Science.gov

Sample records for elliptic random-walk equation

  1. Nonlocal operators, parabolic-type equations, and ultrametric random walks

    SciTech Connect

    Chacón-Cortes, L. F. Zúñiga-Galindo, W. A.

    2013-11-15

    In this article, we introduce a new type of nonlocal operators and study the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated to these operators. Some of these equations are the p-adic master equations of certain models of complex systems introduced by Avetisov, V. A. and Bikulov, A. Kh., “On the ultrametricity of the fluctuation dynamicmobility of protein molecules,” Proc. Steklov Inst. Math. 265(1), 75–81 (2009) [Tr. Mat. Inst. Steklova 265, 82–89 (2009) (Izbrannye Voprosy Matematicheskoy Fiziki i p-adicheskogo Analiza) (in Russian)]; Avetisov, V. A., Bikulov, A. Kh., and Zubarev, A. P., “First passage time distribution and the number of returns for ultrametric random walks,” J. Phys. A 42(8), 085003 (2009); Avetisov, V. A., Bikulov, A. Kh., and Osipov, V. A., “p-adic models of ultrametric diffusion in the conformational dynamics of macromolecules,” Proc. Steklov Inst. Math. 245(2), 48–57 (2004) [Tr. Mat. Inst. Steklova 245, 55–64 (2004) (Izbrannye Voprosy Matematicheskoy Fiziki i p-adicheskogo Analiza) (in Russian)]; Avetisov, V. A., Bikulov, A. Kh., and Osipov, V. A., “p-adic description of characteristic relaxation in complex systems,” J. Phys. A 36(15), 4239–4246 (2003); Avetisov, V. A., Bikulov, A. H., Kozyrev, S. V., and Osipov, V. A., “p-adic models of ultrametric diffusion constrained by hierarchical energy landscapes,” J. Phys. A 35(2), 177–189 (2002); Avetisov, V. A., Bikulov, A. Kh., and Kozyrev, S. V., “Description of logarithmic relaxation by a model of a hierarchical random walk,” Dokl. Akad. Nauk 368(2), 164–167 (1999) (in Russian). The fundamental solutions of these parabolic-type equations are transition functions of random walks on the n-dimensional vector space over the field of p-adic numbers. We study some properties of these random walks, including the first passage time.

  2. Continuous Time Open Quantum Random Walks and Non-Markovian Lindblad Master Equations

    NASA Astrophysics Data System (ADS)

    Pellegrini, Clément

    2014-02-01

    A new type of quantum random walks, called Open Quantum Random Walks, has been developed and studied in Attal et al. (Open quantum random walks, preprint) and (Central limit theorems for open quantum random walks, preprint). In this article we present a natural continuous time extension of these Open Quantum Random Walks. This continuous time version is obtained by taking a continuous time limit of the discrete time Open Quantum Random Walks. This approximation procedure is based on some adaptation of Repeated Quantum Interactions Theory (Attal and Pautrat in Annales Henri Poincaré Physique Théorique 7:59-104, 2006) coupled with the use of correlated projectors (Breuer in Phys Rev A 75:022103, 2007). The limit evolutions obtained this way give rise to a particular type of quantum master equations. These equations appeared originally in the non-Markovian generalization of the Lindblad theory (Breuer in Phys Rev A 75:022103, 2007). We also investigate the continuous time limits of the quantum trajectories associated with Open Quantum Random Walks. We show that the limit evolutions in this context are described by jump stochastic differential equations. Finally we present a physical example which can be described in terms of Open Quantum Random Walks and their associated continuous time limits.

  3. Generalized master equation via aging continuous-time random walks.

    PubMed

    Allegrini, Paolo; Aquino, Gerardo; Grigolini, Paolo; Palatella, Luigi; Rosa, Angelo

    2003-11-01

    We discuss the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density psi(t) that is assumed to be an inverse power law with the power index micro. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where psi(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure we create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations.

  4. Solving the Hodgkin-Huxley equations by a random walk method

    SciTech Connect

    Sherman, A.S.; Peskin, C.S.

    1988-01-01

    A numerical method is describes for solving the Hodgkin-Huxley cable equations, set up as an initial boundary value problem. The method uses a gradient random walk combined with creation and destruction of the diffusing elements. It is designed to be efficient in the presence of sharp wavefronts. Details of implementation are given with particular emphasis on satisfying the boundary conditions without introducing excessive noise.

  5. Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations.

    PubMed

    Sánchez, R; Carreras, B A; van Milligen, B Ph

    2005-01-01

    The fluid limit of a recently introduced family of nonintegrable (nonlinear) continuous-time random walks is derived in terms of fractional differential equations. In this limit, it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales. For this reason, the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems, ranging from the study of self-organized criticality to the modeling of turbulent transport in fluids and plasmas.

  6. Kardar-Parisi-Zhang Equation and Large Deviations for Random Walks in Weak Random Environments

    NASA Astrophysics Data System (ADS)

    Corwin, Ivan; Gu, Yu

    2017-01-01

    We consider the transition probabilities for random walks in 1+1 dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.

  7. Delayed uncoupled continuous-time random walks do not provide a model for the telegraph equation.

    PubMed

    Rukolaine, S A; Samsonov, A M

    2012-02-01

    It has been alleged in several papers that the so-called delayed continuous-time random walks (DCTRWs) provide a model for the one-dimensional telegraph equation at microscopic level. This conclusion, being widespread now, is strange, since the telegraph equation describes phenomena with finite propagation speed, while the velocity of the motion of particles in the DCTRWs is infinite. In this paper we investigate the accuracy of the approximations to the DCTRWs provided by the telegraph equation. We show that the diffusion equation, being the correct limit of the DCTRWs, gives better approximations in L(2) norm to the DCTRWs than the telegraph equation. We conclude, therefore, that first, the DCTRWs do not provide any correct microscopic interpretation of the one-dimensional telegraph equation, and second, the kinetic (exact) model of the telegraph equation is different from the model based on the DCTRWs.

  8. Equations for the distributions of functionals of a random-walk trajectory in an inhomogeneous medium

    SciTech Connect

    Shkilev, V. P.

    2012-01-15

    Based on the random-trap model and using the mean-field approximation, we derive an equation that allows the distribution of a functional of the trajectory of a particle making random walks over inhomogeneous-lattice site to be calculated. The derived equation is a generalization of the Feynman-Kac equation to an inhomogeneous medium. We also derive a backward equation in which not the final position of the particle but its position at the initial time is used as an independent variable. As an example of applying the derived equations, we consider the one-dimensional problem of calculating the first-passage time distribution. We show that the average first-passage times for homogeneous and inhomogeneous media with identical diffusion coefficients coincide, but the variance of the distribution for an inhomogeneous medium can be many times larger than that for a homogeneous one.

  9. Elliptic scattering equations

    NASA Astrophysics Data System (ADS)

    Cardona, Carlos; Gomez, Humberto

    2016-06-01

    Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.

  10. From analytical solutions of solute transport equations to multidimensional time-domain random walk (TDRW) algorithms

    NASA Astrophysics Data System (ADS)

    Bodin, Jacques

    2015-03-01

    In this study, new multi-dimensional time-domain random walk (TDRW) algorithms are derived from approximate one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) analytical solutions of the advection-dispersion equation and from exact 1-D, 2-D, and 3-D analytical solutions of the pure-diffusion equation. These algorithms enable the calculation of both the time required for a particle to travel a specified distance in a homogeneous medium and the mass recovery at the observation point, which may be incomplete due to 2-D or 3-D transverse dispersion or diffusion. The method is extended to heterogeneous media, represented as a piecewise collection of homogeneous media. The particle motion is then decomposed along a series of intermediate checkpoints located on the medium interface boundaries. The accuracy of the multi-dimensional TDRW method is verified against (i) exact analytical solutions of solute transport in homogeneous media and (ii) finite-difference simulations in a synthetic 2-D heterogeneous medium of simple geometry. The results demonstrate that the method is ideally suited to purely diffusive transport and to advection-dispersion transport problems dominated by advection. Conversely, the method is not recommended for highly dispersive transport problems because the accuracy of the advection-dispersion TDRW algorithms degrades rapidly for a low Péclet number, consistent with the accuracy limit of the approximate analytical solutions. The proposed approach provides a unified methodology for deriving multi-dimensional time-domain particle equations and may be applicable to other mathematical transport models, provided that appropriate analytical solutions are available.

  11. Functional equation for the crossover in the model of one-dimensional Weierstrass random walks

    NASA Astrophysics Data System (ADS)

    Rudoi, Yu. G.; Kotel'nikova, O. A.

    2016-12-01

    We consider the problem of one-dimensional symmetric diffusion in the framework of Markov random walks of the Weierstrass type using two-parameter scaling for the transition probability. We construct a solution for the characteristic Lyapunov function as a sum of regular (homogeneous) and singular (nonhomogeneous) solutions and find the conditions for the crossover from normal to anomalous diffusion.

  12. A relativistically covariant random walk

    NASA Astrophysics Data System (ADS)

    Almaguer, J.; Larralde, H.

    2007-08-01

    In this work we present and analyze an extremely simple relativistically covariant random walk model. In our approach, the probability density and the flow of probability arise naturally as the components of a four-vector and they are related to one another via a tensorial constitutive equation. We show that the system can be described in terms of an underlying invariant space time random walk parameterized by the number of sojourns. Finally, we obtain explicit expressions for the moments of the covariant random walk as well as for the underlying invariant random walk.

  13. Fractional random walk lattice dynamics

    NASA Astrophysics Data System (ADS)

    Michelitsch, T. M.; Collet, B. A.; Riascos, A. P.; Nowakowski, A. F.; Nicolleau, F. C. G. A.

    2017-02-01

    We analyze time-discrete and time-continuous ‘fractional’ random walks on undirected regular networks with special focus on cubic periodic lattices in n  =  1, 2, 3,.. dimensions. The fractional random walk dynamics is governed by a master equation involving fractional powers of Laplacian matrices {{L}\\fracα{2}}} where α =2 recovers the normal walk. First we demonstrate that the interval 0<α ≤slant 2 is admissible for the fractional random walk. We derive analytical expressions for the transition matrix of the fractional random walk and closely related the average return probabilities. We further obtain the fundamental matrix {{Z}(α )} , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix {{Z}(α )} relates fractional random walks with normal random walks. We show that the matrix elements of the transition matrix of the fractional random walk exihibit for large cubic n-dimensional lattices a power law decay of an n-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the n-dimensional space, the transition matrix and return probabilities of the fractional random walk are dominated for large times t by slowly relaxing long-wave modes leading to a characteristic {{t}-\\frac{n{α}} -decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen.

  14. Non-Markovian stochastic Liouville equation and its Markovian representation: Extensions of the continuous-time random-walk approach.

    PubMed

    Shushin, A I

    2008-03-01

    Some specific features and extensions of the continuous-time random-walk (CTRW) approach are analyzed in detail within the Markovian representation (MR) and CTRW-based non-Markovian stochastic Liouville equation (SLE). In the MR, CTRW processes are represented by multidimensional Markovian ones. In this representation the probability density function (PDF) W(t) of fluctuation renewals is associated with that of reoccurrences in a certain jump state of some Markovian controlling process. Within the MR the non-Markovian SLE, which describes the effect of CTRW-like noise on the relaxation of dynamic and stochastic systems, is generalized to take into account the influence of relaxing systems on the statistical properties of noise. Some applications of the generalized non-Markovian SLE are discussed. In particular, it is applied to study two modifications of the CTRW approach. One of them considers cascaded CTRWs in which the controlling process is actually a CTRW-like one controlled by another CTRW process, controlled in turn by a third one, etc. Within the MR a simple expression for the PDF W(t) of the total controlling process is obtained in terms of Markovian variants of controlling PDFs in the cascade. The expression is shown to be especially simple and instructive in the case of anomalous processes determined by the long-time tailed W(t) . The cascaded CTRWs can model the effect of the complexity of a system on the relaxation kinetics (in glasses, fractals, branching media, ultrametric structures, etc.). Another CTRW modification describes the kinetics of processes governed by fluctuating W(t) . Within the MR the problem is analyzed in a general form without restrictive assumptions on the correlations of PDFs of consecutive renewals. The analysis shows that fluctuations of W(t) can strongly affect the kinetics of the process. Possible manifestations of this effect are discussed.

  15. Solutions of the integral equation of diffusion and the random walk model for continuous plumes and instantaneous puffs in the atmospheric boundary layer

    NASA Astrophysics Data System (ADS)

    Smith, F. B.; Thomson, D.

    1984-09-01

    The integral equation method is related to the random walk modelling that has proved so effective and popular in recent years. The I.E. method, by using simple probability techniques, avoids the inefficient determination of thousands of trajectories in order to build up concentration profiles. In fact it is so simple and efficient it can be run on a conventional programmable calculator. The method is applied to passive material being released from an elevated source within a neutrally stable surface layer over a uniform surface, and also to an instantaneous release when the effect of wind shear is examined. The latter scenario is also studied using random walk techniques and a comparison of the solutions obtained. Agreement is very good, although downwind spread is shown to be quite sensitive to gridlength size in the I.E. method.

  16. Persistence of random walk records

    NASA Astrophysics Data System (ADS)

    Ben-Naim, E.; Krapivsky, P. L.

    2014-06-01

    We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the mean record, obtained as an average over infinitely many realizations. We term the walk ‘superior’ if the record is always above average, and conversely, the walk is said to be ‘inferior’ if the record is always below average. We find that the fraction of superior walks, S, decays algebraically with time, S ˜ t-β, in the limit t → ∞, and that the persistence exponent is nontrivial, β = 0.382 258…. The fraction of inferior walks, I, also decays as a power law, I ˜ t-α, but the persistence exponent is smaller, α = 0.241 608…. Both exponents are roots of transcendental equations involving the parabolic cylinder function. To obtain these theoretical results, we analyze the joint density of superior walks with a given record and position, while for inferior walks it suffices to study the density as a function of position.

  17. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation.

    PubMed

    Fulger, Daniel; Scalas, Enrico; Germano, Guido

    2008-02-01

    We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Lévy alpha -stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Lévy alpha -stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.

  18. Connecting the dots: Semi-analytical and random walk numerical solutions of the diffusion–reaction equation with stochastic initial conditions

    SciTech Connect

    Paster, Amir; Bolster, Diogo; Benson, David A.

    2014-04-15

    We study a system with bimolecular irreversible kinetic reaction A+B→∅ where the underlying transport of reactants is governed by diffusion, and the local reaction term is given by the law of mass action. We consider the case where the initial concentrations are given in terms of an average and a white noise perturbation. Our goal is to solve the diffusion–reaction equation which governs the system, and we tackle it with both analytical and numerical approaches. To obtain an analytical solution, we develop the equations of moments and solve them approximately. To obtain a numerical solution, we develop a grid-less Monte Carlo particle tracking approach, where diffusion is modeled by a random walk of the particles, and reaction is modeled by annihilation of particles. The probability of annihilation is derived analytically from the particles' co-location probability. We rigorously derive the relationship between the initial number of particles in the system and the amplitude of white noise represented by that number. This enables us to compare the particle simulations and the approximate analytical solution and offer an explanation of the late time discrepancies. - Graphical abstract:.

  19. Quantum random walk polynomial and quantum random walk measure

    NASA Astrophysics Data System (ADS)

    Kang, Yuanbao; Wang, Caishi

    2014-05-01

    In the paper, we introduce a quantum random walk polynomial (QRWP) that can be defined as a polynomial , which is orthogonal with respect to a quantum random walk measure (QRWM) on , such that the parameters are in the recurrence relations and satisfy . We firstly obtain some results of QRWP and QRWM, in which case the correspondence between measures and orthogonal polynomial sequences is one-to-one. It shows that any measure with respect to which a quantum random walk polynomial sequence is orthogonal is a quantum random walk measure. We next collect some properties of QRWM; moreover, we extend Karlin and McGregor's representation formula for the transition probabilities of a quantum random walk (QRW) in the interacting Fock space, which is a parallel result with the CGMV method. Using these findings, we finally obtain some applications for QRWM, which are of interest in the study of quantum random walk, highlighting the role played by QRWP and QRWM.

  20. Random-walk enzymes

    PubMed Central

    Mak, Chi H.; Pham, Phuong; Afif, Samir A.; Goodman, Myron F.

    2015-01-01

    Enzymes that rely on random walk to search for substrate targets in a heterogeneously dispersed medium can leave behind complex spatial profiles of their catalyzed conversions. The catalytic signatures of these random-walk enzymes are the result of two coupled stochastic processes: scanning and catalysis. Here we develop analytical models to understand the conversion profiles produced by these enzymes, comparing an intrusive model, in which scanning and catalysis are tightly coupled, against a loosely coupled passive model. Diagrammatic theory and path-integral solutions of these models revealed clearly distinct predictions. Comparison to experimental data from catalyzed deaminations deposited on single-stranded DNA by the enzyme activation-induced deoxycytidine deaminase (AID) demonstrates that catalysis and diffusion are strongly intertwined, where the chemical conversions give rise to new stochastic trajectories that were absent if the substrate DNA was homogeneous. The C → U deamination profiles in both analytical predictions and experiments exhibit a strong contextual dependence, where the conversion rate of each target site is strongly contingent on the identities of other surrounding targets, with the intrusive model showing an excellent fit to the data. These methods can be applied to deduce sequence-dependent catalytic signatures of other DNA modification enzymes, with potential applications to cancer, gene regulation, and epigenetics. PMID:26465508

  1. Random-walk enzymes.

    PubMed

    Mak, Chi H; Pham, Phuong; Afif, Samir A; Goodman, Myron F

    2015-09-01

    Enzymes that rely on random walk to search for substrate targets in a heterogeneously dispersed medium can leave behind complex spatial profiles of their catalyzed conversions. The catalytic signatures of these random-walk enzymes are the result of two coupled stochastic processes: scanning and catalysis. Here we develop analytical models to understand the conversion profiles produced by these enzymes, comparing an intrusive model, in which scanning and catalysis are tightly coupled, against a loosely coupled passive model. Diagrammatic theory and path-integral solutions of these models revealed clearly distinct predictions. Comparison to experimental data from catalyzed deaminations deposited on single-stranded DNA by the enzyme activation-induced deoxycytidine deaminase (AID) demonstrates that catalysis and diffusion are strongly intertwined, where the chemical conversions give rise to new stochastic trajectories that were absent if the substrate DNA was homogeneous. The C→U deamination profiles in both analytical predictions and experiments exhibit a strong contextual dependence, where the conversion rate of each target site is strongly contingent on the identities of other surrounding targets, with the intrusive model showing an excellent fit to the data. These methods can be applied to deduce sequence-dependent catalytic signatures of other DNA modification enzymes, with potential applications to cancer, gene regulation, and epigenetics.

  2. Random-walk enzymes

    NASA Astrophysics Data System (ADS)

    Mak, Chi H.; Pham, Phuong; Afif, Samir A.; Goodman, Myron F.

    2015-09-01

    Enzymes that rely on random walk to search for substrate targets in a heterogeneously dispersed medium can leave behind complex spatial profiles of their catalyzed conversions. The catalytic signatures of these random-walk enzymes are the result of two coupled stochastic processes: scanning and catalysis. Here we develop analytical models to understand the conversion profiles produced by these enzymes, comparing an intrusive model, in which scanning and catalysis are tightly coupled, against a loosely coupled passive model. Diagrammatic theory and path-integral solutions of these models revealed clearly distinct predictions. Comparison to experimental data from catalyzed deaminations deposited on single-stranded DNA by the enzyme activation-induced deoxycytidine deaminase (AID) demonstrates that catalysis and diffusion are strongly intertwined, where the chemical conversions give rise to new stochastic trajectories that were absent if the substrate DNA was homogeneous. The C →U deamination profiles in both analytical predictions and experiments exhibit a strong contextual dependence, where the conversion rate of each target site is strongly contingent on the identities of other surrounding targets, with the intrusive model showing an excellent fit to the data. These methods can be applied to deduce sequence-dependent catalytic signatures of other DNA modification enzymes, with potential applications to cancer, gene regulation, and epigenetics.

  3. Random Walk Method for Potential Problems

    NASA Technical Reports Server (NTRS)

    Krishnamurthy, T.; Raju, I. S.

    2002-01-01

    A local Random Walk Method (RWM) for potential problems governed by Lapalace's and Paragon's equations is developed for two- and three-dimensional problems. The RWM is implemented and demonstrated in a multiprocessor parallel environment on a Beowulf cluster of computers. A speed gain of 16 is achieved as the number of processors is increased from 1 to 23.

  4. Galerkin Methods for Nonlinear Elliptic Equations.

    NASA Astrophysics Data System (ADS)

    Murdoch, Thomas

    Available from UMI in association with The British Library. Requires signed TDF. This thesis exploits in the nonlinear situation the optimal approximation property of the finite element method for linear, elliptic problems. Of particular interest are the steady state semiconductor equations in one and two dimensions. Instead of discretising the differential equations by the finite element method and solving the nonlinear algebraic equations by Newton's method, a Newton linearisation of the continuous problem is preferred and a sequence of linear problems solved until some convergence criterion is achieved. For nonlinear Poisson equations, this approach reduces to solving a sequence of linear, elliptic, self -adjoint problems, their approximation by the finite element being optimal in a suitably defined energy norm. Consequently, there is the potential to recover a smoother representation of the underlying solution at each step of the Newton iteration. When this approach is applied to the continuity equations for semiconductor devices, a sequence of linear problems of the form -_{nabla }(anabla u - bu) = f must be solved. The Galerkin method in its crude form does not adequately represent the true solution: however, generalising the framework to permit Petrov-Galerkin approximations remedies the situation. For one dimensional problems, the work of Barrett and Morton allows an optimal test space to be chosen at each step of the Newton iteration so that the resulting approximation is near optimal in a norm closely related to the standard L^2 norm. More detailed information about the underlying solution can then be obtained by recovering a solution of an appropriate form. For two-dimensional problems, since the optimal test functions are difficult to find in practice, an upwinding method due to Heinrich et.al. is used at each step of the Newton iteration. Also, a framework is presented in which various upwind methods may be compared. The thesis also addresses the

  5. Noisy continuous time random walks

    NASA Astrophysics Data System (ADS)

    Jeon, Jae-Hyung; Barkai, Eli; Metzler, Ralf

    2013-09-01

    Experimental studies of the diffusion of biomolecules within biological cells are routinely confronted with multiple sources of stochasticity, whose identification renders the detailed data analysis of single molecule trajectories quite intricate. Here, we consider subdiffusive continuous time random walks that represent a seminal model for the anomalous diffusion of tracer particles in complex environments. This motion is characterized by multiple trapping events with infinite mean sojourn time. In real physical situations, however, instead of the full immobilization predicted by the continuous time random walk model, the motion of the tracer particle shows additional jiggling, for instance, due to thermal agitation of the environment. We here present and analyze in detail an extension of the continuous time random walk model. Superimposing the multiple trapping behavior with additive Gaussian noise of variable strength, we demonstrate that the resulting process exhibits a rich variety of apparent dynamic regimes. In particular, such noisy continuous time random walks may appear ergodic, while the bare continuous time random walk exhibits weak ergodicity breaking. Detailed knowledge of this behavior will be useful for the truthful physical analysis of experimentally observed subdiffusion.

  6. Iterative methods for elliptic finite element equations on general meshes

    NASA Technical Reports Server (NTRS)

    Nicolaides, R. A.; Choudhury, Shenaz

    1986-01-01

    Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included.

  7. Theory of Random Walks.

    NASA Astrophysics Data System (ADS)

    Sokol, M.

    1996-11-01

    We develope a mathematical analysis derived from a simple vector-kick model of the evolution of a laser field due to strictly phase diffusion and having arbitrary average photon number barn. We write the exact-coupled, nonlinear equations in two dynamical variables, namely the magnitude of the new field E_0^' and the differential change in angle δ φ. A closed form approximate solution to find the variance in the tangent of phase, for small angles, has yielded the theoretical lower limit for large photon number √n=E_0>> 1. The exact solution to the variance in the tangent of phase angle δ φ was made possible by a trigonometric substitution method, and the transformed argument has been analyzed using residue calculus. There is a double-zero at z=0, simple-poles at z=± i, and double poles at z=± √(E0 +1)(E_0-1) i, in the Argand plane. The variance in the tangent of phase is found to be <(tan δφ)^2> = 2π(√(barn)/(barn-1) -1). An extension of this result would include effects due to amplification and saturation. The general result would include a regime of small photon numbers. Part C of program listing

  8. Directed random walk with random restarts: The Sisyphus random walk

    NASA Astrophysics Data System (ADS)

    Montero, Miquel; Villarroel, Javier

    2016-09-01

    In this paper we consider a particular version of the random walk with restarts: random reset events which suddenly bring the system to the starting value. We analyze its relevant statistical properties, like the transition probability, and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks, and other extreme statistics are also derived; we consider counting problems naturally associated with the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.

  9. Directed random walk with random restarts: The Sisyphus random walk.

    PubMed

    Montero, Miquel; Villarroel, Javier

    2016-09-01

    In this paper we consider a particular version of the random walk with restarts: random reset events which suddenly bring the system to the starting value. We analyze its relevant statistical properties, like the transition probability, and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks, and other extreme statistics are also derived; we consider counting problems naturally associated with the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.

  10. On Convergent Probability of a Random Walk

    ERIC Educational Resources Information Center

    Lee, Y.-F.; Ching, W.-K.

    2006-01-01

    This note introduces an interesting random walk on a straight path with cards of random numbers. The method of recurrent relations is used to obtain the convergent probability of the random walk with different initial positions.

  11. On Convergent Probability of a Random Walk

    ERIC Educational Resources Information Center

    Lee, Y.-F.; Ching, W.-K.

    2006-01-01

    This note introduces an interesting random walk on a straight path with cards of random numbers. The method of recurrent relations is used to obtain the convergent probability of the random walk with different initial positions.

  12. Quantum random walks without walking

    SciTech Connect

    Manouchehri, K.; Wang, J. B.

    2009-12-15

    Quantum random walks have received much interest due to their nonintuitive dynamics, which may hold the key to a new generation of quantum algorithms. What remains a major challenge is a physical realization that is experimentally viable and not limited to special connectivity criteria. We present a scheme for walking on arbitrarily complex graphs, which can be realized using a variety of quantum systems such as a Bose-Einstein condensate trapped inside an optical lattice. This scheme is particularly elegant since the walker is not required to physically step between the nodes; only flipping coins is sufficient.

  13. On an algorithm for solving parabolic and elliptic equations

    NASA Astrophysics Data System (ADS)

    D'Ascenzo, N.; Saveliev, V. I.; Chetverushkin, B. N.

    2015-08-01

    The present-day rapid growth of computer power, in particular, parallel computing systems of ultrahigh performance requires a new approach to the creation of models and solution algorithms for major problems. An algorithm for solving parabolic and elliptic equations is proposed. The capabilities of the method are demonstrated by solving astrophysical problems on high-performance computer systems with massive parallelism.

  14. On the semilinear elliptic equations of electrostatic NEMS devices

    NASA Astrophysics Data System (ADS)

    Zhang, Ruifeng; Cai, Liang

    2014-12-01

    In this paper, we analyze a class of semilinear elliptic equations with boundary value problem based on electrostatic nanoelectromechanical system. First, we will use upper and lower solution method to study the existence of solutions and some properties of minimal solutions for the problem. Then, we will establish the existence of a second solution by variational method in some conditions.

  15. Lipschitz regularity results for nonlinear strictly elliptic equations and applications

    NASA Astrophysics Data System (ADS)

    Ley, Olivier; Nguyen, Vinh Duc

    2017-10-01

    Most of Lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.

  16. Random walks with similar transition probabilities

    NASA Astrophysics Data System (ADS)

    Schiefermayr, Klaus

    2003-04-01

    We consider random walks on the nonnegative integers with a possible absorbing state at -1. A random walk is called [alpha]-similar to a random walk if there exist constants Cij such that for the corresponding n-step transition probabilities , i,j[greater-or-equal, slanted]0, hold. We give necessary and sufficient conditions for the [alpha]-similarity of two random walks both in terms of the parameters and in terms of the corresponding spectral measures which appear in the spectral representation of the n-step transition probabilities developed by Karlin and McGregor.

  17. Excursion set theory for correlated random walks

    NASA Astrophysics Data System (ADS)

    Farahi, Arya; Benson, Andrew J.

    2013-08-01

    We present a new method to compute the first crossing distribution in excursion set theory for the case of correlated random walks. We use a combination of the path integral formalism of Maggiore & Riotto, and the integral equation solution of Zhang & Hui and Benson et al. to find a numerically and convenient algorithm to derive the first crossing distribution. We apply this methodology to the specific case of a Gaussian random density field filtered with a Gaussian smoothing function. By comparing our solutions to results from Monte Carlo calculations of the first crossing distribution we demonstrate that approximately it is in good agreement with exact solution for certain barriers, and at large masses. Our approach is quite general, and can be adapted to other smoothing functions and barrier function and also to non-Gaussian density fields.

  18. Molecular motors: thermodynamics and the random walk.

    PubMed Central

    Thomas, N.; Imafuku, Y.; Tawada, K.

    2001-01-01

    The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors. PMID:11600075

  19. Quantum random walks and decision making.

    PubMed

    Shankar, Karthik H

    2014-01-01

    How realistic is it to adopt a quantum random walk model to account for decisions involving two choices? Here, we discuss the neural plausibility and the effect of initial state and boundary thresholds on such a model and contrast it with various features of the classical random walk model of decision making.

  20. Optimal Control for the Degenerate Elliptic Logistic Equation

    SciTech Connect

    Delgado, M.

    2002-06-05

    We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.

  1. MIB method for elliptic equations with multi-material interfaces

    PubMed Central

    Xia, Kelin; Zhan, Meng; Wei, Guo-Wei

    2011-01-01

    Elliptic partial differential equations (PDEs) are widely used to model real-world problems. Due to the heterogeneous characteristics of many naturally occurring materials and man-made structures, devices, and equipments, one frequently needs to solve elliptic PDEs with discontinuous coefficients and singular sources. The development of high-order elliptic interface schemes has been an active research field for decades. However, challenges remain in the construction of high-order schemes and particularly, for nonsmooth interfaces, i.e., interfaces with geometric singularities. The challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other. High-order methods for elliptic equations with multi-material interfaces have not been reported in the literature to our knowledge. The present work develops matched interface and boundary (MIB) method based schemes for solving two-dimensional (2D) elliptic PDEs with geometric singularities of multi-material interfaces. A number of new MIB schemes are constructed to account for all possible topological variations due to two-material interfaces. The geometric singularities of three-material interfaces are significantly more difficult to handle. Three new MIB schemes are designed to handle a variety of geometric situations and topological variations, although not all of them. The performance of the proposed new MIB schemes is validated by numerical experiments with a wide range of coefficient contrasts, geometric singularities, and solution types. Extensive numerical studies confirm the designed second order accuracy of the MIB method for multi-material interfaces, including a case where the derivative of the solution diverges. PMID:21691433

  2. MIB method for elliptic equations with multi-material interfaces.

    PubMed

    Xia, Kelin; Zhan, Meng; Wei, Guo-Wei

    2011-06-01

    Elliptic partial differential equations (PDEs) are widely used to model real-world problems. Due to the heterogeneous characteristics of many naturally occurring materials and man-made structures, devices, and equipments, one frequently needs to solve elliptic PDEs with discontinuous coefficients and singular sources. The development of high-order elliptic interface schemes has been an active research field for decades. However, challenges remain in the construction of high-order schemes and particularly, for nonsmooth interfaces, i.e., interfaces with geometric singularities. The challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other. High-order methods for elliptic equations with multi-material interfaces have not been reported in the literature to our knowledge. The present work develops matched interface and boundary (MIB) method based schemes for solving two-dimensional (2D) elliptic PDEs with geometric singularities of multi-material interfaces. A number of new MIB schemes are constructed to account for all possible topological variations due to two-material interfaces. The geometric singularities of three-material interfaces are significantly more difficult to handle. Three new MIB schemes are designed to handle a variety of geometric situations and topological variations, although not all of them. The performance of the proposed new MIB schemes is validated by numerical experiments with a wide range of coefficient contrasts, geometric singularities, and solution types. Extensive numerical studies confirm the designed second order accuracy of the MIB method for multi-material interfaces, including a case where the derivative of the solution diverges.

  3. Random walks models with intermediate fractional diffusion asymptotics

    NASA Astrophysics Data System (ADS)

    Saichev, Alexander I.; Utkin, Sergei G.

    2004-05-01

    Random walk process was investigated with PDF of random time intervals similar to fractional exponential law on small times and to regular exponential law on long times. Generalized fractional Kolmogorov-Feller equation was derived for such kind of process. Asymptotics of its PDF in the long time limit and for intermediate times were found. They obey standard diffusion law or fractional diffusion law respectively. Exact solutions of mentioned equations were numerically calculated, demonstrating crossover of fractional diffusion law into the linear one.

  4. Lipschitz Regularity for Elliptic Equations with Random Coefficients

    NASA Astrophysics Data System (ADS)

    Armstrong, Scott N.; Mourrat, Jean-Christophe

    2016-01-01

    We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L ∞-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (for example finite range of dependence). We also prove a quenched L 2 estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.

  5. Singular Solutions of Fully Nonlinear Elliptic Equations and Applications

    NASA Astrophysics Data System (ADS)

    Armstrong, Scott N.; Sirakov, Boyan; Smart, Charles K.

    2012-08-01

    We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of {R^n} , and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén-Lindelöf result as well as a principle of positive singularities in certain Lipschitz domains.

  6. Random walks on simplicial complexes and harmonics†

    PubMed Central

    Steenbergen, John

    2016-01-01

    Abstract In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the k‐dimensional Laplacian for 1 ≤ k ≤ d. We study an example of random walks on simplicial complexes in the context of a semi‐supervised learning problem. Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classification problem on graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 379–405, 2016

  7. Analysis and Numerical Treatment of Elliptic Equations with Stochastic Data

    NASA Astrophysics Data System (ADS)

    Cheng, Shi

    Many science and engineering applications are impacted by a significant amount of uncertainty in the model. Examples include groundwater flow, microscopic biological system, material science and chemical engineering systems. Common mathematical problems in these applications are elliptic equations with stochastic data. In this dissertation, we examine two types of stochastic elliptic partial differential equations(SPDEs), namely nonlinear stochastic diffusion reaction equations and general linearized elastostatic problems in random media. We begin with the construction of an analysis framework for this class of SPDEs, extending prior work of Babuska in 2010. We then use the framework both for establishing well-posedness of the continuous problems and for posing Galerkintype numerical methods. In order to solve these two types of problems, single integral weak formulations and stochastic collocation methods are applied. Moreover, a priori error estimates for stochastic collocation methods are derived, which imply that the rate of convergence is exponential, along with the order of polynomial increasing in the space of random variables. As expected, numerical experiments show the exponential rate of convergence, verified by a posterior error analysis. Finally, an adaptive strategy driven by a posterior error indicators is designed.

  8. On an Elliptic Equation Arising from Composite Materials

    NASA Astrophysics Data System (ADS)

    Dong, Hongjie; Zhang, Hong

    2016-10-01

    In this paper, we derive an interior Schauder estimate for the divergence form elliptic equation D_i (a(x)D_iu) = D_i f_i in R^2,where {a(x)} and {f_i (x)} are piecewise Hölder continuous in a domain containing two touching balls as subdomains. When {f_i ≡ 0} and a is piecewise constant, we prove that u is piecewise smoothwith bounded derivatives.This completely answers a question raised by Li andVogelius (Arch Ration Mech Anal 153(2):91-151, 2000) in dimension 2.

  9. Random walks in the history of life

    PubMed Central

    Cornette, James L.; Lieberman, Bruce S.

    2004-01-01

    The simplest null hypothesis for evolutionary time series is that the observed data follow a random walk. We examined whether aspects of Sepkoski's compilation of marine generic diversity depart from a random walk by using statistical tests from econometrics. Throughout most of the Phanerozoic, the random-walk null hypothesis is not rejected for marine diversity, accumulated origination or accumulated extinction, suggesting that either these variables were correlated with environmental variables that follow a random walk or so many mechanisms were affecting these variables, in different ways, that the resultant trends appear random. The only deviation from this pattern involves rejection of the null hypothesis for roughly the last 75 million years for the diversity and accumulated origination time series. PMID:14684835

  10. Epidemic spreading driven by biased random walks

    NASA Astrophysics Data System (ADS)

    Pu, Cunlai; Li, Siyuan; Yang, Jian

    2015-08-01

    Random walk is one of the basic mechanisms of many network-related applications. In this paper, we study the dynamics of epidemic spreading driven by biased random walks in complex networks. In our epidemic model, infected nodes send out infection packets by biased random walks to their neighbor nodes, and this causes the infection of susceptible nodes that receive the packets. Infected nodes recover from the infection at a constant rate λ, and will not be infected again after recovery. We obtain the largest instantaneous number of infected nodes and the largest number of ever-infected nodes respectively, by tuning the parameter α of the biased random walks. Simulation results on model and real-world networks show that spread of the epidemic becomes intense and widespread with increase of either delivery capacity of infected nodes, average node degree, or homogeneity of node degree distribution.

  11. Relationship between flux and concentration gradient of diffusive particles with the usage of random walk model

    NASA Astrophysics Data System (ADS)

    Ovchinnikov, M. N.

    2017-09-01

    The fundamental solutions of the diffusion equation for the local-equilibrium and nonlocal models are considered as the limiting cases of the solution of a problem related to consideration of the Brownian particles random walks. The differences between fundamental solutions, flows and concentration gradients were studied. The new modified non-local diffusion equation of the telegrapher type with correction function is suggested. It contains only microparameters of the random walk problem.

  12. Quantum random walks using quantum accelerator modes

    SciTech Connect

    Ma, Z.-Y.; Burnett, K.; D'Arcy, M. B.; Gardiner, S. A.

    2006-01-15

    We discuss the use of high-order quantum accelerator modes to achieve an atom optical realization of a biased quantum random walk. We first discuss how one can create coexistent quantum accelerator modes, and hence how momentum transfer that depends on the atoms' internal state can be achieved. When combined with microwave driving of the transition between the states, a different type of atomic beam splitter results. This permits the realization of a biased quantum random walk through quantum accelerator modes.

  13. Solving Schroedinger's equation using random walks

    NASA Astrophysics Data System (ADS)

    Aspuru-Guzik, Alan

    Exact and almost exact solutions for energies and properties of atoms and molecules can be obtained by quantum Monte Carlo (QMC) methods. This thesis is composed of different contributions to various QMC methodologies, as well as applications to electronic excitations of biological systems. We propose a wave function optimization functional that is robust regarding the presence of outliers. Our work, and subsequent applications by others, has shown the convergence properties and robustness of the absolute deviation (AD) functional as compared to the variance functional (VF). We apply the method to atoms from the second row of the periodic table, as well as third-row transition metal atoms, including an all-electron calculation of Sc. In all cases, the AD functional converges faster than the VF. Soft effective core potentials (ECPs) with no divergence at the origin are constructed and validated for second- an third-row atoms of the periodic table. The ECPs we developed have been used by others in several successful studies. As an application of the DMC approach to biochemical problems, we studied the electronic excitations of free-base porphyrin and obtained results in excellent agreement with experiment. These findings validate the use of the DMC approach for these kinds of systems. A study of the role of spheroidene in the photo-protection mechanism of Rhodobacter sphaeroides is described. At the time of writing, calculations for the estimation of excitation energies for the bacteriochlorophyll and spheroidene molecules as well as storage of the random walkers for future prediction of the excitation energy transfer rate are being performed. To date, the calculations mentioned above are the largest all-electron studies on molecules. For the computation of these systems, a sparse linear-scaling DMC algorithm was developed. This algorithm provides a speedup of at least a factor of ten over previously published methods. The method is validated on systems up to 390 electrons. A summary of the Fermion Monte Carlo (FMC) algorithm as well as an application to the Be atom are discussed. The Zori package, a linear-scaling massively-parallel open-source program that uses modern programming libraries, was developed. The program is made available to the public under the GNU/General Public License (GPL). The capabilities of the Zori program are summarized.

  14. Theory of continuum random walks and application to chemotaxis

    NASA Astrophysics Data System (ADS)

    Schnitzer, Mark J.

    1993-10-01

    We formulate the general theory of random walks in continuum, essential for treating a collision rate which depends smoothly upon direction of motion. We also consider a smooth probability distribution of correlations between the directions of motion before and after collisions, as well as orientational Brownian motion between collisions. These features lead to an effective Smoluchowski equation. Such random walks involving an infinite number of distinct directions of motion cannot be treated on a lattice, which permits only a finite number of directions of motion, nor by Langevin methods, which make no reference to individual collisions. The effective Smoluchowski equation enables a description of the biased random walk of the bacterium Escherichia coli during chemotaxis, its search for food. The chemotactic responses of cells which perform temporal comparisons of the concentration of a chemical attractant are predicted to be strongly positive, whereas those of cells which measure averages of the ambient attractant concentration are predicted to be negative. The former prediction explains the observed behavior of wild-type (naturally occurring) cells; however, the latter behavior has yet to be observed, even in cells defective in adaption.

  15. Mild solutions of semilinear elliptic equations in Hilbert spaces

    NASA Astrophysics Data System (ADS)

    Federico, Salvatore; Gozzi, Fausto

    2017-03-01

    This paper extends the theory of regular solutions (C1 in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of G-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into G-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now.

  16. Gravitational lens equation for embedded lenses; magnification and ellipticity

    SciTech Connect

    Chen, B.; Kantowski, R.; Dai, X.

    2011-10-15

    We give the lens equation for light deflections caused by point mass condensations in an otherwise spatially homogeneous and flat universe. We assume the signal from a distant source is deflected by a single condensation before it reaches the observer. We call this deflector an embedded lens because the deflecting mass is part of the mean density. The embedded lens equation differs from the conventional lens equation because the deflector mass is not simply an addition to the cosmic mean. We prescribe an iteration scheme to solve this new lens equation and use it to compare our results with standard linear lensing theory. We also compute analytic expressions for the lowest order corrections to image amplifications and distortions caused by incorporating the lensing mass into the mean. We use these results to estimate the effect of embedding on strong lensing magnifications and ellipticities and find only small effects, <1%, contrary to what we have found for time delays and for weak lensing, {approx}5%.

  17. The Hyperbolic Kepler’s Equation, and the Elliptic Equation Revisited

    DTIC Science & Technology

    1989-07-01

    elliptic equation, several papers having appeared since our previous work, little hes been published on the hyperbolic equation. Burkardt and Danby (1983...universa1 computation, we wish to end with a remark on the recent response by Danby (1987) to our previous comment (Odell and Gooding, 1986) on the...Kepler’s Equation’, Mor. .ot. R. Astron. Soc. 92, 104. Burkardt, T.M. and Danby , J.M.A.: 1983, ’The Solution of Kepler’s Equation, I’, Ceies. Mech. 31

  18. Quantum random-walk search algorithm

    SciTech Connect

    Shenvi, Neil; Whaley, K. Birgitta; Kempe, Julia

    2003-05-01

    Quantum random walks on graphs have been shown to display many interesting properties, including exponentially fast hitting times when compared with their classical counterparts. However, it is still unclear how to use these novel properties to gain an algorithmic speedup over classical algorithms. In this paper, we present a quantum search algorithm based on the quantum random-walk architecture that provides such a speedup. It will be shown that this algorithm performs an oracle search on a database of N items with O({radical}(N)) calls to the oracle, yielding a speedup similar to other quantum search algorithms. It appears that the quantum random-walk formulation has considerable flexibility, presenting interesting opportunities for development of other, possibly novel quantum algorithms.

  19. Random walk of microswimmers: puller and pusher cases

    NASA Astrophysics Data System (ADS)

    Rafai, Salima; Peyla, Philippe; Dyfcom Team

    2014-11-01

    Swimming at a micrometer scale demands particular strategies. Indeed when inertia is negligible as compared to viscous forces (i.e. Reynolds number Re is lower than unity), hydrodynamics equations are reversible in time. To achieve propulsion a low Reynolds number, swimmers must then deform in a way that is not invariant under time reversal. Here we investigate the dispersal properties of self propelled organisms by means of microscopy and cell tracking. Our systems of interest are, on the one hand, the microalga Chlamydomonas Reinhardtii, a puller-type swimmer and on the other hand, Lingulodinium polyedrum, a pusher. Both are quasi-spherical single celled alga. In the case of dilute suspensions, we show that tracked trajectories are well modelled by a correlated random walk. This process is based on short time correlations in the direction of movement called persistence. At longer times, correlations are lost and a standard random walk characterizes the trajectories. Finally we show how drag forces modify the characteristics of this particular random walk.

  20. The random walk of a low-Reynolds-number swimmer

    NASA Astrophysics Data System (ADS)

    Rafaï, Salima; Garcia, Michaël; Berti, Stefano; Peyla, Philippe

    2010-11-01

    Swimming at a micrometer scale demands particular strategies. Indeed when inertia is negligible as compared to viscous forces (i.e. Reynolds number Re is lower than unity), hydrodynamics equations are reversible in time. To achieve propulsion a low Reynolds number, swimmers must then deform in a way that is not invariant under time reversal. Here we investigate the dispersal properties of self propelled organisms by means of microscopy and cell tracking. Our system of interest is the microalga Chlamydomonas Reinhardtii, a motile single celled green alga about 10 micrometers in diameter that swims with two flagellae. In the case of dilute suspensions, we show that tracked trajectories are well modelled by a correlated random walk. This process is based on short time correlations in the direction of movement called persistence. At longer times, correlations are lost and a standard random walk caracterizes the trajectories. Moreover, high speed imaging enables us to show how speed fluctuations at very short times affect the statistical description of the dynamics. Finally we show how drag forces modify the characteristics of this particular random walk.

  1. Quantum Random Walks with General Particle States

    NASA Astrophysics Data System (ADS)

    Belton, Alexander C. R.

    2014-06-01

    A convergence theorem is obtained for quantum random walks with particles in an arbitrary normal state. This unifies and extends previous work on repeated-interactions models, including that of Attal and Pautrat (Ann Henri Poincaré 7:59-104 2006) and Belton (J Lond Math Soc 81:412-434, 2010; Commun Math Phys 300:317-329, 2010). When the random-walk generator acts by ampliation and either multiplication or conjugation by a unitary operator, it is shown that the quantum stochastic cocycle which arises in the limit is driven by a unitary process.

  2. Random Walk Weakly Attracted to a Wall

    NASA Astrophysics Data System (ADS)

    de Coninck, Joël; Dunlop, François; Huillet, Thierry

    2008-10-01

    We consider a random walk X n in ℤ+, starting at X 0= x≥0, with transition probabilities {P}(X_{n+1}=Xn±1|Xn=yge1)={1over2}mp{δover4y+2δ} and X n+1=1 whenever X n =0. We prove {E}Xn˜const. n^{1-{δ over2}} as n ↗∞ when δ∈(1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.

  3. An efficient code to solve the Kepler equation. Elliptic case

    NASA Astrophysics Data System (ADS)

    Raposo-Pulido, V.; Peláez, J.

    2017-05-01

    A new approach for solving Kepler equation for elliptical orbits is developed in this paper. This new approach takes advantage of the very good behaviour of the modified Newton-Raphson method when the initial seed is close to the looked for solution. To determine a good initial seed the eccentric anomaly domain [0, π] is discretized in several intervals and for each one of these intervals a fifth degree interpolating polynomial is introduced. The six coefficients of the polynomial are obtained by requiring six conditions at both ends of the corresponding interval. Thus the real function and the polynomial have equal values at both ends of the interval. Similarly relations are imposed for the two first derivatives. In the singular corner of the Kepler equation, M ≪ 1 and 1 - e ≪ 0 an asymptotic expansion is developed. In most of the cases, the seed generated leads to reach machine error accuracy with the modified Newton-Raphson method with no iterations or just one iteration. This approach improves the computational time compared with other methods currently in use.

  4. Binary Black Hole Initial Data Without Elliptic Equations

    NASA Astrophysics Data System (ADS)

    Winicour, Jeffrey; Racz, Istvan

    2016-03-01

    We describe a radically new method for solving the constraints of Einstein's equations which does not involve elliptic equations. Instead, the constraints are formulated as a symmetric hyperbolic system which can be integrated radially inward from an outer boundary. In this method, the initial metric data for a binary black hole can be freely prescribed, e.g. in a 4-dimensional superimposed Kerr-Schild form for the individual boosted black holes. Two pieces of extrinsic curvature data, which represent the two gravitational degrees of freedom, can also be freely prescribed by superimposing the individual black hole data. The remaining extrinsic curvature data are then determined by the hyperbolic constraint system. Because no puncture or excision boundary conditions are necessary, this approach offers a simple alternative that could provide more physically realistic binary black hole initial data than present methods. Here we present a computational framework for implementing this new method. JW was supported by NSF Grant PHY-1505965 to the University of Pittsburgh. IR was supported in part by the Die Aktion Osterreich-Ungarn, Wissenschafts- und Erziehungskooperation Grant 90ou1.

  5. A Random Walk on a Circular Path

    ERIC Educational Resources Information Center

    Ching, W.-K.; Lee, M. S.

    2005-01-01

    This short note introduces an interesting random walk on a circular path with cards of numbers. By using high school probability theory, it is proved that under some assumptions on the number of cards, the probability that a walker will return to a fixed position will tend to one as the length of the circular path tends to infinity.

  6. Random walk centrality for temporal networks

    NASA Astrophysics Data System (ADS)

    Rocha, Luis E. C.; Masuda, Naoki

    2014-06-01

    Nodes can be ranked according to their relative importance within a network. Ranking algorithms based on random walks are particularly useful because they connect topological and diffusive properties of the network. Previous methods based on random walks, for example the PageRank, have focused on static structures. However, several realistic networks are indeed dynamic, meaning that their structure changes in time. In this paper, we propose a centrality measure for temporal networks based on random walks under periodic boundary conditions that we call TempoRank. It is known that, in static networks, the stationary density of the random walk is proportional to the degree or the strength of a node. In contrast, we find that, in temporal networks, the stationary density is proportional to the in-strength of the so-called effective network, a weighted and directed network explicitly constructed from the original sequence of transition matrices. The stationary density also depends on the sojourn probability q, which regulates the tendency of the walker to stay in the node, and on the temporal resolution of the data. We apply our method to human interaction networks and show that although it is important for a node to be connected to another node with many random walkers (one of the principles of the PageRank) at the right moment, this effect is negligible in practice when the time order of link activation is included.

  7. A Random Walk on a Circular Path

    ERIC Educational Resources Information Center

    Ching, W.-K.; Lee, M. S.

