpha, the slopes of the power-law tails correspond to alpha(PDF)=1.8,
1.75, and 2.5 for Auriga, Carina, and NGC3603 (alpha~1.5-2 is consistent
gravitational collapse). We find that low-mass and high-mass SF clouds display
differences in the overall column density structure. Massive clouds assemble
more gas in smaller cloud volumes than low-mass SF ones. However...

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## The Density Distribution in Turbulent Bi-stable Flows

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/01/2013
Português

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We numerically study the volume density probability distribution function
(n-PDF) and the column density probability distribution function (Sigma-PDF)
resulting from thermally bistable turbulent flows. We analyze three-dimensional
hydrodynamic models in periodic boxes of 100pc by side, where turbulence is
driven in the Fourier space at a wavenumber corresponding to 50pc. At low
densities (n <= 0.6cm^-3) the n-PDF, is well described by a lognormal
distribution for average local Mach number ranging from ~0.2 to ~5.5. As a
consequence of the non linear development of thermal instability (TI), the
logarithmic variance of the distribution for the diffuse gas increases with M
faster than in the well known isothermal case. The average local Mach number
for the dense gas (n >= 7.1cm^-3) goes from ~1.1 to ~16.9 and the shape of the
high density zone of the n-PDF changes from a power-law at low Mach numbers to
a lognormal at high M values. In the latter case the width of the distribution
is smaller than in the isothermal case and grows slower with M. At high column
densities the Sigma-PDF is well described by a lognormal for all the Mach
numbers we consider and, due to the presence of TI, the width of the
distribution is systematically larger than in the isothermal case but follows a
qualitatively similar behavior as M increases. Although a relationship between
the width of the distribution and M can be found for each one of the cases
mentioned above...

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## Multivariate Generalized Gram-Charlier Series Using only Calculus of Several Variables, instead Tensor Calculus

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The article derives multivariate Generalized Gram-Charlier (GGC) series that
expands an unknown joint probability density function (PDF) of a random vector
in terms of the differentiations of the joint PDF of a reference random vector.
Though the higher order differentiations of a multivariate PDF result into
multi-element arrays, the article uses only elementary calculus of several
variables to achieve the extention of an existing derivation for GGC series in
univariate to multivariate. The Tensor calculus is avoided by achieving the
higher order differentiations of a multivariate PDF in vector notations through
application of a specific Kronecker product based differentiation operator. The
article also obtains expressions for the mutual relations between cumulants and
moments of a random vector; integral form of a multivariate PDF; integral form
of the multivariate Hermite polynomials; the multivariate Gram-Charlier A (GCA)
series and others. The derived multivariate expressions are more elementary as
using vector notations compare to coordinatewise tensor notations and more
comprehensive as apparently more nearer to their counterparts for univariate.
The approach has already been used to achieve the same advantages for deriving
higher order cumulants of a random vector and multivariate Hermite polynomials.; Comment: 22 pages; submitted to to the `Journal of Machine Learning Research'
(JMLR); reasons for arXiv update: corrected typos...

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## Ellipsoidal collapse and the redshift space probability distribution function of dark matter

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We use the physics of ellipsoidal collapse to model the probability
distribution function of the smoothed dark matter density field in real and
redshift space. We provide a simple approximation to the exact collapse model
which shows clearly how the evolution can be thought of as a modification of
the spherical evolution model as well as of the Zeldovich Approximation. In
essence, our model specifies how the initial smoothed overdensity and shear
fields can be used to determine the shape and size of the region at later
times. We use our parametrization to extend previous work on the real-space PDF
so that it predicts the redshift space PDF as well. Our results are in good
agreement with measurements of the PDF in simulations of clustering from
Gaussian initial conditions down to scales on which the rms fluctuation is
slightly greater than unity. We also show how the highly non-Gaussian
non-linear redshifted density field in a numerical simulation can be
transformed so that it provides an estimate of the shape of the initial
real-space PDF. When applied to our simulations, our method recovers the
initial Gaussian PDF, provided the variance in the nonlinear smoothed field is
less than 4.; Comment: 10 pages, 7 figures. Accepted for publication in MNRAS

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## Joint probability density function modeling of velocity and scalar in turbulence with unstructured grids

