In this paper we discuss the effect of the complete leading-order QED corrections to the DGLAP equations in the perturbative evolution of parton distribution functions (PDFs). This requires the extension of the purely QCD DGLAP evolution, including a PDF for the photons and, consistently, also for the charged leptons $e^{\pm}$, $\mu^\pm$ and $\tau^\pm$. We present the implementation of the QED-corrected DGLAP evolution in the presence of photon and lepton PDFs in the APFEL program and, by means of different assumptions for the initial scale PDFs, we produce for the first time PDF sets containing charged lepton distributions. We also present phenomenological studies that aim to assess the impact of the presence of lepton PDFs in the proton for some relevant SM (and BSM) processes at the LHC at 13 TeV and the FCC-hh at 100 TeV. The impact of the photon PDF is also outlined for those processes.; Comment: 32 pages, 19 figures, matches published version in JHEP

In the Black-Scholes context we consider the probability distribution function (PDF) of financial returns implied by volatility smile and we study the relation between the decay of its tails and the fitting parameters of the smile. We show that, considering a scaling law derived from data, it is possible to get a new fitting procedure of the volatility smile that considers also the exponential decay of the real PDF of returns observed in the financial markets. Our study finds application in the Risk Management activities where the tails characterization of financial returns PDF has a central role for the risk estimation.

The likelihood calculation of a vast number of particles is the computational bottleneck for the particle filter in applications where the observation information is rich. For fast computing the likelihood of particles, a numerical fitting approach is proposed to construct the Likelihood Probability Density Function (Li-PDF) by using a comparably small number of so-called fulcrums. The likelihood of particles is thereby analytically inferred, explicitly or implicitly, based on the Li-PDF instead of directly computed by utilizing the observation, which can significantly reduce the computation and enables real time filtering. The proposed approach guarantees the estimation quality when an appropriate fitting function and properly distributed fulcrums are used. The details for construction of the fitting function and fulcrums are addressed respectively in detail. In particular, to deal with multivariate fitting, the nonparametric kernel density estimator is presented which is flexible and convenient for implicit Li-PDF implementation. Simulation comparison with a variety of existing approaches on a benchmark 1-dimensional model and multi-dimensional robot localization and visual tracking demonstrate the validity of our approach.; Comment: 42 pages...

We report on a novel stochastic analysis of seismic time series for the Earth's vertical velocity, by using methods originally developed for complex hierarchical systems, and in particular for turbulent flows. Analysis of the fluctuations of the detrended increments of the series reveals a pronounced change of the shapes of the probability density functions (PDF) of the series' increments. Before and close to an earthquake the shape of the PDF and the long-range correlation in the increments both manifest significant changes. For a moderate or large-size earthquake the typical time at which the PDF undergoes the transition from a Gaussian to a non-Gaussian is about 5-10 hours. Thus, the transition represents a new precursor for detecting such earthquakes.; Comment: 12 pages, 5 figures

The dynamics of a tracer particle in a stationary driven granular gas is investigated. We show how to transform the linear Boltzmann equation describing the dynamics of the tracer into a master equation for a continuous Markov process. The transition rates depend upon the stationary velocity distribution of the gas. When the gas has a Gaussian velocity probability distribution function (pdf), the stationary velocity pdf of the tracer is Gaussian with a lower temperature and satisfies detailed balance for any value of the restitution coefficient $\alpha$. As soon as the velocity pdf of the gas departs from the Gaussian form, detailed balance is violated. This non-equilibrium state can be characterized in terms of a Lebowitz-Spohn action functional $W(\tau)$ defined over trajectories of time duration $\tau$. We discuss the properties of this functional and of a similar functional $\bar{W}(\tau)$ which differs from the first for a term which is non-extensive in time. On the one hand we show that in numerical experiments, i.e. at finite times $\tau$, the two functionals have different fluctuations and $\bar{W}$ always satisfies an Evans-Searles-like symmetry. On the other hand we cannot observe the verification of the Lebowitz-Spohn-Gallavotti-Cohen (LS-GC) relation...