    2005-01-01

    This short note introduces an interesting random walk on a circular path with cards of numbers. By using high school probability theory, it is proved that under some assumptions on the number of cards, the probability that a walker will return to a fixed position will tend to one as the length of the circular path tends to infinity.

  8. Mean first return time for random walks on weighted networks

    NASA Astrophysics Data System (ADS)

    Jing, Xing-Li; Ling, Xiang; Long, Jiancheng; Shi, Qing; Hu, Mao-Bin

    2015-11-01

    Random walks on complex networks are of great importance to understand various types of phenomena in real world. In this paper, two types of biased random walks on nonassortative weighted networks are studied: edge-weight-based random walks and node-strength-based random walks, both of which are extended from the normal random walk model. Exact expressions for stationary distribution and mean first return time (MFRT) are derived and examined by simulation. The results will be helpful for understanding the influences of weights on the behavior of random walks.

  9. The Sylvester equation and the elliptic Korteweg-de Vries system

    NASA Astrophysics Data System (ADS)

    Sun, Ying-ying; Zhang, Da-jun; Nijhoff, Frank W.

    2017-03-01

    The elliptic potential Korteweg-de Vries lattice system is a multi-component extension of the lattice potential Korteweg-de Vries equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by allowing the spectral parameter to be a full matrix obeying a matrix version of the equation of the elliptic curve, and for the Cauchy matrix to be a solution of a Sylvester type matrix equation subject to this matrix elliptic curve equation. The construction involves solving the matrix elliptic curve equation by using Toeplitz matrix techniques, and analysing the solution of the Sylvester equation in terms of Jordan normal forms. Furthermore, we consider the continuum limit system associated with the elliptic potential Korteweg-de Vries system, and analyse the dynamics of the soliton solutions, which reveals some new features of the elliptic system in comparison to the non-elliptic case.

  10. A random walk to stochastic diffusion through spreadsheet analysis

    NASA Astrophysics Data System (ADS)

    Brazzle, Bob

    2013-11-01

    This paper describes a random walk simulation using a number cube and a lattice of concentric rings of tiled hexagons. At the basic level, it gives beginning students a concrete connection to the concept of stochastic diffusion and related physical quantities. A simple algorithm is presented that can be used to set up spreadsheet files to calculate these simulated quantities and even to "discover" the diffusion equation. Lattices with different geometries in two and three dimensions are also presented. This type of simulation provides fertile ground for independent investigations by all levels of undergraduate students.

  11. Asymptotic behaviour of random walks with correlated temporal structure

    PubMed Central

    Magdziarz, Marcin; Szczotka, Władysław; Żebrowski, Piotr

    2013-01-01

    We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking. PMID:24204190

  12. Regularity estimates up to the boundary for elliptic systems of difference equations

    NASA Technical Reports Server (NTRS)

    Strikwerda, J. C.; Wade, B. A.; Bube, K. P.

    1986-01-01

    Regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations were proved. The regularity estimates, obtained for boundary fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudo-difference operators to treat systems with variable coefficients. The elliptic systems of difference equations and the boundary conditions which are considered are very general in form. The regularity of a regular elliptic system of difference equations was proved equivalent to the nonexistence of eigensolutions. The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundstrom (1972) and others for hyperbolic difference equations.

  13. From random walks to spin glasses

    NASA Astrophysics Data System (ADS)

    Derrida, B.

    1997-02-01

    The talk was a short review on systems which exhibit non-self-averaging effects: sums of random variables when the distribution has a long tail, mean field spin glasses, random map models and returns of a random walk to the origin. Non-self-averaging effects are identical in the case of sums of random variables and in the spin glass problem as predicted by the replica approach. Also we will see that for the random map models or for the problem of the returns of a random walk to the origin, the non-self-averaging effects coincide with the results of the replica approach when the number n of replica n = - {1}/{2} or n = -1.

  14. Random walk centrality in interconnected multilayer networks

    NASA Astrophysics Data System (ADS)

    Solé-Ribalta, Albert; De Domenico, Manlio; Gómez, Sergio; Arenas, Alex

    2016-06-01

    Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in many fields, from economics to biology and from urban planning to social sciences, to identify the most (or the less) influent nodes in a network using centrality measures. However, defining the centrality of actors in interconnected complex networks is not trivial. In this paper, we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies, and extend two well known random walk centrality measures, the random walk betweenness and closeness centrality, to interconnected multilayer networks. For each of the measures we provide analytical expressions that completely agree with numerically results.

  15. An invariance property of diffusive random walks

    NASA Astrophysics Data System (ADS)

    Blanco, S.; Fournier, R.

    2003-01-01

    Starting from a simple animal-biology example, a general, somewhat counter-intuitive property of diffusion random walks is presented. It is shown that for any (non-homogeneous) purely diffusing system, under any isotropic uniform incidence, the average length of trajectories through the system (the average length of the random walk trajectories from entry point to first exit point) is independent of the characteristics of the diffusion process and therefore depends only on the geometry of the system. This exact invariance property may be seen as a generalization to diffusion of the well-known mean-chord-length property (Case K. M. and Zweifel P. F., Linear Transport Theory (Addison-Wesley) 1967), leading to broad physics and biology applications.

  16. Simulation of pedigree genotypes by random walks.

    PubMed Central

    Lange, K; Matthysse, S

    1989-01-01

    A random walk method, based on the Metropolis algorithm, is developed for simulating the distribution of trait and linkage marker genotypes in pedigrees where trait phenotypes are already known. The method complements techniques suggested by Ploughman and Boehnke and by Ott that are based on sequential sampling of genotypes within a pedigree. These methods are useful for estimating the power of linkage analysis before complete study of a pedigree is undertaken. We apply the random walk technique to a partially penetrant disease, schizophrenia, and to a recessive disease, ataxia-telangiectasia. In the first case we show that accessory phenotypes with higher penetrance than that of schizophrenia itself may be crucial for effective linkage analysis, and in the second case we show that impressionistic selection of informative pedigrees may be misleading. PMID:2589323

  17. A Random Walk Picture of Basketball

    NASA Astrophysics Data System (ADS)

    Gabel, Alan; Redner, Sidney

    2012-02-01

    We analyze NBA basketball play-by-play data and found that scoring is well described by a weakly-biased, anti-persistent, continuous-time random walk. The time between successive scoring events follows an exponential distribution, with little memory between events. We account for a wide variety of statistical properties of scoring, such as the distribution of the score difference between opponents and the fraction of game time that one team is in the lead.

  18. Random walks on generalized Koch networks

    NASA Astrophysics Data System (ADS)

    Sun, Weigang

    2013-10-01

    For deterministically growing networks, it is a theoretical challenge to determine the topological properties and dynamical processes. In this paper, we study random walks on generalized Koch networks with features that include an initial state that is a globally connected network to r nodes. In each step, every existing node produces m complete graphs. We then obtain the analytical expressions for first passage time (FPT), average return time (ART), i.e. the average of FPTs for random walks from node i to return to the starting point i for the first time, and average sending time (AST), defined as the average of FPTs from a hub node to all other nodes, excluding the hub itself with regard to network parameters m and r. For this family of Koch networks, the ART of the new emerging nodes is identical and increases with the parameters m or r. In addition, the AST of our networks grows with network size N as N ln N and also increases with parameter m. The results obtained in this paper are the generalizations of random walks for the original Koch network.

  19. Sunspot random walk and 22-year variation

    USGS Publications Warehouse

    Love, Jeffrey J.; Rigler, E. Joshua

    2012-01-01

    We examine two stochastic models for consistency with observed long-term secular trends in sunspot number and a faint, but semi-persistent, 22-yr signal: (1) a null hypothesis, a simple one-parameter random-walk model of sunspot-number cycle-to-cycle change, and, (2) an alternative hypothesis, a two-parameter random-walk model with an imposed 22-yr alternating amplitude. The observed secular trend in sunspots, seen from solar cycle 5 to 23, would not be an unlikely result of the accumulation of multiple random-walk steps. Statistical tests show that a 22-yr signal can be resolved in historical sunspot data; that is, the probability is low that it would be realized from random data. On the other hand, the 22-yr signal has a small amplitude compared to random variation, and so it has a relatively small effect on sunspot predictions. Many published predictions for cycle 24 sunspots fall within the dispersion of previous cycle-to-cycle sunspot differences. The probability is low that the Sun will, with the accumulation of random steps over the next few cycles, walk down to a Dalton-like minimum. Our models support published interpretations of sunspot secular variation and 22-yr variation resulting from cycle-to-cycle accumulation of dynamo-generated magnetic energy.

  20. Sunspot random walk and 22-year variation

    NASA Astrophysics Data System (ADS)

    Love, Jeffrey J.; Rigler, E. Joshua

    2012-05-01

    We examine two stochastic models for consistency with observed long-term secular trends in sunspot number and a faint, but semi-persistent, 22-yr signal: (1) a null hypothesis, a simple one-parameter log-normal random-walk model of sunspot-number cycle-to-cycle change, and, (2) an alternative hypothesis, a two-parameter random-walk model with an imposed 22-yr alternating amplitude. The observed secular trend in sunspots, seen from solar cycle 5 to 23, would not be an unlikely result of the accumulation of multiple random-walk steps. Statistical tests show that a 22-yr signal can be resolved in historical sunspot data; that is, the probability is low that it would be realized from random data. On the other hand, the 22-yr signal has a small amplitude compared to random variation, and so it has a relatively small effect on sunspot predictions. Many published predictions for cycle 24 sunspots fall within the dispersion of previous cycle-to-cycle sunspot differences. The probability is low that the Sun will, with the accumulation of random steps over the next few cycles, walk down to a Dalton-like minimum. Our models support published interpretations of sunspot secular variation and 22-yr variation resulting from cycle-to-cycle accumulation of dynamo-generated magnetic energy.

  1. Lyapunov-type inequalities for quasilinear elliptic equations with Robin boundary condition.

    PubMed

    Dinlemez Kantar, Ülkü; Özden, Tülay

    2017-01-01

    The aim of this study is to prove Lyapunov-type inequalities for a quasilinear elliptic equation in [Formula: see text]. Also the lower bound for the first positive eigenvalue of the boundary value problem is obtained.

  2. Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations

    NASA Astrophysics Data System (ADS)

    Byun, Sun-Sig; Lee, Mikyoung; Palagachev, Dian K.

    2016-03-01

    We prove global regularity in weighted Lebesgue spaces for the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equations. As a consequence, regularity in Morrey spaces of the Hessian is derived as well.

  3. CLASSICAL AREAS OF PHENOMENOLOGY: Material parameter equation for rotating elliptical spherical cloaks

    NASA Astrophysics Data System (ADS)

    Ma, Hua; Qu, Shao-Bo; Xu, Zhuo; Zhang, Jie-Qiu; Wang, Jia-Fu

    2009-01-01

    By using the coordinate transformation method, we have deduced the material parameter equation for rotating elliptical spherical cloaks and carried out simulation as well. The results indicate that the rotating elliptical spherical cloaking shell, which is made of meta-materials whose permittivity and permeability are governed by the equation deduced in this paper, can achieve perfect invisibility by excluding electromagnetic fields from the internal region without disturbing any external field.

  4. Continuous-time random walks that alter environmental transport properties.

    PubMed

    Angstmann, C; Henry, B I

    2011-12-01

    We consider continuous-time random walks (CTRWs) in which the walkers have a finite probability to alter the waiting-time and/or step-length transport properties of their environment, resulting in possibly transient anomalous diffusion. We refer to these CTRWs as transmogrifying continuous-time random walks (TCTRWs) to emphasize that they change the form of the transport properties of their environment, and in a possibly strange way. The particular case in which the CTRW waiting-time density has a finite probability to be permanently altered at a given site, following a visitation by a walker, is considered in detail. Master equations for the probability density function of transmogrifying random walkers are derived, and results are compared with Monte Carlo simulations. An interesting finding is that TCTRWs can generate transient subdiffusion or transient superdiffusion without invoking truncated or tempered power law densities for either the waiting times or the step lengths. The transient subdiffusion or transient superdiffusion arises in TCTRWs with Gaussian step-length densities and exponential waiting-time densities when the altered average waiting time is greater than or less than, respectively, the original average waiting time.

  5. Random Walks in Model Brain Tissue

    NASA Astrophysics Data System (ADS)

    Grinberg, Farida; Farrher, Ezequiel; Oros-Peusquens, Ana-Maria; Shah, N. Jon

    2011-03-01

    The propagation of water molecules in the brain and the corresponding NMR response are affected by many factors such as compartmentalization, restrictions and anisotropy imposed by the cellular microstructure. Interfacial interactions with cell membranes and exchange additionally come into play. Due to the complexity of the underlying factors, a differentiation between the various contributions to the average NMR signal in in vivo studies represents a difficult task. In this work we perform random-walk Monte Carlo simulations in well-defined model systems aiming at establishing quantitative relations between dynamics and microstructure. The results are compared with experimental data obtained for artificial anisotropic model systems.

  6. Scalable networks for discrete quantum random walks

    SciTech Connect

    Fujiwara, S.; Osaki, H.; Buluta, I.M.; Hasegawa, S.

    2005-09-15

    Recently, quantum random walks (QRWs) have been thoroughly studied in order to develop new quantum algorithms. In this paper we propose scalable quantum networks for discrete QRWs on circles, lines, and also in higher dimensions. In our method the information about the position of the walker is stored in a quantum register and the network consists of only one-qubit rotation and (controlled){sup n}-NOT gates, therefore it is purely computational and independent of the physical implementation. As an example, we describe the experimental realization in an ion-trap system.

  7. Exact Jacobian elliptic function solutions and hyperbolic function solutions for Sawada Kotere equation with variable coefficient

    NASA Astrophysics Data System (ADS)

    Liu, Qing; Zhu, Jia-Min

    2006-03-01

    Variable-coefficient Sawada Kotere equation is researched. By the means of modified mapping method, we establish a mapping relation between the known solutions of elliptic functional equation and the unknown solutions of variable-coefficient Sawada Kotere equation. Based on the relation, we easily deduce abundant exact solutions of Jacobi elliptic function and of hyperbolic function to variable-coefficient Sawada Kotere equation. The merit of our method is that, without much extra effort, we circumvent integration and directly get the above all solutions in an uniform way.

  8. Numerical implementation of the method of fictitious domains for elliptic equations

    NASA Astrophysics Data System (ADS)

    Temirbekov, Almas N.

    2016-08-01

    In the paper, we study the elliptical type equation with strongly changing coefficients. We are interested in studying such equation because the given type equations are yielded when we use the fictitious domain method. In this paper we suggest a special method for numerical solution of the elliptic equation with strongly changing coefficients. We have proved the theorem for the assessment of developed iteration process convergence rate. We have developed computational algorithm and numerical calculations have been done to illustrate the effectiveness of the suggested method.

  9. Clustered continuous-time random walks: diffusion and relaxation consequences

    PubMed Central

    Weron, Karina; Stanislavsky, Aleksander; Jurlewicz, Agnieszka; Meerschaert, Mark M.; Scheffler, Hans-Peter

    2012-01-01

    We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies. PMID:22792038

  10. Local time of Lévy random walks: A path integral approach

    NASA Astrophysics Data System (ADS)

    Zatloukal, Václav

    2017-05-01

    The local time of a stochastic process quantifies the amount of time that sample trajectories x (τ ) spend in the vicinity of an arbitrary point x . For a generic Hamiltonian, we employ the phase-space path-integral representation of random walk transition probabilities in order to quantify the properties of the local time. For time-independent systems, the resolvent of the Hamiltonian operator proves to be a central tool for this purpose. In particular, we focus on the local times of Lévy random walks (Lévy flights), which correspond to fractional diffusion equations.

  11. Convergence of quantum random walks with decoherence

    SciTech Connect

    Fan Shimao; Feng Zhiyong; Yang, Wei-Shih; Xiong Sheng

    2011-10-15

    In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution p(x,t) depends mainly on the spectrum of the superoperator L{sub kk}. We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals 1, then P(({nu}/{radical}(t)),t) converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function p{sub t}(x,t){identical_to}p({radical}(t)x,t), x is an element of Z/{radical}(t), converges in distribution to a continuous convex combination of normal distributions. We give a necessary and sufficient condition for a U(2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk and walks under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of p(x,t) and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.

  12. Random walk of a swimmer in a low-Reynolds-number medium

    NASA Astrophysics Data System (ADS)

    Garcia, Michaël; Berti, Stefano; Peyla, Philippe; Rafaï, Salima

    2011-03-01

    Swimming at a micrometer scale demands particular strategies. When inertia is negligible compared to viscous forces, hydrodynamics equations are reversible in time. To achieve propulsion, microswimmers must therefore deform in a way that is not invariant under time reversal. Here, we investigate dispersal properties of the microalga Chlamydomonas reinhardtii by means of microscopy and cell tracking. We show that tracked trajectories are well modeled by a correlated random walk. This process is based on short time correlations in the direction of movement called persistence. At longer times, correlation is lost and a standard random walk characterizes the trajectories. Moreover, high-speed imaging enables us to show how the back-and-forth motion of flagella at very short times affects the statistical description of the dynamics. Finally, we show how drag forces modify the characteristics of this particular random walk.

  13. Random walk of a swimmer in a low-Reynolds-number medium.

    PubMed

    Garcia, Michaël; Berti, Stefano; Peyla, Philippe; Rafaï, Salima

    2011-03-01

    Swimming at a micrometer scale demands particular strategies. When inertia is negligible compared to viscous forces, hydrodynamics equations are reversible in time. To achieve propulsion, microswimmers must therefore deform in a way that is not invariant under time reversal. Here, we investigate dispersal properties of the microalga Chlamydomonas reinhardtii by means of microscopy and cell tracking. We show that tracked trajectories are well modeled by a correlated random walk. This process is based on short time correlations in the direction of movement called persistence. At longer times, correlation is lost and a standard random walk characterizes the trajectories. Moreover, high-speed imaging enables us to show how the back-and-forth motion of flagella at very short times affects the statistical description of the dynamics. Finally, we show how drag forces modify the characteristics of this particular random walk.

  14. Boundary-value problems for elliptic functional-differential equations and their applications

    NASA Astrophysics Data System (ADS)

    Skubachevskii, A. L.

    2016-10-01

    Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. Applications of these results are found to the theory of non-local elliptic problems, to the Kato square-root problem for an operator, to elasticity theory, and to problems in non-linear optics. Bibliography: 137 titles.

  15. FISHPACK: Efficient FORTRAN Subprograms for the Solution of Separable Elliptic Partial Differential Equations

    NASA Astrophysics Data System (ADS)

    Adams, John C.; Swarztrauber, Paul N.; Sweet, Roland

    2016-09-01

    The FISHPACK collection of Fortran77 subroutines solves second- and fourth-order finite difference approximations to separable elliptic Partial Differential Equations (PDEs). These include Helmholtz equations in cartesian, polar, cylindrical, and spherical coordinates, as well as more general separable elliptic equations. The solvers use the cyclic reduction algorithm. When the problem is singular, a least-squares solution is computed. Singularities induced by the coordinate system are handled, including at the origin r=0 in cylindrical coordinates, and at the poles in spherical coordinates.

  16. Collage-based approaches for elliptic partial differential equations inverse problems

    NASA Astrophysics Data System (ADS)

    Yodzis, Michael; Kunze, Herb

    2017-01-01

    The collage method for inverse problems has become well-established in the literature in recent years. Initial work developed a collage theorem, based upon Banach's fixed point theorem, for treating inverse problems for ordinary differential equations (ODEs). Amongst the subsequent work was a generalized collage theorem, based upon the Lax-Milgram representation theorem, useful for treating inverse problems for elliptic partial differential equations (PDEs). Each of these two different approaches can be applied to elliptic PDEs in one space dimension. In this paper, we explore and compare how the two different approaches perform for the estimation of the diffusivity for a steady-state heat equation.

  17. Infinite Horizon Stochastic Optimal Control Problems with Degenerate Noise and Elliptic Equations in Hilbert Spaces

    SciTech Connect

    Masiero, Federica

    2007-05-15

    Semilinear elliptic partial differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. These results are applied to a stochastic optimal control problem with infinite horizon. Applications to controlled stochastic heat and wave equations are given.

  18. An electric-analog simulation of elliptic partial differential equations using finite element theory

    USGS Publications Warehouse

    Franke, O.L.; Pinder, G.F.; Patten, E.P.

    1982-01-01

    Elliptic partial differential equations can be solved using the Galerkin-finite element method to generate the approximating algebraic equations, and an electrical network to solve the resulting matrices. Some element configurations require the use of networks containing negative resistances which, while physically realizable, are more expensive and time-consuming to construct. ?? 1982.

  19. Random recursive trees and the elephant random walk

    NASA Astrophysics Data System (ADS)

    Kürsten, Rüdiger

    2016-03-01

    One class of random walks with infinite memory, so-called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We use a coupling between the two models to translate results from elephant random walks to the percolation process. We calculate, besides other quantities, exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation. We further introduce another model, the skew elephant random walk, and calculate the first and second moment of this process.

  20. A Branching Random Walk Seen from the Tip

    NASA Astrophysics Data System (ADS)

    Brunet, Éric; Derrida, Bernard

    2011-05-01

    We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. The distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branching Brownian motion shifted by arbitrary amounts are the same as those of a single realization. We discuss the extension of our results to more general branching random walks.

  1. Fragment formation in biased random walks

    NASA Astrophysics Data System (ADS)

    Ramola, Kabir

    2008-10-01

    We analyse a biased random walk on a 1D lattice with unequal step lengths. Such a walk was recently shown to undergo a phase transition from a state containing a single connected cluster of visited sites to one with several clusters of visited sites (fragments) separated by unvisited sites at a critical probability pc (Anteneodo and Morgado 2007 Phys. Rev. Lett. 99 180602). The behaviour of ρ(l), the probability of formation of fragments of length l, is analysed. An exact expression for the generating function of ρ(l) at the critical point is derived. We prove that the asymptotic behaviour is of the form \\rho (l) \\simeq 3/[l(\\log \\ l)^2] .

  2. Background Extraction Using Random Walk Image Fusion.

    PubMed

    Hua, Kai-Lung; Wang, Hong-Cyuan; Yeh, Chih-Hsiang; Cheng, Wen-Huang; Lai, Yu-Chi

    2016-12-23

    It is important to extract a clear background for computer vision and augmented reality. Generally, background extraction assumes the existence of a clean background shot through the input sequence, but realistically, situations may violate this assumption such as highway traffic videos. Therefore, our probabilistic model-based method formulates fusion of candidate background patches of the input sequence as a random walk problem and seeks a globally optimal solution based on their temporal and spatial relationship. Furthermore, we also design two quality measures to consider spatial and temporal coherence and contrast distinctness among pixels as background selection basis. A static background should have high temporal coherence among frames, and thus, we improve our fusion precision with a temporal contrast filter and an optical-flow-based motionless patch extractor. Experiments demonstrate that our algorithm can successfully extract artifact-free background images with low computational cost while comparing to state-of-the-art algorithms.

  3. Lazy random walks for superpixel segmentation.

    PubMed

    Shen, Jianbing; Du, Yunfan; Wang, Wenguan; Li, Xuelong

    2014-04-01

    We present a novel image superpixel segmentation approach using the proposed lazy random walk (LRW) algorithm in this paper. Our method begins with initializing the seed positions and runs the LRW algorithm on the input image to obtain the probabilities of each pixel. Then, the boundaries of initial superpixels are obtained according to the probabilities and the commute time. The initial superpixels are iteratively optimized by the new energy function, which is defined on the commute time and the texture measurement. Our LRW algorithm with self-loops has the merits of segmenting the weak boundaries and complicated texture regions very well by the new global probability maps and the commute time strategy. The performance of superpixel is improved by relocating the center positions of superpixels and dividing the large superpixels into small ones with the proposed optimization algorithm. The experimental results have demonstrated that our method achieves better performance than previous superpixel approaches.

  4. Generalized ruin problems and asynchronous random walks

    NASA Astrophysics Data System (ADS)

    Abad, E.

    2005-07-01

    We consider a gambling game with two different kinds of trials and compute the duration of the game (averaged over all possible initial capitals of the players) by a mapping of the problem to a 1D lattice walk of two particles reacting upon encounter. The relative frequency of the trials is governed by the synchronicity parameter p of the random walk. The duration of the game is given by the mean time to reaction, which turns out to display a different behavior for even and odd lattices, i.e. this quantity is monotonic in p for odd lattices and non-monotonic for even lattices. In the game picture, this implies that the players minimize the duration of the game by restricting themselves to one type of trial if their joint capital is odd, otherwise a non-symmetric mixture of both trials is needed.

  5. Discriminative parameter estimation for random walks segmentation.

    PubMed

    Baudin, Pierre-Yves; Goodman, Danny; Kumrnar, Puneet; Azzabou, Noura; Carlier, Pierre G; Paragios, Nikos; Kumar, M Pawan

    2013-01-01

    The Random Walks (RW) algorithm is one of the most efficient and easy-to-use probabilistic segmentation methods. By combining contrast terms with prior terms, it provides accurate segmentations of medical images in a fully automated manner. However, one of the main drawbacks of using the RW algorithm is that its parameters have to be hand-tuned. we propose a novel discriminative learning framework that estimates the parameters using a training dataset. The main challenge we face is that the training samples are not fully supervised. Specifically, they provide a hard segmentation of the images, instead of a probabilistic segmentation. We overcome this challenge by treating the optimal probabilistic segmentation that is compatible with the given hard segmentation as a latent variable. This allows us to employ the latent support vector machine formulation for parameter estimation. We show that our approach significantly outperforms the baseline methods on a challenging dataset consisting of real clinical 3D MRI volumes of skeletal muscles.

  6. Homogeneous Superpixels from Markov Random Walks

    NASA Astrophysics Data System (ADS)

    Perbet, Frank; Stenger, Björn; Maki, Atsuto

    This paper presents a novel algorithm to generate homogeneous superpixels from Markov random walks. We exploit Markov clustering (MCL) as the methodology, a generic graph clustering method based on stochastic flow circulation. In particular, we introduce a graph pruning strategy called compact pruning in order to capture intrinsic local image structure. The resulting superpixels are homogeneous, i.e. uniform in size and compact in shape. The original MCL algorithm does not scale well to a graph of an image due to the square computation of the Markov matrix which is necessary for circulating the flow. The proposed pruning scheme has the advantages of faster computation, smaller memory footprint, and straightforward parallel implementation. Through comparisons with other recent techniques, we show that the proposed algorithm achieves state-of-the-art performance.

  7. Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients

    PubMed Central

    Xia, Kelin; Zhan, Meng; Wan, Decheng; Wei, Guo-Wei

    2011-01-01

    Mesh deformation methods are a versatile strategy for solving partial differential equations (PDEs) with a vast variety of practical applications. However, these methods break down for elliptic PDEs with discontinuous coefficients, namely, elliptic interface problems. For this class of problems, the additional interface jump conditions are required to maintain the well-posedness of the governing equation. Consequently, in order to achieve high accuracy and high order convergence, additional numerical algorithms are required to enforce the interface jump conditions in solving elliptic interface problems. The present work introduces an interface technique based adaptively deformed mesh strategy for resolving elliptic interface problems. We take the advantages of the high accuracy, flexibility and robustness of the matched interface and boundary (MIB) method to construct an adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. The proposed method generates deformed meshes in the physical domain and solves the transformed governed equations in the computational domain, which maintains regular Cartesian meshes. The mesh deformation is realized by a mesh transformation PDE, which controls the mesh redistribution by a source term. The source term consists of a monitor function, which builds in mesh contraction rules. Both interface geometry based deformed meshes and solution gradient based deformed meshes are constructed to reduce the L∞ and L2 errors in solving elliptic interface problems. The proposed adaptively deformed mesh based interface method is extensively validated by many numerical experiments. Numerical results indicate that the adaptively deformed mesh based interface method outperforms the original MIB method for dealing with elliptic interface problems. PMID:22586356

  8. Numerical Study of Multigrid Methods with Various Smoothers for the Elliptical Grid Generation Equations

    NASA Technical Reports Server (NTRS)

    Golik, W. L.

    1996-01-01

    A robust solver for the elliptic grid generation equations is sought via a numerical study. The system of PDEs is discretized with finite differences, and multigrid methods are applied to the resulting nonlinear algebraic equations. Multigrid iterations are compared with respect to the robustness and efficiency. Different smoothers are tried to improve the convergence of iterations. The methods are applied to four 2D grid generation problems over a wide range of grid distortions. The results of the study help to select smoothing schemes and the overall multigrid procedures for elliptic grid generation.

  9. Random walks in cosmology: Weak lensing, the halo model, and reionization

    NASA Astrophysics Data System (ADS)

    Zhang, Jun

    This thesis discusses theoretical problems in three areas of cosmology: weak lensing, the halo model, and reionization. In weak lensing, we investigate the impact of the intrinsic alignment on the density-ellipticity correlations using the tidal torquing theory. Under the assumption of the Gaussianity of the tidal field, we find that the intrinsic alignment does not contaminate the density-ellipticity correlation even if the source clustering correlations are taken into account. The non-Gaussian contributions to both the intrinsic density-ellipticity and ellipticity- ellipticity correlations are often non-negligible. In a separate work, we discuss a useful scaling relation in weak lensing measurements. Given a foreground galaxy-density field or shear field, its cross-correlation with the shear field from a background population of source galaxies scales with the source redshift in a way that allows us to effectively measure geometrical distances as a function of redshift and thereby constrain dark energy properties without assuming anything about the galaxy-mass/mass power spectrum. Such a geometrical method can yield a ~ 0.03--0.07 [Special characters omitted.] measurement on the dark energy abundance and equation of state, for a photometric redshift accuracy of [Delta] z ~ 0.01--0.05 and a survey with median redshift of ~1. The geometrical method also provides a consistency check of the standard cosmological model because it is completely independent of structure formation. In the excursion set theory of the halo model, we derive the first-crossing distribution of random walks with a moving barrier of a general shape. Such a distribution is shown to satisfy an integral equation that can be solved by a simple matrix inversion, without the need for Monte Carlo simulations, making it useful for exploring a large parameter space. We discuss examples in which common analytic approximations fail, a failure that can be remedied using our method. In reionization, we

  10. The generalized Euler-Poinsot rigid body equations: explicit elliptic solutions

    NASA Astrophysics Data System (ADS)

    Fedorov, Yuri N.; Maciejewski, Andrzej J.; Przybylska, Maria

    2013-10-01

    The classical Euler-Poinsot case of the rigid body dynamics admits a class of simple but non-trivial integrable generalizations, which modify the Poisson equations describing the motion of the body in space. These generalizations possess first integrals which are polynomial in the angular momenta. We consider the modified Poisson equations as a system of linear equations with elliptic coefficients and show that all the solutions of it are single-valued. By using the vector generalization of the Picard theorem, we derive the solutions explicitly in terms of sigma-functions of the corresponding elliptic curve. The solutions are accompanied by a numerical example. We also compare the generalized Poisson equations with the classical third order Halphen equation.

  11. When Human Walking is a Random Walk

    NASA Astrophysics Data System (ADS)

    Hausdorff, J. M.

    1998-03-01

    The complex, hierarchical locomotor system normally does a remarkable job of controlling an inherently unstable, multi-joint system. Nevertheless, the stride interval --- the duration of a gait cycle --- fluctuates from one stride to the next, even under stationary conditions. We used random walk analysis to study the dynamical properties of these fluctuations under normal conditions and how they change with disease and aging. Random walk analysis of the stride-to-stride fluctuations of healthy, young adult men surprisingly reveals a self-similar pattern: fluctuations at one time scale are statistically similar to those at multiple other time scales (Hausdorff et al, J Appl Phsyiol, 1995). To study the stability of this fractal property, we analyzed data obtained from healthy subjects who walked for 1 hour at their usual pace, as well as at slower and faster speeds. The stride interval fluctuations exhibited long-range correlations with power-law decay for up to a thousand strides at all three walking rates. In contrast, during metronomically-paced walking, these long-range correlations disappeared; variations in the stride interval were uncorrelated and non-fractal (Hausdorff et al, J Appl Phsyiol, 1996). To gain insight into the mechanism(s) responsible for this fractal property, we examined the effects of aging and neurological impairment. Using detrended fluctuation analysis (DFA), we computed α, a measure of the degree to which one stride interval is correlated with previous and subsequent intervals over different time scales. α was significantly lower in healthy elderly subjects compared to young adults (p < .003) and in subjects with Huntington's disease, a neuro-degenerative disorder of the central nervous system, compared to disease-free controls (p < 0.005) (Hausdorff et al, J Appl Phsyiol, 1997). α was also significantly related to degree of functional impairment in subjects with Huntington's disease (r=0.78). Recently, we have observed that just as

  12. Random walks on real metro systems

    NASA Astrophysics Data System (ADS)

    Zhu, Yueying; Zhao, Longfeng; Li, Wei; Wang, Qiuping A.; Cai, Xu

    2016-04-01

    In this paper, we investigate the random walks on metro systems in 28 cities from worldwide via the Laplacian spectrum to realize the trapping process on real systems. The average trapping time is a primary description to response the trapping process. Firstly, we calculate the mean trapping time to each target station and to each entire system, respectively. Moreover, we also compare the average trapping time with the strength (the weighted degree) and average shortest path length for each station, separately. It is noted that the average trapping time has a close inverse relation with the station’s strength but rough positive correlation with the average shortest path length. And we also catch the information that the mean trapping time to each metro system approximately positively correlates with the system’s size. Finally, the trapping process on weighted and unweighted metro systems is compared to each other for better understanding the influence of weights on trapping process on metro networks. Numerical results show that the weights have no significant impact on the trapping performance on metro networks.

  13. ISWI remodels nucleosomes through a random walk.

    PubMed

    Al-Ani, Gada; Malik, Shuja Shafi; Eastlund, Allen; Briggs, Koan; Fischer, Christopher J

    2014-07-15

    The chromatin remodeler ISWI is capable of repositioning clusters of nucleosomes to create well-ordered arrays or moving single nucleosomes from the center of DNA fragments toward the ends without disrupting their integrity. Using standard electrophoresis assays, we have monitored the ISWI-catalyzed repositioning of different nucleosome samples each containing a different length of DNA symmetrically flanking the initially centrally positioned histone octamer. We find that ISWI moves the histone octamer between distinct and thermodynamically stable positions on the DNA according to a random walk mechanism. Through the application of a spectrophotometric assay for nucleosome repositioning, we further characterized the repositioning activity of ISWI using short nucleosome substrates and were able to determine the macroscopic rate of nucleosome repositioning by ISWI. Additionally, quantitative analysis of repositioning experiments performed at various ISWI concentrations revealed that a monomeric ISWI is sufficient to obtain the observed repositioning activity as the presence of a second ISWI bound had no effect on the rate of nucleosome repositioning. We also found that ATP hydrolysis is poorly coupled to nucleosome repositioning, suggesting that DNA translocation by ISWI is not energetically rate-limiting for the repositioning reaction. This is the first calculation of a microscopic ATPase coupling efficiency for nucleosome repositioning and also further supports our conclusion that a second bound ISWI does not contribute to the repositioning reaction.

  14. ISWI Remodels Nucleosomes through a Random Walk

    PubMed Central

    2015-01-01

    The chromatin remodeler ISWI is capable of repositioning clusters of nucleosomes to create well-ordered arrays or moving single nucleosomes from the center of DNA fragments toward the ends without disrupting their integrity. Using standard electrophoresis assays, we have monitored the ISWI-catalyzed repositioning of different nucleosome samples each containing a different length of DNA symmetrically flanking the initially centrally positioned histone octamer. We find that ISWI moves the histone octamer between distinct and thermodynamically stable positions on the DNA according to a random walk mechanism. Through the application of a spectrophotometric assay for nucleosome repositioning, we further characterized the repositioning activity of ISWI using short nucleosome substrates and were able to determine the macroscopic rate of nucleosome repositioning by ISWI. Additionally, quantitative analysis of repositioning experiments performed at various ISWI concentrations revealed that a monomeric ISWI is sufficient to obtain the observed repositioning activity as the presence of a second ISWI bound had no effect on the rate of nucleosome repositioning. We also found that ATP hydrolysis is poorly coupled to nucleosome repositioning, suggesting that DNA translocation by ISWI is not energetically rate-limiting for the repositioning reaction. This is the first calculation of a microscopic ATPase coupling efficiency for nucleosome repositioning and also further supports our conclusion that a second bound ISWI does not contribute to the repositioning reaction. PMID:24898619

  15. Random walks in directed modular networks

    NASA Astrophysics Data System (ADS)

    Comin, Cesar H.; Viana, Mateus P.; Antiqueira, Lucas; Costa, Luciano da F.

    2014-12-01

    Because diffusion typically involves symmetric interactions, scant attention has been focused on studying asymmetric cases. However, important networked systems underlain by diffusion (e.g. cortical networks and WWW) are inherently directed. In the case of undirected diffusion, it can be shown that the steady-state probability of the random walk dynamics is fully correlated with the degree, which no longer holds for directed networks. We investigate the relationship between such probability and the inward node degree, which we call efficiency, in modular networks. Our findings show that the efficiency of a given community depends mostly on the balance between its ingoing and outgoing connections. In addition, we derive analytical expressions to show that the internal degree of the nodes does not play a crucial role in their efficiency, when considering the Erdős-Rényi and Barabási-Albert models. The results are illustrated with respect to the macaque cortical network, providing subsidies for improving transportation and communication systems.

  16. Random walk with priorities in communicationlike networks

    NASA Astrophysics Data System (ADS)

    Bastas, Nikolaos; Maragakis, Michalis; Argyrakis, Panos; ben-Avraham, Daniel; Havlin, Shlomo; Carmi, Shai

    2013-08-01

    We study a model for a random walk of two classes of particles (A and B). Where both species are present in the same site, the motion of A's takes precedence over that of B's. The model was originally proposed and analyzed in Maragakis [Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.77.020103 77, 020103(R) (2008)]; here we provide additional results. We solve analytically the diffusion coefficients of the two species in lattices for a number of protocols. In networks, we find that the probability of a B particle to be free decreases exponentially with the node degree. In scale-free networks, this leads to localization of the B's at the hubs and arrest of their motion. To remedy this, we investigate several strategies to avoid trapping of the B's, including moving an A instead of the hindered B, allowing a trapped B to hop with a small probability, biased walk toward non-hub nodes, and limiting the capacity of nodes. We obtain analytic results for lattices and networks, and we discuss the advantages and shortcomings of the possible strategies.

  17. Random walk with chaotically driven bias

    NASA Astrophysics Data System (ADS)

    Kim, Song-Ju; Naruse, Makoto; Aono, Masashi; Hori, Hirokazu; Akimoto, Takuma

    2016-12-01

    We investigate two types of random walks with a fluctuating probability (bias) in which the random walker jumps to the right. One is a ‘time-quenched framework’ using bias time series such as periodic, quasi-periodic, and chaotic time series (chaotically driven bias). The other is a ‘time-annealed framework’ using the fluctuating bias generated by a stochastic process, which is not quenched in time. We show that the diffusive properties in the time-quenched framework can be characterised by the ensemble average of the time-averaged variance (ETVAR), whereas the ensemble average of the time-averaged mean square displacement (ETMSD) fails to capture the diffusion, even when the total bias is zero. We demonstrate that the ETVAR increases linearly with time, and the diffusion coefficient can be estimated by the time average of the local diffusion coefficient. In the time-annealed framework, we analytically and numerically show normal diffusion and superdiffusion, similar to the Lévy walk. Our findings will lead to new developments in information and communication technologies, such as efficient energy transfer for information propagation and quick solution searching.

  18. Random walk with chaotically driven bias.

    PubMed

    Kim, Song-Ju; Naruse, Makoto; Aono, Masashi; Hori, Hirokazu; Akimoto, Takuma

    2016-12-08

    We investigate two types of random walks with a fluctuating probability (bias) in which the random walker jumps to the right. One is a 'time-quenched framework' using bias time series such as periodic, quasi-periodic, and chaotic time series (chaotically driven bias). The other is a 'time-annealed framework' using the fluctuating bias generated by a stochastic process, which is not quenched in time. We show that the diffusive properties in the time-quenched framework can be characterised by the ensemble average of the time-averaged variance (ETVAR), whereas the ensemble average of the time-averaged mean square displacement (ETMSD) fails to capture the diffusion, even when the total bias is zero. We demonstrate that the ETVAR increases linearly with time, and the diffusion coefficient can be estimated by the time average of the local diffusion coefficient. In the time-annealed framework, we analytically and numerically show normal diffusion and superdiffusion, similar to the Lévy walk. Our findings will lead to new developments in information and communication technologies, such as efficient energy transfer for information propagation and quick solution searching.

  19. Random-walk model of homologous recombination

    NASA Astrophysics Data System (ADS)

    Fujitani, Youhei; Kobayashi, Ichizo

    1995-12-01

    Interaction between two homologous (i.e., identical or nearly identical) DNA sequences leads to their homologous recombination in the cell. We present the following stochastic model to explain the dependence of the frequency of homologous recombination on the length of the homologous region. The branch point connecting the two DNAs in a reaction intermediate follows the random-walk process along the homology (N base-pairs). If the branch point reaches either of the homology ends, it bounds back to the homologous region at a probability of γ (reflection coefficient) and is destroyed at a probability of 1-γ. When γ is small, the frequency of homologous recombination is found to be proportional to N3 for smaller N and a linear function of N for larger N. The exponent of the nonlinear dependence for smaller N decreases from three as γ increases. When γ=1, only the linear dependence is left. These theoretical results can explain many experimental data in various systems. (c) 1995 The American Physical Society

  20. Random walk with chaotically driven bias

    PubMed Central

    Kim, Song-Ju; Naruse, Makoto; Aono, Masashi; Hori, Hirokazu; Akimoto, Takuma

    2016-01-01

    We investigate two types of random walks with a fluctuating probability (bias) in which the random walker jumps to the right. One is a ‘time-quenched framework’ using bias time series such as periodic, quasi-periodic, and chaotic time series (chaotically driven bias). The other is a ‘time-annealed framework’ using the fluctuating bias generated by a stochastic process, which is not quenched in time. We show that the diffusive properties in the time-quenched framework can be characterised by the ensemble average of the time-averaged variance (ETVAR), whereas the ensemble average of the time-averaged mean square displacement (ETMSD) fails to capture the diffusion, even when the total bias is zero. We demonstrate that the ETVAR increases linearly with time, and the diffusion coefficient can be estimated by the time average of the local diffusion coefficient. In the time-annealed framework, we analytically and numerically show normal diffusion and superdiffusion, similar to the Lévy walk. Our findings will lead to new developments in information and communication technologies, such as efficient energy transfer for information propagation and quick solution searching. PMID:27929091

  1. Lp gradient estimate for elliptic equations with high-contrast conductivities in Rn

    NASA Astrophysics Data System (ADS)

    Yeh, Li-Ming

    2016-07-01

    Uniform estimate for the solutions of elliptic equations with high-contrast conductivities in Rn is concerned. The space domain consists of a periodic connected sub-region and a periodic disconnected matrix block subset. The elliptic equations have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Suppose ɛ ∈ (0 , 1 ] is the diameter of each matrix block and ω2 ∈ (0 , 1 ] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W 1 , p norm of the elliptic solutions in the connected sub-region is bounded uniformly in ɛ, ω; when ɛ ≤ ω, the Lp norm of the elliptic solutions in the whole space is bounded uniformly in ɛ, ω; the W 1 , p norm of the elliptic solutions in perforated domains is bounded uniformly in ɛ. However, the Lp norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in ɛ, ω.

  2. Anomalous transport in turbulent plasmas and continuous time random walks

    SciTech Connect

    Balescu, R.

    1995-05-01

    The possibility of a model of anomalous transport problems in a turbulent plasma by a purely stochastic process is investigated. The theory of continuous time random walks (CTRW`s) is briefly reviewed. It is shown that a particular class, called the standard long tail CTRW`s is of special interest for the description of subdiffusive transport. Its evolution is described by a non-Markovian diffusion equation that is constructed in such a way as to yield exact values for all the moments of the density profile. The concept of a CTRW model is compared to an exact solution of a simple test problem: transport of charged particles in a fluctuating magnetic field in the limit of infinite perpendicular correlation length. Although the well-known behavior of the mean square displacement proportional to {ital t}{sup 1/2} is easily recovered, the exact density profile cannot be modeled by a CTRW. However, the quasilinear approximation of the kinetic equation has the form of a non-Markovian diffusion equation and can thus be generated by a CTRW.

  3. Stochastic calculus for uncoupled continuous-time random walks

    NASA Astrophysics Data System (ADS)

    Germano, Guido; Politi, Mauro; Scalas, Enrico; Schilling, René L.

    2009-06-01

    The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy α -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.

  4. Stochastic calculus for uncoupled continuous-time random walks.

    PubMed

    Germano, Guido; Politi, Mauro; Scalas, Enrico; Schilling, René L

    2009-06-01

    The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.

  5. On the solution of elliptic partial differential equations on regions with corners

    NASA Astrophysics Data System (ADS)

    Serkh, Kirill; Rokhlin, Vladimir

    2016-01-01

    In this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. We observe that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

  6. On the solution of elliptic partial differential equations on regions with corners

    SciTech Connect

    Serkh, Kirill Rokhlin, Vladimir

    2016-01-15

    In this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. We observe that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

  7. Multibump solutions for quasilinear elliptic equations with critical growth

    SciTech Connect

    Liu, Jiaquan; Wang, Zhi-Qiang; Wu, Xian

    2013-12-15

    The current paper is concerned with constructing multibump solutions for a class of quasilinear Schrödinger equations with critical growth. This extends the classical results of Coti Zelati and Rabinowitz [Commun. Pure Appl. Math. 45, 1217–1269 (1992)] for semilinear equations as well as recent work of Liu, Wang, and Guo [J. Funct. Anal. 262, 4040–4102 (2012)] for quasilinear problems with subcritical growth. The periodicity of the potentials is used to glue ground state solutions to construct multibump bound state solutions.

  8. Phenomenological picture of fluctuations in branching random walks.

    PubMed

    Mueller, A H; Munier, S

    2014-10-01

    We propose a picture of the fluctuations in branching random walks, which leads to predictions for the distribution of a random variable that characterizes the position of the bulk of the particles. We also interpret the 1/sqrt[t] correction to the average position of the rightmost particle of a branching random walk for large times t≫1, computed by Ebert and Van Saarloos, as fluctuations on top of the mean-field approximation of this process with a Brunet-Derrida cutoff at the tip that simulates discreteness. Our analytical formulas successfully compare to numerical simulations of a particular model of a branching random walk.

  9. Phenomenological picture of fluctuations in branching random walks

    NASA Astrophysics Data System (ADS)

    Mueller, A. H.; Munier, S.

    2014-10-01

    We propose a picture of the fluctuations in branching random walks, which leads to predictions for the distribution of a random variable that characterizes the position of the bulk of the particles. We also interpret the 1 /√{t } correction to the average position of the rightmost particle of a branching random walk for large times t ≫1 , computed by Ebert and Van Saarloos, as fluctuations on top of the mean-field approximation of this process with a Brunet-Derrida cutoff at the tip that simulates discreteness. Our analytical formulas successfully compare to numerical simulations of a particular model of a branching random walk.

  10. The RANLUX Generator:. Resonances in a Random Walk Test

    NASA Astrophysics Data System (ADS)

    Shchur, Lev N.; Butera, Paolo

    Using a recently proposed directed random walk test, we systematically investigate the popular random number generator RANLUX developed by Lüscher and implemented by James. We confirm the good quality of this generator with the recommended luxury level. At a smaller luxury level (for instance equal to 1) resonances are observed in the random walk test. We also find that the lagged Fibonacci and Subtract-with-Carry recipes exhibit similar failures in the random walk test. A revised analysis of the corresponding dynamical systems leads to the observation of resonances in the eigenvalues of Jacobi matrix.

  11. Crossover from random walk to self-avoiding walk

    NASA Astrophysics Data System (ADS)

    Rieger, Jens

    1988-11-01

    A one-dimensional n-step random walk on openZ1 which must not visit a vertex more than k times is studied via Monte Carlo methods. The dependences of the mean-square end-to-end distance of the walk and of the fraction of trapped walks on λ=(k-1)/n will be given for the range from λ=0 (self-avoiding walk) to λ=1 (unrestricted random walk). From the results it is conjectured that in the limit n-->∞ the walk obeys simple random walk statistics with respect to its static properties for all λ>0.

  12. Biased random walks on Kleinberg's spatial networks

    NASA Astrophysics Data System (ADS)

    Pan, Gui-Jun; Niu, Rui-Wu

    2016-12-01

    We investigate the problem of the particle or message that travels as a biased random walk toward a target node in Kleinberg's spatial network which is built from a d-dimensional (d = 2) regular lattice improved by adding long-range shortcuts with probability P(rij) ∼rij-α, where rij is the lattice distance between sites i and j, and α is a variable exponent. Bias is represented as a probability p of the packet to travel at every hop toward the node which has the smallest Manhattan distance to the target node. We study the mean first passage time (MFPT) for different exponent α and the scaling of the MFPT with the size of the network L. We find that there exists a threshold probability pth ≈ 0.5, for p ≥pth the optimal transportation condition is obtained with an optimal transport exponent αop = d, while for 0 < p pth, and increases with L less than a power law and get close to logarithmical law for 0 < p

  13. FISHPACK90: Efficient FORTRAN Subprograms for the Solution of Separable Elliptic Partial Differential Equations

    NASA Astrophysics Data System (ADS)

    Adams, John C.; Swarztrauber, Paul N.; Sweet, Roland

    2016-09-01

    FISHPACK90 is a modernization of the original FISHPACK (ascl:1609.004), employing Fortran90 to slightly simplify and standardize the interface to some of the routines. This collection of Fortran programs and subroutines solves second- and fourth-order finite difference approximations to separable elliptic Partial Differential Equations (PDEs). These include Helmholtz equations in cartesian, polar, cylindrical, and spherical coordinates, as well as more general separable elliptic equations. The solvers use the cyclic reduction algorithm. When the problem is singular, a least-squares solution is computed. Singularities induced by the coordinate system are handled, including at the origin r=0 in cylindrical coordinates, and at the poles in spherical coordinates. Test programs are provided for the 19 solvers. Each serves two purposes: as a template to guide you in writing your own codes utilizing the FISHPACK90 solvers, and as a demonstration on your computer that you can correctly produce FISHPACK90 executables.

  14. A scaling law for random walks on networks

    PubMed Central

    Perkins, Theodore J.; Foxall, Eric; Glass, Leon; Edwards, Roderick

    2014-01-01

    The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics. PMID:25311870

  15. FRACTAL DIMENSION RESULTS FOR CONTINUOUS TIME RANDOM WALKS

    PubMed Central

    Meerschaert, Mark M.; Nane, Erkan; Xiao, Yimin

    2013-01-01

    Continuous time random walks impose random waiting times between particle jumps. This paper computes the fractal dimensions of their process limits, which represent particle traces in anomalous diffusion. PMID:23482421

  16. Record statistics of financial time series and geometric random walks

    NASA Astrophysics Data System (ADS)

    Sabir, Behlool; Santhanam, M. S.