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/06/2010
Português

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In probability density function (PDF) methods a transport equation is solved
numerically to compute the time and space dependent probability distribution of
several flow variables in a turbulent flow. The joint PDF of the velocity
components contains information on all one-point one-time statistics of the
turbulent velocity field, including the mean, the Reynolds stresses and
higher-order statistics. We developed a series of numerical algorithms to model
the joint PDF of turbulent velocity, frequency and scalar compositions for
high-Reynolds-number incompressible flows in complex geometries using
unstructured grids. Advection, viscous diffusion and chemical reaction appear
in closed form in the PDF formulation, thus require no closure hypotheses. The
generalized Langevin model (GLM) is combined with an elliptic relaxation
technique to represent the non-local effect of walls on the pressure
redistribution and anisotropic dissipation of turbulent kinetic energy. The
governing system of equations is solved fully in the Lagrangian framework
employing a large number of particles representing a finite sample of all fluid
particles. Eulerian statistics are extracted at gridpoints of the unstructured
mesh. Compared to other particle-in-cell approaches for the PDF equations...

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## Multifractal analysis with the probability density function at the three-dimensional Anderson transition

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/12/2008
Português

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The probability density function (PDF) for critical wavefunction amplitudes
is studied in the three-dimensional Anderson model. We present a formal
expression between the PDF and the multifractal spectrum f(alpha) in which the
role of finite-size corrections is properly analyzed. We show the non-gaussian
nature and the existence of a symmetry relation in the PDF. From the PDF, we
extract information about f(alpha) at criticality such as the presence of
negative fractal dimensions and we comment on the possible existence of
termination points. A PDF-based multifractal analysis is hence shown to be a
valid alternative to the standard approach based on the scaling of general
inverse participation ratios.; Comment: 4 pages, 7 figures

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## Partial Decode-Forward Binning Schemes for the Causal Cognitive Relay Channels

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The causal cognitive relay channel (CRC) has two sender-receiver pairs, in
which the second sender obtains information from the first sender causally and
assists the transmission of both senders. In this paper, we study both the
full- and half-duplex modes. In each mode, we propose two new coding schemes
built successively upon one another to illustrate the impact of different
coding techniques. The first scheme called partial decode-forward binning
(PDF-binning) combines the ideas of partial decode-forward relaying and
Gelfand-Pinsker binning. The second scheme called Han-Kobayashi partial
decode-forward binning (HK-PDF-binning) combines PDF-binning with Han-Kobayashi
coding by further splitting rates and applying superposition coding,
conditional binning and relaxed joint decoding.
In both schemes, the second sender decodes a part of the message from the
first sender, then uses Gelfand-Pinsker binning technique to bin against the
decoded codeword, but in such a way that allows both state nullifying and
forwarding. For the Gaussian channels, this PDF-binning essentializes to a
correlation between the transmit signal and the binning state, which
encompasses the traditional dirty-paper-coding binning as a special case when
this correlation factor is zero. We also provide the closed-form optimal
binning parameter for each scheme.
The 2-phase half-duplex schemes are adapted from the full-duplex ones by
removing block Markov encoding...

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## Analyses of turbulence in a wind tunnel by a multifractal theory for probability density functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Physics - Fluid Dynamics#Condensed Matter - Statistical Mechanics#Physics - Data Analysis, Statistics and Probability#Physics - Plasma Physics

The probability density functions (PDFs) for energy dissipation rates,
created from time-series data of grid turbulence in a wind tunnel, are analyzed
in a high precision by the theoretical formulae for PDFs within multifractal
PDF theory which is constructed under the assumption that there are two main
elements constituting fully developed turbulence, i.e., coherent and incoherent
elements. The tail part of PDF, representing intermittent coherent motion, is
determined by Tsallis-type PDF for singularity exponents essentially with one
parameter with the help of new scaling relation whose validity is checked for
the case of the grid turbulence. For the central part PDF representing both
contributions from the coherent motion and the fluctuating incoherent motion
surrounding the former, we introduced a trial function specified by three
adjustable parameters which amazingly represent scaling behaviors in much wider
area not restricted to the inertial range. From the investigation of the
difference between two difference formulae approximating velocity
time-derivative, it is revealed that the connection point between the central
and tail parts of PDF extracted by theoretical analyses of PDFs is actually the
boundary of the two kinds of instabilities associated respectively with
coherent and incoherent elements.; Comment: 17 pages...