Standard definition of the stochastic Risk-Sensitive Linear-Quadratic (RS-LQ) control depends on the risk parameter, which is normally left to be set exogenously. We reconsider the classical approach and suggest two alternatives resolving the spurious freedom naturally. One approach consists in seeking for the minimum of the tail of the Probability Distribution Function (PDF) of the cost functional at some large fixed value. Another option suggests to minimize the expectation value of the cost functional under constraint on the value of the PDF tail. Under assumption of the resulting control stability, both problems are reduced to static optimizations over stationary control matrix. The solutions are illustrated on the examples of scalar and 1d chain (string) systems. Large Deviation self-similar asymptotic of the cost functional PDF is analyzed.; Comment: 11 pages

In random walks, the path representation of the Green's function is an infinite sum over the length of path probability density functions (PDFs). Here we derive and solve, in Laplace space, the recursion relation for the n order path PDF for any arbitrarily inhomogeneous semi-Markovian random walk in a one-dimensional (1D) chain of L states. The recursion relation relates the n order path PDF to L/2 (round towards zero for an odd L) shorter path PDFs, and has n independent coefficients that obey a universal formula. The z transform of the recursion relation straightforwardly gives the generating function for path PDFs, from which we obtain the Green's function of the random walk, and derive an explicit expression for any path PDF of the random walk. These expressions give the most detailed description of arbitrarily inhomogeneous semi-Markovian random walks in 1D.

At Coulomb blockade valleys inelastic cotunneling processes generate particle-hole excitations in quantum dots (QDs), and lead to energy dissipation. We have analyzed the probability distribution function (PDF) of energy dissipated in a QD due to such processes during a given time interval. We obtained analytically the cumulant generating function, and extracted the average, variance and Fano factor. The latter diverges as $T^3/(eV)^2$ at bias $eV$ smaller than the temperature $T$, and reaches the value $3 eV / 5$ in the opposite limit. The PDF is further studied numerically. As expected, Crooks fluctuation relation is not fulfilled by the PDF. Our results can be verified experimentally utilizing transport measurements of charge.; Comment: 5 pages, 3 figures

This document is intended as a study of benchmark cross sections at the LHC (at 7 TeV) at NLO using modern parton distribution functions currently available from the 6 PDF fitting groups that have participated in this exercise. It also contains a succinct user guide to the computation of PDFs, uncertainties and correlations using available PDF sets. A companion note, also submitted to the archive, provides an interim summary of the current recommendations of the PDF4LHC working group for the use of parton distribution functions and of PDF uncertainties at the LHC, for cross section and cross section uncertainty calculations.; Comment: 35 pages

My goal in producing this document was to create a collection of qubit circuit identities that are used in Quantum Computing. Mathematicians and Physicists may consider it as being analogous to a Table of Integrals or a Mathematical Handbook such as Gradshteyn-Ryzhik or Abramowitz-Stegun. Computer Programmers may think of it as a scrapbook of code snippets that are elegant, instructive, well documented, and useful. Electronics experts may view it as a compendium of circuits for performing a large assortment of tasks. The vast majority of the circuit identities collected in this work were not discovered for the first time by me, and I take no credit for discovering them. In producing this document, I am acting as a collector, not as a discoverer.; Comment: 45 pages (files: 1 .tex, 2 .sty, 2 .eps) A nicer pdf version, with better figure resolution than the arxiv pdf, can be found at www.ar-tiste.com/PaulinesiaVer1.pdf

Signal processing techniques will lean on blind methods in the near future, where no redundant, resource allocating information will be transmitted through the channel. To achieve a proper decision, however, it is essential to know at least the probability density function (pdf), which to estimate is classically a time consumption and/or less accurate hard task, that may make decisions to fail. This paper describes the design of a quantum assisted pdf estimation method also by an example, which promises to achieve the exact pdf by proper setting of parameters in a very fast way.; Comment: 5 pages, 3 figures, will be published by International Journal of Quantum Information

We revise and extend the stochastic approach to cumulative weak lensing (hereafter the sGL method) first introduced in Ref. [1]. Here we include a realistic halo mass function and density profiles to model the distribution of mass between and within galaxies, galaxy groups and galaxy clusters. We also introduce a modeling of the filamentary large-scale structures and a method to embed halos into these structures. We show that the sGL method naturally reproduces the weak lensing results for the Millennium Simulation. The strength of the sGL method is that a numerical code based on it can compute the lensing probability distribution function for a given inhomogeneous model universe in a few seconds. This makes it a useful tool to study how lensing depends on cosmological parameters and its impact on observations. The method can also be used to simulate the effect of a wide array of systematic biases on the observable PDF. As an example we show how simple selection effects may reduce the variance of observed PDF, which could possibly mask opposite effects from very large scale structures. We also show how a JDEM-like survey could constrain the lensing PDF relative to a given cosmological model. The updated turboGL code is available at turboGL.org.; Comment: PRD style: 20 pages...