    2014-09-01

    The study of record statistics of correlated series in physics, such as random walks, is gaining momentum, and several analytical results have been obtained in the past few years. In this work, we study the record statistics of correlated empirical data for which random walk models have relevance. We obtain results for the records statistics of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent α lying in the range 1.5≤α≤1.8. Further, the longest record ages follow the Fréchet distribution of extreme value theory. The records statistics of geometric random walk series is in good agreement with that obtained from empirical stock data.

  17. Record statistics of financial time series and geometric random walks.

    PubMed

    Sabir, Behlool; Santhanam, M S

    2014-09-01

    The study of record statistics of correlated series in physics, such as random walks, is gaining momentum, and several analytical results have been obtained in the past few years. In this work, we study the record statistics of correlated empirical data for which random walk models have relevance. We obtain results for the records statistics of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent α lying in the range 1.5≤α≤1.8. Further, the longest record ages follow the Fréchet distribution of extreme value theory. The records statistics of geometric random walk series is in good agreement with that obtained from empirical stock data.

  18. A scaling law for random walks on networks

    NASA Astrophysics Data System (ADS)

    Perkins, Theodore J.; Foxall, Eric; Glass, Leon; Edwards, Roderick

    2014-10-01

    The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.

  19. Ellipticity and the spurious solution problem of k•p envelope equations

    NASA Astrophysics Data System (ADS)

    Veprek, Ratko G.; Steiger, Sebastian; Witzigmann, Bernd

    2007-10-01

    We present an explanation to the spurious solution problem affecting the k•p envelope function method, indicating that the problem is mathematically caused by loss of ellipticity of the differential operator. Focusing on direct band gap zinc-blende heterostructures, we derive criteria that must be fulfilled by the input parameters in order to establish ellipticity. Using these criteria, we compare symmetrized operators with Burt-Foreman [B. A. Foreman, Phys. Rev. B 56, R12748 (1997)] operator ordering. We substantiate our arguments with numerical results obtained using linear finite elements. We find that the space of stable input parameters is very narrow and demonstrate that Burt-Foreman operator ordering together with experimental k•p input parameters leads to near-elliptic envelope equations in the 4×4 and 6×6 models, whereas symmetrization yields strong nonellipticity. In the k•p 8×8 model, the procedure of renormalizing parameters of the 4×4 model generally yields parameters producing spurious solutions, even for Burt-Foreman operator ordering. We find that this problem can be resolved by using a smaller optical matrix parameter Ep . This suggests that the parametrization of k•p models for heterostructures of any dimensionality must be reviewed, checking against the mathematical ellipticity of the equation system.

  20. The melting phenomenon in random-walk model of DNA

    SciTech Connect

    Hayrapetyan, G. N.; Mamasakhlisov, E. Sh.; Papoyan, Vl. V.; Poghosyan, S. S.

    2012-10-15

    The melting phenomenon in a double-stranded homopolypeptide is considered. The relative distance between the corresponding monomers of two polymer chains is modeled by the two-dimensional random walk on the square lattice. Returns of the random walk to the origin describe the formation of hydrogen bonds between complementary units. To take into account the two competing interactions of monomers inside the chains, we obtain a completely denatured state at finite temperature T{sub c}.

  1. Simple model of a random walk with arbitrarily long memory

    NASA Astrophysics Data System (ADS)

    Berrones, Arturo; Larralde, Hernán

    2001-03-01

    We present a generalization of the persistent random-walk model in which the step at time n depends on the state of the step at time n-T, for arbitrary T. This gives rise to arbitrarily long memory effects, yet by an appropriate transformation the model is tractable by essentially the same techniques applicable to the usual persistent random-walk problem. We apply our results to the specific case of delayed ``step'' persistence, and analyze its asymptotic statistical properties.

  2. Continuous Dependence on Modeling in the Cauchy Problem for Nonlinear Elliptic Equations.

    DTIC Science & Technology

    1987-04-01

    parameter 4. AMON INTRODUCTION A problem in ordinary or partial differential equations is said to properly posed if it has a unique solution in the...problem for second-order nonlinear partial differential equations , Doctoral thesis, Cornell University, Ithaca, N.Y., 1986. [6] J. Conlan and G. N. Trytten...IModeling in the Cauchy Problem for Nonlinear Elliptic Equations by Allan Bennett DT1C A z1t17n m (It C ltd n Inttt " CENTER.FOR.NAVAL.ANALYSFS 4401

  3. Lévy random walks on multiplex networks

    PubMed Central

    Guo, Quantong; Cozzo, Emanuele; Zheng, Zhiming; Moreno, Yamir

    2016-01-01

    Random walks constitute a fundamental mechanism for many dynamics taking place on complex networks. Besides, as a more realistic description of our society, multiplex networks have been receiving a growing interest, as well as the dynamical processes that occur on top of them. Here, inspired by one specific model of random walks that seems to be ubiquitous across many scientific fields, the Lévy flight, we study a new navigation strategy on top of multiplex networks. Capitalizing on spectral graph and stochastic matrix theories, we derive analytical expressions for the mean first passage time and the average time to reach a node on these networks. Moreover, we also explore the efficiency of Lévy random walks, which we found to be very different as compared to the single layer scenario, accounting for the structure and dynamics inherent to the multiplex network. Finally, by comparing with some other important random walk processes defined on multiplex networks, we find that in some region of the parameters, a Lévy random walk is the most efficient strategy. Our results give us a deeper understanding of Lévy random walks and show the importance of considering the topological structure of multiplex networks when trying to find efficient navigation strategies. PMID:27892508

  4. Lévy random walks on multiplex networks

    NASA Astrophysics Data System (ADS)

    Guo, Quantong; Cozzo, Emanuele; Zheng, Zhiming; Moreno, Yamir

    2016-11-01

    Random walks constitute a fundamental mechanism for many dynamics taking place on complex networks. Besides, as a more realistic description of our society, multiplex networks have been receiving a growing interest, as well as the dynamical processes that occur on top of them. Here, inspired by one specific model of random walks that seems to be ubiquitous across many scientific fields, the Lévy flight, we study a new navigation strategy on top of multiplex networks. Capitalizing on spectral graph and stochastic matrix theories, we derive analytical expressions for the mean first passage time and the average time to reach a node on these networks. Moreover, we also explore the efficiency of Lévy random walks, which we found to be very different as compared to the single layer scenario, accounting for the structure and dynamics inherent to the multiplex network. Finally, by comparing with some other important random walk processes defined on multiplex networks, we find that in some region of the parameters, a Lévy random walk is the most efficient strategy. Our results give us a deeper understanding of Lévy random walks and show the importance of considering the topological structure of multiplex networks when trying to find efficient navigation strategies.

  5. Scaling random walks on arbitrary sets

    NASA Astrophysics Data System (ADS)

    Harris, Simon C.; Williams, David; Sibson, Robin

    1999-01-01

    Let I be a countably infinite set of points in [open face R] which we can write as I={ui: i[set membership][open face Z]}, with uirandom-walk, when repeatedly rescaled suitably in space and time, looks more and more like a Brownian motion. In this paper we explore the convergence properties of the Markov chain Y on the set I under suitable space-time scalings. Later, we consider some cases when the set I consists of the points of a renewal process and the jump rates assigned to each state in I are perhaps also randomly chosen.This work sprang from a question asked by one of us (Sibson) about ‘driftless nearest-neighbour’ Markov chains on countable subsets I of [open face R]d, work of Sibson [7] and of Christ, Friedberg and Lee [2] having identified examples of such chains in terms of the Dirichlet tessellation associated with I. Amongst methods which can be brought to bear on this d-dimensional problem is the theory of Dirichlet forms. There are potential problems in doing this because we wish I to be random (for example, a realization of a Poisson point process), we do not wish to impose artificial boundedness conditions which would clearly make things work for certain deterministic sets I. In the 1-dimensional case discussed here and in the following paper by Harris, much simpler techniques (where we embed the Markov chain in a Brownian motion using local time) work very effectively; and it is these, rather than the theory of Dirichlet forms, that we use.

  6. Continuous time random walk with linear force applied to hydrated proteins

    NASA Astrophysics Data System (ADS)

    Fa, Kwok Sau

    2013-08-01

    An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations. Analytical expressions for transition probability density, mean square displacement, and intermediate scattering function are presented. The mean square displacement and intermediate scattering function can fit well the simulation data of the temperature-dependent translational dynamics of nitrogen atoms of elastin for a wide range of temperatures and various scattering vectors. Moreover, the numerical results are also compared with those of a fractional diffusion equation.

  7. Continuous time random walk with linear force applied to hydrated proteins.

    PubMed

    Fa, Kwok Sau

    2013-08-14

    An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations. Analytical expressions for transition probability density, mean square displacement, and intermediate scattering function are presented. The mean square displacement and intermediate scattering function can fit well the simulation data of the temperature-dependent translational dynamics of nitrogen atoms of elastin for a wide range of temperatures and various scattering vectors. Moreover, the numerical results are also compared with those of a fractional diffusion equation.

  8. A new form of the elliptic relaxation equation to account for wall effects in RANS modeling

    NASA Astrophysics Data System (ADS)

    Manceau, Rémi; Hanjalić, Kemal

    2000-09-01

    Different methods for improving the behavior in the logarithmic layer of the elliptic relaxation equation, which enable the extension of Reynolds stress models or eddy viscosity models down to the wall, are tested in a channel flow at Reτ=590 and compared with direct numerical simulation (DNS) data. First, a priori tests are performed in order to confirm the improvement predicted by the theory, either with the Rotta+IP (isotropization of production) model or the Speziale-Sarkar-Gatski (SSG) model as the source term of the elliptic relaxation equation. The best form of the model is then used for full simulations, in Durbin second moment closure or in the frame of the v2¯-f model. It is shown that the results can be significantly improved, in particular by using a formulation based on the refinement of the modeling of the two-point correlations involved in the redistribution term.

  9. Elliptical vortex solutions, integrable Ermakov structure, and Lax pair formulation of the compressible Euler equations.

    PubMed

    An, Hongli; Fan, Engui; Zhu, Haixing

    2015-01-01

    The 2+1-dimensional compressible Euler equations are investigated here. A power-type elliptic vortex ansatz is introduced and thereby reduction obtains to an eight-dimensional nonlinear dynamical system. The latter is shown to have an underlying integral Ermakov-Ray-Reid structure of Hamiltonian type. It is of interest to notice that such an integrable Ermakov structure exists not only in the density representations but also in the velocity components. A class of typical elliptical vortex solutions termed pulsrodons corresponding to warm-core eddy theory is isolated and its behavior is simulated. In addition, a Lax pair formulation is constructed and the connection with stationary nonlinear cubic Schrödinger equations is established.

  10. On removability of singularities on manifolds for solutions of non-linear elliptic equations

    SciTech Connect

    Skrypnik, I I

    2003-10-31

    A precise condition is found for the removability of a singularity on a smooth manifold for solutions of non-linear second-order elliptic equations of divergence form. The condition is stated in the form of a dependence of the pointwise behaviour of the solution on the distance to the singular manifold. The condition obtained is weaker than Serrin's well-known sufficient condition for the removability of a singularity on a manifold.

  11. A GPU parallelized spectral method for elliptic equations in rectangular domains

    NASA Astrophysics Data System (ADS)

    Chen, Feng; Shen, Jie

    2013-10-01

    We design and implement a polynomial-based spectral method on graphic processing units (GPUs). The key to success lies in the seamless integration of the matrix diagonalization technique and the new generation CUDA tools. The method is applicable to elliptic equations in rectangular domains with general boundary condition. We show remarkable speedups of up to 15 times in the 2-D case and more than 35 times in the 3-D case.

  12. Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation.

    PubMed

    Wu, Rengmao; Xu, Liang; Liu, Peng; Zhang, Yaqin; Zheng, Zhenrong; Li, Haifeng; Liu, Xu

    2013-01-15

    We propose an approach to deal with the problem of freeform surface illumination design without assuming any symmetry based on the concept that this problem is similar to the problem of optimal mass transport. With this approach, the freeform design is converted into a nonlinear boundary problem for the elliptic Monge-Ampére equation. The theory and numerical method are given for solving this boundary problem. Experimental results show the feasibility of this approach in tackling this freeform design problem.

  13. Superposition of elliptic functions as solutions for a large number of nonlinear equations

    NASA Astrophysics Data System (ADS)

    Khare, Avinash; Saxena, Avadh

    2014-03-01

    For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ4, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn2(x, m), it also admits solutions in terms of dn^2(x,m) ± sqrt{m} cn(x,m) dn(x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.

  14. Superposition of elliptic functions as solutions for a large number of nonlinear equations

    SciTech Connect

    Khare, Avinash; Saxena, Avadh

    2014-03-15

    For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ{sup 4}, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn{sup 2}(x, m), it also admits solutions in terms of dn {sup 2}(x,m)±√(m) cn (x,m) dn (x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.

  15. Numerical study of hydrogen-air supersonic combustion by using elliptic and parabolized equations

    NASA Technical Reports Server (NTRS)

    Chitsomboon, T.; Tiwari, S. N.

    1986-01-01

    The two-dimensional Navier-Stokes and species continuity equations are used to investigate supersonic chemically reacting flow problems which are related to scramjet-engine configurations. A global two-step finite-rate chemistry model is employed to represent the hydrogen-air combustion in the flow. An algebraic turbulent model is adopted for turbulent flow calculations. The explicit unsplit MacCormack finite-difference algorithm is used to develop a computer program suitable for a vector processing computer. The computer program developed is then used to integrate the system of the governing equations in time until convergence is attained. The chemistry source terms in the species continuity equations are evaluated implicitly to alleviate stiffness associated with fast chemical reactions. The problems solved by the elliptic code are re-investigated by using a set of two-dimensional parabolized Navier-Stokes and species equations. A linearized fully-coupled fully-implicit finite difference algorithm is used to develop a second computer code which solves the governing equations by marching in spce rather than time, resulting in a considerable saving in computer resources. Results obtained by using the parabolized formulation are compared with the results obtained by using the fully-elliptic equations. The comparisons indicate fairly good agreement of the results of the two formulations.

  16. All-time dynamics of continuous-time random walks on complex networks

    NASA Astrophysics Data System (ADS)

    Teimouri, Hamid; Kolomeisky, Anatoly B.

    2013-02-01

    The concept of continuous-time random walks (CTRW) is a generalization of ordinary random walk models, and it is a powerful tool for investigating a broad spectrum of phenomena in natural, engineering, social, and economic sciences. Recently, several theoretical approaches have been developed that allowed to analyze explicitly dynamics of CTRW at all times, which is critically important for understanding mechanisms of underlying phenomena. However, theoretical analysis has been done mostly for systems with a simple geometry. Here we extend the original method based on generalized master equations to analyze all-time dynamics of CTRW models on complex networks. Specific calculations are performed for models on lattices with branches and for models on coupled parallel-chain lattices. Exact expressions for velocities and dispersions are obtained. Generalized fluctuations theorems for CTRW models on complex networks are discussed.

  17. Application of multiquadric method for numerical solution of elliptic partial differential equations

    SciTech Connect

    Sharan, M.; Kansa, E.J.; Gupta, S.

    1994-01-01

    We have used the multiquadric (MQ) approximation scheme for the solution of elliptic partial differential equations with Dirichlet and/or Neumann boundary conditions. The scheme has the advantage to use the data points in arbitrary locations with an arbitrary ordering. Two dimensional Laplace, Poisson and Biharmonic equations describing the various physical processes, have been taken as the test examples. The agreement is found to be very good between the computed and exact solutions. The method also provides an excellent approximation with curve boundary.

  18. A boundary value approach for solving three-dimensional elliptic and hyperbolic partial differential equations.

    PubMed

    Biala, T A; Jator, S N

    2015-01-01

    In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.

  19. A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations

    SciTech Connect

    Mazalov, M Ya

    2008-02-28

    Let X be an arbitrary compact subset of the plane. It is proved that if L is a homogeneous elliptic operator with constant coefficients and locally bounded fundamental solution, then each function f that is continuous on X and satisfies the equation Lf = 0 at all interior points of X can be uniformly approximated on X by solutions of the same equation having singularities outside X. A theorem on uniform piecemeal approximation of a function is also established under weaker constraints than in the standard Vitushkin scheme. Bibliography: 24 titles.

  20. Existence of Large Solutions to Semilinear Elliptic Equations with Multiple Terms

    DTIC Science & Technology

    2006-09-01

    the solutions in bounded domains in Rn was studied by Lazer and McKenna [14]. Bandle and Marcus [3] showed that 4u = g(x, u) has a unique large positive...Wood (Shaker). "Large solutions of sublinear elliptic equations," Nonlinear Analysis, 39: 745-753 (2000). 14. Lazer , A.C. and P.J. McKenna. "Asymptotic...behavior of solutions of boundary blow up problems," Differential Integral Equations, 7:1001-1020 (1994). 15. Lazer , A.C. and P.J. McKenna. "On a

  1. Approximation of the Lévy Feller advection dispersion process by random walk and finite difference method

    NASA Astrophysics Data System (ADS)

    Liu, Q.; Liu, F.; Turner, I.; Anh, V.

    2007-03-01

    In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion process, governed by the Lévy-Feller advection-dispersion differential equation (LFADE). We show that the random walk model converges to LFADE by use of a properly scaled transition to vanishing space and time steps. We propose an explicit finite difference approximation (EFDA) for LFADE, resulting from the Grünwald-Letnikov discretization of fractional derivatives. As a result of the interpretation of the random walk model, the stability and convergence of EFDA for LFADE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

  2. Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

    NASA Astrophysics Data System (ADS)

    Feng, Wenqiang; Salgado, Abner J.; Wang, Cheng; Wise, Steven M.

    2017-04-01

    We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems - including thin film epitaxy with slope selection and the square phase field crystal model - are carried out to verify the efficiency of the scheme.

  3. Current-reinforced random walks for constructing transport networks

    PubMed Central

    Ma, Qi; Johansson, Anders; Tero, Atsushi; Nakagaki, Toshiyuki; Sumpter, David J. T.

    2013-01-01

    Biological systems that build transport networks, such as trail-laying ants and the slime mould Physarum, can be described in terms of reinforced random walks. In a reinforced random walk, the route taken by ‘walking’ particles depends on the previous routes of other particles. Here, we present a novel form of random walk in which the flow of particles provides this reinforcement. Starting from an analogy between electrical networks and random walks, we show how to include current reinforcement. We demonstrate that current-reinforcement results in particles converging on the optimal solution of shortest path transport problems, and avoids the self-reinforcing loops seen in standard density-based reinforcement models. We further develop a variant of the model that is biologically realistic, in the sense that the particles can be identified as ants and their measured density corresponds to those observed in maze-solving experiments on Argentine ants. For network formation, we identify the importance of nonlinear current reinforcement in producing networks that optimize both network maintenance and travel times. Other than ant trail formation, these random walks are also closely related to other biological systems, such as blood vessels and neuronal networks, which involve the transport of materials or information. We argue that current reinforcement is likely to be a common mechanism in a range of systems where network construction is observed. PMID:23269849

  4. A New Random Walk for Replica Detection in WSNs

    PubMed Central

    Aalsalem, Mohammed Y.; Saad, N. M.; Hossain, Md. Shohrab; Atiquzzaman, Mohammed; Khan, Muhammad Khurram

    2016-01-01

    Wireless Sensor Networks (WSNs) are vulnerable to Node Replication attacks or Clone attacks. Among all the existing clone detection protocols in WSNs, RAWL shows the most promising results by employing Simple Random Walk (SRW). More recently, RAND outperforms RAWL by incorporating Network Division with SRW. Both RAND and RAWL have used SRW for random selection of witness nodes which is problematic because of frequently revisiting the previously passed nodes that leads to longer delays, high expenditures of energy with lower probability that witness nodes intersect. To circumvent this problem, we propose to employ a new kind of constrained random walk, namely Single Stage Memory Random Walk and present a distributed technique called SSRWND (Single Stage Memory Random Walk with Network Division). In SSRWND, single stage memory random walk is combined with network division aiming to decrease the communication and memory costs while keeping the detection probability higher. Through intensive simulations it is verified that SSRWND guarantees higher witness node security with moderate communication and memory overheads. SSRWND is expedient for security oriented application fields of WSNs like military and medical. PMID:27409082

  5. The anisotropic Ising correlations as elliptic integrals: duality and differential equations

    NASA Astrophysics Data System (ADS)

    McCoy, B. M.; Maillard, J.-M.

    2016-10-01

    We present the reduction of the correlation functions of the Ising model on the anisotropic square lattice to complete elliptic integrals of the first, second and third kind, the extension of Kramers-Wannier duality to anisotropic correlation functions, and the linear differential equations for these anisotropic correlations. More precisely, we show that the anisotropic correlation functions are homogeneous polynomials of the complete elliptic integrals of the first, second and third kind. We give the exact dual transformation matching the correlation functions and the dual correlation functions. We show that the linear differential operators annihilating the general two-point correlation functions are factorized in a very simple way, in operators of decreasing orders. Dedicated to A J Guttmann, for his 70th birthday.

  6. Some Minorants and Majorants of Random Walks and Levy Processes

    NASA Astrophysics Data System (ADS)

    Abramson, Joshua Simon

    This thesis consists of four chapters, all relating to some sort of minorant or majorant of random walks or Levy processes. In Chapter 1 we provide an overview of recent work on descriptions and properties of the convex minorant of random walks and Levy processes as detailed in Chapter 2, [72] and [73]. This work rejuvenated the field of minorants, and led to the work in all the subsequent chapters. The results surveyed include point process descriptions of the convex minorant of random walks and Levy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander. This chapter is based on [3], which was co-written with Jim Pitman, Nathan Ross and Geronimo Uribe Bravo. Chapter 1 serves as a long introduction to Chapter 2, in which we offer a unified approach to the theory of concave majorants of random walks. The reasons for the switch from convex minorants to concave majorants are discussed in Section 1.1, but the results are all equivalent. This unified theory is arrived at by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant - the path transformation is different from the one discussed in Chapter 1, but this is necessary to deal with a more general case than the standard one as done in Section 2.6. The path transformation of Chapter 1, which is discussed in detail in Section 2.8, is more relevant to the limiting results for Levy processes that are of interest in Chapter 1. Our results lead to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave

  7. Some physical consequences of a random walk in velocity space

    NASA Astrophysics Data System (ADS)

    Herzenberg, Caroline

    2012-03-01

    A simple conceptual model of stochastic behavior based on a random walk process in velocity space is examined. For objects moving at non-relativistic velocities, this leads under asymmetric directional probabilities to acceleration processes that resemble the behavior of objects subject to Newton's second law. For three dimensional space, inverse square law acceleration emerges for sufficiently separated objects. In modeling classical behavior, such non-relativistic random walks would appear to be limited to objects of sufficiently large mass. Objects with smaller mass exhibit more rapid diffusion and less localization, and a relativistic random walk would seem to be required for objects having masses smaller than a threshold mass value. Results suggest that the threshold mass value must be similar in magnitude to the Planck mass, which leads to behavior somewhat comparable to that characterizing an intrinsic quantum classical transition in the microgram mass range.

  8. Superstatistical analysis and modelling of heterogeneous random walks.

    PubMed

    Metzner, Claus; Mark, Christoph; Steinwachs, Julian; Lautscham, Lena; Stadler, Franz; Fabry, Ben

    2015-06-25

    Stochastic time series are ubiquitous in nature. In particular, random walks with time-varying statistical properties are found in many scientific disciplines. Here we present a superstatistical approach to analyse and model such heterogeneous random walks. The time-dependent statistical parameters can be extracted from measured random walk trajectories with a Bayesian method of sequential inference. The distributions and correlations of these parameters reveal subtle features of the random process that are not captured by conventional measures, such as the mean-squared displacement or the step width distribution. We apply our new approach to migration trajectories of tumour cells in two and three dimensions, and demonstrate the superior ability of the superstatistical method to discriminate cell migration strategies in different environments. Finally, we show how the resulting insights can be used to design simple and meaningful models of the underlying random processes.

  9. Random walk of passive tracers among randomly moving obstacles.

    PubMed

    Gori, Matteo; Donato, Irene; Floriani, Elena; Nardecchia, Ilaria; Pettini, Marco

    2016-04-14

    This study is mainly motivated by the need of understanding how the diffusion behavior of a biomolecule (or even of a larger object) is affected by other moving macromolecules, organelles, and so on, inside a living cell, whence the possibility of understanding whether or not a randomly walking biomolecule is also subject to a long-range force field driving it to its target. By means of the Continuous Time Random Walk (CTRW) technique the topic of random walk in random environment is here considered in the case of a passively diffusing particle among randomly moving and interacting obstacles. The relevant physical quantity which is worked out is the diffusion coefficient of the passive tracer which is computed as a function of the average inter-obstacles distance. The results reported here suggest that if a biomolecule, let us call it a test molecule, moves towards its target in the presence of other independently interacting molecules, its motion can be considerably slowed down.

  10. Superstatistical analysis and modelling of heterogeneous random walks

    NASA Astrophysics Data System (ADS)

    Metzner, Claus; Mark, Christoph; Steinwachs, Julian; Lautscham, Lena; Stadler, Franz; Fabry, Ben

    2015-06-01

    Stochastic time series are ubiquitous in nature. In particular, random walks with time-varying statistical properties are found in many scientific disciplines. Here we present a superstatistical approach to analyse and model such heterogeneous random walks. The time-dependent statistical parameters can be extracted from measured random walk trajectories with a Bayesian method of sequential inference. The distributions and correlations of these parameters reveal subtle features of the random process that are not captured by conventional measures, such as the mean-squared displacement or the step width distribution. We apply our new approach to migration trajectories of tumour cells in two and three dimensions, and demonstrate the superior ability of the superstatistical method to discriminate cell migration strategies in different environments. Finally, we show how the resulting insights can be used to design simple and meaningful models of the underlying random processes.

  11. Superstatistical analysis and modelling of heterogeneous random walks

    PubMed Central

    Metzner, Claus; Mark, Christoph; Steinwachs, Julian; Lautscham, Lena; Stadler, Franz; Fabry, Ben

    2015-01-01

    Stochastic time series are ubiquitous in nature. In particular, random walks with time-varying statistical properties are found in many scientific disciplines. Here we present a superstatistical approach to analyse and model such heterogeneous random walks. The time-dependent statistical parameters can be extracted from measured random walk trajectories with a Bayesian method of sequential inference. The distributions and correlations of these parameters reveal subtle features of the random process that are not captured by conventional measures, such as the mean-squared displacement or the step width distribution. We apply our new approach to migration trajectories of tumour cells in two and three dimensions, and demonstrate the superior ability of the superstatistical method to discriminate cell migration strategies in different environments. Finally, we show how the resulting insights can be used to design simple and meaningful models of the underlying random processes. PMID:26108639

  12. MODEL OF THE FIELD LINE RANDOM WALK EVOLUTION AND APPROACH TO ASYMPTOTIC DIFFUSION IN MAGNETIC TURBULENCE

    SciTech Connect

    Snodin, A. P.; Ruffolo, D.; Matthaeus, W. H. E-mail: david.ruf@mahidol.ac.th

    2013-01-01

    The turbulent random walk of magnetic field lines plays an important role in the transport of plasmas and energetic particles in a wide variety of astrophysical situations, but most theoretical work has concentrated on determination of the asymptotic field line diffusion coefficient. Here we consider the evolution with distance of the field line random walk using a general ordinary differential equation (ODE), which for most cases of interest in astrophysics describes a transition from free streaming to asymptotic diffusion. By challenging theories of asymptotic diffusion to also describe the evolution, one gains insight on how accurately they describe the random walk process. Previous theoretical work has effectively involved closure of the ODE, often by assuming Corrsin's hypothesis and a Gaussian displacement distribution. Approaches that use quasilinear theory and prescribe the mean squared displacement ({Delta}x {sup 2}) according to free streaming (random ballistic decorrelation, RBD) or asymptotic diffusion (diffusive decorrelation, DD) can match computer simulation results, but only over specific parameter ranges, with no obvious 'marker' of the range of validity. Here we make use of a unified description in which the ODE determines ({Delta}x {sup 2}) self-consistently, providing a natural transition between the assumptions of RBD and DD. We find that the minimum kurtosis of the displacement distribution provides a good indicator of whether the self-consistent ODE is applicable, i.e., inaccuracy of the self-consistent ODE is associated with non-Gaussian displacement distributions.

  13. Feature Learning Based Random Walk for Liver Segmentation

    PubMed Central

    Zheng, Yongchang; Ai, Danni; Zhang, Pan; Gao, Yefei; Xia, Likun; Du, Shunda; Sang, Xinting; Yang, Jian

    2016-01-01

    Liver segmentation is a significant processing technique for computer-assisted diagnosis. This method has attracted considerable attention and achieved effective result. However, liver segmentation using computed tomography (CT) images remains a challenging task because of the low contrast between the liver and adjacent organs. This paper proposes a feature-learning-based random walk method for liver segmentation using CT images. Four texture features were extracted and then classified to determine the classification probability corresponding to the test images. Seed points on the original test image were automatically selected and further used in the random walk (RW) algorithm to achieve comparable results to previous segmentation methods. PMID:27846217

  14. Quantum random walks do not need a coin toss

    SciTech Connect

    Patel, Apoorva; Raghunathan, K.S.; Rungta, Pranaw

    2005-03-01

    Classical randomized algorithms use a coin toss instruction to explore different evolutionary branches of a problem. Quantum algorithms, on the other hand, can explore multiple evolutionary branches by mere superposition of states. Discrete quantum random walks, studied in the literature, have nonetheless used both superposition and a quantum coin toss instruction. This is not necessary, and a discrete quantum random walk without a quantum coin toss instruction is defined and analyzed here. Our construction eliminates quantum entanglement between the coin and the position degrees of freedom from the algorithm, and the results match those obtained with a quantum coin toss instruction.

  15. Simplified equations of the compliant matrix for right elliptical flexure hinges.

    PubMed

    Fu, Jinjiang; Yan, Changxiang; Liu, Wei; Yuan, Ting

    2015-11-01

    The simplified compliance matrix for right elliptical hinges is presented in this paper by nonlinear curve fitting on the basis of the equations derived by Chen et al. [Rev. Sci. Instrum. 79, 095103 (2008)]. The equations of the rotation stiffness are then confirmed by comparison with results from finite element analysis and experimental measurements. Percentage errors between theoretical predictions and results from both the finite element analysis and experimental testing are within 5% for a range of geometries with the ratio s (b/t) between 1 and 14. The geometric parameter optimization for the purposes of maximizing the rotation stiffness for one universal hinge is utilized to illustrate the application of the simplified equations. The theoretical predictions are in good agreement with both the result of simulation and experiment for the universal hinge: the error between them is within 6.5%.

  16. Solitary and Jacobi elliptic wave solutions of the generalized Benjamin-Bona-Mahony equation

    NASA Astrophysics Data System (ADS)

    Belobo, Didier Belobo; Das, Tapas

    2017-07-01

    Exact bright, dark, antikink solitary waves and Jacobi elliptic function solutions of the generalized Benjamin-Bona-Mahony equation with arbitrary power-law nonlinearity will be constructed in this work. The method used to carry out the integration is the F-expansion method. Solutions obtained have fractional and integer negative or positive power-law nonlinearities. These solutions have many free parameters such that they may be used to simulate many experimental situations, and to precisely control the dynamics of the system.

  17. Singularities for a 2-Dimensional Semilinear Elliptic Equation with a Non-Lipschitz Nonlinearity

    NASA Astrophysics Data System (ADS)

    Bidaut-Véron, Marie-Francoise; Galaktionov, Victor; Grillot, Philippe; Véron, Laurent

    1999-05-01

    We study the limit behaviour of solutions of the semilinear elliptic equationΔu=|x|σ |u|q-1 u in R2, q∈(0, 1), σ∈R,with a non-Lipschitz nonlinearity on the right-hand side. When |σ+2|⩽2 we give a complete classification of the types of singularities asx→0 andx→∞ which in the rescaled form are essentially non-analytic and, even more, notC∞. The proof is based on the asymptotic study of the corresponding evolution dynamical system and the Sturmian argument on zero set analysis.

  18. A simple finite element method for non-divergence form elliptic equation

    DOE PAGES

    Mu, Lin; Ye, Xiu

    2017-03-01

    Here, we develop a simple finite element method for solving second order elliptic equations in non-divergence form by combining least squares concept with discontinuous approximations. This simple method has a symmetric and positive definite system and can be easily analyzed and implemented. We could have also used general meshes with polytopal element and hanging node in the method. We prove that our finite element solution approaches to the true solution when the mesh size approaches to zero. Numerical examples are tested that demonstrate the robustness and flexibility of the method.

  19. Inference of random walk models to describe leukocyte migration

    NASA Astrophysics Data System (ADS)

    Jones, Phoebe J. M.; Sim, Aaron; Taylor, Harriet B.; Bugeon, Laurence; Dallman, Magaret J.; Pereira, Bernard; Stumpf, Michael P. H.; Liepe, Juliane

    2015-12-01

    While the majority of cells in an organism are static and remain relatively immobile in their tissue, migrating cells occur commonly during developmental processes and are crucial for a functioning immune response. The mode of migration has been described in terms of various types of random walks. To understand the details of the migratory behaviour we rely on mathematical models and their calibration to experimental data. Here we propose an approximate Bayesian inference scheme to calibrate a class of random walk models characterized by a specific, parametric particle re-orientation mechanism to observed trajectory data. We elaborate the concept of transition matrices (TMs) to detect random walk patterns and determine a statistic to quantify these TM to make them applicable for inference schemes. We apply the developed pipeline to in vivo trajectory data of macrophages and neutrophils, extracted from zebrafish that had undergone tail transection. We find that macrophage and neutrophils exhibit very distinct biased persistent random walk patterns, where the strengths of the persistence and bias are spatio-temporally regulated. Furthermore, the movement of macrophages is far less persistent than that of neutrophils in response to wounding.

  20. Protein localization prediction using random walks on graphs

    PubMed Central

    2013-01-01

    Background Understanding the localization of proteins in cells is vital to characterizing their functions and possible interactions. As a result, identifying the (sub)cellular compartment within which a protein is located becomes an important problem in protein classification. This classification issue thus involves predicting labels in a dataset with a limited number of labeled data points available. By utilizing a graph representation of protein data, random walk techniques have performed well in sequence classification and functional prediction; however, this method has not yet been applied to protein localization. Accordingly, we propose a novel classifier in the site prediction of proteins based on random walks on a graph. Results We propose a graph theory model for predicting protein localization using data generated in yeast and gram-negative (Gneg) bacteria. We tested the performance of our classifier on the two datasets, optimizing the model training parameters by varying the laziness values and the number of steps taken during the random walk. Using 10-fold cross-validation, we achieved an accuracy of above 61% for yeast data and about 93% for gram-negative bacteria. Conclusions This study presents a new classifier derived from the random walk technique and applies this classifier to investigate the cellular localization of proteins. The prediction accuracy and additional validation demonstrate an improvement over previous methods, such as support vector machine (SVM)-based classifiers. PMID:23815126

  1. Adaptive importance sampling of random walks on continuous state spaces

    SciTech Connect

    Baggerly, K.; Cox, D.; Picard, R.

    1998-11-01

    The authors consider adaptive importance sampling for a random walk with scoring in a general state space. Conditions under which exponential convergence occurs to the zero-variance solution are reviewed. These results generalize previous work for finite, discrete state spaces in Kollman (1993) and in Kollman, Baggerly, Cox, and Picard (1996). This paper is intended for nonstatisticians and includes considerable explanatory material.

  2. Numerical and Analytic Studies of Random-Walk Models.

    NASA Astrophysics Data System (ADS)

    Li, Bin

    We begin by recapitulating the universality approach to problems associated with critical systems, and discussing the role that random-walk models play in the study of phase transitions and critical phenomena. As our first numerical simulation project, we perform high-precision Monte Carlo calculations for the exponents of the intersection probability of pairs and triplets of ordinary random walks in 2 dimensions, in order to test the conformal-invariance theory predictions. Our numerical results strongly support the theory. Our second numerical project aims to test the hyperscaling relation dnu = 2 Delta_4-gamma for self-avoiding walks in 2 and 3 dimensions. We apply the pivot method to generate pairs of self-avoiding walks, and then for each pair, using the Karp-Luby algorithm, perform an inner -loop Monte Carlo calculation of the number of different translates of one walk that makes at least one intersection with the other. Applying a least-squares fit to estimate the exponents, we have obtained strong numerical evidence that the hyperscaling relation is true in 3 dimensions. Our great amount of data for walks of unprecedented length(up to 80000 steps), yield a updated value for the end-to-end distance and radius of gyration exponent nu = 0.588 +/- 0.001 (95% confidence limit), which comes out in good agreement with the renormalization -group prediction. In an analytic study of random-walk models, we introduce multi-colored random-walk models and generalize the Symanzik and B.F.S. random-walk representations to the multi-colored case. We prove that the zero-component lambdavarphi^2psi^2 theory can be represented by a two-color mutually -repelling random-walk model, and it becomes the mutually -avoiding walk model in the limit lambda to infty. However, our main concern and major break-through lies in the study of the two-point correlation function for the lambda varphi^2psi^2 theory with N > 0 components. By representing it as a two-color random-walk expansion

  3. Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation

    SciTech Connect

    Jasra, Ajay; Law, Kody J. H.; Zhou, Yan

    2016-01-01

    Our paper considers uncertainty quantification for an elliptic nonlocal equation. In particular, it is assumed that the parameters which define the kernel in the nonlocal operator are uncertain and a priori distributed according to a probability measure. It is shown that the induced probability measure on some quantities of interest arising from functionals of the solution to the equation with random inputs is well-defined,s as is the posterior distribution on parameters given observations. As the elliptic nonlocal equation cannot be solved approximate posteriors are constructed. The multilevel Monte Carlo (MLMC) and multilevel sequential Monte Carlo (MLSMC) sampling algorithms are used for a priori and a posteriori estimation, respectively, of quantities of interest. Furthermore, these algorithms reduce the amount of work to estimate posterior expectations, for a given level of error, relative to Monte Carlo and i.i.d. sampling from the posterior at a given level of approximation of the solution of the elliptic nonlocal equation.

  4. A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations

    NASA Astrophysics Data System (ADS)

    Vidal-Codina, F.; Nguyen, N. C.; Giles, M. B.; Peraire, J.

    2015-09-01

    We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method.

  5. A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations

    SciTech Connect

    Vidal-Codina, F.; Nguyen, N.C.; Giles, M.B.; Peraire, J.

    2015-09-15

    We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method.

  6. Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

    NASA Astrophysics Data System (ADS)

    Armstrong, Scott; Lin, Jessica

    2017-08-01

    We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto (Ann Probab 39(3):779-856, 2011; Ann Appl Probab 22(1):1-28, 2012) for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a C 1,1 regularity theory down to microscopic scale, which is of independent interest and is inspired by the C 0,1 theory introduced in the divergence form case by the first author and Smart (Ann Sci Éc Norm Supér (4) 49(2):423-481, 2016).

  7. Uniqueness and Long Time Asymptotic for the Keller-Segel Equation: The Parabolic-Elliptic Case

    NASA Astrophysics Data System (ADS)

    Egaña Fernández, Giani; Mischler, Stéphane

    2016-06-01

    The present paper deals with the parabolic-elliptic Keller-Segel equation in the plane in the general framework of weak (or "free energy") solutions associated to initial datum with finite mass M, finite second moment and finite entropy. The aim of the paper is threefold: (1) We prove the uniqueness of the "free energy" solution on the maximal interval of existence [0, T*) with T* = ∞ in the case when M ≦ 8π and T* < ∞ in the case when M > 8π. The proof uses a DiPerna-Lions renormalizing argument which makes it possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical L 4/3 Lebesgue norm similarly to the 2 d vorticity Navier-Stokes equation.

  8. Statistics at the tip of a branching random walk and the delay of traveling waves

    NASA Astrophysics Data System (ADS)

    Brunet, É.; Derrida, B.

    2009-09-01

    We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean-field spin-glasses.

  9. A directed continuous time random walk model with jump length depending on waiting time.

    PubMed

    Shi, Long; Yu, Zuguo; Mao, Zhi; Xiao, Aiguo

    2014-01-01

    In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density function P(x, t) of finding the walker at position x at time t is completely determined by the Laplace transform of the probability density function φ(t) of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived.

  10. Continuous time random walks for non-local radial solute transport

    NASA Astrophysics Data System (ADS)

    Dentz, Marco; Kang, Peter K.; Le Borgne, Tanguy

    2015-08-01

    This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile-immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection-dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer

  11. Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons

    SciTech Connect

    Zhang, Pei; Ren, Xi-Feng; Zou, Xu-Bo; Liu, Bi-Heng; Huang, Yun-Feng; Guo, Guang-Can

    2007-05-15

    Quantum random walks have attracted special interest because they could lead to new quantum algorithms. Photons can carry orbital angular momentum (OAM) thereby offering a practical realization of a high-dimensional quantum information carrier. By employing OAM of photons, we experimentally realized the one-dimensional discrete-time quantum random walk. Three steps of a one-dimensional quantum random walk were implemented in our protocol showing the obvious difference between quantum and classical random walks.

  12. Eigenfunction expansions for a fundamental solution of Laplace’s equation on R3 in parabolic and elliptic cylinder coordinates

    NASA Astrophysics Data System (ADS)

    Cohl, H. S.; Volkmer, H.

    2012-09-01

    A fundamental solution of Laplace’s equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions J0(kr) or K0(kr), r2 = (x - x0)2 + (y - y0)2, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that K0(kr) is a fundamental solution and J0(kr) is the Riemann function of partial differential equations on the Euclidean plane.

  13. On the convergence of the conditional gradient method as applied to the optimization of an elliptic equation

    NASA Astrophysics Data System (ADS)

    Chernov, A. V.

    2015-02-01

    The optimal control of a second-order semilinear elliptic diffusion-reaction equation is considered. Sufficient conditions for the convergence of the conditional gradient method are obtained without using assumptions (traditional for optimization theory) that ensure the Lipschitz continuity of the objective functional derivative. The total (over the entire set of admissible controls) preservation of solvability, a pointwise estimate of solutions, and the uniqueness of a solution to the homogeneous Dirichlet problem for a controlled elliptic equation are proved as preliminary results, which are of interest on their own.

  14. Reweighted ℓ{sub 1} minimization method for stochastic elliptic differential equations

    SciTech Connect

    Yang, Xiu; Karniadakis, George Em

    2013-09-01

    We consider elliptic stochastic partial differential equations (SPDEs) with random coefficients and solve them by expanding the solution using generalized polynomial chaos (gPC). Under some mild conditions on the coefficients, the solution is “sparse” in the random space, i.e., only a small number of gPC basis makes considerable contribution to the solution. To exploit this sparsity, we employ reweighted l{sub 1} minimization to recover the coefficients of the gPC expansion. We also combine this method with random sampling points based on the Chebyshev probability measure to further increase the accuracy of the recovery of the gPC coefficients. We first present a one-dimensional test to demonstrate the main idea, and then we consider 14 and 40 dimensional elliptic SPDEs to demonstrate the significant improvement of this method over the standard l{sub 1} minimization method. For moderately high dimensional (∼10) problems, the combination of Chebyshev measure with reweighted l{sub 1} minimization performs well while for higher dimensional problems, reweighted l{sub 1} only is sufficient. The proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources.

  15. [Some exact results for random walk models with applications].

    PubMed

    Schwarz, W

    1989-01-01

    This article presents a random walk model that can be analyzed without recourse to Wald's (1947) approximation, which neglects the excess over the absorbing barriers. Hence, the model yields exact predictions for the absorption probabilities and all mean conditional absorption times. We derive these predictions in some detail and fit them to the extensive data of an identification experiment published by Green et al. (1983). The fit of the model seems satisfactory. The relationship of the model to existing classes of random walk models (SPRT and SSR; see Luce, 1986) is discussed; for certain combinations of its parameters, the model belongs either to the SPRT or to the SSR class, or to both. We stress the theoretical significance of the knowledge of exact results for the evaluation of Wald's approximation and general properties of the several models proposed derived from this approximation.

  16. A generalized model via random walks for information filtering

    NASA Astrophysics Data System (ADS)

    Ren, Zhuo-Ming; Kong, Yixiu; Shang, Ming-Sheng; Zhang, Yi-Cheng

    2016-08-01

    There could exist a simple general mechanism lurking beneath collaborative filtering and interdisciplinary physics approaches which have been successfully applied to online E-commerce platforms. Motivated by this idea, we propose a generalized model employing the dynamics of the random walk in the bipartite networks. Taking into account the degree information, the proposed generalized model could deduce the collaborative filtering, interdisciplinary physics approaches and even the enormous expansion of them. Furthermore, we analyze the generalized model with single and hybrid of degree information on the process of random walk in bipartite networks, and propose a possible strategy by using the hybrid degree information for different popular objects to toward promising precision of the recommendation.

  17. An Analysis of Random-Walk Cuckoo Hashing

    NASA Astrophysics Data System (ADS)

    Frieze, Alan; Melsted, Páll; Mitzenmacher, Michael

    In this paper, we provide a polylogarithmic bound that holds with high probability on the insertion time for cuckoo hashing under the random-walk insertion method. Cuckoo hashing provides a useful methodology for building practical, high-performance hash tables. The essential idea of cuckoo hashing is to combine the power of schemes that allow multiple hash locations for an item with the power to dynamically change the location of an item among its possible locations. Previous work on the case where the number of choices is larger than two has required a breadth-first search analysis, which is both inefficient in practice and currently has only a polynomial high probability upper bound on the insertion time. Here we significantly advance the state of the art by proving a polylogarithmic bound on the more efficient random-walk method, where items repeatedly kick out random blocking items until a free location for an item is found.

  18. Statistical Modeling of Robotic Random Walks on Different Terrain

    NASA Astrophysics Data System (ADS)

    Naylor, Austin; Kinnaman, Laura

    Issues of public safety, especially with crowd dynamics and pedestrian movement, have been modeled by physicists using methods from statistical mechanics over the last few years. Complex decision making of humans moving on different terrains can be modeled using random walks (RW) and correlated random walks (CRW). The effect of different terrains, such as a constant increasing slope, on RW and CRW was explored. LEGO robots were programmed to make RW and CRW with uniform step sizes. Level ground tests demonstrated that the robots had the expected step size distribution and correlation angles (for CRW). The mean square displacement was calculated for each RW and CRW on different terrains and matched expected trends. The step size distribution was determined to change based on the terrain; theoretical predictions for the step size distribution were made for various simple terrains. It's Dr. Laura Kinnaman, not sure where to put the Prefix.

  19. The random walk of tracers through river catchments

    NASA Astrophysics Data System (ADS)

    Bhattacharya, Atreyee

    2012-08-01

    River catchments play critical roles in regional economies and in the global economy. In addition, rivers carry large volumes of nutrients, pollutants, and several other forms of tracers into the ocean. An intricate system of pathways and channels, both on the surface and in the subsurface of catchments, allows rivers to carry large volumes of tracers. However, scientists do not yet fully understand how pollutants and other tracers travel through the intricate web of channels in the catchment areas of rivers. In a new study, Cvetkovic et al show that the travel path of tracers through channels can be modeled as a random walk, which is mathematically similar to the path an animal would trace when foraging. Previous studies have applied the random walk approach to understand the behavior of fluids flowing through aquifers and soils but not to model the transport mechanism of tracers that travel passively with water flowing through catchments.

  20. Random Walk on the High-Dimensional IIC

    NASA Astrophysics Data System (ADS)

    Heydenreich, Markus; van der Hofstad, Remco; Hulshof, Tim

    2014-07-01

    We study the asymptotic behavior of the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by Kumagai and Misumi (J Theor Probab 21:910-935, 2008). We do this by getting bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander-Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation.

  1. Aggregation is the key to succeed in random walks.

    PubMed

    Hernandez-Suarez, Carlos M

    2016-09-01

    In a random walk (RW) in Z an individual starts at 0 and moves at discrete unitary steps to the right or left with respective probabilities p and 1-p. Assuming p > 1/2 and finite a, a > 1, the probability that state a will be reached before -a is Q(a, p) where Q(a, p) > p. Here we introduce the cooperative random walk (CRW) involving two individuals that move independently according to a RW each but dedicate a fraction of time θ to approach the other one unit. This simple strategy seems to be effective in increasing the expected number of individuals arriving to a first. We conjecture that this is a possible underlying mechanism for efficient animal migration under noisy conditions. Copyright © 2016 Elsevier Inc. All rights reserved.

  2. Universal properties of branching random walks in confined geometries

    NASA Astrophysics Data System (ADS)

    de Mulatier, C.; Mazzolo, A.; Zoia, A.

    2014-08-01

    Characterizing the occupation statistics of random walks through confined geometries amounts to assessing the distribution of the travelled length ℓ and the number of collisions n performed by the stochastic process in a given region, for which remarkably simple Cauchy-like formulas were established in the case of branching Pearson random walks with exponentially distributed jumps. In this letter, we derive two key results: first, we show that such formulas strikingly carry over to the much broader class of branching processes with arbitrary jumps, and have thus a universal character; second, we obtain a stronger version of these formulas relating the travelled length density and the collision density at any point of the phase space. Our results are key to such technological issues as the analysis of radiation flow for nuclear reactor design and medical diagnosis and apply more broadly to physical and biological systems with diffusion, reproduction and death.

  3. Quantization of Random Walks: Search Algorithms and Hitting Time

    NASA Astrophysics Data System (ADS)

    Santha, Miklos

    Many classical search problems can be cast in the following abstract framework: Given a finite set X and a subset M ⊆ X of marked elements, detect if M is empty or not, or find an element in M if there is any. When M is not empty, a naive approach to the finding problem is to repeatedly pick a uniformly random element of X until a marked element is sampled. A more sophisticated approach might use a Markov chain, that is a random walk on the state space X in order to generate the samples. In that case the resources spent for previous steps are often reused to generate the next sample. Random walks also model spatial search in physical regions where the possible moves are expressed by the edges of some specific graph. The hitting time of a Markov chain is the number of steps necessary to reach a marked element, starting from the stationary distribution of the chain.

