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

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...

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...

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...

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

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...

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

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...

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...

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)

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

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

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...

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

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

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...

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...

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

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

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...