memory functions of the additive markov chains applications to complex dynamic systems abstract a new approach to describing correlation properties of complex dynamic systems with long range memory based on a concept of additive markov chains phys rev e 68 2003 061107 is developed an equation connecting the memory and correlation function of the system under study is presented this equation allows reconstructing a memory function using a correlation function of the system effectiveness and robustness of the proposed method is demonstrated by simple model examples memory functions of concrete coarse grained literary texts are found and their universal power law behavior at long distances is revealed

kramers like escape driven by fractional gaussian noise we investigate the escape from a potential well of a test particle driven by fractional gaussian noise with hurst exponent 0 h 1 from a numerical analysis we demonstrate the exponential distribution of escape times from the well and analyze in detail the dependence of the mean escape time on the hurst exponent h and the particle diffusivity d we observe different behavior for the subdiffusive antipersistent and superdiffusive persistent domains in particular we find that the escape becomes increasingly faster for decreasing values of h consistent with previous findings on the first passage behavior approximate analytical calculations are shown to support the numerically observed dependencies

euler s elastica and curvature based inpainting image inpainting is a special image restoration problem for which image prior models play a crucial role euler s elastica was first introduced to computer vision by mumford algebraic geometry and its applications springer verlag new york 1994 pp 491 506 as a curve prior model by functionalizing the elastica energy masnou and morel proceedings of the 5th ieee international conference image processing 3 1998 pp 259 263 proposed an elastica based variational inpainting model the current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of bertalmio sapiro caselles and ballester siggraph 2000 acm press new york 2000 and chan and shen j visual comm image rep 12 2001 pp 436 449 a computational scheme based on numerical pdes is presented which allows the automatic handling of topologically complex inpainting domains

computing optical flow via variational techniques defined as the apparent motion in a sequence of images the optical flow is very important in the computer vision community where its accurate estimation is necessary for many applications it is o

partial differential equations for zooming deinterlacing and dejittering in this paper for imaging applications we introduce partial differential equations pdes which allow for correcting displacement errors for dejittering and for deinterlacing respectively in multi channel data these equations are derived via semi groups for non convex energy functionals as a particular example for gray valued data we find the mean curvature equation and the corresponding non convex energy functional as a further application for correction of displacement errors we study image interpolation in particular zooming of digital color images for actual image zooming the solutions of the proposed pdes are projected onto a space of functions satisfying interpolation constraints a comparison of the test results with standard and state of the art interpolation algorithms shows the competitiveness of this approach

first passage time statistics of a brownian particle driven by an arbitrary unidimensional potential with a superimposed exponential time dependent drift in one dimensional systems the dynamics of a brownian particle are governed by the force derived from a potential as well as by diffusion properties in this work we obtain the first passage time statistics of a brownian particle driven by an arbitrary potential with an exponential temporally decaying superimposed field up to a prescribed threshold the general system analyzed here describes the sub threshold signal integration of integrate and fire neuron models of any kind supplemented by an adaptation like current whereas the first passage time corresponds to the declaration of a spike following our previous studies we base our analysis on the backward fokker planck equation and study the survival probability and the first passage time density function in the space of the initial condition by proposing a series solution we obtain a system of recurrence equations which given the specific structure of the exponential time dependent drift easily admit a simpler laplace representation naturally the present general derivation agrees with the explicit solution we found previously for the wiener process in 2012 j phys a math theor 45 185001 however to demonstrate the generality of the approach we further explicitly evaluate the first passage time statistics of the underlying ornstein uhlenbeck process to test the validity of the series solution we extensively compare theoretical expressions with the data obtained from numerical simulations in different regimes as shown agreement is precise whenever the series is truncated at an appropriate order beyond the fact that both the wiener and ornstein uhlenbeck processes have a direct interpretation in the context of neuronal models given their ubiquity in different fields our present results will be of interest in other settings where an additive state independent temporal relaxation process is being developed as the particle diffuses