  4. Sub-Markov Random Walk for Image Segmentation.

    PubMed

    Dong, Xingping; Shen, Jianbing; Shao, Ling; Van Gool, Luc

    2016-02-01

    A novel sub-Markov random walk (subRW) algorithm with label prior is proposed for seeded image segmentation, which can be interpreted as a traditional random walker on a graph with added auxiliary nodes. Under this explanation, we unify the proposed subRW and other popular random walk (RW) algorithms. This unifying view will make it possible for transferring intrinsic findings between different RW algorithms, and offer new ideas for designing novel RW algorithms by adding or changing auxiliary nodes. To verify the second benefit, we design a new subRW algorithm with label prior to solve the segmentation problem of objects with thin and elongated parts. The experimental results on both synthetic and natural images with twigs demonstrate that the proposed subRW method outperforms previous RW algorithms for seeded image segmentation.

  5. Reheating-volume measure for random-walk inflation

    SciTech Connect

    Winitzki, Sergei

    2008-09-15

    The recently proposed 'reheating-volume' (RV) measure promises to solve the long-standing problem of extracting probabilistic predictions from cosmological multiverse scenarios involving eternal inflation. I give a detailed description of the new measure and its applications to generic models of eternal inflation of random-walk type. For those models I derive a general formula for RV-regulated probability distributions that is suitable for numerical computations. I show that the results of the RV cutoff in random-walk type models are always gauge invariant and independent of the initial conditions at the beginning of inflation. In a toy model where equal-time cutoffs lead to the 'youngness paradox', the RV cutoff yields unbiased results that are distinct from previously proposed measures.

  6. A Random Walk Phenomenon under an Interesting Stopping Rule

    ERIC Educational Resources Information Center

    Chakraborty, S.

    2007-01-01

    In the simple one-dimensional random walk setup, a path is described as follows. Toss a coin. If the result is head, score +1 and move one step forward; otherwise score -1 and move one step backward. One is interested to know the position after a given number of steps. In this paper, once again a coin-tossing experiment is carried out. But this…

  7. Random Walks and Branching Processes in Correlated Gaussian Environment

    NASA Astrophysics Data System (ADS)

    Aurzada, Frank; Devulder, Alexis; Guillotin-Plantard, Nadine; Pène, Françoise

    2017-01-01

    We study persistence probabilities for random walks in correlated Gaussian random environment investigated by Oshanin et al. (Phys Rev Lett, 110:100602, 2013). From the persistence results, we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters. More precisely, we obtain estimates on the tail distribution of its total population size, of its maximum population, and of its extinction time.

  8. Correlated continuous time random walk with time averaged waiting time

    NASA Astrophysics Data System (ADS)

    Lv, Longjin; Ren, Fu-Yao; Wang, Jun; Xiao, Jianbin

    2015-03-01

    In this paper, we study the dynamics of a correlated continuous time random walk with time averaged waiting time. The mean square displacement (MSD) shows this process is subdiffusive and generalized Einstein relation holds. We also get the asymptotic behavior of the probability density function (PDF) of this process is stretched Gaussian. At last, by computing the time averaged MSD, we find ergodicity breaking occurs in this process.

  9. Vibration driven random walk in a Chladni experiment

    NASA Astrophysics Data System (ADS)

    Grabec, Igor

    2017-01-01

    Drifting of sand particles bouncing on a vibrating membrane of a Chladni experiment is characterized statistically. Records of trajectories reveal that bounces are circularly distributed and random. The mean length of their horizontal displacement is approximately proportional to the vibration amplitude above the critical level and amounts about one fourth of the corresponding bounce height. For the description of horizontal drifting of particles a model of vibration driven random walk is proposed that yields a good agreement between experimental and numerically simulated data.

  10. A Random Walk Phenomenon under an Interesting Stopping Rule

    ERIC Educational Resources Information Center

    Chakraborty, S.

    2007-01-01

    In the simple one-dimensional random walk setup, a path is described as follows. Toss a coin. If the result is head, score +1 and move one step forward; otherwise score -1 and move one step backward. One is interested to know the position after a given number of steps. In this paper, once again a coin-tossing experiment is carried out. But this…

  11. Navigation by anomalous random walks on complex networks

    NASA Astrophysics Data System (ADS)

    Weng, Tongfeng; Zhang, Jie; Khajehnejad, Moein; Small, Michael; Zheng, Rui; Hui, Pan

    2016-11-01

    Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Lévy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Lévy walks and the underlying network structure. Moreover, applying our framework to the famous PageRank search, we show how to inform the optimality of the PageRank search. The framework for analyzing anomalous random walks on complex networks offers a useful new paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks.

  12. Random walk with random resetting to the maximum position

    NASA Astrophysics Data System (ADS)

    Majumdar, Satya N.; Sabhapandit, Sanjib; Schehr, Grégory

    2015-11-01

    We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability r , and with probability (1 -r ) , it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability (1 -r )/2 . For r =0 , it reduces to a standard random walk whose typical distance grows as √{n } for large n . In the presence of a nonzero resetting rate 0

  13. Random Walks in Social Networks and their Applications: A Survey

    NASA Astrophysics Data System (ADS)

    Sarkar, Purnamrita; Moore, Andrew W.

    A wide variety of interesting real world applications, e.g. friend suggestion in social networks, keyword search in databases, web-spam detection etc. can be framed as ranking entities in a graph. In order to obtain ranking we need a graph-theoretic measure of similarity. Ideally this should capture the information hidden in the graph structure. For example, two entities are similar, if there are lots of short paths between them. Random walks have proven to be a simple, yet powerful mathematical tool for extracting information from the ensemble of paths between entities in a graph. Since real world graphs are enormous and complex, ranking using random walks is still an active area of research. The research in this area spans from new applications to novel algorithms and mathematical analysis, bringing together ideas from different branches of statistics, mathematics and computer science. In this book chapter, we describe different random walk based proximity measures, their applications, and existing algorithms for computing them.

  14. Navigation by anomalous random walks on complex networks

    PubMed Central

    Weng, Tongfeng; Zhang, Jie; Khajehnejad, Moein; Small, Michael; Zheng, Rui; Hui, Pan

    2016-01-01

    Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Lévy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Lévy walks and the underlying network structure. Moreover, applying our framework to the famous PageRank search, we show how to inform the optimality of the PageRank search. The framework for analyzing anomalous random walks on complex networks offers a useful new paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks. PMID:27876855

  15. Ant-inspired density estimation via random walks.

    PubMed

    Musco, Cameron; Su, Hsin-Hao; Lynch, Nancy A

    2017-09-19

    Many ant species use distributed population density estimation in applications ranging from quorum sensing, to task allocation, to appraisal of enemy colony strength. It has been shown that ants estimate local population density by tracking encounter rates: The higher the density, the more often the ants bump into each other. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using local mixing properties of the underlying graph. Our results extend beyond the grid to more general graphs, and we discuss applications to size estimation for social networks, density estimation for robot swarms, and random walk-based sampling for sensor networks.

  16. A critical nonlinear fractional elliptic equation with saddle-like potential in ℝN

    NASA Astrophysics Data System (ADS)

    O. Alves, Claudianor; Miyagaki, Olimpio H.

    2016-08-01

    In this paper, we study the existence of positive solution for the following class of fractional elliptic equation ɛ 2 s ( - Δ ) s u + V ( z ) u = λ |" separators=" u | q - 2 u + |" separators=" u | 2s ∗ - 2 u in R N , where ɛ, λ > 0 are positive parameters, q ∈ ( 2 , 2s ∗ ) , 2s ∗ = /2 N N - 2 s , N > 2 s , s ∈ ( 0 , 1 ) , ( - Δ ) s u is the fractional Laplacian, and V is a saddle-like potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Benci and Cerami.

  17. Grid generation by elliptic partial differential equations for a tri-element Augmentor-Wing airfoil

    NASA Technical Reports Server (NTRS)

    Sorenson, R. L.

    1982-01-01

    Two efforts to numerically simulate the flow about the Augmentor-Wing airfoil in the cruise configuration using the GRAPE elliptic partial differential equation grid generator algorithm are discussed. The Augmentor-Wing consists of a main airfoil with a slotted trailing edge for blowing and two smaller airfoils shrouding the blowing jet. The airfoil and the algorithm are described, and the application of GRAPE to an unsteady viscous flow simulation and a transonic full-potential approach is considered. The procedure involves dividing a complicated flow region into an arbitrary number of zones and ensuring continuity of grid lines, their slopes, and their point distributions across the zonal boundaries. The method for distributing the body-surface grid points is discussed.

  18. Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation

    NASA Astrophysics Data System (ADS)

    Wright, O. C.

    2016-05-01

    An effective integration method based on the classical solution of the Jacobi inversion problem, using Kleinian ultra-elliptic functions and Riemann theta functions, is presented for the quasi-periodic two-phase solutions of the focusing cubic nonlinear Schrödinger equation. Each two-phase solution with real quasi-periods forms a two-real-dimensional torus, modulo a circle of complex-phase factors, expressed as a ratio of theta functions associated with the Riemann surface of the invariant spectral curve. The initial conditions of the Dirichlet eigenvalues satisfy reality conditions which are explicitly parametrized by two physically-meaningful real variables: the squared modulus and a scalar multiple of the wavenumber. Simple new formulas for the maximum modulus and the minimum modulus are obtained in terms of the imaginary parts of the branch points of the Riemann surface.

  19. A Gas-Kinetic Method for Hyperbolic-Elliptic Equations and Its Application in Two-Phase Fluid Flow

    NASA Technical Reports Server (NTRS)

    Xu, Kun

    1999-01-01

    A gas-kinetic method for the hyperbolic-elliptic equations is presented in this paper. In the mixed type system, the co-existence and the phase transition between liquid and gas are described by the van der Waals-type equation of state (EOS). Due to the unstable mechanism for a fluid in the elliptic region, interface between the liquid and gas can be kept sharp through the condensation and evaporation process to remove the "averaged" numerical fluid away from the elliptic region, and the interface thickness depends on the numerical diffusion and stiffness of the phase change. A few examples are presented in this paper for both phase transition and multifluid interface problems.

  20. Fast solution of elliptic partial differential equations using linear combinations of plane waves

    NASA Astrophysics Data System (ADS)

    Pérez-Jordá, José M.

    2016-02-01

    Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations A x =b , where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O (N logN ) memory and executing an iteration in O (N log2N ) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

  1. Fast solution of elliptic partial differential equations using linear combinations of plane waves.

    PubMed

    Pérez-Jordá, José M

    2016-02-01

    Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations Ax=b, where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O(NlogN) memory and executing an iteration in O(Nlog(2)N) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

  2. A New Family of Solvable Pearson-Dirichlet Random Walks

    NASA Astrophysics Data System (ADS)

    Le Caër, Gérard

    2011-07-01

    An n-step Pearson-Gamma random walk in ℝ d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ℝ d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥ d 0 and any n≥2 when q is either q = d/2 - 1 ( d 0=3) or q= d-1 ( d 0=2) (Le Caër in J. Stat. Phys. 140:728-751, 2010). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type ( n≥2), with q= d=2, was shown recently to be a weighted mixture of 1+ floor( n/2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201-229, 2010). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q= d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q= d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+ floor( n/2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d.

  3. The survival probability of a branching random walk in presence of an absorbing wall

    NASA Astrophysics Data System (ADS)

    Derrida, B.; Simon, D.

    2007-06-01

    A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity vc of the wall with an essential singularity and we characterize the divergences of the relaxation times for vvc. At v=vc the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditioned by the survival of one individual at a later time T>t. Our numerical results indicate that the size of the population diverges like the exponential of (vc-v)-1/2 in the quasi-stationary regime below vc. Moreover for v>vc, our data indicate that there is no quasi-stationary regime.

  4. Random walk theory of a trap-controlled hopping transport process

    PubMed Central

    Scher, H.; Wu, C. H.

    1981-01-01

    A random walk theory of hopping motion in the presence of a periodic distribution of traps is presented. The solution of the continuous-time random walk equations is exact and valid for arbitrary intersite interactions and trap concentration. The treatment is shown to be equivalent to an exact solution of the master equation for this trapping problem. These interactions can be a general function of electric field and are not restricted to nearest neighbors. In particular, with the inclusion of trap-to-trap interactions, as well as trap-to-host interactions, an exact treatment of the change from one hopping channel to another has been obtained. The trap-modulated propagator has been derived in terms of a type of Green's function that is introduced. The results are specialized to spatial moments of the propagator, from which expressions for the drift velocity and diffusion coefficient are obtained. Numerical results for the drift velocity are presented and shown to account for the change in hopping channels in recent transport measurements in mixed molecularly doped polymers. PMID:16592944

  5. A Continuous Time Random Walk Description of Monodisperse, Hard-Sphere Colloids below the Ordering Transition

    NASA Astrophysics Data System (ADS)

    Lechman, Jeremy; Pierce, Flint

    2012-02-01

    Diffusive transport is a ubiquitous process that is typically understood in terms of a classical random walk of non-interacting particles. Here we present the results for a model of hard-sphere colloids in a Newtonian incompressible solvent at various volume fractions below the ordering transition (˜50%). We numerically simulate the colloidal systems via Fast Lubrication Dynamics -- a Brownian Dynamics approach with corrected mean-field hydrodynamic interactions. Colloid-colloid interactions are also included so that we effectively solve a system of interacting Langevin equations. The results of the simulations are analyzed in terms of the diffusion coefficient as a function of time with the early and late time diffusion coefficients comparing well with experimental results. An interpretation of the full time dependent behavior of the diffusion coefficient and mean-squared displacement is given in terms of a continuous time random walk. Therefore, the deterministic, continuum diffusion equation which arises from the discrete, interacting random walkers is presented. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

  6. Subdiffusivity of a Random Walk Among a Poisson System of Moving Traps on {Z}

    NASA Astrophysics Data System (ADS)

    Athreya, Siva; Drewitz, Alexander; Sun, Rongfeng

    2017-03-01

    We consider a random walk among a Poisson system of moving traps on {Z}. In earlier work (Drewitz et al. Springer Proc. Math. 11, 119-158 2012), the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time t in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.

  7. Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

    SciTech Connect

    Casas, E.

    1999-03-15

    In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem.

  8. A numerical technique for linear elliptic partial differential equations in polygonal domains

    PubMed Central

    Hashemzadeh, P.; Fokas, A. S.; Smitheman, S. A.

    2015-01-01

    Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low. PMID:25792955

  9. A numerical technique for linear elliptic partial differential equations in polygonal domains.

    PubMed

    Hashemzadeh, P; Fokas, A S; Smitheman, S A

    2015-03-08

    Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

  10. Gradient estimate for solutions of a class of nonlinear elliptic equations below the duality exponent

    NASA Astrophysics Data System (ADS)

    Cirmi, G. Rita; D'Asero, Salvatore; Leonardi, Salvatore

    2017-07-01

    We study the regularity of a solution of the Dirichlet problem associated to the singular equation -div(a (x )D u )+M |/D u | uθ =f (x )inΩ (1) where Ω is an open bounded subset of ℝN (N ≥ 3) with smooth boundary, a(x) is a L∞ -matrix satisfying the standard ellipticity condition, θ ∈]0, 1[, M is a positive constant and f is sufficiently regular i.e. it belongs to a suitable Morrey space Lq,λ (Ω), with q ≥ 1, to be specified later on. We will be concerned with the regularity of the gradient of a solution in Morrey spaces in correspondence with the regularity properties of the right-hand side of the equation (1). There is a huge literature about the problems with quadratic term in the gradient also for high-order equations whose coefficients satisfy a strengthened ellipticity condition (see [5]). The problem (1) has been studied in the paper [2] by D. Arcoya, J. Carmona, T. Leonori, P. J. Martínez-Aparicio, L. Orsina, F. Petitta and in the paper [3] by L. Boccardo where the source term f belonged to Lq(Ω) with q ≥ 1. In [4] we have extended to the gradient of a solution the Morrey property of the right-hand side f and, in some cases, we have improved some results contained in [2, 3] without increasing the summability of f. Here we will be concerned with an intermediate case, in the sense that the right hand side is not merely a measure and it doesn't belong to the right dual space (see [10, 9]). In obtaining the necessary local estimates we faced the problem of performing the Campanato's decomposition of the solution due to the presence of the degenerate lower order term. We had thus to adopt an alternative method already used in [4]. This is part of a new set of estimates, including potential estimates, as for instance in [6, 7, 8], that would be very interesting to extend in the present setting too.

  11. Boundary behavior of non-negative solutions to degenerate sub-elliptic equations

    NASA Astrophysics Data System (ADS)

    Götmark, Elin; Nyström, Kaj

    Let X={X1,…,Xm} be a system of C∞ vector fields in Rn satisfying Hörmander's finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance d induced by X. We study the boundary behavior of non-negative solutions to the equation Lu=∑i,j=1mXi*(aXju)=∑i,j=1mXi*(x)(a(x)Xj(x)u(x))=0 where Xi* is the formal adjoint of Xi and x∈Ω. Concerning A(x)={a(x)} we assume that A(x) is real, symmetric and that β-1λ(x)|⩽∑i,j=1ma(x)ξiξj⩽βλ(x)|for all x∈Rn, ξ∈Rm, for some constant β⩾1 and for some non-negative and real-valued function λ=λ(x). Concerning λ we assume that λ defines an A2-weight with respect to the metric introduced by the system of vector fields X={X1,…,Xm}. Our main results include a proof of the doubling property of the associated elliptic measure and the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes et al. (1982, 1983) [18-20] (m=n, {X1,…,Xm}={∂,…,∂}, λ is an A2-weight) and Capogna and Garofalo (1998) [6] (X={X1,…,Xm} satisfies Hörmander's finite rank condition and λ(x)≡λ for some constant λ). One motivation for this study is the ambition to generalize, as far as possible, the results in Lewis and Nyström (2007, 2010, 2008) [35-38], Lewis et al. (2008) [34] concerning the boundary behavior of non-negative solutions to (Euclidean) quasi-linear equations of p-Laplace type, to non-negative solutions to certain sub-elliptic quasi-linear equations of p-Laplace type.

  12. ON BOUNDARY VALUES IN L_p, p > 1, OF SOLUTIONS OF ELLIPTIC EQUATIONS

    NASA Astrophysics Data System (ADS)

    Guščin, A. K.; Mihaĭlov, V. P.

    1980-02-01

    The behavior near the boundary of generalized solutions of a second order elliptic equation \\displaystyle \\sum_{i,j=1}^n\\frac{\\partial}{\\partial x_i}\\biggl(a_{ij}(x)\\fra......artial x_j}\\biggr)=f,\\qquad x\\in Q=\\{\\vert x\\vert < 1\\}\\subset\\mathbf{R}_n,in W_p^1(\\mathcal{Q}), p > 1, is studied. It is shown that under a certain condition on the right side of the equation, the boundedness of the function \\Vert x\\Vert_{L_p(\\vert x\\vert=r)}, \\frac{1}{2}\\le r < 1, is necessary and sufficient for the existence of a limit for the solution u(rw), \\frac{1}{2}\\le r < 1, \\vert w\\vert=1, in L_p(\\vert w\\vert=1) as r\\to 1 - 0. Moreover, the summability of the function \\displaystyle (1-\\vert x\\vert)\\vert u(x)\\vert^{p - 2}\\vert\

  13. An†§ efficient code to solve the Kepler equation. Elliptic case.

    NASA Astrophysics Data System (ADS)

    Raposo Pulido, V.; Peláez, J.

    2017-01-01

    A new approach for solving Kepler equation for elliptical orbits is developed in this paper. This new approach takes advantage of the very good behavior of the modified Newton-Raphson method when the initial seed is close to the looked for solution. To determine a good initial seed the eccentric anomaly domain [0, π] is discretized in several intervals and for each one of these intervals a fifth degree interpolating polynomial is introduced. The six coefficients of the polynomial are obtained by requiring six conditions at both ends of the corresponding interval. Thus the real function and the polynomial have equal values at both ends of the interval. Similarly relations are imposed for the two first derivatives. In the singular corner of the Kepler equation, M ≪ 1 and 1 - e ≪ 0 an asymptotic expansion is developed. In most of the cases, the seed generated leads to reach machine error accuracy with the modified Newton-Raphson method with no iterations or just one iteration. This approach improves the computational time compared with other methods currently in use.

  14. Social Aggregation in Pea Aphids: Experiment and Random Walk Modeling

    PubMed Central

    Nilsen, Christa; Paige, John; Warner, Olivia; Mayhew, Benjamin; Sutley, Ryan; Lam, Matthew; Bernoff, Andrew J.; Topaz, Chad M.

    2013-01-01

    From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. The probabilities of motion state transitions, as well as the random walk parameters, depend strongly on distance to an aphid's nearest neighbor. For large nearest neighbor distances, when an aphid is essentially isolated, its motion is ballistic with aphids moving faster, turning less, and being less likely to stop. In contrast, for short nearest neighbor distances, aphids move more slowly, turn more, and are more likely to become stationary; this behavior constitutes an aggregation mechanism. From the experimental data, we estimate the state transition probabilities and correlated random walk parameters as a function of nearest neighbor distance. With the individual-level model established, we assess whether it reproduces the macroscopic patterns of movement at the group level. To do so, we consider three distributions, namely distance to nearest neighbor, angle to nearest neighbor, and percentage of population moving at any given time. For each of these three distributions, we compare our experimental data to the output of numerical simulations of our nearest neighbor model, and of a control model in which aphids do not interact socially. Our stochastic, social nearest neighbor model reproduces salient features of the experimental data that are not captured by the control. PMID:24376691

  15. Social aggregation in pea aphids: experiment and random walk modeling.

    PubMed

    Nilsen, Christa; Paige, John; Warner, Olivia; Mayhew, Benjamin; Sutley, Ryan; Lam, Matthew; Bernoff, Andrew J; Topaz, Chad M

    2013-01-01

    From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. The probabilities of motion state transitions, as well as the random walk parameters, depend strongly on distance to an aphid's nearest neighbor. For large nearest neighbor distances, when an aphid is essentially isolated, its motion is ballistic with aphids moving faster, turning less, and being less likely to stop. In contrast, for short nearest neighbor distances, aphids move more slowly, turn more, and are more likely to become stationary; this behavior constitutes an aggregation mechanism. From the experimental data, we estimate the state transition probabilities and correlated random walk parameters as a function of nearest neighbor distance. With the individual-level model established, we assess whether it reproduces the macroscopic patterns of movement at the group level. To do so, we consider three distributions, namely distance to nearest neighbor, angle to nearest neighbor, and percentage of population moving at any given time. For each of these three distributions, we compare our experimental data to the output of numerical simulations of our nearest neighbor model, and of a control model in which aphids do not interact socially. Our stochastic, social nearest neighbor model reproduces salient features of the experimental data that are not captured by the control.

  16. A Random Walk in the Park: An Individual-Based Null Model for Behavioral Thermoregulation.

    PubMed

    Vickers, Mathew; Schwarzkopf, Lin

    2016-04-01

    Behavioral thermoregulators leverage environmental temperature to control their body temperature. Habitat thermal quality therefore dictates the difficulty and necessity of precise thermoregulation, and the quality of behavioral thermoregulation in turn impacts organism fitness via the thermal dependence of performance. Comparing the body temperature of a thermoregulator with a null (non-thermoregulating) model allows us to estimate habitat thermal quality and the effect of behavioral thermoregulation on body temperature. We define a null model for behavioral thermoregulation that is a random walk in a temporally and spatially explicit thermal landscape. Predicted body temperature is also integrated through time, so recent body temperature history, environmental temperature, and movement influence current body temperature; there is no particular reliance on an organism's equilibrium temperature. We develop a metric called thermal benefit that equates body temperature to thermally dependent performance as a proxy for fitness. We measure thermal quality of two distinct tropical habitats as a temporally dynamic distribution that is an ergodic property of many random walks, and we compare it with the thermal benefit of real lizards in both habitats. Our simple model focuses on transient body temperature; as such, using it we observe such subtleties as shifts in the thermoregulatory effort and investment of lizards throughout the day, from thermoregulators to thermoconformers.

  17. Stability of a Random Walk Model for Fruiting Body Aggregation in M. xanthus

    NASA Astrophysics Data System (ADS)

    McKenzie-Smith, G. C.; Schüttler, H. B.; Cotter, C.; Shimkets, L.

    2015-03-01

    Myxococcus xanthus exhibits the social starvation behavior of aggregation into a fruiting body containing myxospores able to survive harsh conditions. During fruiting body aggregation, individual bacteria follow random walk paths determined by randomly selected runtimes, turning angles, and speeds. We have simulated this behavior in terms of a continuous-time random walk (CTRW) model, re-formulated as a system of integral equations, describing the angle-resolved cell density, R(r, t, θ), at position r and cell orientation angle θ at time t, and angle-integrated ambient cell density ρ(r, t). By way of a linear stability analysis, we investigated whether a uniform cell density R0 will be unstable for a small non-uniform density perturbation δR(r, t, θ). Such instability indicates aggregate formation, whereas stability indicates absence of aggregation. We show that a broadening of CTRW distributions of the random speed and/or random runtimes strongly favors aggregation. We also show that, in the limit of slowly-varying (long-wavelength) density perturbations, the time-dependent linear density response can be approximated by a drift-diffusion model for which we calculate diffusion and drift coefficients as functions of the CTRW model parameters. Funded by the Fungal Genomics and Computational Biology REU at UGA.

  18. Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks

    NASA Astrophysics Data System (ADS)

    Zaburdaev, V.; Fouxon, I.; Denisov, S.; Barkai, E.

    2016-12-01

    It is recognized now that a variety of real-life phenomena ranging from diffusion of cold atoms to the motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.

  19. Non-equilibrium Phase Transitions: Activated Random Walks at Criticality

    NASA Astrophysics Data System (ADS)

    Cabezas, M.; Rolla, L. T.; Sidoravicius, V.

    2014-06-01

    In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including , and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which it is known that there is no fixation at criticality.

  20. Homogeneous Open Quantum Random Walks on a Lattice

    NASA Astrophysics Data System (ADS)

    Carbone, Raffaella; Pautrat, Yan

    2015-09-01

    We study open quantum random walks (OQRWs) for which the underlying graph is a lattice, and the generators of the walk are homogeneous in space. Using the results recently obtained in Carbone and Pautrat (Ann Henri Poincaré, 2015), we study the quantum trajectory associated with the OQRW, which is described by a position process and a state process. We obtain a central limit theorem and a large deviation principle for the position process. We study in detail the case of homogeneous OQRWs on the lattice , with internal space.

  1. Zero Range Process and Multi-Dimensional Random Walks

    NASA Astrophysics Data System (ADS)

    Bogoliubov, Nicolay M.; Malyshev, Cyril

    2017-07-01

    The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of random walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions.

  2. Impact Parameter Dependence of Elliptic Flow: A new Constraint for the Determination of the Equation of State

    NASA Astrophysics Data System (ADS)

    Lacey, Roy

    2000-10-01

    The delimitation of the parameters of the nuclear equation of state (EOS) has been, and continues to be, a major impetus for continued interest in proton elliptic flow measurements. In recent work we have shown that the elliptic flow of protons in relativistic heavy ion collisions can serve as an important probe for the EOS and QGP formation(P. Danielewicz, et al.), Phys. Rev. Let. 81, 2438, (1998),( C. Pinkenburg et al.) (E895 Collaboration), Phys. Rev. Lett. 83, 1295, (1999). Subsequently, we have studied the utility of the impact parameter dependence of elliptic flow as a source for important additional constraints for the determination of the EOS. Results for several beam energies will be presented and compared to model calculations.

  3. Impact Parmeter Dependence of Elliptic Flow: A new Constraint for the Determination of the Equation of State

    NASA Astrophysics Data System (ADS)

    Lacey, Roy A.

    2000-04-01

    The delimitation of the parameters of the nuclear equation of state (EOS) has been, and continues to be, a major impetus for continued interest in proton elliptic flow measurements. In recent work it has been shown that the elliptic flow of protons in relativistic heavy ion collisions can serve as an important probe for the EOS and QGP formation.(P. Danielewicz, et al.), Phys. Rev. Let. 81, 2438, (1998) (C. Pinkenburg et al.) (E895 Collaboration), Phys. Rev. Lett. 83, 1295, (1999) Subsequently, the E895 collaboration has investigated the utility of the impact parameter dependence of elliptic flow as an additional constraint for the EOS. Results for several beam energies will be presented and compared to model calculations.

  4. Simply and multiply scaled diffusion limits for continuous time random walks

    NASA Astrophysics Data System (ADS)

    Gorenflo, Rudolf; Mainardi, Francesco

    2005-01-01

    First a survey is presented on how space-time fractional diffusion processes can be obtained by well-scaled limiting from continuous time random walks under the sole assumption of asymptotic power laws (with appropriate exponents for the tail behaviour of waiting times and jumps). The spatial operator in the limiting pseudo-differential equation is the inverse of a general Riesz-Feller potential operator. The analysis is carried out via the transforms of Fourier and Laplace. Then mixtures of waiting time distributions, likewise of jump distributions, are considered, and it is shown that correct multiple scaling in the limit yields diffusion equations with distributed order fractional derivatives (fractional operators being replaced by integrals over such ones, with the order of differentiation as variable of integration). It is outlined how in this way super-fast and super-slow diffusion can be modelled.

  5. Steady state and mean recurrence time for random walks on stochastic temporal networks

    NASA Astrophysics Data System (ADS)

    Speidel, Leo; Lambiotte, Renaud; Aihara, Kazuyuki; Masuda, Naoki

    2015-01-01

    Random walks are basic diffusion processes on networks and have applications in, for example, searching, navigation, ranking, and community detection. Recent recognition of the importance of temporal aspects on networks spurred studies of random walks on temporal networks. Here we theoretically study two types of event-driven random walks on a stochastic temporal network model that produces arbitrary distributions of interevent times. In the so-called active random walk, the interevent time is reinitialized on all links upon each movement of the walker. In the so-called passive random walk, the interevent time is reinitialized only on the link that has been used the last time, and it is a type of correlated random walk. We find that the steady state is always the uniform density for the passive random walk. In contrast, for the active random walk, it increases or decreases with the node's degree depending on the distribution of interevent times. The mean recurrence time of a node is inversely proportional to the degree for both active and passive random walks. Furthermore, the mean recurrence time does or does not depend on the distribution of interevent times for the active and passive random walks, respectively.

  6. Effects of reciprocity on random walks in weighted networks.

    PubMed

    Zhang, Zhongzhi; Li, Huan; Sheng, Yibin

    2014-12-12

    It has been recently reported that the reciprocity of real-life weighted networks is very pronounced, however its impact on dynamical processes is poorly understood. In this paper, we study random walks in a scale-free directed weighted network with a trap at the central hub node, where the weight of each directed edge is dominated by a parameter controlling the extent of network reciprocity. We derive two expressions for the mean first passage time (MFPT) to the trap, by using two different techniques, the results of which agree well with each other. We also analytically determine all the eigenvalues as well as their multiplicities for the fundamental matrix of the dynamical process, and show that the largest eigenvalue has an identical dominant scaling as that of the MFPT.We find that the weight parameter has a substantial effect on the MFPT, which behaves as a power-law function of the system size with the power exponent dependent on the parameter, signaling the crucial role of reciprocity in random walks occurring in weighted networks.

  7. Effects of reciprocity on random walks in weighted networks

    PubMed Central

    Zhang, Zhongzhi; Li, Huan; Sheng, Yibin

    2014-01-01

    It has been recently reported that the reciprocity of real-life weighted networks is very pronounced, however its impact on dynamical processes is poorly understood. In this paper, we study random walks in a scale-free directed weighted network with a trap at the central hub node, where the weight of each directed edge is dominated by a parameter controlling the extent of network reciprocity. We derive two expressions for the mean first passage time (MFPT) to the trap, by using two different techniques, the results of which agree well with each other. We also analytically determine all the eigenvalues as well as their multiplicities for the fundamental matrix of the dynamical process, and show that the largest eigenvalue has an identical dominant scaling as that of the MFPT.We find that the weight parameter has a substantial effect on the MFPT, which behaves as a power-law function of the system size with the power exponent dependent on the parameter, signaling the crucial role of reciprocity in random walks occurring in weighted networks. PMID:25500907

  8. First Passage Time for Random Walks in Heterogeneous Networks

    NASA Astrophysics Data System (ADS)

    Hwang, S.; Lee, D.-S.; Kahng, B.

    2012-08-01

    The first passage time (FPT) for random walks is a key indicator of how fast information diffuses in a given system. Despite the role of FPT as a fundamental feature in transport phenomena, its behavior, particularly in heterogeneous networks, is not yet fully understood. Here, we study, both analytically and numerically, the scaling behavior of the FPT distribution to a given target node, averaged over all starting nodes. We find that random walks arrive quickly at a local hub, and therefore, the FPT distribution shows a crossover with respect to time from fast decay behavior (induced from the attractive effect to the hub) to slow decay behavior (caused by the exploring of the entire system). Moreover, the mean FPT is independent of the degree of the target node in the case of compact exploration. These theoretical results justify the necessity of using a random jump protocol (empirically used in search engines) and provide guidelines for designing an effective network to make information quickly accessible.

  9. Joint clustering of protein interaction networks through Markov random walk

    PubMed Central

    2014-01-01

    Biological networks obtained by high-throughput profiling or human curation are typically noisy. For functional module identification, single network clustering algorithms may not yield accurate and robust results. In order to borrow information across multiple sources to alleviate such problems due to data quality, we propose a new joint network clustering algorithm ASModel in this paper. We construct an integrated network to combine network topological information based on protein-protein interaction (PPI) datasets and homological information introduced by constituent similarity between proteins across networks. A novel random walk strategy on the integrated network is developed for joint network clustering and an optimization problem is formulated by searching for low conductance sets defined on the derived transition matrix of the random walk, which fuses both topology and homology information. The optimization problem of joint clustering is solved by a derived spectral clustering algorithm. Network clustering using several state-of-the-art algorithms has been implemented to both PPI networks within the same species (two yeast PPI networks and two human PPI networks) and those from different species (a yeast PPI network and a human PPI network). Experimental results demonstrate that ASModel outperforms the existing single network clustering algorithms as well as another recent joint clustering algorithm in terms of complex prediction and Gene Ontology (GO) enrichment analysis. PMID:24565376

  10. Cochlea segmentation using iterated random walks with shape prior

    NASA Astrophysics Data System (ADS)

    Ruiz Pujadas, Esmeralda; Kjer, Hans Martin; Vera, Sergio; Ceresa, Mario; González Ballester, Miguel Ángel

    2016-03-01

    Cochlear implants can restore hearing to deaf or partially deaf patients. In order to plan the intervention, a model from high resolution µCT images is to be built from accurate cochlea segmentations and then, adapted to a patient-specific model. Thus, a precise segmentation is required to build such a model. We propose a new framework for segmentation of µCT cochlear images using random walks where a region term is combined with a distance shape prior weighted by a confidence map to adjust its influence according to the strength of the image contour. Then, the region term can take advantage of the high contrast between the background and foreground and the distance prior guides the segmentation to the exterior of the cochlea as well as to less contrasted regions inside the cochlea. Finally, a refinement is performed preserving the topology using a topological method and an error control map to prevent boundary leakage. We tested the proposed approach with 10 datasets and compared it with the latest techniques with random walks and priors. The experiments suggest that this method gives promising results for cochlea segmentation.

  11. Radio variability and random walk noise properties of four blazars

    SciTech Connect

    Park, Jong-Ho; Trippe, Sascha E-mail: trippe@astro.snu.ac.kr

    2014-04-10

    We present the results of a time series analysis of the long-term radio light curves of four blazars: 3C 279, 3C 345, 3C 446, and BL Lacertae. We exploit the database of the University of Michigan Radio Astronomy Observatory monitoring program which provides densely sampled light curves spanning 32 years in time in three frequency bands located at 4.8, 8, and 14.5 GHz. Our sources show mostly flat or inverted (spectral indices –0.5 ≲ α ≲ 0) spectra, in agreement with optically thick emission. All light curves show strong variability on all timescales. Analyzing the time lags between the light curves from different frequency bands, we find that we can distinguish high-peaking flares and low-peaking flares in accordance with the classification of Valtaoja et al. The periodograms (temporal power spectra) of the observed light curves are consistent with random-walk power-law noise without any indication of (quasi-)periodic variability. The fact that all four sources studied are in agreement with being random-walk noise emitters at radio wavelengths suggests that such behavior is a general property of blazars.

  12. Characteristic times of biased random walks on complex networks

    NASA Astrophysics Data System (ADS)

    Bonaventura, Moreno; Nicosia, Vincenzo; Latora, Vito

    2014-01-01

    We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree k is proportional to kα, where α is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely (i) the time the walker needs to come back to the starting node, (ii) the time it takes to visit a given node for the first time, and (iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of α which minimizes the three characteristic times differs from the value αmin=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of αmin in the range [-1,-0.5], while disassortative networks have αmin in the range [-0.5,0]. We derive an analytical relation between the degree correlation exponent ν and the optimal bias value αmin, which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks by means of an appropriate tuning of the motion bias.

  13. Combinatorial approximation algorithms for MAXCUT using random walks.

    SciTech Connect

    Seshadhri, Comandur; Kale, Satyen

    2010-11-01

    We give the first combinatorial approximation algorithm for MaxCut that beats the trivial 0.5 factor by a constant. The main partitioning procedure is very intuitive, natural, and easily described. It essentially performs a number of random walks and aggregates the information to provide the partition. We can control the running time to get an approximation factor-running time tradeoff. We show that for any constant b > 1.5, there is an {tilde O}(n{sup b}) algorithm that outputs a (0.5 + {delta})-approximation for MaxCut, where {delta} = {delta}(b) is some positive constant. One of the components of our algorithm is a weak local graph partitioning procedure that may be of independent interest. Given a starting vertex i and a conductance parameter {phi}, unless a random walk of length {ell} = O(log n) starting from i mixes rapidly (in terms of {phi} and {ell}), we can find a cut of conductance at most {phi} close to the vertex. The work done per vertex found in the cut is sublinear in n.

  14. Cauchy's formulas for random walks in bounded domains

    NASA Astrophysics Data System (ADS)

    Mazzolo, Alain; de Mulatier, Clélia; Zoia, Andrea

    2014-08-01

    Cauchy's formula was originally established for random straight paths crossing a body B subset {R}n and basically relates the average chord length through B to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in B. Similar results are also obtained for the average number of collisions performed by the walker in B.

  15. Dynamic decoupling in the presence of 1D random walk

    NASA Astrophysics Data System (ADS)

    Chakrabarti, Arnab; Chakraborty, Ipsita; Bhattacharyya, Rangeet

    2016-05-01

    In the recent past, many dynamic decoupling sequences have been proposed for the suppression of decoherence of spins connected to thermal baths of various natures. Dynamic decoupling schemes for suppressing decoherence due to Gaussian diffusion have also been developed. In this work, we study the relative performances of dynamic decoupling schemes in the presence of a non-stationary Gaussian noise such as a 1D random walk. Frequency domain analysis is not suitable to determine the performances of various dynamic decoupling schemes in suppressing decoherence due to such a process. Thus, in this work, we follow a time domain calculation to arrive at the following conclusions: in the presence of such a noise, we show that (i) the traditional Carr-Purcell-Meiboom-Gill (CPMG) sequence outperforms Uhrig’s dynamic decoupling scheme, (ii) CPMG remains the optimal sequence for suppression of decoherence due to random walk in the presence of an external field gradient. Later, the theoretical predictions are experimentally verified by using nuclear magnetic resonance spectroscopy on spin 1/2 particles diffusing in a liquid medium.

  16. Cauchy's formulas for random walks in bounded domains

    SciTech Connect

    Mazzolo, Alain Zoia, Andrea

    2014-08-01

    Cauchy's formula was originally established for random straight paths crossing a body B⊂R{sup n} and basically relates the average chord length through B to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length traveled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in B. Similar results are also obtained for the average number of collisions performed by the walker in B.

  17. Characteristic times of biased random walks on complex networks.

    PubMed

    Bonaventura, Moreno; Nicosia, Vincenzo; Latora, Vito

    2014-01-01

    We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree k is proportional to k(α), where α is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely (i) the time the walker needs to come back to the starting node, (ii) the time it takes to visit a given node for the first time, and (iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of α which minimizes the three characteristic times differs from the value α(min)=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of α(min) in the range [-1,-0.5], while disassortative networks have α(min) in the range [-0.5,0]. We derive an analytical relation between the degree correlation exponent ν and the optimal bias value α(min), which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks by means of an appropriate tuning of the motion bias.

  18. Applications of random walks: From network exploration to cellulose hydrolysis

    NASA Astrophysics Data System (ADS)

    Asztalos, Andrea

    In the first part of the thesis we investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large, but finite graphs. An exact formula for the number of visited nodes and edges as function of time is presented, that is valid for arbitrary graphs and arbitrary walks defined by stationary transition probabilities (STP). We show that for STP walks site and edge exploration obey the same scaling ˜ nnu as function of time n, and therefore edge exploration on graphs with many loops is always lagging compared to site exploration. We then introduce the Edge Explorer Model, presenting a novel class of adaptive walks, that performs faithful network discovery even on dense networks. In the second part of the thesis we present a random walk-based computational model of enzymatic degradation of cellulose. The coarse-grained dynamical model accounts for the mobility and action of a single enzyme as well as for the synergy of multiple enzymes on a homogeneous cellulose surface. The quantitative description of cellulose degradation is calculated on a spatial model by including free and bound states of all enzymes with explicit reactive surface terms (e.g., hydrogen bond reformation) and corresponding reaction rates. The dynamical evolution of the system is based on physical interactions between enzymes and cellulose. We show how the model provides insight into enzyme loading and coverage for the degradation process.

  19. Direct discontinuous Galerkin method and its variations for second order elliptic equations

    SciTech Connect

    Huang, Hongying; Chen, Zheng; Li, Jin; Yan, Jue

    2016-08-23

    In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.

  20. Direct discontinuous Galerkin method and its variations for second order elliptic equations

    DOE PAGES

    Huang, Hongying; Chen, Zheng; Li, Jin; ...

    2016-08-23

    In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662,more » 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.« less

  1. Direct discontinuous Galerkin method and its variations for second order elliptic equations

    SciTech Connect

    Huang, Hongying; Chen, Zheng; Li, Jin; Yan, Jue

    2016-08-23

    In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.

  2. Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation

    DOE PAGES

    Jasra, Ajay; Law, Kody J. H.; Zhou, Yan

    2016-01-01

    Our paper considers uncertainty quantification for an elliptic nonlocal equation. In particular, it is assumed that the parameters which define the kernel in the nonlocal operator are uncertain and a priori distributed according to a probability measure. It is shown that the induced probability measure on some quantities of interest arising from functionals of the solution to the equation with random inputs is well-defined,s as is the posterior distribution on parameters given observations. As the elliptic nonlocal equation cannot be solved approximate posteriors are constructed. The multilevel Monte Carlo (MLMC) and multilevel sequential Monte Carlo (MLSMC) sampling algorithms are usedmore » for a priori and a posteriori estimation, respectively, of quantities of interest. Furthermore, these algorithms reduce the amount of work to estimate posterior expectations, for a given level of error, relative to Monte Carlo and i.i.d. sampling from the posterior at a given level of approximation of the solution of the elliptic nonlocal equation.« less

  3. An analysis of the convergence of Newton iterations for solving elliptic Kepler's equation

    NASA Astrophysics Data System (ADS)

    Elipe, A.; Montijano, J. I.; Rández, L.; Calvo, M.

    2017-09-01

    In this note a study of the convergence properties of some starters E_0 = E_0(e,M) in the eccentricity-mean anomaly variables for solving the elliptic Kepler's equation (KE) by Newton's method is presented. By using a Wang Xinghua's theorem (Xinghua in Math Comput 68(225):169-186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton's method, we obtain for each starter E_0(e,M) a set of values (e,M) \\in [0, 1) × [0, π ] that lead to the q-convergence in the sense that Newton's sequence (E_n)_{n ≥ 0} generated from E_0 = E_0(e,M) is well defined, converges to the exact solution E^* = E^*(e,M) of KE and further \\vert E_n - E^* \\vert ≤ q^{2^n -1} \\vert E_0 - E^* \\vert holds for all n ≥ 0 . This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27-44, 2014) by using Smale's α -test with q=1/2 . Also since in KE the convergence rate of Newton's method tends to zero as e → 0 , we show that the error estimates given in the Wang Xinghua's theorem for KE can also be used to determine sets of q-convergence with q = e^k \\widetilde{q} for all e \\in [0,1) and a fixed \\widetilde{q} ≤ 1 . Some remarks on the use of this theorem to derive a priori estimates of the error \\vert E_n - E^* \\vert after n Kepler's iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.

  4. Symmetric flux continuous positive definite approximation of the elliptic full tensor pressure equation in local conservative form

    SciTech Connect

    Edwards, M.G.

    1995-12-31

    A classical finite volume scheme well known in computational aerodynamics for solving the Transonic full potential equation is imported into reservoir simulation and applied to the full tensor pressure equation. Cell vertex discretization is shown to be a natural framework for approximation. With permeability placed at the cell centers and potential at the vertices (cell corner points), of the grid the scheme is flux continuous and locally conservative. Analysis is presented which proves that the resulting discrete matrix is symmetric positive definite provided the full permeability tensor is symmetric elliptic. The discrete matrix is also diagonally dominant subject to a sufficient elliptically condition. For a diagonal anisotropic tensor the discrete matrix is always symmetric positive definite and the scheme is up to 4th order accurate. A cell centered version of the scheme is indicated.

  5. Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure

    NASA Astrophysics Data System (ADS)

    Cappa, G.; Ferrari, S.

    2016-12-01

    Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ. The associated Cameron-Martin space is denoted by H. Let ν =e-U μ, where U : X → R is a sufficiently regular convex and continuous function. In this paper we are interested in the W 2 , 2 regularity of the weak solutions of elliptic equations of the type

  6. Asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy-Sobolev critical exponent

    NASA Astrophysics Data System (ADS)

    Hashizume, Masato

    2017-02-01

    We investigate the existence, the non-existence and the asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy-Sobolev critical exponent. In the boundary singularity case, it is known that the mean curvature of the boundary at origin plays a crucial role on the existence of the least-energy solutions. In this paper, we study the relation between the asymptotic behavior of the solutions and the mean curvature at origin.

  7. Record statistics of a strongly correlated time series: random walks and Lévy flights

    NASA Astrophysics Data System (ADS)

    Godrèche, Claude; Majumdar, Satya N.; Schehr, Grégory

    2017-08-01

    We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a Lévy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few years, it was indeed realized that random walks are a very useful ‘laboratory’ to test the effects of correlations on the record statistics. We start with the simple one-dimensional random walk with symmetric jumps (both continuous and discrete) and discuss in detail the statistics of the number of records, as well as of the ages of the records, i.e. the lapses of time between two successive record breaking events. Then we review the results that were obtained for a wide variety of random walk models, including random walks with a linear drift, continuous time random walks, constrained random walks (like the random walk bridge) and the case of multiple independent random walkers. Finally, we discuss further observables related to records, like the record increments, as well as some questions raised by physical applications of record statistics, like the effects of measurement error and noise.

  8. Random walk particle tracking simulations of non-Fickian transport in heterogeneous media

    SciTech Connect

    Srinivasan, G. Tartakovsky, D.M. Dentz, M. Viswanathan, H.; Berkowitz, B.; Robinson, B.A.

    2010-06-01

    Derivations of continuum nonlocal models of non-Fickian (anomalous) transport require assumptions that might limit their applicability. We present a particle-based algorithm, which obviates the need for many of these assumptions by allowing stochastic processes that represent spatial and temporal random increments to be correlated in space and time, be stationary or non-stationary, and to have arbitrary distributions. The approach treats a particle trajectory as a subordinated stochastic process that is described by a set of Langevin equations, which represent a continuous time random walk (CTRW). Convolution-based particle tracking (CBPT) is used to increase the computational efficiency and accuracy of these particle-based simulations. The combined CTRW-CBPT approach enables one to convert any particle tracking legacy code into a simulator capable of handling non-Fickian transport.

  9. Random walk in nonhomogeneous environments: A possible approach to human and animal mobility

    NASA Astrophysics Data System (ADS)

    Srokowski, Tomasz

    2017-03-01

    The random walk process in a nonhomogeneous medium, characterized by a Lévy stable distribution of jump length, is discussed. The width depends on a position: either before the jump or after that. In the latter case, the density slope is affected by the variable width and the variance may be finite; then all kinds of the anomalous diffusion are predicted. In the former case, only the time characteristics are sensitive to the variable width. The corresponding Langevin equation with different interpretations of the multiplicative noise is discussed. The dependence of the distribution width on position after jump is interpreted in terms of cognitive abilities and related to such problems as migration in a human population and foraging habits of animals.

  10. Social influencing and associated random walk models: Asymptotic consensus times on the complete graph

    NASA Astrophysics Data System (ADS)

    Zhang, W.; Lim, C.; Sreenivasan, S.; Xie, J.; Szymanski, B. K.; Korniss, G.

    2011-06-01

    We investigate consensus formation and the asymptotic consensus times in stylized individual- or agent-based models, in which global agreement is achieved through pairwise negotiations with or without a bias. Considering a class of individual-based models on finite complete graphs, we introduce a coarse-graining approach (lumping microscopic variables into macrostates) to analyze the ordering dynamics in an associated random-walk framework. Within this framework, yielding a linear system, we derive general equations for the expected consensus time and the expected time spent in each macro-state. Further, we present the asymptotic solutions of the 2-word naming game and separately discuss its behavior under the influence of an external field and with the introduction of committed agents.

  11. History dependent quantum random walks as quantum lattice gas automata

    SciTech Connect

    Shakeel, Asif E-mail: dmeyer@math.ucsd.edu Love, Peter J. E-mail: dmeyer@math.ucsd.edu; Meyer, David A. E-mail: dmeyer@math.ucsd.edu

    2014-12-15

    Quantum Random Walks (QRW) were first defined as one-particle sectors of Quantum Lattice Gas Automata (QLGA). Recently, they have been generalized to include history dependence, either on previous coin (internal, i.e., spin or velocity) states or on previous position states. These models have the goal of studying the transition to classicality, or more generally, changes in the performance of quantum walks in algorithmic applications. We show that several history dependent QRW can be identified as one-particle sectors of QLGA. This provides a unifying conceptual framework for these models in which the extra degrees of freedom required to store the history information arise naturally as geometrical degrees of freedom on the lattice.

  12. Maps of random walks on complex networks reveal community structure.

    PubMed

    Rosvall, Martin; Bergstrom, Carl T

    2008-01-29

    To comprehend the multipartite organization of large-scale biological and social systems, we introduce an information theoretic approach that reveals community structure in weighted and directed networks. We use the probability flow of random walks on a network as a proxy for information flows in the real system and decompose the network into modules by compressing a description of the probability flow. The result is a map that both simplifies and highlights the regularities in the structure and their relationships. We illustrate the method by making a map of scientific communication as captured in the citation patterns of >6,000 journals. We discover a multicentric organization with fields that vary dramatically in size and degree of integration into the network of science. Along the backbone of the network-including physics, chemistry, molecular biology, and medicine-information flows bidirectionally, but the map reveals a directional pattern of citation from the applied fields to the basic sciences.