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## Gravitational Evolution of the Large-Scale Density Distribution: The Edgeworth & Gamma Expansions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/10/1999
Português

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The gravitational evolution of the cosmic one-point Probability Distribution
Function (PDF) can be estimated using an analytic approximation that combines
gravitational Perturbation Theory (PT) with the Edgeworth expansion around a
Gaussian PDF. We present an alternative to the Edgeworth series based on an
expansion around the Gamma PDF, which is more appropriate to describe a
realistic PDF. The Gamma expansion converges when the PDF exhibits exponential
tails, which are predicted by PT and N-body simulations in the weakly
non-linear regime (i.e, when the variance, $\sigma^2$, is small). We compare
both expansions to N-body simulations and find that the Gamma expansion yields
a better overall match to the numerical results.; Comment: 2 pages, Latex using newpasp.sty, 2 figures. To appear in the
proceedings of the IGRAP meeting on "Clustering at high redshift" (Marseille,
June 1999)

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## Evolution of the probability distribution function of galaxies in redshift-space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/06/2000
Português

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We present a new analytic calculation for the redshift-space evolution of the
1-point galaxy Probability Distribution Function (PDF). The nonlinear evolution
of the matter density field is treated by second-order Eulerian perturbation
theory and transformed to the galaxy density field via a second-order local
biasing scheme. We then transform the galaxy density field to redshift space,
again to second order. Our method uses an exact statistical treatment based on
the Chapman-Kolmogorov equation to propagate the probability distribution of
the initial mass field to the final redshifted galaxy density field. We derive
the moment generating function of the PDF and use it to find a new, closed-form
expression for the skewness of the redshifted galaxy distribution. We show that
our formalism is general enough to allow a non-deterministic (or stochastic)
biasing prescription. We demonstrate the dependence of the redshift space PDF
on cosmological and biasing parameters. Our results are compared with existing
models for the PDF in redshift space and with the results of biased N-body
simulations. We find that our PDF accurately models the redshift space
evolution and the nonlinear biasing.; Comment: 15 pages, 7 figures, submitted to MNRAS

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## Evolution of the Cosmological Density Distribution Function: A New Analytical Model

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The one-point probability distribution function (pdf) of the large-scale
density field is an important tool to follow the evolution of cosmological
structures. In this paper we present a new model for this pdf for all regimes
and all densities, that is from linear to highly non-linear scales and from
rare voids up to rare high densities. This is probably the simplest model one
can build which is consistent with normalization constraints and known rigorous
results (the quasi-linear regime and the rare void limit). It is fully
parameterized by the non-linear variance and skewness. We obtain a good
agreement with N-body data from realistic cosmological simulations of the VIRGO
consortium and we find that it works significantly better than previous models
such as the lognormal model or the Extended Perturbation Theory (EPT). We
explain this success as a result of the tight constraints onto the pdf provided
by these consistency conditions. We also point out that while the Lagrangian
dynamics of typical fluctuations is quite complex the statistical outcome seems
rather simple. This simple model should be useful for studies which require a
realistic and convenient description of this pdf.; Comment: 14 pages, published in MNRAS. A few typos are corrected. The Fortran
code used to compute the pdf can be downloaded at
http://www-spht.cea.fr/pisp/valag/codepdf-en.html

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## Fluctuation spectrum and size scaling of river flow and level

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/07/2003
Português

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We describe the statistical properties of two large river systems: the Danube
and the Mississippi. The properties of the two rivers are compared
qualitatively to the general properties of a critical steady state system.
Specifically, we test the recent suggestion by Bramwell, Fennell, Holdsworth
and Portelli [{\it Europhys. Lett.} {\bf 57}, 310 (2002)] that a universal
probability density function (PDF) exists for the fluctuations in river level,
namely the Bramwell-Holdsworth-Pinton (BHP) PDF. The statistical properties
investigated in this paper are: the PDF of the river flow and river level;
moment scaling with basin area; moment to moment scaling or relative scaling;
and power spectral properties of the data. We find that the moments of the flow
scale approximately with basin area and that the seasonally adjusted flows
exhibit relative moment scaling.
Compared to the Mississippi, the Danube shows large deviations from spatial
scaling and the power spectra show considerable dependence on system size. This
might be due to water use and regulations as well as inhomogeneities in the
basin area.
We also find that the PDF of level data in some, but not all, cases can be
qualitatively approximated by the BHP PDF. We discuss why this coincidence
appears to be accidental.; Comment: 17 pages...