construction of high order force gradient algorithms for integration of motion in classical and quantum systems a consequent approach is proposed to construct symplectic force gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations within this approach the basic algorithms are first derived up to the eighth order by direct decompositions of exponential propagators and further collected using an advanced composition scheme to obtain the algorithms of higher orders contrary to the scheme proposed by chin and kidwell phys rev e 62 8746 2000 where high order algorithms are introduced by standard iterations of a force gradient integrator of order four the present method allows one to reduce the total number of expensive force and its gradient evaluations to a minimum at the same time the precision of the integration increases significantly especially with increasing the order of the generated schemes the algorithms are tested in molecular dynamics and celestial mechanics simulations it is shown in particular that the efficiency of the advanced fourth order based algorithms is better approximately in factors 5 to 1000 for orders 4 to 12 respectively the results corresponding to sixth and eighth order based composition schemes are also presented up to the sixteenth order for orders 14 and 16 such highly precise schemes at considerably smaller computational costs allow to reduce unphysical deviations in the total energy up in 100000 times with respect to those of the standard fourth order based iteration approach

bayesian video dejittering by the bv image model line jittering or random horizontal displacement in video images occurs when the synchronization signals are corrupted in video storage media or by electromagnetic interference in wireless video transmission the goal of intrinsic video dejittering is to recover the ideal video directly from the observed jittered and often noisy frames the existing approaches in the literature are mostly based on local or semilocal filtering techniques and autoregressive image models and are complemented by various image processing tools in this paper based on the statistical rationale of bayesian inference we propose the first variational dejittering model based on the bounded variation bv image model which is global clean and self contained and intrinsically combines dejittering with denoising the mathematical properties of the model are studied based on the direct method of calculus of variations we design one effective algorithm and present its computational implementation based on techniques from numeri

cyclization of a polymer first passage problem for a non markovian process we discuss a closure of a polymer chain released from a state with given end to end distance considering the situation as a first passage problem for a non markovian process we express the survival probability through the three time distribution of the end to end distance and evaluate it numerically by solving a volterra equation we also show how the wilemski fixmann approximation emerges within this scheme and discuss its geometrical nature

edge direction preserving image zooming a mathematical and numerical analysis we focus in this paper on some reconstruction restoration methods whose aim is to improve the resolution of digital images the main point here is to study the ability of such methods to preserve one dimensional 1d structures indeed such structures are important since they are often carried by the image edges first we focus on linear methods give a general framework to design them and show that the preservation of 1d structures pleads in favor of the cancellation of the periodization of the image spectrum more precisely we show that preserving 1d structures implies the linear methods to be written as a convolution of the sinc interpolation as a consequence we cannot cope linearly with gibbs effects sharpness of the results and the preservation of the 1d structure second we study variational nonlinear methods and in particular the one based on total variation we show that this latter permits us to avoid these shortcomings we also prove the existence and consistency of an approximate solution to this variational problem at last this theoretical study is highlighted by experiments both on synthetic and natural images which show the effects of the described methods on images as well as on their spectrum

improving the accuracy of the avf method the average vector field avf method is a b series scheme of the second order as a discrete gradient method it preserves exactly the energy integral for any canonical hamiltonian system we present and discuss two locally exact and energy preserving modifications of the avf method avf lex of the third order and avf slex of the fourth order applications to spherically symmetric potentials are given including a compact explicit expression for the avf scheme for the coulomb kepler problem

hamiltonian conserving discrete canonical equations based on variational difference quotients abstract new discrete mechanics based on the assumption of the discrete time is proposed the discrete mechanics does not contain any continuous differentiation but contains only difference quotients resulting discrete hamiltonian s canonical equations are single time step difference equations and exactly conserve the hamiltonian the canonical equations give the numerical results more accurately than the heun scheme and the 4th order runge kutta scheme

on the statistics of generalized gaussian structures collapse and random external fields we consider the statistics of generalized gaussian structures ggs exposed to a random external field a ggs comprises n monomers connected to each other by harmonic potentials when the spectral dimension d of a ggs exceeds the value of two its radius of gyration r becomes independent of its mass n the cross over into this collapse can be treated continuously by cross linking m precursor chains of length n in the stretched state to an object which we call a polymer bundle we demonstrate that an external field f applied to each monomer can unfold such a collapsed state in the case where every monomer has an individual randomly distributed charge the critical spectral dimension for the collapse is raised to four r scales like fnalpha with alpha 4 ds 2ds for ds 4

correlation properties of the random linear high order markov chains abstract the aim of this paper is to study the correlation properties of random sequences with additive linear conditional probability distribution function cpdf and elaborate a reliable tool for their generation it is supposed that the state space of the sequence under examination belongs to a finite set of real numbers the cpdf is assumed to be additive and linear with respect to the values of the random variable we derive the equations that relate the correlation functions of the sequence to the memory function coefficients which determine the cpdf the obtained analytical solutions for the equations connecting the memory and correlation functions are compared with the results of numerical simulation examples of possible correlation scenarios in the high order additive linear chains are given