  13. First-passage properties of bursty random walks

    NASA Astrophysics Data System (ADS)

    Volovik, D.; Redner, S.

    2010-06-01

    We investigate the first-passage properties of bursty random walks on a finite one-dimensional interval of length L, in which unit-length steps to the left occur with probability close to one, while steps of length b to the right—'bursts'—occur with small probability. This stochastic process provides a crude description of the early stages of virus spread in an organism after exposure. The interesting regime arises when b/L\\lesssim 1 , where the conditional exit time to reach L, corresponding to an infected state, has a non-monotonic dependence on initial position. Both the exit probability and the infection time exhibit complex dependencies on the initial condition due to the interplay between the burst length and interval length.

  14. Ergodic transitions in continuous-time random walks

    NASA Astrophysics Data System (ADS)

    Saa, Alberto; Venegeroles, Roberto

    2010-09-01

    We consider continuous-time random walk models described by arbitrary sojourn time probability density functions. We find a general expression for the distribution of time-averaged observables for such systems, generalizing some recent results presented in the literature. For the case where sojourn times are identically distributed independent random variables, our results shed some light on the recently proposed transitions between ergodic and weakly nonergodic regimes. On the other hand, for the case of nonidentical trapping time densities over the lattice points, the distribution of time-averaged observables reveals that such systems are typically nonergodic, in agreement with some recent experimental evidences on the statistics of blinking quantum dots. Some explicit examples are considered in detail. Our results are independent of the lattice topology and dimensionality.

  15. Random walks for spike-timing-dependent plasticity

    NASA Astrophysics Data System (ADS)

    Williams, Alan; Leen, Todd K.; Roberts, Patrick D.

    2004-08-01

    Random walk methods are used to calculate the moments of negative image equilibrium distributions in synaptic weight dynamics governed by spike-timing-dependent plasticity. The neural architecture of the model is based on the electrosensory lateral line lobe of mormyrid electric fish, which forms a negative image of the reafferent signal from the fish’s own electric discharge to optimize detection of sensory electric fields. Of particular behavioral importance to the fish is the variance of the equilibrium postsynaptic potential in the presence of noise, which is determined by the variance of the equilibrium weight distribution. Recurrence relations are derived for the moments of the equilibrium weight distribution, for arbitrary postsynaptic potential functions and arbitrary learning rules. For the case of homogeneous network parameters, explicit closed form solutions are developed for the covariances of the synaptic weight and postsynaptic potential distributions.

  16. Random walks on activity-driven networks with attractiveness

    NASA Astrophysics Data System (ADS)

    Alessandretti, Laura; Sun, Kaiyuan; Baronchelli, Andrea; Perra, Nicola

    2017-05-01

    Virtually all real-world networks are dynamical entities. In social networks, the propensity of nodes to engage in social interactions (activity) and their chances to be selected by active nodes (attractiveness) are heterogeneously distributed. Here, we present a time-varying network model where each node and the dynamical formation of ties are characterized by these two features. We study how these properties affect random-walk processes unfolding on the network when the time scales describing the process and the network evolution are comparable. We derive analytical solutions for the stationary state and the mean first-passage time of the process, and we study cases informed by empirical observations of social networks. Our work shows that previously disregarded properties of real social systems, such as heterogeneous distributions of activity and attractiveness as well as the correlations between them, substantially affect the dynamical process unfolding on the network.

  17. Information filtering via biased random walk on coupled social network.

    PubMed

    Nie, Da-Cheng; Zhang, Zi-Ke; Dong, Qiang; Sun, Chongjing; Fu, Yan

    2014-01-01

    The recommender systems have advanced a great deal in the past two decades. However, most researchers focus their attentions on mining the similarities among users or objects in recommender systems and overlook the social influence which plays an important role in users' purchase process. In this paper, we design a biased random walk algorithm on coupled social networks which gives recommendation results based on both social interests and users' preference. Numerical analyses on two real data sets, Epinions and Friendfeed, demonstrate the improvement of recommendation performance by taking social interests into account, and experimental results show that our algorithm can alleviate the user cold-start problem more effectively compared with the mass diffusion and user-based collaborative filtering methods.

  18. Subdiffusion in time-averaged, confined random walks

    NASA Astrophysics Data System (ADS)

    Neusius, Thomas; Sokolov, Igor M.; Smith, Jeremy C.

    2009-07-01

    Certain techniques characterizing diffusive processes, such as single-particle tracking or molecular dynamics simulation, provide time averages rather than ensemble averages. Whereas the ensemble-averaged mean-squared displacement (MSD) of an unbounded continuous time random walk (CTRW) with a broad distribution of waiting times exhibits subdiffusion, the time-averaged MSD, δ2¯ , does not. We demonstrate that, in contrast to the unbounded CTRW, in which δ2¯ is linear in the lag time Δ , the time-averaged MSD of the CTRW of a walker confined to a finite volume is sublinear in Δ , i.e., for long lag times δ2¯˜Δ1-α . The present results permit the application of CTRW to interpret time-averaged experimental quantities.

  19. Complex networks: when random walk dynamics equals synchronization

    NASA Astrophysics Data System (ADS)

    Kriener, Birgit; Anand, Lishma; Timme, Marc

    2012-09-01

    Synchrony prevalently emerges from the interactions of coupled dynamical units. For simple systems such as networks of phase oscillators, the asymptotic synchronization process is assumed to be equivalent to a Markov process that models standard diffusion or random walks on the same network topology. In this paper, we analytically derive the conditions for such equivalence for networks of pulse-coupled oscillators, which serve as models for neurons and pacemaker cells interacting by exchanging electric pulses or fireflies interacting via light flashes. We find that the pulse synchronization process is less simple, but there are classes of, e.g., network topologies that ensure equivalence. In particular, local dynamical operators are required to be doubly stochastic. These results provide a natural link between stochastic processes and deterministic synchronization on networks. Tools for analyzing diffusion (or, more generally, Markov processes) may now be transferred to pin down features of synchronization in networks of pulse-coupled units such as neural circuits.

  20. Maxima of two random walks: Universal statistics of lead changes

    SciTech Connect

    Ben-Naim, E.; Krapivsky, P. L.; Randon-Furling, J.

    2016-04-18

    In this study, we investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as ${\\pi }^{-1}\\mathrm{ln}t$ in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Lévy flights. We also show that the probability to have at most n lead changes behaves as ${t}^{-1/4}{(\\mathrm{ln}t)}^{n}$ for Brownian motion and as ${t}^{-\\beta (\\mu )}{(\\mathrm{ln}t)}^{n}$ for symmetric Lévy flights with index μ. The decay exponent $\\beta \\equiv \\beta (\\mu )$ varies continuously with the Lévy index when $0\\lt \\mu \\lt 2$, and remains constant $\\beta =1/4$ for $\\mu \\gt 2$.

  1. Spectral coarse graining for random walks in bipartite networks.

    PubMed

    Wang, Yang; Zeng, An; Di, Zengru; Fan, Ying

    2013-03-01

    Many real-world networks display a natural bipartite structure, yet analyzing and visualizing large bipartite networks is one of the open challenges in complex network research. A practical approach to this problem would be to reduce the complexity of the bipartite system while at the same time preserve its functionality. However, we find that existing coarse graining methods for monopartite networks usually fail for bipartite networks. In this paper, we use spectral analysis to design a coarse graining scheme specific for bipartite networks, which keeps their random walk properties unchanged. Numerical analysis on both artificial and real-world networks indicates that our coarse graining can better preserve most of the relevant spectral properties of the network. We validate our coarse graining method by directly comparing the mean first passage time of the walker in the original network and the reduced one.

  2. Knots and Random Walks in Vibrated Granular Chains

    NASA Astrophysics Data System (ADS)

    Ben-Naim, Eli

    2002-03-01

    We study experimentally and theoretically statistical properties of the opening times of knots in vertically vibrated granular chains. Our measurements are in good qualitative and quantitative agreement with a theoretical model involving three random walks interacting via hard-core exclusion in one spatial dimension. In particular, the knot survival probability follows a universal scaling function which is independent of the chain length, with a corresponding diffusive characteristic time scale. Both the large-exit-time and the small-exit-time tails of the distribution are suppressed exponentially, and the corresponding decay coefficients are in excellent agreement with theoretical values. E. Ben-Naim, Z. A. Daya, P. Vorobieff, and R. E. Ecke, Phys. Rev. Lett. 86, 1414 (2001).

  3. Information Filtering via Biased Random Walk on Coupled Social Network

    PubMed Central

    Dong, Qiang; Fu, Yan

    2014-01-01

    The recommender systems have advanced a great deal in the past two decades. However, most researchers focus their attentions on mining the similarities among users or objects in recommender systems and overlook the social influence which plays an important role in users' purchase process. In this paper, we design a biased random walk algorithm on coupled social networks which gives recommendation results based on both social interests and users' preference. Numerical analyses on two real data sets, Epinions and Friendfeed, demonstrate the improvement of recommendation performance by taking social interests into account, and experimental results show that our algorithm can alleviate the user cold-start problem more effectively compared with the mass diffusion and user-based collaborative filtering methods. PMID:25147867

  4. Tests of the random walk hypothesis for financial data

    NASA Astrophysics Data System (ADS)

    Nakamura, Tomomichi; Small, Michael

    2007-04-01

    We propose a method from the viewpoint of deterministic dynamical systems to investigate whether observed data follow a random walk (RW) and apply the method to several financial data. Our method is based on the previously proposed small-shuffle surrogate method. Hence, our method does not depend on the specific data distribution, although previously proposed methods depend on properties of the data distribution. The data we use are stock market (Standard & Poor's 500 in US market and Nikkei225 in Japanese market), exchange rate (British Pound/US dollar and Japanese Yen/US dollar), and commodity market (gold price and crude oil price). We found that these financial data are RW whose first differences are independently distributed random variables or time-varying random variables.

  5. Phase diffusion and random walk interpretation of electromagnetic scattering.

    PubMed

    Bahcivan, Hasan; Hysell, David L; Kelley, Michael C

    2003-08-01

    The relaxation behavior of phase observables for different particle diffusion models is found to establish a ground for radioscience interpretations of coherent backscatter spectra. The characteristic function for a random walk process at twice the incident radiation wave number is associated with the complex amplitude of the scattered field from a medium containing refractive index fluctuations. The phase relaxation function can be connected to the evolution of the characteristic function and may describe the average regression of the scattered field from a spontaneous fluctuation undergoing turbulent mixing. This connection holds when we assume that the stochastic description of particle movements based on a diffusion model is valid. The phase relaxation function, when identified as the generalized susceptibility function of the fluctuation dissipation theorem, is related to the spectral density of the scattered field from steady-state fluctuations.

  6. Statistics of largest loops in a random walk

    NASA Astrophysics Data System (ADS)

    Ertas, Deniz; Kantor, Yacov

    1997-01-01

    We report further findings on the size distribution of the largest neutral segments in a sequence of N randomly charged monomers [D. Ertaş and Y. Kantor, Phys. Rev. E 53, 846 (1996)]. Upon mapping to one-dimensional random walks (RW's), this corresponds to finding the probability distribution for the size L of the largest segment that returns to its starting position in an N-step RW. We focus primarily on the large N, l=L/N<<1 limit, which exhibits an essential singularity. We establish analytical upper and lower bounds on the probability distribution, and numerically probe the distribution down to l~0.04 (corresponding to probabilities as low as 10-15) using a recursive Monte Carlo algorithm. We also investigate the possibility of singularities at l=1/k for integer k.

  7. Conditioned random walks and interaction-driven condensation

    NASA Astrophysics Data System (ADS)

    Szavits-Nossan, Juraj; Evans, Martin R.; Majumdar, Satya N.

    2017-01-01

    We consider a discrete-time continuous-space random walk under the constraints that the number of returns to the origin (local time) and the total area under the walk are fixed. We first compute the joint probability of an excursion having area a and returning to the origin for the first time after time τ. We then show how condensation occurs when the total area constraint is increased: an excursion containing a finite fraction of the area emerges. Finally we show how the phenomena generalises previously studied cases of condensation induced by several constraints and how it is related to interaction-driven condensation which allows us to explain the phenomenon in the framework of large deviation theory.

  8. Maxima of two random walks: Universal statistics of lead changes

    SciTech Connect

    Ben-Naim, E.; Krapivsky, P. L.; Randon-Furling, J.

    2016-04-18

    In this study, we investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as ${\\pi }^{-1}\\mathrm{ln}t$ in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Lévy flights. We also show that the probability to have at most n lead changes behaves as ${t}^{-1/4}{(\\mathrm{ln}t)}^{n}$ for Brownian motion and as ${t}^{-\\beta (\\mu )}{(\\mathrm{ln}t)}^{n}$ for symmetric Lévy flights with index μ. The decay exponent $\\beta \\equiv \\beta (\\mu )$ varies continuously with the Lévy index when $0\\lt \\mu \\lt 2$, and remains constant $\\beta =1/4$ for $\\mu \\gt 2$.

  9. Composition of many spins, random walks and statistics

    NASA Astrophysics Data System (ADS)

    Polychronakos, Alexios P.; Sfetsos, Konstantinos

    2016-12-01

    The multiplicities of the decomposition of the product of an arbitrary number n of spin s states into irreducible SU (2) representations are computed. Two complementary methods are presented, one based on random walks in representation space and another based on the partition function of the system in the presence of a magnetic field. The large-n scaling limit of these multiplicities is derived, including nonperturbative corrections, and related to semiclassical features of the system. A physical application of these results to ferromagnetism is explicitly worked out. Generalizations involving several types of spins, as well as spin distributions, are also presented. The corresponding problem for (anti-)symmetric composition of spins is also considered and shown to obey remarkable duality and bosonization relations and exhibit novel large-n scaling properties.

  10. Correlated continuous time random walk and option pricing

    NASA Astrophysics Data System (ADS)

    Lv, Longjin; Xiao, Jianbin; Fan, Liangzhong; Ren, Fuyao

    2016-04-01

    In this paper, we study a correlated continuous time random walk (CCTRW) with averaged waiting time, whose probability density function (PDF) is proved to follow stretched Gaussian distribution. Then, we apply this process into option pricing problem. Supposing the price of the underlying is driven by this CCTRW, we find this model captures the subdiffusive characteristic of financial markets. By using the mean self-financing hedging strategy, we obtain the closed-form pricing formulas for a European option with and without transaction costs, respectively. At last, comparing the obtained model with the classical Black-Scholes model, we find the price obtained in this paper is higher than that obtained from the Black-Scholes model. A empirical analysis is also introduced to confirm the obtained results can fit the real data well.

  11. The linear Ising model and its analytic continuation, random walk

    NASA Astrophysics Data System (ADS)

    Lavenda, B. H.

    2004-02-01

    A generalization of Gauss's principle is used to derive the error laws corresponding to Types II and VII distributions in Pearson's classification scheme. Student's r-p.d.f. (Type II) governs the distribution of the internal energy of a uniform, linear chain, Ising model, while the analytic continuation of the uniform exchange energy converts it into a Student t-density (Type VII) for the position of a random walk in a single spatial dimension. Higher-dimensional spaces, corresponding to larger degrees of freedom and generalizations to multidimensional Student r- and t-densities, are obtained by considering independent and identically random variables, having rotationally invariant densities, whose entropies are additive and generating functions are multiplicative.

  12. On the elliptic flow for nearly symmetric collisions and nuclear equation of state

    NASA Astrophysics Data System (ADS)

    Kaur, Varinderjit; Kumar, Suneel

    2011-12-01

    We here present the results of elliptical flow for the collision of different asymmetric nuclei (10Ne20 +13 Al27, 18Ar40 +21 Sc45, 30Zn64 +28 Ni58, 36Kr86 +41 Nb93) by using the Quantum Molecular Dynamics (QMD) model. General features of elliptical flow are investigated with the help of theoretical simulations. The simulations are performed at different beam energies between 40 and 105 MeV/nucleon. A significant change can be seen from in-plane to out-of-plane elliptical flow of different fragments with incident energy. A comparison with experimental data is also made. Further, we predict, for the first time that, elliptical flow for different kind of fragments follow power law dependence ? C(Atot)? for asymmetric systems.

  13. DCPT: A dual-continua random walk particle tracker fortransport

    SciTech Connect

    Pan, L.; Liu, H.H.; Cushey, M.; Bodvarsson, G.S.

    2000-04-11

    Accurate and efficient simulation of chemical transport processes in the unsaturated zone of Yucca Mountain is important to evaluate the performance of the potential repository. The scale of the unsaturated zone model domain for Yucca Mountain (50 km{sup 2} area with a 600 meter depth to the water table) requires a large gridblock approach to efficiently analyze complex flow & transport processes. The conventional schemes based on finite element or finite difference methods perform well for dispersion-dominated transport, but are subject to considerable numerical dilution/dispersion for advection-dominated transport, especially when a large gridblock size is used. Numerical dispersion is an artificial, grid-dependent chemical spreading, especially for otherwise steep concentration fronts. One effective scheme to deal with numerical dispersion is the random walk particle method (RWPM). While significant progress has been made in developing RWPM algorithms and codes for single continuum systems, a random walk particle tracker, which can handle chemical transport in dual-continua (fractured porous media) associated with irregular grid systems, is still absent (to our knowledge) in the public domain. This is largely due to the lacking of rigorous schemes to deal with particle transfer between the continua, and efficient schemes to track particles in irregular grid systems. The main objectives of this study are (1) to develop approaches to extend RWPM from a single continuum to a dual-continua system; (2) to develop an efficient algorithm for tracking particles in 3D irregular grids; and (3) to integrate these approaches into an efficient and user-friendly software, DCPT, for simulating chemical transport in fractured porous media.

  14. Geometric random walk of finite number of agents under constant variance

    NASA Astrophysics Data System (ADS)

    Yano, Ryosuke

    2017-05-01

    The characteristics of the 1D geometric random walk of a finite number of agents are investigated by assuming constant variance. Firstly, the characteristics of the steady state solution of the distribution function, which is obtained using the extended geometric Brownian motion (EGBM), are investigated in the framework of the 1D Fokker-Planck type equation. The uniqueness and existence of the steady state solution of the distribution function requires the number of particles to be finite. To avoid the divergence of the steady state solution of the distribution function at the mean value in the 1D Fokker-Planck type equation, the hybrid model, which is a combination of EGBM and normal BM, is proposed. Next, the steady state solution of the distribution function, which is obtained using the geometric Lévy flight, is investigated under constant variance in the framework of the space fractional 1D Fokker-Planck type equation. Additionally, we confirm that the solution of the distribution function obtained using the super-elastic and inelastic (SI-) Boltzmann equation under constant variance approaches the Cauchy distribution, when the power law number of the relative velocity increases. Finally, dissipation processes of the pressure deviator and heat flux are numerically investigated using the 2D space fractional Fokker-Planck type equations for Lévy flight and SI-Boltzmann equation by assuming their linear response relations.

  15. L_p-estimates for the nontangential maximal function of the solution to a second-order elliptic equation

    NASA Astrophysics Data System (ADS)

    Gushchin, A. K.

    2016-10-01

    The paper is concerned with the properties of the solution to a Dirichlet problem for a homogeneous second-order elliptic equation with L_p-boundary function, p>1. The same conditions are imposed on the coefficients of the equation and the boundary of the bounded domain as were used to establish the solvability of this problem. The L_p-norm of the nontangential maximal function is estimated in terms of the L_p-norm of the boundary value. This result depends on a new estimate, proved below, for the nontangential maximal function in terms of an analogue of the Lusin area integral. Bibliography: 31 titles.

  16. Asymptotic Behaviour of the Ground State of Singularly Perturbed Elliptic Equations

    NASA Astrophysics Data System (ADS)

    Piatnitski, Andrey L.

    The ground state of a singularly perturbed nonselfadjoint elliptic operator defined on a smooth compact Riemannian manifold with metric aij(x)=(aij(x))-1, is studied. We investigate the limiting behaviour of the first eigenvalue of this operator as μ goes to zero, and find the logarithmic asymptotics of the first eigenfunction everywhere on the manifold. The results are formulated in terms of auxiliary variational problems on the manifold. This approach also allows to study the general singularly perturbed second order elliptic operator on a bounded domain in Rn.

  17. Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

    SciTech Connect

    Kalchev, D

    2012-04-02

    This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigen-problems (corresponding to eigenvalues close to zero) form the

  18. Random walk approach for dispersive transport in pipe networks

    NASA Astrophysics Data System (ADS)

    Sämann, Robert; Graf, Thomas; Neuweiler, Insa

    2016-04-01

    Keywords: particle transport, random walk, pipe, network, HYSTEM-EXTAN, OpenGeoSys After heavy pluvial events in urban areas the available drainage system may be undersized at peak flows (Fuchs, 2013). Consequently, rainwater in the pipe network is likely to spill out through manholes. The presence of hazardous contaminants in the pipe drainage system represents a potential risk to humans especially when the contaminated drainage water reaches the land surface. Real-time forecasting of contaminants in the drainage system needs a quick calculation. Numerical models to predict the fate of contaminants are usually based on finite volume methods. Those are not applicable here because of their volume averaging elements. Thus, a more efficient method is preferable, which is independent from spatial discretization. In the present study, a particle-based method is chosen to calculate transport paths and spatial distribution of contaminants within a pipe network. A random walk method for particles in turbulent flow in partially filled pipes has been developed. Different approaches for in-pipe-mixing and node-mixing with respect to the geometry in a drainage network are shown. A comparison of dispersive behavior and calculation time is given to find the fastest model. The HYSTEM-EXTRAN (itwh, 2002) model is used to provide hydrodynamic conditions in the pipe network according to surface runoff scenarios in order to real-time predict contaminant transport in an urban pipe network system. The newly developed particle-based model will later be coupled to the subsurface flow model OpenGeoSys (Kolditz et al., 2012). References: Fuchs, L. (2013). Gefährdungsanalyse zur Überflutungsvorsorge kommunaler Entwässerungssysteme. Sanierung und Anpassung von Entwässerungssystemen-Alternde Infrastruktur und Klimawandel, Österreichischer Wasser-und Abfallwirtschaftsverband, Wien, ISBN, 978-3. itwh (2002). Modellbeschreibung, Institut für technisch-wissenschaftliche Hydrologie Gmb

  19. MFPT calculation for random walks in inhomogeneous networks

    NASA Astrophysics Data System (ADS)

    Wijesundera, Isuri; Halgamuge, Malka N.; Nirmalathas, Ampalavanapillai; Nanayakkara, Thrishantha

    2016-11-01

    Knowing the expected arrival time at a particular state, also known as the mean first passage time (MFPT), often plays an important role for a large class of random walkers in their respective state-spaces. Contrasting to ideal conditions required by recent advancements on MFPT estimations, many naturally occurring random walkers encounter inhomogeneity of transport characteristics in the networks they walk on. This paper presents a heuristic method to divide an inhomogeneous network into homogeneous network primitives (NPs) optimized using particle swarm optimizer, and to use a 'hop-wise' MFPT calculation method. This methodology's potential is demonstrated through simulated random walks and with a case study using the dataset of past cyclone tracks over the North Atlantic Ocean. Parallel processing was used to increase calculation efficiency. The predictions using the proposed method are compared to real data averages and predictions assuming homogeneous transport properties. The results show that breaking the problem into NPs reduces the average error from 18.8% to 5.4% with respect to the homogeneous network assumption.

  20. How fast does a random walk cover a torus?

    NASA Astrophysics Data System (ADS)

    Grassberger, Peter

    2017-07-01

    We present high statistics simulation data for the average time that a random walk needs to cover completely a two-dimensional torus of size L ×L . They confirm the mathematical prediction that ˜(LlnL ) 2 for large L , but the prefactor seems to deviate significantly from the supposedly exact result 4 /π derived by Dembo et al. [Ann. Math. 160, 433 (2004), 10.4007/annals.2004.160.433], if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time TN (t )=1(L ) at which the average number of yet unvisited sites is 1, as also predicted previously. This might suggest (wrongly) that and TN (t )=1(L ) scale differently, although the distribution of rescaled cover times becomes sharp in the limit L →∞ . But our results can be reconciled with those of Dembo et al. by a very slow and nonmonotonic convergence of /(LlnL ) 2 , as had been indeed proven by Belius et al. [Probab. Theory Relat. Fields 167, 461 (2017), 10.1007/s00440-015-0689-6] for Brownian walks, and was conjectured by them to hold also for lattice walks.

  1. Ranking Competitors Using Degree-Neutralized Random Walks

    PubMed Central

    Shin, Seungkyu; Ahnert, Sebastian E.; Park, Juyong

    2014-01-01

    Competition is ubiquitous in many complex biological, social, and technological systems, playing an integral role in the evolutionary dynamics of the systems. It is often useful to determine the dominance hierarchy or the rankings of the components of the system that compete for survival and success based on the outcomes of the competitions between them. Here we propose a ranking method based on the random walk on the network representing the competitors as nodes and competitions as directed edges with asymmetric weights. We use the edge weights and node degrees to define the gradient on each edge that guides the random walker towards the weaker (or the stronger) node, which enables us to interpret the steady-state occupancy as the measure of the node's weakness (or strength) that is free of unwarranted degree-induced bias. We apply our method to two real-world competition networks and explore the issues of ranking stabilization and prediction accuracy, finding that our method outperforms other methods including the baseline win–loss differential method in sparse networks. PMID:25517977

  2. IS QUASAR OPTICAL VARIABILITY A DAMPED RANDOM WALK?

    SciTech Connect

    Zu Ying; Kochanek, C. S.; Kozlowski, Szymon; Udalski, Andrzej

    2013-03-10

    The damped random walk (DRW) model is increasingly used to model the variability in quasar optical light curves, but it is still uncertain whether the DRW model provides an adequate description of quasar optical variability across all timescales. Using a sample of OGLE quasar light curves, we consider four modifications to the DRW model by introducing additional parameters into the covariance function to search for deviations from the DRW model on both short and long timescales. We find good agreement with the DRW model on timescales that are well sampled by the data (from a month to a few years), possibly with some intrinsic scatter in the additional parameters, but this conclusion depends on the statistical test employed and is sensitive to whether the estimates of the photometric errors are correct to within {approx}10%. On very short timescales (below a few months), we see some evidence of the existence of a cutoff below which the correlation is stronger than the DRW model, echoing the recent finding of Mushotzky et al. using quasar light curves from Kepler. On very long timescales (>a few years), the light curves do not constrain models well, but are consistent with the DRW model.

  3. Electron avalanche structure determined by random walk theory

    NASA Technical Reports Server (NTRS)

    Englert, G. W.

    1973-01-01

    A self-consistent avalanche solution which accounts for collective long range Coulomb interactions as well as short range elastic and inelastic collisions between electrons and background atoms is made possible by a random walk technique. Results show that the electric field patterns in the early formation stages of avalanches in helium are close to those obtained from theory based on constant transport coefficients. Regions of maximum and minimum induced electrostatic potential phi are located on the axis of symmetry and within the volume covered by the electron swarm. As formation time continues, however, the region of minimum phi moves to slightly higher radii and the electric field between the extrema becomes somewhat erratic. In the intermediate formation periods the avalanche growth is slightly retarded by the high concentration of ions in the tail which oppose the external electric field. Eventually the formation of ions and electrons in the localized regions of high field strength more than offset this effect causing a very abrupt increase in avalanche growth.

  4. Learning Markov Random Walks for robust subspace clustering and estimation.

    PubMed

    Liu, Risheng; Lin, Zhouchen; Su, Zhixun

    2014-11-01

    Markov Random Walks (MRW) has proven to be an effective way to understand spectral clustering and embedding. However, due to less global structural measure, conventional MRW (e.g., the Gaussian kernel MRW) cannot be applied to handle data points drawn from a mixture of subspaces. In this paper, we introduce a regularized MRW learning model, using a low-rank penalty to constrain the global subspace structure, for subspace clustering and estimation. In our framework, both the local pairwise similarity and the global subspace structure can be learnt from the transition probabilities of MRW. We prove that under some suitable conditions, our proposed local/global criteria can exactly capture the multiple subspace structure and learn a low-dimensional embedding for the data, in which giving the true segmentation of subspaces. To improve robustness in real situations, we also propose an extension of the MRW learning model based on integrating transition matrix learning and error correction in a unified framework. Experimental results on both synthetic data and real applications demonstrate that our proposed MRW learning model and its robust extension outperform the state-of-the-art subspace clustering methods.

  5. An effective Hamiltonian approach to quantum random walk

    NASA Astrophysics Data System (ADS)

    Sarkar, Debajyoti; Paul, Niladri; Bhattacharya, Kaushik; Ghosh, Tarun Kanti

    2017-03-01

    In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative (see Chandrashekar, Sci. Rep. 3, 2829 (18)). We showed that in the case of two-step walk, the time evolution operator effectively can have multiplicative form. In the case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the graphene Hamiltonian, the walk has been studied on a graphene lattice and we conclude the preference of additive approach over the multiplicative one.

  6. Peer-to-Peer Topology Formation Using Random Walk

    NASA Astrophysics Data System (ADS)

    Kwong, Kin-Wah; Tsang, Danny H. K.

    Peer-to-Peer (P2P) systems such as live video streaming and content sharing are usually composed of a huge number of users with heterogeneous capacities. As a result, designing a distributed algorithm to form such a giant-scale topology in a heterogeneous environment is a challenging question because, on the one hand, the algorithm should exploit the heterogeneity of users' capacities to achieve load-balancing and, on the other hand, the overhead of the algorithm should be kept as low as possible. To meet such requirements, we introduce a very simple protocol for building heterogeneous unstructured P2P networks. The basic idea behind our protocol is to exploit a simple, distributed nature of random walk sampling to assist the peers in selecting their suitable neighbors in terms of capacity and connectivity to achieve load-balancing. To gain more insights into our proposed protocol, we also develop a detailed analysis to investigate our protocol under any heterogeneous P2P environment. The analytical results are validated by the simulations. The ultimate goal of this chapter is to stimulate further research to explore the fundamental issues in heterogeneous P2P networks.

  7. Scaling properties of random walks on small-world networks.

    PubMed

    Almaas, E; Kulkarni, R V; Stroud, D

    2003-11-01

    Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in d dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.

  8. Maxima of two random walks: Universal statistics of lead changes

    DOE PAGES

    Ben-Naim, E.; Krapivsky, P. L.; Randon-Furling, J.

    2016-04-18

    In this study, we investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows asmore » $${\\pi }^{-1}\\mathrm{ln}t$$ in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Lévy flights. We also show that the probability to have at most n lead changes behaves as $${t}^{-1/4}{(\\mathrm{ln}t)}^{n}$$ for Brownian motion and as $${t}^{-\\beta (\\mu )}{(\\mathrm{ln}t)}^{n}$$ for symmetric Lévy flights with index μ. The decay exponent $$\\beta \\equiv \\beta (\\mu )$$ varies continuously with the Lévy index when $$0\\lt \\mu \\lt 2$$, and remains constant $$\\beta =1/4$$ for $$\\mu \\gt 2$$.« less

  9. Convex hulls of random walks: Large-deviation properties

    NASA Astrophysics Data System (ADS)

    Claussen, Gunnar; Hartmann, Alexander K.; Majumdar, Satya N.

    2015-05-01

    We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A . While the mean perimeter and the mean area have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P (A ) and P (L ) . In this paper we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P (A ) and P (L ) are as small as 10-300. We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T , a universal scaling behavior (independent of the details of the jump distributions) as a function of A /T and L /√{T } , respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

  10. Convex hulls of random walks: Large-deviation properties.

    PubMed

    Claussen, Gunnar; Hartmann, Alexander K; Majumdar, Satya N

    2015-05-01

    We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter 〈L〉 and the mean area 〈A〉 have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P(A) and P(L). In this paper we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P(A) and P(L) are as small as 10(-300). We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T, a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/√[T], respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

  11. Intracellular transport of insulin granules is a subordinated random walk

    PubMed Central

    Tabei, S. M. Ali; Burov, Stanislav; Kim, Hee Y.; Kuznetsov, Andrey; Huynh, Toan; Jureller, Justin; Philipson, Louis H.; Dinner, Aaron R.; Scherer, Norbert F.

    2013-01-01

    We quantitatively analyzed particle tracking data on insulin granules expressing fluorescent fusion proteins in MIN6 cells to better understand the motions contributing to intracellular transport and, more generally, the means for characterizing systems far from equilibrium. Care was taken to ensure that the statistics reflected intrinsic features of the individual granules rather than details of the measurement and overall cell state. We find anomalous diffusion. Interpreting such data conventionally requires assuming that a process is either ergodic with particles working against fluctuating obstacles (fractional Brownian motion) or nonergodic with a broad distribution of dwell times for traps (continuous-time random walk). However, we find that statistical tests based on these two models give conflicting results. We resolve this issue by introducing a subordinated scheme in which particles in cages with random dwell times undergo correlated motions owing to interactions with a fluctuating environment. We relate this picture to the underlying microtubule structure by imaging in the presence of vinblastine. Our results provide a simple physical picture for how diverse pools of insulin granules and, in turn, biphasic secretion could arise. PMID:23479621

  12. Accurate multiple network alignment through context-sensitive random walk

    PubMed Central

    2015-01-01

    Background Comparative network analysis can provide an effective means of analyzing large-scale biological networks and gaining novel insights into their structure and organization. Global network alignment aims to predict the best overall mapping between a given set of biological networks, thereby identifying important similarities as well as differences among the networks. It has been shown that network alignment methods can be used to detect pathways or network modules that are conserved across different networks. Until now, a number of network alignment algorithms have been proposed based on different formulations and approaches, many of them focusing on pairwise alignment. Results In this work, we propose a novel multiple network alignment algorithm based on a context-sensitive random walk model. The random walker employed in the proposed algorithm switches between two different modes, namely, an individual walk on a single network and a simultaneous walk on two networks. The switching decision is made in a context-sensitive manner by examining the current neighborhood, which is effective for quantitatively estimating the degree of correspondence between nodes that belong to different networks, in a manner that sensibly integrates node similarity and topological similarity. The resulting node correspondence scores are then used to predict the maximum expected accuracy (MEA) alignment of the given networks. Conclusions Performance evaluation based on synthetic networks as well as real protein-protein interaction networks shows that the proposed algorithm can construct more accurate multiple network alignments compared to other leading methods. PMID:25707987

  13. Random walk models of worker sorting in ant colonies.

    PubMed

    Sendova-Franks, Ana B; Van Lent, Jan

    2002-07-21

    Sorting can be an important mechanism for the transfer of information from one level of biological organization to another. Here we study the algorithm underlying worker sorting in Leptothorax ant colonies. Worker sorting is related to task allocation and therefore to the adaptive advantages associated with an efficient system for the division of labour in ant colonies. We considered four spatially explicit individual-based models founded on two-dimensional correlated random walk. Our aim was to establish whether sorting at the level of the worker population could occur with minimal assumptions about the behavioural algorithm of individual workers. The behaviour of an individual worker in the models could be summarized by the rule "move if you can, turn always". We assume that the turning angle of a worker is individually specific and negatively dependent on the magnitude of an internal parameter micro which could be regarded as a measure of individual experience or task specialization. All four models attained a level of worker sortedness that was compatible with results from experiments onLeptothorax ant colonies. We found that the presence of a sorting pivot, such as the nest wall or an attraction force towards the centre of the worker population, was crucial for sorting. We make a distinction between such pivots and templates and discuss the biological implications of their difference.

  14. Random Walks and Effective Optical Depth in Relativistic Flow

    NASA Astrophysics Data System (ADS)

    Shibata, Sanshiro; Tominaga, Nozomu; Tanaka, Masaomi

    2014-05-01

    We investigate the random walk process in relativistic flow. In the relativistic flow, photon propagation is concentrated in the direction of the flow velocity due to the relativistic beaming effect. We show that in the pure scattering case, the number of scatterings is proportional to the size parameter ξ ≡ L/l 0 if the flow velocity β ≡ v/c satisfies β/Γ Gt ξ-1, while it is proportional to ξ2 if β/Γ Lt ξ-1, where L and l 0 are the size of the system in the observer frame and the mean free path in the comoving frame, respectively. We also examine the photon propagation in the scattering and absorptive medium. We find that if the optical depth for absorption τa is considerably smaller than the optical depth for scattering τs (τa/τs Lt 1) and the flow velocity satisfies \\beta \\gg \\sqrt{2\\tau _a/\\tau _s}, then the effective optical depth is approximated by τ* ~= τa(1 + β)/β. Furthermore, we perform Monte Carlo simulations of radiative transfer and compare the results with the analytic expression for the number of scatterings. The analytic expression is consistent with the results of the numerical simulations. The expression derived in this study can be used to estimate the photon production site in relativistic phenomena, e.g., gamma-ray burst and active galactic nuclei.

  15. The linking number and the writhe of uniform random walks and polygons in confined spaces

    NASA Astrophysics Data System (ADS)

    Panagiotou, E.; Millett, K. C.; Lambropoulou, S.

    2010-01-01

    Random walks and polygons are used to model polymers. In this paper we consider the extension of the writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n2). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form O(\\sqrt{n}) . Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length.

  16. Generalized solutions for a class of non-uniformly elliptic equations in divergence form

    SciTech Connect

    Gregori, G.

    1997-11-01

    We study a general class of quasilinear non-uniformly elliptic pdes in divergence form with linear growth in the gradient. We examine the notions of BV and viscosity solutions and derive for such generalized solutions various a priori pointwise and integral estimates, including a Harnack inequality. In particular we prove that viscosity solutions are unique (on strictly convex domains), are contained in the space BV{sub loc} and are C{sup 1,{alpha}} almost everywhere. 10 refs.

  17. A Straightforward Random Walk Model for Fast Push-Pull Tracer Test Evaluation.

    PubMed

    Klotzsch, Stephan; Binder, Martin; Händel, Falk

    2017-01-01

    In this article, we present a straightforward random walk model for fast evaluation of push-pull tracer tests. By developing an adaptive algorithm, we overcome the problem of manually defining how many particles have to be used to simulate the transport problem. Beside this, we validate the random walk model by evaluating a push-pull tracer test with drift phase and confirm the results with MT3DMS. The random walk model took less than 1% of computational time of MT3DMS, thus allowing a remarkable faster evaluation of push-pull tracer tests. © 2016, National Ground Water Association.

  18. Quantum random walks on congested lattices and the effect of dephasing

    PubMed Central

    Motes, Keith R.; Gilchrist, Alexei; Rohde, Peter P.

    2016-01-01

    We consider quantum random walks on congested lattices and contrast them to classical random walks. Congestion is modelled on lattices that contain static defects which reverse the walker’s direction. We implement a dephasing process after each step which allows us to smoothly interpolate between classical and quantum random walks as well as study the effect of dephasing on the quantum walk. Our key results show that a quantum walker escapes a finite boundary dramatically faster than a classical walker and that this advantage remains in the presence of heavily congested lattices. PMID:26812924

  19. Correlated Random Walks with a Finite Memory Range

    NASA Astrophysics Data System (ADS)

    Bidaux, Roger; Boccara, Nino

    We study a family of correlated one-dimensional random walks with a finite memory range M. These walks are extensions of the Taylor's walk as investigated by Goldstein, which has a memory range equal to one. At each step, with a probability p, the random walker moves either to the right or to the left with equal probabilities, or with a probability q=1-p performs a move, which is a stochastic Boolean function of the M previous steps. We first derive the most general form of this stochastic Boolean function, and study some typical cases which ensure that the average value of the walker's location after n steps is zero for all values of n. In each case, using a matrix technique, we provide a general method for constructing the generating function of the probability distribution of Rn; we also establish directly an exact analytic expression for the step-step correlations and the variance < R2n > of the walk. From the expression of < R2n >, which is not straightforward to derive from the probability distribution, we show that, for n approaching infinity, the variance of any of these walks behaves as n, provided p>0. Moreover, in many cases, for a very small fixed value of p, the variance exhibits a crossover phenomenon as n increases from a not too large value. The crossover takes place for values of n around 1/p. This feature may mimic the existence of a nontrivial Hurst exponent, and induce a misleading analysis of numerical data issued from mathematical or natural sciences experiments.

  20. Exact and approximate graph matching using random walks.

    PubMed

    Gori, Marco; Maggini, Marco; Sarti, Lorenzo

    2005-07-01

    In this paper, we propose a general framework for graph matching which is suitable for different problems of pattern recognition. The pattern representation we assume is at the same time highly structured, like for classic syntactic and structural approaches, and of subsymbolic nature with real-valued features, like for connectionist and statistic approaches. We show that random walk based models, inspired by Google's PageRank, give rise to a spectral theory that nicely enhances the graph topological features at node level. As a straightforward consequence, we derive a polynomial algorithm for the classic graph isomorphism problem, under the restriction of dealing with Markovian spectrally distinguishable graphs (MSD), a class of graphs that does not seem to be easily reducible to others proposed in the literature. The experimental results that we found on different test-beds of the TC-15 graph database show that the defined MSD class "almost always" covers the database, and that the proposed algorithm is significantly more efficient than top scoring VF algorithm on the same data. Most interestingly, the proposed approach is very well-suited for dealing with partial and approximate graph matching problems, derived for instance from image retrieval tasks. We consider the objects of the COIL-100 visual collection and provide a graph-based representation, whose node's labels contain appropriate visual features. We show that the adoption of classic bipartite graph matching algorithms offers a straightforward generalization of the algorithm given for graph isomorphism and, finally, we report very promising experimental results on the COIL-100 visual collection.

  1. RANDOM WALKS AND EFFECTIVE OPTICAL DEPTH IN RELATIVISTIC FLOW

    SciTech Connect

    Shibata, Sanshiro; Tominaga, Nozomu; Tanaka, Masaomi

    2014-05-20

    We investigate the random walk process in relativistic flow. In the relativistic flow, photon propagation is concentrated in the direction of the flow velocity due to the relativistic beaming effect. We show that in the pure scattering case, the number of scatterings is proportional to the size parameter ξ ≡ L/l {sub 0} if the flow velocity β ≡ v/c satisfies β/Γ >> ξ{sup –1}, while it is proportional to ξ{sup 2} if β/Γ << ξ{sup –1}, where L and l {sub 0} are the size of the system in the observer frame and the mean free path in the comoving frame, respectively. We also examine the photon propagation in the scattering and absorptive medium. We find that if the optical depth for absorption τ{sub a} is considerably smaller than the optical depth for scattering τ{sub s} (τ{sub a}/τ{sub s} << 1) and the flow velocity satisfies β≫√(2τ{sub a}/τ{sub s}), then the effective optical depth is approximated by τ{sub *} ≅ τ{sub a}(1 + β)/β. Furthermore, we perform Monte Carlo simulations of radiative transfer and compare the results with the analytic expression for the number of scatterings. The analytic expression is consistent with the results of the numerical simulations. The expression derived in this study can be used to estimate the photon production site in relativistic phenomena, e.g., gamma-ray burst and active galactic nuclei.

  2. Blow-up rate and uniqueness of singular radial solutions for a class of quasi-linear elliptic equations

    NASA Astrophysics Data System (ADS)

    Xie, Zhifu; Zhao, Chunshan

    We establish the uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem -Δu=λu-b(x)h(u) in B(x) with boundary condition u=+∞ on ∂B(x), where B(x) is a ball centered at x∈R with radius R, N⩾3, 2⩽p<∞, λ>0 are constants and the weight function b is a positive radially symmetrical function. We only require h(u) to be a locally Lipschitz function with h(u)/u increasing on (0,∞) and h(u)˜u for large u with q>p-1. Our results extend the previous work [Z. Xie, Uniqueness and blow-up rate of large solutions for elliptic equation -Δu=λu-b(x)h(u), J. Differential Equations 247 (2009) 344-363] from case p=2 to case 2⩽p<∞.

  3. Determination of Mean Dynamic Topography (MDT) to Bridge Geoid and Mean Sea Surface Height (SSH) with a New Elliptic Equation

    NASA Astrophysics Data System (ADS)

    Chu, P. C.

    2016-12-01

    Mean dynamic topography (MDT, η) bridges the geoid and the mean sea surface (from satellite altimetry) and constrains large scale surface geostrophic circulations. It can be estimated from either satellite or underwater ocean temperature (T) and salinity (S) data. Satellite altimeter measures sea surface height (SSH) with high precision and unique resolution above a reference ellipsoid (not geoid). Two Gravity Recovery and Climate Experiment (GRACE) satellites launched in 2002, provide data to compute the marine geoid [called the GRACE Gravity Model (GGM)] (see website: http://www.csr.utexas.edu/grace/). The MDT is the difference of altimetry-derived mean SSH and the mean marine geoid (using GGM or pre-GRACE gravity model such as EGM96). A major difficulty arises that the spatial variations in mean SSH and marine geoid are approximately two orders of magnitude larger than the spatial variations in η.The second approach (using T, Sdata) is based on geostrophic balance, which is at the minimum energy state in the linear Boussinesq primitive equations with conservation of potential vorticity. In this paper, a new elliptic equation, -[∂x(gh/f2)∂xη+∂y(gh/f2)∂yη]+η = (g/f2)(∂C/∂x-∂B/∂y)is derived to determine MDT with H the water depth, g the gravitational acceleration, and coefficients (B, C) depend on 3D mean temperature (T) and salinity (S) data. Numerical approach transforms the elliptic equation into a set of well-posed linear algebraic equations of η at grid points. The solution for the North Atlantic Ocean (100oW-6oW, 7oN-72oN) on 1oX1ogrids with the coefficients (B, C) calculated from the three-dimensional (T, S) data of the NOAA National Centers for Environmental Information (NCEI) World Ocean Atlas 2013 version 2 (http://www.nodc.noaa.gov/OC5/woa13/woa13data.html) and H from the NOAA ETOPO5 (https://www.ngdc.noaa.gov/mgg/fliers/93mgg01.html), compares well with the difference (also considered as the MDT) between the time-averaged SSH and

  4. Drifting solutions with elliptic symmetry for the compressible Navier-Stokes equations with density-dependent viscosity

    SciTech Connect

    An, Hongli; Yuen, Manwai

    2014-05-15

    In this paper, we investigate the analytical solutions of the compressible Navier-Stokes equations with dependent-density viscosity. By using the characteristic method, we successfully obtain a class of drifting solutions with elliptic symmetry for the Navier-Stokes model wherein the velocity components are governed by a generalized Emden dynamical system. In particular, when the viscosity variables are taken the same as Yuen [M. W. Yuen, “Analytical solutions to the Navier-Stokes equations,” J. Math. Phys. 49, 113102 (2008)], our solutions constitute a generalization of that obtained by Yuen. Interestingly, numerical simulations show that the analytical solutions can be used to explain the drifting phenomena of the propagation wave like Tsunamis in oceans.

  5. Averaging of random walks and shift-invariant measures on a Hilbert space

    NASA Astrophysics Data System (ADS)

    Sakbaev, V. Zh.

    2017-06-01

    We study random walks in a Hilbert space H and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on H. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure λ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in H. We define the Hilbert space H of equivalence classes of complex-valued functions on H that are square integrable with respect to a shift-invariant measure λ. Using averaging of the shift operator in H over random vectors in H with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on H, we define a one-parameter semigroup of contracting self-adjoint transformations on H, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.

  6. Heat transfer in fractured geothermal reservoirs and continuous time random walks (Invited)

    NASA Astrophysics Data System (ADS)

    Emmanuel, S.; Geiger, S.

    2010-12-01

    Solute transport in fractured geological media is often characterized by early breakthrough and long tailing, which cannot be adequately described using the classical advection-dispersion equation. Heat transfer in subsurface systems is also governed by advection and dispersion, and might therefore be expected to demonstrate similar “anomalous” behavior. To test this hypothesis, we ran simulations of heat transfer in two geologically realistic fractured porous domains using a high-resolution finite element-finite volume numerical model. Thermal breakthrough curves were calculated at various locations in the domains and compared with the classical advection-dispersion model. The simulations indicate that heat transfer in well-connected fracture networks conforms to the classical model, despite the presence of highly irregular thermal fronts. By contrast, heat transport in poorly connected networks is not adequately described by the advection-dispersion equation. We also use an alternative mathematical framework - the Continuous Time Random Walk (CTRW) - to describe heat transfer and we show that it successfully matches the breakthrough curves in both the well connected and poorly connected fracture systems. We discuss the implications of the study for industrial and environmental processes ranging from geothermal reservoirs to enhanced oil recovery.

  7. Matrix coefficient identification in an elliptic equation with the convex energy functional method

    NASA Astrophysics Data System (ADS)

    Hinze, Michael; Nhan Tam Quyen, Tran

    2016-08-01

    In this paper we study the inverse problem of identifying the diffusion matrix in an elliptic PDE from measurements. The convex energy functional method with Tikhonov regularization is applied to tackle this problem. For the discretization we use the variational discretization concept, where the PDE is discretized with piecewise linear, continuous finite elements. We show the convergence of approximations. Using a suitable source condition, we prove an error bound for discrete solutions. For the numerical solution we propose a gradient-projection algorithm and prove the strong convergence of its iterates to a solution of the identification problem. Finally, we present a numerical experiment which illustrates our theoretical results.

  8. Preconditioning cubic spline collocation method by FEM and FDM for elliptic equations

    SciTech Connect

    Kim, Sang Dong

    1996-12-31

    In this talk we discuss the finite element and finite difference technique for the cubic spline collocation method. For this purpose, we consider the uniformly elliptic operator A defined by Au := -{Delta}u + a{sub 1}u{sub x} + a{sub 2}u{sub y} + a{sub 0}u in {Omega} (the unit square) with Dirichlet or Neumann boundary conditions and its discretization based on Hermite cubic spline spaces and collocation at the Gauss points. Using an interpolatory basis with support on the Gauss points one obtains the matrix A{sub N} (h = 1/N).

  9. A solution of the variational equations for elliptic orbits in rotating coordinates

    NASA Technical Reports Server (NTRS)

    Jones, J. B.

    1980-01-01

    For elliptic reference orbits, formulas are given for the perturbation state transition matrix of the two-body problem. The formulas relate perturbations expressed in a local vertical rotating coordinate system and are valid for motion in the linear neighborhood of reference orbits with e in the range of 0 to 1. The elements of the state transition matrix are expressed in terms of natural parameters (horizontal and radial velocity, radius, eccentricity, true anomaly, etc.) at the initial and final points. In addition to the general form, a simplified version, valid for small eccentricity orbits, is given.