We discuss two methods for exploring energy diffusion in lattices with finite temperature in this paper. The first one is the energy-kick (EK) method. To apply this method, one adds an external energy kick to a particle in the lattice, and tracks its evolution by evolving the kicked system. The second one is the fluctuation-correlation (FC) method. The formula for calculating the probability density function (PDF) using the canonical ensemble is slightly revised and extended to the microcanonical ensemble. We show that the FC method has advantages over the EK method theoretically and technically. Theoretically, the PDF obtained by the FC method reveals the diffusion processes of the inner energy while the PDF obtained by the EK method represents that of the kick energy. The diffusion processes of the inner energy and the external energy added to the system, i.e., the kick energy, may be different quantitatively and even qualitatively depending on models. To show these facts, we study not only the equilibrium systems but also the stationary nonequilibrium systems. Examples showing that the inner energy and the kick energy may have different diffusion behavior are reported in both cases. The technical advantage enables us to study the long-time diffusion processes and thus avoids the finite-time effect.; Comment: 10 pages;7figure

Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques, such as Markov chain Monte Carlo (MCMC) and particle filters, have become very popular in signal processing over the last years. However, in many problems of practical interest these techniques demand procedures for sampling from probability distributions with non-standard forms, hence we are often brought back to the consideration of fundamental simulation algorithms, such as rejection sampling (RS). Unfortunately, the use of RS techniques demands the calculation of tight upper bounds for the ratio of the target probability density function (pdf) over the proposal density from which candidate samples are drawn. Except for the class of log-concave target pdf's, for which an efficient algorithm exists, there are no general methods to analytically determine this bound, which has to be derived from scratch for each specific case. In this paper, we introduce new schemes for (a) obtaining upper bounds for likelihood functions and (b) adaptively computing proposal densities that approximate the target pdf closely. The former class of methods provides the tools to easily sample from a posteriori probability distributions (that appear very often in signal processing problems) by drawing candidates from the prior distribution. However...

In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker, starting from $x_0 = 0$, after $n$ time steps (taking the $x$-axis vertical). The jumps $\eta_i = x_i - x_{i-1}$ are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), $f(\eta)$, the Fourier transform of which has the small $k$ behavior $1 - \hat f(k) \propto |k|^\mu$, with $0 < \mu \leq 2$. For $\mu=2$, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For $0<\mu<2$, the RW is a L\'evy flight of index $\mu$. We show that the joint PDF of $G_n$ and $L_n$ converges to a well defined stationary bi-variate distribution $p(g,l)$ as the RW duration $n$ goes to infinity. We present a thorough analytical study of the limiting joint distribution $p(g,l)$, as well as of its associated marginals $p_{\rm gap}(g)$ and $p_{\rm time}(l)$, revealing a rich variety of behaviors depending on the tail of $f(\eta)$ (from slow decreasing algebraic tail to fast decreasing super-exponential tail). We also address the problem for a random bridge where the RW starts and ends at the origin after $n$ time steps. We show that in the large $n$ limit...

Parton distributions with QED corrections open new scenarios for high precision physics. We recall the need for accurate and improved predictions which keeps into account higher order QCD corrections together with electroweak effects. We study predictions obtained with the improved Born approximation and the $G_{\mu}$ scheme by using two public codes: DYNNLO and HORACE. We focus our attention on the Drell-Yan Z-boson invariant mass distribution at low- and high-mass regions, recently measured by the ATLAS experiment and we estimate the impact of each component of the final prediction. We show that electroweak corrections are larger than PDF uncertainties for modern PDF sets and therefore such corrections are necessary to improve the extraction of future PDF sets.; Comment: 5 pages, 4 figures, to appear in the proceedings of the Les Rencontres de Physique de la Vall\'ee d'Aoste, La Thuile 2014