discrete approximations on functional classes for the integrable nonlinear schrodinger dynamical system a symplectic finite dimensional reduction approach abstract we investigate discretizations of the integrable discrete nonlinear schrodinger dynamical system and related symplectic structures we develop an effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter we construct a finite set of recurrent algebraic regular relations allowing to generate solutions of the discrete nonlinear schrodinger dynamical system and we discuss the related functional spaces of solutions finally we discuss the fourier transform approach to studying the solution set of the discrete nonlinear schrodinger dynamical system and its functional analytical aspects

an improved stormer verlet method based on exact discretization for nonlinear oscillators abstract motivated by the advantage of exact discretization of a linear differential equation and the importance of symplectic numerical methods for conservative nonlinear oscillators a modified stormer verlet method relying on a parameter is proposed the main idea is firstly based on some analytical approximation strategies relating a linear equation with the corresponding nonlinear equation such that linear equation s frequency approximates the exact frequency of the nonlinear equation secondly forcing the modified stormer verlet method to solve the related linear equation exactly the convergence symplectic and symmetric properties of the new method are analyzed for numerical implementation the cubic duffing equation and the simple pendulum are solved by the new method with some approximate frequencies as the parameter respectively numerical results show that the new method is much more accurate than its classical partner

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

energy preserving numerical schemes of high accuracy for one dimensional hamiltonian systems we present a class of non standard numerical schemes which are modifications of the discrete gradient method they preserve the energy integral exactly up to the round off error the considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order in numerical experiments we compare our integrators with some other numerical schemes including the standard discrete gradient method the leap frog scheme and a symplectic scheme of fourth order we study the error accumulation for a very long time and the conservation of the energy integral

isotropy properties of the multi step markov symbolic sequences abstract a new object of the probability theory the two sided chain of symbols introduced in ref arxiv physics 0306170 is used to study isotropy properties of binary multi step markov chains with the long range correlations established statistical correspondence between the markov chains and certain two sided sequences allows us to prove the isotropy properties of three classes of the markov chains one of them is the important class of weakly correlated additive markov chains which turned out to be equivalent to the additive two sided sequences

highly accurate optic flow computation with theoretically justified warping in this paper we suggest a variational model for optic flow computation based on non linearised and higher order constancy assumptions besides the common grey value constancy assumption also gradient constancy as well as the constancy of the hessian and the laplacian are proposed since the model strictly refrains from a linearisation of these assumptions it is also capable to deal with large displacements for the minimisation of the rather complex energy functional we present an efficient numerical scheme employing two nested fixed point iterations following a coarse to fine strategy it turns out that there is a theoretical foundation of so called warping techniques hitherto justified only on an experimental basis since our algorithm consists of the integration of various concepts ranging from different constancy assumptions to numerical implementation issues a detailed account of the effect of each of these concepts is included in the experimental section the superior performance of the proposed method shows up by significantly smaller estimation errors when compared to previous techniques further experiments also confirm excellent robustness under noise and insensitivity to parameter variations

reversible interpolation of vectorial images by an anisotropic diffusion projection pde in this paper a nonlinear model for the interpolation of vector valued images is proposed this model is based on an anisotropic diffusion pde and performs an interpolation that is reversible the interpolation solution is restricted to the subspace of functions that can recover the discrete input image after an appropriate smoothing and sampling the proposed nonlinear diffusion flow lies on this subspace while its strength and anisotropy adapt to the local variations and geometry of image structures the derived method effectively reconstructs the real image structures and yields a satisfactory interpolation result compared to classic and other existing pde based interpolation methods our proposed method seems to increase the accuracy of the result and to reduce the undesirable artifacts such as blurring ringing block effects and edge distortion we present extensive experimental results that demonstrate the potential of the method as applied to graylevel and color images

memory functions and correlations in additive binary markov chains a theory of additive markov chains with a long range memory proposed earlier in usatenko et al 2003 phys rev e 68 061107 is developed and used to describe statistical properties of long range correlated systems the convenient characteristics of such systems memory functions and their relation to the correlation properties of the systems are examined various methods for finding the memory function via the correlation function are proposed the inverse problem calculation of the correlation function by means of the prescribed memory function is also solved this is demonstrated for the analytically solvable model of the system with a step wise memory function