  10. Origins and applications of the Montroll-Weiss continuous time random walk

    NASA Astrophysics Data System (ADS)

    Shlesinger, Michael F.

    2017-05-01

    The Continuous Time Random Walk (CTRW) was introduced by Montroll and Weiss in 1965 in a purely mathematical paper. Its antecedents and later applications beginning in 1973 are discussed, especially for the case of fractal time where the mean waiting time between jumps is infinite. Contribution to the Topical Issue: "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver.

  11. Robust Multigrid Smoothers for Three Dimensional Elliptic Equations with Strong Anisotropies

    NASA Technical Reports Server (NTRS)

    Llorente, Ignacio M.; Melson, N. Duane

    1998-01-01

    We discuss the behavior of several plane relaxation methods as multigrid smoothers for the solution of a discrete anisotropic elliptic model problem on cell-centered grids. The methods compared are plane Jacobi with damping, plane Jacobi with partial damping, plane Gauss-Seidel, plane zebra Gauss-Seidel, and line Gauss-Seidel. Based on numerical experiments and local mode analysis, we compare the smoothing factor of the different methods in the presence of strong anisotropies. A four-color Gauss-Seidel method is found to have the best numerical and architectural properties of the methods considered in the present work. Although alternating direction plane relaxation schemes are simpler and more robust than other approaches, they are not currently used in industrial and production codes because they require the solution of a two-dimensional problem for each plane in each direction. We verify the theoretical predictions of Thole and Trottenberg that an exact solution of each plane is not necessary and that a single two-dimensional multigrid cycle gives the same result as an exact solution, in much less execution time. Parallelization of the two-dimensional multigrid cycles, the kernel of the three-dimensional implicit solver, is also discussed. Alternating-plane smoothers are found to be highly efficient multigrid smoothers for anisotropic elliptic problems.

  12. A seamless hybrid RANS-LES model based on transport equations for the subgrid stresses and elliptic blending

    NASA Astrophysics Data System (ADS)

    Fadai-Ghotbi, Atabak; Friess, Christophe; Manceau, Rémi; Borée, Jacques

    2010-05-01

    The aim of the present work is to develop a seamless hybrid Reynolds-averaged Navier-Stokes (RANS) large-eddy simulation (LES) model based on transport equations for the subgrid stresses, using the elliptic-blending method to account for the nonlocal kinematic blocking effect of the wall. It is shown that the elliptic relaxation strategy of Durbin is valid in a RANS (steady) as well as a LES context (unsteady). In order to reproduce the complex production and redistribution mechanisms when the cutoff wavenumber is located in the productive zone of the turbulent energy spectrum, the model is based on transport equations for the subgrid-stress tensor. The partially integrated transport model (PITM) methodology offers a consistent theoretical framework for such a model, enabling to control the cutoff wavenumber κc, and thus the transition from RANS to LES, by making the Cɛ2 coefficient in the dissipation equation of a RANS model a function of κc. The equivalence between the PITM and the Smagorinsky model is shown when κc is in the inertial range of the energy spectrum. The extension of the underlying RANS model used in the present work, the elliptic-blending Reynolds-stress model, to the hybrid RANS-LES context, brings out some modeling issues. The different modeling possibilities are compared in a channel flow at Reτ=395. Finally, a dynamic procedure is proposed in order to adjust during the computation the dissipation rate necessary to drive the model toward the expected amount of resolved energy. The final model gives very encouraging results in comparison to the direct numerical simulation data. In particular, the turbulence anisotropy in the near-wall region is satisfactorily reproduced. The contribution of the resolved and modeled fields to the Reynolds stresses behaves as expected: the modeled part is dominant in the near-wall zones (RANS mode) and decreases toward the center of the channel, where the relative contribution of the resolved part increases

  13. Reflecting Solutions of High Order Elliptic Differential Equations in Two Independent Variables Across Analytic Arcs. Ph.D. Thesis

    NASA Technical Reports Server (NTRS)

    Carleton, O.

    1972-01-01

    Consideration is given specifically to sixth order elliptic partial differential equations in two independent real variables x, y such that the coefficients of the highest order terms are real constants. It is assumed that the differential operator has distinct characteristics and that it can be factored as a product of second order operators. By analytically continuing into the complex domain and using the complex characteristic coordinates of the differential equation, it is shown that its solutions, u, may be reflected across analytic arcs on which u satisfies certain analytic boundary conditions. Moreover, a method is given whereby one can determine a region into which the solution is extensible. It is seen that this region of reflection is dependent on the original domain of difinition of the solution, the arc and the coefficients of the highest order terms of the equation and not on any sufficiently small quantities; i.e., the reflection is global in nature. The method employed may be applied to similar differential equations of order 2n.

  14. Upscaling solute transport in naturally fractured porous media with the continuous time random walk method

    SciTech Connect

    Geiger, S.; Cortis, A.; Birkholzer, J.T.

    2010-04-01

    Solute transport in fractured porous media is typically 'non-Fickian'; that is, it is characterized by early breakthrough and long tailing and by nonlinear growth of the Green function-centered second moment. This behavior is due to the effects of (1) multirate diffusion occurring between the highly permeable fracture network and the low-permeability rock matrix, (2) a wide range of advection rates in the fractures and, possibly, the matrix as well, and (3) a range of path lengths. As a consequence, prediction of solute transport processes at the macroscale represents a formidable challenge. Classical dual-porosity (or mobile-immobile) approaches in conjunction with an advection-dispersion equation and macroscopic dispersivity commonly fail to predict breakthrough of fractured porous media accurately. It was recently demonstrated that the continuous time random walk (CTRW) method can be used as a generalized upscaling approach. Here we extend this work and use results from high-resolution finite element-finite volume-based simulations of solute transport in an outcrop analogue of a naturally fractured reservoir to calibrate the CTRW method by extracting a distribution of retention times. This procedure allows us to predict breakthrough at other model locations accurately and to gain significant insight into the nature of the fracture-matrix interaction in naturally fractured porous reservoirs with geologically realistic fracture geometries.

  15. Bloch-like waves in random-walk potentials based on supersymmetry

    PubMed Central

    Yu, Sunkyu; Piao, Xianji; Hong, Jiho; Park, Namkyoo

    2015-01-01

    Bloch's theorem was a major milestone that established the principle of bandgaps in crystals. Although it was once believed that bandgaps could form only under conditions of periodicity and long-range correlations for Bloch's theorem, this restriction was disproven by the discoveries of amorphous media and quasicrystals. While network and liquid models have been suggested for the interpretation of Bloch-like waves in disordered media, these approaches based on searching for random networks with bandgaps have failed in the deterministic creation of bandgaps. Here we reveal a deterministic pathway to bandgaps in random-walk potentials by applying the notion of supersymmetry to the wave equation. Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps. Our approach enables the formation of bandgaps in extremely disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with ‘Bloch-like eigenstates'. PMID:26373616

  16. A time-varying biased random walk approach to human growth.

    PubMed

    Suki, Béla; Frey, Urs

    2017-08-10

    Growth and development are dominated by gene-environment interactions. Many approaches have been proposed to model growth, but most are either descriptive or describe population level phenomena. We present a random walk-based growth model capable of predicting individual height, in which the growth increments are taken from time varying distributions mimicking the bursting behaviour of observed saltatory growth. We derive analytic equations and also develop a computational model of such growth that takes into account gene-environment interactions. Using an independent prospective birth cohort study of 190 infants, we predict height at 6 years of age. In a subset of 27 subjects, we adaptively train the model to account for growth between birth and 1 year of age using a Bayesian approach. The 5-year predicted heights compare well with actual data (measured height = 0.838*predicted height + 18.3; R(2) = 0.51) with an average error of 3.3%. In one patient, we also exemplify how our growth prediction model can be used for the early detection of growth deficiency and the evaluation of the effectiveness of growth hormone therapy.

  17. Novel pseudo-random number generator based on quantum random walks

    PubMed Central

    Yang, Yu-Guang; Zhao, Qian-Qian

    2016-01-01

    In this paper, we investigate the potential application of quantum computation for constructing pseudo-random number generators (PRNGs) and further construct a novel PRNG based on quantum random walks (QRWs), a famous quantum computation model. The PRNG merely relies on the equations used in the QRWs, and thus the generation algorithm is simple and the computation speed is fast. The proposed PRNG is subjected to statistical tests such as NIST and successfully passed the test. Compared with the representative PRNG based on quantum chaotic maps (QCM), the present QRWs-based PRNG has some advantages such as better statistical complexity and recurrence. For example, the normalized Shannon entropy and the statistical complexity of the QRWs-based PRNG are 0.999699456771172 and 1.799961178212329e-04 respectively given the number of 8 bits-words, say, 16Mbits. By contrast, the corresponding values of the QCM-based PRNG are 0.999448131481064 and 3.701210794388818e-04 respectively. Thus the statistical complexity and the normalized entropy of the QRWs-based PRNG are closer to 0 and 1 respectively than those of the QCM-based PRNG when the number of words of the analyzed sequence increases. It provides a new clue to construct PRNGs and also extends the applications of quantum computation. PMID:26842402

  18. Heterogeneous Memorized Continuous Time Random Walks in an External Force Fields

    NASA Astrophysics Data System (ADS)

    Wang, Jun; Zhou, Ji; Lv, Long-Jin; Qiu, Wei-Yuan; Ren, Fu-Yao

    2014-09-01

    In this paper, we study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated memorized waiting times, which involves Reimann-Liouville fractional derivative or Reimann-Liouville fractional integral. We show that the mean squared displacement of the test particle which is dependent on its location of the form (El-Wakil and Zahran, Chaos Solitons Fractals, 12, 1929-1935, 2001) where is the anomalous exponent, the diffusion exponent is dependent on the model parameters. We obtain the Fokker-Planck-type dynamic equations, and their stationary solutions are of the Boltzmann-Gibbs form. These processes obey a generalized Einstein-Stokes-Smoluchowski relation and the second Einstein relation. We observe that the asymptotic behavior of waiting times and subordinations are of stretched Gaussian distributions. We also discuss the time averaged in the case of an harmonic potential, and show that the process exhibits aging and ergodicity breaking.

  19. Natural Organic Matter Transport Modeling with a Continuous Time Random Walk Approach

    PubMed Central

    McInnis, Daniel P.; Bolster, Diogo; Maurice, Patricia A.

    2014-01-01

    Abstract In transport experiments through columns packed with naturally Fe/Al oxide-coated quartz sand, breakthrough curves (BTCs) of natural organic matter (NOM) displayed strong and persistent power law tailing that could not be described by the classical advection–dispersion equation. Tailing was not observed in BTCs for a nonreactive tracer (sulforhodamine B); therefore, the anomalous transport is attributed to diverse adsorptive behavior of the polydisperse NOM sample rather than to physical heterogeneity of the porous medium. NOM BTC tailing became more pronounced with decreases in pH and increases in ionic strength, conditions previously shown to be associated with enhanced preferential adsorption of intermediate to high molecular weight NOM components. Drawing from previous work on anomalous solute transport, we develop an approach to model NOM transport within the framework of a continuous time random walk (CTRW) and show that under all conditions examined, the CTRW model is able to capture tailing of NOM BTCs by accounting for differences in transport rates of NOM fractions through a distribution of effective retardation factors. These results demonstrate the importance of considering effects of adsorptive fractionation on NOM mobility, and illustrate the ability of the CTRW model to describe transport of a multicomponent solute. PMID:24596449

  20. Upscaling solute transport in naturally fractured porous media with the continuous time random walk method

    NASA Astrophysics Data System (ADS)

    Geiger, S.; Cortis, A.; Birkholzer, J. T.

    2010-12-01

    Solute transport in fractured porous media is typically "non-Fickian"; that is, it is characterized by early breakthrough and long tailing and by nonlinear growth of the Green function-centered second moment. This behavior is due to the effects of (1) multirate diffusion occurring between the highly permeable fracture network and the low-permeability rock matrix, (2) a wide range of advection rates in the fractures and, possibly, the matrix as well, and (3) a range of path lengths. As a consequence, prediction of solute transport processes at the macroscale represents a formidable challenge. Classical dual-porosity (or mobile-immobile) approaches in conjunction with an advection-dispersion equation and macroscopic dispersivity commonly fail to predict breakthrough of fractured porous media accurately. It was recently demonstrated that the continuous time random walk (CTRW) method can be used as a generalized upscaling approach. Here we extend this work and use results from high-resolution finite element-finite volume-based simulations of solute transport in an outcrop analogue of a naturally fractured reservoir to calibrate the CTRW method by extracting a distribution of retention times. This procedure allows us to predict breakthrough at other model locations accurately and to gain significant insight into the nature of the fracture-matrix interaction in naturally fractured porous reservoirs with geologically realistic fracture geometries.

  1. Novel pseudo-random number generator based on quantum random walks.

    PubMed

    Yang, Yu-Guang; Zhao, Qian-Qian

    2016-02-04

    In this paper, we investigate the potential application of quantum computation for constructing pseudo-random number generators (PRNGs) and further construct a novel PRNG based on quantum random walks (QRWs), a famous quantum computation model. The PRNG merely relies on the equations used in the QRWs, and thus the generation algorithm is simple and the computation speed is fast. The proposed PRNG is subjected to statistical tests such as NIST and successfully passed the test. Compared with the representative PRNG based on quantum chaotic maps (QCM), the present QRWs-based PRNG has some advantages such as better statistical complexity and recurrence. For example, the normalized Shannon entropy and the statistical complexity of the QRWs-based PRNG are 0.999699456771172 and 1.799961178212329e-04 respectively given the number of 8 bits-words, say, 16Mbits. By contrast, the corresponding values of the QCM-based PRNG are 0.999448131481064 and 3.701210794388818e-04 respectively. Thus the statistical complexity and the normalized entropy of the QRWs-based PRNG are closer to 0 and 1 respectively than those of the QCM-based PRNG when the number of words of the analyzed sequence increases. It provides a new clue to construct PRNGs and also extends the applications of quantum computation.

  2. Novel pseudo-random number generator based on quantum random walks

    NASA Astrophysics Data System (ADS)

    Yang, Yu-Guang; Zhao, Qian-Qian

    2016-02-01

    In this paper, we investigate the potential application of quantum computation for constructing pseudo-random number generators (PRNGs) and further construct a novel PRNG based on quantum random walks (QRWs), a famous quantum computation model. The PRNG merely relies on the equations used in the QRWs, and thus the generation algorithm is simple and the computation speed is fast. The proposed PRNG is subjected to statistical tests such as NIST and successfully passed the test. Compared with the representative PRNG based on quantum chaotic maps (QCM), the present QRWs-based PRNG has some advantages such as better statistical complexity and recurrence. For example, the normalized Shannon entropy and the statistical complexity of the QRWs-based PRNG are 0.999699456771172 and 1.799961178212329e-04 respectively given the number of 8 bits-words, say, 16Mbits. By contrast, the corresponding values of the QCM-based PRNG are 0.999448131481064 and 3.701210794388818e-04 respectively. Thus the statistical complexity and the normalized entropy of the QRWs-based PRNG are closer to 0 and 1 respectively than those of the QCM-based PRNG when the number of words of the analyzed sequence increases. It provides a new clue to construct PRNGs and also extends the applications of quantum computation.

  3. Field Line Random Walk in Isotropic Magnetic Turbulence up to Infinite Kubo Number

    NASA Astrophysics Data System (ADS)

    Sonsrettee, W.; Wongpan, P.; Ruffolo, D. J.; Matthaeus, W. H.; Chuychai, P.; Rowlands, G.

    2013-12-01

    In astrophysical plasmas, the magnetic field line random walk (FLRW) plays a key role in the transport of energetic particles. In the present, we consider isotropic magnetic turbulence, which is a reasonable model for interstellar space. Theoretical conceptions of the FLRW have been strongly influenced by studies of the limit of weak fluctuations (or a strong mean field) (e.g, Isichenko 1991a, b). In this case, the behavior of FLRW can be characterized by the Kubo number R = (b/B0)(l_∥ /l_ \\bot ) , where l∥ and l_ \\bot are turbulence coherence scales parallel and perpendicular to the mean field, respectively, and b is the root mean squared fluctuation field. In the 2D limit (R ≫ 1), there has been an apparent conflict between concepts of Bohm diffusion, which is based on the Corrsin's independence hypothesis, and percolative diffusion. Here we have used three non-perturbative analytic techniques based on Corrsin's independence hypothesis for B0 = 0 (R = ∞ ): diffusive decorrelation (DD), random ballistic decorrelation (RBD) and a general ordinary differential equation (ODE), and compared them with direct computer simulations. All the analytical models and computer simulations agree that isotropic turbulence for R = ∞ has a field line diffusion coefficient that is consistent with Bohm diffusion. Partially supported by the Thailand Research Fund, NASA, and NSF.

  4. Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

    NASA Astrophysics Data System (ADS)

    Schütz, Gunter M.; Trimper, Steffen

    2004-10-01

    We consider a discrete-time random walk where the random increment at time step t depends on the full history of the process. We calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p . At a critical value pc(1)=1/2 where memory effects vanish there is a transition from a weakly localized regime [where the walker (elephant) returns to its starting point] to an escape regime. Inside the escape regime there is a second critical value where the random walk becomes superdiffusive. The probability distribution is shown to be governed by a non-Markovian Fokker-Planck equation with hopping rates that depend both on time and on the starting position of the walk. On large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time-dependent spring constant k=(2p-1)/t . The solution of this problem is a Gaussian distribution with time-dependent mean and variance which both depend on the initiation of the process.

  5. Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 1: One-step method

    NASA Technical Reports Server (NTRS)

    Chang, S. C.

    1986-01-01

    An algorithm for solving a large class of two- and three-dimensional nonseparable elliptic partial differential equations (PDE's) is developed and tested. It uses a modified D'Yakanov-Gunn iterative procedure in which the relaxation factor is grid-point dependent. It is easy to implement and applicable to a variety of boundary conditions. It is also computationally efficient, as indicated by the results of numerical comparisons with other established methods. Furthermore, the current algorithm has the advantage of possessing two important properties which the traditional iterative methods lack; that is: (1) the convergence rate is relatively insensitive to grid-cell size and aspect ratio, and (2) the convergence rate can be easily estimated by using the coefficient of the PDE being solved.

  6. Numerical solution of supersonic three-dimensional free-mixing flows using the parabolic-elliptic Navier-Stokes equations

    NASA Technical Reports Server (NTRS)

    Hirsh, R. S.

    1976-01-01

    A numerical method is presented for solving the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to three-dimensional supersonic laminar jet flow issuing parallel with a supersonic free stream. A coordinate transformation is introduced which maps the boundaries at infinity into a finite computational domain in order to eliminate difficulties associated with the imposition of free-stream boundary conditions. Results are presented for an approximate circular jet, a square jet, varying aspect ratio rectangular jets, and interacting square jets. The solution behavior varies from axisymmetric to nearly two-dimensional in character. For cases where comparisons of the present results with those obtained from shear layer calculations could be made, agreement was good.

  7. ON BOUNDARY VALUES IN \\mathscr{L}_p, p>1, OF SOLUTIONS OF ELLIPTIC EQUATIONS IN DOMAINS WITH A LYAPUNOV BOUNDARY

    NASA Astrophysics Data System (ADS)

    Petrushko, I. M.

    1984-02-01

    Necessary and sufficient conditions are established for the existence of limits in the \\mathscr{L}_p sense, p>1, on the boundary of a domain, of solutions of second order elliptic equations in domains with Lyapunov boundaries.Bibliography: 8 titles.

  8. On the pertinence to Physics of random walks induced by random dynamical systems: a survey

    NASA Astrophysics Data System (ADS)

    Petritis, Dimitri

    2016-08-01

    Let be an abstract space and a denumerable (finite or infinite) alphabet. Suppose that is a family of functions such that for all we have and a family of transformations . The pair ((Sa)a , (pa)a ) is termed an iterated function system with place dependent probabilities. Such systems can be thought as generalisations of random dynamical systems. As a matter of fact, suppose we start from a given ; we pick then randomly, with probability pa (x), the transformation Sa and evolve to Sa (x). We are interested in the behaviour of the system when the iteration continues indefinitely. Random walks of the above type are omnipresent in both classical and quantum Physics. To give a small sample of occurrences we mention: random walks on the affine group, random walks on Penrose lattices, random walks on partially directed lattices, evolution of density matrices induced by repeated quantum measurements, quantum channels, quantum random walks, etc. In this article, we review some basic properties of such systems and provide with a pathfinder in the extensive bibliography (both on mathematical and physical sides) where the main results have been originally published.

  9. A random walk-based method for segmentation of intravascular ultrasound images

    NASA Astrophysics Data System (ADS)

    Yan, Jiayong; Liu, Hong; Cui, Yaoyao

    2014-04-01

    Intravascular ultrasound (IVUS) is an important imaging technique that is used to study vascular wall architecture for diagnosis and assessment of the vascular diseases. Segmentation of lumen and media-adventitia boundaries from IVUS images is a basic and necessary step for quantitative assessment of the vascular walls. Due to ultrasound speckles, artifacts and individual differences, automated segmentation of IVUS images represents a challenging task. In this paper, a random walk based method is proposed for fully automated segmentation of IVUS images. Robust and accurate determination of the seed points for different regions is the key to successful use of the random walk algorithm in segmentation of IVUS images and is the focus of our work. The presented method mainly comprises five steps: firstly, the seed points inside the lumen and outside the adventitia are roughly estimated with intensity information, respectively; secondly, the seed points outside the adventitia are refined, and those of the media are determined through the results of applying random walk to the IVUS image with the roughly estimated seed points; thirdly, the media-adventitia boundary is detected by using random walk with the seed points obtained in the second step and the image gradient; fourthly, the seed points for media and lumen are refined; finally, the lumen boundary is extracted by using random walk again with the seed points obtained in the fourth step and the image gradient. The tests of the proposed algorithm on the in vivo dataset demonstrate the effectiveness of the presented IVUS image segmentation approach.

  10. Boundedness and regularity of solutions of degenerate elliptic partial differential equations

    NASA Astrophysics Data System (ADS)

    Chua, Seng-Kee

    2017-10-01

    We study quasilinear degenerate equations with their associated operators satisfying Sobolev and Poincaré inequalities. Our problems are similar to those in Sawyer and Wheeden (2006) [20], but we consider more general integral equations on abstract Sobolev spaces. Indeed, we provide a unified approach via abstract setting. Our main tool is Moser iteration. We obtain boundedness (assuming a weighted Sobolev embedding) of weak solutions and also Harnack inequalities for nonnegative solutions (assuming further that a weighted Poincaré inequality holds for their logarithm and existence of a sequence of cutoff functions) and hence Hölder's continuity of solutions. Our assumption is weaker than the above mentioned paper and other existing papers even in the case where the setting is on Rn; for example, we do not assume any doubling condition and we are able to deal with weighted estimates. Our results are new even when the leading term in the equations are vector fields that satisfy Hörmander's conditions.

  11. On domain decomposition preconditioner of BPS type for finite element discretizations of 3D elliptic equations

    NASA Astrophysics Data System (ADS)

    Korneev, V. G.

    2012-09-01

    BPS is a well known an efficient and rather general domain decomposition Dirichlet-Dirichlet type preconditioner, suggested in the famous series of papers Bramble, Pasciak and Schatz (1986-1989). Since then, it has been serving as the origin for the whole family of domain decomposition Dirichlet-Dirichlet type preconditioners-solvers as for h so hp discretizations of elliptic problems. For its original version, designed for h discretizations, the named authors proved the bound O(1 + log2 H/ h) for the relative condition number under some restricting conditions on the domain decomposition and finite element discretization. Here H/ h is the maximal relation of the characteristic size H of a decomposition subdomain to the mesh parameter h of its discretization. It was assumed that subdomains are images of the reference unite cube by trilinear mappings. Later similar bounds related to h discretizations were proved for more general domain decompositions, defined by means of coarse tetrahedral meshes. These results, accompanied by the development of some special tools of analysis aimed at such type of decompositions, were summarized in the book of Toselli and Widlund (2005). This paper is also confined to h discretizations. We further expand the range of admissible domain decompositions for constructing BPS preconditioners, in which decomposition subdomains can be convex polyhedrons, satisfying some conditions of shape regularity. We prove the bound for the relative condition number with the same dependence on H/ h as in the bound given above. Along the way to this result, we simplify the proof of the so called abstract bound for the relative condition number of the domain decomposition preconditioner. In the part, related to the analysis of the interface sub-problem preconditioning, our technical tools are generalization of those used by Bramble, Pasciak and Schatz.

  12. On the Solution of Elliptic Partial Differential Equations on Regions with Corners

    DTIC Science & Technology

    2015-07-09

    nature. It might be a Lipschitz or Hölder continuous curve, or a fractal , etc. While during the last fifty years, such environments have been studied in...45 [14] Jones, P., J. Ma, and V. Rokhlin. “A fast direct algorithm for the solution of the Laplace equation on regions with fractal boundaries.” J

  13. A dynamical system approach to the construction of singular solutions of some degenerate elliptic equations

    NASA Astrophysics Data System (ADS)

    Huentutripay, Jorge; Jazar, Mustapha; Véron, Laurent

    We study the existence of singular separable solutions to the 2-dimensional quasilinear equation -∇·(|∇ u| p-2 ∇ u)+| u| q-1 u=0 under the form u( r, θ)= r- βω( θ). We obtain the full description of the set of such solutions by combining a 2-dimensional shooting method with a phase plane analysis approach.

  14. Scaling behavior for random walks with memory of the largest distance from the origin

    NASA Astrophysics Data System (ADS)

    Serva, Maurizio

    2013-11-01

    We study a one-dimensional random walk with memory. The behavior of the walker is modified with respect to the simple symmetric random walk only when he or she is at the maximum distance ever reached from his or her starting point (home). In this case, having the choice to move farther or to move closer, the walker decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold, otherwise he or she is timorous. We investigate the asymptotic properties of this bold-timorous random walk, showing that the scaling behavior varies continuously from subdiffusive (timorous) to superdiffusive (bold). The scaling exponents are fully determined with a new mathematical approach based on a decomposition of the dynamics in active journeys (the walker is at the maximum distance) and lazy journeys (the walker is not at the maximum distance).

  15. Random walk study of electron motion in helium in crossed electromagnetic fields

    NASA Technical Reports Server (NTRS)

    Englert, G. W.

    1972-01-01

    Random walk theory, previously adapted to electron motion in the presence of an electric field, is extended to include a transverse magnetic field. In principle, the random walk approach avoids mathematical complexity and concomitant simplifying assumptions and permits determination of energy distributions and transport coefficients within the accuracy of available collisional cross section data. Application is made to a weakly ionized helium gas. Time of relaxation of electron energy distribution, determined by the random walk, is described by simple expressions based on energy exchange between the electron and an effective electric field. The restrictive effect of the magnetic field on electron motion, which increases the required number of collisions per walk to reach a terminal steady state condition, as well as the effect of the magnetic field on electron transport coefficients and mean energy can be quite adequately described by expressions involving only the Hall parameter.

  16. Some recent variations on the expected number of distinct sites visited by an n-step random walk

    NASA Astrophysics Data System (ADS)

    Weiss, George H.; Dayan, Ido; Havlin, Shlomo; Kiefer, James E.; Larralde, Hernan; Stanley, H. Eugene; Trunfio, Paul

    1992-12-01

    Asymptotic forms for the expected number of distinct sites visited by an n-step random walk, being calculable for many random walks, have been used in a number of analyses of physical models. We describe three recent extensions of the problem, the first replacing the single random walker by N→∞ random walkers, the second to the study of a random walk in the presence of a trapping site, and the third to a random walk in the presence of a trapping hyperplane.

  17. The elliptic sinh-Gordon equation and the construction of toroidal soap bubbles

    SciTech Connect

    Spruck, J.

    1987-10-01

    In this paper we study all positive solutions to the nonlinear eigenvalue problem ..delta.. u + lambda sinh u = 0 on a symmetric domain D. We characterize the limit solution as lambda tends to zero. For a rectangle we prove that the solutions are unique and have a hidden additional symmetry property. This equation figures prominently in recent work on the construction of compact soap bubbles of genus 1. 11 refs.

  18. The expected number of distinct sites visited by N biased random walks in one dimension

    NASA Astrophysics Data System (ADS)

    Larralde, Hernan; Weiss, George H.; Eugene Stanley, H.

    1994-09-01

    We calculate the asymptotic form of the expected number of distinct sites visited by N random walkers moving independently in one dimension. It is shown that to lowest order and at long times, the leading term in the asymptotic result is that found for the random walk of a single biased particle, which implies that the bias is strong enough a factor to dominate the many-body effects in that regime. The lowest order correction term contains the many-body contribution. This is essentially the result for the unbiased random walk.

  19. A multiple step random walk Monte Carlo method for heat conduction involving distributed heat sources

    NASA Astrophysics Data System (ADS)

    Naraghi, M. H. N.; Chung, B. T. F.

    1982-06-01

    A multiple step fixed random walk Monte Carlo method for solving heat conduction in solids with distributed internal heat sources is developed. In this method, the probability that a walker reaches a point a few steps away is calculated analytically and is stored in the computer. Instead of moving to the immediate neighboring point the walker is allowed to jump several steps further. The present multiple step random walk technique can be applied to both conventional Monte Carlo and the Exodus methods. Numerical results indicate that the present method compares well with finite difference solutions while the computation speed is much faster than that of single step Exodus and conventional Monte Carlo methods.

  20. Random-Walk Type Model with Fat Tails for Financial Markets

    NASA Astrophysics Data System (ADS)

    Matuttis, Hans-Geors

    Starting from the random-walk model, practices of financial markets are included into the random-walk so that fat tail distributions like those in the high frequency data of the SP500 index are reproduced, though the individual mechanisms are modeled by normally distributed data. The incorporation of local correlation narrows the distribution for "frequent" events, whereas global correlations due to technical analysis leads to fat tails. Delay of market transactions in the trading process shifts the fat tail probabilities downwards. Such an inclusion of reactions to market fluctuations leads to mini-trends which are distributed with unit variance.

  1. Central limit theorem and related results for the elephant random walk

    NASA Astrophysics Data System (ADS)

    Coletti, Cristian F.; Gava, Renato; Schütz, Gunter M.

    2017-05-01

    We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on ℤ with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior. We prove a law of large numbers and a central limit theorem. Remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive. Inside the superdiffusive regime, the ERW converges to a non-degenerate random variable which is not normal. We also obtain explicit expressions for the correlations of increments of the ERW.

  2. Multiscale modeling of interwoven Kevlar fibers based on random walk to predict yarn structural response

    NASA Astrophysics Data System (ADS)

    Recchia, Stephen

    Kevlar is the most common high-end plastic filament yarn used in body armor, tire reinforcement, and wear resistant applications. Kevlar is a trade name for an aramid fiber. These are fibers in which the chain molecules are highly oriented along the fiber axis, so the strength of the chemical bond can be exploited. The bulk material is extruded into filaments that are bound together into yarn, which may be chorded with other materials as in car tires, woven into a fabric, or layered in an epoxy to make composite panels. The high tensile strength to low weight ratio makes this material ideal for designs that decrease weight and inertia, such as automobile tires, body panels, and body armor. For designs that use Kevlar, increasing the strength, or tenacity, to weight ratio would improve performance or reduce cost of all products that are based on this material. This thesis computationally and experimentally investigates the tenacity and stiffness of Kevlar yarns with varying twist ratios. The test boundary conditions were replicated with a geometrically accurate finite element model, resulting in a customized code that can reproduce tortuous filaments in a yarn was developed. The solid model geometry capturing filament tortuosity was implemented through a random walk method of axial geometry creation. A finite element analysis successfully recreated the yarn strength and stiffness dependency observed during the tests. The physics applied in the finite element model was reproduced in an analytical equation that was able to predict the failure strength and strain dependency of twist ratio. The analytical solution can be employed to optimize yarn design for high strength applications.

  3. A Local Bifurcation Theorem for Degenerate Elliptic Equations With Radial Symmetry

    NASA Astrophysics Data System (ADS)

    García-Melián, J.; Sabina de Lis, J.

    2002-02-01

    In this work we provide local bifurcation results for equations involving the p-Laplacian in balls. We analyze the continua Cn of radial solutions emanating from (λn, p, 0), {λn, p} being the radial eigenvalues of -Δp. First, we show that the only nontrivial solutions close to (λn, p, 0) lie on a continuous curve, thus extending the Crandall-Rabinowitz theorem. Second, it is proved that Cn{(λn, p, 0)} splits into two unbounded connected pieces, characterized by their nodal properties thus sharpening previous results.

  4. A comparison of iterative methods for a model coupled system of elliptic equations

    SciTech Connect

    Donato, J.M.

    1993-08-01

    Many interesting areas of current industry work deal with non-linear coupled systems of partial differential equations. We examine iterative methods for the solution of a model two-dimensional coupled system based on a linearized form of the two carrier drift-diffusion equations from semiconductor modeling. Discretizing this model system yields a large non-symmetric indefinite sparse matrix. To solve the model system various point and block methods, including the hybrid iterative method Alternate Block Factorization (ABF), are applied. We also employ GMRES with various preconditioners, including block and point incomplete LU (ILU) factorizations. The performance of these methods is compared. It is seen that the preferred ordering of the grid variables and the choice of iterative method are dependent upon the magnitudes of the coupling parameters. For this model, ABF is the most robust of the non-accelerated iterative methods. Among the preconditioners employed with GMRES, the blocked ``by grid point`` version of both the ILU and MILU preconditioners are the most robust and the most time efficient over the wide range of parameter values tested. This information may aid in the choice of iterative methods and preconditioners for solving more complicated, yet analogous, coupled systems.

  5. A general algorithm for the solution of Kepler's equation for elliptic orbits

    NASA Technical Reports Server (NTRS)

    Ng, E. W.

    1979-01-01

    An efficient algorithm is presented for the solution of Kepler's equation f(E)=E-M-e sin E=0, where e is the eccentricity, M the mean anomaly and E the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics in M, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20,000 pairs of values of e and M show that for single precision, 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision one additional iteration is required.

  6. A Probabilistic Approach to Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations in Smooth Domains

    SciTech Connect

    Zhou Wei

    2013-06-15

    We consider the value function of a stochastic optimal control of degenerate diffusion processes in a domain D. We study the smoothness of the value function, under the assumption of the non-degeneracy of the diffusion term along the normal to the boundary and an interior condition weaker than the non-degeneracy of the diffusion term. When the diffusion term, drift term, discount factor, running payoff and terminal payoff are all in the class of C{sup 1,1}( D-bar ) , the value function turns out to be the unique solution in the class of C{sub loc}{sup 1,1}(D) Intersection C{sup 0,1}( D-bar ) to the associated degenerate Bellman equation with Dirichlet boundary data. Our approach is probabilistic.

  7. On Lambert’s problem and the elliptic time of flight equation: A simple semi-analytical inversion method

    NASA Astrophysics Data System (ADS)

    Wailliez, Sébastien E.

    2014-03-01

    In the two-body model, time of flight between two positions can be expressed as a single-variable function and a variety of formulations exist. Lambert’s problem can be solved by inverting such a function. In this article, a method which inverts Lagrange’s flight time equation and supports the problematic 180° transfer is proposed. This method relies on a Householder algorithm of variable order. However, unlike other iterative methods, it is semi-analytical in the sense that flight time functions are derived analytically to second order vs. first order finite differences. The author investigated the profile of Lagrange’s elliptic flight time equation and its derivatives with a special focus on their significance to the behaviour of the proposed method and the stated goal of guaranteed convergence. Possible numerical deficiencies were identified and dealt with. As a test, 28 scenarios of variable difficulty were designed to cover a wide variety of geometries. The context of this research being the orbit determination of artificial satellites and debris, the scenarios are representative of typical such objects in Low-Earth, Geostationary and Geostationary Transfer Orbits. An analysis of the computational impact of the quality of the initial guess vs. that of the order of the method was also done, providing clues for further research and optimisations (e.g. asteroids, long period comets, multi-revolution cases). The results indicate fast to very fast convergence in all test cases, they validate the numerical safeguards and also give a quantitative assessment of the importance of the initial guess.

  8. Branching random walk with step size coming from a power law

    NASA Astrophysics Data System (ADS)

    Bhattacharya, Ayan; Subhra Hazra, Rajat; Roy, Parthanil

    2015-09-01

    In their seminal work, Brunet and Derrida made predictions on the random point configurations associated with branching random walks. We shall discuss the limiting behavior of such point configurations when the displacement random variables come from a power law. In particular, we establish that two prediction of remains valid in this setup and investigate various other issues mentioned in their paper.

  9. Expected number of sites visited by a constrained n-step random walk

    NASA Astrophysics Data System (ADS)

    Larralde, Hernan; Weiss, George H.

    1995-08-01

    We develop a formalism based on generating functions for calculating the expected number of sites visited by a lattice random walk constrained to visit a fixed point at the nth step. Explicit results are given in the large-n limit when the target point is not too far from the origin.

  10. Comment on ’Corrected Diffusion Approximations in Certain Random Walk Problems’.

    DTIC Science & Technology

    1984-05-01

    This paper is concerned with extensions to the nonexponential family case of two problems considered in an earlier work. The first problem is to find the...expected value of the maximum of a random walk with small, negative drift, and the second is to find the distribution of the same quantity.

  11. Energy difference space random walk to achieve fast free energy calculations.

    PubMed

    Min, Donghong; Yang, Wei

    2008-05-21

    A method is proposed to efficiently obtain free energy differences. In the present algorithm, free energy calculations proceed by the realization of an energy difference space random walk. Thereby, this algorithm can greatly improve the sampling of the regions in phase space where target states overlap.

  12. Asymptotic normality in a two-dimensional random walk model for cell motility

    SciTech Connect

    Stadje, W.

    1988-05-01

    We prove the asymptotic normality of a two-dimensional random walk describing the locomotion of cells on planar surfaces and calculate the asymptotic covariance matrix. The trajectories of the walk are random broken lines covered with constant speed, where the time intervals between turns as well as the turn angles are random and stochastically independent.

  13. Limit Theorem and Applications of the Pauli Open Quantum Random Walk on Z

    NASA Astrophysics Data System (ADS)

    Ampadu, Clement

    2013-04-01

    Following the recent talk in the ``Workshop of Quantum Dynamics and Quantum Walks'' held at Okazaki Conference Center, Okazaki, Japan. This talk clarifies the relationship between the convergent behavior of the Pauli quantum walk on the line, and the open quantum random walk obtained from the Pauli quantum walk.

  14. Continuous-time random-walk model for anomalous diffusion in expanding media

    NASA Astrophysics Data System (ADS)

    Le Vot, F.; Abad, E.; Yuste, S. B.

    2017-09-01

    Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium

  15. Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation

    NASA Astrophysics Data System (ADS)

    Gushchin, A. K.

    2015-10-01

    We consider a statement of the Dirichlet problem which generalizes the notions of classical and weak solutions, in which a solution belongs to the space of (n-1)-dimensionally continuous functions with values in the space L_p. The property of (n-1)-dimensional continuity is similar to the classical definition of uniform continuity; however, instead of the value of a function at a point, it looks at the trace of the function on measures in a special class, that is, elements of the space L_p with respect to these measures. Up to now, the problem in the statement under consideration has not been studied in sufficient detail. This relates first to the question of conditions on the right-hand side of the equation which ensure the solvability of the problem. The main results of the paper are devoted to just this question. We discuss the terms in which these conditions can be expressed. In addition, the way the behaviour of a solution near the boundary depends on the right-hand side is investigated. Bibliography: 47 titles.

  16. Modeling of Line Shapes using Continuous Time Random Walk Theory

    NASA Astrophysics Data System (ADS)

    Capes, H.; Christova, M.; Boland, D.; Bouzaher, A.; Catoire, F.; Godbert-Mouret, L.; Koubiti, M.; Mekkaoui, S.; Rosato, J.; Marandet, Y.; Stamm, R.

    2010-11-01

    In order to provide a general framework where the Stark broadening of atomic lines in plasmas can be calculated, we model the plasma stochastic electric field by using the CTRW approach [1,2]. This allows retaining non Markovian terms in the Schrödinger equation averaged over the electric field fluctuations. As an application we consider a special case of a non separable CTRW process, the so called Kangaroo process [3]. An analytic expression for the line profile is finally obtained for arbitrary waiting time distribution functions. An application to the hydrogen Lyman α line is discussed.

  17. Random walks in weighted networks with a perfect trap: an application of Laplacian spectra.

    PubMed

    Lin, Yuan; Zhang, Zhongzhi

    2013-06-01

    Trapping processes constitute a primary problem of random walks, which characterize various other dynamical processes taking place on networks. Most previous works focused on the case of binary networks, while there is much less related research about weighted networks. In this paper, we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node. By utilizing the spectral graph theory, we provide an exact formula for mean first-passage time (MFPT) from one node to another, based on which we deduce an explicit expression for average trapping time (ATT) in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the weighted graph, where ATT is the average of MFPTs to the trap over all source nodes. We then further derive a sharp lower bound for the ATT in terms of only the local information of the trap node, which can be obtained in some graphs. Moreover, we deduce the ATT when the trap is distributed uniformly in the whole network. Our results show that network weights play a significant role in the trapping process. To apply our framework, we use the obtained formulas to study random walks on two specific networks: trapping in weighted uncorrelated networks with a deep trap, the weights of which are characterized by a parameter, and Lévy random walks in a connected binary network with a trap distributed uniformly, which can be looked on as random walks on a weighted network. For weighted uncorrelated networks we show that the ATT to any target node depends on the weight parameter, that is, the ATT to any node can change drastically by modifying the parameter, a phenomenon that is in contrast to that for trapping in binary networks. For Lévy random walks in any connected network, by using their equivalence to random walks on a weighted complete network, we obtain the optimal exponent characterizing Lévy random walks, which have the minimal average of ATTs taken over all

  18. On the asymptotics of the mean sojourn time of a random walk on a semi-axis

    NASA Astrophysics Data System (ADS)

    Lotov, V. I.; Tarasenko, A. S.

    2015-06-01

    We find asymptotic expansions for the expectation of the sojourn time above an increasing level of a trajectory of a random walk with zero drift. The Cramér condition on the existence of an exponential moment is imposed on the distribution of jumps of the random walk.

  19. Covering Ground: Movement Patterns and Random Walk Behavior in Aquilonastra anomala Sea Stars.

    PubMed

    Lohmann, Amanda C; Evangelista, Dennis; Waldrop, Lindsay D; Mah, Christopher L; Hedrick, Tyson L

    2016-10-01

    The paths animals take while moving through their environments affect their likelihood of encountering food and other resources; thus, models of foraging behavior abound. To collect movement data appropriate for comparison with these models, we used time-lapse photography to track movements of a small, hardy, and easy-to-obtain organism, Aquilonastra anomala sea stars. We recorded the sea stars in a tank over many hours, with and without a food cue. With food present, they covered less distance, as predicted by theory; this strategy would allow them to remain near food. We then compared the paths of the sea stars to three common models of animal movement: Brownian motion, Lévy walks, and correlated random walks; we found that the sea stars' movements most closely resembled a correlated random walk. Additionally, we compared the search performance of models of Brownian motion, a Lévy walk, and a correlated random walk to that of a model based on the sea stars' movements. We found that the behavior of the modeled sea star walk was similar to that of the modeled correlated random walk and the Brownian motion model, but that the sea star walk was slightly more likely than the other walks to find targets at intermediate distances. While organisms are unlikely to follow an idealized random walk in all details, our data suggest that comparing the effectiveness of an organism's paths to those from theory can give insight into the organism's actual movement strategy. Finally, automated optical tracking of invertebrates proved feasible, and A. anomala was revealed to be a tractable, 2D-movement study system.

  20. Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and states and with controls in the coefficients multiplying the highest derivatives

    NASA Astrophysics Data System (ADS)

    Lubyshev, F. V.; Fairuzov, M. E.

    2016-07-01

    Mathematical formulations of nonlinear optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions and with controls in the coefficients multiplying the highest derivatives are studied. Finite difference approximations of optimization problems are constructed, and the approximation error is estimated with respect to the state and the cost functional. Weak convergence of the approximations with respect to the control is proved. The approximations are regularized in the sense of Tikhonov.

  1. Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems, and KZB equations

    NASA Astrophysics Data System (ADS)

    Levin, A. M.; Olshanetsky, M. A.; Zotov, A. V.

    2016-08-01

    We construct twisted Calogero-Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker-Phong and Bordner-Corrigan-Sasaki-Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of Lie algebras.

  2. Optimization of the deflated Conjugate Gradient algorithm for the solving of elliptic equations on massively parallel machines

    NASA Astrophysics Data System (ADS)

    Malandain, Mathias; Maheu, Nicolas; Moureau, Vincent

    2013-04-01

    The discretization of Partial Differential Equations often leads to the need of solving large symmetric linear systems. In the case of the Navier-Stokes equations for incompressible flows, solving the elliptic pressure Poisson equation can represent the most important part of the computational time required for the massively parallel simulation of the flow. The need for efficiency that this issue induces is completed with a need for stability, in particular when dealing with unstructured meshes. Here, a stable and efficient variant of the Deflated Preconditioned Conjugate Gradient (DPCG) solver is first presented. This two-level method uses an arbitrary coarse grid to reduce the computational cost of the solving. However, in the massively parallel implementation of this technique for very large linear systems, the coarse grids generated can count up to millions of cells, which makes direct solvings on the coarse level impossible. The solving on the coarse grid, performed with a Preconditioned Conjugate Gradient (PCG) solver for this reason, may involve a large number of communications, which reduces dramatically the performances on massively parallel machines. To this effect, two methods developed in order to reduce the number of iterations on the coarse level are introduced, that is the creation of improved initial guesses and the adaptation of the convergence criterion. The design of these methods make them easy to implement in any already existing DPCG solver. The structural requirements for an efficient massively parallel unstructured solver and the implementation of this solver are described. The novel DPCG method is assessed for applications involving turbulence, heat transfers and two-phase flows, with grids up to 17.8 billion elements. Numerical results show a two- to 12-fold reduction of the number of iterations on the coarse level, which implies a reduction of the computational time of the Poisson solver up to 71% and a global reduction of the proportion

  3. Correlated random walks caused by dynamical wavefunction collapse

    NASA Astrophysics Data System (ADS)

    Bedingham, D. J.; Ulbricht, H.

    2015-08-01

    Wavefunction collapse models modify Schrödinger’s equation so that it describes the collapse of a superposition of macroscopically distinguishable states as a dynamical process. This provides a basis for the resolution of the quantum measurement problem. An additional generic consequence of the collapse mechanism is that it causes particles to exhibit a tiny random diffusive motion. Here it is shown that for the continuous spontaneous localization (CSL) model—one of the most well developed collapse models—the diffusions of two sufficiently nearby particles are positively correlated. An experimental test of this effect is proposed in which random displacements of pairs of free nanoparticles are measured after they have been simultaneously released from nearby traps. The experiment must be carried out at sufficiently low temperature and pressure in order for the collapse effects to dominate over the ambient environmental noise. It is argued that these constraints can be satisfied by current technologies for a large region of the viable parameter space of the CSL model. The effect disappears as the separation between particles exceeds the CSL length scale. The test therefore provides a means of bounding this length scale.

  4. Correlated random walks caused by dynamical wavefunction collapse

    PubMed Central

    Bedingham, D. J.; Ulbricht, H.

    2015-01-01

    Wavefunction collapse models modify Schrödinger’s equation so that it describes the collapse of a superposition of macroscopically distinguishable states as a dynamical process. This provides a basis for the resolution of the quantum measurement problem. An additional generic consequence of the collapse mechanism is that it causes particles to exhibit a tiny random diffusive motion. Here it is shown that for the continuous spontaneous localization (CSL) model—one of the most well developed collapse models—the diffusions of two sufficiently nearby particles are positively correlated. An experimental test of this effect is proposed in which random displacements of pairs of free nanoparticles are measured after they have been simultaneously released from nearby traps. The experiment must be carried out at sufficiently low temperature and pressure in order for the collapse effects to dominate over the ambient environmental noise. It is argued that these constraints can be satisfied by current technologies for a large region of the viable parameter space of the CSL model. The effect disappears as the separation between particles exceeds the CSL length scale. The test therefore provides a means of bounding this length scale. PMID:26303388

  5. Lattice Boltzmann simulation of solute transport in heterogeneous porous media with conduits to estimate macroscopic continuous time random walk model parameters

    SciTech Connect

    Anwar, S.; Cortis, A.; Sukop, M.

    2008-10-20

    Lattice Boltzmann models simulate solute transport in porous media traversed by conduits. Resulting solute breakthrough curves are fitted with Continuous Time Random Walk models. Porous media are simulated by damping flow inertia and, when the damping is large enough, a Darcy's Law solution instead of the Navier-Stokes solution normally provided by the lattice Boltzmann model is obtained. Anisotropic dispersion is incorporated using a direction-dependent relaxation time. Our particular interest is to simulate transport processes outside the applicability of the standard Advection-Dispersion Equation (ADE) including eddy mixing in conduits. The ADE fails to adequately fit any of these breakthrough curves.

  6. Diffusion of epicenters of earthquake aftershocks, Omori's law, and generalized continuous-time random walk models.

    PubMed

    Helmstetter, A; Sornette, D

    2002-12-01

    The epidemic-type aftershock sequence (ETAS) model is a simple stochastic process modeling seismicity, based on the two best-established empirical laws, the Omori law (power-law decay approximately 1/t(1+theta) of seismicity after an earthquake) and Gutenberg-Richter law (power-law distribution of earthquake energies). In order to describe also the space distribution of seismicity, we use in addition a power-law distribution approximately 1/r(1+mu) of distances between triggered and triggering earthquakes. The ETAS model has been studied for the last two decades to model real seismicity catalogs and to obtain short-term probabilistic forecasts. Here, we present a mapping between the ETAS model and a class of CTRW (continuous time random walk) models, based on the identification of their corresponding master equations. This mapping allows us to use the wealth of results previously obtained on anomalous diffusion of CTRW. After translating into the relevant variable for the ETAS model, we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock. Specifically, we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks. The different regimes are fully characterized by the two exponents theta and mu. Our predictions are checked by careful numerical simulations. We stress the distinction between the "bare" Omori law describing the seismic rate activated directly by a mainshock and the "renormalized" Omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock, and so on. In particular, we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed Omori exponent is less than 1, because this signals the operation of the renormalization of the bare Omori law, also at the

  7. A random walk on water (Henry Darcy Medal Lecture)

    NASA Astrophysics Data System (ADS)

    Koutsoyiannis, D.