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## Properties and applications of the prime detecting function: infinitude of twin primes, asymptotic law of distribution of prime pairs differing by an even number

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/09/2011
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The prime detecting function (PDF) approach can be effective instrument in
the investigation of numbers. The PDF is constructed by recurrence sequence -
each successive prime adds a sieving factor in the form of PDF. With built-in
prime sieving features and properties such as simplicity,
integro-differentiability and configurable capability for a wide variety of
problems, the application of PDF leads to new interesting results. As an
example, in this exposition we present proofs of the infinitude of twin primes
and the first Hardy-Littlewood conjecture for prime pairs (the twin prime
number theorem). On this example one can see that application of PDF is
especially effective in investigation of asymptotic problems in combination
with the proposed method of test and probe functions.; Comment: 16 pages

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## Validity of the factorization approximation and correlation induced by nonextensivity in $N$-unit independent systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We have discussed the validity of the factorization approximation (FA) and
nonextensivity-induced correlation, by using the multivariate $q$-Gaussian
probability distribution function (PDF) for $N$-unit independent nonextensive
systems. The Tsallis entropy is shown to be expressed by $S_q^{(N)} =
S_{q,FA}^{(N)}+ \Delta S_q^{(N)}$ where $q$ denotes the entropic index,
$S_{q,FA}^{(N)}$ a contribution in the FA, and $\Delta S_q^{(N)}$ a correction
term. It is pointed out that the correction term of $\Delta S_q^{(N)}$ is
considerable for large $| q-1 |$ and/or large $N$ because the multivariate PDF
cannot be expressed by the factorized form which is assumed in the FA. This
implies that the pseudoadditivity of the Tsallis entropy, which is obtained
with PDFs in the FA, does not hold although it is commonly postulated in the
literatures. We have calculated correlations defined by $C_m= < (\delta x_i
\:\delta x_j)^m >_q -< (\delta x_i)^m >_q\: < (\delta x_j)^m >_q$ for $i \neq
j$, where $\delta x_i=x_i -< x_i >_q$ and $<\cdot >_q$ stands for $q$-average
over the escort PDF. It has been shown that $C_1$ expresses the intrinsic
correlation and that $C_m$ with $m \geq 2$ signifies correlation induced by
nonextensivity whose physical origin is elucidated within the superstatistics.
PDFs calculated for the classical ideal gas and harmonic oscillator are
compared with the $q$-Gaussian PDF. A discussion on the $q$-product PDF is
presented also.; Comment: 23 pages (incl. 4 figures): changed title and added references

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## Lattice dynamics and correlated atomic motion from the atomic pair distribution function

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/09/2002
Português

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The mean-square relative displacements (MSRD) of atomic pair motions in
crystals are studied as a function of pair distance and temperature using the
atomic pair distribution function (PDF). The effects of the lattice vibrations
on the PDF peak widths are modelled using both a multi-parameter Born
von-Karman (BvK) force model and a single-parameter Debye model. These results
are compared to experimentally determined PDFs. We find that the near-neighbor
atomic motions are strongly correlated, and that the extent of this correlation
depends both on the interatomic interactions and crystal structure. These
results suggest that proper account of the lattice vibrational effects on the
PDF peak width is important in extracting information on static disorder in a
disordered system such as an alloy. Good agreement is obtained between the BvK
model calculations of PDF peak widths and the experimentally determined peak
widths. The Debye model successfully explains the average, though not detailed,
natures of the MSRD of atomic pair motion with just one parameter. Also the
temperature dependence of the Debye model largely agrees with the BvK model
predictions. Therefore, the Debye model provides a simple description of the
effects of lattice vibrations on the PDF peak widths.; Comment: 9 pages...

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## The Density Probability Distribution Function in Turbulent, Isothermal, Magnetized Flows in Slab Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/11/2000
Português