Dispersion of a passive scalar from concentrated sources in fully developed turbulent channel flow is studied with the probability density function (PDF) method. The joint PDF of velocity, turbulent frequency and scalar concentration is represented by a large number of Lagrangian particles. A stochastic near-wall PDF model combines the generalized Langevin model of Haworth & Pope with Durbin's method of elliptic relaxation to provide a mathematically exact treatment of convective and viscous transport with a non-local representation of the near-wall Reynolds stress anisotropy. The presence of walls is incorporated through the imposition of no-slip and impermeability conditions on particles without the use of damping or wall-functions. Information on the turbulent timescale is supplied by the gamma-distribution model of van Slooten et al. Two different micromixing models are compared that incorporate the effect of small scale mixing on the transported scalar: the widely used interaction by exchange with the mean (IEM) and the interaction by exchange with the conditional mean (IECM) model. Single-point velocity and concentration statistics are compared to direct numerical simulation and experimental data at Re_\tau=1080 based on the friction velocity and the channel half width. The joint model accurately reproduces a wide variety of conditional and unconditional statistics in both physical and composition space.; Comment: Accepted in Physics of Fluids...

We study the effects of substructure in the Galactic halo on direct detection of dark matter, on searches for energetic neutrinos from WIMP annihilation in the Sun and Earth, and on the enhancement in the WIMP annihilation rate in the halo. Our central result is a probability distribution function (PDF) P(\rho) for the local dark-matter density. This distribution must be taken into account when using null dark-matter searches to constrain the properties of dark-matter candidates. We take two approaches to calculating the PDF. The first is an analytic model that capitalizes on the scale-invariant nature of the structure--formation hierarchy in order to address early stages in the hierarchy (very small scales; high densities). Our second approach uses simulation-inspired results to describe the PDF that arises from lower-density larger-scale substructures which formed in more recent stages in the merger hierarchy. The distributions are skew positive, and they peak at densities lower than the mean density. The local dark-matter density may be as small as 1/10th the canonical value of ~ 0.4 GeV/cm^3, but it is probably no less than 0.2 GeV/cm^3.; Comment: 11 pages, 3 figures, Replaced with version accepted for publication in Phys. Rev. D

A sample of eight quasars observed at high resolution and signal-to-noise is used to determine the probability distribution function (PDF), the power spectrum, and the correlation function of the transmitted flux in the \lya forest, in three redshift bins centered at z=2.41, 3.00, and 3.89. All the results are presented in tabular form, with full error covariance matrices to allow for comparisons with any numerical simulations and with other data sets. The observations are compared with a numerical simulation of the \lya forest of a Lambda-CDM model with Omega=0.4, known to agree with other large-scale structure observational constraints. There is excellent agreement for the PDF, if the mean transmitted flux is adjusted to match the observations. A small difference between the observed and predicted PDF is found at high fluxes and low redshift, which may be due to the uncertain effects of fitting the spectral continuum. Using the numerical simulation, we show how the flux power spectrum can be used to recover the initial power spectrum of density fluctuations. From our sample of eight quasars, we measure the amplitude of the mass power spectrum to correspond to a linear variance per unit ln(k) of $\Delta^2_\rho(k)=0.72\pm0.09$ at k=0.04(km/s)^{-1} and z=3...

The class of orthogonal relay channels in which the orthogonal channels connecting the source terminal to the relay and the destination, and the relay to the destination, depend on a state sequence, is considered. It is assumed that the state sequence is fully known at the destination while it is not known at the source or the relay. The capacity of this class of relay channels is characterized, and shown to be achieved by the partial decode-compress-and-forward (pDCF) scheme. Then the capacity of certain binary and Gaussian state-dependent orthogonal relay channels are studied in detail, and it is shown that the compress-and-forward (CF) and partial-decode-and-forward (pDF) schemes are suboptimal in general. To the best of our knowledge, this is the first single relay channel model for which the capacity is achieved by pDCF, while pDF and CF schemes are both suboptimal. Furthermore, it is shown that the capacity of the considered class of state-dependent orthogonal relay channels is in general below the cut-set bound. The conditions under which pDF or CF suffices to meet the cut-set bound, and hence, achieve the capacity, are also derived.; Comment: This paper has been accepted by IEEE Transactions on Information Theory