    2009-04-01

    . Experimentation with this toy model demonstrates, inter alia, that: (1) for short time horizons the deterministic dynamics is able to give good predictions; but (2) these predictions become extremely inaccurate and useless for long time horizons; (3) for such horizons a naïve statistical prediction (average of past data) which fully neglects the deterministic dynamics is more skilful; and (4) if this statistical prediction, in addition to past data, is combined with the probability theory (the principle of maximum entropy, in particular), it can provide a more informative prediction. Also, the toy model shows that the trajectories of the system state (and derivative properties thereof) do not resemble a regular (e.g., periodic) deterministic process nor a purely random process, but exhibit patterns indicating anti-persistence and persistence (where the latter statistically complies with a Hurst-Kolmogorov behaviour). If the process is averaged over long time scales, the anti-persistent behaviour improves predictability, whereas the persistent behaviour substantially deteriorates it. A stochastic representation of this deterministic system, which incorporates dynamics, is not only possible, but also powerful as it provides good predictions for both short and long horizons and helps to decide on when the deterministic dynamics should be considered or neglected. Obviously, a natural system is extremely more complex than this simple toy model and hence unpredictability is naturally even more prominent in the former. In addition, in a complex natural system, we can never know the exact dynamics and we must infer it from past data, which implies additional uncertainty and an additional role of stochastics in the process of formulating the system equations and estimating the involved parameters. Data also offer the only solid grounds to test any hypothesis about the dynamics, and failure of performing such testing against evidence from data renders the hypothesised dynamics worthless

  8. Random-walk-based stochastic modeling of three-dimensional fiber systems.

    PubMed

    Altendorf, Hellen; Jeulin, Dominique

    2011-04-01

    For the simulation of fiber systems, there exist several stochastic models: systems of straight nonoverlapping fibers, systems of overlapping bending fibers, or fiber systems created by sedimentation. However, there is a lack of models providing dense, nonoverlapping fiber systems with a given random orientation distribution and a controllable level of bending. We introduce a new stochastic model in this paper that generalizes the force-biased packing approach to fibers represented as chains of balls. The starting configuration is modeled using random walks, where two parameters in the multivariate von Mises-Fisher orientation distribution control the bending. The points of the random walk are associated with a radius and the current orientation. The resulting chains of balls are interpreted as fibers. The final fiber configuration is obtained as an equilibrium between repulsion forces avoiding crossing fibers and recover forces ensuring the fiber structure. This approach provides high volume fractions up to 72.0075%.

  9. Optimization of goal-directed movements in the cerebellum: a random walk hypothesis.

    PubMed

    Kitazawa, Shigeru

    2002-08-01

    Voluntary goal-directed movements, such as arm reaching, are nearly optimized in terms of smoothness over the entire movement. Such smoothness is lost with cerebellar dysfunction, suggesting the essential role of the cerebellum in optimizing movement. However, it is still not clear how the cerebellum contributes to achieving smoothness over an entire movement. A recent study has shown that such smoothness of movement can be achieved by reducing the variance of errors at the end of the movement. Here, I hypothesize that the terminal errors conveyed by climbing fibers in the cerebellum serve to reduce not only the mean error, but also the variance of the error, through a process analogous to the random walk through movement control candidates. In the random walk, the direction of each step is randomly determined, but the size of each step is determined by the error at the end of each trial.

  10. Random-walk model simulation of air pollutant dispersion in atmospheric boundary layer in China.

    PubMed

    Wang, Peng; Mu, Hailin

    2011-01-01

    In this study, the land-sea breeze circulation model coupled with a random-walk model is developed by the analysis of the formation and the mechanism of the land-sea breeze. Based on the data of the land-sea circulation in Dalian, China, the model simulated the diurnal variation of pressure, flow, temperature, and turbulent kinetic energy field and also provides a basis for solving the air pollutant concentration in the land-sea breeze circulation so as to estimate the economic cost attributable to the atmospheric pollution. The air pollutant concentration in the background of land-sea circulation is also simulated by a Gaussian dispersion model, and the results revealed that the land-sea circulation model coupled with the random-walk model gives a reasonable description of air pollutant dispersion in coastal areas.

  11. The Length Scale of 3-Space Knots, Ephemeral Knots, and Slipknots in Random Walks

    NASA Astrophysics Data System (ADS)

    Millett, K. C.

    The probability that a random walk or polygon in the 3-space or in the simple cubic lattice contains a small knot, an ephemeral knot, or a slipknot goes to one as the length goes to infinity. The probability that a polygon or walk contains a ``global'' knot also goes to one as the length goes to infinity. What immerges is a highly complex picture of the length scale of knotting in polygons and walks. Here we study the average scale of knots, ephemeral knots, and slipknots in 3-space random walks, paying special attention to the probability of their occurance and to the growth of their average sizes as a function of the length of the walk.

  12. Random walks in Rindler spacetime and string theory at the tip of the cigar

    NASA Astrophysics Data System (ADS)

    Mertens, Thomas G.; Verschelde, Henri; Zakharov, Valentin I.

    2014-03-01

    In this paper, we discuss Rindler space string thermodynamics from a thermal scalar point of view as an explicit example of the results obtained in [1]. We discuss the critical behavior of the string gas and interpret this as a random walk near the black hole horizon. Combining field theory arguments with the random walk path integral picture, we realize (at genus one) the picture put forward by Susskind of a long string surrounding black hole horizons. We find that thermodynamics is dominated by a long string living at stringscale distance from the horizon whose redshifted temperature is the Rindler or Hawking temperature. We provide further evidence of the recent proposal for string theory at the tip of the cigar by comparing with the flat space orbifold approach to Rindler thermodynamics. We discuss all types of closed strings (bosonic, type II and heterotic strings).

  13. A partially reflecting random walk on spheres algorithm for electrical impedance tomography

    SciTech Connect

    Maire, Sylvain; Simon, Martin

    2015-12-15

    In this work, we develop a probabilistic estimator for the voltage-to-current map arising in electrical impedance tomography. This novel so-called partially reflecting random walk on spheres estimator enables Monte Carlo methods to compute the voltage-to-current map in an embarrassingly parallel manner, which is an important issue with regard to the corresponding inverse problem. Our method uses the well-known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions. We analyze the global bias and the variance of the new estimator both theoretically and experimentally. Subsequently, the variance of the new estimator is considerably reduced via a novel control variate conditional sampling technique which yields a highly efficient hybrid forward solver coupling probabilistic and deterministic algorithms.

  14. Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

    PubMed Central

    Skliros, Aris; Park, Wooram; Chirikjian, Gregory S.

    2010-01-01

    This paper presents an efficient group-theoretic approach for computing the statistics of non-reversal random walks (NRRW) on lattices. These framed walks evolve on proper crystallographic space groups. In a previous paper we introduced a convolution method for computing the statistics of NRRWs in which the convolution product is defined relative to the space-group operation. Here we use the corresponding concept of the fast Fourier transform for functions on crystallographic space groups together with a non-Abelian version of the convolution theorem. We develop the theory behind this technique and present numerical results for two-dimensional and three-dimensional lattices (square, cubic and diamond). In order to verify our results, the statistics of the end-to-end distance and the probability of ring closure are calculated and compared with results obtained in the literature for the random walks for which closed-form expressions exist. PMID:21037950

  15. Elephant random walks and their connection to Pólya-type urns

    NASA Astrophysics Data System (ADS)

    Baur, Erich; Bertoin, Jean

    2016-11-01

    In this paper, we explain the connection between the elephant random walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the former. The ERW model was introduced in [Phys. Rev. E 70, 045101 (2004), 10.1103/PhysRevE.70.045101] to study memory effects in a highly non-Markovian setting. More specifically, the ERW is a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a memory parameter p between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of p . Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.

  16. Parsimonious Continuous Time Random Walk Models and Kurtosis for Diffusion in Magnetic Resonance of Biological Tissue

    NASA Astrophysics Data System (ADS)

    Ingo, Carson; Sui, Yi; Chen, Yufen; Parrish, Todd; Webb, Andrew; Ronen, Itamar

    2015-03-01

    In this paper, we provide a context for the modeling approaches that have been developed to describe non-Gaussian diffusion behavior, which is ubiquitous in diffusion weighted magnetic resonance imaging of water in biological tissue. Subsequently, we focus on the formalism of the continuous time random walk theory to extract properties of subdiffusion and superdiffusion through novel simplifications of the Mittag-Leffler function. For the case of time-fractional subdiffusion, we compute the kurtosis for the Mittag-Leffler function, which provides both a connection and physical context to the much-used approach of diffusional kurtosis imaging. We provide Monte Carlo simulations to illustrate the concepts of anomalous diffusion as stochastic processes of the random walk. Finally, we demonstrate the clinical utility of the Mittag-Leffler function as a model to describe tissue microstructure through estimations of subdiffusion and kurtosis with diffusion MRI measurements in the brain of a chronic ischemic stroke patient.

  17. Envelope periodic solutions for a discrete network with the Jacobi elliptic functions and the alternative (G'/G)-expansion method including the generalized Riccati equation

    NASA Astrophysics Data System (ADS)

    Tala-Tebue, E.; Tsobgni-Fozap, D. C.; Kenfack-Jiotsa, A.; Kofane, T. C.

    2014-06-01

    Using the Jacobi elliptic functions and the alternative ( G'/ G-expansion method including the generalized Riccati equation, we derive exact soliton solutions for a discrete nonlinear electrical transmission line in (2+1) dimension. More precisely, these methods are general as they lead us to diverse solutions that have not been previously obtained for the nonlinear electrical transmission lines. This study seeks to show that it is not often necessary to transform the equation of the network into a well-known differential equation before finding its solutions. The solutions obtained by the current methods are generalized periodic solutions of nonlinear equations. The shape of solutions can be well controlled by adjusting the parameters of the network. These exact solutions may have significant applications in telecommunication systems where solitons are used to codify or for the transmission of data.

  18. First passage time: Connecting random walks to functional responses in heterogeneous environments (Invited)

    NASA Astrophysics Data System (ADS)

    Lewis, M. A.; McKenzie, H.; Merrill, E.

    2010-12-01

    In this talk I will outline first passage time analysis for animals undertaking complex movement patterns, and will demonstrate how first passage time can be used to derive functional responses in predator prey systems. The result is a new approach to understanding type III functional responses based on a random walk model. I will extend the analysis to heterogeneous environments to assess the effects of linear features on functional responses in wolves and elk using GPS tracking data.

  19. On the genealogy of branching random walks and of directed polymers

    NASA Astrophysics Data System (ADS)

    Derrida, Bernard; Mottishaw, Peter

    2016-08-01

    It is well known that the mean-field theory of directed polymers in a random medium exhibits replica symmetry breaking with a distribution of overlaps which consists of two delta functions. Here we show that the leading finite-size correction to this distribution of overlaps has a universal character which can be computed explicitly. Our results can also be interpreted as genealogical properties of branching Brownian motion or of branching random walks.

  20. Dynamics of technological evolution: Random walk model for the research enterprise.

    PubMed

    Montroll, E W; Shuler, K E

    1979-12-01

    Technological evolution is a consequence of a sequence of replacements. The development of a new technology generally follows from model testing of the basic ideas on a small scale. Traditional technologies such as aerodynamics and naval architecture involved feasibility experiments on systems characterized by only one or two dimensionless constants. Technologies of the "future" such as magnetically confined fusion depend upon many coupled dimensionless constants. Research and development is modeled and analyzed in terms of random walks in appropriate dimensionless constant space.

  1. Record statistics for biased random walks, with an application to financial data

    NASA Astrophysics Data System (ADS)

    Wergen, Gregor; Bogner, Miro; Krug, Joachim

    2011-05-01

    We consider the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff [Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.101.050601 101, 050601 (2008)] and is well understood. Unlike the case of symmetric jump distributions, in the asymmetric case the statistics of records depends on the choice of the jump distribution. We compute the record rate Pn(c), defined as the probability for the nth value to be larger than all previous values, for a Gaussian jump distribution with standard deviation σ that is shifted by a constant drift c. For small drift, in the sense of c/σ≪n-1/2, the correction to Pn(c) grows proportional to arctan(n) and saturates at the value (c)/(2σ). For large n the record rate approaches a constant, which is approximately given by 1-(σ/2πc)exp(-c2/2σ2) for c/σ≫1. These asymptotic results carry over to other continuous jump distributions with finite variance. As an application, we compare our analytical results to the record statistics of 366 daily stock prices from the Standard & Poor's 500 index. The biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices, and after detrending the number of upper records is in good agreement with the symmetric random walk. However the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified.

  2. Random Walk Models for the Spike Activity of a Single Neuron

    PubMed Central

    Gerstein, George L.; Mandelbrot, Benoit

    1964-01-01

    Quantitative methods for the study of the statistical properties of spontaneously occurring spike trains from single neurons have recently been presented. Such measurements suggest a number of descriptive mathematical models. One of these, based on a random walk towards an absorbing barrier, can describe a wide range of neuronal activity in terms of two parameters. These parameters are readily associated with known physiological mechanisms. ImagesFigure 3 PMID:14104072

  3. Random-walk analysis of displacement statistics of particles in concentrated suspensions of hard spheres

    NASA Astrophysics Data System (ADS)

    van Megen, W.

    2006-01-01

    Mean-squared displacements (MSDs) of colloidal fluids of hard spheres are analyzed in terms of a random walk, an analysis which assumes that the process of structural relaxation among the particles can be described in terms of thermally driven memoryless encounters. For the colloidal fluid in thermodynamic equilibrium the magnitude of the stretching of the MSD is able to be reconciled by a bias in the walk. This description fails for the under-cooled colloidal fluid.

  4. Random walk expectancies for recent global climate, and in an enhanced Greenhouse warming

    NASA Astrophysics Data System (ADS)

    Gordon, Adrian H.; Bye, John A. T.

    1993-11-01

    We partition the United Kingdom Meteorological Office Global Temperature Series ( Tk) using an exponential decay filter into a filtered series ( T̂k) and a difference series ( T' k = T k - T̂k). For a decay time constant, τ ≈ 0.85 years, T̂k is shown to be agood approximation to a random walk generated by a cumulation of normally distributed interannual temperature transitions, and hence ' k contains the predictable temperature signal. The standard deviation of the T̂k series, σ = 0.083K, which is about 1 1/2 that of the T' k series. From this partition, it is argued that τ is the decay time costant (e-folding time) for the global temperature series, and also by the elementary theory of damped oscillations, that the global cimate system (as represented by the global temperature) can only support free oscillations of natural period less than T = 2 πτ ≈ 5 years, i.e. the QBO and ENSO signals. On assuming that σ does not vary significantly over periods up to 20,000 B.P. we find that the expected maximum excursions of the random walks are consistent with the actual inferred temperature variability. On the other hand, the projected temperature rise due to the enhanced Greenhouse effect possibly cannot be supported as a random walk by σ. This suggests that the interannual structure of the climate system would change under these conditions. This conjecture can be tested adequately only with climate models which correctly reproduced random walk behaviour. This is inhibited in published simulated temperature series from coupled models, possibly because of flux correction. An assessment of the likelihood of a change in the interannual variance, and of the ratio between its predictable and random proportions is clearly of utmost significance in the Greenhouse debate, yet it appears to have received very little discussion.

  5. Estimating Mean First Passage Time of Biased Random Walks with Short Relaxation Time on Complex Networks

    PubMed Central

    Lee, Zhuo Qi; Hsu, Wen-Jing; Lin, Miao

    2014-01-01

    Biased random walk has been studied extensively over the past decade especially in the transport and communication networks communities. The mean first passage time (MFPT) of a biased random walk is an important performance indicator in those domains. While the fundamental matrix approach gives precise solution to MFPT, the computation is expensive and the solution lacks interpretability. Other approaches based on the Mean Field Theory relate MFPT to the node degree alone. However, nodes with the same degree may have very different local weight distribution, which may result in vastly different MFPT. We derive an approximate bound to the MFPT of biased random walk with short relaxation time on complex network where the biases are controlled by arbitrarily assigned node weights. We show that the MFPT of a node in this general case is closely related to not only its node degree, but also its local weight distribution. The MFPTs obtained from computer simulations also agree with the new theoretical analysis. Our result enables fast estimation of MFPT, which is useful especially to differentiate between nodes that have very different local node weight distribution even though they share the same node degrees. PMID:24699325

  6. Mean first-passage time for maximal-entropy random walks in complex networks.

    PubMed

    Lin, Yuan; Zhang, Zhongzhi

    2014-06-20

    We perform an in-depth study for mean first-passage time (MFPT)--a primary quantity for random walks with numerous applications--of maximal-entropy random walks (MERW) performed in complex networks. For MERW in a general network, we derive an explicit expression of MFPT in terms of the eigenvalues and eigenvectors of the adjacency matrix associated with the network. For MERW in uncorrelated networks, we also provide a theoretical formula of MFPT at the mean-field level, based on which we further evaluate the dominant scalings of MFPT to different targets for MERW in uncorrelated scale-free networks, and compare the results with those corresponding to traditional unbiased random walks (TURW). We show that the MFPT to a hub node is much lower for MERW than for TURW. However, when the destination is a node with the least degree or a uniformly chosen node, the MFPT is higher for MERW than for TURW. Since MFPT to a uniformly chosen node measures real efficiency of search in networks, our work provides insight into general searching process in complex networks.

  7. Self-organized anomalous aggregation of particles performing nonlinear and non-Markovian random walks

    NASA Astrophysics Data System (ADS)

    Fedotov, Sergei; Korabel, Nickolay

    2015-12-01

    We present a nonlinear and non-Markovian random walks model for stochastic movement and the spatial aggregation of living organisms that have the ability to sense population density. We take into account social crowding effects for which the dispersal rate is a decreasing function of the population density and residence time. We perform stochastic simulations of random walks and discover the phenomenon of self-organized anomaly (SOA), which leads to a collapse of stationary aggregation pattern. This anomalous regime is self-organized and arises without the need for a heavy tailed waiting time distribution from the inception. Conditions have been found under which the nonlinear random walk evolves into anomalous state when all particles aggregate inside a tiny domain (anomalous aggregation). We obtain power-law stationary density-dependent survival function and define the critical condition for SOA as the divergence of mean residence time. The role of the initial conditions in different SOA scenarios is discussed. We observe phenomenon of transient anomalous bimodal aggregation.

  8. Distributed clone detection in static wireless sensor networks: random walk with network division.

    PubMed

    Khan, Wazir Zada; Aalsalem, Mohammed Y; Saad, N M

    2015-01-01

    Wireless Sensor Networks (WSNs) are vulnerable to clone attacks or node replication attacks as they are deployed in hostile and unattended environments where they are deprived of physical protection, lacking physical tamper-resistance of sensor nodes. As a result, an adversary can easily capture and compromise sensor nodes and after replicating them, he inserts arbitrary number of clones/replicas into the network. If these clones are not efficiently detected, an adversary can be further capable to mount a wide variety of internal attacks which can emasculate the various protocols and sensor applications. Several solutions have been proposed in the literature to address the crucial problem of clone detection, which are not satisfactory as they suffer from some serious drawbacks. In this paper we propose a novel distributed solution called Random Walk with Network Division (RWND) for the detection of node replication attack in static WSNs which is based on claimer-reporter-witness framework and combines a simple random walk with network division. RWND detects clone(s) by following a claimer-reporter-witness framework and a random walk is employed within each area for the selection of witness nodes. Splitting the network into levels and areas makes clone detection more efficient and the high security of witness nodes is ensured with moderate communication and memory overheads. Our simulation results show that RWND outperforms the existing witness node based strategies with moderate communication and memory overheads.

  9. Distributed Clone Detection in Static Wireless Sensor Networks: Random Walk with Network Division

    PubMed Central

    Khan, Wazir Zada; Aalsalem, Mohammed Y.; Saad, N. M.

    2015-01-01

    Wireless Sensor Networks (WSNs) are vulnerable to clone attacks or node replication attacks as they are deployed in hostile and unattended environments where they are deprived of physical protection, lacking physical tamper-resistance of sensor nodes. As a result, an adversary can easily capture and compromise sensor nodes and after replicating them, he inserts arbitrary number of clones/replicas into the network. If these clones are not efficiently detected, an adversary can be further capable to mount a wide variety of internal attacks which can emasculate the various protocols and sensor applications. Several solutions have been proposed in the literature to address the crucial problem of clone detection, which are not satisfactory as they suffer from some serious drawbacks. In this paper we propose a novel distributed solution called Random Walk with Network Division (RWND) for the detection of node replication attack in static WSNs which is based on claimer-reporter-witness framework and combines a simple random walk with network division. RWND detects clone(s) by following a claimer-reporter-witness framework and a random walk is employed within each area for the selection of witness nodes. Splitting the network into levels and areas makes clone detection more efficient and the high security of witness nodes is ensured with moderate communication and memory overheads. Our simulation results show that RWND outperforms the existing witness node based strategies with moderate communication and memory overheads. PMID:25992913

  10. Random Walk Based Segmentation for the Prostate on 3D Transrectal Ultrasound Images.

    PubMed

    Ma, Ling; Guo, Rongrong; Tian, Zhiqiang; Venkataraman, Rajesh; Sarkar, Saradwata; Liu, Xiabi; Nieh, Peter T; Master, Viraj V; Schuster, David M; Fei, Baowei

    2016-02-27

    This paper proposes a new semi-automatic segmentation method for the prostate on 3D transrectal ultrasound images (TRUS) by combining the region and classification information. We use a random walk algorithm to express the region information efficiently and flexibly because it can avoid segmentation leakage and shrinking bias. We further use the decision tree as the classifier to distinguish the prostate from the non-prostate tissue because of its fast speed and superior performance, especially for a binary classification problem. Our segmentation algorithm is initialized with the user roughly marking the prostate and non-prostate points on the mid-gland slice which are fitted into an ellipse for obtaining more points. Based on these fitted seed points, we run the random walk algorithm to segment the prostate on the mid-gland slice. The segmented contour and the information from the decision tree classification are combined to determine the initial seed points for the other slices. The random walk algorithm is then used to segment the prostate on the adjacent slice. We propagate the process until all slices are segmented. The segmentation method was tested in 32 3D transrectal ultrasound images. Manual segmentation by a radiologist serves as the gold standard for the validation. The experimental results show that the proposed method achieved a Dice similarity coefficient of 91.37±0.05%. The segmentation method can be applied to 3D ultrasound-guided prostate biopsy and other applications.

  11. Comparing Algorithms for Graph Isomorphism Using Discrete- and Continuous-Time Quantum Random Walks

    DOE PAGES

    Rudinger, Kenneth; Gamble, John King; Bach, Eric; ...

    2013-07-01

    Berry and Wang [Phys. Rev. A 83, 042317 (2011)] show numerically that a discrete-time quan- tum random walk of two noninteracting particles is able to distinguish some non-isomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuous-time quantum walks of two noninteracting parti- cles cannot. We show analytically and numerically that even single-particle discrete-time quantum random walks can distinguish some strongly regular graphs, though not as many as two-particle noninteracting discrete-time walks. Additionally, we demonstrate how, given the same quantum random walk, subtle di erencesmore » in the graph certi cate construction algorithm can nontrivially im- pact the walk's distinguishing power. We also show that no continuous-time walk of a xed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certi cates we consider. We extend this constraint to discrete-time walks of xed numbers of noninteracting particles for one kind of graph certi cate; it remains an open question as to whether or not this constraint applies to the other graph certi cates we consider.« less

  12. Comparing Algorithms for Graph Isomorphism Using Discrete- and Continuous-Time Quantum Random Walks

    SciTech Connect

    Rudinger, Kenneth; Gamble, John King; Bach, Eric; Friesen, Mark; Joynt, Robert; Coppersmith, S. N.

    2013-07-01

    Berry and Wang [Phys. Rev. A 83, 042317 (2011)] show numerically that a discrete-time quan- tum random walk of two noninteracting particles is able to distinguish some non-isomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuous-time quantum walks of two noninteracting parti- cles cannot. We show analytically and numerically that even single-particle discrete-time quantum random walks can distinguish some strongly regular graphs, though not as many as two-particle noninteracting discrete-time walks. Additionally, we demonstrate how, given the same quantum random walk, subtle di erences in the graph certi cate construction algorithm can nontrivially im- pact the walk's distinguishing power. We also show that no continuous-time walk of a xed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certi cates we consider. We extend this constraint to discrete-time walks of xed numbers of noninteracting particles for one kind of graph certi cate; it remains an open question as to whether or not this constraint applies to the other graph certi cates we consider.

  13. Empirical scaling of the length of the longest increasing subsequences of random walks

    NASA Astrophysics Data System (ADS)

    Mendonça, J. Ricardo G.

    2017-02-01

    We provide Monte Carlo estimates of the scaling of the length L n of the longest increasing subsequences of n-step random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate that, barring possible logarithmic corrections, {{L}n}∼ {{n}θ} with the leading scaling exponent 0.60≲ θ ≲ 0.69 for the heavy-tailed distributions of step lengths examined, with values increasing as the distribution becomes more heavy-tailed, and θ ≃ 0.57 for distributions of finite variance, irrespective of the particular distribution. The results are consistent with existing rigorous bounds for θ, although in a somewhat surprising manner. For random walks with step lengths of finite variance, we conjecture that the correct asymptotic behavior of L n is given by \\sqrt{n}\\ln n , and also propose the form for the subleading asymptotics. The distribution of L n was found to follow a simple scaling form with scaling functions that vary with θ. Accordingly, when the step lengths are of finite variance they seem to be universal. The nature of this scaling remains unclear, since we lack a working model, microscopic or hydrodynamic, for the behavior of the length of the longest increasing subsequences of random walks.

  14. Random Walk Based Segmentation for the Prostate on 3D Transrectal Ultrasound Images

    PubMed Central

    Ma, Ling; Guo, Rongrong; Tian, Zhiqiang; Venkataraman, Rajesh; Sarkar, Saradwata; Liu, Xiabi; Nieh, Peter T.; Master, Viraj V.; Schuster, David M.; Fei, Baowei

    2016-01-01

    This paper proposes a new semi-automatic segmentation method for the prostate on 3D transrectal ultrasound images (TRUS) by combining the region and classification information. We use a random walk algorithm to express the region information efficiently and flexibly because it can avoid segmentation leakage and shrinking bias. We further use the decision tree as the classifier to distinguish the prostate from the non-prostate tissue because of its fast speed and superior performance, especially for a binary classification problem. Our segmentation algorithm is initialized with the user roughly marking the prostate and non-prostate points on the mid-gland slice which are fitted into an ellipse for obtaining more points. Based on these fitted seed points, we run the random walk algorithm to segment the prostate on the mid-gland slice. The segmented contour and the information from the decision tree classification are combined to determine the initial seed points for the other slices. The random walk algorithm is then used to segment the prostate on the adjacent slice. We propagate the process until all slices are segmented. The segmentation method was tested in 32 3D transrectal ultrasound images. Manual segmentation by a radiologist serves as the gold standard for the validation. The experimental results show that the proposed method achieved a Dice similarity coefficient of 91.37±0.05%. The segmentation method can be applied to 3D ultrasound-guided prostate biopsy and other applications. PMID:27660383

  15. Random walk with long-range interaction with a barrier and its dual: Exact results

    NASA Astrophysics Data System (ADS)

    Huillet, Thierry

    2010-03-01

    We consider the random walk on , with up and down transition probabilities given the chain is in state x[set membership, variant]{1,2,...}: Here [delta]>=-1 is a real tuning parameter. We assume that this random walk is reflected at the origin. For [delta]>0, the walker is attracted to the origin. The strength of the attraction goes like for large x and so is long-ranged. For [delta]<0, the walker is repelled from the origin. This chain is irreducible and periodic; it is always recurrent, either positive or null recurrent. Using Karlin-McGregor's spectral representations in terms of orthogonal polynomials and first associated orthogonal polynomials, exact expressions are obtained for first return time probabilities to the origin (excursion length), eventual return (contact) probability, excursion height and spatial moments of the walker. All exhibit power-law decay in some range of the parameter [delta]. In the study, an important role is played by the Wall duality relation for birth and death chains with reflecting barrier. Some qualitative aspects of the dual random walk (obtained by interchanging px and qx) are therefore also included.

  16. The adaptive dynamic community detection algorithm based on the non-homogeneous random walking

    NASA Astrophysics Data System (ADS)

    Xin, Yu; Xie, Zhi-Qiang; Yang, Jing

    2016-05-01

    With the changing of the habit and custom, people's social activity tends to be changeable. It is required to have a community evolution analyzing method to mine the dynamic information in social network. For that, we design the random walking possibility function and the topology gain function to calculate the global influence matrix of the nodes. By the analysis of the global influence matrix, the clustering directions of the nodes can be obtained, thus the NRW (Non-Homogeneous Random Walk) method for detecting the static overlapping communities can be established. We design the ANRW (Adaptive Non-Homogeneous Random Walk) method via adapting the nodes impacted by the dynamic events based on the NRW. The ANRW combines the local community detection with dynamic adaptive adjustment to decrease the computational cost for ANRW. Furthermore, the ANRW treats the node as the calculating unity, thus the running manner of the ANRW is suitable to the parallel computing, which could meet the requirement of large dataset mining. Finally, by the experiment analysis, the efficiency of ANRW on dynamic community detection is verified.

  17. Random walk based segmentation for the prostate on 3D transrectal ultrasound images

    NASA Astrophysics Data System (ADS)

    Ma, Ling; Guo, Rongrong; Tian, Zhiqiang; Venkataraman, Rajesh; Sarkar, Saradwata; Liu, Xiabi; Nieh, Peter T.; Master, Viraj V.; Schuster, David M.; Fei, Baowei

    2016-03-01

    This paper proposes a new semi-automatic segmentation method for the prostate on 3D transrectal ultrasound images (TRUS) by combining the region and classification information. We use a random walk algorithm to express the region information efficiently and flexibly because it can avoid segmentation leakage and shrinking bias. We further use the decision tree as the classifier to distinguish the prostate from the non-prostate tissue because of its fast speed and superior performance, especially for a binary classification problem. Our segmentation algorithm is initialized with the user roughly marking the prostate and non-prostate points on the mid-gland slice which are fitted into an ellipse for obtaining more points. Based on these fitted seed points, we run the random walk algorithm to segment the prostate on the mid-gland slice. The segmented contour and the information from the decision tree classification are combined to determine the initial seed points for the other slices. The random walk algorithm is then used to segment the prostate on the adjacent slice. We propagate the process until all slices are segmented. The segmentation method was tested in 32 3D transrectal ultrasound images. Manual segmentation by a radiologist serves as the gold standard for the validation. The experimental results show that the proposed method achieved a Dice similarity coefficient of 91.37+/-0.05%. The segmentation method can be applied to 3D ultrasound-guided prostate biopsy and other applications.

  18. Self-organized anomalous aggregation of particles performing nonlinear and non-Markovian random walks.

    PubMed

    Fedotov, Sergei; Korabel, Nickolay

    2015-12-01

    We present a nonlinear and non-Markovian random walks model for stochastic movement and the spatial aggregation of living organisms that have the ability to sense population density. We take into account social crowding effects for which the dispersal rate is a decreasing function of the population density and residence time. We perform stochastic simulations of random walks and discover the phenomenon of self-organized anomaly (SOA), which leads to a collapse of stationary aggregation pattern. This anomalous regime is self-organized and arises without the need for a heavy tailed waiting time distribution from the inception. Conditions have been found under which the nonlinear random walk evolves into anomalous state when all particles aggregate inside a tiny domain (anomalous aggregation). We obtain power-law stationary density-dependent survival function and define the critical condition for SOA as the divergence of mean residence time. The role of the initial conditions in different SOA scenarios is discussed. We observe phenomenon of transient anomalous bimodal aggregation.

  19. Planar elliptic growth

    SciTech Connect

    Mineev, Mark

    2008-01-01

    The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for areapreserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integration for closed form solutions are identified as the singularities of the Schwarz function, which are located both inside and outside the moving contour. Well-posedness of the recovery of the elliptic operator governing the process from the continuum of interfaces parametrized by time is addressed and two examples of exact solutions of elliptic growth are presented.

  20. Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in ℝN

    NASA Astrophysics Data System (ADS)

    Liang, Sihua; Zhang, Jihui

    2016-11-01

    In this paper, we deal with the existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical nonlinearity: -ε4 Δ2 u + ε4 a + b ∫ ℝN |∇ u|)2 d x Δ u + V ( x ) u = |u| 2* * - 2 u + h ( x , u ) , (t, x) ∈ ℝ × ℝN. By using Lions' second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PS) condition holds locally and by variational method, we prove that it has at least one solution and for any m ∈ ℕ, it has at least m pairs of solutions.

  1. Application of random walk concept to the cyclic diffusion mechanisms for self-diffusion in intermetallic compounds

    NASA Astrophysics Data System (ADS)

    Tiwari, G. P.; Mehrotra, R. S.; Iijima, Y.

    2014-02-01

    Huntington-Elcock-McCombie (HEM) mechanism involving six consecutive and correlated jumps, a triple-defect mechanism (TDM) involving three correlated jumps and an anti-structure bridge (ASB) mechanism invoking the migration of an anti-structure atom are the three mechanisms currently in vogue to explain the self- and solute diffusion in intermetallic compounds. Among them, HEM and TDM are cyclic in nature. The HEM and TDM constitute the theme of the present article. The concept of random walk is applied to them and appropriate expressions for the diffusion coefficient are derived. These equations are then employed to estimate activation energies for self-diffusion via HEM and TDM processes and compared with the available experimental data on activation energy for self-diffusion in intermetallic compounds. The resulting activation energies do not favour HEM and TDM for the self-diffusion in intermetallic compounds. A comparison of the sum of experimentally determined activation energies for vacancy formation and migration with the activation energies for self-diffusion determined from radioactive tracer method favours the conventional monovacancy-mediated process for self-diffusion in intermetallic compounds.

  2. A polymer, random walk model for the size-distribution of large DNA fragments after high linear energy transfer radiation

    NASA Technical Reports Server (NTRS)

    Ponomarev, A. L.; Brenner, D.; Hlatky, L. R.; Sachs, R. K.

    2000-01-01

    DNA double-strand breaks (DSBs) produced by densely ionizing radiation are not located randomly in the genome: recent data indicate DSB clustering along chromosomes. Stochastic DSB clustering at large scales, from > 100 Mbp down to < 0.01 Mbp, is modeled using computer simulations and analytic equations. A random-walk, coarse-grained polymer model for chromatin is combined with a simple track structure model in Monte Carlo software called DNAbreak and is applied to data on alpha-particle irradiation of V-79 cells. The chromatin model neglects molecular details but systematically incorporates an increase in average spatial separation between two DNA loci as the number of base-pairs between the loci increases. Fragment-size distributions obtained using DNAbreak match data on large fragments about as well as distributions previously obtained with a less mechanistic approach. Dose-response relations, linear at small doses of high linear energy transfer (LET) radiation, are obtained. They are found to be non-linear when the dose becomes so large that there is a significant probability of overlapping or close juxtaposition, along one chromosome, for different DSB clusters from different tracks. The non-linearity is more evident for large fragments than for small. The DNAbreak results furnish an example of the RLC (randomly located clusters) analytic formalism, which generalizes the broken-stick fragment-size distribution of the random-breakage model that is often applied to low-LET data.

  3. A polymer, random walk model for the size-distribution of large DNA fragments after high linear energy transfer radiation

    NASA Technical Reports Server (NTRS)

    Ponomarev, A. L.; Brenner, D.; Hlatky, L. R.; Sachs, R. K.

    2000-01-01

    DNA double-strand breaks (DSBs) produced by densely ionizing radiation are not located randomly in the genome: recent data indicate DSB clustering along chromosomes. Stochastic DSB clustering at large scales, from > 100 Mbp down to < 0.01 Mbp, is modeled using computer simulations and analytic equations. A random-walk, coarse-grained polymer model for chromatin is combined with a simple track structure model in Monte Carlo software called DNAbreak and is applied to data on alpha-particle irradiation of V-79 cells. The chromatin model neglects molecular details but systematically incorporates an increase in average spatial separation between two DNA loci as the number of base-pairs between the loci increases. Fragment-size distributions obtained using DNAbreak match data on large fragments about as well as distributions previously obtained with a less mechanistic approach. Dose-response relations, linear at small doses of high linear energy transfer (LET) radiation, are obtained. They are found to be non-linear when the dose becomes so large that there is a significant probability of overlapping or close juxtaposition, along one chromosome, for different DSB clusters from different tracks. The non-linearity is more evident for large fragments than for small. The DNAbreak results furnish an example of the RLC (randomly located clusters) analytic formalism, which generalizes the broken-stick fragment-size distribution of the random-breakage model that is often applied to low-LET data.

  4. Applications of a general random-walk theory for confined diffusion

    NASA Astrophysics Data System (ADS)

    Calvo-Muñoz, Elisa M.; Selvan, Myvizhi Esai; Xiong, Ruichang; Ojha, Madhusudan; Keffer, David J.; Nicholson, Donald M.; Egami, Takeshi

    2011-01-01

    A general random walk theory for diffusion in the presence of nanoscale confinement is developed and applied. The random-walk theory contains two parameters describing confinement: a cage size and a cage-to-cage hopping probability. The theory captures the correct nonlinear dependence of the mean square displacement (MSD) on observation time for intermediate times. Because of its simplicity, the theory also requires modest computational requirements and is thus able to simulate systems with very low diffusivities for sufficiently long time to reach the infinite-time-limit regime where the Einstein relation can be used to extract the self-diffusivity. The theory is applied to three practical cases in which the degree of order in confinement varies. The three systems include diffusion of (i) polyatomic molecules in metal organic frameworks, (ii) water in proton exchange membranes, and (iii) liquid and glassy iron. For all three cases, the comparison between theory and the results of molecular dynamics (MD) simulations indicates that the theory can describe the observed diffusion behavior with a small fraction of the computational expense. The confined-random-walk theory fit to the MSDs of very short MD simulations is capable of accurately reproducing the MSDs of much longer MD simulations. Furthermore, the values of the parameter for cage size correspond to the physical dimensions of the systems and the cage-to-cage hopping probability corresponds to the activation barrier for diffusion, indicating that the two parameters in the theory are not simply fitted values but correspond to real properties of the physical system.

  5. Generalized Hammersley Process and Phase Transition for Activated Random Walk Models

    NASA Astrophysics Data System (ADS)

    Rolla, Leonardo T.

    2008-12-01

    * ACTIVATED RANDOM WALK MODEL * This is a conservative particle system on the lattice, with a Markovian continuous-time evolution. Active particles perform random walks without interaction, and they may as well change their state to passive, then stopping to jump. When particles of both types occupy the same site, they all become active. This model exhibits phase transition in the sense that for low initial densities the system locally fixates and for high densities it keeps active. Though extensively studied in the physics literature, the matter of giving a mathematical proof of such phase transition remained as an open problem for several years. In this work we identify some variables that are sufficient to characterize fixation and at the same time are stochastically monotone in the model's parameters. We employ an explicit graphical representation in order to obtain the monotonicity. With this method we prove that there is a unique phase transition for the one-dimensional finite-range random walk. Joint with V. Sidoravicius. * BROKEN LINE PROCESS * We introduce the broken line process and derive some of its properties. Its discrete version is presented first and a natural generalization to the continuum is then proposed and studied. The broken lines are related to the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented paths. For a class of passage time distributions there is a family of boundary conditions that make the process stationary and reversible. One application is a simple proof of the explicit law of large numbers for last passage percolation with exponential and geometric distributions. Joint with V. Sidoravicius, D. Surgailis, and M. E. Vares.

  6. Applications of a general random-walk theory for confined diffusion.

    PubMed

    Calvo-Muñoz, Elisa M; Selvan, Myvizhi Esai; Xiong, Ruichang; Ojha, Madhusudan; Keffer, David J; Nicholson, Donald M; Egami, Takeshi

    2011-01-01

    A general random walk theory for diffusion in the presence of nanoscale confinement is developed and applied. The random-walk theory contains two parameters describing confinement: a cage size and a cage-to-cage hopping probability. The theory captures the correct nonlinear dependence of the mean square displacement (MSD) on observation time for intermediate times. Because of its simplicity, the theory also requires modest computational requirements and is thus able to simulate systems with very low diffusivities for sufficiently long time to reach the infinite-time-limit regime where the Einstein relation can be used to extract the self-diffusivity. The theory is applied to three practical cases in which the degree of order in confinement varies. The three systems include diffusion of (i) polyatomic molecules in metal organic frameworks, (ii) water in proton exchange membranes, and (iii) liquid and glassy iron. For all three cases, the comparison between theory and the results of molecular dynamics (MD) simulations indicates that the theory can describe the observed diffusion behavior with a small fraction of the computational expense. The confined-random-walk theory fit to the MSDs of very short MD simulations is capable of accurately reproducing the MSDs of much longer MD simulations. Furthermore, the values of the parameter for cage size correspond to the physical dimensions of the systems and the cage-to-cage hopping probability corresponds to the activation barrier for diffusion, indicating that the two parameters in the theory are not simply fitted values but correspond to real properties of the physical system.

  7. An Elliptic Garnier System from Interpolation

    NASA Astrophysics Data System (ADS)

    Yamada, Yasuhiko

    2017-09-01

    Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painlevé equation.

  8. Intra-fraction motion of the prostate is a random walk

    NASA Astrophysics Data System (ADS)

    Ballhausen, H.; Li, M.; Hegemann, N.-S.; Ganswindt, U.; Belka, C.

    2015-01-01

    A random walk model for intra-fraction motion has been proposed, where at each step the prostate moves a small amount from its current position in a random direction. Online tracking data from perineal ultrasound is used to validate or reject this model against alternatives. Intra-fraction motion of a prostate was recorded by 4D ultrasound (Elekta Clarity system) during 84 fractions of external beam radiotherapy of six patients. In total, the center of the prostate was tracked for 8 h in intervals of 4 s. Maximum likelihood model parameters were fitted to the data. The null hypothesis of a random walk was tested with the Dickey-Fuller test. The null hypothesis of stationarity was tested by the Kwiatkowski-Phillips-Schmidt-Shin test. The increase of variance in prostate position over time and the variability in motility between fractions were analyzed. Intra-fraction motion of the prostate was best described as a stochastic process with an auto-correlation coefficient of ρ = 0.92  ±  0.13. The random walk hypothesis (ρ = 1) could not be rejected (p = 0.27). The static noise hypothesis (ρ = 0) was rejected (p < 0.001). The Dickey-Fuller test rejected the null hypothesis ρ = 1 in 25% to 32% of cases. On average, the Kwiatkowski-Phillips-Schmidt-Shin test rejected the null hypothesis ρ = 0 with a probability of 93% to 96%. The variance in prostate position increased linearly over time (r2 = 0.9  ±  0.1). Variance kept increasing and did not settle at a maximum as would be expected from a stationary process. There was substantial variability in motility between fractions and patients with maximum aberrations from isocenter ranging from 0.5 mm to over 10 mm in one patient alone. In conclusion, evidence strongly suggests that intra-fraction motion of the prostate is a random walk and neither static (like inter-fraction setup errors) nor stationary (like a cyclic motion such as breathing, for example). The prostate tends to drift away from the

  9. Continuous Time Random Walk and Migration-Proliferation Dichotomy of Brain Cancer

    NASA Astrophysics Data System (ADS)

    Iomin, A.

    A theory of fractional kinetics of glial cancer cells is presented. A role of the migration-proliferation dichotomy in the fractional cancer cell dynamics in the outer-invasive zone is discussed and explained in the framework of a continuous time random walk. The main suggested model is based on a construction of a 3D comb model, where the migration-proliferation dichotomy becomes naturally apparent and the outer-invasive zone of glioma cancer is considered as a fractal composite with a fractal dimension Dfr < 3.

  10. Continuous Time Random Walk and Migration-Proliferation Dichotomy of Brain Cancer

    NASA Astrophysics Data System (ADS)

    Iomin, A.

    2015-10-01

    A theory of fractional kinetics of glial cancer cells is presented. A role of the migration-proliferation dichotomy in the fractional cancer cell dynamics in the outer-invasive zone is discussed and explained in the framework of a continuous time random walk. The main suggested model is based on a construction of a 3D comb model, where the migration-proliferation dichotomy becomes naturally apparent and the outer-invasive zone of glioma cancer is considered as a fractal composite with a fractal dimension Dfr < 3.

  11. Transport properties of a two-dimensional ``chiral'' persistent random walk

    NASA Astrophysics Data System (ADS)

    Larralde, H.

    1997-11-01

    The usual two-dimensional persistent random walk is generalized by introducing a clockwise (or counterclockwise) angular bias at each new step direction. This bias breaks the reflection symmetry of the problem, giving the walker a tendency to ``loop,'' and gives rise to unusual transport properties. In particular, there is a resonantlike enhancement of the diffusion constant as the parameters of the system are changed. Also, in response to an external field, the looping tendency can resist or enhance the drift along the field and gives rise to a drift transverse to the field. These results are obtained analytically, and, for completeness, compared with Monte Carlo simulations of the walk.

  12. Comparing quantum versus Markov random walk models of judgements measured by rating scales

    PubMed Central

    Wang, Z.; Busemeyer, J. R.

    2016-01-01

    Quantum and Markov random walk models are proposed for describing how people evaluate stimuli using rating scales. To empirically test these competing models, we conducted an experiment in which participants judged the effectiveness of public health service announcements from either their own personal perspective or from the perspective of another person. The order of the self versus other judgements was manipulated, which produced significant sequential effects. The quantum and Markov models were fitted to the data using the same number of parameters, and the model comparison strongly supported the quantum over the Markov model. PMID:26621984

  13. Tracking Random Walk of Individual Domain Walls in Cylindrical Nanomagnets with Resistance Noise

    NASA Astrophysics Data System (ADS)

    Singh, Amrita; Mukhopadhyay, Soumik; Ghosh, Arindam

    2010-08-01

    The stochasticity of domain-wall (DW) motion in magnetic nanowires has been probed by measuring slow fluctuations, or noise, in electrical resistance at small magnetic fields. By controlled injection of DWs into isolated cylindrical nanowires of nickel, we have been able to track the motion of the DWs between the electrical leads by discrete steps in the resistance. Closer inspection of the time dependence of noise reveals a diffusive random walk of the DWs with a universal kinetic exponent. Our experiments outline a method with which electrical resistance is able to detect the kinetic state of the DWs inside the nanowires, which can be useful in DW-based memory designs.

  14. Experimental implementation of a quantum random-walk search algorithm using strongly dipolar coupled spins

    NASA Astrophysics Data System (ADS)

    Lu, Dawei; Zhu, Jing; Zou, Ping; Peng, Xinhua; Yu, Yihua; Zhang, Shanmin; Chen, Qun; Du, Jiangfeng

    2010-02-01

    An important quantum search algorithm based on the quantum random walk performs an oracle search on a database of N items with O(phN) calls, yielding a speedup similar to the Grover quantum search algorithm. The algorithm was implemented on a quantum information processor of three-qubit liquid-crystal nuclear magnetic resonance (NMR) in the case of finding 1 out of 4, and the diagonal elements’ tomography of all the final density matrices was completed with comprehensible one-dimensional NMR spectra. The experimental results agree well with the theoretical predictions.

  15. Dynamics of technological evolution: Random walk model for the research enterprise

    PubMed Central

    Montroll, Elliott W.; Shuler, Kurt E.

    1979-01-01

    Technological evolution is a consequence of a sequence of replacements. The development of a new technology generally follows from model testing of the basic ideas on a small scale. Traditional technologies such as aerodynamics and naval architecture involved feasibility experiments on systems characterized by only one or two dimensionless constants. Technologies of the “future” such as magnetically confined fusion depend upon many coupled dimensionless constants. Research and development is modeled and analyzed in terms of random walks in appropriate dimensionless constant space. PMID:16592727

  16. The exact probability distribution of a two-dimensional random walk

    NASA Astrophysics Data System (ADS)

    Stadje, W.

    1987-01-01

    A calculation is made of the exact probability distribution of the two-dimensional displacement of a particle at time t that starts at the origin, moves in straight-line paths at constant speed, and changes its direction after exponentially distributed time intervals, where the lengths of the straight-line paths and the turn angles are independent, the angles being uniformly distributed. This random walk is the simplest model for the locomotion of microorganisms on surfaces. Its weak convergence to a Wiener process is also shown.

  17. A boundary element-Random walk model of mass transport in groundwater

    USGS Publications Warehouse

    Kemblowski, M.

    1986-01-01

    A boundary element solution to the convective mass transport in groundwater is presented. This solution produces a continuous velocity field and reduces the amount of data preparation time and bookkeeping. By combining this solution and the random walk procedure, a convective-dispersive mass transport model is obtained. This model may be easily used to simulate groundwater contamination problems. The accuracy of the boundary element model has been verified by reproducing the analytical solution to a two-dimensional convective mass transport problem. The method was also used to simulate a convective-dispersive problem. ?? 1986.

  18. Slower deviations of the branching Brownian motion and of branching random walks

    NASA Astrophysics Data System (ADS)

    Derrida, Bernard; Shi, Zhan

    2017-08-01

    We have shown recently how to calculate the large deviation function of the position X\\max(t) of the rightmost particle of a branching Brownian motion at time t. This large deviation function exhibits a phase transition at a certain negative velocity. Here we extend this result to more general branching random walks and show that the probability distribution of X\\max(t) has, asymptotically in time, a prefactor characterized by a non trivial power law. Dedicated to John Cardy on the occasion of his 70th birthday.

  19. Random-walk approach to the d-dimensional disordered Lorentz gas.

    PubMed

    Adib, Artur B

    2008-02-01

    A correlated random walk approach to diffusion is applied to the disordered nonoverlapping Lorentz gas. By invoking the Lu-Torquato theory for chord-length distributions in random media [J. Chem. Phys. 98, 6472 (1993)], an analytic expression for the diffusion constant in arbitrary number of dimensions d is obtained. The result corresponds to an Enskog-like correction to the Boltzmann prediction, being exact in the dilute limit, and better or nearly exact in comparison to renormalized kinetic theory predictions for all allowed densities in d=2,3 . Extensive numerical simulations were also performed to elucidate the role of the approximations involved.

  20. Experimental implementation of a quantum random-walk search algorithm using strongly dipolar coupled spins

    SciTech Connect

    Lu Dawei; Peng Xinhua; Du Jiangfeng; Zhu Jing; Zou Ping; Yu Yihua; Zhang Shanmin; Chen Qun

    2010-02-15

    An important quantum search algorithm based on the quantum random walk performs an oracle search on a database of N items with O({radical}(phN)) calls, yielding a speedup similar to the Grover quantum search algorithm. The algorithm was implemented on a quantum information processor of three-qubit liquid-crystal nuclear magnetic resonance (NMR) in the case of finding 1 out of 4, and the diagonal elements' tomography of all the final density matrices was completed with comprehensible one-dimensional NMR spectra. The experimental results agree well with the theoretical predictions.