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We investigate the behavior of the magnetic pressure, $b^2$, in fully
turbulent MHD flows in ``1+2/3'' dimensions by means of its effect on the
probability density function (PDF) of the density field. We start by reviewing
our previous results for general polytropic flows, according to which the value
of the polytropic exponent $\gamma$ determines the functional shape of the PDF.
A lognormal density PDF appears in the isothermal ($\gamma=1$) case, but a
power-law tail at either large or small densities appears for large Mach
numbers when $\gamma >1$ and $\gamma < 1$, respectively. In the isothermal
magnetic case, the relevant parameter is the field fluctuation amplitude,
$\dbb$. A lognormal PDF still appears for small field fluctuations (generally
the case for {\it large mean fields}), but a significant low-density excess
appears at large fluctuation amplitudes ({\it weak mean fields}), similar to
the behavior at $\gamma > 1$ of polytropic flows. We interpret these results in
terms of simple nonlinear MHD waves, for which the magnetic pressure behaves
linearly with the density in the case of the slow mode, and quadratically in
the case of the fast wave. Finally, we discuss some implications of these
results, in particular the fact that the effect of the magnetic field in
modifying the PDF is strongest when the mean field is weak.; Comment: To appear in "Computational Fluid Dynamics": Proceedings of the
Fourth UNAM Supercomputing Conference...

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## Tumbling of Polymers in a Random Flow with Mean Shear

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/11/2004
Português

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A polymer placed in chaotic flow with large mean shear tumbles, making
a-periodic flips. We describe the statistics of angular orientation, as well as
of tumbling time (separating two subsequent flips) of polymers in this flow.
The probability distribution function (PDF) of the polymer orientation is
peaked around a shear-preferred direction. The tails of this angular PDF are
algebraic. The PDF of the tumbling time, $\tau$, has a maximum at the value
estimated as inverse Lyapunov exponent of the flow. This PDF shows an
exponential tail for large $\tau$ and a small-$\tau$ tail determined by the
simultaneous statistics of velocity PDF.; Comment: submitted to Journal of Fluid Mechanics

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## On the Gap and Time Interval between the First Two Maxima of Long Continuous Time Random Walks

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/09/2015
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#Condensed Matter - Statistical Mechanics#Condensed Matter - Disordered Systems and Neural Networks#Mathematical Physics#Mathematics - Probability

We consider a one-dimensional continuous time random walk (CTRW) on a fixed
time interval $T$ where at each time step the walker waits a random time
$\tau$, before performing a jump drawn from a symmetric continuous probability
distribution function (PDF) $f(\eta)$, of L\'evy index $0 < \mu \leq 2$. Our
study includes the case where the waiting time PDF $\Psi(\tau)$ has a power law
tail, $\Psi(\tau) \propto \tau^{-1 - \gamma}$, with $0< \gamma < 1$, such that
the average time between two consecutive jumps is infinite. The random motion
is sub-diffusive if $\gamma < \mu/2$ (and super-diffusive if $\gamma > \mu/2$).
We investigate the joint PDF of the gap $g$ between the first two highest
positions of the CTRW and the time $t$ separating these two maxima. We show
that this PDF reaches a stationary limiting joint distribution $p(g,t)$ in the
limit of long CTRW, $T \to \infty$. Our exact analytical results show a very
rich behavior of this joint PDF in the $(\gamma, \mu)$ plane, which we study in
great detail. Our main results are verified by numerical simulations. This work
provides a non trivial extension to CTRWs of the recent study in the discrete
time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).; Comment: 36 pages, 10 figures. arXiv admin note: text overlap with
arXiv:1405.1222

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## Understanding star formation in molecular clouds III. Probability distribution functions of molecular lines in Cygnus X

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/09/2015
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Column density (N) PDFs serve as a powerful tool to characterize the physical
processes that influence the structure of molecular clouds. Star-forming clouds
can best be characterized by lognormal PDFs for the lower N range and a
power-law tail for higher N, commonly attributed to turbulence and self-gravity
and/or pressure, respectively. We report here on PDFs obtained from
observations of 12CO, 13CO, C18O, CS, and N2H+ in the Cygnus X North region and
compare to a PDF derived from dust observations with the Herschel satellite.
The PDF of 12CO is lognormal for Av~1-30, but is cut for higher Av due to
optical depth effects. The PDFs of C18O and 13CO are mostly lognormal up for
Av~1-15, followed by excess up to Av~40. Above that value, all CO PDFs drop,
most likely due to depletion. The high density tracers CS and N2H+ exhibit only
a power law distribution between Av~15 and 400, respectively. The PDF from dust
is lognormal for Av~2-15 and has a power-law tail up to Av~500. Absolute values
for the molecular line column densities are, however, rather uncertain due to
abundance and excitation temperature variations. Taken the dust PDF face value,
we 'calibrate' the molecular line PDF of CS to the one of the dust and
determined an abundance [CS]/[H2] of 10^-9. The slopes of the power-law tails
of the CS...

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