locally exact modifications of numerical schemes we present a new class of exponential integrators for ordinary differential equations locally exact modifications of known numerical schemes local exactness means that they preserve the linearization of the original system at every point in particular locally exact integrators preserve all fixed points and are a stable we apply this approach to popular schemes including euler schemes the implicit midpoint rule and the trapezoidal rule we found locally exact modifications of discrete gradient schemes for symmetric discrete gradients and coordinate increment discrete gradients preserving their main geometric property exact conservation of the energy integral for arbitrary multidimensional hamiltonian systems in canonical coordinates numerical experiments for a two dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium much higher than suggested by the order of new schemes locally exact modification sometimes increases the order but in many cases leaves it unchanged

competition between two kinds of correlations in literary texts a theory of additive markov chains with long range memory is used to describe the correlation properties of coarse grained literary texts the complex structure of the correlations in the texts is revealed anti persistent correlations at small distances l approximately 300 define this non trivial structure for some concrete examples of literary texts the memory functions are obtained and their power law behavior at long distances is disclosed this property is shown to be a cause of self similarity of texts with respect to the decimation procedure

unfold dynamics of generalized gaussian structures we consider the unfold dynamics of generalized gaussian structures ggss exposed to different kinds of external forces a ggs consists of n monomers connected by harmonic springs into a network when its spectral dimension d s exceeds the critical value of 2 the ggs is in a collapsed state sommer and blumen j phys a 28 6669 1995 showed that collapsed structures can be unfolded under external forces they demonstrate this for the case where each monomer is exposed to a force with a random direction then networks with a spectral dimension up to 4 become unfolded in the present paper we focus on the dynamics of such unfold processes we investigate ggss exposed to different kinds of external forces pulling one monomer uncorrelated forces long range correlated forces and diblocklike forces we show that external perturbations that act only on a few monomers are not able to unfold a collapsed structure on the other hand more general kinds of forces lead to a stretching of ggss even for d s g2 as long as d s l d c where d c depends on the kind of the force field in general during the unfold process the size r of the ggs grows via a power law r ensuremath propto t ensuremath alpha 0l ensuremath alpha l 1 where ensuremath alpha depends on d s as well as on the kind of force field that is applied

mean first passage times in confined media from markovian to non markovian processes we review recent theoretical works that enable the accurate evaluation of the mean first passage time mfpt of a random walker to a target in confinement for markovian memory less and non markovian walkers for the markovian problem we present a general theory which allows one to accurately evaluate the mfpt and its extensions to related first passage observables such as splitting probabilities and occupation times we show that this analytical approach provides a universal scaling dependence of the mfpt on both the volume of the confining domain and the source target distance in the case of general scale invariant processes this analysis is applicable to a broad range of stochastic processes characterized by length scale invariant properties and reveals the key role that can be played by the starting position of the random walker we then present an extension to non markovian walks by taking the specific example of a tagged monomer of a polymer chain looking for a target in confinement we show that the mfpt can be calculated accurately by computing the distribution of the positions of all the monomers in the chain at the instant of reaction such a theory can be used to derive asymptotic relations that generalize the scaling dependence with the volume and the initial distance to the target derived for markovian walks finally we present an application of this theory to the problem of the first contact time between the two ends of a polymer chain and review the various theoretical approaches of this nonmarkovian problem

improving the accuracy of the discrete gradient method in the one dimensional case we present two numerical schemes of high accuracy for one dimensional dynamical systems they are modifications of the discrete gradient method and keep its advantages including stability and conservation of the energy integral however their accuracy is higher by several orders of magnitude

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

long time behaviour of discretizations of the simple pendulum equation we compare several discretizations of the simple pendulum equation in a series of numerical experiments the stress is put on the long time behaviour the chosen numerical schemes are either symplectic maps or integrable energy preserving maps or both therefore they preserve qualitative features of solutions such as periodicity we describe characteristic periodic time dependences of numerical estimates of the period and the amplitude and explain them as systematic numerical by effects produced by any method finally we propose a new numerical scheme which is a modification of the discrete gradient method this modified discrete gradient method preserves almost exactly the period of small oscillations for any time step