  1. Random walk in degree space and the time-dependent Watts-Strogatz model.

    PubMed

    Casa Grande, H L; Cotacallapa, M; Hase, M O

    2017-01-01

    In this work, we propose a scheme that provides an analytical estimate for the time-dependent degree distribution of some networks. This scheme maps the problem into a random walk in degree space, and then we choose the paths that are responsible for the dominant contributions. The method is illustrated on the dynamical versions of the Erdős-Rényi and Watts-Strogatz graphs, which were introduced as static models in the original formulation. We have succeeded in obtaining an analytical form for the dynamics Watts-Strogatz model, which is asymptotically exact for some regimes.

  2. Random walk in degree space and the time-dependent Watts-Strogatz model

    NASA Astrophysics Data System (ADS)

    Casa Grande, H. L.; Cotacallapa, M.; Hase, M. O.

    2017-01-01

    In this work, we propose a scheme that provides an analytical estimate for the time-dependent degree distribution of some networks. This scheme maps the problem into a random walk in degree space, and then we choose the paths that are responsible for the dominant contributions. The method is illustrated on the dynamical versions of the Erdős-Rényi and Watts-Strogatz graphs, which were introduced as static models in the original formulation. We have succeeded in obtaining an analytical form for the dynamics Watts-Strogatz model, which is asymptotically exact for some regimes.

  3. Parrondo-like behavior in continuous-time random walks with memory

    NASA Astrophysics Data System (ADS)

    Montero, Miquel

    2011-11-01

    The continuous-time random walk (CTRW) formalism can be adapted to encompass stochastic processes with memory. In this paper we will show how the random combination of two different unbiased CTRWs can give rise to a process with clear drift, if one of them is a CTRW with memory. If one identifies the other one as noise, the effect can be thought of as a kind of stochastic resonance. The ultimate origin of this phenomenon is the same as that of the Parrondo paradox in game theory.

  4. Describing the motion of a body with an elliptical cross section in a viscous uncompressible fluid by model equations reconstructed from data processing

    NASA Astrophysics Data System (ADS)

    Borisov, A. V.; Kuznetsov, S. P.; Mamaev, I. S.; Tenenev, V. A.

    2016-09-01

    From analysis of time series obtained on the numerical solution of a plane problem on the motion of a body with an elliptic cross section under the action of gravity force in an incompressible viscous fluid, a system of ordinary differential equations approximately describing the dynamics of the body is reconstructed. To this end, coefficients responsible for the added mass, the force caused by the circulation of the velocity field, and the resisting force are found by the least square adjustment. The agreement between the finitedimensional description and the simulation on the basis of the Navier-Stokes equations is illustrated by images of attractors in regular and chaotic modes. The coefficients found make it possible to estimate the actual contribution of different effects to the dynamics of the body.

  5. Interpolating between random walks and optimal transportation routes: Flow with multiple sources and targets

    NASA Astrophysics Data System (ADS)

    Guex, Guillaume

    2016-05-01

    In recent articles about graphs, different models proposed a formalism to find a type of path between two nodes, the source and the target, at crossroads between the shortest-path and the random-walk path. These models include a freely adjustable parameter, allowing to tune the behavior of the path toward randomized movements or direct routes. This article presents a natural generalization of these models, namely a model with multiple sources and targets. In this context, source nodes can be viewed as locations with a supply of a certain good (e.g. people, money, information) and target nodes as locations with a demand of the same good. An algorithm is constructed to display the flow of goods in the network between sources and targets. With again a freely adjustable parameter, this flow can be tuned to follow routes of minimum cost, thus displaying the flow in the context of the optimal transportation problem or, by contrast, a random flow, known to be similar to the electrical current flow if the random-walk is reversible. Moreover, a source-targetcoupling can be retrieved from this flow, offering an optimal assignment to the transportation problem. This algorithm is described in the first part of this article and then illustrated with case studies.

  6. BRWLDA: bi-random walks for predicting lncRNA-disease associations

    PubMed Central

    Yu, Guoxian; Fu, Guangyuan; Lu, Chang; Ren, Yazhou; Wang, Jun

    2017-01-01

    Increasing efforts have been done to figure out the association between lncRNAs and complex diseases. Many computational models construct various lncRNA similarity networks, disease similarity networks, along with known lncRNA-disease associations to infer novel associations. However, most of them neglect the structural difference between lncRNAs network and diseases network, hierarchical relationships between diseases and pattern of newly discovered associations. In this study, we developed a model that performs Bi-Random Walks to predict novel LncRNA-Disease Associations (BRWLDA in short). This model utilizes multiple heterogeneous data to construct the lncRNA functional similarity network, and Disease Ontology to construct a disease network. It then constructs a directed bi-relational network based on these two networks and available lncRNAs-disease associations. Next, it applies bi-random walks on the network to predict potential associations. BRWLDA achieves reliable and better performance than other comparing methods not only on experiment verified associations, but also on the simulated experiments with masked associations. Case studies further demonstrate the feasibility of BRWLDA in identifying new lncRNA-disease associations.

  7. BRWLDA: bi-random walks for predicting lncRNA-disease associations.

    PubMed

    Yu, Guoxian; Fu, Guangyuan; Lu, Chang; Ren, Yazhou; Wang, Jun

    2017-09-01

    Increasing efforts have been done to figure out the association between lncRNAs and complex diseases. Many computational models construct various lncRNA similarity networks, disease similarity networks, along with known lncRNA-disease associations to infer novel associations. However, most of them neglect the structural difference between lncRNAs network and diseases network, hierarchical relationships between diseases and pattern of newly discovered associations. In this study, we developed a model that performs Bi-Random Walks to predict novel LncRNA-Disease Associations (BRWLDA in short). This model utilizes multiple heterogeneous data to construct the lncRNA functional similarity network, and Disease Ontology to construct a disease network. It then constructs a directed bi-relational network based on these two networks and available lncRNAs-disease associations. Next, it applies bi-random walks on the network to predict potential associations. BRWLDA achieves reliable and better performance than other comparing methods not only on experiment verified associations, but also on the simulated experiments with masked associations. Case studies further demonstrate the feasibility of BRWLDA in identifying new lncRNA-disease associations.

  8. IRWRLDA: improved random walk with restart for lncRNA-disease association prediction

    PubMed Central

    Chen, Xing; You, Zhu-Hong; Yan, Gui-Ying; Gong, Dun-Wei

    2016-01-01

    In recent years, accumulating evidences have shown that the dysregulations of lncRNAs are associated with a wide range of human diseases. It is necessary and feasible to analyze known lncRNA-disease associations, predict potential lncRNA-disease associations, and provide the most possible lncRNA-disease pairs for experimental validation. Considering the limitations of traditional Random Walk with Restart (RWR), the model of Improved Random Walk with Restart for LncRNA-Disease Association prediction (IRWRLDA) was developed to predict novel lncRNA-disease associations by integrating known lncRNA-disease associations, disease semantic similarity, and various lncRNA similarity measures. The novelty of IRWRLDA lies in the incorporation of lncRNA expression similarity and disease semantic similarity to set the initial probability vector of the RWR. Therefore, IRWRLDA could be applied to diseases without any known related lncRNAs. IRWRLDA significantly improved previous classical models with reliable AUCs of 0.7242 and 0.7872 in two known lncRNA-disease association datasets downloaded from the lncRNADisease database, respectively. Further case studies of colon cancer and leukemia were implemented for IRWRLDA and 60% of lncRNAs in the top 10 prediction lists have been confirmed by recent experimental reports. PMID:27517318

  9. Seed-weighted random walk ranking for cancer biomarker prioritisation: a case study in leukaemia.

    PubMed

    Huan, Tianxiao; Wu, Xiaogang; Bai, Zengliang; Chen, Jake Y

    2014-01-01

    A central focus of clinical proteomics for cancer is to identify protein biomarkers with diagnostic and therapeutic application potential. Network-based analyses have been used in computational disease-related gene prioritisation for several years. The Random Walk Ranking (RWR) algorithm has been successfully applied to prioritising disease-related gene candidates by exploiting global network topology in a Protein-Protein Interaction (PPI) network. Increasing the specificity and sensitivity ofbiomarkers may require consideration of similar or closely-related disease phenotypes and molecular pathological mechanisms shared across different disease phenotypes. In this paper, we propose a method called Seed-Weighted Random Walk Ranking (SW-RWR) for prioritizing cancer biomarker candidates. This method uses the information of cancer phenotype association to assign to each gene a disease-specific, weighted value to guide the RWR algorithm in a global human PPI network. In a case study of prioritizing leukaemia biomarkers, SW-RWR outperformed a typical local network-based analysis in coverage and also showed better accuracy and sensitivity than the original RWR method (global network-based analysis). Our results suggest that the tight correlation among different cancer phenotypes could play an important role in cancer biomarker discovery.

  10. Generalized essential energy space random walks to more effectively accelerate solute sampling in aqueous environment.

    PubMed

    Lv, Chao; Zheng, Lianqing; Yang, Wei

    2012-01-28

    Molecular dynamics sampling can be enhanced via the promoting of potential energy fluctuations, for instance, based on a Hamiltonian modified with the addition of a potential-energy-dependent biasing term. To overcome the diffusion sampling issue, which reveals the fact that enlargement of event-irrelevant energy fluctuations may abolish sampling efficiency, the essential energy space random walk (EESRW) approach was proposed earlier. To more effectively accelerate the sampling of solute conformations in aqueous environment, in the current work, we generalized the EESRW method to a two-dimension-EESRW (2D-EESRW) strategy. Specifically, the essential internal energy component of a focused region and the essential interaction energy component between the focused region and the environmental region are employed to define the two-dimensional essential energy space. This proposal is motivated by the general observation that in different conformational events, the two essential energy components have distinctive interplays. Model studies on the alanine dipeptide and the aspartate-arginine peptide demonstrate sampling improvement over the original one-dimension-EESRW strategy; with the same biasing level, the present generalization allows more effective acceleration of the sampling of conformational transitions in aqueous solution. The 2D-EESRW generalization is readily extended to higher dimension schemes and employed in more advanced enhanced-sampling schemes, such as the recent orthogonal space random walk method. © 2012 American Institute of Physics

  11. Random Walks on Digraphs, the Generalized Digraph Laplacian and the Degree of Asymmetry

    NASA Astrophysics Data System (ADS)

    Li, Yanhua; Zhang, Zhi-Li

    In this paper we extend and generalize the standard random walk theory (or spectral graph theory) on undirected graphs to digraphs. In particular, we introduce and define a (normalized) digraph Laplacian matrix, and prove that 1) its Moore-Penrose pseudo-inverse is the (discrete) Green's function of the digraph Laplacian matrix (as an operator on digraphs), and 2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs. Using these results, we derive new formula for computing hitting and commute times in terms of the Moore-Penrose pseudo-inverse of the digraph Laplacian, or equivalently, the singular values and vectors of the digraph Laplacian. Furthermore, we show that the Cheeger constant defined in [6] is intrinsically a quantity associated with undirected graphs. This motivates us to introduce a metric - the largest singular value of Δ:=(tilde{\\cal L}-tilde{\\cal L}^T)/2 - to quantify and measure the degree of asymmetry in a digraph. Using this measure, we establish several new results, such as a tighter bound (than that of Fill's in [9] and Chung's in [6]) on the Markov chain mixing rate, and a bound on the second smallest singular value of tilde{\\cal L}.

  12. Identify the diversity of mesoscopic structures in networks: A mixed random walk approach

    NASA Astrophysics Data System (ADS)

    Ma, Yifang; Jiang, Xin; Li, Meng; Shen, Xin; Guo, Quantong; Lei, Yanjun; Zheng, Zhiming

    2013-10-01

    Community or cluster structure, which can provide insight into the natural partitions and inner connections of a network, is a key feature in studying the mesoscopic structure of complex systems. Although numerous methods for community detection have been proposed ever since, there is still a lack of understanding on how to quantify the diversity of pre-divided community structures, or rank the roles of communities in participating in specific dynamic processes. Inspired by the Law of Mass Action in chemical kinetics, we introduce here the community random walk energy (CRWE), which reflects a potential based on the diffusion phase of a mixed random walk process taking place on the network, to identify the configuration of community structures. The difference of CRWE allows us to distinguish the intrinsic topological diversity between individual communities, on condition that all the communities are pre-arranged in the network. We illustrate our method by performing numerical simulations on constructive community networks and a real social network with distinct community structures. As an application, we apply our method to characterize the diversity of human genome communities, which provides a possible use of our method in inferring the genetic similarity between human populations.

  13. An improved label propagation algorithm based on node importance and random walk for community detection

    NASA Astrophysics Data System (ADS)

    Ma, Tianren; Xia, Zhengyou

    2017-05-01

    Currently, with the rapid development of information technology, the electronic media for social communication is becoming more and more popular. Discovery of communities is a very effective way to understand the properties of complex networks. However, traditional community detection algorithms consider the structural characteristics of a social organization only, with more information about nodes and edges wasted. In the meanwhile, these algorithms do not consider each node on its merits. Label propagation algorithm (LPA) is a near linear time algorithm which aims to find the community in the network. It attracts many scholars owing to its high efficiency. In recent years, there are more improved algorithms that were put forward based on LPA. In this paper, an improved LPA based on random walk and node importance (NILPA) is proposed. Firstly, a list of node importance is obtained through calculation. The nodes in the network are sorted in descending order of importance. On the basis of random walk, a matrix is constructed to measure the similarity of nodes and it avoids the random choice in the LPA. Secondly, a new metric IAS (importance and similarity) is calculated by node importance and similarity matrix, which we can use to avoid the random selection in the original LPA and improve the algorithm stability. Finally, a test in real-world and synthetic networks is given. The result shows that this algorithm has better performance than existing methods in finding community structure.

  14. Magnetic field line random walk in models and simulations of reduced magnetohydrodynamic turbulence

    SciTech Connect

    Snodin, A. P.; Ruffolo, D.; Oughton, S.; Servidio, S.; Matthaeus, W. H.

    2013-12-10

    The random walk of magnetic field lines is examined numerically and analytically in the context of reduced magnetohydrodynamic (RMHD) turbulence, which provides a useful description of plasmas dominated by a strong mean field, such as in the solar corona. A recently developed non-perturbative theory of magnetic field line diffusion is compared with the diffusion coefficients obtained by accurate numerical tracing of magnetic field lines for both synthetic models and direct numerical simulations of RMHD. Statistical analysis of an ensemble of trajectories confirms the applicability of the theory, which very closely matches the numerical field line diffusion coefficient as a function of distance z along the mean magnetic field for a wide range of the Kubo number R. This theory employs Corrsin's independence hypothesis, sometimes thought to be valid only at low R. However, the results demonstrate that it works well up to R = 10, both for a synthetic RMHD model and an RMHD simulation. The numerical results from the RMHD simulation are compared with and without phase randomization, demonstrating a clear effect of coherent structures on the field line random walk for a very low Kubo number.

  15. The continuous time random walk, still trendy: fifty-year history, state of art and outlook

    NASA Astrophysics Data System (ADS)

    Kutner, Ryszard; Masoliver, Jaume

    2017-03-01

    In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Phys. J. B Special Issue [http://epjb.epj.org/open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on] containing articles which show incredible possibilities of the CTRWs. Contribution to the Topical Issue "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver.

  16. Self organization of social hierarchy and clusters in a challenging society with free random walks

    NASA Astrophysics Data System (ADS)

    Fujie, Ryo; Odagaki, Takashi

    2010-04-01

    Emergence of social hierarchy and clusters in a challenging equal-right society is studied on the basis of the agent-based model, where warlike individuals who have own power or wealth perform random walks in a random order on a lattice and when meeting others the individuals challenge the strongest among the neighbors. We assume that the winning probability depends on the difference in their wealth and after the fight the winner gets and the loser loses a unit of the wealth. We show that hierarchy is self organized when the population exceeds a critical value and the transition from egalitarian state to hierarchical state occurs in two steps. The first transition is continuous to the society with widespread winning-probability. At the second transition the variance of the winning fraction decrease discontinuously, which was not observed in previous studies. The second hierarchical society consists of a small number of extreme winners and many individuals in the middle class and losers. We also show that when the relaxation parameter for the wealth is large, the first transition disappears. In the second hierarchical society, a giant cluster of individuals is formed with a layered structure in the power order and some people stray around it. The structure of the cluster and the distribution of wealth are quite different from the results of the previous challenging model [M. Tsujiguchi and T. Odagaki, Physica A 375 (2007) 317] which adopts the preassigned order for random walk.

  17. Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

    NASA Astrophysics Data System (ADS)

    Boyer, D.; Romo-Cruz, J. C. R.

    2014-10-01

    Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random-walk model with long-range memory for which not only the mean-square displacement (MSD) but also the propagator can be obtained exactly in the asymptotic limit. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anticorrelated Lévy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1 /t .

  18. Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?

    NASA Astrophysics Data System (ADS)

    Czégel, Dániel; Palla, Gergely

    2015-12-01

    Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications of the transition matrix describing the random walk process. In addition, the tests on real world networks provided very intuitive results, e.g., the trophic levels obtained from our approach on a food web were highly consistent with former results from ecology.

  19. Phylogeography Takes a Relaxed Random Walk in Continuous Space and Time

    PubMed Central

    Lemey, Philippe; Rambaut, Andrew; Welch, John J.; Suchard, Marc A.

    2010-01-01

    Research aimed at understanding the geographic context of evolutionary histories is burgeoning across biological disciplines. Recent endeavors attempt to interpret contemporaneous genetic variation in the light of increasingly detailed geographical and environmental observations. Such interest has promoted the development of phylogeographic inference techniques that explicitly aim to integrate such heterogeneous data. One promising development involves reconstructing phylogeographic history on a continuous landscape. Here, we present a Bayesian statistical approach to infer continuous phylogeographic diffusion using random walk models while simultaneously reconstructing the evolutionary history in time from molecular sequence data. Moreover, by accommodating branch-specific variation in dispersal rates, we relax the most restrictive assumption of the standard Brownian diffusion process and demonstrate increased statistical efficiency in spatial reconstructions of overdispersed random walks by analyzing both simulated and real viral genetic data. We further illustrate how drawing inference about summary statistics from a fully specified stochastic process over both sequence evolution and spatial movement reveals important characteristics of a rabies epidemic. Together with recent advances in discrete phylogeographic inference, the continuous model developments furnish a flexible statistical framework for biogeographical reconstructions that is easily expanded upon to accommodate various landscape genetic features. PMID:20203288

  20. Multiple random walks on complex networks: A harmonic law predicts search time.

    PubMed

    Weng, Tongfeng; Zhang, Jie; Small, Michael; Hui, Pan

    2017-05-01

    We investigate multiple random walks traversing independently and concurrently on complex networks and introduce the concept of mean first parallel passage time (MFPPT) to quantify their search efficiency. The mean first parallel passage time represents the expected time required to find a given target by one or some of the multiple walkers. We develop a general theory that allows us to calculate the MFPPT analytically. Interestingly, we find that the global MFPPT follows a harmonic law with respect to the global mean first passage times of the associated walkers. Remarkably, when the properties of multiple walkers are identical, the global MFPPT decays in a power law manner with an exponent of unity, irrespective of network structure. These findings are confirmed by numerical and theoretical results on various synthetic and real networks. The harmonic law reveals a universal principle governing multiple random walks on networks that uncovers the contribution and role of the combined walkers in a target search. Our paradigm is also applicable to a broad range of random search processes.

  1. Three-dimensional cell migration does not follow a random walk.

    PubMed

    Wu, Pei-Hsun; Giri, Anjil; Sun, Sean X; Wirtz, Denis

    2014-03-18

    Cell migration through 3D extracellular matrices is critical to the normal development of tissues and organs and in disease processes, yet adequate analytical tools to characterize 3D migration are lacking. Here, we quantified the migration patterns of individual fibrosarcoma cells on 2D substrates and in 3D collagen matrices and found that 3D migration does not follow a random walk. Both 2D and 3D migration features a non-Gaussian, exponential mean cell velocity distribution, which we show is primarily a result of cell-to-cell variations. Unlike in the 2D case, 3D cell migration is anisotropic: velocity profiles display different speed and self-correlation processes in different directions, rendering the classical persistent random walk (PRW) model of cell migration inadequate. By incorporating cell heterogeneity and local anisotropy to the PRW model, we predict 3D cell motility over a wide range of matrix densities, which identifies density-independent emerging migratory properties. This analysis also reveals the unexpected robust relation between cell speed and persistence of migration over a wide range of matrix densities.

  2. Propagators and related descriptors for non-Markovian asymmetric random walks with and without boundaries.

    PubMed

    Berezhkovskii, Alexander M; Weiss, George H

    2008-01-28

    There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRW's in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t-->infinity, and the effective diffusion constant. We show that in the infinite space, the propagator satisfies a relation that can be interpreted as a generalized fluctuation theorem since it reduces to the conventional fluctuation theorem at large times. Based on the Laplace transform of the propagator in the presence of an absorbing boundary, we derive the Laplace transform of the survival probability of the random walker, which is then used to find the mean lifetime for terminated trajectories of the random walk.

  3. The relative importance of directional change, random walks, and stasis in the evolution of fossil lineages.

    PubMed

    Hunt, Gene

    2007-11-20

    The nature of evolutionary changes recorded by the fossil record has long been controversial, with particular disagreement concerning the relative frequency of gradual change versus stasis within lineages. Here, I present a large-scale, statistical survey of evolutionary mode in fossil lineages. Over 250 sequences of evolving traits were fit by using maximum likelihood to three evolutionary models: directional change, random walk, and stasis. Evolution in these traits was rarely directional; in only 5% of fossil sequences was directional evolution the most strongly supported of the three modes of change. The remaining 95% of sequences were divided nearly equally between random walks and stasis. Variables related to body size were significantly less likely than shape traits to experience stasis. This finding is in accord with previous suggestions that size may be more evolutionarily labile than shape and is consistent with some but not all of the mechanisms proposed to explain evolutionary stasis. In general, similar evolutionary patterns are observed across other variables, such as clade membership and temporal resolution, but there is some evidence that directional change in planktonic organisms is more frequent than in benthic organisms. The rarity with which directional evolution was observed in this study corroborates a key claim of punctuated equilibria and suggests that truly directional evolution is infrequent or, perhaps more importantly, of short enough duration so as to rarely register in paleontological sampling.

  4. Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?

    PubMed Central

    Czégel, Dániel; Palla, Gergely

    2015-01-01

    Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications of the transition matrix describing the random walk process. In addition, the tests on real world networks provided very intuitive results, e.g., the trophic levels obtained from our approach on a food web were highly consistent with former results from ecology. PMID:26657012

  5. Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?

    PubMed

    Czégel, Dániel; Palla, Gergely

    2015-12-10

    Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications of the transition matrix describing the random walk process. In addition, the tests on real world networks provided very intuitive results, e.g., the trophic levels obtained from our approach on a food web were highly consistent with former results from ecology.

  6. Mean first passage time for random walk on dual structure of dendrimer

    NASA Astrophysics Data System (ADS)

    Li, Ling; Guan, Jihong; Zhou, Shuigeng

    2014-12-01

    The random walk approach has recently been widely employed to study the relations between the underlying structure and dynamic of complex systems. The mean first-passage time (MFPT) for random walks is a key index to evaluate the transport efficiency in a given system. In this paper we study analytically the MFPT in a dual structure of dendrimer network, Husimi cactus, which has different application background and different structure (contains loops) from dendrimer. By making use of the iterative construction, we explicitly determine both the partial mean first-passage time (PMFT, the average of MFPTs to a given target) and the global mean first-passage time (GMFT, the average of MFPTs over all couples of nodes) on Husimi cactus. The obtained closed-form results show that PMFPT and EMFPT follow different scaling with the network order, suggesting that the target location has essential influence on the transport efficiency. Finally, the impact that loop structure could bring is analyzed and discussed.

  7. Random walks on finite lattices with multiple traps: Application to particle-cluster aggregation

    NASA Astrophysics Data System (ADS)

    Evans, J. W.; Nord, R. S.

    1985-11-01

    For random walks on finite lattices with multiple (completely adsorbing) traps, one is interested in the mean walk length until trapping and in the probability of capture for the various traps (either for a walk with a specific starting site, or for an average over all nontrap sites). We develop the formulation of Montroll to enable determination of the large-lattice-size asymptotic behavior of these quantities. (Only the case of a single trap has been analyzed in detail previously.) Explicit results are given for the case of symmetric nearest-neighbor random walks on two-dimensional (2D) square and triangular lattices. Procedures for exact calculation of walk lengths on a finite lattice with a single trap are extended to the multiple-trap case to determine all the above quantities. We examine convergence to asymptotic behavior as the lattice size increases. Connection with Witten-Sander irreversible particle-cluster aggregation is made by noting that this process corresponds to designating all sites adjacent to the cluster as traps. Thus capture probabilities for different traps determine the proportions of the various shaped clusters formed. (Reciprocals of) associated average walk lengths relate to rates for various irreversible aggregation processes involving a gas of walkers and clusters. Results are also presented for some of these quantities.

  8. Random-walk mobility analysis of Lisbon's plans for the post-1755 reconstruction

    NASA Astrophysics Data System (ADS)

    de Sampayo, Mafalda Teixeira; Sousa-Rodrigues, David

    2016-11-01

    The different options for the reconstruction of the city of Lisbon in the aftermath of the 1755 earthquake are studied with an agent-based model based on randomwalks. This method gives a comparative quantitative measure of mobility of the circulation spaces within the city. The plans proposed for the city of Lisbon signified a departure from the medieval mobility city model. The intricacy of the old city circulation spaces is greatly reduced in the new plans and the mobility between different areas is substantially improved. The simulation results of the random-walk model show that those plans keeping the main force lines of the old city presented less improvement in terms ofmobility. The plans that had greater design freedom were, by contrast, easier to navigate. Lisbon's reconstruction followed a plan that included a shift in the traditional notions of mobility. This affected the daily lives of its citizens by potentiating an easy access to the waterfront, simplifying orientation and navigability. Using the random-walk model it is shown how to quantitatively measure the potential that synthetic plans have in terms of the permeability and navigability of different city public spaces.

  9. Random walk model of subdiffusion in a system with a thin membrane.

    PubMed

    Kosztołowicz, Tadeusz

    2015-02-01

    We consider in this paper subdiffusion in a system with a thin membrane. The subdiffusion parameters are the same in both parts of the system separated by the membrane. Using the random walk model with discrete time and space variables the probabilities (Green's functions) P(x,t) describing a particle's random walk are found. The membrane, which can be asymmetrical, is characterized by the two probabilities of stopping a random walker by the membrane when it tries to pass through the membrane in both opposite directions. Green's functions are transformed to the system in which the variables are continuous, and then the membrane permeability coefficients are given by special formulas which involve the probabilities mentioned above. From the obtained Green's functions, we derive boundary conditions at the membrane. One of the conditions demands the continuity of a flux at the membrane, but the other one is rather unexpected and contains the Riemann-Liouville fractional time derivative P(x(N)(-),t)=λ(1)P(x(N)(+),t)+λ(2)∂(α/2)P(x(N)(+),t)/∂t(α/2), where λ(1),λ(2) depending on membrane permeability coefficients (λ(1)=1 for a symmetrical membrane), α is a subdiffusion parameter, and x(N) is the position of the membrane. This boundary condition shows that the additional "memory effect," represented by the fractional derivative, is created by the membrane. This effect is also created by the membrane for a normal diffusion case in which α=1.

  10. Rare events statistics of random walks on networks: localisation and other dynamical phase transitions

    NASA Astrophysics Data System (ADS)

    De Bacco, Caterina; Guggiola, Alberto; Kühn, Reimer; Paga, Pierre

    2016-05-01

    Rare event statistics for random walks on complex networks are investigated using the large deviation formalism. Within this formalism, rare events are realised as typical events in a suitably deformed path-ensemble, and their statistics can be studied in terms of spectral properties of a deformed Markov transition matrix. We observe two different types of phase transition in such systems: (i) rare events which are singled out for sufficiently large values of the deformation parameter may correspond to localised modes of the deformed transition matrix; (ii) ‘mode-switching transitions’ may occur as the deformation parameter is varied. Details depend on the nature of the observable for which the rare event statistics is studied, as well as on the underlying graph ensemble. In the present paper we report results on rare events statistics for path averages of random walks in Erdős-Rényi and scale free networks. Large deviation rate functions and localisation properties are studied numerically. For observables of the type considered here, we also derive an analytical approximation for the Legendre transform of the large deviation rate function, which is valid in the large connectivity limit. It is found to agree well with simulations.

  11. Multiple random walks on complex networks: A harmonic law predicts search time

    NASA Astrophysics Data System (ADS)

    Weng, Tongfeng; Zhang, Jie; Small, Michael; Hui, Pan

    2017-05-01

    We investigate multiple random walks traversing independently and concurrently on complex networks and introduce the concept of mean first parallel passage time (MFPPT) to quantify their search efficiency. The mean first parallel passage time represents the expected time required to find a given target by one or some of the multiple walkers. We develop a general theory that allows us to calculate the MFPPT analytically. Interestingly, we find that the global MFPPT follows a harmonic law with respect to the global mean first passage times of the associated walkers. Remarkably, when the properties of multiple walkers are identical, the global MFPPT decays in a power law manner with an exponent of unity, irrespective of network structure. These findings are confirmed by numerical and theoretical results on various synthetic and real networks. The harmonic law reveals a universal principle governing multiple random walks on networks that uncovers the contribution and role of the combined walkers in a target search. Our paradigm is also applicable to a broad range of random search processes.

  12. Random walk to describe diffusion phenomena in three-dimensional discontinuous media: Step-balance and fictitious-velocity corrections

    NASA Astrophysics Data System (ADS)

    Maruyama, Yutaka

    2017-09-01

    In this paper, we show that diffusion phenomena in three-dimensional discontinuous media can be described as a random walk by two simple interface-correction methods, namely step-balance and fictitious-velocity corrections, which are completely different in a physical picture but equivalent in that the continuity of the random walk at interfaces is considered. In both corrections, asymmetric interface permeability of a random walker, which comes from ensuring the continuity, causes apparent confinement of the walker in higher-diffusivity layers for benchmark tests on heat diffusion in two-phase multilayered systems. Effective thermal conductivities (walker diffusivities) computed from the trajectories are in excellent agreement with the series and parallel conduction formulas, indicating the equivalence of the two corrections and the importance of ensuring the continuity of a random walk at interfaces.

  13. A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths

    NASA Astrophysics Data System (ADS)

    Le Caër, Gérard

    2010-08-01

    A constrained diffusive random walk of n steps in ℝ d and a random flight in ℝ d , which are equivalent, were investigated independently in recent papers (J. Stat. Phys. 127:813, 2007; J. Theor. Probab. 20:769, 2007, and J. Stat. Phys. 131:1039, 2008). The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1,2,4. Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D. The previous walk is generalized by considering step lengths which have independent and identical gamma distributions with a shape parameter q>0. Given the total walk length being equal to 1, the step lengths have a Dirichlet distribution whose parameters are all equal to q. The walk and the flight above correspond to q=1. Simple analytical expressions are obtained for any d≥2 and n≥2 for the endpoint distributions of two families of walks whose q are integers or half-integers which depend solely on d. These endpoint distributions have a simple geometrical interpretation. Expressed for a two-step planar walk whose q=1, it means that the distribution of the endpoint on a disc of radius 1 is identical to the distribution of the projection on the disc of a point M uniformly distributed over the surface of the 3D unit sphere. Five additional walks, with a uniform distribution of the endpoint in the inside of a ball, are found from known finite integrals of products of powers and Bessel functions of the first kind. They include four different walks in ℝ3, two of two steps and two of three steps, and one walk of two steps in ℝ4. Pearson-Liouville random walks, obtained by distributing the total lengths of the previous Pearson-Dirichlet walks according to some

  14. Uniform and C^1-approximability of functions on compact subsets of \\mathbb R^2 by solutions of second-order elliptic equations

    NASA Astrophysics Data System (ADS)

    Paramonov, P. V.; Fedorovskii, K. Yu

    1999-02-01

    Several necessary and sufficient conditions for the existence of uniform or C^1-approximation of functions on compact subsets of \\mathbb R^2 by solutions of elliptic systems of the form c_{11}u_{x_1x_1}+2c_{12}u_{x_1x_2}+c_{22}u_{x_2x_2}=0 with constant complex coefficients c_{11}, c_{12} and c_{22} are obtained. The proofs are based on a refinement of Vitushkin's localization method, in which one constructs localized approximating functions by "gluing together" some special many-valued solutions of the above equations. The resulting conditions of approximation are of a topological and metric nature.

  15. General mapping between random walks and thermal vibrations in elastic networks: fractal networks as a case study.

    PubMed

    Reuveni, Shlomi; Granek, Rony; Klafter, Joseph

    2010-10-01

    We present an approach to mapping between random walks and vibrational dynamics on general networks. Random walk occupation probabilities, first passage time distributions and passage probabilities between nodes are expressed in terms of thermal vibrational correlation functions. Recurrence is demonstrated equivalent to the Landau-Peierls instability. Fractal networks are analyzed as a case study. In particular, we show that the spectral dimension governs whether or not the first passage time distribution is well represented by its mean. We discuss relevance to universal features arising in protein vibrational dynamics.

  16. Collapse transition of a hydrophobic self-avoiding random walk in a coarse-grained model solvent.

    PubMed

    Gaudreault, Mathieu; Viñals, Jorge

    2009-08-01

    In order to study solvation effects on protein folding, we analyze the collapse transition of a self-avoiding random walk composed of hydrophobic segments that is embedded in a lattice model of a solvent. As expected, hydrophobic interactions lead to an attractive potential of mean force among chain segments. As a consequence, the random walk in solvent undergoes a collapse transition at a higher temperature than in its absence. Chain collapse is accompanied by the formation of a region depleted of solvent around the chain. In our simulation, the depleted region at collapse is as large as our computational domain.

  17. On the temporal order of first-passage times in one-dimensional lattice random walks

    NASA Astrophysics Data System (ADS)

    Sanders, J. B.; Temme, N. M.

    2005-10-01

    A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing modified Bessel functions of the first kind. By using several transformations, simpler integrals are obtained from which for two and three particles asymptotic approximations are derived for large values of the parameters. Expressions of the probability for n particles are also derived.I returned and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all. George Orwell, Politics and the English Language, Selected Essays, Penguin Books, 1957. (The citation is from Ecclesiastes 9:11.)

  18. Note: Network random walk model of two-state protein folding: Test of the theory

    NASA Astrophysics Data System (ADS)

    Berezhkovskii, Alexander M.; Murphy, Ronan D.; Buchete, Nicolae-Viorel

    2013-01-01

    We study two-state protein folding in the framework of a toy model of protein dynamics. This model has an important advantage: it allows for an analytical solution for the sum of folding and unfolding rate constants [A. M. Berezhkovskii, F. Tofoleanu, and N.-V. Buchete, J. Chem. Theory Comput. 7, 2370 (2011), 10.1021/ct200281d] and hence for the reactive flux at equilibrium. We use the model to test the Kramers-type formula for the reactive flux, which was derived assuming that the protein dynamics is described by a Markov random walk on a network of complex connectivity [A. Berezhkovskii, G. Hummer, and A. Szabo, J. Chem. Phys. 130, 205102 (2009), 10.1063/1.3139063]. It is shown that the Kramers-type formula leads to the same result for the reactive flux as the sum of the rate constants.

  19. Unbinding of mutually avoiding random walks and two-dimensional quantum gravity

    NASA Astrophysics Data System (ADS)

    Carlon, Enrico; Baiesi, Marco

    2004-12-01

    We analyze the unbinding transition for a two-dimensional lattice polymer in which the constituent strands are mutually avoiding random walks. At low temperatures the strands are bound and form a single self-avoiding walk. We show that unbinding in this model is a strong first order transition. The entropic exponents associated with denaturated loops and end-segment distributions show sharp differences at the transition point and in the high temperature phase. Their values can be deduced from some exact arguments relying on a conformal mapping of copolymer networks into a fluctuating geometry, i.e., in the presence of quantum gravity. An excellent agreement between analytical and numerical estimates is observed for all cases analyzed.

  20. Degree distributions of the visibility graphs mapped from fractional Brownian motions and multifractal random walks

    NASA Astrophysics Data System (ADS)

    Ni, Xiao-Hui; Jiang, Zhi-Qiang; Zhou, Wei-Xing

    2009-10-01

    The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent α is a linear function of the Hurst index H of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from three Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the α∼H linear relationship.

  1. RecRWR: a recursive random walk method for improved identification of diseases.

    PubMed

    Arrais, Joel Perdiz; Oliveira, José Luís

    2015-01-01

    High-throughput methods such as next-generation sequencing or DNA microarrays lack precision, as they return hundreds of genes for a single disease profile. Several computational methods applied to physical interaction of protein networks have been successfully used in identification of the best disease candidates for each expression profile. An open problem for these methods is the ability to combine and take advantage of the wealth of biomedical data publicly available. We propose an enhanced method to improve selection of the best disease targets for a multilayer biomedical network that integrates PPI data annotated with stable knowledge from OMIM diseases and GO biological processes. We present a comprehensive validation that demonstrates the advantage of the proposed approach, Recursive Random Walk with Restarts (RecRWR). The obtained results outline the superiority of the proposed approach, RecRWR, in identifying disease candidates, especially with high levels of biological noise and benefiting from all data available.

  2. Random-Walk Monte Carlo Simulation of Intergranular Gas Bubble Nucleation in UO2 Fuel

    SciTech Connect

    Yongfeng Zhang; Michael R. Tonks; S. B. Biner; D.A. Andersson

    2012-11-01

    Using a random-walk particle algorithm, we investigate the clustering of fission gas atoms on grain bound- aries in oxide fuels. The computational algorithm implemented in this work considers a planar surface representing a grain boundary on which particles appear at a rate dictated by the Booth flux, migrate two dimensionally according to their grain boundary diffusivity, and coalesce by random encounters. Specifically, the intergranular bubble nucleation density is the key variable we investigate using a parametric study in which the temperature, grain boundary gas diffusivity, and grain boundary segregation energy are varied. The results reveal that the grain boundary bubble nucleation density can vary widely due to these three parameters, which may be an important factor in the observed variability in intergranular bubble percolation among grain boundaries in oxide fuel during fission gas release.

  3. A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

    NASA Astrophysics Data System (ADS)

    Rösler, Margit; Voit, Michael

    2015-02-01

    We consider compact Grassmann manifolds G/K over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type BC. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of G/K, which are constructed by successive decompositions of tensor powers of spherical representations of G. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.

  4. Random Walks with Preferential Relocations to Places Visited in the Past and their Application to Biology

    NASA Astrophysics Data System (ADS)

    Boyer, Denis; Solis-Salas, Citlali

    2014-06-01

    Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the central limit theorem but from an unusual balance between random and long-range memory steps. In single trajectories, occupation patterns are heterogeneous and have a scale-free structure. The model exhibits good agreement with data of free-ranging capuchin monkeys.

  5. Monotonic continuous-time random walks with drift and stochastic reset events

    NASA Astrophysics Data System (ADS)

    Montero, Miquel; Villarroel, Javier

    2013-01-01

    In this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes. We focus our attention on monotonic continuous-time random walks with a constant drift: The process increases between the reset events, either by the effect of the random jumps, or by the action of the deterministic drift. As a result of all these combined factors interesting properties emerge, like the existence (for any drift strength) of a stationary transition probability density function, or the faculty of the model to reproduce power-law-like behavior. General formulas for two extreme statistics, the survival probability, and the mean exit time are also derived. To corroborate in an independent way the results of the paper, Monte Carlo methods were used. These numerical estimations are in full agreement with the analytical predictions.

  6. Random walks with efficient search and contextually adapted image similarity for deformable registration.

    PubMed

    Tang, Lisa Y W; Hamarneh, Ghassan

    2013-01-01

    We develop a random walk-based image registration method that incorporates two novelties: 1) a progressive optimization scheme that conducts the solution search efficiently via a novel use of information derived from the obtained probabilistic solution, and 2) a data-likelihood re-weighting step that contextually performs feature selection in a spatially adaptive manner so that the data costs are based primarily on trusted information sources. Synthetic experiments on three public datasets of different anatomical regions and modalities showed that our method performed efficient search without sacrificing registration accuracy. Experiments performed on 60 real brain image pairs from a public dataset also demonstrated our method's better performance over existing non-probabilistic image registration methods.

  7. SLE on Doubly-Connected Domains and the Winding of Loop-Erased Random Walks

    NASA Astrophysics Data System (ADS)

    Hagendorf, Christian; Le Doussal, Pierre

    2008-10-01

    Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLE κ with parameter κ=2. In this note, some properties of an SLE κ trace on doubly-connected domains are studied and a connection to passive scalar diffusion in a Burgers flow is emphasised. In particular, the endpoint probability distribution and winding probabilities for SLE2 on a cylinder, starting from one boundary component and stopped when hitting the other, are found. A relation of the result to conditioned one-dimensional Brownian motion is pointed out. Moreover, this result permits to study the statistics of the winding number for SLE2 with fixed endpoints. A solution for the endpoint distribution of SLE4 on the cylinder is obtained and a relation to reflected Brownian motion pointed out.

  8. When human walking becomes random walking: fractal analysis and modeling of gait rhythm fluctuations.

    PubMed

    Hausdorff, J M; Ashkenazy, Y; Peng, C K; Ivanov, P C; Stanley, H E; Goldberger, A L

    2001-12-15

    We present a random walk, fractal analysis of the stride-to-stride fluctuations in the human gait rhythm. The gait of healthy young adults is scale-free with long-range correlations extending over hundreds of strides. This fractal scaling changes characteristically with maturation in children and older adults and becomes almost completely uncorrelated with certain neurologic diseases. Stochastic modeling of the gait rhythm dynamics, based on transitions between different "neural centers", reproduces distinctive statistical properties of the gait pattern. By tuning one model parameter, the hopping (transition) range, the model can describe alterations in gait dynamics from childhood to adulthood including a decrease in the correlation and volatility exponents with maturation.

  9. Estimating Genomic Distance from DNA Sequence Location in Cell Nuclei by a Random Walk Model

    NASA Astrophysics Data System (ADS)

    van den Engh, Ger; Sachs, Rainer; Trask, Barbara J.

    1992-09-01

    The folding of chromatin in interphase cell nuclei was studied by fluorescent in situ hybridization with pairs of unique DNA sequence probes. The sites of DNA sequences separated by 100 to 2000 kilobase pairs (kbp) are distributed in interphase chromatin according to a random walk model. This model provides the basis for calculating the spacing of sequences along the linear DNA molecule from interphase distance measurements. An interphase mapping strategy based on this model was tested with 13 probes from a 4-megabase pair (Mbp) region of chromosome 4 containing the Huntington disease locus. The results confirmed the locations of the probes and showed that the remaining gap in the published maps of this region is negligible in size. Interphase distance measurements should facilitate construction of chromosome maps with an average marker density of one per 100 kbp, approximately ten times greater than that achieved by hybridization to metaphase chromosomes.

  10. Modeling spreading of oil slicks based on random walk methods and Voronoi diagrams.

    PubMed

    Durgut, İsmail; Reed, Mark

    2017-02-19

    We introduce a methodology for representation of a surface oil slick using a Voronoi diagram updated at each time step. The Voronoi cells scale the Gaussian random walk procedure representing the spreading process by individual particle stepping. The step length of stochastically moving particles is based on a theoretical model of the spreading process, establishing a relationship between the step length of diffusive spreading and the thickness of the slick at the particle locations. The Voronoi tessellation provides the areal extent of the slick particles and in turn the thicknesses of the slick and the diffusive-type spreading length for all particles. The algorithm successfully simulates the spreading process and results show very good agreement with the analytical solution. Moreover, the results are robust for a wide range of values for computational time step and total number of particles.

  11. Exact Statistics of Record Increments of Random Walks and Lévy Flights.

    PubMed

    Godrèche, Claude; Majumdar, Satya N; Schehr, Grégory

    2016-07-01

    We study the statistics of increments in record values in a time series {x_{0}=0,x_{1},x_{2},…,x_{n}} generated by the positions of a random walk (discrete time, continuous space) of duration n steps. For arbitrary jump length distribution, including Lévy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of n for large n, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability Q(n) that the record increments decrease monotonically up to step n. Remarkably, Q(n) is universal (i.e., independent of the jump distribution) for each n, decaying as Q(n)∼A/sqrt[n] for large n, with a universal amplitude A=e/sqrt[π]=1.53362….

  12. The random walk function in the analysis of time-activity curves from dynamic radionuclide studies.

    PubMed

    Hart, G C; Bunday, B; Kiri, V

    1987-04-01

    The random walk function is a mathematical function derived from studies of the mass transport and flow of diffusible materials through tubes. Approximations to the function were first used some time ago in the field of cardiac tracer dilution curves, but in the absence of rapid and reproducible curve fitting the method never became commonplace. The current study uses the latest curve-fitting techniques and shows how the method may be used with precision in the analysis of time-activity curves from dynamic oesophageal and blood flow studies. The physiological basis of the method is given and parameters obtained which relate to both the rate of flow and the local dispersion of the bolus.

  13. Observations of Random Walk of the Ground In Space and Time

    SciTech Connect

    Shiltsev, Vladimir; /Fermilab

    2010-01-01

    We present results of micron-resolution measurements of the ground motions in large particle accelerators over the range of spatial scales L from several meters to tens of km and time intervals T from minutes to several years and show that in addition to systematic changes due to tides or slow drifts, there is a stochastic component which has a 'random-walk' character both in time and in space. The measured mean square of the relative displacement of ground elements scales as dY{sup 2} {approx} ATL over broad range of the intervals, and the site dependent constant A is of the order of 10{sup -5{+-}1} {micro}m{sup 2}/(s{center_dot}m).

  14. Random-walk model to study cycles emerging from the exploration-exploitation trade-off

    NASA Astrophysics Data System (ADS)

    Kazimierski, Laila D.; Abramson, Guillermo; Kuperman, Marcelo N.

    2015-01-01

    We present a model for a random walk with memory, phenomenologically inspired in a biological system. The walker has the capacity to remember the time of the last visit to each site and the step taken from there. This memory affects the behavior of the walker each time it reaches an already visited site modulating the probability of repeating previous moves. This probability increases with the time elapsed from the last visit. A biological analog of the walker is a frugivore, with the lattice sites representing plants. The memory effect can be associated with the time needed by plants to recover its fruit load. We propose two different strategies, conservative and explorative, as well as intermediate cases, leading to nonintuitive interesting results, such as the emergence of cycles.

  15. Sedimentary bed evolution as a mean-reverting random walk: Implications for tracer statistics

    NASA Astrophysics Data System (ADS)

    Martin, Raleigh L.; Purohit, Prashant K.; Jerolmack, Douglas J.

    2014-09-01

    Sediment tracers are increasingly employed to estimate bed load transport and landscape evolution rates. Tracer trajectories are dominated by periods of immobility ("waiting times") as they are buried and reexcavated in the stochastically evolving river bed. Here we model bed evolution as a random walk with mean-reverting tendency (Ornstein-Uhlenbeck process) originating from the restoring effect of erosion and deposition. The Ornstein-Uhlenbeck model contains two parameters, a and b, related to the particle feed rate and range of bed elevation fluctuations, respectively. Observations of bed evolution in flume experiments agree with model predictions; in particular, the model reproduces the asymptotic t-1 tail in the tracer waiting time exceedance probability distribution. This waiting time distribution is similar to that inferred for tracers in natural gravel streams and avalanching rice piles, indicating applicability of the Ornstein-Uhlenbeck mean-reverting model to many disordered transport systems with tracer burial and excavation.

  16. Random walk-based similarity measure method for patterns in complex object

    NASA Astrophysics Data System (ADS)

    Liu, Shihu; Chen, Xiaozhou

    2017-04-01

    This paper discusses the similarity of the patterns in complex objects. The complex object is composed both of the attribute information of patterns and the relational information between patterns. Bearing in mind the specificity of complex object, a random walk-based similarity measurement method for patterns is constructed. In this method, the reachability of any two patterns with respect to the relational information is fully studied, and in the case of similarity of patterns with respect to the relational information can be calculated. On this bases, an integrated similarity measurement method is proposed, and algorithms 1 and 2 show the performed calculation procedure. One can find that this method makes full use of the attribute information and relational information. Finally, a synthetic example shows that our proposed similarity measurement method is validated.

  17. Magnetic field line random walk in two-dimensional dynamical turbulence

    NASA Astrophysics Data System (ADS)

    Wang, J. F.; Qin, G.; Ma, Q. M.; Song, T.; Yuan, S. B.

    2017-08-01

    The field line random walk (FLRW) of magnetic turbulence is one of the important topics in plasma physics and astrophysics. In this article, by using the field line tracing method, the mean square displacement (MSD) of FLRW is calculated on all possible length scales for pure two-dimensional turbulence with the damping dynamical model. We demonstrate that in order to describe FLRW with the damping dynamical model, a new dimensionless quantity R is needed to be introduced. On different length scales, dimensionless MSD shows different relationships with the dimensionless quantity R. Although the temporal effect affects the MSD of FLRW and even changes regimes of FLRW, it does not affect the relationship between the dimensionless MSD and dimensionless quantity R on all possible length scales.

  18. Effective-medium approximation for lattice random walks with long-range jumps

    NASA Astrophysics Data System (ADS)

    Thiel, Felix; Sokolov, Igor M.

    2016-07-01

    We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.

  19. Lattice statistical theory of random walks on a fractal-like geometry.

    PubMed

    Kozak, John J; Garza-López, Roberto A; Abad, Enrique

    2014-03-01

    We have designed a two-dimensional, fractal-like lattice and explored, both numerically and analytically, the differences between random walks on this lattice and a regular, square-planar Euclidean lattice. We study the efficiency of diffusion-controlled processes for flows from external sites to a centrosymmetric reaction center and, conversely, for flows from a centrosymmetric source to boundary sites. In both cases, we find that analytic expressions derived for the mean walk length on the fractal-like lattice have an algebraic dependence on system size, whereas for regular Euclidean lattices the dependence can be transcendental. These expressions are compared with those derived in the continuum limit using classical diffusion theory. Our analysis and the numerical results quantify the extent to which one paradigmatic class of spatial inhomogeneities can compromise the efficiency of adatom diffusion on solid supports and of surface-assisted self-assembly in metal-organic materials.

  20. Branching and annihilating random walks: exact results at low branching rate.

    PubMed

    Benitez, Federico; Wschebor, Nicolás

    2013-05-01

    We present some exact results on the behavior of branching and annihilating random walks, both in the directed percolation and parity conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the nontrivial pure annihilation (PA) model, whose correlation and response function we compute exactly. With this, the nonuniversal threshold value for having a phase transition in the simplest system belonging to the directed percolation universality class is found to coincide with previous nonperturbative renormalization group (RG) approximate results. We also show that the parity conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.